PSEUDOPOTENTIALS FOR BAND STRUCTURE CALCULATIONS

Transcription

1 TMCSIII: Jan 2012, Leeds PSEUDOPOTENTIALS FOR BAND STRUCTURE CALCULATIONS Rita Magri Physics Department, University of Modena and Reggio Emilia, Modena, Italy CNR-Nano -S 3, Modena, Italy

2 OUTLINE Evolution of the Pseudopotential Concept First-Principles Pseudopotentials Empirical Pseudopotential Method Construction, Use and Results

3 EVOLUTION OF THE PSEUDOPOTENTIAL CONCEPT The Beginning - OPW formalism - Herring, Phys. Rev. 57, Phillips and Kleinman, Phys. Rev. 116, Cohen and Heine, Phys. Rev. 122, Empirical Pseudopotentials - Cohen and Bergstresser, Phys. Rev. 141, Cheliowsy and Cohen, Phys. Rev. B 14, Model Pseudopotentials - Abarenov and Heine, Phil. Mag. 12,

4 WHY PSEUDOPOTENTIALS? All-electron true Wave function The fundamental idea of a pseudopotential is to replace one problem with another. R. Martin, Electronic Structure, Cambridge All-electron true potential Figura Replace the strong Coulomb potential of the nucleus and tightly bound core electrons by an effective ionic potential acting on the valence electrons.

5 THE ENESIS OF THE PSEUPODOTENTIAL CONCEPT Taing the Fourier transforms of the periodic part of the Bloch function u r and the periodic crystal potential Vr which become series in and substituting into the Schrödinger Equation we obtain: Reciprocal Space Representation Equation we obtain: u V u E m r i r i e u r u e r Slowly Convergent u and V

6 HOW TO CALCULATE CRYSTAL BLOCH FUNCTIONS FOR VALENCE ELECTRONS? The first problem is that the crystal potential V is a highly varying function of real space. Also wavefunctions are expected to change a lot, with atomic-lie behavior near the nuclei and a more plane wave-lie behavior in the interstitial regions, where V is weaer. It is not possible to express u r as a simple superposition of plane-waves. Too many would be required!!! > 10 5 plane waves per atom!! Core Wiggles

7 FIRST STEP The lower lying bands come from the atomic core levels, while the electrons in higher bands feel a weaer potential screened by the core electrons. The main idea is to divide the bands into two groups, in the first group are the low-lying bands of core electrons, in the second group are the valence and conduction bands. Valence states Large Energy difference Core states Atomic C Z = 6 Atomic Cu Z = 29

8 OPW FORMALISM HERRIN, 1940 We assume the narrow lower bands are unchanged by the atom environment Frozen Core Approximation: so we can approximate these states with the core states of the free atom or other appropriately chosen localized functions φ. We are interested in describing valence electron bands core electrons are chemically inert True valence function r r n, n, Smooth function n, Localized function Core States It can be shown that ψ n, is orthogonal to all φ

9 THE PSEUDOPOTENTIAL PROTOTYPE PSP We now insert the expression n n n r r,,, into the Schrödinger equation and obtain: n n n n n E H H ˆ ˆ n n n n n E E E r V H ˆ 0 an equation for the smooth function χ. We have an effective potential: n ps r E E r V r V, Nuclear potentials

10 PROPERTIES OF PSEUDOPOTENTIAL V ps >0 r, V r Attractive long-ranged E n E r Repulsive short-ranged V ps is a much weaer potential than V: the attractive and repulsive parts partly compensate in the core region Cancellation Theorem, Cohen and Heine, 1961 V ps depends generally on the angular momentum φ = φ lm and is a non-local energy-dependent integral operator. No effect if there are no core functions with angular momentum l. The true valence function has no nodes.

11 WHAT IS MORE IMPORTANT. We got rid of the core states/electrons. Valence electrons feel a pseudopotential wea in the core region instead of the nucleus + core electrons SAME EFFECTS ON THE VALENCE ELECTRONS Core Electrons Pseudo Core Valence Electron

12 THE PSEUDOFUNCTION FOR THE VALENCE ELECTRON χ being slowly varying can be approximated by a few terms of a superposition of plane waves. Core region Bonding region

13 PROPERTIES OF THE SMOOTH PSEUDOFUNCTION χ is not the valence electron wavefunction but only its smooth part no wiggles in the core. The equation for χ has the same eigenvalues E n for the valence electrons as the original Hamiltonian. χ is not uniquely defined. The pseudofunction χ n, : 1 n, n, n, 2 Norm is not unity! Note: if we now the smooth pseudofunction we can always build the corresponding true valence function.

