Resonant field emission from noble-metal/graphene heterostructures

Abstract

Field emission from metals underpinned early vacuum-tube technology, and recent nanoscale engineering made field-emission devices compatible with modern silicon platforms. However, the limited tunability of electron transport in metals has restricted their applicability. Here, we show that noble metals coated with graphene exhibit clean non-monotonic $I-V$ characteristics arising from resonant tunneling through graphene's electronic states, enabled by graphene's atomic thinness and weak electronic hybridization with noble metals. Our approach combines ab-initio interface parameters with exact solutions of the Schrödinger equation for electron transmission across the interface. We analyze two experimentally relevant geometries: a vertical configuration with a flat suspended emitter and a coplanar configuration with sharp electrodes allowing for strong field enhancement and gating. These results establish a practical route to tunable electron transport in metal heterostructures, positioning them as competitive components for air-channel field-emission nanoelectronics.

Figures (9)

• Figure 1: Resonant field-emission by metal-graphene interface. (a) Out-of-plane potential profile of isolated graphene exhibits extremely narrow electron confinement, with the work function $W_0=4.55$ eV. (b) Upon physisorption on a metal, the work function difference leads to a redistribution of surface charge, shifting the Fermi level by $\Delta E_F$ in graphene and creating an interfacial barrier of height $\Delta V$ and thickness $d$. (c) Field electron emission from a metal covered by graphene is governed by tunneling through a composite potential barrier consisting of triangular and trapezoid sections, which supports a resonant state. (d) Fowler-Nordheim representation of the field-emission current density $J$ clearly demonstrates deviation from the conventional linear relation between $\ln(J/{\cal E}_\mathrm{ext}^2)$ and $1/{\cal E}_\mathrm{ext}$ when $e{\cal E}_\mathrm{ext}\sim \Delta V/d$ (shown by dashed lines) and the resonant level enters the energy window between $-E_F$ and $-V_0$, see Fig. \ref{['fig2']}. The exact position of the emission maximum also depends on charge doping, which can be either $n-$ or $p-$type (negative or positive $\Delta E_F$; see Table \ref{['tab1']}), depending on a metal employed (Ag, Au, and Pt). The finite-size channel effects are neglected here (no direct tunneling, see Fig. \ref{['fig3']}). (e) Field emission current density computed for the device shown in (f) and normalized to $J_0$ computed for the same geometry without graphene. The curves are averaged over 100 channels of randomized lengths between $L_\mathrm{min}=4.5$ nm and $L_\mathrm{max}=5.5$ nm to take into account the collector surface roughness. The model neglects the field enhancement factors (e.g. the image-charge effects), which are addressed in Fig. \ref{['fig4']}. (f) Possible realization of the resonance field-emission on a heavily doped Si/SiO$_2$ wafer with a bow-tie gold bridge and SiO$_2$ etched away underneath, see Ref. tayari2015tailoring. Doped Si is usually used as a gate contact but if SiO$_2$ is removed completely under the nanobridge, it plays an electron collector role.
• Figure 2: Resonant features in transmission (red) and reflection (blue) probabilities with (solid) and without (dashed) graphene. Parameters are chosen such to make the resonance clearly visible, $W_m=6.2$ eV, $W=4.5$ eV, $\Delta E_F=0$, $V_0=9.5$ eV, $d=0.17$ nm, and they do not correspond to a specific metal. (a) For $e{\cal E}_\mathrm{ext} < \Delta V/d$, the resonance lies above the Fermi level; emission from occupied states occurs via resonance broadening. (b) For $e{\cal E}_\mathrm{ext} = \Delta V/d$, the resonance matches the Fermi level, producing a dramatic increase in emission. (c) For $e{\cal E}_\mathrm{ext} > \Delta V/d$, the resonance falls within the occupied energy window between the metal conduction-band bottom and the Fermi level. At such high fields, however, non-resonant emission without graphene also becomes efficient, reducing $J/J_0$. With further increase of ${\cal E}_\mathrm{ext}$ the resonance exit the occupied energy window and the emission current drops to the non-resonant values.
• Figure 3: Effects of finite channel length $L$. Parameters are the same as in Fig. \ref{['fig2']} and do not correspond to a specific metal. (a,b) Potential profiles for FN ($e{\cal E}_\mathrm{ext}L > V_0$) and direct tunneling ($e{\cal E}_\mathrm{ext}L<W$) regimes, respectively. The intermediate field range $W\leq e{\cal E}_\mathrm{ext}L\leq V_0$ corresponds to a transitional regime in which part of the occupied states tunnel via FN and the remainder via direct tunneling. (c) Distinction between FN and direct tunneling. In direct tunneling, the emission current decreases exponentially with increasing $L$, whereas in the FN regime it is essentially independent of $L$. Moreover, graphene reduces the direct-tunneling current compared with bare metal, while enhancing emission in the FN regime. (d) Transmission (red) and reflection (blue) probabilities with (solid) and without (dashed) graphene for a finite channel length of 1 nm, showing additional resonance-like features due to Fabry-Perot interference, cf. Fig. \ref{['fig2']}(b). (e) Field-emission current ratio $J/J_0$ versus ${\cal E}_\mathrm{ext}/{\cal E}_\mathrm{res}$, computed for different $L$, where $e{\cal E}_\mathrm{res}=\Delta V/d$ and $J_0$ is the current for the same channel without graphene. In the direct-tunneling regime, the resonance maximum is distorted by backscattering between the electrodes.
• Figure 4: Mapping the model to a coplanar device geometry with pointy contacts. (a) Possible schematics of a resonant field-emission device. The graphene layer is not to scale. The doped Si/SiO$_2$ substrate can be used for electrostatic gating nirantar2018metal. (b) Current density versus applied voltage for a gold/graphene emitter, computed for different $\beta$ values; other parameters are taken from Table \ref{['tab1']}. (c) Current-voltage characteristics for a specific device with $L=10.1$ nm and $\beta=70$. Gray curve is obtained by fitting the field-emission model to experimental data (blue dots) for bare gold electrodes wang2024enhancing. Red curve is computed using the same geometric parameters but with a graphene-coated emitter. The model predicts a resonance near $V \sim 2.3$ V.
• Figure S1: Out-of-plane confinement of electrons in graphene modeled by a delta-function potential. (a) Electrons in an isolated delta-function potential have a single bound state at energy $-E_0$ corresponding to the difference between the vacuum level and the charge-neutrality (band-crossing) point in intrinsic (undoped) graphene. (b,c) An applied electrostatic potential shifts $-E_0$ up or down depending on its sign. The potential gradient $\Delta V/d$ may arise from a built-in field at the graphene-metal interface or from an external field. (d,e) Solutions of Eqs. (\ref{['DV1']}) and (\ref{['DV2']}) showing the shift of $-E_0$ for different $\Delta V$ and $d$. The shift is maximized at $d\sim l_0$ of about an angstrom, reaching a few tens of meV for a potential drop on the order of $0.1$ eV.