PHYSICAL REVIEW B VOLUME 47, NUMBER 11 15 MARCH 1993-I

Phonon sidebands of excitons bound to isoelectronic impurities in semiconductors

Yong Zhang, Weikun Cxe, and M. D. SturgeDepartment of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755 3528-

Jiansheng Zheng and Boxi WuPhysics Department, Xiamen University, Xiamen 361005, The People's Republic of China

(Received 14 July 1992; revised manuscript received 17 November 1992)

The configuration coordinate (CC) and momentum conservation (MC) models have been widely usedto explain the phonon sidebands of impurity spectra in semiconductors. In this paper, the distinction be-tween the CC and MC models is discussed. We conclude that the MC model only'applies to shallowCoulombic impurities; in other cases, such as isoelectronic traps, the CC model is more appropriate. Weshow that the Huang-Rhys parameters for bulk phonon modes coupling to a bound electron or excitoncan be calculated from the bound-state wave function in k space if the phonon-induced intervalley andintravalley electron scattering processes of the pure crystal are known. We study in detail the phononsidebands of nitrogen-bound excitons in GaP, giving the selection rules for electron-phonon coupling in

the CC model, and showing that their strength can be well accounted for by the CC model. The ap-parently anomalous "X"peak of the LO-phonon sideband in GaP:N is shown to be associated with inter-valley scattering in the conduction band. The MC model, which has been used in an attempt to explainthe phonon sidebands of GaP:N in some previous work, is shown to be inapplicable to this case.

I. INTRODUCTION

It has been known for a long time that impurity spec-tra in semiconductors may show two kinds of phononsidebands (PS's) which can be interpreted by either the"configuration coordinate" (CC) or the "momentum con-servation" (MC) models. ' The CC model is a simplifiedversion of the multiphonon optical transition theory pro-posed by Huang and Rhys and Pekar about 40 yearsago. The MC model, on the other hand, is a natural ex-tension to impurity spectra of the theory of indirecttransitions in a perfect semiconductor. These two mod-els have been used to explain the PS's in many impurityassociated spectra, but the relationship between them hasnot been adequately discussed. For this discussion, wefirst need to know if these two models correspond to twocoexisting mechanisms of electron-phonon interactions orif they are just two different approximations to describethe interactions. In the former case, they could be takenas two independent processes, and the total transitioncross section of the PS's would be the sum of the contri-bution from the two processes, or in other words, thetwo models could be combined to explain the observedPS's. But the problem is not that simple since, as hasbeen pointed out before and will be discussed in furtherdetail in this paper, they are, in fact, two different ap-proximations to the same basic method for calculatingthe phonon-assisted optical transition probabilities in-volving an impurity. Therefore, the two models must becombined with caution.

While the PS's of excitons bound to shallow Coulombicimpurities (donors and acceptors), have been interpretedin terms of the MC model, ' the PS's of excitons bound tonitrogen and other isolectronic impurities in CxaP havebeen consistently interpreted in terms of the CC mod-

el. ' ' However, Snyder and coworkers and Dai et al.have argued that all the PS's of the isolated N-bound ex-citon except the LO(I ) peak should also be interpretedby the MC model. They showed that they could accountfor the spectral shape of the PS's in this model, but didnot discuss its absolute intensity. Hong, Zhang, andDou' have also applied the MC model in an attempt toaccount for an apparently anomalous temperature depen-dent of the PS's. ' ' However, this anomaly has beenshown to be an experimental artifact. ' ' ' ' In Ref. 8 itwas argued that the CC model can be applied only toLO(I ) phonons, which couple to the exciton via theFrohlich interaction. Other phonon features could not beexplained by the CC model because "the k-dependenteffects of both the impurity wave function and the pho-non energy are not included in the model. " This is in-correct: such effects have been included in the CC model,for instance by Stoneham. ' In this paper we will formu-late the CC model in a way that includes such k-dependent effects. The model accounts semiquantitative-ly for all the PS's, including the so-called "L" sidebandwhose interpretation has been controversial. We will alsoshow that the MC model accounts quantitatively for thePS intensity of excitons bound to Coulombic impurities,but fails by many orders of magnitude to account for thatof excitons bound to isoelectronic traps such as nitrogen.

In Sec. II of this paper, we will discuss the distinctionbetween the CC and the MC models, how they are relat-ed, and how they differ in their predictions of the opticaltransition rate for the PS's and for the no-phonon (NP)line. We show how for bulk-mode phonons the Huang-Rhys parameters S, which represent the strength of theexciton-phonon coupling and determine the intensities ofthe PS's in the CC model, relate to the intravalley and in-tervalley electron-phonon scattering processes of the in-

47 6330 1993 The American Physical Society

PHONON SIDEBANDS OF EXCITONS BOUND TO. . . 6331

trinsic semiconductor. Based on these scattering process-es, an (in principle) accurate way of calculating the S pa-rameters is given. In Sec. III, we apply our calculationsto excitons bound to an isoelectronic trap, using theHopfield-Thomas-Lynch (HTL) model, as developed re-cently by one of us, to model the exciton state, andshow that while the MC model cannot explain the ob-served intensities of the PS s of nitrogen-bound excitonsin GaP, the CC model does. Symmetry selection rules forthe phonons which can be involved in the optical transi-tion are also discussed. Section IV gives a summary ofthe work.

