Excitonic relaxation processes in quantum well structures

Journal of Luminescence 44 (1989) 347—366 North-Holland, Amsterdam

347

EXCITONIC RELAXATION PROCESSES IN QUANTUM WELL STRUCTURES T. TAKAGAHARA NTT Basic Research Laboratories, Musashino-shi, Tokyo 180. Japan Received 8 July 1989 Accepted 8 August 1989

Excitonic relaxation processes in quantum well structures are reviewed, focusing attention on the localized and the weakly delocalized regimes. In the localized regime, the acoustic phonon-assisted exciton transfer among the localized sites due to the interface roughness plays an important role in determining the energy and phase coherence relaxation at low temperatures. The effect of magnetic field on the exciton localization is also discussed. In the weakly delocalized regime the possible mechanisms of the dephasing relaxation are the acoustic phonon-mediated intra- and inter-subband scattering, the Fano-type resonance between the light hole exciton and the heavy hole exciton continuum and the elastic scattering by the interface roughness. Under the high intensity excitation, the interactions between excitons and free carriers become increasingly important in determining the dephasing relaxation.

1. Introduction Since the exciton energy is sensitive to the layer thickness, the interface roughness gives rise to inhomogeneous broadening of the exciton absorption line. It is now well-known that the quantum well interface has island-like structure with a height of one monolayer and a lateral size of several hundred A [1—3].A more recent study based on the cathodoluminescence reports a larger lateral size for the islands of the order of a few ~m [4,5]. On the other hand, the transport measurements find that the lateral size of islands is of the order of several tens of A [6,7]. Thus the optical measurements always give a larger size for the islands than the transport measurements. Most recently, the chemical mapping technique has successfully measured the compositional change across GaAs/A1GaAs interfaces with near-atomic resolution and has revealed the substantial roughness on atomic scales [8]. The difference in the lateral size of the interface roughness estimated from the transport and optical measurements can be interpreted as follows. The electrons are scattered significantly by the interface roughness with size of the order of inverse the Fermi wave-vector. Then the transport measurements give information on the interface roughness of the order of 10—100 A. On the other hand, what is relevant in the optical measurements is the excitonic state with a Bohr radius of the order of 100 A. The excitons feel the random potential averaged over the atomic scale roughness within the exciton Bohr radius and can only probe the change arising from macroscopic roughness larger in size than the exciton Bohr radius. Thus the estimated lateral size of the interface islands depends on the method of measurement. In any case, in the low-energy region of the spectrum the excitons are considered to be localized at one of these island-like structures since the exciton energy changes by several meV due to the one monolayer difference in the quantum well thickness. The excitons localized at such islands are in the local minima in energy and migrate among islands towards lower energy sites emitting acoustic phonons. The typical magnitude of the energy and the wave-vector of participating acoustic phonons are 0.01—0.1 meV and i05—i06 cm~. This exciton migration process plays a key role in determining the energy [9—11]and dephasing [12,13] relaxation rates in the low energy region of the exciton spectrum. In the study of the temperature dependence of the homogeneous linewidth in GaAs/GaA1As quantum wells, the activation 0022-2313/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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type behavior was observed [12]. The estimated activation energy decreases linearly as the energy is increased from below the exciton resonance and vanishes at the center of the absorption line. Hence the absorption line center was interpreted as the mobility edge for the exciton motion. On the other hand, in InGaAs/InP alloy quantum wells all the excitonic states seem to be localized and the mobility edge is not observed [14]. This difference can be attributed to the strong localization due to alloy disordering in the latter case. Hence the presence of mobility edge at the center of the adsorption line does not seem to be a universal phenomenon. The application of a magnetic field perpendicular to the quantum well layer shrinks the exciton Bohr radius and enhances the exciton localization. This expectation was confirmed by the absence of activation-type temperature dependence of the homogeneous linewidth up to 20 K at high magnetic fields [15]. In the weakly delocalized regime, the exciton is mobile and experiences many scattering processes, e.g., acoustic-phonon scattering and elastic scattering from the interface roughness [13]. In the higher energy side, the inter-subband transitions among the conduction and valence subbands and the Fano-type resonance between the heavy hole exciton continuum and the light hole exciton [16] begin to contribute to the homogeneous linewidth. Furthermore under high intensity excitation, the exciton—exciton and exciton—free-carrier collisions become important in determining the homogeneous linewidth [17]. 2. Phonon-assisted transfer of localized excitons The time-resolved photoluminescence of is heavy hole excitons in GaAs/AlAs quantum well structures was studied and the anomalously slow relaxation of the average energy of luminescence and its nonexponential behavior were found [9,10]. In that experiment the excitons were created selectively in GaAs layers. After photoexcitation, the generated electron—hole pairs quickly lose their energy and form excitons with emission of a number of phonons. At the next stage the kinetic energy relaxation on the dispersion curve of the quasi-two-dimensional exciton takes place with a relaxation rate which was estimated to be about one order of magnitude larger [ii] than the observed rate [9]. After these processes are completed, the anomalously slow energy relaxation begins showing nonexponential behavior. In this stage the lowest is exciton in the GaAs layer can be considered to be localized at some island-like structure. The localized excitons then migrate among the local minimum sites in search of the lower energy sites with emission of acoustic phonons. This intralayer migration of localized excitons is the key process in explaining the anomalously slow energy relaxation [11]. Let us now formulate the quasi-two-dimensional exciton transfer among island-like structures. The localization energy is dependent on the lateral size and the height in the growth direction and was estimated variationally to be of the order of meV [18]. In the process of exciton transfer the energy mismatch of excitons is compensated for by acoustic phonons. Since the typical energy mismatch is about I meV or less as will be shown later, we need to take into account only the one-phonon assisted processes at temperatures lower than about 10 K. The exciton state localized at site Ra will be denoted by I Ra), assuming the lowest is state for the electron—hole relative motion. The relevant Hamiltonians for the phonon-assisted exciton transfer are the exciton—phonon interaction Hamiltonian denoted by Hex.~ph and the inter-site transfer Hamiltonian denoted by H5~. Then there are three possibilities for the exciton transfer from site (a) Ra;nQ)

to site

Ra

I

—*

namely Rb;nQ ±1), Rb,

(1)

ph

(b)

IRa;nQ)—s

(c)

I Ra;nQ)

—~

I

IRa;nQ±l)—s Rb;nQ)—s

I

Rb;nQ±i),

(2)

