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Cooling of hot electron-hole plasmas in the presence of screened electron-phonon interactions
Solid State Communications, Vol. 38, pp. 531-536 Pergamon Press Ltd. 1981. Printed in Great Britain.
0038-1098/81/180531-06$02.00/0
COOLING OF HOT ELECTRON-HOLE PLASMAS IN THE PRESENCE OF SCREENED E LECTRON-PHONON INTERACTIONS M. Pugnet, J. Collet and A. Cornet Laboratoire de Physique des Solides, associd au CNRS, INSA, D4partement de Physique, Avenue de Rangueil, 31077 Toulouse Cedex, France
(Received 7 November 1980 by F. Bassani) We have computed for several semiconductors, (GaAs, CdSe, CdS), the energy loss-rate in hot electron-hole plasmas, at high density, when occurs the screening of the electron-phonon interactions. We show that the long range interactions (piezoelectrical and polar) are quickly reduced when the plasma density is raised, and that the energy loss-rate diminishes therefore significantly (one order of magnitude). I. INTRODUCTION
interaction, and piezoelectrical interaction) in the presence of electron-electron interaction screening.
IN THE LAST YEARS, the formation of hot electron-hole plasmas has been achieved in various semiconductors [1-3] and the problem of energy relaxation in such hot plasmas was investigated in the limit of low density of carriers, and for Boltzmann distribution of electrons and holes. The plasma cooling is due to the emission of phonons by the non thermalized carriers. in the low density limit, this problem is now relatively simple, since analytical formulas have been already derived for the energy losses due to the different electron-phonon interactions (Polar interaction with longitudinal optical phonon, piezoelectrical interaction, and interaction with acoustical phonon or non polar optical phonons described in the model of deformation potential) [4, 5]. Let us emphasize that in this limit, the total loss rate of energy per electron hole pair is indepen. dent of the plasma density. Recently, with the occurence of lasers generating high power picopulses, high density electron-hole plasma could be generated, and the dynamics of the plasma cooling was investigated by means of time resolved (picosecond or subpicosecond) spectroscopy [6, 7]. As evidenced by the experiments at low temperature [6], the cooling becomes less effective in high density plasmas, than in the plasmas at low density. This is consecutive to a lowering of the phonon emission rate by the hot carriers, in the high density regime. Two changes must be taken into account to explain this effect :
In the present paper, we are mainly concerned with this effect, which we have just indicated above. We have computed the energy loss rate per electron-hole pair, in the static limit of the random phase approximation (RPA). In our calculation, the phonon distributions are Bose distributions at bath temperature. To our knowledge, few accurate results are available concerning the non-equilibrium LO phonons in picosecond experiments, so that we attempt to describe this LO phonon distribution, by a Bose distribution with an effective temperature TLo, which can be easily accounted in our calculations. 2. MODEL In the presence of electron-electron interactions, occurs a screening of the electron-phonon interactions. This effect has been discussed in former publications (see for ex~,mple [9, 10]). To summarize these results, the electron phonon matrix element must be divided by the dielectric constant of the electron gas, consecutive to electron-electron interactions, considered here in the RPA. The Fourier transform is therefore:
gs ere(q, co)
(I) The appearance of non equilibrium distributions of phonons (specially longitudinal optical (LO) phonons) [8], so that phonon absorption becomes possible and slakens the plasma cooling due to phonons emission. (2) The screening of the electron phonon interactions (in particular the long range interactions: polar
v(q) 6RPA(q,CO)"
(l)
V(q) is the usual matrix element of the bare electron phonon interaction. In the static RPA, for small q, the dielectric constant reduces to the classical DebyeHiJckel expression: eRPA(q, O) = I + q---b-~ q~. In an electron-hole plasma, the Debye screening length qD is given by:
531
(2)
532
Vol. 38, No. 6
