A computational study of short-channel effects in double-gate junctionless graphene nanoribbon field-effect transistors

Abstract

As the channel length shrinks below the 10-nm regime, emerging materials, junctionless technology, and multiple-gate geometries provide an excellent combination to continue progress towards lower-cost high-performance ultrascaled devices. In this study, the double-gate junctionless (JL) graphene nanoribbon field-effect transistor (GNRFET) and its conventional counterpart (C-GNRFET) are compared in terms of short-channel effects (SCEs) using a quantum simulation. The computational approach is based on solving the Schrödinger equation using the mode-space nonequilibrium Green’s function formalism coupled self-consistently with a Poisson equation in the ballistic limit. The analysis of gate length downscaling shows that the JL GNRFET exhibits better leakage current, subthreshold swing (SS), drain-induced barrier lowering, and threshold voltage roll-off in comparison with the conventional GNRFET. In addition, we reveal that a decrease in the n-type doping concentration can enhance the above-mentioned characteristics of both devices. The results indicate that the JL GNRFET can mitigate critical issues and enhance the immunity to SCEs of the GNRFET, making it a promising candidate for high-performance ultrascaled (sub-5-nm) technology.

Appendix

The appendix of this paper is devoted to presenting the main equations of the quantum simulation described herein to solve the MS NEGF–Poisson system self-consistently. More details regarding the computational approach applied can be found in Refs. [41,42,43,44]. The NEGF-based atomistic simulation is principally based on the numerical computation of the famous retarded Green’s function, which can be determined by [27]

$$G\left( E \right) = \left[ {\left( {E + \mathrm{i}0^{ + } } \right)I - H - \varSigma_{\text{S}} - \varSigma_{\text{D}} } \right]^{ - 1},$$

(1)

where I, E, H, and Σ_S(D) are the identity matrix, energy, Hamiltonian matrix (H is based on the atomistic nearest neighbor p_z-orbital TB estimation [22]), and the source (drain) self-energy, respectively. Note that 0⁺ is an infinitesimal value, positively fixed. It is worth indicating that the Hamiltonian has a size of (M × M), where M is the number of columns (lines) of carbon atoms in the x direction [22, 25]. Note also that the self-energy matrices have the same size as the Hamiltonian and only the (1, 1) element of Σ_S and the (M, M) element of Σ_D are nonzero [26]. As seen in Eq. (1), ballistic transport is normally assumed. It is worth indicating that the self-energies can be computed analytically [22]. The local density of states (LDOS) resulting from the source (drain) injected states is computed using [22, 27]

$$D_{{{\text{S}}({\text{D}})}} = G\varGamma_{{{\text{S}}({\text{D}})}} G^{\dag }.$$

(2)

Note that the energy level broadening Γ_S(D) due to the source (drain) contact is given by [22, 27]

$$\varGamma_{{{\text{S}}({\text{D}})}} = i(\varSigma_{{{\text{S}}({\text{D}})}} - \varSigma_{{{\text{S}}({\text{D}})}}^{\dag } ).$$

(3)

Based on the source/drain LDOS, the charge density is then computed as [22]

$$\begin{aligned} Q(x) & = ( - q)\int_{ - \infty }^{ + \infty } {{\text{d}}E \cdot \text{sgn} \left[ {E - E_{\text{N}} (x)} \right]} \\ & \quad \times \left\{ {D_{\text{S}} (E,x)} \right.f(\text{sgn} \left[ {E - E_{\text{N}} (x)} \right](E - E_{\text{FS}} )) \\ & \quad \quad \left. { +\, D_{\text{D}} (E,x)f(\text{sgn} \left[ {E - E_{\text{N}} (x)} \right](E - E_{\text{FD}} ))} \right\}, \\ \end{aligned}$$

(4)

where q, sgn, f, E_FS(FD), and E_N(x) are the electron charge, sign function, Fermi function, source (drain) Fermi level, and charge neutrality level, respectively. Note that the charge neutrality level E_N(x) lies at the middle of the bandgap because the conduction and valence band of an armchair-edge graphene nanoribbon are symmetric [38]. In the Poisson solver, the self-consistent electrostatic potential is calculated from the electrode potentials and the GNR charge density using the Poisson equation [26, 37]

$$\nabla^{2} U = \frac{ - q}{\varepsilon }\rho,$$

(5)

where U, ρ, and ε are the electrostatic potential within the GNRFET, the distribution of the charge density including the doping density [22, 25, 28], and the dielectric constant, respectively. Note that the discrete nature of the dopants is not taken into account and could be a subject for further investigations. The finite difference method (FDM) is used to solve Eq. (5), where the grid spacing used (in the 2D FDM discretization) is taken to be equal to 1 Å in both the x- and y-direction [25, 41]. After the convergence of the self-consistent system, the source–drain current is calculated as [22]

$$I = \frac{2q}{\hbar }\int\limits_{ - \infty }^{ + \infty } {{\text{d}}E\;T(E)[f(E - E_{\text{FS}} ) - f(E - E_{\text{FD}} )]},$$

(6)

where ħ, q, and T(E) = Tr[Γ_SGΓ_DG^†] are the Planck constant, electron charge, and transmission coefficient, respectively.