- Email: [email protected]
Electron-exciton collision line in highly excited semiconductors
J. Phw
0022-3697/84 $3.00 + .I0 Pergamon Press Ltd.
Chm~. SO/M/QVol. 45. No. I l/12, pp. 1165-l 173. 1984
Printed in the U.S.A.
ELECTRON-EXCITON HIGHLY EXCITED
COLLISION LINE IN SEMICONDUCTORS
S. G.
ELKOMOSS
Laboratoire de Spectroscopic et d’optique du Corps Solide,t Institut de Physique, Strasbourg, France (Received
21 September
1983; accepted in revisedform
1 March 1984)
Abstrac&-The
derivation of the Boltzmann equation at the steady state given by Morse, Allis and Lamar where only elastic scattering has been considered is generalized to include both elastic and inelastic collisions. This treatment is simpler than that carried out by the different authors working with gas discharges. In the absence of an external electric field the solution of this equation which corresponds to the energy distribution functionfgiven in terms of the temperature T has been applied to the case of the electron-exciton collision in the highly excited semiconductor CdS. Knowing the values offat different temperatures T, the half-width and the strong shifts of the electron+xciton collision line to lower quantum energies as T increases have been calculated for this semiconductor. The results which are independent
of any adjustable parameter are in good agreement with experiment. In these calculations, the elastic collison and the variation of the energy gap with temperature, which have been neglected previously by different authors, have been included and are found to play an important role. INTRODUCITON
In highly excited semiconductors such as CdS, ZnO, CdSe and CuX where the symbol X could be Cl, B or I, an emission line interpreted by different authors to be due to the electron-exciton collision, has been observed[l-71. This line lies at the low energy side of the exciton and is usually present with other lines. One is the so-called I, line corresponding to the bound exciton with neutral donor[l] or the I, line of a bound exciton with neutral acceptor[l]. The other is interpreted as being due to the excitonexciton [ 1,3,6,8,9] collision. The exciton-exciton line appears at low temperature T while that of electron+xciton is observed as T increases. To each semiconductor there corresponds a specific minimum critical temperature T, at which the electron-exciton collision process becomes predominant and for T > T, the electron-exciton collision line is observed. In ZnO[q, CdSe[3], CdS[l, 2,5-71 and CuX[4] the respective critical temperatures T, = 70 K, 90 K, 100 K and 150 K are reported. The position of this line is displaced towards lower energies and the observed line width increases with an increase in T and the excitation intensity J[5]. The line shape in CdS has been studied theoretically by Benoit A La Guillaume et a[.[ 11. In the other theoretical work[5] only the question of the line displacement with temperature has been treated using a Maxwellian distribution and not a more appropriate Boltzmann energy distribution function as has been already mentioned by the author[lO] for such system. Moreover the line is interpreted[5,7] as arising only from electron-exciton inelastic collisions. In the present paper, these assumptions will appear to be inconsistent with the Boltzmann equation where both elastic TResearch group associated with the Centre National de la Recherche Scientifique, France.
and inelastic electron-exciton collisions have to be considered in order to calculate a correct energy distribution function f for a particular value of T. This means there is no sense for the Boltzmann equation in which only the inelastic collisions are considered. On the other hand this equation can be solved where only elastic collisions are taken into account. The consideration of the inelastic collisions has to be accompanied by the inclusion of the elastic scattering. For the solution of the Boltzmann equation the values of the inelastic Q,, and the momentum-transfer Q, collision cross-sections corresponding to a given semiconductor are necessary. Unfortunately, experimental cross-sections are not available in the literature although they would present a particular and great interest. For this reason, the elastic scattering cross-sections QR as a function of the energy k2 for different semiconductors have been calculated [ 1l] theoretically in the central field and exchange approximations. These calculations are in good agreement with the cyclotron resonance measurements given by Otsuka et aZ.[12] for Si. The Q, have also been already calculated [ lo] in the different semiconductors using Glauber approximation [13, 141 for the transitions ls-2.s, ls-3s, ls-2p and 1~3~. This approximation [ 13, 141is valid for the different values of k2. Using the central field or the exchange approximations, the cross-sections Q, as a function of k* can be calculated for the different semiconductors as discussed in the next section. Usually these crosssection calculations for different semiconductors are a function of the mass ratio u = M t/m X, where m T and rnf are, respectively, the electron and the hole effective masses. In this paper, to economise computer time, the cross-sections Q, have been calculated for o = 0.21
1165
1166
S. G. ELKOMOS
and 0.5 in the central field approximation neglecting the exchange effect. Also in this paper the theory for the energy distribution function where the electron-atom or electron-exciton collision process is predominant has been developed in detail and in the quite simpler manner than that already given for gas discharges[lS, 161. This treatment where both elastic and inelastic electron-atom collisions are included is a generalization to that of Morse, Allis and Lamar[l7] in which only elastic scattering has been considered. It should also be noticed that the electron-exciton scattering corresponds exactly to that of electron-atom except for different values of 0, or in other words different masses[lO, 111. This energy distribution function f corresponding to the particular case under study where no external electric field is present has been solved completely for the semiconductor CdS with a = 0.21. Knowing the values offat the different temperatures T, the displacements of the electron-exciton collision line and the haif-width as a function of T and excitation intensity J have been calculated for the semiconductor. The agreement with experiment is reasonably good. In these calculations the elastic collisions and the variation with temperature of the ~miconductor energy gap which have been completely neglected by the previous authors[l-71 are seen to play an important role. MOMENTUM-TRANSFER CROSS-SECTIONS Q,
In the phase shift method of central potential scattering, the total scattering QR and the momentum-transfer QD cross-sections are given[lS, 191, respectively, by eqns (1) and (2): QR = $ f (2i+ 1) sin2 a1
(1)
110
QD = $
,go[(21 + 1) sin2 a1- 2(1 + 1) cos (a,‘-- a,+ ,)
x sin a1sin a,+ ,]
(2)
where 6, is the phase shift of order 1. Substituting eqn (1) into eqn (2), the expression for Q, becomes: QD=Q,-G
E 2(1+l)[cosS,cos6,+,sin& t-0
x
sin6,+, + sin2 f$ sin2 6! +.,I.
(3)
In eqns (l)-(3) the cross-sections QR and Q, are given in terms of xuoz where &I + o) is the first Bohr exciton radius. From eqns (1) and (3), knowing the total scattering cross-section QR and the phase shifts 6,, one can calculate the momentum-transfer or the diffusion cross-section Q, as function of the energy k2. In this central field approximation the phase shifts 6, are the solutions of a second-order differential equation[ll]. In the exchange approximation the
expressions for QR and QD are given by eqns (4) and (5), respectively, QR = $ i (2Z + 1)[3 sin* 6,- + sin a,+]
(4)
/SO
Q, = $ [zo(21 + I)(3 sin2 6,- + sin aft) - $ m
x c (1 + 1)[3(cos a,- cos a,;, I=0 x
sin 6; sin 6,,
+ sin’ 6,- sin2 6,; ,)
+ (cos 6,+ cos St+ , sin 6,+ sin 6 T+, + sin* 6,+ sin* S&. ,)I. Substituting
(5)
eqn (4) into eqn (5) one gets:
x (I + 1)[3(cos a,- cos a;+, sin &- sin a;+, + sin* 6; sin* ST+,) + (cos ST cos S,Z+I x sin 6 :. , sin a,+ + sin* a,+ sin* 6 t+, 1.
(6)
From eqns (4) and (6) knowing the cross-sections Q, and the phase shifts 6{- and S,’ one can ealcuiate in the exchange approximation the momentum-transfer cross-sections Q, as a function of the energy k2. In this approximation the phase shifts 15~~and &+ are the solutions of a second order in~gro-diffe~ntia1 equation[ll]. In this case the calculations of Q, carried out using eqns (4) and (6) are much more tedious and require more computer time than those cor~~nding to eqns (1) and (3) where the exchange effect has been neglected. The values of QR decrease[l l] as CTincreases. This decrease of the values of QK as u increases is much more pronounced in the central field method than that in the exchange approximation[l 11. For the semiconductors with large values of 6, the values of QR in the central field and exchange approximations approach each other and the use of the central field method is justified. This is not the case for electron-hydrogen atom[l 1,201 with u x 0 where the exchange effect is important. BO~TZ~N
EQUATION
The theoretical study of the Boltzmann equation including both elastic and inelastic electron-atom collisions has been developed for the high frequency gas discharges]15 161, The dis~bution function is the solution of this Boltzmann equation. The different theories have been carried out for the general case and in complicated manner [ 151. It is of considerable importance to know the dist~bution function as accurately as possible. Assuming the steady state case and taking account of the elastic and inelastic
1167
Electron-exciton collision line electron-atom collisions, it was possible to derive the same Boltzmann equation in a very simple manner. It has to be mentioned that this treatment is a genera~~~on of the one already developed by Morse et a1.[17] where only elastic collisions have been considered. Furthermore, the case for which the external electric field is absent and the electron-exciton elastic and inelastic collisions are predominant has been considered. In this case the distribution function is given in terms of the temperature T. Referring to electron-atom collisions in Fig. 1, the electric field E is along the x-axis, dr and dy’ are the initial and final volume elements before and after collision in the phase space, e is the angle of scattering, and cii and ~5,’are the angles that the elements dy and dy ’ make, respectively, with the x-axis. In phase space the element of volume dz dy is dr = dx dy dz dy=v2sinG
dvdo?drp
dy’=v’2sin&Vdv’d&5’dqo’
eE
c d7 dy =
,cos~~+~~-&~~, [ tJ %
ah
+vaxcoso3
+5z
(11)
where m is the electron mass. In elastic scattering the velocities v and 0’ before and after collision, respectively, differ slightly v’=u$Av
(12)
and $=;(l-co&)
(13)
where A4 is the mass of the atom. The distribution function d(u’, J) can be written as
(7) f(v’,~‘)=f(v,ui’)+Av~f(v,6’).
