Experimental verification of reciprocity relations in quantum thermoelectric transport

Symmetry relations are manifestations of fundamental principles and constitute cornerstones of modern physics. An example are the Onsager relations between coefficients connecting thermodynamic fluxes and forces, central to transport theory and experiments. Initially formulated for classical systems, these reciprocity relations are also fulfilled in quantum conductors. Surprisingly, novel relations have been predicted specifically for thermoelectric transport. However, whereas these thermoelectric reciprocity relations have to date not been verified, they have been predicted to be sensitive to inelastic scattering, always present at finite temperature. The question whether the relations exist in practice is important for thermoelectricity: whereas their existence may simplify the theory of complex thermoelectric materials, their absence has been shown to enable, in principle, higher thermoelectric energy conversion efficiency for a given material quality. Here we experimentally verify the thermoelectric reciprocity relations in a four-terminal mesoscopic device where each terminal can be electrically and thermally biased, individually. The linear response thermoelectric coefficients are found to be symmetric under simultaneous reversal of magnetic field and exchange of injection and emission contacts. Intriguingly, we also observe the breakdown of the reciprocity relations as a function of increasing thermal bias. Our measurements thus clearly establish the existence of the thermoelectric reciprocity relations, as well as the possibility to control their breakdown with the potential to enhance thermoelectric performance

