Statistics of charge carriers of quantum semiconductor film in the presence of strong lateral electrostatic field

: The influence of an external strong electrostatic field on the statistical characteristics of equilibrium free charge carriers in a quasi-two-dimensional structure is considered using an example of an InAs semiconductor quantum film. The analysis is carried out for the case when a strong quantization regime for charge carriers is realized in the film and only the first and second film sub-bands of their quantized motion are occupied. The general analytic expressions for the energy of the quantized motion of charge carriers in a film in the presence of a strong electrostatic field are given. An explicit dependence of the width of the band gap of the sample on the value of the external field is obtained. It is shown that with the increase in the external field, the width of band gap of the sample increases. The calculations and corresponding analytical expressions for the chemical potential, concentration, energy and heat capacity of free charge carriers of an InAs quantum film in the presence of a strong electrostatic field are also presented. It is shown that with an increase in the external electrostatic field, the chemical potential of the system retains its negative sign, but increases by the magnitude. For a given value of the external field, the chemical potential of the system increases linearly with the increase in temperature. It is shown that at a given temperature the carriers’ density decreases with the increase in the external field. At a fixed value of the field, the concentration increases with the increase in temperature. The behavior of the energy and heat capacity of the charge carriers is also determined by the same dependence on the temperature of the system and the magnitude of the external field. It is also shown that the main contribution to the indicated characteristics of the sample is made namely by the first filled sub-band


Introduction
For more than five decades, quasi-two-dimensional structures in the form of quantum semiconductor films (quantum wells) have been intensively studied both theoretically and experimentally [1][2][3][4][5][6][7][8][9][10]. They very quickly found a variety of instrumental applications and occupy their solid place in modern solid-state electronics [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. Such a wide application of these quasitwo-dimensional quantum structures is mainly due to two factors, the presence of which is fundamentally impossible in a three-dimensional sample. This is the phenomenon of the size quantization of charge carriers in one of the directions of motion  and corresponding to this phenomenon the specificity of the quasi-two-dimensional Fermi distribution of electrons and holes in the film [1,2,[29][30][31][32][33][34][35][36][37][38]. In reality, being a three-dimensional system, a quantum semiconductor film exhibits a number of properties characteristic of 2D systems in the field of multiparticle phenomena (in fact, the two-dimensional nature of the Coulomb coupling in the sample, the two-dimensional dispersion law and the two-dimensional density of states for carriers, etc.).
The object of our consideration in this paper will be precisely questions of the statistics of charge carriers in a quantum semiconductor film. The general questions of the properties of the Fermi-Dirac distribution in a semiconductor film, when dimensional quantization of carriers in one direction takes place are considered in Refs. [1,2,[29][30][31]36,37]. In these papers, the dependence of the chemical potential of the electronic subsystem on the thickness of the film is also considered. The papers [32][33][34][35] are devoted to the determination of the chemical potential of the electronic subsystem in quasi-two-dimensional systems, and in particular, in quantum semiconductor films.
Chemical potential and internal energy of a noninteracting Fermi gas at low temperature are evaluated using the Sommerfeld method in the fractional-and quasi-two-dimensional spaces in Ref. [34]. The effects of low dimensionality on the thermodynamics of a Fermi gas trapped by isotropic power law potentials are analyzed in Ref. [38].
Simultaneously it is well known that along with size quantization, one of the powerful external modulating factors is an external homogeneous electrostatic field. In the case of a semiconductor film, the situation where the field is directed along the quantization direction is of greatest physical interest. In this case, the states of the discrete spectrum of charge carriers undergo an additional, field modulation. This leads to a change in the kinetic, optical, and a number of other properties of the semiconductor film [39][40][41][42][43][44]. Naturally, the external electrostatic field exerts its significant influence also on the statistical properties of a quantum semiconductor film.
In the present paper, the influence of a strong external lateral homogeneous electrostatic field on the statistical properties of the electronic subsystem in a quantum narrow-band semiconductor film is theoretically considered, when a regime of "strong quantization" is realized for charge carriers in a film.