COOLING OF HOT ELECTRON-HOLE PLASMAS
Table 1 CdSe
60
Phonon LO energy (meV) Conduction mass Mean valence band mass Density (g cm -3) Acoustical conduction band deformation potential (eV) Acoustical valence band deformation potential (eV) Optical deformation potential (eV cm -t ) Mean sound velocity (cm sec -t) Piezoacoustic factor
h~Lo t/q c
(,%) P Ec
& D c
F/c a
q2D = 87re____~:l_ E f x ( k, #i) -- {fx( k,/ai)}:. e~,kT v k.h
GaAs
10.63 12.56 36.7 0.0665
[181 [181 []91 [20]
0.8 [13] 5.68 [14] 3.7 [15]
0.57 5.36 7
[2] [15] [2l]
5.28 [15] 8.8 [15l 36.81 [15] 0.17 [15] 1.3S [15] 6.75 [151 14.5 [151
5.7
[161
3.5
[221
I0
109
[17]
10 9
10 9
2.3 x 10 s
3.57 × l0 s [23]
3.54 x IO s [151
2.6 x 10-2
4.24 x 10-3
5 × lO-z
6.3 9.4 26 0.13
[91 [91 [91 [12]
(3)
fX(k, #i) is the Fermi function of electron (hole) in the band h, with Fermi level/Je ~h)- The sum over X includes the conduction band and all the valence bands f'dled by holes, v is the crystal volume. The total power (dE/dt), that a gas of fermions releases to the lattice by phonon interaction has been computed by Kogan [5]: _ m2kT[" d3q hw a
d/( \
\dr~,
~s
j (27r)3 Iqt
{ N q ( T ) - - Nq(Tp)}
x [V~¢~e(q)l2
•
(4)
1 + exp ~
+
+
where hcoq is the phonon energy; Tis the fermions temperature (we assume the same temperature for holes and electrons), To is the phonon temperature, rn is the effective mass of fermions, the index i refers to electrons or holes, Nq(T) is the phonon distribution, and
No(T ) =
I
exp ( h w q / k T ) -
CdS
The calculation of the energy losses in a dense electron-hole plasma is achieved, by means of a computer, in three steps: (a) For each pair of parameters (n the plasma density and T the plasma temperature), we compute the Fermi levels (,%, tan) of electrons and holes. For zincblende structure, the two valence bands are roughly replaced by a sin~e valence band mass [2]. For wurzite structure, we account the two valence bands (so called A and B) in the hole Fermi level calculation. (b) We compute the Debye-Htickel screening length given by equation (3). (c) We calculate the energy losses for each kind of carrier, and various electron phonon interactions. All the parameters which have been used in these computations are listed in Table 1. The net plasma loss rate (dE/clt)plas is the sum of all the preceding losses. Finally we report the energy loss rate per electron-hole pair, that means
(1/n)(dE/dt)p~. We distinguish between long range electron-phonon interactions, sensitive to screening, and short range interactions, tess affected by the screening.
I"
The expression (4) is valid for degenerate or nondegenerate gas, and includes spontaneous and stimulation emission of phonons, as well as phonon absorption. The energy losses of the plasma are the sum of the losses of the conduction band, and valence bands given by equation (4):
2.1. Long range interactions Polar interaction. The matrix element for electronphonon interaction is: 4n'e2 ( 1
l__~(heOLo/
IVLo(q)l:- q~ ~e.. ~o1\ 2 j
533
COOLING OF HOT ELECTRON-HOLE PLASMAS
Vol. 38, No. 6
GoAs
1o2 -
GoAs T~=4 2K
;0 ' --
n : 104cm 3
y
:
=IO~cm"3
I
/ I~
iO c
,"~
//','
/%
v
7
:0 ~
iO7
-I=
IOG
{05
I
!
tO
20
X..:T-~ ,e If
.
#"
yl
AI,"
~ Z :~
T~:42K
n
/
;
•
IO0
! 500"
/
I0
TK
100
500
TK
Fig. 1. Energy losses for the four processes vs plasma temperature, with n = 10'~cm -3, in GaAs. • Polar Longitudinal optic interation; • Non-polar optic interaction; • Acoustical deformation potential; ? Piezoelectric interaction;- Total e n e r ~ loss rate.
Fig. 3. Energy losses for the four processes vs plasma temperature, with n = 10tScm -3, in GaAs. • Polar longitudinal optic interaction; • Non-polar optic interaction;• Acoustical deformation potential; re Piezoelectric interaction; - Total energy loss rate.
GaAs
T~=42K
!0 ! ' --
n = 10"cm
lOIr _
3
T==42 K n=2x IO'%m3
iO !O _
A,S
o ~0~
~ III
S
7; 108
2_/'.,
3
v I
kJ
_ i~o"
--I=
~ / I ;• ::
g
iI/
o !09
ht
~
GoAs
i I --Ic
~0"~
,, t" ~e
d:'i'
~C 5
7,o" // _.//.j;'
II~
b'-
/"
10 4
~
o
'0
i
20
~OO
/"
I
500
TK
tOa "f~"
/
.~'" ,Itt h g
I IO
I
I
lOO
500
TK
Fig. 2. Energy losses for the four processes vs plasma temperature with n = 1017cm -3, in GaAs. • Polar longitudinal optic interaction; • Non-polar optic interaction: • Acoustical deformation potential ; re Piezoelectric interaction;- Total energy loss rate.
