Perturbation theory, linear response, and finite ... - VASP · VASP: Dielectric response...

VASP:DielectricresponsePerturbationtheory,linearresponse,

andfiniteelectricfields

UniversityofVienna,FacultyofPhysicsandCenterforComputationalMaterialsScience,

Vienna,Austria

Outline

Dielectricresponse

Frequencydependentdielectricproperties

Thestaticdielectricresponse

ResponsetoelectricfieldsfromDFPT

Themacroscopicpolarization

SCFresponsetofiniteelectricfields

Ioniccontributionstodielectricproperties

Experiment: Staticandfrequencydependentdielectricfunctions:measurementofabsorption,reflectanceandenergylossspectra.(Opticalpropertiesofsemiconductorsandmetals.)

Thelong-wavelengthlimitofthefrequencydependentmicroscopicpolarizabilityanddielectricmatricesdeterminetheopticalpropertiesintheregimeaccessibletoopticalandelectronicprobes.

Theory: Thefrequencydependentpolarizabilitymatrixisneededinmanypost-DFTschemes,e.g.:

GW)frequencydependentmicroscopicdielectricresponseneededtocomputeW.)frequencydependentmacroscopicdielectrictensorrequiredfortheanalyticalintegrationoftheCoulombsingularityintheself-energy.

Exact-exchangeoptimized-effective-potentialmethod(EXX-OEP). Bethe-Salpeter-Equation(BSE)

)dielectricscreeningoftheinteractionpotentialneededtoproperlyincludeexcitonic effects.

Frequencydependent

Frequencydependentmicroscopicdielectricmatrix)IntheRPA,andincludingchangesintheDFTxc-potential.

Frequencydependentmacroscopicdielectrictensor)Imaginaryandrealpartofthedielectricfunction.)In- orexcludinglocalfieldeffects.)IntheRPA,andincludingchangesintheDFTxc-potential:

Static

Staticdielectrictensor,Borneffectivecharges,andPiezo-electrictensor,in- orexcludinglocalfieldeffects.)Fromdensity-functional-perturbation-theory(DFPT).LocalfieldeffectsinRPAandDFTxc-potential.)Fromtheself-consistentresponsetoafiniteelectricfield(PEAD).LocalfieldeffectsfromchangesinaHF/DFThybridxc-potential,aswell.

Macroscopiccontinuumconsiderations Themacroscopicdielectrictensorcouplestheelectricfieldinamaterialto

anappliedexternalelectricfield:

E = 1Eext

where isa33tensor

Foralongitudinalfield,i.e.,afieldcausedbystationaryexternalcharges,thiscanbereformulatedas(inmomentumspace,inthelong-wavelengthlimit):

vtot

= 1vext

vtot

= vext

+ vind

with

Theinducedpotentialisgeneratedbytheinducedchangeinthechargedensity#$%.Inthelinearresponseregime(weakexternalfields):

ind

= vext

ind

= Pvtot

where isthereducible polarizability

whereP istheirreducible polarizability

Itmaybestraightforwardlyshownthat:

1 = 1 + = 1 P = P + P (aDysoneq.)

where istheCoulombkernel.Inmomentumspace: = 4e2/q2

MacroscopicandmicroscopicquantitiesThemacroscopicdielectricfunctioncanbeformallywrittenas

E(r,!) =

Zdr01

mac

(r r0,!)Eext

(r0,!)

orinmomentumspaceE(q,!) = 1

mac

(q,!)Eext

(q,!)

Themicroscopicdielectricfunctionentersas

e(r,!) =

Zdr01(r, r0,!)E

ext

(r0,!)

andinmomentumspace

e(q+G,!) =X

G0

1G,G0(q,!)Eext(q+G0,!)

Themicroscopicdielectricfunctionisaccessiblethroughab-initiocalculations.Macroscopicandmicroscopicquantitiesarelinkedthrough:

E(R,!) =1

Z

(R)e(r,!)dr

MacroscopicandmicroscopicquantitiesAssumingtheexternalfieldvariesonalengthscalemuchlargerthattheatomicdistances,onemayshowthat

E(q,!) = 10,0(q,!)Eext(q,!)

and

1mac(q,!) = 10,0(q,!)

mac(q,!) =10,0(q,!)