±1),

(3)

Rb;nQ

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where n ~ represents the occupation number of phonons relevant to the exciton transfer. Term (a) arises from the first-order perturbation process with respect to Hex_ph, whereas terms (b) and (c) are the contributions from the second-order perturbation process using both Hex ph and H5~once for each. As will -

be seen later, term (a) is possible through the overlap of exciton wave-functions and is short-ranged in nature, while terms (b) and (c) are effective over a long range. The transition amplitude of the exciton transfer for each process in eqs. (i)—(3) is given as follows: (a) , (4) KRb;nQ ±11 H55 I Ra;nQ ±l>KRa;nQ ±11

Hex_ph

I Ra;nQ>

(b)

(5)

‘

(Rb;nQ ±il

Hex_ph

I Rb;nQ>(Rb;nQ I

H5~I Ra;nQ>

±h~.Q

(c)

(6)

,

where is the phonon frequency for the wave vector Q. The inter-site exciton transfer Hamiltonian H55 arises from the dipole—dipole interaction when the inter-site distance is larger than the exciton Bohr radius and the localization length. Now the exciton—phonon interaction Hamiltonian in the quasi-two-dimensional system will be discussed. It is well-known that in the quantum-well structure the acoustic phonon dispersion becomes folded due to the periodic potential and new branches appear in the reduced Brillouin zone [19]. However, the typical phonon energy and wave-vector which are relevant in the exciton migration process are rather small and we can confine ourselves to the lowest branch of the acoustic phonons. Furthermore the lattice properties of GaAs and AlAs, e.g., the lattice constant and elastic moduli are in close proximity and thus the acoustic phonons which interact with the quasi-two-dimensional excitons in GaAs/AlAs quantum wells can be considered as having a three-dimensional character. We derive the interaction Hamiltonian of the quasi-two-dimensional exciton with acoustic phonons starting from the three-dimensional exciton-phonon interaction Hamiltoman. Two types of the exciton—phonon interaction are considered; namely, deformation potential coupling and piezoelectric coupling. Then the Hamiltonian of the deformation potential coupling for the quasi-two-dimensional exciton is obtained as r.j-DF(Q2D) ex_ph

h[(K — —

—

2

_K~)2+q2]/ II Z 2nuV

II

K~1,K1~,q

1/2

1”2$L D~G((4+b~) x

2, q2L~) 2 3/2

lis,

K1c>(is, K11

2 3/2

—

[i+(bh/2)J X

1”2/3L~, q 2L~)

D~G((4+b~)

/I(2f3L~)

[i+(be/2)1

I(bK~_K,q, + b~_K1~,_q~),

(7)

with bh=ahIKII—KI1 I/a,

b~=a~IKii—Ki1 I/a,

(8)

where ae = me/(me + mh), ah = mh/(m~+ mh), D~(Dy) is the deformation potential for the conduction (valence) band, a and $ are variational parameters in the exciton envelope function, I X K11> denotes the quasi-two-dimensional exciton state with a total wave-vector K11 and the quantum state label X with respect to the electron—hole relative motion and G and I are defined by 2 G(y,ô)= ~~l/2 dZeIrl/ ~ J_l/

2

J_l/2

Xc0s

2(iTze)cos2(iTzh),

(9)

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350

and 1/2

pl/2

—y z —z

~t(’i)211/2 Ir dz~g dzh(1+yIzezhI)e J_i/

Ic

hi

cos

2

2 (rrze)cos (TTZh).

(10)

2

For the lowest is exciton state, the envelope function is chosen as [20] FlS(rH,

Ze, Zh)

—Nfl exp{—[ar~+$

2 1/2 TTZ ITZh (zC—zh)] }cos(~)cos(~).

(ii)

The variable q~in the summation of eq. (7) is the phonon wave-vector in the direction perpendicular to the layer. This wave-vector can be arbitrary in the two-dimensional exciton—phonon scattering process and this arbitrariness leads to the enhanced scattering probability compared with the three-dimensional case. The piezoelectric coupling arises from the longitudinal electric field induced by the strain field associated with acoustic phonon modes. The piezoelectric exciton—phonon interaction Hamiltonian for the zincblende type crystal with Td symmetry was derived in ref. [ii]. As will be shown later, the contribution from the piezoelectric coupling to the phonon-assisted exciton migration is not negligible in GaAs. Now the matrix element J(R) of the inter-site transfer Hamiltonian H55 is discussed. When the separation between island-like structures is much larger than the exciton Bohr radius and the localization length of the exciton envelope function, the contribution to the exciton transfer matrix element comes dominantly from the dipole—dipole interaction. Then we obtain J( I1?a1~bI) 2~(~) ~

(i2)

=

where n (Ra Rb)/IRa Rb I~1(y) is the normalization integral defined by eq. (10) and j.t is the dipole moment of the excitonic transition. In the calculation of the exciton transfer rate, j( I Ra Rb I) appears always in the squared form. Thus we can drop the angular dependence and reduce J( R a Rb I) to an isotropic form as =

—

—

— —

A 3j ~\2 3 ~ hJ~-~-a~aç) j~I1a1~bI) 8I(2/3L~) I1a~hI ___________

—

IRa1~bI3’

i

—

with A=

(14)

[((i_3n.n)2>]l/2=(~)1/2,

where ~ LT is the LT splitting of the exciton energy and ~ is the localization length of the exciton. By choosing the values ~LT 0.08 meV, ~ 136 A, ~ 150 A, a~ 100 A, and $L 2 0.37, we have 3. (15) f(theor) 5.3 X 10~eVA So far, we have discussed the behavior of J( R) at long distances. In the intermediate range in which the intersite distance is comparable to the exciton Bohr radius and/or the localization length, the wave-function overlap between the localized excitons is not negligible and the tunneling type transfer becomes relevant. Thus some interpolation between tunneling-type transfer and dipole—dipole type transfer may be appropriate to simulate the actual behavior of J( R). In order to calculate the transition amplitude of the exciton transfer, we must specify the localization envelope of the localized exciton. Its functional form may depend on the details of microscopic configuration of the localized site. However, the dynamical properties of the system, such as the energy relaxation, are expected to be not very sensitive to the microscopic details of localization, but may be characterized by only a few parameters, such as the localization length Two typical cases of exciton =

=

=

=

=

=

~.

