We derive the steady-state electron distribution function for a semiconductor
driven far from equilibrium by the inter-band photoexcitation assumed
homogeneous over the nanoscale sample. Our analytical treatment is based on the
generalization of a stochastic model known for a driven dissipative granular
gas. The generalization is physically realizable in a semiconducting sample
where electrons are injected into the conduction band by photoexcitation, and
removed through the electron-hole recombination process at the bottom of the
conduction band. Here the kinetics of the electron-electron and the
electron-phonon (bath) scattering processes, as also the partitioning of the
total energy in the inelastic collisions, are duly parametrized by certain rate
constants. Our analytical results give the steady-state-energy distribution of
the classical (non-degenerate) electron gas as function of the phonon (bath)
temperature and the rates of injection (cw pump) and depletion (recombination).
Interestingly, we obtain an accumulation of the electrons at the bottom of the
conduction band in the form of a delta-function peak $-$ a non-equilibrium
classical analogue of condensation. Our model is specially appropriate to a
disordered, indirect band-gap, polar semiconducting sample where energy is the
only state label, and the electron-phonon coupling is strong while the
recombination rate is slow. A possible mechanism for the dissipative inelastic
collisions between the electrons is also suggested.Comment: 4 page

Kumar, N.

Singh, Navinder

English

arXiv

Generalizing the stochastic model known for a driven dissipative granular gas, we have derived the steady-state non-degenerate electron distribution for a semiconductor driven far from equilibrium by the interband photo-excitation assumed uniform over the nanoscale sample. Partitioning of the total inelastic electron scattering into dissipative electron-electron and electron-phonon components is included. The model is applicable to a photo-excited semiconducting sample with fast removal of the electrons by electron-hole recombination from the bottom of the conduction band

Indian Academy of Sciences

Dissipative electron–phonon system photo-excited far from equilibrium
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Home Search Collections Journals About Contact us My IOPscience
J.S
tat.M
ech.
(2005)
L06001
ournal of Statistical Mechanics:
An IOP and SISSA journalJ Theory and Experiment
LETTER
Dissipative electron–phonon system
photo-excited far from equilibrium
Navinder Singh and N Kumar
Raman Research Institute, Bangalore 560080, India
E-mail: navinder@rri.res.in and nkumar@rri.res.in
Received 21 January 2005
Accepted 16 June 2005
Published 28 June 2005
Online at stacks.iop.org/JSTAT/2005/L06001
doi:10.1088/1742-5468/2005/06/L06001
Abstract. Generalizing the stochastic model known for a driven dissipative
granular gas, we have derived the steady-state non-degenerate electron
distribution for a semiconductor driven far from equilibrium by the interband
photo-excitation assumed uniform over the nanoscale sample. Partitioning
of the total inelastic electron scattering into dissipative electron–electron and
electron–phonon components is included. The model is applicable to a photo-
excited semiconducting sample with fast removal of the electrons by electron–hole
recombination from the bottom of the conduction band.
Keywords: dissipative systems (theory), disordered systems (theory)
ArXiv ePrint: cond-mat/0501479
c©2005 IOP Publishing Ltd 1742-5468/05/L06001+6$30.00
J.S
tat.M
ech.
(2005)
L06001
Dissipative electron–phonon system photo-excited far from equilibrium
Contents
1. Introduction 2
2. The model 2
3. Discussion 5
References 6
1. Introduction
The kinetics of evolution of the electron distribution function for an electron–phonon
system driven far from equilibrium by photo-excitation is of considerable current interest,
experimentally [1] (in pump–probe experiments) and theoretically [2]–[5] (e.g., the two-
temperature model). In the following, we develop a general stochastic approach to this
kinetic problem as an alternative to the commonly used two-temperature model. Most
such studies have been for metallic nanometric particles. More specifically, we derive the
steady-state electron distribution function for a semiconducting sample driven far from
equilibrium by the interband photo-excitation assumed homogeneous over the sample.