1

Formaterialsthatarehomogeneousonthemicroscopicscale,theoff-diagonalelementsof,*

+, (, ),(i.e.,for )arezero,and

mac(q,!) = 0,0(q,!)

Thiscalledtheneglectoflocalfieldeffects

Thelongitudinalmicroscopicdielectricfunction

Themicroscopic(symmetric)dielectricfunctionthatlinksthelongitudinalcomponentofanexternalfield(i.e.,thepartpolarizedalongthepropagationwavevectorq)tothelongitudinalcomponentofthetotalelectricfield,isgivenby

1G,G0(q,!) := G,G0 +4e2

|q+G||q+G0|@

ind

(q+G,!)

@vext

(q+G0,!)

G,G0(q,!) := G,G0 4e2

|q+G||q+G0|@

ind

(q+G,!)

@vtot

(q+G0,!)

andwith G,G0(q,!) :=@

ind

(q+G,!)

@vext

(q+G0,!)PG,G0(q,!) :=

@ind

(q+G,!)

@vtot

(q+G0,!)

sG,G0(q) :=4e2

|q+G||q+G0|and

oneobtainstheDysonequationlinkingP and

G,G0(q,!) = PG,G0(q,!) +X

G1,G2

PG,G1(q,!)sG1,G2(q)G2,G0(q,!)

Approximations

Problem: WeknowneitherP and

Problem: ThequantitywecaneasilyaccessinKohn-ShamDFTisthe:irreduciblepolarizabilityintheindependentparticlepicture3 (or45)

0G,G0(q,!) :=@ind(q+G,!)

@ve(q+G0,!)

AdlerandWiserderivedexpressionsfor3 which,intermsofBlochfunctions,canbewrittenas(inreciprocalspace):

0G,G0(q,!) =1

X

nn0k

2wk(fn0k+q fn0k)

h n0k+q|ei(q+G)r| nkih nk|ei(q+G

0)r0 | n0k+qin0k+q nk ! i

Approximations(cont.)

FortheKohn-Shamsystem,thefollowingrelationscanbeshowntohold

= 0 + 0( + fxc

)

P = 0 + 0fxc

P

= P + P

where istheCoulombkerneland89 istheDFTxc-kernel: fxc = @vxc/@ |=0

1 = 1 + = 1 P

Random-Phase-Approximation(RPA): P = 0

G,G0(q,!) := G,G0 4e2

|q+G||q+G0|0G,G0(q,!)

IncludingchangesintheDFTxc-potential: P = 0 + 0fxc

P

Calculationofopticalproperties

Thelong-wavelengthlimit( )ofthedielectricmatrixdeterminestheopticalpropertiesintheregimeaccessibletoopticalprobes.

Themacroscopicdielectrictensor

Frequencydependent(neglectinglocalfieldeffects)

LOPTICS=.TRUE.q 1(!) q lim

q!00,0(q,!)

Theimaginarypartof

ExamplesGajdo etal.,Phys.Rev.B73,045112(2006).

ThefrequencydependentdielectricfunctioniswrittentotheOUTCAR file.Searchfor

frequency dependent IMAGINARY DIELECTRIC FUNCTION (independent particle, no local field effects)

frequency dependent REAL DIELECTRIC FUNCTION (independent particle, no local field effects)

and

Frequencydependent(includinglocalfieldeffects)

FortheKohn-Shamsystem,thefollowingrelationscanbeshowntohold

= 0 + 0( + fxc

)

P = 0 + 0fxc

P

= P + P

where istheCoulombkerneland89 istheDFTxc-kernel: fxc = @vxc/@ |=0

1 = 1 + = 1 P

Random-Phase-Approximation(RPA): P = 0

G,G0(q,!) := G,G0 4e2

|q+G||q+G0|0G,G0(q,!)

IncludingchangesintheDFTxc-potential: P = 0 + 0fxc

P

IrreduciblepolarizabilityintheIPpicture:3

ThequantitywecaneasilyaccessinKohn-ShamDFTisthe:irreduciblepolarizabilityintheindependentparticlepicture3 (or45)

0G,G0(q,!) :=@ind(q+G,!)

