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Cooling of high density electron-hole plasma
Solid State Communications, Vol. 42, No. 12, pp. 883-887, 1982. Printed in Great Britain.
0038-1098/82/240883-05,$03.00/0 Pergamon Press Ltd.
COOLING OF HIGH DENSITY ELECTRON-HOLE PLASMA J. Collet, A. Cornet, M. Pugnet and T. Amand Laboratoire de Physique des Solides, associ6 au C.N.R.S., I.N.S.A., D6partement de Physique Avenue de Rangueil, 31077 Toulouse-Cedex, France
(Received 22 January 1982 by F. Bassani) We generalize for high density electron-hole plasma, the previous theories [ 12] of temperature cooling of non-equilibrium hot plasma. Especially we take into account the cooling by emission of mixed longitudinal optical phonon and plasmon modes, these quasiparticles described by a nonequilibrium distribution function. We show that a strong slowing of the plasma cooling occurs, at high electron-hole density. We calculate for CdSe the temperature kinetics of plasma created by Yag laser (pulse duration 30 ps). 1. INTRODUCTION THE COOLING OF optically created carriers in semiconductors has been widely investigated both theoretically [1, 2] and especially experimentally [3, 4, 5, 6] during the last decade. However, so long as the excitation power was low, the researches were focussed on nondegenerate plasma, characterized by a temperature Tp. In this range of low plasma density, analytical expressions have been already derived by several authors [1, 2] for the energy loss rate of the plasma. Recently, with the development of high power picosecond lasers, the optical generation of hot plasma at high density become possible [7, 8]. The previously proposed calculations of plasma cooling must be extended at high electron hole density to account for the following effects: (1) The increase of the plasmon energy gives rise to a mixing of the longitudinal optical (LO) phonons with plasmons. Therefore the plasma cooling is due to the emission of mixed LO phonon-plasmon quasiparticles. (Here after we note LO-PL these quasiparticles). (2) A non.equilibrium distribution of LO-PL quasiparticles occurs, which cannot be described by a BoseEinstein function, even with an effective temperature. (3) The interaction of electrons with acoustical phonons is screened. In the present communication, we have developed a general theory to include the phenomena mentioned above [ 14]. In the last paragraph, the calculations are specialized to CdSe, for the usual following conditions: two photon excitation by Yag laser (k = 1.064#m, pulse duration 30 ps) so that: (1) The plasma temperature and density are homogeneous in the sample (no diffusion);
(2) Surface effects are neglectible. No energy exchange occurs between the bath (thermosta0 and the sample, which is justified for time interval 5t after the excitation pulse checking: 8t ~ I/c (l sample thickness; c is sound velocity). 2. GENERAL MODEL
2.1. Introduction The electron-hole plasma can emit or absorb several quasiparticles: acoustical phonons, transverse optical phonons (TO), LO-PL quasiparticles. The total energy loss rate (dE/dt) of the electron-hole plasma is given by:
f
.
~ dt ]~HVd3q
(1)
hcoqx: Energy of the emitted quasiparticle. The index runs over all the quasiparticles which can be emitted. We use the following notation: • = + or X = -- for the two branches of LO-PL quasiparticles, X = ac for the acoustical phonon, X = TO for the optical phonons. (dnXa/dt)v.np is the emission rate of quasiparticle by the plasma. In the following we neglect the optical phonon dispersion vs the wavevector. The acoustical dispersion curve is linear 60~e = c.q. For the LO-PL quasiparticles, we recall the standard result: 2
883
Vol. 42, No. 12
COOLING OF HIGH DENSITY ELECTRON-HOLE PLASMA
884
LO phonon exchange: (Fr61ich interaction)
4rrNe2( l--~ --~v) 60~) = e~.mo \me + , N is the plasma density 60~(q) = 60g / 1 4- ~~ }
qTf is the standard Thomas Fermi screening length. The problem is therefore reduced to the knowledge of (dn~q/dt)EnP and has been already resolved [ 1]. With a straightforward generalization of [ 1], we can write the emission of quasiparticles by the plasma as follows:
d nXqt [ V~(q) Is dt ]Eri,:--i=~c,v C ~ql x {nX(t)
l
nXq,Tp}d~x(m *, Tp, Pl, q)
(3)
The summation i runs over the conduction and valence bands. I V~(q) [2 is the square of the matrix element of the electron interaction (or hole interaction) with the emitted quasiparticle. (I)x, has been computed in [1]: dPx(m*, Tp, pi, q)
(6)
Vs(q, 60) = If(q) 12BoA(q, 60) (7)
Acoustical phonon exchange:
DoA(q, 60) = 260at(q)/(60 2 -- 60~(q)) is the bare acoustical phonon Green function. Let us note 111+ 1124- V3 by the dashed line . . . . . . Vr(q, 60) is obtained by summing the following diagrammatic series, computed with the usual rules of diagrammatic technics: ~
(q'
~)
. . . .
