Advances in Condensed Matter Physics

Volume 2019, Article ID 8317278, 6 pages

https://doi.org/10.1155/2019/8317278

## On the Heat Capacity of a Quasi-Two-Dimensional Electron Gas

^{1}Namangan State University, Namangan, 160119, Uzbekistan^{2}Physical-Technical Institute, Uzbek Academy of Sciences, Tashkent, 100084, Uzbekistan

Correspondence should be addressed to A. G. Gulyamov; ur.liam@vomaylug.lusarudba

Received 11 October 2018; Accepted 14 November 2018; Published 10 January 2019

Academic Editor: Luis L. Bonilla

Copyright © 2019 P. J. Baymatov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Numerical and analytical results of the investigation of the thermodynamic properties of a quasi-2D electron gas are presented. The density of states, the temperature derivative of the chemical potential, and the heat capacity of the gas at the resonance points and away from it are analyzed. It is shown that, in the dependence of the heat capacity on the chemical potential, there are additional steps at the resonance points. The width of the additional steps increases with temperature. With the increase in temperature, when the main steps are practically blurred, there are still visible marks from the additional steps.

#### 1. Introduction

A quasi-two-dimensional electron gas in heterostructural quantum wells causes both fundamental and applied interest, since the spatial quantization of the carrier energy leads to the manifestation of interesting low-dimensional effects [1–8]. Of particular interest is the behavior of a two-dimensional electron gas in a magnetic field [9–15], since the latter can significantly change the properties of the system.

Under certain conditions, the effects of size quantization appear in the observed thermodynamic quantities even in the absence of a magnetic field [16–23].

It is known that the main reason for the manifestation of size-quantum effects in a gas is a stepwise change in the density of states at critical points, i.e., at the points of resonance, where the chemical potential* μ* is compared with the energy levels of spatial quantization [1]. Such a change in the density of states manifests itself in one form or another in all the observed thermodynamic quantities. The change in the chemical potential relative to the energy levels of spatial quantization can be observed depending on the concentration

*n*

_{s}or width of the quantum well

*L*.

In [18], the entropy, which is incident on a single particle in a quasi-two-dimensional electron gas, is studied as a function of the chemical potential* μ*. It turned out that at low temperatures it has narrow peaks at the resonance points

*= (*

*μ**n*= 1, 2, 3...), and the height and width of these peaks depend on the temperature and the scattering event (degree of disorder).

In this paper, the heat capacity of a quasi-two-dimensional ideal gas is studied as a function of the chemical potential at different temperatures. Based on known thermodynamic relationships, the density of states, the temperature derivative of the chemical potential, and the heat capacity of a two-dimensional electron gas are analyzed. For these quantities, low-temperature asymptotic formulas are obtained.

Numerical simulations show that, in the dependence of the heat capacity on the chemical potential, at the resonance points* μ* = (

*n*= 1, 2, 3...) there are additional steps. The width of the additional steps is proportional to the temperature and they are connected by sharp changes in the value at the resonance point. In conclusion, the results of the work are briefly summarized.

#### 2. Basic Relationships

Let us consider an ideal electron gas with a simple parabolic spectrum. Energy is measured from the bottom of the zone of a massive semiconductor. In a heterostructural quantum well, the energy of the transverse motion is quantized, and the electron gas becomes quasi-two-dimensional. One-particle spectrum can be represented in the form [1]where -wave vector, are the energy levels of the transverse motion, and is the effective mass of the electron. The total two-dimensional concentration iswhere and is the Fermi-Dirac distribution function. The thermodynamic density of a state [9] is defined as

Expressions for the heat capacity of a gas can be obtained by differentiating the total energy

by temperature, i.e., . We represent it in the form

where

here . A temperature derivative of the chemical potential can be found from the conditions of constancy of the number of particles (2), where

First, let us analyze the low-temperature asymptotics of these formulas.

(i) Let the chemical potential* μ* be near the level , and the temperature is sufficiently low

*~ ,*

*μ**T*<< − . Then in the expressions (6) and (7) all the summands with

*~ < exponentially small and*

*μ**N*−1 the term with

*~ > give a finite contribution. By retaining also the*

*μ**N*- term (since

*~ ), from (6) and (7) we obtain (*

*μ**N*> 1)

The first integrals in these expressions are calculated analytically. Then

The remaining integrals in (11) and (12) can also be calculated analytically, but the results are cumbersome. These integrals can be calculated numerically with good accuracy.

Similarly, near the resonance point under the condition of (8) and (9) we haveThus, near the resonance point under the condition the quantity is equal* d μ* /

*dT*toIn the approximation considered, for the density of states (3) we can obtain the following formula:The quantity also determines the entropy per electron , and the low-temperature dependence is studied in detail in [18]. Using (11), (12), and (14), according to formula (5), we can find the specific heat value near the resonance point , and also the line shape near .

(ii) Suppose that the chemical potential* μ* is between the levels and , i.e., <

*< and the temperature is low enough*

*μ**T*<< − . Then in the expressions (11) and (12) the values of the integrals are exponentially small. Neglecting them, we obtain

Under the conditions considered, the quantity in formula (14) is exponentially small. Thus, according to (5), the specific heat in this limit has the form

and the density of state (15) reduces to the form

It can be seen from the comparisons (18) and (19) that when the condition , and , both the density of the state and the heat capacity of a quasi-two-dimensional ideal gas , vary abruptly with increasing chemical potential* μ* At low temperatures at the resonance points and from (11) and (12) we have

The integrals in these expressions are calculated analytically. Then

Similarly, it follows from (14) thatThus, at the resonance point the heat capacity (5) isFor* N*=2, the second term in the curly bracket is approximately equal 0,065 but decreases rapidly with growth. It is approximately equal to half the sum of the successive steps of the heat capacity (18) obtained under the condition . In a similar way, from (15) we can find an expression for the density of states at the resonance pointAnd in this limit the jumps in the specific heat (26) are due to a change in the density of states (27)

#### 3. Numerical Results and Discussion

At not too low temperatures, the dependence of the heat capacity on the chemical potential can be obtained only numerically, on the basis of relations (5) - (9). The explicit form of the electron spectrum in a heterostructural quantum well can be determined by solving the quantum-mechanical problem. In the future, in numerical calculations, we use its simple approximation ,

First, consider the dependence of the density of states and the derivative of the chemical potential . The results of numerical simulations of these quantities using formulas (3) and (8) and (9) are shown in Figure 1.