Statement by the proposer highlighting major scientific...
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Transcript of Statement by the proposer highlighting major scientific...
Statement by the proposer highlighting major scientific contributions made by thenominee (NOT TO EXCEED 100 WORDS)
The nominee is a condensed matter theorist with deep insight into the physics of real materials.He uses numerical and analytic techniques to study contemporary problems in the electronicstructure of bulk as well as nano-materials. He specializes in methods based on first principlesdensity functional theory where chemical realism may be introduced in the description ofstrongly correlated systems by deriving realistic low energy model Hamiltonians and solvingthem numerically using mean field theories as well as non-perturbative approaches. Hisimportant contributions are (i) derivation of material specific spin Hamiltonians to clarify theorigin of novel ground states in low dimensional quantum spin systems, (ii) explanation of novelphases in strongly correlated oxides with spin-orbit coupling, (iii) origin of ferroelectricpolarization in multiferroic materials (MnWO4, FeTe2O5Br), (iv) defect induced magnetism inoxide based diluted magnetic semiconductors and prediction of novel magnetism in dopedcluster assembled solids, (v) discovery of chemical trends in high Tc cuprates and its correlationwith Tc,max . He has also significantly contributed in the development of efficient real spacemethods based on recursion technique to study superconductivity in disordered systems. Hehas exceptional skill as teacher and articulation of exposition.
Areas of specialisation : (mention three)
Nomination to be consideredby the Sectional Committee for : PHYSICS
(Please choose the most appropriate subject listed below)
List of Sectional Committees
I. Mathematics IV. Engineering & Technology VII. Animal/Plant Sciences
II. Physics V. Medicine VIII. General Biology
III. Chemistry VI. Earth & Planetary Sciences
1. Electronic Structure of Strongly Correlated Systems
2. Magnetism 3. Computational Condensed Matter Physics
BIOGRAPHICAL INFORMATION(of the nominee)
Name (expand initials) INDRA DASGUPTA
Present position SENIOR PROFESSOR Gender M
Date of birth 20-th October, 1965
Address Department of Solid State Physics
Indian Association for the Cultivation of Science
Jadavpur Kolkata 700 032
Phone: Off. +91-33-24734971 EXT:1319 Res. +91-3323241232
Mobile+91 9903053248
Fax+91-33 -24732805
[email protected] / [email protected]
PersonalHomepage
http://www.iacs.res.in/ssp/sspid/
Academic qualificationsYear Degree University/Institution
1987 (Results declared in 1988)
B.Sc (Physics Hons) Calcutta University, Presidency College
1990 M Sc (Physics) IIT Kanpur
1996 PhD Calcutta University, S.N. National Centrefor Basic Sciences
Positions held (in chronological order)Year(s) University/Institution Position held
1995 -1999 Max Planck Institute for Solid State Research, Stuttgart, Germany
Postdoctoral Research Associate
1999-2010 Department of Physics, IIT Bombay
Lecturer (1999-2001), Assistant Professor (2001-2005), Associate Profesor (2005-2010)
2003 Department of Physics andMeteorology, IIT Kharagpur(on lien from IIT Bombay)
Assistant Profesor
2007-2008 Indian Association for the Cultivation of Science ( on lien from IIT Bombay)
Associate Professor
2008-present Indian Association for the Cultivation of Science
Professor (2008-2013), Senior Professor (2013-present)
Awards and Honours
1. National Merit scholarship (1988-1990)
2. Max Planck Stipendium (1995-1999)
3. Award for excellence in teaching from IIT Bombay, (2004)
List of the five most important papers published by the nominee from his/her independent career:
Please give: 1) title of the paper
2) names of all authors in the same sequence as it appears in the papers
3) full journal reference.
1. Effect of spin orbit coupling and Hubbard U on the electronicstructure of IrO2
S.K. Panda, S. Bhowal, A. Delin, O. Eriksson and I. Dasgupta Phys. Rev B 89, 155102 (2014)
2. First-principles study of the electronic structure of CdS/ZnSecoupled quantum dotsN. Ganguli, S. Acharya, and I. DasguptaPhys. Rev B 89, 245423 (2014)
3. Role of Te in the low-dimensional multiferroic material FeTe2O5BrJ. Chakraborty, N. Ganguli, T. Saha-Dasgupta, and I. DasguptaPhys Rev B 88, 094409 (2013)
4. First-principles study of the spin-gap system Sr2Cu(BO3)2
Jayita Chakraborty and Indra DasguptaPhys Rev B 86, 054434 (2012)
5. Superconductivity in multiband disordered systems: A vector recursion approachShreemoyee Ganguly, Indra Dasgupta and Abhijit MookerjeePhys. Rev. B 84, 174508 (2011)
Summary of the Scientific Contributions
The nominee uses numerical and analytic techniques as well as interpretation of experiments to study contemporaryproblems in the electronic structure of materials. Research work is devoted to study both weakly and stronglycorrelated materials where the dominant electronic energy is the kinetic energy and the Coulomb repulsionrespectively. The study of weakly correlated systems is based on density functional theory (DFT) implemented invariety of basis sets. For strongly correlated systems, the main objective is to introduce material dependence in themodel Hamiltonians employed to study such systems. The nominee has pioneered in the development of methodswhere chemical realism may be introduced in the description of strongly correlated systems by deriving realisticHubbard Hamiltonians and solving them using static and dynamical mean field approximation as well asnon-perturbative approaches. The nominee has also contributed to the development of efficient real space methodsbased on the recursion technique to study phase stability, quantum transport and superconductivity in disorderedsystems. The important contribution of the nominee in the study of strongly correlated systems is the derivation of materialdependent Hubbard Hamiltonians using a basis set of Wannier functions obtained from the N-th order muffin tinorbital (NMTO) downfolding method. The Hubbard model at half-filling in the limit of strong correlation reduces tothe Heisenberg model. The NMTO downfolding method was employed to understand variety of low dimensionalquantum spin systems by deriving material specific Heisenberg Hamiltonians. These calculations not only providedaccurate estimate of the exchange interactions but also clarified the exchange paths and identified the relevant spinHamiltonians necessary to interpret experimental data (Phys. Rev. B 76, 085104, 2007; Phys. Rev. B 76, 052402,2007; Phys. Rev. B 77, 012410, 2008; Phys. Rev B 90, 035141, 2014). In some cases the spin Hamiltonians weresolved using quantum Monte Carlo (QMC) technique with the stochastic series expansion (SSE) algorithm. TheNMTO downfolding coupled with QMC-SSE was applied to study the electronic structure of the spin gap compoundSr2Cu(BO3)2. This calculation illustrated that a careful analysis of the electronic structure plays a key role for theidentification of the correct low energy model Hamiltonian for this system. The validity of the model was checked bycalculating the magnetic susceptibility as a function of temperature and magnetization both as a function oftemperature as well as field using QMC-SSE technique and comparing the calculated results with the availableexperimental data (Phys. Rev. B 86, 054434 2012). Recent research work of the nominee is also devoted to systems where Coulomb correlation and spin-orbit coupling(SOC) are of comparable strengths resulting in novel and emergent properties that are quite distinct from eitherspin-orbit coupled systems or strongly correlated systems. In d5 Ir based oxides, due to a large crystal field splittingand a strong SOC the t2g orbitals are renormalized into doubly degenerate Jeff = ½ and quadruply degenerate Jeff = 3/2states leading to a narrow band of half-filled Jeff = ½ states. Inclusion of a moderate Coulomb interaction in the spinorbit entangled Jeff = ½ manifold opens up a gap explaining the insulating property of some of these iridates. Veryrecently it has been speculated that in contrast to the orbital ordered states in 3d insulating oxides, the spin-orbitalentangled Jeff = ½ state is robust and does not melt away even in itinerant metallic systems. Very recently the nomineeand co-workers established that IrO2 is an example of Jeff = ½ metal where the Ir t2g states at the Fermi level largelyretain the complex Jeff = ½ character (Phys. Rev. B 89 , 155102 2014). While spin-orbit coupling is expected to beweak for 3d systems, recently it was shown that it may have important consequences even for such systems. This wasillustrated by explaining the origin of insulating state in the spinel compound FeCr2S4 (Phys. Rev. B 80, RapidComm. 201101, 2009) due to Coulomb enhanced spin orbit splitting. Using DFT + Hubbard (U) and DFT + Dynamical Mean Field Theory (DMFT) the nominee and co-workersaddressed the several decade long controversy whether the low temperature (LT) phase of hexagonal NiS is a metalor an insulator. Detailed calculations conclusively established that all experimental data for the low temperature (LT)phase of NiS can be understood in terms of a rather unusual ground state of NiS that is best described as a self-dopednearly compensated, antiferromagnetic metal (Scientific Report 3, 2995 2013; New Journal of Physics 16 093049,2014 ). In an earlier work the nominee and co-workers have discovered chemical trends in hole doped high Tc cuprates andits correlation with Tc at optimal doping (Tc,max) (PRL 87, 043007, 2001). Since the discovery of high Tc
superconductivity, there have been large body of experimental work showing Tc,max is material dependent.Interestingly, the Neel temperatures are hardly material dependent. This material dependence of Tc,max in High Tc
cuprates is unraveled by analyzing the electronic structure for a large number of hole doped High T c cuprates usingNMTO downfolding method. It is identified that the range of the intra-layer hopping as the essential materialdependent parameter and is controlled by the axial orbital, which is a hybrid between Cu 4s, apical oxygen pz andfarther orbitals. Materials with higher Tc,max have larger hopping ranges and the axial orbitals more localized in the
CuO2 planes. This work emphasized the importance of material dependence in understanding strongly correlatedsystems and till date received close to 500 citations. Another direction of nominee's research is the investigation of novel multi-functional materials both in bulk and innano-crystalline form with a goal to predict new systems with desirable properties. Some of these work are carried outin close collaboration with experimentalists. In the growing field of spintronics materials, there are considerabletheoretical and experimental activity in search for half-metallic magnets (HM) and diluted magnetic semiconductors(DMS) with high Curie temperature. In this respect, the nominee along with his graduate student investigated indetails the half-Heusler alloys. By careful and accurate DFT calculations on the family of half-Heusler alloys theywere able to identify several generic features in the electronic structure that are responsible for the half metallicbehavior. The detailed analysis presented in this work (JPCM 15, 7307, 2003) continues to be a benchmark forresearch in half-Heusler alloys and related systems as evidenced in close to 100 citations to this work. Based on theinsights obtained from these calculations they suggested that transition metal doped semi conducting half-Heusleralloys may be a new class of DMS material with very high Curie temperature (JPCM 17, 5037 2005; APL 89,212502, 2006). This theoretical prediction was validated by experiments thereby opening the possibility to considerthis system as a potential candidate for use in room-temperature spintronic device. In the area of DMS materials another important contribution of the nominee has been to understand the origin ofmagnetism in transition metal (TM) doped nano-crystalline ZnO. In a joint theoretical and experimental work (PRB75, 144404 2007) on ferromagnetic Fe doped ZnO nanocrystals, the nominee along with co-workers discovered thepresence of dopant Fe in an unusual valence state (Fe 3+) and suggested that the unusual valence is due to possiblehole doping in the system by cation (Zn) vacancies. This work that established the important role of defects inmediating ferromagnetism in Fe doped ZnO nanocrystals is well accepted by the DMS community as evidenced bymore than 200 citations of this work. A detailed theoretical investigation (APL 94, 192503, 2009) by the nominee ofthe energetics and magnetic interactions in Fe doped ZnO nano-clusters revealed that defects under suitable conditionscan induce ferromagnetic interactions between the dopant Fe atoms whereas the anti ferromagnetic couplingdominates in a neutral defect-free cluster. Another class of multi functional materials investigated by the nominee are multiferroic materials. Of particularimportance are improper multiferroics where ferroelectricity (FE) is induced by an inversion symmetry breakingmagnetic ordering resulting in strong coupling between the two order parameters. Some spiral magnets belong tothese class of materials. The correlation between the FE polarization and the cycloidal-spiral spin structure is due toDzyaloshinskii-Moriya (DM) interaction where the presence of the spin orbit coupling is indispensable in generatingdipole moments. In this context, the origin of ferroelectric polarization in the spiral magnetic structure of MnWO 4 isintriguing with a nominally d5 L=0 orbitally quenched state. In a recent work (PRB 81, 212406, 2010), the nomineeand his collaborators with the aid of detailed ab-initio electronic structure calculations and X-ray absorptionspectroscopy resolved this puzzle and provided a microscopic understanding of the phenomena. In another project, theelectronic structure of the low-dimensional multiferroic compound FeTe2O5Br was studied to investigate the origin ofthe magnetoelectric (ME) effect and the role of Te ions in this system. The exchange striction within the structural Fetetramers as well as between them is found to be responsible for the ME effect in FeTe 2O5Br. Further the Te4+ ions arefound to play an important role in the inter tetramer exchange striction as well as contributing to the electricpolarization once the polarization is triggered by the magnetic ordering (Phys. Rev. B 88, 094409, 2013). Materials at nano scale offer an unique possibility of tuning properties by tailoring sizes and shapes. Recentlycoupled quantum dots of ZnSe and CdS were synthesized that provided an alternative route to tune the electronicproperties via. band off-set engineering. Electronic structure calculations carried out by the nominee clarified thechemical bonding at the interface of these coupled dots and also provided an estimate of the band offsets that played akey role in understanding and controlling long-range visible fluorescence tunability of these coupled dots.(Advanced Materials, 23 1998, 2011, highlighted by Nature India) . The strain profiles of coupled quantum dotsare calculated using a multiscale approach where the parameters for an atomistic model for elasticity are calculatedusing ab-initio electronic structure calculations. Calculations revealed that the effect of strain is minimum in thecoupled quantum dots thereby providing an opportunity of designing interface as a novel quantum device (Phys. Rev.B, 89, 245423 , 2014). The nominee also clarified the important role of ligands in controlling the crystal structure ofnano-systems ( J. Phys. Chem Lett. 2, 706, 2011, J. Phys. Chem. C 116 6507 2012 ). The magnetic properties ofunique cluster assembled solids namely Mn Doped Ge46 and Ba8Ge46 clathrates was investigated and predicted tobe ferromagnetic (JPCM 24, 505501 2012). The nominee has also contributed to the development of an efficient real space approach (PRB 79, 224204, 2009) tosolve Bogoliubov de-Gennes (BdG) equations in systems modeled by disordered attractive Hubbard model. Thismethod proved to be important to understand the effect of disorder on superconductivity in single band as well asmulti-band systems (PRB B 84, 174508, 2011).
CURRICULUM VITAE
Name INDRA DASGUPTA
Date of Birth 20-th October, 1965Marital Status MarriedNationality Indian
Residential Address: LX-20504, USTA Condiville,New Town, RajarhatKolkata 700 156
Address For Department of Solid State Physics,Correspondence: Indian Association for the Cultivation of Science
Jadavpur,Kolkata 700 032India
E-mail [email protected] (0091)-(33)-24734971 Ext-1319, Off.
(0091)-(33)-23241232, Res.Fax (0091)-(33)-24732805
Present Position Senior ProfessorDepartment of Solid State Physics,IACS, Kolkata
University Education:
Ph.D, (1996) Theoretical Condensed Matter PhysicsCalcutta University, Kolkata, IndiaThesis title: Electronic Structure andTransport in Quantum Disordered Solids.
M.Sc,(1990) PhysicsIIT KANPUR
B.Sc,(1987*) Physics (Honours),Mathematics and StatisticsPresidency CollegeCalcutta University* results declared in 1988
Post doctoral Research Experience:
Institution Max Planck Institut fur FestkorperforschungStuttgart, GermanyDepartment of Prof. O.K. Andersen
Duration 4 years, (April 1995-April 1999)
Positions held:
(1) Junior Research Fellow, S.N. Bose National Center For Basic Sciences, Kolkata, (1991-1993).
(2) Senior Research Fellow, S.N. Bose National Center For Basic Sciences, Kolkata, (1993-1995).
(3) Research Associate Max- Planck Institute fuer Festkoerperforschung, Stuttgart, (April 1995-April 1999).
(4) Lecturer, Department of Physics, IIT Bombay, (June 1999-Jan-2001).
(5) Assistant Professor, Department of Physics, IIT Bombay, (Feb-2001-Feb-2005).
(6) Assistant Professor, Department of Physics and Meteorology, IIT Kharagpur, (on lien fromIIT Bombay), (July-Dec, 2003)
(7) Associate Professor, Department of Physics, IIT Bombay, (Mar-2005-June 2010).
(8) Associate Member, Centre for Computational Materials Science, JNCASR, Bangalore,(2006- present)
(9) Associate Professor, Department of Solid State Physics, IACS, Kolkata, (on lien from IITBombay), (June 2007 - Dec. 2008)
(10) Associate Member, Center for Advanced Materials, IACS, Kolkata, (June 2008 - present)
(11) Professor, Department of Solid State Physics, IACS, Kolkata,(Dec 2008- Dec 2013)
(12) Professor, Department of Physics, IIT Bombay (Feb 2009, promoted while on lien at IACS,Kolkata)
(13) Senior Professor, Department of Solid State Physics, IACS, Kolkata,(Dec 2013- present)
Visiting Positions:
(1) Visitor, International Centre for Theoretical Physics (ICTP), Trieste, Summer 1993, 1994,1995 and 1996.
(2) Visiting Scientist, Max Planck Institut fur Festkorperforschung, Stuttgart, Germany, (May- June, 2001), (May-June, 2006).
(3) Visitor, University of Missouri Columbia, USA, July-2005
(4) Visitor, Laboratoire de Physique des Solides, Universit Paris XI, Orsay, (June-July, 2008)
(5) Visitor, Uppsala University, Uppsala, Sweden (2010, 2011, 2012)
Scholarships and Awards:
(1) National merit scholarship awarded by Govt. of India on the basis of B.Sc Physics Honoursresult, (1988-1990).
(2) Qualified for the award of J.R.F. /Lecturership by Council of Scientific and IndustrialResearch (C.S.I.R.), (1989).
(3) Qualified in Graduate Aptitude Test in Engineering (G.A.T.E ,1990), securing 98.78 per-centile.
(4) Max-Planck Stipendium (1995-1999)
(5) Award for excellence in teaching from IIT Bombay, (2004)
Membership:
1. Member of American Physical Society.
2. Life member of Materials Research Society of India.
3. Life member of Indian Physics Association.
4. Life member of Indian Association for Cultivation of Science.
Field of Interest: Theoretical and Computational Condensed Matter PhysicsAreas Of Interest
1. Electronic structure calculations of complex crystalline solids, novel magnetic systemsincluding multiferroic and spintronic materials.
2. Electronic structure and magnetism in nanomaterials.
3. Extraction of low energy model Hamiltonians using ab-initio approaches.
4. Study of electronic structure of strongly correlated systems and low dimensional quantumspin systems.
5. Multiscale simulations of novel functional materials.
6. Study of electronic structure and phase stability of random alloys.
7. Study of superconductivity in disordered systems.
8. Quantum transmittance in two and three dimensional disordered media.
PhD Thesis Supervision:
Serial Student/ Title of Thesis Doctorate Year ofNo. research scholar Master’s level of Completion
1. Arup Chakraborty Electronic Structure and Ph.D OngoingTransport in Functional
Nanomaterials
2. Sayantika Bhowal Spin-Orbit Interaction Ph.D Ongoingin Strongly Correlated Systems
3. Atanu Paul Structural and Electronic Ph.D OngoingProperties of
Functional Materials
4. Atasi Chakraborty Electronic Structure of Ph.D OngoingOxide Interfaces
5. Subhadeep Bandyopadhyay Superconductivity in Ph.D OngoingDisordered Systems
Ph.D Thesis Completed:
1. Dr. Birabar Ranjit Kumar NandaThesis Title: First Principles Study of Novel Magnetic SystemsYear of Completion: Sept 2006Present Position: Assistant Professor, IIT Madras, Chennai
2. Dr. Sarita S. SalunkeThesis Title: Electronic Structure of Quantum Spin SystemsYear of Completion: March 2008Present Position: Petrophysicist, BP America Inc, Houston, USA
3. Dr. K.V. ShanavasThesis Title: Classical and Quantum Simulation of Novel Functional MaterialsYear of Completion: April 2011Present Position: Postdoctoral Research Associate, Oak Ridge National Laboratory, USA
4. Dr. Nirmal GanguliThesis Title: Electronic Structure of Functional NanomaterialsYear of Completion: April 2012Present Position: Postdoctoral Research Associate, Max Planck Institute for Solid StatePhysics, Stuttgart, Germany
5. Dr. Vijay R. SinghThesis Title: First Principles Study of Electronic Structure and Magnetism in Functional Ma-terials.
Year of Completion: May 2014Present Position: Postdoctoral Research Associate, Bar-Ilan University, Ramat Gan, Israel
6. Dr. Jayita ChakrabortyThesis Title: Electronic Structure of Novel Magnetic Systems.Year of Completion: June 2014Present Position: Postdoctoral Research Associate, University of Stuttgart, Stuttgart, Ger-many
7. Dr. Swarup Kumar PandaThesis Title: Study of Electronic Structure of Strongly Correlated Systems.Year of Completion: Nov. 2014Present Position: Postdoctoral Research Associate, Uppsala University, Sweden
Post Doctoral Research Associate Supervision:
1. Dr. Chol Sam Jong, Sept 2010-August 2011 (funded by IACS-TWAS programme)2. Dr. Bhaskar Kamble, March 2011- Sept 2011 (funded by DST)3. Dr. Shreemoyee Ganguly, Feb 2015 - present (funded by IACS)
Sponsored Projects Undertaken (as Principal Investigator):
Sponsoring Agency Title of Project Amount of grant Period
C.S.I.R First Principles Study of Rs. 5,94,000 2001-04Complex Crystalline Solids.
DST Ab initio Calculation of Rs. 16,20 000 2005-2008Electronic Properties of
Novel Magnetic Materials.
MHRD Modernisation of Computational Rs. 10,00 000 2005-2007Physics Laboratory
Swedish Research Theoretical and Experimental SEK 600 000 2007-2010Council Study of DMS Materials
DST and Modelling Nanostructured Advanced Rs. 4,00,00000 2009-2013European Union Materials Intelligently
DST and Electronic Structure and Rs. 4,84,000 2015-2017DAAD Transport in Functional Nano-Materials
Other Sponsored Projects Undertaken :
(a) Co-PI of an experiment-theory Indo-French project entitled Novel magnetic ground
state of low dimension system, in collaboration with Prof. A.V. Mahajan, IIT Bombayand the group of Prof. H. Alloul, Laboratorie de Physique des Solides, Universite de Paris-Sud, Orsay (2007-2010).
(b) Co-PI of an experiment-theory Indo-Japan project entitled Understanding novel mag-netic oxide nano-materials: Spectroscopy and ab-initio theories in collaborationwith Prof. D.D. Sarma, IISc. Bangalore and Prof. A. Fujimori, University of Tokyo(2008-2010).
(c) Co-PI of an experiment-theory BRNS project entitled Probing magneto structuraltransition in ferromagnetic shape Memory alloys and related materials incollaboration with Prof. S. Majumdar, IACS Kolkata and Dr. A. Das BARC Mumbai(2013-2015).
(d) Co-Investigator of Indo-Russian project entitled Spin-dependent phenomena in quasi- 2D structures and films with magnetic impurities in collaboration with Dr. V.Tripathi TIFR, Mumbai and Prof. B. Aronzon , Kurchatov Institute, Moscow Russia(2013-2015).
Other Research Activities:
Referee work for Physical Review Letters, Physical Review B, Physical Review E, ScientificReport (Nature Publishing Group), Solid State Communications, Journal of Applied Physics,Journal of Magnetism and Magnetic Materials, Journal of Physics D, International Journal ofModern Physics B, Modern Physics Letters, Pramanna and Indian Journal of Physics
PhD thesis examiner for Dhaka University, JNC Bangalore, University of Mumbai, JadavpurUniversity, SRM University, Chennai.
Honorary Associate Editor, Indian Journal of Physics, 2008-present
Conferences/Scientific Meetings Organized
1. Convener, Asia-Sweden workshop and conference on electronic structure theory: methodsand applications (IACS, Kolkata, Jan 2008).
2. Organizing committee member, for the 3-rd Indo-Japan conference on ferroics and multi-ferroics ( IACS, Kolkata Feb 2008)
3. Convener, 1-st IACS-APCTP Conference on Recent Trends in Strongly Correlated Sys-tems (IACS, Kolkata, March 2009)
4. Member, Organizing Committee for Magnetism, Superconductivity and Phase Transitionsin Novel and Complex Materials : MSM09 Meeting (S N Bose Centre Kolkata, Nov 2009)
5. Convener, Kick-off Meeting for the Indo-EU project Modeling Nanoscaled Advanced Ma-terials Intelligently (MONAMI) (Max Planck Institute, Stuttgart, Nov 2009)
6. Member, Organizing Committee for the 2-nd APCTP conference on Strongly CorrelatedSystems, ( Pohang Korea, July 2010)
7. Convener, Meeting for the Indo-EU project Modeling Nanoscaled Advanced Materials In-telligently (MONAMI) (IISc. Bangalore, January 2011)
8. Convener, 3-rd IACS-APCTP Conference on Novel Functional Materials, (IACS, Kolkata,Nov. 2011)
9. Convener, 4-th IACS-APCTP Conference on Novel and Emergent Materials, (APCTPHeadQuarters, Pohang Korea, Oct. 2012)
10. Co-organizer, 5-th IACS-APCTP Conference on Novel Oxide Materials and Low Dimen-sional Systems, (IISc Bangalore, Dec. 2013)
11. Co-organizer, ICTS School and Conference on Strongly Correlated Systems: From Modelsto Materials, (IISc Bangalore, Jan. 2014)
12. Member, Symposium Organizing Committee for DAE BRNS Symposium on MultiscaleModeling of Materials and Devices (MMMD - 2014) (BARC, Mumbai Nov 2014)
13. Convener, 6-th IACS-APCTP Conference on Novel Oxide Materials and Low DimensionalSystems, (Seoul National University, Seoul, Korea, Dec. 2014)
Invited Talks in School/Workshops/Conferences
1. A cluster of topical meetings on Current Trends in Condensed Matter Physics (CTCMP2015), NISER Bhubaneswar, Feb. 2015Invited Talk: Electronic Structure and Magnetism in Ir-based Oxides.
2. The 9th International Conference on Computational Physics (ICCP9), National Universityof Singapore, Singapore Jan. 2015Invited Talk: Superconductivity in Multiband Disordered Systems
3. 6-th IACS-APCTP Conference on Novel Oxide Materials and Low Dimensional Systems,Seoul National University, Seoul Korea, Dec. 2014Invited Talk: Electronic structure and magnetism in Ir-based oxides
4. DAE BRNS Symposium on Multiscale Modelling of Materials and Devices, BARC Mum-bai, Nov 2014Invited Talk: Electronic Structure of coupled quantum dots: A novel heterostructure atnano-scale
5. Summer School on Materials Simulations Theory and Numerics, IISER Pune, July 2014Invited Talk: Electronic Structure of Novel Magnetic Systems: Insights from Spin-PolarizedDensity Functional Theory Calculation
6. 7-th India-Singapore Symposium on Condensed Matter Physics, IIT Bombay, Feb 2014Invited Talk: Electronic Structure and Magnetism in low dimensional and cluster assembledsolids
7. 5-th APCTP-IACS Joint Conference on Novel Oxide Materials and Low Dimensional Sys-tems, IISC Bangalore Dec 2013Invited Talk: Electronic Structure and Magnetism in Novel Oxides
8. Physics colloquium at IIT Kanpur, October 2013.Title: First Principles Calculations: The glue that binds materials, models and mechanism.
