PSEUDOPOTENTIALS FOR BAND

STRUCTURE CALCULATIONS

TMCSIII: Jan 2012, Leeds

Rita Magri

Physics Department, University of Modena andReggio Emilia, Modena, Italy

CNR-Nano -S3 , Modena, Italy

OUTLINE

Evolution of the Pseudopotential Concept

First-Principles Pseudopotentials First-Principles Pseudopotentials

Empirical Pseudopotential Method

Construction, Use and Results

EVOLUTION OF THEPSEUDOPOTENTIAL CONCEPT

• The Beginning- OPW formalism - Herring , Phys. Rev. 57, 1169 (1940)- Phillips and Kleinman, Phys. Rev. 116, 287 (1959)- Cohen and Heine, Phys. Rev. 122, 1821 (1961)

• Empirical Pseudopotentials- Cohen and Bergstresser, Phys. Rev. 141, 789 (1966)- Chelikowsky and Cohen, Phys. Rev. B 14, 556 (1976)

• Model Pseudopotentials- Abarenkov and Heine, Phil. Mag. 12, 529 (1965)

WHY PSEUDOPOTENTIALS?

The fundamental idea ofa pseudopotential is toreplace one problem withanother. (R. Martin,“Electronic Structure”,Cambridge)

All-electron (true)Wave function

Replace the strongCoulomb potential of thenucleus and tightlybound core electrons byan effective ionicpotential acting on thevalence electrons.

FiguraAll-electron (true)potential

THE GENESIS OF THE PSEUPODOTENTIAL

CONCEPT

Taking the Fourier transforms of the periodic part of the Blochfunction uk(r) and the periodic crystal potential V(r)(which become series in G) and substituting into the SchrödingerEquation we obtain:

Reciprocal Space Representation

Equation we obtain:

Gkk

GuGGVGukEGkm

0)()()()(2

22

G

rGki

kk

rki

keGuruer

)()()()(

SlowlyConvergent uk and V

HOW TO CALCULATE CRYSTAL BLOCH

FUNCTIONS FOR VALENCE ELECTRONS?

• The first problem is that the crystal potential V is a highlyvarying function of real space. Also wavefunctions are expectedto change a lot, with atomic-like behavior near the nuclei and amore plane wave-like behavior in the interstitial regions, whereV is weaker.

• It is not possible to express uk (r) as a simple superposition ofplane-waves. Too many would be required!!!plane-waves. Too many would be required!!!

> 105 plane wavesper atom!! Core Wiggles

FIRST STEP

The lower lying bands come from the atomic core levels,while the electrons in higher bands feel a weaker 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

Atomic C (Z = 6) Atomic Cu (Z = 29)

Valence states

Core states

LargeEnergydifference

OPW FORMALISM (HERRING, 1940)

• We assume the narrow lower bands are unchanged bythe atom environment (Frozen CoreApproximation): so we can approximate thesestates with the core states of the free atom or otherappropriately chosen localized functions φj.

• We are interested in describing valence electronbands (core electrons are chemically inert)

j

jknjknknrr

,,,)()(

CoreStates

Smoothfunction

bands (core electrons are chemically inert)

Localized function

Truevalencefunction

It can be shown that ψn,k is orthogonal to all φj

THE PSEUDOPOTENTIAL (PROTOTYPE PSP)

• We now insert the expression

j

jknjknknrr

,,,)()(

into the Schrödinger equation and obtain:

jj

jnjnnjnjn kEHH )(ˆˆ

jnnnjjjnn

j j

kEEkErVH )()()(ˆ0

• an equation for the smooth function χ. Wehave an effective potential:

jjj

jnjps rEkErVrV )()()(),(

Nuclear potentials

PROPERTIES OF PSEUDOPOTENTIAL

V is a much weaker potential than V: the attractive and

>0

jjj

jnjps rEkErVrV )()()(),(

Attractivelong-ranged

Repulsiveshort-ranged

Vps is a much weaker potential than V: the attractive andrepulsive parts partly compensate in the core region(Cancellation Theorem, Cohen and Heine, 1961)

• Vps depends generally on the angular momentum φj = φlmj

and is a non-local energy-dependent integraloperator.

No effect if there are no core functions with angularmomentum l. The true valence function has no nodes.

WHAT IS MORE IMPORTANT ….

