First principles calculations – project

Tomasz Wozniak

winter semester 2019/2020

Contents

1 Introduction 11.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Databases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Structure construction 32.1 Bulk crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Monolayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Running the calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Geometry optimization 8

4 Convergence tests 10

5 Electronic band structure 125.1 Orbital composition of bands . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6 Density of states 15

7 Effective mass 16

A Control questions 17

B Tips and hints 18

C Scripts 19

D Report contents 20

E Literature 22

i

Chapter 1

Introduction

1.1 Preliminaries

This textbook is dedicated to ’Quantum Engineering’ students attending the ’Obliczeniaz zasad pierwszych’ computer lab at Faculty of Fundamental Problems of Technology atWroclaw University of Science and Technology. It contains description and guidelines forprojects realized by students in the second part of the course.

The aim of the project is to learn how to perform density functional theory (DFT)based ab initio studies of fundamental structural and electronic properties of chosen atomicsystems. Among them, at the first place are hexagonal layered crystals: transition metaldichalcogenides (TMDCs) and carbon, boron and nitrogen based compounds. Apart fromthat, some original ideas of students can be considered as projects, but they have to beaccepted by the tutor.

Calculations are performed with Abinit program. Students are assumed to have fullyaccomplished and understood first three basic tutorials of Abinit. The ’Control questions’Appendix can be used to check if the students are properly prepared to realize their owncomputational projects. Practical tips for calculations are given in ’Tips and hints’ Ap-pendix.

The textbook contains chapters describing subsequent computational tasks. Each ofthem consists of: preparation of input, performing calculations, data post-processing andanalysis. The class time (20 ZZU hours) should be maximally devoted to perform calcula-tions. The reading, literature research, input files preparation, output data post-processingand analysis are to be done in the self study hours (40 CNPS hours in total).

The obligatory tasks should be done in the following order:

1. Construction of bulk unit cell and initial calculation (Chapter 2)

2. Initial geometry optimization in bulk (Chapter 3)

3. Convergence tests in bulk (Chapter 4)

4. Construction of monolayer unit cell (Chapter 2)

5. Final geometry optimization in bulk and monolayer (Chapter 3)

6. Band structure calculations in bulk and monolayer (Chapter 5)

7. Density of states calculations in bulk and monolayer (Chapter 6)

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After completing the obligatory tasks, one of the following supplementary tasks shouldbe performed:

1. Orbital composition of bands calculations in bulk and monolayer (Chapter 5)

2. Effective mass calculations in bulk and monolayer (Chapter 7)

1.2 Databases

Theoretical and experimental data of various properties of crystals can be found in thefollowing databases:

1. Crystallography Open Database (COD) - database of experimental crystallo-graphic data (crystal structure, lattice parameters and atom positions) for bulk crys-tals. These information is collected in CIF files which can be downloaded and visual-ized in MP Crystal Toolkit (https://www.materialsproject.org/apps/xtaltoolkit). Inthe search engine one should enter the names of elements forming the compound in”1 to 8 elements” field and 1 and 2 in ”number of distinct elements min and max”fields.

http://www.crystallography.net/cod/search.html

2. Materials Project (MP) - database of computational data (geometrical parameters,electronic structure) obtained in VASP with GGA-PBE functional for bulk crystals.In the search engine one should enter the compound formula. The crystal should beidentified against COD by its space group number/symbol, number of atomic sites inprimitive cell and structure visualization. Lattice constants in MP (especially c) canbe significantly different from the experimental values. The CIF files can be visualizedin Crystal Toolkit.

https://www.materialsproject.org/#search/materials/

3. Computational 2D materials database (C2DB) - database of computationaldata (GGA-PBE geometrical parameters, GGA-PBE, HSE06 and GW electronicstructure) obtained in GPAW for monolayers. In the search engine one should enterthe compound formula. The prototype for 2H crystals is MoS2, 1T - CdI2, boroncompounds - BN.

https://cmrdb.fysik.dtu.dk/c2db

Experimental electronic properties (energy band gaps) can be found in scientific articles.