II. FORMALISM

In a crystal which contains an impurity, within theBorn-Oppenheimer approximation, the wave function forthe system has the form

4 „=C& (x,Q)y „(Q),where N~(x, Q) and y~„(Q) are the electronic part andthe phonon part, and x and Q are coordinates of elec-trons and lattice ions, respectively.

For the optical transitions with phonon participation,there are two basic approximations, i.e., the CC and theMC models, in calculating the transition rates.

In the CC model, we first consider the electron-phononinteraction Hzl to the uncoupled states of the total sys-tem I@ y„ I, and obtain the coupled states I@jgj„I. Ingeneral, both the electron states 4. and the phonon states

are modified from the uncoupled states. However,here we will adopt the Condon approximation, in whichthe electronic state is unaffected by the phonon interac-tion so that the optical transition is between two coupledstates 4;y;„(subscript i stands for the initial state) and

4fgf ~ (f stands for the final state).In the MC model, the optical transition rate is usually

calculated by applying second-order time-dependent per-turbation between the initial state 4;y„and the finalstate N~y„. . This is, in fact, equivalent to another pro-cedure: first apply HEI to the electronic parts I@J.I ofthe uncoupled states I N y„} to yield coupled states

0 J 7f

I4 y„ I (note that the phonon parts are unchanged); andthen consider the optical transition between the initialstate N;g„and the final state Nf/„.

The CC and the MC models, as used here, both assumelinear electron-phonon coupling, but differ as follows:the CC model considers the electron-phonon interactionto infinite-order perturbation in the phonon states, butonly to zeroth-order perturbation in the electron states(the Condon approximation). The MC model treats theelectron-phonon interaction as a perturbation in the elec-tron states, but not in the phonon states. In this sense,they are two distinct mechanisms. However, in principle,the CC model can go beyond the Condon approximationby taking into account the effect of the electron-phononinteraction on the electron wave functions, as, for exam-ple, in the theory of multiphonon nonradiative. transi-tions. If this is done, there will be an MC-like contribu-tion in the optical transition matrix element; but this is ahigher-order effect. We are not going to pursue this ex-

According to the theory of Huang-Rhys and Pe-kar, '"' with linear electron-phonon interaction, the op-tical transition rate for the NP line is

F«( W;r) =(2m/fi) IM. ;r Ie (2)

where 8';& is the separation of the electronic states in therelaxed lattice; i.e., the transition energy of the NP line,M;r =

& 4&lHzz l @, ) is the transition matrix element inthe Condon approximation, and Hz~ is the interaction ofthe electrons with the external electromagnetic field. Forthe transition emitting one phonon fico, the correspond-ing transition rate is

F«( W;, r~, ) =(2~xX—) lM, , l'e 'S, -

where S=XqS . S is the so-called Huang-Rhys parame-ter for the vibration mode Am and is defined as

S, =[I&(&~,)']I & +IIU, Ic~ &—&@', IU, Ic', &I', (4)

where U is the potenti. al energy of the electron-phononinteraction.

In general, the PS structure is determined by the densi-ty of states of the phonons g(co) and the spectrum of S~.For an explicit calculation of S, we must be able to cal-culate the matrix elements &N l U~l@J ) for both initialand final states, which requires the knowledge of the im-purity state wave function and of the electron-phonon in-teraction. The selection rules for the possible phononmodes involved in the PS's are determined by whether&4~l Uq l@~ ) —

& 4; l Uq l+; ) is nonzero. Up to thispoint, we have talked about transitions of the total sys-tem, and the transition rates are given in terms of many-electron quantities. In most cases, however, they can becalculated in terms of one-electron quantities, as we do inthis paper.

Now we are going to apply the theory to an excitonbound to an isoelectronic (i.e., electrically neutral) impur-ity. To do so we use the HTL model of the exciton. Inthis model the impurity is assumed to exert a short-rangeattractive potential on one type of carrier: in the case ofnitrogen in GaP this is the electron. The electron boundby this potential produces an attractive Coulomb field forthe hole, which thus occupies an acceptorlike state. Thismodel has been shown to account quantitatively for theenergy-level structure of excitons bound to nitrogen pairsin GaP, ' and has been used to obtain the wave func-tion of the exciton bound to isolated nitrogen.

The electron bound state in the HTL model can begenerally represented by a sum of Bloch states y„k,

00 i ~nk nkvd nk (5)

tension in this paper, but we must point out that such anextension might in fact be necessary to explain some finestructure in the observed spectra, such as the very weaksharp peak at the TA(X) phonon energy riding on thebroad PS's in the excitation spectrum of GaP:N. In ei-ther model, we may have the same mechanisms ofelectron-phonon interaction [Frohlich (polar), deforma-tion potential, piezoelectric].

