Mechanisms of charge carriers nonequilibrium in transport processes in bipolar semiconductors

Current Applied Physics 16 (2016) 191e196

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Mechanisms of charge carriers nonequilibrium in transport processes in bipolar semiconductors I.N. Volovichev a, Yu.G. Gurevich b, * a b

A. Usikov Institute for Radiophysics and Electronics, National Academy of Sciences of Ukraine, 12 Ac. Proscura St., Kharkov 61085, Ukraine Departamento de Fisica, CINVESTAV-IPN, Apdo Postal 14-740, Distrito Federal 07000, Mexico

a r t i c l e i n f o

a b s t r a c t

Article history: Received 27 August 2015 Received in revised form 11 November 2015 Accepted 17 November 2015 Available online 2 December 2015

The interplay between physical origins of the nonequilibrium and their influence on the linear steady state transport processes in bipolar semiconductors are under investigation. Particular attention is paid to the influence of the energy nonequilibrium on the generation-recombination processes under various conditions. It is shown that in the case of the same (even if coordinate-dependant) temperature of the charge carriers and the phonons the volume recombination rate of the charge carriers in the steady state is completely determined by the splitting of the quasi-Fermi levels. Particular emphasis has been placed on the manifestation of the energy nonequilibrium in the presence of hot charge carriers in a semiconductor. It is shown that in this case the generation-recombination balance shifts, being completely equivalent to the appearance of an additional external generation of electron-hole pairs. The two-temperature model (with electron temperature being different from the single temperature of holes and phonons) of the Dember photovoltaic effect is used to illustrate that the electromotive force (emf) may differ significantly from its corresponding values with no hot electrons. This additional contribution to the emf does not depend neither on the Seebeck coefficient nor on the temperature gradient and the electron-hole pair generation rate. This contribution to the emf is exclusively determined by the magnitude of the electron heating. © 2015 Elsevier B.V. All rights reserved.

Keywords: Nonequilibrium charge carriers Lifetime Recombination Hot electrons Photovoltaic effect

1. Introduction Semiconductor materials are widely used in the electronics industry because of their nonlinear properties, which in turn, are due to the fact that the nonequilibrium charge carriers and phonons readily appear even with relatively weak external excitation [1,2]. The electric field, light, ionizing radiation, heat and other external influences easily disturb the thermodynamic equilibrium in the semiconductor system, leading to the emergence of the nonequilibrium charge carriers. Since a bipolar semiconductor is a system of three interacting subsystems (electrons, holes and phonons), the emergence of nonequilibrium in one of them disturbs the equilibrium state of the rest. In general, to study transport processes in semiconductors one needs to find a nonequilibrium distribution function of the charge carriers and phonons, e.g., by solving the Boltzmann equation [3,4].

* Corresponding author. E-mail addresses: [email protected] (I.N. Volovichev), gurevich@fis.cinvestav. mx (Yu.G. Gurevich). http://dx.doi.org/10.1016/j.cap.2015.11.013 1567-1739/© 2015 Elsevier B.V. All rights reserved.

However, in most cases of practical importance it is possible to introduce the nonequilibrium thermodynamic characteristics: the nonequilibrium chemical potentials, the electrochemical potentials (so called the quasi-Fermi levels or imrefs) and the temperature for each subsystem of the charge carriers and phonons. Accordingly, special cases of the nonequilibrium can be distinguished, depending on which parameter differs from its value in the thermodynamic equilibrium. For example, in the cases of the injection of charge carriers into the semiconductor from the contacting medium (as well as in the case of the generation of charge carriers by lighting or by heating the sample) the concentration of the charge carriers changes [5]. As a result, the carrier concentration usually becomes nonuniform in space. This type of the nonequilibrium is appropriately called the concentration nonequilibrium. In general, due to the physical nature of the semiconductor, the concentration nonequilibrium disturbs the dynamic balance of the other processes; in particular, the ratio of the diffusion and drift components of the electron and hole currents changes [6]. Also, the balance between the capture of electrons and holes by impurity levels (i.e., recombination of electrons) and the thermal generation