(8) (9
where v and u’ are the velocities before and after collision, respectively. The number of particles in the volume dz where velocities fall in the range dy is f(x, y, z, 6, q, 0 dT dy. For reasons of simplicity fis homogeneous and isotropic in the yz plane so that it is only a function of (x, v, W) or (x, v, 4). Assume that f can be expanded in a series of Legendre functions of cosrs =l$/v
(14)
In this case the net number of particles after scattering coming into the volume dz dr is given by [ 17] - v“J(v, ~53)cos ~5
(b - a) dt dy = (2nN/v3) S[
x (1 - cos fQa(v, e> + .4v;
x {df (v, G’))g(v’,
(10)
fo(x, v) determines the random distribution velocity and f, gives the drift velocity. The higher order terms that simply modify the distribution function, do not correspond to any simple physical property of the distribution and are always nearly very small[l7]. Using eqn (10) the number of particles leaving the element dz dy due to an applied electric field E and due to diffusion is[17]
Ep
dir’ \
1
(15)
0) sin 0 d6 dz dy,
f (x, v, 6) = Cf”@, v)~,(cos 6)
= f@(.x,v) + f,(x, v) cos (3 + . . .
dTdy, 1
where N is the number density of gas atoms and ~(v, 0) the scattering differential cross-section. This elastic scattering process developed by Morse, Allis and Lamar[ll can be generalized for the inelastic collision. In the inelastic process the energy $rnvn2 transferred during collision corresponds to the electronic excitation energy for the /I level. In this case the relation equivalent to eqn (12) is [15] V'2 =v2+v,2.
(16)
The number of particles leaving the element dt dy due to inelastic collisions is g dz dy = 2n c NV f(v, h
cii)uh(v, 0)
sin 0 de dT dy.
s
(17) The number coming into the element dz dr is I2 j
Fig. 1. The phase space of the electron-atom collisions; dr and dy’ are the initial and final volume elements before and after collision.
dz dr = 2n C NV f (v’, cii‘)~~(t;),v’) b h f 0 x
sin 0 de d z dy.
(18)
1168
S. G. ELKOMOSS
The net number coming into the element dz dy due to inelastic collisions is (j -g)
f(v’, w’)a,(e, u’)
dz dy = 2n 1 NV h
+CKu + u,JQ,(u+ df(U + 4 - uQr(u)f(u)l= 0.
S[
h
(24) XV(:I
-f(u,
w)(T,,(u, 011
sin B d0 dr dy.
(19)
where u,,(u, 0) is the excitation differential crosssection. The eqn (19) corresponds to eqn (13) of Holstein paper[l5] written in different form and without any explanation. The function f (u', (5') in eqn (19) cannot be expanded in a Taylor’s series as is done in eqn (14). In eqn (19) one may simply replace f (u’, W’), f (u, 6) and f (u, 6’) by fo(u’, (s’), fo(u, (5) and_f,(u, W’), respectively. From eqns (15) and (19) the net number of particles coming into the element of dr dy after suffering both elastic and inelastic collisions is
Equation follows:
+ &
(24) can be written in two equations
+ n,,)Q,,(u + u,+)f (u+ u,,) =0
(26)
where
- u4f,(u,6)cos 6
Q,(u) =
Jl
x O(U,e)(i - cose) + dua(u, e) xg
u > ~1
- uQ&)f(u) =O
rr
u < U, (25)
and
[b + j - (a + g)] dr dy = (27rNIu’)
as
{u”f (u, 6’))
1
sin 0 dfI dr dy
x ~,w 0) 3x0, ad4
11
0)
x sin tI de dr dy.
(20)
T Q/n(U).
(27)
Equations (25) and (26) are exactly the same equations that one obtains from Holstein’s equations[l5] in the steady state condition and with a dc electric field, assuming both elastic and inelastic electronic collisions. Following then the same procedure used by Morse et al. [17] where only elastic scattering is considered Boltzmann equation (24) for elastic and inelastic excitation collisions has been derived in simpler and clear manner than that given by Holstein[ 151. Substituting t = (1/2)mu in eqns (25) and (26), one gets
At steady state eqns (11) and (20) are equivalent; i.e. +$k(r’Qof)+ cdrdy=[b+j-(a+g)ldrdy.
Assuming f independently of x, carrying out the integrals in eqn (20) and equating terms in cos CZand without cos W in both sides of eqn (21), one gets
eE% --= m au
3
(21)
-uQd
x ; (e + $)Q& + $,)f (6+ E,,) = 0 C < CI (28)
= 3tQcJ
(22)
t > t,.
(29)
The existence and uniqueness of the solution for eqns (28) and (29) have already been studied in detail by Sherman[21].
+Ch
K;>
1
. (23) h( u’,(3)/7,(u’) -.M~,~)/%l(u)
where zhm’(u) = uQh(u) and Qh(u) is the excitation cross-section. Making a change of variables u = u* and substituting f; of eqn (22) into eqn (23) one obtains
METHOD OF SOLUTION WITH E = 0
In the absence of any external electric field E, eqns (28) and (29) are:
$;(c2Q.f)+x@
+Gt)Q/r(~+G)f(C:
+Ch)=o
h
E
(30)
1169
Electron-exciton collision line
$$(~~Q~f)=treex
E.>E*.
(3l)
The same two eqns (30) and (31) can be obtained for the case where 6 = (1/2)Mu. Considering the effect of the temperature T, the energy distribution function f(t) becomes j(r + kBT), where kB is the Boltzmann constant. In this case the integrations of eqns (30) and (3 1) give, respectively,
&
f(c)+k
’
Taf=!t% z 2mc2Q,
6
function fN corresponding to a particular temperature Tis
fN=
m f(') s0
de.
(40)
(~)‘l?r(c) dc
If one considers the distribution function to be (~)“~f, instead of simply fN of eqn (40) one then can write
(32)
>&,
(33) The mean energies c,r and c2r corresponding, respectively, to fNand (c)lnfN of eqns (40) and (40’) are
where rt
G(6) = J0
c (E + c&f (6 + ~,)Q,,(E + CAdc
e < E,
(41)
(34) CZT=
For the electron-exciton can be rewritten as
collision, eqns (32) and (33)
The line half-width relations
(41’) l/r can be obtained
N&Q)
(42)
N, = (N,NJ”
(43)
(vQ> =(;~'k)'"e)
(W
l/7 =
f(c)+&,T$,-i!!& 2mz .s2QD
e > Cl.
(33’)
In eqns (32’) and (33’), rn: and M, are, respectively, the electron effective mass and M, = rn: + m $ the total exciton mass. Considering only elastic collisions, eqns (32’) and (33’) reduce into one equation F,(e) + k,Tz
= 0.
(36)
To carry out the integrations of eqns (32’) and (33’) let us put f = F,F,. In this case the solutions of these eqns (32’) and (33’) are, respectively,
f=++&T~'-$&d+
n
where N,, and N, are the exciton and electron density, respectively. The quantity Q represents either the inelastic or the momentum-transfer cross-section. Following eqns (40) and (40’), respectively, the term ((e)‘“Q) can be given either by eqns (45) and (45’)
((~)"'Q> = om(rY2QfN(c)k s
cc, (37)
(38)
where F, = e-&r
(45)
or ((E)“~Q) = om(c)3/2Qfdc) ds. s
f=F,[l+~TJ-.:&d+~,
using the
(39)
is the solution of the homogeneous eqn (36). The solutions given in eqns (37) and (38) for the energy distribution function where both elastic and inelastic collisions are considered have been evaluated numerically using a computer program in double precision for UNIVAC 1110. The normalized distribution
(45’)
Considering the solution given in eqn (39) that takes into account only the elastic collision for the energy distribution function, the mean energies r,, and ErTEcorresponding, respectively, to eqns (41) and (41’) at a temperature T can be evaluated analytically as k,T
;,;,=2 ( <2TE=
‘fl
y WPW.