@ve(q+G0,!)

AdlerandWiserderivedexpressionsfor3 which,intermsofBlockfunctions,canbewrittenas

0G,G0(q,!) =1

X

nn0k

2wk(fn0k+q fn0k)



AndintermsofBlockfunctions3 canbewrittenas

0G,G0(q,!) =1

X

nn0k

2wk(fn0k+q fn0k)



TheIP-polarizability:3

W = + 0 + 00 + 000 + ... = (1 0)1| {z }1

Oncewehave3 thescreenedCoulombinteraction(intheRPA)iscomputedas:

1.ThebareCoulombinteractionbetweentwoparticles

2.Theelectronicenvironmentreactstothefieldgeneratedbyaparticle:inducedchangeinthedensity3,thatgivesrisetoachangeintheHartreepotential:3.

3.Theelectronsreacttotheinducedchangeinthepotential:additionalchangeinthedensity,33,andcorrespondingchangeintheHartree potential:33.

andsoon,andsoon

geometricalseries

Expensive:computingtheIP-polarizabilityscalesasN4

TheOUTPUT

Informationconcerningthedielectricfunctionintheindependent-particlepictureiswrittenintheOUTCAR file,aftertheline

HEAD OF MICROSCOPIC DIELECTRIC TENSOR (INDEPENDENT PARTICLE)

Perdefault,forALGO=CHI,localfieldeffectsareincludedattheleveloftheRPA(LRPA=.TRUE.),i.e.,limitedtoHartree contributionsonly.SeetheinformationintheOUTCAR file,after

INVERSE MACROSCOPIC DIELECTRIC TENSOR (including local field effects in RPA (Hartree))

ToincludelocalfieldeffectsbeyondtheRPA,i.e.,contributionsfromDFTexchangeandcorrelation,onhastospecifyLRPA=.FALSE. intheINCAR file.InthiscaselookattheoutputintheOUTCAR file,after

INVERSE MACROSCOPIC DIELECTRIC TENSOR (test charge-test charge, local field effects in DFT)

Virtualorbitals/emptystatesProblem:theiterativematrixdiagonalizationtechniquesconvergerapidlyforthelowesteigenstatesoftheHamiltonian.Highlying(virtual/emptystates)tendtoconvergemuchslower.

Doagroundstate calculation(i.e.,DFTorhybridfunctional).BydefaultVASPwillincludeonlyaverylimitednumberofemptystates(lookforNBANDS andNELECT intheOUTCAR file).

Toobtainvirtualorbitals(emptystates)ofsufficientquality,wediagonalizethegroundstate Hamiltonmatrix(intheplanewavebasis: )exactly.FromtheFFG eigenstatesofthisHamiltonian,wethenkeeptheNBANDSlowest.

YourINCAR fileshouldlooksomethinglike:

..ALGO = ExactNBANDS = .. #set to include a larger number of empty states..

Typicaljobs:3steps1. Standardgroundstate calculation

2. RestartfromtheWAVECAR fileofstep1,andtoobtainacertainnumberofvirtualorbitalsspecify:

inyourINCAR file.

3. Computefrequencydependentdielectricproperties:restartfromtheWAVECAR ofstep2,withthefollowinginyourINCAR file:

..ALGO = ExactNBANDS = .. #set to include a larger number of empty states..

ALGO = CHI or LOPTICS = .TRUE.

N.B.:InthecaseofLOPTICS=.TRUE. step2and3canbedoneinthesamerun(simplyaddLOPTICS=.TRUE. toINCAR ofstep2).

TheGWpotentials:*_GWPOTCARfiles

ThestaticdielectricresponseThefollowingquantities:

Theion-clampedstaticmacroscopicdielectrictensor< = 0(orsimply

ResponsetoelectricfieldfromDFPTLEPSILON=.TRUE.Insteadofusingasumoverstates(perturbationtheory)tocompute|?,onecansolvethelinearSternheimer equation:

[H(k) nkS(k)] |rkunki = @ [H(k) nkS(k)]

@k|unki

for|?.