+
""
" ~
" " " +
l "" ~
l-
" ~
" " -- +
. . .
We obtain after few algebra:
Vr(q, 60) = +
v,+v2+v3
E+v2
] - ( v , + v2 + v3).
] - ( v l + v2).
nA(q, 60)
If(q) Is
(8)
[1 - - ( V 1 -J- V2) 7r] 2
h2 14-exp
1+
4rre2[1 1~ 60~ V2(q, 60) = --q--i-~-~--~]w~_~2 L
(2)
2mi60 q--
exp - -
8m~krp
q+
+ Pi
lg/ ]
hq ]
k-T
(4)
Here, DA(q, 60) is the renormalized acoustical green function:
DA(q, 60) =
+
l - [ i Lf(q)l~D°al~ .] - ( v , + v2).J
equations (3, 4) hold for degenerate or non-degenerate
The polarization part It(q, 60) is calculated in the I, plasmon pole approximation:
plasma.
n~q,Tp : 1/[eht°hqIkTp- 1] The central task of this work is displayed in expression (3). To compute the loss rate (dE/dr), we must: (1) Determine the matrix element I V~(q)I s of the electron interaction (hole interaction) with the emitted quasiparticles, at high plasma density. Particularly, we must calculate the electron interaction (resp. hole interaction) with the mixed LO-PL quasiparticles, and acoustical phonons. (2) Determine the quasiparticle distribution function nXq(t). These two problems are elucidated in the following paragraphs. 2.2. Electron-phonons interactions at high density To compute I V~(q)12, we calculate the Fourier transform Vr(q, 60) of the electron-electron interaction, taking into account the three following interaction processes: Coulomb interaction:
DAo (q, 60)
4zre2
Vl(q, w) - e~q2
(5)
e:.q 2 -- CO2o n(q, 60) = 41re2 x 60~ _ 60~ _ 602 We obtain therefore for the first term in the right side of equation (8):
VI + II2
41re2
27re2Ax
260xq
- + 2 k=+,~ eo.q260x602_(60ax) 2 (9) 1 --(V 1 + V2)rr= -e..q Here the summation runs over the two LO-PL modes 6%x given by equations (2). Equation (9) describes the electron-electron interaction by Coulomb interaction, and exchange of one LO-PL quasiparticle. The square of the interaction matrix element of electrons with LO-PL modes is simply: [ Vx(q ) [2 = A~,=
2~re2A x
e**q260~
? (60~__ 60x)(60q2 = - 60o2- 60x)2?, = + o r 2 -60_) 2 (60+
(10)
h60L (respect h60T): Energy of the longitudinal (respect transverse) optical phonon. In the low density limit, we recover the standard
COOLING OF HIGH DENSITY ELECTRON-HOLE PLASMA
Vol. 42, No. 12
Fr6hlich interaction for the upper LO-PL branch and the electron-plasmon interaction for the lower branch. Let us calculate now the second term in the right side of equation (8). Since for acoustical phonons: co~ ~ coT < coL and: The renormalization of the acoustical phonon spectrum due to electron-electron interaction and LO phonon is very small. So we write: 2co
co2 _
1 _(V1 + V2)~r ~ 1 + COT \coL!
+ CC To the first order perturbation theory, r~reh~(q) is given by:
-- cog
1
So we obtain for the second term in the right side of equation (8) ]f(q) ]2
[q
D A"
i1
q2
tq, co) =
]2 2
2 + ( coT1 q~F] \ coL] x
If(q)
J
12Doa(q,co)
The interaction matrix element of electrons with acoustical phonons is therefore statically renormalized:
I V~(q)12 =
q2 2 q2 +
If(q) 12 q~,
=-
nqh(t)
x 6 (Eo -- E~ -- fico~)
(13)
4), 4)' are quantum states of the external system responsible of LO phonon destruction, mainly acoustical phonons. In principle, Ti~eaax(q)Can be computed from equation (13) if h(q) is known. ( 4)] h(q) 14)' ) and (4) Ih÷(- q) [ 4)' ) cannot be simultaneously different from zero so we rewrite:
1
r/ea~,,(q) = IS/(q)
12 21r 4), --~-~ 1(4)lh(q)l )12 27r
(14)
The second term is neglectible. (12)
([d/dt] nX(t))EaP is given by equation (3) and describes the emission of quasiparticles by the hot plasma. The non-electronical processes of quasiparticles destructions are represented by (dnXq/dt)reaa~.This term is negligible for acoustical phonons, but plays a central part for LO-PL modes. We write:
dnxa(t)l
1(4)lh(q)Xi(q)+h+(--q)yi(--q)l¢>l 2
l(4)lh+(--q)[4)')12~(E~--E#- hco~).
