Bloch-Zener oscillations in graphene and topological insulators

We show that superlattices based on zero-gap semiconductors such as graphene and mercury telluride exhibit characteristic Bloch-Zener oscillations that emerge from the coherent superposition of Bloch oscillations and multiple Zener tunneling between the electron and hole branch. We demonstrate this mechanism by means of wave-packet dynamics in various spatially periodically modulated nanoribbons subject to an external bias field. The associated Bloch frequencies exhibit a peculiar periodic bias dependence, which we explain within a two-band model. Supported by extensive numerical transport calculations, we show that this effect gives rise to distinct current oscillations observable in the $I$-$V$ characteristics of graphene and mercury telluride superlattices.

Figure 1

Exemplary setup for Bloch-Zener oscillations in a graphene nanoribbon. (a) Sketch of a Gaussian wave packet in the presence of a periodic mass potential $M\left(x\right)=M_0\sin (2\pi x/a)$ and an electrostatic drift potential $V\left(x\right)=-e{E}_{D}x$. (b) Band structure of the superlattice with small avoided crossing at $k=0$ (nanoribbon width $W=10a_0$, $a=10\sqrt{3}{a}_0$, $M_0=0.1t$). Thick and dashed lines show the first and second Bloch band from the metallic armchair mode. The gray lines represent higher transversal modes.

Figure 2

Snapshots of a wave packet in the course of Bloch-Zener oscillations. (a) Center-of-mass motion of a wave packet on a graphene nanoribbon ($W=10a_0$, $a=10\sqrt{3}{a}_0$, $M_0=0.1t$). (b) Snapshots of the probability distribution of the wave packet for the corresponding times marked with crosses in panel (a). Please note the video of the dynamics in the online version of the Supplemental Material (Ref. 37).

Figure 3

Frequency spectra $E=\hslash \omega$ from the center-of-mass motion of a wave packet for varying drift field $E_D$ for (a) moderate ($M_0=0.1t$) and (b) stronger ($M_0=0.2t$) periodic potential. Dark colors represent strong intensities. The dashed lines correspond to {1/2, 1, 3/2} times the conventional Bloch frequency.

Figure 4

(a) Band structure of the Dirac model Hamiltonian (6) for $v=1$, $\hslash =1$, $a=1/10$, and $g=1/2$. The shaded (yellow) area denotes the integral (9). (b) Frequency spectrum of the Bloch oscillations for different drift accelerations $\alpha =e{E}_D/\hslash$. Solid lines show the frequencies $n{\omega }^{+}+m{\omega }^{-}$ given by Eq. (16); dotted (dashed) lines show the strong (weak) tunneling limit.

Figure 5

(a) Polarization dependence on the phase difference between the electron and hole branch of the scattering eigenstate ${\chi }^{+}$ (solid line shows the upper spinor entry; dashed line shows the lower spinor entry). (b),(c) Center-of-mass motion of an initially electron-polarized wave packet on a graphene nanoribbon superlattice $M\left(x\right)=M_0+V\left(x\right)$ for different drift fields $E_D$ ($M_0=50\mathrm{meV}$, $V\left(x\right)=300\mathrm{meV}\sin (2\pi x/a)$, $a=10\sqrt{3}{a}_0$). Blue dots depict regions with a negative amplitude, corresponding to a wave packet with strong hole character.

Figure 6

Current-voltage characteristics for graphene nanoribbon superlattices ($L=3000\sqrt{3}{a}_0$, $W=10a_0$, $a=30\sqrt{3}{a}_0$, $V_0=500$ meV) for (a) different Fermi energies ($M_0=20$ meV, $T=20$ K) and (b) different temperatures ($M_0=50$ meV, $E_F=0$) showing pronounced signatures of Bloch-Zener oscillations at higher bias. Arrows mark expected peak positions from phase condition (17). Upper inset: Band structure (for $M_0=20$ meV). Lower inset: Transmission map $T(E,V_{\mathrm{SD}})$ used in Eq. (18) to get the current of panel (b); dark colors represent high transmissions.

Figure 7

Bloch and Bloch-Zener oscillations in spatially modulated two-dimensional HgTe nanoribbons. (a) Band structure for a HgTe nanoribbon with periodically modulated width $W\left(x\right)$ [Eq. (20)] ranging from $W_0=300$ to $W_1=50\mathrm{nm}$ and periodicity $a=200\mathrm{nm}$. (b) Frequency spectrum $E=\hslash \omega$ of the wave-packet center-of-mass motion as a function of drift field $E_D$. Dashed lines indicate the frequencies of the Bloch oscillations. (c) $I$-$V_{\mathrm{SD}}$ characteristics of a nanoribbon with constant width $W=150\mathrm{nm}$ and electrostatic modulation $V\left(x\right)=V_0\sin (2\pi x/a)$. Small vertical arrows mark the expected maxima from phase condition (17). Inset: Corresponding miniband structure.