Valley filtering in strain-induced $\alpha \text{−}{\mathcal{T}}_3$ quantum dots

We test the valley-filtering capabilities of a quantum dot inscribed by locally straining an $\alpha \text{−}{\mathcal{T}}_3$ lattice. Specifically, we consider an out-of-plane Gaussian bump in the center of a four-terminal configuration and calculate the generated pseudomagnetic field having an opposite direction for electrons originating from different valleys, the resulting valley-polarized currents, and the conductance between the injector and collector situated opposite one another. Depending on the quantum dot's width and width-to-height ratio, we detect different transport regimes with and without valley filtering for both the $\alpha \text{−}{\mathcal{T}}_3$ and dice lattice structures. In addition, we analyze the essence of the conductance resonances with a high valley polarization in terms of related (pseudo-) Landau levels, the spatial distribution of the local density of states, and the local current densities. The observed local charge and current density patterns reflect the local inversion symmetry breaking by the strain, besides the global inversion symmetry breaking due to the scaling parameter $\alpha$. By this way we can also filter out different sublattices.

Figure 1

(a) Pristine $\alpha \text{−}{\mathcal{T}}_3$ lattice with basis $\{A,B,C\}$ and Bravais lattice vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$. Neighboring sites are connected by vectors ${\mathit{\delta }}_{A,j}$ ($j=1,2,3$); the transfer amplitudes on $A\text{−}B$ and $B\text{−}C$ bonds are $t$ and $\alpha t$, respectively. (b) Four-terminal configuration with a quantum dot generated by the Gaussian deformation (16), where $H=17.9\mathrm{nm}, \sigma =20\mathrm{nm}$, and $W=50\mathrm{nm}$. (c) Strain-induced pseudomagnetic field calculated for electrons residing in the $\mathbf{K}$ valley. These electrons can pass the quantum dot from L to R whereas electrons stemming from the ${\mathbf{K}}^{\prime }$ valley will be reflected.

Figure 2

Valley polarization by a strain-induced graphene-based quantum dot. The contour plot shows ${\tau }^{[\mathbf{K}]}$, depending on $H$ and $\sigma$, for the four-terminal configuration in Fig. 1. The Fermi energy of the injected particles is $E_{\text{F}}\simeq 0.22\mathrm{eV}$.

Figure 3

Valley polarization by a strain-induced $\alpha \text{−}{\mathcal{T}}_3$ quantum dot. The contour plots give ${\tau }^{[\mathbf{K}]}$ as a function of $H$ and $\sigma$ for the four-terminal configurations with $\alpha =1/\sqrt{3}$ (top) and $\alpha =1$ (bottom), where $E_{\text{F}}\simeq 0.22\mathrm{eV}$. We included the linear regime boundaries of Fig. 2 for comparison.

Figure 4

(a), (b) Conductance $G/G_0$ and valley polarization ${\tau }^{\left[\mathbf{K}\right]}$ as a function of $E_{\mathrm{F}}$ for a strain-induced $\alpha \text{−}{\mathcal{T}}_3$ quantum dot with $H=17.9\mathrm{nm}$ and $\sigma =20\mathrm{nm}$. For the marked resonances (1) at $E_{\text{F}}=0.219\mathrm{eV}$ and (2) at $E_{\text{F}}=0.387\mathrm{eV}$ (lattice with $\alpha =1/\sqrt{3}$) we have ${\tau }^{\left[\mathbf{K}\right]}=0.86$ and ${\tau }^{\left[\mathbf{K}\right]}=0.76$, respectively. For resonances ($1^{\prime }$) at $E_{\text{F}}=0.219\mathrm{eV}$ and ($2^{\prime }$) at $E_{\text{F}}=0.344\mathrm{eV}$ (dice lattice, $\alpha =1$) we find ${\tau }^{\left[\mathbf{K}\right]}=0.98$ and ${\tau }^{\left[\mathbf{K}\right]}=0.86$, respectively. (c)–(f) Zoomed-in LDOS at resonances ($1^{(\prime )}$) and ($2^{(\prime )}$). (g)–(j) Corresponding intensity-coded current densities $|J_{ij}|$ (here, the lines and arrows are a guide to the eye.)

Figure 5

Transport properties of a strained $\alpha \text{−}{\mathcal{T}}_3$ quantum dot with $A=17.9\mathrm{nm}, \sigma =20\mathrm{nm}$. (a) Contour plot of the conductance $G$ in the $\alpha \text{−}{E}_{\text{F}}$ plane. Dashed lines mark the lowest pseudo-LLs, according to Eq. (15) with $n=0$, for five equidistant $B_{\text{s}}$ fields ranging from 100 to 200 T and $\gamma =1$ (yellow), $-1$ (magenta). (b) Zoom-in of the contour plot. (c) LDOS at $E=0.335\mathrm{eV}$ and $\alpha =0.95$ [marked in (b) by the red circle].