>
(46) (46’)
1170
S. G. ELKOMOSS RESULTS
To calculate the energy distribution function at a particular temperature from eqns (37) and (38), it is necessary to know the variation of the momentum-transfer cross-section QD with the energy k2. The values of QD as a function of k2 calculated using eqns (1) and (2) and which correspond to g = 0.21 and 0.5 are given in Figs. 2 and 3. These values of Q. decrease as u increases. For CdS[22] with u = 0.21 and taking into account the inelastic collison crosssection Qls_2s that corresponds to the transition ls2s only, the normalized energy distribution function fN or (t)i’tfN at different temperatures T have been calculated using eqns (34), (35) and (37)-(40’). Consequently, the mean energies FIT and F2T or the displacements of the electron-exciton collision line corresponding to these temperatures have been obtained using eqns (40)-(41’). The mean energies FITE and izTE of eqns (46) and (46’) where only the elastic collisions have been considered are also evaluated at different temperatures T. In solid state semiconductors the energy gap E, between the valence and the conduction bands varies with the temperature T. Two main contributions are responsible for the temperature dependence of the optical edge[23]. First, the band structure is affected by the thermal expansion of the lattice. The second contribution arises from the electron-phonon interaction. In 1959 Dexter[24] reported the temperature shift of the absorption spectrum in CdS. Recently Yacobi[23] and Yacobi et a1.[25] have studied the temperature shift of the edge-emission-band maximum E,(T) in ZnS and CdS. This study[23,25] is based on the phonon-generated microfield unified theory of Dow and Redfield[26] for the absorption exponential edge through the Franz-Keldysh effect[27] in alkali halides, II-VI compounds, com-
Fig. 3. The monentum-transfer cross-section for CdS having the available value of O
pensated semiconductors, or ionic crystals. The physical model[26] is the electric field ionization of the exciton, i.e. the electron tunnels through the Coulomb barrier away from the hole. The sources of these internal “microfields” may involve phonons, LO and LA, ionized impurities, dislocations, surfaces and other defects [26]. Only phonons whose wavelength is larger than a, the exciton radius, have to be considered[26]. As the temperature is increased, the shorter-wavelength phonons[26] will eventually give rise to a sufficiently deep potential to induce. tunneling. In this case the shorter-wavelength optical phonons become important. The fact that both impurities and phonons must be taken into account, that absorption and emission bands are closely similar [25], Yacobi[23] and Yacobi et al. [25] have shown that the temperature shift of the edge-emission-band maximum E,(T) in CdS varies linearly with coth(Rw/2k,T). For the II-VI compounds the main contribution to the variation of EAT) with temperature is the electron-phonon interaction. The origin of this treatment is the Friilich polaron theory (Appendix C of Ref. [26]). In CdS the edge emission bands were less affected by impurities than those of ZnS[25]. At all temperatures, the band shape parameters were unambiguously determined for CdS[25]. In the range 140-340K, it is the effective LO phonon energy hiwro = 38 mev to be used. For temperatures in the range 78-140 K, one has to consider the effective LA phonon energy ftaLA = 18.5mev instead of hw,[25]. Yacobi et al. [25] have given the two experimental curves corresponding to the variation of E,(T) as function of coth (hw/2k,T) in the range 10&340 K using both ftoLA and AU,. As the theory of phonon-coupled emission predicts[25,28,29], these two curves[25] can be represented by the equation E,,(T) = G + H
Fig. 2. The momentum-transfer for
CdS
having
the
cross-section Q, calculated available value of (r = 0.21 with 0 < k* < 3.
Q, calculated u = 0.5 with
coth @w/2k,T)
(47)
where H = - 0.15 ev[23]. In eqn (47) using the value of EA292 K) = 2.432 ev given in the Table of Yacobi et a1.[25] and ho, = 38 mev, one can find G = 2.6667 ev. With these values of G and H, eqn (47) reproduces exactly the experimental values of EdT)[25] at any temperature T between 140 and
1171
Electron-exciton collision line Table 1. The values of the experimental results of &(T) calculated from eqn (47) with G = 2.6667 and hv given by Yacobi et al.[25] and Fischer et ul.[5], respectively, as well as the calculated values of&,< and EAT) - C,, at the temperatures 160, 200, 250 and 300 K
E
2TE
in ev
Ep(T) -
c2TE
hV 153
Temperature T
Ep(T) 1251 in ev
1 60°K
2.4965
0.0207
2.475,
2.468
200'K
2.4797
0.0258
2.453y
2.442
250'K
2.454y
0.0322
2.422
2.409
3OO'K
2.4275
0.0387
2.3888
340 K. Using eqn (47), the experimental values[25] of E&r’) corresponding to 160, 200, 250 and 300 K are given in Table 1. The values of &, calculated from eqns (40’) and (41’) at the temperatures 160,200,250 and 300 K are, respectively, 0.02,0.025,0.03 and 0.035 ev. The corresponding values of E2rE= (3/2)kJ of eqn (46’) where only elastic collisions have been considered are 0.0207, 0.0258, 0.0322 and 0.0387 ev, respectively. The values of Zrr do not differ much from those of &. This means that the effect of the inelastic collisions is still small at 300 K. The differences between the values of EAT) given by eqn (47) with the constant G = 2.6667 and the calculated values of &, given above at the different temperatures can be compared with the experimental photon-energy values given by Fischer et al.[5] as a function of temperature for the emission electron-xc&on collision line. In Table 1 the values of EAT) calculated from eqn (47) and which correspond to the experimental values of Yacobi et al.[25], czTE, EAT) - EzTE,and experimental hv results given in Ref.[5] for the temperatures 160, 200, 250 and 300 K are given. In spite of the different simplifications and the fact that the results of hv are taken from the figure given by Fischer et al.[5] and not from actual numbers, columns 4 and 5 of Table 1 show the good agreement between theory and experiment. It is also easy to notice that the calculated results of E2rcorresponding to the temperatures 160, 200, 250 and 300 K and where both elastic and inelastic collisions are considered, give less agreement between the values of hv given in Table 1 and those of E,(T) - &. This again indicates the important contribution of the elastic collisions. Instead of using the experimental values of EAT), one may try to find an energy distribution function
7
in ev
2.375
to fit directly the experimental results hv of Fischer et al. [5]. In this case instead of eqn (41’), one can write m E=
CfN(e)dc.
I0
(48)
Considering the energy gap to be 2.582 ev, the values of n corresponding to the best fit between F of eqn (48) and the results of Ref. [5] vary between 1.035 and 0.9473 in the range 16&300 K. The values of n of eqn (48) that fit the experimental results of Fischer et a1.[5] at different temperatures are given in Fig. 4. A mean value of n = 1 may then give good agreement between theory and experiment. As a matter of fact this corresponds to eqn (41) with the energy distribution function fdc) given in eqn (40). For only elastic collisions, the corresponding mean energies c,, are given in eqn (46). In Fig. 5 as a function of temperature the solid curve and the broken curves C, A and B correspond, respectively, to the experimental hv results of Fischer et al. [5], E,(T) - & given in
Fig. 4. The variation of the power n of eqn (48) with the
temperature T.
1172
S. G. ELKOM~SS 2.551
I
0 2.35’
Fig.
’ 100
’
I ’ ’ ’ 200 300 Temperature (K)
’ 400
’
5.
The experimental and calculated values of the photon energy corresponding to the positions of the electron+xciton collision line at different temperatures T. The solid curve represents the experimental hv results of Fischer et al. [5]. The broken curves C, A and B correspond, respectively, to the values of E#‘) - .&.s of Table 1, the calculated values 2.582 - E,, and 2.582 - r,,.