9. Telluride Workshop on Physics of Emergent Correlated Materials, Telluride Colorado,USA, June 2013.Invited Talk: Electronic Structure of Correlated Oxides and Sulphides
10. School and Workshop on Electronic Structure Calculations with HPC Systems, Naukuchi-atal, Uttarakhand, (Organized by IUC New Delhi), May 2013Invited Lectures: Density Functional Theory: What do we get and how do we analyze?
11. First National Conference on Mapping the Materials Genome, Shiv Nadar University,Tehsil Dadri, March 2013Invited Talk: First Principles Study of Functional Nanomaterials
12. International Symposium on Science of Clusters, Nanoparticles and Nanoscale-Materials,Jaipur March 2013Invited Talk: First Principles Study of Functional Nanomaterials
13. Advances in Computational Physics 2013 (ACP2013), Central University of Tamil Nadu.Thiruvarur, Feb 2013Invited Talk: Electronic structure of strongly correlated systems
14. International Workshop on Computational Materials Design and Engineering, IIT Jodh-pur, Jodhpur Feb 2013Invited Talk: First Principles Study of Novel Materials
15. National workshop on electron dynamics in magnetic materials (EDMM-2013) Chandipur,Odisha, Jan 2013Invited Talk: Electronic Structure of Multiferroic Materials
16. Workshop on electronic structure approaches to atoms, molecules, clusters and solids, Univof Hyderabad, Jan 2013Invited Talk: Electronic Structure and Novel Functionalities in low dimensional and clusterassembled solids
17. Workshop on Novel Materials: Adding material-specific reality in physicists’ models, Natal, Brazil, Dec 2012Invited Talk: Spin-Orbit Effects in Oxides and Sulphides
18. 4-th IACS-APCTP conference on Novel and Emergent Materials, Pohang, South Korea,Oct 2012Invited Talk: Understanding the microscopic origin of magnetic ordering in FeTe
19. Indo Japan Conference on Functional Materials, IISc Bangalore, Oct 2012Invited Talk: Electronic Structure and Novel Functionalities in low dimensional and clusterassembled solids
20. 4-th International Conference on Advanced Nano-materials, ANM 2012, IIT Madras, Oct2012Invited Talk: Functional Nano-Materials insights from first principles calculations
21. ACCMS-Theme Meeting on 2D Nanostructures, Graphene and beyond, IISc Bangalore,July 2012Invited Lecture: Functional Nano-Materials insights from first principles calculations
22. School on Nano-Materials, Institute of Radio Physics, Calcutta University, Kolkata, June2012Invited Lecture: Functional Nano-Materials insights from first principles calculations
23. MONAMI Conference on Functional Materials, Uppsala University, Uppsala, June 2012Invited Lecture: Functional Nano-Materials insights from first principles calculations
24. School on Complex Oxides (ATHENA-2012), S.N. Bose Centre, Kolkata April 2012Invited Talk: Electronic structure of strongly correlated systems: Role of spin-orbit inter-action
25. Conference on Physics and Chemistry of Spintronics Materials, Coorg, February 2012Invited Talk: Understanding the microscopic origin of magnetic ordering in FeTe
26. International Conference on Advanced Functional Materials, Coimbatore, Dec 2011Invited Talk: Electronic structure of Improper Multiferroics
27. ICTS Condensed Matter Program, IISc Bangalore, Dec 2011Invited Talk: Understanding the microscopic origin of magnetic ordering in FeTe
28. Refresher Course in Physics, Calcutta University, Kolkata, July 2011A course of four lectures on Electronic properties of materials
29. Indo-EU Network Meeting, Uppsala , June 2011Invited Talk: Electronic structure of nanomaterials: Insights from first principles calcula-tions
30. ICTS Condensed Matter Program, Mysore, Dec 2010Invited Talk: Electronic structure of strongly correlated systems: Role of spin orbit inter-action.
31. Current Trends in Condensed Matter Physics, Dec 2010, NISER , BhubaneswarInvited Talk: Electronic structure of functional nanomaterials
32. L’ Aquilla initiatives in ferroics, L’Aquilla Italy, Sept 2010Invited Talk: Origin of ferroelectric polarization in spiral magnetic structure of MnWO4
33. Recent Trends in Strongly Correlated Systems, APCTP, Pohang, Korea July 2010Invited Talk: Origin of ferroelectric polarization in spiral magnetic structure of MnWO4
34. Conference on electronic structure of functional materials, Uppsala, June 2010Invited Talk: Electronic structure of functional nanomaterials
35. National Workshop on electron dynamics in quantum systems, Digha Feb 2010Invited Talk: First principles modeling of materials: Application to diluted magnetic semi-concuctors
36. Frontiers in electronic structure calculations: Techniques and Applications, Pune Feb 2010Invited Talk: Electronic structure of strongly correlated systems: Role of spin orbit inter-action
37. Indo-French Workshop on spintronics, Varanasi Jan 2010Invited Talk: Electronic structure of spintronic materials: Insights from first principlescalculations
38. Recent Advances in Correlated electron systems, IIT Gauhati Jan 2010Invited Talk: Strongly Correlated systems: Role of spin-orbit interaction
39. ICTS Condensed Matter Programme, Mahabaleshwar Dec 2009Invited Talk: Strongly Correlated systems: Insights from first principles calculations
40. Startup Meeting of Modeling Nanostructered Advanced Materials Intelligently, StuttgartGermany Nov 2009Invited Talk: Electronic structure of novel magnetic materials: Insights from first princi-ples calculations
41. International Workshop on High Performance Computing and Modeling of Complex Sys-tems at the Peiking University Beijing, March 2009Invited Talk: Understanding Novel Materials: Insights from First Principles Density Func-tional Calculations
42. Recent Trends in strongly correlated systems, IACS, Kolkata March 2009Invited Talk: Electronic structure of novel oxides
43. Meeting on Physics and Chemistry of Oxide Materials, S.N. Bose Centre, Feb 2009.Invited Talk: Multifunctional Oxides: Insights from First Principles Calculations
44. Homi Bhabha Centenary DAE-BRNS School on Spintronic and Magnetoelectric Materialsand Devices, Toshali Sands, Puri, Jan 2009Invited Talk: Electronic structure, Model Hamiltonians, and Mechanisms in Half-metallicand DMS systems
45. Indian Condensed Matter Workshop (ICMW08), Mahabaleshwar, Dec 2008.Invited Talk: Electronic structure of Novel Oxides
46. 3-rd Indo Japan Conference on Ferroics and Multiferroics, IACS and S.N. Bose CentreCalcutta, Feb 2008.Invited Talk: Electronic structure of oxide based diluted magnetic semiconductor
47. Workshop and Conference on electronic structure theory: Methods and Applications, SNBose Centre, Calcutta, Jan 2008Invited Talk: Magnetism in low dimensional oxides: A perspective from first principlescalculations.
48. Symposium on Atomic, Molecular and Optical Sciences, Vedic Village, Calcutta, Jan 2008Invited Talk: Magnetism in low dimensional oxides: A perspective from first principlescalculations.
49. Conference on Correlated Electrons and Frustrated Magnetism, Goa, Dec 2007Invited Talk: First principles study of strongly correlated systems
50. International Conference on Materials for Advanced Technologies (ICMAT), Singapore,July 2007Invited Talk: First Principles Electronic Structure of Novel Oxides
51. Symposium on Trends in Computational Materials Science, CCMS, JNC, Bangalore, Feb2007Invited Talk: Magnetism in novel spin chain oxides: A perspective from first principlescalculations
52. Workshop on Electronic Structure of Emerging Materials: Theory and Experiment, Lon-avala, Feb, 2007Invited Talk: First principles electronic structure of diluted magnetic semiconductors withhigh Curie temperature
53. Asia Sweden Research Links Conference on Magnetism in Materials, Calcutta, Jan 2007Invited Talk: Electronic structure of novel one dimensional oxides.
54. Workshop on Correlated Systems and Novel Materials (CSNM-07), CTS, IIT Kharagpur,Jan 2007Invited Talk: First principles electronic structure of diluted magnetic semiconductors withhigh Curie temperature
55. Conference on simulations bridging length scales, Mumbai, Oct 2006Invited Talk: Electronic structure of half-metallic magnets
56. Workshop on Computational Materials Theory, July 2006, BangaloreInvited Talk: First Principles electronic structure and design of half-metallic magnets.
57. Discussion Meeting on Materials and Molecular Modelling, March 2006, CalcuttaInvited Talk: Going beyond LDA for strongly correlated systems: LDA+U approach
58. A meeting on condensed matter physics, March 2006, CalcuttaInvited Talk: A real space approach to study superconductivity in disordered alloys
59. Indo-EU Thematic Meeting on Computational Material Science, Feb 2006, IISc BangaloreInvited Talk: Electronic Structure of half-metallic magnets: Issues and Prospects
60. International Conference on Advanced Materials Design and Development, Goa, Dec 2005Invited Talk: Electronic structure of half-metallic magnets
61. Telluride Workshop on the Physics of the Magnetic Oxides, Telluride Colorado (USA) ,August 2005Invited Talk: First principles study of electronic structure and magnetism in novel spinchain oxides
62. Second Conference of Asian Consortium for Computational Materials Science (ACCMS-2),Novosibirsk, Russia, July 2004Invited Talk: First Principles Study of Novel Magnetic and Strongly Correlated Systems
63. Workshop on physics of novel materials: Electronic and Magnetic properties, S.N. BoseCentre, Calcutta Jan 2004.Invited Talk: A tale of two gaps
64. DAE-BRNS theme meeting on spintronics materials, BARC Bombay Dec 2003Invited Talk: Electronic structure of half-metallic systems: Semi-Heusler alloys a casestudy
65. Electronic Structure of Condensed Matter -II, Ringberg, Germany, June 2003Invited Talk: Superconductivity in disordered alloys: a real space approach
66. Discussion meeting on strongly correlated systems, Harish Chandra Institute, Allahabad,Dec 2002.Invited Talk: Electronic Structure of strongly correlated systems.
67. D.A.E. Solid State Physics Symposium, Panjab University, Chandigarh, Dec, 2002Invited Talk: Electronic Structure of Magnetic Superconductors
68. Electronic Structure of Condensed Matter -1, Ringberg, Germany, May 2002Invited Talk: Electronic Structure of Magnetic Superconductor RuSr2GdCu2O8
69. First Conference of Asian Consortium for Computational Materials Science (ACCMS-1) ,Bangalore, 29 Nov to 1 Dec, 2001.Invited Talk: Electronic Structure of Magnetic Superconductor RuSr2GdCu2O8
70. Research Perspectives and Projections in Condensed Matter Physics, Jan 2-4 , 2001, Cal-cuttaInvited Talk: Band-structure trend in cuprates and correlation with Tc,max.
71. The national conference on computational material science (NCCMS-2000), Bombay July.2000Invited Talk: Tight-binding modelling of materials.
72. QIP course on Modern magnetic materials, IIT Bombay, May 2000Invited Lectures (nos. 4) : First principles study of bulk and surface magnetism
73. D.A.E. Solid state physics symposium, Kalpakkam, Dec. 1999Invited Talk: Electronic structure of High Tc cuprates.
74. S.E.R.C school on electronic structure and physics of materials, Calcutta, Nov. (1999)Guest Lecturer: Trends in LDA derived model Hamiltonians for High Tc cuprates.
75. International conference on physics of materials, Calcutta, Nov. (1999)Invited Talk: Electronic Structure and Exchange interactions in Ladder vanadates.
76. University of Miami conference on High Temperature Superconductivity, Miami, Florida,U.S.A., Jan. (1999)Invited Talk: Trends in LDA derived model Hamiltonians for High Tc cuprates.
77. Workshop on the Bogoliubov-de-Gennes Equations for Superconductors, Bristol, Nov.(1998).Invited Talk: Trends in LDA derived model Hamiltonians for High Tc cuprates
78. Workshop on the Bogoliubov-de-Gennes Equations for Superconductors, Bristol, July(1997)Invited Talk: Low Energy Model Hamiltonians for High Tc Superconductors.
79. Network Workshop On Disordered Alloys, University Of Dhaka, Bangladesh, June (1994)Invited Talk: Short-Range Ordering Effect in Alloys.
Teaching and Lecture/Seminar Courses given :
1. Advanced Solid State Physics (Course Work for PhD Program, IACS) : A course of 25lectures (2015)
2. Mathematics for Chemists (CH 411, Post BSc Integrated PhD, IACS) : A course of 15Lectures on Linear Vector Spaces and Complex Analysis (2010,2011, 2012-2015)
3. Physics and Chemistry of Materials (CH 521, Post BSc Integrated PhD, IACS) : A courseof 25 lectures (2010).
4. Theoretical Condensed Matter Physics (Course Work for PhD Program, IACS) : A courseof 25 lectures (2012, 2013, 2014).
5. Material Physics (PH 522, Advance II Physics, M.Sc. University of Calcutta) : A courseof 10 lectures on Computational Material Science (2012, 2013, 2014, 2015).
6. Theoretical Condensed Matter Physics (PH 522, M.Sc Ramakrishna Mission Narendrapur,Kolkata): A course of 10 lectures on Density Functional Theory and Optical Properties ofSolids (2008-2013).
7. Refresher Course in Physics ( Department of Physics, Calcutta University, Kolkata): Acourse of four lectures on Electronic properties of materials (2011)
8. Classical Mechanics (EP-206), B.Tech Engg. Physics, IIT Bombay (2000)
9. Mathematical Methods (PH-407), M.Sc. IIT Bombay (2000,2001,2002)
10. Theoretical Condensed Matter Physics (PH 522) B.Tech Engg. Phys and M.Sc, IIT Bom-bay (2001, 2002, 2004-2007)
11. Advanced Simulation Techniques (PH 810) B.Tech Engg. Phys, M.Sc and PhD, IIT Bom-bay (2003)
12. Electronic Theory of Solids, MSc, IIT Kharagpur, (2003)
13. Mechanics (PH-101) Institute Core, IIT Bombay, (2004-2006)
Other Academic Activities:
Associated with IIT JEE, JAM and GATE examinations at various levels. Co-ordinating headexaminer (IIT Bombay) for IIT JEE (2005).Reviewer for a web based book on physics for undergraduate students.
Prof. Indra Dasgupta
1. Research Publications(Total Citations 1710, Source Google Scholar, April 2015)
1. Exchange striction induced giant ferroelectric polarization in copper based multiferroicmaterial α-Cu2V2O7
J Sannigrahi, S Bhowal, S Giri, S Majumdar, I DasguptaSubmitted for Publication ( arXiv preprint arXiv:1501.00809) (2015)
2. First-principles study of the electronic structure of CdS/ZnSe coupled quantum dots.N. Ganguli, S. Acharya, and I. DasguptaPhys. Rev B 89 245423, (2014)
3. A charge self-consistent LDA+DMFT study of the spectral properties of hexagonal NiS.S K Panda, P Thunstrom, I Di Marco, J Schott, A Delin, I Dasgupta, O Eriksson andD D SarmaNew Journal of Physics 16 , 093049, (2014)
4. Effect of spin orbit coupling and Hubbard U on the electronic structucture of IrO2.S.K. Panda, S. Bhowal, A. Delin, O. Eriksson, and I. DasguptaPhys. Rev B 89, 155102, (2014)
5. Magnetic properties and heat capacity of the three-dimensional frustrated S=1/2 antifer-romagnet PbCuTe2O6. B. Koteswararao, R. Kumar, P. Khuntia, Sayantika Bhowal, S. K.Panda, M. R. Rahman, A. V. Mahajan, I. Dasgupta, M. Baenitz, Kee Hoon Kim, andF. C. ChouPhys. Rev. B 90, 035141 (2014)
6. NiS -An unusual self-doped, nearly compensated antiferromagnetic metal.S.K. Panda, I. Dasgupta, E. Sasioglu, S. Bluegel, and D.D. SarmaScientific Report (Nature Publishing Group) 3, 2995 (2013)
7. Role of Te in the low-dimensional multiferroic material FeTe2O5Br.J. Chakraborty, N. Ganguli, T. Saha-Dasgupta, and I. DasguptaPhysical Review B 88, 094409, (2013)
8. PbCu3TeO7: an S = 12 staircase kagome lattice with significant intra-plane and inter-plane
couplings.B Koteswararao,, R Kumar, Jayita Chakraborty, Byung-Gu Jeon, A V Mahajan, I Das-gupta, Kee Hoon Kim, and F C ChouJ. Phys.: Condens. Matter 25 336003 (2013)
9. Electronic Structure and Magnetism in Ir based double perovskite Sr2CeIrO6Swarup Kr. Panda and Indra DasguptaMod. Phys. Lett 27, 1350041, (2013)
10. Magneto-structural transitions: Molecular dynamics simulation of a united atom meso-scopic model.J. Bhattacharya, V. Singh, S. Sengupta, and I. DasguptaMod. Phys. Lett 27, 1450047, (2013)
11. First-principles study of the spin-gap system Sr2Cu(BO3)2Jayita Chakraborty and Indra DasguptaPhysical Review B 86 , 054434 (2012)
12. Magnetic properties of Mn-doped Ge46 and Ba8Ge46 clathratesNirmal Ganguli, K.V. Shanavas, and Indra DasguptaJournal of Physics: Condensed Matter 24, 505501 (2012)
13. How Crucial Are Finite Temperature and Solvent Effects on Structure and AbsorptionSpectra of Si10?NA Murugan, I Dasgupta, A Chakraborty, N Ganguli, J Kongsted, H AgrenThe Journal of Physical Chemistry C 116 26618, (2012)
14. X-Ray Absorption Spectroscopy and X-Ray Magnetic Circular Dichroism Studies of Transition-Metal-Codoped ZnO Nano-Particles.T. Kataoka, Y. Yamazaki, V. R. Singh, Y. Sakamoto, K. Ishigami, V. K. Verma, A. Fuji-mori, F.-H. Chang, H.-J. Lin, D. J. Huang, C. T. Chen, D. Asakura,T. Koide, A. Tanaka,D. Karmakar, S.K. Mandal, T.K. Nath, and I. Dasguptae-J. Surf. Sci. Nanotech. 10 , 594 (2012)
15. LaSrVMoO6: A case study for A-site covalency-driven local cationic order in double per-ovskitesSomnath Jana, Vijay Singh, Abhishek Nag, Carlo Meneghini, Indra Dasgupta, GiulianaAquilanti, Sugata RayPhysical Review B 86, 014203, (2012)
16. Magnetic behavior of Ba3Cu3Sc4O12.B. Koteswararao, A V Mahajan, F. Bert, P Mendels, J. Chakraborty, V. R. Singh, I.Dasgupta, S. Rayaprol, V. Siruguri, A Hoser, and S.D. KaushikJournal of Physics: Condensed Matter 24 (23), 236001, (2012)
17. First-Principles Study of the Effect of Organic Ligands on the Crystal Structure of CdSNanoparticlesK V Shanavas, S M Sharma, I. Dasgupta, A. Nag, A. Hazarika and D.D. SarmaJournal of Physical Chemistry C 116 (11), 6507, (2012)
18. A theoretical and experimental study of magnetism in Gd2InV. Singh, A. Bhattacharyya, S. Majumdar and I. DasguptaJ. Appl. Phys. 111, 053709 , (2012)
19. Electronic and magnetic structures of bilayer La3Ni2O6 and trilayer La4Ni3O8 nickelatesfrom first principles.Soumyajit Sarkar, I. Dasgupta, Martha Greenblatt, and T. Saha-DasguptaPhys. Rev. B 84, 180411(R) (2011)
20. Tuning of Long range Visible Emissions Using Coupled Quantum DotsSucheta Sengupta, Nirmal Ganguli, I. Dasgupta, D. D. Sarma and Somobrata AcharyaAIP Proceedings, 1349, 277 (2011)
21. Superconductivity in multiband disordered systems: A vector recursion approach.Shreemoyee Ganguly, Indra Dasgupta and Abhijit MookerjeePhys. Rev. B 84, 174508 (2011)
22. Long Range Visible Fluorescence Tunability Using Component Modulated Coupled Quan-tum Dots.Sucheta Sengupta, Nirmal Ganguli, I. Dasgupta, D.D. Sarma, and Somobrata AcharyaAdvanced Materials, 23 1998, (2011)
23. Crystal structure engineering by fine tuning the surface energy : The case of CdE (E=S/Se)A. Nag, A. Hazarika, K,V. Shanavas, S.M. Sharma, I. Dasgupta and D.D. SarmaJournal of Physical Chemistry Letters, 2 706, (2011)
24. Electronic structure and magnetism of transition metal doped Zn12O12 clusters: Role ofdefectsN. Ganguli, I. Dasgupta, and B. SanyalJournal of Applied Physics 108 , 123911, ( 2010 )
25. Atomic-scale chemical fluctuation in LaSrVMoO6, a proposed half-metallic antiferromag-netS. Jana, V. Singh, S.D. Kaushik, C. Meneghini, P. Pal, R. Knut, O. Karis, I. DasguptaV. Siruguri, and S. RayPhys. Rev. B Rapid Communication, 82, 180407 (2010)
26. Origin of ferroelectric polarization in spiral magnetic structure of MnWO4K.V. Shanavas, D. Choudhury, I. Dasgupta, S.M. Sharma and D.D. SarmaPhys. Rev. B 81, 212406 ( 2010)
27. A real space approach to study the effect of off-diagonal disorder on superconductivityS. Ganguly, I. Dasgupta and A. MookerjeePhysica C- Superconductivity and Applications 470, 640 , (2010)
28. Surface- and bulk-sensitive x-ray absorption study of the valence states of Mn and Co ionsin Zn1−2xMnxCoxO nanoparticles.T. Kataoka, Y. Yamazaki, Y. Sakamoto, A. Fujimori, F. H. Chang, H.J. Lin, D.J. Huang,C.T. Chen, A. Tanaka, S.K. Mandal, T.K. Nath, D. Karmakar and I. DasguptaAppl. Phys. Lett 96, 252502 (2010)
29. Electronic structure and magnetic properties of (Fe,Co)-codoped ZnO: Theory and exper-imentD. Karmakar, T.V.C. Rao, J.V. Yakhmi, A. Yaresko, V.N. Antonov, R.M. Kadam, S.K.Mandal, R. Adhikari, A.K. Das, T.K. Nath, N. Ganguli, I. Dasgupta and G.P. DasPhys. Rev. B 81 , 184421, (2010)
30. Electronic structure and magnetism of the diluted magnetic semiconductor Fe-doped ZnOnanoparticles.T. Kataoka T, M. Kobayashi, Y. Sakamoto, G.S. Song, A. Fujimori, F.H. Chang, H.J. Lin,D.J. Huang, C.T. Chen, T. Ohkochi, Y. Takeda, T. Okane, Y. Saitoh, H. Yamagami, A.Tanaka, S.K. Mandal, T.K. Nath, D. Karmakar, and I. DasguptaJournal of Applied Physics 107 , 033718 ( 2010 )
31. Magnetic properties of Ni2+xMn1-xIn Heusler alloys: Theory and experiment.S. Chatterjee,V.R. Singh, A.K. Deb, S. Giri, S.K. De, I. Dasgupta, and S. MajumdarJournal of Magnetism and Magnetic Materials, 322, 102 ( 2010)
32. Electronic structure of FeCr2S4: Evidence of Coulomb enhanced spin-orbit splittingS. Sarkar, M. De Raychaudhury, I. Dasgupta, and T. Saha-DasguptaPhys. Rev. B 80, 201101, (2009)
33. Augmented space recursion study of the effect of disorder on superconductivityS. Ganguli, A. Venkatasubramanian, K. Tarafdar, I. Dasgupta and A. MookerjeePhys. Rev B 79 , 224204 (2009)
34. The making of magnetic ZnO nano-clusters.Nirmal Ganguli, I. Dasgupta, and B. SanyalApp. Phys Lett 94, 192503 (2009)
35. Quantum transmittance through random media.A. Mookerjee, T. Saha-Dasgupta and I. DasguptaQuantum and semi-classical percolation and breakdown in disordered solids ed. A.K.Sen, K.K. Bardhan and B.K. Chakrabarti (Lecture Notes in Physics, Springer-VerlagHeidelberg), 83, (2009)
36. Reply to ”Comment on ’Electronic structure of spin-(1)/(2) Heisenberg antiferromagneticsystems: Ba2Cu(PO4)(2) and Sr2Cu(PO4)(2)’ ”S. Salunke, M.A.H. Ahsan, R. Nath, A.V. Mahajan, and I. DasguptaPhys. Rev B 79, 127102, ( 2009 )
37. X-ray Magnetic Circular Dichroism investigations of the origin of room temperature fer-romagnetism in Fe doped ZnO nano-particles.T. Kataoka, M. Kobayashi, G. S, Song, Y. Sakamoto, A. Fujimori, F.H. Chang, Hong-JiLin, Di Jing Huang, C T Chen, S.K. Mandal T.K. Nath, D. Karmakar and I. DasguptaJpn. J. Appl. Phys. 48, 04C200 (2009)
38. Electronic structure of Na2CuP2O7: A nearly 2D Heisenberg antiferromagnetic system.S. Salunke, V.R. Singh, A.V. Mahajan, and I. DasguptaJ. Phys: Condens. Matter 21, 025603 (2009)
39. Magnetic properties and electronic structure of S=12 spin gap compound BaCu2V2O8.
S. Salunke, A.V. Mahajan, and I. DasguptaPhys. Rev. B 77, 012410 (2008)
40. Spintronics- A Revolution in Materials Science and Device PhysicsGour P. Das and Indra DasguptaPhysics News 38, 46 (2008)
41. Spin gap behaviour in a 2-leg spin-ladder BiCu2PO6
B. Koteswararao, S. Salunke, A.V. Mahajan, I. Dasgupta and J. BobroffPhys. Rev. B 76, 052402 (2007)
42. Electronic structure of spin 1/2 Heisenberg antiferromagnetic systems: Ba2Cu(PO4)2 andBa2Cu(PO4)2Sarita S. Salunke, M.A.H. Ahsan, R. Nath, A.V. Mahajan and I. DasguptaPhys. Rev. B 76, 085104 (2007)
43. High temperature ferromagnetism in Fe-doped ZnO : a Density Functional InvestigationD. Karmakar, I. Dasgupta, G.P. Das and Y. KawazoeMaterials Trasactions, 48, 2119 (2007).