The quasiparticle rate equation is:
(dnX~(t-----~) ~ dt IlEaP + ( d nX(t))relax
2It
x ~(e,--E~,--hw~) + l Yi(--q)l 2 "-~ Z
Quasiparticle rate equation
dnx(t)
-
(11)
\ COL] 2.3.
~ [h(q)X~(q) +h+(--q) Yi(--q)lai(q)
i=+, - q
('02 co
Xi(q)ai(q) + Yi(q)ot~(--q) -
HL°ax = ~
_ coo _ co2
For acoustical frequency co ~ coT < col ; c%,(q) so:
co
a~o(q) is the LO phonon creator. We introduce the mixed modes creator ai(q) (i = + or --). Using a Bogoliubov-Hopfield transform [9]:
LO we rewrite Hrela.,, in the form:
cog
,02 _
LOx = ~ h(q)aLo(q) + h+(q)a~,o(q) H~ea, q
/=+,
Moreover: (plasmon pole approximation) 1 --(V1 + V2)lr- 1 + - -
Hamiltonian responsible of LO-PL quasiparticle destruction can be written:*
aLo(q ) = ~
(q)
DA(q, co) = DoA(q' co) -- co2__ co2ae(q)"
885
(14)
1 2~r = IXi(q) 12 --fi--~ 1(4)lh(q)l 4), )12 r~elax(q) x 6(Eo -- Eed -- hw~) Since few is known on the LO phonon destruction into acoustical phonons (this is a process of order hcoLO/(XWae)maxi) or other processes, we use the following "Ansatz": 2rr ~ 1<4)lh(q)14)' >I2~(E~,--E~0' --h6o~) ~ 1 h TLO
dt
The problem is to calculate Tr~ax(q). The interaction
* We neglect the plasmon damping due to non-electronical processes.
COOLING OF HIGH DENSITY ELECTRON-HOLE PLASMA
886
Using equations (3), (14) and (15) the rate equation can be solved. The final calculations are achieved with the help of a computer, following the algorithm reported in Fig. 1. N is the plasma density, g(t) the electron-hole pair generation rate by the laser pulse. Recombinations are neglected during the first hundred picoseconds, but could be included if necessary.
rniLiot conditions I To:4.2 K; TLO /~c ,/Zv <
Disptay Tp
Tp , ~-c , ~-v
d X
d X
[~
]
nx
Vol. 42, No. 12
-~.nq=(-~-nq)EHP__TLO Ixx(q)12 [i
3. RESULTS
l DispLay (~'>
(-~>=
X
I
T(K~j~) tdnXq' d 3 ~dT]EHP q
I
t DispLay phononLi~ distribution
I~
x (,~+~)= x +(dn~% 1
nq
nq
~ dt /
p ( t + ~ ) = p (t)+gdE (t) 8 AE +8)>:+<~-~> B +g (t)8
Deduce from p end E new / Tp (t+8), /zc (t+8) #~ (t+8)
I /
Fig. 1. Program.
,5oI 100
coso ~I°O~PO'%X
N
~
TL0 (ps)
",,.%
0
20
40
60
t,
80
100
ps
Fig. 2. Dynamical cooling of high density plasma (N = 2 x 10 is cm -3) for four different models: Low density model [4]. -- . . . . Screened interactions model [ 12]. - Screened interactions with nonequilibrium population of L O - P L quasiparticles. The different curves correspond to different LO phonon lifetimes. ~ Temporal profile of exciting laser pulse intensity. rLO is here an experimental optical lifetime. Typically in semiconductors rLO ~-- 10 ps [ 10, 11 ]. So we obtain: 1
1
rlea~(q)
rLO
I Xi(q) 12.
