Título:  Thermoelectricity in QuantumHall Corbino structures 
Fuente:  Physical Review Applied, 14 
Autor/es:  Real, Mariano A.; Gresta, Daniel; Reichl, Christian; Weis, Jürgen; Tonina, Alejandra; Giudici, Paula; Arrachea, Liliana; Wegscheider, Werner; Dietsche, Werner 
Materias:  Termoelectricidad; Estructuras; Electrodinámica cuántica; Voltaje; Conductividad eléctrica; Electrones; Fonones; Enfriamiento; Mediciones; Campo magnético 
Editor/Edición:  American Physical Society; 2020 
Licencia:  info:eurepo/semantics/openAccess; https://creativecommons.org/licenses/byncsa/4.0/ 
Afiliaciones:  Real, Mariano A. Instituto Nacional de Tecnología Industrial (INTI); Argentina Gresta, Daniel. Universidad Nacional de San Martín (UNSAM); Argentina Reichl, Christian. ETH Zürich; Suiza Weis, Jürgen. MaxPlackInstitut für Festkörperforschung; Alemania Tonina, Alejandra. Instituto Nacional de Tecnología Industrial (INTI); Argentina Giudici, Paula. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Nanociencia y Nanotecnología (INNCNEACONICET); Argentina Arrachea, Liliana. Universidad Nacional de San Martín (UNSAM); Argentina Wegscheider, Werner. ETH Zürich; Suiza Dietsche, Werner. ETH Zürich; Suiza 


Resumen:  We measure the thermoelectric response of Corbino structures in the quantum Hall effect 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 indicating that they originate predominantly from the electron diffusion. Electronphonon interaction does not lead to a phonondrag contribution in contrast to earlier Hallbar experiments. This implies a description of the Onsager coefficients on the basis of a single transmission function, from which both thermovoltage and conductivity can be predicted with a single fitting parameter. It furthermore let us predict a figure of merit for the efficiency of thermoelectric cooling which becomes very large for partially filled Landau levels (LL) and high magnetic fields. 
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Thermoelectricity in QuantumHall Corbino Structures Mariano Real,1 Daniel Gresta,2 Christian Reichl,3 Ju¨rgen Weis,4 Alejandra Tonina,1 Paula Giudici,5 Liliana Arrachea,2 Werner Wegscheider,3 and Werner Dietsche3, 4 1Instituto Nacional de Tecnolog´ıa Industrial, INTI and INCALINUNSAM, Av. Gral. Paz 5445, (1650) Buenos Aires, Argentina 2International Center for Advanced Studies, ECyTUNSAM, 25 de Mayo y Francia, 1650 Buenos Aires, Argentina 3Solid State Physics Laboratory, ETH Zu¨rich, CH8093 Zu¨rich, Switzerland 4MaxPlackInstitut fu¨r Festk¨orperforschung, Heisenbergstrasse 1, D70569 Stuttgart, Germany 5INN CNEACONICET, Av. Gral. Paz 1499 (1650) Buenos Aires, Argentina We measure the thermoelectric response of Corbino structures in the quantum Hall eﬀect 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 indicating that they originate predominantly from the electron diﬀusion. Electronphonon interaction does not lead to a phonondrag contribution in contrast to earlier Hallbar experiments. This implies a description of the Onsager coeﬃcients on the basis of a single transmission function, from which both thermovoltage and conductivity can be predicted with a single ﬁtting parameter. It furthermore let us predict a ﬁgure of merit for the eﬃciency of thermoelectric cooling which becomes very large for partially ﬁlled Landau levels (LL) and high magnetic ﬁelds. arXiv:2003.01748v4 [condmat.meshall] 10 Aug 2020 I. INTRODUCTION The quantum Hall eﬀect (QHE) which occurs in twodimensional electron systems (2DES) exposed to quantizing magnetic ﬁelds is one of the most prominent examples 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 nonabelian statistics2–4. At the same time, these complex properties are precisely the reason for its robustness and appeal for practical applications. It is nowadays at the heart of the deﬁnition 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 importance in verifying the theoretically expected quantum states, particularly of the nonabelian 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 obtained with Hallbars, Fig. 1 (a), with theories based upon electron diﬀusion. The overwhelming eﬀect of phonondrag was invoked as one reason9,10,12, but more recently inherent problems connected with the topology of the Hallbar geometry, aﬀecting both phonondrag and electron diﬀusion, have been realized8. The longitudinal thermopower (or Seebeck coeﬃcient) measured along a Hall bar resembles closely the longitudinal resistance, both in the phonondrag and in the diﬀusive regime while an oscillating behavior with sign changes was expected by the theory. It was suggested that the longitudinal thermopower could be measured correctly in Corbino geometry, 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 eﬀects, Hallbar (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 Hallbar. 