44. Ferromagnetism in Fe-doped ZnO nanocrystals: experimental and theoretical investiga-tions.D. Karmakar, S.K. Mandal, R.M. Kadam. P.L. Paulose, A.K. Rajarajan, T.K. Nath, A.K.Das, I. Dasgupta and G.P. DasPhys. Rev. B 75, 144404 (2007)
45. Ferromagnetism in Mn doped half-Heusler NiTiSn: theory and experimentB. Sanyal, O. Eriksson, K.G. Suresh, I. Dasgupta , A.K. Nigam, and P Nordb ladAppl. Phys. Lett. 89, 212502 (2006)
46. Electronic structure of half-metallic magnets.B.R.K. Nanda and I. DasguptaJ. Compt. Mat. Sci. 36, 96 (2006)Invited Paper
47. Electronic structure and magnetism in doped semiconducting half-Heusler co mpounds.B.R.K. Nanda and I. DasguptaJ. Phys: Condens. Matter, 17, 5037 ( 2005)
48. Symmetry reduction in the augmented space recursion formalism for random b inary alloys.K.K. Saha, T. Saha-Dasgupta, A. Mookerjee, and I. DasguptaJ. Phys: Condens. Matter ,16 , 1409, (2004)
49. A first principles thermodynamic approach to ordering in disordered alloys.A. Mookerjee, T. Saha-Dasgupta, I. Dasgupta, G.P. Das, A. Arya, and S. Ban erjeeBull. of Mater. Sci. 26, 79, (2003)
50. Electronic structure and magnetism in half-Heusler compounds.B.R.K. Nanda, and I. DasguptaJ. Phys: Condens. Matter , 15 , 7307, (2003)
51. Nature of insulating state in NaV2O5 above charge-ordering transition: a cluster DMFTstudy.V.V. Mazurenko, A.I. Lichtenstein, M.I. Katsnelson, I. Dasgupta, T. Saha-D asgupta,V.I. Anisimov.Phys. Rev. B 66, Rapid Commun. 081104(R) (2002)
52. Electronic structure of Chevrel phase compounds, SnxMo6Se7.5: Photoemission spec-troscopy and Band structure calculations.K. Kobayashi, A. Fujimori, T. Ohtani, I. Dasgupta, O. Jepsen and O.K. AndersenPhys Rev. B 63, 195109, (2001).
53. Band-structure trend in cuprates and correlation with Tc,max.E. Pavarini, I. Dasgupta, T. Saha-Dasgupta, O. Jepsen and O.K. AndersenPhys Rev. Lett. 87, 047003, (2001)
54. A first principles thermodynamic approach to ordering in Ni-Mo alloys.A. Arya, S. Banerjee, G.P. Das, I. Dasgupta , T. Saha-Dasgupta and A. MookerjeeActa mater, 49 , 3575, (2001)
55. Electronic structure and exchange interactions of the ladder vanadates CaV2O5 and MgV2O5.M.A. Korotin, V.I. Anisimov, T. Saha-Dasgupta, and I. Dasgupta.J. Phys Condens Matter 12, 113 (2000)
56. Electronic structure of High Tc cuprates.I. Dasgupta and O.K. AndersenProceedings of the DAE Solid State Physics Symposium, 42, 26, (1999) Invited Paper.
57. Buckling and d-wave pairing in High Tc cuprates.O. Jepsen, O.K. Andersen, Indra Dasgupta, and S.Y. SavrasovJ. Phys. Chem. of Solids 59, 1718 (1998)
58. Electronic structure of Ladder Cuprates.T. Muler, V. Anisimov, T.M. Rice, I. Dasgupta, and T. Saha-Dasgupta.Phys Rev. B 57, R12655 (1998)
59. Third-Generation TB-LMTOO.K. Andersen, C Arcangeli, R.W. Tank, T. Saha Dasgupta, G. Krier, O. Jepsen and I.DasguptaTight-Binding Approach to Computational Materials Science, P.E.A. Turchi, A. Gonis,and L. Colombo, eds., MRS Proceedings 491, Materials Research Society, Warrendale,PA, (1998).
60. Study of Transition metal aluminide alloys.Indra Dasgupta, Tanusri Saha-Dasgupta, Abhijit Mookerjee, and G P DasJ. Phys Condens Matter 9, 3529 (1997)
61. Augmented space recursion method for the calculation of electronic structure of randomalloys.Indra Dasgupta, Tanusri Saha-Dasgupta and Abhijit MookerjeeProperties of Complex Inorganic Solids, Plenum, ed. A Gonis, A. Meike and P.E.A Turchi,63 (1997)
62. Augmented space recursive approach for alloy phase stability.Tanusri Saha-Dasgupta , Indra Dasgupta and Abhijit MookerjeeProperties of Complex Inorganic Solids, Plenum ed. A Gonis, A. Meike and P.E.A Turchi,25 (1997)
63. Electronic structure of random binary alloys.Tanusri Saha , Indra Dasgupta and Abhijit MookerjeeJ. Phys Condens Matter 8, 1979 (1996 )
64. An Augmented Space Recursive Method For The First Principles Study of concentrationprofiles at CuNi Alloy Surfaces.Indra Dasgupta and Abhijit Mookerjee.J. Phys Condens Matter 8, 4125 (1996 )
65. An Augmented Space Recursive Technique for the analysis of alloy phase stability inrandom binary alloys.Indra Dasgupta, Tanusri Saha and Abhijit MookerjeePhys. Rev. B 51, 3413 (1995)
66. Quantum Percolation (Review Article)Abhijit Mookerjee, Indra Dasgupta, and Tanusri SahaInt. J. Mod. Phys. B 9, 2989 (1995)
67. Study of electrons in contact with non-markovian baths.Indra Dasgupta, and Abhijit MookerjeeJ. Phys. Condens Matter 6, 1545 (1994).
68. Stochastic Resonances and the Mobility Edge in the three dimensional tight binding An-derson Model .Indra Dasgupta, Tanusri Saha, and Abhijit MookerjeePhys. Rev. B 50, 4867 (1994) .
69. An Augmented Space recursive Technique for the calculation of electronic structure ofrandom binary alloys.Tanusri Saha, Indra Dasgupta, and Abhijit MookerjeeJ. Phys. Condens. Matter 6, L245 (1994)
70. An Augmented Space Recursive Method for the study of short range ordering effects inbinary alloysTanusri Saha, Indra Dasgupta and Abhijit MookerjeePhys. Rev. B 50, 13267 (1994).
71. Quantum Transmittance And Resonance In 2-D Quantum Percolation ModelIndra Dasgupta, Tanusri Saha , Abhijit Mookerjee, and B. K. ChakrabartiA.I.P Conference Proceedings 286 , 249 (1994)
72. Analysis of Stochastic Resonances in two dimensional Quantum Percolation Model.Indra Dasgupta, Tanusri Saha, and Abhijit MookerjeePhys. Rev. B 47, 3097 (1993).
73. Is There A Delocalization Transition In a Two-dimensional Model For Quantum Percola-tion ?Indra Dasgupta, Tanusri Saha, Abhijit Mookerjee, and B. K. ChakrabartiMod. Phys. Lett. B 6 , 817 (1992).
74. Quantum Percolation and breakdown. Absence of the delocalization transition in twodimensions.Abhijit Mookerjee, Bikas K. Chakrabarti , Indra Dasgupta, and Tanusri SahaPhysica A 186 , 258 (1992)
75. Scaling of resistance in two dimensional tight binding Anderson Model.Indra Dasgupta , Tanusri Saha, and Abhijit MookerjeeJ. Phys. Condens Matter 4, 7865 (1992).
2. Patent
1. Asymmetric heterostructures for synergistic tunability of visible emissionS. Sengupta, N. Ganguli, I. Dasgupta, D.D. Sarma, and S. AcharyaIndian Patent application no.1050/KOL/2010, priority date 21st september, 2010
3. Other Publications:
1. Electronic Structure and Magnetism of Ir based Oxides.Swarup K. Panda and I. DasguptaBulletin of the American Physical Society, 59 (2014)
2. Spin spiral state in hexagonal NiSR. Lizarraga, E. Holmstorm, L. Nordstrom, O. Eriksson, S.K. Panda, I. Dasgupta andD.D. SarmaBulletin of the American Physical Society, (2012)
3. Ab-initio Electronic Structure Calculation of Hexagonal NiSS. K. Panda, I. Dasgupta, and D. D. SarmaProceedings of the 53rd DAE Solid State Physics Symposium (2008)
4. First principles study of the multiferroic: MnWO4.K.V. Shanavas, I. Dasgupta , and S.M. SharmaProceedings of the DAE Solid State Physics Symposium,52, 775 (2007)
5. Electronic structure of S=1/2 1D Heisenberg Antiferromagnetic systems: (Sr,Ba)2Cu(PO4)2Sarita Salunke, R. Nath, A.V. Mahajan and I. DasguptaProceedings of the DAE Solid State Physics Symposium,50, 595 (2005)
6. Electronic Structure of the spin chain compound Ca3Co2O6.B.R.K. Nanda, and I. DasguptaProceedings of the DAE Solid State Physics Symposium,49, 592, (2004)
7. Electronic structure of half-Heusler compounds: Narrow-gap thermoelectric materials.B.R.K. Nanda, K.Deepa, and I. DasguptaProceedings of the DAE Solid State Physics Symposium,44, 313,(2001).
8. Electronic structure of divalent and trivalent hexaborides.G.P. Das, I. Dasgupta, and A. AryaProceedings of the DAE Solid State Physics Symposium,42, 433, (1999)
9. A New Mechanism Of Transport In Quantum Percolation ModelTanusri Saha, Indra Dasgupta, and Abhijit MookerjeeProceedings of the DAE Solid State Physics Symposium, 35 C , (1992)
10. Fine Structure Of Anderson Transition.Indra Dasgupta, Tanusri Saha and Abhijit MookerjeeProceedings of the DAE Solid State Physics Symposium, 36 C , (1993)
11. Fermi-Liquid to Non Fermi-Liquid Transition In A Dynamical Generalization Of The CPAIn A Disordered Hubbard Model.Indra Dasgupta and Abhijit MookerjeeICTP Preprint IC 93, 213 (1993)
12. An augmented space recursion technique for the calculation of electronic structure of ran-dom binary alloys: IIB. Sanyal, P. Biswas, I Dasgupta, and A MookerjeeICTP preprint, IC 96, 121 (1996)
PHYSICAL REVIEW B 89, 155102 (2014)
Effect of spin orbit coupling and Hubbard U on the electronic structure of IrO2
S. K. Panda,1 S. Bhowal,2 A. Delin,3,4,5 O. Eriksson,5 and I. Dasgupta1,2,*
1Centre for Advanced Materials, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India2Department of Solid State Physics, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India
3Department of Materials and Nanophysics, School of Information and Communication Technology, Electrum 229,Royal Institute of Technology (KTH), SE-16440 Kista, Sweden
4SeRC (Swedish e-Science Research Center), KTH, SE-10044 Stockholm, Sweden5Department of Physics and Astronomy, Uppsala University, P.O. Box 516, SE-751 20 Uppsala, Sweden
(Received 4 December 2013; revised manuscript received 28 February 2014; published 2 April 2014)
We have studied in detail the electronic structure of IrO2 including spin orbit coupling (SOC) and electron-electron interaction, both within the generalized gradient approximation plus Hubbard U (GGA+U) and GGAplus dynamical mean field theory (GGA+DMFT) approximations. Our calculations reveal that the Ir t2g states atthe Fermi level largely retain the Jeff = 1
2 character, suggesting that this complex spin orbit entangled state maybe robust even in metallic IrO2. We have calculated the phase diagram for the ground state of IrO2 as a functionof U and find a metal insulator transition that coincides with a magnetic phase change, where the effect of SOCis only to reduce the critical values of U necessary for the transition. We also find that dynamic correlations, asgiven by the GGA+DMFT calculations, tend to suppress the spin-splitting, yielding a Pauli paramagnetic metalfor moderate values of the Hubbard U . Our calculated optical spectra and photoemission spectra including SOCare in good agreement with experiment, demonstrating the importance of SOC in IrO2.
DOI: 10.1103/PhysRevB.89.155102 PACS number(s): 71.20.−b, 71.30.+h
I. INTRODUCTION
In recent years, 5d based oxides have attracted considerableattention where a combined influence of band-structure,electron correlation, and spin orbit coupling lead to emergentquantum phenomena [1–8]. Until a few years ago, the commonbelief has been that, due to the extended nature of the 5d or-bitals, the ratio between effective electron-electron interactionand bandwidth, U/W (where U is the Coulomb interactionand W is the bandwidth) is quite small in 5d transition metaloxides (TMO) and density functional theory (DFT) withinlocal density approximation (LDA) or generalized gradientapproximation (GGA) can explain the metallic ground stateof these systems. Contrary to this expectation there are recentreports of an insulating antiferromagnetic ground state in 5d
TMO, e.g., Sr2IrO4, Ba2IrO4, and Na2IrO3 [2,3,9], where, inaddition to the crystal field and Coulomb repulsion, strongspin orbit coupling plays a key role. In d5 Ir oxides, due tolarge crystal field splitting and strong SOC the t2g orbitals arerenormalized into doubly degenerate Jeff = 1/2 and quadruplydegenerate Jeff = 3/2 states, leading to a narrow band of halffilled Jeff = 1/2 states [1]. Inclusion of moderate Coulombinteraction in the spin orbit entangled Jeff = 1/2 manifold opensup a gap explaining the insulating property of some of the iri-dates [1]. Very recently it has been speculated that, in contrastto the orbitally ordered states in 3d insulating oxides, the spinorbital entangled Jeff = 1/2 state is robust and does not meltaway even in itinerant metallic systems [10]. This possibilitywas recently suggested for metallic IrO2 using resonant x-raydiffraction experiments [10] where it was argued that Ir 5d t2g
orbitals at the Fermi level are fairly close to the Jeff = 1/2 statedue to strong spin orbit coupling. The importance of spin orbitcoupling in IrO2 is also manifested by the recent observation
*Corresponding author: [email protected]
of large spin Hall effect [11], which is a novel topologicaltransport phenomena caused by spin orbit interaction.
IrO2 crystallizes in the rutile type structure with twoformula units per unit cell. The electronic structure aswell as optical properties of IrO2 have been investigatedby several groups [12–14] in the past but none of thesecalculations analyzed the possibility of the Jeff = 1/2 state inthis system. Further, there are no studies where the combinedrole of spin orbit interaction and Coulomb correlation isanalyzed in detail. In the present paper, we have investigatedthe electronic structure of IrO2 using density functionaltheory (DFT) in the framework of GGA + SOC + HubbardU (GGA+SOC+U) as well as GGA + SOC + dynamicalmean field theory (GGA+SOC+DMFT) calculations. OurGGA+SOC+U calculations as a function of U reveal thatnonmagnetic metallic IrO2 transforms to an antiferromagneticmetal and eventually into an antiferromagnetic Slater insulator.The GGA+SOC+DMFT calculations result in a suppressedexchange splitting, for moderate values of the Hubbard U .We have analyzed the nonmagnetic metallic state in detailand examined the suggestions for the Jeff = 1/2 state. Inaddition we have also calculated the optical conductivityand the photoemission spectra including SOC and Coulombcorrelation and compared with available experiments. Theremainder of the paper is organized as follows. In Sec. II,we discuss the crystal structure and the computational details.Section III is devoted to results and discussions followed byconclusions in Sec. IV.
II. COMPUTATIONAL DETAILAND CRYSTAL STRUCTURE
The density functional theory calculations have beenperformed using three different methods, namely (a) theplane wave based method as implemented in the Vienna abinitio simulation package (VASP) [15,16], (b) the full potential
1098-0121/2014/89(15)/155102(7) 155102-1 ©2014 American Physical Society
PANDA, BHOWAL, DELIN, ERIKSSON, AND DASGUPTA PHYSICAL REVIEW B 89, 155102 (2014)
linearized augmented plane wave (FP-LAPW) method [17],and (c) the full potential linearized muffin-tin orbital (FP-LMTO) method [18]. We have checked that all the threemethods yield essentially identical band structures for IrO2. Inorder to find out the ground state, the plane wave calculationswere performed within the local (spin) density approximation(LSDA), with generalized gradient correction (GGA) ofPerdew-Burke-Ernzerhof, with and without including Hub-bard U [19] and SOC [20]. The kinetic energy cutoff of theplane wave basis was chosen to be 600 eV. Brillouin-zoneintegration have been performed using a 14 × 14 × 20 k mesh.For the calculation of the optical spectra corresponding to thenonmagnetic metallic state, we have employed the all-electronFP-LAPW method. The muffin-tin radii (RMT ) of Ir and Oare chosen to be 1.09 and 0.87 A, respectively. To achieveenergy convergence of the eigenvalues, the wave functions inthe interstitial region were expanded in plane waves with acutoff RMT kmax = 7, where RMT denotes the smallest atomicsphere radius and kmax represents the magnitude of the largestk vector in the plane wave expansion. The valence wavefunctions inside the spheres are expanded up to lmax = 10,while the charge density is Fourier expanded up to Gmax = 12.
All the GGA+DMFT calculations have been carried outusing a full potential linear muffin-tin orbital (FP-LMTO)method [18] as implemented in the RSPT code. In this imple-mentation of GGA+DMFT the many-body corrections appearin a form which depends on a self-consistently calculateddensity matrix and on the correlated orbitals [18,21]. In thepresent case the correlated orbitals are 5d states on the Iratoms. Hence, the calculations treat in equal footing spin-orbiteffects, crystal field splittings, band formation, as well aselectron correlations. The effective impurity problem in theGGA+DMFT calculations has been solved through the spinpolarized T -matrix fluctuation-exchange (SPTF) solver [22].The SPTF solver has been chosen as it is known to be veryefficient for moderately correlated systems (U � W ) and hasbeen successfully applied to various materials [21–24]. TheSPTF solver is based on a perturbation expansion in theCoulomb interaction, where the Hubbard U is considered tobe smaller than the bandwidth. In the past it has been usedwith success for heavy elements [22] where, as for IrO2, thespin-orbit effects are important.
IrO2 crystallizes within a tetragonal rutile structure, havingspace group P 42/mnm. Each unit cell contains two Ir ions,at (0,0,0) and ( 1
2 , 12 , 1
2 ), and four O ions with coordinates(u, − u,0), (−u,u,0), ( 1
2 − u, 12 − u, 1
2 ), and (− 12 + u, − 1
2 +u, − 1
2 ), where u = 0.3077 [25]. Each Ir ion is surrounded bysix O ions in a distorted octahedron environment. NeighboringIrO6 octahedral units share edges along the c axis andvertices in the basal plane. Each Ir atom has two O atoms asneighbors at a distance d1 = √
2ua and four at d2 = [2( 12 −
u)2 + ( c2a
)2]1/2a. All the Ir-O bond lengths are equal if theparameter u has the value u∗ = 1
4 [1 + 12 ( c
a)2]. The octahedral
coordination of each Ir atom is ideal if ( ca
)ideal = 2 − √2
and uideal = 12 (2 − √
2). For IrO2, u∗ = 0.312 and thereforeu < u∗. All the calculations have been carried out with theexperimental structure and the antiferromagnetic ordering hasbeen simulated by considering an antiparallel alignment of
Γ X M Γ Z R A M-4
-2
0
2
4
Ener
gy (e
V)
-4 -2 0 2 4Energy (eV)
0
5
10
15
DO
S (S
tate
s/eV
-Cel
l)
(a) (b)
eg
t2g
FIG. 1. Non-spin-polarized (a) band dispersion along varioushigh symmetry directions and (b) density of states, computed withinthe GGA approximation.
the spin of two Ir ions in the unit cell. In addition, we havealso carried out electronic structure calculations for the idealstructure in order to assess the impact of distortion on theelectronic structure.
III. RESULTS AND DISCUSSION
To begin with we have analyzed the non-spin-polarizedband structure and DOS of rutile IrO2 obtained using theGGA method. The results of our calculations are presentedin Fig. 1. Both the DOS and the band structure are in goodagreement with the earlier calculation [14] on the same system.As discussed earlier, in a rutile structure each Ir atom issurrounded by a nearly octahedral array of six O atoms andthe site symmetry of the Ir atom may be considered as a sumof a large octahedral term plus a small orthorhombic distortion[12]. In such a crystal field the Ir d orbitals split into threefolddegenerate t2g states and twofold degenerate eg states. Theseorbitals for the rutile structure are a linear combination of thed orbitals expressed along the crystallographic a, b, and c axes[12], as the O octahedron around each Ir is not aligned alongthe crystallographic a, b, and c axes. The degeneracy of the t2g
and eg orbitals is, however, lifted by the orthorhombic term.As a consequence, the Fermi level is dominated by six Ir t2g
states arising from the two Ir atoms in the unit cell. The eg
states are completely empty and lie above the Fermi level. Thetwelve O p states are below the Ir t2g manifold, where againthe degeneracy of the O p states is lifted by the orthorhombicterm.
Next we have calculated the crystal field splitting at the Ir d
site. For this purpose, the N th order muffin-tin orbital (NMTO)down-folding calculations [26] were carried out keeping onlythe Ir d states in the basis and down-folding the O p states.The onsite block of the real space Hamiltonian provide thecrystal field splitting at the Ir site where the O covalency effectis also taken into account. The crystal field splitting of the Ird state for the experimental structure is shown in Fig. 2(a) andis consistent with the D2h symmetry of the Ir site, where thedegeneracy of all the d orbitals is completely removed. Ourcalculations reveal that the eg block is separated from the t2g
complex by 3.6 eV. Since the eg block is completely empty,we shall concentrate on the t2g block, and the crystal field term
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EFFECT OF SPIN ORBIT COUPLING AND HUBBARD U . . . PHYSICAL REVIEW B 89, 155102 (2014)
FIG. 2. (Color online) Crystal field splitting, obtained fromNMTO calculation for (a) experimental and (c) ideal structure. Effectof SOC on the t2g states of (b) experimental structure and (d) idealstructure.
for the t2g block may be written as⎛⎜⎝
−2ε 0 0
0 ε t
0 t ε
⎞⎟⎠,
where we obtain ε = 0.17 eV and t = 0.08 eV from ourNMTO calculation. In order to assess the role of distortionon the crystal field splitting, we have also carried out a GGAcalculation for the ideal structure of IrO2 with a = 5.3919 A,c = 3.1586 A, and u = 0.2929. While we find substantialsplitting between the eg and the t2g states, the intra-t2g splittingis now appreciably reduced and the crystal field term for thet2g block is calculated to be ε = t = 0.075 eV. The details ofthe crystal field splitting are shown in Fig. 2(c).
We have next considered the spin orbit interaction in ourcalculations, where the magnetization direction was chosenalong (001). In addition to spin orbit coupling, we have alsoincluded a Hubbard U = 2 eV in order to consider the effect ofelectron correlations in IrO2. These calculations were done onthe GGA+U level, as well as employing dynamical mean fieldtheory. Recently the electronic structures of several iridiumbased oxides have been investigated both in the framework ofGGA+U+SOC [1,27–29] and GGA+DMFT+SOC [30,31].The result of our calculation for IrO2 is displayed in Fig. 3.The spin orbit coupling (SOC) leads to important changes inthe band structure in the t2g manifold near the Fermi level. Asa result of SOC the t2g states are further split in such a way thatthe Fermi level now lies on a pair of bands separated from therest. Spin orbit effects are in general dependent on the degreeof hybridization as well as on the symmetry of the eigenstates,which varies across the Brillouin zone [12]. In particular, alongthe direction ZRA the degeneracy of the bands is removedin the relativistic limit where spin orbit coupling induces asplitting of approximately 0.5 eV. As a consequence of SOC,the t2g states are grouped in such a way that there is a pairof bands forming a quartet. These bands are fully occupiedand are close to each other while the Fermi level is on a
Γ X M Γ Z R A M-4
-2
0
2
4
Ener
gy (e
V)
(b)(a)
FIG. 3. (Color online) (a) Band dispersion computed withinGGA+U+SOC approximation and (b) the k-resolved total spectralfunction A(k,ω) along the high-symmetry directions of the Brillouinzone, computed within GGA+DMFT+SOC approximation. Thevertical color scale shows the intensity of A(k,ω).
band forming a doublet. The former is reminiscent of Jeff =3/2 states and the latter of Jeff = 1/2 states that have beendiscussed in the literature to understand the physics of iridates[1]. Recently x-ray absorption spectroscopy emphasized theimportance of j quantum states due to strong SOC in IrO2 [32].The division of the t2g orbitals with an effective quantum stateLeff = 1 forming a Jeff = 3/2 quartet and a Jeff = 1/2 doubletnot only requires large SOC but also completely degeneratet2g states that are well separated from the eg states. Any kindof mixing between the t2g and eg states or the breaking of thethreefold degeneracy of the t2g states will lead to a deviationfrom the Jeff = 1/2 state. As the orthorhombic distortion liftsthe degeneracy of the t2g states, the existence of the Jeff = 1/2state even in metallic IrO2, as suggested by the resonant x-raydiffraction experiment [10], therefore requires further scrutiny.
In view of the above, we have examined the validity ofthe Jeff = 1/2 state in IrO2 based on a model Hamiltonianwith realistic crystal field splitting and SOC. We have firstconsidered the onsite term of the down-folded Hamiltonianfor the t2g block in the presence of spin orbit coupling whichmay be written as
H =(
H+ 00 H−
), (1)
where
H± =
⎛⎜⎝
−2ε ± λ2 −i λ
2
± λ2 ε t ∓ i λ
2
i λ2 t ± i λ
2 ε
⎞⎟⎠. (2)
Here for the representation of H+ and H− we have employedthe basis functions (|xy+〉, |yz−〉, |zx−〉) and (|xy−〉, |yz+〉,|zx+〉), respectively. The spin orbit interaction is representedby the parameter λ. Each eigenstate of H+ has its counterpartin an eigenstate of H− for the Kramer’s doublet. The higheststate becomes the Jeff = 1/2 state given by
ψ±Jeff=1/2 = 1√
3[|xy±〉 + |yz∓〉 + i|zx∓〉]. (3)
In Figs. 2(b) and 2(d), we show the Ir t2g levels in thepresence of spin orbit coupling both for the experimentalstructure and the idealized structure for λ = 0.5 eV, a value
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PANDA, BHOWAL, DELIN, ERIKSSON, AND DASGUPTA PHYSICAL REVIEW B 89, 155102 (2014)
0 0.4 0.8 1.2 1.6 2λ (eV)
0.4
0.6
0.8
1
Proj
ectio
n of
Ψ3 o
n J ef
f = 1
/2 st
ate
Experimental StructureIdeal Structure
Ener
gy (e
V)
-4
-3
-2
-1
0
1
2
3
4
Γ X M Γ Z R A M
J =1/2eff
FIG. 4. (Color online) Projection of |ψ±3 〉 on the Jeff = 1/2 state.
The inset shows the low-energy bands for the t2g states decorated withJeff = 1/2 character.
typical for the iridates [33]. Figures 2(b) and 2(d) clearly revealthat the t2g levels are renormalized upon spin orbit coupling.Next we have calculated the projection |〈ψ±
Jeff=1/2|ψ±3 〉|2 where
|ψ±3 〉 is the eigenstate corresponding to the highest eigenvalue
of the Hamiltonian H± [see Eq. (2)]. The results of ourcalculation for the experimental as well as for the idealizedstructure are shown as a function of λ, in Fig. 4. We gatherfrom Fig. 4 that the Jeff = 1/2 character depends cruciallyon the strength of spin orbit coupling λ, and |ψ±
3 〉 has about90% Jeff = 1/2 character for the experimental structure andabout 96% for the idealized structure for λ = 0.5 eV. It isinteresting to note that the Jeff = 1/2 character largely surviveseven for nondegenerate t2g states in the presence of strongspin orbit coupling. As the bare width of the t2g state in IrO2
is larger than the SOC (λ ∼ 0.5 eV), the Jeff = 3/2 levelsare mixed with the Jeff = 1/2 ones, and the highest Kramer’sdoublet may deviate from the pure Jeff = 1/2 character. Inorder to clarify that, in addition to the analysis based on theon-site term as described above, we have derived a low-energytight-binding model including spin orbit coupling for the t2g
states of IrO2 where the hopping parameters are obtained fromour NMTO down-folding calculations. The projection of theJeff = 1/2 character on the tight-binding band structure (seeFig. 4 inset) clearly reveal that there is small hybridizationbetween Jeff = 1/2 and Jeff = 3/2 bands in some regions of theBrillouin zone, but importantly the pair of bands at the Fermilevel largely retain the Jeff = 1/2 character and it does not meltaway in the metallic state as suggested in Ref. [10], indicatingthe robustness of the Jeff = 1/2 state for large enough valuesof SOC.
