1. Introduction.
Question about transport of nonequilibrium carriers many decades is one of the central problems of Solid State Physics [1]. At the same time overwhelming majority of works is devoted to the transport of nonequilibrium charge carriers [2]. Nevertheless, as it is well known, a phonon subsystem can be easy to turn out from the equilibrium thermodynamic state too. In these cases fluxes of different nature occur in this subsystem, and these fluxes essentially result in the transport of charge carriers. Generally, the interaction between nonequilibrium charge carriers and nonequilibrium phonons is carried out with transferring of energy as well as with transferring of directed momentum [3, 4].
It is well known that different external agents can readily produce in a semiconductor an essential increase in the average energy of charge carriers and phonons (energy nonequilibrium) [4], which can be most conveniently described in terms of heating of the charge carrier and phonon gas (i.e. increase of their respective temperatures Te and Tp ). Since heating is known to exert a pronounced influence on transport phenomena in semiconductors, being largely dependent on the character of transport itself [4], it is quite understandable that the heating of electrons and phonons should be essentially interdependent. Note that the possibility of phonon deviation from equilibrium was first considered by R. Peierls [5] (see also [6]).
The heat generated by hot carriers can be harvested to drive a wide range of physical and chemical processes. Their kinetic energy can be used to harvest solar energy or create sensitive photodetectors and spectrometers. Photoejected charges can also be used to electrically dope two- dimensional materials [7].
Hot (nonequilibrium) phonon effects on electron transport in rectangular GaAs/AlAs quantum wires have been investigated by a self-consistent Monte Carlo simulation [8]. Authors have demonstrated that at room temperature hot optical phonons lead to a significant increase in electron drift velocity.
Steady state and modulated heat conduction in layered systems predicted by the analytical solution of the phonon Boltzmann transport equation was presented in [9] and has widely been used as the theoretical framework for the development of photoacoustic and photothermal techniques.
In work [10] the electron and phonon temperature distribution functions in semiconductors are calculated. The electron and phonon temperature distributions in the sample are given as a function of both, time and position valid for a wide range of the modulation frequency of the incident light.
The term "hot carrier injection" usually refers to the effect in MOSFETs, where a carrier is injected from the conducting channel in the silicon substrate to the gate dielectric, which usually is made of silicon dioxide (SiO2). [11]. In some semiconductor devices, the energy dissipated by hot electron phonons represents an inefficiency as energy is lost as heat. For instance, some solar cells rely on the photovoltaic properties of semiconductors to convert light to electricity. In such cells, the hot electron effect is the reason that a portion of the light energy is lost to heat rather than converted to electricity [12].
The physical peculiarities of the thermoelectric cooling phenomenon by an electric current in p-n structures of thermally thick structure in the linear approximation are investigated in [13].
It is shown the possibility of the realization of an exotic distribution of the temperatures of electrons, holes, and phonons.
The aim of the present paper is to present a short review of rigorous kinetic approach to energy interaction between nonequlibrium charge carriers and nonequilibrium phonons.
2. Kinetics of nonequilibrium electrons and phonons in semiconductors
Let us consider for simplicity an unipolar semiconductor (for example with electron type conductivity) with a quadratic and isotropic dispersion law for charge carriers,
where ε,
The acoustic phonons dispersion law is
Here ω is the acoustic phonon frequency,
Here
We suppose that all conditions ensuring applicability of the kinetic equation method are to be satisfied (see, Ref.[ 15 ]). We use the energy system units throughout the paper, so temperature has the energy dimension.
As was shown in Ref.[16] the electron-acoustic phonon scattering is quasielastic at temperatures just above 1
The made assumption about the quasielestic scattering on the phonons gives possibility to represent the electron distribution function in diffusion approximation [4, 16],
Where
Using the same approximation the phonon distribution function can be written in the following form,
Where
In this paper later we will consider only the stationary case.
It is well known that the electron distribution function form essentially depends on ratio between the electron-electron collisions rate Vee (ε), the electron momentum relaxation rate v(ε), and the electron energy relaxation rate vε(ε) [17]. We will suppose that these rates satisfy the following correlations in this work,
Approximation (7) is often called the partial or energy control [18]. In this case the electron subsystem can be described by means of the temperature approximation with the isotropic part of distribution function like the Fermi-Dirac (Maxwell) distribution function [18],
where
The brief proof of the stated above consists in the following. The largest term in Eq.(3) is the electron-electron collision integral which is the order of
Later we will consider only nondegenerate semiconductors, and will assume that the Debye length
Now let us to turn to the phonon subsystem, and first of all examine the collision integral Spe which can be written as [4],
Here
To transform the collision integral (9) to more convenient form, we shall use the identity being true for the Fermi distribution function (8),
Where
is the Planck distribution function with the temperature Te .
