Materials Theory and Computation S. V. Khare 1.Department of Physics and Astronomy University of...

Materials Theory and Computation

S. V. Khare

1. Department of Physics and AstronomyUniversity of Toledo, Ohio

2. Department of Electrical Engineering and Computer Science

University of Toledo, Ohio

http://astro1.panet.utoledo.edu/~khare/

Funding: DARPA, Air Force, NSF, DoE, State of Ohio

General theme of research

My research involves the application of appropriate theoretical and computational techniques to understand condensed matter systems of significant experimental interest. This work involves predictions for new phenomena, explanation of existing data, and collaborations with experimentalists on their current experiments. It has involved a variety of thin film and bulk materials from metals to semiconductors, crystalline to disordered materials, and nano- to micro- length scales. Varied theoretical techniques utilized are density functional theory based computations, classical molecular dynamics, Monte Carlo simulations, and continuum analytical equations.

Papers with students I• Effect of structure, surface passivation, and doping on the electronic and optical properties

of GaAs nanowires: A first principles study V. Gade, N. Shi, D. Medaboina, S. V. Khare, R. Ramprasad (Submitted to journal)

• Structural and Electronic properties of β-In2X3 (X = O, S, Se, Te) using ab initio calculations S. Marsillac, N. S. Mangale, V. Gade, S. V. Khare (Submitted to journal)

• Super Hard Cubic Phases of Period VI Transition Metal Nitrides: A First Principles Investigation S. K. R. Patil, N. S. Mangale, S. V. Khare, and S. Marsillac Accepted in Thin Solid Films 2008.

• Effect of structure, surface passivation, and doping on the electronic properties of Ge nanowires: A first-principles study D. Medaboina, V. Gade, S. K. R. Patil, and S. V. Khare Phys. Rev. B 76, 205327 (2007).

• Impact of Structure Relaxation on the Ultimate Performance of a Small Diameter, n-Type <110> Si-Nanowire MOSFET G. Liang, D. Kienle, S. K. R. Patil, J. Wang, A. W. Ghosh, and S. V. Khare IEEE Trans. Nano. Tech. 6, 225 (2007).

Papers with students II

• Mechanical stability of possible structures of PtN investigated using first-principles calculations S. K. R. Patil, S. V. Khare, B. R. Tuttle, J. K. Bording, and S. Kodambaka Phys. Rev. B 73, 104118 (2006).

• Ab Initio calculations for Properties of MAX phases Ti2TlC, Zr2TlC, and Hf2TlC J. A. Warner, S. K. R. Patil, S. V. Khare, and R. S. Masiuliniec Appl. Phys. Lett. 88, 101911 (2006).

Ab initio computations of structural and electronic properties of doped and undoped

Ge nanowires

S. V. Khare1, D. Medaboina2, V. Gade2, and S. K. R. Patil3

1. Department of Physics and AstronomyUniversity of Toledo, Ohio

2. Department of Electrical Engineering and Computer Science

University of Toledo, Ohio

3. Department of Mechanical and Industrial EngineeringUniversity of Toledo, Ohio

http://www.physics.utoledo.edu/~khare/

Outline

• Experimental motivation

• Ab initio methods

• Structural properties

• Band structures of doped and undoped nanowires

• Band gaps of Si and Ge nanowires

• Conclusions

Introduction

• Diameter (d) of NWs range from 1 nm – 100 nm.

• Length (ℓ) varies from 10nm – 1µm

• Different names to NWs in literature:– Nanowires: Wires with large aspect ratios (ℓ/d > 20)– Nanorods: Wires with small aspect ratios (ℓ/d)– Nanocontacts: Short wires bridged between two larger

electrodes.

ℓ

Experimental methods for preparing Ge nanowires

Laser ablation Vapor transport Low-temperature CVD Supercritical fluid–liquid–solid synthesis : In this method thermal evaporation of Ge

powder at 950C onto silicon wafer and ceramic (alumina) substrate using Au catalyst via a vapour–liquid–solid (VLS) process. Diameters up to 30 nm and length tens of micro meters. Preferred growth direction for the nanowires is [111].

Nanowires developed by Kamanev et al†., of 40 nm diameter along [111] growth direction grown on silicon substrate.