Our analytical treatment is based on the generalization of a stochastic model known
for a driven dissipative granular gas [6]. The generalization is physically realizable in a
semiconducting sample where electrons are injected into the conduction band by photo-
excitation and removed at the bottom of the conduction band through the electron–hole
recombination process. Here the kinetics of the electron–electron and the electron–phonon
(bath) scattering processes, as also the partitioning of the total energy in the inelastic
collisions, are duly parameterized by certain rate constants. Our analytical results give
the steady-state electron distribution function, and the mean energy of the classical non-
equilibrium electron gas as function of the phonon (bath) temperature and the rates of
injection (continuous wave pump) and depletion (recombination). Interestingly, we obtain
an accumulation of the electrons at the bottom of the conduction band in the form of a
delta-function. This is an analytical result obtained for a non-degenerate photo-excited
electron gas. It is to be physically interpreted as broadened into a peak, with an area under
it equal to the strength of the delta-function. We will return to this point later. Our model
is specially appropriate to a disordered, indirect bandgap, polar semiconducting sample
where energy is the only state label, and the electron–phonon coupling is strong whereas
the recombination rate is slow. A possible mechanism for the dissipative inelastic binary
collisions between electrons is also suggested.
2. The model
Let ne(E) dE be the number of electrons lying in the energy range ±dE/2 centred about
E in the conduction band of a semiconducting sample of volume Ω. The electron–electron
collisions, assumed inelastic in general, are described by the process Ei +E
′
i −→ Ef +E ′f =
α(Ei + E
′
i) with α ≤ 1, in which the tagged electron of energy Ei collides with another
doi:10.1088/1742-5468/2005/06/L06001 2
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Dissipative electron–phonon system photo-excited far from equilibrium
electron of energy E ′i lying in the energy shell E
′
i ± 12∆E ′i , and is scattered to the final state
Ef . The scattering rate for this inelastic process is taken to be (1−f)Γn(E ′i) dE ′i . Similarly,
the electron–phonon scattering rate is given by fΓnph(E
′
i) dE
′
i , with nph(E
′
i) dE
′
i as the
number of thermal phonons in the phonon-energy shell E ′i ± 12∆E ′i . Here, the fraction
0 ≤ f ≤ 1 determines the relative strengths of the binary electron–electron and the
electron–phonon collisions. Also, let the electrons be injected through photo-excitation
into the conduction band at energy Eex at the rate gexδ(E − Eex), and then be removed
(depleted) from the bottom of the conduction band through recombination. This depletion
rate can be modelled by a term −gdδ(E)ne(0). Here the phonons are assumed to remain in
thermal equilibrium at temperatures T . In our model sample we assume a uniform density
of states for the electrons and the energy to be the only label for the single particle states.
The photo-excitation is taken to be homogenous over the sample, which is reasonable for
a nanoscale disordered semiconducting sample. For the above dissipative model driven
far from equilibrium, the kinetics for the non-equilibrium electron number density ne(E)
is given by the rate equation
∂ne(E)
∂t
= −ne(E)
∫
dE ′ [ne(E
′)(1 − f) + nph(E ′)f ]Γ
+
∫ 1
0
dz p(z)
∫
dE ′
∫
dE ′′ δ(E − zα(E ′ + E ′′))ne(E ′)ne(E ′′)(1 − f)Γ
+
∫ 1
0
dz p(z)
∫
dE ′
∫
dE ′′ δ(E − z(E ′ + E ′′))ne(E ′)nph(E ′′)fΓ
+ gex(t)δ(E − Eex) − gdδ(E)ne(0). (1)
In the above, we have assumed the total energy (E ′ + E ′′) for a binary collision to
be partitioned such that a fraction z, with probability density p(z), goes to the tagged
electron of initial energy E ′, and 1 − z to the colliding particle (electron or phonon of
initial energy E ′′). The inclusion of α in the electron–electron collision takes care of the
possibility of inelastic electron–electron collisions. Note that we have suppressed the time
argument (t) in the non-equilibrium electron-number density ne(E). Taking the energy
Laplace transform
ñe(s) =
∫ ∞
0
e−sEne(E) dE, (2)
of equation (1), we obtain
∂
∂t
ñ(s) = −Γñe(s)[(1 − f)Ne + fNph] + (1 − f)Γ
∫ 1
0
p(z) dz ñ2e(αzs)
+ fΓ
∫ 1
0
dz p(z) ñe(zs)ñph(zs) + gex(t)e
−sEex − gdne(0). (3)
In the following, we will consider for simplicity the steady-state condition under
constant (cw) photo-excitation, gex(t) = gex. A pulsed excitation can, of course, be
considered in general. Accordingly, we set (∂/∂t)ñe(s) = 0 above, and all quantities on
the rhs of equation (3) are then independent of time.
