Low-bias negative differential resistance in graphene nanoribbon superlattices

We theoretically investigate negative differential resistance (NDR) for ballistic transport in semiconducting armchair graphene nanoribbon (aGNR) superlattices (5 to 20 barriers) at low bias voltages $V_{\mathrm{SD}}<500$ mV. We combine the graphene Dirac Hamiltonian with the Landauer-Büttiker formalism to calculate the current $I_{\mathrm{SD}}$ through the system. We find three distinct transport regimes in which NDR occurs: (i) a “classical” regime for wide layers, through which the transport across band gaps is strongly suppressed, leading to alternating regions of nearly unity and zero transmission probabilities as a function of $V_{\mathrm{SD}}$ due to crossing of band gaps from different layers; (ii) a quantum regime dominated by superlattice miniband conduction, with current suppression arising from the misalignment of miniband states with increasing $V_{\mathrm{SD}}$; and (iii) a Wannier-Stark ladder regime with current peaks occurring at the crossings of Wannier-Stark rungs from distinct ladders. We observe NDR at voltage biases as low as 10 mV with a high current density, making the aGNR superlattices attractive for device applications.

Figure 1

(a) Metal-aGNR junctions and the modulated chemical shift $\Delta {\varepsilon }_{\mathrm{F}}$ of the Dirac point across the aGNR (Refs.31, 32, 33, 34). ${\Delta }_0$ (shaded regions) denotes the barrier and valley band gaps. Here we consider square potentials, solid line. The dashed line shows the numerical results of Ref.33. (b) Additional electrodes modulate the Dirac cone shift into a superlattice potential. The bias voltage $V_{\mathrm{SD}}$ is also shown. (c) Doped layers of a semiconductor superlattice can also modulate the local potential. (d) Schematic of the $\varepsilon -V_{\mathrm{SD}}$ diagram of the source-drain transmission coefficient showing crossings of the band gaps ${\Delta }_0$ (black lines). The shaded regions delimit the energy range between the source ${\mu }_{\mathrm{S}}=\Delta {\varepsilon }_{\mathrm{F}}$ and drain ${\mu }_{\mathrm{D}}={\mu }_{\mathrm{S}}-e{V}_{\mathrm{SD}}$ chemical potentials.

Figure 2

(a) Energy-voltage diagram of $T_{\mathrm{SD}}$ for $N=5$ barriers showing the evolution of the $N-1$ hybridized modes [panels (b)–(d)] into Wannier-Stark ladders. Labels A, B, and C show the zero-bias hybridized modes in panels (a) and (d). Crossings of ladders’ rungs from distinct minibands increases $T_{\mathrm{SD}}$ near $V_{\mathrm{SD}}=30$ and 50 mV. (b) Schematic of the modulated Dirac point (dashed line), band gaps ${\Delta }_0\sim 28$ meV (gray area), and confined mode $B^{\prime }$. In the transmission coefficient $T_{\mathrm{SD}}$ across two barriers ($a=b=50$ nm) (c), the confined mode $B^{\prime }$ shows up as a resonant spike near 230 meV. For (d) $N=5$, and (e) $N=20$ barriers the confined modes hybridize into $N-1$ spikes, building up a miniband. Similar resonances lead to minibands at energies away from the band-gap region ${\Delta }_0$.

Figure 3

Current and energy-voltage diagram of the transmission coefficient for $\mathrm{five}$-barrier superlattice with $a=b=100$ nm [(a) and (b)] and $a=b=50$ nm [(c) and (d)]. The current-voltage characteristics are shown for $T=300$ K and 0 K. For wide barriers (a) and (b) the current follows closely the limiting “classical” case of $T_{\mathrm{SD}}$ either 0 across band gaps, or 1 otherwise (dashed line).

Figure 4

(a) Current-voltage characteristics and (b) $T_{\mathrm{SD}}$ diagram of a 20-barrier aGNR superlattice with $a=b=50$ nm. In (a) the currents for 0 and 300 K in the range $0\le V_{\mathrm{SD}}\le 125$ mV are multiplied by 6 and 10, respectively, for clarity. As the voltage increases the miniband near 230 meV, Fig. 2e, breaks up as the resonant levels misalign, leading to the pronounced spike near 10 mV for 0 K. Near 50 mV the resonant levels return as resonant crossings of Wannier-Stark ladder rungs [see also Fig. 2a]. At the crossings $T_{\mathrm{SD}}$ increases, showing current spikes at both 0 and 300 K for $V_{\mathrm{SD}}<230$ mV. For $V_{\mathrm{SD}}>230$ mV the current spikes arise from crossings of rungs at the coincidence region.