Berry phase effects on electronic properties

Ever since its discovery the notion of Berry phase has permeated through all branches of physics. Over the past three decades it was gradually realized that the Berry phase of the electronic wave function can have a profound effect on material properties and is responsible for a spectrum of phenomena, such as polarization, orbital magnetism, various (quantum, anomalous, or spin) Hall effects, and quantum charge pumping. This progress is summarized in a pedagogical manner in this review. A brief summary of necessary background is given and a detailed discussion of the Berry phase effect in a variety of solid-state applications. A common thread of the review is the semiclassical formulation of electron dynamics, which is a versatile tool in the study of electron dynamics in the presence of electromagnetic fields and more general perturbations. Finally, a requantization method is demonstrated that converts a semiclassical theory to an effective quantum theory. It is clear that the Berry phase should be added as an essential ingredient to our understanding of basic material properties.

Figure 1

Brillouin zone as a torus. (a) A torus with its surface parametrized by $(q,t)$. The control parameter $\mathit{R}\left(t\right)$ is periodic in $t$. (b) The equivalence of a torus: a rectangle with periodic boundary conditions: $\overline{AB}=\overline{DC}$ and $\overline{AD}=\overline{BC}$. To make use of Stokes’s theorem, we relax the boundary condition and allow the wave functions on parallel sides to have different phases.

Figure 2

(Color online) Polarization as a function of $\Delta$ and $\delta$ in the Rice-Mele model. The unit is $ea$ with $a$ the lattice constant. The line of discontinuity can be chosen anywhere depending on the particular phase choice of the eigenstate.

Figure 3

(Color online) Fermi surface in (010) plane (solid lines) and the integrated Berry curvature $-{\Omega }_z\left(\mathit{k}\right)$ in atomic units (color map) of fcc Fe. From 245.

Figure 4

(Color online) Energy bands (top panel) and Berry curvature of the conduction band (bottom panel) of a graphene sheet with broken inversion symmetry. The first Brillouin zone is outlined by the dashed lines, and two inequivalent valleys are labeled as $K_1$ and $K_2$. Details are presented in 240.

Figure 5

Magnetic flux going through the holes of the torus.

Figure 6

Anomalous Hall effect in a simple two-band model. (a) Energy dispersion of spin-split bands. (b) The Hall conductivity $-{\sigma }_{xy}$ in the units of $e^2∕h$ as a function of Fermi energy.

Figure 7

(Color online) ${\sigma }_{ah}$ vs ${\sigma }_{xx}{\left(T\right)}^2$ in Fe thin films with different film thickness over the temperature range of $5–300\mathrm{K}$. From 217.

Figure 8

Electron energy $\tilde{\epsilon }$ in a slowly varying confining potential $V\left(\mathit{r}\right)$. In addition to the self-rotation, wave packets near the boundary will also move along the boundary due to the potential $V$. Level spacings between different bulk $\mathit{q}$ states are exaggerated; they are continuous in the semiclassical limit. The inset shows directions of the Berry curvature, the effective force, and the current carried by a wave packet on the left boundary.

Figure 9

Sampling function and a wave packet at ${\mathit{r}}_c$. The width $L$ of the sampling function is sufficiently small so that it can be treated as a $\delta$ function at the macroscopic level and is sufficiently large so that it contains a large number of wave packets of width $l$ inside its range. Equation (4.15) is indeed a microscopic average over the distance $L$ around the point $\mathit{r}$. See Sec. 6.6 in 84 for an analogy in macroscopic electromagnetism.

Figure 10

The wave-packet description of a charge carrier whose center is $({\mathit{r}}_c,{\mathit{q}}_c)$. A wave packet generally possesses two kinds of motion: the center-of-mass motion and the self-rotation around its center. From 241.

Figure 11

(Color online) Electron density of the Rice-Mele model with a spatial varying dimerization parameter. The parameters used are $\Delta =0.5$, $t=2$, and $\delta =\tanh \left(0.02x\right)$. Inset: The profile of $\delta \left(x\right)$. From 236.

Figure 12

(Color online) When the magnetic flux per plaquette is ${\varphi }_0∕3$, the magnetic unit cell is composed of three plaquettes. The magnetic Brillouin zone is three times smaller than the usual Brillouin zone. Furthermore, the magnetic Bloch states are threefold degenerate.

Figure 13

(Color online) Reduced magnetic Brillouin zone. (a) The phases of the MBS in the reduced MBZ can be assigned using the parallel transport conditions, first along the $k_1$ axis, then along the paths parallel to the $k_2$ axis. (b) Hyperorbits in a reduced MBZ. Their sizes are quantized following the Bohr-Sommerfeld quantization condition. The orbit enclosing the largest area is indicated by solid lines.

Figure 14

A parent magnetic Bloch band at magnetic field $B_r$ splits to $D_r$ subbands ($D_r=5$ here) due to a perturbation $\delta B_{r+1}$. The subbands near the band edges of the parent band are usually originated from closed hyperorbits. The subband in the middle is from an open hyperorbit.

Figure 15

(Color online) Schematic of the semiconductor band structure near the fundamental gap. The wave packet in the conduction band is formed from a two-component spinor.

Figure 16

The extent of wave-packet space, parent space, and full space.