Nanowires developed by Nguyen et al*., grown along [110] on heavily doped Si.

* Nguyen, P.; Ng, H. T.; Meyyappan, M. Adv. Mater. 2005, 17, 5. † Kamanev, B. V.; Sharma, V.; Tsybeskov, L.; Kamins, T. I. Phys. Stat. Sol. (a) 2005, 202, 2753.

Tip of nanowires generated using supercritical fluid–liquid–solid (SLFS) method by Hanrath et al*.,

Orientation of Ge nanowires generated using SLFS method

* Hanrath, T.; Korgel, B. A. Small 2005, 1, 7.

[111]

[211]

[111]

[110]

HRTEM image of [110] growth direction developed by Hanrath et al*., representing the faceted cap structure of nanowire.

Faceting of Ge nanowires

* Hanrath, T.; Korgel, B. A. Small 2005, 1, 7.

Fourier transform of image representing the [110] pole axis of the wire [110]

Tapered end of nanowire showing the facets

Crystallographic model of nanowire showing the facets of nanowire.

HRTEM image of nanowire along [110] growth direction showing the length of nanowire.

A SEM image of a p-n diode. Diode obtained by simply crossing p- and n-type NW.*

Diode made of NWs

* Duan et al., Nature 2001, 409, 66, Harvard University, Cambridge.

Ŧ Huang et al., Pure Appl. Chem. 2004, 76, 2051, Harvard University, Cambridge.

n p p-n

Schematics illustrating the crossed NW-FET concept.Ŧ

FET made of NWs

1 μm

Ab initio method• Powerful predictive tool to calculate properties of materials

• Fully first principles – (1) no fitting parameters, use only fundamental constants

(e, h, me, c) as input– (2) Fully quantum mechanical for electrons

• Thousands of materials properties calculated to date

• Used by biochemists, drug designers, geologists, materials scientists, and even astrophysicists!

• Evolved into different varieties for ease of applications

• Awarded chemistry Nobel Prize to W. Kohn and H. Pople 1998

Pros and Cons of ab initio methodPros:• Very good at predicting structural properties:

(1) Lattice constant good to 0-3%.(2) Bulk modulus good to 1-10%.(3) Very robust relative energy ordering between structures.(4) Good pressure induced phase changes.

• Good band structures, electronic properties.• Used to study the properties of materials at unstable conditions.

Cons:• Computationally intensive.• Excited electronic states: difficult to compute.• Band gaps are under estimated by 50%.

Ab initio method details

• LDA, Ceperley-Alder exchange-correlation functional as parameterized by Perdew and Zunger

• Generalized ultra-soft Vanderbilt pseudo-potentials and plane wave basis set

• Supercell approach with periodic boundary conditions in all three dimensions

• Wires are infinite along their axis

Theoretical and experimental comparison of lattice constant and

bulk modulus of GeLattice

constant (nm)Bulk mudulus

(GPa)

Theoretical calculations 0.5638 72.57

Experimental calculations* 0.5658 75.00

* Kittel, C. Introduction to Solid State Physics, 2nd ed., (John Wiley & Sons, Inc., New York, 1976), p. 40.

Nomenclature used for describing a nanowire

( )03.2)44,89(

]001[

-- HGeNW

NanowireOrientation of the nanowire

Number of Ge atoms in the nanowire

Number of H atoms in the nanowire

Diameter of the nanowire in nm

Structural Properties of Ge nanowires

)44,89(]001[

HGeNW )44,89(]001[

HGeNW

)44,89(]001[

HGeNW

[001] [110] [111]

03.2)44,89(]001[

HGeNW 12.2)32,69(]110[

HGeNW 11.2)66,170(]111[

HGeNW

[001] [110] [111]

03.3)60,185(]001[

HGeNW 3.3)40,133(]110[

HGeNW 03.3)90,326(]111[

HGeNW

All results in this talk are with DFT-LDA, VASP.