(15)
In a previous paper [12], we have studied the energy loss rate equation (1) of the hot plasma, in the static approximation of the screened electron-phonon interaction. The plasmon-LO phonon resonance, as well as the occurrence of non-equilibrium distributions of LO phonon-plasmon modes, were not taken into account. With such approximations, we have deduced a uniform and strong decrease of the energy loss rate, as a function of the plasma density. However, as emphasized in Section 2, the non-equilibrium population of L O - P L mixed modes must be considered and has a strong influence on the temperature kinetics. The general calculations have been carried out in CdSe, for a plasma density n ~_ 2 × 10 is cm -3, higher than the plasmon-LO phonon resonance density n R 3.4 x 1017 cm -3. We report in the Figure (2), the kinetics of the plasma temperature expected for three models: (1) Our general calculation developed in Section 2; (2) Static approximation [ 12]. Thermalized population of L O - P L quasiparticles at 4.2 K; (3) No screening. Thermalized phonons [4] at 4.2K. It is clear that the cooling is strongly slowed by the screening and the occurrence of non-equilibrium L O - P L mixed modes which is obviously depending on the LO phonon lifetime rLO- For 5 ps < TLO < 10 ps, which is very reasonable, the dynamical cooling is strongly reduced. The effect is important. In Fig. 3, we report the calculated L O - P L distribution in the Brillouin zone during the first 100 ps. The mixed modes remain localized near the F point. It must be emphasized, that in any case, the slowing of the temperature cooling is a general feature, as a consequence of a "self regulation" of the plasma cooling! As displayed in equation (3), the energy loss rate is the product of the emission probability I V x 12 by the excess density of quasiparticles [nan(t)- nan,Tv]. If [ V x 12 is enhanced, a higher population nan(t) of nonequilibrium L O - P L modes appears, which slows still more the cooling! The "self regulation" is due to the balance between the emission rate of L O - P L mixed modes, and the occurrence of large non-equilibrium population.
Vol.42, No. 12 Io-m I
COOLING OF HIGH DENSITY ELECTRON-HOLE PLASMA
/''''~ # I
i
5.10-2
[
~o-~k 10-3 m
Detay~ps •
1. 2.
•
,~-..... 3o
;I., I ~-~,," 106
107 crrl-I
REFERENCES
CdSe Tb-4.2 k
I##O4"~% % • ~ - - J ~ - - - •- - - 2 0 |I • •• oe • ---4,--50
2.107
qt
Fig. 3. Dynamicalbehaviourof the non-equilibrium LO-phonondistributionin the conditionof Fig. 2. 4. CONCLUSION We have developed a calculation which describes the cooling in plasma at high electron-hole density. We included the renormalization of the electron-phonon interactions, the occurrence of the LO-PL resonance, and particularly the appearance of non-equilibrium population of LO-PL mixed modes. We show that a slower cooling of the plasma temperature is expected, with respect to the model formerly developed for low density plasmas. Such a slowing of the temperature kinetics has been reported in several experiments [10, 13]. Acknowledgement - We wish to thank Dr A. Klochikin (Nuclear Physics Institute, Leningrad) for stimulating discussions and a careful reading of the manuscript.
887
Sh. Kogan, Soy. Phys. Solid State 4, 1813 (1963). E.M. ConweU, Solid State Phys. Suppl. No. 7, Academic Press (1967). 3. E.O. Gobel & O. Hildebrand, Phys. Status Solidi (b) 88,645 (1978). 4. R. Ulbrich, Phys. Rev. 8, 5719 (1973). 5. J. Shah, Phys. Rev. B9, 562 (1974). 6. R.F. Leheny, J. Shah, R.L. Fork, C.V. Shank & A. Migus, Solid State Commun. 26, 567 (1978). 7. M. Pugnet, A. Cornet, J. Collet, M. Brousseau, B.S. Razbirin & G.V. Michailov, Solid State Commun. 36, 85 (1980). 8. A. Comet, J. Collet, M. Pugnet, T. Amand, B.S. Razbirin & G.V. Michailov, Phys. Status Solidi (to be published). 9. J.J. Hopfield, Phys. Rev. 112, 1555 (1958). 10. D. yon der Linde, J. Kuhl & H. Klingenberg, Phys. Rev. Lett. 44, 1505 (1980). 11. A. Laubereau, D. vonder Linde & W. Kaiser, Phys. Rev. Lett. 27,802 (1971). 12. M. Pugnet, J. Collet & A. Comet, Solid State Commun. 38, 531 (1981). 13. S. Tanaka, H. Kobayashi, H. Saito & S. Shionoya, J. Phys. Soc. Japan 49, 1051 (1980). 14. Part of this work has been presented at the International co n/erence o[ excited states and multiresonant non-linear optical processes in solids, Aussois, France, 18-20 March 1981. (Unpublished).
