“Bottleneck” in Electron-Electron Interactions and Anomalies in the Landau Quantization Damping

A.B. DUBOIS, R.V. LYSOV

Mathematical Department

Ryazan Institute of Open Education

2 Bronnaya Str., Ryazan, 390010

RUSSIA

Abstract: - The report is aimed at clarifying the contribution of intrasubband and intersubband electron – electron interactions to Landau quantization damping of transverse magnitoresistance oscillations. Expressions are derived for the time of electron-electron interaction, matrix elements of the full screening potential and dynamic dielectric function in a 2D electron system. The calculated dependences provide a good description of the experimental times of Landau levels collisional broadening.

Key-Words: - Nanostructures, Dynamic dielectric function, 2D electron system

1 Introduction

Starting from the pioneering works [1] and up to the present time [2-4] the electron interactions are the subject of ever growing interest because of their important role in kinetic phenomena. Also known are the anomalies in the low-temperature magnitotransport arising when 2D electrons fill several size-quantized subbands. In particular, authors of Ref. [5] predicted non-monotonous behavior of kinetic coefficients as the density of 2D electrons is changed and several size-quantized subbands in a 2D system are filled. When the doping level of the nanostructure is high enough for concentration to reach , the quantum well contains two size-quantized subbands. In present paper we report the results of the study of electron-electron processes in a system of highly degenerate 2D electrons with the fine structure of energy spectrum and the electron density spatial distribution. Expressions for the times of electron-electron intrasubband and intersubband interaction are derived and matrix elements of the full screening potential and dynamic dielectric function for in the approximation far from the long wave limit are calculated.

2 Theory of Electron-Electron Interactions

One of the important points in the derivation of expression for and is the calculation of the full screening potential matrix elements which, within the perturbation theory approach, implies the transformation of the potential into . To within the second order in external potential in the perturbation theory expansion, the time required for the e-e interaction to change the state into is given by a well-known expression

(1)

Where indices i, j, k, l run over the set consisting of symbols m (first subband of dimensional quantization) and n, d (second subband of dimensional quantization) which label the electron transition type. When performing summation over k and p we transform (1) to

(2)

with matrix elements

and dielectric function

(3)

In the form convenient for calculations the relaxation times for intrasubband transitions are written as

(4)

while for the third type

(5)

where , , - polynomials in Eq. (5) are

with the coefficients defined by the Riemann zeta-function .

Non-monotonous behavior of is determined by the uniformly converging sums , and multiplied by the zeta-function

The products of , and with zeta-function are rather sensitive to the electron concentration in the size-quantized subbands. For example, for ...…the factor in Eq. (5) results only in some smoothening of the

non-monotonous behavior while at the curve does not contain any non-monotonous parts at all.

3 Results and Discussion

Presented for comparison in Figs. 1a and 1b are the calculated and experimentally measured curves for several nanostructure samples where 2D electrons are certainly known to fill only the lowest size-quantized subband (see Refs. [6,7] for the details of the analysis of the experiment).

Fig.1a

4 Conclusion

In conclusion, It should be noted that a similar problem for 2D electrons seems to have been considered in [8] and since then numerous attempts have been undertaken [9-11] to study this problem for 2D electron system where several size-quantized subbands are filled at in the long wavelength limit. However, the plasma oscillation spectrum has not been obtained in any of these works. Characteristic features of 2D electron systems, such as the amplitude – frequency modulation, beatings, and sharp bends in the oscillation, and sharp bends in the oscillation amplitude magnetic field dependence make the description of Landau quantization damping in terms of the Dingle temperature rather problematic. Another point to be pointed out is the fact that in the magnetic field range where a strong amplitude – frequency modulation takes place the second – subband electrons are in the state close to the quantum limit and one can only speak of the oscillations period in a rather limited sense.

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