PHYSICAL REUIE%' B VOLUME 37, NUMBER 9 15 MARCH 1988-II

Phonon-assisted exciton tunneling in GaAs„p, „:ND. Gershoni' and E. Cohen

Department ofPhysics and Solid State Institute, Technion Isr-ael Institute of Technology, Haifa 32 000, Israel

Arza RonDepartment of Chemistry and Solid State Institute, Technion Isra-el Institute of Technology, Haifa 32 000, Israel

(Received 10 June 1987)

The low-temperature photoluminescence of GaAs„Pl „.N (x &0.07) crystals has been studiedunder selective excitation into the nitrogen-related exciton band (Nz). The spectra consist ofseveral features whose relative intensity depends on the excitation energy. Excitons created on non-transferring (terminal) states give rise to a sharp line (8 ) 0.9 meV below the exciting-laser line. Ex-citons in transferring states tunnel to lover-energy states by acoustic-phonon-assisted processes.This results in broad phonon sidebands. We introduce a model for exciton transfer and fit both theband shapes and their relative intensity. %e thus 6nd that the most probable tunneling process in-volves single-phonon emission and that the dominant interaction mechanism between excitons andzone-center acoustic phonons is the deformation potential. The fit to the experimental spectra ex-cited at various energies in the Nz band yields the ratio between the densities of transferring andterminal states.

I. INTRODUCTION

The band of 1S excitons bound to impurities in alloysemiconductors is inhomogeneously broadened, re6ectingthe distribution of exciton binding energies. ' This dis-tribution is due to the random potential fiuctuations pro-duced by the local configurations of atoms around the im-

purity centers. It has been well established that excitonscan tunnel between the localizing impurity sites. Thisgives rise to a shift between the absorption band and theluminescence band at low temperatures, as was first dis-cussed by Mariette, Chevallier, and Leroux-Hugon forexcitons bound to nitrogen in GaAs„P, „(Nxband).Time-resolved experiments (in the subnanosecond range}confirmed that tunneling takes place, ' and led Kash topropose that it ceases when the exciton is trapped on aterminal state (where it can only recombine within its ra-diative lifetime}. The density of terminal states (whichare a subgroup of the total density of nitrogen-bound ex-citon states) has been measured by selective excitation ofluminescence and resonant Raman scattering forGaAs Pl „.N.

In the present study we address ourselves to the energydependence of the tunneling within the Nz band inindirect-gap GaAs„Pi „(x(0.07). The analysis of aseries of selectively excited luminescence spectra indi-cates that the tunneling is accompanied by the emissionof acoustic phonons. These are mainly zone-center pho-nons which interact with the exciton by the deformationpotential (DP). An additional contribution comes fromTAz phonons. %hile the spectral shape of the densitiesof N-bound excitons and the subgroup of terminal stateswere previously determined, to obtain their relative mag-nitudes we need an analysis of. the tunneling processes.

The paper is laid out as follows. Section II describesthe experimental procedure and results, Sec. III describes

the model for exciton tunneling within the Nz band andthe comparison between the calculated and observedspectra. A summary and comparison with other semi-conductor alloys is given in Sec. IV.

II. KXPKRIMKNTAL PROCEDURE AND RESULTS

Two bulk GaAs, P& „crystals were studied. One,with x=0.02, was doped with nitrogen to a level of about10' cm . The other, with x=0.07, was nominally un-

doped but contained background nitrogen impurities atan estimated concentration of 10' cm . Luminescencespectra were excited with a dye laser, having a linewidthof 0.12 A. The laser intensity impinging on the sampledid not exceed 0.1 W/cm . The emission spectra weremonitored with a double monochromator having a reso-lution of 0.05 A. The crystals were immersed in pumpedliquid He, Figure 1 shows the transmission spectrum ofthe GaAso o2Po 98 crystal. Also shown is its photo-luminescence spectrum excited above the band gap. TheNz band and its acoustic- and optic-phonon sidebandsare typical of all indirect gap GaAs„P& .'N cIystals. '