Table 1, the calculated values 2.582 - E;r, and 2.582 -E,rE. From this figure it is seen that the agreement between the solid curve and the broken curve C is as good as that between the solid curve and the curve B. On the other hand less agreement is obtained between the solid curve and that of A where both elastic and inelastic collisions are included. From the broken curves A and B of Fig. 5 the effect of inelastic collisions appears for T > 120 K. From Fig. 5 one can see the dominant contribution due to the elastic collisions. From this figure one can also notice that instead of considering the temperature shift of the optical edge and using an energy distribution function E’/‘~AE),one may use directly an energy distribution function fN(e) that takes into account this effect and gives comparable results. In the following, the energy distribution function fN(t) of eqn (40) will be used. The mean values of (E r’*&,) and (E ‘/*Q,& of eqn (45) have been calculated at different temperatures. The values of (~1/2Q,p2J) at these temperatures are too small compared with those obtained from experiment. For the case J = 50 and T = 160 K, eqn (45) gives $‘i2~$=~~532rra,-,2. With N,, - lOI cm-’ and 3 eqns (42)-(45) give a value eN l/r = 163.45 cmm2. This calculated value of l/r is in agreement with the value 197.16 cm-’ experimentally obtained for Cd8 The experimental value [5] for l/r corresponds to N, = IO” cme3 considered for the theoretical value. For the case J = 150 and T = 300 K, the calculated value (t’/*QD) = 3.5818*~u,,* is obtained from eqn (45). With the values of N,, and N, given above and applying Klingshim’s[30] assumption that N,,z J while Nea J”* the calculated value of (t”*QD) = 3.58 187r~* for such a case leads to a value of l/7 = 232.5 cm-‘. Again this calculated value of l/r is in good agreement
with that 285.9 cm-’ obtained experimentally[5]. This experimental value of l/7 corresponds to N, 2.8 X 10” cmm3 while the value N, - 2.3 X 10” cmm3 has been used for the evaluation of the theoretical l/7 value. From the broken curve C of Fig. 5 one should expect at this temperature of 300 K a better agreement between the calculated and experimental values of l/7. On the other hand, at this temperature the calculated value of (cO.~~‘~Q~) = 4.5949~%* corresponds to n = 0.9473 of eqn (48) that fits the experimental results of Fischer et al.[5]. In this case the corresponding calculated value of l/7 = 298.26 cm-’ is in excellent agreement with that obtained experimentally[5]. DISCUSSION
The electronexciton collision line studied in this paper has been interpreted by different authors to be due only to inelastic collisions between the two constituents of the system, the electron and the exciton, without any further consideration of the contributing mechanisms. This interpretation is not consistent with eqns (37) and (38) developed in this paper. These two equations show that it is impossible to consider the inelastic collisions without including the elastic scattering. On the other hand one can consider the elastic scattering without taking into account the inelastic collisions. In neglecting the elastic scattering and considering only the inelastic collisions eqns (37) and (38) become nonsense. In highly excited ZnO Magde and Mahr[31] have explained the exciton-exciton collision line as being due to the scattering of one of the constituents to higher energy and the other to lower energy. The lower energy exciton is photon like. This mechanism is consistent with the general model treated in the present paper where both elastic and inelastic collisions between the two constituents, the electron and the exciton, have been considered. But the calculations given in the present paper show that the elastic scattering and the variation of the energy gap with temperature play the important role. Table 1 and Fig. 5 indicate that the elastic scattering contributes a larger part of the electron-exciton collision line shift with temperature and that the role of the inelastic collisions is negligible compared to the one of elastic scattering. It is also only the elastic collisions that contribute to the line half-width and its broadening with temperature and exciton intensity. It is the first time that the variation of the electron-exciton momentum-transfer cross-section QD as function of energy is reported for 0 = 0.21 and 0.5. Without knowing these values of Q, given in Fig. 2 for B = 0.21 it would have been impossible to carry out such calculations reported in the present paper particularly those concerning the line half-width. It is worth mentioning that the shift of the electron-exciton collision line with temperature has been given by different authors to be (1/2)(M,,/m:)(k,T). In this case the mass ratio (M&n :) is taken to be constant and the variation of
1173
Electron-exciton collision line the edge-emission-band maximum E&P) with temperature has not been considered. The interpretation is not adequate since it gives misleading results. For instance, if one considers the shift to be (1/2)(M,,/m f)(k,T) and the experimental results of EAT) and hv given, respectively, by Yacobi et al. [25] and Fischer et al. [S] for CdS in the range 160-300 K one finds that (h&,/m:) is no longer constant but varies from 4.64 to 4.03. For this reason one should be very careful about the effective masses obtained experimentally from highly excited semiconductors at high temperatures. Again without taking into account the variation of Ep(T) and considering the shift to be (3/2)(M,,/m t)&T) one gets complete disagreement between theory and experimental results of Refs. [25] and [5] unless much lower values of (M&n:) are considered at high temperatures. For instance, in the range 160-300 K the values of (M&Z z) vary from 1.52 to 1.34, respectively. It should also be mentioned that it is the first time that the variation Ep( T) has been considered in this paper for the treatment of the shift of the electron-exciton collision line with temperature. To avoid misleading results this variation of E,(T), which is quite important, has to be considered in any treatment concerning semiconductors at high temperatures. It appears that considering the variation of EAT)[25] and taking into account only the elastic collisions for which the corresponding mean energy is simply (3/2)(k,T) given in eqn (46’) a very good agreement between theory and experiment [S] is obtained. Better agreement with experiment[5] may be obtained at high temperature by including excitonimpurity or even exciton-exciton elastic collisions. However, this will require far more computer time. It is worth mentioning that considering eqn (47) for the variation of the energy gap with temperature and using an energy distribution function cf (6) instead of f (6) or c”‘f (c), the inclusion of only the elastic scattering gives an excellent agreement with column 5 in Table 1. In this case the mean energy c corresponding to the elastic collisions is (2k,T). It would be worthwhile to pursue the experimental study of the external electric field effect on such electron+xciton collision line, as this seems to be of some interest. From the theoretical point of view this would correspond to the elaboration of eqns (28) and (29). In this case the inelastic collisions due to the other transitions ls-3s, ls-2p, Is-3p, and probably the ionization cross-sections to the conduction band,
as well as the calculations of the momentum-transfer cross-section Q, in the exchange approximation have to be considered. Acknowledgements-I am very grateful to profs. C. Benoit A La Guillaume and C. Cribier for stimulating discussions. The computations have been carried out at the Computer Center Cronenbourg-Strasbourg. I wish to thank Professor G. Monsonego, Director of the Computer Center, for his valuable encouragement. REFERENCES
1. Benoit A La Guillaume C., Debever I. M. and Sulvan F., Phys. Rev. 177, 567 (1969).
2. Iwai S. and Namba S.., ADDI. *. Phvs. Letters 19.41 (1971). 3. Braun W., Bille J., Fischer T: and Huber’ G.,‘ Phyi. Status Solidi (b) 58, 759 (1973). 4. Yu C. I., Goto T. and Ueta M., J. Phys. Sot. Japan 34, 693 (1973).
5. Fischer T. and Bille J., J. Appl. Phys. 45, 3937 (1974). 6. Levy R. and Grun J. B., Phys. Status Solidi (a) 22, 11 (1971). 7. Honerlage H., Klingshim C. and Grun J. B., Phys. Status Solidi (b) 78, 599 (1976). 8. Magde D. and Mahr H., Phys. Rev. Letters 24, 890 (1970).
9. Hvam J. M., Phys. Rev. B4, 4459 (1971). 10. Elkomoss S. G. and Munschy G., J. Phys. Chem. Solids 40, 431 (1979). 11. Elkomoss S. G. and Munschy G., J. Phys. Chem. Solids 38, 557 (1977).
12. Ohyama T., Sanada T. and Otsuka E., J. Phys. Sot. Japan 35, 822 (1977).
13. Glauber R. J., Phys. Rev. 100, 242 (1955). 14. Franc0 V. and Glauber R. J., Phys. Rev. 142, (1966). 15. Holstein T., Phys. Rev. 70, 367 (1946). 16. Margenau H., Phys. Rev. 73, 297 (1948). 17. Morse P. M., Allis W. P. and Lamar E. S., Phys. 48, 412 (1935). 18. Massey H. S. W. and Mohr C. B. O., Proc. Roy. (London) A141, 434 (1933). 19. Elkomoss S. G., Phys. Rev. 172, 153 (1968). 20. Seaton M. J., Proc. Roy. Sot. (London) A241,
1195
Rev. Sot.
522
(1957).
21. Sherman B., J. Math. Anal. Appl. 1, 342 (1960). 22. Reynolds D. C., Litton C. W. and Collins T. C., Phys. Status Solidi 12, 3 (1965).
23. Yacobi B. G., Phys. Rev. B22, 1007 (1980). 24. Dexter D. L., J. Phys. Chem. Solids 8, 473 (1959). 25. Yacobi B. G., Datta S. and Holt D. B., Phil. Msg. 35, 145 (1977).
26. 27. 28. 29.
Dow J. D. and Redfield D., Phys. Rev. 5B, 594 (1972).
Franz W., Z. Naturforsch A13, 484 (1958).
Markham J. J., Rev. Mod. Phys. 31, 956 (1959). Klick C. C. and Schulman, J. H., Solid State Physics, Vol. 5 (Edited by F. Seitz and D. Tumbull), p. 97. New York, Academic Press (1957). 30. Klingshim C., Phys. Status Solidi (b) 71, 547 (1975). 31. Magde D. and Mahr H., Phys. Rev. B2, 4098 (1970).