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of charge carriers shifts. This means that e.g. the nonequilibrium electrons, appearing in the conduction band, give rise to the generation-recombination nonequilibrium, which in turn generates nonequilibrium holes in the valence band [5]. At the same time, such processes as the Peltier effect, Joule heating and nonradiative recombination disturb the energy equilibrium as well, making the temperatures of electrons, holes and phonons spatially nonuniform. From the point of view of the thermodynamic characteristics of the semiconductor both the concentration nonequilibrium and the generation-recombination nonequilibrium are caused by the deviation of the electrochemical potential from its equilibrium value: the concentration nonequilibrium corresponds to the spatial inhomogeneity of the electrochemical potential, and the generationrecombination nonequilibrium corresponds to the difference in the electrochemical potential of electrons and holes (to the splitting of the quasi-Fermi levels) [6]. Similarly, the spatial inhomogeneity of the temperature and the differences in temperature between the three subsystems (charge carriers and phonons) are the physical mechanisms of the energy nonequilibrium in transport processes in semiconductors [4]. In turn, the energy nonequilibrium rarely can be analyzed independently of the other mechanisms of the nonequilibrium. For instance, the energy nonequilibrium caused by spatial nonuniformity of the charge carrier temperature disturbs the drift-diffusion balance due to the appearance of the thermoelectric field [7]. Additionally the temperature dependence of the cross sections of the rate of electron capture by impurities leads to a modification of the generation-recombination balance that is directly reflected on the charge carrier concentration, i.e. it leads to the concentration nonequilibrium [8]. This paper deals with the peculiarities and the interrelation between various mechanisms of the nonequilibrium in bipolar semiconductors in the steady state. Note that in the steady state, when the nonequilibrium space charge in the semiconductor does not depend on time, the mathematical description of transport processes is simplified. However, the physics of nonequilibrium transport processes is more complex and richer. This is due to the fact that in the steady state we have an additional relation div j ¼ 0 (here, j is the total electric current density in the circuit), therefore an additional mechanism for the interaction among the nonequilibrium of different types appears. For non-stationary (e.g., highfrequency) processes this interaction does not have enough time to manifest itself, and it is possible to analyze some mechanisms of the nonequilibrium independently [9]. 2. Energy nonequilibrium and generation-recombination balance in uniform bipolar semiconductor Evidently a nonuniform heating of an uniform sample of a bipolar semiconductor gives rise to nonequilibrium electron-hole pairs whose concentration profile will be non-uniform and, therefore, modifying the original homogeneity of the sample. There are two physical reasons for this: the appearance of thermoelectrical currents (causing the spatial redistribution of the charge carrier concentration) and thermal generation. The same result is obtained by the action of light and ionizing radiation, as well as by the injection of charge carriers from contacting media. Are these sources of nonequilibrium charge carriers equivalent in terms of the formation of the currentevoltage characteristics (CVC) and the possibility of generating electromotive forces (emf)? To answer this question, we shall take a closer look at the energy nonequilibrium and its impact on the generation-recombination balance in bipolar semiconductors. Mathematically, to do this one needs to calculate the displacement of the dynamic equilibrium between the thermal

generation and the recombination of electrons and holes; and then to determine the concentration of nonequilibrium charge carriers caused by this displacement. For this purpose we use the drift-diffusion approximation, that is widely used and adequately describes the operation of the most modern solid-state microelectronics devices [10]. The continuity equation for the electron and hole current densities jn,p, describing the stationary transport processes are as follows [11]:

divjn ¼ eðRn  Gn Þ;


 divjp ¼ e Gp  Rp ;

