Thermoelectricity in Quantum-Hall Corbino Structures
Mariano Real,1 Daniel Gresta,2 Christian Reichl,3 J urgen Weis,4 Alejandra Tonina,1
Paula Giudici,5 Liliana Arrachea,2 Werner Wegscheider,3 and Werner Dietsche3, 4
1Instituto Nacional de Tecnolog a Industrial, INTI and INCALIN-UNSAM,
Av. Gral. Paz 5445, (1650) Buenos Aires, Argentina
2International Center for Advanced Studies, ECyT-UNSAM,
25 de Mayo y Francia, 1650 Buenos Aires, Argentina
3Solid State Physics Laboratory, ETH Z urich, CH-8093 Z urich, Switzerland
4Max-Plack-Institut f ur Festk orperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
5INN CNEA-CONICET, Av. Gral. Paz 1499 (1650) Buenos Aires, Argentina
We measure the thermoelectric response of Corbino structures in the quantum Hall e ect regime
and compare it with a theoretical analysis. The measured thermoelectric voltages are qualitatively
and quantitatively simulated based upon the independent measurement of the conductivity indicat-
ing that they originate predominantly from the electron di usion. Electron-phonon interaction does
not lead to a phonon-drag contribution in contrast to earlier Hall-bar experiments. This implies a
description of the Onsager coe cients on the basis of a single transmission function, from which
both thermovoltage and conductivity can be predicted with a single tting parameter. It further-
more let us predict a gure of merit for the e ciency of thermoelectric cooling which becomes very
large for partially lled Landau levels (LL) and high magnetic elds.
I. INTRODUCTION
The quantum Hall e ect (QHE) which occurs in two-
dimensional electron systems (2DES) exposed to quan-
tizing magnetic elds is one of the most prominent ex-
amples of the synergy between fundamental physics and
quantum technologies1. It is topological in nature and
intrinsically related to exotic properties of matter, like
fractionalization and non-abelian statistics2{4. At the
same time, these complex properties are precisely the
reason for its robustness and appeal for practical appli-
cations. It is nowadays at the heart of the de nition
of the electrical metrological standards5, while it is also
a promising platform for the development of topological
quantum computation6.
Measurements of the entropy would be of great im-
portance in verifying the theoretically expected quan-
tum states, particularly of the non-abelian ones. One
possibility to access entropy in a 2DES is measuring
thermoelectricity7,8. But, although thermoelectricity
had been studied both experimentally and theoretically
since the discovery of the integer QHE,9{11 it had not
been possible to reconcile the experimental results ob-
tained with Hall-bars, Fig. 1 (a), with theories based
upon electron di usion. The overwhelming e ect of
phonon-drag was invoked as one reason9,10,12, but more
recently inherent problems connected with the topology
of the Hall-bar geometry, a ecting both phonon-drag and
electron di usion, have been realized8. The longitudi-
nal thermopower (or Seebeck coe cient) measured along
a Hall bar resembles closely the longitudinal resistance,
both in the phonon-drag and in the di usive regime while
an oscillating behavior with sign changes was expected by
the theory. It was suggested that the longitudinal ther-
mopower could be measured correctly in Corbino geom-
etry, Fig. 1 (b), where due to the circular geometry, the
thermal bias is applied radially, hence, any thermal and
FIG. 1. The two sample designs to investigate thermoelectric
e ects, Hall-bar (a) and Corbino (b). The dark gray areas
are the 2DES. The hot and the cold contacts for measuring
the thermovoltage are at two ends of the rectangular shaped
Hall-bar. For the Corbino, the hot contact is in the center of
the 2DES which is surrounded by the cold one.
electrical transport is induced along the radial direction13
and the transport takes place through the bulk.
Early thermopower experiments in Corbino geometry
failed to observe the expected sign-changing behavior14.
It was reported however in other experimental works us-
ing the Corbino geometry15. The latter used rf-heating of
the 2DES to produce the temperature gradient directly
in the 2DES claiming that no phonons are involved in
the measured thermopower.
In this article we report Corbino thermopower mea-
surements in the QHE regime by using a conventional
heater in the center of the device. This way, a temper-
ature gradient is set up in both the substrate and the
2DES. Results at temperatures from about 300 mK to
2 K are presented and compared with theoretical results
where both electrical conductance and thermopower are
modeled based upon the same transmission function. A