Figures 2 and 3 show a series of photoluminescence spec-tra (solid lines) which were selectively excited into the Nxband. In each series the exciting-laser intensity was keptconstant and the spectra were drawn so that their relativeintensity was preserved. There are three distinct spectralfeatures (which we analyze in the following section). Justbelow the exciting-laser line is the 8 line which is the ra-diative recombination of N-bound excitons in the J=2state. These were resonantly excited into their J= 1 state(the A' and 8' lines are split by EEt,n ——0.9 meV). Theband denoted FF is due to excitons which have tunneledfrom the resonantly excited states while emitting a low-qacoustic phonon. The band denoted Nz is due to a simi-

Qc1988 The American Physical Society

D. GHUSHONI, E. COHEN, AND ARZA RON 37

GOAsoua1

l1

I

l

Po98: N

T&2K

Ah~aCI

L, i

G~A~ouv Po.oa 'N

~Nx1

I1 I1 I

I 1

/ ~Slt

r I/

Lor+Lox -'LAx TAx

I

2.24 2.28 2.32PHOTON KNKRG& teV)

FIG. l. The transmission and luminescence spectra ofnitrogen-doped GaAsp p2pp 98. The luminescence was cw excitedat 2.35 eV.

ha

hao

FF„I

S

llI)t~

ItI

II I

I

I

1 g

8'

~o.ol Po.ol ' Nl

2.240 2.2' 2~~0PHOTON ENERGY (yV)

tI

2.280

FIG. 3. Same as Fig. 2 but for GaAsp p7pp 93 (nominally un-

doped).

heO

~~Chhea

LLl(Jx'lal

Vl

l

2.2 80

FFI i

FF

I I

2.290 2.300PHOTON ENFRGY (eV)

8'."

2.3 'lO

lar process but the emitted acoustic phonons are from theBrillouin-zone edge (TAz ).

III. MODEL AND DISCUSSION

According to our present understanding of N-bound excitons in GaAs„P& „,an exciton selectivelycreated at an energy E, (within the Nx band) can be ei-ther in a terminal or in a transferring state. (A schematicdescription is shown in Fig. 4.) An exciton in a terminalstate with E, will recombine radiatively with a lifetime~&. The probability of it tunne1ing out of a terminal statewithin v& is negligible. At the same energy E, there arestates from which the exciton can transfer into 1ower-energy states Eb, while emitting an acoustic phonon (atT=0). The tunneling rate is given by9

4m, I~. «.» )

I

'f «.b ~a«b )W,b —— J (R,b) E2

FIG. 2. Luminescence spectra of GaAsoo2Pp». N selectively.*xcited within the N& band (solid curves). The gain in all spec-tra is the same, The results of the calculations detailed in Sec.III are shown in dashed curves. N& denotes transfer assisted byTAz phonons.

E,b —E, Eb, J(R,b ) is—the o—verlap integral between thetwo exciton sites separated by R,b, f (E,b ) is the pho-non density of states at energy E,b, and p(Eb ) is the den-sity of excitonic states. The form factor of theexciton-acoustic-phoQOQ DP lnte1action ls given by

PHONON-ASSISTED EXCITON TUNNELING IN GaAs„P, „:N 4579

OENQTY OF STATESTABLE I. The parameters used in calculating the

exciton-acoustic-phonon interaction.

GaPGaAs„Pl

x =0.02 x=0.07

gTEANNAL TRANSFERRINS

STATES STATES

c, jc,V, (10' cm/s)V, (10' cm/s)

7?l q

+ Pflg

aq (A)

—1.38'6.3423.748'

0.60'

40'

—1.32'5.142'3.027'

0 70'

—1.386.3183.734

0.6

33+4

—1.386.2583.698

0.6

33+4

FIG. 4. A schematic description of the densities of excitonstates: p(E), solid curve; p'(E), dashed curve. Also shown arethe dynamic processes which the exciton undergoes in atransferring and a terminal state. The spectrum on the rightshows the 8' line and FF band, which are observed under selec-tive excitation.