0038-1098/82/240883-05,$03.00/0 Pergamon Press Ltd.
COOLING OF HIGH DENSITY ELECTRON-HOLE PLASMA J. Collet, A. Cornet, M. Pugnet and T. Amand Laboratoire de Physique des Solides, associ6 au C.N.R.S., I.N.S.A., D6partement de Physique Avenue de Rangueil, 31077 Toulouse-Cedex, France
(Received 22 January 1982 by F. Bassani) We generalize for high density electron-hole plasma, the previous theories [ 12] of temperature cooling of non-equilibrium hot plasma. Especially we take into account the cooling by emission of mixed longitudinal optical phonon and plasmon modes, these quasiparticles described by a nonequilibrium distribution function. We show that a strong slowing of the plasma cooling occurs, at high electron-hole density. We calculate for CdSe the temperature kinetics of plasma created by Yag laser (pulse duration 30 ps). 1. INTRODUCTION THE COOLING OF optically created carriers in semiconductors has been widely investigated both theoretically [1, 2] and especially experimentally [3, 4, 5, 6] during the last decade. However, so long as the excitation power was low, the researches were focussed on nondegenerate plasma, characterized by a temperature Tp. In this range of low plasma density, analytical expressions have been already derived by several authors [1, 2] for the energy loss rate of the plasma. Recently, with the development of high power picosecond lasers, the optical generation of hot plasma at high density become possible [7, 8]. The previously proposed calculations of plasma cooling must be extended at high electron hole density to account for the following effects: (1) The increase of the plasmon energy gives rise to a mixing of the longitudinal optical (LO) phonons with plasmons. Therefore the plasma cooling is due to the emission of mixed LO phonon-plasmon quasiparticles. (Here after we note LO-PL these quasiparticles). (2) A non.equilibrium distribution of LO-PL quasiparticles occurs, which cannot be described by a BoseEinstein function, even with an effective temperature. (3) The interaction of electrons with acoustical phonons is screened. In the present communication, we have developed a general theory to include the phenomena mentioned above [ 14]. In the last paragraph, the calculations are specialized to CdSe, for the usual following conditions: two photon excitation by Yag laser (k = 1.064#m, pulse duration 30 ps) so that: (1) The plasma temperature and density are homogeneous in the sample (no diffusion);
(2) Surface effects are neglectible. No energy exchange occurs between the bath (thermosta0 and the sample, which is justified for time interval 5t after the excitation pulse checking: 8t ~ I/c (l sample thickness; c is sound velocity). 2. GENERAL MODEL
2.1. Introduction The electron-hole plasma can emit or absorb several quasiparticles: acoustical phonons, transverse optical phonons (TO), LO-PL quasiparticles. The total energy loss rate (dE/dt) of the electron-hole plasma is given by:
f
.
~ dt ]~HVd3q
(1)
hcoqx: Energy of the emitted quasiparticle. The index runs over all the quasiparticles which can be emitted. We use the following notation: • = + or X = -- for the two branches of LO-PL quasiparticles, X = ac for the acoustical phonon, X = TO for the optical phonons. (dnXa/dt)v.np is the emission rate of quasiparticle by the plasma. In the following we neglect the optical phonon dispersion vs the wavevector. The acoustical dispersion curve is linear 60~e = c.q. For the LO-PL quasiparticles, we recall the standard result: 2
883
Vol. 42, No. 12
COOLING OF HIGH DENSITY ELECTRON-HOLE PLASMA
884
LO phonon exchange: (Fr61ich interaction)
4rrNe2( l--~ --~v) 60~) = e~.mo \me + , N is the plasma density 60~(q) = 60g / 1 4- ~~ }
qTf is the standard Thomas Fermi screening length. The problem is therefore reduced to the knowledge of (dn~q/dt)EnP and has been already resolved [ 1]. With a straightforward generalization of [ 1], we can write the emission of quasiparticles by the plasma as follows:
d nXqt [ V~(q) Is dt ]Eri,:--i=~c,v C ~ql x {nX(t)
l
nXq,Tp}d~x(m *, Tp, Pl, q)
(3)
The summation i runs over the conduction and valence bands. I V~(q) [2 is the square of the matrix element of the electron interaction (or hole interaction) with the emitted quasiparticle. (I)x, has been computed in [1]: dPx(m*, Tp, pi, q)
(6)
Vs(q, 60) = If(q) 12BoA(q, 60) (7)
Acoustical phonon exchange:
DoA(q, 60) = 260at(q)/(60 2 -- 60~(q)) is the bare acoustical phonon Green function. Let us note 111+ 1124- V3 by the dashed line . . . . . . Vr(q, 60) is obtained by summing the following diagrammatic series, computed with the usual rules of diagrammatic technics: ~
(q'
~)
. . . .
