Thermoelectric property of a one dimensional channel in the presence of a transverse magnetic field

TL;DR

This study investigates how a transverse magnetic field affects thermal conduction and thermo-voltage behavior in a quantum point contact within a GaAs-AlGaAs heterostructure, revealing field-dependent shifts and calibration needs.

Contribution

It provides new insights into the magnetic field influence on thermoelectric properties of quantum point contacts, highlighting the need for calibration in magnetic environments.

Findings

Thermo-voltage peak shifts with increasing magnetic field.

The shift follows the Cutler-Mott relation at low fields.

Divergence from the relation occurs at moderate fields, saturating at high fields.

Abstract

We studied the thermal conduction through a quantum point contact (QPC), defined in GaAs-AlGaAs heterostructure, in the presence of a transverse magnetic field. A shift in the position of thermo-voltage peak is observed with increasing field. The position of the thermo-voltage peak follows the Cutler-Mott relation in the small field regime (B < 0.5 T); it starts diverging from the Cutler-Mott relation in the moderate field regime, where a cubic magnetic field term dominates over the trivial quadratic term; eventually the shift saturates in the large field regime (B > 3.0 T). Our results suggest that additional calibration is necessary when using QPC as thermometry, especially when the transverse magnetic field is applied.

Equations

$$\frac{V_{th,i}}{T_i-T_0}=-\frac{{\pi }^2{k}_B}{3e}(T_i+T_0)\frac{\partial ln{G}_i}{\partial {\mu }_i}$$

Full text

††preprint: APS/123-QED

Thermal and electric conduction, in a conducting system, are generally strongly coupled to each other, the mode that carries charge is the same one that carries energy (heat). The coupling is found to be well established in two-dimensional electron gas (2DEG) in both the low magnetic field and integer quantum Hall regimePFC03 ; PFT04 ; ZCK09 ; GJJ09 , as well as one-dimensional electron gasSTR89 ; MVB90 ; ANP00 . Some recent studies on the fractional quantum Hall regime (FQHE), where electron-electron interaction dominates, show different results regarding thermal conductionKFP94 ; BOI10 ; GG16 ; SGR17 . In all these works, quantum point contacts (QPCs) or quantum dots (QDs) have been widely employed to probe the local temperature and reveal the heat flow through the system. However, the role of QPCs or QDs in terms of heat flow has not been well characterized, especially in the presence of a transverse magnetic field.

In the present work, we focus on the impact of a transverse magnetic field on thermal conduction through a QPC. It is noticed the observations are well captured by Cutler-Mott relation at zero field; however, a noticeable discrepancy between Cutler-Mott relation and experimental results occurs at a finite magnetic field. A cubic magnetic field term, in terms of the position of the thermo-voltage peaks, dominates over the trivial quadratic term with a moderate magnetic field. Meanwhile, an anomalous lattice temperature dependence associated with the field is also revealed. Our results suggest that additional calibration is necessary when using QPC as thermometry.

The thermoelectric property of a system is closely associated with a thermal gradient across the system. In the experiment setup [see note 1 of supplementary information for detailed setup] shown in Fig. 1(a) the whole system consists of three sections, once the split gates that define the quantum point contacts (QPCs) deplete electrons underneath the gates, namely the large 2DEG area with temperature $T_{0}$ and the relatively small area enclosed by QPC1 (QPC2) with temperature $T_{1}$ ($T_{2}$). The QPCs serve as thermal valve and local thermometryMVB90 ; ANS98 ; ANP00 . A heating current $I_{H}$ (at frequency $f_{0}$, 33 Hz; 1 $\mu$A) is fed to the 2DEG to increase $T_{0}$. Since $T_{0}>T_{1,2}$, a thermo-voltage $V_{th}$ is built between the large and small area (measured at 2$f_{0}$, because heating power $P\propto I_{H}^{2}$). $V_{th}$ depends on the conductance of the QPCs, it vanishes if the QPCs are transparent (i.e. the QPCs are set to $n\times\frac{2e^{2}}{h}$, $n$ is an integer), otherwise it takes a finite value. This behavior is well characterized by Cutler-Mott relationCM69 ; SI86

[TABLE]

where $V_{th,i}$ is the thermo-voltage between $i$th QPC and common 2DEG, $k_{B}$ is the Boltzmann constant, $e$ is the elementary electron charge, $G_{i}$ is the electric conductance of $i$th QPC and $\mu_{i}$ is the chemical potential within the QPCs. It is necessary to comment that the usage of two QPCs minimizes the effect due to temperature fluctuation in the large 2DEG regionANS98 ; ANP00 . As a result, we focus on the net thermo-voltage drop between the two small area $V_{th}$ = $V_{th,1}$ - $V_{th,2}$. However, most of the results can be obtained with a single-QPC setup.

