Binding probability of free electrons and free holes into Wannier-Mott exciton in non-polar semiconductors

J. Phys. Chem. Solid& 1973, Vol. 34. pp. 1 5 6 7 - 1 5 7 7 .

Pe rgamon Press.

Printed in G r e a t Britain

B I N D I N G P R O B A B I L I T Y OF F R E E E L E C T R O N S A N D F R E E H O L E S I N T O W A N N I E R - M O T T E X C I T O N IN NON-POLAR SEMICONDUCTORS J. BARRAU, M. HECKMANN, J. COLLET and M. BROUSSEAU Laboratoire de Physique des Solides, Associ6 au C.N.R.S., Universit6 Paul Sabatier et 1.N.S.A.T., 118, Route de Narbonne 31077. Toulouse, Cedex, France

(Received 12 January 1973) Abstraet--A free electron and a free hole in a crystal may be bound together due to the attractive coulomb field: in semiconductors such an electron-hole pair is similar to the Wannier-Mott exciton. The probability of binding into the fundamental state has been calculated by Lipnik [ 1]. Lipnik's model does not predict our recent low temperature measurements [2]. Actually the binding probabilities of excited states of excitons must be considered. This is the purpose of the present paper: a quantum model, (adequate in the low temperature limi0 and a semi-classical model (adequate when the temperature rises) are presented.

electron (klc) + hole (k2v) ' W,K~k,,',k,v )

1. INTRODUCTION

IN A FIRST section we develop the quantum treatment. We present a statistical study of the multilevel recombination problem. We deduce the condition for having a bimolecular kinetic. The associated binding coefficient ye is obtained directly as the ratio of two determinants in terms of transition probabilities: This result is a closed form of the result predicted by Ascarelli's theory of 'sticking probability'. We then calculate the transition probabilities which enter in the expression for Ye according to the quantum perturbation theory. In a second section we develop a semiclassical model by a method which is derived from the Lax's procedure for cross-section calculations of electron on hydrogen-like centers.

Wk~c.k2v~nK

Wannier-Mott exciton nK Wannier-Mott e x c i t o n nK1 < Wannier-Mott exciton n' Kz

(1)

n is the internal quantum number of the exciton (n = 1, 2, 3 . . . . ) and K its wave vector. A quantity such as Wk.c,k. . . . K represents the probability that, in a second, a dissociated electron (klc)-hole (kzv) pair is bound into a Wannier-Mott exciton (nK). All the transitions probabilities are supposed to keep their thermal equilibrium value. We shall design by N, P, N" the densities of electrons, holes and excitons in the state n. The contribution of those electronic transitions to the time-derivatives of population dN densities are s p e c i f i e d - ~ - - ] , -~-t ], d N " ] as dt 1" They are defined as:

2. THE QUANTUM MODEL

dNJd_t.= - = ~ -d P ] - - N P ~ W a n + ~ N ' W n a

2.1 Binding probability of electrons and holes in an excitonic recombination: the statistical problem in the low temperature limit We shall consider the following electronic transitions in a crystal:

n

n

dN" l (2) dt ]=NPWan+ Z Nn'Wn'n--Nn2 Wn,,,

1567

nl#yt

~t#~t and rr

1568

J. BARRAU, M. HECKMANN, J. COLLET and M. BROUSSEAU energy before undergoing a transition into other quantum state n' # n [3]. The application of the principle of detailed balance yields

We have taken: (a) Wd. = f W(k,c,k.~--.r) f ( ' k , ) [ 1--f('k~)]dz~ dr~ drK1

Wn,n = Wnn, exp--

f W (nKv'--~n K2)f(aK, ) drK, d~-K, ] ,. I i f ( % , )dzK, J

En, - En ksT

and

(b) Wnn,= "

Wnd=[Wdnexp--b]