Substituting (10) into (9) and integrating over the solid angle (the axis oz is directed along the vector
Here
The first item in Eq.(12) has a very clear sense, only those phonons interact with electrons (with energy exchange) which have the temperature different from the electron temperature Te
. Thus, the factor at the difference
Since
and
Thus,
and the phonon-electron collision integral takes the form,
Eq. (13) determines the lowest energy for electron being able to interact with a phonon with the quasimomentum q. The corresponding lowest electron momentum is
Since the momentums of electrons and phonons are of the same order at quasielastic scattering, the second item in the right-hand side is the order of
These phonons are known as the long-wave phonons (LW). The phonons with momentum
Naturally, contribution of phonons to kinetic phenomena essentially depends on its number. Let Tp
is the characteristic phonon temperature (see below in detail). In this case the momentum
1. LW-phonons occupy a small phase space volume,
This equation leads to the following limitation of the electron temperature,
Let us note that Eq.(18) does not contradict the strong electron heating
2. LW-phonons occupy a large phase space volume,
In this case electrons interact with all phonons, and the electron temperature satisfy the condition
It´s easy to see from Eq. (14) that the rate vpe(q) rapidly decreases for
always takes place for SW-phonons. Here vpp is the SW-phonons-LW-phonons rate interaction.
At the same time both correlations
and
can take place for LW-phonons.
Since the phonon-phonon collision rate depends on phonon temperature, Eqs.(21) and (22) determine the temperature region for phonons and electrons respectively at which the inequalities (21) and (22) are hold true. Thus, for example Eq.(22) holds at T˂50K for n-GE (deformation acoustic electron-phonon scattering) and n-GaAs (piezoelectric acoustic electron-phonon scattering) at the electron concentration n≈1014 cm -3 [19].
3. Strong phonon-phonon interaction
If the condition vpp > > vpe holds true for all q then N 0 is the Planck distribution function with the temperature Tp ,
In this case one could obtain by usual technique [3] the energy balance equation for phonon subsystem,
Here
is the energy flux from the electron to phonon subsystem;
is the heat flux in the phonon subsystem.
It follows from definition of the value Ppe that
and
Thus, if the criterion
Let us obtain now the energy balance equation for electron subsystem. For this purpose, it is necessary to multiply equation (3) by
Here
is the thermal flux in the electron subsystem,
is the electric current, and
is the term describing the energy transferring from the electrons to phonons. Here
Let us note that the item
4. Strong Phonon-Electron Interaction
Let us now consider the situation represented by inequality (22). In this case the phonon-phonon collisions cannot provide a full energetic control in phonon subsystem, and its description becomes more complicated. Since the inequality (22) holds true only for LW-phonons then the parameters which characterize the phonon subsystem essentially depend on the relationship between
The LW-phonons occupy a small volume in the phase space in comparison with the SW-phonons if Eq. (17) takes place. In this case SW-phonons have enough time to redistribute energy received from LW-phonons among themselves, and the symmetric part of its distribution function is the Planck function with the temperature Tp . Just this temperature appears in Eq. (17).
All phonons interact actively with electrons if Eq. (19) holds, and fragmentation all phonons to LW- and SW- phonons have no sense. In this case all phonons are characterized by the temperature
Let us turn to solution of the kinetic equations in the case when conditions (17) and (22) take place.
In the region
Since the energy of all phonon subsystem is conserved at phonon-phonon scattering, then the following integral is equal to zero,
Integration in (33) is made over all values of
The left-hand side of this equation determines the energy transfer from the LW- to SW-phonons PLS
, and the righthand of this equation determines the energy transfer from the SW- to LW-phonons. These values are equal,
The expression for
In the region
where
It is impossible to represent the phonon-phonon collision integral in the region
Thus, the kinetic equation (4) for N
0
in the region
Eq. (36) can be solved by the perturbation method with respect to parameter
The zero approximation is
The function N o (1) determining the second approximation can be found from equation,
Multiplying Eq. (38) by
where
and
Eq. (41) remains indefinite because it contains the unknown value N 0 (1) . Naturally, that the value P ep containing in Eq. (28) is indefinite too. By this reason, it is necessary to consider the equations (28) and (41) as the set of equations for obtaining both the temperature T e and the parameter P ep . Eliminating the unknown term P ep from this system we find equation for obtaining the temperature T e ,
Thus, electrons and LW-phonons behave themselves as a common subsystem with the temperature T
e
under conditions discussed above. This common subsystem relaxes it energy to SW-phonons with the characteristic rate
Electrons and all phonons behave themselves as a common system with the temperature T
e
if
Conclusion
In this paper we have considered the energy nonequilibrium electrons and phonons which interact each with other. Nevertheless, as it is known, it is easy to realize the momentum nonequilibrium. This momentum nonequilibrium results in the momentum exchange among charge carriers and phonons in addition to the energy exchange between subsystems. In semiconductors this momentum exchange is known as electron-phonon drag [3, 4].
The electron and phonon nonequilibrium considered in this paper is space-nonuniform in real semiconductors and semiconductor structures. So, this space-nonuniform energy nonequilibrium produces in semiconductors fluxes of different nature. Thus, the energy nonequilibrium has to result in the momentum nonequilibrium of these qusiparticles. The discussion of some aspects of this problem one can find in Refs. [21-25].