Calculation of the biexciton shift in nanocrystals of inorganic perovskites
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Calculation of the biexciton shift in nanocrystals of inorganic perovskites
T. P. T. Nguyen, ∗ S. A. Blundell, † and C. Guet
2, 3, ‡ Univ. Grenoble Alpes, CEA, CNRS, IRIG, SyMMES, F-38000 Grenoble, France Energy Research Institute, Nanyang Technological University, 637141 Singapore School of Materials Science and Engineering, Nanyang Technological University, 639798 Singapore (Dated: February 19, 2020)We calculate the shift in emission frequency of the trion and biexciton (relative to that of the singleexciton) for nanocrystals (NCs) of inorganic perovskites CsPbBr and CsPbI . The calculations usean envelope-function k · p model combined with self-consistent Hartree-Fock and a treatment of theintercarrier correlation energy in the lowest (second) order of many-body perturbation theory. Thecarriers in the trion and biexciton are assumed to have relaxed nonradiatively to the ground stateat the band edge before emission occurs. The theoretical trion shifts for both CsPbBr and CsPbI are found to be in fair agreement with available experimental data, which include low-temperaturesingle-dot measurements, though are perhaps systematically small by a factor of order 1.5, whichcan plausibly be explained by a combination of a slightly overestimated dielectric constant andomitted third- and higher-order terms in the correlation energy. Taking this level of agreementinto account, we estimate that the ground-state biexciton shift for CsPbBr is a redshift of order10–20 meV for NCs with an edge-length of 12 nm. This value is intermediate among the numeroushigh-temperature measurements on NCs of CsPbBr , which vary from large redshifts of order 100meV to blueshifts of several meV. Keywords: perovskite, nanocrystal, exciton, biexciton, trion, correlation
I. INTRODUCTION
Hybrid organic-inorganic lead halide perovskites suchas CH NH PbX (X = Cl, Br, or I) attracted widespreadattention several years ago on account of their ex-cellent properties for photovoltaic applications [1, 2].The reported power-conversion efficiencies have increasedrapidly since then and now reach 23.7% [3]. These highefficiencies are possible in part because the materials havea high defect tolerance [4] and very long carrier diffusionlengths [5].More recently, nanocrystals (NCs) of all-inorganic leadhalide perovskites CsPbX (X = Cl, Br, or I) were shownto be outstanding candidates for light-emitting applica-tions [6]. The NCs fluoresce strongly, with the emissionfrequency tunable over the entire visible range by vary-ing the size of the NCs and their composition (halide X,including mixtures of different halides) [6]. The quan-tum yields obtained are close to 100% [7]. This has ledto important applications of inorganic perovskite NCsto light-emitting diodes [8, 9], lasers [10, 11], and room-temperature single-photon sources [12], among others.An important quantity in many light-emitting appli-cations using NCs is the strength of the exciton-excitoninteraction, which causes a shift in the frequency of lightemitted by a biexciton (two confined excitons) comparedto a single exciton. The presence of biexcitons (or, moregenerally, of multiexcitons) under device conditions canreduce the frequency purity of the emitted light, depend-ing on the size of the shift. The biexciton shift plays ∗ [email protected] † [email protected] ‡ [email protected] a critical role in lasers based on NCs of II-VI semicon-ductors such as CdSe, where the small biexciton red-shift is instrumental in creating a population inversionon the biexciton-to-exciton transition where lasing occurs[13]. It might also be possible to generate polarization-entangled photon pairs from the biexciton-exciton cas-cade | XX i → | X i → | i in NCs of CsPbBr [12], forwhich it would help to understand the energetics of thebiexciton decay.However, the biexciton shift in NCs of CsPbBr is atpresent poorly understood. Many measurements exist[14–19] that largely contradict one another for reasonsthat are still controversial, with reported values of thebiexciton shift varying from large redshifts [16] of order100 meV to a recently reported small blueshift [19] oforder a few meV.To help understand this issue, we present here cal-culations of the biexciton shift in NCs of CsPbI andCsPbBr using a multiband k · p envelope-function ap-proach, combined with many-body perturbation theory(MBPT). We assume that the biexciton has relaxed non-radiatively (by rapid phonon emission) to its ground stateat the band edge before emitting, which enables us toconstruct a detailed microscopic theory of the multicar-rier correlations responsible for the shift. Our resultssuggest that the biexciton shift under these conditions isa redshift having a value that is intermediate among theavailable measurements on NCs of CsPbBr .The plan of the paper is the following. In Sec. II weoutline our formalism. We treat the confined carriers asan ‘artificial atom’ using methods of MBPT from atomicphysics and quantum chemistry [20, 21]. The first stepis a self-consistent Hartree-Fock (HF) model of the con-fined carriers; then we apply the leading correlation cor-rection from second-order MBPT. Our basic envelope-function model is discussed in Sec. II A, the HF methodin Sec. II B, and the correlation energy in Sec. II C. Forreasons of computational efficiency, we use a spherical ba-sis set in the MBPT calculations. This leads to extensiveformulas for the various terms involving radial integralsand angular factors, which can be