In order to calculate the steady-state electron distribution for the system in terms
of the bath (phonon) temperature and other rate parameters, we expand ñe(s) in powers of
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Dissipative electron–phonon system photo-excited far from equilibrium
the Laplace transform parameter s as
ñe(s) = Ne − s〈Ee〉 + s2〈E2e 〉/2 . . . , (4)
and equate the coefficients of like powers of s. Thus, from the zeroth power of s, we obtain
at once
ne(0) = (gex/gd). (5)
Similarly, from the first power of s, we get
〈ee〉 =
(f/2)〈eph〉
ρe−ph(1 − α)(1 − f) + f/2
+
gexEex/Γ
N2phρe−ph[ρe−ph(1 − α)(1 − f) + f/2]
. (6)
In the above, we have taken a uniform limit for the energy partition: p(z) = 1.
Here, we have defined 〈ee〉 ≡ 〈Ee〉/Ne = mean electron energy; 〈eph〉 ≡ 〈Eph〉/Nph =
mean phonon energy (=kBTB); and ρe−ph = Ne/Nph ≡ electron-to-phonon number ratio.
It is to be noted that in the limit α = 1 (i.e., for elastic electron–electron collisions as is
usually expected for an electronic system unlike the case of the granular gas), and gex = 0
(i.e., no photo-excitation), we recover 〈ee〉 = 〈eph〉, i.e., the electrons and the phonons
are at the same temperature, as is physically expected under equilibrium conditions. In
general, however, the mean electron energy in the steady state is not the same as the
mean phonon energy, and the former depends on the excitation rate (the drive gex).
We will now consider the specific case of the extreme partition limit p(z) = 1
2
(δ(z) +
δ(z − 1)) to illustrate our treatment. An analytic form for ñe(s) in the steady state will
be obtained for the case of elastic scattering (α = 1). It can be readily verified that for
this insertion of p(z) in equation (3), the equations (5) and (6) remain unchanged. With
this choice of p(z), equation (3) now becomes
ñe(s)[(1 − f)Ne + fNph] = (1 − f)
∫ 1
0
(1
2
δ(z) + 1
2
δ(z − 1)) dz ñ2e(zs)
+ f
∫ 1
0
(1
2
δ(z) + 1
2
δ(z − 1)) dz ñe(zs)ñph(zs) + gex[e−sEex − 1]/Γ. (7)
This can be readily solved to give
ñe(s) =
2[(1 − f)Ne + fNph] − fñph(s)
2(1 − f)
±
√
[fñph(s) − 2[(1 − f)Ne + fNph]]2 − 4(1 − f)C
2(1 − f) (8)
with
C = (1 − f)N2e + fNeNph + 2gex(e−sEex − 1)/Γ.
We have to choose the −ve sign in the above solution so as to satisfy ñe(0) = Ne, ñph(0) =
Nph. Here, we have assumed that the bath phonons obey the equilibrium Boltzmann
distribution (i.e., ñph(s) = Nph/(1 + 〈Eph〉s)). Now, our boundary condition for the case
of fast electron–hole recombination at the bottom of the band demands that the electron
number density at the bottom of the conduction band be zero, i.e., ne(0) = 0. But, from
the steady-state analysis, we have gex = gdne(0). Thus the above boundary condition is
doi:10.1088/1742-5468/2005/06/L06001 4
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Dissipative electron–phonon system photo-excited far from equilibrium
implemented mathematically by formally demanding that the drain coefficient gd tends to
infinity, so that the product gdne(0) remains finite = gex. With this boundary condition,
and using the initial-value theorem for Laplace transforms, we get a relation between the
electron-to-phonon number ratio (ρe−ph) and the excitation parameter η as
ρe−ph =
1
2
[√
β2 +
8η
1 − f − β
]
, η =
gex
ΓN2ph
. (9)
The electron number density distribution ne(E) (in the energy E-domain) is now obtained
by numerically inverting ñe(s) (in the s-domain) in equation (8), and this is plotted in
figure 1 for various values of electron–phonon coupling parameter, f . (The noise seen on
the lowest curve is an artefact of the numerical inversion.)