Here f ( % ), 1 - f(Ek~), f(~K) are the respective occupation probabilities of electron • (21rkBT (m~m2/m2~h + m2) 3/2 (4) states, hole states and (n,K) exciton states. We have also d~-K= (V/877"Z) dlr~, dzK= En is given by the Bohr relation En = + EJrt 2, ( V/87r3) dK. m~ and m 2 are the effective masses for elecIt is easy to see from formula (3a) and (3b) tron and hole respectively. that the transition probabilities Wan and Wn,n For solving the recombination problem we have their thermal equilibrium values for consider here the low temperature limit: occupation probability having a Maxwellspecifically, we suppose that the temperature Boltzmann form characterized by t h e equiliis low enough to have no reexcitation from the brium temperature T. fundamental exciton level into higher levels We consider then in the following that (W~n = W~a = 0). With such conditions the electrons in their conduction band, holes in system (2) may be solved as a linear one their valence band, excitons in the n-states N 2. . . . . as have such Maxwell-Boltzmann distributions. regarding --~--j For electrons and holes such a condition is the unknown quantities; we obtain: usual but for excitons this supposes that an exciton which appear in the (n,K) state has dN dP d--i-= dt NPTe with (5) a high probability for relaxing its kinetic Z Wan n __ W d 2 .~ l

W2d d_N 2]

NP dt J

--Wda-~ 1 d N 3]

NP dt ]

[ i I

-- ~

rF~2 and n*=d

W3d

W3z

Wz.,

W23

L I I

--E W3n t n'~3 and n'=d

(6) -- ~ W2n, nr~2 and nP=d

W32

W42

W~

- E W~, ntr

W43

and nt~d W24 I I I

W34 I I I

-~

nt#4 and nP~d

I

W~n,

WANNIER-MOTT

EXCITON IN NON-POLAR SEMICONDUCTORS

Ye is the probability that, in a second, an electron and a hole recombine (net binding probability). An interesting case is that for which the d N n] quantities dt J' for n > 1, are zero. T h e n y~ is a constant which depends on transition probabilities only. Expression (6) may be developed and written as follows:

(6')

Ye =-- ~, Ten. ?g

We have taken: Yen = Wdn Sn

with

f~ IminorofWa~[ S n ~ ~.1+ (- 1) denominator

for n = 1 for n > 1

this form for Ye is a closed formula which is equivalent to the result predicted by Ascarelli's theory of 'sticking probabilities'J4,5] (see Appendix A for the proof). Consider experimental conditions in which electrons and holes recombine according to the model (1). T h e criteria for having a constant ye may be obtained from the preceding analysis (in Appendix A we illustrate the method for the case of a three-levels exciton). Qualitatively we can say that a constant value for Ye implies an adiabatic evolution o f the exciton gas population which follows the variations o f electron and hole populations. Such an adiabatic evolution is possible, due to the high transition probabilities which determine the re-arrangement o f excitonic levels populations: it will no longer hold for a fast evolution of c a r d e r population, such as takes place during a transient regime at very high carrier densities. But transient recombination from a not too high level at low temperatures is consistent with such an adiabatic evolution of exciton population and a constant Ye is effectively observed.

tion, the probability o f binding the electronhole dissociated pair into the (n, K) exciton state: W (klc, k2v ~ n, K). Then, using formula (3a) we shall deduce the binding probabilities Wd, which enter in the expression (6) for y~. We consider a conduction electron and a valence hole having isotropic effective masses. Also, the excitonic states are taken to be hydrogen-like. Only wave functions corresponding to s-states will be used since binding into such states is more probable than into states having higher angular momentum. T h e wave function of a crystal with a dissociated pair (the initial state of the system) is:

qJa = M V -vzei(~x+k') F (in, 1, i(kr -- kr) Ha ~bHq with R = / g i r l + ~2r2

y = k l -I- k2

k =/z2k~ --/~1k2 /xj-

m~ mex

r = r~ -- rz , /x-

mirn2 mex

,

(7)