Symmetry relations are manifestations of fundamental principles and constitute cornerstones of modern physics.An example are the Onsager relations [1] between coefficients connecting thermodynamic fluxes and forces, central to transport theory and experiments.Initially formulated for classical systems, these reciprocity relations are also fulfilled in quantum conductors [2,3].Surprisingly, novel relations have been predicted [4] specifically [5] for thermoelectric transport.However, whereas these thermoelectric reciprocity relations have to date not been verified, they have been predicted to be sensitive to inelastic scattering [6,7], always present at finite temperature.The question whether the relations exist in practice is important for thermoelectricity: whereas their existence may simplify the theory of complex thermoelectric materials, their absence has been shown to enable, in principle, higher thermoelectric energy conversion efficiency for a given material quality [8][9][10].Here we experimentally verify the thermoelectric reciprocity relations in a four-terminal mesoscopic device where each terminal can be electrically and thermally biased, individually.The linear response thermoelectric coefficients are found to be symmetric under simultaneous reversal of magnetic field and exchange of injection and emission contacts.Intriguingly, we also observe the breakdown of the reciprocity relations as a function of increasing thermal bias.Our measurements thus clearly establish the existence of the thermoelectric reciprocity relations, as well as the possibility to control their breakdown with the potential to enhance thermoelectric performance.
The Onsager relations between coefficients connecting thermodynamic fluxes and forces derive from the fundamental principle of microreversibility.In addition to the Onsager relations, reciprocity relations for thermoelectric transport coefficients have been predicted [4,5]: reversing the magnetic field and simultaneously exchanging the injection and emission contacts is expected to leave the coefficients invariant.These thermoelectric (TE) reci-procity relations are similar in nature to the ones for electrical resistance [2] observed in multi-terminal mesoscopic systems [3,11], but are concerned with coupled thermal and electric transport.
In addition to their fundamental interest, they are of practical importance.On the one hand, the existence of symmetry relations could simplify the theory of improved, future TE materials, such as in nanoscale, anisotropic [12] or hybrid materials [13] where nonlocal effects may play a role.On the other hand, the absence of symmetries could be equally important: asymmetric thermopower was recently shown to allow, in principle, for improved TE performance [8][9][10] in the maximum power regime.
However, to date the reciprocity relations have not been tested experimentally, and the extent to which they can be observed is unclear.Recent works [6,7] theoretically investigated the robustness of magnetic field symmetries in the thermopower, which are directly related to the thermoelectric reciprocity relations.In contrast to Onsager's relations [14], it was predicted that inelastic electron scattering (always present at finite temperature), in combination with a breakdown of the Wiedemann-Franz law can break the thermopower symmetries.The Wiedemann-Franz law is known to break down in low-dimensional structures due to their strongly energy-dependent density of states [15] -the same property that makes them interesting candidates for TE-materials [16].
A fundamental question is thus: can TE reciprocity relations be observed in practice and can they be controlled in experiment?Such a test of the TE reciprocity relations requires a multi-terminal normal conductor where each terminal can be electrically and thermally biased, individually, while subjected to an applied magnetic field.
Here we present such an experimental test in a fourterminal mesoscopic device (Fig. 1), and establish that the TE reciprocity relations manifest themselves in real devices.We also find evidence for a breakdown of the relations when we increase the thermal bias, indicating that the symmetries can be experimentally controlled, either by inelastic scattering or by non-linear thermal transport, analogous to the symmetry-breakdown in purely electronic transport at finite voltages in mesoscopic systems [17][18][19][20][21][22][23][24].This motivates further investigations on the symmetry breaking properties and relative role of inelastic scattering and non-linear thermal transport.
We begin by defining the thermoelectric coefficients and their expected symmetry, and describe how they can be determined in experiments.The linear response of the electrical current flowing in the α'th terminal, I α , of a multi-terminal, mesoscopic junction is where V α and θ α are the voltage and temperature, respectively, at terminal α, and G αβ and L αβ are the electrical conductance and thermoelectric coefficients, respectively, between terminals α and β.The L αβ are directly related to the thermopower, or Seebeck coefficients where the sum runs over all terminals [4].
Scattering theory provides [4] the magnetic-field symmetry relations for the thermoelectric coefficients (Supplementary Information).Writing out the diagonal and off-diagonal relations separately, we have To experimentally test these symmetries, we first determine G αβ through electric bias measurements with no thermal bias (∆θ αβ = 0) (Supplementary Information).Thereafter, the thermoelectric coefficients are investigated by thermally biasing the system under zero-electric-current conditions (with floating terminals), measuring the resulting potentials in all reservoirs, and using Eq. ( 1) as explained in the following.
The induced temperature increase at terminal α can be written as a Fourier sum, ∆θ α (t) ≡ θ α (t) − θ 0 = n=0 ∆θ (n) α sin(nωt), where ω is the frequency of the heating current (see Methods).This allows us to write the different Fourier components of the linear response current expression in Eq. ( 1) as where β .When using Joule heating (quadratic in heating current), one expects the second harmonic to give the strongest contribution to the thermoelectric response.Indeed, in our experiment we find that n = 2 gives the largest signal and provides the clearest data to determine the range of the linear response regime (see below); in the following, we only consider the second harmonics in Eq. ( 3).Heating the γ'th terminal, and making the assumption that the unheated terminals remain cold, we can make use of the sum rules [4] α L αβ = β L αβ = 0 and write Here, V αβ represents the measured values when heating terminal γ.Since the G αβ elements depend weakly on magnetic field up to B ∼ 50 mT, we can use Eq. ( 4) and G αβ [see Supplementary Information, Eq. ( 8)] to test the magnetic-field symmetries of L αγ ∆θ αβ .In the following, we also assume that ∆θ (2) γ is independent of B, so that all of the B-field dependence in L αγ ∆θ The symmetry relations predicted by Eq. ( 2) are clearly visible in the representative magnetic field traces for L αγ ∆θ (2) γ presented in Fig. 2b),c).We also find that there is no significant symmetry relation between L αβ (B) and L γδ (−B) for αβ = γδ (see Fig. 2d) for an example).The L αγ ∆θ (2) γ typically oscillate around zero, a signature of quantum interference effects [25][26][27].
To quantify the symmetry of two data sets L αβ and L γδ , we define a symmetry parameter Σ αβ,γδ that goes from +1 for complete symmetry, to -1 for complete antisymmetry (see Supplementary Information).Deviations from the perfect symmetries predicted in Eq. ( 2) are seen in our measurements.Supplemental Information), demonstrating the high repeatability of these fluctuations.We attribute the limited symmetry in Fig. 2b),c) mainly to the same mechanisms that limit the observed conductance symmetry (Supplementary Information).In addition, however, we offer two other possible mechanisms: i) the unheated terminals do not remain cold, which would modify Eq. ( 4); and ii) inelastic scattering, which at finite temperature can lead to asymmetries in the thermopower even in the linear response regime [6,7].One can expect the symmetry of L αβ to break down for finite heating voltage, analogous to the well-established breakdown of symmetries in the differential conductance [18,19] observed at finite bias voltage in mesoscopic systems [17,[20][21][22][23][24].In Fig. 3, all ten symmetry relations defined by Eq. ( 2) are plotted as a function of heating voltage V H .At low V H , all symmetries described by Eq. ( 2) manifest themselves, with Σ αβ,βα 0.5.As V H is increased though, the trend in the diagonal elements, α = β, is towards decreased symmetry, while the offdiagonal elements, α = β, remain fairly symmetric with a slight trend to decrease.
The overall tendency is for the B-field symmetries of L αβ to be suppressed with increasing thermal bias.From further analysis of our measured data (Supplementary information), we establish that the linear-response regime extends to about V H ≈ 1 mV.The decreasing symmetry observed in L αα , Fig. 3a, is then consistent with symmetry-breaking due to non-linear thermoelectric behavior, analogous to non-linear electronic effects.Increased inelastic scattering due to heating effects may also play a role.We have verified that the TE reciprocity relations predicted more than 20 years ago [4] manifest themselves in a mesoscopic device in the linear transport regime.The relations were observed at low temperatures, where inelastic scattering (predicted to suppress the symmetries [6,7]) can be expected to be small.At finite thermal bias we observe a breakdown of the reciprocity relations, tentatively due to a combination of inelastic scattering and non-linear thermal transport.Further investigations are needed to quantify the robustness of the reciprocity relations with respect to these mechanisms.Of particular interest will be the role of sharp features in the transmission function or density of states commonly used for energy filtering to enhance thermoelectric performance, for example in low dimensional coolers [28,29] and highly efficient thermoelectric generators [30,31].The possibility to experimentally control the absence or presence of the TE symmetry relations opens for exciting and fundamentally new opportunities in increasing TE energy efficiency [8][9][10].
We acknowledge financial support from NSF IGERT grant No. DGE-0549503, the National Science Foundation Grant No. DGE-0742540, ARO Grant No. W911NF0720083, Energimyndigheten Grant No. 32920-1, nmC@LU, the ESF Research Network EPSD, the Foundation for Strategic Research (SSF), and MINECO Grant No. FIS2011-23526.Effort sponsored by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant number FA8655-11-1-3037.The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.

