Thermoelectric Scanning-Gate Interferometry on a Quantum Point Contact

We introduce a scanning probe technique derived from scanning gate microscopy (SGM) to investigate thermoelectric transport in two-dimensional semiconductor devices. Thermoelectric scanning gate microscopy (TSGM) consists in measuring the thermoelectric voltage induced by a temperature difference across a device while scanning a polarized tip that locally changes the potential landscape. We apply this technique to perform interferometry of the thermoelectric transport in a quantum point contact (QPC). We observe an interference pattern in both SGM and TSGM images, and evidence large differences between the two signals in the low-density regime of the QPC. In particular, a large phase jump appears in the interference fringes recorded by TSGM, which is not visible in the interference fringes recorded by SGM. We discuss this difference of sensitivity using a microscopic model of the experiment, based on the contribution from a resonant level inside or close to the QPC. This work demonstrates that combining scanning gate microscopy with thermoelectric measurements offers new information as compared with SGM alone, and provides direct access to the derivative of the device transmission with respect to energy, both in amplitude and in phase.

Figure 1

(a) The TSGM experiment: one side of the device is brought to higher temperature and the thermovoltage is recorded while scanning the polarized tip. (b) Seebeck effect across a barrier (black region): red and blue bars illustrate the energy distribution of charge carriers on the hot and cold sides. If transmission is energy dependent, fluxes of hot and cold carriers are imbalanced. At equilibrium, a charge accumulation on the cold side restores the total balance of fluxes: this increases the thermovoltage $V_{\mathrm{th}}$. (c) Differential conductance $G$ measured with a four-probe technique at 25 mK (blue) and thermovoltage measured with a heating current of 180 nA (red) versus gate voltage $V_g$. (d) Theoretical transmission (blue) and Seebeck coefficient (red) calculated from the saddle-point model [15] versus energy for ${\omega }_x/{\omega }_y=0.6$.

Figure 2

(a) The black curve represents $-S^M/T$ calculated with Mott’s law from the $G(V_g)$ curve at 25 mK (dashed line). The colored curves represent $-S/T=(V_{\mathrm{th}}/\mathrm{\Delta }T)/T_{\mathrm{average}}$ scaled to the black curve at the third peak summit with $\mathrm{\Delta }T$ as the only fitting parameter. The heating currents range from 50 to 240 nA (blue to red). (b) Temperature difference $\mathrm{\Delta }T$ as a function of the heating current. The inset shows an electron-microscope image of the device. The scale bar is 5 $\mu \mathrm{m}$. (c) $S/T-S^M/T$ for temperature differences from 160 to 570 mK (blue to red) as a function of $V_g$. Same data as in (a).

Figure 3

(a) SGM image recorded when the QPC is open on the first plateau ($V_g=-0.66$ V): conductance $G$ as a function of tip coordinates. The QPC center is located at ($-150$; 0) nm on the left side of the image. (b) TSGM image of the thermopower in the same conditions, but recorded in a second pass while the reservoir is being heated on the opposite side of the scanning area. (c) Conductance and (d) thermovoltage as a function of gate voltage and tip position along the red line drawn in (a),(b) recorded in the same conditions but in separate tip scans. (e),(f) Derivative of (a),(b) with respect to tip position.

Figure 4

(a) Conductance and (b) thermovoltage as a function of gate voltage and tip position along the red line drawn in Fig. 3, where $d_{\mathrm{tip}}$ here denotes tip position between 250 and 650 nm in Figs. 3–3. (c),(d) Derivative of (a),(b) with respect to tip position. (e) Phase of the interference fringes as a function of gate voltage extracted from the conductance (purple) and thermovoltage (green) following the pinch-off lines shown in (c),(d) to account for a cross-talk effect. (f) Line profiles of $G$, $-S^M/T$, and $-S/T$ extracted along the red and blue lines in (a),(b).

Figure 5

(a) The top scheme shows a typical energetic potential that could lead to the proposed scenario: a localized state is located on the tip side of the main QPC barrier. The bottom scheme shows the one-dimensional model including, from left to right, the QPC barrier (${\mathrm{\Gamma }}_L$), the localized state (${ϵ}_0$ and ${\mathrm{\Gamma }}_R$), and the tip-induced cavity ($D_{\mathrm{tip}}$ and $T_{\mathrm{tip}}$). (b) Conductance of the resonant level as a function of its energy relative to $\mu$ for fixed ${\mathrm{\Gamma }}_R=0.2$ and ${\mathrm{\Gamma }}_L=0.2$ (blue) to 0.01 (red). (c) Seebeck coefficient in the same configuration. (d) Conductance evolution with ${\mathrm{\Gamma }}_L$ assuming that the resonant level, perfectly coupled to the right lead (${\mathrm{\Gamma }}_R=1$), crosses $\mu$ at ${\mathrm{\Gamma }}_L=0.1$ and then stays just below the Fermi level while the QPC opens. (e) Seebeck coefficient in the exact same scenario. (f),(g) Evolution of the Fabry-Perot interference in this scenario: conductance (f) and Seebeck coefficient (g) as a function of tip distance and ${\mathrm{\Gamma }}_L$. (h),(i) Derivative of $G$ and $S$ with respect to $D_{\mathrm{tip}}$.