We got rid of the core states/electrons. Valence electronsfeel a “pseudopotential “ weak in the core regioninstead of the nucleus + core electrons

SAME EFFECTSON THE

Valence Electron

Core Electrons Pseudo Core

ON THEVALENCE ELECTRONS

THE PSEUDOFUNCTION FOR THE

VALENCE ELECTRON

• χ being slowly varying can be approximated by afew terms of a superposition of plane waves.

Core region

Bonding region

PROPERTIES OF THE SMOOTH

PSEUDOFUNCTION

• χ is not the valence electron wavefunction but only itssmooth part (no wiggles in the core).

• The equation for χ has the same eigenvalues En(k) forthe valence electrons as the original Hamiltonian.

• χ is not uniquely defined.

Note: if we know the smooth “pseudofunction” wecan always build the corresponding true valencefunction.

• The “pseudofunction” χn,k:

j

knjknkn

2

,,,1 Norm is not unity!

• χ is not uniquely defined.

MODEL PSEUDOPOTENTIALS,EMPIRICAL PSEUDOPOTENTIALS,FIRST-PRINCIPLES PSEUDOPOTENTIALS

In practice the expression for Vps isapproximated.

Vps can be built to satisfy our needs, forVps can be built to satisfy our needs, forexample we can require it to be the smoothestand weakest possible and maintaining thesame scattering properties of the core potentialon the valence electrons (ab-initio approaches),or reproduce some measured quantities(empirical approaches)

TERMINOLOGY: LOCAL PSEUDOPOTENTIAL

The pseudopotential can be local, semilocal, non-local.

Please note ALL the pseudopotentials are sphericallysymmetric. (thus given on a radial mesh)

LOCAL s, p, d electronsall feel the same

LOCALPseudopotential(the less accurate)

EMPIRICAL andMODELPseudopotentials)(ˆ rVV LL

s, p, d electronsall feel the samepotential

TERMINOLOGY: SEMILOCAL PSEUDOPOTENTIAL

SEMILOCALPseudopotential

lm

lmllmSL YrVYV )(ˆ

It is non-local in the angularvariables, local in the radialvariable Pseudo

Wavefunctions

dSphericalharmonics

Mo – from Haman, Schluter and ChangPRL 43, 1494 (1979)

s, p, d electronsfeel differentpotentials

s

p

TERMINOLOGY: NON-LOCAL PSEUDOPOTENTIAL

NON LOCAL SEPARABLEPseudopotential

lm

lmllmNL EV ˆ Fully non local in angles θ

and φAND radius r

Functions of r, θ, φ

In position representation ),( rrVNL

Note: this PSP form is closer to the prototype PSP

THE L-DEPENDENT TERMS ARE SHORT-RANGED

The l-dependent terms of V are different only inside thecore region (radius rc). A common long-ranged localpotential Vloc (r) is subtracted

In this way the semilocal, non-local terms of thepseudopotential are zero outside rc. rc

lmlm

llmps

locps ErVV )(ˆ

Vanishesoutside

rc

r

eZV ionps

loc

at large r

MODEL PSEUDOPOTENTIALS

The model potential of Abarenkov and Heine (1965)

The core is a black box. Any core potential whichyields the correct logarihmic derivative at rc is OK.

Usually Al(E) is often a constantvalue

fitted to reproduce theatomic eigenvalues and data

Al

Al

FROM THE IONIC (OR ATOMIC)PSEUDOPOTENTIAL TO THE CRYSTAL

POTENTIAL

Valence electrons move in the crystal potential

Pseudopotentials describe the interaction of the valenceelectrons with a single ionic core.

In the crystal the valence electrons interact one each In the crystal the valence electrons interact one eachother

We have screened pseudopotentials (usually inempirical calculations) or ionic (bare)pseudopotentials (unscreened, usually in first-principles calculations).

The crystal potential is built as a superposition of atomicpseudopotentials

CRYSTAL POTENTIAL

Let’s suppose the pseudopotential is local, then it canbe written simply as Vα(r). The electron Hamiltonianis:

)()(

ion RrVrVα is the ion kindR is the lattice vector

ionHxc VVTH ˆˆˆˆ screenedionVTH ˆˆˆ or(AB-INITIO) (EPM)

,,

Rion R is the lattice vector

τα is the basis vector

Whose Fourier expansion is:

)()( GVeerV Gi

G

rGiion

G is a reciprocalvector

StructureFactor Sα(G)

FormFactor

EXAMPLE: GAAS BULK

Two atoms in the unit cell:

Ga in τGa = τ = (1/8,1/8,1/8)a, As in τAs = -(1/8, 1/8,1/8)a =-τ, we obtain:

)()()sen()()()cos()( GVGVGiGVGVGerV GaAsGaAsrGi

)()()sen()()()cos()( GVGVGiGVGVGerV GaAsGaAs

G

VS(G) VA(G)

Gkk

GuGGVGukEGkm

0)()()()(2

22

And solve:

The form factors are treated as adjustable parameters(empirical approach). Only those corresponding to fewG vectors are needed.