(1)

where e is the electron charge, Rn,p are the electron (n) and hole (p) recombination rates, and Gn,p are the external generation rates of charge carriers (the photo-generation rates). Note that the recombination rates Rn,p are defined as the difference between the capture rate of a conduction electron or hole by impurity levels (or by holes and electrons, respectively, in interband recombination) and the rate of the reverse process d the thermal generation of free charge carriers. In interband recombination [13] and recombination via an impurity level in the Shockley-Read-Hall (SRH) model in the steady state [14] the recombination rate can be written as Rn ¼ Rp ¼ cðnp  n2i Þ, where c is the capture factor, n and p are the concentrations of electrons and holes, respectively; ni is the intrinsic charge carrier concentration [15]. Note that in the stationary SRH recombination the capture factor c is a function of impurity level parameters and the charge carrier concentration [15]. However, this dependence is not relevant in the linear approximation, since, as will be shown below, drops out of the expression for the recombination rate (see Eq. (2), as well as the discussion at the end of the Section 3). Following Ref. [15], we write the expression for the recombination rate in the linear approximation (with respect to the perturbation of any type) for stationary processes, as follows (recall, that according to the Ref. [15] this expression has been developed for the interband recombination and can be extended to the trapassisted recombination only if Gn ¼ Gp):

Rn ¼ Rp ¼

  1 p0 n0 dn þ dp  bdT þ hðdTn  dTÞ ; T n0 þ p0 n0 þ p0 (2)

where dn ¼ nn0, dp ¼ pp0; n0, p0 are respectively the electron and hole concentrations in the state of the thermodynamic equilibrium at the temperature T0; dTn ¼ TnT0, dT ¼ TT0, Tn is the electron temperature, T is the temperature of phonons and holes, t ¼ ½cðT0 Þðn0 þ p0 Þ1 is the lifetime of the nonequilibrium charge carriers [15],

b¼

  εg n0 p0 1 3þ ; n0 þ p0 T0 T0

(3)

h¼

1 vcðT0 Þ n0 p0 ; cðT0 Þ vTn n0 þ p0

(4)

εg is the semiconductor band gap. Let us note that for simplification when deriving Eq. (2) heating was assumed to affect only the conduction electrons while holes quickly transfer excess energy to phonons (inelastic scattering of holes by phonons [3], mn ≪ mp, where mn and mp are the effective masses of electrons and holes, respectively), so that the temperatures of the hole subsystem and the lattice match. The densities of electron and hole currents in the nonequilibrium case are conveniently expressed by the quasi-Fermi level

I.N. Volovichev, Yu.G. Gurevich / Current Applied Physics 16 (2016) 191e196

recombination and the SRH recombination in the steady state.

(electrochemical potential) [6,16]:

jn ¼ sn ðVdjn  an VdTÞ;

(5)

 jp ¼ sp Vdjp  ap VdT ;

(6)




where sn,p are the electron and hole electrical conductivities, djn,p ¼ jn,pjn0,p0, jn0,p0 are respectively the electrochemical potential of the electrons and holes at the thermodynamic equilibrium, an,p are the electron and hole Seebeck coefficients. The nonequilibrium variations of the electrochemical potentials of the charge carriers djn,p are related to the nonequilibrium variations of the chemical and electric potentials by simple expressions:

djn ¼ d4  dmn =e;

 djp ¼ d4 þ dmp e;

(7)

where d4 ¼ 440, dmn,p ¼ mn,pmn0,p0, mn0,p0 are the chemical potentials of the charge carriers at thermodynamic equilibrium, 4 is the electric potential in the nonequilibrium situation, 40 is the potential of the built-in electric field in the state of the thermodynamic equilibrium. Moreover, because of the uniqueness of the Fermi level at the thermodynamic equilibrium, the equilibrium chemical and electrochemical potentials of electrons and holes must fulfill the following relationships: mn0 þ mp0 ¼ εg and e(jp0jn0) ¼ εg. Note that this condition of uniqueness of the Fermi level implies that in non-degenerate semiconductors, the equality n0 p0 ¼ n2i holds [6]. For this reason, in the SRH model the dependence of the capture rate c on the concentration of the nonequilibrium charge carriers and the impurity level population is only represented by the second order and the higher order terms in the serial expansion of the recombination rate Rn ¼ Rp ¼ cðnp  n2i Þ with respect to the nonequilibrium charge carrier concentration. Thus, in the linear approximation with respect to the nonequilibrium charge carrier concentration it may be assumed that c is constant (or explicitly depending only on the equilibrium temperature and the equilibrium concentrations). In turn, this assumption simplifies the study of the interband recombination and the recombination via impurity level in a unified manner, since the steady state linear forms of their rates are identical. In the linear approximation the nonequilibrium concentrations are expressed in terms of the variation of the chemical potential as follows [7]:

dn ¼ n0 dmn =T0 ;

 dp ¼ p0 dmp T0 :

(8)

Then, taking into account Eqs. (7) and (8) the recombination rate in Eq. (2) can be expressed in terms of the quasi-Fermi levels and the nonequilibrium temperatures as:

Rn ¼ Rp ¼

   en0 p0 1 3 mn0 n þ p0  djp  djn þ þ hT0 0 e 2 tðn0 þ p0 ÞT0 T0 n0 p0   ðdTn  dTÞ :

193

(9)

Note that in the expression (9) the term containing b drops out (see Eq. (2)). Thus, nonuniform heating, preserving the single (even if spatially nonuniform) temperature of the charge carriers and phonons, has no effect on the recombination rate. In other words, the energy nonequilibrium between the charge carrier and phonon subsystems affects the generation-recombination balance, while the energy nonequilibrium within the charge carrier and phonon subsystems leaves the generation-recombination balance intact. This conclusion is equally valid for both the interband

3. Recombination and energy nonequilibrium in the singletemperature approximation Assuming that the external excitation is such that no heating of the electron subsystem occurs (dTn ¼ dT), i.e. the whole system (semiconductor) is at a single (possibly, spatially nonuniform) temperature. Then:

Rn ¼ Rp ¼


 en0 p0 djp  djn : tðn0 þ p0 ÞT0

(10)

In this case, the recombination rate is determined entirely by the splitting of the quasi-Fermi levels. From (10) it follows that in the limit of infinitely strong recombination (t/0) for any spatial energy nonequilibrium there is a single Fermi level of both electrons and holes in the system. Moreover, in the quasi-neutral approximation [5,17] the charge carrier concentration at each point of the sample is equal to the concentration value that in an uniform semiconductor kept at the same uniform temperature T ¼ T(x). Indeed, the occurrence of a single Fermi level gives the first relation between the electron and hole concentrations. The condition of the quasi-neutrality dr(x) ¼ 0 (where dr(x) is the space charge in the semiconductor) gives the second relation between the electron and hole concentrations. These two relations allow to define dn and dp unambiguously. However, the quasi-neutrality condition in each point of the sample x¼xi coincides with the neutrality condition of the uniform semiconductor in the thermodynamic equilibrium at the uniform temperature Ti ¼ T(xi). Thus in the case of an infinite recombination rate the charge carrier concentration is equal to the charge carrier concentration in thermal equilibrium at the temperature Ti ¼ T(xi). Conversely, when the quasi-neutrality conditions are not met, the charge carrier concentration always differs from its local value in equilibrium. It is worth noting that in any case we can talk about nonequilibrium carriers, since the charge carrier concentration differs from the equilibrium value corresponding to the temperature of the reference (thermostat) T0 if T(x) s T0. We emphasize once again the non-obvious at first glance result: the single Fermi level, resulting from the zero charge carrier lifetime, does not mean the absence of the nonequilibrium charge carriers. From physical point of view, this result manifests the general condition for equilibrium between two subsystems of a macroscopic system with respect to the particle transitions between them [12]. Besides, in contrast to the state of the thermodynamic equilibrium the electrochemical potential (the single Fermi level) in the case of infinite recombination rate is generally not constant in space. Its spatial nonuniformity in these conditions may account for the existence of nonequilibrium charge carriers and the electric current flowing as a result of the displacement of the dynamic balance of the diffusion and the drift of charge carriers in an initially inhomogeneous semiconductor system. The physical reason for the fact that at a single temperature of phonons and charge carriers the recombination rate depends solely on the splitting of the quasi-Fermi levels, is as follows. Eq. (9), as well as the results of Ref. [15] stem from the choice of a microscopic model of recombination. The latter actually postulates that (i) the recombination rate depends on the concentration and the mean energy of charge carriers (the charge carriers temperature), but not on the concentration and the temperature gradients and so on, i.e. it is fully determined by the symmetric part of the charge carrier distribution function. (ii) the thermal generation of the charge carriers is independent of the state of the charge carrier subsystem at all, and it is determined only by the lattice temperature (but