/.zeo2

m j , k ~ = kj/aex,

rj=

rjaex a r e

the

effective

masses, wave vectors and radius vectors for an electron (j = 1) and a hole (j = 2); R ' = Raex is the radius vector for the exciton; X is the dielectric constant; q ' = q/aex and K ' = K/aex are the wave vectors for a phonon and an exciton; tOyq are the wave functions for the lattice states. F u r t h e r m o r e the normalization constant M is expressed by: M 2=

27r~/V-1 , with "O 1 1 - e x p - - 2~r7/ = k"

T h e wave function for a crystal with one exciton in the (n, K) state (the final state of the system) is: ~Jn v ~

2.2 Binding probability o f a free electron and a free hole into the n-exciton states We compute in the first Born approxima-

1569

/.rr~3fl3 V ~--I/2~iKR -r

• F ( - - n + 1,2, ~s

~ ~bHq.

(8)

1570

J. B A R R A U , M. H E C K M A N N ,

Here, v =

J. C O L L E T and M. B R O U S S E A U

+__I (for emission o f a phonon) 1 (for absorption of a phonon) 2

F ( a , b , Z ) which appears in (7), (8) is the confluent hypergeometric function[6]. We shall use the following forms: (a) F ( i v , 1, Z) = ~

• ~ (-- 1)JCj exp -- iajvrqdr J=l

Also, we have taken here: a j = (-- 1)J(1--m).

1 f ~t'O--1(~:-- 1) -'n

• ( e x p + Z ~ ) (d~).

(9a)

The closed contour of integration, in the complex plane, contains the points ~ = 0 and ~=1.

(b) F ( - - n + l , 2 , ~ ) = l = ~ - l ( - - 2 )

l

W (Tk --> nK, vq) dK' = (2rr)-2 8 (7 -- K -- vq) hEn

1=0

•

Here, p is the crystal density and c is the sound velocity. The probability that in one second the specified pair (7 k) is bound into a Mott exciton (n, K), after having emitted (having absorbed) a phonon (q) is equal to:

(9b)

•

( l + 1)!/! The operator for interaction of an electron and a hole with acoustic phonons, according ing to the deformation potential theory, is [3,7]: (__~h ~''2 H T = i \ 2 p V ] ~ (t%) -v2--q el

~ k = l-k-Ek-i-E ~,

h2(k')2 h2(y)2 h2(K')2 (12) ek = 2lzE n ; e r - 2mexE,' ~ = 2m~xEn

aex

where C1 and C2 are the deformation potential constants; bq and bq+ are the operators for annihilation and generation of a phonon. The matrix element for binding process follows:

f on*~Hr~ddr'dR ' = 8 (7 -- K - vq) Mq,,Mdn

(m)

in this formula 8 ( 7 - K - v q ) express the wave vector conservation we have taken:

J h q (Nq + 8,,1) Mq ~ = ~ with

E n

~

/ze04 1 2x2h 2 n 2 -,

_ _ =

hz 1 2/~a~ n 2 _

_

.

Here, as in [1], we neglect the contribution of the phonon energy to the energy balance. The matrix element Mdn may be evaluated in the low temperature limit (Appendix B): we observe that the emission of a phonon is the only possibility in this case (v = 1). The probability that, in a second, the specified pair (7, k) is bound in a Mott exciton (n) is then defined as: W(7, k--> n ) = f ~ W (7 k ---> nK; lq) dK'. We obtain (Appendix B): W(,:,,k~n) ~

1] -1

(11)

We have taken:

• [C~e ~q'~ -- C2e iqr~] bq+ Complex conjugate

N q = [exp knTae~ hcq

'.

