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!