A. CC model

6332 ZHANG, GE, STURGE, ZHENG, AND WU

or in k space~k I ~/, I'&y.k luqlq, k & I' (7a)

S, = [1/(&p/q )']I&„„&ka„"k+qa„k& q. k+, lu, Iq „k &

—&k I &k I'& q „/, Iu, lq.k & I' (7b)

where A & is the Fourier transform of the envelope func-tion F(r ) for the hole motion, /p, k is the valence-bandstate„H, „and uq are the one-electron operators corre-sponding to HE+ and Uq for many electrons.

From Eq. (7) we can see how the electron and the holeof a bound exciton contribute to the exciton-phonon cou-pling. Because the matrix element & p, k I uq I g,k ) is zerounless q =0, the hole contribution is negligible for a deepimpurity center ' and S is only determined by theelectron bound state. Equation (7) has taken into accountthe Coulomb correlation between electron and hole, so itis Inore general than the well-known result ' where theCoulomb correlation is neglected.

If we use the one-band approximation for the electronbound state, and neglect the contribution from the hole,we have

Sq =[1/«p/q)')I&kak+qak &/p k+qluq I'P k ) I

In practice, in order to calculate the Huang-Rhys pa-rameter S, we need to know the detailed wave function,given by a„&, as well as all the scattering matrix elements& y„.k+q luq I/p„k ). Therefore the shape of the PS's arenot only determined by the density of states (DOS) of the

I

If the acceptorlike bound exciton state is taken as the ini-tial state, and the ground state of the crystal as the finalstate, we have

M'/ ~ kafka k& p klH„lq. k )

S, = [ 1 /( A'p/, )']I & y',

I u, I @,'

&

In the MC model, H~ =HEL +Hz~ is taken as the per-turbation to the uncoupled state [ N,.y„]. The first-orderperturbation theory gives the transition rate of the NPline

FMC ( Wq ) = (2~/R ) I Mr I(9)

where 8'of is the separation of the uncoupled electronicstates, and the matrix element M;f is the same as that inthe CC model. The most obvious difference between (9)and (2) is the exponential factor e . They are equivalentonly when S && 1 (very weak lattice relaxation). Also, theNP line energy is different. The CC model includes thelattice relaxation and gives a polaron correction, whilethe MC does not. This is important in determining theposition of the NP line, because not only the Frohlich in-teraction but also the other interactions (e.g. , deforma-tion potential) contribute to the polaron correction.While the correction due to the Frohlich interaction isnot very much different for band-edge states and boundstates, that due to the deformation interaction can bevery different and can affect the binding energysignificantly.

The second-order perturbation gives the transition rateof the one-phonon transition between the acceptorlikebound exciton state and the ground state (see the Appen-dix),

phonons, ' but also by the electronic wave function andthe scattering matrix elements. For interaction with abulk mode, uq is the same as in the perfect crystal, sincethe change of bulk modes induced by introducing a finitenumber of impurities is vanishingly small. ' For a local orquasilocal mode, however, u is determined by eachspecific impurity center. In the above theory, u is notlimited to any specific interaction mechanism.

B. MC model

FMC(E hp/ )= —IM„ I g A,„(k) g A,*„(k)Ak g a„+k2' & p, k I u, I /p. ,+ k &

(10)

where M,„=& y, plH, „I@,p ), E is the bound exciton state,E„is the ground state of the free exciton at I associatedwith the first conduction band, and A,„(k ) is the Fouriertransform of the envelope function F,„(r ) for the free ex-citon. We have assumed that the I state is the onlysignificant intermediate state. For phonons with a wavevector near the conduction-band minimum k o, the transi-tion rate given by (10) cannot exceed the transition rate ofthe indirect excitons, and will not greatly change fromone kind of impurity to another.

In calculating the transition rate of a PS rnicroscopical-ly, the MC model only considers the scattering processesbetween the k =q point and I . The CC Inodel includesthe contribution from scattering between two generalpoints k and k', as long as k' —k =q and the process is al-lowed by the symmetry selection rules. For the MC mod-el, the weaker the localization of an impurity state (i.e.,

the more concentrated in k space), the greater the ratio ofthe PS to the NP line in an indirect-gap semiconductor.For the CC model, the tendency is the other way: themore localized the state, the stronger the lattice relaxa-tion will usually be, and thus the stronger the PS be-comes On the other hand, for the limiting case of a verydelocalized state a &

-5& &0, the transition rates of the NPand the PS are zero in the CC model (within the Condonapproximation) because a„is zero, while in the MC mod-el, they are nonzero for the PS as a is not zero. In theCC model, the NP line and all the PS's have the samematrix element M;f, and the total transition rate is distri-buted among the NP line and the PS's. In the MC mod-el, the NP line and the PS's are independent, since PS'sare possible regardless whether or not the NP line ap-pears, as long as a q and the scattering matrix element& Ip pl uq I p q & are not zero.