2
very good agreement over our range of magnetic elds
and temperatures is found. This demonstrates that the
substantial disagreement which was typical for Hall bars
can be removed by using the Corbino topology. In con-
trast to Hall-bar studies, it is not necessary to consider
phonon-drag in the theory.
II. EXPERIMENTAL DETAILS
A. Setup
Our setup is sketched in Fig. 2. An AuPd thin- lm
heater is inserted in the center of the Corbino samples
and heated with an AC current with a frequency f of
a few Hertz producing a temperature oscillation of 2f .
In this way, a radial thermal gradient is induced be-
tween the center and the external edge of the sample,
which is assumed to be close to the temperature T of
the bath. The device used here consists of ve con-
centic ohmic contact rings with diameters ranging from
0:4 mm to 3:2 mm made by alloying Au-Ge-Ni into the
2DES structure forming four independent Corbino rings.
Under the heater and outside of the rings the 2DES is re-
moved. It is assumed that the local temperature over the
2DES follows the one of the underlying GaAs substrate.
This was already veri ed by Chickering et al.16,17 down
to much lower temperatures than the ones used here. We
neglect possible anisotropies in the heat conductivity of
the substrate due to the ballistic nature of the phonons
because these become only relevant if the dimension of
the heater and the contacts are much smaller than the
substrate thickness18.
The four Corbino rings of this device allow not only to
measure thermopower at four di erent radial distances
from the heater but also to determine, in a di erent
experiment, the temperatures at the di erent ring po-
sitions. This can be done by using the conductances of
the four Corbinos as thermometers. Measuring the con-
ductance as function of bath temperature without any
heat applied is used for calibration. With the heater on,
the temperatures at the di erent rings can be measured.
As a response to the thermal bias between the cen-
ter and the edge of the sample, charges di use across
the Corbino ring which is compensated by generating a
voltage with frequecy 2f between the inner and the outer
circumferences. The sign and the magnitude of this ther-
movoltage is determined by the transmission function as
discussed in the theory section. The thermoelectric re-
sponse in this device is much simpler than the one in
the Hall bar geometry, where the transport takes place
along longitudinal and transverse directions with respect
to the applied biases. In fact, in the Corbino geome-
try, the thermovoltage develops along the direction of
the temperature bias.
The samples were grown by molecular beam-epitaxy
on GaAs wafers having a single 2DES located in a
30 nm wide quantum well with Si-doped doped lay-
FIG. 2. Scheme of experimental setup. (a) Cross-section of
the sample, notice that the heater element is over the sub-
strate outside the 2DES. (b) Measurement con gurations for
the conductance and the thermovoltage are shown in light-
grey (magenta online) and black (blue online), respectively.
LIA denotes lock-in ampli er. The two type of experiments
were done in separate runs. Only two of the four Corbino
rings are labeled in the gure.
ers on both sides. Data from two samples from two
wafers, A and B are presented here. Separate test pieces
from these wafers in van-der-Pauw geometry had mobil-
ities of 21 106 cm2 V 1 s 1 and 18 106 cm2 V 1 s 1
at electron densities of ne = 3:06 1011 cm 2 and ne =
2:0 1011 cm 2, respectively, measured at 1:3 K in the
dark.
The Corbino samples were glued in a standard com-
mercial ceramic holder with gold-plated pins and base
and a 3 mm diameter hole drilled in the middle to reduce
thermal contact to the samples. The measurements were
performed in vacuum in a 3He cryostat equipped with a
14 T magnet being able to achieve a base temperature of
250 mK.
Fig. 2 also shows the con gurations used for the mea-
surements of the conductance (light-gray, magenta on-
line) and of the thermovoltage (black, blue online). The
conductance G was measured by applying an AC volt-
age through a voltage divider and measuring the cur-
rent with an ampli er (IUAmp). Thermovoltage Vtp was
measured in separate runs by passing an AC current of
frequency f to the central heater having a resistance of
about 650 . The thermopower induced in the sample
was measured by using a 1000 di erential DC voltage
ampli er (DCamp). The input impedance of this ampli-
er must be very high because the internal resistance of
the Corbino device diverges in the quantum-Hall states.
We used an ampli er with an iput impedance of about
1 T 19. Very little frequency dependence of the thermo-
voltage was found between f = 3 Hz and 100 Hz. Most
measurements were done at f = 13:8 Hz. To avoid e ects
of time-dependent magnetic uxes, the waiting time for

