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TRANSCRIPT
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VASP:DielectricresponsePerturbationtheory,linearresponse,
andfiniteelectricfields
UniversityofVienna,FacultyofPhysicsandCenterforComputationalMaterialsScience,
Vienna,Austria
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Outline
Dielectricresponse
Frequencydependentdielectricproperties
Thestaticdielectricresponse
ResponsetoelectricfieldsfromDFPT
Themacroscopicpolarization
SCFresponsetofiniteelectricfields
Ioniccontributionstodielectricproperties
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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.
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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.
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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
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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
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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
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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,!)
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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
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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
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Calculationofopticalproperties
Thelong-wavelengthlimit( )ofthedielectricmatrixdeterminestheopticalpropertiesintheregimeaccessibletoopticalprobes.
Themacroscopicdielectrictensor
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Frequencydependent(neglectinglocalfieldeffects)
LOPTICS=.TRUE.q 1(!) q lim
q!00,0(q,!)
Theimaginarypartof
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First-orderchangeintheorbitals
Expandinguptofirstorderinq
|unk+qi = |unki+ q |rkunki+ ...
andusingperturbationtheorywehave
|rkunki =X
n 6=n0
|un0kihun0k|@[H(k)nkS(k)]@k |unkink n0k
whereH(k)andS(k)aretheHamiltonianandoverlapoperatorforthecell-periodicpartoftheorbitals.
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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
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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
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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)
h n0k+q|ei(q+G)r| nkih nk|ei(q+G
0)r0 | n0k+qin0k+q nk ! i
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AndintermsofBlockfunctions3 canbewrittenas
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
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
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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)
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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..
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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).
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TheGWpotentials:*_GWPOTCARfiles
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ThestaticdielectricresponseThefollowingquantities:
Theion-clampedstaticmacroscopicdielectrictensor< = 0(orsimply
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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
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ResponsetoelectricfieldsfromDFPT(cont.)
Thestaticmacroscopicdielectricmatrixisthengivenby
q 1 q = 18e2
X
vk
2wkhq rkunk|nki
wherethesumoverv runsoveroccupiedstatesonly.
TheBorneffectivechargesandpiezo-electrictensormaybeconvenientlycomputedfromthechangeintheHellmann-Feynmanforcesandthemechanicalstresstensor,duetoachangeinthewavefunctionsinafinitedifferencemanner:
|u(1)nki = |unki+s|nki
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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.).
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Examples
Gajdo etal.,Phys.Rev.B73,045112(2006).
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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)
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=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)
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kj = k? + jBiJ
, j = 1, ..., J and u()nk?+Bi(r) = eiBiru()nk?(r)
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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).
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PEAD(cont.)
Lc
ZEg
VB
CB
L
Souzaetal.,Phys.Rev.Lett.89,117602(2002).
e|Ec Ai| Eg/NiNiAi < LZ
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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.
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Example:ion-clamped< usingtheHSEhybrid
J.Paier,M.Marsman,andG.Kresse,Phys.Rev.B78,121201(R)(2008).
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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|?.
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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)
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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
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TheEnd
Thankyou!