47 PHONON SIDEBANDS OF EXCITONS BOUND TO. . . 6333

From the above argument we conclude that in the caseof impurity-involved optical transitions, the MC model isonly applicable for very delocalized impurity states,where S ((1 for all the vibration modes %co .

A concrete calculation of S is rather difficult. Rid-ley did a simplified calculation for some materialswhich shows how S depends on the degree of localizationof the impurity states. In the next section, we will calcu-late S for the PS's of the A line in GaP:N.

III. THE PS's OF W-BOUND EXCITONS IN GaP

A. Is the MC model applicable to GaP:N?

Photoluminescenee Intensity

0 20 40 60 K 100 120 140

120

- 2 LO (F')

A - X - LO (I ) 100

A-LO-LA

-LO- TA

A-LO(1 )

A-X

The phonon sidebands of N-bound excitons have beenmeasured several times since the first investigation ofGaP:N (Ref. 33) and are shown in Fig. 1 (Ref. 11) (thisspectrum is taken at 4.2 K, where 3-line emission ispredominant). Those PS's which are believed to be asso-ciated with bulk modes are labeled LO(I ), X, TO(I ),LA, and TA. TA and LA are broadbands, while LO( I ),X, and TO(I ) are sharp, TO(I ) being very weak. Com-paring the PS spectrum to the phonon DOS, we findthat all the PS peaks are close to, if not exactly at, thepeaks of the DOS. The relative intensities, however, arequite different from the DOS. Furthermore, not all the

peaks of the DOS show as peaks in the spectrum. The Xband at 48.4+0. 1 meV (Ref. 13) is close to a DOS peak atabout 48.6 meV, and the LA band at 26.7+0.4 meV(Ref. 13) is close to a DOS peak at about 27 meV. TheTA band at 13.3+0.2 meV (Ref. 13) is slightly higherthan the nearest DOS peak at about 11.4 meV, but theDOS corresponding to this PS peak is still quite large.The same bulk phonons are seen for both isolated N andNN, pair centers. ' ' A local mode sideband has alsobeen observed for most NN, pair centers.

As we have already discussed in Sec. II, the MC modelshould be only applicable to very delocalized impuritystates. For the localized impurity states associated withnitrogen in GaP, in some key aspects, the MC model can-not be consistent with the experimental results for thefollowing reasons, some of which have been mentionedbefore.

(1) The PS's of excitons bound to neutral shallowdonors and acceptors ( A X and D X) provide good ex-amples of MC transitions. ' In the first row of Table I,we give the spontaneous emission probabilities (in orderof magnitude) of the NP lines of A X, D X, and N-boundexcitons in GaP, obtained from the absorption strengthof the NP line through the Einstein relations. " Inthe second row we give the spontaneous emission proba-bilities for the PS of each exciton, obtained from that ofthe NF by measuring the relative contributions to thephotoluminescence intensity, and also an estimated resultfor the indirect free exciton which is taken from a calcu-lated value for Si. The probabilities are all in the orderof 10 —10 sec ', except for the N-bound exciton whichis higher by a factor of over 10 . A numerical estimatebased on the MC model, made in the Appendix, showsthat the predicted spontaneous emission probability forthe PS of the N-bound exciton is of the order of 10sec

(2) The "X" band at 48.4 meV (Ref. 13) cannot beidentified with the MC LO phonon at the X point, 'whose energy is 46.6+0.2 meV. ' ' Furthermore, cou-pling to the LO(X) phonon is forbidden by symmetryselection rules which we will discuss later. Even if wetake into account the fact that the conduction-bandminimum is not exactly at X (Ref. 42) so that the MCphonon is on the 6, line, the energies do not agree,since the dispersion along 6& near X is very small.

(3) In the MC model calculation, the line shape of theTA and LA PS's is quite sensitive to the electron bindingenergy. The "near-resonant effect" for some com-ponents a& of the bound electron wave function is impor-tant in the explanation of Dai et aI. of the strong

A-TA

0

TABLE I. Spontaneous emission probabilities of NP and to-tal PS for A X, D X, and N-bound excitons in GaP, and the in-direct free exciton in Si (in sec ').

FIG. 1. Photoluminescence spectrum of the isolatednitrogen-bound exciton in GaP {T=4.2 K, A line at 2.317 eV.Excitation: 2.328 eV, [N]=10' cm 3) (Ref. 11).