3
each data point was set to a few seconds to guarantee the
stabilization of the magnetic elds at a constant value.
B. Thermovoltage measurement
Experimental results for both the thermal voltage
(solid blue) and the conductance (solid orange) of sample
A are shown in Fig. 3 at a base temperature T of 269 mK
and an average heater power P of 277 nW. The magnetic
eld is swept from 0.3 Tesla to 5 Tesla. The conduc-
tance shows the typical Shubnikov-de-Haas (SdH) oscil-
lations with the spin splitting becoming visible at about
0.9 Tesla and the conductance minima aproaching zero
at even lling factors less than 20. The thermovoltage
Vtp shows numerous features. At small magnetic elds,
it oscillates with a similar periodicity as the conductance
changing sign both at the conductance maxima and min-
ima. At higher magnetic elds additional features appear
in the regions of the conductance minima which become
very signifcant and chaotic at even larger magnetic elds
where the conductance minima are wider. Between the
conductance minima, the thermovoltage Vtp now changes
to saw-tooth like behavior, still changing signs at both
the maxima and the minima of the conductance. Such
sign changes had not been observed in the earlier Hall-
bar experiments but were already seen in the previous
rf-based Corbino experiments15 and had been expected
theoretically8.
We have observed the sign change behavior in a similar
way on several samples with di erent densities and mo-
bilities. Data of the sample B are presented in Fig. 4
showing Vtp measured across the three di erent outer
Corbino rings at a temperature of 600 mK. The oscil-
latory behavior of Vtp is again clearly visible as are the
large signals in the regions which correspond to the con-
ductance minima.
The large signals have not been reported before. These
are no spurious signals. They are reproducible, they per-
sist if the magnetic eld is stopped and kept constant
or if the sweep direction is reversed. They are de nitely
thermally induced signals and are not produced by an
electromagnetic crosstalk. The large signals vanish by
applying a dc current ontop of a square AC current. This
leads to a constant heating and thus a vanishing temper-
ature oscillation but would leave any suspected crosstalk
unchanged. We will speculate at the end of this paper
about possible origins of the large signals.
In the following we will concentrate on the analysis of
the thermal voltages outside of the SdH minima. We
show that the magnetic eld trace of both Vtp and con-
ductance G can be tted using only charge di usion. The
same transmission function based upon model Landau
levels is used for calculating both conductance and ther-
movoltage. The only tting parameter will be the tem-
perature gradient. The resulting ts are already shown
in Fig. 3 as dashed lines.
FIG. 3. Conductance G and thermovoltage Vtp as a function
of the magnetic eld B for the ring 2 in Fig. 2 at tempera-
ture T with power P supplied at the heater. Experimental
data is plotted in solid lines. Theoretical (dashed) plots are
based on the calculation of Eq. (2) with the respective inferred
transmission functions.
FIG. 4. Vtp response of sample B at di erent rings at a bath
temperature of 600 mK and a heater power of 213 nW. ring 2
and ring 3 present a greater temperature gradient and hence
a larger voltage response than ring 4.
III. THERMOELECTRIC RESPONSE
A. Onsager coe cients
We consider the Corbino geometry in Fig. 1 (b) to
describe the thermoelectric transport. The Corbino ring
acts as a conductor in radial direction between hot and
cold reservoirs with a temperature bias of T . In linear
response, the corresponding charge and heat currents for

