Current modulation by voltage control of the quantum phase in crossed graphene nanoribbons

A relative rotation of 90${}^{\circ }$ between two graphene nanoribbons (GNRs) creates a crossbar with a nanoscale overlap region. Calculations, based on the first principles density functional theory (DFT) and the nonequilibrium Green's function (NEGF) formalism, show that the electronic states of the individual GNRs of an unbiased crossbar are decoupled from each other similar to the decoupling that occurs in twisted bilayer graphene. Analytical calculations, based on Fermi's golden rule, reveal that the decoupling is a consequence of the cancellation of quantum phases of the electronic wave functions of the individual GNRs. As a result, the inter-GNR transmission is strongly suppressed over a large energy window. An external bias applied between the GNRs changes the relative phases of the wave functions resulting in modulation of the transmission and current by several orders of magnitude. A built-in potential between the two GNRs can lead to a large peak-to-valley current ratio $(>1000)$ resulting from the strong electronic decoupling of the two GNRs that occurs when they are driven to the same potential. Current switching by voltage control of the quantum phase in a graphene crossbar structure is a novel switching mechanism. It is robust even with an overlap of $\sim$$1.8\text{nm}\times 1.8\text{nm}$ that is well below the smallest horizontal length scale envisioned in the international technology roadmap for semiconductors (ITRS).

Figure 1

Atomistic geometry of the crossbar GNR (xGNR) consisting of two H-passivated armchair GNRs with one placed on top of the other rotated by ${90}^{\circ }$. Each GNR is 14-C atomic layers wide ($\sim$$1.8$ nm) with a band gap of 130 meV. The contacts are modeled by the self-energies of semi-infinite leads. The region bounded by the broken lines is used as a supercell for the band structure calculations.

Figure 2

Band structure of the xGNR supercell calculated using fireball (a) as a function of $k_x$ and $k_y$ and (b) as a function of $k_x$ only, at $k_y=0$. The energy $E=0$ eV is set at the Fermi level. The bands shown in (a) appear as a superposition of bands of two isolated GNRs with one aligned in the $x$ and the other in $y$ direction. The bands indicated by 1 and 2 in (b) are degenerate at $\Gamma$ indicating that they are decoupled.

Figure 3

Three-dimensional isosurface of the eigenstate corresponding to (a) band 1 and (b) band 2 in Fig. 2b at $\Gamma$. The eigenstate of band 1 is localized on the top GNR, and the eigenstate of band 2 is localized on the bottom GNR.

Figure 4

Current voltage (I-V) characteristic of the intrinsic xGNR.

Figure 5

Transmission as a function of energy for different bias voltages. The energy $E=0$ eV is set at the equilibrium Fermi level. The vertical lines represent the chemical potentials of the top and the bottom contacts. The dips in the transmission near the vertical lines correspond to energies lying inside the band gap of either the top or the bottom GNR. The transmission does not go to zero at these energies as a result of the finite energy broadening used to calculate the surface self-energies of the contacts.

Figure 6

Simulated I-V characteristics of xGNR p-n junctions with different built-in potentials. Inset: Transmission of the xGNR p-n junction for ${\phi }_{\mathrm{bi}}=0.25$ V as a function of energy for different bias voltages. The energy $E=0$ eV is set at the equilibrium Fermi level. The vertical lines represent the chemical potentials of the top and bottom contacts.

Figure 7

Transmission as a function of energy for different bias voltages calculated using Eq. (5). The energy $E=0$ eV is set at the equilibrium Fermi level. The dashed vertical lines represent the chemical potentials of the top and the bottom contacts. The gaps in the transmission near the chemical potentials correspond to energies lying inside the band gap of either the top or the bottom GNR. Since the analytical calculations include no energy broadening, the transmission is zero at those energies.

Figure 8

Magnitude squared of the matrix element and its four components considering only the fundamental modes. The total matrix element squared and its components are indicated according to the legend in (a). (a) $V=0$ V. For $E\gtrsim 0$ eV, $m=10$ and $n=10$ and for $E\lesssim 0$ eV, $m=-10$ and $n=-10$. (b) $V=0.25$ V. For $-0.06<E<0.06$ eV, $m=-10$ and $n=10$; for $E<-0.19$ eV, $m=-10$ and $n=-10$; and for $E>0.19$ eV, $m=10$ and $n=10$. The AB and BA components are very small at low energies near $E=0$ compared to the AA and BB components in both cases. The vertical lines represent the chemical potentials of the contacts.

Figure 9

Phasor plots of $M_{mn}^{\alpha \beta }$ (a,b), $H_{mn}^{\alpha \beta }$ (c,d), and $C_{mn}^{\alpha \beta }$ (e,f). The AA, BB, AB, and BA components are indicated according to the legend at the top of the figure. (a), (c), and (e) $V=0$ V and $E=0.064$ eV (the conduction band edge). (b), (d), and (f) $V=0.25$ V and $E=0$ eV. The lengths and the directions of the arrows represent the magnitude and the angle of the corresponding complex quantities, respectively. Since the AB and BA components of $H$ and $M$ are very small, they are magnified several orders of magnitude and shown in the insets.