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Chapter 2

Structure construction

All compounds in the projects have hexagonal crystal lattice. We will model them usingprimitive unit cell. Use values of rprim proposed in kptbounds (see the variable description)in order to have reduced coordinates of high symmetry points in Brilloun zone compatiblewith proposed in kptbounds. Initial values of acell for LDA and GGA+D3 calculationsshould be taken from COD and for GGA - from MP.

2.1 Bulk crystals

We will use xred to define the atomic positions. It is convenient to place one atom (metal incase of TMDCs) in the origin. Figure 2.1 shows top view of TMDCs and III-V compoundsstructures. Figures 2.2, 2.3 and 2.4 show side view of 2H, 1T and III-V compoundsstructures, respectively. Figure 4.1 shows top and side view of graphite structure. Variablesznucl, natom, ntypat and typat should be set accordingly.

Figure 2.1: Crystal structure of 2H TMDCs and III-V compounds - top view.

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Figure 2.2: Crystal structure of 2H TMDCs - side view. u ≈ 18 .

M (0, 0, 0)

X2 (2/3, 1/3, +u)

X1 (-2/3, -1/3, -u)

r1’

r3’

c

Figure 2.3: Crystal structure of 1T TMDCs - side view. u ≈ 14 .

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B1 (0, 0, 0), N1 (2/3, 1/3, 0)r1’

r3’

cB2 (0, 0, 1/2), N2 (2/3, 1/3, 1/2)

Figure 2.4: Crystal structure of III-V compounds - side view.

C1 (0, 0, 0), C2 (2/3, 1/3, 0)r1’

r3’

c C3 (0, 0, 1/2), C2 (1/3, 2/3, 1/2)

Figure 2.5: Crystal structure of graphite - top and side view.

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Generally layered crystals can exist in multiple structures and politypes. For a givencompound only the structure indicated by the teacher should be considered in the project.

2.2 Monolayers

Monolayers can be obtained from bulk crystals by removing atoms of the second layer fromthe primitive cell (this is not the case in 1T structures, where the bulk unit cell contains onlyone monolayer) and increasing lattice constant c so that there is at least 12A of vacuumbetween periodic images of monolayers, see Fig. 2.6. Use atomic positions and latticeconstants obtained from final optimization of bulk crystals. Since reduced coordinates xredare scaled by acell, find the z components which conserve ’bulk’ thickness of the monolayer.

r3’

c 10 Angstr.

Figure 2.6: Monolayer in a supercell - side view.

2.3 Running the calculations

In the project we will use PAW datasets instead of pseudopotentials. PAWs for LDA andGGA can be downloaded from Abinit website. In the header of PAW file there are threesuggested values of cutoff energy, for example: <pw ecut low="12.00" medium="15.00"

high="15.00"/>. Choose medium value for the initial ecut. If this energy is different forthe other atom, choose the bigger value of these two. The value of pawecutdg should besubject to a convergence test. We will, however, omit this step and always keep it twice ashigh as ecut.

The convergence criterion for the SCF cycle can be kept at default value toldfe 1d-6.Increase the maximum number of SCF steps nstep to 40. Use the diemac value recom-mended for semiconductors.

Set the k point grid sampling to ngkpt 6 6 ngkpt(3), where ngkpt(3) is a natural

number such that ngkpt(1)ngkpt(3) ≈

ca , where a and c are lattice constants. For 2D systems

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ngkpt(3) is 1 since in Z direction the BZ integration is over flat bands and thus only onespecial k-point is needed. Set kptopt to take into account all symmetries. Use the shift ofk grid proposed in shiftk for hexagonal lattices.

D3 van der Waals correction is switched on by an appropriate value of vdw_xc.Suppress writing output files with wavefunction, charge density, eigenvalues and geom-

etry setting prtwf, prtden, prteig, prtebands and prtgeo to 0.

