Graphene Quantum Strain Transistors

A.C. McRae, G. Wei, and A.R. Champagne

Phys. Rev. Applied 11, 054019 – Published 7 May 2019

Abstract

There is a wide range of science and applications accessible via the strain engineering of quantum transport in 2D materials. We propose a realistic experimental platform for uniaxial strain engineering of ballistic charge transport in graphene. We focus on high-aspect-ratio mesoscopic devices ( $W = 1000$ nm, $L = 100$ nm) whose conductivity is well defined even without atomically precise crystal edges. We develop an applied theoretical model, based on this platform, to calculate charge conductivity and demonstrate realistic graphene quantum strain transistors (GQSTs). We define GQSTs as mechanically strained ballistic graphene transistors with on:off conductivity ratios $> 10^{4}$ , which can be operated via modest gate voltages. Such devices would permit excellent transistor operations in pristine graphene, where there is no band gap. We consider all dominant uniaxial strain effects on conductivity, while including experimental considerations to guide the realization of the proposal. We predict multiple strain-tunable transport signatures, and demonstrate that a broad range of experimentally accessible device parameters lead to robust GQSTs. These devices could find applications in flexible electronic transistors, strain sensors, and valleytronics.

Received 11 October 2018
Revised 14 December 2018

DOI:https://doi.org/10.1103/PhysRevApplied.11.054019

Physics Subject Headings (PhySH)

Ballistic transport Elastic deformation Quantum interference effects Quantum transport

Flexible electronics Graphene Transistors

Strain engineering

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

A.C. McRae, G. Wei, and A.R. Champagne^*

Department of Physics, Concordia University, Montréal, Québec H4B 1R6, Canada

^*a.champagne@concordia.ca

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Vol. 11, Iss. 5 — May 2019

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Images

Figure 1
Platform for uniaxial QTSE in graphene. (a) Top-down view of the proposed ballistic graphene transistor geometry. Outset: the graphene lattice, showing the crystal orientation $θ$ with respect to the $x$ axis. (b) Side views of the proposed graphene device and mechanical-strain instrumentation. The mechanical assembly bends the substrate, which strains the suspended channel. (c) The three sources of strain in the channel: the mechanical motion (black, top axis) of the push screw, the thermal contraction (red) at 1 K, and electrostatic strain (blue, bottom axis) from $V_{G}$ . Insets: visualizing the strain imparted by the thermal contraction of the gold cantilevers (top left), and electrostatic pulling (bottom right). (d) Conductivity versus charge density, $n$ (bottom axis), or $V_{G}$ (top axis). The data are for an unstrained channel of $L = 100$ nm, $W = 1000$ nm, with contact doping $Δ μ_{contact} = - 0.12$ eV (black), $\pm \infty$ (red), and $Δ μ_{contact} = Δ μ_{G}$ (blue).
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Figure 2
Applied theory for uniaxial QTSE in graphene. (a) Dirac cone and Fermi circle in the unstrained source and drain graphene contacts, and (b) in the strained graphene channel. (c) Unstrained (black) and uniaxially strained (red) first Brillouin zone (FBZ) of graphene when $θ = 0^{\circ}$ . The strain value in this figure is exaggerated, $ε_{total} = 20 %$ , to make its effects clearly visible. (d) Under strain, the Dirac point shifts define gauge-vector potentials (blue arrows), $A_{i} = A_{lat, i} + A_{hop}$ . The outset shows that the corner of the FBZ does not coincide with the Dirac point under strain. (e) Charge-carrier wave vectors in the source, channel, and drain of the uniaxially strained ballistic transistor. The ballistic modes are labeled with their $y$ -component wave number $q_{n}$ . The $A_{i, y}$ in the channel modifies the propagation angle, $ϕ$ , and transmission probability, $T$ , of the carriers. (f),(g) $T$ of the conduction modes for $A_{i, y} = 0$ and $A_{i, y} = k_{F}$ , respectively. The circles represent the Fermi surfaces in the contacts (big circles) and channel (small circles), while the solid curves show the $T$ (origin $= 0$ , outer circle $= 1$ ) versus the incidence angle on the channel (polar axis), when $θ = 15^{\circ}$ and $Δ μ_{contact} = - 0.12$ eV. The boundary condition in $y$ imposes transversal momentum conservation (dashed lines), and explains the zero transmission at some angles.
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Figure 3
Conductivity signatures of uniaxial QTSE in graphene. (a) $σ - V_{G}$ for $ε_{mech} = 0$ , 0.5, 1.0, and 1.5% (black, red, blue, gold) at $θ = 0^{\circ}$ . The strain-induced scalar potential shifts the $σ - V_{G}$ curves. The inset shows the gate shift of the Dirac point, $V_{D}$ , as a function of $ε_{total}$ . (b) $σ - (V_{G} - V_{D})$ at $θ = 15^{\circ}$ . There is a rapid decrease of $σ$ with strain. FP resonances are clearly visible. (c) $σ - (V_{G} - V_{D})$ data at $θ = 30^{\circ}$ show a complete suppression of $σ$ under strain. There is an asymmetry between hole ( $σ_{h}$ ) and electron ( $σ_{e}$ ) conductivities at opposite $(V_{G} - V_{D})$ . (d) $σ - ε_{total}$ for $θ = 15^{\circ}$ , with a linear fit (red) to extract $(d σ / d ε)_{\max}$ . Inset: $(d σ / d ε)_{\max} - θ$ with a sinusoidal fit (red). (e) Color map of $d σ / d (Δ μ_{G})$ versus $Δ μ_{G}$ and $ε_{total}$ . Clear vertical FP resonances are visible. The dashed lines identify the three conductivity features $k_{F} = {\tilde{k}}_{F} + A_{i, y}$ , where the vector potentials turn off the propagation of conduction modes. (f) Relative electron-hole asymmetry, $η = 2 (σ_{h} - σ_{e}) / (σ_{h} + σ_{e})$ , as a function of $ε_{total}$ at $θ = 15^{\circ}$ . Insets: $σ - V_{G}$ shows $η > 0$ at $ε_{total} = 1$ % (bottom left), and $η < 0$ at $ε_{total} = 3$ % (top right).
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Figure 4
GQSTs. (a) $σ - V_{G}$ with a gate-channel spacing $t_{vac} = 50$ (black), 100 (red), and 200 nm (blue) and under strain $ε_{total} = 5.1 %$ , $θ = 30^{\circ}$ . The 50-nm gate spacing gives rise to a robust GQST effect with $σ_{on:off} ≫ 10^{4}$ . The inset shows a subthreshold slope of 240 mV/dec when $t_{vac} = 50$ nm. (b) $σ - ε_{mech}$ shows the extreme strain sensitivity of $σ$ versus uniaxial strain at fixed $V_{G} = 0$ V (black), $- 5$ V (red), and $- 10$ V (blue). Inset: resistance-based strain sensitivity at $V_{G} = - 8$ V (gold) and $- 10$ V (blue). (c) $σ_{on:off} - Δ μ_{contact} - θ$ color maps at $ε_{total} = 5.1 %$ when $Δ μ_{contact} < 0$ ( $p$ doping of contacts), and (d) when $Δ μ_{contact} > 0$ ( $n$ doping of contacts). The color scales show $σ_{on:off}$ calculated with $Δ V_{G} = 10$ V (i.e., at 0 V and $- 10$ V). The contour lines mark the boundary where $σ_{on:off} > 10^{4}$ , corresponding to a strong GQST effect. The solid black, dashed white, and dashed black contours are for $Δ V_{G} = 10$ , 5, and 10 V but for a lower $ε_{total} = 2.6 %$ , respectively.
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