T. Takagahara

/ Excitonic relaxation in quantum well structures

351

localization are the Gaussian case and the exponential case. For the case of Gaussian localization, the transition amplitude of the exciton transfer is calculated as Q

2

=

ex~( —

e~i~/

+

!-~~

—

J( IRa~RbI) (e’Qii~

—

e

(Ra~Rb)2

QiiRa)

)

Hex ph(QI]~Qz)

ex~(

—

Hex ph(Qii,

Q~)~

(16)

where the subscript Q denotes the wave-vector of acoustic phonons participating in the exciton transfer process. The first term on the right-hand side is the contribution from process (a) in (4) and the second term combines the contribution from processes (b) and (c) in (5) and (6), respectively. The first term contains the Gaussian factor exp[—(Ra Rb)2/4~2] arising from the overlap integral between two localized exciton states and has a short-range character. On the other hand, the second term in eq. (16) depends on the distance IRa Rb I through the function J and the coherence factor exp(—iQii Ri,) exp(—iQii Rn), which are generally extended over a long range. The common factor exp(—~2Q~/4) implies that the magnitude of the wave-vector of phonons which can interact with the localized exciton is limited to within a few times the inverse localization length. When the transition amplitude is obtained, the exciton transfer rate can be calculated by —

—

.

—

(17)

T(IEa~EbI, IRa_RbI)=~~I(RbITIRa>QI2~(Ea_Eb±h~jQ).

In the absolute square of the transition amplitude, there appears the interference between two terms in eq. (16). However, in the study of the rate equation in section 3, the relevant quantity is the spatial integral of the exciton transfer rate (17). Then the contribution from the square of the second term of (16) is predominant because of its long-range character and the contribution from the square of the first term and the interference term can be safely neglected. In this way we can obtain the exciton transfer rates TDF( I Ea Eb I~IRa RbI) and T~~( I Eb I~I1~a Rb I) for the deformation-potential coupling and for the piezoelectric coupling, respectively. For illustrative purposes we show in fig. 1 the spatially integrated transfer rate for a typical case of Gaussian localization and tunneling-type transfer; namely —

—

—

—

TDF_JdRTDF(E~ R),

(18)

R).

(19)

ipzfd2RTpz(E,

I,

1.61E

Energy(meV)

Fig. 1. Energy dependence of the spatially integrated one-phonon-assisted transfer rate of excitons for deformation-potential coupling (DF) and piezoelectric coupling (PZ). (From ref. [11].)

‘—c.—.--

TIME DELAY (ps)

Fig. 2. Comparison of the theoretical calculation of the average energy of luminescence with the experimental data cited from ref. [10].The dashed line is a linear fit to the experimental data to show the non-exponential behavior. (From ref. [ill).

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Excitonic relaxation in quantum well Structures

The relevant parameters are chosen here as D~ —6.5 eV, D~ 3.1 eV and e14 1.6 X iO~ C/cm2. It is seen that the contribution from the piezoelectric coupling is smaller than that from the deformation-potential coupling, but it is not negligible. The energy dependence is similar for both cases. Roughly speaking, the peak position is determined by the localization length as E hu~1, where u is a typical sound velocity. In fact, this estimate gives the right order of 0.3—0.4 meV. =

=

=

3. Rate equation for exciton distribution function and energy relaxation Now that the transition probability of the exciton transfer is derived, we can set up the rate equation for the exciton distribution function. In the following the inter-layer transfer of excitons through the AlAs barrier layer is neglected and only the exciton transfer within a GaAs quantum well will be taken into account. For simplicity an assumption will be introduced that the line broadening is microscopic, i.e., that there is no correlation between the energy of the localized exciton and its position in space [21]. Thus the energy distribution of the localized exciton at any particular site depends only on the overall density of states but not on the nearby configuration of the localized site. In addition, the density of states of localized excitons is assumed to be proportional to the absorption spectrum at low temperatures. This assumption is reasonable because in the low-energy tail of the absorption spectrum the contribution from the localized excitons is dominant. Under these assumptions the rate equation for the distribution function f(E,t) of the localized exciton with energy E is derived as t)=

— 70f(E,

t)_~1J

d2RF(R)f dE’D(E’)f(E,

+[i+n(E—E’)]O(E—E’)}T(IE—E’I,

t){n(E’—E)O(E’—E)

R)

+a~1fd2RF(R)fdE’D(E’)f(E’, t){n(E—E’)e(E—E’) E)}T( I E— E’ I~ R), (20) 1the areal number density of island-like structures, n the where y~is the lifetime of localized excitons, a~ phonon occupation number, D(E) is the density of states of localized excitons normalized as JdED(E) +

[1 + n(E’

—

E)]e(E’

—

1 and 0 is the Heaviside step function, respectively. This equation is a reduced form of eq. (35) which includes the spatial variation of the exciton distribution. In fact, by putting =

f(E,

t)

J d2Rf(E, R,

=

(21)

t),

we can derive eq. (20) from eq. (35). The exciton transfer rate T(E,R) the deformation-potential and the piezoelectric coupling, namely T(E, R)

=

TDF(E,

R)

+ TPZ(E,

is

a sum of contributions from both

R).

(22)

F( R) is the distribution function of nearest-neighbor island-like structures and is given by F(R)

=

exp[_.rrR2/ao].

(23)

Solving the above rate equation with an initial distribution taken from the experimental data, we can calculate the average energy of photoluminescence by =

f dEEf(E,

t).

(24)

T. Takagahara / Excitonic relaxation in quantum well structures

353

\r~t ,

2 >

E

\

5

i

~

.4,

.‘~j

d

/

~

Ia

\-

2

‘~

1,605

1.610

.

1615

1.615

ftw, ev Fig. 3. Absorption coefficient

a

(dashed line), homogeneous

linewidth hE5 (solid line) and activation energies ~E (squares) as functions of the exciton energy for GaAs quantum wells with L~= 51 A at 5 K (From ref. [12]).

~ 6 .5

~“

: ~‘~I 4

I

E

A

i

—

\~\ J4~ \\ ‘H3~

~H ~ 1.620

1.625

Energy(eV)

Fig. 4. Calculated homogeneous linewidth hE

5 and activation energy ~E as functions of exciton energy. The dashed line indicates the assumed exciton absorption spectrum. The region indicated by a double arrow is the supposed transition region between the localized and the delocalized regimes (From ref. [13].)