Cohen andBergstresser

Phys. Rev.141, 789-796(1966)

Using the empirical approach the bandstructures, reflectivity spectra and photoemissionspectra of bulk IV, III-V, and II-VIsemiconductors were calculated (Cohen andChelikowsky “Electronic Structure and OpticalProperties of Semiconductors”, Ed. Springer)

It is difficult to apply this method for systems It is difficult to apply this method for systemswith hundreds or thousands of atoms per unitcell because the fitting parameters (the formfactors for each G-shell) would become too many!

-New Atomistic Pseudopotentials (AEPM)

FORM FACTORS FOR SEMI-LOCAL PSP

If the pseudopotential is semi-local still we havethe structure factor and the form factor is morecomplicated:

)(),,( , GGVGGkV loc

,loc

drrrGkjrGkjrVPl

llll2

0)()()()(cos

with: GkGk

GkGk

cos

It depends on k and requires a double loop over the Gvectors or in real space a radial integral for each pair ofbasis functions computationally very expensive!

NON-LOCAL (SEPARABLE) PSPS DO IT BETTER

Non-local pseudopotentials make thecomputation of the Hamiltonian matrix elementsless expensive. Instead of

drrrGkjrGkjrVP llll2)()()()(cos

drrrGkjrGkjrVP llll

0)()()()(cos

We have:

drrrGkjrdrrrGkjrP lllll2

0 0

2 )()()()()(cos

Factorized into a product of integrals for each basisfunction separately, in plane-wave calculations onlysingle loops over G are involved.

HOW TO GENERATE AN ATOMIC

PSEUDOPOTENTIAL

Pseudopotentials for first-principlescalculations

Unscreened (bare) pseudopotential(ionic psp)

Extracted from an all-electron

Pseudopotentials for semiempiricalcalculations

Extracted from an all-electroncalculation on the free atom.

Extracted fitting experimental data of(one or more) compounds containingthe atom.

It is assumed to be screened.

PSEUDOPOTENTIALS FOR FIRST-PRINCIPLES CALCULATIONS

Hamann, Schlüter, and Chang, PRL 43, 1494 (1979) – Norm-Conserving Pseudopotentials

Kleinman and Bylander, PRL 48, 1425 (1982) – Separable

Main steps in development

Kleinman and Bylander, PRL 48, 1425 (1982) – SeparablePseudopotentials

Louie, Froyen, and Cohen, PRB 26, 1738 (1982) – Non linearcore correction

Vanderbilt, PRB (RC) 41, 7892 (1990) – UltrasoftPseudopotentials

Blöchl, PRB (RC) 41, 5414 (1990) – Generalized SeparablePseudopotentials

Blöchl, PRB 50, 17953 (1994) - PAW

REQUIREMENS FOR CONSTRUCTING A GOOD

NORM-CONSERVING PSEUDOPOTENTIAL

Choose an atomic reference configuration

Use an atomic code to calculate the all-electron valencewavefunctions (AE).

Hamann et al. PRL 43, 1494 (1979)

Example: Si 3s2 3p2

Impose that thepseudo-wavefunction(PS) agrees with the(AE) wave-functionbeyond a chosen cutoffradius rc (l-dependent)

Also the l-channel pseudo-potential (PS) has toagree with the AE potential for r > rc.

0)()(2

)1(

2

1 .

22

2

rrV

mr

ll

dr

d

mnlmAEnl

scrAE

0)()(2

)1(

2

1 .

22

2

rrV

mr

ll

dr

d

mnlmPSl

scrPS

for the same eigenvalue εnl = εl

22 22

mrdrm

NORM-CONSERVATION REQUIREMENT

The integrals from 0 to rc of the real and pseudocharge densities agree for each valence state.

c cr

nlmAE

rlPS drrdrr

0

2

0

222 rc

The charge containedin this region is thesame for AE and PSwave-functions

SCATTERING PROPERTIES

0)()(2

)1(

2

1 .

22

2

rrV

mr

ll

dr

d

mnlmAEnl

scrAE

0)()(2

)1(

2

1 .