Fig. 1 (b) demonstrates the excellent agreement between Cutler-Mott relation (upper panel) and measured $V_{th}$ (lower panel), which can be used to calibrate the system. The thermo-voltage was recorded by sweeping gate voltage applied to QPC2 whereas QPC1 was set to 3G0. Tuning $V_{1}$ and $V_{2}$ (gate voltage applied to QPC1 and 2) independently, we could extract a detailed $V_{th}$ spectrum, as illustrated in Fig. 1(c). Well organized patterns were observed. The bright horizontal segments happened when QPC2 was transparent, and QPC1 was not; vice versa for the vertical segments. The dark square region corresponded to the situation when both QPC1 and QPC2 were transparent. A complete spectrum, including the 2D regime, can be found in supplementary material, S1.

After calibrating the measurement system at zero magnetic field, a transverse magnetic field $B_{\perp}$ was applied to the sample. $B_{\perp}$ is expected to influence the thermal property of the system because it affects the electron cooling and heat generation processes. At zero field, the temperature reaches maximal at the center of the sample then decreases to reach the lattice temperature at the contacts; in the presence of a transverse magnetic field, the heat is generated when the edge channels connect to the contacts, creating two hot spots close to each contact and located on opposite sides (top and bottom) of the sampleKLA91 .

It is necessary to comment that the position of thermo-voltage peak with respect to gate voltage $V_{i}$ is mainly determined by the ith QPC within the range of magnetic field in the present work. Using a single QPC or double QPCs leads to a similar result, as illustrated by Fig. S1. On the other hand, the amplitude of the peak could be sensitive to the arrangement of the QPCs. For instance, the amplitude of thermo-voltage peak with the QPCs locate at the same edge may differ from that with the QPCs locate at the opposite edges.

Here we present a detailed magnetic field dependence of trace in Fig. 1(b) (i.e. QPC1 is set to 3G0 at zero field). It was found that the field had a twofold impact, as demonstrated in Fig. 2$:$ first, the magnitude of $V_{th}$ peaks was enhanced noticeably by $B_{\perp}$, which is consistent with the predicted increase in thermopower (hence thermo-voltage) if size quantization owing to QPC is strongMS03 ; this enhancement is also related to the fact the hot spot moves towards the edge of the sample; second and more important, a shift in the position of $V_{th}$ peaks was introduced by $B_{\perp}$ as shown in Fig. 2(a) and (b) [also supplementary material, Fig.S2]. The position of $V_{th}$ peak changed rapidly when -3 T $<B_{\perp}<$ 3 T, and then saturated (when bulk Landau filling factor approaches $\nu$ = 2). The peak position follows such a relation $V_{pos}=c_{3}|B|^{3}+c_{2}B^{2}+c_{0}$ [red solid trace in Fig.3(b)]. The $B^{2}$ term dominated in small field regime whereas $|B|^{3}$ became the leading factor when $B_{\perp}\geqslant$ 0.5 T. In addition, it was also noticed that the $V_{th}$ peak position agreed well with that predicted by $\frac{\partial lnG_{2}}{\partial V_{2}}$ (i.e., Cutler-Mott relation) in small field ($B_{\perp}<$ 0.6 T); however, it diverged significantly from the prediction in a large field regime [Fig. 2 (c)]. Besides, $\frac{\partial lnG_{2}}{\partial V_{2}}$ split into two spin-resolved branches with increasing field due to Zeeman splitting whereas there was no sign of such splitting in $V_{th}$ peak.