0022-3697/84 $3.00 + .I0 Pergamon Press Ltd.
Chm~. SO/M/QVol. 45. No. I l/12, pp. 1165-l 173. 1984
Printed in the U.S.A.
ELECTRON-EXCITON HIGHLY EXCITED
COLLISION LINE IN SEMICONDUCTORS
S. G.
ELKOMOSS
Laboratoire de Spectroscopic et d’optique du Corps Solide,t Institut de Physique, Strasbourg, France (Received
21 September
1983; accepted in revisedform
1 March 1984)
Abstrac&-The
derivation of the Boltzmann equation at the steady state given by Morse, Allis and Lamar where only elastic scattering has been considered is generalized to include both elastic and inelastic collisions. This treatment is simpler than that carried out by the different authors working with gas discharges. In the absence of an external electric field the solution of this equation which corresponds to the energy distribution functionfgiven in terms of the temperature T has been applied to the case of the electron-exciton collision in the highly excited semiconductor CdS. Knowing the values offat different temperatures T, the half-width and the strong shifts of the electron+xciton collision line to lower quantum energies as T increases have been calculated for this semiconductor. The results which are independent
of any adjustable parameter are in good agreement with experiment. In these calculations, the elastic collison and the variation of the energy gap with temperature, which have been neglected previously by different authors, have been included and are found to play an important role. INTRODUCITON
In highly excited semiconductors such as CdS, ZnO, CdSe and CuX where the symbol X could be Cl, B or I, an emission line interpreted by different authors to be due to the electron-exciton collision, has been observed[l-71. This line lies at the low energy side of the exciton and is usually present with other lines. One is the so-called I, line corresponding to the bound exciton with neutral donor[l] or the I, line of a bound exciton with neutral acceptor[l]. The other is interpreted as being due to the excitonexciton [ 1,3,6,8,9] collision. The exciton-exciton line appears at low temperature T while that of electron+xciton is observed as T increases. To each semiconductor there corresponds a specific minimum critical temperature T, at which the electron-exciton collision process becomes predominant and for T > T, the electron-exciton collision line is observed. In ZnO[q, CdSe[3], CdS[l, 2,5-71 and CuX[4] the respective critical temperatures T, = 70 K, 90 K, 100 K and 150 K are reported. The position of this line is displaced towards lower energies and the observed line width increases with an increase in T and the excitation intensity J[5]. The line shape in CdS has been studied theoretically by Benoit A La Guillaume et a[.[ 11. In the other theoretical work[5] only the question of the line displacement with temperature has been treated using a Maxwellian distribution and not a more appropriate Boltzmann energy distribution function as has been already mentioned by the author[lO] for such system. Moreover the line is interpreted[5,7] as arising only from electron-exciton inelastic collisions. In the present paper, these assumptions will appear to be inconsistent with the Boltzmann equation where both elastic TResearch group associated with the Centre National de la Recherche Scientifique, France.
and inelastic electron-exciton collisions have to be considered in order to calculate a correct energy distribution function f for a particular value of T. This means there is no sense for the Boltzmann equation in which only the inelastic collisions are considered. On the other hand this equation can be solved where only elastic collisions are taken into account. The consideration of the inelastic collisions has to be accompanied by the inclusion of the elastic scattering. For the solution of the Boltzmann equation the values of the inelastic Q,, and the momentum-transfer Q, collision cross-sections corresponding to a given semiconductor are necessary. Unfortunately, experimental cross-sections are not available in the literature although they would present a particular and great interest. For this reason, the elastic scattering cross-sections QR as a function of the energy k2 for different semiconductors have been calculated [ 1l] theoretically in the central field and exchange approximations. These calculations are in good agreement with the cyclotron resonance measurements given by Otsuka et aZ.[12] for Si. The Q, have also been already calculated [ lo] in the different semiconductors using Glauber approximation [13, 141 for the transitions ls-2.s, ls-3s, ls-2p and 1~3~. This approximation [ 13, 141is valid for the different values of k2. Using the central field or the exchange approximations, the cross-sections Q, as a function of k* can be calculated for the different semiconductors as discussed in the next section. Usually these crosssection calculations for different semiconductors are a function of the mass ratio u = M t/m X, where m T and rnf are, respectively, the electron and the hole effective masses. In this paper, to economise computer time, the cross-sections Q, have been calculated for o = 0.21
1165
1166
S. G. ELKOMOS
and 0.5 in the central field approximation neglecting the exchange effect. Also in this paper the theory for the energy distribution function where the electron-atom or electron-exciton collision process is predominant has been developed in detail and in the quite simpler manner than that already given for gas discharges[lS, 161. This treatment where both elastic and inelastic electron-atom collisions are included is a generalization to that of Morse, Allis and Lamar[l7] in which only elastic scattering has been considered. It should also be noticed that the electron-exciton scattering corresponds exactly to that of electron-atom except for different values of 0, or in other words different masses[lO, 111. This energy distribution function f corresponding to the particular case under study where no external electric field is present has been solved completely for the semiconductor CdS with a = 0.21. Knowing the values offat the different temperatures T, the displacements of the electron-exciton collision line and the haif-width as a function of T and excitation intensity J have been calculated for the semiconductor. The agreement with experiment is reasonably good. In these calculations the elastic collisions and the variation with temperature of the ~miconductor energy gap which have been completely neglected by the previous authors[l-71 are seen to play an important role. MOMENTUM-TRANSFER CROSS-SECTIONS Q,
In the phase shift method of central potential scattering, the total scattering QR and the momentum-transfer QD cross-sections are given[lS, 191, respectively, by eqns (1) and (2): QR = $ f (2i+ 1) sin2 a1
(1)
110
QD = $
,go[(21 + 1) sin2 a1- 2(1 + 1) cos (a,‘-- a,+ ,)
x sin a1sin a,+ ,]
(2)
where 6, is the phase shift of order 1. Substituting eqn (1) into eqn (2), the expression for Q, becomes: QD=Q,-G
E 2(1+l)[cosS,cos6,+,sin& t-0
x
sin6,+, + sin2 f$ sin2 6! +.,I.
(3)
In eqns (l)-(3) the cross-sections QR and Q, are given in terms of xuoz where &I + o) is the first Bohr exciton radius. From eqns (1) and (3), knowing the total scattering cross-section QR and the phase shifts 6,, one can calculate the momentum-transfer or the diffusion cross-section Q, as function of the energy k2. In this central field approximation the phase shifts 6, are the solutions of a second-order differential equation[ll]. In the exchange approximation the
expressions for QR and QD are given by eqns (4) and (5), respectively, QR = $ i (2Z + 1)[3 sin* 6,- + sin a,+]
(4)
/SO
Q, = $ [zo(21 + I)(3 sin2 6,- + sin aft) - $ m
x c (1 + 1)[3(cos a,- cos a,;, I=0 x
sin 6; sin 6,,
+ sin’ 6,- sin2 6,; ,)
+ (cos 6,+ cos St+ , sin 6,+ sin 6 T+, + sin* 6,+ sin* S&. ,)I. Substituting
(5)
eqn (4) into eqn (5) one gets:
x (I + 1)[3(cos a,- cos a;+, sin &- sin a;+, + sin* 6; sin* ST+,) + (cos ST cos S,Z+I x sin 6 :. , sin a,+ + sin* a,+ sin* 6 t+, 1.
(6)
From eqns (4) and (6) knowing the cross-sections Q, and the phase shifts 6{- and S,’ one can ealcuiate in the exchange approximation the momentum-transfer cross-sections Q, as a function of the energy k2. In this approximation the phase shifts 15~~and &+ are the solutions of a second order in~gro-diffe~ntia1 equation[ll]. In this case the calculations of Q, carried out using eqns (4) and (6) are much more tedious and require more computer time than those cor~~nding to eqns (1) and (3) where the exchange effect has been neglected. The values of QR decrease[l l] as CTincreases. This decrease of the values of QK as u increases is much more pronounced in the central field method than that in the exchange approximation[l 11. For the semiconductors with large values of 6, the values of QR in the central field and exchange approximations approach each other and the use of the central field method is justified. This is not the case for electron-hydrogen atom[l 1,201 with u x 0 where the exchange effect is important. BO~TZ~N
EQUATION
The theoretical study of the Boltzmann equation including both elastic and inelastic electron-atom collisions has been developed for the high frequency gas discharges]15 161, The dis~bution function is the solution of this Boltzmann equation. The different theories have been carried out for the general case and in complicated manner [ 151. It is of considerable importance to know the dist~bution function as accurately as possible. Assuming the steady state case and taking account of the elastic and inelastic
1167
Electron-exciton collision line electron-atom collisions, it was possible to derive the same Boltzmann equation in a very simple manner. It has to be mentioned that this treatment is a genera~~~on of the one already developed by Morse et a1.[17] where only elastic collisions have been considered. Furthermore, the case for which the external electric field is absent and the electron-exciton elastic and inelastic collisions are predominant has been considered. In this case the distribution function is given in terms of the temperature T. Referring to electron-atom collisions in Fig. 1, the electric field E is along the x-axis, dr and dy’ are the initial and final volume elements before and after collision in the phase space, e is the angle of scattering, and cii and ~5,’are the angles that the elements dy and dy ’ make, respectively, with the x-axis. In phase space the element of volume dz dy is dr = dx dy dz dy=v2sinG
dvdo?drp
dy’=v’2sin&Vdv’d&5’dqo’
eE
c d7 dy =
,cos~~+~~-&~~, [ tJ %
ah
+vaxcoso3
+5z
(11)
where m is the electron mass. In elastic scattering the velocities v and 0’ before and after collision, respectively, differ slightly v’=u$Av
(12)
and $=;(l-co&)
(13)
where A4 is the mass of the atom. The distribution function d(u’, J) can be written as
(7) f(v’,~‘)=f(v,ui’)+Av~f(v,6’).