At this stage, it is in order to comment briefly on our use of the extreme partition
P (z) = 1
2
δ(z) + 1
2
δ(1 − z). First, let us note that our approach, based on equation (1),
is of course valid for any general form of p(z). It is, however, only for the above extreme
limiting form that it admits an analytical solution as presented here in order to illustrate
the model. The above form can now be justified in the following physical terms. It is to be
noted that for any given collision, the partitioning of the total incoming energy between
the outgoing particles necessarily depends on the kinematic parameters of the collision,
for example, on the scattering angles, that vary from one collision to the other. Thus, for
a fine-grained treatment of the successive collisions under the above extreme form, the
energy of a ‘tagged’ electron colliding with others in the gas will fluctuate between extreme
values. However, a coarse-graining over even a small number of successive collisions will
suppress these fluctuations. Thus, in a coarse-grained statistical sense the partition is
expected to become effectively a smooth and broad function. This physically justifies
our use of the simple, though admittedly extreme, form for the partition p(z) without
resulting in unphysical features, as validated a posteriori from figure 1.
3. Discussion
We have treated here the problem of energy distribution of photo-excited electrons in a
semiconducting sample as a generalization of the model for dissipative granular gas driven
far from equilibrium. A feature of the above non-equilibrium distribution of the cw photo-
excited electrons is the peak appearing at the excitation energy (=1 eV) as is indeed
expected. However, a notable feature of the steady-state electron distribution in this case
is that, as the electron–phonon interaction strength (f) increases, (1) the peak height of
the distribution decreases, as does also (2) the total area under the distribution (the total
number of steady-state photo-excited electrons in the conduction band). This is physically
understandable as follows. Inasmuch as the increased electron–phonon interaction implies
fast removal of energy from the non-equilibrium photo-excited electron distribution to
the phonon bath, the electrons get pushed to lower energies towards the bottom of the
conduction band, from where they get removed by the fast electron–hole recombination.
Hence the decrease in the height of the distribution, and also the reduction of the area
under the curve. It should be possible to probe these steady-state features through a cw
pump–probe experiment. As for our assumption of non-degeneracy for the photo-excited
electron gas, it is readily verified that the peak occupation number of one-particle states
is of the order of (Ne/total number of atoms in the nanoscale sample)  10−4  1,
doi:10.1088/1742-5468/2005/06/L06001 5
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Dissipative electron–phonon system photo-excited far from equilibrium
10 f = 0.3
f = 0.5
f = 0.8
x1017
8
6
4
2
0.8 1 1.2 1.61.4
E(eV )
n e
(E
) 
(e
V
-1
 c
m
-3
)
Figure 1. Plot of the steady-state photo-excited electron distribution function
ne(E) (eV−1 cm−3) for a typical nanoscale semiconducting sample of radius
∼10 nm with ∼105 atoms, as a function of the electron energy E (eV), for
the following choice of parameters: the excitation parameter η = 10−7 and the
electron–phonon coupling parameter f = 0.3 (top line), 0.5 (middle line), 0.8
(bottom line). In all cases, the peak occurs at E = 1 eV, at which the electrons
are being photo-injected in the conduction band. Here the phonons are assumed
to remain at room temperature (300 K).
thus validating the assumed non-degeneracy. Finally, the dissipative electron–electron
collisions (α < 1) can physically derive from the dissipative polarization of the dielectric
medium that screens the electron–electron Coulombic interaction [7].
One of the authors (NS) would like to thank Yashodhan Hatwalne and A M Jayannavar
for discussions.
References
[1] Arbouet A et al , 2003 Phys. Rev. Lett. 90 177401
[2] Kaganov M I, Lifshitz I M and Tanatarov L V, 1957 Zh. Eksp. Teor. Fiz. 31 232
Kaganov M I, Lifshitz I M and Tanatarov L V, 1974 Sov. Phys. JETP 39 375 (translation)
Singh N, 2004 Mod. Phys. Lett. 18 1261
[3] Sun C-K, Vallee F, Acioli L H, Ippen E P and Fujimoto J G, 1994 Phys. Rev. B 50 15337
[4] Singh N, 2005 Pramana J. Phys. 64 111
[5] Kuhn T and Rossi F, 1992 Phys. Rev. B 46 7496
[6] Srebro Y and Levine D, 2004 Phys. Rev. Lett. 93 240601
See also, Pöschel T and Luding S (ed), 2001 Granular Gases (Berlin: Springer)
Pöschel T and Brilliantov N (ed), 2003 Granular Gas Dynamics (Berlin: Springer)
[7] Shimizu A and Yamanishi M, 1994 Phys. Rev. Lett. 72 3343
doi:10.1088/1742-5468/2005/06/L06001 6


Dissipative electron-phonon system photo-excited far from equilibrium

Publications of the IAS Fellows

Dissipative electron–phonon system photo-excited far from equilibrium
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
J. Stat. Mech. (2005) L06001
(http://iopscience.iop.org/1742-5468/2005/06/L06001)
Download details:
IP Address: 122.179.17.200
The article was downloaded on 25/10/2010 at 06:41
Please note that terms and conditions apply.