14 MODEL PSEUDOPOTENTIALS, EMPIRICAL PSEUDOPOTENTIALS, FIRST-PRINCIPLES PSEUDOPOTENTIALS In practice the expression for V ps is approximated. V ps can be built to satisfy our needs, for example we can require it to be the smoothest and weaest possible and maintaining the same scattering properties of the core potential on the valence electrons ab-initio approaches, or reproduce some measured quantities empirical approaches

15 TERMINOLOY: LOCAL PSEUDOPOTENTIAL The pseudopotential can be local, semilocal, nonlocal. Please note ALL the pseudopotentials are spherically symmetric. thus given on a radial mesh LOCAL Pseudopotential the less accurate s, p, d electrons all feel the same potential Vˆ L V L r EMPIRICAL and MODEL Pseudopotentials

16 TERMINOLOY: SEMILOCAL PSEUDOPOTENTIAL SEMILOCAL Pseudopotential V ˆ SL Ylm Vl r lm It is non-local in the angular variables, local in the radial variable Y lm Spherical harmonics Pseudo Wave functions d s, p, d electrons feel different potentials p s Mo from Haman, Schluter and Chang PRL 43,

17 TERMINOLOY: NON-LOCAL PSEUDOPOTENTIAL NON LOCAL SEPARABLE Pseudopotential Vˆ NL lm lm E l lm Fully non local in angles θ and φ AND radius r Functions of r, θ, φ In position representation r, r V NL Note: this PSP form is closer to the prototype PSP

18 Vˆ THE L-DEPENDENT TERMS ARE SHORT-RANED The l-dependent terms of V are different only inside the core region radius r c. A common long-ranged local potential V loc r is subtracted ps In this way the semilocal, non-local terms of the pseudopotential are zero outside r c. r c V r ps loc lm lm E l lm ps Zione Vloc r at large r Vanishes outside r c

19 MODEL PSEUDOPOTENTIALS The model potential of Abarenov and Heine 1965 The core is a blac box. Any core potential which yields the correct logarihmic derivative at r c is OK. A l A l Usually A l E is often a constant value fitted to reproduce the atomic eigenvalues and data

20 FROM THE IONIC OR ATOMIC PSEUDOPOTENTIAL TO THE CRYSTAL POTENTIAL Valence electrons move in the crystal potential Pseudopotentials describe the interaction of the valence electrons with a single ionic core. In the crystal the valence electrons interact one each other We have screened pseudopotentials usually in empirical calculations or ionic bare pseudopotentials unscreened, usually in firstprinciples calculations. The crystal potential is built as a superposition of atomic pseudopotentials

21 CRYSTAL POTENTIAL Let s suppose the pseudopotential is local, then it can be written simply as V α r. The electron Hamiltonian is: AB-INITIO V Hˆ Tˆ Vˆ Hxc Vˆ ion screened V ion α is the ion ind V r ion V r R R is the lattice vector, R, τ α is the basis vector Whose Fourier expansion is: or ir i ion r e e V Hˆ Tˆ ˆ is a reciprocal vector EPM Structure Factor S α Form Factor

22 EXAMPLE: AAS BULK Two atoms in the unit cell: a in τ a = τ = 1/8,1/8,1/8a, As in τ As = -1/8, 1/8,1/8a =- τ, we obtain: sen cos V V i V V e r V a As a As i r sen cos V V i V V e r V a As a As V S V A u V u E m And solve: The form factors are treated as adustable parameters empirical approach. Only those corresponding to few vectors are needed.

23 Cohen and Bergstresser Phys. Rev. 141,

24 Using the empirical approach the band structures, reflectivity spectra and photoemission spectra of bul IV, III-V, and II-VI semiconductors were calculated Cohen and Cheliowsy Electronic Structure and Optical Properties of Semiconductors, Ed. Springer It is difficult to apply this method for systems with hundreds or thousands of atoms per unit cell because the fitting parameters the form factors for each -shell would become too many! -New Atomistic Pseudopotentials AEPM

25 FORM FACTORS FOR SEMI-LOCAL PSP If the pseudopotential is semi-local still we have the structure factor and the form factor is more complicated:,,, V V loc,loc dr r r r r V P l l l l l 2 0 cos with: cos It depends on and requires a double loop over the vectors or in real space a radial integral for each pair of basis functions computationally very expensive!

26 NON-LOCAL SEPARABLE PSPS DO IT BETTER Non-local pseudopotentials mae the computation of the Hamiltonian matrix elements less expensive. Instead of dr r r r r V P l l l l 2 cos dr r r r r V P l l l l 0 cos We have: dr r r r dr r r r P l l l l l cos Factorized into a product of integrals for each basis function separately, in plane-wave calculations only single loops over are involved.