In principle, both $V_{th}$ and $\frac{\partial lnG_{2}}{\partial V_{2}}$ monitor the position of the 1D subbands within the QPC, which determine the heat and charge transport through the QPC. The fact that $V_{th}$ peak correlates with $\frac{\partial lnG_{2}}{\partial V_{2}}$ in a small field regime where $B^{2}$ dominated, indicates that the $B^{2}$ term arises from magnetic depopulation induced 1D subbands rearrangementBTN86 ; VKW91 . The origin of the $|B|^{3}$ term, on the other hand, was unclear. Higher-order $B$ dependence can be induced if the electrostatic confinement is not parabolic, for instance, due to disorderBTN86 ; however, the impact of non-parabolic confinement should also be revealed in electrical measurement $\frac{\partial lnG_{2}}{\partial V_{2}}$. Besides, the parabolicity of electrostatic confinement is double-checked by fitting the conductance to the saddle point model [note 2 and Fig. S3 of supplementary information]. Therefore the $|B|^{3}$ term is a unique characteristic of heat transport.

Anomalous temperature dependence associated with the magnetic field was also observed, as shown in Fig. 3 [see note 3 of supplementary information for details on the fitting]. It was found that the magnitude of thermo-voltage peak attenuated rapidly with increasing lattice temperature at low magnetic field similar to the previous reportANS98 ; ANP00 as seen in Fig. 3(a), and exhibited a universal scaling [Fig. 3(b) and (c)]; on the other hand, it was rather insensitive to temperature change in the regime where $|B|^{3}$ term dominates, as exemplified by results at B = 0.8 T in Fig. 3(d). It is intuitive to think that the observed temperature might be relevant with the edge channels because the QPC is attached to the hot spot near the Ohmic contact via the medium of edge channels. This scenario is somehow not supported by the data shown in Fig. 3(e). It is particularly interesting to note $V_{th}$ obtained at 500 (Shubnikov$–$de Haas dip), 530 (peak) and 550 mT (dip) showed similar behavior, whereas it was well known that electrical measurements at SdH dip and peak had rather distinguishable temperature dependenceWCT85 ; LCE99 ; ZTS05 .

The results at zero magnetic field can be well described by Cutler-Mott relation, which is a single-electron framework. Our results suggest this framework does not hold in the presence of a finite transverse magnetic field. It is helpful to discuss several mechanisms beyond single-electron framework which are known to affect thermal conduction$:$

I. Phonon drag. Phonon drag augments thermopower (also thermo-voltage) with increasing magnetic fieldFDS88 ; ROJ88 . However, phonon drag usually freezes out below 0.6 K in the GaAs heterojunctionROJ88 . In our experiment, there was no sign of phonon drag switching on/off at the finite magnetic field.

II. A spin density wave (SDW). An SDW can be induced by a transverse magnetic fieldYL83 ; KSC86 ; NCY88 ; KFS96 , and it is separated from the single-particle mode by an energy gapPSD89 ; SGP94 . The melting temperature of an SDW is enhanced by the magnetic fieldNCY88 ; YCU90 , which might result in the observed evolution in temperature dependence. However, observations on an SDW in GaAs heterojunction usually involve excitation between several 2D subbandsPSD89 ; SGP94 , which is unlikely to be found in the current experiment.

III. Electron-electron (e-e) interaction. It is known that e-e interaction can make thermal conduction of a 1D system diverge from Cutler-Mott relation, such as the finite thermopower at ’0.7 conductance anomaly’ANP00 ; BMF19 . However, a detailed theory on QPC-mediated thermal conduction that incorporates electron-electron interaction, especially in the presence of a transverse magnetic field, is lacking. More specifically, it is difficult to comment at the moment why the e-e interaction prevents the occurrence of spin splitting in $V_{th}$.

In conclusion, we have observed unexplained thermal conduction through a 1D channel in the presence of a transverse magnetic field. The magnetic field and temperature dependence indicate that the thermal conduction at the finite magnetic field is beyond the single-particle picture. The results are important when employing the QPC to probe energy flow in integer/fractional quantum Hall or other exotic systems, where the magnetic field plays an important role.

See the supplementary material for the results under different experimental conditions and details on the fitting of temperature dependence data.

The authors gratefully acknowledge fruitful discussions with Karl-Fredrik Berggren, Kalarikad Thomas, and James Nicholls. The work was funded by United Kingdom Research and Innovation (UKRI).