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neither its gradient nor mean quasi-momentum of the phonons, etc.), i.e. again by the symmetric part of the distribution function of the phonons. Thus, both the interband recombination model and the SRH model originally lack dependencies on the gradients of the physical quantities and on the asymmetric part of the distribution functions. Hence it constitutes the physical basis of the independence of the recombination rate from the gradient of the quasiFermi levels, which is particularly reflected in the fact that the quasi-Fermi levels coincide in the limit of a zero lifetime, regardless of the degree of the nonequilibrium in the systems. On the other hand, such models of recombination are generally accepted, and the underlying independence of the transition probability of electrons from the anisotropic part of the distribution function [13] (provided, of course, a fixed mean energy, i.e. a fixed temperature) is in good agreement with the experimental results. The appearance of the dependence of the transition probability of electrons from the distribution function is possible only when the concept of temperature itself becomes incorrect [18]. 4. Two-temperature model If the energy nonequilibrium goes beyond the spatial inhomogeneity of the temperature and there exist hot charge carriers (e.g., electrons: dTn s dT) in a semiconductor, the situation changes substantially. Note that the term in Eq. (9), proportional to dTndT, describes the shift of the generation-recombination balance caused by the temperature difference between electrons and holes. Recall that here we assume no external generation in the semiconductor. This term does not include the quasi-Fermi levels. In other words, now the thermal generation of charge carriers is completely equivalent to the external charge carrier generation with the rate ðTÞ

ðTÞ

Gn ¼ Gp ¼

n0 p0 tðn0 þ p0 ÞT0

 

 3 mn0 n þ p0 þ ðdTn  dTÞ:  hT0 0 2 T0 n0 p0

semiconductor sample, one side of which (at the coordinate x ¼ 0) is illuminated by strongly absorbed light with absorption depth of g1. We assume the length of the sample in the direction of light incidence (Ox) large enough to consider the sample in this direction to be infinite. Besides, assume all the physical quantities in the transverse directions being uniform. In this case, the problem under consideration becomes one-dimensional. If the photon energy substantially exceeds the semiconductor band gap, the light absorption is accompanied by the generation of electron-hole pairs at the rate Gph ¼ gI Iph g expðgxÞ (where Iph is the intensity of the incident light, gI is the quantum yield) as well as by the charge carriers heating. The latter is due to two factors. First, the excess energy of photons Zuεg (where u is the frequency of the photon, Z is Planck's constant) is redistributed within the charge carrier subsystems, affecting their temperatures and causing the difference in temperature between the charge carrier subsystems and the lattice. Second, the direct heating of conduction electrons due to the light absorption by free charge carriers without photo-generation is also possible [25]. This second way of the electron gas heating exists even when the photon energy is smaller than the semiconductor band gap, so no photo-generation of the electron-hole pairs occurs. Simultaneous solution of the charge transport and energy equations is a cumbersome task, even in the linear approximation. Because our goal is to show qualitatively the interrelation between the concentration nonequilibrium and the energy nonequilibrium in the emf generation, we will introduce two additional assumptions to allow a separate analysis of the heat transfer and the formation of the emf. First, it is assumed that the electron cooling length [26] is considerably larger than the light absorption depth, i.e. there is the only superficial heating of the electron subsystem. Second, the Joule heating is negligible, this is the usual convention when studying the photovoltaic effects in the open circuit conditions in the linear theory. Then, the equation for heat transfer in the electron subsystem may be written in the following form [3,4]:

(11) Mathematically, this means that the system of differential equations of the current continuity (Eqs. (1), (5), (6) and (9)), as well as the system of equations for determining the integration constants from the relevant boundary conditions (which are the conditions for matching the electrochemical potential and currents at the interfaces, in details see. Refs. [5,19,20]), are heterogeneous, even if there are no temperature gradients or if the Seebeck coefficients are negligible an,p ¼ 0. Thus, in this case, even a uniform heating of the electrons leads to both the concentration nonequilibrium and the recombination nonequilibrium [21]. In other words, in the case of charge carriers at different temperatures the thermal generation can result in emf generation by the mechanisms like the Dember photovoltaic effect, the bulk photovoltaic effect and so on. As this takes place, the role of the external photo-generation rate is played by the value determined by the Eq. (11). 5. Energy nonequilibrium as source of emf It is common knowledge that the energy nonequilibrium within a quasiparticle subsystem (i.e. nonuniform temperature) results in the thermoelectric power (the Seebeck effect [22]), but it's trivial. Much more interesting is the possibility of the emf occurrence in response to the nonequilibrium thermal generation. Here we will illustrate it by the example of the Dember effect [23,24]. Let us study the Dember photo-emf in the one-dimensional two-temperature model. Consider a uniform bar-shaped

dQn  ne ndTn ¼ 0; dx

(12)

where Qn is the heat flow in the electron subsystem, e n is the energy relaxation frequency. In the linear approximation e n ¼ const and n ¼ n0. We consider temperatures of lattice and hole subsystems being equal and spatially uniform (e.g. T ¼ T0). In turn, the heat flow in the electron subsystem may be written as follows [27,28]:

Qn ¼ Pn jn  kn

ddTn ; dx

(13)

where Pn is the Peltier coefficient, kn is the electron thermal conductivity. Note that even in the open circuit (when the total electrical current density j0 ¼ 0), the partial electron and hole currents are non vanishing due to volume recombination. To further simplify the calculations, we suppose that the electron thermal conductivity is constant and high enough to omit in Eq. (13) the term proportional to the electron current density. Below we discuss the criteria for such an assumption in more detail. Then the Eq. (13) takes the following form:

Qn ¼ kn

ddTn : dx

(14)

The solution of the differential Eq. (12), taking into account the above assumptions has the ffiform dTn ¼ dTn0exp(lTx), where pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dTn0 ≡ dTn(0), lT ¼ n0en=kn is the electron cooling length.

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According to Eq. (11) due to the shift of the generationrecombination balance such distribution of the electron temperature is equivalent to the external photo-generation of the electronhole pairs with the rate GT ¼ GTdTn0exp(lTx), where

GT ¼

  n0 p0 mn0 3 T0 vc0 ;   2 c0 vTn ðn0 þ p0 ÞT0 t T0

(15)

c0 ≡ c(T0). So, the external generation rates in Eq. (1) are Gn ¼ Gp ¼ Gph þ GT. When studying the emf formation in bounded semiconductors, the Dember effect in particular, the correct choice of boundary conditions (BCs) is of decisive importance [24,29], which in turn are determined by the type of contacts and physical processes at the semiconductor interface. Consider the semiconductor is illuminated through a thin (transparent) metal contact. For simplicity, let us assume that at the “metal-semiconductor” interface there is neither surface recombination nor surface generation of charge carriers. This physical situation is adequately described by the following BCs [5,19,20]: jn ð0Þ ¼ sns ðdjn ð0Þ  4m ð0ÞÞ;

(16)

jp ð0Þ ¼ 0:

(17)