In order to check the reliability of the GGA+U+SOCcalculations for metallic IrO2, we have also carried outGGA+DMFT+SOC calculations, and the resulting k-resolved spectral density is shown in Fig. 3(b). We findthat the quasiparticle features are protected and the basicstructure of the k-resolved spectral density is very similar tothe GGA+U+SOC calculations [Fig. 3(a)].
Next we have investigated whether IrO2 can be mademagnetic as well as insulating upon increasing the value of U .
0 1 2 3 4 5 6 7U
0
0.2
0.4
0.6
0.8
1
Mom
ent (
μ B/Ir
) and
Ban
d G
ap (e
V)
Spin momentOrb Moment Band Gap
NM Metal AFM Metal AFM Insulator
FIG. 5. (Color online) The phase diagram for the ground state ofIrO2 as a function of U within the GGA+U+SOC approach. Changeof the spin and orbital moment per Ir site and the band gap in theantiferromagnetic phase as a function of U are also shown.
In practical experimental situations, this can be done by tuningthe ratio between U and the bandwidth, W . This can be doneexperimentally in several ways, e.g., by means of alloying thatnarrows the bandwidth either by reducing the direct overlapbetween the Ir 5d orbitals, or by means of negative chemicalpressure. Furthermore, the relative importance of U can beenhanced by reducing the t2g bandwidth by intercalation. Theresults of our GGA+U+SOC calculations are shown in Fig. 5,where we plot the magnetic moment and the band gap. Figure 5clearly establishes a phase diagram for the ground state of IrO2,showing that it is essentially a nonmagnetic metal for U <
1.6 eV, an antiferromagnetic metal for 1.6 � U < 4.3 eV andan antiferromagnetic insulator for U � 4.3 eV. As expected,the magnetic moment increases monotonously and the bandgap increases nearly linearly with U , beyond two criticalvalues of U , Uc1 � 1.6 eV and Uc2 � 4.3 eV for the magnetictransition and the metal-insulator transition, respectively. It isinteresting to note that SOC is neither necessary for stabilizingmagnetism nor for the existence of metal insulator transition.The role of SOC is merely to reduce the critical value of Uc1 andUc2. The ratio 〈lz〉/〈mz〉 (where 〈mz〉 = 2〈sz〉) deviates fromthe ideal value of 2 due to the deviation from the Jeff = 1/2character. However, for the ideal structure the Jeff = 1/2behavior is restored.
The data in Fig. 5 show that, already for a rather modestvalue of U , one finds a nonmagnetic state. For instance, inthe GGA+U calculation, a value of U = 2 eV results in aspin-polarized ground state, and one may ask how accurate theGGA+U calculation is in reproducing the ground state proper-ties of this material, especially since a value of U = 1.5–2.0 eVis rather realistic for the 5d orbitals in IrO2. Experimentally it isknown that IrO2 is a Pauli paramagnet, with a spin-degenerateground state configuration, and the data in Fig. 5 suggestthat only U values less than 1.6 eV are consistent withthis experimental fact. To further analyze this situation wecarried out spin-polarized DMFT calculations for the casewhen U = 2 eV. In this case we found that a spin-degeneratesolution is stable, and hence that the experimental groundstate is reproduced for realistic values of U . The dynamic
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EFFECT OF SPIN ORBIT COUPLING AND HUBBARD U . . . PHYSICAL REVIEW B 89, 155102 (2014)
FIG. 6. (Color online) (a) A comparison of the experimental and theoretical optical conductivity as discussed in the text. Experimental dataare taken from Ref. [34]. The inset shows that the small peak around 0.4 eV is only captured when SOC is included. (b) Interband part of theoptical conductivity.
correlations embodied within the GGA+DMFT calculationtend to suppress the exchange splitting for moderate values ofthe Hubbard U . It is expected that the DMFT results wouldgive qualitatively the same results as those shown in Fig. 5,albeit with slightly large values of the critical U values for thetransition to the AFM metal and AFM insulating phase. Thiswas not pursued further here.
Finally we have calculated the optical properties and thephotoemission spectra of IrO2 including the effect of SOC.The optical conductivity has two components because of thetetragonal symmetry of the Bravais lattice, one parallel andthe other perpendicular to the c axis. In order to comparewith experimental results on polycrystalline samples [34],we calculate the average of the two theoretically com-puted interband components of the optical conductivity as[2 × σxx(ω) + σzz(ω)]/3. The interband part of the opticalconductivity is broadened by a Lorentzian function. Theintraband Drude component of the optical conductivity iscalculated from the bare plasma frequencies and has beenadded to each interband part separately to obtain the full opticalconductivity. Figure 6(a) displays the experimental opticalconductivity along with the computed optical conductivitywith and without SOC. The small peak seen around 0.4 eVis only captured when SOC is included [see inset of Fig. 6(a)].The amplitude of the real part of the optical conductivity in theentire energy range is sightly different when SOC is included.The calculated spectra within the GGA+U+SOC approachshows a feature-for-feature similarity when compared withthe experimental results, except that the main experimentalpeak located around 6.5 eV appears at higher photon energy inour calculation. Similar disagreement has also been reportedfor fcc Ni [35] and for some of the spectral properties ofCeN [36]. The fact that the calculated peak appears slightlyhigher than in experiment may indicate that a non-Hermitianenergy-dependent self-energy is important for the correctdescription of the optical properties. This has been shownto be the case for other moderately correlated systems [37].To understand the origin of this high-energy peak, we havecomputed the optical conductivity corresponding to interbandtransitions from the Ir t2g valence states to all the conductionstates and also O p valence states to the conduction states. Our
results, as presented in Fig. 6(b) clearly shows that the mainpeak arise due to interband transition from O p states to theconduction states.
The calculated DOS is usually compared with the pho-toemission spectra (PES); however, we know that the DOSand PES are not strictly comparable because of the differentphotoelectric cross section of the Ir d and O p states.In the following we have computed the PES within theso-called single-scatterer final-state approximation [39,40].Here the photocurrent is a sum of local (atomic-like) andpartial (l-like) density of states weighted by the correspondingcross-sections. In order to take into account the lifetimebroadening which increases in proportion to the square of thebinding energy within the Fermi liquid theory, the spectrumhas been broadened by a Lorentzian function. The broadeningparameter of the Lorentzian (�) is taken to be dependent onthe photon energy (E), having the form: � = 0.04 + 0.03E2.The computed spectrum shows a very good agreement with theexperimental data (see Fig. 7) as far as the main peak positionand the bandwidth are concerned. Finally, to investigate theimportance of dynamical correlation over the static correlationon the electronic structure of IrO2, we have carried out DMFTcalculation including SOC and computed the photoemission
-8 -4 0Binding energy (eV)
Inte
nsity
(arb
. uni
ts)
ExperimentGGA+SOCGGA+U+SOCDMFT+SOC
FIG. 7. (Color online) A comparison of the PES. The experimen-tal data has been taken from Ref. [38].
155102-5
PANDA, BHOWAL, DELIN, ERIKSSON, AND DASGUPTA PHYSICAL REVIEW B 89, 155102 (2014)
spectrum. Figure 7 displays the computed spectrum withinthe DMFT approach on top of the experimental spectra andthe spectra obtained within GGA+U+SOC approach. The twomethods yield very similar spectra and hence we conclude thatthe static correlation is sufficient to describe the nonmagneticmetallic phase of IrO2.
IV. CONCLUSIONS
In conclusion, our detailed study of the electronic structureand spectroscopic properties of IrO2 reveal that it is essentiallyan uncorrelated material with strong SOC. As a result of strongSOC, the Ir 5d t2g states at the Fermi level largely retain theJeff = 1/2 character even in the metallic state. We show, asa function of U , that IrO2 transforms from a nonmagneticmetal to an antiferromagnetic metal and eventually into anantiferromagnetic insulator, where the role of SOC is toreduce the critical values of U necessary for the transitions.This shows that it is possible to tune the properties of IrO2
by means of correlation effects. We have discussed waysto realize this experimentally, by means of tuning the ratiobetween Coulomb U over bandwidth W , e.g., by alloying andnegative chemical pressure. The optical and photoemissionspectra calculated including SOC are in good agreement withexperiment, suggesting the importance of SOC to understandthe electronic structure of IrO2.
ACKNOWLEDGMENTS
The authors thank Department of Science and Technology,Government of India for financial support. Support fromthe EU-INDIA collaboration program MONAMI is acknowl-edged. O.E. acknowledges support from the Swedish ResearchCouncil (VR), the KAW foundation, eSSENCE, and theERC (Project No. 247062-ASD). A.D. acknowledges financialsupport from VR, the Royal Swedish Academy of Sciences(KVA), and the Knut and Alice Wallenberg trust (KAW).
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PHYSICAL REVIEW B 89, 245423 (2014)
First-principles study of the electronic structure of CdS/ZnSe coupled quantum dots
Nirmal Ganguli,1,* S. Acharya,2 and I. Dasgupta1,2,†1Department of Solid State Physics, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India
2Centre for Advanced Materials, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India(Received 4 April 2014; revised manuscript received 4 June 2014; published 17 June 2014)
We have studied the electronic structure of CdS/ZnSe coupled quantum dots, a novel heterostructure at thenanoscale. Our calculations reveal CdS/ZnSe coupled quantum dots are type II in nature where the anion p statesplay an important role in deciding the band offset for the highest occupied molecular orbitals (HOMO). We showthat the offsets of HOMO as well as the lowest unoccupied molecular orbitals (LUMO) can be tuned by changingthe sizes of the components of the coupled quantum dots, thereby providing an additional control parameterto tune the band gap and the optical properties. Our investigations also suggest that the formation of an alloynear the interface has very little influence on the band offsets, although it affects the spatial localization of thequantum states from the individual components. Comparing the influence of strain on coupled quantum dots andcore-shell nanowires, we find strain practically has no role in the electronic structure of coupled quantum dotsas the small effective area of the interface in a coupled quantum dot helps a large part of the structure remainfree from any substantial strain. We argue that in contrast to core-shell nanowires, quantum confinement is thekey parameter that controls the electronic properties of coupled quantum dots and should therefore be an idealcandidate for the design of a quantum device.
DOI: 10.1103/PhysRevB.89.245423 PACS number(s): 73.22.−f, 73.40.Lq
I. INTRODUCTION
Semiconductor heterostructures [1] at the nanoscale haveattracted considerable attention recently because novel func-tionalities may be obtained not only by tailoring the size andshape of the individual components but also by exploiting acombination of the properties of both semiconductors, therebyincreasing their applicability far beyond the limits imposedby the individual nanoparticles [2]. Modern colloidal tech-niques allow fabrication of various types of heterostructuressuch as core-shell nanocrystals (NC) [3], multicomponentheteronanorods [4], tetrapods [5], and, very recently, het-erodimers [6] and coupled quantum dots [7]. Semiconductorheterostructures are typically classified either as type I or typeII, depending on the relative alignment of the conduction andthe valence-band edges of the materials that constitute theinterface. In a type-I heterostructure, the alignment of thebands is such that both conduction- and valence-band edgesof semiconductor A (smaller band gap) are located withinthe energy gap of semiconductor B (larger band gap), so thatthe electron and hole pairs excited near the interface tend tolocalize in semiconductor A. For a type-II heterostructure,the relative alignment of the conduction and valence bandsof the constituent materials offer a spatially indirect band gapresulting in an optical transition energy smaller than the bandgap of either of the constituent materials. As a consequenceof this staggered alignment of bands, the lowest-energy statesfor the electrons and the holes are in different semiconductors,which is highly attractive for applications in photovoltaics,where such charge separation is desirable [1,5,8].
Tuning the optical properties of semiconductingnanoheterostructures can be achieved by selecting the con-
*Present address: Faculty of Science and Technology and MESA+
Institute for Nanotechnology, University of Twente, P.O. Box 217,7500 AE Enschede, The Netherlands; [email protected]
stituent materials and taking advantage of additional parame-ters such as size-dependent quantum confinement exhibitedby the systems at nanometer scale. In addition, type-IIheterostructures offer an attractive possibility to control theeffective band gap by engineering the band offsets at the inter-face [7]. Another parameter that has profound impact on theelectronic structure and band offsets in nanoheterostructuresis the strain resulting due to sharp lattice mismatch of theconstituents at the interface [9,10]. While nanoscale hetero-junctions can tolerate larger lattice mismatch in comparisonto their bulk counterpart, the resulting strain may further shiftenergy levels and band offsets in a nontrivial way [11]. It hasbeen shown that the strain-induced change in the band gapmay be comparable to that induced by quantum confinementin highly lattice mismatched nanoscale heterojunctions [11].Recently, it has been illustrated that strain can be advantageousin tuning the optical properties of core-shell nanocrystals [9].Epitaxial deposition of a compressive shell (ZnS, ZnSe, ZnTe,CdS, or CdSe) onto a nanocrystalline core (CdTe) producesstrain that changes standard type-I band alignment to type-IIbehavior, ideal for application in photovoltaics [9]. On theother hand, in some cases strains produced at the interfacemay be relieved by creating dislocations at the interface,giving rise to nonradiative decay channels, thereby provingto be highly detrimental for applications [12]. In this respect,recently suggested type-II nanoheterostructures obtained bycoupling semiconductor quantum dots are interesting as theypossibly rely on controlling the effective transition energy gapby engineering the band offsets at the interface primarily byquantum confinement as the effect of strain in such systems isexpected to be small. This is due to the fact that the actualarea of the interface is much smaller in coupled quantumdots in comparison to core-shell nanosystems due to theirdifference in geometry, which substantially reduces the stressin the former. As the effect of strain is expected to be minimal,quantum confinement is of prime importance in coupledquantum dots, providing an ideal opportunity to design an
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NIRMAL GANGULI, S. ACHARYA, AND I. DASGUPTA PHYSICAL REVIEW B 89, 245423 (2014)
interface as a quantum device that may find application eitherin optoelectronic devices (e.g., photovoltaic device) or inthe realization of qubits for quantum information processing[13]. It is interesting to note that a recent report on coupledsemiconductor quantum dots of CdS/ZnSe demonstrated thetuning of the photoluminescence wavelength (a manifestationof the effective gap) over a large range of ∼100 nm simply bychanging the ratio of the component sizes constituting thecoupled quantum dots [7], which clearly demonstrates thetunability of optical properties via controlling band offsetsprimarily due to quantum confinement.
The electronic structure at the interface of nanoheterostruc-tures plays a crucial role in tailoring the band gap and bandoffsets. In the present paper, using density functional theoryin the framework of the generalized gradient approximation(GGA), we have investigated in detail the electronic structureof coupled quantum dots and compared them with core-shellnanowires. A well-known limitation of GGA is that it tendsto underestimate the band gaps and does not provide areliable estimate of band offsets between chemically dissimilarmaterials [14]. However, that is not a matter of concern inthis paper as the materials considered here (CdS, CdSe, ZnSe)exhibit very similar quasiparticle shifts [14], so the value of theoffsets may not change significantly. Further we shall discussthe trends and the physical origin of band offsets which are notdependent on the actual value of the band gap and band offsets.We have studied in detail the coupled quantum dots of CdSand ZnSe from first-principles density functional calculationsto understand the nature of the chemical interaction at theinterface that leads to the type-II nature of the heterojunction.We have also calculated the band offsets and investigated howthey change with variation of the component size. As the roleof strain has remained unexplored in coupled quantum dots,we have calculated the strain profile and the impact of strain onband offsets and compared our results with CdScore/ZnSeshell
nanowires. In the following we shall argue that band offsets incoupled quantum dots are primarily dictated by the interactionbetween the anion p states along with quantum confinement,making them ideal for a quantum device.
II. COMPUTATIONAL DETAILS AND SIMULATEDSTRUCTURE
All the electronic structure calculations presented here areperformed using ab initio density functional theory (DFT) asimplemented in the Vienna ab initio simulation package (VASP)[15]. The projector augmented wave (PAW) method [16] alongwith the plane-wave basis set are used for our calculations.PAW potentials with 12 valence electrons (4d10 5s2) for Cd,12 valence electrons (3d10 4s2) for Zn, 6 valence electrons(3s2,3p4) for S, and 6 valence electrons (4s2,4p4) for Se withan energy cutoff of 500 eV for the plane-wave expansion ofthe PAWs were employed in our calculations. The exchange-correlation (XC) part is approximated through GGA dueto Perdew, Burke, and Ernzerhof (PBE) [17]. The danglingbonds at the surface of the clusters as well as the nanowiresare saturated using fictitious hydrogen atoms with fractionalcharges, as proposed by Huang et al. [18]. Our calculations areperformed in the framework of periodic boundary conditions,and the periodic images of the clusters and the nanowires
along the transverse directions are separated by vacuum layersof sufficient width (∼10 A). In view of the large size of thesimulation cell (tiny Brillouin zone) we have employed onlyone k point (� point) for the coupled quantum dots and a�-centered k mesh of 1 × 1 × 8 for the nanowires. The atomicpositions were relaxed to minimize the Hellman-Feynmanforce on each atom with a tolerance of 0.01 eV/A. The latticestrain in the heterostructures was calculated following themethod proposed by Pryor et al. [19].
III. RESULTS AND DISCUSSION
A. Coupled quantum dots of similar size
To begin with, we have simulated the coupled quantumdot formed by coupling CdS and ZnSe quantum dots ofsimilar sizes. Following the experimental observations [7],we have taken the CdS cluster in the wurtzite (hexagonal)phase and the ZnSe cluster in the zinc-blende (cubic) phase.In our simulation, the CdS cluster consists of 45 Cd atoms and51 S atoms, whereas the ZnSe cluster comprises 44 Zn and46 Se atoms. The diameters of both clusters are ∼1.6 nm. Theheterostructure is formed by attaching the polar (0001) facet ofthe CdS cluster in wurtzite structure with the polar (111) facetof the ZnSe cluster in cubic structure, where the Cd-terminatedpolar facet of the CdS cluster binds to the Se-terminated polarfacet of the ZnSe dot as shown in the inset of Fig. 1(a). Thetotal density of states (DOS) corresponding to the coupledquantum dot of CdS-ZnSe is shown in Fig. 1(a). The DOSsuggests that the gap between the highest occupied molecularorbital (HOMO) and the lowest unoccupied molecular orbital(LUMO) for the coupled quantum dot is 2.35 eV. This value
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FIG. 1. (Color online) (a) The total density of states for coupledCdS/ZnSe quantum dots of similar size. The inset shows the structureof the coupled quantum dots, where red, blue, green, maroon, andlight blue balls indicate Cd, S, Zn, Se, and the fictitious passivatoratoms, respectively (this convention has been followed throughoutthe paper). (b) The energy-resolved charge density has been plottedas a function of the distance from the interface.
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TABLE I. The band offsets (in eV) for coupled quantum dots calculated (i) using energy-resolved charge-density plots and (ii) using theaverage electrostatic potential at the atomic PAW spheres as suggested by Hinuma et al. [20].
Charge-density method Electrostatic potential method
HOMO LUMO HOMO LUMOHeterostructure offset offset offset offset
CdS/ZnSe 1:1 0.27 0.45 0.29 0.74CdSe/ZnSe 0.00 0.64 0.06 0.96CdS/ZnSe 2:1 0.30 0.94 0.37 1.20CdS/ZnSe diffused (first bilayer) 0.26 0.46 0.34 0.79CdS/ZnSe diffused (second bilayer) 0.21 0.42 0.34 0.79
of the gap is smaller than the calculated gaps for both of itscomponents, namely, the CdS cluster (∼2.65 eV) and thatof the ZnSe cluster (∼3.10 eV). The trend in the calculatedgap of the components is consistent with the experimentalband gap of bulk CdS (2.42 eV) and ZnSe (2.70 eV). Thecalculated gaps for the individual dots are larger comparedto the bulk experimental values due to quantum confinementbut are possibly underestimated due to the usual limitation ofGGA. The effective gap of the heterostructure being less thaneither of the components indicates that the band alignment atthe interface may be type II.
To obtain further insights on the nature of the bandalignment and the character of the HOMO and the LUMOstates and to estimate the offsets for HOMO and LUMO atthe interface we have calculated the energy-resolved chargedensity along the direction perpendicular to the interface.In order to evaluate the energy-resolved charge density, theband-decomposed charge density corresponding to a particularenergy eigenvalue is calculated for each k point. The resultingcharge density is averaged over planes parallel to the interface.This averaged charge density for a given energy at a particulark point scaled by an arbitrary constant (the same constant isused for all calculations) is plotted as a function of the distancefrom the interface. Such spatially averaged charge densitiesreflect the spatial distribution of every state perpendicular tothe interface within a suitable range of energy. This energy-resolved charge density is particularly useful for visualizingthe band alignment of nanoscale heterostructures where eitherone or only a few k points are used for the calculation.The energy-resolved charge density for CdS/ZnSe coupledquantum dots is shown in Fig. 1(b). We gather from thefigure that the highest occupied state is primarily confinedto the ZnSe part of the coupled quantum dot with a shorttail extended to the CdS part. On the other hand, the lowestunoccupied state is confined to the CdS part. Further, Fig. 1offers us a clear view of the localization of all the states inthe energy range of interest. From Fig. 1, we have obtainedvalues of the offsets for HOMO and LUMO of 0.27 and0.45 eV respectively. In addition, we have also calculatedthe valence-band offset following the method suggested byHinuma et al. [20]. In this approach an electrostatic potentialaveraged within a PAW sphere at an atomic site is taken asthe reference level for the evaluation of the valence-bandoffsets. The valence-band offsets calculated using this methodare shown in Table I. We find both methods provide nearlyidentical values for the HOMO offsets. The LUMO offsets
calculated using the latter method, i.e., adding the HOMOoffsets and the difference of energy gaps of the individualpristine clusters, show a trend similar to that obtained from theenergy-resolved charge-density method but have substantiallylarger values. The reason for this discrepancy may be attributedto the fact that the energy gaps of CdS and ZnSe obtainedfrom the CdS and ZnSe part of the coupled quantum dot aresmaller than their values in the respective pristine clusters dueto the mitigation of quantum confinement. Upon formation ofthe heterostructure, we find the energy gap of the ZnSe partbecomes substantially smaller, while the gap of the CdS partis marginally reduced. This effect is not accounted for withinthe latter method for calculating the LUMO offsets, leading toa systematic overestimation of the same.
We have next investigated the role of anion p states on thevalence-band offset. We have plotted the partial DOS for theanion p states and cation s states (see Fig. 2) to understand theinteraction at the interface of the heterojunction that leads tothe offset. The partial DOS indicates that the occupied statesnear the gap are primarily anion-p-like, which also has someadmixture with the cation d states, whereas the unoccupiedstates near the gap mainly show cation-s-like character, witha little admixture with anion p states. Notably, the offsetsbetween the p states of S and Se and the s states of Cd and Znturn out to be the same as the calculated HOMO and LUMOoffsets, respectively. This observation corroborates the picturethat the interaction between the anion p states admixed with
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FIG. 2. (Color online) The density of states projected onto(a) Cd s, (b) S p, (c) Zn s, and (d) Se p states for coupled CdS/ZnSequantum dots of similar size.
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cation d states is crucial for the HOMO offset [21] with onlyan implicit role of the cation d states. The coupling betweenthe anion p and cation d states, however, plays an importantrole for bulk semiconductor heterostructures, as suggested byWei and Zunger [22].
To ascertain further the role of anion p and cation d
states in determining the HOMO (valence band) offsets, wehave simulated a similar heterostructure comprising CdSeand ZnSe quantum dots: i.e., the same anion for both thecomponents. The CdSe/ZnSe heterostructure is simulated byreplacing all the sulfur atoms of the CdS/ZnSe quantum-dotheterostructure with selenium atoms followed by optimizationof the atomic positions. Our calculated band gap (2.20 eV)for the CdSe cluster is found to be smaller compared to theZnSe cluster (3.10 eV) and is consistent with the experimentalbulk band gaps of CdSe (1.73 eV) and ZnSe (2.70 eV). Theeffective gap is found to decrease marginally (by 0.14 eV) uponformation of the heterojunction. The band offsets calculatedusing the energy-resolved charge density are listed in Table Iand show no offset for HOMO and a large offset for LUMO.Only a small value for the HOMO offset is obtained fromthe method of Hinuma et al. [20] (See Table I). Hence thereduction in the effective HOMO-LUMO gap upon formationof the heterojunction may be attributed to the increase in thesystem size and thereby reduction of the band gap due tomitigation of quantum confinement. The quasi-type-II natureof the heterojunction (i.e., no offset for HOMO and substantialoffset for LUMO) supports the common-anion rule [21], whichargues that no offset for the valence band should be found forheterojunctions with a common anion for both components,confirming the important role played by the anion p states indetermining the valence-band offset.
B. Variation of component size
Having confirmed that an ideal CdS-ZnSe coupled dotleads to type-II heterostructures, next, we have explored theeffect of varying the component size on the offsets of HOMOand LUMO. In this context, recent experiments suggest thatthe photoluminescence wavelength increases with increasingconcentration of CdS (i.e., increasing the size of the CdSquantum dot) [7], which may be attributed to the change in theband offset. Engineering the band offset by modifying the sizesof the components in a coupled quantum dot is an attractivefeature that may find application for device fabrication. In viewof the above, we have simulated a coupled quantum dot wherethe number of atoms in the CdS cluster is nearly double thenumber of atoms in the ZnSe cluster.
We have therefore considered a heterojunction of CdS/ZnSeclusters where a CdS cluster consisting of 93 Cd atoms and 96S atoms is coupled to a ZnSe cluster consisting of 44 Zn and46 Se atoms. We shall refer to this system as the 2:1 system,whereas the system with similar component sizes studied ear-lier will be referred to as the 1:1 system. The densities of statesfor the 1:1 system and the 2:1 system are displayed in Figs. 3(a)and 3(b), respectively; a comparison between them shows thatthe effective gap between HOMO and LUMO decreases by0.38 eV upon increasing the size of the CdS cluster. It isinteresting to note that the gap between HOMO and LUMOfor pristine CdS clusters also decreases by the same amount
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FIG. 3. (Color online) Comparison of the density of states for (a)CdS/ZnSe 1:1 dots and (b) CdS/ZnSe 2:1 dots along with the densityof states for (c) the small and (d) large CdS quantum dots.
(0.38 eV) upon increasing the size, as seen from Figs. 3(c) and3(d). The reduction of the effective gap for the heterojunction isa result of mitigation of quantum confinement due to the largesize of the CdS cluster. The band offsets calculated using twodifferent methods are listed in Table I. Hence increasing thesize of one of the components (here CdS) of a coupled quantumdot reduces the HOMO-LUMO gap for that component dueto quantum confinement, which primarily changes the LUMOoffset, whereas the offset between the HOMO states does notchange much. The above discussion points to the fact thatvarying the size of the components for a coupled quantum dotheterojunction is a novel control parameter that provides anopportunity to tune the offsets for suitable applications. Moreimportantly, the range of the effective gap thereby accessiblemay be far beyond the range accessible by manipulating thesize of an individual quantum dot.
