Materials Research Science and Engineering Center
Fluctuating Spin g-tensor in Small Metal Grains

The upper part of the figure shows the computed electron wave flow over the potential shown in the lower part (high potential is white, low is green).

The ground state of a small metal particle with an odd number of electrons is required by quantum mechanics to be doubly degenerate (Kramer's degeneracy). Application of a weak magnetic field B splits the degeneracy linearly, by an amount g B, which is observable in experiments that measure the addition spectrum for electrons tunneling onto the particle. For a material such as gold, with strong spin-orbit scattering, the value of g may be reduced considerably relative to the value which holds for free electrons, and may vary in a chaotic way from one energy level to another in a particle. (This may be contrasted with the value which is observed in spin-resonance experiments in Au films.)

The value of g depends on the direction of the applied magnetic field, and may be written as where b is a unit vector in the direction of B, and G is a symmetric tensor characteristic of the energy level. The three eigenvalues of G are nonnegative, and may be denoted where are the "principal g-values" of the energy level. We show that with no loss of generality, any two of these value may be chosen to be positive; however, the sign of the third is then determinable, at least in principle, by a spin resonance experiment, and can be positive or negative. We have calculated analytically the joint probability distribution for the three principal values, in the case of strong spin-orbit scattering, and have obtained numerical results for typical g-values and for the mean g-value, for intermediate values of the spin-orbit scattering.

P. W. Brouwer, X. Waintal, and B. I. Halperin, Phys. Rev. Letters 85, 369 (2000).