(8) (9
where v and u’ are the velocities before and after collision, respectively. The number of particles in the volume dz where velocities fall in the range dy is f(x, y, z, 6, q, 0 dT dy. For reasons of simplicity fis homogeneous and isotropic in the yz plane so that it is only a function of (x, v, W) or (x, v, 4). Assume that f can be expanded in a series of Legendre functions of cosrs =l$/v
(14)
In this case the net number of particles after scattering coming into the volume dz dr is given by [ 17] - v“J(v, ~53)cos ~5
(b - a) dt dy = (2nN/v3) S[
x (1 - cos fQa(v, e> + .4v;
x {df (v, G’))g(v’,
(10)
fo(x, v) determines the random distribution velocity and f, gives the drift velocity. The higher order terms that simply modify the distribution function, do not correspond to any simple physical property of the distribution and are always nearly very small[l7]. Using eqn (10) the number of particles leaving the element dz dy due to an applied electric field E and due to diffusion is[17]
Ep
dir’ \
1
(15)
0) sin 0 d6 dz dy,
f (x, v, 6) = Cf”@, v)~,(cos 6)
= f@(.x,v) + f,(x, v) cos (3 + . . .
dTdy, 1
where N is the number density of gas atoms and ~(v, 0) the scattering differential cross-section. This elastic scattering process developed by Morse, Allis and Lamar[ll can be generalized for the inelastic collision. In the inelastic process the energy $rnvn2 transferred during collision corresponds to the electronic excitation energy for the /I level. In this case the relation equivalent to eqn (12) is [15] V'2 =v2+v,2.
(16)
The number of particles leaving the element dt dy due to inelastic collisions is g dz dy = 2n c NV f(v, h
cii)uh(v, 0)
sin 0 de dT dy.
s
(17) The number coming into the element dz dr is I2 j
Fig. 1. The phase space of the electron-atom collisions; dr and dy’ are the initial and final volume elements before and after collision.
dz dr = 2n C NV f (v’, cii‘)~~(t;),v’) b h f 0 x
sin 0 de d z dy.
(18)
1168
S. G. ELKOMOSS
The net number coming into the element dz dy due to inelastic collisions is (j -g)
f(v’, w’)a,(e, u’)
dz dy = 2n 1 NV h
+CKu + u,JQ,(u+ df(U + 4 - uQr(u)f(u)l= 0.
S[
h
(24) XV(:I
-f(u,
w)(T,,(u, 011
sin B d0 dr dy.
(19)
where u,,(u, 0) is the excitation differential crosssection. The eqn (19) corresponds to eqn (13) of Holstein paper[l5] written in different form and without any explanation. The function f (u', (5') in eqn (19) cannot be expanded in a Taylor’s series as is done in eqn (14). In eqn (19) one may simply replace f (u’, W’), f (u, 6) and f (u, 6’) by fo(u’, (s’), fo(u, (5) and_f,(u, W’), respectively. From eqns (15) and (19) the net number of particles coming into the element of dr dy after suffering both elastic and inelastic collisions is
Equation follows:
+ &
(24) can be written in two equations
+ n,,)Q,,(u + u,+)f (u+ u,,) =0
(26)
where
- u4f,(u,6)cos 6
Q,(u) =
Jl
x O(U,e)(i - cose) + dua(u, e) xg
u > ~1
- uQ&)f(u) =O
rr
u < U, (25)
and
[b + j - (a + g)] dr dy = (27rNIu’)
as
{u”f (u, 6’))
1
sin 0 dfI dr dy
x ~,w 0) 3x0, ad4
11
0)
x sin tI de dr dy.
(20)
T Q/n(U).
(27)
Equations (25) and (26) are exactly the same equations that one obtains from Holstein’s equations[l5] in the steady state condition and with a dc electric field, assuming both elastic and inelastic electronic collisions. Following then the same procedure used by Morse et al. [17] where only elastic scattering is considered Boltzmann equation (24) for elastic and inelastic excitation collisions has been derived in simpler and clear manner than that given by Holstein[ 151. Substituting t = (1/2)mu in eqns (25) and (26), one gets
At steady state eqns (11) and (20) are equivalent; i.e. +$k(r’Qof)+ cdrdy=[b+j-(a+g)ldrdy.
Assuming f independently of x, carrying out the integrals in eqn (20) and equating terms in cos CZand without cos W in both sides of eqn (21), one gets
eE% --= m au
3
(21)
-uQd
x ; (e + $)Q& + $,)f (6+ E,,) = 0 C < CI (28)
= 3tQcJ
(22)
t > t,.
(29)
The existence and uniqueness of the solution for eqns (28) and (29) have already been studied in detail by Sherman[21].
+Ch
K;>
1
. (23) h( u’,(3)/7,(u’) -.M~,~)/%l(u)
where zhm’(u) = uQh(u) and Qh(u) is the excitation cross-section. Making a change of variables u = u* and substituting f; of eqn (22) into eqn (23) one obtains
METHOD OF SOLUTION WITH E = 0
In the absence of any external electric field E, eqns (28) and (29) are:
$;(c2Q.f)+x@
+Gt)Q/r(~+G)f(C:
+Ch)=o
h
E
(30)
1169
Electron-exciton collision line
$$(~~Q~f)=treex
E.>E*.
(3l)
The same two eqns (30) and (31) can be obtained for the case where 6 = (1/2)Mu. Considering the effect of the temperature T, the energy distribution function f(t) becomes j(r + kBT), where kB is the Boltzmann constant. In this case the integrations of eqns (30) and (3 1) give, respectively,
&
f(c)+k
’
Taf=!t% z 2mc2Q,
6
function fN corresponding to a particular temperature Tis
fN=
m f(') s0
de.
(40)
(~)‘l?r(c) dc
If one considers the distribution function to be (~)“~f, instead of simply fN of eqn (40) one then can write
(32)
>&,
(33) The mean energies c,r and c2r corresponding, respectively, to fNand (c)lnfN of eqns (40) and (40’) are
where rt
G(6) = J0
c (E + c&f (6 + ~,)Q,,(E + CAdc
e < E,
(41)
(34) CZT=
For the electron-exciton can be rewritten as
collision, eqns (32) and (33)
The line half-width relations
(41’) l/r can be obtained
N&Q)
(42)
N, = (N,NJ”
(43)
(vQ> =(;~'k)'"e)
(W
l/7 =
f(c)+&,T$,-i!!& 2mz .s2QD
e > Cl.
(33’)
In eqns (32’) and (33’), rn: and M, are, respectively, the electron effective mass and M, = rn: + m $ the total exciton mass. Considering only elastic collisions, eqns (32’) and (33’) reduce into one equation F,(e) + k,Tz
= 0.
(36)
To carry out the integrations of eqns (32’) and (33’) let us put f = F,F,. In this case the solutions of these eqns (32’) and (33’) are, respectively,
f=++&T~'-$&d+
n
where N,, and N, are the exciton and electron density, respectively. The quantity Q represents either the inelastic or the momentum-transfer cross-section. Following eqns (40) and (40’), respectively, the term ((e)‘“Q) can be given either by eqns (45) and (45’)
((~)"'Q> = om(rY2QfN(c)k s
cc, (37)
(38)
where F, = e-&r
(45)
or ((E)“~Q) = om(c)3/2Qfdc) ds. s
f=F,[l+~TJ-.:&d+~,
using the
(39)
is the solution of the homogeneous eqn (36). The solutions given in eqns (37) and (38) for the energy distribution function where both elastic and inelastic collisions are considered have been evaluated numerically using a computer program in double precision for UNIVAC 1110. The normalized distribution
(45’)
Considering the solution given in eqn (39) that takes into account only the elastic collision for the energy distribution function, the mean energies r,, and ErTEcorresponding, respectively, to eqns (41) and (41’) at a temperature T can be evaluated analytically as k,T
;,;,=2 ( <2TE=
‘fl
y WPW.