AUTHOR CONTRIBUTIONS
HL and PS conceived the study.JM fabricated the device and performed all measurements.JM, FB, PS, and DS analyzed the data.All authors contributed to interpreting the data and writing the manuscript.

METHODS
Device fabrication.The four-terminal junction was defined by patterning the two-dimensional electron gas (2DEG) formed in an InP/Ga 0.23 In 0.77 As heterostruc-ture by using electron-beam lithography and shallow wet etching (for details, see Ref. [12]).The wafer has a carrier concentration of n = 1.1 × 10 12 cm −2 and an electron mobility of µ = 3.2 × 10 5 cm 2 /Vs at 4.2 K.
Measurement of transport coefficients.We individually use electric or thermal biases to determine the elements of the electrical conductance matrix, G αβ , and the thermoelectric matrix, L αβ , respectively.Electric biases are generated by applying a 37 Hz drive current between any two of the four central terminals.The measured G αβ are discussed in the Supplementary Information.A representative circuit configuration used for thermal-bias measurements is shown to the left in Fig. 1.The thermal bias is generated by individually heating one terminal by applying two 180 • out-of-phase, 37 Hz voltages, denoted by ν ± H in Fig. 1, to the two channel contacts (in Fig. 1, contacts A and B are used to heat terminal 1).This heating configuration is designed to eliminate any electric bias of the terminal due to the heating current.Additionally, a DC shift was applied to the thermal bias to cancel out residual DC offsets measured at the respective terminal's voltage probe.We note that only one terminal is heated at a time, and that negligible electric current is drawn through the junction during thermovoltage measurements.For further details on the thermal bias measurements, see Ref. [12].Under each type of bias, all four-terminal voltages, {V 1 , V 2 , V 3 , V 4 }, were simultaneously measured using lock-in detection.All measurements were made at a background temperature of θ 0 = 240 mK.50 mT, consistent with the slightly asymmetric conductance coefficients G αβ observed in Eq (8).The origin of this asymmetry is not clear.We can rule out noise and nonlinear effects [32] as the cause by comparing to a second measurement taken at much higher bias current (Fig. 2a), which essentially shows the same asymmetry.A magnetic sample holder can also be ruled out, as care was taken to use nonmagnetic materials.Leakage currents due to the voltage probes are also found to be negligibly small.We speculate that magnetic impurities may play a role.

