Ab Initio modelling of structural and electronic. Matt Probert University of York

Transcription

1 Ab Initio modelling of structural and electronic properties of semiconductors Matt Probert University of York

2 Overview of Talk What is Ab Initio? What can we model? How does it all work? Semiconductor examples Summary

3 Ab Initio = From the beginning What is Ab Initio? i.e. first principles (quantum mechanics) no fitting to experimental data In quantum chemistry, usually taken to mean Hartree-Fock theory or beyond (eg MP2, CI, etc) In physics, usually taken to mean DFT or beyond (eg GW, QMC, etc)

4 Advantages of Ab Initio High accuracy and predictive ability not limited by the fitting data-set can cope with unusual environments, and bond breaking/making wide range of properties can be calculated BUT computationally expensive limited in size of system can study need powerful computers

5 Which Ab Initio? In this talk, will focus in particular on DFT Density Functional Theory widely used see many of the talks in rest of workshop good balance of accuracy and speed has some limitations one uncontrolled approximation at its heart GW uses many-body perturbation theory to improve excited state properties QMC avoids this altogether

6 Quantum Mechanics and Density Functional Theory

7 Quantum Mechanics General Time Dependent Schrödinger Equation: Atomic Time Independent Schrödinger Equation: t i V z y x m Atomic Time Independent Schrödinger Equation: where N=Nuclei at positions {R}, e=electrons at positions {r}, e is the eigenenergy and i J e N e N i J e N e N H r R r R,, ˆ,,,, e e e N N N e N e N V V V T T H ˆ ˆ ˆ ˆ ˆ ˆ,

8 Density Functional Theory (I) Uses Born-Oppenheimer Approximation decouples electron and nuclear problems assumes electrons respond instantly to the current nuclear coordinates Uses charge density n(r) instead of manybody wavefunction Y(r 1,r 2 r N ) Hohenberg-Kohn proved 2 key theorems: 1. total energy is a unique functional of n(r) 2. the density which minimises the energy is the ground state density and the minimum energy is the ground state energy

9 Density Functional Theory (II) Hohenberg-Kohn (1964) proved that there was a universal functional E[n] which could be minimised to obtain the exact groundstate density and energy but not what the form of E[n] was! Kohn-Sham (1965) derived an expression for E[n] with one key unknown the exchange-correlation functional E xc [n] and introduced the Local Density Approximation (LDA)

10 Do not know the unknown functional DFT Problems write in terms of things we do know, e.g. electronelectron interaction and hope the things we do not know are small and hence can be easily approximated hence the LDA etc. Do not know how to calculate the Kinetic Energy T of a density! only for a wavefunction so introduce fictitious single-particle orbitals that give the correct density and for which a Kinetic Energy can be calculated.

11 Kohn-Sham Equations Write density in terms of KS orbitals Hence KS equations: ) ( ) ( ) ( ' ) ' ( ) ( 1 2 r r r r r r V d n V N i i n 2 ) ( ) ( r r where and E xc is the Exchange-Correlation functional the only approximation! ) ( ) ( ) ( ' ' ) ' ( ) ( r r r r r r r r i i i ext V xc d n V ) ( ] [ ) ( r r n n E V xc xc N atoms ext Z V R r

12 Exchange-Correlation Functional DFT is formally exact but in practice we need to approximate E xc the LDA assumes that E xc [n(r)] at some point r is the same as if every point had the same density hence LDA is fitted to Homogenous Electron Gas data calculated with high level QM methods expected to be good for metals but actually works well for many systems including semiconductors tends to overbind energies, shorten bonds can also use density gradient information, i.e. Generalised Gradient Approximation (GGA) various GGAs available, all tend to underbind can also use exact exchange from HF theory to create hybrid functionals but not so rigorous see talks on Thursday

13 Many-body QM is hard Why Bother? It is impossible to solve the Schrödinger equation exactly for all but the most simple problems Numerical approaches expand unknown wavefunction in terms of known basis functions and unknown coefficients a simple spin system with 20 particles needs 2 20 coefficients (spin up & down) exponential scaling is bad! DFT is easy uses a 3D scalar field instead much better!