22

2

rrV

mr

ll

dr

d

mnlmPSl

scrPS

. By construction, we know that at energy ε = εnl, thesolution ψPS(r ) coincides with the ψAE(r ) for r > rc.But what about other energies? The transferabilityof the pseudopotential depends on the fact that ψPS(r )reproduces ψAE(r ) over a certain range of energiesabout εnl. We are interested in the energy range ofvalence bands in solid.

22 mrdrm

The logarithmic derivatives of the real and pseudowave function and their first energy derivativesagree for r > rc.

Logarithmicderivative

),(ln),(

),(),( r

dr

dr

r

rrrD l

l

ll

The first energyderivative of thelogarithmicderivatives of the all-electron and pseudowave-functions agreesat rc, and therefore forall r > rc.

),( rDd

dl

SCATTERING PROPERTIES

The fundamental advance of Hamann, Schlüterand Chang, 1979, is to have shown that:

If norm conservation is imposed, then pseudoDl(ε,r) matches all-electron Dl(ε,r) to secondorder in (ε –εl)

This means that the norm-conservingpseudopotential has the same scattering phaseshifts as the all-electron atom to linear order inenergy around the chosen energy εl.

l

These properties however leaves plenty offreedom in the form of the pseudopotential and inits construction.

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:

VAE(r),ψAE,nl(r)VAE(r),ψAE,nl(r)

Step 3: construct the pseudo wavefunction that satifiesrules (nodeless, matching to AE wavefunction, norm-conservation, etc)

Step 4: Invert the Schrödinger equation to get VPS,l(r)which is a screened potential

Step 5: Unscreening the potential to obtain the bareVPS,l,ion

PSEUDOPOTENTIAL GENERATION

VERY IMPORTANT STEP: THE PSEUDO

TEST

(1) Tests on excited configurations”

Example: Reference configuration for Si [core]s2p2

we compare AE and PS results for other configurations:

[core]sp3

[core] s2p1 ….. and many others[core] s2p1 ….. and many others

We compare

- Total energies

- Energy Eigenvalues

- Logarithmic derivatives

Then calculate small (well-known) systems and check…..

OPTIMIZATION OF A PSEUDOPOTENTIAL

Pseudopotentials are optimized with regard to:

1. Accuracy and trasferability – leads tochoose small cutoff radius rc and harderpseudopotentials

2. Smoothness – leads to choose a larger cutoffradius r and softer pseudopotentialsradius rc and softer pseudopotentials

DifferentAuthorshaveProposeddifferentRecipes

CONSTRUCTION RECIPES FOR SMOOTH

AND ACCURATE NORM-CONSERVING PSPS

Bachelet, Hamann, Schlüter, PRB 26, 4199 (1982)

Vanderbilt, PRB 32, 8412 (1985)

Kerker, J. Phys. C 13,L189 (1980)

Troullier and Martins, PRB 43, 1993 (1991) Troullier and Martins, PRB 43, 1993 (1991)

Rappe, Rabe, Kaxiras, and Joannopoulos, PRB 41, 1227 (1990)

……….. Cu

Troullier-Martins Kerker HSC Vanderbilt

From Troullier and Martins, PRB 43, 1993 (1991)

UNSCREENING THE PSEUDOPOTENTIAL

The inversion of the Schrödinger equation gives thescreened pseudopotential. We need to unscreen it.

To unscreen:

Pseudo valence charge densityl

lPSlPS rfrn

2

)()(

)],([],[)( rnVrnVrV PS

)()()( .,, rVrVrV PSHxc

PSlscr

PSlion

)],([],[)( rnVrnVrV PSxcPSHPS

Hxc

However, Vxc is a non-linear functional of n so it is ambiguousto separate the effects of core and valence charge if there is asignificant overlap of the two densities.

This leads to errors and reduced transferability.

NON-LINEARCORE CORRECTIONS

)],([ )( rnV corePStotalxc

IMPROVEMENTS ON THE METHOD: SEPARABLE

PSEUDOPOTENTIALS

Separable Pseudopotentials (Kleinman-Bylander

We separate the semi-local pseudopotential in a long-range local part and one short-range l-dependent part

lm

PSl

PSlocal

PSion mlrVmlrVrV ,)(,)()(

Separable Pseudopotentials (Kleinman-BylanderTrasform) 1982

For each l-channel

mlPSlm

PSl

PSlm

PSlm

PSl

PSl

PSlmPS

localNLV

VVrVV

,

)(ˆ

VNL acts on the reference state ψlm as thesemilocal pseudopotentialΔVl

Possible presence of bound ghost states at lowerenergies requires some care.