View the table of contents for this issue, or go to the journal homepage for more
Home Search Collections Journals About Contact us My IOPscience
J.Stat
.M
ech.(2005)L06001
ournal of Statistical Mechanics:
An IOP and SISSA journalJ Theory and Experiment
LETTER
Dissipative electron–phonon system
photo-excited far from equilibrium
Navinder Singh and N Kumar
Raman Research Institute, Bangalore 560080, India
E-mail: navinder@rri.res.in and nkumar@rri.res.in
Received 21 January 2005
Accepted 16 June 2005
Published 28 June 2005
Online at stacks.iop.org/JSTAT/2005/L06001
doi:10.1088/1742-5468/2005/06/L06001
Abstract. Generalizing the stochastic model known for a driven dissipative
granular gas, we have derived the steady-state non-degenerate electron
distribution for a semiconductor driven far from equilibrium by the interband
photo-excitation assumed uniform over the nanoscale sample. Partitioning
of the total inelastic electron scattering into dissipative electron–electron and
electron–phonon components is included. The model is applicable to a photo-
excited semiconducting sample with fast removal of the electrons by electron–hole
recombination from the bottom of the conduction band.
Keywords: dissipative systems (theory), disordered systems (theory)
ArXiv ePrint: cond-mat/0501479
c©2005 IOP Publishing Ltd 1742-5468/05/L06001+6$30.00
J.Stat
.M
ech.(2005)L06001
Dissipative electron–phonon system photo-excited far from equilibrium
Contents
1. Introduction 2
2. The model 2
3. Discussion 5
References 6
1. Introduction
The kinetics of evolution of the electron distribution function for an electron–phonon
system driven far from equilibrium by photo-excitation is of considerable current interest,
experimentally [1] (in pump–probe experiments) and theoretically [2]–[5] (e.g., the two-
temperature model). In the following, we develop a general stochastic approach to this
kinetic problem as an alternative to the commonly used two-temperature model. Most
such studies have been for metallic nanometric particles. More specifically, we derive the
steady-state electron distribution function for a semiconducting sample driven far from
equilibrium by the interband photo-excitation assumed homogeneous over the sample.
Our analytical treatment is based on the generalization of a stochastic model known
for a driven dissipative granular gas [6]. The generalization is physically realizable in a
semiconducting sample where electrons are injected into the conduction band by photo-
excitation and removed at the bottom of the conduction band through the electron–hole
recombination process. Here the kinetics of the electron–electron and the electron–phonon
(bath) scattering processes, as also the partitioning of the total energy in the inelastic
collisions, are duly parameterized by certain rate constants. Our analytical results give
the steady-state electron distribution function, and the mean energy of the classical non-
equilibrium electron gas as function of the phonon (bath) temperature and the rates of
injection (continuous wave pump) and depletion (recombination). Interestingly, we obtain
an accumulation of the electrons at the bottom of the conduction band in the form of a
delta-function. This is an analytical result obtained for a non-degenerate photo-excited
electron gas. It is to be physically interpreted as broadened into a peak, with an area under
it equal to the strength of the delta-function. We will return to this point later. Our model
is specially appropriate to a disordered, indirect bandgap, polar semiconducting sample
where energy is the only state label, and the electron–phonon coupling is strong whereas
the recombination rate is slow. A possible mechanism for the dissipative inelastic binary
collisions between electrons is also suggested.