'D. L. Camphausen, G. A. Nevill-Connel, and W. Paul, Phys.Rev. Lett. 26, 184 (1971).R. Weil and W. Groves, J. Appl. Phys. 39, 4049 (1968).

'S. Adachi, J. Appl. Phys. 58, 871 (1985).G. Beni and T. M. Rice, Solid State Commun. 23, 871 (1977);

P, Lawaetz, Phys. Rev. B 4, 3460 (1971).'These are the efFective-mass values. The value quoted forGaAs„P, , is obtained by our fit to the data.

(q)~

'= (s,a, —e„ab)2,

q =E,b/hv& is the wave vector of the emitted zone-centeracoustic phonon and v& is its velocity. s, is the (averaged)conduction-band DP (s„is the corresponding one for thevalence band). g is the crystal density.

'2 —2m&, qa&+

2MX

where m, (mb ) is the electron (hole) effective mass, M» isthe exciton translatory mass, and az is its Bohr radius.

For piezoelectric interaction (PE) between the excitonand the zone-center acoustic phonon, the form factor is'

2e)4

(a, ab)-2gqU,

e,~ is the piezoelectric coef6cient. The parameters usedin the calculations detailed below are summarized inTable I. Those for the alloys were interpolated from thevalues reported for GaAs and GaP. Since the exciton in-teracts mainly with LA phonons via the DP, and mainlywith TA phonons via the PE interaction, we take the cor-responding sound velocities in Eqs. (1) and (4).

Now, the exciton can tunnel between transferringstates until it is trapped on a terminal state. %e describethese dynamic processes by considering the rate equa-tions for the probabilities P, (t) and PI(t) of finding theexciton on a transferring state (a) or on a terminal state(f) at time t:

dPI(t) PI(t)+C g WbfPI(t)

dt b

(E pE )

dP, (t)dt

P.(t)

W„+ g W,Z P.(t}b

'f

(E )Eb) (E yE~)

+C g Wb, Pb(t) .b

(Eb)E )

The constant C contains all the common factors whichappear in the previous equations, so that it actually mea-sures the tunneling rate in units of rti . In writing rateequations for the probabilities P, (t) of finding the excitonin a state with a given energy E, we make the simplifyingassumption that E,b and R,b are uncorrelated ("micro-scopic disorder" ). This means that W,b dependence on

E,b comes only through the variation of MD and MPE

with Egb ~

There are two densities of exciton states in the set ofcoupled-rate equations (5) and (6}: the density of all N-bound exciton states p(E) and that of terminal statesp'(E). We know their spectral shapes from absorptionmeasurements [which yield p(E)] and from the energydependence of the 8' line intensity and resonant Rarnanscattering by LO» phonons [which yield p'(E)]. p(E}

TABLE II. The Gaussian parameters of the nitrogen-bound exciton densities of states.

0.020.07

2.306+0.0012.271+0.004

hp(meV)

6.920.715+2

Mean p'(e&)

2.302+0.0012.269+0.001

hp'(meV)

5.4+0.5

10.0+ 1

D. GEINHONI, E. COHEN, AND ARZA RON

60Apop Pg96 I N

T ~ 2K

4JQxLLIQCALLIgx

Fxperimen&

—--Form foe)or model

(b)

k

2.28l I

2.29FNFRGY (eV)

I I

2.50

FIG. 5. (a) A comparison between the measured FF band

and that calculated by solving the set of coupled-rate equations.