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 signchanging behavior14. It was reported however in other experimental works using the Corbino geometry15. The latter used rfheating 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 measurements in the QHE regime by using a conventional heater in the center of the device. This way, a temperature 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 ﬁelds 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 contrast to Hallbar studies, it is not necessary to consider phonondrag in the theory. II. EXPERIMENTAL DETAILS A. Setup Our setup is sketched in Fig. 2. An AuPd thinﬁlm 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 between 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 ﬁve concentic ohmic contact rings with diameters ranging from 0.4 mm to 3.2 mm made by alloying AuGeNi into the 2DES structure forming four independent Corbino rings. Under the heater and outside of the rings the 2DES is removed. It is assumed that the local temperature over the 2DES follows the one of the underlying GaAs substrate. This was already veriﬁed 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 diﬀerent radial distances from the heater but also to determine, in a diﬀerent experiment, the temperatures at the diﬀerent ring positions. This can be done by using the conductances of the four Corbinos as thermometers. Measuring the conductance as function of bath temperature without any heat applied is used for calibration. With the heater on, the temperatures at the diﬀerent rings can be measured. As a response to the thermal bias between the center and the edge of the sample, charges diﬀuse 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 thermovoltage is determined by the transmission function as discussed in the theory section. The thermoelectric response 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 geometry, the thermovoltage develops along the direction of the temperature bias. The samples were grown by molecular beamepitaxy on GaAs wafers having a single 2DES located in a 30 nm wide quantum well with Sidoped doped lay FIG. 2. Scheme of experimental setup. (a) Crosssection of the sample, notice that the heater element is over the substrate outside the 2DES. (b) Measurement conﬁgurations for the conductance and the thermovoltage are shown in lightgrey (magenta online) and black (blue online), respectively. LIA denotes lockin ampliﬁer. The two type of experiments were done in separate runs. Only two of the four Corbino rings are labeled in the ﬁgure. ers on both sides. Data from two samples from two wafers, A and B are presented here. Separate test pieces from these wafers in vanderPauw geometry had mobilities 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 commercial ceramic holder with goldplated 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 conﬁgurations used for the measurements of the conductance (lightgray, magenta online) and of the thermovoltage (black, blue online). The conductance G was measured by applying an AC voltage through a voltage divider and measuring the current with an ampliﬁer (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 Ω. The thermopower induced in the sample was measured by using a ×1000 diﬀerential DC voltage ampliﬁer (DCamp). The input impedance of this ampliﬁer must be very high because the internal resistance of the Corbino device diverges in the quantumHall states. We used an ampliﬁer with an iput impedance of about 1 TΩ19. Very little frequency dependence of the thermovoltage was found between f = 3 Hz and 100 Hz. Most measurements were done at f = 13.8 Hz. To avoid eﬀects of timedependent magnetic ﬂuxes, the waiting time for 3 each data point was set to a few seconds to guarantee the stabilization of the magnetic ﬁelds 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 ﬁeld is swept from 0.3 Tesla to 5 Tesla. The conductance shows the typical ShubnikovdeHaas (SdH) oscillations with the spin splitting becoming visible at about 0.9 Tesla and the conductance minima aproaching zero at even ﬁlling factors less than 20. The thermovoltage Vtp shows numerous features. At small magnetic ﬁelds, it oscillates with a similar periodicity as the conductance changing sign both at the conductance maxima and minima. At higher magnetic ﬁelds additional