+
""
" ~
" " " +
l "" ~
l-
" ~
" " -- +
. . .
We obtain after few algebra:
Vr(q, 60) = +
v,+v2+v3
E+v2
] - ( v , + v2 + v3).
] - ( v l + v2).
nA(q, 60)
If(q) Is
(8)
[1 - - ( V 1 -J- V2) 7r] 2
h2 14-exp
1+
4rre2[1 1~ 60~ V2(q, 60) = --q--i-~-~--~]w~_~2 L
(2)
2mi60 q--
exp - -
8m~krp
q+
+ Pi
lg/ ]
hq ]
k-T
(4)
Here, DA(q, 60) is the renormalized acoustical green function:
DA(q, 60) =
+
l - [ i Lf(q)l~D°al~ .] - ( v , + v2).J
equations (3, 4) hold for degenerate or non-degenerate
The polarization part It(q, 60) is calculated in the I, plasmon pole approximation:
plasma.
n~q,Tp : 1/[eht°hqIkTp- 1] The central task of this work is displayed in expression (3). To compute the loss rate (dE/dr), we must: (1) Determine the matrix element I V~(q)I s of the electron interaction (hole interaction) with the emitted quasiparticles, at high plasma density. Particularly, we must calculate the electron interaction (resp. hole interaction) with the mixed LO-PL quasiparticles, and acoustical phonons. (2) Determine the quasiparticle distribution function nXq(t). These two problems are elucidated in the following paragraphs. 2.2. Electron-phonons interactions at high density To compute I V~(q)12, we calculate the Fourier transform Vr(q, 60) of the electron-electron interaction, taking into account the three following interaction processes: Coulomb interaction:
DAo (q, 60)
4zre2
Vl(q, w) - e~q2
(5)
e:.q 2 -- CO2o n(q, 60) = 41re2 x 60~ _ 60~ _ 602 We obtain therefore for the first term in the right side of equation (8):
VI + II2
41re2
27re2Ax
260xq
- + 2 k=+,~ eo.q260x602_(60ax) 2 (9) 1 --(V 1 + V2)rr= -e..q Here the summation runs over the two LO-PL modes 6%x given by equations (2). Equation (9) describes the electron-electron interaction by Coulomb interaction, and exchange of one LO-PL quasiparticle. The square of the interaction matrix element of electrons with LO-PL modes is simply: [ Vx(q ) [2 = A~,=
2~re2A x
e**q260~
? (60~__ 60x)(60q2 = - 60o2- 60x)2?, = + o r 2 -60_) 2 (60+
(10)
h60L (respect h60T): Energy of the longitudinal (respect transverse) optical phonon. In the low density limit, we recover the standard
COOLING OF HIGH DENSITY ELECTRON-HOLE PLASMA
Vol. 42, No. 12
Fr6hlich interaction for the upper LO-PL branch and the electron-plasmon interaction for the lower branch. Let us calculate now the second term in the right side of equation (8). Since for acoustical phonons: co~ ~ coT < coL and: The renormalization of the acoustical phonon spectrum due to electron-electron interaction and LO phonon is very small. So we write: 2co
co2 _
1 _(V1 + V2)~r ~ 1 + COT \coL!
+ CC To the first order perturbation theory, r~reh~(q) is given by:
-- cog
1
So we obtain for the second term in the right side of equation (8) ]f(q) ]2
[q
D A"
i1
q2
tq, co) =
]2 2
2 + ( coT1 q~F] \ coL] x
If(q)
J
12Doa(q,co)
The interaction matrix element of electrons with acoustical phonons is therefore statically renormalized:
I V~(q)12 =
q2 2 q2 +
If(q) 12 q~,
=-
nqh(t)
x 6 (Eo -- E~ -- fico~)
(13)
4), 4)' are quantum states of the external system responsible of LO phonon destruction, mainly acoustical phonons. In principle, Ti~eaax(q)Can be computed from equation (13) if h(q) is known. ( 4)] h(q) 14)' ) and (4) Ih÷(- q) [ 4)' ) cannot be simultaneously different from zero so we rewrite:
1
r/ea~,,(q) = IS/(q)
12 21r 4), --~-~ 1(4)lh(q)l )12 27r
(14)
The second term is neglectible. (12)
([d/dt] nX(t))EaP is given by equation (3) and describes the emission of quasiparticles by the hot plasma. The non-electronical processes of quasiparticles destructions are represented by (dnXq/dt)reaa~.This term is negligible for acoustical phonons, but plays a central part for LO-PL modes. We write:
dnxa(t)l
1(4)lh(q)Xi(q)+h+(--q)yi(--q)l¢>l 2
l(4)lh+(--q)[4)')12~(E~--E#- hco~).
