Mesoscopic diffusion thermopower in two-dimensional electron gases

The diffusion of energy that is locally deposited into two-dimensional electron gases by Joule heating generates transverse voltages across devices with broken symmetry. For mesoscopic structures characterized by device dimensions comparable to the energy diffusion length, the resulting thermopower strongly depends on details of the potential profile defined by electric gates. We discuss these mesoscopic features within a diffusion thermopower model and propose schemes to measure the energy diffusion length and its dependence on gate voltage.

Figure 1

Schematic picture of a transverse thermoelectric rectifier. The carrier density of a 2DEG (blue region) is given by the potentials $V_{\mathrm{mc}}^{\mathrm{L}},V_{\mathrm{mc}}^{\mathrm{R}}$, and $V_{\mathrm{hc}}$ locally controlled by voltages applied to top gates. An applied current $I_x$ heats up the electrons in the heating channel (hc). The transverse, thermoelectric output voltage $U_y$ depends on the gate voltages applied to the gates on top of the modulation channels (mc) on the left- (L) and right- (R) hand side.

Figure 2

Setup consisting of one heating (hc) and one modulation (mc) channel in a distance $d=l_0/2$ from the heated region. The width of both is $w_{\mathrm{hc}}=w_{\mathrm{mc}}=l_0/2$. The red, solid line marks the temperature difference to the base temperature, $T_0$. The local Fermi energy is depicted by the blue, dashed line.

Figure 3

Dependence of $s_0$ on the electrostatic potential $V_{\mathrm{mc}}$ in the modulation channel for three different values of the potential $V_{\mathrm{hc}}$ in the heating channel. Referring to the setup in Fig. 2, we set $w_{\mathrm{hc}}=w_{\mathrm{mc}}=l_0/2$ and $d=0$.

Figure 4

Dependence of $s_0$ on the electrostatic potential in the modulation channel $V_{\mathrm{mc}}$ comparing different ${\alpha }_{\mathrm{e}}$ and ${\alpha }_{\mathrm{i}}$. Referring to the setup in Fig. 2, we set $w_{\mathrm{hc}}=w_{\mathrm{mc}}=l_0/2$ and $d=0$.

Figure 5

Dependence of $s_0$ on the width of the heating channel $w_{\mathrm{hc}}$ for the potential $V_{\mathrm{mc}}$ in the modulation channel. Referring to the setup in Fig. 2, we set $w_{\mathrm{mc}}=l_0/2$ and $d=0$.

Figure 6

Dependence of $s_0$ on the width of the modulation channel $w_{\mathrm{mc}}$ for the potential $V_{\mathrm{mc}}$ in the modulation channel. Referring to the setup in Fig. 2, we set $w_{\mathrm{hc}}=l_0/2$ and $d=0$.

Figure 7

Dependence of $s_0$ on the distance $d$ between heating and modulation channel for different potentials $V_{\mathrm{mc}}$ in the modulation channel. Referring to the setup in Fig. 2, we set $w_{\mathrm{hc}}=w_{\mathrm{mc}}=l_0/2$.

Figure 8

Dependence of $s_0$ on the sharpness of the potential step between modulation and heating channel. The width of the channels is set to $w_{\mathrm{hc}}=w_{\mathrm{mc}}=6l_0$ and the distance $d=0$. (a) The step of the Fermi energy at the heating-/modulation-channel interface is given by ${\epsilon }_{\mathrm{F}}\left(y\right)={\epsilon }_0-V\left(y\right)$ in the range $-6l_0<y<6l_0$ shown by the blue dashed line. Here, the potential $V\left(y\right)$ is taken from Eq. (15). The red line marks the temperature difference $\delta T\left(y\right)$ to the base temperature $T_0$. For (a), we use $V_{\mathrm{mc}}=0.2{\epsilon }_0$ and $\lambda =l_0/2$. (b) The dashed lines mark $s_0$ for a sharp step potential for different potentials $V_{\mathrm{mc}}$ in the modulation channel. The solid lines depict the influence of the step width $\lambda$.

Figure 9

The solid lines mark $s_0$ for $n$ devices as a function of $a$. The black, dashed line show the minimum temperature value between the devices, also as a function of $a$. The inset depicts an example consisting of three devices (with $w_{\mathrm{hc}}=w_{\mathrm{mc}}=l_0/2$ each), separated by a distance $a$.

Figure 10

Results for a setup consisting of one heating channel of width $w_{\mathrm{hc}}=60\mu \mathrm{m}$ and two neighboring modulation channels of widths $w_{\mathrm{hc}}=300\mu \mathrm{m}$ each. The calculations are done for $T_0=2\mathrm{K}$ and the momentum relaxation time ${\tau }_{\mathrm{e},0}=0.126\mathrm{ns}$ (extracted from the mobility in Ref. [35]). (a) The red, solid line marks the profile of $\delta T$ and the local Fermi energy is depicted by the blue, dashed line. We choose ${\tau }_{\mathrm{i},0}=100{\tau }_{\mathrm{e},0}$ for the energy-relaxation time. (b) The black, dashed and the blue, dotted lines represent the theoretical and measured results for thermopower divided by the lattice temperature obtained by Chickering et al., respectively [35]. The red, solid line accounts for mesoscopic effects.