features appear in the regions of the conductance minima which become very signifcant and chaotic at even larger magnetic ﬁelds where the conductance minima are wider. Between the conductance minima, the thermovoltage Vtp now changes to sawtooth 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 Hallbar experiments but were already seen in the previous rfbased Corbino experiments15 and had been expected theoretically8. We have observed the sign change behavior in a similar way on several samples with diﬀerent densities and mobilities. Data of the sample B are presented in Fig. 4 showing Vtp measured across the three diﬀerent outer Corbino rings at a temperature of 600 mK. The oscillatory behavior of Vtp is again clearly visible as are the large signals in the regions which correspond to the conductance minima. The large signals have not been reported before. These are no spurious signals. They are reproducible, they persist if the magnetic ﬁeld is stopped and kept constant or if the sweep direction is reversed. They are deﬁnitely 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 temperature 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 ﬁeld trace of both Vtp and conductance G can be ﬁtted using only charge diﬀusion. The same transmission function based upon model Landau levels is used for calculating both conductance and thermovoltage. The only ﬁtting parameter will be the temperature gradient. The resulting ﬁts are already shown in Fig. 3 as dashed lines. FIG. 3. Conductance G and thermovoltage Vtp as a function of the magnetic ﬁeld B for the ring 2 in Fig. 2 at temperature 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 diﬀerent 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 coeﬃcients 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 Lˆ is the Onsager matrix. The electrical and thermal conductances are, respectively, G = e2L11/T , and κ = DetLˆ/ T 2L11 . S = L12/L11 deﬁnes the Seebeck and Π = L21/L11 is the Peltier coeﬃcient. For ballistic or diﬀusive transport Lij depends only on the quantum dynamics of the electrons in the presence of the magnetic ﬁeld and the disorder of the sample. They are described by a transmission function T (ε), Lij = −T dε ∂f (ε) (ε − µ)i+j−2 T (ε), h ∂ε (2) where f (ε) = 1/(e(ε−µ)/kBT +1) is the Fermi distribution function, µ is the chemical potential and T is the temperature of the electrons. In the presence of disorder and absence of electronelectron interactions T (ε) was originally calculated by Jonson and Girvin21. At high temperatures, electronphonon interaction gives rise to an additional component to the transport coeﬃcients Lij. B. Conductance and thermovoltage. Our goal is to accurately describe the electronic component of the Onsager coeﬃcients obtained from the experimental data. At ﬁrst we measure the conductance G(B) as a function of the applied magnetic ﬁeld B. The thermovoltage Vtp is measured separately and corresponds to the voltage for which IC = 0 in Eq. (1), ∆T Vtp(B) = −S(B) T . (3) Here S(B) is the Seebeck coeﬃcient as a function of the magnetic ﬁeld, T is the temperature of the bath (cold ﬁnger in our case) and ∆T is the temperature diﬀerence between the two contacts of the Corbino ring under investigation. From the data of G(B) we infer the transmission function T (ε) entering Eq. (2). Given T (ε), we can evaluate the electrical component of the other Onsager coeﬃcients, 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 calculation of T (ε): (i) At low magnetic ﬁelds, where the diﬀerent Landau levels are not clearly resolved, we calculate the transmission function with the model introduced in Ref.8,21. The latter is based on a singleparticle picture for the 2DES in the presence of a magnetic ﬁeld and elastic scattering introduced by impurities. (ii) For higher magnetic ﬁelds, where the diﬀerent ﬁlled 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 (µ) ∼ G(µ)/e. C. Transmission function 1. Low magnetic ﬁeld Here we consider the transmission function8,21 T (ε) = Λ (n + 1) 8πh ωc2 An,σ (ε)An+1,σ (ε), (4) n,σ where Λ is a geometric factor relating the conductance to the conductivity, while An,σ(ε) = Im Gn,σ(ε) , being Gn,σ(ε) = ε − εn,σ − Σ(ε) −1 the Green function calculated within the selfconsistent Born approximation. εn,σ = (n + 1/2)ωc ± µBB/2 is the energy of the Landau levels, including the Zeeman splitting, with ± corre sponding, respectively, to σ =↑, ↓. Here, µB is the Born magneton, ωc = eB/m∗ is the cyclotron frequency, and m∗ = 0.067me is the eﬀective mass of the electrons in the structure and me is the electron mass. The eﬀect of disorder due to impurities introduces a widening Γ in the Landau levels, which is accounted for the selfenergy Σ(ε) = (ω − εL)/2 − iΓ 1 − (ε − εL)2/(4Γ2). Here εL is the energy of the Landau level which is closest to ε. This model has two ﬁtting parameters: Λ and Γ, which we adjust to ﬁt the data of the conductance G, through Eq. (2). This model fails to reproduce G(B) for high magnetic ﬁelds (B > 1 T). Thus a diﬀerent model has to be used in this regime. 