Fig. 4. Energy losses for the four processes vs plasma temperature, with n = 2 x 1019cm -3, in GaAs. • Polar longitudinal optic interaction; • Non-polar optic interaction; • Acoustical deformation potential; re Piezoelectric interaction;- Total energy loss rate.
534
Non-polar optical interaction [4] hD z
For holes, this expression must be divided by a numerical factor K, to account for the reduced overlap of p-like wave functions of the valence band, compared to s-like functions of the conduction band [22]. In the case of zincblende structures, the appropriate value of K is between 2 and 2.5 [22]. We used K = 2 for zincblende and wurzite structures. The same discussion holds concerning the h o l e phonon matrix elements of the further h o l e - p h o n o n interactions we shall consider, and wilt not be repeated. After a little algebra, we obtain for electrons:
dE
e2m2kT 1
i x
q3dq [qZ + q ~ ] z L ° g
1
I V(q)l: -
IV(q)[2
--h z
2
-~-t/yp O - 47r3h3-------~
47rh3e0
+ ~-~ ~
l+exp
qSdq [q2 + q~]2 {Nq(T)-Nq(Tb)} _ h2
l +exp
2
#~
+
We used hcoq = h c.q. For zincblende and wurzite structures we have respectively: 1767r e~4
Peo
g - 350eo [ ~ 31
33
+
' +
+
+
3 ]]
~e~s
eh].
Following Mahan [9], the numerical values of e u are those given in [11].
2.2. Short range interactions The short range interactions are described in the framework of deformation potential theory.
/ +E/
x Log' 1 + exp
+
Acoustical interaction
nq
[5 ]
iV(q)l 2 _ hE~q'- .
2iO¢OqU
hcqwe get: m"kTE~ ~ q7dq =-47r3h30 ~ [q2 + q~]2{Nq(T)--Nq(To)}
Using h ~ q =
/dE~ \~t/acou s
x Log 1 + exp
35
{NTo(T ) _,VTo(To) }
qS dq [q2 + q ~ l ~
x
LO~2
The index X, refers to transverse acoustical (TA) or longitudinal acoustical (LA) phonons, co(q) is the static dielectric constant. We shall use a mean static value for eo(q).~(X, q) is the unit vector of polarisation. We obtain for electrons: i~o -
D~2 k T D 2
dE\
1 + exp
~ qkek,ij~i(x, q)qi ow(q)q%2o(q) ~k,i.s
e2m2kT i
2 vpCOTO"
2
8~2e2h
d~-~-/1 p
(NPO)
For electrons:
t~'ezoacoustical interactions [9].
F-
Vol. 38, No. 6
COOLING OF HOT E L E C T R O N - H O L E PLASMAS
~
+
+
~e ]
"
3. RESULTS We have calculated the loss rates for GaAs, CdSe and CdS, at low temperature T 0 = 4.2 K. We discuss in detail the results for GaAs. Our conclusions hold also for the other compounds. In a first step, all the phonons are supposed thermalized at the bath temperature T o = 4.2 K. This can be questionable at high excitation for the LO phonons, and we shall therefore introduce later an effective temperature TLo to describe LO phonons non thermalized at To. In Figs. 1 - 4 are reported the different energy losses (corresponding to the four interactions which we have considered in the previous section) for several plasma density (1014 cm -3 < n < 2 x 1019cm -3) in GaAs. Let us emphasize that the energy losses are mainly due to
Vol. 38, No. 6
COOLING OF HOT ELECTRON-HOLE PLASMAS
~0'Z
:1
3
10~2
~I
Cd Se
535
T:
~
~2K
i~,
fo%,, #
T~: 4 2 K
• Is,F ,~,',
I0 !~
op s It
o
;o,O
/2/
io~
°l
! V
~
p
_o 10~0
_/
•
#I III
lu
,,,,i."
ot
j
/
•/ 10:)
¢1"
///
i0 z
#
,
/2'
P vii i
i06
,,7" I
I
I
I
I0
I00
500
~o5
TK
hole-phonon interaction. Generally
NLO(T) --NLo(Tb)
I
I
I
10
I00
500
TK
Fig 5 Total energy loss vs plasma temperature for different densities in CdSe (- n = 10~4cm-3; • n = 1017 cm -3", ~ n = 10tScm-3;~Tn = 2 x 10tgcm-3).
