## Abstract

An investigation into the Landau diamagnetism of an electron gas in superlattices is presented. We find that the magnetization of a strongly degenerate electron gas changes its sign depending on the degree of band filling and magnetic field magnitude.

## 1. Introduction

The quantum behaviour of a two-dimensional electron gas immersed in a perpendicular magnetic field continues to demand attention owing to its technological importance. Of special interest are the studies of statistical properties of the system, in particular magnetic ones (Vagner *et al*. 1983; Skudralski & Vignale 1991; Lin & Nori 1996; Prado & de Aguiar 1996; Wilde 2005; Tao & Vignale 2006). Misra *et al*. (1971) developed an expression for the diamagnetic susceptibility of an electron gas with a complicated band structure, and predicted a positive diamagnetism if there was a gap over most of the original Fermi surface. Prado & de Aguiar (1996) revealed, for a strictly two-dimensional electron gas in weak enough magnetic fields (up to 2 T), that the diamagnetic susceptibility grows with increasing field and takes on small positive values. Lin & Nori (1996), in a quasi-two-dimensional electron gas, observed a non-monotonic crossover of the susceptibility from a large negative value to a relatively small positive value as the electron filling factor raises. In this paper, from the general expression for the grand thermodynamic potential, the diamagnetic magnetization of a quasi-two-dimensional electron gas with a cosine dispersion law is calculated. An external quantizing magnetic field is assumed to be applied normal to the layer plane. Therefore, the electron motion in the layer plane occurs in a circle and, being finite, turns out to be quantized. An analytical dependence of the magnetization on the magnetic field magnitude and degree of band filling in the case of a strongly degenerate electron gas in the quantum limit has been deduced. The magnetic properties of a quasi-two-dimensional electron gas are dependent on temperature, the degree of degeneracy of the electron gas and the degree of mini-band filling. It has been shown that, in certain ranges of the magnetic field and the electron filling factor, the magnetization becomes positive.

## 2. Magnetization of a two-dimensional electron gas

The magnetization of an electron gas, *M*, can be found, proceeding from the grand thermodynamic potential (Landau & Lifshitz 2003),(2.1)where the grand thermodynamic potential has the appearance (Askerov 1994)(2.2)where *ζ* is the chemical potential of the electron gas; *N* is the Landau level number; *k*_{z} is the quasi-impulse component along the *z*-axis; *k*_{0} is the Boltzmann constant; and *T* is the temperature.

We employed the energy spectrum of a quasi-two-dimensional electron gas in a quantizing magnetic field in a superlattice neglecting the spin splitting(2.3)where , , *a* is the lattice constant along the *z*-axis; , is the Bohr magneton, *m*_{0} is the free electron mass, *m*_{⊥} is the electron mass in the layer plane; *B* is the magnetic field induction; and *ϵ*_{0} is the mini-band half-width in the *k*_{z}-direction.

Substituting (2.3) in (2.2), the grand thermodynamic potential in the general form may be written as(2.4)where is the magnetic length, *e* is the electron's charge; ; and the upper limit of the integral is defined as(2.5)

Integrating the expression for the grand thermodynamic potential (2.4) by parts and taking the derivative with respect to the magnetic field induction, according to formula (2.1), for the diamagnetic magnetization at any degree of degeneracy of an electron gas we get(2.6)where *f*_{0}(*ϵ*) is the Dirac distribution function.

Formulae (2.4) and (2.6) hold true for cases of both a strongly degenerate and a non-degenerate electron gas.

### (a) Non-degenerate electron gas

In this case, it is easily seen that the grand thermodynamic potential is of the shape(2.7)where *n* is the electron gas concentration, which in this case reads as follows:(2.8)where ; is the dimensionless quantization parameter; is the modified Bessel function of zero order; ; and . Then, for the chemical potential of a non-degenerate quasi-two-dimensional electron gas in a quantizing magnetic field from (2.8), we obtain(2.9)

From (2.1), (2.7) and (2.8), the diamagnetic magnetization of a non-degenerate electron gas in an arbitrary quantizing magnetic field with regard to *μB*>*k*_{0}*T* is given by(2.10)If we substitute the expression of the chemical potential (2.8) into (2.9), the magnetization of a non-degenerate quasi-two-dimensional electron gas is conclusively derived to be(2.11)

This simple result, obtained for the quasi-two-dimensional electronic system, coincides with the expression of the magnetization of a free electron gas with the parabolic dispersion law in the quantum limit, when the magnetic field satisfies the condition *μB*≫*k*_{0}*T*.