=

W ( k l , k2 ~

n)

16zrme~(Cl2 + C2z) nB,. h2cpVK,

(13)

and Mdn = M

~

e ira~-(r/") F(i~, 1, i(kr--kr))

We deduce the probability that, in a second, an electron-hole dissociated pair is bound

WANNIER-MOTT EXCITON IN NON-POLAR SEMICONDUCTORS

./hq(Nq+8~,)

into the n-exciton state according to 3(a): Mq~=

16 2V~--~mex(C12k-C22)n B , . Wdn =

(14)

T he calculated values of B, in the case ma = m 2 , - Ca = C2 are B 3 = 0.127

B2=0"141

B 4= 0. 121.

I/InK' = V-a/2(Tgn3a3)-a/2eiK,R

X

2pVca~

f d r e iajqr-r(l/n+l/n')

• F \(- - n ' +

(15)

2.3 T r a n s i t i o n p r o b a b i l i t y b e t w e e n e x c i t o n i c levels We compute in the first Born approximation the transition probability between exciton states: W K_..,K~ . Then, using formula 3b, we shall deduce the transition probability between excitonic levels W._.., which enter in the expression for Ye. Initial (nKa) and final (n'K2) exciton states are described by the following waves function:

xt

Mnnp __ ~ ( n n ,1) a / 2 ~ ( - - 1 ) J C J #

hcp %/-I~kn T

B1 = 0.125

1571

1 , 2 ,-~7]. 2r]

(18)

The transition probability between the specified exciton-states is then written as: (27r) -2

W~.~.,K, ) d K~'-

hE.., dK~

x ~ tMgl21M.., 128(K~

-

K~--vq)

/J

X ~(~.-- EK2 )

(19)

with

h---- 1-k-EK, h2Ka 2 e.K, -- 2mexa2exEnn '

e-tin and (16)

lira,K, = V-al2( 7.fn3a3)-a/2 e,X,R e-r/n' h2K22 Ek2 -- 2me:~a~xE,n,"

(20)

The matrix element for transition is:

f

I~ *n'K2H Tl~/nKtdR'dr'

= 8(KI --K~-- v q ' ) M , , n , M q " with

Wnnt

16m2e~:E.n, ( C. --

(17)

We have neglected the contribution of the phonon to the energy balance. Now, in the low temperature limit (eK ~ l) and in the simple case of identical masses for electron and hole (ma = m2) we derive the transition probability between excitonic levels according to formula 3b:

C2)211.., 12for lajq[ r t p c h 4 ( n n , )3

= %~1In 2 - 1In '2

1572

J. B A R R A U ,

M. H E C K M A N N ,

J. C O L L E T a n d M. B R O U S S E A U

with

l,n,=a_~f~rdr(sinajqr) exp_(ln+_~)1

• F(--n'+

1,2,~)

2.4 The net binding probability W e h a v e considered the contribution of n = 1, 2, 3, 4 exciton levels to the recombination of electrons and holes. In T a b l e 1 we give the values of Wd, obtained f r o m (14). (W,a is deduced according to (4)). I n T a b l e 2

o,I o

Table 1. Wal=l'14•

"a

~

1 (Units: 1015 c m 3 s -~) Wan = 2"41 • ~ VT

Wda=3"4• Wa4=

4"5 x

1

~

% 7")

1

v

"~?~~

;5 -

oK

Fig. 1. T h e contribution o f the f o u r lower level to t h e net binding probability v s temperature.

1

X/I' lg Na I0-'

Table 2. (Units: 101~ . sec -1)

W21 = 10"2 W31 = 4.25 W32 = 2"48

5

W41 = 0.64 W42 = 2.6 W o = 0"99

to u

i0-5

we give the values of W,n, (with n ' < n) obtained from (21). (W.,, with n ' > n is deduced from (4)). W e have t a k e n for silicon: c = 5.5 10s cm sec -t, p = 2.3 # c m -3, E+ = 15 m e V , C1 = C2 = 10 eV. T h e contribution o f the n level to the net binding probability ~/e appears in (6) as ye,. T h e curves of Fig. 1 show the contributions of n = 1, 2, 3, 4 levels. W e see that is convenient to neglect the contribution o f levels n > 4 in the t e m p e r a t u r e range 4 - 1 5 ~ Finally the net binding probability Ye, according to the q u a n t u m theory, is obtained

5

16 +

I

I

1

4

6

8

T,

I 10

I

I

12

~

Fig. 2. T h e n e t binding probability Ye, vs temperature: I from our experiment[2]: Ye exp; II from our q u a n t u m multilevel model: ye*"; III from L i p n i k ' s model: y Li),,m; I V from our semi-classical model: ye ct.