O,TLO
i
• s I S •T,o~
2"
" ,~I" 6
s
,oil #
v ~03 -Ic
r o¢:." ,0 ~ ~"
/
I " °','y
C iII i i e" l*It o; u
I
i
II
J
/I # #
~09
"
•
o"
A similar formula is valid for non-polar optical energy loss. If the energy loss due to polar LO phonon interaction is strongly reduced, the non-polar optic loss may become dominant (in GaAs) at high temperature. In these extreme conditions, a reduction of the energy loss by one order of magnitude may occur for any temperature (4.2 K < T < 400 K). For CdSe and CdS, we report directly the net loss rate (1/n)t 10tTcm -3.
Fig. 6. Total energy loss vs plasma temperature for different densities in CdS. (mn = 1014cm-3; • n = 10 t7 cm-3;w n = 101Scm-3; ~zn = 2 x 10tgcm-3). 4. CONCLUSION The understanding of cooling kinetics of hot nonequilibrium electron-hole plasmas remains an unelucidated problem at high plasma density. Its resolution, requires investigations in several subjects, for example: (1) Hot optical phonon-distribution, opticalphonon lifetime; (2) Lattice heating due to electron-hole cooling; (3) Electron-hole recombination processes at high plasma density in direct gap semiconductors; (4) Screening of the electron-phonon interaction. In this paper, we have contributed to clarify the influence of the electron-electron interactions on the energy loss rate at high density. Further calculations are in progress to include dynamical screening by plasmons and phonons, as well as non-equilibrium distributions of phonons [24].
REFERENCES 1.
2. 3. 4. 5.
E.O. Goebel & O. Hildebrand, Phys. Status Solidi (b) 88,645 (1978). R. Ulbrich, Phys. Rev. 8, 5719 (1973). J. Shah, Phys. Rev. B9,562(1974);J. Shah& R.F. Leheny, Phys. Rev. BI6, 1577 (1977). E.M. ConweU, High Field Transport in Semiconductors. Academic Press, New York (1967). S.M. Kogan,Sov. Phys.-SolidState 4, 1813 (1963).
536 6. 7. 8. 9. 10. I 1. 12. 13. 14. 15. 16.
COOLING OF HOT ELECTRON-HOLE PLASMAS R.F. Leheny, J. Shah, R.L. Fork, C.V. Shank, 17. A. Migus, Solid State Commun. 31,809 (1979). C.V. Shank, D.H. Auston, E.P. Ippen & O. Teschke, Solid State Commun. 26,567 (1978). 18. D. yon Der Linde, J. Kuhl & H. Kltngenberg, Phys. Rev. Left. 44, 1505 (1980). G. Mahan, Polarons in Ionic Crystals and Polar 19. Semiconductors (Edited by J. Devreese). North Holland, Amsterdam (1972). P. VoW, NATO, Advanced Studies Institutes Series, 20. S6;rie B: Physics, Vol. 52 (Edited by D.K. Ferry, J.R. Barker & C.J. Jacobini) (1980). D. Berlincourt, H. Jaffe & L.R. Shizawa, Phys. Rev. 21. 22. 129, 1009 (1963). M.V. Kurik,Phys. Lett. 24A, 742 (1967). R.G. W]aeeler & J.O. Dimmock, Phys. Rev. 125, 23. 1805 (1962). K.F. Cline, H.L. Dunegan & G.W. Handersen,.L 24. Appl. Phys. 38, 1944 (1967). B.R. Nag, Electron Transport in Compound Semiconductors. Springer Verlag, Berlin (1980). M. Grinberg, Phys. Status Solidi (b ) 27,255 (1968).
Vol. 38, No. 6
Deduced from the following reference: Semiconductors atzd Semimetals (Edited by Kenneth Zanio), Vol. 13. Academic Press, New York (1978). G.E. Stillmann, D.M. Larsen, C.M. Wolfe & R.C. Brandt, Solid State Commun. 9, 2245 (1971). A. Mooradian & G.B. Wright, Solid State Commutz. 4, 431 (1966). H.R. Fetterman, D.M. Larsen, G.E. Stillmann, P.E. Tannenwald & J. Waldman, Phys. Rev. Lett. 26,975 (1971). D.L. Rode, Phys. Rev. B2, 1012 (1970). J.D. Wiley, Semiconductors andSemimetals (Edited by R.K. Willardson & A.C. Beer), Vol. 10. Academic Press, New York, (1975). T.B. Bateman, H.J. McSkimin & J.M. Whelan, J. AppL Phys. 30, 544 (1959). M. Pugnet, J. Collet & A. Cornet, Communication accepted by the International Conference on Excited States and Multiresonant Non Linear Optical Processes in Solids, Aussais, France, March 1981.
