Phonon-Assisted Resonant Tunneling of Electrons in Graphene–Boron Nitride Transistors

We observe a series of sharp resonant features in the differential conductance of graphene-hexagonal boron nitride-graphene tunnel transistors over a wide range of bias voltages between 10 and 200 mV. We attribute them to electron tunneling assisted by the emission of phonons of well-defined energy. The bias voltages at which they occur are insensitive to the applied gate voltage and hence independent of the carrier densities in the graphene electrodes, so plasmonic effects can be ruled out. The phonon energies corresponding to the resonances are compared with the lattice dispersion curves of graphene–boron nitride heterostructures and are close to peaks in the single phonon density of states.

Figure 1

Differential conductance of Device 1 at $T=4\mathrm{K}$. (a) $G(V_b)$ for $V_g=-30\mathrm{V}$ (red) to 30 V (blue) and intervals $\mathrm{\Delta }{V}_g=2\mathrm{V}$. Inset: Schematic band diagram of Device 1 with bias $V_b$ and gate $V_g$ voltages applied to the monolayer graphene electrodes, which are separated by an hBN barrier of thickness $d=0.9\mathrm{nm}$. Here, ${\mu }_{b,t}$ are the chemical potentials of the bottom and top electrodes and $F$ is the electric field across the barrier. A phonon-assisted tunnel process is shown schematically. (b) Color map of $dI/d{V}_b$ for a range of $V_g$ and $V_b$.

Figure 2

(a) Measured plots $\alpha dG/d{V}_b$ for Device 1 and (b) calculated gray-scale map of $\alpha dG/d{V}_b$ for $T=4\mathrm{K}$, best fit to data in Fig. 2 using the model described in the text and in Table 1. Yellow dashed curves show where the chemical potential in a graphene layer intersects with the Dirac point in that layer.

Figure 3

(a) The three top curves (blue): $dG/d{V}_b$ averaged over all gate voltages for our model (dashed) and measured data for Devices 1 and 2 (both solid). Lower curves: Total density of phonon states for monolayer graphene (red, monolayer Gr) and a graphene-hBN heterostructure [green, Gr-hBN (total)]. Lower three curves: The partial density of states associated with the carbon (red, C), boron (blue, B), and nitrogen (black, N) atoms of a graphene-hBN heterostructure. Solid and dashed curves show contributions by in- and out-of-plane phonons, respectively. (b) Phonon dispersion of a graphene-hBN heterostructure [17]. Vertical dotted curves are guides to the eye, highlighting the alignments.

Figure 4

Temperature dependence of $|d^{2}I/d{V}_b^2|$ vs $V_b$ in Device 2 measured from $T=2.3\mathrm{K}$ to $T=180\mathrm{K}$ (blue to red) for $V_g=40\mathrm{V}$. Inset: Peak amplitude vs $T$. Curve colors match peaks marked by correspondingly colored arrows in the main plot.