NPPS

D'X

—10—10

N-boundexciton

9.8X10'1.1X10

Indirectfree exciton

5X 10

6334 ZHANG, GE, STURGE, ZHENG, AND WU

B. CC model in GaP:N

The symmetry selection rules for phonons involved inthe CC model are mainly determined by the matrix ele-ment (y;iu~ig, ) in Eq. (7) for a localized bound state.For an A

&symmetry state of P-site impurity in GaP,

only LA(X), LO(L ), and LA(I. ) phonons are allowed ifonly phonons at X and L points are considered. How-ever, phonons elsewhere in the Brillouin zone are al-lowed (Table II). These phonons could make asignificant contribution to bound electron-phonon cou-pling for a deep impurity state since its wave function iswell extended in k space. Also, when the conduction-band minima are not located at the zone boundary, as inGaP, particular phonons, not precisely at a symmetrypoint, could be important. From Eq. (48) we can see thatan intervalley or intravalley scattering which connectpoints k and k+q with large electron wave-function am-plitudes of ak and ak+ will give a large contribution tothe coupling constant S . For the N-bound exciton thewave function of the bound electron is 80—90% in the vi-cinity of X, the remainder being near L, with less than1% near I . ' Hence X—X and X i. scattering will b—edominant, while the contribution from X—I or L —Iscattering will be negligible. In contrast, in the MC mod-el only X—I scattering is considered.

The origin of the "X"PS's at 48.4 meV in GaP:N hasbeen unclear since the earliest studies. Here we will dis-cuss a few possible origins of this PS. Our general con-clusion is that while the PS at 50.1 meV labeled LO(l )

comes from small-q LO phonons coupled by the Frohlichinteraction, "X" includes contributions from the entireLO-phonon branch, and, like the TA and LA PS's (con-sidered below), is due to deformation potential interac-tion.

From Table II, the LO(L ) and LA(L ) phonons are al-

TABLE II. Symmetry allowed bulk phonons in coupling to aP site. A

&electron bound state for CC model in GaP (the origin

of coordinates is assumed at P site) (Refs. 27 and 45).

LOTOLATA

L,

Li

XIXiXiXi

LO(X) band (i.e., the X band) in terms of the small elec-tron binding energy ( ( 10 meV) of the A line. However,all NN; pair centers have almost the same PS struc-ture, ' ' while their electron binding energies are verydifferent, being as large as 125 meV for NN„so thenear-resonant e8'ect cannot be of any importance in thesecases.

(4) In GaAs:N under pressure, a similar doublet struc-ture of the LO PS was observed for NN, (Ref. 43) andisolated N (Ref. 44) centers, with a PS/NP ratio compa-rable to that in GaP, even in the pressure range where theband structure is direct. In this case, the I -X mixing isvery strong and the MC model would require the NP lineto be strongly enhanced relative to the PS's.

lowed for an A&

state. If we consider the selection rulesfor deformation potential scattering, " ' the LO(L) andLA(L ) phonons can be coupled by intervalley scatteringbetween the X, and L, conduction-band states. LA(X) isallowed and LO(X) is forbidden both by the A, impuritysymmetry and by the selection rules for intervalleyscattering between two nonequivalent X& points belong-ing to two "stars". In GaAs, on the other hand, LO(X)is allowed, and LA(X) forbidden, since M~, )Mo, )M~.However, coupling to an LO(X) phonon, which can con-nect two nonequivalent valleys on 5&, is allowed underboth selection rules. This could be important because ofthe spread of the wave function in k space, and theconduction-band minima are not precisely at X but at(0,0, 1 —5) points on the b, line, ' with 5=0.08, which wecall "b.5." A phonon connecting two such minima (theso-called f process ' ) has a wave vectorq=(1 —5, 1 —5,0) which is equivalent to (5, 5, 1), i.e., aphonon with X, symmetry and q 6.45' off [001]. Thephonon energy at that point is about 47.8 meV (Ref. 35)which is only 0.6 meV from that of the X band. Howev-er, the peak position could be mainly determined by theDOS of the phonons. Also, the X band is close to the en-ergy of the LO(L) phonon, and it is likely that bothh5 —h5 and 65—L intervalley scattering contribute to it.A5 —A5 scattering is favored by the electron wave func-tion, but near (5,5, 1) the phonon DOS is relatively low,so that b5 —L scattering is favored by the DOS. Notethat the A5 —h5 scattering matrix element is not zeroeven though it is zero between exact X points, and thematrix elements may increase rather fast when leavingthe critical points. '

Another type of possible scattering between two X val-leys is the so-called g process, ' which is allowed andrequires phonons on b with q near (0,0,0. 16). Howeversuch near zone-center LO phonons are not appreciablyshifted from LO(l ) at 50. 1 meV, so the contribution ofthese phonons would be indistinguishable from theLO(I") band. As neither the phonon energies at generalk points nor the deformation potential constants for twogeneral points are available, we are not able to make amore quantitative judgment at the present stage.