4
small T and bias voltage V can be expressed as20

IC=e
IQ

=

L11 L12
L21 L22

X1
X2

; (1)
where X1 = eV=kBT and X2 = T=kBT 2 and L^ is
the Onsager matrix. The electrical and thermal con-
ductances are, respectively, G = e2L11=T , and =
DetL^=

T 2L11

. S = L12=L11 de nes the Seebeck and
= L21=L11 is the Peltier coe cient. For ballistic or
di usive transport Lij depends only on the quantum dy-
namics of the electrons in the presence of the magnetic
eld and the disorder of the sample. They are described
by a transmission function T ("),
Lij = T
Z
d"
h
@f(")
@"
(" )i+j 2 T ("); (2)
where f(") = 1=(e(" )=kBT +1) is the Fermi distribution
function, is the chemical potential and T is the tem-
perature of the electrons. In the presence of disorder and
absence of electron-electron interactions T (") was origi-
nally calculated by Jonson and Girvin21. At high tem-
peratures, electron-phonon interaction gives rise to an
additional component to the transport coe cients Lij .
B. Conductance and thermovoltage.
Our goal is to accurately describe the electronic com-
ponent of the Onsager coe cients obtained from the ex-
perimental data. At rst we measure the conductance
G(B) as a function of the applied magnetic eld B.
The thermovoltage Vtp is measured separately and corre-
sponds to the voltage for which IC = 0 in Eq. (1),
Vtp(B) = S(B)
T
T
: (3)
Here S(B) is the Seebeck coe cient as a function of the
magnetic eld, T is the temperature of the bath (cold
nger in our case) and T is the temperature di erence
between the two contacts of the Corbino ring under in-
vestigation. From the data of G(B) we infer the trans-
mission function T (") entering Eq. (2). Given T ("), we
can evaluate the electrical component of the other On-
sager coe cients, in particular L12(B). Through Eq. (3),
this leads to a theoretical prediction for the behavior of
Vtp(B) resulting from the electrical transport, which can
be directly contrasted with the experimental data.
There are two regimes to be considered for the cal-
culation of T ("): (i) At low magnetic elds, where the
di erent Landau levels are not clearly resolved, we calcu-
late the transmission function with the model introduced
in Ref.8,21. The latter is based on a single-particle pic-
ture for the 2DES in the presence of a magnetic eld
and elastic scattering introduced by impurities. (ii) For
higher magnetic elds, where the di erent lled LL are
clearly distinguished, and separated by a gap, we use
the fact that in the limit of T ! 0, Eq. (2) leads to
T ( ) G( )=e.
C. Transmission function
1. Low magnetic eld
Here we consider the transmission function8,21
T (") =
X
n;
(n+ 1)!2c
8 h
An; (")An+1; ("); (4)
where is a geometric factor relating the conductance
to the conductivity, while An; (") = Im

Gn; (")