0038-1098/81/180531-06$02.00/0
COOLING OF HOT ELECTRON-HOLE PLASMAS IN THE PRESENCE OF SCREENED E LECTRON-PHONON INTERACTIONS M. Pugnet, J. Collet and A. Cornet Laboratoire de Physique des Solides, associd au CNRS, INSA, D4partement de Physique, Avenue de Rangueil, 31077 Toulouse Cedex, France
(Received 7 November 1980 by F. Bassani) We have computed for several semiconductors, (GaAs, CdSe, CdS), the energy loss-rate in hot electron-hole plasmas, at high density, when occurs the screening of the electron-phonon interactions. We show that the long range interactions (piezoelectrical and polar) are quickly reduced when the plasma density is raised, and that the energy loss-rate diminishes therefore significantly (one order of magnitude). I. INTRODUCTION
interaction, and piezoelectrical interaction) in the presence of electron-electron interaction screening.
IN THE LAST YEARS, the formation of hot electron-hole plasmas has been achieved in various semiconductors [1-3] and the problem of energy relaxation in such hot plasmas was investigated in the limit of low density of carriers, and for Boltzmann distribution of electrons and holes. The plasma cooling is due to the emission of phonons by the non thermalized carriers. in the low density limit, this problem is now relatively simple, since analytical formulas have been already derived for the energy losses due to the different electron-phonon interactions (Polar interaction with longitudinal optical phonon, piezoelectrical interaction, and interaction with acoustical phonon or non polar optical phonons described in the model of deformation potential) [4, 5]. Let us emphasize that in this limit, the total loss rate of energy per electron hole pair is indepen. dent of the plasma density. Recently, with the occurence of lasers generating high power picopulses, high density electron-hole plasma could be generated, and the dynamics of the plasma cooling was investigated by means of time resolved (picosecond or subpicosecond) spectroscopy [6, 7]. As evidenced by the experiments at low temperature [6], the cooling becomes less effective in high density plasmas, than in the plasmas at low density. This is consecutive to a lowering of the phonon emission rate by the hot carriers, in the high density regime. Two changes must be taken into account to explain this effect :
In the present paper, we are mainly concerned with this effect, which we have just indicated above. We have computed the energy loss rate per electron-hole pair, in the static limit of the random phase approximation (RPA). In our calculation, the phonon distributions are Bose distributions at bath temperature. To our knowledge, few accurate results are available concerning the non-equilibrium LO phonons in picosecond experiments, so that we attempt to describe this LO phonon distribution, by a Bose distribution with an effective temperature TLo, which can be easily accounted in our calculations. 2. MODEL In the presence of electron-electron interactions, occurs a screening of the electron-phonon interactions. This effect has been discussed in former publications (see for ex~,mple [9, 10]). To summarize these results, the electron phonon matrix element must be divided by the dielectric constant of the electron gas, consecutive to electron-electron interactions, considered here in the RPA. The Fourier transform is therefore:
gs ere(q, co)
(I) The appearance of non equilibrium distributions of phonons (specially longitudinal optical (LO) phonons) [8], so that phonon absorption becomes possible and slakens the plasma cooling due to phonons emission. (2) The screening of the electron phonon interactions (in particular the long range interactions: polar
v(q) 6RPA(q,CO)"
(l)
V(q) is the usual matrix element of the bare electron phonon interaction. In the static RPA, for small q, the dielectric constant reduces to the classical DebyeHiJckel expression: eRPA(q, O) = I + q---b-~ q~. In an electron-hole plasma, the Debye screening length qD is given by:
531
(2)
532
Vol. 38, No. 6
COOLING OF HOT ELECTRON-HOLE PLASMAS
Table 1 CdSe
60