27 HOW TO ENERATE AN ATOMIC PSEUDOPOTENTIAL Pseudopotentials for first-principles calculations Unscreened bare pseudopotential ionic psp Extracted from an all-electron calculation on the free atom. Pseudopotentials for semiempirical calculations Extracted fitting experimental data of one or more compounds containing the atom. It is assumed to be screened.

28 PSEUDOPOTENTIALS FOR FIRST- PRINCIPLES CALCULATIONS Main steps in development Hamann, Schlüter, and Chang, PRL 43, Norm- Conserving Pseudopotentials Kleinman and Bylander, PRL 48, Separable Pseudopotentials Louie, Froyen, and Cohen, PRB 26, Non linear core correction Vanderbilt, PRB RC 41, Ultrasoft Pseudopotentials Blöchl, PRB RC 41, eneralized Separable Pseudopotentials Blöchl, PRB 50, PAW

29 REQUIREMENS FOR CONSTRUCTIN A OOD NORM-CONSERVIN PSEUDOPOTENTIAL Hamann et al. PRL 43, Choose an atomic reference configuration Example: Si 3s 2 3p 2 Use an atomic code to calculate the all-electron valence wavefunctions AE. Impose that the pseudo-wavefunction PS agrees with the AE wave-function beyond a chosen cutoff radius r c l-dependent

30 Also the l-channel pseudo-potential PS has to agree with the AE potential for r > r c. d 2m dr 2 1 scr. V 2 AE l l 1 2 2mr r nl nlm AE r d l l 1 scr. V r PS l m dr 2 mr nlm PS r 0 for the same eigenvalue ε nl = ε l

31 NORM-CONSERVATION REQUIREMENT The integrals from 0 to r c of the real and pseudo charge densities agree for each valence state. r c 2 r c 0 r 2 l PS dr 0 r 2 nlm AE 2 dr r c The charge contained in this region is the same for AE and PS wave-functions

32 SCATTERIN PROPERTIES d 2m dr 2 1 scr. V 2 AE l l 1 2 2mr r nl nlm AE r d l l 1 scr. V r 2 2 PS l 2 m dr 2 mr nlm PS r 0. By construction, we now that at energy ε = ε nl, the solution ψ PS r coincides with the ψ AE r for r > r c. But what about other energies? The transferability of the pseudopotential depends on the fact that ψ PS r reproduces ψ AE r over a certain range of energies about ε nl. We are interested in the energy range of valence bands in solid.

33 The logarithmic derivatives of the real and pseudo wave function and their first energy derivatives agree for r > r c. Logarithmic derivative l, r d Dl, r r r ln l, r, r dr l The first energy derivative of the logarithmic d d derivatives of the allelectron and pseudo wave-functions agrees at r c, and therefore for all r > r c. D l, r

34 SCATTERIN PROPERTIES The fundamental advance of Hamann, Schlüter and Chang, 1979, is to have shown that: If norm conservation is imposed, then pseudo D l ε,r matches all-electron D l ε,r to second order in ε ε l l This means that the norm-conserving pseudopotential has the same scattering phase shifts as the all-electron atom to linear order in energy around the chosen energy ε l. These properties however leaves plenty of freedom in the form of the pseudopotential and in its construction.

35 STEPS FOR PSEUDOPOTENTIAL CONSTRUCTION Step 1: choose a reference configuration F : 1s 2 2s 2 2p 5 Si: 1s 2 2s 2 2p 6 3s 2 3p 2 Step 2: solve the all-electron problem: V AE r,ψ AE,nl r Step 3: construct the pseudo wavefunction that satifies rules nodeless, matching to AE wavefunction, normconservation, etc Step 4: Invert the Schrödinger equation to get V PS,l r which is a screened potential Step 5: Unscreening the potential to obtain the bare V PS,l,ion PSEUDOPOTENTIAL ENERATION

36 VERY IMPORTANT STEP: THE PSEUDO TEST 1 Tests on excited configurations Example: Reference configuration for Si [core]s 2 p 2 we compare AE and PS results for other configurations: [core]sp 3 [core] s 2 p 1.. and many others We compare - Total energies - Energy Eigenvalues - Logarithmic derivatives Then calculate small well-nown systems and chec..

