Effect of contact induced states on minimum conductivity in graphene

Roksana Golizadeh-Mojarad and Supriyo Datta

Phys. Rev. B 79, 085410 – Published 9 February 2009

Abstract

The objective of this paper is to point out that contact induced states can help explain the structure dependence of the minimum conductivity observed experimentally. Contact induced states are similar to the well-known metal induced gap states in metal-semiconductor Schottky junctions, which typically penetrate a few atomic lengths into the semiconductor, while the depth of penetration decreases with increasing band gap. However, in graphene we find that these states penetrate a much longer distance of the order of the width of the contacts. As a result, ballistic graphene samples with a length less than their width at Dirac points can exhibit a length-dependent resistance that is not “Ohmic” in origin but arises from a reduced role of contact induced states. While earlier theoretical works have shown that ballistic graphene samples can exhibit a minimum conductivity, our numerical results demonstrate that this minimum conductivity depends strongly on the structure and configuration of the channel and contacts. In diffusive samples, our results still show that the contact induced states effect needs to be taken into account in explaining minimum conductivity and its dependence on the structure (two terminal vs four terminal) and configuration used.

Received 18 November 2008

DOI:https://doi.org/10.1103/PhysRevB.79.085410

Authors & Affiliations

Roksana Golizadeh-Mojarad and Supriyo Datta

School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47906, USA

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Vol. 79, Iss. 8 — 15 February 2009

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Images

Figure 1
Effect of contact induced states on an armchair graphene channel in a two-terminal geometry. (a) General schematic illustration used for NEGF quantum transport calculation. Local density of state (LDOS) averaged over the width plotted against the length $(x)$ for (b) semiconducting channel with $L / W = 0.5$ , (c) semiconducting channel with $L / W = 2$ , (d) metallic channel with $L / W = 0.5$ , and (e) metallic channel with $L / W = 2$ . Dashed line is for when contact DOS is the same as the channel; ∗-line for contact DOS of $8 \times 10^{13} / (eV {cm}^{2})$ and ●-line for contact DOS of $22 \times 10^{13} / (eV {cm}^{2})$ . Also shown in insets is the quasi-Fermi-level profile across the channel normalized to “0” and “1” in the right and left contacts.Reuse & Permissions
Figure 2
Effect of the contact induced states on the resistance of a two-terminaled graphene strip. Resistance as a function of the channel length for (a) a metallic armchair channel and (b) a semiconducting armchair channel. Derived conductivity as a function of the channel length for (c) a metallic armchair channel and for (d) a semiconducting armchair channel. ∗-solid line when $T / T_{W} < 0.5$ , $★$ -solid line when $T / T_{W} \approx 3$ , and ●-solid line when $T / T_{W} \approx 6$ .Reuse & Permissions
Figure 3
Effect of contact induced states in Hall bar geometry with full width voltage probes. (a) Schematic illustration of structure. (b) Calculated resistance as a function of the distance between voltage probes $(L_{P})$ . $★$ -line for weakly coupled voltage probes; ∗-line for strongly coupled voltage probes with DOS of $22 \times 10^{13} / (eV {cm}^{2})$ ; ●-line for strongly coupled voltage probes with DOS of $8 \times 10^{13} / (eV {cm}^{2})$ . Two-dimensional (2D) local-density-of-state profile inside a metallic armchair channel for fixed $L$ and $W$ , when (c) the voltage probes are weakly coupled, (d) the voltage probes are strongly coupled for $L_{P} < W$ , and (e) the voltage probes are strongly couple for $L_{P} > W$ . Also shown in insets is the quasi-Fermi-level profile across the channel.Reuse & Permissions
Figure 4
Effect of contact induced states in four-probe structure with side voltage probes. (a) Schematic illustration of structure. (b) Calculated resistance as a function of width of the side voltage probes $(W_{P})$ ; solid line for contact DOS of $8 \times 10^{13} / (eV {cm}^{2})$ and dashed line for contact DOS of $22 \times 10^{13} / (eV {cm}^{2})$ . 2D local density of state profile inside a metallic armchair channel for fixed $L$ , $L_{P}$ , and $W$ ; (c) $W_{P} = 0.4 W$ and (d) $W_{P} = 1.6 W$ . Also shown in insets is the quasi-Fermi-level profile across the channel.Reuse & Permissions
Figure 5
Comparison between calculated $σ_{min}$ and measured $σ_{min}$ for Hall bar settings of the experiment in Ref. 1. Since the Hall bar geometry used in the experiment is a combination of side probes and probes with widths less than the channel width, we find a range that $σ_{min}$ can fall in for this type of structure considering two limits; side probes [dashed line in Fig. 4b] and full probes [∗-line in Fig. 3b]. ●-line: measured $σ_{min}$ ; △-line: upper limit of calculated $σ_{min}$ ; and ▽ line: lower limit of calculated $σ_{min}$ . We assumed $W_{P} \sim 250 nm$ as estimated from Fig. 1b in Ref. 1. Note that $T / T_{W} < 0.5$ for the all considered devices.Reuse & Permissions

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