First-row elements have valence states with angularmomentum l without l core state. Already nodeless!

IMPROVEMENTS ON THE METHOD: ULTRA-SOFT PSEUDOPOTENTIALS

AE

PS O: 1s2 2s2 2p4

core valenceno p states in core

O 2p wavefunction

Highly localized states in firstrow and transition-metal atoms

Difficult convergence in aplane wave basis

NORM-CONSERVATION makes PS AE

New “core”Radius for UltraSoft

D. Vanderbilt, Phys. Rev. B 41, 7892 (1990)

Release the norm conservation criteria to obtainsmoother pseudo wave functions. This is done bysplitting the pseudo wave functions into two parts:

)(rUS

1. The ultrasoft valence wave function that do notfulfill the norm conservation criteria:

)(rUSi

)()()()()( ,, rrrrrQ USj

USi

AEj

AEiij

The Ultra-Soft Pseudopotential takes the NL form

2. Plus a core augmentation charge (charge deficitin the core region):

ij

jiijlocalUS DrVV )(

ilocii VT

An overlap operator S is introduced:

ij

jiijQS 1

i lkiklilki rQrrn )()()(

2

Main Properties:

0 in case of norm-conservation

Main Properties:

1 Changed orthonormalization:

2 Generalized eigenvalue problem to be solved

2 The NL Pseudopotential is updated during theiterative procedure

0)ˆˆ( nknk SH

jiji S ,ˆ

)()(0 rQrVrdDD mnHxcmnItmn

A PSEUDOPOTENTIAL FOR ALL SEASONS

Many different PSPs and Pseudo GeneratorCodes provided in packages:

http://www.quantum-espresso.org/pseudo.php

Plane-waves pseudopotential codes

http://www.quantum-espresso.org/pseudo.php

On-The-Fly Pseudopotential Generation in CASTEP -a 164 kB pdf tutorial.

http://cms.mpi.univie.ac.at/vasp/vasp/

http://www.abinit.org/downloads/atomic-data-files

PSEUDO-ELEMENT TABLES

http://www.quantum-espresso.org/pseudo.php

Name: OxygenSymbol: OAtomic number: 8Atomic configuration: [He] 2s2 2p4Atomic mass: 15.9994 (3)

Available pseudopotentials:O.pz-mt.UPF (details)Perdew-Zunger (LDA) exch-corr Martins-TroullierO.blyp-van_ak.UPF (details)Becke-Lee-Yang-Parr (BLYP) exch-corrVanderbilt ultrasoft author: ak

And many other…….Vanderbilt ultrasoft author: akO.pbe-van_gipaw.UPF (details)Perdew-Burke-Ernzerhof (PBE) exch-corrVanderbilt ultrasoft author: gipawO.blyp-mt.UPF (details)Becke-Lee-Yang-Parr (BLYP) exch-corr Martins-TroullierO.pz-kjpaw.UPF (details)Perdew-Zunger (LDA) exch-corr Projector AugmentedWaves (Kresse-Joubert paper)O.pbe-van_ak.UPF (details)Perdew-Burke-Ernzerhof (PBE) exch-corr Vanderbiltultrasoft author: akO.pbe-rrkjus.UPF (details)Perdew-Burke-Ernzerhof (PBE) exch-corr Rabe RappeKaxiras Joannopoulos (ultrasoft)

And many other…….

PSEUDOPOTENTIALS FOR SEMIEMPIRICAL

CALCULATIONS

Chelikowsky and Cohen, PRB 14, 556 (1976)

Atomistic Empirical Pseudopotential

Main steps in development

Atomistic Empirical Pseudopotential

Mader and Zunger, PRB 50, 17393 (1994) –

Wang and Zunger, PRB 51, 17398 (1995) – LDA derivedsemiempirical pseudopotentials

LDA-DERIVED EMPRICAL PSEUDOPOTENTIALS

Problems with first-principles methods

1. Difficult to apply to systems with thousands-

million atoms (nanostructured materials)

2. Problem with excited states: the band gap is2. 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 bandenergies, optical spectra, etc, and at the same time, LDA-quality wavefunctions and related quantities.