2. High magnetic ﬁeld For higher magnetic ﬁelds, satisfying kBT ωc, and Γ ωc, we can infer the transmission function more eﬃciently from the behavior of the conductance within a range of magnetic ﬁelds in the neighborhood of a given ﬁlling fraction ν. 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 (ε)/∂ε → δ(ε − µ). Therefore, for low temperatures, such that kBT ωc, we have T (µν ) ∼ G(µν ) , e eB µν = 2m∗ , Bν+1 < B < Bν , (5) where Bν = neh/(eν) is the magnetic ﬁeld corresponding to the ﬁling fraction ν, while µν is the Fermi energy for the range of B within two consecutive integer ﬁlling factors. IV. RESULTS A. Thermoelectric response Results for the conductance and the thermovoltage are shown in Fig. 3 for the temperature 269 mK. The experimental data for G and Vtp within the regime of low 5 FIG. 5. Thermovoltage Vtp for a ﬁxed temperature and different powers P applied at the heater, assuming ∆T (P ) = P /P 1.08 mK. P and other details are the same as in Fig. 3. magnetic ﬁeld is shown in the upper panel of the ﬁgure along with the theoretical description based on the transmission function of Eq. (4). In the case of high magnetic ﬁeld, shown in the lower panel, the theoretical description was based in the transmission function of Eq. (5). Given T (ε), we calculate the Onsager coeﬃcients of Eq. (2) and the Seebeck coeﬃcient S = L12/L11. The ratio ∆T /T has been adjusted in order to ﬁt 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 ﬁelds, respectively. Overall, in particular for high magnetic ﬁelds, the agreement between experiment and theory is excellent within the range of B corresponding to partially ﬁlled LL, for which G = 0. Taking into account the good agreement between the experimental and theoretical estimates of the temperature diﬀerence ∆T found in the analysis of the data of Fig. 3, we now analyze the relation between the electrical power supplied at the heater and ∆T . In Fig. 5 we show experimental data for the thermovoltage at a ﬁxed temperature and diﬀerent heater powers. We have assumed a linear dependence between these quantities. Therefore, we have ﬁtted the experimental data with the same Seebeck coeﬃcient S(B) calculated for Fig. 3 and the following values of the temperature diﬀerence, ∆T (P ) = P /P 1.08 mK, being P the power corresponding 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 ﬁeld 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 ﬁelds corresponding to partially ﬁlled LL within a wide range of tempera FIG. 6. Thermovoltage Vtp, as function of the magnetic ﬁeld for diﬀerent 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 (ε) as explained before, from where we can calculate the Onsager coeﬃcients Lij. We recall that the thermovoltage is deﬁned in Eq. (3). Given the calculation of S(B), we need to adjust the parameter ∆T /T in order to ﬁt 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 ﬁlled and we ﬁt a linear function to obtain the slope. Examples are shown in Fig. 7 for two diﬀerent Landau levels, with bath temperature T = 269 mK and a power of 277 nW. The corresponding ﬁts 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. 8. Bottom: Transmission function T (ε). Top: Electron contribution to the ﬁgure of merit ZT . FIG. 7. Measured Vtp signal vs calculated −S within the range of ﬁelds 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% conﬁdence 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 gradient will be nearly 10 times smaller and, correspondingly, a much smaller temperature diﬀerence is expected at the higher temperature. C. Experimental estimate of ∆T most and the outermost rings in Fig. 2 were used for this measurement. The conductance minimum at ﬁlling factor 9 in Fig. 3 was used because it showed a pronounced temperature dependence. Actual temperatures at these rings were found by comparing the respective conductances with the heater on and oﬀ. 