Symmetry parameter
To quantify the symmetry of L αβ as a function of V H , we first introduce the normalized thermoelectric coefficients, where L αβ (B) is the average of L αβ (B) between ±50 mT.We calculate Eq. ( 9) using L αβ ∆θ (2) in place of L αβ , since we have assumed that ∆θ (2) is independent of B and thus cancels out.We then define the symmetry parameter as The sum over B in both equations runs from −50 mT to 50 mT.Σ αβ,γδ goes from +1 for complete symmetry, to −1 for complete anti-symmetry.It is also well suited to quantify and compare the symmetry properties of functions that are typically of different magnitude and oscillate around zero.

Reproducibility
To investigate the repeatability of the magnetic field traces L αβ (B), two traces measured almost two weeks apart are shown in Fig. 4. The degree of correlation between the two traces is quantified by the modified symmetry parameter Σ + αβ,γδ = B L αβ (B)L γδ (B), reaching 1 for perfect correlation L αβ (B) = L γδ (B).

Linear thermal bias response
To establish the range over V H where we expect a linear-in-temperature response, we used the solution of the quasi-one-dimensional heat diffusion equation [28] to estimate the temperature rise in the heated channel, labelled below as the α'th terminal, as a function of V H , where V C is a heating channel dependent parameter.Using Eq. ( 11), we can estimate the predicted Fourier components of L αγ ∆θ γ and compare them to our measured data, β G αβ V (2) αβ .In this way, we have clearly established that the linear-response regime extends to about V H ≈ 1 mV, which corresponds to ∆θ (2) γ /θ 0 ≈ 0.21 to 0.75 depending on which terminal is heated.

4 FIG. 1 :
FIG. 1: (Main figure) Scanning electron micrograph of a device with a geometry identical to the one measured on here, featuring a junction of four ballistic micro-channels (terminals) in a cross configuration, with an asymmetric scatterer in the central junction.The eight surrounding contacts, {A,B,...H}, are used to apply a thermal or electrical bias.Four probes are used to measure the terminal voltages: {V1, V2, V3, V4}.The regions between contact pairs, tinted red, can be electrically heated to thermally bias the junction.In the configuration shown, the channel between contacts A and B are heated through two out-of-phase heating voltages, ν ± H (see Methods).(Inset) Close up image of the central region.
directly by analyzing the B-field dependence of the measured V(2)

L 24
FIG. 2: a) Representative four-terminal resistances R αβ,γδ (B) as a function of magnetic field.The symmetry R αβ,γδ (B) = R γδ,αβ (−B) is clearly visible; small deviations are discussed in the Supplementary Information.b)-d) Magnetic field traces of the thermoelectric coefficient L αβ (B).Each panel also displays the corresponding quantitative symmetry parameter Σ αβ,γδ [Supplementary Information, Eq. (10)] calculated for the range −50 mT < B < 50 mT, for the respective pair of traces shown.The symmetry of the diagonal terms Lαα(B) = Lαα(−B) is clearly visible in b), as well as the symmetry relation L αβ (B) = L βα (−B), in c).For comparison, an example of the expected absence of symmetries, here between L14(B) and L23(−B), is illustrated in d) and manifested by a Σ14,13 value near zero.The green and red curves in a), b), and d) are offset for clarity.All measurements were performed at a cryostat temperature of θ0 = 240 mK.

FIG. 3 :
FIG.3: Heating voltage dependence of the symmetries described in Eq. (2) for the (a) diagonal, α = β, and (b) offdiagonal, α = β, elements of L αβ .In panel a), a clear trend for decreasing symmetry with increasing thermal bias is seen for the diagonal elements.This same trend is only present in three of the six curves in panel b).

FIG. 4 :
FIG. 4: Magnetic field traces of the thermoelectric coefficient L44(B) taken almost two weeks apart.The modified symmetry parameter Σ + 44,44 (see text), calculated for the range −50 mT < B < 50 mT, gives a value close to unity, demonstrating a high degree of repeatability.The measurements were performed at a cryostat temperature of θ0 = 240 mK.