Confinement-Induced Berry Phase and Helicity-Dependent Photocurrents

The photocurrent in an optically active metal is known to contain a component that switches sign with the helicity of the incident radiation. At low frequencies, this current depends on the orbital Berry phase of the Bloch electrons via the “anomalous velocity” of Karplus and Luttinger. We consider quantum wells in which the parent material, such as GaAs, is not optically active and the relevant Berry phase only arises as a result of quantum confinement. Using an envelope approximation that is supported by numerical tight-binding results, it is shown that the Berry-phase contribution is determined for realistic wells by a cubic Berry phase intrinsic to the bulk material, the well width, and the well direction. These results for the Berry-phase effect suggest that it may already have been observed in quantum well experiments.

Figure 1

Snapshot of electron distribution in a circularly polarized wave. The electron distribution is pushed away from the ground state (solid circle) by an amount $\delta k=eE\tau /\hslash$ that is small relative to the Fermi wave vector $k_F$ for accessible fields. Here we show the component of $d\mathbf{k}/dt$ that dominates the photocurrent for small $\omega \tau$, which switches with the sense of circular polarization. At the time shown, the conventional velocity ${\mathbf{v}}_0$ and anomalous velocity ${\mathbf{v}}_1$ are in opposite directions. After a half-period, the conventional velocity changes sign but the anomalous velocity does not. The directions assume $\mathbf{\Omega }=\beta k_x\widehat{\mathbf{z}}$ with $\beta >0$.

Figure 2

Berry phase in (110) quantum wells. (a) Positions of atoms (Ga gold, As blue) within (110) planes of GaAs showing inversion asymmetry. Shadowed atoms are in the next plane above or below. (b) Strength of Berry phase factor $\beta =\partial \Omega /\partial k$ for (110) quantum wells: square wells of depth 0.25 eV and 0.5 eV, Gaussian well of depth 0.25 eV, and triangular well of depth 0.5 eV. The wave function width $2\sqrt{⟨{\tilde{x}}^2⟩}$ (horizontal axis), $\tilde{x}=(x+y)/\sqrt{2}$ changes with well potential and determines $\beta$ almost independently of well shape.