††thanks: to whom correspondence should be addressed<EMAIL_ADDRESS>
# Differential thermopower images as a probe of many-body effects in quantum
point contacts

Yuan Ren  Joshua A. Folk  Department of Physics and Astronomy, University of
British Columbia, Vancouver, BC V6T 1Z1, Canada  Stewart Blusson Quantum
Matter Institute, University of British Columbia, Vancouver, BC V6T 1Z4,
Canada  Yigal Meir  Department of Physics and the Ilse Katz Institute of
Nanotechnology, Ben-Gurion University, Beer Sheva 84105, Israel  Tomaž Rejec
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19,
SI-1000 Ljubljana, Slovenia  Jožef Stefan Institute, Jamova 39, SI-1000
Ljubljana, Slovenia  Werner Wegscheider  Department of Physics, ETH Zurich,
CH-8093 Zurich, Switzerland  Quantum Center, ETH Zurich, CH-8093 Zurich,
Switzerland

Mesoscopic circuit elements such as quantum dots and quantum point contacts
(QPCs) offer a uniquely controllable platform for engineering complex quantum
devices, whose tunability makes them ideal for generating and investigating
interacting quantum systems. However, the conductance measurements commonly
employed in mesoscopics experiments are poorly suited to discerning correlated
phenomena from those of single-particle origin. Here, we introduce non-
equilibrium thermopower measurements as a novel approach to probing the local
density of states (LDOS), offering an energy-resolved readout of many-body
effects. We combine differential thermopower measurements with non-equilibrium
density functional theory (DFT) to both confirm the presence of a localized
state at the saddle point of a QPC and reveal secondary states that emerge
wherever the reservoir chemical potential intersects the gate-induced
potential barrier. These experiments establish differential thermopower
imaging as a robust and general approach to exploring quantum many-body
effects in mesoscopic circuits.

Thermal measurements are currently emerging as powerful probes for the
investigation of strongly correlated quantum states in low-dimensional
systems. For instance, thermal conductance provides strong evidence for non-
abelian quasiparticles in fractional quantum Hall samples Banerjee  _et al._
(2018), and chemical-potential measurements have demonstrated the emergence of
Pomeranchuk-like local moments in twisted bilayer graphene Rozen  _et al._
(2021); Saito  _et al._ (2021). With a view to experimental implementations,
thermopower is of particular interest, as it can be read out as easily as
conductance while still probing thermodynamic characteristics. Thermopower has
been proposed as a metric to characterize the non-Fermi liquid nature of
strongly interacting quantum states in, for example, multi-channel charge
Kondo circuits Nguyen and Kiselev (2020) and condensed matter realizations of
the SYK model Kruchkov  _et al._ (2020). However, experimental investigations
of mesoscopic thermodynamics are still in their infancy. Typically,
comparisons are made to single-particle predictions of the Mott formula Cutler
and Mott (1969) as a way of testing for the presence of interactions, but
without offering microscopic insight into what role those interactions might
play. Here, we show that non-equilibrium thermopower measured in a mesoscopic
system offers an energy-resolved readout of many-body effects in the density
of states, that can be compared directly with state-of-the-art modelling.

As an initial test for this approach, we turn our attention to the most
prototypical of mesoscopic devices, the quantum point contact (QPC). QPCs are
short saddle-point constrictions in a 2D sheet of high-mobility electrons
whose transmission can be tuned by a gate voltage van Wees  _et al._ (1988);
Wharam  _et al._ (1988); Büttiker (1990). Near pinchoff, interactions dominate
the potential landscape seen by electrons in a QPC, and it has been proposed
that localized states emerge in and near the constriction due to interactions
Hirose  _et al._ (2003); Rejec and Meir (2006); Ihnatsenka and Zozoulenko
(2009); Güçlü  _et al._ (2008); Qian and Halperin (2008). Such localized
states can, for example, explain the phenomenology of an extra plateau — the
famed 0.7 structure — that appears in the conductance of the cleanest QPCs
Thomas  _et al._ (1998); Cronenwett  _et al._ (2002). Beyond their ability to
explain the conductance characteristics of 0.7 structure, however, direct
evidence for localized states remains scarce Iqbal  _et al._ (2013); Brun  _et
al._ (2014), and what evidence there is leaves open the possibility of
different interpretations Bauer  _et al._ (2013). Here, we demonstrate that
non-equilibrium thermopower data offers a complete spectroscopic map of the
energy of localized states in a QPC and their effect on the local density of
states, enabling a direct comparison to DFT (or other) models as a test of
their accuracy.