2. The model
Let ne(E) dE be the number of electrons lying in the energy range ±dE/2 centred about
E in the conduction band of a semiconducting sample of volume Ω. The electron–electron
collisions, assumed inelastic in general, are described by the process Ei+E
′
i −→ Ef +E ′f =
α(Ei + E
′
i) with α ≤ 1, in which the tagged electron of energy Ei collides with another
doi:10.1088/1742-5468/2005/06/L06001 2
J.Stat
.M
ech.(2005)L06001
Dissipative electron–phonon system photo-excited far from equilibrium
electron of energy E ′i lying in the energy shell E
′
i± 12∆E ′i , and is scattered to the final state
Ef . The scattering rate for this inelastic process is taken to be (1−f)Γn(E ′i) dE ′i . Similarly,
the electron–phonon scattering rate is given by fΓnph(E
′
i) dE
′
i , with nph(E
′
i) dE
′
i as the
number of thermal phonons in the phonon-energy shell E ′i ± 12∆E ′i . Here, the fraction
0 ≤ f ≤ 1 determines the relative strengths of the binary electron–electron and the
electron–phonon collisions. Also, let the electrons be injected through photo-excitation
into the conduction band at energy Eex at the rate gexδ(E − Eex), and then be removed
(depleted) from the bottom of the conduction band through recombination. This depletion
rate can be modelled by a term −gdδ(E)ne(0). Here the phonons are assumed to remain in
thermal equilibrium at temperatures T . In our model sample we assume a uniform density
of states for the electrons and the energy to be the only label for the single particle states.
The photo-excitation is taken to be homogenous over the sample, which is reasonable for
a nanoscale disordered semiconducting sample. For the above dissipative model driven
far from equilibrium, the kinetics for the non-equilibrium electron number density ne(E)
is given by the rate equation
∂ne(E)
∂t
= −ne(E)
∫
dE ′ [ne(E ′)(1− f) + nph(E ′)f ]Γ
+
∫ 1
0
dz p(z)
∫
dE ′
∫
dE ′′ δ(E − zα(E ′ + E ′′))ne(E ′)ne(E ′′)(1− f)Γ
+
∫ 1
0
dz p(z)
∫
dE ′
∫
dE ′′ δ(E − z(E ′ + E ′′))ne(E ′)nph(E ′′)fΓ
+ gex(t)δ(E − Eex)− gdδ(E)ne(0). (1)
In the above, we have assumed the total energy (E ′ + E ′′) for a binary collision to
be partitioned such that a fraction z, with probability density p(z), goes to the tagged
electron of initial energy E ′, and 1 − z to the colliding particle (electron or phonon of
initial energy E ′′). The inclusion of α in the electron–electron collision takes care of the
possibility of inelastic electron–electron collisions. Note that we have suppressed the time
argument (t) in the non-equilibrium electron-number density ne(E). Taking the energy
Laplace transform
n˜e(s) =
∫ ∞
0
e−sEne(E) dE, (2)
of equation (1), we obtain
∂
∂t
n˜(s) = −Γn˜e(s)[(1− f)Ne + fNph] + (1− f)Γ
∫ 1
0
p(z) dz n˜2e(αzs)
+ fΓ
∫ 1
0
dz p(z) n˜e(zs)n˜ph(zs) + gex(t)e
−sEex − gdne(0). (3)
In the following, we will consider for simplicity the steady-state condition under
constant (cw) photo-excitation, gex(t) = gex. A pulsed excitation can, of course, be
considered in general. Accordingly, we set (∂/∂t)n˜e(s) = 0 above, and all quantities on
the rhs of equation (3) are then independent of time.
In order to calculate the steady-state electron distribution for the system in terms
of the bath (phonon) temperature and other rate parameters, we expand n˜e(s) in powers of
doi:10.1088/1742-5468/2005/06/L06001 3
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.M
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Dissipative electron–phonon system photo-excited far from equilibrium
the Laplace transform parameter s as
n˜e(s) = Ne − s〈Ee〉+ s2〈E2e 〉/2 . . . , (4)
and equate the coefficients of like powers of s. Thus, from the zeroth power of s, we obtain
at once
ne(0) = (gex/gd). (5)
Similarly, from the first power of s, we get
〈ee〉 = (f/2)〈eph〉
ρe−ph(1− α)(1− f) + f/2 +
gexEex/Γ
N2phρe−ph[ρe−ph(1− α)(1− f) + f/2]
. (6)
In the above, we have taken a uniform limit for the energy partition: p(z) = 1.
