Current-induced atomic forces in gated graphene nanoconstrictions

Electronic current densities can reach extreme values in highly conducting nanostructures where constrictions limit current. For bias voltages on the 1 V scale, the highly nonequilibrium situation can influence the electronic density between atoms, leading to significant interatomic forces of the order of 1 nN. An easy interpretation of the nonequilibrium forces is currently not available, to our knowledge. In this work, we present an ab initio study based on density functional theory of bias-induced atomic forces in gated graphene nanoconstrictions consisting of junctions between graphene electrodes and graphene nanoribbons in the presence of current. We find that current-induced bond forces and bond charges are correlated, while bond forces are not simply correlated to bond currents. We discuss, in particular, how the forces are related to induced charges and the electrostatic potential profile (voltage drop) across the junctions. For long current-carrying junctions we may separate the junction into a part with a voltage drop, and a part without voltage drop. The latter situation can be compared to a nanoribbon in the presence of current using an ideal ballistic velocity-dependent occupation function. This shows how the combination of voltage drop and current give rise to the strongest current-induced forces in nanostructures.

Figure 1

Graphene nanoconstrictions with electrostatic bottom gate. (a) Represents the unit cell of a generic constriction, with the blue square indicating the position and shape of the electrostatic gate. Edge atoms of the graphene are saturated by hydrogen in order to avoid dangling bonds. (b) Short constriction and (c) semi-infinite GNR constrictions considered in this work.

Figure 2

(a) Zero-bias transmission for different gate charges $g$ (gray without gate), (b) total current as a function of bias voltage for selected gate charges, (c) real-space bond currents at $1\mathrm{V}$, $g=-2$, and (d) maximal absolute force over bias voltage for different gate charges.

Figure 3

(a) Energy scheme of the electrode DOS for negative $g$ and positive bias voltage, (b) electrostatic potential profile, and (c) bias-induced charge redistribution at $1\mathrm{V}$ for $g=-2$. (d), (e), (f) DOS, potential, and charges at $1\mathrm{V}$ for $g=+2$.

Figure 4

(a) Induced bond forces, $F_b$, and change in overlap population $\mathrm{\Delta }\mathrm{OP}$ at 1.0 V, $g=-2$. The forces $F_b$ are shown as vectors in light red and blue. Force vectors are pointing inwards (outwards) to indicate bond compressing (stretching), while the vector thickness corresponds to the force strength. The change in OP is depicted as in-/decreasing density along the bond, in red for positive and blue for negative $\mathrm{\Delta }\mathrm{OP}$. Induced forces/charges below a cutoff of $|F_b|=0.004\mathrm{nN}/\mathrm{\Delta }\mathrm{OP}=1\times {10}^{-4}$ e are set to zero. (b) Correlation between bond forces $F_b$ and $\mathrm{\Delta }\mathrm{OP}$, (c) correlation between bond forces $|F_b|$ and bond currents $J_{nm}$, (d), (e), (f) equivalent pictures for $g=+2$.

Figure 5

(a) Chemical bond of atom $m$ and $n$ with compressive bond force (red), and bond of atom $n$ and $k$ with repulsive bond force (blue) at $1\mathrm{V}$ and $g=-2$. (b) DOS of left-/right-incoming states, ${\mathbf{A}}^L$ and ${\mathbf{A}}^R$, on atoms $m$ and $n$ (left panel) and atoms $n$ and $k$ (right panel) at $1\mathrm{V}$. States in ${\mathbf{A}}^L$ (${\mathbf{A}}^R$) below ${\mu }_L=0.5$ eV (${\mu }_R=-0.5\mathrm{eV}$) are occupied. (c) COOP analysis for bond $m\text{−}n$ (left panels) and bond $n\text{−}k$ (right panels). Bond $m\text{−}n$ is strengthened by current, as at $1\mathrm{V}$ bonding states get populated in ${\mathbf{A}}^L$ (red shaded area in left COOP). The DOS that is depleted in ${\mathbf{A}}_R$ is small. Bond $n\text{−}k$ is weakened by current, as antibonding states get filled (blue shaded area).

Figure 6

(a) Electrostatic potential profile (b) bias-induced charge redistribution, and (c) bond currents in the wide-narrow GNR constriction at 0.75 V, $g=-2$.

Figure 7

(a) Distribution of induced bond forces $F_b$ and $\mathrm{\Delta }\mathrm{OP}$ in the GNR constriction, (b) maximal values of $F_b$ along the transport direction $z$, (c) correlation between $F_b$ and $\mathrm{\Delta }\mathrm{OP}$, at 0.75 V, $g=-2$.

Figure 8

Bulk-bias applied to GNR. Shown is the band structure, where a positive bias of 0.75 V along the GNR direction changes the occupation of right-moving states with positive velocity such that states are filled up to ${\mu }_L=eV/2$. Similarly are the left moving states with negative velocity emptied above ${\mu }_R=-eV/2$. Filled bands are indicated in red.

Figure 9

(a) Force pattern far away from the potential drop at $0.75\mathrm{V}$ from TranSiesta calculation, (b) forces, and (c) $\mathrm{\Delta }\mathrm{OP}$ from bulk calculation at $0.75\mathrm{V}$.