, being
Gn; (") =

" "n; (")
1
the Green function cal-
culated within the self-consistent Born approximation.
"n; = ~(n + 1=2)!c BB=2 is the energy of the Lan-
dau levels, including the Zeeman splitting, with corre-
sponding, respectively, to ="; #. Here, B is the Born
magneton, !c = eB=m is the cyclotron frequency, and
m = 0:067me is the e ective mass of the electrons in
the structure and me is the electron mass. The e ect
of disorder due to impurities introduces a widening in
the Landau levels, which is accounted for the self-energy
(") = (! "L)=2 i
p
1 (" "L)2=(4 2). Here "L
is the energy of the Landau level which is closest to ".
This model has two tting parameters: and , which
we adjust to t the data of the conductance G, through
Eq. (2). This model fails to reproduce G(B) for high
magnetic elds (B > 1 T). Thus a di erent model has to
be used in this regime.
2. High magnetic eld
For higher magnetic elds, satisfying kBT ~!c,
and ~!c, we can infer the transmission function
more e ciently from the behavior of the conductance
within a range of magnetic elds in the neighborhood
of a given lling fraction . Notice that in the limit
of T ! 0, the derivative of the Fermi function enter-
ing Eq. (2) of the main text, has the following behavior,
@f(")=@"! (" ). Therefore, for low temperatures,
such that kBT ~!c, we have
T ( )
G( )
e
; =
~eB
2m
; B +1 < B < B ; (5)
where B = neh=(e ) is the magnetic eld correspond-
ing to the ling fraction , while is the Fermi energy
for the range of B within two consecutive integer lling
factors.
IV. RESULTS
A. Thermoelectric response
Results for the conductance and the thermovoltage are
shown in Fig. 3 for the temperature 269 mK. The ex-
perimental data for G and Vtp within the regime of low

5
FIG. 5. Thermovoltage Vtp for a xed temperature and dif-
ferent powers P 0 applied at the heater, assuming T (P 0) =
P 0=P 1:08 mK. P and other details are the same as in Fig. 3.
magnetic eld is shown in the upper panel of the g-
ure along with the theoretical description based on the
transmission function of Eq. (4).
In the case of high magnetic eld, shown in the lower
panel, the theoretical description was based in the trans-
mission function of Eq. (5). Given T ("), we calculate the
Onsager coe cients of Eq. (2) and the Seebeck coe cient
S = L12=L11. The ratio T=T has been adjusted in or-
der to t the experimental measurements with Eq. (3).
The estimates for the temperature bias were T = 1 mK
and 1:08 mK, for low and high magnetic elds, respec-
tively. Overall, in particular for high magnetic elds, the
agreement between experiment and theory is excellent
within the range of B corresponding to partially lled
LL, for which G 6= 0.
Taking into account the good agreement between the
experimental and theoretical estimates of the tempera-
ture di erence T found in the analysis of the data of
Fig. 3, we now analyze the relation between the elec-
trical power supplied at the heater and T . In Fig. 5
we show experimental data for the thermovoltage at a
xed temperature and di erent heater powers. We have
assumed a linear dependence between these quantities.
Therefore, we have tted the experimental data with
the same Seebeck coe cient S(B) calculated for Fig. 3
and the following values of the temperature di erence,
T (P 0) = P 0=P 1:08 mK, being P 0 the power corre-
sponding to the experimental data and P the power used
in the data of Fig. 3. We see a very good agreement
between the theoretical prediction and the experimental
data. In Fig. 6 we discuss the evolution of Vtp as the
temperature grows, focusing on the high magnetic eld
region. The experimental data is presented along with
the theoretical prediction obtained by following the same
procedure of the previous Figs, and taking into account
the linear dependence of T with P explained in Fig. 5.
The agreement between the theoretical predictions and
the experimental data for magnetic elds corresponding
to partially lled LL within a wide range of tempera-
FIG. 6. Thermovoltage Vtp, as function of the magnetic eld
for di erent temperatures. In the case of 269 mK to 680 mK
a power of 277 nW was used, while for 1:37 K to 1:5 K the
heater power was 433 nW. Other details are the same as in
previous Figs. The scale for Vtp is the same in all panels.
ture is overall very good, improving as the temperature
decreases.
B. Theoretical estimate of T
From the behavior of the conductance we can infer
the transmission function T (") as explained before, from
where we can calculate the Onsager coe cients Lij . We
recall that the thermovoltage is de ned in Eq. (3). Given
the calculation of S(B), we need to adjust the parameter
T=T in order to t the data. Since the latter enters as
the slope in the linear function V (S), we analyze plots
of the measured Vtp vs the calculated S for values of B
within which the Landau levels are partially lled and
we t a linear function to obtain the slope. Examples
are shown in Fig. 7 for two di erent Landau levels, with
bath temperature T = 269 mK and a power of 277 nW.
The corresponding ts cast T = (1:01 0:06) mK in
the region from B = 2:21 T to 2:46 T (upper panel), and
T = (1:33 0:06) mK in the region from B = 2:625 T