(b) The same spectrum with a St obtained with the spectralshape of the deformation potential interaction form factor.

and p'(E) are approximated by Gaussians with parame-ters given in Table II. Their ratio

r), =Ip(E)dE Jp'(E)dE

will be determined by the fit to the experimental data.Our goal is to calculate the spectral shapes of,the series

of spectra shown in Figs. 2 and 3 with a single set of pa-rameters. Due to the computational complications intro-duced by such a problem, we Srst solved the set of Eqs.(5) and (6) to fit a single spectrum obtained by selective

excitation into the Nx band. Figure 5(a) shows an exam-

ple of such a fit to the FF band. The separate contribu-tions of the terminal and transferring states to the emis-sion intensity are shown in the figure. It is clear that thecontribution of transferring states to the luminescence in-tensity is negligible. From the values of parameterswhich gave the best fit it became clear to us that (1) mostof the tunneling occurs through the DP coupling mecha-nism, and (2) the transfer rate C is very fast in compar-ison with the inverse radiative lifetime (C~„&10),which is equivalent to only one phonon-assisted exciton-tunneling process. In other words, excitons created byselective excitation into a subset of transferring states andtunnel directly to terminal states by emission of zone-center acoustic phonon give rise to the FF band. Theabove conclusions allow to greatly simplify the calcula-tion procedure. There is essentially no need to solve thecoupled-rate equations. In Fig. 5(b) we present such a fitusing the simplified approach. It can be seen that al-though this fitting procedure required only one free pa-rameter, the exciton Bohr radius az, no considerable de-gradation in the St quality is observed. %e obtained avalue of 33%4 A for this parameter in excellent agree-ment with Leroux-Hugon and Marriete" who obtained avalue of 32 A by fitting the transfer rates of excitonsbound to nitrogen in heavily doped GaP:N. %e wouldlike to remark also that the fit was extremely insensitiveto the parameter e, js, .

Using the above approximation we now turn to calcu-late simultaneously the spectral shapes of the series ofspectra shown in Figs. 2 and 3. In considering this largerspectral range one has to take into account yet anothertransfer mechanism, namely, zone-edge phonon-assistedexciton tunnehng into lower-energy terminal states. Thismechanism manifests itself in the line denoted Nz and itevolves tunneling of excitons into lower-lying states byemitting zone-edge transverse-acoustic (TAx) phonon.While we know the explicit dependence of the exciton-phonon interaction on the phonon wave vector q forzone-center acoustic phonons, we do not known it forTA& phonons. %e, therefore, describe their contributionto the transfer processes by a Gaussian function denotedNz(E). The parameters defining this function are themean TAx-phonon energy ( AcoT~) and the width of thisband. Both are obtained by the fitting procedures.

The explicit expression used for the fluorescence inten-sity at photon energy F. when selectively excited into theNz band at energy E& is given by

I(E,EI )= Jp'(E')dE' p'(Ei )8(E (Ei AE„a),o's )— —

+[N(Ei)] C «i) S'«i) 'p—'(E)

[F (EI E)+i)2F (EI E—) + i)3N~ (EI —E—) ][p(E') p'(E')]&E' p'—(E')&E'

in Eq. (7). We denote by 8 (E,cr~ ) the line shape of the 8' line: a Gaussian centered at E and width of o ii. The nor-malized spectral shapes of the DP interaction form factor are given by

~MD'(E., ) ~'(E )= (8)I (M (E,i, )

~dEgi,

37 PHQNON-ASSISTE~W EXCITON TUNNELING IN GaAs„P) .N

TABLE III. The parameters obtained from the best At to the selective excitation luminescence spec-

tra.

0.020.07

6.0+2.27.0+3.0

0.01+0.010.07+0.07

fl3

2.7+1.33.9+1.4

17.0+3.018.6+2.2

7.1+2.87.6+3.0

and a similar expression for E . g2 is the ratio between the two interactions coupling strengths and q3 is the TAX cou-

pling strength relative to DP coupling strength at q -0. The function N(EI ) is an overall normalizing factor for all

three types of tunneling processes:

DpN (E, ) = fp'(E)dE f [I' D'(E, E)+—ri,FPE(E, E)+—rI,XX(E, E)]—p'(E)dE . (9)0

Note that the two exciton densities of states are intro-duced normalized.

%e have Gtted all the spectra shown in Figs. 2 and 3with the parameters given in Table III. %e must notethat for excitation energies into the lower part of the Nzband (the two lowest energies in Figs. 2 and 3), the TAXphonon sideband is due to phonon-assisted radiativerecombination and not to tunneling followed by emission.The reason is that (RcorA) is larger than the width ofp(E). Therefore, these sidebands have not been includedin the model 6tting.

