This file contains supplementary material for J. L. Movilla, J. Planelles and J. I. Climente, “Excitons in metal halide perovskite nanoplatelets: an effective mass description of polaronic, dielectric and quantum confinement effects,” Nanoscale Advances, 2023, 5, 6093, DOI: https://doi.org/10.1039/D3NA00592E

Calculation of the ground state energy of single excitons in nanoplatelets including 3D spatial confinement, quantum-well dielectric confinement, short range polaronic effects and 2D electronic correlation variationally

Jose L. Movilla (a), Josep Planelles (b), Juan I. Climente (b)

(a) Dept. d’Educació i Didàctiques Específiques, Universitat Jaume I, 12080, Castelló, Spain
(b) Dept. de Química Física i Analítica, Universitat Jaume I, 12080, Castelló, Spain

This code implements a model which extends the effective mass theory description of excitons in quasi-2D systems for the case of hybrid metal halide perovskites. This is done by considering not only quantum and dielectric confinement, but also short- and long-range polaron screening –through the inclusion of Haken-like potentials–. The technical novelty lies in the conjugation of dielectric confinement and short range polaronic effects. Both are conveniently expressed using the image charge method for quantum well-like structures, which we adapt here to account for the Yukawa (non-Coulomb) term included in the Haken and Bajaj potentials.

Written with Mathematica 8.0

(*Clearmemory*)ClearAll["Global`*"](*ParametersconvertingfromeVandlengthinputunitstoatomicunits*)elv=27.211385;ban=RootApproximant[5.29177249,2];(*Writingbanascombinationofintegersallowsanalyticalintegrationbelow*)(*=====================*)(*INPUTPARAMETERS*)(*=====================*)(*NPLdimensions-inunitsofAngs*10-;useintegernumbers*)Lx=3000;Ly=3000;Lz=126;(*here(30x30x1.26)nm(thicknessequalto2layersofMAPbI3)*)(*=====================*)(*Inner(1)andouter(2)staticdielectricconstants*)eps1=22;eps2=2;(*Inneropticaldielectricconstant*)eps1∞=5.6;(*======================*)(*Effectivemasses[timesm0]*)mep=0.19;mez=0.19;mhp=0.22;mhz=0.22;(*ISOTROPICMASSESareconsidered*)(*======================*)(*Opticalphononenergy(eV)*)ELO=16.5*10^-3;(*======================*)(*OPTIONS*)diel=1;(*Options:diel=1takesintoaccountdielectricconfinement.Otherwise,onlydirecttermsarecosidered*)Yukawa=1;(*Options:Yukawa=1includesYukawatermsinthehamiltonian.Otherwise,Yukawatermsareignored*)Vtype="Haken";(*Options:Vtype="Haken"orVtype="Bajaj"*)(*======================*)(*Variationalparameterrange*)(*3scansareperformed:Afirstonewithastepwidth=αstep,andtworefinations,withαstep1andαstep2*)αmin=0.5;αmax=5;αstep=0.3;αstep1=0.1;αstep2=0.02;(*======================*)(*CutoffsinthecalculationofCoulombandYukawaintegrals*)nC=10;nY=10;(*=======================*)(*ENDOFINPUTPARAMETERS*)(*=======================*)(*Internaldefinitionsandunitconversion*)If[Yukawa==1,Yukawa="ON",Yukawa="OFF"];Lx=Lx/ban;Ly=Ly/ban;Lz=Lz/ban;kx=Pi/Lx;ky=Pi/Ly;kz=Pi/Lz;mup=mep*mhp/(mep+mhp);muz=mez*mhz/(mez+mhz);(*reducedmasses*)ELO=ELO/elv;(*Calculateelectronandholepolaronradii*)le=Sqrt[1/(2*me*ELO)]/.me->mez;lh=Sqrt[1/(2*mh*ELO)]/.mh->mhz;(*Definepseudo-dielectricconstantsforYukawa-typepotentials*)eQW=Which[Vtype=="Haken",eps1*eps1∞/(eps1-eps1∞),Vtype=="Bajaj",(eps1∞/eps1)^(-3/5)*eps1*eps1∞/(eps1-eps1∞),True,Print["Warning: Vtype must be 'Haken' or 'Bajaj'"]];eout=eps2;(*======================*)

(*Calculations*)(*ExpandtheYukawaexponentialintermsof(5)Gaussianfunctions*)fYp[rp_]=Exp[-rp];(*Hererpisinunitsofthepolaronradiuslp(i.e.,rp=r/lp).Otherwise,thefittingdoesnotconvergeefficiently.*)m=1;sto={};n=1000;rf=10;For[i=0,i<=n,i++,rr=i*rf/n;sto=AppendTo[sto,{rr,fYp[rr]}]];sol=FindFit[sto,a1*Exp[-b1*rp^2]+a2*Exp[-b2*rp^2]+a3*Exp[-b3*rp^2]+a4*Exp[-b4*rp^2]+a5*Exp[-b5*rp^2],{a1,a2,a3,a4,a5,{b1,1},{b2,2},{b3,5},{b4,2},{b5,7}},rp];alist={a1,a2,a3,a4,a5}/.sol;blist={b1,b2,b3,b4,b5}/.sol;bliste=blist/le^2;blisth=blist/lh^2;(*========================*)(*COULOMBSelf-energy-sameforelectronorhole-seeTakagaharaPRB40,12359,atomicunits*)Ifdiel==1,q[n_]:=((eps1-eps2)/(eps1+eps2))^Abs[n];term[n_,z_]:=If[n==0,0,q[n]/(2*eps1*Sqrt[(z-z*(-1)^n-n*Pi)^2])];nt=50;(*numberoftermsintheself-energypotentialexpansion*)Pself[z_]=

nt

∑

n=-nt

term[n,z];delta=10^-12;self=NIntegrate[Cos[z]^2Pself[z],{z,-Pi/2+delta,Pi/2-delta}];pot=self(Lz/2);,pot=0;;(*========================*)(*YUKAWAelectronandholeself-energies,atomicunits*)Ifdiel==1&&Yukawa=="ON",q[n_]:=((eQW-eout)/(eQW+eout))^Abs[n];term[n_,z_,lp_]:=If[n==0,0,q[n]*Exp[-(1/(kz*lp))*Sqrt[(z-z*(-1)^n-n*Pi)^2]]/(2*eQW*Sqrt[(z-z*(-1)^n-n*Pi)^2])];nt=50;(*numberoftermsintheself-energypotentialexpansion*)Pself[z_]=

nt

∑

n=-nt

term[n,z,le];delta=10^-12;self=NIntegrate[Cos[z]^2Pself[z],{z,-Pi/2+delta,Pi/2-delta}];potYe=self(Lz/2);Pself[z_]=

nt

∑

n=-nt

term[n,z,lh];delta=10^-12;self=NIntegrate[Cos[z]^2Pself[z],{z,-Pi/2+delta,Pi/2-delta}];potYh=self(Lz/2);,potYe=0;potYh=0;;

Calculation of the ground state energy of single excitons in nanoplatelets including 3D spatial confinement, quantum-well dielectric confinement, short range polaronic effects and 2D electronic correlation variationally

​Written with Mathematica 8.0

Written with Mathematica 8.0