### (b) Degenerate electron gas

In formula (2.4), it is impossible to conduct the summation with respect to the quantum number *N*; therefore, we consider the quantum limit (*N*=0). In the quantum limit for a degenerate electron gas in formula (2.6), the integration with respect to *Z* is carried out and in the case of a quasi-two-dimensional electron gas, when the Fermi level is positioned inside the mini-band, *ζ*_{F}<2*ϵ*_{0}, in the first approximation with respect to degeneracy from (2.6) for the diamagnetic magnetization, we have(2.12)where(2.13)(2.14)is the Fermi level in the magnetic field at *T*=0 (Askerov *et al*. 2006).

Taking into account the relationship between the Fermi level, concentration and magnetic field magnitude (2.14) in (2.13) and thereupon substituting the obtained dependence *Z*_{0} into formula (2.12), for the diamagnetic magnetization of a degenerate electron gas, we get the following dependence on the concentration and the magnetic field magnitude:(2.15)

In figures 1 and 2, we have plotted the magnetization against the degree of band filling and the magnetic field magnitude through formula (2.15) with the parameters *ϵ*_{0}=1 meV, *a*=10 nm, *n*=10^{23} m^{−3} and *m*_{⊥}=0.1*m*_{0} appropriate to superlattices.

From the figures and from formulae (2.12) and (2.15), it is evident that the magnetic response of the electron gas in superlattices is significantly distinct from the familiar Landau diamagnetism (where the susceptibility has its largest value at zero field *B*=0 and its vicinity, and declines monotonically as *B* is increased further); in a strictly two-dimensional electron gas (*ζ*_{F}>2*ϵ*_{0}) in superlattices, the diamagnetic magnetization of a degenerate two-dimensional electron gas is positive. The magnetization of a degenerate quasi-two-dimensional electron gas according to the degree of band filling changes its sign, at large degrees of band filling, when the Fermi level lies above the mini-band top, the magnetization goes positive. The magnetization also reverses the sign in a magnetic field. The magnetization oscillates in the magnetic field; frequencies and amplitudes of these oscillations tend to decrease with a rise in the field. The shape of magnetization oscillations (peak positions, amplitude and frequency) is determined by the relationship between the Fermi level, Landau level and mini-band width of the superlattice. It might be pointed out that, depending on a specific model of the band structure, de Haas–van Alphen oscillations in the quasi-two-dimensional metals have wave forms from the rare sawtooth and inverse-sawtooth to the usual symmetrical ones (Itskovsky *et al*. 2000).

The behaviour of the magnetization is probably associated with the behaviour of an electron gas (electrons behave as if they were holes) due to the existence of a region of negative electron mass in the conduction band (Romanov 2003). The similar counter-intuitive result—electrons move in the opposite direction to the free electron—was reported for galvanomagnetic phenomena in superlattices by Fleischmann *et al*. (1994).

Outcomes of the paper, together with those of Misra *et al*. (1971), Lin & Nori (1996) and Prado & de Aguiar (1996) for other low-dimensional electronic systems, show that the availability of the positive diamagnetism is related to the motion confinement of an electron gas.

Turning the diamagnetic magnetization into zero at specified values of the degree of band filling and magnetic field magnitude can be used when experimentally determining other components of the magnetic susceptibility.

## 3. Summary

In conclusion, we give an analytical calculation of the magnetization of an electron gas in superlattices subjected to a transverse magnetic field. We demonstrate that the magnetization, depending on the relationship between the Fermi level, Landau level and mini-band width, may become positive. The magnetization in the degenerate two-dimensional case is always positive and in the quasi-two-dimensional case, it may be either positive or negative. The magnetization oscillates in the magnetic field, fluctuating less around zero with the field. The positive diamagnetism is attributed to an inadequate behaviour of the electron gas owing to the existence of a negative effective mass in the electron mini-band region.

## Footnotes

- Received March 3, 2008.
- Accepted July 16, 2008.

- © 2008 The Royal Society