The theoretical results are shown in fig. 2. The best fit is obtained by adjusting .1 in eq. (13) as 3, (25) f= 11.7 X i0~eV(A) when the Gaussian localization envelope and dipole—dipole type transfer function are employed. The agreement within a factor of 2 or 3 with the theoretical value in eq. (15) is quite satisfactory in view of ambiguities in the material parameters. The discrepancy in the low energy region or in the later stage of relaxation may come from the Gaussian approximation of the density of excitonic states. Actually the density of states remains finite even in the low energy tail. Because the Gaussian approximation gives a rapidly decreasing density of states in the low energy region, the calculated energy loss rate becomes smaller than the experimental value. Anyway, the good agreement between the theory and the experiment confirms the adequacy of our model for the localized excitons and our theory of the energy transfer.

4. Dephasing relaxation of quasi-two-dimensional excitons Recently, Hegarty and coworkers have measured the homogeneous linewidth and the diffusion constant of quasi-two-dimensional excitons in GaAs/GaAlAs quantum well structures with various methods, such as resonant Rayleigh scattering [22], hole burning [12], transient grating [12] and photon echo [23] method. Their measurements revealed the salient features of the energy and temperature dependences of the homogeneous linewidth of quasi-two-dimensional excitons. They found that the homogeneous linewidth increases sharply as the exciton energy increases through the center of the absorption line and that below the line center the homogeneous linewidth is thermally activated as shown in fig. 3. These experimental results suggest the existence of the mobility edge for the quasi-two-dimensional excitons. The experimental details of measurement of the homogeneous linewidth and the diffusion constant of excitons are given in the recent review articles by Hegarty and Sturge .[24,25]. In the localized regime the phonon-assisted migration of excitons among localized sites within a quantum well layer determines the homogeneous linewidth Fh and gives rise to the characteristic temperature dependence. As another mechanism contributing to the homogeneous linewidth, we can consider the phonon-assisted transition to the delocalized exciton states. The latter mechanism is effective in the intermediate temperature range ( 10 K) because the transition is associated with phonon absorption. Obviously this mechanism leads to the activation-type

354

T. Takagahara / Excitonic relaxation in quantum wellstructures

behavior of the temperature dependence of the homogeneous linewidth. On the other hand, the tunneling mechanism is working even at low temperatures (= 1 K) and gives rise to a very weak temperature dependence. In the delocalized regime, the dephasing relaxation is caused by acoustic phonon-mediated intra- and inter-subbands scattering. In fact, the phonon scattering rate is found to be enhanced by two orders of magnitude over that for the three-dimensional case because the phonon momentum perpendicular to the quantum well layer can be arbitrary in the scattering. Another mechanism of dephasing relaxation of the delocalized exciton is elastic scattering by the potential fluctuation due to the interface roughness and by impurities. The homogeneous linewidth of the localized exciton state with energy E due to the phonon-assisted tunneling is calculated by

=J dE’D(E’) i~(

r~(E)

IE—E’ I)(n(E’—E)0(E’—E)

+

where the same notations as in eq. (20) are employed and T( I E

[1 +n(E—E’)]o(E—E’)}, —

E’

I) is

(26)

defined by

i~(IE_E~I)=a~hfd2RT(IE_E~I, R)F(R).

(27)

In the same way, the dephasing relaxation rate due to the activation process is given by ~

(28) K

11

Q

where I Ku> is the delocalized exciton state with a wave-vector K11 and I R a> is the localized exciton state at site R a~As for the envelope function of the localized exciton, it is found that a Gaussian form yields too sharp an energy dependence of the calculated Fh [13] to explain the experimental results. Although the macroscopic broadening [21] actually smears the sharp mobility edge and will change the sharp energy dependence into a moderate one, an exponential function is adopted here for the localization envelope. The calculated dephasing relaxation constants are shown in fig. 4. The quantum well thickness taken as Th is ofisthe order 80 0.1 A and temperature is 5 K. centerThis of the absorption line, thewith calculated of meVtheand increases with the Below excitontheenergy. is in good agreement the experimental results. The temperature dependence of “h is plotted in fig. 5. In the temperature region around 10 K there occurs

0.1 ie2~

h

~i.622t~d

~ ~1)

Fig. 5. Calculated homogeneous linewidth hE 5 as a function of inverse temperature at various exciton energies in the localized regime for the activation mechanism (E~)and for the phonon-assisted tunneling (F~’).(From ref. [13].)

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/ Excitonic relaxation in quantum

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355

a crossover in the temperature dependence from the thermal activation type to the phonon-assisted tunneling type because the latter mechanism is effective even at low temperatures. From the least-squares fit in the temperature range between 2 and 0.5 K, T’~is found to obey the temperature dependence [13] F~(T)=T~exp[BT~],

(29)

where B is positive and the exponent a is estimated to be about 1.6 to 1.7, depending weakly on the exciton energy. This form of temperature dependence is rather different from that of the variable range hopping. The electronic conduction in the localized regime is usually interpreted in terms of the variable range hopping [26]. However, in the case of exciton transfer, the transfer probability cannot be simply written in an exponential form as in the Mott’s argument since the transfer process includes necessarily the long-range dipole—dipole interaction. Thus it is not strange that we obtain another exponent different from that of the variable range hopping in the temperature dependence of the homogeneous linewidth. Next the dephasing relaxation rate in the delocalized regime will be discussed. As mentioned before, one of the typical dephasing mechanisms is the acoustic phonon scattering on the two-dimensional dispersion curve and the homogeneous linewidth due to this mechanism is calculated by ~ f
r~(K~~) =

x [nQ~(EK+

~,,

—

EK

ph

—

I

KiI>Q 12

hwQ) + (1

+ flQ)~(EK

1~_ Q

—

EK

+

hwQ)1.