>
(46) (46’)
1170
S. G. ELKOMOSS RESULTS
To calculate the energy distribution function at a particular temperature from eqns (37) and (38), it is necessary to know the variation of the momentum-transfer cross-section QD with the energy k2. The values of QD as a function of k2 calculated using eqns (1) and (2) and which correspond to g = 0.21 and 0.5 are given in Figs. 2 and 3. These values of Q. decrease as u increases. For CdS[22] with u = 0.21 and taking into account the inelastic collison crosssection Qls_2s that corresponds to the transition ls2s only, the normalized energy distribution function fN or (t)i’tfN at different temperatures T have been calculated using eqns (34), (35) and (37)-(40’). Consequently, the mean energies FIT and F2T or the displacements of the electron-exciton collision line corresponding to these temperatures have been obtained using eqns (40)-(41’). The mean energies FITE and izTE of eqns (46) and (46’) where only the elastic collisions have been considered are also evaluated at different temperatures T. In solid state semiconductors the energy gap E, between the valence and the conduction bands varies with the temperature T. Two main contributions are responsible for the temperature dependence of the optical edge[23]. First, the band structure is affected by the thermal expansion of the lattice. The second contribution arises from the electron-phonon interaction. In 1959 Dexter[24] reported the temperature shift of the absorption spectrum in CdS. Recently Yacobi[23] and Yacobi et a1.[25] have studied the temperature shift of the edge-emission-band maximum E,(T) in ZnS and CdS. This study[23,25] is based on the phonon-generated microfield unified theory of Dow and Redfield[26] for the absorption exponential edge through the Franz-Keldysh effect[27] in alkali halides, II-VI compounds, com-
Fig. 3. The monentum-transfer cross-section for CdS having the available value of O
pensated semiconductors, or ionic crystals. The physical model[26] is the electric field ionization of the exciton, i.e. the electron tunnels through the Coulomb barrier away from the hole. The sources of these internal “microfields” may involve phonons, LO and LA, ionized impurities, dislocations, surfaces and other defects [26]. Only phonons whose wavelength is larger than a, the exciton radius, have to be considered[26]. As the temperature is increased, the shorter-wavelength phonons[26] will eventually give rise to a sufficiently deep potential to induce. tunneling. In this case the shorter-wavelength optical phonons become important. The fact that both impurities and phonons must be taken into account, that absorption and emission bands are closely similar [25], Yacobi[23] and Yacobi et al. [25] have shown that the temperature shift of the edge-emission-band maximum E,(T) in CdS varies linearly with coth(Rw/2k,T). For the II-VI compounds the main contribution to the variation of EAT) with temperature is the electron-phonon interaction. The origin of this treatment is the Friilich polaron theory (Appendix C of Ref. [26]). In CdS the edge emission bands were less affected by impurities than those of ZnS[25]. At all temperatures, the band shape parameters were unambiguously determined for CdS[25]. In the range 140-340K, it is the effective LO phonon energy hiwro = 38 mev to be used. For temperatures in the range 78-140 K, one has to consider the effective LA phonon energy ftaLA = 18.5mev instead of hw,[25]. Yacobi et al. [25] have given the two experimental curves corresponding to the variation of E,(T) as function of coth (hw/2k,T) in the range 10&340 K using both ftoLA and AU,. As the theory of phonon-coupled emission predicts[25,28,29], these two curves[25] can be represented by the equation E,,(T) = G + H
Fig. 2. The momentum-transfer for
CdS
having
the
cross-section Q, calculated available value of (r = 0.21 with 0 < k* < 3.
Q, calculated u = 0.5 with
coth @w/2k,T)
(47)
where H = - 0.15 ev[23]. In eqn (47) using the value of EA292 K) = 2.432 ev given in the Table of Yacobi et a1.[25] and ho, = 38 mev, one can find G = 2.6667 ev. With these values of G and H, eqn (47) reproduces exactly the experimental values of EdT)[25] at any temperature T between 140 and
1171
Electron-exciton collision line Table 1. The values of the experimental results of &(T) calculated from eqn (47) with G = 2.6667 and hv given by Yacobi et al.[25] and Fischer et ul.[5], respectively, as well as the calculated values of&,< and EAT) - C,, at the temperatures 160, 200, 250 and 300 K
E
2TE
in ev
Ep(T) -
c2TE
hV 153
Temperature T
Ep(T) 1251 in ev
1 60°K
2.4965
0.0207
2.475,
2.468
200'K
2.4797
0.0258
2.453y
2.442
250'K
2.454y
0.0322
2.422
2.409
3OO'K
2.4275
0.0387
2.3888
340 K. Using eqn (47), the experimental values[25] of E&r’) corresponding to 160, 200, 250 and 300 K are given in Table 1. The values of &, calculated from eqns (40’) and (41’) at the temperatures 160,200,250 and 300 K are, respectively, 0.02,0.025,0.03 and 0.035 ev. The corresponding values of E2rE= (3/2)kJ of eqn (46’) where only elastic collisions have been considered are 0.0207, 0.0258, 0.0322 and 0.0387 ev, respectively. The values of Zrr do not differ much from those of &. This means that the effect of the inelastic collisions is still small at 300 K. The differences between the values of EAT) given by eqn (47) with the constant G = 2.6667 and the calculated values of &, given above at the different temperatures can be compared with the experimental photon-energy values given by Fischer et al.[5] as a function of temperature for the emission electron-xc&on collision line. In Table 1 the values of EAT) calculated from eqn (47) and which correspond to the experimental values of Yacobi et al.[25], czTE, EAT) - EzTE,and experimental hv results given in Ref.[5] for the temperatures 160, 200, 250 and 300 K are given. In spite of the different simplifications and the fact that the results of hv are taken from the figure given by Fischer et al.[5] and not from actual numbers, columns 4 and 5 of Table 1 show the good agreement between theory and experiment. It is also easy to notice that the calculated results of E2rcorresponding to the temperatures 160, 200, 250 and 300 K and where both elastic and inelastic collisions are considered, give less agreement between the values of hv given in Table 1 and those of E,(T) - &. This again indicates the important contribution of the elastic collisions. Instead of using the experimental values of EAT), one may try to find an energy distribution function
7
in ev
2.375
to fit directly the experimental results hv of Fischer et al. [5]. In this case instead of eqn (41’), one can write m E=
CfN(e)dc.
I0
(48)
Considering the energy gap to be 2.582 ev, the values of n corresponding to the best fit between F of eqn (48) and the results of Ref. [5] vary between 1.035 and 0.9473 in the range 16&300 K. The values of n of eqn (48) that fit the experimental results of Fischer et a1.[5] at different temperatures are given in Fig. 4. A mean value of n = 1 may then give good agreement between theory and experiment. As a matter of fact this corresponds to eqn (41) with the energy distribution function fdc) given in eqn (40). For only elastic collisions, the corresponding mean energies c,, are given in eqn (46). In Fig. 5 as a function of temperature the solid curve and the broken curves C, A and B correspond, respectively, to the experimental hv results of Fischer et al. [5], E,(T) - & given in
Fig. 4. The variation of the power n of eqn (48) with the
temperature T.
1172
S. G. ELKOM~SS 2.551
I
0 2.35’
Fig.
’ 100
’
I ’ ’ ’ 200 300 Temperature (K)
’ 400
’
5.
The experimental and calculated values of the photon energy corresponding to the positions of the electron+xciton collision line at different temperatures T. The solid curve represents the experimental hv results of Fischer et al. [5]. The broken curves C, A and B correspond, respectively, to the values of E#‘) - .&.s of Table 1, the calculated values 2.582 - E,, and 2.582 - r,,.