Phonon LO energy (meV) Conduction mass Mean valence band mass Density (g cm -3) Acoustical conduction band deformation potential (eV) Acoustical valence band deformation potential (eV) Optical deformation potential (eV cm -t ) Mean sound velocity (cm sec -t) Piezoacoustic factor
h~Lo t/q c
(,%) P Ec
& D c
F/c a
q2D = 87re____~:l_ E f x ( k, #i) -- {fx( k,/ai)}:. e~,kT v k.h
GaAs
10.63 12.56 36.7 0.0665
[181 [181 []91 [20]
0.8 [13] 5.68 [14] 3.7 [15]
0.57 5.36 7
[2] [15] [2l]
5.28 [15] 8.8 [15l 36.81 [15] 0.17 [15] 1.3S [15] 6.75 [151 14.5 [151
5.7
[161
3.5
[221
I0
109
[17]
10 9
10 9
2.3 x 10 s
3.57 × l0 s [23]
3.54 x IO s [151
2.6 x 10-2
4.24 x 10-3
5 × lO-z
6.3 9.4 26 0.13
[91 [91 [91 [12]
(3)
fX(k, #i) is the Fermi function of electron (hole) in the band h, with Fermi level/Je ~h)- The sum over X includes the conduction band and all the valence bands f'dled by holes, v is the crystal volume. The total power (dE/dt), that a gas of fermions releases to the lattice by phonon interaction has been computed by Kogan [5]: _ m2kT[" d3q hw a
d/( \
\dr~,
~s
j (27r)3 Iqt
{ N q ( T ) - - Nq(Tp)}
x [V~¢~e(q)l2
•
(4)
1 + exp ~
+
+
where hcoq is the phonon energy; Tis the fermions temperature (we assume the same temperature for holes and electrons), To is the phonon temperature, rn is the effective mass of fermions, the index i refers to electrons or holes, Nq(T) is the phonon distribution, and
No(T ) =
I
exp ( h w q / k T ) -
CdS
The calculation of the energy losses in a dense electron-hole plasma is achieved, by means of a computer, in three steps: (a) For each pair of parameters (n the plasma density and T the plasma temperature), we compute the Fermi levels (,%, tan) of electrons and holes. For zincblende structure, the two valence bands are roughly replaced by a sin~e valence band mass [2]. For wurzite structure, we account the two valence bands (so called A and B) in the hole Fermi level calculation. (b) We compute the Debye-Htickel screening length given by equation (3). (c) We calculate the energy losses for each kind of carrier, and various electron phonon interactions. All the parameters which have been used in these computations are listed in Table 1. The net plasma loss rate (dE/clt)plas is the sum of all the preceding losses. Finally we report the energy loss rate per electron-hole pair, that means
(1/n)(dE/dt)p~. We distinguish between long range electron-phonon interactions, sensitive to screening, and short range interactions, tess affected by the screening.
I"
The expression (4) is valid for degenerate or nondegenerate gas, and includes spontaneous and stimulation emission of phonons, as well as phonon absorption. The energy losses of the plasma are the sum of the losses of the conduction band, and valence bands given by equation (4):
2.1. Long range interactions Polar interaction. The matrix element for electronphonon interaction is: 4n'e2 ( 1
l__~(heOLo/
IVLo(q)l:- q~ ~e.. ~o1\ 2 j
533
COOLING OF HOT ELECTRON-HOLE PLASMAS
Vol. 38, No. 6
GoAs
1o2 -
GoAs T~=4 2K
;0 ' --
n : 104cm 3
y
:
=IO~cm"3
I
/ I~
iO c
,"~
//','
/%
v
7
:0 ~
iO7
-I=
IOG
{05
I
!
tO
20
X..:T-~ ,e If
.
#"
yl
AI,"
~ Z :~
T~:42K
n
/
;
•
IO0
! 500"
/
I0
TK
100
500
TK
Fig. 1. Energy losses for the four processes vs plasma temperature, with n = 10'~cm -3, in GaAs. • Polar Longitudinal optic interation; • Non-polar optic interaction; • Acoustical deformation potential; ? Piezoelectric interaction;- Total e n e r ~ loss rate.
Fig. 3. Energy losses for the four processes vs plasma temperature, with n = 10tScm -3, in GaAs. • Polar longitudinal optic interaction; • Non-polar optic interaction;• Acoustical deformation potential; re Piezoelectric interaction; - Total energy loss rate.
GaAs
T~=42K
!0 ! ' --
n = 10"cm
lOIr _
3
T==42 K n=2x IO'%m3
iO !O _
A,S
o ~0~
~ III
S
7; 108
2_/'.,
3
v I
kJ
_ i~o"
--I=
~ / I ;• ::
g
iI/
o !09
ht
~
GoAs
i I --Ic
~0"~
,, t" ~e
d:'i'
~C 5
7,o" // _.//.j;'
II~
b'-
/"
10 4
~
o
'0
i
20
~OO
/"
I
500
TK
tOa "f~"
/
.~'" ,Itt h g
I IO
I
I
lOO
500
TK
Fig. 2. Energy losses for the four processes vs plasma temperature with n = 1017cm -3, in GaAs. • Polar longitudinal optic interaction; • Non-polar optic interaction: • Acoustical deformation potential ; re Piezoelectric interaction;- Total energy loss rate.