WANNIER-MOTT

EXCITON IN NON-P OL AR S E MICONDUCT ORS

from (6) in the adiabatic approximation for the exciton gas population. The curve (Fig. 2:11) is deduced and is compared to the experimental curve (Fig. 2:1). Good agreement is obtained for the net binding probability, but the temperature dependence is not well verified (experimental y~x,, .=._0-9 lO-3/T 2 cm3/sec; theoretical ye~ --- 2IT 10 -4 cm3/sec. 3. THE SEMI.CLASSICAL MODEL

The quantum calculation of binding probability Wan has been carried out in the low temperature approximation k s T ~ En [see Appendix B]. When this condition is not well satisfied it seems more suitable to use a semiclassical treatment: therefore we make an impact parameter calculation of the crosssection. The method is derived from Lax's procedure for calculating the cross-section of electrons on hydrogen-like centers [8]. The cross-section or(pc, Pn) for binding an electron-hole dissociated pair characterized by the momenta Pc (for center of mass motion) and PR (for relative motion) is obtained as: or(Pc, PR) =

f 2~rbdbPcap(Po pn, b)

(23)

b is the impact parameter; Pcap (Pc, PR, b) is the probability that an electron-hole dissociated pair, with impact parameter b and momentum Pc, Pa will undergo a collision with capture somewhere along the trajectory. This formulation emphasizes a single important collision. With this formalism, the net binding probability 'Yeis then obtained as an average:

1573

It must be emphasized that hto depends on u, Pc, Pn. W [ER(t), hto(u, Pc, PR)] dhto is the transition probability per unit time (with binding energy u) of the specified electron-hole pair for a collision with the emission of a phonon with energy between hto and hoJ+dhto. P(u) is the probability that the formed exciton will not later be dissociated (sticking probability). The kinetic energy ER (t) of relative motion is: En(t) = En ~

V ( r ) = ER~

X" e~ r(t)" (26)

The function r(t) is to be obtained by solving the classical equation of relative motion. So (25) can be rewritten: V~R Peau (Pc, Pn, b) = 2 f [ER (r) -ER~(r) •

2] 1/2 dr

P(u).

(27)

VR

J

If (27) is inserted into (23) the integration over b can first be performed. Then, if the integration over hoJ is saved for last, the result may be written in the form: or (Pc, Pn) = with

fo-(E~,ho~). P ( u ) d h t o

Or(ER, hw) dhto =

f?

41rr2d r ER(t) ER ~

dhto x l(ER(r), hto)"

f or (Pc, PR) (pR/lX) f ( P c ) f(PR) dl)pcdI~PRPc2PR 2 dpcdpR ")/e =

(28)

(29)

(o-(pc, pn) . vn) (24)

f f (Pc) f (PR) dI'IpcdlIPRPc2PR2dpcdPR We have taken VR = PR/IX. The probability (23) of capture along the trajectory will be written in the form:

f f

Pca,(pc, pn, b) = dt (dhw) • W [En(t),hto]P[u(hto)].