Several conclusions can be drawn from this fit. (i) Fortunneling assisted by zone-center phonons (FF bands),the main contribution comes from the DP interaction(r12-0). (ii) The FF bands have a comparable intensity tothe Xz bands. Yet the density of zone-center acousticphonons is much smaller than that of zone-edge phonons.Therefore, the exciton DP interaction with zone-centerphonons is much stronger than that with zone-edge pho-nons. (iii) The fit yields ri&-6 —7, which is comparable tothat obtained by Lash, namely, g&-15. Figure 6 showsthe appropriate densities of states for the two crystals.

I l I l l l l l

GOAlxp~x: N

haC}ala

g}C}

LL.

chKleCh

x =0.07/ 4II

'l

ii 'i

ii

II

(b)x"- 0.02

Ii

I

III \

2.2+I

2.28I l

2.28 2.50ENERGY (eV)

—-- hlonterminol StatesTyrminol St4tO

IV. SUMMARY

In this study we describe a model for the tunnelingprocesses of excitons bound to N impurities in

GaAs„P~ . The model is based on the existence oftransferring and terminal excitation states and on the en-

ergy dependence of the exciton-acoustic phonon interac-tion. Using this model we calculated the spectra obtainedby selective excitation into the Nz band. An adequate Gt

is obtained both for the spectral shapes and for the rela-tive intensities of the observed 8' line which results fromexeitons created on terminal states and the bands whichare due to exciton tunneling. These results, together withthe direct measurement of the density of terminal statesand the radiative decay curves, ' provide a comprehen-sive understanding of exciton dynamics in indirect-gapGaAs, P, , In particular they show that the most prob-able transfer process involves single-phonon emission,and they determine the relative abundance of transferringand terminal states.

It should be noted that the mechanism of exciton trap-ping into terminal states may account also for some un-resolved problems in direct-gap CdS„Se, . ' In thesecrystals, multiple-phonon-assisted tunneling processeshave been observed. Under pulse excitation the decaycurves were nonexponential. This observation mayreAect a distribution of transfer rates into terminal states.

FIG. 6. The transferring (broken curve) and terminal (solidcurve) density of exciton states obtained by 5tting the series ofluminescence spectra shown in Figs. 2 and 3 for GaAs„P&with x =0.02 and 0.07, respectively.

This study was supported by the US-Israel BinationalScience Foundation, Jerusalem, Israel.

D. GERSHONI, E. COHEN, AND ARZA RON 37

Present address: AT8rT Bell Laboratories, Murray Hill, NJ07974.

'R. J. Nelson, in Excitons, edited by E. I. Rashba and M. D.Sturge {North-Holland, Amsterdam, 1982), p. 319.

~0. Goede, D. Henning, and L. John, Phys. Status Solidi 8 96,671 (1979).

~H. Mariette, J. Chevallier, and P. Leroux-Hugon, Phys. Rev. 821, 5706 (1980).

~J. H. Collet, J. A. Kash, D. J. %olford, and J. Thompson, J.Phys. C 16, 1283 {1983).

5H. Mariette, J. A. Kash, D. J. %olford, and A. Marbeuf, Phys.Rev. 8 31, 5217 {1985).

6J. A. Kash, Phys. Rev. 8 29, 7069 (1984).

7D. Gershoni, E. Cohen, and Arza Ron, Phys. Rev. Lett. 56,2211 {1986).

~D. J. %'olford, B. G. Streetman, S. T. Lai, and M. V. Klein,Solid State Commun. 32, 51 {1979).

~E. Cohen and M. D. Sturge, Phys. Rev. 8 25, 3828 (1982}.~0%. Ulbrich and C. %'eisbuch, in Light Scattering in Solids III,

edited by M. Cardona and G. Guntherodt (Springer-Uerlag,New York, 1982), p. 207.

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'~J. A. Kash, Arza Ron, and E. Cohen, Phys. Rev. B 28, 6147(1983).