This is the minimum input for a self-consistent (SCF) calculation, which will be modifiedin the next tasks. Run the calculation:

mpirun -np 4 abinit < *.files > log &

You can see what is currently written to the log or output file:tail -f log

You can check if Abinit is running using top command. If you need to kill the calculation,type k in it and confirm. Quit with q. You can use killall abinit command instead.

If the calculation breaks down, there should be error messages in log or MPIABORT-FILE - read them and correct the input. If the calculation ends successfully, there is aCalculation completed message at the end of output file and --- !FinalSummary at theend of log file. Look through the files and answer the following questions:

1. What is the space group number spgroup? Compare it to literature value.

2. How many k points are there in the grid in the full Brillouin zone and how is thisnumber reduced in irreducible Brillouin zone (nkpt)?

3. In how many SCF steps did the calculation converge?

4. What are the forces fcart acting on atoms?

5. What is the stress tensor and pressure acting on unit cell?

6. How do the answers for these questions differ for different xc functionals?

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Chapter 3

Geometry optimization

Material covered in this chapter corresponds to Base tutorial 1: Computation of the inter-atomic distance (method 2) and Base tutorial 3: Determination of the lattice parameters.

The aim of geometry optimization is to find the atomic positions and lattice parametersthat correspond to the ground state and are called the optimal geometry. Ideally, forcesand stresses vanish and the crystal total energy is minimal. In practice, one searches for ageometry for which forces and stresses are lower than given criterion. Automatic geometryoptimization will be used for this project. Since the geometrical parameters in databasesare obtained using different codes, xc functionals, pseudopotentials/PAWs and technicalparameters, they are certainly not optimal for our calculations. Thus, initial geometryoptimization should be performed prior to convergence tests. The method described belowis to be used in LDA and GGA+D3 calculations.

Automatic optimization uses a minimization algorithm to update geometric parametersin the consecutive steps. We use BFGS algorithm: ionmov 2. The criterion on maximalforce tolmxf can be set to default value 5d-5 (Ha/Bohr). The SCF tolerance criterionshould be changed to toldff and the value should be at least one order of magnitude lowerthan tolmxf (1d-6 is enough). This is, however, uneffective for structures in which allforces vanish due to symmetry. In that case toldfe should be used with significantly lowervalue (1d-10). Set the maximal number of optimization steps ntime to 30.

First, only the atomic positions should be optimized - set an appropriate value ofoptcell.

Run the calculations. After finishing, check if the optimization converged (At Broyd/MD

step ..., gradients are converged...) and if the SCF cycle converged in every opti-mization step. Check the values of forces at the end of calculations.

Copy the optimized atomic positions to the input and prepare it for the full optimizationof cell shape, volume and atomic positions (choose appropriate optcell value). If Abinitcorrectly recognized the space group, the shape of cell will not change in optimization. Sincethe volume of the cell can change, the following input variables should be set: ecutsm 0.5

and dilatmx 1.15 - read their descriptions. Remember that LDA and GGA+D3 calcula-tions should start from experimental acell values (from COD).

Run the calculations. While running and after finishing, check whether the optimization

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converged and whether the SCF loop converged in every optimization step. Check the valuesof forces and stresses at the end of calculations.

Copy the optimized atomic positions and lattice constants to the input and use themfor the next tasks.

Try to optimize the crystal using GGA and converged values of ecut and ngkpt (fromChapter 3). If no success, use experimental lattice constants and optimize atomic positionsonly.

For monolayer start from geometrical parameters from final optimization of bulk (afterconvergence tests) and fix the third lattice vector (use appropriate optcell value). Afterfinishing, answer the questions from the end of Chapter 2.