For the LA PS, the f process A5 —b,5 scattering by a(5,5, 1) LA phonon would give a peak at —30 meVwhich is not observed. The peak of the LA PS at26.7+0.4 meV (Ref. 13) agrees with the LA(i. ) phonon26.7+0. 1 meV. From the electron wave function, theintervalley scattering matrix elements, and the phononDOS, the contribution of A5 —h5 scattering to the LAPS's should be an order of magnitude larger than that ofX—L scattering by L phonons. The fact that this is notthe case remains to be explained, as does the fact that theLA PS is much weaker than TA and X PS's, when theoverlapping contribution of two TA phonons is subtract-ed. ' In general, LA PS's associated with localized im-purities in GaP have lower energies than the 31.7 meVLA(X) phonon, which appears in shallow impurity spec-tra. ' For example, in the spectra of donor-acceptor(D-2 ) pairs involving deep impurities the LA PS are allin the range 27.7—29.5 meV. For the TA PS of the Aline, which peaks at 13.3 meV, ' Table II show that TA

47 PHONON SIDEBANDS OF EXCITONS BOUND TO. . . 633S

where a is the Frohlich constant, m* is the effectivemass, and V is the crystal volume. From (7a) we thenhave

S = (1/E —1/c,o)F(q)/qAcoLo V

(12)

phonons from high symmetry points and lines are forbid-den except those from the X, branch. The fact that thisslightly exceeds the TA(X) phonon energy (13.0+0. 1

meV (Ref. 37) again suggests that b,5—b,5 scattering isimportant.

The situation in GaAs:N under hydrostatic pressure isvery similar to that in GaP:N. All the main features,LO(I ), X, TO(I ), LA and TA, appear in the spectra forthe isolated N center and the NN; centers. The pho-non energy for the X band is 33.5+0.2 meV at atmos-pheric pressure. This is close to a peak in the DOS of theGaAs phonons at 33.7 meV which appears to corre-spond to a maximum in the LO(X) branch. Again the fprocess intervalley scattering between nonequivalent Xva11eys apparently makes a significant contribution to theexciton-phonon coupling.

The Huang-Rhys parameter can be calculated either byusing (7a) where a bound-state wave function in realspace is used, or using (7b) where the wave function in kspace is used. Some calculations have been done for vari-ous types of bound electron wave functions in real space:billiard-ball-like, hydrogenic, "5 function, " quantumdefect and Gaussian, ' or spherical square well. Theso-called "5-function-type" is actually an approximationto the spherical square well, obtained by ignoring thecontribution to the coupling of the wave function withinthe well (there is no bound state for a true 5-function po-tential in three dimensions). There are some disadvan-tages in using a real-space bound-state wave function.For example, one has to use a single effective mass in allthese wave functions, which is not a good approximationwhen one has a deep impurity state with its wave func-tion well extended in k space. On the other hand, to per-form the calculation in k space is usually more difficultbecause one needs to know the detailed properties of theelectron bound state a„k and all the scattering matrix ele-ments (q „„+,I u,. I q „„).

We will consider two kinds of electron-phonon interac-tions: Frohlich and deformation potential, and then ap-ply them to the N-bound exciton in GaP. For theFrohlich interaction, we will calculate the S parametertwo ways: in real space and k space. For the deforma-tion potential, we will work only in real space.

For the Frohlich interaction, we have1/4

u =i(AcoLo/q) &4vra/V exp(iq. r),2m *cuLo

qP, =(13/2m. )'~ exp( P—r )/r, (14)

where P=(2m*E)'~ /fi, E is the electron binding energy.F(q ) is given explicitly as

F„(q ) = [~/2 —arctan(2P/q ) j (2P/q ) (15)

If the k-space electron wave function is used, sinceonly modes with small q are important for the Frohlichinteraction, F(q ) is approximately given by

Fk(q)= ls(q)l (16)

where s(q)=deka *kaz q, which is roughly the Fouriertransform of the bound electron probability density

s(q ) was calculated in Ref. 23 by using a Koster-Slater one-band one-site model and in this case it de-pends only on Iql.

The contribution from all LO phonons is

2

7TAco i-(1/E„—1/eo) I F(q)dq,

0

where F(q) is given by F„(q) or Fk(q). Q=(6m N/V)'~(=1.14 A ' for GaP) is taken as the cutoff wave vector.Only contributions from small q are important in (17).

Introducing a bound-polaron radius rp, so that

2S=

26r0 AcoLo

where rp is defined as

ro ' =2/m I F(q)dq

and 1/e= 1/c, —1/eo. For GaP, e =9.04, a~= 11.1;m ' in P takes the average value for the X valley,m*=0.365mp; A~Lo takes its I point value of 50.1

meV. '

We have calculated S for the Frohlich interaction fordifferent electron binding energies by using F„(q) orFk(q) in (17), as shown in Fig. 2. For example, if theelectron binding energy for an isolated N-bound excitonis taken as 6 meV, we obtain an S parameter of 0.195 or0.175, respectively, which agrees well with the experi-mental value for the LO(l ) sideband of 0.20+0.02."This good agreement supports the result for the electronwave function obtained in Ref. 23. The correspondingbound polaron radius given by (19) is 15 or 17 A, which isabout the same as the polaron radius for a conduction-band state at X point defined byr~=(A/2m*coLo)' —14.5 A, since the bound electronwave function is, in fact, mostly concentrated in the Xvalleys.