Here, we have defined 〈ee〉 ≡ 〈Ee〉/Ne = mean electron energy; 〈eph〉 ≡ 〈Eph〉/Nph =
mean phonon energy (=kBTB); and ρe−ph = Ne/Nph ≡ electron-to-phonon number ratio.
It is to be noted that in the limit α = 1 (i.e., for elastic electron–electron collisions as is
usually expected for an electronic system unlike the case of the granular gas), and gex = 0
(i.e., no photo-excitation), we recover 〈ee〉 = 〈eph〉, i.e., the electrons and the phonons
are at the same temperature, as is physically expected under equilibrium conditions. In
general, however, the mean electron energy in the steady state is not the same as the
mean phonon energy, and the former depends on the excitation rate (the drive gex).
We will now consider the specific case of the extreme partition limit p(z) = 1
2
(δ(z) +
δ(z − 1)) to illustrate our treatment. An analytic form for n˜e(s) in the steady state will
be obtained for the case of elastic scattering (α = 1). It can be readily verified that for
this insertion of p(z) in equation (3), the equations (5) and (6) remain unchanged. With
this choice of p(z), equation (3) now becomes
n˜e(s)[(1− f)Ne + fNph] = (1− f)
∫ 1
0
(1
2
δ(z) + 1
2
δ(z − 1)) dz n˜2e(zs)
+ f
∫ 1
0
(1
2
δ(z) + 1
2
δ(z − 1)) dz n˜e(zs)n˜ph(zs) + gex[e−sEex − 1]/Γ. (7)
This can be readily solved to give
n˜e(s) =
2[(1− f)Ne + fNph]− fn˜ph(s)
2(1− f)
±
√
[fn˜ph(s)− 2[(1− f)Ne + fNph]]2 − 4(1− f)C
2(1− f) (8)
with
C = (1− f)N2e + fNeNph + 2gex(e−sEex − 1)/Γ.
We have to choose the −ve sign in the above solution so as to satisfy n˜e(0) = Ne, n˜ph(0) =
Nph. Here, we have assumed that the bath phonons obey the equilibrium Boltzmann
distribution (i.e., n˜ph(s) = Nph/(1 + 〈Eph〉s)). Now, our boundary condition for the case
of fast electron–hole recombination at the bottom of the band demands that the electron
number density at the bottom of the conduction band be zero, i.e., ne(0) = 0. But, from
the steady-state analysis, we have gex = gdne(0). Thus the above boundary condition is
doi:10.1088/1742-5468/2005/06/L06001 4
J.Stat
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ech.(2005)L06001
Dissipative electron–phonon system photo-excited far from equilibrium
implemented mathematically by formally demanding that the drain coefficient gd tends to
infinity, so that the product gdne(0) remains finite = gex. With this boundary condition,
and using the initial-value theorem for Laplace transforms, we get a relation between the
electron-to-phonon number ratio (ρe−ph) and the excitation parameter η as
ρe−ph =
1
2
[√
β2 +
8η
1− f − β
]
, η =
gex
ΓN2ph
. (9)
The electron number density distribution ne(E) (in the energy E-domain) is now obtained
by numerically inverting n˜e(s) (in the s-domain) in equation (8), and this is plotted in
figure 1 for various values of electron–phonon coupling parameter, f . (The noise seen on
the lowest curve is an artefact of the numerical inversion.)
At this stage, it is in order to comment briefly on our use of the extreme partition
P (z) = 1
2
δ(z) + 1
2
δ(1 − z). First, let us note that our approach, based on equation (1),
is of course valid for any general form of p(z). It is, however, only for the above extreme
limiting form that it admits an analytical solution as presented here in order to illustrate
the model. The above form can now be justified in the following physical terms. It is to be
noted that for any given collision, the partitioning of the total incoming energy between
the outgoing particles necessarily depends on the kinematic parameters of the collision,
for example, on the scattering angles, that vary from one collision to the other. Thus, for
a fine-grained treatment of the successive collisions under the above extreme form, the
energy of a ‘tagged’ electron colliding with others in the gas will fluctuate between extreme
values. However, a coarse-graining over even a small number of successive collisions will
suppress these fluctuations. Thus, in a coarse-grained statistical sense the partition is
expected to become effectively a smooth and broad function. This physically justifies
our use of the simple, though admittedly extreme, form for the partition p(z) without
resulting in unphysical features, as validated a posteriori from figure 1.