1. How many optimization steps were required to reach the criterion?

2. How do the optimized lattice constants and atom positions differ from the initial ones?

3. How do the monolayer geometrical parameters differ from the bulk ones?

4. How do the answers for these questions differ for different xc functionals?

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Chapter 4

Convergence tests

Material covered in this chapter corresponds to Base tutorial 2: The convergence in ecut(I) and (II) and Base tutorial 3: Convergence study with respect to k-points.

The aim of the convergence tests is to find the lowest values of control parameters thatprovide converged calculations results, i.e. the results do not improve with increasing thecalculation parameters. Often convergence tests of crystal total energy are performed (likein Tutorials). The correct approach is, however, to check the convergence of physical, mea-surable quantities such as lattice constants, band structure or energy gaps, with arbitrarilychosen precision. For example, one can converge lattice constants and energy gap (num-bers which one intends to publish) up to 0.001A and 1 meV, respectively. Additionally,convergence of these quantities can be different than the total energy. Here, we will testthe convergence of lattice constants with precision of four significant digits for LDA andGGA+D3.

The convergence calculations with respect to both ecut and ngkpt will be done in a dou-ble loop, similar to Abinit Tutorial 2.3 (read https://docs.abinit.org/guide/abinit/

#loop). Full geometry optimization will be performed simultaneously. We will test:

• 6 values of ecut, starting from value from initial calculations and increasing it by 2Ha in every step

• 5 values of ngkpt, starting from initial value from previous calculations and increasing

it by 1 1 X, choosing X so that ngkpt(1)ngkpt(3) ≈

ca in every step

An example input:ndtset 30

udtset 6 5

ecut:? 10

ecut+? 2

ngkpt?1 6 6 2

ngkpt?2 7 7 2

ngkpt?3 8 8 3

ngkpt?4 9 9 3

ngkpt?5 10 10 4

Use the geometry parameters obtained from the initial optimization. Set pawecutdg tobe two times larger than the largest ecut value in the loop.

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After finishing check if the SCF cycle and geometry optimization converged in everydataset.

Create three tables with lattice constants and etotal values where rows are indexed byecut and columns by ngkpt values.

In the first table find a value of total energy that differs no more than 0.001 Ha fromthe neighbouring cells and corresponds to the lowest ecut and ngkpt., e.g.:

Figure 4.1: Expemplary results of total energy convergence tests.

Analogically, in the second and third table find values of lattice constants that do notdiffer in the fourth significant digit from the neighbouring cells and correspond to the lowestecut and ngkpt.

Converged values of ecut and ngkpt can be different for both lattice constants. In suchcase, take the higher values. If no convergence is obtained in this range of parameters, doadditional calculations for higher values of ecut and ngkpt.

Copy the geometry parameters obtained for the converged values of ecut and ngkpt.Set pawecutdg twice bigger than ecut. Perform the final full geometry optimization usingthese three parameters. Obtained lattice constants and atom positions will be used withoutchange for all next tasks.

Convergence tests were omitted for GGA, since automatic optimization of lattice con-stants often fails with this functional, due to underestimation of chemical bonding by thisfunctional and thus inherently weak interlayer interactions. Try to optimize geometry withGGA using ecut, pawecutdg and ngkpt from GGA+D3. If it fails, use lattice constantsfrom MP.

For monolayer use the same ecut, pawecutdg and ngkpt (ngkpt(3)=1) as for bulk.

1. How does the total energy change when increasing cutoff energy at fixed k grid andwhen increasing k grid at fixed cutoff energy?

2. How do the lattice constants change when increasing cutoff and k grid separately?

3. Is the convergence with respect to cutoff energy independent on k grid and vice versa?

4. Are the convergence criteria for total energy and lattice constants fulfilled at the samecutoff energy and k grid?

5. How do the final converged lattice constants and atom positions differ from the initialones and experimental ones?

6. How do the answers to these questions differ for LDA and GGA+D3?

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Chapter 5

Electronic band structure

Material covered in this chapter corresponds to Base tutorial 3: Computing the band struc-ture.