37 OPTIMIZATION OF A PSEUDOPOTENTIAL Pseudopotentials are optimized with regard to: 1. Accuracy and trasferability leads to choose small cutoff radius r c and harder pseudopotentials 2. Smoothness leads to choose a larger cutoff radius r c and softer pseudopotentials Different Authors have Proposed different Recipes

38 CONSTRUCTION RECIPES FOR SMOOTH AND ACCURATE NORM-CONSERVIN PSPS Bachelet, Hamann, Schlüter, PRB 26, Vanderbilt, PRB 32, Kerer, J. Phys. C 13,L Troullier and Martins, PRB 43, Rappe, Rabe, Kaxiras, and Joannopoulos, PRB 41, Cu

39 Troullier-Martins Kerer HSC Vanderbilt From Troullier and Martins, PRB 43,

40 UNSCREENIN THE PSEUDOPOTENTIAL The inversion of the Schrödinger equation gives the screened pseudopotential. We need to unscreen it. To unscreen: n PS V PS Hxc V PS ion l r fl l PS r 2 Pseudo valence charge density [ n ], r V [ n ], r V r H PS xc PS PS, l r Vscr., l r VHxc r However, Vxc is a non-linear functional of n so it is ambiguous to separate the effects of core and valence charge if there is a significant overlap of the two densities. This leads to errors and reduced transferability. NON-LINEAR V [ ], CORE CORRECTIONS xc ntotal PS core r PS

41 IMPROVEMENTS ON THE METHOD: SEPARABLE PSEUDOPOTENTIALS We separate the semi-local pseudopotential in a longrange local part and one short-range l-dependent part l, m PS PS PS V r V r V r l, m ion local lm Separable Pseudopotentials Kleinman-Bylander Trasform 1982 PS PS PS lm Vl Vl PS Vˆ NL Vlocal r PS PS PS V For each l-channel l, m V NL acts on the reference state ψ lm as the semilocal pseudopotential ΔV l Possible presence of bound ghost states at lower energies requires some care. l lm l lm PS lm

42 IMPROVEMENTS ON THE METHOD: ULTRA- SOFT PSEUDOPOTENTIALS First-row elements have valence states with angular momentum l without l core state. Already nodeless! AE PS O: 1s 2 2s 2 2p 4 core valence no p states in core O 2p wavefunction NORM-CONSERVATION maes PS AE Highly localized states in first row and transition-metal atoms New core Radius for UltraSoft Difficult convergence in a plane wave basis

43 D. Vanderbilt, Phys. Rev. B 41, Release the norm conservation criteria to obtain smoother pseudo wave functions. This is done by splitting the pseudo wave functions into two parts: 1. The ultrasoft valence wave function that do not fulfill the norm conservation criteria: Q i r, AE i US i r r 2. Plus a core augmentation charge charge deficit in the core region: AE r, US i r US r The Ultra-Soft Pseudopotential taes the NL form US V V local r Di i T V i i i loc i

44 An overlap operator S is introduced: S 1 n r i Q i 2 i r Ql r i l i i Main Properties: 1 Changed orthonormalization: l i 0 in case of norm-conservation 2 eneralized eigenvalue problem to be solved Hˆ ˆ ns 2 The NL Pseudopotential is updated during the iterative procedure D It mn n 0 ˆ i S i, 0 D drv r Q r mn Hxc mn

45 A PSEUDOPOTENTIAL FOR ALL SEASONS Many different PSPs and Pseudo enerator Codes provided in pacages: Plane-waves pseudopotential codes On-The-Fly Pseudopotential eneration in CASTEP - a 164 B pdf tutorial.

46 PSEUDO-ELEMENT TABLES

47 Name: Oxygen Symbol: O Atomic number: 8 Atomic configuration: [He] 2s2 2p4 Atomic mass: Available pseudopotentials: O.pz-mt.UPF details Perdew-Zunger LDA exch-corr Martins-Troullier O.blyp-van_a.UPF details Bece-Lee-Yang-Parr BLYP exch-corr Vanderbilt ultrasoft author: a O.pbe-van_gipaw.UPF details Perdew-Bure-Ernzerhof PBE exch-corr Vanderbilt ultrasoft author: gipaw O.blyp-mt.UPF details Bece-Lee-Yang-Parr BLYP exch-corr Martins-Troullier O.pz-paw.UPF details Perdew-Zunger LDA exch-corr Proector Augmented Waves Kresse-Joubert paper O.pbe-van_a.UPF details Perdew-Bure-Ernzerhof PBE exch-corr Vanderbilt ultrasoft author: a O.pbe-rrus.UPF details Perdew-Bure-Ernzerhof PBE exch-corr Rabe Rappe Kaxiras Joannopoulos ultrasoft And many other.