derived using stan-dard methods of angular-momentum theory [20, 22, 23].These detailed formulas will be presented elsewhere.These methods are then applied to NCs of inorganicperovskites in Sec. III. A difficulty with these materials,which have only recently become the subject of intensiveresearch, is that many of their properties are at presentpoorly understood. Even some basic properties, such asthe effective masses of the valence and conduction bands,are uncertain. We discuss the available data and the pa-rameters that we assume in our model in Sec. III A. Next,in Sec. III B, we apply our approach to the trion and biex-citon shift in NCs of CsPbI and CsPbBr . Both theseshifts are dominated by intercarrier correlation effects,the mean-field (HF) contribution largely canceling [24].As we shall see, the calculations of the correlation energyfor trions and biexcitons are very closely related, so thatthe data on trion shifts provide a very useful additionalcheck on our calculation of the biexciton shift. Our con-clusions are given in Sec. IV. II. FORMALISMA. Model
Our approach is based on an envelope-function formal-ism [25] for a system of carriers (holes and electrons) con-fined in a potential V ext , with the bulk band structure de-scribed by a k · p Hamiltonian h k · p and screened Coulombinteractions among the carriers. The total Hamiltonianin the space of envelope functions is H = X ij { i † j }h i | h k · p + V ext | j i + 12 X ijkl { i † j † lk }h ij | g | kl i , (1)where { i † i † . . . j j . . . } is a normally ordered productof creation (and absorption) operators for electron en-velope states i , i , . . . (and j , j , . . . ), which span theconduction bands (CBs) and valence bands (VBs) in-cluded in the calculation. The Coulomb interaction g inenvelope-function approaches is given generally by a sumof long-range (LR) and short-range (SR) terms [26, 27].Here we will consider only the LR part (we use atomicunits throughout) g = 1 ε in | r − r | , (2)where ε in is the dielectric constant of the NC materialappropriate to the length scale L dot of the nanostruc-ture (see Sec. III A). The LR Coulomb interaction is in principle modified by the mismatch with the dielectricconstant ε out of the surrounding medium, which leads toinduced polarization charges at the interface, althoughwe will not consider this effect in the present paper.Even though perovskite NCs are generally cuboid, weuse a basis of envelope states i, j, . . . , etc., in Eq. (1) ap-propriate to spherical symmetry. This is done for reasonsof computational efficiency. In a spherical basis, the an-gular integrals can be carried out analytically and theremaining radial integrals are one-dimensional. It is alsopossible to sum over the magnetic substates of the ba-sis states analytically [20, 22], which effectively reduces(very substantially) the size of the basis required in cor-relation calculations. Although we will not do so in thispaper, nonspherical terms in the Hamiltonian (for exam-ple, arising from the crystal lattice or from the overallshape of the NC) can in principle be included in laterstages of the formalism as perturbations.To generate a spherical basis, we take the confining po-tential to be spherically symmetric. We choose a spheri-cal well with infinite walls, V ext ( r ) = (cid:26)
0, if r < R ∞ , otherwise . (3)If the NC is a cube with edge-length L , the radius R canbe conveniently chosen to satisfy R = L/ √ . (4)To motivate this choice of R , we note that at effective-mass level the eigenvalues of noninteracting electrons ina cubic box are given by ǫ cube λ ( n x , n y , n z ) = π m ∗ λ L ( n x + n y + n z ) , (5)where ( n x , n y , n z ) are integers and m ∗ λ is the band effec-tive mass. Thus, the condition (4) ensures that the entirespectrum of ‘ S -like’ states in a cube ( n x = n y = n z = n )coincides exactly with the spectrum of nS states in asphere, ǫ sph λ ( n ) = π n m ∗ λ R . (6)One can also show that the lowest ‘ P -like’ state in acube ( n x = 2, n y = n z = 1, together with the two otherpermutations, n x ↔ n y and n x ↔ n z [28]) has an en-ergy within 2.3% of that of the 1 P state in the equiva-lent sphere (4), and that higher-lying ‘ P -like’ states alsohave energies within several percent of their analog in thesphere.Even though the single-particle energies are in closeagreement, wave functions and therefore matrix ele-ments can still differ between cubic and spherical con-finement. However, in Sec. II B we show that the first-order Coulomb energy of the ground-state exciton differsby only about 1.5% in the two cases, and the HF energyby about 0.04%. In Sec. II C, we estimate that the errorin the correlation energy from using a spherical basis isabout 5%. Therefore, for the purposes of this paper, thenonspherical correction term arising from the NC shapeis expected to be unimportant.We consider two k · p models. The first is a 4 × s -like VB and p / -like CBaround the R point of the Brillouin zone in inorganicperovskite compounds [29, 30]. The other is an 8 × p / -like CB, which liesabout 1 eV above the p / -like CB at the R point [29–31]. Including the p / -like CB in this way leads to asmall correction to correlation energies at the 1% level(see Sec. II C).For spherical confinement, the angular part of an enve-lope function with orbital angular momentum l couples toa Bloch function with Bloch angular momentum J (here J = 1 / F, m F ) [32], which we denote by a basis vector | ( l, J ) F m F i . In the 8 × | ηF m F i = g s ( r ) r | ( l + 1 , / F m F i + ¯ g p ( r ) r | (¯ l, / F m F i + g p ( r ) r | ( l, / F m F i + f p ( r ) r | ( l + 2 , / F m F i . (7)Here g s ( r ) and ¯ g p ( r ) are the radial envelope functions forthe s -like and p / -like bands, respectively, while g p ( r )and f p ( r ) apply to the p / -like band. These last twoterms are absent in the 4 × l and ¯ l follow from angular-momentum and parity selection rules [32]. We solvefor the radial functions and eigenvalues of the single-particle states in the presence of a Hartree-Fock poten-tial (Sec. II B) using a generalization of the method ofRef. [32].For states in the s -like VB, the term involving g s ( r )in Eq. (7) is typically the large component of the wavefunction, while the other terms are small components rep-resenting the admixture of CB states into the VB statesdue to the finite range of the confining potential V ext and the k · p interaction. In the CB states, the role ofthe small and large components are interchanged. Thepresence of the small components allows the formalismto pick up the leading k · p corrections arising from thecoupling of the VB and CB. B. Hartree-Fock
The first step in the correlation calculation for a gen-eral excitonic system with N e electrons and N h holes isto solve the self-consistent HF equations including exactexchange [20, 21]. The HF potential will then be used to define the single-particle states of the many-body proce-dure discussed in Sec. II C.For an occupied state | a i (either a hole or an electron),the HF equation is( h k · p + V ext + V avHF ) | a i = ǫ a | a i , (8)where the HF potential V avHF is given by a sum of directand exchange terms, V avHF = V dir + V exc , with h i | V dir | a i = occ X b e b q ab h ib | g | ab i , (9) h i | V exc | a i = − occ X b e b q ab h ib | g | ba i , (10)where the sum is over all occupied (or partially occupied)states. Here e b is a charge-related parameter, with e b = 1for electrons and e b = − ǫ a refer to electron states,even though the states may be ‘occupied’ by a hole withan energy − ǫ a .)The usual HF potential with q ab = 1 in Eqs. (9)and (10) is generally only a scalar operator for closed-shell systems. Since we wish to create a sphericalbasis for open-shell systems as well, we employ in-stead a configuration-averaged HF [20], in which theconfiguration-averaging weights q ab are given by q ab = (cid:26) n B /g B b / ∈ A ( n B − / ( g B − b ∈ A . (11)Here A or B denotes the shell containing the states a or b , respectively, n B is the occupation number of shell B , and g B is the degeneracy (maximum occupation) ofshell B . For a closed-shell system, n B = g B for all shellsand then all weights q ab = 1. The configuration-averagedHF equations (8) for a spherically symmetric V ext cannow be reduced to a set of radial HF equations followingstandard procedures [20].The configuration-averaged HF energy of the excitonicsystem is E avHF = occ X a e a q a h a | h k · p + V ext | a i + 12 occ X a e a q a h a | V avHF | a i , (12)where q a = n A /g A (13)is the fractional occupation of shell A (where a ∈ A ).Conventionally we define the zero of the band-structureenergy to be the VB maximum. Then we can decompose E avHF into different physical contributions as E avHF = E band + E conf + E Coul , (14) TABLE I. Hartree-Fock calculation for a ground-state singleexciton ( X ), negative trion ( X − ), and biexciton ( XX ) con-fined in a NC of CsPbBr with edge-length L = 9 nm, us-ing the material parameters in Table III (and E P = 20 eV). E band is the band energy, E conf the confinement energy, E dir and E exc are the direct and exchange Coulomb energy, respec-tively, E Coul is the total Coulomb energy, E Coul = E dir + E exc ,and E HF = E band + E conf + E Coul is the total HF energy. X (eV) X − (eV) XX (eV) E band . . . E conf . . . E dir − . − . . E exc . − . − . E Coul − . − . − . E HF . . . where E band = N e E g is the ‘band energy’ ( E g is the gapbetween the s -like VB and the p / -like CB) and E conf isthe confinement energy, E conf = occ X a e a q a h a | h k · p | a i − E band . (15)One can also define an energy of interaction with theexternal potential, E ext = P occ a e a q a h a | V ext | a i , althoughhere E ext ≡ E Coul = 12 occ X a e a q a h a | V avHF | a i , (16)which can be further decomposed into direct and ex-change terms using Eqs. (9) and (10). Example calcu-lations showing these energy contributions for a NC ofCsPbBr are given in Table I. Note that the exchangeenergy for a single exciton is very small; this contribu-tion can be shown to be formally of order ( L atom /L dot ) ,where L atom is the interatomic length scale.To study the dependence of the HF energy on the shapeof the NC (sphere or cube), consider the 1 S e -1 S h groundstate of a single exciton. In the effective-mass limit, thenoninteracting 1 S states (electron or hole) have wavefunctions ψ cube1 S ( r ) = r L cos (cid:16) πxL (cid:17) cos (cid:16) πyL (cid:17) cos (cid:16) πzL (cid:17) (17)for cubic confinement, and ψ sph1 S ( r ) = 1 √ πR r sin (cid:16) πrR (cid:17) (18)for spherical confinement. The confinement (kinetic) en-ergy of the the 1 S e -1 S h exciton at this level of approxi-mation follows from Eqs. (4)–(6) to be E (1)conf = 3 π L (cid:18) m ∗ e + 1 m ∗ h (cid:19) (19) FIG. 1. Closed-shell second-order correlation energy: direct(on the left) and exchange (on the right). for both the cube and the equivalent sphere (4). Thefirst-order Coulomb energy can be obtained by insertingthe wave functions (17) and (18) into Eqs. (9) and (16),and neglecting the exchange term. This gives E (1)Coul = − ξε in L , (20)where, after numerical integration, we find ξ ≈ .
389 eV nm (for a cube) and ξ ≈ .