Figure 1:  (a) Device geometry and measurement setup showing currents for
heating ($I_{\rm H}$), monitoring QPC conductance ($I_{\rm G}$) and offsetting
chemical potentials of hot and cold reservoirs ($I_{\rm off}$), and the
$V_{\rm ds}$ measurement. Light blue: cold 2DEG; red: hot 2DEG; grey:
electrostatic gates. Reference QPC is discussed in Methods. Inset: scanning
electron micrograph shows QPC geometry. (b) Differential thermopower
$\widetilde{S}\equiv S\cdot G$ over the first two conductance steps for a
typical QPC (QPC1 shown here); line plots for $V_{\rm off}=0$ (right axis)
show peaks in $\widetilde{S}$ (solid) as well as $G^{\prime}$ (dashed) for
each step in $G$. (c) Transconductance ($G^{\prime}\equiv dG/dV_{\rm g}$) for
QPC1; Colorscale data plotted as a function of $V_{\rm g}$ (bottom axis) and
$V_{\rm off}$ (left axis). Line plot (right axis) of conductance $G$ for
$V_{\rm off}=0$ shows characteristic steps in $G$ and clear 0.7 structure.

Thermopower, $S$, quantifies the excess voltage that appears across a device
due to a temperature difference, $T_{\rm ds}\equiv T_{\rm d}-T_{\rm s}$,
between drain and source reservoirs when the current through the device fixed:
$S=-\partial V_{\rm ds}/\partial T_{\rm ds}$. $S$ typically reflects an
imbalance between the electron flow above the chemical potential, $\mu$, and
hole flow below $\mu$. That imbalance yields a temperature-driven current that
must be counteracted by a voltage to keep the net current constant.
Multiplying by the differential conductance, $G$, the product
$\widetilde{S}\equiv S\cdot G$ is often more useful to analyze than $S$ itself
as it provides direct access to the temperature-driven current rather than the
voltage that builds up to oppose it. $\widetilde{S}$ is connected to the
derivative of the energy-dependent transmission through the device at $\mu$,
$dt/d\varepsilon|_{\varepsilon=\mu}$, making it sensitive to fine features in
the density of states and offering therefore an important advantage over
conductance, which depends on $t$. Early measurements of QPC thermopower
confirmed peaks in $\widetilde{S}$ at the entry of each new subband, where
$dt/d\varepsilon$ is large Kozub and Rudin (1994); Houten  _et al._ (1999),
whereas later experiments identified significant deviations from the
thermopower that would be expected in a simple picture of non-interacting 1D
subbands, and inspired the measurements presented here Appleyard  _et al._
(2000).

Although $\widetilde{S}$ is often measured with the current fixed at zero,
differential measurements of $\widetilde{S}$ may also be performed with a bias
current that offsets source and drain chemical potentials by $eV_{\rm
off}\equiv\mu_{\rm d}-\mu_{\rm s}$ ($V_{\rm off}$ is defined in the limit
$T_{\rm ds}=0$). The temperature of one of the reservoirs is then oscillated
to measure $\partial V_{\rm ds}/\partial T_{\rm ds}$. In mesoscopic systems
such as QPCs, $\widetilde{S}$($V_{\rm off}$) offers a spectroscopic readout of
thermopower that quantifies $dt/d\varepsilon$ at the chemical potential of the
reservoir whose temperature changes Appleyard  _et al._ (2000). This
represents a further distinct advantage of $\widetilde{S}$ over standard
measurements of differential conductance at finite bias, which superimpose
features at both source and drain potentials. (See, for example, Fig. 1b vs
1c.)

In this paper, we analyze $\widetilde{S}(V_{\rm off})$ spectra of QPCs
covering weakly to strongly interacting regimes [Fig. 1b]. The data are
broadly consistent with published experiments Appleyard  _et al._ (2000), but
a careful examination reveals robust signatures that are inconsistent with
existing theoretical models. The new realization described in this paper is
that those signatures are in fact a direct readout of many-body modifications
to the density of states in the QPC. In order to clarify this connection, we
extend density functional theory (DFT) previously developed to describe the
microscopic state of QPCs at equilibrium, to incorporate non-equilibrium
effects on the potential due to finite bias and different source/drain
temperatures. Although exact steady-state DFT has, in principle, been
formulated for non-equilibrium conditions Stefanucci and Kurth (2015); Sobrino
_et al._ (2019), a Landauer+DFT approach Hirose and Tsukada (1995); Lang
(1995) is used here (see the SI) due to the lack of a corresponding exchange-
correlation functional.

