Viscous corrections to the resistance of nanojunctions: A dispersion relation approach

It is well known that the viscosity of a homogeneous electron liquid diverges in the limits of zero frequency and zero temperature. A nanojunction breaks translational invariance and necessarily cuts off this divergence. However, the estimate of the ensuing viscosity is far from trivial. Here, we propose an approach based on a Kramers-Kronig dispersion relation, which connects the zero-frequency viscosity $\eta (0)$ to the high-frequency shear modulus ${\mu }_{\infty }$ of the electron liquid via $\eta (0)={\mu }_{\infty }\tau$, with $\tau$ the junction-specific momentum relaxation time. By making use of a simple formula derived from time-dependent current density functional theory we then estimate the many-body contributions to the resistance for an integrable junction potential and find that these viscous effects may be much larger than previously suggested for junctions of low conductance.

Figure 1

The comparison between the viscosity calculated by Abrikosov and Khalatnikov, ${\eta }_{\text{AK}}$, and the viscosity calculated using the dispersion relation (3.2) for different values of $r_s$. The connecting lines are a guide to the eye.

Figure 2

A schematic of the potential (4.1), for a constant value of $z$, used in this work to represent the scattering off a nanojunction.

Figure 3

Longitudinal momentum relaxation time ${\tau }_{\mathbf{k}}^{\left|\right|}$ as a function of the transmission coefficient $T_{k_x}$ along the $x$ direction for $r_s=3$. Inset shows the corresponding transverse momentum relaxation time ${\tau }_{\mathbf{k}}^{\perp }$.

Figure 4

Zero-frequency xc viscosity ${\eta }_{\text{xc}}$ in a nanojunction as a function of the transmission coefficient for $r_s=3$.

Figure 5

Dynamical viscous resistance as a function of transmission coefficient for the potential (4.1). The inset shows the averaged density $\langle n(x)\rangle$ across the nanojunction for the longitudinal transmission coefficient 0.1 and 0.9. Other parameters are in the text.