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

where𝜖 isa3⨉3tensor

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

vtot

= ✏�1vext

vtot

= vext

+ vind

with

• Theinducedpotentialisgeneratedbytheinducedchangeinthechargedensity𝜌#$%.Inthelinearresponseregime(weakexternalfields):

⇢ind

= �vext

⇢ind

= Pvtot

where𝜒 isthereducible polarizability

whereP istheirreducible polarizability

• Itmaybestraightforwardlyshownthat:

✏�1 = 1 + ⌫� ✏ = 1� ⌫P � = P + P⌫� (aDysoneq.)

where𝜈 istheCoulombkernel.Inmomentumspace: ⌫ = 4⇡e2/q2

MacroscopicandmicroscopicquantitiesThemacroscopicdielectricfunctioncanbeformallywrittenas

E(r,!) =

Zdr0✏�1

mac

(r� r0,!)Eext

(r0,!)

orinmomentumspaceE(q,!) = ✏�1

mac

(q,!)Eext

(q,!)

Themicroscopicdielectricfunctionentersas

e(r,!) =

Zdr0✏�1(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-diagonalelementsof𝜖𝐆,𝐆*

+, (𝐪, 𝜔),(i.e.,for𝐆 ≠ 𝐆′)arezero,and

✏mac(q,!) = ✏0,0(q,!)

Thiscalledthe“neglectoflocalfieldeffects”

Thelongitudinalmicroscopicdielectricfunction

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

✏�1G,G0(q,!) := �G,G0 +4⇡e2

|q+G||q+G0|@⇢

ind

(q+G,!)

@vext

(q+G0,!)

✏G,G0(q,!) := �G,G0 �4⇡e2

|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) :=4⇡e2

|q+G||q+G0|and

oneobtainstheDysonequationlinkingP and𝜒

�G,G0(q,!) = PG,G0(q,!) +X

G1,G2

PG,G1(q,!)⌫sG1,G2(q)�G2,G0(q,!)

Approximations

Problem: WeknowneitherP and𝜒

Problem: ThequantitywecaneasilyaccessinKohn-ShamDFTisthe:“irreduciblepolarizabilityintheindependentparticlepicture”𝜒3 (or𝜒45)

�0G,G0(q,!) :=@⇢ind(q+G,!)

@ve↵(q+G0,!)

AdlerandWiserderivedexpressionsfor𝜒3 which,intermsofBlochfunctions,canbewrittenas(inreciprocalspace):

�0G,G0(q,!) =1

⌦

X

nn0k

2wk(fn0k+q � fn0k)

⇥ h n0k+q|ei(q+G)r| nkih nk|e�i(q+G

0)r0 | n0k+qi✏n0k+q � ✏nk � ! � i⌘

Approximations(cont.)

FortheKohn-Shamsystem,thefollowingrelationscanbeshowntohold

� = �0 + �0(⌫ + fxc

)�

P = �0 + �0fxc

P

� = P + P⌫�

where𝜐 istheCoulombkerneland𝑓89 istheDFTxc-kernel: fxc = @vxc/@⇢ |⇢=⇢0

✏�1 = 1 + ⌫� ✏ = 1� ⌫P

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

✏G,G0(q,!) := �G,G0 �4⇡e2

|q+G||q+G0|�0G,G0(q,!)

IncludingchangesintheDFTxc-potential: P = �0 + �0fxc

P

Calculationofopticalproperties

Thelong-wavelengthlimit(𝐪 ⟶ 𝟎)ofthedielectricmatrixdeterminestheopticalpropertiesintheregimeaccessibletoopticalprobes.

Themacroscopicdielectrictensor𝜖

Frequencydependent(neglectinglocalfieldeffects)

LOPTICS=.TRUE.q̂ · ✏1(!) · q̂ ⇡ lim

q!0✏0,0(q,!)

Theimaginarypartof 𝜖

First-orderchangeintheorbitals

Expandinguptofirstorderinq

|unk+qi = |unki+ q · |rkunki+ ...

andusingperturbationtheorywehave

|rkunki =X

n 6=n0

|un0kihun0k|@[H(k)�✏nkS(k)]@k |unki✏nk � ✏n0k

whereH(k)andS(k)aretheHamiltonianandoverlapoperatorforthecell-periodicpartoftheorbitals.

ExamplesGajdoš 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⌫�

where𝜐 istheCoulombkerneland𝑓89 istheDFTxc-kernel: fxc = @vxc/@⇢ |⇢=⇢0

✏�1 = 1 + ⌫� ✏ = 1� ⌫P

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

✏G,G0(q,!) := �G,G0 �4⇡e2

|q+G||q+G0|�0G,G0(q,!)

IncludingchangesintheDFTxc-potential: P = �0 + �0fxc

P

IrreduciblepolarizabilityintheIPpicture:𝜒3

ThequantitywecaneasilyaccessinKohn-ShamDFTisthe:“irreduciblepolarizabilityintheindependentparticlepicture”𝜒3 (or𝜒45)

�0G,G0(q,!) :=@⇢ind(q+G,!)

@ve↵(q+G0,!)