48 PSEUDOPOTENTIALS FOR SEMIEMPIRICAL CALCULATIONS Main steps in development Cheliowsy and Cohen, PRB 14, Atomistic Empirical Pseudopotential Mader and Zunger, PRB 50, Wang and Zunger, PRB 51, LDA derived semiempirical pseudopotentials

49 LDA-DERIVED EMPRICAL PSEUDOPOTENTIALS Problems with first-principles methods 1. Difficult to apply to systems with thousandsmillion atoms nanostructured materials 2. Problem with excited states: the band gap is often severely understimated comparison with experiments spectroscopies not goood Transferable screened pseudopotentials The idea: reproduce experimentally determined band energies, optical spectra, etc, and at the same time, LDAquality wavefunctions and related quantities.

50 SEPM FROM LDA CALCULATIONS 1. LDA SLDA Form factors - Calculate LDA for structure V LDA V loc +V Hxc - Spherical Average of the Screened Local Potential V LDA S - Structural average v SLDA v,, SLDA v SLDA r Local Potential -The points vs are fitted by the continous function : N v, SLDA r

51 2. SLDA SEPM - Only the Coefficients C SLDA are adusted to fit the experimental or quasiparticle calculated excitation properties Unlie standard EPM, which produces only discrete form factors and is hence suitable only for a particular crystal structure and lattice constant, the new SEPM or AEPM can be used for different structures and volumes with good transferability. The form factors for each particular structure are extracted from a Universal continous function of q N 4,5,6 2 2 c qb q a n n v q ane or v q a0 2 a3 q a e n

52 PROPERTIES OF AEPMS ood band structure Accurate effective masses Accurate band gaps ood elastic properties Bul modulus, deformation potentials Transferable Low Energy cut-off ~5 Ryd Simple analytic form few parameters

53 AVAILABLE ATOMISTIC EMPIRICAL PSEUDOPOTENTIALS Only certain combinations are available IV, III-V, II- VI

54 AN EXAMPLE: THE INAS/ASB SYSTEM Broen ap System Semiconductor because of the e 1 and h 1 confinement Possibility of tuning the band gap between mev Type II: short periods SLs to increase the radiative ricombination efficiency

55 aas InAs aas CA As a The single 001 interface has C2v symmetry a NCA asb As In In a Sb As In In InAs a asb

56 IN-PLANE POLARIZATION ANISOTROPY I p [1 1 0 ] 1 e h I p [ ] e h InAs/AlSb superlattice Y=[-110] X=[110] Fuchs et al. in Antimonide-Related Strained-Layer Heterostructures Wavenumber cm-1

57 ATOMISTIC EMPIRICAL PSEUDOPOTENTIAL ELECTRONIC STRUCTURE Solve the Schrödinger equation FULLY ATOMISTICALLY, v r R n r r 2m n 2 each atom individually described not no strain minimizing atomic positions self-consistent LDA errors plane wave expansion of ψr Folded spectrum method The spectrum at the left is the original spectrum of H. The spectrum at the right is the folded spectrum of H-E_ref^2

58 FORM FACTORS vq continous function of q v r e i q r v q 1 vn vn a4 Tr q v q a 0 q 2 a1 2 a e a 3 q 2 1 Parameters fit to reproduce: 1. aps Eg and effective masses m* 2. Hydrostatic ag and biaxial b deformation potentials 3. Band offsets and spin-orbit splitting so 4. LDA-predicted single band edge deformation potentials av, ac for ALL 4 binaries

59 FIT: RESULTS

60 HANDLIN OF BIAXIAL STRAIN Explicit with strain dependence in v q, v n a 4 Tr EPM LAPW IF specific offsets IF specific bonds

61 CRYSTAL POTENTIAL FROM A SUPERPOSITION OF ATOMIC POTENTIALS Interfaces or Disorder n 4 n vin Asn Sb 4 n vin InAs vin InSb 4 n

62 FIRST HEAVY-HOLE CHARE DENSITY

63 The method predicts the positive band bowings parameters of the ternary alloys in agreement with experiment!

64 RESULTS FOR THE INAS6/ASBM AND INAS8/ASBN SUPERLATTICES Eg with increasing n

65 CORRECT TREND! InAs8/aSbn Number of asb monolayers n

66 OVERLAP OF THE ELECTRON STATES InAs asb

67 LON-PERIOD INAS/ASB SLS InAs46aSb14

68 MORE TO BE SEEN FRIDAY MORNIN THANK YOU FOR YOUR ATTENTION

69 If we choose a plane wave for χn, we call the ψn an OPW orthogonalized plane wave OPW q 1 1 q r r e V q OPW OPWs were used as basis functions for expansion: n, r i q OPW,i i l,m, r Dependence on l,m nl,,m r n, r l, m, n, l, m,