For a deformation potential interaction, we have

where F(q) is defined as u =[I/(2NMcu ))'~ D exp(iq. r), (20)

F(q) =I & q';I exp(iq r)lq'; & I' . (13)

We take the real-space wave function as the 5-functiontype2 1

where D is the deformation potential, N is the totalnumber of unit cells, and M is the total mass in the unitcell. To perform the calculation in real space, we have

6336 ZHANG, GE, STURGE, ZHENG, AND WU 47

0.30—

R 0.25-O

O~ 0.20-O

6"O.l5-

~ 0.10—

0.05-

I I I

4 6 8Electron Binding Energy (mev)

I

10

FIG. 2. Calculated Huang-Rhys parameters of LO-phononsidebands vs the electron binding energy for the isolated nitro-gen bound exciton in GaP. Full line: Frohlich interaction, k-space wave function. Dotted line: Frohlich interaction, real-space wave function. Dashed line: deformation-potential in-teraction, real-space wave function.

X[D, /'$, = ', F(q) .

(2NMco )(fico )(21)

If a 5-function type wave function is used, F(q) is thesame F„(q) given by (15). Using the k-space wave func-tion, we have

2 g aj', +qa&D(q, k), (22)(2NMco~ )(ficoq )

where (k+q~uq ~k) =[A'/(2XMco )]' D(q, k), andD (q, k ) is the deformation-potential constant defined inRef. 53.

In principle, (22) is more accurate than (21). However,it is much more difficult to do the calculation in k spacefor deformation-potential interaction than for theFrohlich interaction, because the summation over k spacein (22) includes both intervalley and intravalley scatteringprocesses and the small-q approximation cannot be made.To evaluate D(q, k) for each scattering process is alsovery tedious. Here we will estimate S for LO, LA, andTA PS's, using the real-space wave function. From (21),we have

$=2 3 f F(q)q dq,

4m Mfgco0(23)

where 0 is the volume of unit cell, and we have assumedan average value ~D

~coo for ~Dq ~ co& . D is an

"effective" deformation-potential constant and ~0 is thepeak frequency of the PS. The approximation is expectedto be rough, especially for acoustic phonons. The peakenergies for X, LA, and TA sidebands are 48.4, 26.7, and13.3 meV, respectively. ' We choose DLo=6. 5 eV/A,DLA=1 eV/A, and DT'S=1 eV/A, which are typicalvalues of deformation-potential constants in III-V semi-conductors. The calculated result for LO or X PS isshown in Fig. 2 for the range of possible binding energies.Specifically, for a 6-meV binding energy, we have$(X)=0.19, $(LA) =0.03 and $(TA) =0.22. Thesevalues are in reasonable agreement with experiment. "

The above calculation assumes that the f process b,5—b,5intervalley scattering dominates, since the wave function(14) is for a single valley and we have given m * its X val-ley value. For the reasons discussed above this approxi-mation might not be valid for all the PS's, but the resultsindicate that the CC model does give the correct order ofmagnitude for the transition rates of the PS's.

C. Discussion

Based on some general arguments and comparisonsamong different impurities, we have shown that in theMC model the PS transition rate of an impurity boundexciton is limited to the corresponding intrinsic indirecttransition rate, and the model cannot account for the ob-served PS of localized excitons. It is not surprising thatthe MC (Refs. 5 and 8) model gave a spectral shape of thePS in reasonable agreement with the experimental resultsfor GaP:N because this shape is mainly determined bythe DOS of the phonons (in Refs. 5 and 8 the overallPS/NP ratio was scaled to fit the experimental data).

In the MC model, the absolute transition rate of the PSwill not change very much for different impurity centers,but the ratio of PS to NP can change by orders of magni-tude, since the ratio of PS to NP is directly related to theratio ~a~~ /~ar~ and ar is strongly dependent on the im-

purity. On the other hand, in the CC model, ~a~ ~ /~ar ~

may change without significantly affecting the S parame-ters and hence the PS/NP ratio, so long as the contribu-tion to the bound-state wave function from valleys withlarge density of states, usually X valleys, does not changemuch. This explains qualitatively why the S parametersdo not vary significantly from the isolated X center to theNX& pair center.

Although the application of the MC model is done inthe one-band approximation, the PS transition rate willnot change by orders of magnitude if the multiband mod-el is used, even if the contribution from the higher con-duction bands is of the same order of the magnitude, ascan be seen from expression (10). In fact, as pointed outin Ref. 8, the contribution from higher conduction bandsis negligible because the transitions involving the higherconduction bands are very weak and the major contribu-tion to the impurity wave function comes from the lowestconduction band. On the other hand, the NP transitionrate, and hence the PS/NP ratio, depends on az, whichwill change a lot if the multiband model is used. In themultiband model ar is larger than in the one-band mod-el, ' thus increasing the discrepancy between the MCmodel and experiment.