C. Diffused interface
In the preceding discussion we have considered the idealinterface, but it is quite likely that the interface of the coupleddots may be a diffused alloy of CdS and ZnSe. In light of theabove, we have examined the influence of interlayer diffusionof the atoms near the interface by simulating two differentalloyed heterostructures, namely, (i) where two Cd (S) atomsreplace two Zn (Se) atoms and vice versa at the first cationic(anionic) interlayer, i.e., the diffusion is restricted to the firstbilayers near the interface [see Fig. 4(a)], and (ii) wherein addition to heterostructure (i) one Cd (S) atom replaces
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FIG. 4. (Color online) (top) The structure and (bottom) theenergy-resolved charge density for (a) and (b) the diffused upto firstbilayer and (c) and (d) diffused upto second bilayer systems.
one Zn (Se) atoms and vice versa at the second cationic(anionic) interlayer, i.e., the diffusion extends up to the secondbilayer near the interface [see Fig. 4(c)]. The energy-resolvedcharge-density plots for these diffused interfaces are shown inFigs. 4(b) and 4(d), respectively. The calculated band offsetsare listed in Table I. These values compare well with theideal 1:1 interface. Our observations indicate that the offsetsare not very sensitive to the diffusion at the interface as theinterlayer diffusion possibly does not influence the interactionbetween anion p and cation s states significantly, but itaffects the spatial localization of the states, which may bedetrimental for carrier separation required for photovoltaicapplications. We do not observe localized interface-inducedstates for perfectly passivated coupled quantum dots. However,the lack of fictitious H atoms near the interface may lead tosuch localized (dangling bond) states. Coupled dots preparedusing the colloidal technique usually have long-chain organicmolecules passivating the dangling bonds.
D. CdScore/ZnSeshell nanowire heterojunction
After investigating the coupled quantum dot heterojunc-tions in detail, we have studied CdScore/ZnSeshell nanowireheterojunctions, where we anticipate a significant differencein the electronic structure because of a larger interfacial areaat the interface. In order to simulate the heterojunction wehave assumed two rings of CdS in the wurtzite structure asthe core, surrounded by two rings of ZnSe in the wurtzitestructure as the shell, as shown in the cross-sectional viewof the nanowire [see Figs. 5(a) and 5(b)]. The radius of thiscylindrical heterostructure is ∼13 A. The dangling bonds atthe surface are properly saturated by fictitious hydrogen atomswith fractional charge [18]. The crystallographic c directionof the wurtzite structure has been assumed to be the growthdirection of the nanowire. Unlike coupled quantum dots, herethe interface is not formed by attaching polar facets.
The density of states for the CdScore/ZnSeshell nanowireheterojunction is shown in Fig. 5(c). We note that the effectiveband gap for this system is calculated to be 1.90 eV. The smallervalue of the effective band gap for the core-shell nanowiremay be attributed to the absence of confinement along the c
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direction, which reduces the band gaps for both componentsconstituting the nanowire.
The charge densities corresponding to valence-band max-imum (VBM) and conduction-band minimum (CBM) areshown in Figs. 5(a) and 5(b), respectively. From Fig. 5 wefind the VBM and the CBM to be confined in the shell andthe core region, respectively, confirming the type-II nature ofthe heterojunction. The energy-resolved charge-density plotfor this system is displayed in Fig. 6, which indicates VBMand the CBM offsets of 0.20 and 0.44 eV, respectively. Incomparison to the offsets calculated for the coupled quantumdots, the VBM offset is smaller in this case, while the CBMoffset is nearly the same.
E. Effect of strain
As discussed earlier, the band alignment of nanoscaleheterojunctions should be very sensitive to the lattice strain.
FIG. 6. (Color online) The energy-resolved charge density asa function of the radial coordinate of the cylindrical nanowireheterojunction.
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CdS ZnSe
(a) (b)
FIG. 7. (Color online) The trace of the strain tensor plotted as a function of distance along the direction perpendicular to the interface for(a) coupled quantum dots with varying size and (b) the CdScore/ZnSeshell nanowire.
In terms of the effect of strain, the coupled quantum dots andcore-shell nanowires are expected to be very different. Wehave calculated the strain profiles for coupled quantum dotsand core-shell nanowires using an atomistic model [19] forelasticity where the parameters of the model are calculatedusing ab initio electronic structure calculations within densityfunctional theory. We have calculated the trace of the straintensor that represents the volumetric strain for the system.Our results for the volumetric strain for coupled dots (for 1:1and 2:1) and core-shell nanowrires are displayed in Figs. 7(a)and 7(b), respectively. We note from Fig. 7(a) that the strainprofiles for coupled quantum dots do not change significantlywith variation of component size. As expected, the strain isquite large near the interface, and it sharply decreases as wemove away from the interface. On the other hand, for thecore-shell structure the strain profiles are shown for the cationicand the anionic planes separately in Fig. 7(b), and we find anoscillatory nature of the strain field where the value of the strainmay be significantly large even far away from the interface.Comparing the strain profiles for both systems we gatherthat the core-shell nanowire is more strained compared to thecoupled quantum dots due to the large interface of the latter.
In order to quantify the effect of strain on the alignmentof bands, we have calculated the band offsets for the unre-laxed (discretely strained at the interface) heterostructures ofCdS/ZnSe in a coupled dot as well as core-shell nanowire
geometry. A similar model was employed earlier to study theimpact of strain on band gaps in core-shell nanostructures [11].The model for unrelaxed coupled quantum dots comprisesa CdS quantum dot and a ZnSe quantum dot with bondlengths and bond angles matching the corresponding bulkstructures in wurtzite and zinc-blende forms, respectively. Theheterojunction is formed by bringing the Cd-terminated (0001)plane of CdS close to the Se-terminated (111) plane of ZnSe.The separation between the Cd and Se planes is 2.44 A.The dangling bonds are passivated by fictitious hydrogenatoms with fractional charge, located at a distance of 1.25 A.On the other hand, the unrelaxed CdScore/ZnSeshell nanowireheterostructure consists of both of its components in wurtziteform, with their respective bulk bond length and bond anglevalues. The dangling bonds of the ZnSeshell part are passivatedby fictitious hydrogen atoms with fractional charge, locatedat a distance of 1.25 A. The corresponding energy-resolvedcharge-density plots for the unrelaxed structures are shown inFig. 8. The band offsets for the unrelaxed 1:1 coupled quantumdots (HOMO offset: 0.29 eV, LUMO offset: 0.47 eV, type II)are nearly identical to those obtained for the relaxed coupledquantum dots (HOMO offset: 0.27 eV, LUMO offset: 0.45 eV),indicating the effect of relaxation (i.e., distribution of strainover the structure) is negligible for coupled quantum dots.On the other hand, the impact of the distribution of strainis appreciable in the core-shell nanowire. In the core-shell
FIG. 8. (Color online) The energy-resolved charge-density plots for unrelaxed systems: (a) CdS/ZnSe coupled quantum dots (HOMOoffset: 0.29 eV, LUMO offset: 0.47 eV, type II) and (b) CdScore/ZnSeshell nanowire (VBM offset: 0.09 eV, CBM offset: 0.33 eV, type I). Theinsets show the schematic band alignments.
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FIRST-PRINCIPLES STUDY OF THE ELECTRONIC . . . PHYSICAL REVIEW B 89, 245423 (2014)
nanowire, upon relaxation, not only does the value of theband offset change appreciably, but also the band alignmentbecomes type II (see Fig. 6) for the relaxed system, whereasthe unrelaxed system [see Fig. 8(b)] shows the type-I natureof alignment. The band offset for the unrelaxed (relaxed)core-shell nanowire is calculated to be 0.09 eV (0.20 eV)for the VBM offset and 0.33 eV (0.44 eV) for the CBMoffset, where both the VBM and CBM offsets of the unrelaxedstructure are substantially different from the relaxed structure.From the quantitative estimate of the volumetric strain andthe comparison of the effect of strain on the band offsetsfor the coupled quantum dots and core-shell nanowire weunderstand that as opposed to coupled quantum dots, theelectronic structure is very sensitive to strain for core-shellnanowires due to the large interfacial area of the latter. Whilestrain is an important factor in determining the band offsetsin core-shell nanowires, quantum confinement is the only keydeciding factor for band offsets in coupled quantum dots.
IV. CONCLUSION
In conclusion, we have studied the electronic structure ofcoupled quantum dots consisting of CdS/ZnSe clusters in detailto understand the origin and nature of band offset. We have alsoexplored in detail the impact of varying the component sizeand lattice strain on the band offset of the coupled quantumdots. We have found that the band alignment of CdS/ZnSecoupled quantum dots is type II in nature, where the effectivegap is smaller than the gap of either of its components. Wehave analyzed in detail the nature of chemical bonding at the
interface; in particular, the calculation of the energy-resolvedcharge density not only clarified the alignment of the bands atthe interface but also provided a direct estimate of the bandoffsets. Our calculations also indicate the important role of theanion p states in deciding the HOMO offset. The importanceof the anion p states was further clarified by consideringCdSe/ZnSe coupled quantum dots with a common anion Se,and the calculations revealed the absence of the HOMO offsetwith a quasi-type-II band alignment. We have illustrated thatthe offsets of HOMO and LUMO can be tuned by changing thesizes of the components of the coupled quantum dot, therebyproviding an additional control parameter to tune band gapand optical properties. Our investigations also suggest thatformation of an alloy near the interface does not change theband offsets substantially but affects the spatial localizationof the states. Comparing the influence of strain on coupledquantum dots and core-shell nanowires of CdS/ZnSe, weconclude that the strain at the interface plays a crucial rolein the electronic structure of core-shell nanowires and hardlyaffects the electronic structure at the interface of a coupledquantum dot. This is due to the fact that the effective area ofinterface of the coupled dots is small, and as a consequence,a small lattice mismatch does not lead to much stress. Wehave illustrated that quantum confinement primarily controlsthe properties of coupled quantum dots and should thereforebe an ideal candidate for the design of a quantum device.
ACKNOWLEDGMENTS
I.D. and S.A. thank the Department of Science andTechnology, government of India for financial support.
[1] S. S. Lo, T. Mirkovic, C.-H. Chuang, C. Burda, and G. D.Scholes, Adv. Mater. 23, 180 (2011).
[2] A. M. Smith and S. Nie, Acc. Chem. Res. 43, 190 (2010).[3] S. Kim, B. Fisher, H.-J. Eisler, and M. Bawendi, J. Am. Chem.
Soc. 125, 11466 (2003).[4] S. Kumar, M. Jones, S. Lo, and G. Scholes, Small 3, 1633
(2007).[5] D. J. Milliron, S. M. Hughes, Y. Cui, L. Manna, J. Li, L.-W.
Wang, and A. P. Alivisatos, Nature (London) 430, 190 (2004).[6] T. Teranishi and M. Sakamoto, J. Phys. Chem. Lett. 4, 2867
(2013).[7] S. Sengupta, N. Ganguli, I. Dasgupta, D. D. Sarma, and S.
Acharya, Adv. Mater. 23, 1998 (2011).[8] S. A. Ivanov, A. Piryatinski, J. Nanda, S. Tretiak, K. R. Zavadil,
W. O. Wallace, D. Werder, and V. I. Klimov, J. Am. Chem. Soc.129, 11708 (2007).
[9] A. M. Smith, A. M. Mohs, and S. Nie, Nat. Nanotechnol. 4, 56(2009).
[10] T. Sadowski and R. Ramprasad, J. Phys. Chem. C 114, 1773(2010).
[11] S. Yang, D. Prendergast, and J. B. Neaton, Nano Lett. 10, 3156(2010).
[12] D. D. Sarma, A. Nag, P. K. Santra, A. Kumar, S. Sapra, andP. Mahadevan, J. Phys. Chem. Lett. 1, 2149 (2010).
[13] E. A. Stinaff, M. Scheibner, A. S. Bracker, I. V. Ponomarev,V. L. Korenev, M. E. Ware, M. F. Doty, T. L. Reinecke, andD. Gammon, Science 311, 636 (2006).
[14] V. Stevanovic, S. Lany, D. S. Ginley, W. Tumas, and A. Zunger,Phys. Chem. Chem. Phys. 16, 3706 (2014); A. Gruneis, G.Kresse, Y. Hinuma, and F. Oba, Phys. Rev. Lett. 112, 096401(2014).
[15] G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993); G. Kresseand J. Furthmuller, ibid. 54, 11169 (1996).
[16] P. E. Blochl, Phys. Rev. B 50, 17953 (1994).[17] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77,
3865 (1996).[18] X. Huang, E. Lindgren, and J. R. Chelikowsky, Phys. Rev. B 71,
165328 (2005).[19] C. Pryor, J. Kim, L. W. Wang, A. J. Williamson, and A. Zunger,
J. Appl. Phys. 83, 2548 (1998).[20] Y. Hinuma, F. Oba, Y. Kumagai, and I. Tanaka, Phys. Rev. B 88,
035305 (2013).[21] W. R. Frensley and H. Kroemer, Phys. Rev. B 16, 2642 (1977).[22] S.-H. Wei and A. Zunger, Appl. Phys. Lett. 72, 2011 (1998).
245423-7
PHYSICAL REVIEW B 88, 094409 (2013)
Role of Te in the low-dimensional multiferroic material FeTe2O5Br
Jayita Chakraborty,1 Nirmal Ganguli,2 Tanusri Saha-Dasgupta,3 and Indra Dasgupta1,*
1Department of Solid State Physics, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India2Faculty of Science and Technology and MESA + Institute for Nanotechnology, University of Twente,
P.O. Box 217, 7500 AE Enschede, The Netherlands3S. N. Bose National Center for Basic Sciences, JD-III, Salt Lake City, Kolkata 700098, India
(Received 28 June 2013; revised manuscript received 21 August 2013; published 9 September 2013)
Using first principles density functional calculations, we study the electronic structure of the low-dimensionalmultiferroic compound FeTe2O5Br to investigate the origin of the magnetoelectric (ME) effect and the role of Teions in this system. We find that without magnetism, even in the presence of Te 5s lone pairs, the system remainscentrosymmetric due to the antipolar orientation of the lone pairs. Our study shows that the exchange strictionwithin the Fe tetramers as well as between them is responsible for the ME effect in FeTe2O5Br. We also find thatthe Te4+ ions play an important role in the intertetramer exchange striction as well as contributing to the electricpolarization in FeTe2O5Br, once the polarization is triggered by the magnetic ordering.
DOI: 10.1103/PhysRevB.88.094409 PACS number(s): 71.20.−b, 75.30.Et, 75.80.+q, 77.80.−e
I. INTRODUCTION
Multiferroic materials with the simultaneous presenceof ferroelectricity and magnetism have been the focus ofattention in recent times.1,2 Based on the microscopic originof ferroelectricity (FE) multiferroic materials can be classifiedinto two different classes, namely, type-I (proper) and type-II(improper) multiferroic materials. In type-I multiferroics,ferroelectricity and magnetism stem from an independentorigin and the coupling between magnetism and ferroelec-tricity is usually weak. In these materials, ferroelectricitytypically appears at higher temperatures than magnetism,and the magnitude of spontaneous electric polarization (P)is often large (∼10–100 μC/cm2). One possible mechanismfor ferroelectricity in a type-I multiferroic material is lone-pair driven. It is well known that cations containing highlypolarizable 5s or 6s lone pairs of valence electrons have astrong tendency to break the local inversion symmetry of thecrystal. This lone-pair driven mechanism was identified asthe source of ferroelectric instability in BiFeO3.3 In contrast,type-II multiferroics, where ferroelectricity may arise due to aparticular kind of magnetic ordering that breaks the inversionsymmetry, are more interesting from an application pointof view due to the strong coupling between magnetism andFE.4,5 However, the magnitude of electric polarization in thesematerials is usually very small (∼10−2 μC/cm2). For type-IImultiferroics, nonsymmetric lattice distortion and ferroelectricorder may be induced through exchange striction,6,7 a spincurrent mechanism,8 or inverse Dzyaloshinskii-Moriya (DM)interactions.9 In particular, the exchange striction is consideredto induce ferroelectricity in some collinear antiferromagnetssuch as HoMnO3 (Ref. 6) and Ca3CoMnO6.7,10,11 Whilestrong coupling between the magnetic and ferroelectric orderparameters makes them attractive, their real applications havebeen restricted by the small magnitude of the polarizationvalues. A possible way to overcome this difficulty could be tocombine the best features of type-I and type-II multiferroics.In this context, the transition metal (TM) selenium (Se) andtellurium (Te) oxihalides may offer an attractive possibilityas they exhibit exotic magnetic properties driven by thegeometric frustration in low dimensions and they also contain
a stereochemically active lone pair in Te4+ and Se4+ that mayresult in lone-pair driven ferroelectricity as in the case of type-Imutiferroics. Interestingly, some of these systems exhibit mul-tiferroic behavior. An example of such a system is FeTe2O5Br.It adopts a layered structure, where individual layers consist ofgeometrically frustrated iron tetramer units [Fe4O16] linked bythe [Te4O10Br2]6− groups.12 However, the structure remainscentrosymmetric even in the presence of Te-5s2 lone pairs.The high-temperature fit to the susceptibility data shows anegative Curie-Weiss temperature (θCW = −98 K), indicatingstrong antiferromagnetic interactions between the Fe3+(d5)ions.12 The system develops long range magnetic order ata considerably low temperature TN1 = 11 K, followed bya second magnetic transition at TN2 = 10.5 K.13 The firsttransition at TN1 is a paramagnetic to a high-temperatureincommensurate magnetic state (HT-IC) with a constant wavevector qIC1 = (0.5,0.466,0.0) and is immediately followed byanother transition at TN2 = 10.5 K into the low-temperatureincommensurate (LT-IC) multiferroic state. The amplitudemodulated magnetic order in the LT-IC phase is described withthe wave vector q = (0.5,0.463,0) and concomitantly with themagnetic order a ferroelectric polarization (P = 8 μC/m2)is induced perpendicular to q and the direction of the Fe3+moments.14 As the polarization is found to be triggered bymagnetic ordering, the resulting small value of the polarizationprovides direct evidence that FeTe2O5Br is an example ofa type-II (improper) multiferroic, contrary to the originalexpectation of combining the features of type-I and type-IImultiferroics. A recent study15 on the magnetic orderingin the HT-IC phase of FeTe2O5Br showed that while theinversion symmetry is already broken in the HT-IC phase,the ferroelectricity is only realized in the LT-IC phase. Thedifference in the orientation of the magnetic moments andphase shift of the amplitude modulated waves between thetwo magnetic structures is suggested to be responsible for therealization of ferroelectricity in the LT-IC phase. In addition,there is evidence of minute displacements of the Te4+ ionsin the LT-IC phase, and these subtle displacements may beimportant for the electric polarization in this phase.15 In viewof the above, it is suggested that polarization is possibly driven
094409-11098-0121/2013/88(9)/094409(7) ©2013 American Physical Society
CHAKRABORTY, GANGULI, SAHA-DASGUPTA, AND DASGUPTA PHYSICAL REVIEW B 88, 094409 (2013)
by exchange striction on the interchain bond containing highlypolarizable Te lone-pair electrons. In the search for a suitablespin Hamiltonian, magnetic susceptibility was analyzed byvarious groups. An early report suggested that magneticsusceptibility can be explained by considering the dominantinteractions within the Fe tetramers.12 A recent study, however,shows that the system should be described as a systemof alternating antiferromagnetic S = 5/2 chains with strongFe-O-Te-O-Fe bridges weakly coupled by two-dimensionalfrustrated interactions.16
The preceding discussion suggests that it will be importantto clarify the role of Te ions in the multiferroic propertyof FeTe2O5Br. In particular, it will be interesting to under-stand the interplay of magnetic interaction and the activityof the Te4+ lone pairs and eventually their combined rolein the ferroelectric polarization. In the present paper wehave examined this issue in detail using ab initio electronicstructure calculations. The remainder of the paper is organizedas follows. In Sec. II we describe the crystal structure alongwith the computational details. Section III is devoted to adetailed discussion of our results on the electronic structurecalculations. Finally, a summary and conclusions are given inSec. IV.
II. CRYSTAL STRUCTURE AND COMPUTATIONALDETAILS
FeTe2O5Br crystallizes in the monoclinic space groupP 21/c. The crystallographic unit cell has an inversion center.The lattice parameters for FeTe2O5Br are a = 13.396 A,b = 6.597 A, c = 14.289 A, and β = 108.12◦.12 The unit cell(depicted in Fig. 1) contains 72 atoms.
There are two crystallographically inequivalent Fe3+ ionsin the structure which are in a distorted [FeO6] octahedralenvironment. Four such octahedra share their edges with eachother and form a [Fe4O16] iron tetramer cluster (see the insetof Fig. 1). These iron tetramers are linked by [Te4O10Br2]6−units forming a layered structure in the bc plane. The layersare weakly connected via van der Waals forces as they stackalong the monoclinic a axis.
The first principles density functional theory (DFT)calculations have been performed using the plane-wave basedprojector augmented wave (PAW)17 method as implemented
c
a
Fe1
Fe2
BrTe
O
Fe2
O1
O7
O2O2
Fe1
FIG. 1. (Color online) Layered structure of FeTe2O5Br. The insetshows one tetramer unit.
in the Vienna ab initio simulation package (VASP).18 We haveused a local density approximation (LDA) to the exchangecorrelation functional. The localized Fe-d states weretreated in the framework of local spin-density approximation(LSDA) + U method,19 where calculations were done forseveral values of Ueff = U − J in the range 0 (LDA)–5 eV. Thecalculations for the unit cell were performed with a (4 × 8 × 4)� centered k point mesh and 550 eV as the plane-wave cutoffenergy. In order to simulate the magnetic structure we haveneglected the amplitude modulation and have approximatedthe incommensurate wave vector q ∼ (0.5,0.463,0) by acommensurate one (0.5,0.5,0), and have generated a supercell(2 × 2 × 1) of the original unit cell containing 288 atoms. Forthe calculations with the supercell, a plane-wave cutoff energyof 500 eV was used along with a (1 × 2 × 2) � centered k
point mesh. All structural relaxations were carried out untilthe Hellman-Feynman forces became less than 0.01 eV/A.
For the derivation of the low energy model Hamilto-nian and identification of various exchange paths we haveemployed the Nth order muffin-tin orbital (NMTO) down-folding method.20,21 The NMTO downfolding method is anefficient ab initio scheme to construct a low energy, fewband, tight-binding model Hamiltonian. The low energymodel Hamiltonian is constructed by the energy selectivedownfolding method, where high energy degrees of freedomare integrated out from the all orbital LDA calculations. TheFourier transform of the resulting low energy Hamiltonianyields the effective hopping parameters which can be utilizedto identify the dominant exchange paths.
III. RESULTS AND DISCUSSIONS
A. Non-spin-polarized calculation
To begin with, we have investigated the electronic structureof FeTe2O5Br without magnetic order. The non-spin-polarizedtotal and partial density of states are shown in Fig. 2. The den-sity of states (DOS) is consistent with the Fe3+Te2
4+O52−Br1−
nominal ionic formula for the system. Figure 2 reveals that O-pand Br-p states are completely occupied while the Fermi level(EF ) is dominated by the Fe-d states. The occupied Te-5s
states lie far below the EF . The empty Te-5p states lie abovethe Fermi level, spreading on an energy range 2–6 eV withrespect to the Fermi level. There is a significant admixture ofTe-5s and Te-5p states with the O-p states, which suggests thehybridization between Te and O, which in turn hybridizes withFe-d states crossing the Fermi level (see the insets in Fig. 2).
The presence of Te in a 4+ oxidation state suggests thepossibility of the stereochemical activity of a Te lone pairformed from 5s2 electrons. In order to visualize the lone pairsarising from 5s2 electrons of Te4+ ions, we have calculated theelectron localization function (ELF).22,23 The ELF is definedas follows:
ELF =[
1 +(
D
Dh
)2]−1
, (1)
where
D = 1
2
∑i
|∇φi |2 − 1
8
|∇ρ|2ρ
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ROLE OF Te IN THE LOW-DIMENSIONAL . . . PHYSICAL REVIEW B 88, 094409 (2013)
0
60
120
180Total
0
80
160Fe-d
0102030405060
DO
S (
Sat
es/e
V C
ell)
Te-sTe-p
-12 -9 -6 -3 0 3 6
E - EF (eV)
0
60O-pBr-p
-0.4 -0.2 0 0.2 0.40
50
100
150
-0.4 -0.2 0 0.2 0.40
4
-0.4-0.2 0 0.2 0.40
20
(a)
(c)
(b)
(d)
FIG. 2. (Color online) The non-spin-polarized density of statesfor FeTe2O5Br. (a) The total DOS, orbital-projected density of statesfor (b) Fe-d , (c) Te-s and Te-p, and (d) O-p and Br-p. The insetsshow the orbital characters near the Fermi level.
and
Dh = 3
10(3π2)5/3ρ5/3.
ρ is the electron density and φi are the Kohn-Sham wavefunctions. The ELF is defined in such a way that its value liesbetween 0 and 1. The values are close to 1 when, in the vicinityof one electron, no other electron with the same spin may befound. For instance, this would occur in bonding pairs or lonepairs.24 From the plot of the electron localization function,in the experimental centrosymmetric structure12 displayedin Fig. 3, we find that the electron density around Te isasymmetric and forms a usual lobe shape arising from the5s lone pair of Te. It has been pointed out by Watson andParker that the hybridization with anion p orbitals (oxygen2p) plays an important role in the formation of an asymmetric
a
Fe2
Fe1O
Te
c
FIG. 3. (Color online) Electron localization function within a unitcell. The isosurfaces are visualized for a value of 0.9.
lobe shaped isosurface of the electron localization function forsterically active lone pairs.25 We gather from the DOS shownin Fig. 2 that the occupied Te s and O p orbitals hybridizeto form a pair of occupied bonding and antibonding states.This Te-5s–O-2p mixed state further hybridizes with emptyTe-5p states. As a consequence both the Te-5s and Te-5p
states are involved in the formation of the asymmetric electrondistribution where empty Te-5p orbitals are able to interact dueto the presence of Te-s–O-p occupied antibonding states. Thisemphasizes the importance of the O-p states in the formationof lone pairs.
In order to quantify the hybridization, we have calculatedthe hybridization index defined as follows:
HI−l,J−l′ =∑
k
(∑i
hi,kI−l,J−l′
)× weight(k),
where
hi,kI−l,J−l′ =
∑I,J,m,m′
w(I )lm,i,kw
(J )l′m′,i,k;
w(I )lm,i,k are the coefficients in the spherical harmonic decom-
position of the local (partial) charge, associated with the ithKohn-Sham orbital,26 around the I th atom. l,m indicates theorbital and the magnetic quantum numbers, respectively, I andJ are atom indices, I ∈ {Te atoms} and J ∈ {O atoms}, andi and k stand for the band index and k points, respectively.Weight(k) is the weight on each k point in the irreducibleBrillouin zone that is necessary for the integration. Ourcalculations find that the hybridization index between Te-pand O-p is 6.13 and that between Te-s and O-p it is 3.80 for theexperimental centrosymmetric structure,12 indicating a sizablehybridization between Te and O. It is interesting to note thatthese lone pairs, however, do not promote structural distortionand the structure remains centrosymmetric, as the pair of lobesare arranged in an opposite manner, resulting in cancelingpolarization of the structure, as is evident from Fig. 3.