Electronic Properties: Band Structures of Ge nanowires

[001] [110] [111]

03.2)44,89(]001[

HGeNW 12.2)32,69(]110[

HGeNW 11.2)66,170(]111[

HGeNW

[001] [110] [111]

03.3)60,185(]001[

HGeNW 3.3)40,133(]110[

HGeNW 03.3)90,326(]111[

HGeNW

Band Structures of doped and undoped Ge nanowires

[100]

[110]

[111]

11.2)1,66,169(]111[

PHGeNW 11.2)66,170(]111[

HGeNW 11.2)1,66,169(]111[

BHGeNW

03.2)1,44,88(]001[

PHGeNW 03.2)44,89(]001[

HGeNW 03.2)1,44,88(]001[

BHGeNW

n-doped undoped p-doped

12.2)1,32,68(]110[

PHGeNW 12.2)32,69(]110[

HGeNW 12.2)1,32,68(]110[

BHGeNW

Plot of Energy gap (eV) versus Diameter (nm)

Axis

Dia (nm)

Comparison of band gap of Ge and Si nanowires along different diameter and

axes

0.5 1.0 1.5 2.0 2.5 3.0

[001] D D I I I I

[110] D D D D D D

[111] I I I I I I

D = Direct band gap, I = Indirect band gap

Axis0.5 1.0 1.5 2.0 2.5 3.0

[001] I I I I I I

[110] D D D D D D

[111] D D D D I I

Dia (nm)

Si nanowires*

Ge nanowires

* Zhao, X.; Wei, C. M.; Yang, L.; Chou, M.Y. PRL 2004, 92, 23.

Conclusions of work on Ge nanowires

1. Study of structural, energetic, and electronic properties of hydrogen-passivated doped and undoped germanium nanowires along [001], [110], and [111] directions with diameter d up to 3 nm, using ab initio methods.

2. The electronic band structure shows a significant response to changes in surface passivation with hydrogen.

3. Doping of wires with n and p type atoms produced a response in the band structure similar to that in a doped bulk crystal.

4. Quantum confinement has a substantial effect on the electronic band structure and hence the band gap, which increases with decreasing diameter.

5. Wires oriented along [110] are found to have a direct band gap while the wires along [111] are found to have an indirect band gap. Wires along [001] show a crossover from a direct to an indirect band gap as diameter increases above the critical diameter for the transition being 1.3 nm.

Institutional Support

• University of Toledo Parallel Computing Cluster

• Ohio Supercomputer Cluster

• National Center for Supercomputing Applications (NCSA)

Thank you!

Ab initio method details

• LDA, Ceperley-Alder exchange-correlation functional as parameterized by Perdew and Zunger

• Used the VASP code with generalized ultra-soft Vanderbilt pseudo-potentials and plane wave basis set

• Supercell approach with periodic boundary conditions in all three dimensions

• Energy cut-offs of 150.00 eV for H-terminated Ge nanowires,

dense k-point meshes

• Forces converged till < 0.01 eV/ Å

• Used supercomputers of NCSA and OSC

Structural and Electronic Properties of Doped and Undoped GaN Nanowires:

A First Principles Investigation

Shandeep Voggu(MS Thesis Candidate)

Department of EECSUniversity of Toledo

AcknowledgementsPeople

• Prof. Sanjay V. Khare (Thesis advisor)

• Prof. Daniel Georgiev (Committee member)

• Prof. Vijay Devabhaktuni (Committee member)

• Varun Gade, Dayasagar Medaboina, Sunil K. R. Patil, Nikhil Mangale, Ashok Kolagatla, Kausthuba Ippagunta, Abbas Naseem, Krishnakanth Ganguri (Prof. Khare’s group)

Institutional support

• Ohio Supercomputer Center (OSC)

• National Center for Supercomputing Applications (NCSA)

Outline

• Introduction• Experimental motivation and applications• Crystal structures • Generation of nanowires• Ab initio methods• Properties: Doped and undoped nanowires

1) Structural

2) Electronic • Conclusions• Future work

Introduction

• Diameter (d) of NWs range from 1 nm – 100 nm.

• Length (ℓ) varies from 10nm – 1µm

• Different names to NWs in literature:– Nanowires: Wires with large aspect ratios (ℓ/d > 20)– Nanorods: Wires with small aspect ratios (ℓ/d)– Nanocontacts: Short wires bridged between two larger

electrodes.

ℓ

Outline


1) Structural


Growth of GaN NWs using the Metalorganic Chemical Vapour Deposition (MOCVD)

* J. C. Johnson et al., Nature Materials 1, 106–110 (2002), University of California, Berkeley.