Electronic band structure calculation (BS) is performed in two steps. In the first step,charge density is calculated on a k points grid in SCF loop. In the second step, startingfrom the previously obtained charge density, eigenenergies are evaluated on a given k pointpath in BZ in a non-SCF calculation. The calculations will be done for LDA, GGA andGGA+D3.

We define two datasets. First dataset is a usual SCF calculation with a more restrictivestop criterion toldfe1 1d-8 and printing the charge density file: prtden1 1.

Second dataset is a non-SCF calculation (iscf2 -2) starting from charge density fromprevious dataset (getden2 -1). The convergence criterion for non-SCF is tolwfr2 1d-12.nstep should be increased to 50, since the wavefunction converges slower than total energy.

Standard k path in hexagonal BZ is presented of Fig. 5.1. We shall calculate BS onΓ−M −K − Γ−A− L−H −A path.

Figure 5.1: Standard k path in hexagonal BZ.

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Reduced coordinates of high symmetry points are given in kptbounds variable descrip-tion. One should set a negative value of kptopt2 equal to number of path segments. ndivk2provides numbers of points on every segment. The ratios of these (natural) numbers shouldbe close to the ratios of distances in reciprocal space (formulas for the lengths are givenin kptbounds variable description). The total number of k points on the path should notbe between 100 and 200. As no k point shifts should be applied in the second dataset, setnshiftk2 1 and shiftk2 0 0 0.

Number of bands is automatically determined by Abinit as number of valence electronsin PAWs divided by two (all occupied valence bands) plus one (empty band) in the case ofnon-magnetic calculations. For a band structure calculation nband2 should be increased byabout 16 (and even), as we wish to see a part of the conduction band in the BS plot.

Take care that the proper files are produced (or not): prtden1 1, prtden2 0, prteig1 0,prteig2 1, prtebands1 0, prtwf 0, prtgeo 0. prteband2 1 will produce a BS plotwhich can be opened in xmgrace to quickly check the overall corectness of the result. Setpawfatbnd2 2 (see Sec. 5.1).

Run the calculation. Check if the SCF cycle converged in first dataset. During the sec-ond dataset calculation several warnings can be produced. They correspond to the highestband, which is never fully converged, and can be ignored. After finishing the calculationsinspect the produced band structure plot in the vicinity of band gap and compare it toliterature. Data for the ”final” plot can be obtained from the output file using a scriptdescribed in Appendix C.

All input variables for monolayer calculations are the same, except kptopt2, kptbounds2and ndivk2, since the BZ is two dimensional and the k path is Γ−M −K − Γ. Comparethe result to literature.

The second calculation for bulk and monolayer will be performed with spin-orbit cou-pling switched on: pawspnorb 1 (read https://docs.abinit.org/tutorial/spin/#5-the-spin-orbit-coupling). All input variables remain unchanged, except kptopt1 and nband2 (statesare now singly occupied, so twice more bands are needed). Do the bulk calculation only forGGA+D3.

5.1 Orbital composition of bands

It is possible to calculate contributions of atomic orbitals to a specific band at a specifick point, projecting the calculated wavefunction inside the PAW augmentation sphere ontoreal spherical harmonics with L (angular quantum number) and M (magnetic quantumnumber) resolution. See the description of pawfatbnd variable.

Perform band structure calculations for monolayer using GGA+D3 without spin-orbitand setting pawfatbnd2 2. Make a band structure plot with atomic orbitals contributions(so called ’fatbands’ plot), following the example on Fig. 5.2. The correspondence betweencomplex and real spherical harmonics is the following:

|l = 0,m = 0〉 ←→ s

|1,−1〉 ←→ px, |1, 0〉 ←→ pz, |1,+1〉 ←→ py

|2,−2〉 ←→ dxy, |2,−1〉 ←→ dyz, |2, 0〉 ←→ dz2 , |2,+1〉 ←→ dxz, |2,+2〉 ←→ dx2−y2

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Plot the following orbital contributions:

• 2H and 1T: dz2 , dxz + dyz, dxy + dx2−y2 orbitals of metal and p orbitals of chalcogeneatoms

• III-V compounds and graphite/graphene: s, pz and px + py orbitals of all atoms

Figure 5.2: Band structure with orbital contributions.