B. Spin-polarized calculation
We next consider magnetism and its impact on the crystalstructure and ferroelectric polarization. In order to simulate thelow-temperature magnetic order found in the LT-IC phase, wehave made a (2 × 2 × 1) supercell which contains 288 atoms.As mentioned before, in our calculation we have neglected theamplitude modulation. We consider various antiferromagnetic(AFM) configurations (see Fig. 4), depending on the arrange-ment of Fe spins within each tetramer as well as betweenthe neighboring tetramers. In the AFM1 configuration, notonly are Fe1 spins aligned antiparallel to Fe2 within eachtetramer [see the inset of Fig. 4(a)], but also each tetramer isantiferromagnetically coupled along the a and b directions,leading to q = (0.5,0.5,0). The AFM2 configuration differsfrom the AFM1 configuration only in the arrangement of spinswithin each tetramer [see the inset of Fig. 4(b)], where a pairof Fe1 spins in a tetramer are antiparallel and the same is truefor a pair of Fe2 spins. Finally, in the AFM3 configuration, thearrangement of Fe1 and Fe2 spins in each tetramer is identicalto AFM1 but the tetramers are coupled ferromagneticallyalong the a, b, and c directions, leading to q = (0,0,0). The
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CHAKRABORTY, GANGULI, SAHA-DASGUPTA, AND DASGUPTA PHYSICAL REVIEW B 88, 094409 (2013)
(a)
(c)
Fe2Fe1
(b)
Fe2
Fe1
c
b
b
c
c
b
FIG. 4. (Color online) Various antiferromagnetic configurations:(a) AFM1, (b) AFM2, and (c) AFM3.
results of our calculations are displayed in Table I. The resultsreveal that among the magnetic configurations considered here,AFM1 has the lowest energy. All magnetic states are foundto be insulating and the magnetic moment at the Fe site ismFe ∼ 4.2μB . The rest of the moments are at the oxygen(mO ∼ 0.13μB ) and bromine (mBr ∼ 0.09μB ) sites, arisingdue to the Fe-O and Fe-Br hybridization effect.
-200
0
200 Total
-4
0
4 Fe-d
-1.5
0
1.5DO
S
Te-pTe-s
-12 -10 -8 -6 -4 -2 0 2 4 6E - E
F (eV)
-1
0
1 O-pBr-p
(a)
(b)
(c)
(d)
FIG. 5. (Color online) The density of states for FeTe2O5Br in theAFM1 configuration with an experimental structure. (a) Total DOS(states/eV cell). Orbital projected DOS (states/eV atom) for (b) Fe-d ,(c) Te-s and Te-p, and (d) O-p and Br-p states.
The total density of states as well as its projection ontodifferent atomic orbitals for the AFM1 phase are shownFigs. 5(a)–5(d). Focusing on Fig. 5(b), we find that Fe-d statesin the majority spin channel are completely occupied while theminority states are completely empty, which is consistent withthe Fe3+ valence state of Fe with a 3d5 configuration. Such ahalf filled configuration promotes the AFM order.
Next, we have identified the dominant exchange paths andthe relevant spin Hamiltonian using the Nth order muffin-tinorbital (NMTO) downfolding method.20,21 In order to derive alow energy effective model Hamiltonian, we have retainedthe isolated Fe band complex near the Fermi level for anon-spin-polarized calculation and downfolded the rest withthe choice of two energy points E0 and E1. The downfoldedbands in comparison to the all orbital LDA band structureis shown in Fig. 6, and we note that the agreement is verygood. The Fourier transform of the low energy HamiltonianHk → HR [where HR is given by HR = ∑
ij tij (c†i cj + H.c.)]gives the effective hopping parameters between the variousFe atoms. The various hopping integrals can be utilized toidentify the dominant exchange paths. For strongly correlatedsystems, the antiferromagnetic contribution to the exchange
integral can be computed using J AFM = 4∑
i,j t2ij
U, where U is
the effective on-site Coulomb interaction and tij correspondsto the hopping via superexchange paths. The ratio of thevarious exchange interactions are displayed in Table II and
TABLE I. The relative energies, magnetic moments, and band gaps for different magnetic configurations are listed here.
Ueff = 3 eV Ueff = 5 eV
Magnetic �E Band gap mFe1 mFe2 �E Band gap mFe1 mFe2
config. (meV) (eV) (μB ) (μB ) (meV) (eV) (μB ) (μB )
FM 49.7 1.3 4.1 4.1 34.1 1.5 4.2 4.2AFM1 0.0 1.2 4.1 4.1 0.0 1.6 4.2 4.2AFM2 15.7 1.4 4.1 4.1 10.1 1.6 4.2 4.2AFM3 9.0 1.5 4.1 4.1 5.8 1.7 4.2 4.2
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ROLE OF Te IN THE LOW-DIMENSIONAL . . . PHYSICAL REVIEW B 88, 094409 (2013)
-0.5
0
0.5
1
1.5
Γ Z B D Y C A E
Ene
rgy
(eV
)
E0
E1
FIG. 6. (Color online) Downfolded band structure (red dottedline) compared with a full orbital LDA band structure (black solidline) of FeTe2O5Br.
the various exchange paths are indicated in Fig. 7. In lasttwo columns we have also reproduced the ratio of exchangeinteractions obtained in Ref. 16 using the total energy method.In Ref. 16, it is reported that the alternating spin chain model ismore appropriate instead of the tetramer model suggested forthis system as the intertetramer superexchange (J4) mediatedby Fe-O-Te-O-Fe bridges is appreciable. The values of theexchange interactions obtained from the NMTO downfoldingmethod reveal that, in addition to the intracluster exchangeinteractions J1, J2, J3, the intercluster exchange interactionJ4 is substantial, supporting the suggestion made in Ref. 16.However, the quantitative values of the exchange interactions,specifically the values of J1
J2and J4
J2, differ in the two studies,
possibly due to the different calculation schemes adopted inthese two independent investigations.
We next investigated the impact of magnetism on crystalstructure, viz., exchange striction. We have carried out thestructure optimization with nonmagnetic, ferromagnetic, andAFM1 magnetic configurations. In this optimization, the cellparameters were fixed to the experimental values, but the po-sitions of the atoms were allowed to relax. The change in bondlengths with respect to the unrelaxed (experimental) structurecorresponding to various exchange paths are displayed in
Fe1
Fe2
J
Te1
Te3
J
23J
J1
6
J5
J4
a
c
b
FIG. 7. (Color online) Structure of FeTe2O5Br; exchange pathsare indicated.
Table III for the AFM1, FM, and non-spin-polarized cases.The bond lengths hardly change due to the ionic relaxations fornonmagnetic and ferromagnetic cases, indicating negligibleexchange striction. The maximum change in ionic positionsoccurs in the relaxed structure with AFM1 magnetic ordering.The dominant changes correspond to the exchange path J3
involving oxygens and J5 involving the Te ions (marked in boldin Table III). Our calculations provide a direct evidence that theexchange paths J3 and J5 are responsible for the spin-phononcoupling in this compound. The importance of the exchangepath J5 was also anticipated in Ref. 16.
To obtain an estimate of the impact of structural distortionon the lone pairs, we computed the hybridization index forthe relaxed structure in the AFM1 phase. The H indicesfor the relaxed structure are found to be 17.305 and 11.015between Te-p and O-p, and Te-s and O-p, respectively, asopposed to 17.00 and 10.99 in the AFM1 phase for the
TABLE II. Exchange interactions along different exchange paths obtained from the NMTO downfolding method and energy method(Ref. 16) have been tabulated here.
Exchange paths, Ji/J2 Ji/J2 Ji/J2
bond lengths, from NMTO in Ref. 16 in Ref. 16Exchange Distance (A) and angles (Ueff = 3 eV) (Ueff = 3 eV) (Ueff = 4 eV)
J1 3.16 � Fe1-O1-Fe2 = 101.8◦ 0.89 0.46 0.35� Fe1-O2-Fe2 = 99.5◦
J2 3.34 � Fe1-O7-Fe2 = 110.2◦ 1 1 1� Fe1-O2-Fe2 = 95.79◦
J3 3.43 � Fe1-O2-Fe1 = 101.7◦ 0.44 0.33 0.34J4 4.76 Fe1-O-Te1-O-Fe2 0.26 0.62 0.59
Fe1-O-Te4-O-Fe2J5 4.77 Fe2-O-Te3-O-Fe2 0.05 0.04 0.0J6 5.10 Fe1-O-Te1-O-Fe1 0.15 0.27 0.26J7 5.52 O-O ∼ 2.81 0.02
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CHAKRABORTY, GANGULI, SAHA-DASGUPTA, AND DASGUPTA PHYSICAL REVIEW B 88, 094409 (2013)
TABLE III. The bond lengths between the Fe atoms in the experimental structure and the change in the Fe-Fe bond lengths upon relaxationwithin different magnetic configurations have been listed here. +(−) signs indicate the increment (decrement) of the distance.
Change in the bond length upon relaxation (A)with respect to the experimental structure
Exchange Bond length (A)paths expt. structure AFM1 FM NM
J1 (Fe1-Fe2) 3.16 −0.04 −0.01 −0.01J2 (Fe1-Fe2) 3.34 −0.03 0.00 0.00J3 (Fe1-Fe1) 3.43 −0.11 −0.04 −0.02J4 (Fe1-Fe2) 4.76 0.02 0.00 0.00J5 (Fe2-Fe2) 4.77 0.05 0.02 0.01J6 (Fe2-Fe2) 5.10 0.00 0.00 0.00
centrosymmetric experimental structure.12 This indicates thatthe Te-O hybridization increases as a result of the structuraldistortion, pointing to the importance of Te lone pairs. Finally,to access the asymmetry between two neighboring lobe shapedcharge distributions of the lone pairs, we have calculated themoment of the electron localization function ( Mi
ELF) for theith Te atom as follows:
MiELF =
∫ R
|r|=0d3rELF(r)r, (2)
where r is the position vector assuming the ith Te atom at theorigin and R is a suitably chosen radius of a sphere that coversthe range of ELF around the ith Te atom. We find that the sumof Mi
ELF’s vanishes for a pair of suitably chosen Te atoms inthe centrosymmetric experimental structure,12 whereas it has afinite value for the same pairs of atoms in the relaxed structure.This observation suggests that, unlike the centrosymmetricexperimental structure where the local dipole moments cancelpairwise, leading to no net polarization, in the relaxed magneticstructure they do not cancel out. (The average ELF momentfor a pair of Te atoms in a relaxed magnetic structure is 7.2 A.)This calculation hints at the activation of the stereochemicalactivity of the Te ions once the polarization is triggered by themagnetic ordering, as elaborated in the next section. In fact,the minute displacements of the Te4+ ions below TN2 in themultiferroic LT-IC phase has been corroborated by the nuclearquadrupolar resonance (NQR) results.15
C. Polarization
We have calculated the ferroelectric polarization using theBerry phase method27 as implemented in the Vienna ab initiosimulation package (VASP).28 The polarization calculationsare carried out with the idealized magnetic configurationAFM1 for several Ueff values. Our results are summarized inTable IV. The direction of polarization is the same withdifferent Ueff values, but the magnitude decreases with theincreasing value of Ueff . The calculated polarization forFeTe2O5Br is large compared to the experimental value. Suchan overestimation is also reported for other systems,29,30
and may be attributed to the idealized magnetic structureconsidered in our calculation. In view of the fact that upon ionicrelaxation the bond lengths corresponding to the exchange pathJ3 and J5 change substantially, we have investigated the impact
of the change in bond length on the exchange interaction andhence on the values of the polarization.
The exchange interaction J3 involves Fe-O-Fe, the su-perexchange pathway, and therefore obeys the Anderson-Goodenough-Kanamori rules. When the Fe-Fe distance in theJ3 exchange path is reduced, not only does the J3 increase,but the value of the polarization also increases, indicatingthe importance of this superexchange path on polarization.High resolution synchrotron x-ray diffraction, however, didnot detect significant structural changes for this bond.16 Nextwe have investigated the J5 exchange path involving the Teions. In Ref. 16, it is reported that the only sizable change inthe LT-IC phase corresponds to the shortening of the Fe2-Te3distance in the J5 exchange pathway. In order to see how thedisplacement of Te3 ions affects the exchange interaction J5
and in turn its effect on the electric polarization, we havechanged the distance between Fe2-Te3 (d1) (also the distanced2 between Fe2-Te3) (see the inset of Fig. 8) and computedthe exchange interaction J5 and the ferroelectic polarization.In Fig. 8, we have plotted the polarization as a function ofthe change in exchange interaction �J5 (between the distortedand the experimental structure). �J5 may be considered asa measure of the spin-phonon interaction mediated by theTe ions. Polarization increases as �J5 is increased, andthis polarization originates from the spin-phonon couplingcorresponding to the J5 exchange pathway. Our calculationsreveal that polarizable lone pairs enhance the spin-phononcoupling upon exchange striction in the AFM1 phase, whichin turn leads to ferroelectric polarization. In order to checkthe role of Te ions in the polarization, we have carried out aconstrained ionic relaxation calculation in which the positionsof the Te ions were kept fixed and other ionic positions were
TABLE IV. Calculated electric polarizations with an AFM1 mag-netic configuration with different values of the Coulomb interactionparameter U for the relaxed structure are listed here.
Ueff values (eV) Polarization (μC/m2)
1 217.72 208.03 198.04 187.85 177.7
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ROLE OF Te IN THE LOW-DIMENSIONAL . . . PHYSICAL REVIEW B 88, 094409 (2013)
0 0.5 1 1.5Δ J
5 (K)
0
100
200
300
400P
olar
izat
ion
(μC
/m2 )
Fe1
Fe2
Te3dd2
1J2
J5
FIG. 8. (Color online) Variation of polarization with �J5. Theinset shows the J5 exchange path involving Te3 ions.
allowed to relax for the AFM1 configuration with Ueff = 4 eV.The value of polarization is calculated to be 102 μC/m2,substantially reduced from the polarization (187.8 μC/m2)calculated for the relaxed structure where the Te ions arealso moved from their centrosymmetric positions. This resultsuggests that exchange striction within the Fe tetramers, aswell as between them mediated by Te ions, are responsible forthe magnetoelectric (ME) effect in FeTe2O5Br. Interestingly,
the magnetic ordering also triggers the stereochemical activityof Te ions, giving rise to a feedback mechanism.
IV. CONCLUSIONS
We have investigated the electronic properties of a mul-tiferroic compound FeTe2O5Br by using density functionaltheory to elucidate the role of Te ions on the ferroelectricpolarization of this system. We find that, in the absence ofmagnetism, the system remains centrosymmetric due to theantipolar orientation of the Te lone pairs that does not promotestructural distortion. The results from our calculations revealthat FeTe2O5Br is an improper multiferroic where exchangestriction within the Fe tetramers as well as between them isresponsible for the magnetoelectric (ME) effect. We find thatthe electric polarization is very sensitive to the J5 exchangepath involving the polarizable Te4+ lone pairs. Te-5s lone pairsshow stereochemical activity only when the polarization istriggered by the magnetic ordering.
ACKNOWLEDGMENTS
I.D. thanks the Department of Science and Technology,Government of India for financial support, and J.C. thanksCSIR, India [Grant No. 09/080(0615)/2008-EMR-1] for fund-ing through a research fellowship.
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17P. E. Blochl, Phys. Rev. B 50, 17953 (1994).18G. Kresse and J. Furthmuller, Phys. Rev. B 54, 11169 (1996).19S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and
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094409-7
PHYSICAL REVIEW B 86, 054434 (2012)
First-principles study of the spin-gap system Sr2Cu(BO3)2
Jayita Chakraborty and Indra Dasgupta*
Department of Solid State Physics and Center for Advanced Materials, Indian Association for the Cultivation of Science,Jadavpur, Kolkata 700032, India
(Received 9 February 2012; revised manuscript received 7 August 2012; published 20 August 2012)
We have employed first-principles electronic structure calculations based on the N th-order muffin-tin orbitaldownfolding method to derive a low energy spin model for the spin-gap compound Sr2Cu(BO3)2. Our calculationsreveal that this compound is a coupled dimer system with the strongest Cu-Cu interaction mediated by a pairof BO3 triangular units. The appreciable interdimer interactions are mediated via super-super exchange due toshort O-O distances in the exchange pathway. The validity of the model is checked by calculating the magneticsusceptibility as a function of temperature and magnetization both as a function of temperature as well asfield using quantum Monte Carlo technique and comparing them with the available experimental data. Thiscomparison establishes the suitability of the coupled dimer model for the description of the low energy physicsof Sr2Cu(BO3)2.
DOI: 10.1103/PhysRevB.86.054434 PACS number(s): 71.20.−b, 75.30.Et, 75.10.Jm
I. INTRODUCTION
Low dimensional quantum spin systems continue to enjoyconsiderable attention both theoretically as well as experi-mentally due to the wealth of fascinating properties exhibitedby them.1 Of particular importance are systems that exhibitspin gap.2 Exotic features related to the ground state andexcitations of such gapped systems form a subject matter ofcurrent interest. Copper based compounds have received aspecial interest due to their proximity to the superconductingtwo-dimensional cuprates. In this respect Cu based boratesalso attracted considerable attention. In particular, the quasi-two-dimensional spin S = 1/2 compound SrCu2(BO3)2
3 wassuggested to be the experimental realization of the Shastry-Sutherland4 model exhibiting spin-gap behavior and mag-netization plateaus at 1/8, 1/4, and 1/3 of the saturatedmagnetization. Following this discovery, there have beenattempts to tune the magnetic couplings in such systemsupon proper substitutions in order to explore the rich phasediagram of the Shastry-Sutherland model. One such attempthas been the realization of CdCu2(BO3)2 by replacement ofSr2+ by another divalent cation Cd2+.5 This compound exhibitslong range magnetic order and 1/2 magnetization plateau.A recent theoretical work6 uncovers that CdCu2(BO3)2 isa possible realization of the spin 1/2 decorated anisotropicShastry-Sutherland lattice.
In this borate family, the magnetic properties of the recentlydiscovered spin-gap compound Sr2Cu(BO3)2 is particularlyintriguing.7 The structure of Sr2Cu(BO3)2 consists of CuO4
square plaquettes and CuO6 octahedra which are connectedto each other by BO3 triangular units and run along thecrystallographic c direction as illustrated in Fig. 1. Thiscomposition is very similar to the Shastry-Sutherland4 com-pound SrCu2(BO3)2,8 but it has a very different magneticlattice and the high field and low temperature magneticbehavior is particularly interesting. The magnetization data7
for Sr2Cu(BO3)2 at low field is consistent with the fact thatthe material has a singlet ground state comprising of dimers,with intradimer coupling J ≈ 100 K. As expected in thepresence of higher applied field the triplet excitations areobserved. Interestingly, the applied field where excitation of
singlet into triplet state takes place is found to be significantlysmaller than that predicted for the isolated dimers. This inturn indicated that the interdimer couplings may be importantand possibly responsible for the triplet states at smaller fields.The importance and the nature of the interdimer couplingsare not always obvious from the structural considerations.It is therefore important to establish a connection betweenthe underlying chemical complexity of the compound andthe corresponding spin lattice. In this context ab initioelectronic structure calculations have played an important rolein analyzing and understanding such low dimensional quantumspin systems.9–12
In this paper, we shall examine the electronic structureof Sr2Cu(BO3)2 in some detail and in particular identify thedominant exchange paths and the relevant spin Hamiltonian.This spin Hamiltonian will be employed to compute themagnetic susceptibility as a function of temperature andmagnetization both as a function of field as well as temperatureusing quantum Monte Carlo simulation (stochastic seriesexpansion).13–16 We shall compare our results with availableexperimental data, in order to clarify the importance of theinterdimer coupling as anticipated in the experiments. Theremainder of this paper is organized as follows. In Sec. IIwe discuss the crystal structure and the computational details.Section III is devoted to the analysis of the electronic structure,derivation of the relevant spin Hamiltonian, and its solutionusing the QMC method. Finally, the summary and conclusionsare given in Sec. IV.
II. CRYSTAL STRUCTURE AND COMPUTATIONALDETAILS
Sr2Cu(BO3)2 exists in two structural phases. The hightemperature β-Sr2Cu(BO3)2 phase studied here crystallizes inthe orthorhombic space group Pnma with the lattice parametera = 7.612 A, b = 10.854 A, and c = 13.503 A.17 All theelectronic structure calculations in the present work are carriedout using the lattice parameters and the atomic positionsas reported in Ref. 17. The structure of Sr2Cu(BO3)2 isshown in Fig. 1. The unit cell has eight formula units with88 atoms in the unit cell as shown in Fig. 1(a). It has a
054434-11098-0121/2012/86(5)/054434(7) ©2012 American Physical Society
JAYITA CHAKRABORTY AND INDRA DASGUPTA PHYSICAL REVIEW B 86, 054434 (2012)
FIG. 1. (Color online) (a) Unit cell of Sr2Cu(BO3)2. (b) One Cu(I)octahedron is connected with two Cu(II)O4 square planar units andtwo other Cu(I)O6 octahedra. The dimer unit is indicated by the dottedcircle.layered structure (ac plane), where each layer is built ofdistorted Cu(I)O6 octahedra, square planar Cu(II)O4 units,and triangular B(1,2,3)O3 units, as shown in Fig. 1(b). Thisdistorted Cu(I)O6 octahedron has four inequivalent oxygensO1, O2, O3, and O7 surrounding each Cu(I) ion. Each Cu(I)O6
octahedron is elongated along the O1-O2 axis, with distancesdCu(I)-O1 = 2.42 A and dCu(I)-O2 = 2.49 A. The equilateraloxygens are at dCu(I)-O7 = 1.92 A and dCu(I)-O3 = 1.99 A. Theselayers are stacked along the crystallographic b axis with Sr ionsin between them. As can be seen in Fig. 1(b), each Cu(I)O6
octahedron is connected to two Cu(I)O6 octahedra and a pairof square planar Cu(II)O4 units with the aid of BO3 triangularunits. Among the two square planar units one is connectedto the Cu(I)O6 octahedron by a pair of BO3 units and theCu(I) and Cu(II) residing on the octahedral and square planarunit respectively defines the structural dimer, as indicated inFig. 1(b).
In order to analyze the electronic structure of Sr2Cu(BO3)2
we have carried out density functional theory (DFT) calcula-tions within local density approximation (LDA) by employingthe Stuttgart TB-LMTO-47 code,18 based on the tight bindinglinearized muffin-tin orbital (TB-LMTO) method in the atomicsphere approximation (ASA). The space filling in the ASAis obtained by inserting appropriate empty spheres in theinterstitial regions. For the TB-LMTO-ASA calculation thebasis set for the self-consistent electronic structure calculationfor Sr2Cu(BO3)2 includes Sr(s,d), Cu (s,p,d), O(s,p), andB (s,p) and the rest are downfolded. A (4 × 2 × 2) k mesh hasbeen used for self-consistency. In addition, the total energycalculations necessary for the evaluation of the exchangeintegrals are carried out in the plane wave basis along with theprojected augmented wave (PAW)19 method as implementedin the Vienna ab initio simulation package (VASP).20 Theexchange-correlation (XC) term in DFT was treated within
the GGA due to Perdew-Burke-Ernzerhof (PBE).21 We used aplane-wave energy cutoff of 500 eV and k-space sampling ona 4 × 2 × 2 Monkhorst-Pack grid.
In order to derive a low energy effective model Hamiltonianthat will serve as the single electron part of the many bodyHamiltonian necessary to model the system, we have em-ployed the N th-order muffin-tin orbital (NMTO) downfoldingmethod.22–24 A strong on-site Coulomb interaction (U ) isadded to the noninteracting Hamiltonian obtained from theNMTO downfolding method to construct a Hubbard modelfor the many body description of Sr2Cu(BO3)2. This model inthe limit of half filling reduces to the Heisenberg model andthereby provides the necessary spin Hamiltonian. The resultingspin Hamiltonian has been solved to calculate susceptibilityas a function of temperature and magnetization as a functionof temperature and field using QMC with the aid of stochasticseries expansion (SSE) algorithm.13–16
III. RESULTS AND DISCUSSIONS
A. Electronic structure and low energy model Hamiltonian
The non-spin-polarized band structure for Sr2Cu(BO3)2
obtained by TB-LMTO ASA method is shown in Fig. 2. Thebands are plotted along the various high symmetry pointsof the Brillouin zone corresponding to the orthorhombiclattice of Sr2Cu(BO3)2. All the energies are measured withrespect to the Fermi level of the compound. The characteristicfeature of the band structure is the isolated manifold of eightbands crossing the Fermi level (EF ) which arises from theeight copper atoms in the unit cell. These eight bands arepredominantly of Cu-dx2−y2 character in the local frame ofreference where Cu is at the origin and the x and y axispoint along the oxygens residing either on the basal planeof the octahedron or the square planar unit. These isolatedeight bands are responsible for the low energy physics of thematerial. These bands are half filled and separated from the
-6
-4
-2
0
2
4
Γ X S Y Γ Z U R T Z
Ene
rgy
(eV
)
Cu−d
B−s
x−y
O−p
Cu−dxy Cu−dyzCu−dzx Cu−d,
,
3z−1
FIG. 2. LDA band structure of Sr2Cu(BO3)2 plotted along varioussymmetry directions of the orthorhombic lattice. The zero of theenergy has been set up at the LDA Fermi energy. The dominantorbital contributions in various energy ranges are shown in boxes onthe right-hand side. Inset shows the total density of states close to theFermi level.
054434-2
FIRST-PRINCIPLES STUDY OF THE SPIN-GAP SYSTEM . . . PHYSICAL REVIEW B 86, 054434 (2012)
-6 -4 -2 0 2 40
50100150200
Total
-0.2 0 0.20
30
60Cu-d
-0.2 0 0.20
5
DO
S (S
tate
s/eV
Cel
l)
B
-0.2 0 0.20
30 O
-0.2 0 0.2E - EF (eV)
0
1
2
Sr
FIG. 3. (Color online) Total density of states of Sr2Cu(BO3)2.Inset shows the partial density of states for Cu-d, B, O and Sr in theenergy range close to EF .
low lying O p and other non-Cu-dx2−y2 valence bands bya gap of about 1.8 eV. The system is insulating (see insetof Fig. 2) and the magnitude of the gap is calculated tobe 10 meV in LDA indicating interactions in the Cu-dx2−y2
manifold. Figure 3 displays the total DOS and the partial DOScontribution from Cu, B, O, and Sr near the Fermi level. Wenote that in addition to the Cu and oxygens the bands at theFermi level have non-negligible admixture with the B stateswhich are expected to participate in the superexchange process.In order to ascertain the accuracy of our ASA calculationswe also performed the electronic structure calculation usinga projected augmented wave (PAW)19 method encoded in theVienna ab initio simulation package (VASP).20 The density ofstates calculated by these two different approaches are foundto agree well with each other.