6
FIG. 7. Measured Vtp signal vs calculated S within
the range of elds B = 2:21 T to 2:46 T (upper panel) and
B = 2:625 T to 3 T (lower panel). The slope of this relation
is T=T .
to 3 T (lower panel), with uncertainties corresponding
to a 95% con dence probability. Also notice that the
intercept, which was taken as a free parameter of the
regression is zero within the error in both cases.
The data of the other sample measured at 600 mK
shown in Fig. 4 can be analyzed similar for the several
rings. For the ring 2 we obtain T = (60 3) µK, while
for the ring 3 we get ( T = (110 10) µK and for ring 4
T = (74 50)µK. These values are considerably lower
than the ones of the sample A measured at 269 mK . The
reason is that the thermal conductivity of the substrate
increases with T 3 and at 600 mK the temperature gradi-
ent will be nearly 10 times smaller and, correspondingly,
a much smaller temperature di erence is expected at the
higher temperature.
C. Experimental estimate of T
The exact determination of T in a Corbino device
turns out to be challenging. The reason is the high
phonon-heat conductivity in the GaAs substrate leads to
small temperature gradients between the center and the
edges of the sample. One consequence is that the thermal
resistance from the sample to the ceramic carrier can no
longer be neglected. Using the temperature dependent
conductance of the Corbino rings as thermometers, we
were nevertheless able to make estimates. The inner-
FIG. 8. Bottom: Transmission function T ("). Top: Electron
contribution to the gure of merit ZT .
most and the outermost rings in Fig. 2 were used for this
measurement. The conductance minimum at lling fac-
tor 9 in Fig. 3 was used because it showed a pronounced
temperature dependence. Actual temperatures at these
rings were found by comparing the respective conduc-
tances with the heater on and o . The temperature rise
at both rings could be close to 50 mK with the cryostat
at 269 mK and heater powers reaching 300 nW. The tem-
perature di erence between the two rings was found to
be less than 9 mK at the highest power. This number
is only an estimate because the precision and the repro-
ducibility of the calibration procedure was limited by the
temperature control of the cryostat.
The determination of T could have been improved
by thinning the sample and thereby decreasing its ther-
mal conductance. We have done this for several samples
but the thermovoltage data as function of magnetic eld
became erratic. We suspect that the thinning led to inho-
mogeneities in the 2DES making the Vtp measurements
useless. Doping of the substrate with Cr would be an-
other way to decrease the thermal conductivity.
Alternatively, we estimated the temperature pro le
using literature values of the thermal conductivity .
From16 we deduced a of about 0.01 W/mK at 300 mK.
Using the simulation software Comsol a temperature dif-
ference of 2:5 mK between the center and edge of our sam-
ple was found from the heat- ow equation. This would
lead to a temperature di erence across ring 2 of about
1 mK in good agreement with the used t values.
V. THERMOELECTRIC PERFORMANCE
The quality of the thermoelectric performance, i.e.
the e ciency (for a heat engine), or coe cient of per-
formance (for a refrigerator) has found great interest
in recent years, particularly in the context of the bal-
listic transport along edge channels and nano sized
devices22{30. We nd that the theoretical performance