For bulk and monolayer:

1. At wich k points VBM and CBM are located? What is the fundamental gap value?What are direct gaps at K and H points? What is the energy difference betweenconduction band minima at K and Q? What is the energy difference between valenceband maxima at K and Γ?

2. How does the spin-orbit affect interatomic forces and stress tensor? (first dataset)

3. How does the spin-orbit affect the band structure? What are the spin splittings ofVBM and CBM at K?

4. How do the answers to these questions differ for LDA, GGA and GGA+D3?

5. What is the orbital composition of VBM and CBM?

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Chapter 6

Density of states

Density of states (DOS) calculation is performed in two steps, similarly to BS. In thefirst step, charge density is calculated on a k point grid in an SCF cycle. In the secondstep, starting from the previously obtained charge density, eigenenergies are calculated ona given k point grid in BZ in a non-SCF cycle. The calculations will be done for GGA+D3functional.

We will define two datasets. First dataset is a usual SCF calculation with tighter stopcriterion toldfe1 1d-8 and printing the charge density file: prtden1 1.

Second dataset is a non-SCF calculation (iscf2 -3) starting from charge density fromprevious dataset (getden2 -1). The convergence criterion for non-SCF is tolwfr2 1d-12.nstep should be increased to 50, as the wavefunction converges slower than total energy.

DOS for gapped systems is calculated using tetrahedron method. As we are interestedin l and m quantum numbers projections, set prtdos2 3 and prtdosm2 1. It should becalculated on a denser k point grid - double the values used in SCF (keep ngkpt(3)=1 formonolayer). As no k point shifts should be applied, set nshiftk2 1 and shiftk2 0 0 0.

Use the same number of bands as in the BS calculation. Switch off the spin-orbitcoupling. Take care that the proper files are produced (or not): prtden1 1, prtden2 0,prteig 0, prteband 0, prtwf 0, prtgeo 0.

Run the calculation. Check if the SCF cycle converged in first dataset. During thesecond dataset calculation several warnings can be produced. They correspond to thehighest band, which is never fully converged, and can be ignored.

A ” DOS” output file for every atom is produced. The first column is the energy scale(in Ha), then there are columns with l projections and integrated l projections. In the nextcolumns there are lm projections. Plot the following orbital contributions:

• 2H and 1T: dz2 , dxz + dyz, dxy + dx2−y2 orbitals of metal and p orbitals of chalcogeneatoms

• III-V compounds and graphite/graphene: s, pz and px + py orbitals of all atoms

Compare the result to MP, C2DB and experimental STS spectra (if available).

1. Which orbitals compose the bands close to Fermi level?

2. What is the fundamental energy gap from DOS? Compare it to value obtained fromband structure.

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Chapter 7

Effective mass

Close to some special points in Brilloun zone electronic bands dispersion can be approxi-mated by

E(k) = E0 +h2k2

2m∗ , (7.1)

where E(k) is the energy of a carrier at wavevector k in that band, E0 is a constant and m∗

is the effective mass of holes in valence bands or electrons in conduction bands. One canthen evaluate specific m∗ fitting a parabola to the band: E(k) = E0 + a · k2, from where

m∗ = h2

2a .

Calculate the effective masses of VBM(-1) and CBM(+1) bands at points specifiedby the tutor in bulk and monolayer system for GGA+D3 with spin orbit coupling. Forthat purpose perform band structure calculations on a k path including different in-planedirections, 0.05 Bohr−1 away from the specified point. Use 20 k points in every direction.

Fit two separate parabolas for two directions, e.g ΓK and KM . Reduce the fitting rangeto the parabolic region if necessary.