50 nm 5 nm

Electron microscopy images of synthesized GaN nanowires.

(a)Scanning electron microscopy (SEM) images of GaN nanowires grown on sapphire substrate. Scale bar, 3μm.

(b)High-resolution transmission electron microscopy image of GaN nanowire.Scale bar, 1 nm.

(c)SEM image of single GaN wire after dispersing onto sapphire substrate.Scale bar, 5μm.

* Kuykendall et al., Nature Materials 3, 524–528 (2004), University of California, Berkeley.

Growth of GaN NWs using the Metalorganic Chemical Vapour Deposition (MOCVD)

TEM images of the GaN nanowires.a–c,Wires grown on (100) γ-LiAlO2.The inset in a is an electron-diffraction pattern recorded along [001] axis.

d–f,Wires grown on (111) MgO substrates.The insets in d show the hexagonal cross-section of the wire and an electron-diffraction pattern recorded along the [100] axis.

c and f show space-filling structural models for the nanowires with triangular and hexagonal cross-sections.

Advantages of NWs

• NW devices can be assembled in a rational and predictable way because:

– NWs can be precisely controlled for structure and chemical composition during synthesis.

• NW building blocks can be combined in ways not possible in conventional electronics.

• Series of electronic devices are being assembled using semiconductor NWs:

– Crossed NW p-n diodes,– Crossed NW-FETs,– Nanoscale logic gates,– Optoelectronic devices

A SEM image of a p-n diode. Diode obtained by simply crossing p- and n-type NW.*

Diode made of NWs

* Duan et al., Nature 2001, 409, 66, Harvard University, Cambridge.

Ŧ Huang et al., Pure Appl. Chem. 2004, 76, 2051, Harvard University, Cambridge.

n p p-n

Schematics illustrating the crossed NW-FET concept.Ŧ

FET made of NWs

1 μm

Far-field image of a single GaN nanolaser*

GaN nanowire laser

(a) SEM image of a GaN nanowire connected with two electrodes for the

transport study. The inset is an illustration of the GaN transistor layout.

(b) Current-voltage measurement at different gating voltages for the GaN

nanowire. Ŧ

GaN Nanowire Transistor: n-type1 μm

*J. C. Johnson et al., Nature Materials 1, 106–110 (2002). Ŧ Kuykendall et al., Nano. Lett. 3, 1063, 2003.

University of California, Berkeley.

Outline


1) Structural


Definition of a crystal

a

Basis atomic position:

(0.0, 0.0, 0.0)

Bases atomic positions:

(0.0, 0.0, 0.0)

(0.0, 0.5, 0.5)

(0.5, 0.0, 0.5)

(0.5, 0.5, 0.0)

x

zy

x

zy

• Crystal atomic position = Bravais lattice position + Basis vector

• Bravais lattice is regular arrangement of points.

• Vectors determining the position of the atom from every Bravais lattice point are called basis vectors.

• Basis vector = 1 – basis atom 4 – basis atoms

Hexagonal Bravais lattice structures

Wurtzite unit cellHexagonal Bravais lattice structure

The wurtzite lattice.

A1 = ½ a X - ½ 31/2 a Y

A2 = ½ a X + ½ 31/2 a Y

A3 = c Z

B1 = ½ a X + ½ 3-1/2 a Y (Ga) (2b)

B2 = ½ a X - ½ 3-1/2 Y + ½ c Z (Ga) (2b)

B3 = ½ a X + ½ 3-1/2 a Y + u c Z (N) (2b)

B4 = ½ a X - ½ 3-1/2 a Y + (½ + u) c Z (N) (2b)

Lattice Vectors Basis Vectors

Wurtzite structure

Structure representing the wurtzite lattice. N atoms

Ga atoms

Outline


1) Structural


Objective of making NW structures

• Periodically repeating unit along arbitrary direction (m n o) in a crystal.

- For example consider a [001] axis wire

z

xy

- Indicate Ga atoms

- Indicate N atoms

Objective of making NW structures

• Periodically repeating unit along arbitrary direction (m n o) in a crystal.