455 eV nm (fora sphere). Thus, the Coulomb energy differs by about1.5% between the cube and the equivalent sphere. FromEqs. (14), (19), and (20) we then find that, for the param-eters used in Table I, the HF energy of the single excitonat this level of approximation is E HF = 2 . E HF = 2 . k · p corrections (with E P = 20 eV)and iterating the HF equations to self-consistency. C. Correlation energy
From the point of view of MBPT [20, 21], the HFenergy of a closed-shell system is correct through firstorder, E HF = E (0) + E (1) , where E (0) = P occ a e a ǫ a isthe sum of the single-particle eigenvalues of the occu-pied HF states, and E (1) is the first-order correctionof the residual Coulomb interaction. The configuration-averaged HF energy (12) of an open-shell system is sim-ilar, but gives the energy of the center of gravity of theconfiguration multiplet, again correct through first orderin MBPT [20]. The higher-order corrections to the en-ergy, E corr = E (2) + E (3) + . . . , are referred to as the correlation energy .In this paper we will consider only the second-orderenergy, E corr ≈ E (2) . For atoms and molecules, E (2) typically accounts for about 75% or more of the totalcorrelation energy (depending on the system)[20, 21] andusually E (2) /E corr < . This approximation has the meritof simplicity. Using the spherical basis (7), E (2) for theexcitonic systems considered here can be converged to anaccuracy of a fraction of a percent in about 1 s or less ona single processing core.The second-order energy for a closed-shell atom ormolecule in a HF potential is given [20, 21] by the many-body diagrams in Fig. 1. To apply this approach to anexcitonic system of holes and electrons, we will effectivelyconsider the electrons and holes to be different speciesof particle and evaluate the diagram for the mixed sys-tem [33]. Thus, the lines directed downward in Fig. 1 ( a and b ) correspond to occupied states (either holes or elec-trons) while upward-directed lines ( r and s ) correspondto unoccupied states (either holes or electrons). The totalsecond-order energy is given by E (2) = 12 X abrs D (2) abrs , (21)where D (2) abrs = h ab | g | rs i ( h rs | g | ab i − h rs | g | ba i ) ω a + ω b − ω r − ω s , (22)and ω i = ǫ i for electrons and ω i = − ǫ i for holes, since thesingle-particle energies must now apply to each particletype. Decomposing E (2) explicitly into electron and holecontributions gives E (2) = E (2) ee + E (2) hh + E (2) eh , (23)where E (2) ee = 12 elec X abrs D (2) abrs , E (2) hh = 12 hole X abrs D (2) abrs ,E (2) eh = X ar (elec) bs (hole) D (2) abrs . (24)The terms E (2) ee and E (2) hh correspond to the correlationenergy of the separate electron and hole subsystems, re-spectively. The third term E (2) eh is a cross-term, involvingsingle excitations of both the electron and hole subsys-tems.A general excitonic system with N e electrons and N h holes may contain open shells. An approximate formulafor the correlation energy in this case may be found by in-serting configuration-averaging weights for the occupiedor partially occupied shells of a and b into the closed-shellformula (21), following the same argument used for theconfiguration-averaged HF energy [20]. Equation (21) isthen modified by D (2) abrs → q a q ab D (2) abrs , (25)where q ab and q a are given by Eqs. (11) and (13), respec-tively.To evaluate the sums over states in Eq. (21), we createa basis set of single-particle states in the HF potential (8)up to a high energy cutoff. This basis set contains theoccupied or partially occupied states a and b , which con-tribute to the HF potential, together with unoccupied(excited) states r and s . For the calculations on singleexcitons, trions, and biexcitons presented in this paper,we take q ab for any unoccupied state a to be q cb , where c is chosen to be the 1 S e state (for electrons) or the 1 S h TABLE II. Second-order correlation energy for a ground-state single exciton ( X ), negative trion ( X − ), and biexci-ton ( XX ) confined in a NC of CsPbBr with edge-length L = 9 nm, using the material parameters in Table III (and E P = 20 eV). E (2) ee , E (2) hh , and E (2) eh are the electron term,the hole term, and the electron-hole cross-term, respectively,given by Eq. (24). Direct (dir) and exchange (exc) terms areshown separately; the exchange term from E (2) eh is negligible.First three columns: 4 × k · p model; last column: 8 × k · p model. Units: meV. 4 × k · p × k · p X X − XX XE (2) ee (dir) 0 . − . − .
41 0 . E (2) ee (exc) 0 .
00 4 .
21 4 .
20 0 . E (2) hh (dir) 0 .
00 0 . − .
41 0 . E (2) hh (exc) 0 .
00 0 .
00 4 .
20 0 . E (2) eh (dir) − . − . − . − . E (2) (total) − . − . − . − . state (for holes). This choice forces all S -wave excitedstates in the basis set to be orthogonal to the occupied1 S e and 1 S h states (as required).Example calculations of the second-order correlationenergy are given in Table II. Note that only the cross-term E (2) eh contributes for a single exciton, since theconfiguration-averaging weights in Eq. (25) vanish for theother terms. Also, the electron E (2) ee and hole E (2) hh termshere contribute equally for the biexciton, because we as-sume VB-CB symmetry in the material parameters; thisis not true in general. The sums over the intermedi-ate states r and s are quite rapidly convergent: about10% of E (2) arises from the S -wave channel, 70% fromthe P -wave channel, and 18% from the D - and F -wavechannels. In addition, the first three principal quantumnumbers of each angular channel are sufficient to obtainabout 98% of E (2) . We note that the contributions to E (2) presented contain small k · p corrections of about2%.From Table II, we see that using the 8 × k · p modelmodifies the single-exciton correlation energy by onlyabout 4% compared to the 4 × k · p correctionsby the presence of the p / -like band. If the calculationsare repeated in the effective-mass limit ( E P → × × p / -like band are not very significant in themselves (ow-ing to their relatively high excitation energy). This jus-tifies the use of the 4 × k · p model for perovskite NCsfor the calculation of the correlation energy.Noting that the dominant intermediate channel is P -wave, we estimate the error in E (2) from using a spherical(not cubic) basis to be about 5%, which is the error inthe energy denominator associated with the 1 S → nP excitations for n = 1–3 (see Sec. II A).For an alternative approach to correlation in a confined TABLE III. Parameters used in the calculations. E (1) P is theKane parameter estimated from the 4 × k · p model, E (2) P fromthe 8 × k · p model. For further explanation, see Sec. III A.CsPbBr CsPbI E g (eV) 2 . a . a µ ∗ ( m ) 0 . a . a m ∗ e , m ∗ h ( m ) 0 .
252 0 . soc (eV) 1 . b . b ε eff . a . a ε opt . c . d E (1) P (eV) 27 . . E (2) P (eV) 16 . . a Ref. [35] b Ref. [31] c Ref. [36], at a wavelength of 600 nm. d Ref. [37], at a wavelength of 600 nm. excitonic system with spherical symmetry, see Ref. [34].