The result of our analysis is a more complete understanding of QPCs as
strongly interacting mesoscopic elements: the convenient but naive picture in
which gate voltages uniquely define a saddle-point potential barrier evolves
to one in which the self-consistent barrier that determines transmission is
additionally a result of charge and spin degrees of freedom at the barrier
responding to non-equilibrium conditions imposed by the reservoirs. Our data
highlight the critical role of localized states that emerge when the bottom of
a 1D subband crosses the chemical potential. These states were previously
invoked to explain 0.7 structure in QPC conductance Meir  _et al._ (2002);
Meir (2008); here they explain striking thermopower signatures that defy other
explanation.

QPCs were defined in the wall of a micron-wide channel (Fig. 1a). The interior
of the channel formed the grounded drain of the QPC, which could be Joule-
heated to $T_{\rm d}\sim 150\,$mK by passing current $I_{\rm H}$ while the
source remained at base temperature ($T_{\rm s}=100\,$mK) due to its low-
resistance connection to an ohmic contact. Independent AC currents $I_{\rm H}$
and $I_{\rm G}$ enabled simultaneous lock-in measurement of $S$ and $G$, while
$I_{\rm off}$ at DC was used to offset $\mu_{\rm s}$ and $\mu_{\rm d}$, see
Methods. The thermal voltage across the QPC was dominated by diffusion
thermopower, with phonons nearly absent at $100\,$mK.

Figure 1b shows $\widetilde{S}$ for a typical QPC. For comparison, Fig. 1c
shows the trans-conductance, $G^{\prime}\equiv dG/dV_{\rm g}\propto dG/d\mu$,
another metric that is related to $dt/d\varepsilon$ when $V_{\rm g}$ controls
$\mu$. This connection is, however, indirect in interacting systems where
$t(\varepsilon)$ itself depends on $\mu$, and when the shape of the potential
changes with $V_{\rm g}$ as is often the case in mesoscopic devices. Because
$G^{\prime}$ is based on conductance, Fig. 1c is symmetric around $V_{\rm
off}=0$, superimposing features at the source chemical potential (lower left
to upper right in the image) on others at the drain (upper left to lower
right). For the discussion that follows, the lack of symmetry in
$\widetilde{S}$ [Fig. 1b] is greatly clarifying, distinguishing responses at
negative energy from those at positive energy (below vs above the average
$\mu$). This will, for example be key to identifying the role of localized
states in the analysis to come.

Figure 2:  (a-d) Evolution of thermopower ($\widetilde{S}$, colorscale) with
$V_{\rm off}$ across the entry of the second subband into the QPC.
Experimental data from QPC1 (a) is compared to simulations (b,c,d) of
$\widetilde{S}$ calculated using the screened potential from local gates. The
potential in (b) [DFT] is calculated at $T_{\rm s}=T_{\rm d}=100\,$mK and in
equilibrium; in (c) [$\Delta\mu$-DFT], at $T_{\rm s}=T_{\rm d}=100\,$mK but
considering different source/drain potentials; and in (d) [$\Delta\mu$,$\Delta
T$-DFT], considering different source/drain potentials as well as different
temperatures. Dotted line in (c) traces the maximum of $\widetilde{S}$; the
identical line brought to (d) illustrates the spreading of large
$\widetilde{S}$ to the left due to temperature-dependent screening. (d, inset)
shows the numerical difference between $\widetilde{S}$ from (c) and (d), main
panels.

We first consider measurements in a regime of relatively weak interactions, at
the $2e^{2}/h\rightarrow 4e^{2}/h$ transition where the second subband drops
below the chemical potential. Figure 2 compares experimental data to three
levels of numerical simulation. Fig. 2b [DFT] represents an equilibrium
calculation: the QPC potential and LDOS are calculated self-consistently for
each $V_{\rm g}$, assuming $V_{\rm off}=0$ and $T_{\rm s}=T_{\rm d}=100\,$mK,
with $t(\varepsilon)$ obtained from Kohn–Sham scattering statesHirose and
Tsukada (1995); Lang (1995). $\widetilde{S}(V_{\rm off})$ is determined from
the simulation by comparing the current through the device for {$T_{\rm
d}=150\,$mK, $T_{\rm s}=100\,$mK} and $T_{\rm d}=T_{\rm s}=100\,$mK, where
current is calculated by integrating $t(\varepsilon)$ over thermally broadened
source and drain reservoirs offset by $V_{\rm off}$.