- For example consider a [001] axis wire

x

zy

• Surfaces should be passivated

- Indicate Ga atoms

- Indicate N atoms

Generation of nanowires

Three major steps in generation of nanowires:

1) Generate a large cube of bulk material using lattice and basis vectors of wurtzite lattice.

2) Cut a wire of given length and diameter from the bulk material using a separate algorithm.

3) Identify the missing neighbors and passivate the dangling bonds with hydrogen atoms.

Generation of Bulk material

• represents the lattice vectors for i = 1, 2, and 3;

represent the basis atoms.

• The generated bulk material has square cross-section.

jii b)an(R

ai

• Position vector of any atom in bulk material is given by

bj

- Indicate Ga atoms

- Indicate N atoms

Extracting the nanowire

For each atom in the bulk material:

1. Cross product of its position vector with the normal along the axis of wire < radius of the wire.

2. Dot product of the position vector of atom and normal along the axis of wire lies in the range -(wire-length)/2 to +(wire-length)/2

3. Wire-length determined from the crystal

Axis of the NW

* Goldstein. H, Poole. C, Safko. J, Classical Mechanics, 3rd Edition, Addison Wesley.

• The generated nanowires will have dangling bonds left on the surface of wire due to the cutting.

• These dangling bonds create states in bandstructure.

Nanowire cut from bulk material.

Generated nanowire

ℓ- Indicate Ga atoms

- Indicate N atoms

d

Termination with Hydrogen

• Each atom in the wire is checked to see four neighbors. The atoms without four neighbors are identified and the missing neighbors are replaced with hydrogen atoms.

Top view

H passivated GaN NW.

- Indicate Ga atoms

- Indicate N atoms

- Indicate H atoms

Outline


1) Structural


Ab initio method• Powerful predictive tool to calculate properties of materials

• Fully first principles – (1) no fitting parameters, use only fundamental constants

(e, h, me, c) as input– (2) Fully quantum mechanical for electrons

• Thousands of materials properties calculated to date

• Used by biochemists, drug designers, geologists, materials scientists, and even astrophysicists!

• Evolved into different varieties for ease of applications

• Awarded chemistry Nobel Prize to W. Kohn and H. Pople 1998

Pros and Cons of ab initio methodPros:• Very good at predicting structural properties:

(1) Lattice constant good to 0-3%.(2) Bulk modulus good to 1-10%.(3) Very robust relative energy ordering between structures.(4) Good pressure induced phase changes.

• Good band structures, electronic properties.• Used to study the properties of materials at unstable conditions.

Cons:• Computationally intensive.• Excited electronic states: difficult to compute.• Band gaps are under estimated by 50%.

• Different codes:

• SIESTA • VASP • CASTEP• Abinit• CRYSTAL

• VASP - Vienna Ab initio Simulation Package

Ab initio codes

VASP

OUTCAROSZICARCONTCARCHGCARWAVECAREIGENVALPROCARXDATCARLOCPOTDOSCAR

POSCARPOTCARKPOINTSINCAR

Input files Output files

• Implementing ab initio quantum mechanical molecular dynamics.

VASP input files

• POSCAR: Positions of ions Bravais lattice

Periodic boundary conditions

• POTCAR: Pseudopotentials from VASP

• KPOINTS: Would be used for parallelization

• INCAR: Different parameters for different properties

GaN-bulk5.602 0.000000000 0.500000000 0.50000000 0.500000000 0.000000000 0.50000000 0.500000000 0.500000000 0.000000002 2Selective dynamicsDirect0.33333333 0.66666667 0.000 T T T0.66666667 0.33333333 0.500 T F T 0.33333333 0.66666667 0.385 T T F0.66666667 0.33333333 0.885 F F T

Ge Bulk 5.6435 0.00000000 0.50000000 0.50000000 0.50000000 0.00000000 0.50000000 0.50000000 0.50000000 0.00000000 2 Direct 0.000000000000000 0.000000000000000 0.000000000000000 0.250000000000000 0.250000000000000 0.250000000000000

POSCAR

(Å )

n atom

n1, n2

The ordering must be consistent with the POTCAR

321 a,a,a

VASP output files

• OUTCAR: Complete information of the simulation- Number of irreducible points- Final position of ions and forces- Time take to complete simulation

• OSZICAR: It contains the information about free energy (E0)

and about convergence speed.