Here sns is the surface conductivity of the “metal-semiconductor” interface, 4m(0) is the electric potential of the left metal contact. Eq. (17) represents a unipolar (most common - electronic) type of the metal conductivity. In other words, in the absence of surface recombination the holes can not move from the semiconductor to the metal, for details, see Refs. [20,30]. For an open circuit from Eq. (17) it follows that jn(0) ¼ 0. For the finite value of the surface conductivity sns in view of Eq. (16) it means that jn(0) ¼ 4m(0). Deep within the semi-infinite sample there is almost no nonequilibrium charge carriers, and the second pair of the BCs becomes:

djn;p /0;

if x/∞:

(18)

With above choice of the BCs the emf is equal to the value of the electrochemical potential at the left edge of the sample: ε ¼ djn(0) [25]. The system of Eq. (1) for the linear approximation should present no particular difficulties. Using standard methods to solve systems of linear differential equations with the BCs Eqs. 16e18 in the limit of the strongly absorbed light (g/∞) one obtains the following expressions for the quasi-Fermi levels:

lT CT dTn0 þ eIph lx CT  an sn djn ¼  e þ dTn0 elT x ; lsn sn djp ¼

lT CT dTn0 þ eIph lx CT e  dT elT x ; lsp sp n0

(19)

(20)

where CT ¼ ðl2n an sn  eGT Þ=ðl2  l2T Þ, l2 ¼ l2n þ l2p is the inverse ambipolar diffusion length, ln;p ¼ =e2 n0 =ðT0 tn;p sn;p Þ. Then, the photo-emf is as follows:

E ¼

llT þ l2p e eGT an dTn0  dT : Iph  lsn lðl þ lT Þ sn lðl þ lT Þ n0

(21)

The first term in Eq. (21) represents the Dember photo-emf in the open circuit for the given sample geometry in the absence of electron heating. It must be emphasized that the presence of metal electrodes leads to a Dember emf different from the electric

195

potential difference, that is often identified with the Dember emf in the literature. A detailed discussion of this issue can be found in Refs. [24,29]. The second term in Eq. (21) corresponds to the electron thermoelectric power in a nonuniform temperature field . If the cooling length is sufficiently small (lT [ l), when all the tempe dTn ¼ dTn0 expðlT xÞ rature change takes place actually on the metal-semiconductor contact, this term adopts a typical form for the thermal emf (andTn0) and contains only the properties of the electron subsystem (its Seebeck coefficient an). In the opposite limiting case lT ≪ l, holes begin to influence the thermoelectric contribution: ðlp =lÞ2 an dTn0 . Thus, if the diffusion length of the holes substantially exceeds the diffusion length of the electrons lp ≪ ln, then l z ln ≪ lp and the thermoelectric contribution becomes negligible. Physically, this means that despite of the considerable large cooling length, the electrons recombine near the contact, failing to “feel” the temperature difference and generate a thermo-emf. Note that in the case of a small cooling length the initial assumption of the negligible influence of the electric current on the electron heat flow Eq. (13), i.e. the validity of Eq. (14), may be broken. Indeed, the condition for a predominance of the contribution from the thermal conductivity can be written as n jn =ddT ≪kn =Pn , that after substituting the expressions for the curdx rent and temperature gives the following requirement for the applicability of Eq. (14):



l2 a s  eG  eIph ðlT lÞx

n n n T ðlT lÞx e kn =Pn [

1 þ e

:

l2  l2 lT dTn0 T

(22)