14 What Do You Get? Key output of any DFT calculation is the ground state energy and density of the electrons hence can get total energy of system at T=0 can extend to finite temperature With other ab initio methods can also get spectrum of excited states

15 What Can We Model?

16 Ground State Properties Many experimental measurements can be derived from ground state total energy and how it varies with some parameter

17 Forces and Stresses Classically, can derive forces, stresses, from derivative of energy Hellman-Feynman theorem shows how to R F U U do this with QM: and so for a position-independent basis set R R F H E R R R R H H H E

18 Implications If we have analytic forces then it is much more efficient to calculate optimal bonding arrangements Ditto stresses and cell parameters Can also extend this to molecular dynamics time and/or temperature variation And can extend general approach to do linear response (e.g. phonons, electric field response, magnetic field response, etc)

19 Introducing CASTEP CASTEP is an example of a general purpose DFT code uses plane-wave basis set (position independent, easy to improve accuracy) use pseudopotentials (replaces nuclei and inner electrons with pseudo-ion) can calculate wide range of properties will feature in some talks tomorrow freely available to UK academics

20 Total energies CASTEP Abilities forces and stresses, with LDA/GGA/sX/EXX/LDA+U/ etc. Electronic structure electronic charge, potential, band structure, DOS, atomic populations Geometry Optimisation atomic positions, cell parameters, external pressure/stress Molecular dynamics finite temperature, zero-point and non-equilibrium properties Transition state searches chemical reaction pathways, diffusion barriers Phonons perturbation theory, finite differences Electric field response polarisability, dielectric constants, Born charges, LO/TO splitting Magnetic Response NMR, Chemical shifts, electric field gradients, hyperfine constants, etc. ELNES, EELS, Raman, Wannier Functions, and more

21 2eV indirect bandgap CaS 2 TeP

22 How Does It All Work? So what is going on in CASTEP or other plane-wave DFT codes? To get accurate answers from such codes, we need to understand a little about what is going on. There are several key parameters that must be carefully converged before good quality answers will emerge!

23 Basis Sets, Periodic Systems, Bloch s Theorem, k-points and Supercells

24 Plane Waves The Kohn-Sham equations can be written as 2 V r r 1 2 We are interested in periodic systems eff hence expand the K-S orbitals in terms of a periodic basis set: i i i ( r) j cg, j g e ig.r

25 Basic Crystallography The atomic structure of a crystal can be described by the unit cell periodically repeated along the lattice vectors. The smallest cell is spanned by the primitive vectors: a 1, a 2 and a 3 that connect the nearest neighbors. Any point in the lattice is connected to another point by a lattice vector: R=n 1 a 1 + n 2 a 2 +n 3 a 3 The unit cell of face centered cubic (FCC) system, with the primitive cell also shown.

26 The potential in the crystal is periodic V(r + R) =V(r) so any crystal Hamiltonian is periodic: 1 2 V ( r ) 2 k The wave functions Y nk in a crystal are then quasi-periodic and can be written as: nk ( r) u ( r) e nk k i kr k Bloch s Theorem where u nk (r) = u nk (r + R) is a periodic function that fits inside the unit cell and e ik.r is a complex phase-factor. FCC unit cell

27 Reciprocal Space The set of wave vectors {g} that give plane waves with the wavelength having the periodicity of a given lattice with lattice vectors {R} is defined as the reciprocal space: ig. R e 1 The reciprocal basis vectors are defined by: b l 2 a a ( a l m m a a A reciprocal lattice vector can be written as a linear combination of the basis vectors: g g b g b n n ) g3 b 3

28 The Brillouin Zone The region in reciprocal space that is closer to the reference lattice point than any other reciprocal lattice point is defined as the first Brillouin Zone: The first Brillouin zone in a 2D rectangular lattice. The first Brillouin zone for the 3D FCC lattice, Bouckaert et al., Phys. Rev 50, 58 (1938).

29 Plane Waves Revisited Hence the set of plane waves is finite: r i( gk). r k( ) ck, ge g with the smallest g given by the Brillouin zone, and the longest g determined by the planewave cutoff energy, E cut : g k 2 2 E cut which therefore defines a length-scale for smallest features captured in the calculation: E cut

30 Si8 Convergence with E cut Tota al Energy (ev) Cut-off Energy (ev) Variational principle monotonic decrease in E total with E cut

31 Band Structure The solutions to the free electron Schrödinger equation: 1 2 are plane waves: 2 k ( r) k ) e r i k. r k k k (k) (ev) with the eigenvalues ( k) k 2 2 k (Å -1 ) The energy of the electrons has a quadratic dispersion (k dependence).