III. APPLICATION TO PEROVSKITENANOCRYSTALSA. Parameters
The material parameters that we use for CsPbBr andCsPbI are summarized in Table III. The bulk parame-ters µ ∗ and ε eff are taken from Ref. [35] and apply to theorthorhombic phase of CsPbBr and the cubic phase ofCsPbI at cryogenic temperatures [38–40]. Although thereduced mass µ ∗ = m ∗ e m ∗ h / ( m ∗ e + m ∗ h ) has been measured[35] by magneto-transmission techniques, the individualeffective masses of electron m ∗ e and hole m ∗ h are unknown.Evidence from experiment [41] and first-principles calcu-lations [6, 30, 42] suggests, however, that m ∗ e and m ∗ h areapproximately equal. Here we will assume m ∗ e = m ∗ h ex-actly ( m ∗ e applies to the p / -like CB, and m ∗ h to the s -likeVB, around the R point of the Brillouin zone). The spin-orbit splitting ∆ soc between the p / -like and the higher-lying p / -like band has been measured in Ref. [31].The ‘effective’ dielectric constant ε eff in Table III is de-rived [35] from the measured binding energy of the bulkexciton. We also give for comparison values of the op-tical dielectric constant ε opt at a wavelength of 600 nm,which are somewhat smaller than ε eff . The constant ε eff applies to a length scale of order the bulk Bohr radius a B , which is quite close to the size of the NCs that weconsider (2 a B = 6 . and 2 a B = 9 . , using the parameters in Table III). Thereforewe shall use ε in = ε eff to screen the LR Coulomb inter-action (2) in the main parts of our calculations of thecorrelation and exchange energy.The Kane parameter E P of CsPbBr and CsPbI hasnot been measured directly. An estimate of E P can bemade by assuming that the contribution to m ∗ e and m ∗ h from remote bands is zero, which in the 8 × k · p model ✺ (cid:0)✵ ✶✁ ✷✂✄☎✼✆✝✽✞✟✾✠✡☛☞✌✍✎ ❡✏❣✑ ❧✒✓✔t❤ ▲ ✭♥♠✮✕✖✐s✗✘♦✙✚✛✜r✢②✣✤❱✥ ❨✦✧★✩✪ ✫✬ ❛✯✳❉✰✱✲ ✴✸ ✹✻✿P❀❁ ❂❃ ❄❅❆❇❈❊❋● ❍■ ❏❑▼◆❖✉ ◗❘ ❙❚❯❲❳❩ ❬❭ ❪❫❴❵❜❝❞❢❥ ❦ ♣q✈✇① ③④ ⑤⑥⑦⑧⑨⑩❶❷❸❹❺❻❼❽❾ ❿➀➁➂ ➃➄➅➆➇➈ ➉ ➊➋➌➍➎➏➐➑➒➓➔→➣↔↕➙➛➜➝➞➟➠ ➡➢➤➥➦➧➨➩➫➭ ➯➲ ➳➵➸➺➻➼➽ ➾➚ ➪➶➹➘➴➷➬➮➱✃❐ ❒❮ ❰ÏÐÑÒÓÔÕÖ× ØÙ ÚÛÜÝÞßàáâ ã äåæç !" *$ !( *$ FIG. 2. Measured photoluminescence peak energies of NCs of(a) CsPbI and (b) CsPbBr (triangles/squares/circles), andtheoretical single-exciton energy using HF (dashed curve) andHF plus second-order correlation energy (full curve). Yumoto et al ., Ref. [43]; Dong et al ., Ref. [44]; Pan et al ., Ref. [45];Dutta et al ., Ref. [46]; Liu et al ., Ref. [47]; Yin et al ., Ref. [48];Protesescu et al ., Ref. [6]; Canneson et al ., Ref. [49]; Brennan et al ., Ref. [50]. implies [51] 1 µ ∗ = 23 (cid:18) E P E g + E P E g + ∆ soc (cid:19) , (26)where E g is the gap energy. The Kane parameter here isdefined by E P = 2 |h S | p z | Z i| , (27)where | S i is the Bloch state of the s -like band and | Z i is the z -component of the Bloch state of the (spin-uncoupled) p -like band [52]. Equation (26) can now besolved for E P . The corresponding equation [29, 35] forthe 4 × k · p model is obtained by allowing ∆ soc → ∞ .The values of E P inferred in this way for the two modelsare summarized in Table III.We take the view that E P is uncertain. A conserva-tive range would be 10 eV ≤ E P ≤
32 eV for CsPbBr and 8 eV ≤ E P ≤
26 eV for CsPbI . Note that the un-certainty in E P is not critical for the calculation of theenergy, since E P determines only the rather small k · p corrections to E avHF and E corr (see Secs. II B and II C).For illustrative purposes, we choose a central value of E P = 20 eV for CsPbBr in Tables I and II.An overall assessment of the parameters and the modelcan be made by comparing the theoretical single-excitonenergy with the energy of the emission peak [53], asshown in Fig. 2. The data in the figure correspond to avariety of experimental conditions. Most of the measure-ments were made at room temperature, although Yin etal . [48] (CsPbI ) and Canneson et al . [49] (CsPbBr ) wereat cryogenic temperatures, as were the measurements [35]used to determine our parameters (Table III), which aretherefore more appropriate to low temperatures. Thisexplains part of the apparent small discrepancy at largesizes L , as the bandgap increases at room temperature byabout 60 meV (CsPbBr ) to 80 meV (CsPbI ) [35]. Also,the measurement of Liu et al . [47] (CsPbI ) is ligand-dependent, as indicated by the multiple data points.It is clear from Fig. 2 that the contribution of corre-lation to the total emission frequency is not significant,but the role of correlation is greatly enhanced in mea-surements of the trion and biexciton shifts, which arediscussed in the next section. B. Bi-exciton and trion shifts