Figures 2c [$\Delta\mu$-DFT] and 2d [$\Delta\mu$,$\Delta T$-DFT] bring in the
non-equilibrium character of the experiment at increasing levels. First, in
Fig. 2c [$\Delta\mu$-DFT] the LDOS and $t(\varepsilon)$ are recalculated for
each {$V_{\rm g}$, $V_{\rm off}$} pair, including different reservoir
potentials but still assuming $T_{\rm s}=T_{\rm d}=100\,$mK. As before,
$\widetilde{S}$ is determined for each $t(\varepsilon)$ by comparing the
currents at $T_{\rm d}=150$ and $100\,$mK. Fig. 2d [$\Delta\mu$,$\Delta
T$-DFT] extends this to include different values of $T_{\rm d}$ from the
beginning, that is, including the effect of the elevated drain temperature on
the LDOS and thereby on $t(\varepsilon)$.

All simulations capture the diagonal motion of the subband, with a kink in the
slope at $V_{\rm off}=0$ due to the divergence in the 1D density of states at
the bottom of of the subband. But Figs. 2b,c miss the broadening of the
subband thermopower after crossing $V_{\rm off}=0$, which only appears once
finite temperature difference is taken into account (Fig. 2d). The difference
between Fig. 2c and 2d, shown in the inset of Fig. 2d, highlights one of the
most important insights to be gained from this work: temperature affects
mesoscopic transport, including thermopower, not only by broadening the
reservoir Fermi distributions (Figs. 2b,c), but also by changing the screened
QPC potential itself, the LDOS, and thereby $t(\varepsilon)$ (Fig. 2d). We
take advantage of this sensitivity to probe sharp features in the LDOS, which
are more strongly affected by a change in temperature.

For insight into this effect, consider the full electrostatic potential seen
by an incoming electron just at the entry of the second subband, including the
potential defined by electrostatic gates as well as repulsion from electrons
occupying the first or second subbands. With $\mu$ at the bottom of second
subband, the density of states is significantly larger just above $\mu$ than
below, so thermal broadening around $\mu$ leads to a net increase in electron
density (more electrons occupying the second subband). This extra negative
charge creates a more repulsive potential for the incoming electron, reducing
the total (positive) current and suppressing $\widetilde{S}$ when $V_{\rm
off}>0$. The contribution is opposite when the potentials are reversed,
yielding the broad positive feature in $\widetilde{S}$ when $V_{\rm off}<0$.
For further discussion of Fig. 2d, see the SI.

Figure 3:  (a) Characteristic signatures at the entry of the first subband
include a significant weakening of the subband thermopower just before it
crosses the chemical potential (‘A’) and a broad region of strong thermopower
at negative $V_{\rm off}$, extending into a diagonally moving feature deep in
the pinchoff regime (‘B’). Main panel data are from QPC2; inset shows
equivalent data from QPC1. (b) The $\Delta\mu$,$\Delta T$-DFT simulation
reflects all signatures of the data (saturated colorscale); inset shows the
$\Delta\mu$-DFT simulation over region surrounded by dotted line in main
panel. (c-g) LDOS along the length of the QPC (horizontal axis), versus energy
(vertical axis), for $V_{\rm off}=-0.8\,$mV and gate voltages marked ‘Xi’ in
panel (b). Dashed lines show $\mu_{\rm s}=0.4\,$meV and $\mu_{\rm
d}=-0.4\,$meV. Localized states are indicated by arrows. (h) Equivalent LDOS
map for point ‘Y’ in panel (b).

Building off this insight, we turn to the entry of the first subband into the
QPC, the $0\rightarrow 2e^{2}/h$ transition, where interactions are widely
believed to dominate conductance phenomenology (Fig. 3a). As in the case of
the second-subband data, the feature moving diagonally down from the upper
left reflects the subband lowering in energy with more positive gate voltage.
In contrast to Fig. 2a, however, this feature reduces significantly in
strength just before crossing $V_{\rm off}=0$ (location ‘A’ in Fig. 3a), then
after crossing $V_{\rm off}=0$ it broadens and strengthens much more
dramatically than for the second subband. In fact, the extension of this
feature to more negative $V_{\rm g}$ is so extreme as to create an extra ‘leg’
of strong $\widetilde{S}$ that moves down and to the left, as if associated
with the movement of the source instead of the drain (‘B’ in Fig. 3a).
Analogous features were observed in all QPCs measured.