AdlerandWiserderivedexpressionsfor𝜒3 which,intermsofBlockfunctions,canbewrittenas

�0G,G0(q,!) =1

⌦

X

nn0k

2wk(fn0k+q � fn0k)

⇥ h n0k+q|ei(q+G)r| nkih nk|e�i(q+G

0)r0 | n0k+qi✏n0k+q � ✏nk � ! � i⌘

AndintermsofBlockfunctions𝜒3 canbewrittenas

�0G,G0(q,!) =1

⌦

X

nn0k

2wk(fn0k+q � fn0k)

⇥ h n0k+q|ei(q+G)r| nkih nk|e�i(q+G

0)r0 | n0k+qi✏n0k+q � ✏nk � ! � i⌘

TheIP-polarizability:𝜒3

W = ⌫ + ⌫�0⌫ + ⌫�0⌫�0⌫ + ⌫�0⌫�0⌫�0⌫ + ... = ⌫ (1� �0⌫)�1| {z }✏�1

Oncewehave𝜒3 thescreenedCoulombinteraction(intheRPA)iscomputedas:

1.ThebareCoulombinteractionbetweentwoparticles

2.Theelectronicenvironmentreactstothefieldgeneratedbyaparticle:inducedchangeinthedensity𝜒3𝜐,thatgivesrisetoachangeintheHartreepotential:𝜐𝜒3𝜐.

3.Theelectronsreacttotheinducedchangeinthepotential:additionalchangeinthedensity,𝜒3𝜐𝜒3𝜐,andcorrespondingchangeintheHartree potential:𝜐𝜒3𝜐𝜒3𝜐.

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: 𝐆 𝐻 𝐆′ )exactly.Fromthe𝑁FFG 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-clampedstaticmacroscopicdielectrictensor𝜖< 𝜔 = 0(orsimply𝜖

ResponsetoelectricfieldfromDFPTLEPSILON=.TRUE.Insteadofusingasumoverstates(perturbationtheory)tocompute|𝛻𝐤𝑢?𝐤⟩,onecansolvethelinearSternheimer equation:

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

@k|unki

for|𝛻𝐤𝑢?𝐤⟩.

Thelinearresponseoftheorbitalstoanexternallyappliedelectricfield|𝜉?𝐤⟩,canbefoundsolving

[H(k)� ✏nkS(k)] |⇠nki = ��HSCF(k)|unki � q̂ · |rkunki

where∆𝐻5QF(𝐤) isthemicroscopiccellperiodicchangeintheHamiltonian,duetochangesintheorbitals,i.e.,localfieldeffects(!):thesemaybeincludedattheRPAlevelonly(LRPA=.TRUE.)ormayincludechangesintheDFTxc-potentialaswell

ResponsetoelectricfieldsfromDFPT(cont.)

• Thestaticmacroscopicdielectricmatrixisthengivenby

q̂ · ✏1 · q̂ = 1�8⇡e2

⌦

X

vk

2wkhq̂ ·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

Gajdoš 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)eik·r|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

Vdru⇤k(r)(R+ r)uk0(r)e

�i(k�k0)·(R+r)

=eV 2

(2⇡)6

Z

⌦k

dk

Z

⌦k0

dk0X

R

Z

Vdru⇤k(r)uk0(r)(�irk0)e�i(k�k

0)·(R+r)

= �i eV(2⇡)3

Z

⌦k

dkhuk|rk|uki

whereweusedintegrationbypartstolet𝛻𝐤R workon𝑢𝐤R insteadofontheexponent,andthefollowingrelation

X

R

e�i(k�k0)·R =

(2⇡)3

V�(k� k0)

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

Bi ·P(�) =f |e|V

AiNk?

X

Nk?

={lnJ�1Y

j=0

det|hu(�)nkj |u(�)mkj+1

i|}

where

kj = k? + jBiJ

, j = 1, ..., J and u(�)nk?+Bi(r) = e�iBi·ru(�)nk?(r)

kj = k? + jBiJ

, j = 1, ..., J and u(�)nk?+Bi(r) = e�iBi·ru(�)nk?(r)

Self-consistentresponsetofiniteelectricfields(PEAD)†

Addtheinteractionwithasmallbutfiniteelectricfieldℇ totheexpressionforthetotalenergy

E[{ (E)}, E ] = E0[{ (E)}]� ⌦E ·P[{ (E)}]

where𝑃 𝜓(ℇ) isthemacroscopicpolarizationasdefinedinthe“moderntheoryofpolarization”‡

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 𝜓(ℇ) 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 𝜓(ℇ) 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/Å].)LRPA=.FALSE. Skipthecalculationswithoutlocalfieldeffects(Default).LSKIP_NSCF=.TRUE. idem.LSKIP_SCF=.TRUE. Skipthecalculationswithlocalfieldeffects.

Example:ion-clamped𝜖< usingtheHSEhybrid

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

PEAD:Hamiltonianterms

TheadditionaltermsintheHamiltonian,arisingfromtheΩℇ V 𝐏 𝜓 ℇ termintheenthalpyareofthefrom

�P

j ={ln det|S(kj ,kj+1)|}�u⇤nkj

=

� i2

NX

m=1

⇥|unkj+1iS�1mn(kj ,kj+1)� |unkj�1iS�1mn(kj ,kj�1)

⇤

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

Byanalogywehave

@|unkj i@k

⇡ 12�k

NX

m=1

⇥|unkj+1iS�1mn(kj ,kj+1)� |unkj�1iS�1mn(kj ,kj�1)

⇤

i.e.,afinitedifferenceexpressionfor|𝛻𝐤𝑢?𝐤⟩.

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 =4⇡e2

⌦

X

ss0

X

kl

Z⇤sik (�ss0)�1kl Z

⇤s0lj

andtothepiezo-electrictensor

eionil = eX

ss0

X

jk

Z⇤sij (�ss0)�1jk ⌅

s0

kl

TheEnd

Thankyou!