The N th-order muffin-tin orbital (NMTO) downfoldingmethod22–24 has been established to be an efficient ab initoscheme to construct a low energy, few band, tight bindingmodel Hamiltonian. This method generates the basis setwhich describes an isolated band or group of bands. Thelow energy model Hamiltonian is constructed by selectivedownfolding method via integration process. The high energydegrees of freedom are integrated out from the all orbital LDAcalculations. The number of energy points (N ) used for thedownfolding technique is very important for the accuracy ofthe calculation. For an isolated set of bands, an atom centeredand localized set of Wannier functions may be generated bysymmetrical orthonormalization of the NMTOs.25
In order to extract the low energy model Hamiltonian,we have retained the isolated eight band complex near theFermi level and downfolded the rest with the choice of fourenergy points E0, E1, E2, and E3. The downfolded bands in
-0.4
-0.2
0
0.2
0.4
Γ X S Y Γ Z U R T Z
Ene
rgy
(eV
)
E0
E1
E2
E3
FIG. 4. (Color online) Downfolded band structure (shown indotted line) compared to the full orbital band structure (shown inblack line) for Sr2Cu(BO3)2. E0,E1,E2,E3 mark the energy pointsused in the NMTO calculation.
comparison to the all orbital LDA band structure is shownin Fig. 4 and we note that the agreement is remarkable. TheFourier transform of the low energy Hamiltonian Hk → HR
[where HR is given by HR = ∑ij tij (c†i cj + H.c.)] gives
the effective hopping parameters between the various Cuatoms. The exchange interactions can be expressed as sumof antiferomagnetic (AFM) and ferromagnetic (FM) contri-butions J = J FM + J AFM. For strongly correlated systems,the antiferromagnetic contributions can be calculated by the
following relation: J AFMn = 4t2
n
Ueff, where Ueff is the effective on-
site Coulomb interaction and tn corresponds to the hopping viasuperexchange paths. The various hopping integrals (>1 meV)extracted for Sr2Cu(BO3)2 are listed in Table I and the notationis indicated in Fig. 1.
The various Cu-Cu hoppings in the a-c plane [see Fig. 1(b)]are primarily mediated via oxygens that are shared by theCu(I)O6 octahedra, Cu(II)O4 square planar units, and the BO3
units. The equitorial oxygens of an Cu(I)6 octahedron (e.g.,O3 and O7) hybridize strongly with the Cu(I)-dx2−y2 orbitalwhile the apical oxygens (O1 and O2) hardly hybridize with it.As a consequence the effective Cu-dx2−y2 –Cu-dx2−y2 hoppingis strong provided the equitorial oxygens of an octahedron arelinked by B either to the oxygens in the square planar unit orto the equitorial oxygens of a neighboring octahedron. This isprecisely the case for the structural dimer thereby accountingfor its appreciable hopping, t2 = 112 meV. The hopping t3 andt4 [see Fig. 1(b)] proceed with the aid of the apical oxygensand therefore are expected to be small. Our calculations indeedreveal that t3 is negligible and t4 ≈ 4 meV. In addition to theabove hoppings there is a small direct hopping t1 betweena pair of Cu(II) residing on neighboring square planar units,along crystallographic a axis.
Our above argument is further substantiated by a plotof Cu-dx2−y2 Wannier function, where we have plotted theCu-dx2−y2 Wannier function for Cu(II) residing on the squareplanar unit and connected to the Cu(I)O6 octahedron by
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JAYITA CHAKRABORTY AND INDRA DASGUPTA PHYSICAL REVIEW B 86, 054434 (2012)
TABLE I. Hopping integrals and exchange interactions (in meV).
Cu-Cu O-O Hopping Total exchange interactionsdistance distance parameters for U = 6 eV
Types of Cu Hopping (A) (A) (meV) J AFMi /J AFM
2 = (ti/t2)2 (meV)
Cu(II)-Cu(II) t1 4.03 3.39 (O5-O6) 13.6 0.015 0.25Cu(I)-Cu(II) t2 4.27 2.37 (O6-O7) 112 1 −10.75Cu(I)-Cu(I) t4 6.12 2.40 (O1-O3) 4.08 0.001 −0.60Cu(I)-Cu(II) t5 6.25 3.07 (O7-O6) 40 0.127 −2.40Cu(I)-Cu(II) t6 6.35 3.1 (O3-O5) 9.52 0.007 1.78
a pair of BO3 units [see Fig. 5 (left)]. The plot revealsthat the Cu(II)-dx2−y2 orbital forms strong pdσ antibondswith the neighboring O-px and O-py orbitals. From the tailof the Wannier function we find that it strongly hybridizeswith the BO3 network and also with the Cu(I)-dx2−y2 orbital.This hybridization mediated by a pair of BO3 units as arguedabove is responsible for the strong t2 hopping. The plot of theWannier function for the Cu(II) which is connected to Cu(I)by a single BO3 unit via the apical oxygen [see Fig. 5 (right)]reveals that although Cu(II) hybridizes strongly with the BO3
unit the tail of the Wannier function does not have any weight atthe Cu(I) site because this hybridization is weak as it proceedsvia the apical oxygen, thereby accounting for its negligiblecoupling.
In order to clarify the role of B in mediating the stronghopping t2, we have plotted in Fig. 6 the Wannier functioncorresponding to B(3)-px in an energy range above the Fermilevel belonging to the antibonding part of the B -O hybriddominated by the B-p states. As expected the plot reveals thatB(3) px orbital forms antibonds with the neighboring oxygens
B3
B1
CuII
O2
O3 CuI
CuI
FIG. 5. (Color online) Wannier function of Cu-dx2−y2 , placed attwo Cu(II) sites residing on the square planar units. One is connectedto the Cu(I)O6 octahedron by a pair of BO3 units (in left) and theother Cu(II) is connected to the Cu(I)O6 octahedron by one BO3 unit(in right). Lobes of orbitals placed at different Cu(II) sites are coloreddifferently.
in the triangular unit which in turn form an antibonding linearcombination with Cu(I) and Cu(II). The main role of the B is tobring the oxygens together in the BO3 unit and strengthen theintradimer coupling. The participation of B in the intradimercoupling is reflected by the appearance of finite weight bothat the CuI and the CuII sites in the tail of the B(3)-px Wannierfunction.
Next we have considered the hoppings along the crys-tallographic b direction particularly t5, which is the secondlargest interaction. Figure 7 shows the Wannier function plotsof Cu-dx2−y2 corresponding to t5 hopping. From the Wannierfunction, we gather that the Cu-Cu interdimer hopping alongthe b direction primarily proceeds via oxygens. Although theCu-Cu distance is large, the relatively strong hopping is dueto short O-O distances (see Table I) (comparable to van derWaals distance) in the hopping pathway.12 This hopping pathwill be responsible for the interdimer exchange interaction.
Table I reveals that the dominant hoppings (>10 meV) aret1, t2, and t5. However, a tight binding (TB) analysis revealsthat excluding t1 a reasonable fit to the low energy NMTO bandstructure is obtained except for the small splitting along (�X),(S Y), and (R T) directions. Using second order perturbation
expression J AFMn = 4t2
n
Ueff, the intradimer exchange interaction is
calculated to be 12.5–10.03 meV, taking the standard value ofUeff = 4.0–5.0 eV.6,26 The ratio of the exchange integrals arelisted in Table I, where we have J AFM
5 = 0.127J AFM2 , and as
expected the other alternation parameters are negligibly small.In addition to the above estimate for the AFM contribution
to the exchange interactions, we have employed a complemen-tary approach to calculate the total exchange interactions. Wehave performed total energy calculations in the framework ofGGA +U 27 method for various ordered spin configurations.
B3
O4
CuI
O7
CuII
O6
O5
FIG. 6. (Color online) Effective orbital corresponding to thedownfolded NMTOs, placed at B(3) site.
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FIRST-PRINCIPLES STUDY OF THE SPIN-GAP SYSTEM . . . PHYSICAL REVIEW B 86, 054434 (2012)
B1
O2
CuIO7O6
CuII
CuI
CuII
FIG. 7. (Color online) Effective orbital corresponding to thedownfolded NMTOs, placed at Cu(I) and Cu(II) sites correspondingto the t5 interaction. Lobes of orbitals placed at different Cu sites arecolored differently.
In order to extract the various exchange interactions, therelative energies of these ordered spin state determined fromthe GGA +U calculations, are mapped onto the correspondingenergies obtained from the total spin exchange energies of theHeisenberg spin Hamiltonian. For the GGA +U calculationsthe on-site Coulomb interaction (U ) for Cu is taken to beU = 6 eV and the on-site exchange interaction J = 1 eV.28,29
The results of our calculations are displayed in the last columnof Table I. We gather from Table I that the dominant exchangeinteractions are J2 and J5 which are antiferromagnetic. J1 andJ6 are weakly ferromagnetic. It is interesting to note that theestimate of J2 and J5 using the second order perturbationexpression is −11.15 meV and −1.42 meV, respectively,for Ueff = 4.5 eV in good agreement with the exchangeinteractions obtained from the total energy calculations.
Table I reveals that the only relevant exchange interactionsare the intradimer coupling J2, and the interdimer exchangeJ5. In contrast to our ab initio estimate, J3 correspondingto hopping path t3 and J4 were anticipated to be the dom-inant interdimer exchange interactions based on structuralconsiderations.7 This in turn suggests that the identificationof the exchange paths based on structural consideration maybe deceptive and emphasizes the importance of ab initiocalculations in identifying the dominant exchange paths.
The resulting spin model for Sr2Cu(BO3)2 therefore turnsout to be a system of coupled dimers as indicated in Fig. 8.The figure clearly reveals that the spin lattice is a system ofdecoupled spin ladders running along crystallographic b axis.The effective spin 1/2 Heisenberg model for the two leg ladder(N × 2 lattice) may be written as
H =N∑
i=1
J2Si,1 · Si,2 +N∑
i=1
∑
j=1,2
J5Si,j · Si+1,j . (1)
Cu1
Cu2
J
j
i
5J2
FIG. 8. Spin lattice of Sr2Cu(BO3)2.
B. Susceptibility and magnetization
In order to check how the proposed spin model works,we have calculated the magnetic susceptibility as a functionof temperature and magnetization both as a function oftemperature and field and compared our results with theavailable experimental data.7 We have used the stochasticseries expansion (SSE) method to study the finite temperatureproperties of the Heisenberg antiferromagnet. The SSE isa finite-temperature QMC technique based on importancesampling of the diagonal matrix elements of the densitymatrix exp(−β H).13–15 The susceptibility calculated by theSSE method is
χ th = β(〈M2〉 − 〈M〉2), with M =∑
i
Szi ,
χ = Ng2μ2Bχ th
kBJ2,
0 100 200 300Temperature (K)
0
0.001
0.002
0.003
0.004
Sus
cept
ibil
ity
(em
u/m
ole
Oe)
Experimental DataDimer modelCoupled dimer model
10 20 30
Temperature (K)
0
0.0005
0.001
0.0015
Sus
cept
ibil
ity(
emu/
mol
e O
e)
FIG. 9. (Color online) Temperature dependence of magneticsusceptibility.
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JAYITA CHAKRABORTY AND INDRA DASGUPTA PHYSICAL REVIEW B 86, 054434 (2012)
0 20 40 60 80H (T)
0
0.1
0.2
0.3
0.4
M(
μ Β/ C
u)
0 10 20 30 40 50T (K)
0
0.1
0.2
0.3
0.4
0.5
M(
μ Β/ C
u)
45 50 55 60H (T)
0
0.1
M(
μ Β/ C
u)
(a)
(b)
FIG. 10. Magnetization plotted (a) as a function of appliedmagnetic field at temperature T = 1.5 K and (b) as a function oftemperature in an applied magnetic field strength H = 60 T fordimer model (dotted line) and coupled dimer model (solid line) ofSr2Cu(BO3)2. The symbols indicate the experimental data.
where N is the Avogadro number. The QMC simulation ofthe spin Hamiltonian was carried out by considering a finitelattice of 96 × 2 sites with periodic boundary conditions,using the stochastic series expansion algorithm.13–16 We haveused 50 000 steps for thermalization and 500 000 steps afterthermalization to ensure low statistical errors. To simulatethe low-temperature region of the susceptibility data, wehave also included the Curie contribution from impurities asχCW = Cimp/T , where Cimp = 0.000 93 emu K/mol Oe. Thebest fit susceptibility for the dimer model (J2 = 106.66 K andg = 2.13) and the coupled dimer model (J2 = 106.66 K, α =0.125, and g = 2.146) and a comparison with experimentalsusceptibility is shown in Fig. 9. While the overall agreementof the dimer model as well as the coupled dimer model with theexperimental data is good, at the low temperature region (seeinset of Fig. 9) the calculated susceptibility with the coupleddimer model has a better agreement with the experimentaldata. Also the g = 2.146 value for the coupled dimer model isin agreement with the average g value obtained from the ESRexperiment.7
The stochastic series expansion implementation ofthe quantum Monte Carlo method also allows one tosimulate quantum spin models in an external magnetic field.The importance of interdimer coupling is particularly visiblein the magnetization versus field and magnetization versus
temperature, shown in Figs. 10(a) and 10(b), respectively.Figure 10(a) displays magnetization as a function of field at avery low temperature of 1.5 K. We find that our simulation withthe coupled dimer model where J2 = 106.66 K, α = 0.125,and g = 2.146 is in excellent agreement with the experimentalresults. We clearly see the upturn in the magnetization occursat a lower magnetic field than that predicted for a dimer model(Hc = J2/gμB = 74.6 T for J2 = 106.66 K and g = 2.13).The characteristic cusplike singularity expected for 1Dsystems and spin ladders30,31is clearly seen in our simulation[see inset of Fig. 10(a)]. This feature is however absent in theexperimental data possibly due to anisotropies. The simulationusing the isolated dimer model J2 = 106.66 and g = 2.13hardly agrees with the experimental data except for the lowfield region. Figure 10(b) displays the magnetization as afunction of temperature for H = 60 T < J2/gμB (74.6 T). Inthe same figure we have shown the results obtained from theisolated dimer model and the experimental data. We again findexcellent agreement of the result obtained from the coupleddimer model with the experiment. The figure reveals thatthe magnetization does not fall to zero at low temperature.At higher temperature the M(T ) curve rises sharply andcoincides with the isolated dimer model. These calculationsindicate the reliability of our spin Hamiltonian derived fromfirst-principles electronic structure calculation.
IV. CONCLUSIONS
We have employed ab initio density functional calculationusing the TB-LMTO and NMTO downfolding method tostudy the electronic structure of Sr2Cu(BO3)2. Calculatingvarious effective hopping integrals we find that the intradimerexchange interaction is dominant and is responsible for thespin gap seen in the system. In contrast to the structuralconsiderations, our ab initio calculations reveal that theinterdimer interaction is along the crystallographic b directionand is mediated by super-super exchange due to short O-Odistances in the exchange pathway. We have derived therelevant spin model for Sr2Cu(BO3)2 and the spin modelis a system of decoupled spin ladders running along thecrystallographic b direction. This spin model was employed tocompute magnetic susceptibility as a function of temperatureand magnetization as a function of temperature as well as field.At low temperature and high magnetic field our simulationswith the coupled dimer model show excellent agreementwith the experimental data. Our calculations support that thecoupled dimer model is the appropriate model to describe thephysics of Sr2Cu(BO3)2.
ACKNOWLEDGMENTS
J.C. thanks CSIR, India (Grant No. 09/080(0615)/2008-EMR -1) for research fellowship. I.D. thanks Department ofScience and Technology, Govt. of India for financial support.
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html.17R. Smith and D. A. Keszler, J. Solid State Chem. 81, 305 (1989).18O. K. Andersen and O. Jepsen, Phys. Rev. Lett. 53, 2571 (1984).19P. E. Blochl, Phys. Rev. B 50, 17953 (1994).20G. Kresse and J. Furthmuller, Phys. Rev. B 54, 11169 (1996).
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23O. K. Andersen, T. Saha-Dasgupta, R. W. Tank, C. Arcangeli,O. Jepsen, and G. Krier, in Electronic Structure and PhysicalProperties of Solids. The Uses of the LMTO Method, edited byH. Dreysse, Springer Lecture Notes in Physics, Vol. 535 (Springer,Berlin, 2000), pp. 3–84.
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PHYSICAL REVIEW B 84, 174508 (2011)
Superconductivity in multiband disordered systems: A vector recursion approach
Shreemoyee GangulyDepartment of Materials Science, S.N. Bose National Centre for Basic Sciences, JD-III Salt Lake City, Kolkata 700098, India
Indra DasguptaDepartment of Solid State Physics and Centre for Advanced Materials, Indian Association for the Cultivation of Science,
Jadavpur, Kolkata 700032, India
Abhijit MookerjeeDepartment of Materials Science and Advanced Materials Research Unit, S.N. Bose National Centre for Basic Sciences,
JD-III Salt Lake City, Kolkata 700098, India(Received 26 May 2011; revised manuscript received 19 October 2011; published 10 November 2011)
We present a vector recursion based approach to study the effect of disorder on superconductivity in a systemmodeled by the two-band attractive Hubbard model. We use the augmented space formalism for the disorderaveraging. In the presence of only intraband pairing in a two-band disordered system with disorder in either orboth bands, our calculations reveal that the gap survives in the quasiparticle spectrum; similar to single bandsystems. However, for interband pairing the gap in the quasiparticle spectrum ceases to exist beyond a criticalvalue of the disorder strength. In the presence of both interband and intraband pairing interaction, dependingon the relative magnitude of the pairing strength, only a particular kind of pairing is possible for a half filledtwo-band system.
DOI: 10.1103/PhysRevB.84.174508 PACS number(s): 71.10.−w, 71.23.−k, 74.20.−z
I. INTRODUCTION
The study of superconductivity in multiband systemshas received considerable interest recently because of thediscovery of superconducting materials where the Fermisurface is dominated by several bands. Examples includeMgB2 where the Fermi surface is determined by the σ
and π bands arising from the B-p orbitals. It is nowconfirmed that the superconductivity in this material can beexplained with the Bardeen-Cooper-Schrieffer (BCS) theorywith two different superconducting gaps in agreement withexperiments.1 A description of unusual p-wave superconduc-tivity in Sr2RuO4 also necessitates a multiband model forsuperconductivity.2,3 Very recently the discovery of super-conductivity in Fe pnictides, whose Fermi surface is builtout of the t2g orbitals of Fe, has again emphasized theimportance of the study of superconductivity in multibandsystems.4,5
The complex problem of superconductivity in multibandsystems was first studied by Suhl et al.6 using a tight-bindingmodel Hamiltonian with two bands. The model includedintraband pairing and also the interband hopping of pairsof electrons belonging to the same band. They showed thatpairing could occur in each band and, because electron-phononinteractions may have different strengths in different bands,this can give rise to two different superconducting gaps. Butin the special case of only interband scattering, a single gapwas found to be present in the density of states unless the banddispersion of the two bands had different shapes.7 A similarmodel was also investigated by Machida et al.8 for the studyof superconductivity in multiband systems. Recently Moreoet al.9 revisited the theory of superconductivity in multibandsystems in the context of Fe pnictides. In particular they haveemphasized the importance of interband pairing in multiband
systems in which, in contrast to earlier studies,6,8 Cooperpairs are formed by electrons belonging to two differentbands. The calculations by Moreo et al.9 revealed that threedifferent regions can result from a purely interband pairing asa function of the interaction parameter: (i) a normal regimewhere the ground state is not superconducting; (ii) an exoticsuperconducting “breached” regime where one of the bandsis gapped at the Fermi level while the other is not, and (iii)a superconducting regime resembling the BCS states, at largeattractive coupling. The existence of an exotic superconducting“breached” regime with both gapped and gapless quasiparticleexcitations was also discussed by Liu and Wilczek10 inthe context of cold atoms and quantum chromodynamicsystems.
The preceding discussion suggests that superconductivityin multiband systems is not only interesting but markedlydifferent from its single-band counterpart. In this context itwill also be important to understand the role of disorderin multiband superconducting systems since disorder is animportant factor that has a profound impact on superconduc-tivity. While the effect of disorder on superconductivity insingle-band systems have been actively investigated, there arevery few systematic studies of the role of disorder in multibandsystems.
The effect of disorder in single-band systems is usuallydiscussed within the framework of Anderson’s theorem.11 Fors-wave superconductors Anderson’s theorem guarantees thesurvival of an absolute gap in the quasiparticle spectrumof the system provided the perturbation due to disorderpreserves time-reversal invariance and the coherence lengthis long enough to ensure that the pairing amplitude �
does not fluctuate. There exists a body of work where theBogoliubov–de Gennes (BdG) equations,12 which provide
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GANGULY, DASGUPTA, AND MOOKERJEE PHYSICAL REVIEW B 84, 174508 (2011)
a natural framework for a fully microscopic descriptionof the phenomena of superconductivity, have been solvedin conjunction with the mean-field single-site coherent po-tential approximation (CPA),13–15 in order to understandthe physics of superconductivity in single-band disorderedsystems.
Recently we have proposed an efficient real-space schemeto solve the BdG equations for single-band disordered attrac-tive Hubbard models.16 The aim of this paper is to proposea real space, vector recursion based approach to study theeffect of disorder on a multiband attractive-U Hubbard modelwhere the configuration averaging, as in our earlier study, willbe based on the augmented space recursion (ASR) formalismintroduced by one of us.17 The ASR gives us the flexibility ofintroducing the effects of random configuration fluctuations inthe local environment of a site. It does not violate analyticalproperties of the configuration-averaged Green’s function,which form an essential ingredient of the solution. It candeal easily with the effect of either off-diagonal disorder orinhomogeneous disorder such as clustering, segregation, andshort-ranged ordering, which usually occur intrinsically inmost disordered materials due to different chemical affinitiesof the constituents.
We shall begin by studying superconductivity in an or-dered two-band, tight-binding, attractive-U Hubbard model,using our vector recursion technique. Then, having satisfiedourselves with the reliability of our methodology, we shallproceed to study the effect of disorder on the same model.The rest of the paper is organized as follows: in Sec. II weshall discuss our method in some detail. Section III will bedevoted to results and discussions for multiband ordered anddisordered systems. Finally in Sec. IV we will summarize ourstudy.
II. METHODOLOGY
A. Multiband attractive-U Hubbard model
To study the effect of disorder on a multiband s-wavesuperconducting system we shall begin with the simplestmodel, namely, the two-band attractive Hubbard Hamiltonianin model lattices. The Hamiltonian is given by
H = −∑〈i,j〉
∑m,m′,σ
tim,jm′ c†imσ cjm′σ +
∑i,m,σ
(εim − μ)nimσ
−∑i,m
|Umm(i)|nim↑nim↓
−∑
i
∑m,m′,σ,σ
′|Umm′(i)|nimσnim′σ ′ . (1)
Here m, m′ are the band index. This Hamiltonian is ageneralization of the single-band Hubbard Hamiltonian and
is similar to earlier studies by Annett and co-workers.2,3
Our model Hamiltonian allows for both intraband as wellas interband pairing. The interband pairing term is similarto that of Annett and co-workers2,3 and Moreo et al.9 whichallows Cooper pairs to be formed by electrons belonging totwo different bands. The earlier studies by Suhl et al.6 andMachida et al.8 did not consider the pairing of electronsbelonging to two different bands but a pair tunneling termgiven by
−∑
i
∑m,m′,σ,σ
′
∣∣Utmm′ (i)
∣∣(cimσ cimσ ′ )†cim′σ cim′σ ′ . (2)
This term allowed for the tunneling of the Cooper pairs fromone band to the other with a tunneling strength given byUt
mm′ .In Eq. (1) {c†imσ },{cimσ } are the usual electron creation and
annihilation operators for orbital m with spin σ on site labeledi of a square or cubic lattice. The index m runs over the twobands labeled s and l, μ is the chemical potential, and εim isthe local on-site energy at the site labeled i in the band m. Thehopping integral tim,jm′ has four components:: tis,js = ts is thehopping integral in the s band from a site i to one of its nearestneighbors j and til,j l = tl is that in the l band from a site toone of its nearest neighbors. The interband hopping integralsare tis,il = tsl , which is the hopping integral from a site inthe s band to the same site in the l band (or vice versa) andtis,j l = tnn
sl , which is the hopping integral from a site i in thes band to one of its nearest neighbors j in the l band (or viceversa). In this work we have not included the interband intersitehopping integral tnn
sl . However, we do consider the effect ofon-site interband hopping integrals tsl in some of our analysis.As we will see subsequently, tsl will not alter the qualitativefeatures of our results. In this model, Uss = −|Us | correspondsto a local Hubbard parameter leading to a pairing interactionpotential for s-band electrons and Ull = −|Ul| correspond toa local Hubbard parameter for l-band electrons. Here, both theattractive interactions give rise to s-wave pairing since theyare local. The interband pairing interaction Umm′ = −|Usl | isthe local attractive potential between electrons in the s and l
band.The BdG mean-field decomposition12 of the interaction
terms give expectation values to the intra- and interband pairingamplitudes,
�m = −|Um| 〈cim↓cim↑〉; �sl = −|Usl | 〈cil↓cis↑〉, (3)
and also to the intra- and interband “densities,”
〈nimσ 〉 = 〈cimσ c†imσ 〉 ; 〈nislσ 〉 = 〈cilσ c
†isσ 〉. (4)
The effective quadratic BdG Hamiltonian becomes
Heff = −∑〈i,j〉
∑m,m′,σ
tim,jm′ c†imσ cjm′σ +
∑imσ
(εim − μim) nimσ −∑
im,m′,σ
|Umm′ | 〈nimm′σ 〉2
c†imσ cjm′σ
+∑im
(�mc†im↑c
†im↓ − �∗
mcim↑cim↓) +∑
i,m,m′(�mm′c
†im↑c
†im′↓ − �∗
mm′cim↑cim′↓), (5)
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SUPERCONDUCTIVITY IN MULTIBAND DISORDERED . . . PHYSICAL REVIEW B 84, 174508 (2011)
where μim = μ − |Umm|〈nim〉/2 incorporates the site depen-dent Hartree shift.
This effective Hamiltonian can be diagonalized by usingthe Hartree-Fock-Bogoliubov (HFB)18 transformation,
cim↑ =∑
n
[βn↑um(ri,E) − β†n↓v∗
m(ri,E)],
(6)cim↓ =
∑n
[βn↓um(ri,E) + β†n↑v∗
m(ri,E)],
where β and β† are quasiparticle operators, andum(ri,E), vm(ri,E) are the quasiparticle amplitudes associatedwith an eigenenergy En.
In the Hartree-Fock mean-field approximation incorporat-ing charge-order and superconducting decoupling along withthe above canonical transformation we have⎛⎜⎝ Hss �s −Nsl �sl
�∗s −Hss �∗
sl Nsl
−Nls �ls Hll �l
�∗ls −Nls �∗
l −Hll
⎞⎟⎠⎛⎜⎝us(ri,E)
vs(ri,E)ul(ri,E)vl(ri,E)
⎞⎟⎠
= E
⎛⎜⎝us(ri,E)vs(ri,E)ul(ri,E)vl(ri,E)
⎞⎟⎠ , (7)
where (the excitation eigenvalue E � 0)
Hmmum(ri,E) = (εim − μim)um(ri,E) −∑
j
tmum(rj ,E),
Nmm′um′ (ri,E) ={
1
2|Umm′ |〈nmm′ 〉 + tmm′
}um′ (ri,E)
+∑
j
tnnmm′um′(rj ,E). (8)
Here j is the nearest neighbor of i. We can express theparticle densities and the pairing amplitudes in terms of thequasiparticle amplitude as
〈nim〉 = 2∫
dE|um(ri,E)|2f (E)
+ |vm(ri,E)|2[1 − f (E)],
〈nimm′ 〉 = 2∫
dEum′ (ri,E)u∗m(ri,E)f (E)
+ v∗m′ (ri,E)vm(ri,E)[1 − f (E)],
(9)�m = |Um|
∫dEv∗
m(ri,E)um(ri,E)f (E)
−um(ri,E)v∗m(ri,E)[1 − f (E)],
�mm′ = |Umm′ |∫
dEv∗m(ri,E)um′ (ri,E)f (E)
−um(ri,E)v∗m′ (ri,E)[1 − f (E)],
where f (E) is the Fermi function. A fully self-consistentsolution of Eq. (7) can be obtained provided all the normalpotentials (|Um|nim and |Umm′ |nimm′) and anomalous potentials(�im and �imm′ ) are determined self-consistently from Eq. (9).The self-consistency criteria is set to 10−6 for calculation ofall self-consistent parameters throughout the present study.