32 Zone Schemes All properties of the crystal can be described in the first Brillouin zone. Fold the bands outside the first Brillouin zone back inside the first Brillouin zone. The extended zone scheme The folded zone scheme (k) (ev) First BZ Second BZ First BZ

33 BZ Integration Many observables are calculated as an integral over all k- points within the 1st Brillouin zone, for example: Etot E( k) d k 3 n( r) nk( r) d k V V BZ 1stBZ 1stBZ So must make sure have enough sampling points in the 1 st BZ (k-points) for convergence a non-variational parameter so E total may go up as well as down as the density of points is increased standard method is a regular Monkhorst-Pack grid must be dense for metals to capture discontinuity in occupation of bands at E=E f hence use of smearing can use crystal symmetry to reduce the number of points BZ

34 Band Structure of Si Silicon has diamond structure. Bouckaert et al., Phys. Rev 50, 58 (1938). High symmetry points in the Brillouin zone: G=center of the Brillouin zone L=mid point on the zone boundary plane in the {111}-directions K=mid point on the edge between two hexagons {110}-direction X= mid point on the zone boundary plane in the {100}-direction

35 Band Structure of Si Primitive cell of Si has 2 atoms and hence four valence electrons: four bands below the Fermi level. Indirect bandgap.

36 Supercell Approximation What if want to calculate properties of a crystal defect? or an isolated molecule? or a surface? Use a supercell e.g. put 1 defect into a 2x2x2 cell e.g. add vacuum around molecule e.g. add vacuum above surface

37 Nanotube Supercell

38 Nanotube Primitive Cell

39 Nanotube Charge Density

40 Key Ideas Summary DFT replaces many-body wavefunction with density and Kohn-Sham equation LDA the simplest approximation to the unknown physics in the Kohn-Sham equation Bands single particle solutions to K-S equation Plane wave basis set bands represented in reciprocal space with cut-off energy K-points integration grid in 1st Brillouin zone Pseudopotentials a smooth representation of the nucleus and core electrons Supercells enable you to study non-periodic systems

41 Example 1 Si(100) Surface Reconstruction

42 Set up Supercell Si(100) supercell with vacuum gap with added hydrogen passivation 9 layers of Silicon and 9 Ǻ vacuum

43 Convergence Tests Converge cut-off energy 370 ev Converge k-point sampling 9 k-points Converge number of bulk layers 9 layers Converge vacuum gap 9 Å only then see asymmetric dimerisation: (x)=best lit. value

44 Si(100) The Movie

45 Example 2 Si Vacancy

46 Si Vacancy Simplest example of a point defect But there have been many different values given in literature, using DFT, for neutral defect formation energy: N 1 EV EN 1 EN 1 N Why? A detailed study showed various factors responsible, including long-range relaxations need large supercells shallow PES around vacancy high sensitivity to noise in forces and k-point sampling

47 System Size Dependence Clear effect of supercell symmetry due to (spurious) longrange vacancy interaction between periodic images. Comparison is with all atoms in unrelaxed configuration. Convergence confirmed with 864 atom BCC supercell

48 Long-range Interaction Charge density difference iso-surface for unrelaxed 216 and 215 atom SC supercells Clear effect of supercell symmetry on electronic structure spurious!

49 Structural Optimisation Spontaneous symmetry breaking during relaxation Jahn-Teller distortion lowering energy by 1.2 ev Initial state has Td-point symmetry, final has D2d

50 Long-Ranged Relaxation Convergence of the ionic displacement of successive shells of atoms, centred on the vacancy site, as move across the supercell.

51 Example 3 Si Bandstructure

52 XC Functionals CASTEP can do standard LDA and GGA functionals (PW91, PBE, rpbe, WC) fast and reasonably accurate for structures Lattice parameters Bulk modulus But details depend on choice of XC functional

53 LDA Overbinding Charge density difference plot of Si8 r(lda)-r(pw91)

54 XC Functionals But still the traditional problems with band structures LDA/GGA typically underestimate band-gap whereas Hartree-Fock overestimates gap hence demand for hybrid functionals Fundamental science what is true nature of XC-hole? Can we go beyond local/semi-local approximations?

