Symmetry of Two-Terminal Nonlinear Electric Conduction

The well-established symmetry relations for linear transport phenomena cannot, in general, be applied in the nonlinear regime. Here we propose a set of symmetry relations with respect to bias voltage and magnetic field for the nonlinear conductance of two-terminal electric conductors. We experimentally confirm these relations using phase-coherent, semiconductor quantum dots.

Figure 1

Schematic electron trajectories for positive and negative magnetic fields. In the absence of symmetry in a mesoscopic device, the conductance is not expected to be symmetric with respect to the direction of a magnetic field, when a bias voltage defines a source and a drain contact.

Figure 2

Illustration of the symmetry relations expected for a rigid device in the nonlinear regime and at finite magnetic field. The upper and lower rows show the two possible lead configurations $G_{12}$ and $G_{21}$, distinguished by the position of the grounding point relative to the device. Different classical electron trajectories illustrate the difference in transmission probability that results when the potential depends on the sign of the voltage applied to the source contact. Positive magnetic field is taken to be into the page.

Figure 3

Magnetoconductance fluctuations for an UD-symmetric quantum dot measured in the eight possible different configurations of lead orientation, sign of bias voltage, and sign of magnetic field (capital letters refer to the panels in Fig. 2): (a) $g_{12}(V=+1\mathrm{m}\mathrm{V},\pm B)$, (b) $g_{12}(V=-1\mathrm{m}\mathrm{V},\pm B)$, (c) $g_{21}(V=+1\mathrm{m}\mathrm{V},\pm B)$, and (d) $g_{21}(V=-1\mathrm{m}\mathrm{V},\pm B)$. The lower trace in each panel has been offset by $-0.5e^2/h$ for clarity. The inset in (a) shows $d_{\mathrm{L}\mathrm{R}}/d_0$, $d_{\mathrm{U}\mathrm{D}}/d_0$, and $d_{\mathrm{C}\mathrm{I}\mathrm{A}}/d_0$ as a function of $V$ (lines are guides to the eye).

Figure 4

Magnetoconductance fluctuations $g_{12}(V,B)$ of a LR-symmetric device for $V=0$, $V=+2\mathrm{m}\mathrm{V}$, and $V=-2\mathrm{m}\mathrm{V}$ (capital letters in each measurement configuration refer to the corresponding panel in Fig. 2). Data are offset for clarity. Note that $g_{12}(V,B)\ne g_{12}(V,-B)$, while $g_{12}(V,B)$ and $g_{12}(-V,-B)$ show very similar features, as predicted by Eq. (2) (see, e.g., the marked features). Inset: $d_{\mathrm{L}\mathrm{R}}/d_0$ and $d_{\mathrm{U}\mathrm{D}}/d_0$ as a function of $V$ ($d_0=4.21\times {10}^{-2}{e}^2/h$ for this device). Lines are guides to the eye.