Emission from trions or biexcitons in NCs is usuallyobserved to occur at a slightly lower frequency than froma single exciton [16, 17, 24, 54, 55]. The trion ∆ X − andbiexciton ∆ XX redshifts, relative to the single-excitonemission frequency, can be found by taking the differenceof the initial and final energies,∆ XX = 2 E X − E XX , (28)∆ X − = E X + E e − E X − , (29)where E X , E X − , E XX , and E e are the total energies ofthe single exciton, the negative trion, the biexciton, anda single confined electron, respectively. We assume herethat the excitonic systems relax nonradiatively under ex-perimental conditions before emitting, so that these totalenergies will be taken to refer to the ground state. InEqs. (28) and (29), we have anticipated that the shiftsare redshifts by defining ∆ X − and ∆ XX as minus thechange in energy relative to a single exciton. Because weassume VB-CB symmetry of effective-mass parameters(see Table III), the positive trion will have an identicalshift to the negative trion, ∆ X + = ∆ X − .Remarkably, the biexciton [Eq. (28)] and trion[Eq. (29)] shifts are dominated by intercarrier correlationeffects, as the mean-field contribution largely cancels [24].This phenomenon for a NC of CsPbBr is illustrated inTable IV. Note that for large edge-lengths L & XX and ∆ X − have a quite weak size dependence.This can be understood by noting that a Coulomb matrixelement scales approximately as h ab | g | rs i ∼ /L , while TABLE IV. Calculated biexciton ∆ XX and trion ∆ X − red-shifts in NCs of CsPbBr with edge-lengths 4 nm ≤ L ≤
12 nm, assuming the material parameters in Table III (and E P = 20 eV). The contributions to Eqs. (28) and (29) fromHartree-Fock (HF) and correlation (Corr) are shown sepa-rately; ∆ XX and ∆ X − are the sum of the HF and Corr terms.Units: meV. 4 nm 6 nm 9 nm 12 nmHF − . − . − . − . .
16 14 .
68 11 .
58 9 . XX .
19 13 .
11 11 .
00 9 . − .
05 1 .
03 1 .
41 1 . .
84 9 .
15 7 .
61 6 . X − .
79 10 .
18 9 .
02 7 . ! " )*+) ,)-+./ 0 1-23 ’’’ . - ’ / ’’’’’’ (cid:0) ’’’’’ ’’’’’ : ’’’’’ ; ’’’’’ < ’’’’’’ ) = ’’’ ’’’’’&’’’’’ -* ’’’’’<64*)4’>?@A’’’’’’’’BC5-6’).’C,D’’’’’’’’EF’).’C,D’’’’’’’’GCHC/C4C’).’C,D’’’ ✹ ✻ ✽ ✶✁ ✶✂ ✶✹ ✶✻✸✹✺✻✼✽✾✶✁ ❡✄☎❡ ✆❡✝☎✞✟ ✠ ✡✝☛☞t✌✍✎✏✑✒✍✓t ∆ ❳✰ ✴❳✲✭✔✕✖✗ ✷♥✘✙✚✛✜✢✛ ✣✤✥✦❨✧★ ✢✩ ✪✫✬ !" *$ !( *$ FIG. 3. Measured trion redshift ∆ X − of NCs of (a) CsPbI and (b) CsPbBr (squares/circles/diamonds). Solid line: the-ory (second-order MBPT). Yin et al ., Ref. [48]; Rain`o et al .,Ref. [55]; Fu et al ., Ref. [56]; Nakahara et al ., Ref. [54]. the energy denominator in Eq. (22) scales approximatelyas ω a + ω b − ω r − ω s ∼ /L owing to the confinementeffect (5), so that E (2) is approximately independent of L . In fact, both ∆ XX and ∆ X − become slightly largerat the smaller sizes in Table IV, an effect that has beenobserved experimentally in perovskite NCs [16].Our theoretical trion shifts are compared with theavailable experimental data in Fig. 3. The agreementwith the trion data is fair, although the theoretical val-ues are perhaps systematically too small (by a factor ! " ()*( +(,*-. / 0,12 ’’’ ( - , ’ . ’’’’’’ (cid:0) ’’’’’ :: ’’’’’’ ( ; ’’’ ’’’’’&’’’’’ ,) ’’’’’<7=)(=’>?@A’’’’’’’’B,((8.’(-’C+D’’’’’’’’EC8-C,()C’(-’C+D’’’’’’’’FC,*’(-’C+D’’’’’’’’B8.,(=’(-’C+D’’’ ✹ ✻ ✽ ✶✁ ✂✄ ☎✆ ✝✞✟✠✡☛☞✌✍✎✏✑✒ ❡✓✔✕ ✖✗✘✙✚✛ ✜ ✢✣✤✥❜✦✧★✩✪✫✬✭✮✯✰✱✲✦ ❳✳✴✵✷✸✺ ✼♥✾✿❀❁❂❃❄ ❅❆❇❈❨❉❊ ❋● ❍■❏▼❑▲◆❖P◗ ❘❙ ❚❯❱ !" *$ !( *$ FIG. 4. Measured biexciton redshift ∆ XX ofNCs of (a) CsPbI and (b) CsPbBr (trian-gles/squares/circles/diamonds). Solid line: theory (second-order MBPT). Yin et al ., Ref. [48]; Makarov et al ., Ref. [15];Aneesh et al ., Ref. [17]; Castaneda et al ., Ref. [16]; Wang etal . Ref. [14]; Ashner et al ., Ref. [19]. of order 1.3–1.8). Turning to the data on the biexci-ton shift, shown in Fig. 4, we see that a similar com-ment holds for CsPbI , where the theoretical values aresmaller than the few available measurements by a fac-tor of about 1.8–2.0. The situation is rather unclear forthe biexciton shift in CsPbBr , however, where there aremore data available. The measured values of ∆ XX forCsPbBr range from large redshifts [16] of about 40–100 meV (which is comparable to the HF Coulomb energygiven in Table I) to a recently reported small blueshift[19], of order ∆ XX = − L ≈
10 nm. Oursecond-order MBPT approach predicts a redshift for allsizes considered for both CsPbBr and CsPbI , with avalue ∆ XX = 10 meV for CsPbBr for L ≈
10 nm.Before commenting on the experimental data, let usfirst review some leading sources of theoretical error inour second-order MBPT approach. These are:(i)
Correction terms due to fine-structure splittings .We have neglected the fine structure (FS) of the exci-tonic states, basing our formalism on a configuration-averaged approach (25), which yields the center of grav-ity of the FS multiplet. FS splittings in emission linesof inorganic perovskite NCs are observed to vary fromseveral hundred µ eV (e.g., Ref. [48]) to a few meV (e.g., Ref. [30]). Single-dot spectroscopy reveals that they canvary quite markedly from dot to dot, both in magnitudeand sometimes also in the number of FS components ob-served [30, 48, 56]. In the few cases that FS splittingshave been observed experimentally in the measurementsrelevant to Figs. 3 and 4, the shifts plotted in the figurescorrespond to the values obtained by averaging over theFS (e.g., Ref. [48]). Because of this and the relativelysmall size of the FS splittings, the error in Figs. 3 and 4due to FS seems likely to be at the level of 1–2 meV orless.Let us consider the role of the FS of the single exci-ton in greater detail. In perovskite NCs, the ground-state1 S e -1 S h single exciton consists of electron and hole stateswith angular momentum F = 1 /
2, in the notation ofEq. (7), which can couple to a total angular momentum F tot = 0 or 1 (singlet or triplet, respectively). The tripletstate has an allowed electric-dipole radiative decay and isa bright exciton state; the singlet is a dark state [30, 56].Similarly, the ground-state biexciton in perovskite NCshas closed-shell electron and hole states, 1 S e -1 S h , whichmust therefore couple to F tot = 0, and the negative trionhas a 1 S e -1 S h ground state with F tot = 1 /