55 sx-lda Fully non-local XC-functional Uses Thomas-Fermi screened exchange: E XNL * * r r 1 ktf r r ik ik jq r jq r drdr e 2 ikjq r r with LDA correlation Expensive with plane-waves! Requires a double-sum over bands, double-sum over k-points and a triple-sum over plane-waves! Clever FFT trick NB2NK2 NP log NP

56 Improved Bandstructure Si band-structure Solid=sX-LDA Dash=LDA See later talks for more details

57 Example 4 H in Si

58 Hydrogen Structures Stable / Metastable sites BC two-fold coordinated T four-fold coordinated Possible saddlepoint sites AB antibonding site C half-way to T H hexagonal (6fold) site

59 Spin Density Plot for Site Planes of silicon atoms seen edge-on BC site Spin density iso-surface due to single extra electron

60 Experimental Input Traditional T=0 calculations suggest is most stable but with small binding energy H is light and hence might be quite mobile even at room temperatures must include thermal effects Key experimental probe is msr which uses Muonium (Mu) instead of H with mass Mu ~ 1/9 Mass H and ZPM ~ 1/sqrt(mass) must include quantum effects such as zero-point motion and tunnelling etc. Yse Path Integral Molecular Dynamics (PIMD)

61 Molecular Dynamics ab initio Molecular Dynamics Use classical mechanics to move the atoms Born-Oppenheimer approximation decouples nucleus and electrons and have electrons always relaxed onto the instantaneous B-O surface but using forces and stresses derived from the electronic wavefunction hence ab initio MD can use to study dynamical properties or to simulate a statistical ensemble (e.g. NVE, NVT, NPH or NPT) with various thermostats and/or barostats, etc. But the nucleus is always treated classically hence no quantum fluctuations, tunneling, zero point motion, etc.

62 Path Integral MD Use Feynman Path Integral formulation of Quantum Mechanics for the nucleus now includes ZPM etc important for light defects and/or low temperatures beads on springs view with imaginary time axis computationally expensive! Path integral view of a single quantum particle.

63 PIMD Movie of H in Silicon

64 More Examples Briefly

65 Defect Level of H0 in Al2O3 CASTEP can also calculate electronic density of states across whole Brillouin Zone Or projected DOS onto any given atom

66 Magnetic Systems Can also do magnetic systems including some that are not natural! Example shows AFM state of BCC Iron

67 Optical Properties Can also calculate optical properties, such as dielectric constant and refractive index but best if apply a scissors to get good band gap!

68 Phonon Spectra of NaCl And hence can use harmonic approximation to calculate zeropoint energy, Helmholtz free energy, heat capacity, etc.

69 Equation of State TiO2 Which phase is stable at which pressure? calculate E(V) curve Forsterite is most stable of all the phases considered then Wadsleyite and then Ringwoodite as increase pressure Can use common-tangent to find transition pressure.

70 More Atomic/Bond Population analysis Transition state searches Electric field response polarisability, dielectric constants, Born charges, LO/TO splitting Magnetic Response chemical reaction pathways, diffusion barriers NMR, Chemical shifts, electric field gradients, hyperfine constants, etc. Spectroscopy ELNES, EELS, Raman, and more

71 Summary

72 Disadvantages of DFT It only applies to the electronic groundstate Have to use approximations to the true density functional or an electronic system in thermal equilibrium not possible to predict error in the value of any particular property not possible to systematically improve accuracy of calculation Might have to use a more accurate (and much more expensive) method such as GW or QMC good for excited states as well as ground state but no forces so have to use DFT structures

73 Advantages of DFT Kohn was awarded the Nobel prize for DFT makes it hard to criticise! It offers very good scaling of computational cost with system size. It allows calculations to be performed on large and complex systems. Given the very large number of DFT calculations, the likely property accuracy for many properties/systems is known.

74 Summary of CASTEP CASTEP is a robust and reliable implementation of DFT for periodic systems uses plane wave basis and ultrasoft pseudopotentials can be used to calculate ground state electron density and energy and hence many derived properties See for more

75 Useful References Hohenberg & Kohn, Phys. Rev. B 136, 864 (1964) Kohn & Sham, Phys. Rev. A 140, 1133 (1965) MC Payne et al., Rev. Mod. Phys 64, 1045 (1992) RM Martin, Electronic Structure: basic theory and practical methods, Cambridge University Press (2004) SJ Clark, MD Segall, CJ Pickard, PJ Hasnip, MIJ Probert, K Refson and MC Payne, First principles methods using CASTEP, Zeitschrift für Kristallographie 220, 567 (2005)