Magnetotunneling spectroscopy of chiral two-dimensional electron systems

We present a theoretical study of momentum-resolved tunneling between parallel two-dimensional conductors whose charge carriers have a (pseudo)spin-1/2 degree of freedom that is strongly coupled to their linear orbital momentum. Specific examples are single- and bilayer graphene as well as single-layer molybdenum disulfide. Resonant behavior of the differential tunneling conductance exhibited as a function of an in-plane magnetic field and bias voltage is found to be strongly affected by the (pseudo)spin structure of the tunneling matrix. We discuss ramifications for the direct measurement of electronic properties such as Fermi surfaces and dispersion curves. Furthermore, using a graphene double-layer structure as an example, we show how magnetotunneling transport can be used to measure the pseudospin structure of tunneling matrix elements, thus enabling electronic characterization of the barrier material.

Figure 1

Schematics of the vertical tunneling structure considered in this work. Two parallel chiral two-dimensional electron systems are separated by a uniform barrier. A magnetic field applied parallel to the barrier is used to tune resonances in the tunneling conductance that arise from the requirement of simultaneous energy and momentum conservation.

Figure 2

Visualization of constraints due to combined energy and momentum conservation for chiral electrons. An applied in-plane magnetic field results in a shift by $\mathrm{Q}$ of the Fermi circles associated with the vertically separated 2D conductors. In the zero-bias limit, tunneling can occur only for states at intersection points of the Fermi surfaces. For one of the latter, the kinetic wave vectors and $\mathrm{K}$-valley pseudospin states are indicated for the case of tunneling between (a) two $n$-doped single-layer graphene sheets, (b) a $p$-doped and an $n$-doped graphene layer, and (c) two $n$-doped bilayer graphene sheets.

Figure 3

Linear magnetotunneling conductances between parallel ordinary 2D electron systems (blue solid curves), single-layer graphene sheets (green dashed curves), and bilayer-graphene sheets (red dotted curves) through a pseudospin-conserving barrier. $Q=d/{\ell }_{B_{\parallel }}^2$ is the wave-vector boost induced by a magnetic field of magnitude $B_{\parallel }$ parallel to the two 2D systems, with $d$ denoting the latter's vertical separation and ${\ell }_{B_{\parallel }}=\sqrt{\hslash /|e{B}_{\parallel }|}$ the magnetic length. (a) [(b)] shows results for the case when tunneling occurs between two $n$-type layers (between an $n$-type and a $p$-type layer) with equal densities. The pseudospin structure of chiral electron states in single-layer and bilayer graphene is the origin of the strongly modified magnetic-field dependences of the tunneling conductance as compared with the ordinary 2D-electron case.

Figure 4

Magnetotunneling between parallel single-layer graphene sheets at finite bias. (a) The two systems' Dirac-cone dispersions are shifted with respect to each other by $Q$ ($eV$) in the wave-vector (energy) direction due to an applied in-plane magnetic field (bias voltage). Tunneling is possible for states where the conical surfaces intersect, if the state is occupied in one layer and unoccupied in the other. (b) Differential tunneling conductance $G\left(V\right)$ for the case when both layers have equal $n$-type carrier density, plotted as a function of $\tilde{Q}=Q-eV/\left(\hslash v\right)$ for $eV=0$ (black solid curve), $0.2\hslash v{\overline{k}}_{\text{F}}$ (green dot-dashed curve), $0.4\hslash v{\overline{k}}_{\text{F}}$ (red dotted curve), and $0.6\hslash v{\overline{k}}_{\text{F}}$ (blue dashed curve). $G\left(V\right)$ diverges at $\tilde{Q}=0$ and exhibits square-root-like features at $\tilde{Q}=2{\overline{k}}_{\text{F}}-2eV/\left(\hslash v\right)$ and $\tilde{Q}=2{\overline{k}}_{\text{F}}$. (c) Logarithmic gray-scale plot of the differential tunneling conductance. The divergence at $eV=\hslash vQ$ and the conical feature with apex at $Q=2{\overline{k}}_{\text{F}}$ constitute direct measures for the energy dispersion of charge carriers in single-layer graphene.

Figure 5

Form factors for tunneling between graphene layers spaced at distance $d$ in a tilted magnetic field $\mathrm{B}=({\mathrm{B}}_{\parallel },B_{\perp })$, plotted as a function of the parameter ${\xi }_d=(d/{\ell }_{B_{\perp }})(B_{\parallel }/B_{\perp })$. See Eq. (24). (a) [(b)] shows ${\mathcal{F}}_{\nu }^{(+)}\left({\xi }_d\right)$ (blue solid curve), ${\mathcal{F}}_{\nu }^{(-)}\left({\xi }_d\right)$ (green dashed curve), and ${\mathcal{F}}_{\nu }^{(\perp )}\left({\xi }_d\right)$ (red dotted curve) for $\nu =1$ ($\nu =6$). Note the limiting behavior for ${\xi }_d\to 0$ and the oscillatory behavior for cases with $\nu >1$.