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 temperature diﬀerence 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 reproducibility 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 thermal conductance. We have done this for several samples but the thermovoltage data as function of magnetic ﬁeld became erratic. We suspect that the thinning led to inhomogeneities in the 2DES making the Vtp measurements useless. Doping of the substrate with Cr would be another way to decrease the thermal conductivity. Alternatively, we estimated the temperature proﬁle using literature values of the thermal conductivity κ . From16 we deduced a κ of about 0.01 W/mK at 300 mK. Using the simulation software Comsol a temperature difference of 2.5 mK between the center and edge of our sample was found from the heatﬂow equation. This would lead to a temperature diﬀerence across ring 2 of about 1 mK in good agreement with the used ﬁt values. The exact determination of ∆T in a Corbino device turns out to be challenging. The reason is the high phononheat 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 V. THERMOELECTRIC PERFORMANCE The quality of the thermoelectric performance, i.e. the eﬃciency (for a heat engine), or coeﬃcient of performance (for a refrigerator) has found great interest in recent years, particularly in the context of the ballistic transport along edge channels and nano sized devices22–30. We ﬁnd that the theoretical performance 7 of a Corbino device in the QuantumHall regime is surprisingly large being comparable to the highest predicted values in devices based upon ballistic transport. The performance is parameterized as the ﬁgure of merit20, ZT = L221/DetLˆ, or S2 times the ratio of the electric and thermal conductivity. The optimal Carnot eﬃciency/coeﬃcient of performance is achieved for ZT → ∞. The highest reported values in real, usually semiconducting materials are between 1 ≤ ZT ≤ 2.720,31 while optimistic theoretical predictions in the ballistic edgechannel regime are ZT ∼ 432 or lower. In Fig. 8 we show the transmission function T (ε) used for the Corbino in this work to ﬁt the experimental data of Fig. 3 within the highmagnetic ﬁeld regime. We see that the sequence of sharp features at the LL realize energy ﬁlters, leading to large values of ZT ∼ 6. Thus, diﬀusive transport across the bulk of a Corbino device has a potentially higher performance than the evisioned edgechannel 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 thermoelectric 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 ﬁgure 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 speciﬁc purposes. Replacing the heater by the object to be cooled could already be suﬃcient to form a realistic device. contribution of the phonondrag mechanism to the thermoelectric coeﬃcient L12 should be zero in the heatﬂow direction10 which is, simply put, a consequence of the Lorentz force. It appears that only in Corbino devices the vanishing contribution of phonondrag is reﬂected in the thermovoltage measurement. Within the diﬀusive model applicable for Corbino rings, we were able to accurately ﬁt the temperature diﬀerence producing the thermopower based on the measured conductance traces and ﬁnd that it to be consistent with both our experimental temperature estimates and the one derived from independent thermal conductivity data. The calculated ﬁgure of merit ZT is remarkably high for high magnetic ﬁelds indicating that this system is very promising as a lowtemperature cooling device or a heat engine. Future work needs to clarify the origin the large voltage signals at the conductance minima, i.e. in the quantized state where both the electric conductance and the thermal conductance values of the Onsager equation vanish. Also diﬀerent mechanisms might be relevant in these regimes, like temperature driven magnetic ﬂux13 or temperature dependent contact potentials which cannot equilibrate in the conductance minima33. Another important direction would be the extension of the experiment to lower temperatures. Determining entropy in the fractional quantum Hall regime could answer some urgent questions about the entropy of the suspected nonabelian states. VII. ACKNOWLEDGEMENTS VI. CONCLUSIONS We analyzed the thermoelectric response of a Corbino structure in the quantum Hall regime. For partially ﬁlled 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 diﬀusion of electrons while electronphonon drag does not inﬂuence the thermovoltage in temperature range from 300 mK to 2 K. Clearly, the electronphonon interaction does not vanish in the Corbino geometry, but the transfer of momentum from the phonons to the electrons does not lead to a measurable voltage. 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