Table 1, the calculated values 2.582 - E;r, and 2.582 -E,rE. From this figure it is seen that the agreement between the solid curve and the broken curve C is as good as that between the solid curve and the curve B. On the other hand less agreement is obtained between the solid curve and that of A where both elastic and inelastic collisions are included. From the broken curves A and B of Fig. 5 the effect of inelastic collisions appears for T > 120 K. From Fig. 5 one can see the dominant contribution due to the elastic collisions. From this figure one can also notice that instead of considering the temperature shift of the optical edge and using an energy distribution function E’/‘~AE),one may use directly an energy distribution function fN(e) that takes into account this effect and gives comparable results. In the following, the energy distribution function fN(t) of eqn (40) will be used. The mean values of (E r’*&,) and (E ‘/*Q,& of eqn (45) have been calculated at different temperatures. The values of (~1/2Q,p2J) at these temperatures are too small compared with those obtained from experiment. For the case J = 50 and T = 160 K, eqn (45) gives $‘i2~$=~~532rra,-,2. With N,, - lOI cm-’ and 3 eqns (42)-(45) give a value eN l/r = 163.45 cmm2. This calculated value of l/r is in agreement with the value 197.16 cm-’ experimentally obtained for Cd8 The experimental value [5] for l/r corresponds to N, = IO” cme3 considered for the theoretical value. For the case J = 150 and T = 300 K, the calculated value (t’/*QD) = 3.5818*~u,,* is obtained from eqn (45). With the values of N,, and N, given above and applying Klingshim’s[30] assumption that N,,z J while Nea J”* the calculated value of (t”*QD) = 3.58 187r~* for such a case leads to a value of l/7 = 232.5 cm-‘. Again this calculated value of l/r is in good agreement
with that 285.9 cm-’ obtained experimentally[5]. This experimental value of l/7 corresponds to N, 2.8 X 10” cmm3 while the value N, - 2.3 X 10” cmm3 has been used for the evaluation of the theoretical l/7 value. From the broken curve C of Fig. 5 one should expect at this temperature of 300 K a better agreement between the calculated and experimental values of l/7. On the other hand, at this temperature the calculated value of (cO.~~‘~Q~) = 4.5949~%* corresponds to n = 0.9473 of eqn (48) that fits the experimental results of Fischer et al.[5]. In this case the corresponding calculated value of l/7 = 298.26 cm-’ is in excellent agreement with that obtained experimentally[5]. DISCUSSION
The electronexciton collision line studied in this paper has been interpreted by different authors to be due only to inelastic collisions between the two constituents of the system, the electron and the exciton, without any further consideration of the contributing mechanisms. This interpretation is not consistent with eqns (37) and (38) developed in this paper. These two equations show that it is impossible to consider the inelastic collisions without including the elastic scattering. On the other hand one can consider the elastic scattering without taking into account the inelastic collisions. In neglecting the elastic scattering and considering only the inelastic collisions eqns (37) and (38) become nonsense. In highly excited ZnO Magde and Mahr[31] have explained the exciton-exciton collision line as being due to the scattering of one of the constituents to higher energy and the other to lower energy. The lower energy exciton is photon like. This mechanism is consistent with the general model treated in the present paper where both elastic and inelastic collisions between the two constituents, the electron and the exciton, have been considered. But the calculations given in the present paper show that the elastic scattering and the variation of the energy gap with temperature play the important role. Table 1 and Fig. 5 indicate that the elastic scattering contributes a larger part of the electron-exciton collision line shift with temperature and that the role of the inelastic collisions is negligible compared to the one of elastic scattering. It is also only the elastic collisions that contribute to the line half-width and its broadening with temperature and exciton intensity. It is the first time that the variation of the electron-exciton momentum-transfer cross-section QD as function of energy is reported for 0 = 0.21 and 0.5. Without knowing these values of Q, given in Fig. 2 for B = 0.21 it would have been impossible to carry out such calculations reported in the present paper particularly those concerning the line half-width. It is worth mentioning that the shift of the electron-exciton collision line with temperature has been given by different authors to be (1/2)(M,,/m:)(k,T). In this case the mass ratio (M&n :) is taken to be constant and the variation of
1173
Electron-exciton collision line the edge-emission-band maximum E&P) with temperature has not been considered. The interpretation is not adequate since it gives misleading results. For instance, if one considers the shift to be (1/2)(M,,/m f)(k,T) and the experimental results of EAT) and hv given, respectively, by Yacobi et al. [25] and Fischer et al. [S] for CdS in the range 160-300 K one finds that (h&,/m:) is no longer constant but varies from 4.64 to 4.03. For this reason one should be very careful about the effective masses obtained experimentally from highly excited semiconductors at high temperatures. Again without taking into account the variation of Ep(T) and considering the shift to be (3/2)(M,,/m t)&T) one gets complete disagreement between theory and experimental results of Refs. [25] and [5] unless much lower values of (M&n:) are considered at high temperatures. For instance, in the range 160-300 K the values of (M&Z z) vary from 1.52 to 1.34, respectively. It should also be mentioned that it is the first time that the variation Ep( T) has been considered in this paper for the treatment of the shift of the electron-exciton collision line with temperature. To avoid misleading results this variation of E,(T), which is quite important, has to be considered in any treatment concerning semiconductors at high temperatures. It appears that considering the variation of EAT)[25] and taking into account only the elastic collisions for which the corresponding mean energy is simply (3/2)(k,T) given in eqn (46’) a very good agreement between theory and experiment [S] is obtained. Better agreement with experiment[5] may be obtained at high temperature by including excitonimpurity or even exciton-exciton elastic collisions. However, this will require far more computer time. It is worth mentioning that considering eqn (47) for the variation of the energy gap with temperature and using an energy distribution function cf (6) instead of f (6) or c”‘f (c), the inclusion of only the elastic scattering gives an excellent agreement with column 5 in Table 1. In this case the mean energy c corresponding to the elastic collisions is (2k,T). It would be worthwhile to pursue the experimental study of the external electric field effect on such electron+xciton collision line, as this seems to be of some interest. From the theoretical point of view this would correspond to the elaboration of eqns (28) and (29). In this case the inelastic collisions due to the other transitions ls-3s, ls-2p, Is-3p, and probably the ionization cross-sections to the conduction band,
as well as the calculations of the momentum-transfer cross-section Q, in the exchange approximation have to be considered. Acknowledgements-I am very grateful to profs. C. Benoit A La Guillaume and C. Cribier for stimulating discussions. The computations have been carried out at the Computer Center Cronenbourg-Strasbourg. I wish to thank Professor G. Monsonego, Director of the Computer Center, for his valuable encouragement. REFERENCES
1. Benoit A La Guillaume C., Debever I. M. and Sulvan F., Phys. Rev. 177, 567 (1969).
2. Iwai S. and Namba S.., ADDI. *. Phvs. Letters 19.41 (1971). 3. Braun W., Bille J., Fischer T: and Huber’ G.,‘ Phyi. Status Solidi (b) 58, 759 (1973). 4. Yu C. I., Goto T. and Ueta M., J. Phys. Sot. Japan 34, 693 (1973).
5. Fischer T. and Bille J., J. Appl. Phys. 45, 3937 (1974). 6. Levy R. and Grun J. B., Phys. Status Solidi (a) 22, 11 (1971). 7. Honerlage H., Klingshim C. and Grun J. B., Phys. Status Solidi (b) 78, 599 (1976). 8. Magde D. and Mahr H., Phys. Rev. Letters 24, 890 (1970).
9. Hvam J. M., Phys. Rev. B4, 4459 (1971). 10. Elkomoss S. G. and Munschy G., J. Phys. Chem. Solids 40, 431 (1979). 11. Elkomoss S. G. and Munschy G., J. Phys. Chem. Solids 38, 557 (1977).
12. Ohyama T., Sanada T. and Otsuka E., J. Phys. Sot. Japan 35, 822 (1977).
13. Glauber R. J., Phys. Rev. 100, 242 (1955). 14. Franc0 V. and Glauber R. J., Phys. Rev. 142, (1966). 15. Holstein T., Phys. Rev. 70, 367 (1946). 16. Margenau H., Phys. Rev. 73, 297 (1948). 17. Morse P. M., Allis W. P. and Lamar E. S., Phys. 48, 412 (1935). 18. Massey H. S. W. and Mohr C. B. O., Proc. Roy. (London) A141, 434 (1933). 19. Elkomoss S. G., Phys. Rev. 172, 153 (1968). 20. Seaton M. J., Proc. Roy. Sot. (London) A241,
1195
Rev. Sot.
522
(1957).
21. Sherman B., J. Math. Anal. Appl. 1, 342 (1960). 22. Reynolds D. C., Litton C. W. and Collins T. C., Phys. Status Solidi 12, 3 (1965).
23. Yacobi B. G., Phys. Rev. B22, 1007 (1980). 24. Dexter D. L., J. Phys. Chem. Solids 8, 473 (1959). 25. Yacobi B. G., Datta S. and Holt D. B., Phil. Msg. 35, 145 (1977).
26. 27. 28. 29.
Dow J. D. and Redfield D., Phys. Rev. 5B, 594 (1972).
Franz W., Z. Naturforsch A13, 484 (1958).
Markham J. J., Rev. Mod. Phys. 31, 956 (1959). Klick C. C. and Schulman, J. H., Solid State Physics, Vol. 5 (Edited by F. Seitz and D. Tumbull), p. 97. New York, Academic Press (1957). 30. Klingshim C., Phys. Status Solidi (b) 71, 547 (1975). 31. Magde D. and Mahr H., Phys. Rev. B2, 4098 (1970).