Single-electron states in the film in the presence of strong electrostatic field
As already noted, in this paper we investigate the case in which a regime of "strong quantization" is realized in a semiconductor film for charge carriers. For this, as is known, at least the following two important physical conditions must be satisfied [1,7,43]: Here B k is Boltzmann constant and T is the absolute temperature of the sample.
The conditions (1) and (2)  Here e m  is the effective mass of electron, 0 m is the mass of a free electron, g E is the width of the band gap of a bulk sample (data are taken from Refs. [45][46][47][48]). At a film thickness 10 L nm, the condition (1) for such semiconductors begins with sufficient accuracy even at relatively high temperatures, when the temperature of the system T varies, for example, from 100 to 300 K. The energy of the thermal motion of the electrons varies in this case, respectively, in the range from 10 to 25 meV.
We now consider single-particle states in a quantizing film in the presence of a strong transverse electrostatic field. Тhe quantizing film is approximated by the model of an infinitely deep rectangular potential well along the direction of quantization (z) [1,7,42]. Such a choice of the form of the model potential of a film is justified by the following physical factors. Firstly, a strong external field creates a deep triangular trap for the particle in the lower part of the well. Secondly, we are interested in the lower levels of quantization. But those levels in this case will be located precisely in the triangular part of the quantum well.
Then, we can write for the potential energy of the electron in the presence of an external electric field   Here e is the charge of an electron, L is the thickness of a film.
In the absence of an external field the energy spectrum of one-electron states in the approximation of an isotropic effective mass, is given by the following expression [1,42]: Here , xy pp are the components of the quasi-momentum of the particle in the plane of the film.
We regard the external field as strong. Physically, this implies that the energy received by the particle from the field within the film ( e FL ) is much larger than the particle's size quantization energy ( 1  ): It is shown in Ref. [42], that the strong-field condition with sufficient accuracy can be written as follows 1 10 In the case of an InAs film 10 L  nm, we have for 1  : 1 0.1635  eV. Accordingly, the condition (6) for the InAs film will be satisfied if the potential difference between the planes of the film is as follows: When the condition (5) is satisfied, the energy spectrum of a particle in a film in the presence of potential Eq 3 can be determined by a variation method [42]. Taking into account that it is the lower sub-bands that are important for the manifestation of the quantum size effect, we give the expressions obtained for the first two electronic levels in the film in the presence of a strong electrostatic field:   23 (8) Note that the conditions (5) and (6) are, in fact, also the quasi-classical conditions for the potential Eq 3. In view of this circumstance, the energy spectrum of the particle motion in the field (Eq 3) under the conditions (5) and (6) can also be determined from the Bohr-Sommerfeld quantization conditions [49]: Hence we obtain: It is easy to see that both approaches in this case give very similar results. We obtain from the Eq 9:     We do not write out the shape of the envelope wave functions, since in the future they will not be needed.
Let us now consider the statistical characteristics of the electron-hole subsystem in a film in the presence of an external field.
All calculations and estimates will be made using the example of an InAs  (1) and (2) are satisfied with great accuracy.
The melting temperature of the InAs crystal is equal to 0 1215 T  K, so that over the entire temperature range of this crystal ( ) this semiconductor is non-degenerate: even when We also note that the properties of InAs quantum films are being intensively studied at the present time (for example, [49,50]) and the study proposed here is of real interest from this point of view.

The statistical characteristics of charge carriers in a semiconductor film in the presence of a strong electrostatic field
In the presence of an external field with a strength F for the energy spectrum of an electron in general form, we can now write: Here the energy   n F  is given by the Eqs 7-10.
The general form of Fermi-Dirac distribution function for charge carriers in this case is as follows: Here  is the chemical potential of charge carriers' sub-system.
The effect of size quantization, as already noted, manifests itself most effectively in the conditions when the fewest possible sub-bands are filled. Therefore, we confine ourselves here to considering the filling of only the first two size quantization sub-bands in the presence of an external field.
In the case of the dispersion law from Eq 12, the normalization condition for the distribution function in Eq 13 for electrons and holes is represented in the following forms, respectively: Let us now turn to the calculation of the statistical characteristics of the quasi-two-dimensional sample in question.

Chemical potential and concentration of charge carriers in the film in the presence of external field
We will consider an intrinsic semiconductor with a standard parabolic dispersion law for carriers of the form Eq 12 and confine ourselves to the framework of a simple two-band model.In this case, the splitting of the valence band into bands of heavy (hh) and light (lh) holes will be taken into account. It takes place in the semiconductors cited above, including in InAs.     As we see, the results of solving Eq 19 clearly show the following: (с) At a fixed value of the temperature, the value of the chemical potential is the greater in magnitude, the greater is the intensity of the external field.

AIMS Materials Science
(d) In the presence of a field, the chemical potential also increases linearly with increasing the temperature, again approaching the bottom of the conduction band. The same results can also be obtained analytically.For this it is necessary to take into account that over the entire temperature range of the existence of the InAs crystal it can be regarded as a nondegenerate semiconductor with great accuracy. Using the approximate formula from Eq 19 we arrive to the following analytical expression for the chemical potential: It is easy to see that for the film thickness L = 10 nm, the results in Table 1 and, accordingly, the graphs of Figure 1 are also obtained from Eq 21. In the intrinsic semiconductor, as is known, the chemical potential of electrons and holes is not a function of the concentration (density) of charge carriers. .
However, knowing the chemical potential of the system, it is possible to determine the dependence of the carrier concentration (density) on temperature and in this case, also the dependence on the magnitude of the external field.
As we see, under the considered conditions and in the approximation Eq 20, the contribution of light holes to the total concentration of positive charge carriers is close to zero, when only the first sub-band of size quantization is filled.
So, if special accuracy is not required, then in calculations one can put: The third term in square brackets can simply be neglected because of its apparent smallness with respect to unity, even in the absence of an external field.
The remaining exponent in the last term makes sense only when F = 0. In the presence of a strong field, this term also tends to zero, i.e., we arrive at the following result: For carriers' concentrations in the film, when the first and second sub-bands are filled we obtain, respectively: Comparison of the expressions for 1  and 2  shows that the concentration of each type of carrier in the second sub-band of size quantization is less than the corresponding concentration in the first sub-band.