B. Treatment of disorder: Augmented space formalism
The class of systems which we shall study here willbe binary substitutionally disordered alloys. We shall studyrandomness in the diagonal site energies, either in one of thetwo bands, say the l band ({εil}); or in both the bands ({εis}and {εil}). We shall introduce site occupation variables {ni}(this should not be confused with the number operator nimσ )which take values 1 or 0 according to whether the site labeledi is occupied by an A type or a B type of atom,
εim = εAm ni + εB
m (1 − ni) = εBm + δεmni, (10)
where, m = s or l and εAs , εB
s and εAl , εB
l are the possibleon-site energies corresponding to the s and l band, respectively.We define the strength of disorder in the band labeled m byDm = |δεm| = |εA
m − εBm|.
If the concentrations of A- and B-type atoms in the solidare x and y, then the probability density of ni , in the absenceof short-range order, is given by
p(ni) = xδ(ni − 1) + yδ(ni). (11)
The “configuration space” of ni , �i , has rank 2 and is spannedby the states |Ai〉 and |Bi〉 in which the parameter εim take thevalues εA
m and εBm, respectively.
The augmented space formalism associates with eachrandom variable ni an operator Ni acting on its configurationspace �i and whose spectral density is its probability density.That is,
p(ni) = − 1
πlimδ→0
Im 〈∅i |[(ni + iδ)I − Ni]−1|∅i〉, (12)
where |∅i〉 = √x|Ai〉 + √
y|Bi〉 is the so-called “reference”state. This nomenclature arises from the fact that the aug-mented space theorem19 states that the matrix element in thisstate is the configuration average. The other basis member is|1i〉 = √
y|Ai〉 − √x|Bi〉 which is a state with one “fluctua-
tion” about the reference state at the site i. Alternatively, it isdenoted by |{i}〉 where {i} is the “cardinality sequence” of sitesat which there are fluctuations. The configuration states |Ai〉and |Bi〉 are the eigenkets of Ni corresponding to eigenvalues1 and 0. The representation of the operator Ni in the basis{|∅i〉,|ii〉} is
Ni = xP∅i+ yP1i
+ √xy
[T∅i ,1i
+ T1i ,∅i
]= xI + (y − x)P1i
+ √xy
[T∅i ,1i
+ T1i ,∅i
]. (13)
Here, I is the identity operator, PX are the projectionoperators |X〉〈X|, and TXY are the transfer operators |X〉〈Y |,and X,Y are either ∅i or 1i .
Let us define a configuration fluctuation creation operatorat the site labeled i as γ
†i |∅i〉 = |1i〉. Since each site can
either be ∅ or 1, this is a fermionlike creation operator withγ†i |1i〉 = 0. Similarly we define a configuration fluctuation
annihilation operator γi |1i〉 = |∅i〉 and γi |∅i〉 = 0. In terms ofthese operators P1i
= γ†i γi counts the number of configuration
fluctuations at the site i, and of the transfer operators:T∅i ,1i
= γi annihilates and T1i ,∅i= γ
†i creates a configuration
fluctuation at the site i.The operator Ni in this new representation is
Ni = xI + (y − x) γ†i γi + √
xy (γ †i + γi) (14)
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GANGULY, DASGUPTA, AND MOOKERJEE PHYSICAL REVIEW B 84, 174508 (2011)
So,
εim = εBm + δεm ni has associated with it an operator,
εim = 〈εm〉I + (y − x)δεm γ†i γi + √
xy δεm (γ †i + γi)
(15)
obtained by replacing ni with its operator form Ni [seeEq. (14)] where 〈εm〉 refers to the average:
〈εm〉 = xεAm + yεB
m (16)
with m = s or l, δεs = εAs − εB
s , and Ds = |δεs |, δεl = εAl −
εBl , and Dl = |δεl|.
The augmented space theorem19 states that the configura-tion average of a function of a set of independent randomvariables A({ni}) can be expressed as a matrix elementin the full configuration space of the disordered system
� = ∏⊗�i ,
〈〈A({ni})〉〉 = 〈{∅}|A({Ni})|{∅}〉, (17)
where |{∅}〉 = ∏⊗i |∅i〉 and A({Ni}) is the representation of
the operator A in the configuration space �, constructed byreplacing all random variables ni by their correspondingoperators Ni . A compact way of representing a basis inconfiguration space is to denote it by the set of sites wherewe have a configuration fluctuation. This set is called thecardinality set and the meaning of the empty cardinalityset {∅} then becomes obvious. For the present system theHamiltonian contains the random variables {εis} and{εil}. Sowe need to construct the Hamiltonian in the augmented space� = H ⊗ ∏⊗
i �i by replacing all the random variables εis
and εil by the corresponding operators shown in Eq. (15). Theeffective augmented space Hamiltonian becomes
Heff = −∑
〈i,j〉,m,m′,σ
tim,jm′ c†imσ cjm′σ ⊗ I +
∑imσ
(〈εm〉 − μim)nimσ ⊗ I +∑imσ
δεm nimσ ⊗ {(y − x)γ †i γi + √
xy(γ †i + γi)} · · ·
−∑
im,m′,σ
|Umm′ | 〈nimm′σ 〉2
c†imσ cjm′σ ⊗ I +
∑im
(�mc†im↑c
†im↓ − �∗
mcim↑cim↓) ⊗ I · · ·
+∑
i,m,m′(�mm′c
†im↑c
†im′↓ − �∗
mm′cim↑cim′↓) ⊗ I. (18)
In the special case when there is randomness in just one of thebands (say l), in Eq. (18) we put δεs = 0 and 〈εs〉 = εs .
After constructing the Hamiltonian in augmented space theaugmented space theorem then automatically ensures that theconfiguration average is a projection onto the state with no“fluctuations,”19
〈〈G(i,i,E)〉〉 = 〈∅|G(i,i,E)|∅〉,where G = (EI − H
eff)−1. All operators here are 4 × 4 ma-
trices (here double underbar indicates 4 × 4 matrices) in thespace spanned by the two bands and the electron-hole degreesof freedom20 arising in BdG formalism.
The Green’s functions are obtained using the vectorrecursion technique introduced by Haydock and Godin.21,22
The vector recursion has been described in great detail in thegiven references and in our earlier work.16 We shall indicatethe main points and the interested reader may refer to thequoted references for details. Once the BdG Hamiltonianis set up as in Eq. (7) and the effective augmented spacetransformation carried out as in Eq. (18), the vector recursiontechnique essentially changes the basis in order to block tridi-agonalize the effective Hamiltonian. The basis is recursivelygenerated,
|1〉〉 =
⎛⎜⎝us(�ri,E) ⊗ {∅}vs(�ri,E) ⊗ {∅}ul(�ri,E) ⊗ {∅}vl(�ri,E) ⊗ {∅}
⎞⎟⎠B†
n+1|n + 1〉〉 = H |n〉〉 − An|n〉〉 − B n|n − 1〉〉.
The coefficients An and B n are matrices and obtained fromthe orthogonality of the generated basis and between rowsof the same basis. The configuration averaged diagonal matrixelement of the Green’s function then follows as a matrixcontinued fraction,
〈〈G(�ri�ri ; E)〉〉 = 〈〈1|G|1〉〉 = G0(E),
Gn(E) = [zI − An − B†n+1Gn+1(E)B n+1]−Pn−1 ,
n = 0,1,2, . . . N2 − 1,
where A−Pn denotes inverse in the subspace spanned by thebasis {|n + 1〉〉,|n + 2〉〉 . . .}. The matrix continued fraction isterminated in two steps. The matrix coefficients {A
n,B
n} are
calculated exactly for n < N1, then: first, by putting An = AN1
and B n = B N1 for all N1 � n < N2 and second, GN2 (E) =(E + iη)−1I .
The physical quantities of interest [Eq. (9)] relevant to thestudy can be expressed as appropriate matrix elements of theGreen’s function,
〈nm〉 = − 1
πlimη→0
Im∫ ∞
−∞[G++
mm(i,i,E + iη)fn
+ G−−mm(i,i,E + iη)(1 − fn)]dE,
�m = − 1
πlimη→0
Im∫ +Ec
−Ec
[G+−mm(i,i,E + iη)fn
+ G−+mm(i,i,E + iη)(1 − fn)]dE,
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SUPERCONDUCTIVITY IN MULTIBAND DISORDERED . . . PHYSICAL REVIEW B 84, 174508 (2011)
〈nmm′ 〉 = − 1
πlimη→0
Im∫ ∞
−∞[G++
mm′(i,i,E + iη)fn
+ G−−mm′ (i,i,E + iη)(1 − fn)]dE,
�mm′ = − 1
πlimη→0
Im∫ +Ec
−Ec
[G+−mm′(i,i,E + iη)fn
+ G−+mm′ (i,i,E + iη)(1 − fn)]dE, (19)
where + and − refer to electron and hole spaces of the BdGformalism20 and the energy interval [−Ec, + Ec] is the shortinterval around the Fermi energy of the system where theinteraction has its effect.
III. RESULTS AND DISCUSSION
A. Ordered systems
In this section we shall present results on ordered two-bandsuperconductors (both the bands having s-orbital character)on square and cubic lattices with both local intra- andinterband Hubbard parameters. The system is kept fixed athalf filling unless otherwise stated. Since these results are wellknown from other approaches, a comparison with them willascertain the viability and numerical accuracy of our proposedmethodology.
For our model system the hopping integrals are chosenas follows: in Figs. 1(a)–1(d) the intraband nearest-neighborhopping elements are ts = 1.0 and tl = 0.5 and the interbandon-site hopping is tsl = 0.0.
The s- and l-band partial densities of states (PDOS) forthe case when Us = Ul = Usl = 0 for the ordered systemare shown in Figs. 1(a) and 1(c) for the square and cubiclattices, respectively. The two sets of PDOS exactly matchthe standard calculations using Bloch’s theorem. One canclearly see in Fig. 1(a) the band-center integrable Van Hovesingularity, the two flanking kink singularities, and the square-root singularities at the band edges that are characteristic ofa square lattice. The cubic lattice PDOS [see Fig. 1(c)] ischaracterized by constant DOS at the band center and terminatein kink singularities on both sides. The s band with greaterintraband hopping integral is wider, as expected.
Next we investigate the situation in the presence ofintraband pairing, i.e., Hubbard parameter Us and Ul are onlyfinite. This corresponds to the system studied by Suhl et al.6
in the absence of interband tunneling of electrons. Thus Usl
in Eq. (1) is set to zero. In Figs. 1(b) and 1(d) we considerthe cases where Us = Ul = 4.0 and the system is kept fixedat half filling. The BdG equations are solved recursively andself-consistently as described earlier. After self-consistencythe superconducting order parameters �s and �l are found tobe nonzero. The s and l configuration averaged PDOS for thesystem are calculated by using the relation
〈〈nm(E)〉〉 = − 1
πlimη→0
Im 〈〈G++mm(1,1,E + iη)〉〉,
where m = s or l, η is an infinitesimal positive imaginary partof the energy, and + refer to the electron states in the BdGformalism.
-5 -4 -3 -2 -1 0 1 2 3 4 5Energy
0
0.2
0.4
0.6
PD
OS
s-bandl-band
(a)
-5 -4 -3 -2 -1 0 1 2 3 4 5Energy
0
0.5
1
1.5
2
PD
OS
s-bandl-band
(b)
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7Energy
0
0.1
0.2
0.3
PD
OS
s-bandl-band
(c)
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6Energy
0
0.5
1
1.5
PD
OS
s-bandl-band
(d)
FIG. 1. (Color online) Study of superconductivity in an ordered square lattice [(a) and (b)] and cubic lattice [(c) and (d)] having two bandss and l. (1) Intraband hopping integrals ts = 1.0 and tl = 0.5, and (2) Hubbard parameters for (a) and (c) are Us = Ul = Usl = 0.0 and for (b)and (d) are Us = Ul = 4.0 and Usl = 0.
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GANGULY, DASGUPTA, AND MOOKERJEE PHYSICAL REVIEW B 84, 174508 (2011)
The PDOS shown in Figs. 1(b) and 1(d) reveal that inspite of the parameters Us = Ul , the superconducting pairingamplitude �s and �l are different. This is due to the differencein bandwidth (W ) as ts �= tl , and the observation that theeffective parameters Um/W (m = s or l) are responsible forthe magnitude of the gap seen in the local DOS.
In view of the above we have also investigated the situationonly with intraband Hubbard parameters such that Us �= Ul .We have considered Us = 3.0 and Ul = 1.0. Since the effectiveparameter Us/W = 0.75 > Ul/W = 0.5 we did find �s >�l .The earlier study by Suhl et al.6 had also found two differentband gaps arising in a two-band model system. Two differentsuperconducting gaps were later realized in MgB2.23–29
Next in addition to the intraband pairing we have alsoincluded interband pairing of electrons. In the presence ofboth inter- and intraband Hubbard parameters an interestingcompetitive effect sets in, as can be seen from Fig. 2(a).We keep the intraband attractive Hubbard parameter fixed(Us = Ul = 2.0), and vary the interband Hubbard parameterUsl . The intraband hopping integrals are chosen to be ts = 1.0and tl = 0.5 and interband on-site hopping integral is tsl = 0.2.We see [from Fig. 2(a)] when Us = Ul � Usl then it is theintraband pairing amplitude that is only finite and the interbandpairing amplitude vanishes. On the other hand, when Us =Ul < Usl then it is only the interband pairing amplitude that isnonzero. Our calculations shows for momentum independentpairing in s-like bands depending on the strength of theattractive interaction, only a particular kind of pairing, eitherintraband or interband, is possible for two-band half filledsystems when both bands have s-wave character.
Finally, we examine the effect of the interband (on-site)hopping integral tsl on the pairing amplitude � for a half filledsystem. Figures 2(b) and 2(c) display the case for dominantintraband pairing (Us = Ul = 3.5 > Usl = 2.0) and dominantinterband pairing (Us = Ul = 2.0 < Usl = 3.5), respectively.We find from the figures that inclusion of intraband on-sitehopping term tsl does not change the qualitative picture for atwo-band system except to reduce the magnitude of the gap.
B. Homogeneously disordered systems
We shall now study an attractive-U Hubbard model of atwo-band, disordered, binary substitutional alloy on a squarelattice. First we consider randomness in the on-site energy inone of the two channels, namely the l channel, and study itseffect on the other channel. We introduce randomness in theon-site energy using Eq. (15) and our Hamiltonian takes theform given in Eq. (18). The concentrations are x = y = 0.5and the system is half filled throughout the study.
To begin with, we study the systems in a situation similarto those under which we had investigated the correspondingordered system. We keep ts = 1.0 and tl = 0.5 and thestrength of disorder Dl = |εA
l − εBl | = 1 throughout the cases
considered in Fig. 3.First we study the case when the system is nonsupercon-
ducting (Us = Ul = Usl = 0.0). From Fig. 3(a) we find dueto the absence of hybridization between the s and l bandsthe s PDOS is not affected by randomness in the l channel.The l PDOS [Fig. 3(b)], however, has characteristic features ofdisordered DOS: namely increase in bandwidth and smoothing
0 1 2 3 4Usl
0
0.5
1
1.5
2
2.5
Δ
ΔΔΔ
s(a)
Us = Ul = 2.0
lsl
0 0.2 0.4 0.6 0.8tsl
1
1.2
1.4
1.6
1.8
Δ
ΔΔ
s
l
(b)
Us=Ul=3.5 ; Usl=2.0
0 0.2 0.4 0.6 0.8tsl
0.5
1
1.5
2Δ
ΔslUs=Ul=2.0 ; Usl=3.5
(c)
FIG. 2. (Color online) Variation of � for a square lattice whenboth intra- and interband interaction potentials are nonzero. Herethe intraband hopping integrals are ts = 1.0 and tl = 0.5 for the sand l bands, respectively. In (a) the intraband pairing potentials |Us |and |Ul | are kept fixed at 2.0 and Usl is varied. In (b) and (c) thepairing potentials are kept fixed [(b) Us = Ul > Usl = 2.0 and (c)Us = Ul < Usl = 3.5] and the effect of variation of interband on-sitehopping integral tsl is studied.
out of Van Hove singularities. The total DOS [Fig. 3(c)]therefore carries the signatures of disorder as well.
Next, we investigate the DOS of the same system consider-ing only the intraband Hubbard parameters to be nonzero, i.e.,Us = Ul = 4.0 and Usl = 0.0 [Figs. 3(d)–3(f)]. In this caseonly the intraband pairing amplitudes �s and �l are nonzero[see Eq. (3)]. We see that the s PDOS remains unaffectedby randomness in the l channel [comparing Fig. 3(d) withFig. 1(b)], disorder, however, influences the l PDOS [compar-ing Fig. 3(e) with Fig. 1(b)]. Since both the s PDOS and l PDOSare gapped, the total DOS remains gapped [Fig. 3(f)]. Similarbehavior also prevails with the inclusion of attractive interband
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SUPERCONDUCTIVITY IN MULTIBAND DISORDERED . . . PHYSICAL REVIEW B 84, 174508 (2011)
-4 -2 0 2 4Energy
0
0.1
0.2
0.3
0.4
0.5
PD
OS
s-band
Us=Ul=0.0Usl=0.0
D=1.0
(a)
-4 -2 0 2 4Energy
0
0.2
0.4
0.6
0.8
PD
OS
s-band
Us=Ul=4.0
Usl=0.0
D=1.0
(d)
-4 -2 0 2 4Energy
0
0.1
0.2
0.3
0.4
PD
OS
l-band
Us=Ul=0.0
Usl=0.0
D=1.0
(b)
-4 -2 0 2 4Energy
0
0.5
1
1.5
2
PD
OS
l-band
Us=Ul=4.0
Usl=0.0
D=1.0
(e)
-4 -2 0 2 4Energy
0
0.2
0.4
0.6
0.8
TD
OS
TDOS
Us=0.0 ; Ul=0.0
Usl=0.0D=1.0
(c)
-4 -2 0 2 4Energy
0
0.2
0.4
0.6
0.8
1
TD
OS
TDOS
Usl=0.0
D=1.0
Us=Ul=4.0
(f)
FIG. 3. Study of a two-band superconducting system in a square lattice with disorder in the l channel with strength of disorder D = 1.0.While (a)–(c) study the s, l PDOS and total DOS, respectively, for the nonsuperconducting case (where intra- and interband Hubbard potentialUs = Ul = Usl = 0.0), (d)–(f) study the effect of disorder on the corresponding superconducting system with only intraband interaction.
interaction Usl , provided the intraband pairing dominates, i.e.,Usl < Ul and Us .
The variation of the zero-temperature superconductingorder parameters �s , �l , and �sl are plotted as a function ofthe strength of disorder in Fig. 4(a) where Us = Ul = 2.0 >
Usl = 1.0. As expected for momentum independent pairingonly the intraband pairings are finite. �s does not change as afunction of disorder strength as it does not register the effectof the disorder in the l channel. As the strength of disorder(D) is increased �l reduces but remains finite even for D = 3.Therefore in the chosen parameter regime for the two-bandsystem the situation is similar to that predicted by Andersontheorem11 for the single-band system, where the gap survivesin the quasiparticle spectrum even in the presence of disorder.
Suhl et al.6 using a generalized BCS Hamiltonian for thetwo-band superconductor proposed a generalized expressionfor critical temperature Tc and temperature-dependent pairingamplitude. As stated earlier, our two-band Hubbard Hamilto-nian without the interband pairing term is identical to that ofSuhl et al. The expression for Tc for the s and l bands (T s
c andT l
c , respectively) can be generalized to
1 = |Um|∫ ∞
−∞dE
〈〈Nm(E)〉〉2E
tanh
(E
2kBT mc
), (20)
where m = s or l, while 〈〈Ns(E)〉〉 and 〈〈Nl(E)〉〉 are thes- and l-band configuration averaged density of states in thenormal state at energy E. Setting Us = Ul = 3.5, Usl = 0 and
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GANGULY, DASGUPTA, AND MOOKERJEE PHYSICAL REVIEW B 84, 174508 (2011)
1 1.5 2 2.5 3Disorder Strength (D)
0
0.15
0.3
0.45
0.6
ΔΔ
slUs=Ul=2.0
Usl=1.0
(a)
0 0.5 1 1.5 2 2.5 3Disorder Strength D
0.5
0.6
0.7
0.8
Tc
Tcs
Tcl
(b)
0.2 0.4 0.6 0.8kBT
0
0.5
1
1.5
2
2.5
Δ
D=0D=1D=2D=3D=0 Δ (Τ)
Δ (Τ)Δ (Τ)Δ (Τ)Δ (Τ)
sllll
(c)
FIG. 4. (Color online) (a) Variation of � as a function of disorderstrength (D) in the l band when Us = Ul > Usl . (b) Variations ofs-band and l-band critical temperatures T s
c and T lc as a function of
disorder strength D when only intraband pairing occurs in a two-bands-wave superconductor in a square lattice. (c) Variation of �s(T ) and�l(T ) with T for various strengths of disorder D in the l band.
x = y = 0.5 and keeping the system fixed at half filling, weobtain the corresponding T s
c and T lc for different values of D
[see Fig. 4(b)]. As seen from this figure, T sc remains constant
with increasing disorder strength D since randomness in thel band does not affect the s band in the presence of intrabandpairing alone. T l
c is, however, suppressed with increasing D. Atthis point, however, it must be noted that only the higher of thetwo critical temperatures (T s
c and T lc ) is physically significant
in this respect. So in the present case, Tc first decreases withdisorder and then becomes constant when T s
c > T lc .
These conclusions are further strengthened by a study ofthe pairing amplitude as a function of temperature, and the
expressions for the temperature-dependent pairing amplitudesare
1 = |Um|∫ ∞
−∞dE
〈〈Nm(E)〉〉2(E2 + �2
m
)1/2 tanh
((E2 + �2
m
)1/2
2kBT
)
for the m = s or l bands.We see that with the increase in disorder strength D in
the l band the temperature-dependent pairing amplitude �l
reduces much like the zero-temperature pairing amplitude [seeFig. 4(c)]. Since randomness in the l channel does not affectthe s band thus �s(T ) is not affected by D so we have plotted�s(T ) vs T only at D = 0 [see Fig. 4(c)]. We conclude fromFigs. 4(b) and 4(c) that for temperatures below the criticaltemperatures though disorder (D) suppresses �(T ), but doesnot reduce it to zero.
1 1.5 2 2.5 3Disorder Strength (D)
0
0.1
0.2
0.3
0.4
0.5
Δ
Δ sl
Us=Ul=1.0Usl=2.0
(a)
-4 -2 0 2 4Energy
0
0.1
0.2
0.3
0.4
PD
OS
D=1.0D=1.5D=3.0
s-band
(b)
-4 -2 0 2 4Energy
0
0.1
0.2
0.3
0.4
PD
OS
D=1.0D=1.5D=3.0
l-band
(c)
FIG. 5. (Color online) (a) Variation of � with disorder strength(D) in the l band when Us = Ul < Usl . (b),(c) Studies DOS for asquare-lattice superconducting system with disorder in the l bandwhen Us = Ul < Usl .
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SUPERCONDUCTIVITY IN MULTIBAND DISORDERED . . . PHYSICAL REVIEW B 84, 174508 (2011)
1 1.5 2 2.5D
0
0.1
0.2
0.3
0.4
0.5
0.6
Δ
ΔΔs
Us=U
l=2.0
l
Usl=1.0
(a)
1 1.5 2 2.5D
0
0.1
0.2
0.3
0.4
0.5
Δ
Δsl
Us=Ul=1.0Usl=2.0
(b)
FIG. 6. (Color online) A study of � as a function of disorderstrength (D) in the s and l band for (a) Us = Ul > Usl and (b) Us =Ul < Usl for a two-dimensional (2D) superconducting system havingtwo bands. Here the intraband hopping integrals ts = 1.0 and tl = 0.5and the interband hopping integral tsl = 0.0.
The next set of studies is the investigation of the increasingstrength of the disorder D on a two-band attractive-U Hubbardmodel with dominant interband attractive interaction Usl >
Us,Ul . In the parameter regime Us = Ul = 1.0 < Usl thedominant pairing is the interband pairing Usl and it affects boththe bands. In contrast to the case of only intraband pairing,here for a critical strength of disorder D > 2 the pairingamplitude �sl vanishes indicating the possible disappearanceof superconductivity [see Fig. 5(a)]. This is further illustratedin the DOS plot for the s and l channels in Figs. 5(b) and 5(c),respectively. Here the presence of randomness in the l channelaffects �sl and this in turn affects both s and l PDOS. Withincreasing disorder D in the l channel the gaps both in the s
PDOS and l PDOS reduces. Eventually finite DOS at the Fermilevel is realized indicating the absence of superconductivity.
Finally we address the situation when disorder is introducedin both s and l channels. When the interaction is such thatUs = Ul > Usl [Fig. 6(a)], then only �s and �l are nonzeroeven for strength of disorder as large as D = 2.5 indicatingthe presence of superconductivity. However, in the limitUsl > Us = Ul [Fig. 6(b)], we see that �sl decreases rapidlywith disorder and finally vanishes. These features are verysimilar to the case when disorder was introduced in only onechannel.
C. Summary
In this paper we have developed a real-space approach tostudy the effect of disorder on multiband superconductivityusing a two-band Hubbard Hamiltonian to model our systemand augmented space vector-recursion21,22 method to treatrandomness in our system. We have established the accuracyof our method by comparing our results in ordered systemswith those obtained earlier using other techniques. For orderedsystems we have seen gaps in both bands in the presenceof intraband pairing. In the presence of both intraband andinterband momentum independent pairing, depending on therelative magnitude of the pairing strength, only a particularkind of pairing is possible for a half filled s-like two-bandsystem.
We have then studied the effect of randomness in one ofthe bands. When only intraband pairing occurs, randomnessin one channel does not affect the other. But in the presence ofinterband pairing both the bands are affected by randomness.By increasing the strength of disorder, superconductivitysurvives in the presence of intraband pairing although thepairing amplitudes decrease with disorder. However, forinterband pairing the gap in the quasiparticle spectrum ceasesto exist beyond a critical value of the disorder strength. Inthe case of interband pairing, where the Cooper pairs areformed by electrons belonging to two different bands, wespeculate that phase coherence of the superconducting stateis more sensitive to disorder. The lack of phase coherencedue to disorder is probably responsible for the disappearanceof superconductivity. The same conclusion holds good whendisorder is introduced in both the bands. Our calculationindicates that interband pairing in multiband systems is notonly interesting but opens up a paradigm beyond Ander-son’s theorem11 to understand superconductivity in disorderedsystems.
ACKNOWLEDGMENT
This work was done under the Hydra Collaboration betweenour groups.
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