(30)

The calculated results are shown by p~hin fig. 4. The dispersion curve of the delocalized exciton state is assumed to begin from 2.5 meV above the absorption line center. This choice is rather arbitrary and only for looking into the qualitative features of the energy dependence of Fh. The dephasing relaxation rate due to elastic scattering by the potential fluctuation is calculated as follows. The fluctuation of the exciton energy due to the fluctuation of the quantum well thickness ~ is given by 2~2~L~ ~ L~ (L~+~L~)2 ~E—~—~-— 1 h p~L~ —

(31)

—

~

where j~t is the exciton reduced mass. Assuming the scattering potential due to the exciton energy fluctuation to be a cylindrical one with radius ~, the dephasing relaxation rate is calculated as [13]

=

81TM~2(6E)2 j~/2

dO( J 1(2K~0s9)

)2,

(32)

where M is the exciton total mass and J1 is the first-order1are Bessel function. The calculated are both of the order of meV in results agreement depicted as F~’in fig. results. 4. The However, absolute values of F~and F~ with the experimental F 1~tends to decrease in the higher energy region. This is because the magnitude of wave-vector of the participating phonons increases with the exciton energy and the exciton—phonon coupling constant decreases. Actually the phonon-mediated inelastic scattering among the higher subband states and the excited states of the electron—hole relative motion begins to contribute to the homogeneous linewidth in the higher energy side. Furthermore, in the higher energy region, the multi-phonon processes become more and more important and F~would not decrease but increase with the exciton energy. On the other hand,inthe linewidth T~~1 decreases in the higher energy 2dependence eq. homogeneous (32). region because of the K~ An additional mechanism for the homogeneous linewidth is the elastic scattering by impurities. The elastic scattering cross-section of excitons by neutral impurities was calculated to be 0.1 to 1.0 times ‘rra~

356

T. Takagahara 1

/ Excitonic relaxation in

quantum well structures

r 1.27)

0.1

11

~~22J

1

2

20

3

11,,, 5K

1.520 1.525 PHOTON ENERGY, .5

Fig. 6. Dependence of the homogeneous linewidth hEh on the quantum well thickness. The relevant exciton energies are indicated in parentheses and oR is the exciton Bohr radius in the three-dimension. (From ref. [13].)

in the three-dimensional case [27], where a B the impurity scattering is calculated by

is

Fig. 7. Exciton diffusion constant D as a function of the exciton energy for GaAs quantum wells with L~= 205 A at 5 K. The full line is the absorption spectrum. (From ref. [12].)

the exciton Bohr radius. The homogeneous linewidth due to (33)

where

the velocity of the exciton center-of-mass motion and N1 is the impurity number density. This is estimated typically as s

is the total cross-section,

v

~

(34)

and this value corresponds to about 0.002 meV. Thus the impurity scattering gives rise to a small background of the homogeneous linewidth. The dependence of the homogeneous linewidth on the quantum well thickness is quite important in identifying the dephasing mechanisms. In the localized regime, both F~Cand F1~depend on Lz through the matrix element of the exciton—phonon interaction. In the delocalized regime, F~’in eq. (32) is found to be inversely proportional to the sixth power of L~,while F~ depends on Lz through the exciton—phonon matrix element. Typical variations of Fh at various energies at 5 K are plotted in fig. 6. It is seen that the L~dependence of F1~or F1~in the localized regime is rather sensitive to the exciton energy. On the other hand, Fh~in the delocalized regime is insensitive to the exciton energy and is dependent on L~only weakly. These features will be useful in identifying the mechanisms of dephasing relaxation. However, as mentioned before, in the delocalized regime the homogeneous linewidth is possibly determined by multiphonon processes among higher subband states. Under this situation the dependence on the quantum well thickness of the homogeneous linewidth may be different from the above result which was obtained on the basis of the intra-subband transition within only the lowest subband.

5. Exciton diffusion in the localized regime The exciton diffusion was studied extensively in molecular crystals and their mixed crystals and various features of the exciton localization were clarified [28]. Thus the spatial diffusion constant can be a nice

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Excitonic relaxation in quantum well structures

357

probe of the exciton localization. The diffusion coefficient of excitons was measured by the transient grating method in GaAs/GaA]As quantum wells [12] as shown in fig. 7. In the localized regime, the phonon-assisted exciton migration among island-like structures can be considered as incoherent motion since the excitons are spatially well localized and the coherent interconnection is almost negligible between localized excitons. The exciton diffusion in the localized regime can be formulated in terms of the transfer rate T( I Ea Eb I~IRa Rb I) in eq. (22). The probability that the island-like structure at site R is occupied by a localized exciton with energy E will be denoted by f(E,R,t). The rate equation for f(E,R,t) is given by —

R,

t)

—

2R’ F(IR R’I)J dE’D(E’)f(E, R, t) —y07(E, R, t) d x(n(E’—E)O(E’—E)+ [1 +n(E—E’)]0(E—E’)}T(IE—E’I,

=

—

a~~J

d2R’ F( I R

—

—

R’ I)f dE’D(E’)f(E’,

x(n(E—E’)O(E—E’)

R’,

t)

[1 +n(E’—E)]0(E’—E)}T(

+

IR—R’I)

IE—E’I, IRR’l), (35)

where the notation is the same as in eq. (20). In general, solving this equation with an appropriate initial condition we can describe the exciton diffusion in energy space and in real space. However, if we do not discriminate the exciton energy and consider only the occupation probability of localized excitons at site R defined by P(R,

t)

=

f dED(E) f(E,

R,

(36)

t),

we obtain an equation of motion for P(R, t)

=

—y 0P(R, {n(E’

t) —

—

t)

as

a~fd2R’ F(IR

E)e(E’

—

E) + [i

—

+

R’I)f dED(E)fdE’D(E’)f(E,

n(E— E’)] e(E— E’)}T( I E

+a~’Jd2R~F(IR_R/I)f dED(E)f x(n(E—E’)O(E—E’) Here changing the variable R f(E’, R

—

R’,

t)

—

+

R’,

[1 +n(E’—E)]0(E’—E)}T(

E’

~‘

R

t) —

R’

I)

t)

IEE’I,

IRR’I).

(37)

R’ to R’ and introducing the expansion

=

exp[ —R’vR]f(E’, R,

t)

=

(i

+

—

dE’D(E’)f(E’,

—

R,

R’VR +

~(R’)2v~

...