Energy and heat capacity of charge carriers in a film in the presence of an external field
For the energy of the system of charge carriers we can write in general form: Here, the particle energy   Let us proceed to a concrete calculation of the energy and heat capacity of charge carriers in a film in the presence of an external field.

Only the first sub-band is filled
It was shown above Eq 22 that the concentration of light holes is negligible compared to the concentrations of electrons and heavy holes. Accordingly, we will confine ourselves here to calculating the energy and the heat capacity only for electrons and heavy holes. From Eq 28, taking into account Eqs 7, 8, and 14-17, for the energy (E) and heat capacity (C) of these carriers, we obtain, respectively: The total energy and total heat capacity of electrons and holes in the sample will be expressed by the sum of Eqs 29-32, respectively. Let us discuss now the results obtained in this paper.

Results and discussions
In the quantized film, as it is well known, due to the quantum size effect, an effective broadening of band gap of the sample takes place: For a fixed film thickness, under the influence of the external field, there is an additional broadening of band gap of the sample with respect to its previous broadening.
As in the semiconductor InAs, hh lh mm  , from the valence band only heavy holes will contribute to this broadening: According to Eqs 7, 8, 10 and 11 with the increase in the external field, this broadening increases according to the law у~2 3 x : As we can see, the action of the external field actually substitutes the process of reducing the thickness of the film. This process naturally has a reasonable technological limit. By superimposing an external field, it becomes possible for a given thickness of the film to vary the energy of the quantized motion of the charge carriers in a controlled manner and, accordingly, to vary the width of the band gap of the sample in a controlled manner.
The chemical potential of a system of electrons and holes depends explicitly on the external field.
According to the Eqs 21, 25 and 26 remaining negative in sign, with the increase in the field it increases in absolute value ( 23 F ). Both in the absence and in the presence of an external field, the chemical potential of the electrons and holes decrease by absolute value with the increase in the temperature. In the system under consideration, the contribution of light holes in the statistical characteristics of electrons and holes is negligible. This is explained by the fact that in the system under consideration, between the first levels of heavy and light holes, there is a significant energy gap ( , ( 0) 0.1359 lh hh F eV    ) and, in the presence of size quantization, light holes do not participate in the formation of a new forbidden band. Physically, this is quite understandable: the particles tend to occupy the lowest possible energy states, and such are the levels of the size quantization of heavy holes in this case. We note that in the case of a bulk sample, where the energies of the sub-bands of heavy and light holes in the center of the Brillouin zone coincide, the secondary role of light holes is not so strongly pronounced.
For example, for the chemical potential of a bulk InAs sample, we obtain: Hence it is clear that in the 3D case the contribution of light holes is not as small as in the quasitwo-dimensional case.
It follows from Eqs 22 and 27 that the concentration of free charge carriers increases with the increase in temperature. At fixed values of the temperature and the external field, the carriers' concentration is maximal for the first sub-band. As the sub-band number increases, the concentration decreases. At a fixed temperature, an increase in the external field leads to a decrease in the concentration of free carriers. This is due to an increase in the width of the band gap under the action of the external field. This makes it possible, by changing the external field, to regulate the concentration of free carriers in the quasi-two-dimensional system under consideration. I.e., with the aid of an external field, we are able to control the degree of population of the sub-bands of the quantized motion of free charge carriers. Since the energy and heat capacity of a system are determined by the number of particles, the concentration behavior mentioned above also determines the behavior of the energy and heat capacity of the system of charge carriers in the film in the presence of an external field.
The contribution of the second sub-band to the energy and heat capacity of electrons and holes is much less (respectively,     The dependence of the energy of the electron-hole subsystem for the bulk InAs semiconductor (Figure 2a) and for the InAs film (L = 10 nm) in the temperature range from 0 to 600 K for the different values of the external field is shown (Figures 2b-2d).
A similar dependence also holds for the heat capacity of the electron-hole sub-system in a semiconductor film.
In the end, we note that all the calculations and conclusions that have been made are also applicable to semiconductor quasi-two-dimensional structures noted at the beginning of this article.

Conclusions
In terms of the effect on single-electron states, the presence of an external field makes it possible, for a fixed film thickness and a certain temperature, to obtain for the charge carriers energies unattainable by reducing the film thickness. Simultaneously, by using an external field, the position of the size quantization levels can be moved in a controlled manner.
In terms of the effect on the thermodynamic properties of the electron-hole subsystem, the presence of an external field makes it possible to vary the width of the forbidden band of the sample in a controlled manner. That is, an external strong electrostatic field makes it possible to control the values of such fundamental quantities of a system of free charge carriers in a film as their concentration, energy and heat capacity.