3. Discussion
We have treated here the problem of energy distribution of photo-excited electrons in a
semiconducting sample as a generalization of the model for dissipative granular gas driven
far from equilibrium. A feature of the above non-equilibrium distribution of the cw photo-
excited electrons is the peak appearing at the excitation energy (=1 eV) as is indeed
expected. However, a notable feature of the steady-state electron distribution in this case
is that, as the electron–phonon interaction strength (f) increases, (1) the peak height of
the distribution decreases, as does also (2) the total area under the distribution (the total
number of steady-state photo-excited electrons in the conduction band). This is physically
understandable as follows. Inasmuch as the increased electron–phonon interaction implies
fast removal of energy from the non-equilibrium photo-excited electron distribution to
the phonon bath, the electrons get pushed to lower energies towards the bottom of the
conduction band, from where they get removed by the fast electron–hole recombination.
Hence the decrease in the height of the distribution, and also the reduction of the area
under the curve. It should be possible to probe these steady-state features through a cw
pump–probe experiment. As for our assumption of non-degeneracy for the photo-excited
electron gas, it is readily verified that the peak occupation number of one-particle states
is of the order of (Ne/total number of atoms in the nanoscale sample)  10−4  1,
doi:10.1088/1742-5468/2005/06/L06001 5
J.Stat
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ech.(2005)L06001
Dissipative electron–phonon system photo-excited far from equilibrium
10 f = 0.3
f = 0.5
f = 0.8
x1017
8
6
4
2
0.8 1 1.2 1.61.4
E(eV )
n
e
(E
) (
e
V
-
1  
cm
-
3 )
Figure 1. Plot of the steady-state photo-excited electron distribution function
ne(E) (eV−1 cm−3) for a typical nanoscale semiconducting sample of radius
∼10 nm with ∼105 atoms, as a function of the electron energy E (eV), for
the following choice of parameters: the excitation parameter η = 10−7 and the
electron–phonon coupling parameter f = 0.3 (top line), 0.5 (middle line), 0.8
(bottom line). In all cases, the peak occurs at E = 1 eV, at which the electrons
are being photo-injected in the conduction band. Here the phonons are assumed
to remain at room temperature (300 K).
thus validating the assumed non-degeneracy. Finally, the dissipative electron–electron
collisions (α < 1) can physically derive from the dissipative polarization of the dielectric
medium that screens the electron–electron Coulombic interaction [7].
One of the authors (NS) would like to thank Yashodhan Hatwalne and A M Jayannavar
for discussions.
References
[1] Arbouet A et al , 2003 Phys. Rev. Lett. 90 177401
[2] Kaganov M I, Lifshitz I M and Tanatarov L V, 1957 Zh. Eksp. Teor. Fiz. 31 232
Kaganov M I, Lifshitz I M and Tanatarov L V, 1974 Sov. Phys. JETP 39 375 (translation)
Singh N, 2004 Mod. Phys. Lett. 18 1261
[3] Sun C-K, Vallee F, Acioli L H, Ippen E P and Fujimoto J G, 1994 Phys. Rev. B 50 15337
[4] Singh N, 2005 Pramana J. Phys. 64 111
[5] Kuhn T and Rossi F, 1992 Phys. Rev. B 46 7496
[6] Srebro Y and Levine D, 2004 Phys. Rev. Lett. 93 240601
See also, Po¨schel T and Luding S (ed), 2001 Granular Gases (Berlin: Springer)
Po¨schel T and Brilliantov N (ed), 2003 Granular Gas Dynamics (Berlin: Springer)
[7] Shimizu A and Yamanishi M, 1994 Phys. Rev. Lett. 72 3343
doi:10.1088/1742-5468/2005/06/L06001 6


Restricted Access.Generalizing the stochastic model known for a driven dissipative granular gas, we have derived the steady-state non-degenerate electron distribution for a semiconductor driven far from equilibrium by the interband photo-excitation assumed uniform over the nanoscale sample. Partitioning of the total inelastic electron scattering into dissipative electron–electron and electron–phonon components is included. The model is applicable to a photo-excited semiconducting sample with fast removal of the electrons by electron–hole recombination from the bottom of the conduction band

Dissipative electron-phonon system photoexcited far from equilibrium

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