SEPM FROM LDA CALCULATIONS

- Spherical Average of the Screened Local Potential

1. LDA SLDA

)()()( ,, GvGSGV

- Calculate LDA for structure )( GVLDA

Form factorsLocal Potential

Vloc+VHxc

-The points vs are fitted by the continousfunction :

)()()( ,,

GvGSGV SLDALDA

- Structural average

)()( , rvrv SLDASLDA

)( GvSLDA

G

N

- Only the Coefficients CSLDA are adjusted to fitthe experimental or quasiparticle calculatedexcitation properties

Unlike standard EPM, which produces only discreteform factors and is hence suitable only for aparticular crystal structure and lattice constant, the

2. SLDA SEPM

particular crystal structure and lattice constant, thenew SEPM or AEPM can be used for differentstructures and volumes with good transferability.

The form factors for each particular structureare extracted from a “Universal” continousfunction of q

)6,5,4(

1

)( 2

)(N

n

bqcn

nneaqv

1)( 2

3

2

12

0

qaea

aqaqv

or

PROPERTIES OF AEPMS

• Good band structure

• Accurate effective masses

• Accurate band gaps

• Good elastic properties(Bulk modulus, deformation potentials)

• Transferable

• Low Energy cut-off (~5 Ryd)

• Simple analytic form (few parameters)

AVAILABLE ATOMISTIC EMPIRICAL

PSEUDOPOTENTIALS

Only certainOnly certaincombinationsare available(IV, III-V, II-

VI)

AN EXAMPLE: THE INAS/GASB SYSTEM

Broken Gap System

Semiconductor because of the e1

and h1 confinement1

Possibility of tuning the bandgap between 0 ↔ 400 meV

Type II: short periods SLs toincrease the radiativericombination efficiency

InAsGaAs GaAs

Ga

As

In In

As

Ga

CA

InAsGaSb GaSb

The single (001)interface has C2v

symmetry

a

GaAs

InIn

Sb

GaNCA

IN-PLANE POLARIZATION ANISOTROPY

I p

I p

e h

e h

( [ ])

( [ ])

1 1 0

1 1 01

InAs/AlSbsuperlattice

X=[110]

Y=[-110]

Fuchs et al. in“Antimonide-RelatedStrained-LayerHeterostructures”

Wavenumber (cm-1)

superlattice

ATOMISTIC EMPIRICAL PSEUDOPOTENTIAL

ELECTRONIC STRUCTURE

)()(2

2

rrRrvm n

n

• Solve the Schrödinger equation FULLY ATOMISTICALLY,

each atom strain

plane waveexpansionof ψ(r)

each atomindividuallydescribed

strainminimizingatomic positions

• no LDA errors

• not self-consistentFoldedspectrummethod

The spectrum at the left is the originalspectrum of H. The spectrum at the right isthe folded spectrum of (H-E_ref)^2

qn

rqivqverv 1|)(|)(

)(

)( 12

aq

aqv

• v(q) continous function of q

)()( 4 Travn

FORM FACTORS

Parameters fit to reproduce:

1)( 2

3

2

10

qaea

aqaqv

1. Gaps Eg and effective masses m*

2. Hydrostatic ag andbiaxial b deformation potentials

3. Band offsets and spin-orbit splitting so

4. LDA-predicted single band edgedeformation potentials av, ac

for ALL 4binaries

FIT: RESULTS

HANDLING OF BIAXIAL STRAIN

• Explicit strain dependence in v(q,)

)()( 4 Trav n with

EPM

IF specific offsets IF specific bonds

LAPW

CRYSTAL POTENTIAL FROM A

SUPERPOSITION OF ATOMIC POTENTIALS

vn

vn

nvn nIn In InAs Sb InAs InSb( ) ( ) ( )4

4

4

Interfaces orDisorder

FIRST HEAVY-HOLE CHARGE DENSITY

The method predicts the positive band bowings parametersof the ternary alloys in agreement with experiment!

RESULTS FOR THE (INAS)6/(GASB)M AND

(INAS)8/(GASB)N SUPERLATTICES

with increasing n

Eg

CORRECT TREND!(InAs)8/(GaSb)n

Number of GaSb monolayers n

OVERLAP OF THE ELECTRON STATES

GaSbInAs

LONG-PERIOD INAS/GASB SLS

(InAs)46(GaSb)14

MORE TO BE SEEN FRIDAY MORNING ……

THANK YOU FOR YOUR ATTENTION

• If we choose a plane wave for χn,k we call the ψnkan OPW (orthogonalized plane wave)

jjj

rqOPWq qe

Vr

11)(

• OPWs were used as basis functions for expansion:

OPW

j

jmlknjmlkn

ml

knrr ,,,,,,

,

,)()(

• OPWs were used as basis functions for expansion:

i

OPWiqikn

r ,,)(

)(,, rjmlj

Dependence on l,m