From Eq. (22) it follows that if lT > l the condition of smallness of the influence of the electron current on the total heat flux in the electron subsystem can be violated sufficiently far away from the left contact at the expense of exponential factors exp½ðlT  lÞx, and the Eq. (14) becomes incorrect. On the other hand, in the depth of the sample the concentration of the nonequilibrium charge carriers, as well as the related currents and heat fluxes are small and do not have a pronounced effect on the emf value. At the same time, to neglect the impact of the electron current on the heat transport at the contact (i.e. where the nonequilibrium charge carrier concentration is the highest) it is sufficient to met the requirement knlTdTn0 [ ePnIph. In other words, the proposed theory is valid in the case of a sufficiently strong heating. Nevertheless, at weak heating the contribution from the Seebeck emf and the nonequilibrium thermal generation of the charge carriers to the magnitude of the photovoltaic effect occurs as well. Finally, the last term in Eq. (21) corresponds directly to the emf developed by the nonequilibrium thermal generation of the electron-hole pairs, and thus reflects the influence of the energy nonequilibrium on the concentration nonequilibrium and on the diffusion-drift nonequilibrium. It is obvious that for the small cooling length lT [ l this influence is not significant because the volume of the semiconductor, where the nonequilibrium heat generation in the two-temperature condition takes place, is reduced as the cooling length decreases. However, as mentioned above, in this case, the assumption of smallness of the Thomson heat [31] (i.e. the validity of Eq. (14)) becomes incorrect everywhere in the semiconductor except in the vicinity of the contact (x ¼ 0). On the contrary, when the cooling length is large enough the most favorable conditions for the manifestation of the thermal generation are realized, and the contribution from the thermal generation to the emf magnitude reaches its maximum value eGT dTn0 =ðl2 sn Þ. We note that the contribution of hot carriers to the photo-emf does not depend explicitly on the rate of the external

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I.N. Volovichev, Yu.G. Gurevich / Current Applied Physics 16 (2016) 191e196

photo-generation of electron-hole pairs. That is, the emf, according to the mechanism considered, exists in the absence of charge carrier generation by the incident light as well, exclusively due to the heating of the electron subsystem when photons are absorbed by conduction electrons. Also note that the expression for the emf caused by heat generation, does not contain the Seebeck coefficient. In other words, since the contribution of thermal generation to the emf is not associated with terms of the form anVTn in the electron current density, the emf could significantly exceed the thermopower determined by the thermoelectric properties of the semiconductor (an,p). Moreover, the effect occurs even with a uniform heating of the electron subsystem, when VT n≡ 0. 6. Conclusions In summary, in the case of a spatially nonuniform single temperature of the charge carriers and phonons the rate of the volume recombination of charge carriers in the steady state is completely determined by the splitting of the quasi-Fermi levels. If an extremely high recombination rate exists the energy inhomogeneity never leads to the appearance (split) of the quasiFermi levels. A single, but, in general, spatially nonuniform (in contrast to the state of the thermodynamic equilibrium) Fermi level remains in the semiconductor sample. However, the spatial inhomogeneity of the common Fermi level may provide the existence of nonequilibrium charge carriers and electric current flowing as a result of the shift of the dynamic balance between the charge carrier diffusion and the charge carrier drift processes in an (initially) inhomogeneous semiconductor system. The energy nonequilibrium associated with the hot carriers appearance manifests itself by shifting the generationrecombination dynamic balance so that the thermal generation of the nonequilibrium charge carriers is completely equivalent to the external generation of electron-hole pairs (e.g. by light) as a source of the emf. According to this, the value of the photovoltaic Dember effect (given as an illustration of the interrelation between various types of the nonequilibrium in a bipolar semiconductor) may differ significantly from its equilibrium (in terms of energy) value. The shift of the thermal generation balance is capable to give rise to an thermopower through a physical mechanism similar to the one behind photo-emf formation, even when the Seebeck coefficient of the semiconductor vanishes and there is no temperature gradient across the sample. Moreover, the emf originated by the mechanism considered may arise even when the photon energy is lower than the semiconductor band gap, but the electron subsystem heating due to the light absorption by free charge carriers takes place. Of course, this case is methodically correct to be denoted as the thermoelectric effect rather than as the photovoltaic one. The theory above not only gives a better insight into the physics of the charge transport in bipolar semiconductors, but also may be useful in the development of semiconductor devices in highly nonlinear modes. In the future the two-temperature photoelectric effect considered could form the basis of experimental methods for studying the properties of hot carriers and the electron-phonon interaction parameters. Acknowledgments The authors thank Prof. Jesus Enrique Velazquez Perez (Salamanca University, Spain) for valuable discussions. One of the

authors (Yu.G.G.) is grateful to CONACYT (Mexico) for partial financial support.

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