)f(E’, R,

(38)

t),

into eq. (37), we can obtain approximately as t)=

—y 0P(R, t)+Ddv~P(R,

(39)

t),

with Dd

=

~—fd2R’ F(IR’ I)(R’)2f dED(E) x(n(E—~)e(E—E)+(1+n(~—E))e(E—E)}T(IE—~I,

IR’I),

(40)

358

T. Takagahara

/ Excitonic relaxation in quantum

well structures

where E is the representative energy of the localized exciton and can be identified with the initial energy of the localized exciton if the distribution in the energy space does not broaden very much. This is nothing but a diffusion equation and Dd is the diffusion coefficient. Using the same parameter values as in eq.(24), we estimated Dd to be of the order of 1 cm2/s in agreement with the experimental value. To discuss the energy dependence of the diffusion constant, we must, in principle, resort to the original equation of motion, eq. (35). However, the increasing trend of the diffusion constant with the exciton energy can be understood from eq. (40) where E can be interpreted as the relevant exciton energy. In the delocalized regime, the diffusion constant of excitons is determined by the inelastic scattering from delocalized phonons and by the elastic scattering by impurities and interface roughness. In this case the diffusion constant should be calculated from the velocity—velocity autocorrelation function by [29] Dd

=

urn E’O

f~dte~t(v(0)v(t)>.

(41)

o

6. Homogeneous linewidth of excitons in InGaAs/InP alloy quantum wells Recently, Hegarty et al. [14] measured the homogeneous linewidth of excitons in In 0 53Ga047As/InP alloy quantum wells by spectral hole-burning and pump-probe spectroscopy. In this sample the excitons are more easily localized due to the alloy disordering than in GaAs/GaAlAs quantum wells. The much larger homogeneous linewidth than that in GaAs quantum wells was attributed to a much stronger coupling of excitons to acoustic phonons in InGaAs. The weak dependence of ~h on the energy position within the inhornogeneous line indicates that all excitons are localized and the mobility edge is absent. The temperature dependence of Fh is shown in fig. 8 and can be fitted by the theoretical curve (29) employing a 1.68. The agreement of the exponent with the predicted one implies that the phonon-assisted tunneling is the relevant scattering mechanism in the localized regime. =

7. Resonant Raman scattering as a probe of homogeneous linewidth and effects of magnetic field The exciton localization can be probed by resonant Rayleigh [22] or resonant Raman [30] spectroscopy since the Raman intensity is inversely proportional to the homogeneous linewidth. The 1 LO Raman intensity can be written as 2

(h~L—E—1F(E))(h~L—hwLo—E —iF(E

eop (B

IE .J

0.05

))

42)

).2)

1~~67s1o12,,,,///

10

20

TEMPERATURE (K)

30

40

Fig. 8. The measured spectral relaxation rate as a function of temperature in InGaAs/lnP alloy quantum wells with L~= 80 A along with a fit by the phonon-assisted tunneling model. The excitation wavelength is 1470.5 nm. (From ref. [14].)

T Takagahara

/ Excitonic relaxation in T

000 0.10 0.20

quantum well structures

359

(K)

0.30 0.40 0.50 0.60

In (ie~)

Fig. 9. (a) Temperature dependence of the Raman intensity resonant with the n = 1 (circles) and n = 2 (squares) heavy hole excitons in GaAs/GaAlAs quantum wells with L~= 96 A. (b) Temperature dependence of the luminescence efficiency of the n = 1 heavy hole exciton. (From ref. 30.)

where E is the exciton energy almost resonant with the incident laser light (hw L) and E’ is the energy of the intermediate exciton state, F( E) and F( E’) are the homogeneous linewidth of the corresponding exciton states and h w LO is the LO phonon energy. For the nearly resonant case, we have 1 2

(43)

2~

(hcoL—E) +1(E)

In order to include the inhomogeneous broadening, we add up the contributions from each homogeneous exciton line taking into account a distribution function of exciton energies p(E) as IR(h~L)~

J dE (h~oL—E) +1(E)

Since

and 1(E) are rather smooth functions, we can approximate as

p(E)

IR(~L)

~ p(h~L)f dE

(44)

2~

2

(h~L—E) +1(E)

~p(hwL)

F(hc.iL)

(45)

Thus the Raman intensity is inversely proportional to the homogeneous linewidth. This is the basis of the Raman spectroscopy of the homogeneous linewidth. Zucker et al. [30] measured the temperature dependence of the resonant Raman and luminescence intensities as shown in fig. 9. The luminescence intensity can be simply interpreted in terms of exciton trapping. On the other hand, the Raman intensity continues to rise as the temperature is lowered. The cross-over between two different schemes of temperature dependence is clearly observed. The experimental

360

F Takagahara

/

Excitonic relaxation in quantum well structures T (K)

2 -4.0)

5 I

7

10

I

15

I

T (K)

20 -4.0)’

10 I

5

3

I

I

2

0

0

to)

0

—-5.0-

0

z-7°(”

1~~IlI

035

0.45

0.55

0.65

075

0.85

0.0

1/T1~3(~-1~3)

(bI

22.5

45.0

67.5

90.0

718 (K1.8)

3 Fig. 10. Temperature dependence of the homogeneous linewidth of the n = I heavy hole exciton plotted as a function of (a) 1/TV and (b) T1’5. The line drawn between 7 K and 15 K in the original figure of ref. [30] is erased in (b) to avoid misunderstanding. (From ref. [30].)

temperature dependence of 1h and the theoretical dependence of both the models based on the variable range hopping and the phonon-assisted tunneling are shown in fig. 10(a) and (b), respectively. The discrepancy between the theory and the experiment above 10 K seems to be more pronounced in fig. 10(b) than in fig. 10(a). However, this is only an artifact of plotting due to the difference in the power of T in the abscissa. We should compare the experimental data with the theoretical models below about 10 K. From 10 K down to 2 K, the agreement between the theory and the experiment is rather good for both models. However, the existence of a third temperature regime was suggested below 2 K from the plot according to the phonon-assisted tunneling model. On the other hand, for the n 2 exciton, the Raman intensity does not show any activation type behavior within the temperature range of fig. 9. The homogeneous linewidth of the n 2 exciton is considered to be determined mainly by the Fano-type resonance with the n I exciton continuum states and thus the temperature dependence is rather weak. The effect of magnetic field on the exciton localization is quite interesting because the magnetic field induces the cyclotron motion of electrons and holes and is expected to enhance the localization features of excitons. Zucker et al. [15] studied the exciton dynamics under transverse magnetic fields by the resonant Raman spectroscopy and found a significant enhancement in the intensity of Raman scattering. This enhancement comes from two separate magnetic-field-induced effects. The first effect is due to a reduction in the homogeneous exciton linewidth 1h~ The second is the increase in the excitonic oscillator strength due to the magnetic-field-induced shrinkage of the electron—hole relative motion. The second effect can be seen in fig. 11 which depicts the Raman intensity versus the temperature. Below 2 K, the excitons are fully localized and the localization properties are considered to be not much affected by application of a magnetic field. Thus the difference in the Raman intensity below 2 K can be attributed to the magnetic-field-induced enhancement in the exciton oscillator strength since the Raman intensity is proportional to the square of the oscillator strength. The Raman intensity is plotted in fig. 12 as a function of inverse temperature. The activated behavior 1h a exp[ iX E/k BT] is seen clearly at relatively low magnetic fields and the activation energy is observed to increase with the magnetic field strength. On the other hand, at higher magnetic fields of 13 and 17 T, the activated behavior is not observed for temperatures up to 20 K. These behaviors can be understood qualitatively in terms of the magnetic-fieldinduced shrinkage of the exciton Bohr radius and the enhanced exciton localization. The magnetic length =