This mean free-path for acoustic scattering (25)

JPCS- Vo134No. 9--H

Here, we have: (a) l(En(r),hto), the differential mean freepath, expressed as: 1 W (ER(r), hto) l( En( r) , hto ) vR is:

1574

J. BARRAU, M. H E C K M A N N , J. C O L L E T and M. B R O U S S E A U

l(En(r), heo) = En(r)

1 (hto)2 x

1

]-1

81XC21eq kBT 1 -- exp-- (hto/kBT)

(30)

leq being the mean free-path for an equipartition phonon statistics, i.e.:

Before continuing it is interesting to examine the low temperature limit of this classical binding model for comparison with the quantum one. In such a situation we have 1 2., also P(u) = 1. knT, E o En ~ u and u >>~m~xc So (23) becomes: ho9 ~- V~mexC2U and (32) becomes:

leq = 7rh4pc2ll.t2( C1 -- C2)2kB T

46

(b) For acoustic phonons, conservation of energy and momentum dictates that an energy ER(r) greater than E m = 89 2(1 + 88(hto/89 2) ) 2 is necessary to permit a phonon emission hto. Considering (26) this condition requires as an upper limit rM for r: e02 rM =

1

x 89

+~ho~/89 c~]~-E~ o"

(31)

Now, equation (29) may be integrated. We obtain:

cr(~R, x) dx = c r ~ dx with ~ ' ( e ~ In 189 Orl -~- - ~ \~-~-~C2/ leq kBT"

It is then possible to obtain the probability that, in a second, a specified electron-hole dissociated pair (~n, CC) be bound in lower levels defined by u > u0 = (izeo4/2xZh 2) (1/nZ) :

(o"'fL, = .0,o ' ( E . , x )

4~rM 3 ( h ~ ) 2 ~(ER, h ~ ) d h ~ = 3 ER ~ kBT8ftc2leq

-

dhto • 1 -- e x p - (hto/kBT)" (32) Taking in account energy and momentum conservation in the transition, we deduce:

16V2--~m~z(C12+ C2 2) n z hcp'CtxknT

97r"

(35)

This result should be compared with the quantum one obtained from (14) in the form: nt=~

16 N / ~ m e x (C12 + C22) E Wdn,= hcp%/izknT E nBn. hc~

1-pcu~me~c/

+4(1_

n r ~ 11

(36)

pcU "~2-I "-r~ER~ u ]I mexC 29

mexC/

nt=l

~mea:C I

(33)

We have taken u = q/q, q being the wavevector of the emitted phonon. At this stage we see that y~ will be evaluated if (33) is inserted into (32), then (32) into (28) and finally (28) into (24). We have to consider the reduced quantities: Ek , x lht02, v knT ~n = 89 2 = ~/z-----b-- = 89 2.

(34)

We observe that the ratio (quantum result/ classical result) equals 3,5 for n---- 1 and is between ] and ~- for n = 2, 3, 4. This is a surprising and very interesting result. We carry out now the calculation of the net binding probability ye. F o r this we take a cutoff approximation for the sticking probability e(u):

P(u)=l

for

u>kBT

P(u)=O

for

U
W A N N I E R - M O T T E X C I T O N IN N O N - P O L A R S E M I C O N D U C T O R S

Taking Or 1 = 5-45.10 -1~ cm 2 in'silicon from Lax[8] c = 5.5.105 cm.sec-1; 89 2 = 0.25. 10-4 eV we deduce:

and thus obtain: oo

\

1575

V E R J v ( . f k B T ) [(1 + 88

3

dx ). • 1 - - e x p - - (x/u)

(37)

yeCt = 2 ~10-z c m 3, /sec. On Fig. (2) are reported yeq~, yec~, yeexp, ~ t e L i p n i k

Here, the average has the meaning defined by (24). To make easier the integrations, we approximate, as follows:

+f

Te ~CtTI

~

X2

xo [ ( 1 _.l_ I X ) 2 _ _ ~ b , ] 3

dx • 1 - e x p - (x/v)

(38)

with

hto(u= kBT, ER = (En), Pc = (Pc)) X0 ~

89

2

and w i t h

(En) = ]kBT, (Pc) = 0. The remaining integral is performed numerically and may be fitted by 2 • (4/T) 1"6 as Fig. 3 shows.