Thelinearresponseoftheorbitalstoanexternallyappliedelectricfield|?,canbefoundsolving

[H(k) nkS(k)] |nki = HSCF(k)|unki q |rkunki

where5QF() isthemicroscopiccellperiodicchangeintheHamiltonian,duetochangesintheorbitals,i.e.,localfieldeffects(!):thesemaybeincludedattheRPAlevelonly(LRPA=.TRUE.)ormayincludechangesintheDFTxc-potentialaswell

ResponsetoelectricfieldsfromDFPT(cont.)

Thestaticmacroscopicdielectricmatrixisthengivenby

q 1 q = 18e2

X

vk

2wkhq rkunk|nki

wherethesumoverv runsoveroccupiedstatesonly.

TheBorneffectivechargesandpiezo-electrictensormaybeconvenientlycomputedfromthechangeintheHellmann-Feynmanforcesandthemechanicalstresstensor,duetoachangeinthewavefunctionsinafinitedifferencemanner:

|u(1)nki = |unki+s|nki

TheOUTPUT Thedielectrictensorintheindependent-particlepictureisfoundintheOUTCAR

file,afterthelineHEAD OF MICROSCOPIC STATIC DIELECTRIC TENSOR (INDEPENDENT PARTICLE, excluding Hartree and local field effects)

ItscounterpartincludinglocalfieldeffectsintheRPA(LRPA=.TRUE.)after:MACROSCOPIC STATIC DIELECTRIC TENSOR (including local field effects in RPA (Hartree))

MACROSCOPIC STATIC DIELECTRIC TENSOR (including local field effects in DFT)

andincludinglocalfieldeffectsinDFT(LRPA=.FALSE.)after:

ThepiezoelectrictensorsarewrittentotheOUTCAR immediatelyfollowing:PIEZOELECTRIC TENSOR for field in x, y, z (e Angst)

c.q.,PIEZOELECTRIC TENSOR for field in x, y, z (C/m^2)

TheBorneffectivechargetensorsareprintedafter:BORN EFFECTIVE CHARGES (in e, cummulative output)

(butonlyforLRPA=.FALSE.).

Examples

Gajdo etal.,Phys.Rev.B73,045112(2006).

Moderntheoryofpolarization(Resta,Vanderbilt,etal.)

Thechangeinthepolarizationinducedbyanadiabaticchangeinthecrystallinepotentialisgivenby

P =

Z 2

1

@P

@d = P(2)P(1)

whereP() = fie

(2)3

Z

k

dkhu()nk |rk|u()nk i

Toillustratethis,consideraWannier function

k(r) = uk(r)eikr|wi = V

(2)3

Z

k

dk| ki

withwell-defineddipolemoment

ehri = eZ

drhw|r|wi

=eV 2

(2)6

Z

k

dk

Z

k0

dk0X

R

Z

Vdruk(r)(R+ r)uk0(r)e

i(kk0)(R+r)

=eV 2

(2)6

Z

k

dk

Z

k0

dk0X

R

Z

Vdruk(r)uk0(r)(irk0)ei(kk

0)(R+r)

= i eV(2)3

Z

k

dkhuk|rk|uki

whereweusedintegrationbypartstoletR workonR insteadofontheexponent,andthefollowingrelation

X

R

ei(kk0)R =

(2)3

V(k k0)

Thefinalexpression,proposedbyKing-SmithandVanderbilt,toevaluatethepolarizationonadiscretemeshofk-points,reads:

Bi P() =f |e|V

AiNk?

X

Nk?

={lnJ1Y

j=0

det|hu()nkj |u()mkj+1

i|}

where

kj = k? + jBiJ

, j = 1, ..., J and u()nk?+Bi(r) = eiBiru()nk?(r)

kj = k? + jBiJ

, j = 1, ..., J and u()nk?+Bi(r) = eiBiru()nk?(r)

Self-consistentresponsetofiniteelectricfields(PEAD)

Addtheinteractionwithasmallbutfiniteelectricfield totheexpressionforthetotalenergy

E[{ (E)}, E ] = E0[{ (E)}] E P[{ (E)}]

where () isthemacroscopicpolarizationasdefinedinthemoderntheoryofpolarization

P[{ (E)}] = 2ie(2)3

X

n

Z

BZdkhu(E)nk |rk|u

(E)nk i

AddingacorrespondingtermtoHamiltonian

H| (E)nk i = H0| (E)nk i E

P[{ (E)}]h (E)nk |

allowsonetosolvefor () bymeansofadirectoptimizationmethod(iterateintil self-consistencyisachieved).

