1. Nonlinearresponseofsolids withinthe GWplusBetheSalpeterapproximation ClaudioAttaccalite,InstitutNeel,Grenoble(FR) MyrtaGrning,Queen'sUniversity,Belfast(UK) GDR-CORR Marseille - 09-07-2015GDR-CORR Marseille - 09-07-2015 2. Whatisnonlinearoptics? P(r ,t)=P0+ (1) E+ (2) E 2 +O(E 3 ) Firstexperimentsonlinearoptics byP.Franken1961 Ref:NonlinearOpticsand Spectroscopy TheNobelPrizeinPhysics1981 NicolaasBloembergen 3. Ref:supermaket Whynonlinearoptics? ..applications.. ..research.. linear nonlinear Ref:X.Yinetal. Science,344,488(2014) 4. ...andmore... photonentanglement invivoimaging Ref:PPantazisetal. PNAS107,14535(2007) 5. Howtocalculatenonlinear response? 1)DirectevaluationofX(1) ,X(2) .... 2)Sternheimerequation R.M.Sternheimer,Phys.Rev.96,951(1954) 3)Realtimepropagation P(t)=P0+ (1) E+ (2) E 2 +O(E 3 ) (HKS 0 n 0 )n 1 =(H KS 1 n 1 )n 0 = 0 + 0 (v+f xc) it =Hks 6. Noflexibility:oneeq.foreachresponsefunction, difficulttoincludemorefields Complexity:equationsbecomemoreandmorecomplex withtheresponseorder Ref:KS.Virketal.PRB80,165318(2009), H.Hubener,PRA83,062122(2011), X.Andreadeetal.JCP126,184106(2007) DirectevaluationofX(1) ,X(2) .... Disadvantages: Sternheimerequation 7. Advantages Ref: Y.Takimotoetal. JCP,127,154114(2007) C.Attaccaliteetal. PRB,89,081102(2014) Realtimepropagation Nocomplexity:Thesameequationforallresponsefunctions Flexibility:Candealwithcomplexspectroscopictecniques (SFG,FWM,etc...) Disadvantages Resultsaremoredifficulttoanalize 8. Correlation Ref:C.Attaccaliteetal.PRB84,245110(2011), L.P.Kadanoff&G.A.Baym, Quantumstatisticalmechanics,Benjamin(1962). G.Strinati,LaRivistadelNuovoCimento,11,186(1998) Correlationeffectsarederived fromnonequilibriumGreen's theorywithintheGWapproximation (seeX.Blasetalk) 9. Coupling withtheexternalfield un k|r|unk= Dipoleisilldefinedin periodicsystems WecanuseModernTheoryof Polarization.ButthePolarization becomesamanybodyoperator P= e 2 log e 2 L ^xi P= 2ie (2) 3 BZ d kn=1 nb un k k unk ButcorrelationfromGW+BSEcanbemappedinan effectiveonebodyHamiltonianandsowecanuse KingSmithandVanderbiltformula PRB47,1651(1993) R.RestaPRL80,1800(1998) 10. Ourcomputationalsetup Correlation: Green'sfunction theory(GW+BSE) Coupling: ModernTheoryof Polarization Ref:C.Attaccaliteetal.PRB88,235113(2013) 11. X(2) results:semiconductors CdTe Ref: E.Luppielal.,PRB82,235201(2010) E.Ghahramanietal.,PRB43,9700(1991) I.Shoji,etal.J.Opt.Soc.Am.B14,2268(1997) J.I.Jang,etal.J.Opt.Soc.Am.B30,2292(2013) M.Grningatal.PRB88,235113(2013) AlAs Localfieldeffectsare moreimportantthanin thelinearreposonse 12. X(2) results:monolayers Ref:M.GrningetalPRB89,081102(2014) L.M.Malardetal.PRB87,201401(2014) 13. X(3) insilicon Ref: D.J.Mossetal. PRB41,1542(1990) D.J.Mossetal. Opticalletters,14,57(1989) 14. X(3) innanostructures Interferencebetweenexcitons isnottrivialthecaseofANGR7 Ref:R.Denk,NatureComm.5,4253(2014) A.Maeda,PRL94,047404(2005) 15. Conclusions Developanapproachthat 'mports'successfulGW+BSE recipeintoversatilerealtime approach: correlationinnonlinearoptics elaserinteractiontreated withinmodernpolarizationtheory ehinteractionkeyto understand SHGspectraof2Dmaterials (suchashBN,MoS2) 16. Acknowledgement: MyrtaGrning, Queen'sUniversityBelfast Relatedwork: ImplementationofdynamicalBerryphase: C.Attaccalite,M.Grning,PRB88,235113(2013) ApplicationtoSHGof2Dmaterials: M.Grning.C.Attaccalite,PRB89,081102(2014) C.Attaccaliteetal.,PCPC17,9533(2015) TimedependentBSEtheory: C.Attaccalite,M.Grning.A.MariniPRB84,245110(2011) Thecodes: A.Marinietal.Comp.Phys.Comm.180,1392(2009) P.Giannozzietal.J.ofPhys.:Cond.Matter,21,395502(2009) 17. Toseeinvisibleexcitations TheOpticalResonancesin Carbon NanotubesArise fromExcitons FengWang,etal. Science308,838(2005); 18. Doesthevelocitygaugemake thelifeeasy? ItistrueifyouhavealocalHamiltonian Lengthgauge: H = p2 2 m +r E+V (r) (r ,t ) Velocitygauge: H = 1 2 m ( pe A)2 +V (r) e i rA(t) (r ,t) Analiticdemostration: K.RzazewskiandR.W.Boyd, J.ofMod.optics51,1137(2004) W.E.Lamb,etal. Phys.Rev.A36,2763(1987) Welldonevelocitygauge: M.Springborg,andB.Kirtman Phys.Rev.B77,045102(2008) V.N.GenkinandP.M.Mednis Sov.Phys.JETP27,609(1968) ...butwhenyouhavenonlocalpseudo,exchange, dampingtermetc.. theydonotcommutewiththephasefactor 19. TheKingSmithVanderbilt polarization KingSmithandVanderbiltformula Phys.Rev.B47,1651(1993) P= 2i e (2) 3 BZ d kn=1 nb un k| k |unk Berry'sconnection!! 1)itisabulkquantity 2)timederivativegivesthecurrent 3)reproducesthepolarizabilitiesatallorders 4)isnotanHermitianoperator 20. FromPolarizationtothe EquationsofMotion L= i N n=1 M k vknvknH 0 v P i t vk n=Hk 0 vk n+i e k vk n k Itisanobjectdifficulttocalculatenumerically duetothegaugefreedomoftheBlochfunctions vk mn occ Uk ,nmvkn I.Souza,J.IniguezandD.Vanderbilt,Phys.Rev.B69,085106(2004) 21. Postprocessingrealtimedata P(t)isaperiodicfunctionofperiodTL =2p/wL pn isproportionalto n bythenthorderoftheexternalfield Performingadiscretetimesignal samplingwereducetheproblemto thesolutionofasystemsoflinearequations Ref:C.Attaccaliteetal.PRB88,235113(2013) F.Dingetal.JCP138,064104(2013) 22. Let'saddsomecorrelationin4steps hk 1) We start from the Kohn-Sham Hamiltonian: hk+hk universal, parameter free approach 2) Single-particle levels are renormalized within the G0 W0 approx. hk+hk+V H [] 3) Local-field effects are included in the response function hk+hk+V H []+sex [ ] Time-Dependent Hartree 4) Excitonic effects included by means of the Screened-Exchange