Título/s: | Thermoelectricity in Quantum-Hall Corbino structures |

Autor/es: | Real, Mariano A.; Gresta, Daniel; Reichl, Christian; Weis, Jürgen; Tonina, Alejandra; Giudici, Paula; Arrachea, Liliana; Wegscheider, Werner; Dietsche, Werner |

Editor: | American Physical Society |

Palabras clave: | Termoelectricidad; Estructuras; Electrodinámica cuántica; Voltaje; Conductividad eléctrica; Electrones; Fonones; Enfriamiento; Mediciones; Campo magnético |

Idioma: | eng |

Fecha: | 2020 |

Ver+/- 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. 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