• CONTCAR: It contains the positions of ion at the final ionic step in relaxations.

Test of Pseudopotentials

Lattice constants (nm)

Bulk modulus (GPa)

Theoretical calculations

a=0.3118

c=0.5132183

Experimental measurement *

a=0.3189

c=0.5185187

* http://www.phys.ksu.edu/area/GaNgroup/gparametm.html

Outline


1) Structural


Nomenclature used for describing a nanowire

Diameter (d) of the nanowire in nmNanowire

Orientation of the nanowire

“a” number of Ga atoms in the nanowire

“c” number of H atoms in the nanowire

NW(Ga – a, N – b, H – c,..)

[100]

(d)

“b” number of N atoms in the nanowire

Structural Properties

[001]

c-axis

[100]

a-axis

All results in this presentation are obtained using ab initio method.

).1.09(NW )48H,24N,12Ga(]001[

).37 1(NW )32H,27N,19Ga(]100[

).1.94(NW )96H.96N,72Ga(]001[

).2.18(NW )48H,56N,44Ga(]100[

).25(2NW )96H,120N,96Ga(]001[

)80.2(NW )60H,84N,69Ga(]100[

~ 1.0 nm ~ 2.0 nm > 2.0 nm

Band structures[001] [100]

K (2π/ℓ) K (2π/ℓ)

K (2π/ℓ)K (2π/ℓ)

K (2π/ℓ) K (2π/ℓ)

~ 1.0 nm

~ 2.0 nm

> 2.0 nm

).1.09(NW )48H,24N,12Ga(]001[

).1.94(NW )96H.96N,72Ga(]001[

).25(2NW )96H,120N,96Ga(]001[

).37 1(NW )32H,27N,19Ga(]100[

).2.18(NW )48H,56N,44Ga(]100[

)80.2(NW )60H,84N,69Ga(]100[

Band Structures of doped and undoped GaN nanowires

[001]

[100]

n-doped undoped p-doped

K (2π/ℓ) K (2π/ℓ) K (2π/ℓ)

K (2π/ℓ)K (2π/ℓ)K (2π/ℓ)

NW( 72, 95, 96, 1)[001]Ga N H C

NW( 44, 55, 48, 1)[100]Ga N H C

(2.02)

(2.02) NW( 72, 96, 96)[001]Ga N H (2.02)

NW( 44, 56, 48)[100]Ga N H

(2.02)

NW( 71, 96, 96, 1)[001]Ga N H C

(2.02)

NW( 43, 56, 48, 1)[110]Ga N H C

(2.02)

Comparison of band gap of GaN and Ge nanowires

D = Direct band gap, I = Indirect band gap

Axis0.5 1.0 1.5 2.0 2.5 3.0

[001] D D D D D D

[100] D D D D D D

GaN nanowiresDia (nm)

Axis

Dia (nm) 0.5 1.0 1.5 2.0 2.5 3.0

[001] D D I I I I

[110] D D D D D D

[111] I I I I I I

Ge nanowires by Medaboina et al.,*

* Phys. Rev. B 76, 205327 (2007).

Band gap, Eg of GaN nanowires

Wire axisd (nm)

No. of Ga atoms

No. of N atoms

No. of H* atoms

Eg (eV)[h k l]

[001]

0.83 12 24 48 3.081.08 24 36 48 2.931.37 36 54 72 2.741.84 54 72 72 2.682.20 72 96 96 2.67

[100]

0.75 07 12 20 3.001.00 10 16 24 2.901.30 19 27 32 2.741.60 30 41 44 2.632.18 44 56 48 2.59

Plot of band gap (eV) versus Diameter (nm)

2.5

2.6

2.7

2.8

2.9

3

3.1

3.2

0.5 1 1.5 2 2.5

d (nm)

Eg

(e

V)

[001] axis

[100] axis

Outline


1) Structural


Conclusions of work on GaN nanowires

1. Successfully studied the structural and electronic properties of hydrogen-passivated doped and undoped GaN nanowires along [001] and [100] directions with diameter d up to 3 nm, using ab initio methods.

2. Doping of wires with n and p type atoms produced a response in the band structure similar to that in a doped bulk crystal.