Contrary to the suggestion of Ref. 15, neither CC norMC models predict that the ratio of PS's to the NP linewill change significantly with temperature ( T ( 150 K forGaP:N), apart from the effect of phonon occupation num-ber 20

IV. SUMMARY

We have clarified the distinction between the CC andMC models in describing impurity-related optical transi-tions and the relationship between them. We conclude

47 PHONON SIDEBANDS OF EXCITONS BOUND TO. . . 6337

that the MC model can only be used for shallow Coulom-bic impurities, otherwise the CC model is a better ap-proximation. In the CC model, Huang-Rhys parametersfor bulk phonon modes coupling to a bound electron orexciton can be calculated in terms of the bound-statewave function in k space and the phonon-induced inter-valley and intravalley electron scattering processes of theintrinsic crystal.

We have studied in detail the PS's of nitrogen-boundexcitons in GaP and their selection rules, and calculatedthe Huang-Rhys parameters, within the CC model. Weaccount for the position and strength of the four mainfeatures of the PS s: LO(I ) is due to the Frohlich in-teraction and its calculated S parameter agrees well withthe experimental results; X, LA, and TA are all due tothe deformation-potential interaction, and their estimatedS parameters are also in agreement with experimental re-sults. We have shown that the MC model, which hasbeen used in an attempt to explain the PS of nitrogen-bound excitons in some previous work, is inapplicable tothis case.

ACKNOWLEDGMENTS

We thank Professor J. S. Pan, Professor R. Z. Wang,and Professor B. S. Wang for very helpful discussions andcritical comments on an early draft of the manuscript.Y.Z. would like to thank L. L. Yan Voon for valuablediscussions. The work at Dartmouth College was sup-ported by the U.S. Department of Energy under GrantNO. DEFG 0287ER45330 and that at Xiamen Universityby the National Science Foundation of China.

FMc( 8';t —A'coq ) =Q 2'J

Mj(m+((, f(m+1Wim, j(m+()8;—Am

(A 1)

where

=&e,'x' IH I@,'x' &,

wj j,.=&@,''y'. .la„le,'y'. & .

If a bound exciton state with energy level E= 8',f isconsidered, only the first term in (Al) is important, sincethe second term tends to have a much larger denornina-tor. Then we have

&eflH,„le,'&&@,'lU, le,'& '+Mc«=&~, ) = 8'.; —Ace

(A2)

In this case, we choose a free-exciton state 4 as the in-termediate state, where j= ( m, lt,„)and m is the quantumnumber of the exciton states. It can be shown that theground state of the I free exciton is the most effective in-terrnediate state. Therefore, we get Eq. (10) in the maintext for the transition rate or, correspondingly, the spon-taneous emission probability ' is given as

APPENDIX

In the MC model, the PS transition rate can be calcu-lated by second-order perturbation theory, i.e.,

~~q)Ep &q' i l "qlq' q+i )O'Mc(E —A'co )= g A,„(k) Q A,'„(k)A(, ga„q+(, (A3)

where m is free-electron mass, n is the fine-structure constant, E is the energy constant associated with the momentummatrix element between the conduction band and the valence band, and all the other parameters are defined in the maintext. Using the one-band approximation, and assuming that aq+i, and & y, i, lu li((~ +i, ) are relatively slow varying func-tions of k compared to A,„(k ) and Az, for q near the conduction-band minimum, we have approximately

2a(E —m~, )E. . . & g „lu, lq, )~Mc(E —&~q )=, g A,„(k )

' g A,*„(k) A„' a

3a(E frco0)E-O'Mc =

26A'c m &ex 2M boa

ID(q0) I'

(E,„E—Acta)— (A5)

We want to estimate the O'Mc for the isolated nitrogencenter ( A line) in GaP:N. For simplicity, we only consid-er the LO phonons. The total spontaneous emissionprobability WMc is the summation of (A4) over allmodes, and it is given approximately as

where Cz is the summation ofla

lin X valleys,

/=X(, A,*„(k) Ai„a is the lattice constant, a,„ is the ra-dius of the free exciton, D(q0) is the deformation poten-tial for k=qp to k=0 intervalley scattering process asused in (22), and coa is the frequency of the LO phonon.For the A line, Cx=0.94 and /=0. 63. a,„=72.6 A[the direct free-exciton binding energy is about 9 meV(Ref. 62)] and Ep =22.3 eV. fi/(2Mco0)=8. 3X10 ~ Aand D(qa) =4 eV/A. With these values, we get8'Mc(LO-1. 5X10 sec '. This value is two orderssmaller than the experimental result, but of the same or-der as that for shallow impurities, which is what we ex-

6338 ZHANG, GE, STURGE, ZHENG, AND WU 47

pected for the MC model. For the ratio of PS/NP, theone-band model gives a value a factor of 10 smaller thanexperiment, and may be overestimated due to the inaccu-racy of aI-. In fact, with az from the one-band model,the oscillator strength of the 3 line is about one ordersmaller than the experimental result. By considering this,

the ratio will be of the order of 1X10, which againgives the spontaneous emission probability of MC PS inthe order of 1X10 (sec '). If at. from the multibandmodel is used, we should expect that the ratio is in the or-der of 1X10 since az is about three orders larger thanthe one-band model. '

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