7
of a Corbino device in the Quantum-Hall regime is sur-
prisingly large being comparable to the highest predicted
values in devices based upon ballistic transport. The
performance is parameterized as the gure of merit20,
ZT = L221=DetL^, or S
2 times the ratio of the electric
and thermal conductivity.
The optimal Carnot e ciency/coe cient of perfor-
mance is achieved for ZT ! 1. The highest reported
values in real, usually semiconducting materials are be-
tween 1 ZT 2:720,31 while optimistic theoretical pre-
dictions in the ballistic edge-channel regime are ZT 432
or lower. In Fig. 8 we show the transmission function
T (") used for the Corbino in this work to t the ex-
perimental data of Fig. 3 within the high-magnetic eld
regime. We see that the sequence of sharp features at
the LL realize energy lters, leading to large values of
ZT 6. Thus, di usive transport across the bulk of
a Corbino device has a potentially higher performance
than the evisioned edge-channel devices. Fig. 8 suggests
that even higher ZT values should be possible at lower
temperatures. We stress that this analysis is based on
the assumption that the main contribution to the ther-
moelectric and thermal transport is due to the electrons.
Phononic thermal transport in the substrate would tend
to decrease the performance but would die out at even
lower temperatures with T 3 while the gure of merit
would probably increase. Thus one could envision that
the Corbino device could be used as a thermoelectric
cooler in the low mK regime for speci c purposes. Re-
placing the heater by the object to be cooled could al-
ready be su cient to form a realistic device.
VI. CONCLUSIONS
We analyzed the thermoelectric response of a Corbino
structure in the quantum Hall regime. For partially lled
Landau levels, we found an excellent agreement between
the experimental data and the theoretical description
based on the assumption that the thermoelectric response
originates in the di usion of electrons while electron-
phonon drag does not in uence the thermovoltage in tem-
perature range from 300 mK to 2 K. Clearly, the electron-
phonon interaction does not vanish in the Corbino geom-
etry, but the transfer of momentum from the phonons to
the electrons does not lead to a measurable voltage. Ac-
tually, it had been already noted long time ago that the
contribution of the phonon-drag mechanism to the ther-
moelectric coe cient L12 should be zero in the heat- ow
direction10 which is, simply put, a consequence of the
Lorentz force. It appears that only in Corbino devices
the vanishing contribution of phonon-drag is re ected
in the thermovoltage measurement. Within the di u-
sive model applicable for Corbino rings, we were able to
accurately t the temperature di erence producing the
thermopower based on the measured conductance traces
and nd that it to be consistent with both our experi-
mental temperature estimates and the one derived from
independent thermal conductivity data. The calculated
gure of merit ZT is remarkably high for high magnetic
elds indicating that this system is very promising as a
low-temperature cooling device or a heat engine.
Future work needs to clarify the origin the large volt-
age signals at the conductance minima, i.e. in the quan-
tized state where both the electric conductance and the
thermal conductance values of the Onsager equation van-
ish. Also di erent mechanisms might be relevant in
these regimes, like temperature driven magnetic ux13
or temperature dependent contact potentials which can-
not equilibrate in the conductance minima33. Another
important direction would be the extension of the exper-
iment to lower temperatures. Determining entropy in the
fractional quantum Hall regime could answer some urgent
questions about the entropy of the suspected non-abelian
states.
VII. ACKNOWLEDGEMENTS
We thank Klaus von Klitzing for his constant inter-
est and support, Achim G uth and Marion Hagel for
the wafer lithography, Mirko Lupatini, Luca Alt, and
Simon Parolo for help with the experiments. Peter
M arki provided the ampli ers used in this work while
Lars Tiemann contributed the measurement software
"Nanomeas" (www.nanomeas.com). We received use-
ful comments on the manuscript from Peter Samuelsson.
We acknowledge support from INTI and CONICET, Ar-
gentina. We are sponsored by PIP-RD 20141216-4905 of
CONICET, PICT-2017- 2726 and PICT-2018 from Ar-
gentina, Swiss National Foundation (Schweizerischer Na-
tionalfonds) NCCR "Quantum Science and Technology",
as well as the Alexander von Humboldt Foundation, Ger-
many.
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2 D. C. Tsui, H.L. Stormer, A. C. Gossard, Phys. Rev. Lett.
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