3. Quantum confinement has a substantial effect on the electronic band structure and hence the band gap, which increases with decreasing diameter.

4. All wires studied have direct bandgaps.

Outline


1) Structural


Future work (preliminary stages)

• Optical properties of GaN nanowires are being determined.

ir i

Real and Imaginary plots of the dielectric function of NW( 72, 96, 96)[001]Ga N H (2.02)

Thank you!

Density Functional Theory (DFT)DFT states that the ground state energy of a system of particles moving in a potential

can be consistently expressed as a function of the density of the particles, n(r).

We look for the self-consistent solution to the equations that minimize the expression for total energy within a unit cell as a function of n(r) to find the groundstate n(r).

We assume that the valence electrons experience the effects from nuclei and core electrons as a non-interacting pseudopotential.

The density of electrons in a unit cell is then given by the sum of the probability densities from a set of orthonormal one-electron orbitals.

Below: solving the Kohn-Sham energy minimization equations self-consistently.

*Formatted Equations taken from Wikipedia.org: Density Functional Theory; Content: Michael J. Mehl et al, First Principles Calculations of Elastic Properties of Metals(1993).

Resulting ground state density n(r) substituted into initial expression for energy gives the ground state energy for a unit cell.

Ab initio techniques and approximations

• Techniques:

1. Density functional theory 2. Pseudopotential theory 3. Iterative diagonalization method

• Approximations:

• Local density approximation • Generalized gradient approximation

• Different codes like SIESTA, VASP, CASTEP are used.

VASP - Vienna Ab initio Simulation Package

Graph showing the comparison of wave function and ionic potential in Pseudopotential theory.

Supercell geometry for a molecule

Evolution of theoretical techniques

• The physical properties of any material are found to be related to the total energy or difference between total energies.

• Total energy calculation methods which required specification of number of ions in the material are referred to as ab initio methods.

• Ab initio make use of fundamental properties of material. No fitting parameters are involved.

Practical AlgorithmEffective Schrodinger equation for non-interactng electrons

),()()]([2

22

rrrvm nnneff

,)()(

1

2

N

nn rr

Implementation:

1. Guess an initial charge density for N electrons2. Calculate all the contributions to the effective potential3. Solve the Schrodinger equation and find N electron states4. Fill the eigenstates with electrons starting from the bottom5. Calculate the new charge density6. Calculate all the contributions to the effective potential and

iterate until the charge density and effective potential are self-consistent.

7. Then calculate total energy.

N

nnrE

1

)]([

Density Functional Theory (DFT)

Hohenberg Kohn Theorems (1964)

(1)The external potential of a quantum many body system is uniquely determined by the r), so the total energy is a unique functional of the particle density E = Er)].

(2) The density that minimizes the energy is the ground state density and the energy is the ground state energy,

Min{Er)]} = E0

Synonyms: DFT = Ab initio = First Principles

Kohn Sham Theory (1965)The ground state density of the interacting system of particles can

be calculated as the ground state density of non-interacting particles moving in an effective potential veff r)].

),()()]([2

22

rrrvm nnneff

N

nn rr

1

2)()(

)]([)(

)()]([ 3. rvrd

rr

rrvrv xcnuceff

,)(

)]([)]([

r

rErv xc

xc

Coulomb potential of nuclei

Hartree electrostatic potential

)]([ rExc is universal!

Exchange correlation potential

national laboratory for advanced Tecnologies and nAnoSCience

Self catalitic growth of GaN NWs

Stach et al, Nano Lett. 3, 867 (2003)

•self standing GaN layer•thinned for TEM (≤ 300 nm)•heated at 1050° C in a TEM

Above 850 in high vacuumGaN(s) ―›

Ga (l) + 0.5 N (g) + 0.25 N2 (g)GaN(s) ―›

GaN (g) or [GaN]x (g)

in-situ study of the decomposition and resulting nanostructure evolution

national laboratory for advanced Tecnologies and nAnoSCience

room temperature analysisof the nanostructures:

•single crystal GaN NWs•[0001] oriented•av diameter 50 nm•gr rate 300 nm/s

self catalytic process could be important to avoid undesired contamination from foreign metal atom (catalyst)

Materials Theory and Computation S. V. Khare 1.Department of Physics and Astronomy University of...

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