R.W.Nunes andX.Gonze,Phys.Rev.B63,155107(2001).R.D.King-SmithandD.Vanderbilt,Phys.Rev.B47,1651(1993).

PEAD(cont.)

Lc

ZEg

VB

CB

L

Souzaetal.,Phys.Rev.Lett.89,117602(2002).

e|Ec Ai| Eg/NiNiAi < LZ

PEAD(cont.)Oncetheself-consistentsolution () hasbeenobtained: thestaticmacroscopicdielectricmatrixisgivenby

(1)ij = ij + 4(P[{ (E)}]P[{ (0)}])i

Ej andtheBorneffectivechargesandion-clampedpiezo-electrictensormay

againbeconvenientlycomputedfromthechangeintheHellman-Feynmanforcesandthemechanicalstresstensor.

ThePEADmethodisabletoincludelocalfieldeffectsinanaturalmanner(the self-consistency).

INCAR-tags

LCALPOL =.TRUE. Computemacroscopicpolarization.LCALCEPS=.TRUE. Computestaticmacroscopicdielectric-,Borneffectivecharge-,

andion-clampedpiezo-electrictensors,includinglocalfieldeffects.EFIELD_PEAD = E_x E_y E_z ElectricfieldusedbythePEADroutines.

(DefaultifLCALCEPS=.TRUE.:EFIELD_PEAD=0.010.010.01[eV/].)LRPA=.FALSE. Skipthecalculationswithoutlocalfieldeffects(Default).LSKIP_NSCF=.TRUE. idem.LSKIP_SCF=.TRUE. Skipthecalculationswithlocalfieldeffects.

Example:ion-clamped< usingtheHSEhybrid

J.Paier,M.Marsman,andG.Kresse,Phys.Rev.B78,121201(R)(2008).

PEAD:Hamiltonianterms

TheadditionaltermsintheHamiltonian,arisingfromthe V termintheenthalpyareofthefrom

P

j ={ln det|S(kj ,kj+1)|}unkj

=

i2

NX

m=1

|unkj+1iS1mn(kj ,kj+1) |unkj1iS1mn(kj ,kj1)

Snm(kj ,kj+1) = hunkj |umkj+1iwhere

Byanalogywehave

@|unkj i@k

12k

NX

m=1

|unkj+1iS1mn(kj ,kj+1) |unkj1iS1mn(kj ,kj1)

i.e.,afinitedifferenceexpressionfor|?.

TheOUTPUT ThedielectrictensorincludinglocalfieldeffectsiswrittentotheOUTCAR

file,afterthelineMACROSCOPIC STATIC DIELECTRIC TENSOR (including local field effects)

ThepiezoelectrictensorsarewrittentotheOUTCAR immediatelyfollowing:PIEZOELECTRIC TENSOR (including local field effects) (e Angst)

c.q.,PIEZOELECTRIC TENSOR (including local field effects) (C/m^2)

TheBorneffectivechargetensorsarefoundafter:BORN EFFECTIVE CHARGES (including local field effects)

ForLSKIP_NSCF=.FALSE.onewilladditionallyfinf thecounterpartsoftheabove:

... (excluding local field effects)

Ioniccontributions

Fromfinitedifferenceexpressionsw.r.t.theionicpositions(IBRION=5 or6)orfromperturbationtheory(IBRION=7 or8)weobtain

ss0

ij = @F si@us

0j

sil = @l@usi

theforce-constantmatricesandinternalstraintensors,respectively.

TogetherwiththeBorneffectivechargetensors,thesequantitiesallowforthecomputationof theioniccontributiontothedielectrictensor

ionij =4e2

X

ss0

X

kl

Zsik (ss0)1kl Z

s0lj

andtothepiezo-electrictensor

eionil = eX

ss0

X

jk

Zsij (ss0)1jk

s0

kl

TheEnd

Thankyou!

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