Superballistic boundary conductance and hydrodynamic transport in microstructures

It is shown that the ideal boundary between a perfectly conducting electrode and electron liquid state acts as a contact whose conductance per unit area is higher than the fundamental Sharvin conductance by a numerical coefficient $2\alpha$, where $\alpha$ is slightly smaller than unity and depends on the dimensionality of the system. If the boundary has a finite curvature, an additional correction to the boundary conductance appears, which is parametrically small as a product of the curvature by the electron-electron mean free path length. The relation of the normal current density to the voltage between the electrode and electron liquid represents itself a hydrodynamic boundary condition for current-penetrable boundary. Calculations of the conductance and potential distribution in microstructures by means of numerical solution of the Boltzmann equation show that the concept of boundary conductance works very good when the hydrodynamic transport regime is reached. The superballistic transport, when the device conductance is higher than the Sharvin conductance, can be realized in Corbino disk devices not only in the hydrodynamic regime, although requires that the electron-electron scattering rate must be higher than the momentum-relaxing scattering rate. The theoretical results for Corbino disks are consistent with recent experimental findings.

Figure 1

(a) A thin conducting layer, where electrons move ballistically, is sandwiched between two perfectly conducting electrodes with fixed electrochemical potentials, forming the Sharvin contact. (b) The contact between the electrode and the electron liquid staying at electrochemical potential $V$ spreads over the Knudsen layer (bright yellow), where the potential changes rapidly. The normal current flowing through the contact is indicated by the red arrow. [(c),(d)] Devices of flat geometry and Corbino geometry with the contacts described above.

Figure 2

Distribution of electrochemical potential in the Knudsen layer for the systems of different dimensionalities. The drop of the potential across the Knudsen layer and the boundary conductance are indicated.

Figure 3

Distribution of electrochemical potential in the Knudsen layer for 2D systems in the case of modified relaxation-time approximation for the collision integral described by the two-time model of Eq. (21). The inset shows the dependence of the boundary conductance on the ratio of relaxation lengths for antisymmetric and symmetric parts of the distribution function.

Figure 4

Distribution of electrochemical potential along the 2D device with flat boundaries shown in Fig. 1, in the absence of momentum-relaxing scattering. The inset shows the dependence of the conductance on the Knudsen number $\mathrm{K}\equiv l/L$.

Figure 5

Bulk potential profiles for Corbino disk devices with different ratios $b=R_0/R_1$ from 1.5 to 6 for $\mathrm{K}\equiv l/R_0=0.1$ in the absence of momentum-relaxing scattering. For comparison, thin red line shows the potential profile for purely ballistic transport, $V\left(r\right)=(U/\pi )arcsin(R_1/r)$, at $b=6$. The horizontal lines show the magnitude of the flat part $V_{\mathrm{bulk}}$ according to Eq. (36). The vertical lines indicate the positions of the inner boundary, $r=R_1$.

Figure 6

Conductance of Corbino disk devices as a function of the ratio $R_0/R_1$. Three upper plots (black-solid lines) are calculated in the absence of momentum-relaxing scattering, $l_{tr}\to \infty$, and the lower plot corresponds to $l_{tr}=l_e=2l=2R_0$. The dashed lines show the approximate results according to Eq. (33): $\mathrm{K}=0.1$ and $\alpha =0.96$ (red); $\mathrm{K}=0.1$ and $\alpha =1$ (blue); and $\mathrm{K}=0$ and $\alpha =0.96$ (black).

Figure 7

The borders of superballistic behavior for Corbino disks with different ratios $b=R_0/R_1$ (indicated) in the space of parameters $l_e$ and $l_{tr}$. The lines correspond to $G=G_S$. In the regions above the lines the transport is superballistic, $G>G_S$. Red and blue dashed lines show the dependence of $l_{tr}/R_0$ on $l_e/R_0$ when temperature is changed, plotted by using the experimental details of two devices with $b=4.5$ and $b=6$ from Ref. [35]. The intersection points of these plots with the lines $G=G_S$ for $b=4.5$ and $b=6$ are shown.

Figure 8

Calculated temperature dependence of the normalized resistance $G_S/G$ for two Corbino disk devices studied in Ref. [35]. For the device with $R_0=9\mu \mathrm{m}$ and $R_1=2\mu \mathrm{m}$, the plots with $\gamma =0.5$ and $\gamma =2$ are included to show sensitivity of the resistance to the strength of electron-electron interaction. Squares show the experimental data extracted from Ref. [35] as described in the text.