Fig. 4. Energy losses for the four processes vs plasma temperature, with n = 2 x 1019cm -3, in GaAs. • Polar longitudinal optic interaction; • Non-polar optic interaction; • Acoustical deformation potential; re Piezoelectric interaction;- Total energy loss rate.
534
Non-polar optical interaction [4] hD z
For holes, this expression must be divided by a numerical factor K, to account for the reduced overlap of p-like wave functions of the valence band, compared to s-like functions of the conduction band [22]. In the case of zincblende structures, the appropriate value of K is between 2 and 2.5 [22]. We used K = 2 for zincblende and wurzite structures. The same discussion holds concerning the h o l e phonon matrix elements of the further h o l e - p h o n o n interactions we shall consider, and wilt not be repeated. After a little algebra, we obtain for electrons:
dE
e2m2kT 1
i x
q3dq [qZ + q ~ ] z L ° g
1
I V(q)l: -
IV(q)[2
--h z
2
-~-t/yp O - 47r3h3-------~
47rh3e0
+ ~-~ ~
l+exp
qSdq [q2 + q~]2 {Nq(T)-Nq(Tb)} _ h2
l +exp
2
#~
+
We used hcoq = h c.q. For zincblende and wurzite structures we have respectively: 1767r e~4
Peo
g - 350eo [ ~ 31
33
+
' +
+
+
3 ]]
~e~s
eh].
Following Mahan [9], the numerical values of e u are those given in [11].
2.2. Short range interactions The short range interactions are described in the framework of deformation potential theory.
/ +E/
x Log' 1 + exp
+
Acoustical interaction
nq
[5 ]
iV(q)l 2 _ hE~q'- .
2iO¢OqU
hcqwe get: m"kTE~ ~ q7dq =-47r3h30 ~ [q2 + q~]2{Nq(T)--Nq(To)}
Using h ~ q =
/dE~ \~t/acou s
x Log 1 + exp
35
{NTo(T ) _,VTo(To) }
qS dq [q2 + q ~ l ~
x
LO~2
The index X, refers to transverse acoustical (TA) or longitudinal acoustical (LA) phonons, co(q) is the static dielectric constant. We shall use a mean static value for eo(q).~(X, q) is the unit vector of polarisation. We obtain for electrons: i~o -
D~2 k T D 2
dE\
1 + exp
~ qkek,ij~i(x, q)qi ow(q)q%2o(q) ~k,i.s
e2m2kT i
2 vpCOTO"
2
8~2e2h
d~-~-/1 p
(NPO)
For electrons:
t~'ezoacoustical interactions [9].
F-
Vol. 38, No. 6
COOLING OF HOT E L E C T R O N - H O L E PLASMAS
~
+
+
~e ]
"
3. RESULTS We have calculated the loss rates for GaAs, CdSe and CdS, at low temperature T 0 = 4.2 K. We discuss in detail the results for GaAs. Our conclusions hold also for the other compounds. In a first step, all the phonons are supposed thermalized at the bath temperature T o = 4.2 K. This can be questionable at high excitation for the LO phonons, and we shall therefore introduce later an effective temperature TLo to describe LO phonons non thermalized at To. In Figs. 1 - 4 are reported the different energy losses (corresponding to the four interactions which we have considered in the previous section) for several plasma density (1014 cm -3 < n < 2 x 1019cm -3) in GaAs. Let us emphasize that the energy losses are mainly due to
Vol. 38, No. 6
COOLING OF HOT ELECTRON-HOLE PLASMAS
~0'Z
:1
3
10~2
~I
Cd Se
535
T:
~
~2K
i~,
fo%,, #
T~: 4 2 K
• Is,F ,~,',
I0 !~
op s It
o
;o,O
/2/
io~
°l
! V
~
p
_o 10~0
_/
•
#I III
lu
,,,,i."
ot
j
/
•/ 10:)
¢1"
///
i0 z
#
,
/2'
P vii i
i06
,,7" I
I
I
I
I0
I00
500
~o5
TK
hole-phonon interaction. Generally
NLO(T) --NLo(Tb)
I
I
I
10
I00
500
TK
Fig 5 Total energy loss vs plasma temperature for different densities in CdSe (- n = 10~4cm-3; • n = 1017 cm -3", ~ n = 10tScm-3;~Tn = 2 x 10tgcm-3).
