Application of Berry's phase to the effective mass of Bloch electrons

Abstract

The Berry's phase, although well-known since 1984, has received little attention among textbook authors of solid state physics. We attempt to address this lack by showing how the presence of the Berry's phase significantly changes a standard concept (effective mass) found in most solid state texts. Specifically, we show that the presence of a non-zero Berry curvature in Bloch systems makes the traditional concept of an inverse effective mass tensor M −1 problematic, since a routine application of Newton's 2nd Law leads to a circular definition. As a consequence, the related concept of cyclotron effective mass m * also requires modification. It is shown that m * is magnetic-field dependent in non-inversion symmetric systems. This has important ramifications for cyclotron resonance experiments, since such experiments yield m * and thereby purportedly give the components of M −1 . This work represents a "case study" in how Berry's phase effects can modify "standard" solid-state topics in ways that students and instructors may find surprising.

Berry phase oscillations in a simple model

“…In arriving at this result, we used the fact that for any complex number z a b i = + , where b?a, z a b Im log 2 p » -. From equation (12) we see that the Zak phase exhibits a sort of spatial beat phenomenon, oscillating precisely n times about π as the perturbation sweeps across the unit cell, which is in full agreement with earlier work [16]. If the perturbation is located at the centre of the unit cell (that is, x 1 2 = ˜), the system is inversion symmetric and n 1 2 g p = ( ) for all bands, as expected.…”

“…for all bands, as expected. Figure 2 shows a comparison between numerical results obtained from equation (2) and the perturbative results obtained in equation (12). In the former case, we partitioned the Brillouin zone into D=100 k-values; increasing this number did not appreciably change the numerical results.…”