These new features in the first-subband thermopower are clearly reproduced in
the $\Delta\mu,\Delta T$-DFT (Fig. 3b), but are missing from simulations that
leave temperature difference out of the self-consistent potential calculation
(inset in Fig. 3b). This demonstrates that they originate in the temperature-
dependent screening effect described previously. In order to understand this
phenomenology more fully, we examine the LDOS at several points in Fig. 3b.
Consider, first, the row of points X${}_{1}\rightarrow$X5 at $V_{\rm
off}=-0.8\,$mV, for which energy-dependent LDOS maps along the axis of the QPC
are illustrated in Fig. 3c–g. Localized enhancements of the LDOS (localized
states) are clearly visible anywhere $\mu_{\rm s}$ or $\mu_{\rm d}$ intersect
the potential barrier, especially noticeable in Fig. 3c–e. The fact that these
states are pinned close to $\mu_{\rm s}$ or $\mu_{\rm d}$ is key to their
strong effect on the thermopower.

The simplest cases are the extremes, X1 and X5. At X1, $\widetilde{S}$ is zero
simply because the barrier is well above $\mu_{\rm s}$ and $\mu_{\rm d}$, so
that no current can flow ($dt/d\varepsilon=0$ as $t=0$ at both $\mu_{\rm s}$
and $\mu_{\rm d}$). At $X_{5}$, $\widetilde{S}=0$ again because
$dt/d\varepsilon=0$ in the fully transmitting first subband, where both
chemical potentials lie. Next simplest is X4, where large $\widetilde{S}$ is
expected from a conventional thermopower picture, due to the alignment of
$\mu_{\rm d}$ with the bottom of the subband, where $dt/d\varepsilon$ is
maximum.

Points X2 and X3 are more interesting. At X2, for example, the conventional
picture would predict $\widetilde{S}=0$, because $t$ and $dt/d\varepsilon$ are
both zero at $\mu_{\rm d}$, and the large $dt/d\varepsilon$ at $\mu_{\rm s}$
does not contribute to the thermopower because $T_{\rm s}$ is not varied.
However, significant $\widetilde{S}$ is observed, in both simulation and
experiment. A closer analysis of the simulation results shows that
$\widetilde{S}$ appearing at this location is due to temperature-induced
charging of the localized state at $\mu_{\rm d}$, which affects the barrier
potential and thereby the current. This effect is even stronger at $X_{3}$,
where the localized state is sharper and the barrier maximum is far from
either reservoir potential, free to shift as the localized state charges.

Thermopower observed at $X_{3}$ is analogous to, but much stronger than, the
broadening of second-subband thermopower in Fig. 2d, compared to Fig. 2c. In
both cases, temperature-induced charging of the LDOS where $\mu_{\rm d}$
intersects the bottom of the subband shifts the entire subband up in energy,
tending to reduce the current. This broadening is stronger for the first
subband due to the localized state at $\mu_{\rm d}$, as seen from the
significantly wider extent of large $\widetilde{S}$ in Fig. 3b compared to
Fig. 2d.

The fact that a large $\widetilde{S}$ remains all the way to $X_{2}$ in Fig.
3b can be traced, in the simulation, to the second localized state at
$\mu_{\rm s}$ (see Fig. 3d). The exchange term included in the DFT is what
allows for the emergence of localized states; analogous simulations without
the exchange term retain the thermopower at $X_{3}$, but the extension to
$X_{2}$ (which maps to feature ‘B’ in the data) is gone. The localized state
at $\mu_{\rm s}$ makes the potential at the barrier top especially sensitive
to charging of the state at $\mu_{\rm d}$, just as quantum dots at a Coulomb
blockade transition charge sharply in response to a small changes in gate
voltage (see the SI).

The weakening of thermopower just above $V_{\rm off}=0$, marked $Y$ in Fig.
3b, is again a signature that appears only when temperature-induced screening
is taken into account, and can be motivated by the same qualitative arguments
as above. As seen in Fig. 3h, a localized state is pinned at $\mu_{\rm d}$,
but in this case $\mu_{\rm d}$ is above $\mu_{\rm s}$. Thermally induced
charging of the localized state (which raises the QPC barrier) therefore
reduces the current from drain to source, opposing the conventional sign of
thermopower and leading to the reduction seen at $Y$.