2. From selec-tion rules, the allowed biexciton emission must proceedvia the bright single-exciton state, XX → X , where thesubscript indicates the value of F tot .Now, the center of gravity of the bright-dark FS mul-tiplet in the single exciton is given by¯ E X = (1 / E X + (3 / E X , (30)from which it follows that E X − ¯ E X = ∆ /
4, where∆ = E X − E X is the bright-dark FS splitting. (Weare assuming that any FS in the bright state, whichis due to nonspherical or noncubic symmetry-breakinginteractions [30, 56–58], has been experimentally aver-aged.) For the biexciton, the configuration-averaged en-ergy ¯ E XX = E XX , since there is only one state. There-fore, the observed biexciton shift is given by∆ XX = 2 E X − E XX = (2 ¯ E X − ¯ E XX ) + ∆ / , (31)and we see that our calculated result in Fig. 4 acquiresa correction term ∆ /
2. An analogous argument leadsto a correction term ∆ / | ∆ | is expected to be of order a few meV [58], weconclude again that any error in Figs. 3 and 4 from thissource is likely to be of order at most 1–2 meV.(ii) Uncertainty in the value of the dielectric constant .The biexciton ∆ XX and trion ∆ X − shifts are dominatedby correlation or E (2) , so that they are both approx-imately proportional to 1 /ε , where ε in is the dielec-tric constant of the material (2). However, a more com-plete treatment of dielectric effects than considered inthe present paper would take into account the space- andfrequency-dependent bulk dielectric function ε ( k , ω ). Inthe instantaneous approximation ω = 0, the dielectricconstant ε in in Eq. (2) would then be replaced by a space-dependent function ε ( r , r ). A more general treatmentincluding also the frequency-dependence of ε ( k , ω ) wouldrequire a retarded Coulomb interaction (and, for exam-ple, the use of Feynman propagators [59]). Inorganicperovskites present the complication that the dielectricfunction is rapidly varying; for instance, the effective andoptical dielectric constants given in Table III are quitedifferent.Another dielectric effect, which we have neglected here,arises from the mismatch of the dielectric constant ofthe NC with that of the surroundings, which modifiesthe effective LR Coulomb interaction to take account ofpolarization charges induced at the dielectric boundary[60].In our calculations, we have assumed a dielectric con-stant ε in = ε eff , where ε eff is derived from the mea-sured binding energy of the bulk exciton (see Sec. III A).Formally, ε eff corresponds to length scales of order theBohr radius k ∼ π/a B and to a frequency ω ≈
0, sincethe exciton binding energy is dominated by the directCoulomb energy (9) and (16), in which the energy flow-ing through the Coulomb propagator in the Feynmanrules is zero. The second-order energy E (2) , on theother hand, involves a nonzero average excitation en-ergy δω av = h ω a + ω b − ω r − ω s i in Eq. (25), whichimplies a nonzero average energy flowing through theCoulomb propagators. We find δω av ≈ . ≤ L ≤
12 nm for NCs of CsPbBr . In addition,although the size of our NCs is comparable to the Bohrradius (see Sec. III A), this is not exactly true. It followsthat the appropriate value of the dielectric constant ε in to use in calculations of E (2) might differ from ε eff . Forinstance, it seems likely that the frequency-dependencewill shift the appropriate value of ε in from ε eff toward aslightly smaller value, closer to ε opt (see Table III). Thiswould increase E (2) and could explain part of the discrep-ancy between theory and experiment observed in Figs. 3and 4(a).(iii) Higher-order correlation . Usually in atoms andmolecules, E (2) underestimates the all-order correlationenergy [20, 21]. Unfortunately, it is hard to estimate thehigher-order correlation E (3+) = E (3) + E (4) + . . . with-out explicit calculation, although we note that typicalvalues of E (3+) for atoms and molecules can vary up to25% or so of E (2) , depending on the system. Each orderof MBPT brings in one extra Coulomb interaction g andan energy denominator ∆ ǫ , which scale approximatelyas g/ ∆ ǫ ∼ L . Therefore the contribution of higher-orderMBPT is expected to become more important for largerdots, and this could explain a large part of the discrepan-cies noted in Figs. 3 and 4(a) for the case of intermediateconfinement encountered in perovskite NCs.Table II makes it clear that the calculations of E (2) for the trion and the biexciton are very closely related.The term E (2) ee in Eq. (23) can be seen to have almost thesame value for each. This happens because both systemscontain two electrons, so that the configuration-averagingfactors in Eq. (25) are the same (although the basis setsdiffer slightly, because different states are occupied in the HF potential of the two systems). Similarly, mostof the difference in the other two terms E (2) eh and E (2) hh in Table II is due simply to the different configuration-averaging weights for the trion and biexciton. Becauseof this, we expect that the errors in ∆ XX and ∆ X − dueto both dielectric effects [(ii) above] and omitted higher-order MBPT [(iii) above] should be comparable. Thetrion data in Fig. 3 can therefore serve as an additionalcheck on the biexciton data in Fig. 4.Turning to the experimental data, we note first thatit is useful to distinguish between measurements on sin-gle dots at cryogenic temperatures (e.g., using time-resolved photoluminescence) and high-temperature mea-surements on ensembles of NCs (e.g., using transient ab-sorption). The low-temperature measurements typicallygive narrow well-separated peaks, from which the shiftscan be extracted directly, while the high-temperaturemeasurements typically require extensive fits to side-features on overlapping peaks, or other indirect analy-sis methods. Low-temperature single-dot measurementshave been performed on the trion (Fig. 3) by Fu et al .[56] for CsPbBr and by Yin et al . [48] for CsPbI , andthe latter also measured the biexciton shift for CsPbI (Fig. 4a). No low-temperature measurements are avail-able of the biexciton shift in CsPbBr .We observe that our agreement with all these low-temperature measurements in the trion shift (Fig. 3) isfair. Based on this, and the observation that the the-oretical errors for the trion and the biexciton shift areexpected to be similar, we believe that the present re-sults provide quite strong theoretical evidence that theground-state biexciton shift in NCs of CsPbBr is a red-shift of order ∆ XX = 10–20 meV for L ≈
12 nm (afterallowing for a phenomenological increase in the second-order MBPT values given in Table IV by a factor of upto 2).According to Shulenberger et al . [18], who performedexperiments on NCs of CsPbBr , the fast red-shifted fea-tures often attributed to biexciton emission are actuallyan artifact of the exposure of the sample to air, whichthey claim causes the formation of larger bulk-like par-ticles in the ensemble with a red-shifted single-excitonpeak. Shulenberger et al . [18] placed an upper limiton the true biexciton shift of 20 meV, which is consis-tent with our theoretical prediction. However, the samegroup later inferred [19] a small biexciton blueshift of or-der ∆ XX = − L ≈
10 nm after extensive datafitting, which seems to be inconsistent with our theoret-ical value.Another experimental issue is whether the biexcitonhas truly relaxed to the ground state, as we have assumedin our calculation. Yumoto et al . [43] studied ‘hot’ biex-citons in a transient absorption experiment on NCs ofCsPbI by observing the induced absorption signal im-mediately after the pump excitation. They concludedthat a hot biexciton, composed of one exciton at theband edge and a second excited exciton, had a substan-tially increased exciton-exciton interaction. They found0that ∆ XX for CsPbI could be as large as 60 meV forexcitation energies E ex of the second exciton of order E ex & . et al . [15] measured∆ XX = 12 meV for NCs of CsPbI and obtained almostthe same value ∆ XX = 11 meV for NCs of the mixedperovskite CsPbI . Br . , which would imply a biexci-ton shift for CsPbBr in agreement with our theoreticalvalue. IV. CONCLUSIONS
We have presented a calculation of the trion and biex-citon shifts in NCs of CsPbI and CsPbBr using second-order MBPT. The agreement with the available data forthe biexciton shift in CsPbI and the trion shift in bothCsPbI and CsPbBr is fair, although the theoretical val-ues seem to be systematically slightly smaller than themeasurements, a result that can be plausibly understoodin terms of a slightly overestimated dielectric constantand omitted higher-order terms in MBPT. After takingthis level of agreement between theory and experimentinto account, we infer that the ground-state biexcitonshift in NCs of CsPbBr is a redshift with a value of or-der 10–20 meV (for a size L = 12 nm). This value isintermediate in the large range of measured values forCsPbBr .The theoretical approach used can be improved in var-ious ways in future work. It is possible to include higher- order MBPT for excitonic systems with few carriers bymeans of all-order procedures such as full configurationinteraction [21]. A better understanding of the dielectricfunction in perovskites could perhaps be obtained using ab intio atomistic codes [61]. The LR Coulomb interac-tion (2) can also be generalized to take account of thedielectric mismatch with the surrounding medium [60].Although envelope-function methods naturally work bet-ter on larger NCs, where atomistic effects are relativelyless significant, an important atomistic effect can be in-cluded straightforwardly by assuming a diffuse finite sur-face barrier instead of an abrupt infinite barrier (3). Also,explicit nonspherical corrections for the cubic NC shapecould be added as perturbations.Finally, it should be possible to generalize the methodspresented here to study hot biexcitons, in which one orboth excitons are excited. It would also be interestingto study thermal effects on the biexciton shift at hightemperature, a regime that is more relevant to the con-ditions found in practical devices. The present paperassumes that the excitonic systems are in their quantumground state, so that it is perhaps natural to expect bet-ter agreement with the low-temperature data. ACKNOWLEDGMENTS
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