Valley-dependent optoelectronics from inversion symmetry breaking

Inversion symmetry breaking allows contrasted circular dichroism in different $k$-space regions, which takes the extreme form of optical selection rules for interband transitions at high symmetry points. In materials where band edges occur at noncentral valleys, this enables valley-dependent interplay of electrons with light of different circular polarizations, in analogy to spin dependent optical activities in semiconductors. This discovery is in perfect harmony with the previous finding of valley contrasted Bloch band features of orbital magnetic moment and Berry curvatures from inversion symmetry breaking [D. Xiao, W. Yao, and Q. Niu, Phys. Rev. Lett. 99, 236809 (2007)]. A universal connection is revealed between the $k$-resolved optical oscillator strength of interband transitions, the orbital magnetic moment and the Berry curvatures, which also provides a principle for optical measurement of orbital magnetization and intrinsic anomalous Hall conductivity in ferromagnetic systems. The general physics is demonstrated in graphene where inversion symmetry breaking leads to valley contrasted optical selection rule for interband transitions. We discuss graphene based valley optoelectronics applications where light polarization information can be interconverted with electronic information.

Figure 1

(Color online) (a) Left (right): phase winding of the conduction (valance) band Bloch function at ${\mathit{K}}_1=-\frac{4\pi }{3a}\hat{\mathit{x}}$, showing the intercellular current circulations. (b) Valley optical selection rules: $\sigma +$ $\left(\sigma -\right)$ circularly polarized light couples only to bandedge transitions in valley ${\text{K}}_2$ $\left({\text{K}}_1\right)$.

Figure 2

(Color online) Optical properties of interband transitions in biased graphene bilayer. Energy dispersions are given in (a). $k$-resolved interband oscillator strength averaged over polarization is shown in (b). The degrees of circular polarization for the interband transitions are shown in (c). Note the range of horizontal axis corresponds to valley ${\text{K}}_2$. The values in valley ${\text{K}}_1$ can be obtained by noting that the oscillator strength is even function while the degrees of circular polarization is odd function of $\mathit{k}$. Different line style and color are used for transitions between different pairs of bands as indicated by arrowed lines in (a). The parameters used are $t=2.82\text{eV}$, $\Delta =0.3\text{eV}$, and $t_{\perp }=0.4\text{eV}$.

Figure 3

(Color online) Schematic device geometry of optoelectronics based on valley contrasting optical properties of graphene. The electrons (holes) in valley ${\text{K}}_2$ are denoted by white ‘$-$’ (‘$+$’) symbol in dark circles and their counterparts in valley ${\text{K}}_1$ are denoted by inverse color. (a) Photoinduced anomalous Hall effect. (b) Valley light emitting diode. See text for the explanation of operation mechanisms.