It is remarkable that this qualitative picture, involving a localized state
becoming more occupied (more negatively charged) at higher temperatures, so
consistently explains the sign of thermopower modifications throughout the
pinchoff region. These modifications depend on the derivative of 1D subband
LDOS with respect to energy: when $\mu_{\rm d}$ is just below a peak in the
LDOS, the state will charge with $T$; when it is just above, the state will
discharge, and the opposite sign of thermopower will be observed. The
consistency of the sign observed in both experiment and simulation is
therefore a clear indication that the localized state is pinned to just above
$\mu$, regardless of the details of the QPC electrostatic potential.

This pinning of the localized state just above $\mu$ was a characteristic
specifically of the unpolarized solution of the DFT equations, even though
there was another, polarized, solution that, at equilibrium, was lower in free
energy. Polarized DFT solutions are ones in which the LDOS may be different
between the two spin species; the thermopower calculated from this solution
did not match the experiment, see the SI. The unpolarized solution, presented
throughout this paper, forces them to remain the same. The fact that the
unpolarized solution more closely matches the data supports the role of Kondo
physics in the low-energy spectrum of QPCs: Kondo correlations would tend to
lower the energy of the spin-singlet (unpolarized) state, but these
correlations are not captured in the local density functional used here.

In conclusion, we have discovered a new route to probing the LDOS in
mesoscopic structures, by differential thermopower measurements at finite
bias. Whereas conventional thermopower is a single-particle effect that probes
energy-dependent scattering, this new mechanism is based on interactions and
reflects thermal screening of the self-consistent potential landscape due to
an energy-dependent density of states. As 0.7 structure in QPC conductance is
difficult to explain without a localized state, our data reflect robust
signatures in QPC thermopower, covering a large fraction of parameter space
near pinchoff, that also cannot be explained without this state. Remarkably, a
non-equilibrium DFT calculation that includes mean-field Coulomb interaction,
spin via an exchange potential, _and_ finite temperatures in the leads,
reproduces nearly all features of the thermopower data with no fine-tuning of
parameters.

METHODS: Multiple bias currents are used in this experiment to measure $S$ and
$G$ independently, and to control the chemical potential offset $V_{\rm off}$
between source and drain reservoirs. The measurement of $G$ is analogous to
nonlinear conductance data reported by many groups in the past, albeit in a
less-conventional current biased configuration. A small oscillating $I_{\rm
G}$, at frequency $f_{\rm G}$, is added to a DC $I_{\rm off}$. Locking in to
oscillations in $V_{\rm ds}$ at $f_{\rm G}$ then yields the differential
conductance $G$ at a $V_{\rm off}$ that is fixed by $I_{\rm off}$. Although
$V_{\rm off}$ can in principle be measured directly for a given $I_{\rm off}$
as the DC component of $V_{\rm ds}$, in practice we find it by a numerical
integration of $G^{-1}$ over $I_{\rm off}$, an approach that was found to
yield less measurement noise while giving consistent values.

The measurement of $S$ proceeds in a similar way: temperature is oscillated by
Joule heating due to current bias $I_{\rm H}$ through the channel. With
$I_{\rm H}$ oscillated at frequency $f_{\rm H}$, the thermal voltage is
detected by locking in to $V_{\rm ds}$ at $2f_{\rm H}$ as $I_{\rm off}$ is
scanned (as above) to investigate $S$ for varying source and drain chemical
potentials. The combination of two AC bias currents $I_{\rm H}$ and $I_{\rm
G}$ and one DC current $I_{\rm off}$ thus enables a simultaneous measurement
of $S$ and $G$ while scanning (and measuring) $V_{\rm off}$. $I_{\rm H}$ on
the order of $10\,$nA elevated the temperature by $60\pm 20\,$mK, estimated
using the magnitude of $S$ for $I_{\rm off}=0$ at higher subband transitions,
where thermopower in QPCs is well understood.

In practice the measured $V_{\rm ds}$ includes contributions both from QPC
thermopower and from the 2DEG in the channel. The extra 2DEG contribution can
be quantified by measuring the thermal voltage across the reference QPC set at
on a condutance plateau, $G_{\rm ref}=2e^{2}/h$ where the thermopower is zero,
but in practice this contribution was found to be negligible.

ACKNOWLEDGEMENTS: This project has received funding from European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme under grant agreement No 951541. YM acknowledges support by the
Israel Science Foundation (grant 3523/2020). TR acknowledges support by
Slovenian Research Agency under Contract No. P1-0044. JF acknowledges support
from the Stewart Blusson Quantum Matter Institute, the Natural Sciences and
Engineering Research Council of Canada, the Canada Foundation for Innovation,
the Canadian Institute for Advanced Research, and the Canada First Research
Excellence Fund, Quantum Materials and Future Technologies Program.

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