Failure of the Wiedemann-Franz law in mesoscopic conductors

We study the effect of mesoscopic fluctuations on the validity of the Wiedemannn-Franz (WF) law for quasi-one-dimensional metal wires and open quantum dots. At temperatures much less than the generalized Thouless energy, $E_c$, the WF law is satisfied for each specific sample, but as the temperature is raised through $E_c$, a sample-specific correction to the WF law of order 1/g appears ($g$ is the dimensionless conductance) and then tends to zero again at $k_{B}T⪢E_c$. The mesoscopic violation of the Weidemann-Franz law is even more pronounced in a ring geometry for which the Lorenz number exhibits $h∕e$ flux-periodic Aharonov-Bohm oscillations.

Figure 1

(Color online) Upper panel shows the thermal weight functions for the electrical current, $I_{\mathrm{e}}$, and for the heat current, $I_{\mathrm{h}}$. For constant electron transmissivity $\sim G\left(\epsilon \right)$ on the scale of $k_{B}T$, the ratio of areas below these functions leads to the WF law, which thus always holds as $T\to 0$. However, if $\delta G\left(\epsilon \right)$ fluctuates on the scale of $E_{\mathrm{c}}\sim k_{B}T$ (lower panel), then $\delta G\left(\epsilon \right)$ is unequally weighted for $I_{\mathrm{e}}$ and $I_{\mathrm{h}}$ and sample-specific corrections to the WF law appear.

Figure 2

(Color online) Temperature dependence of the $\mathrm{rms}\left(\eta \right)$ in quasi-one-dimensional wires (solid line) and in open quantum dots with $N_{\mathrm{l}}=N_{\mathrm{r}}⪢1$ (dashed line). In both cases a broad maximum in the violation of the WF law occurs at $k_{B}T\sim E_c$ and persists to much higher temperatures.

Figure 3

(Color online) (Top) Temperature dependence of$F_{G,\Xi }^{(k=1)}\left(T\right)$ for two values of $\gamma ∕E_R$. At high temperatures, $k_{\mathrm{B}}T⪢E_R$ amplitudes $F_{G,\Xi }^{(k=1)}\left(T\right)$ decrease only as $T^{-1}$. (Bottom) The ratio $F_{\Xi }^{\left(1\right)}∕F_G^{\left(1\right)}$ as a function of temperature. At $T\gtrsim E_R$ the ratio saturates to $0.55{\left(l_{0}T\right)}^2$ (dotted line); see Eq. (11).

Figure 4

(Color online) Temperature dependence of the rms(B) in quasi-one-dimensional wires (solid line) and in open quantum dots with $N_{\mathrm{l}}=N_{\mathrm{r}}⪢1$ (dashed line). The maximum in rms(B) occurs at $k_{B}T\sim E_c$ and vanishes at low $(k_{B}T⪡E_c)$ and high $(k_{B}T⪢E_c)$ temperatures.