=

=

—

T Takagahara

/ Excitonic relaxation in

quantum well structures

361

T 2

10000

I

1

~~

1~4-f

0.75

0.50

17TESLA+

-

~iooo~(~TES~

1SLA.

0.50

0.00

1.00

1 50

200

iTT

Fig. 11. Resonant Raman intensity as a function of inverse temperature between 0.5 and 20 K at transverse magnetic fields 0. 13 and 17 Tin GaAs/GaAIA5 quantum wells with L, = 96 A. (From ref. 115].)

which is the classical radius of the ground Landau orbit is given by —

[J~256.5A ~

46

lcVJj

where the magnetic field H is measured in units of Tesla. When this magnetic length is less than the size of the island-like structures under a high magnetic field, all the excitonic states would be localized by potential fluctuations and the mobility edge between the localized states and the delocalized states may disappear. As a result, the activated behavior may not be seen in the temperature dependence of ~ In this respect, the interplay between the magnetic confinement and the localization by potential fluctuations is quite interesting to investigate.

F~~7T

000

015

0 30

045

l/T

Fig. 12. ln(IR) as a function of inverse temperature between 2 and 50 K for magnetic fields 0, 8, 10. 13 and 17 T. The sample is the same as in fig. 11. (From ref. [15].)

362

T. Takagahara

/ Excitonic relaxation in quantum

well structures

8. Dephasing relaxation of excitons in delocalized regime So far we have been discussing the dephasing relaxation of excitons in the vicinity of mobility edge which separates the localized and delocalized regimes. On the other hand, the dephasing relaxation of fully delocalized free excitons gives fruitful information about the exciton—exciton, exciton—free carriers and exciton—phonon interactions. The phase coherence and orientational relaxation of excitons created resonantly in optically thin (~190 nm) GaAs layers [31] and in a 12 nm thick GaAs single quantum well [32] were explored by means of time-resolved degenerate-four-wave mixing (DFWM). Three configurations for polarization of incident laser beams are depicted in the inset of fig. 13 that were adopted in the latter case in order to probe the exciton lifetime, orientational relaxation time and phase coherence relaxation (dephasing) time, respectively. The orientational relaxation is related to the degradation of the coherent macroscopic polarization of excitonic states. Scattering by phonons and impurities induces the change in wave-vector and polarization of the valence band states composing the exciton state. These scattering processes give rise to the orientational relaxation as well as to the dephasing relaxation. In fact, the observed time constants of the orientational relaxation and the phase coherence relaxation are of the same order of magnitude. The measured relaxation times of the order of ten picoseconds can be interpreted in terms of the acoustic-phonon scattering. At the same time, Schultheis et al. [17] found the excitation intensity dependence of the dephasing relaxation time. The dephasing relaxation time of excitons was measured by two-pulse DFWM which are subjected to collisions with free carriers and incoherent excitons that are independently created by a third synchronized light pulse. It was found that the exciton-free-carrier scattering is about ten times more

pOCUIOtIOfl

~\

*2 • *3

A ~

~

101

—.

grotmg

~

.0-2

fl ~

*4

t°185!2Ops

C I

.o

U. U. .,~O .

0

50

0

50

100

100 ~,

150

I

~

150 200 2-P~Asss.4f-diffroctjon

(bI

(C)

DELAY I (psI

Fig. 13. Diffraction curves of time-resolved DFWM on is heavy hole excitons in a 12 nm GaAs single quantum well at 2 K. (a) Depicts the diffracted intensity of a population grating, (b) an orientational grating and (c) a two-pulse self-diffraction experiment. The observed relaxation times are (a) the recombination lifetime r, (b) the orientational relaxation time T 1 and (c) the phase coherence time T2. (From ref. [32].)

F Takagahara

/

Excitonic relaxation in quantum well structures I

0.8

I

I

I

I

I

I

X-eh

I

I

363

I

TboIh~2K

E 3D:L

2~194nm

06

.~

~

(20 \X-X

z

\

20

\X-eh

04

0: INTER PARTICLE DISTANCE rb-a--

Fig. 14. Line broadening of two-dimensional (circles and squares) and three-dimensional excitons (triangles) subjected to collisions with excitons (X—X) or free carriers (X—eh) as a function of the normalized inter-particle distance. (From ref. 135).)

efficient than the exciton—exciton scattering in enhancing the homogeneous linewidth. The carrier density dependent homogeneous linewidth is defined by 1h=1~+yxxNx, =

(47)

1~+ YxehNeh,

(48)

where NX(eh) is the number density of excitons or free carriers. The collision efficiency y can be roughly estimated by (49)

y=av,

where a is the exciton—exciton or exciton—free carrier collision cross-section and v is the relative velocity of two colliding particles. Using the theoretical values for the cross-section of elastic scattering [33,34], we can estimate the collision efficiency y in1’~eh a induced favorableby agreement the orexperimental values. incoherent with excitons free carriers is plottedThe in incremental line broadening N,~ or YXeh fig. 14 as a function of the dimensionless inter-particle distance defined by [35] ~

rb

=

(49Ta~(3D)N/3) 1/3

(50)

(i~a~(2D)N)”2,

(51)

and rb

=

for the three-dimensional (3D) and two-dimensional (2D) cases, respectively. The incremental line broadening is larger in the 2D case than in the 3D case for both the exciton—exciton and the exciton—free carrier collisions. This tendency is consistent with the argument that the screening of the Coulomb interaction is much weaker in the 2D case than in the 3D case [36]. As a consequence, the effect of the background excitons or free carriers on the probe exciton appears stronger for the 2D case. Another interesting feature is the temperature dependence of the homogeneous linewidth of the free excitons. Approximating the phonon occupation number n ( w) in eq. (30) by 1 (h/kT)1

k ~

8T

(52)

we have T’h(T)

=

F~+ ~‘PhT.