The quasiparticle rate equation is:
(dnX~(t-----~) ~ dt IlEaP + ( d nX(t))relax
2It
x ~(e,--E~,--hw~) + l Yi(--q)l 2 "-~ Z
Quasiparticle rate equation
dnx(t)
-
(11)
\ COL] 2.3.
~ [h(q)X~(q) +h+(--q) Yi(--q)lai(q)
i=+, - q
('02 co
Xi(q)ai(q) + Yi(q)ot~(--q) -
HL°ax = ~
_ coo _ co2
For acoustical frequency co ~ coT < col ; c%,(q) so:
co
a~o(q) is the LO phonon creator. We introduce the mixed modes creator ai(q) (i = + or --). Using a Bogoliubov-Hopfield transform [9]:
LO we rewrite Hrela.,, in the form:
cog
,02 _
LOx = ~ h(q)aLo(q) + h+(q)a~,o(q) H~ea, q
/=+,
Moreover: (plasmon pole approximation) 1 --(V1 + V2)lr- 1 + - -
Hamiltonian responsible of LO-PL quasiparticle destruction can be written:*
aLo(q ) = ~
(q)
DA(q, co) = DoA(q' co) -- co2__ co2ae(q)"
885
(14)
1 2~r = IXi(q) 12 --fi--~ 1(4)lh(q)l 4), )12 r~elax(q) x 6(Eo -- Eed -- hw~) Since few is known on the LO phonon destruction into acoustical phonons (this is a process of order hcoLO/(XWae)maxi) or other processes, we use the following "Ansatz": 2rr ~ 1<4)lh(q)14)' >I2~(E~,--E~0' --h6o~) ~ 1 h TLO
dt
The problem is to calculate Tr~ax(q). The interaction
* We neglect the plasmon damping due to non-electronical processes.
COOLING OF HIGH DENSITY ELECTRON-HOLE PLASMA
886
Using equations (3), (14) and (15) the rate equation can be solved. The final calculations are achieved with the help of a computer, following the algorithm reported in Fig. 1. N is the plasma density, g(t) the electron-hole pair generation rate by the laser pulse. Recombinations are neglected during the first hundred picoseconds, but could be included if necessary.
rniLiot conditions I To:4.2 K; TLO /~c ,/Zv <
Disptay Tp
Tp , ~-c , ~-v
d X
d X
[~
]
nx
Vol. 42, No. 12
-~.nq=(-~-nq)EHP__TLO Ixx(q)12 [i
3. RESULTS
l DispLay (~'>
(-~>=
X
I
T(K~j~) tdnXq' d 3 ~dT]EHP q
I
t DispLay phononLi~ distribution
I~
x (,~+~)= x +(dn~% 1
nq
nq
~ dt /
p ( t + ~ ) = p (t)+gdE (t) 8 AE +8)>:
Deduce from p end E new / Tp (t+8), /zc (t+8) #~ (t+8)
I /
Fig. 1. Program.
,5oI 100
coso ~I°O~PO'%X
N
~
TL0 (ps)
",,.%
0
20
40
60
t,
80
100
ps
Fig. 2. Dynamical cooling of high density plasma (N = 2 x 10 is cm -3) for four different models: Low density model [4]. -- . . . . Screened interactions model [ 12]. - Screened interactions with nonequilibrium population of L O - P L quasiparticles. The different curves correspond to different LO phonon lifetimes. ~ Temporal profile of exciting laser pulse intensity. rLO is here an experimental optical lifetime. Typically in semiconductors rLO ~-- 10 ps [ 10, 11 ]. So we obtain: 1
1
rlea~(q)
rLO
I Xi(q) 12.