O,TLO
i
• s I S •T,o~
2"
" ,~I" 6
s
,oil #
v ~03 -Ic
r o¢:." ,0 ~ ~"
/
I " °','y
C iII i i e" l*It o; u
I
i
II
J
/I # #
~09
"
•
o"
A similar formula is valid for non-polar optical energy loss. If the energy loss due to polar LO phonon interaction is strongly reduced, the non-polar optic loss may become dominant (in GaAs) at high temperature. In these extreme conditions, a reduction of the energy loss by one order of magnitude may occur for any temperature (4.2 K < T < 400 K). For CdSe and CdS, we report directly the net loss rate (1/n)
Fig. 6. Total energy loss vs plasma temperature for different densities in CdS. (mn = 1014cm-3; • n = 10 t7 cm-3;w n = 101Scm-3; ~zn = 2 x 10tgcm-3). 4. CONCLUSION The understanding of cooling kinetics of hot nonequilibrium electron-hole plasmas remains an unelucidated problem at high plasma density. Its resolution, requires investigations in several subjects, for example: (1) Hot optical phonon-distribution, opticalphonon lifetime; (2) Lattice heating due to electron-hole cooling; (3) Electron-hole recombination processes at high plasma density in direct gap semiconductors; (4) Screening of the electron-phonon interaction. In this paper, we have contributed to clarify the influence of the electron-electron interactions on the energy loss rate at high density. Further calculations are in progress to include dynamical screening by plasmons and phonons, as well as non-equilibrium distributions of phonons [24].
REFERENCES 1.
2. 3. 4. 5.
E.O. Goebel & O. Hildebrand, Phys. Status Solidi (b) 88,645 (1978). R. Ulbrich, Phys. Rev. 8, 5719 (1973). J. Shah, Phys. Rev. B9,562(1974);J. Shah& R.F. Leheny, Phys. Rev. BI6, 1577 (1977). E.M. ConweU, High Field Transport in Semiconductors. Academic Press, New York (1967). S.M. Kogan,Sov. Phys.-SolidState 4, 1813 (1963).
536 6. 7. 8. 9. 10. I 1. 12. 13. 14. 15. 16.
COOLING OF HOT ELECTRON-HOLE PLASMAS R.F. Leheny, J. Shah, R.L. Fork, C.V. Shank, 17. A. Migus, Solid State Commun. 31,809 (1979). C.V. Shank, D.H. Auston, E.P. Ippen & O. Teschke, Solid State Commun. 26,567 (1978). 18. D. yon Der Linde, J. Kuhl & H. Kltngenberg, Phys. Rev. Left. 44, 1505 (1980). G. Mahan, Polarons in Ionic Crystals and Polar 19. Semiconductors (Edited by J. Devreese). North Holland, Amsterdam (1972). P. VoW, NATO, Advanced Studies Institutes Series, 20. S6;rie B: Physics, Vol. 52 (Edited by D.K. Ferry, J.R. Barker & C.J. Jacobini) (1980). D. Berlincourt, H. Jaffe & L.R. Shizawa, Phys. Rev. 21. 22. 129, 1009 (1963). M.V. Kurik,Phys. Lett. 24A, 742 (1967). R.G. W]aeeler & J.O. Dimmock, Phys. Rev. 125, 23. 1805 (1962). K.F. Cline, H.L. Dunegan & G.W. Handersen,.L 24. Appl. Phys. 38, 1944 (1967). B.R. Nag, Electron Transport in Compound Semiconductors. Springer Verlag, Berlin (1980). M. Grinberg, Phys. Status Solidi (b ) 27,255 (1968).
Vol. 38, No. 6
Deduced from the following reference: Semiconductors atzd Semimetals (Edited by Kenneth Zanio), Vol. 13. Academic Press, New York (1978). G.E. Stillmann, D.M. Larsen, C.M. Wolfe & R.C. Brandt, Solid State Commun. 9, 2245 (1971). A. Mooradian & G.B. Wright, Solid State Commutz. 4, 431 (1966). H.R. Fetterman, D.M. Larsen, G.E. Stillmann, P.E. Tannenwald & J. Waldman, Phys. Rev. Lett. 26,975 (1971). D.L. Rode, Phys. Rev. B2, 1012 (1970). J.D. Wiley, Semiconductors andSemimetals (Edited by R.K. Willardson & A.C. Beer), Vol. 10. Academic Press, New York, (1975). T.B. Bateman, H.J. McSkimin & J.M. Whelan, J. AppL Phys. 30, 544 (1959). M. Pugnet, J. Collet & A. Cornet, Communication accepted by the International Conference on Excited States and Multiresonant Non Linear Optical Processes in Solids, Aussais, France, March 1981.