Mind the units of energy and wavevector. Effective mass is usually given in units of freeelectron mass m0, so provide m∗

m0.

Compare the calculated m∗ to literature values (calculated and experimental).

1. Do the effective masses of VBM-1 and VBM differ? (the same for CBM and CBM+1)

2. Do the effective masses of the same band differ for different in-plane directions?

3. Do the conduction bands cross close to point K?

4. How do the answers to these questions differ for bulk and monolayer?

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Appendix A

Control questions

1. What are ab initio methods and what can be calculated using them?

2. What are the fundamental theorems of DFT?

3. What is the central quantity treated by DFT?

4. What are the three main equations/formulas of DFT? Why are they refered to asself-consistent equations system? How are they solved?

5. What are Kohn-Sham equations? Compare them to one-particle Schroedinger equa-tion.

6. What is the meaning of Kohn-Sham orbitals?

7. What is the Born-Oppenheimer approximation?

8. What is a functional? What are the most important functionals of DFT?

9. What are the two basic approximations of exchange-correlation functional? Whaterrors do they yield for geometrical parameters and electronic band gap?

10. What is a pseudopotential and a PAW?

11. How is the irreducible Brilloun zone constructed and what is a Monkhorst-Pack kpoint grid?

12. What is the meaning of convergence?

13. Why electronic band gaps are underestimated in DFT? How can they be improved?

14. Which unit system is used in Abinit? What are the units of length and energy?

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Appendix B

Tips and hints

• Read the task chapters and prepare input files before classes. You can base ontutorial input files.

• Read the descriptions of all input variables that you use. If the value of a variable isnot given in description of the task, you should find it yourself on Abinit webpage.

• If you don’t remember or were absent on some basic tutorials, go back to their de-scriptions on Abinit webpage.

• Organize your calculations in separate directories.

• After preparation of input file, double-check it (especially for typos) and then show itto the tutor before starting calculations.

• Abinit doesn’t recognize tabulation signs.

• Abinit requires 10 significant digits in acell, rprim and xred to recognize the sym-metry. You can use simple fractions.

• After finishing calculations, check if they converged.

• Always run the calculations in parallel: mpirun -np 4 abinit < *files > log &

• Do not run multiple instances of Abinit at the same time.

• You can check if Abinit is running using top command. If you need to kill thecalculation, type k in it and confirm. Quit with q. You can use killall abinit

command instead.

• After finishing calculations remove DEN and WF files if they are produced.

• At the end of every class copy your files to your external drive or cloud.

• Post-process and insert the results into the report after every class. Do no leavewriting the whole report to the end of the semester. Show your progress to the tutor.

• Answer the questions stated at the end of every chapter.

• Look for answers for your questions and doubts in this script and on Abinit webpagefirst. If you don’t find them - ask the Tutor.

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Appendix C

Scripts

All the following scripts are provided only for the use of students carrying out the project.Information about authors and copyrights are included in the scripts headers.

The scripts are Linux bash shell codes and use standard Linux tools, e.g. awk. Theyare compatible with Abinit 8.10.x. No tests for other versions were performed.

Before using the scripts one needs to modify the access permissions:chmod +x script name.sh

1. bands.sh

Reads in Abinit output file from band structure calculation on a path in reciprocalspace. It is assumed that a two-dataset calculation was performed. From the firstdataset it reads in the reciprocal lattice vectors and Fermi energy. From the seconddataset reduced coordinates of k points and eigenenergies are read in. It calculatesthe distance in reciprocal space (in Bohr−1) along the k points path and prints it tothe output text file along with eigenergies shifted by the Fermi energy. The resultingoutput is readible for Gnuplot, Libre Office Calc, Excel and Origin.

Execution: ./bands.sh abinit output bands output

If an error is displayed:

bash: ./bands.sh: /bin/bashM: bad interpreter: No such file or directory,

run

dos2unix bands.sh

or

sed -i -e ’s/\r$//’ bands.sh

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Appendix D

Report contents

The report in a pdf file should be sent to the tutor via e-mail. It should contain:

1. Basic information about the crystal: compound formula, politype, space group, num-ber of atoms in unit cell, information about centrosymmetry of bulk and monolayer.