(15)
In a previous paper [12], we have studied the energy loss rate equation (1) of the hot plasma, in the static approximation of the screened electron-phonon interaction. The plasmon-LO phonon resonance, as well as the occurrence of non-equilibrium distributions of LO phonon-plasmon modes, were not taken into account. With such approximations, we have deduced a uniform and strong decrease of the energy loss rate, as a function of the plasma density. However, as emphasized in Section 2, the non-equilibrium population of L O - P L mixed modes must be considered and has a strong influence on the temperature kinetics. The general calculations have been carried out in CdSe, for a plasma density n ~_ 2 × 10 is cm -3, higher than the plasmon-LO phonon resonance density n R 3.4 x 1017 cm -3. We report in the Figure (2), the kinetics of the plasma temperature expected for three models: (1) Our general calculation developed in Section 2; (2) Static approximation [ 12]. Thermalized population of L O - P L quasiparticles at 4.2 K; (3) No screening. Thermalized phonons [4] at 4.2K. It is clear that the cooling is strongly slowed by the screening and the occurrence of non-equilibrium L O - P L mixed modes which is obviously depending on the LO phonon lifetime rLO- For 5 ps < TLO < 10 ps, which is very reasonable, the dynamical cooling is strongly reduced. The effect is important. In Fig. 3, we report the calculated L O - P L distribution in the Brillouin zone during the first 100 ps. The mixed modes remain localized near the F point. It must be emphasized, that in any case, the slowing of the temperature cooling is a general feature, as a consequence of a "self regulation" of the plasma cooling! As displayed in equation (3), the energy loss rate is the product of the emission probability I V x 12 by the excess density of quasiparticles [nan(t)- nan,Tv]. If [ V x 12 is enhanced, a higher population nan(t) of nonequilibrium L O - P L modes appears, which slows still more the cooling! The "self regulation" is due to the balance between the emission rate of L O - P L mixed modes, and the occurrence of large non-equilibrium population.
Vol.42, No. 12 Io-m I
COOLING OF HIGH DENSITY ELECTRON-HOLE PLASMA
/''''~ # I
i
5.10-2
[
~o-~k 10-3 m
Detay~ps •
1. 2.
•
,~-..... 3o
;I., I ~-~,," 106
107 crrl-I
REFERENCES
CdSe Tb-4.2 k
I##O4"~% % • ~ - - J ~ - - - •- - - 2 0 |I • •• oe • ---4,--50
2.107
qt
Fig. 3. Dynamicalbehaviourof the non-equilibrium LO-phonondistributionin the conditionof Fig. 2. 4. CONCLUSION We have developed a calculation which describes the cooling in plasma at high electron-hole density. We included the renormalization of the electron-phonon interactions, the occurrence of the LO-PL resonance, and particularly the appearance of non-equilibrium population of LO-PL mixed modes. We show that a slower cooling of the plasma temperature is expected, with respect to the model formerly developed for low density plasmas. Such a slowing of the temperature kinetics has been reported in several experiments [10, 13]. Acknowledgement - We wish to thank Dr A. Klochikin (Nuclear Physics Institute, Leningrad) for stimulating discussions and a careful reading of the manuscript.
887
Sh. Kogan, Soy. Phys. Solid State 4, 1813 (1963). E.M. ConweU, Solid State Phys. Suppl. No. 7, Academic Press (1967). 3. E.O. Gobel & O. Hildebrand, Phys. Status Solidi (b) 88,645 (1978). 4. R. Ulbrich, Phys. Rev. 8, 5719 (1973). 5. J. Shah, Phys. Rev. B9, 562 (1974). 6. R.F. Leheny, J. Shah, R.L. Fork, C.V. Shank & A. Migus, Solid State Commun. 26, 567 (1978). 7. M. Pugnet, A. Cornet, J. Collet, M. Brousseau, B.S. Razbirin & G.V. Michailov, Solid State Commun. 36, 85 (1980). 8. A. Comet, J. Collet, M. Pugnet, T. Amand, B.S. Razbirin & G.V. Michailov, Phys. Status Solidi (to be published). 9. J.J. Hopfield, Phys. Rev. 112, 1555 (1958). 10. D. yon der Linde, J. Kuhl & H. Klingenberg, Phys. Rev. Lett. 44, 1505 (1980). 11. A. Laubereau, D. vonder Linde & W. Kaiser, Phys. Rev. Lett. 27,802 (1971). 12. M. Pugnet, J. Collet & A. Comet, Solid State Commun. 38, 531 (1981). 13. S. Tanaka, H. Kobayashi, H. Saito & S. Shionoya, J. Phys. Soc. Japan 49, 1051 (1980). 14. Part of this work has been presented at the International co n/erence o[ excited states and multiresonant non-linear optical processes in solids, Aussois, France, 18-20 March 1981. (Unpublished).