^{1,a)}, M. Koesters

^{1}, K. G. Libbrecht

^{1}and E. D. Black

^{1}

### Abstract

We present an advanced undergraduate experiment on weak localization in thin silver films with a thickness between 60–200 Å, a mesoscopic length scale. At low temperatures, the inelastic dephasing length for electrons exceeds the film thickness, and the film becomes quasi-two-dimensional. In this limit, theory predicts corrections to the Drude conductivity due to the coherent interference between the wave functions of the conducting electrons, a macroscopically observable effect known as weak localization. This correction can be destroyed by the application of a magnetic field, and the resulting magnetoresistance curve provides information about electron transport in the film.

This work was supported by the National Science Foundation under Grant No. DUE-0088658.

I. INTRODUCTION

II. THEORY

A. Drude conductivity

B. Coherent backscattering

C. Weak localization

III. EXPERIMENTAL APPARATUS

A. Samples and their preparation

B. Detection of magnetoresistance

IV. TYPICAL RESULTS

V. SUMMARY

### Key Topics

- Weak localization
- 23.0
- Electron scattering
- 18.0
- Magnetoresistance
- 13.0
- Magnetic fields
- 12.0
- Localization effects
- 10.0

## Figures

Most partial waves have random relative phases, and add incoherently on average. In this figure, two scattering paths are considered, one shown by a solid line and one by a dashed line. Notice that the wave fronts for both scattering paths are separated by the same radial length. This separation is the Fermi wavelength, . For these randomly scattered partial waves, the wave fronts do not align, so on average there will be no net interference.

Most partial waves have random relative phases, and add incoherently on average. In this figure, two scattering paths are considered, one shown by a solid line and one by a dashed line. Notice that the wave fronts for both scattering paths are separated by the same radial length. This separation is the Fermi wavelength, . For these randomly scattered partial waves, the wave fronts do not align, so on average there will be no net interference.

Each partial wave that returns to the origin has another partial wave with which it is in phase in the backscatter direction. Again, we consider two partially scattered waves. Unlike Fig. 1, the two paths are the same except that they are traversed in opposite order. That is, if we apply time reversal to one path, we obtain the second path. This similarity, and the fact that their net scattering direction is in the backscattering direction, causes the wave fronts of the two paths to align on average, and there is a net interference effect. This interference means on average the backscatter probability is modified from the predictions of classical physics.

Each partial wave that returns to the origin has another partial wave with which it is in phase in the backscatter direction. Again, we consider two partially scattered waves. Unlike Fig. 1, the two paths are the same except that they are traversed in opposite order. That is, if we apply time reversal to one path, we obtain the second path. This similarity, and the fact that their net scattering direction is in the backscattering direction, causes the wave fronts of the two paths to align on average, and there is a net interference effect. This interference means on average the backscatter probability is modified from the predictions of classical physics.

The weak localization magnetoconductance at a fixed temperature. The different curves represent different relative contributions from spin effects.

The weak localization magnetoconductance at a fixed temperature. The different curves represent different relative contributions from spin effects.

The weak localization magnetoconductance in a sample where spin effects are negligible, at several temperatures .

The weak localization magnetoconductance in a sample where spin effects are negligible, at several temperatures .

A four-wire arrangement. No current flows through the contact resistances, so when we measure a voltage, it is due to the sample only.

A four-wire arrangement. No current flows through the contact resistances, so when we measure a voltage, it is due to the sample only.

An example of how a ground loop may develop in a four-wire measurement. The lock-in inputs we use are floated to prevent this condition.

An example of how a ground loop may develop in a four-wire measurement. The lock-in inputs we use are floated to prevent this condition.

The setup used to measure weak localization. It is a four-wire resistance bridge that utilizes a lock-in, a preamplifier, and a decade transformer to resolve the magnetoresistance, (1) decade transformer, (2) resistor, (3a) sample resistance, (3b) contact resistances, (4) SR60 preamplifier, (5) SR830 lock-in amplifier, (6) internal lock-in reference, (7) lock-in inputs, (8) 10 V proportional output, (9) oscilloscope.

The setup used to measure weak localization. It is a four-wire resistance bridge that utilizes a lock-in, a preamplifier, and a decade transformer to resolve the magnetoresistance, (1) decade transformer, (2) resistor, (3a) sample resistance, (3b) contact resistances, (4) SR60 preamplifier, (5) SR830 lock-in amplifier, (6) internal lock-in reference, (7) lock-in inputs, (8) 10 V proportional output, (9) oscilloscope.

Typical data for the magnetoresistance versus field. The field scale is a log scale. Weak localization, with spin effects, is evident from 2 K to about 14 K, as a positive magnetoresistance at low fields and a negative magnetoresistance at larger fields. The data at temperatures above 2 K is offset for clarity; the magnetoresistance goes to 0 when . The lines are fits to Eq. (16).

Typical data for the magnetoresistance versus field. The field scale is a log scale. Weak localization, with spin effects, is evident from 2 K to about 14 K, as a positive magnetoresistance at low fields and a negative magnetoresistance at larger fields. The data at temperatures above 2 K is offset for clarity; the magnetoresistance goes to 0 when . The lines are fits to Eq. (16).

Square of the coherence length versus temperature. The linear portion above 14 K has a slope between and . If two-dimensional thermal phonons dominate inelastic scattering, the slope should be 2.

Square of the coherence length versus temperature. The linear portion above 14 K has a slope between and . If two-dimensional thermal phonons dominate inelastic scattering, the slope should be 2.

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