2. Figure of top and side view of the structure with geometrical parameters symbols.

3. Results of convergence tests: tables, exemplary plots of lattice constants and totalenergy in function of ecut at fixed ngkpt and in function of ngkpt at fixed ecut;converged values of ecut and ngkpt

4. Table with calculated (final) and experimental geometrical parameters: lattice con-stants, layer thickness and interlayer spacing; relative differences between calculatedand experimental values with signs, e.g.:

5. BS plots with and without spin-orbit coupling in energy range [VBM-2, CBM+2] eVwith k points labelled. Follow the standards of plots in C2DB.

6. DOS plots. Valence band edges in BS and DOS should be aligned. Follow the stan-dards of plots in C2DB.

7. Table with calculated and experimental values of band gaps (2H: Γ-Q, K-Q, Γ-K,K-K, H-H; 1T: Γ-M, Γ-L, A-L, Γ-Γ); relative differences between calculated and ex-perimental values with signs.

8. Table with calculated and experimental values of valence band maximum and con-duction band minimum splittings; relative differences between calculated and experi-mental values with signs.

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9. Additional : Table with calculated and experimental values of effective masses ofVBM(-1) and CBM(+1); relative differences between calculated and experimentalvalues with signs.

10. Additional : BS plot with orbital contributions.

11. Conclusions. This is the most important part of the report. Questions at the end ofall chapters may be helpful.

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Appendix E

Literature

1. Pawel Scharoch, Jerzy Peisert, Metody obliczeniowe ab initio w fizyce struktur atom-owych, http://www.if.pwr.wroc.pl/~scharoch/Abinitio/skrypt.pdf

2. Materials on http://www.if.pwr.wroc.pl/~scharoch/Abinitio/Abinitio.html

3. Kieron Burke and friends, The ABC of DFT, http://dft.uci.edu/doc/g1.pdf

4. https://www.cecam.org/workshop-9-1326.html

5. http://dft.uci.edu/learnDFT.php

6. Alexander V. Kolobov, Junji Tominaga, Two-Dimensional Transition-Metal Dichalco-genides, Springer 2016

7. Gang Wang, Alexey Chernikov, Mikhail Glazov, Tony Heinz, Xavier Marie, ThierryAmand, Bernhard Urbaszek, Colloquium: Excitons in atomically thin transition metaldichalcogenides, Reviews of Modern Physics 90 21001 (2018)

8. Andor Kormanyos, Guido Burkard, Martin Gmitra, Jaroslav Fabian, Viktor Zolyomi,Neil D Drummond, Vladimir Falko, k.p theory for two-dimensional transition metaldichalcogenide semiconductors, 2D Materials 2 022001 (2015)

9. Rafael Roldan, Jose A. Silva-Guillen, M. Pilar Lopez-Sancho, Francisco Guinea, Em-manuele Cappelluti, Pablo Ordejon, Electronic properties of singlelayer and multi-layer transition metal dichalcogenides MX2 (M = Mo, W and X = S, Se), Annalender Physik 526, pp. 347-357 (2014)

10. H. Sahin, S. Cahangirov, M. Topsakal, E. Bekaroglu, E. Akturk, R. T. Senger, S.Ciraci, Monolayer honeycomb structures of group-IV elements and III-V binary com-pounds: First-principles calculations, Physical Review B 80 155453 (2009)

11. Xue-Fei Liu, Zi-Jiang Luo, Xun Zhou, Jie-Min Wei, Yi Wang, Xiang Guo, Bing Lvand Zhao Ding, Structural, mechanical, and electronic properties of 25 kinds of III-Vbinary monolayers: A computational study with first-principles calculation, ChinesePhysics B 28 086105 (2019)

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