.

4. CONCLUSION

The striking feature of the semi-classical and quantum multilevel theories is their agreement in the low temperature limit. It must be outlined that the approximations used in the quantum multilevel theory, especially justified in the low temperature limit, are less and less good as the temperature rises: calculations of transition probabilities towards, or from, the n-level are wrong when k B T >~ E,,. This explains why the semiclassical theory gives a better temperature dependance. Finally, the good agreement of the cascade capture model (according to the semi-classical or quantum multilevel theories) with experiment elucidate the binding process of free electrons and free holes into Wannier-Mott exciton in non-polar semiconductors. REFERENCES

I-0

m

I;

~ NI

8

,,,

o

\.\.\.\

1. L I P N I K A. A., Soviet Phys. solid State 2, 1835 (1960). 2. BARRAU J., H E C K M A N N M. and BROUSSEAU M., J. Phys. Chem. Solids 34, 381 (1973). 3. AMSEL'M A. I. and FIRSOV In. A., Soviet Phys. J E T P 1, 139 (1955). 4. A S C A R E L L I G. and R O D R I G U E Z S., Phys. Rev. 124, 1321 (1961). 5. BROWN R. A. and R O D R I G U E Z S., Phys. Rev. 153, 890 (1967). 6. For instance: L A N D A U L. et L I F C H I T Z E., Mdcanique Quantique, edition Mir. 7. B A R D E E N J. and S H O C K L E Y W., Phys. Rev. 80, 102 (1950). 8. LAX M., Phys. Rev. 119, 1502 (1960). APPENDIX A

0"12

.

~

dN ~ ]

(1) The determinant form (6) of Ye taken for --dt-J = 0, T, oK Fig. 3. The integral of equation (38) well fitted by the function 2 • (4/T) vs. We have taken u = knT[89 c ~

n > 1 is equivalent to the result which is obtained from Ascarelli's theory of sticking probability. According to

1576

J. B A R R A U , M. H E C K M A N N ,

the theory of sticking probability[4, 5] the net binding probability of an electron and a hole is obtained from:

J. C O L L E T and M. B R O U S S E A U N 2= NP x

y~ = ]~ WartS..

(A1)

n

(A6)

S . is the probability that an exciton formed into the n level, from a dissociated electron hole pair, is not reionized. S . is defined as: S . = 1--]~ P.~.

(A2)

v

P.~ is the probability of ionization after a v transition from the state of formation n into other bound states (n' # n) of the exciton. So we have: Pn~ = ~ Pnn,Pn, ~-1 f o r v > nl~n

(N 3 is derived by changing 3 by 2 and 2 by 3). The criterion for constant T~ is obtained from (6) as 1 dN~ ~---~-]

~ Wan. for n = 2 and n = 3.

So, using (A5) and (A6) we derive the conditions: Te(N-F P) (W21-~- W23-~- W2d ) (W31-~- W32-~- Wad ) -- W32W23 Wd 2 (W31 + W32 -[- W3d) W a 2 + Wd3W32

1

(A7)

with Wnn,

Pnn' :

n'~n and n'ffid

W..,

and Pn 1 =

Wnd

Y~W...

b

(A3)

n'~n and nlffid

So, using (A 1), (A2), (A3) it is theoretically possible to obtain T~ in term of transition probabilities W in the form of an infinite series. Our purpose here is to show that this infinite series is summable in the determinant expression (6) taken for

dN__" I dt J =

(W31 + W32 + W3d ) Wd2-]- Wd3W32 ( W21 ~- W23- ~- W2d ) ( W31 -~- W32 -~- W3d ) -- W32W23