(53)

364

T Takagahara

/ Excitonic relaxation in

quantum well structures 6 III, i... L

2~I3.5nm

1.2 E1,,~

~

Lo277nm

X~

10

~

~

277 nm

::

Z 04

-J

0.2

210

TEMPERATURE (K) Fig. 15. Homogeneous linewidth (FWHM of the absorption peak) vs temperature for the heavy hole (hh) and light hole (lh) exciton transitions of two single quantum wells with L~= 13.5 nm and 27.7 nm. (From ref. [3].)

This linear temperature dependence was clearly observed in single quantum wells [37] as shown in fig. 15 and this confirms that the optical dephasing of the delocalized two-dimensional excitons is caused by one-phonon scattering. For the 27.7 nm thick quantum well, the homogeneous linewidth is similar in magnitude and the temperature coefficient Yph is also similar for the heavy- and light-hole excitons. On the other hand, for the 13.5 nm thick quantum well, the homogeneous linewidth of the light hole exciton is much larger than that of the heavy hole exciton. This was interpreted in terms of the Fano-type resonance between the light hole exciton and the continuum states of the heavy hole exciton. This point was discussed in more detail in relation to the spectral width of the photoluminescence excitation spectra [16]. It is noted that the temperature coefficient for the 27.7 nm thick quantum well is well aboutthickness twice as islarge as 1h on the quantum rather that of the 13.5 nm thick quantum well. The dependence of complicated as shown in fig. 6 and it is not easy to predict the dependence without a numerical calculation. The calculated p~h which includes only the intra-subband transition within the lowest subband shows a weak dependence on L~.Thus we expect that the temperature coefficient Yph is also weakly dependent on L~. As mentioned before, the realistic evaluation of 1~ should include the inter-subband transitions among higher subbands and also the multi-phonon processes. In the thicker quantum wells the inter-subband energy spacing is smaller and the acoustic-phonon-mediated inter-subband transitions become more efficient than in the thinner quantum wells and hence a larger temperature coefficient Yph would be obtained for thicker quantum wells. Thus the above experimental result can be understood qualitatively in this way, although a more quantitative calculation is necessary to explain the enhancement factor ~Yph(L~ 27.7 nm)/yPh(L~ 13.5 nm). =

=

9. Summary and prospects The present status of understanding of the exciton relaxation processes in quantum well structures are reviewed especially for the localized and weakly delocalized regimes. In the localized regime, the acoustic-phonon-assisted exciton transfer is the key process that determines the energy relaxation rate and

T. Takagahara

/

Excitonic relaxation in quantum well structures

365

the dependences of the homogeneous linewidth on the exciton energy and the temperature. Interesting features of the interplay between the magnetic field confinement and the exciton localization have been revealed and they hold a promise of further development. In the delocalized regime there are many relaxation channels, i.e., acoustic phonon-mediated intra- and inter-subband scattering, Fano-type resonance with the continuum states of excitons associated with the lower index subbands, elastic scattering by impurities and interface roughness and many-body effects under high intensity excitation. These subjects are still under extensive investigation. Very recently, the high-resolution frequency-domain four-wave-mixing spectroscopy was applied to the is heavy hole exciton in GaAs/GaAIAs quantum well structures and the non-Lorentzian homogeneous exciton lineshape was measured directly [38]. This measurement is complementary to the previous time-resolved measurements of the homogeneous linewidth and is, I believe, not providing an interpretation of the excitonic relaxation processes that is contradictory to the previous ones. For example, the asymmetric, namely non-Lorentzian profiles in the four-wave-mixing spectra is most likely to be explained in terms of the onset of spectral diffusion in the vicinity of the mobility edge. However, this frequency-domain measurement can be carried out under a very weak excitation condition and is sensitive enough to detect the slow relaxation processes which were overlooked in the previous time-resolved measurements. These newly revealed relaxation processes are left for the future study.

Acknowledgements The author would like to thank Dr. A. Honold for enlightening discussions on his Ph. D. thesis and unpublished materials.

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Excitonic relaxation in quantum well structures

[25] J. Hegarty and M.D. Sturge, Surf. Sci. 196 (1988) 555. [26] N.F. Mott and E.A. Davis, Electronic Processes in Noncrystalline Materials, 2nd ed. (Oxford Univ. Press, New York, 1979). [27]J. Lee, E.S. Koteles, MO. Vassell and J.P. Salerno, J. Lumin. 34 (1985) 63. [28] R. Kopelman, in Spectroscopy and Excitation Dynamics of Condensed Molecular Systems, eds. V.M. Agranovich and R.M. Hochstrasser (North-Holland, Amsterdam, 1983) p. 139; M.D. Fayer, ibid. p. 185. [29] R. Silbey, in Spectroscopy and Excitation Dynamics of Condensed Molecular Systems, eds. V.M. Agranovich and R.M. Hochstrasser (North-Holland, Amsterdam, 1983) p. 1. [30] J.E. Zucker, A. Pinczuk, D.S. Chenila and A.C. Gossard, Phys. Rev. B35 (1987) 2892. [31] L. Schultheis, J. Kuhl, A. Honold and C.W. Tu, Phys. Rev. Lett. 57 (1986) 1797. [32] A. Honold, L. Schultheis, J. Kuhl and C.W. Tu, Appl. Phys. Lett. 52 (1988) 2105. [33]5G. Elkomoss and G. Munschy, J. Phys. Chem. Solids 38 (1977) 557. [34]5G. Elkomoss and G. Munschy, J. Phys. Chem. Solids 42 (1981) 1. [35] A. Honold, L. Schultheis, J. KuhI and C.W. Tu, Phys. Rev. B40 (1989) 6442. [36] 5. Schmitt-Rink, D.S. Chemla and D.A.B. Miller, Phys. Rev. B32 (1985) 6601. [37] L. Schultheis, A. Honold, J. Kuhl, K. Kbhler and C.W. Tu, Phys. Rev. B34 (1986) 9027. [38] J.T. Remillard, H. Wang, D.G. Steel, J. Oh, J. Pamulapati and P.K. Bhattacharya, Phys. Rev. Lett. 62 (1989) 2861.