(and the similar condition in which 3 and 2 have been exchanged). F o r example, taking the values given in Tables (1 and 2) for 4~ we can conclude that T~ is constant for N, P ~ 1.4 10a5cm -3. APPENDIX B (1) Calculation o f the matrix element Ma~ The hypergeometric function F ( - - n + 1; 2; 2r/n) being considered in its polynomial form (9b), the matrix element Md~ appears as a linear combination of successive derivatives (with respect to h for h = 1) o f the integral:

0 (n > 1). F o r simplicity, consider a three level

Jn~ (h) = f r-~dr exp i[ k -- ajq] r

model. Firstly, using definitions given above we deduce:

--hr F(bq, 1, i(kr--kr). n

~/e = Wdl "~- Wd2 ~- Wd3 -- Wd2P21{ 1 + P23P32 --t- (P23P32) 2 ..j_... -- Wd2P31P23{1 dr- P23P32-[-- (P23P32) 2" - . -- Wd3P21P32{ 1 -~- P23P32 -}- (P23P32) 2 . . . -- Wd3P31 { | -~- P23P32 -I- (P23P32) 2 ..[_...

Now, this integral may be calculated by considering F ( i ~ , I , Z ) in its integral form (9a). Then, using the residue theorem [5] we obtain:

(n > 1):

tcn=nk

Qn=nq

It then follows that: t=,-1

2t

{ n - 1 ~ l .~t+,

J=l

0

(A5)

(2) Criterion f o r constant Te We consider the low temperature situation W ~ , = W~d = 0. We examine the three level model for simplicity. Firstly, we obtain N 2 (and N 3) from (2) in the approximaj

(B2)

with

a~__~ a

We see that the series which appears in A 4 is summed in A5 to give the factor 1/(1 --P~aPa2). So, the two forms are equivalent.

tl/

X [aj2Qn 2 .-k (h - iKn)2] -in

Md.=M ,~,.~(--I)JC' ~ (l+1)!~ I /~'

Te = Wdl + Wa2 + Wd3

d N ~] tion--zT/=0

J ~ = 4~'n2[h2 + ( K n - a~Q~)2]t'-1

(A4)

(Then, using A3b, we can obtain y~ in terms of the transition probability W). Secondly, using the expression (6) we develop the determinants and divide each term by the quantity (W2~ + W23 -4- W2d) ( W31 -~- W32 -+- W3d) . Then, using the preceding definition A3b we obtain:

Wd2 (P21 q- P31P23) + Wa3 (P31 q- P21P32) 1 -- P23Pz2

(B1)

H e r e 1,~~~ is the Ith derivative ofJ,~ (h) in respect o f h and taken at h = 1. In the simple model in which al = - a2 and C1 = - Cz (electron and holes having identical masses, the deformation potential constants have opposite values), we obtain: a3

n-1

2t

N o w , in the low temperature limit k T ,~ En (we have then ~k E~,<~ 1 or "0 ~ ao and Kn = (n/~) --->0), we shall

WANNIER-MOTT

EXCITON

IN NON-POLAR

consider the equivalent form of (B2):

J~ (h) =

4~rn2 h~+aj2Qnzexp

1577

T h e n an angular integration gives (13), in which:

2hn 2niajQn cos O h2+a~Qn~exp- hl+aflQn ~

0 = (Qn, K~).

._1

B.= r~0 sin,,<,op X L ~=0

,,t

(1+ I) !L"

<,+1,(cos 0)

]

(B4)

(2) Binding probability of an electron-hole pair Wan S u m m i n g with respect to q, K in the c a s e u = 1 we oblain:

w(~,, k - . n) = f d.,, f ~ W(~,,k - ' ) n ,

f

SEMICONDUCTORS

K, lq)K'2dK '

f mexQ~2 Ma.l~l t8pc(~hnaex) JQ.= ~viT"ff~l~

with

Ln(COsO, h)--

2e -zxn/cl+x2~ 2n cos 0 l+X2 )
H e r e L. (~ is the I th derivative o f Ln(cos O, h) in respect of and taken at h = 1.