## Cold Thermoelectrics

Thermoelectric effects—such as the creation of a voltage drop in response to a thermal gradient (known as the Seebeck effect)—can be used for a number of applications, including converting wasted heat into power. However, especially in solids that exhibit electronic interactions, this type of behavior is not well understood. **Brantut et al.** (p. 713, published online 24 October; see the Perspective by

**Heikkilä**) studied the Seebeck effect in the very controllable setting of cold atomic gases. Two initially identical reservoirs of

^{6}Li atoms were connected using a quasi–two-dimensional channel, and the particle current after heating one of the reservoirs was measured. The atoms moved from the warmer to the cooler reservoir, the extent of which fit with theoretical predictions as the disorder in the channel and its geometry were varied.

## Abstract

Thermoelectric effects, such as the generation of a particle current by a temperature gradient, have their origin in a reversible coupling between heat and particle flows. These effects are fundamental probes for materials and have applications to cooling and power generation. Here, we demonstrate thermoelectricity in a fermionic cold atoms channel in the ballistic and diffusive regimes, connected to two reservoirs. We show that the magnitude of the effect and the efficiency of energy conversion can be optimized by controlling the geometry or disorder strength. Our observations are in quantitative agreement with a theoretical model based on the Landauer-Büttiker formalism. Our device provides a controllable model system to explore mechanisms of energy conversion and realizes a cold atom–based heat engine.

Heat and charge transport in materials are often coupled processes (*1*). This coupling leads to thermoelectric effects: A temperature gradient may lead to a voltage drop, or vice versa. These effects are important for probing elementary excitations in materials and have practical applications to refrigeration and power generation from waste-heat recovery (*2*, *3*). Recently, there has also been interest in thermoelectric effects in nano- and molecular-scale electronic devices (*4*, *5*). The progress in modeling solid-state physics with cold atoms (*6*, *7*) raises the question of whether thermoelectricity can be observed in such a controlled setting (*8*–*10*), where setups analogous to mesoscopic transport devices have been realized (*11*–*13*). Although the thermodynamic interplay between thermal and density modes has been seen in a second sound experiment (*14*) and studied in the theory of the fountain effect (*15*, *16*), thermoelectric transport has so far not been investigated.

Here, we demonstrate a cold atoms device in which a temperature bias generates a neutral atom current, analogous to a charge current in conductors. A schematic view of the experimental setup is shown in Fig. 1A. It is based on our previous work (*13*, *17*). Initially we prepare *N*_{tot} = 3.1(4) × 10^{5} weakly interacting ^{6}Li atoms at a temperature of 250(9) nK in an elongated trap, where the Fermi temperature of the cloud is *T*_{F} = 931(44) nK. A repulsive laser beam (not shown) having a nodal line at its center separates the cloud into two identical reservoirs connected by a quasi–two-dimensional channel. Tuning the power of the beam allows one to adjust the trap frequency ν* _{z}* in the channel up to 10 kHz. We then raise a gate potential in the channel, preventing any exchange between the reservoirs. A heating beam heats the left reservoir in a controlled way, increasing its temperature by typically 200 nK. Afterwards, the gate potential is removed abruptly, allowing heat and particle exchange between the reservoirs for a variable time. Before determining the particle number and temperature in each reservoir, they are separated by raising the gate potential. The power of the laser beam creating the channel is then ramped to zero in 400 ms, and each reservoir is left to equilibrate independently for 100 ms (

*18*).

We measure the temperatures *T*_{h} (*T*_{c}) and atom number *N*_{h} (*N*_{c}) in the hot (cold) reservoir using absorption images and reconstruct the time evolution of the number imbalance ∆*N* = *N*_{c} − *N*_{h} and temperature bias ∆*T* = *T*_{c} − *T*_{h}. Figure 1, B and C, show typical results. The temperatures equilibrate fast, similar to the equilibration of atom numbers observed in the case of pure atomic flow (*13*). In contrast, the atom number difference in Fig. 1C starts at zero and shows an initial build-up driven by the thermopower of the channel. At later times, when temperatures have equilibrated, the imbalance results in a chemical potential bias, which brings the imbalance back to zero. This transient atomic current, created in response to a temperature gradient, is the fingerprint of the intrinsic thermoelectric power of the channel. Because the compressible cloud in the hot reservoir expands, one naïvely expects an initial particle flow from the cold, denser side to the hot one. In contrast, we observe the opposite effect: a net particle current initially directed from the hot to the cold side.

To explain quantitatively our observations, we model the channel with its conductance *G*, thermal conductance *G*_{T}, and thermopower α_{ch}, which are the linear response coefficients for the current and entropy current to temperature and chemical potential bias. In this framework, the particle and entropy currents flowing in the channel are given by
(1)

In this expression, *I _{N}* = ∂∆

*N*/∂

*t*and

*I*= ∂∆

_{S}*S*/∂

*t*are the particle and entropy currents, with ∆

*S*=

*S*

_{c}−

*S*

_{h}. Further, μ

_{h}and μ

_{c}are the chemical potentials of the hot and cold reservoirs; is the Lorenz number (

*19*) of the channel, which measures the ratio of entropy to particle current noise at equilibrium; and . Combining Eq. 1 with the thermodynamics of the reservoirs leads to the equation for the time evolution of ∆

*N*and ∆

*T*: (2)

Here, , , and are, respectively, the compressibility, specific heat, and dilatation coefficient of each reservoir, calculated at the average temperature and particle number (*N*_{c}* _{}*+

*N*

_{h})/2. is an analog of the Lorenz number for the reservoirs, measuring the relative magnitude of thermal fluctuations of entropy and atom number, and τ

_{0}= κ

*G*

^{−1}is the particle transport time scale, analogous to a capacitor’s discharge time (

*13*).

Equation 2 shows that the thermoelectric response results from the competition between the entropy transported through the channel, described by α_{ch}, and the entropy created by removing one atom from one reservoir and adding it to the other, described by α_{r}. In our system, τ_{0} (typically ~0.5 s) is longer than both the longest oscillation period in the trap and the expected collision time in the reservoirs (~50 ms). Hence, we assume that reservoirs maintain thermal equilibrium at any time, and calculate *G*, *G*_{T}, and α_{ch} using the Landauer-Büttiker formalism (*18*, *20*, *21*). Within this description, the thermopower of the channel is controlled by the energy dependence of the transmission rate. Because the rate increases with energy, there is an excess current of high-energy particles flowing from hot to cold over low-energy particles flowing from cold to hot.

The energy dependence of the transmission coefficient, and hence α_{ch}, can be modified by changing the geometry of the channel (*22*). We measured the transient imbalance and the temperature evolution for various confinements in a ballistic channel. Two examples for ν* _{z}* = 3.5 and 9.3 kHz are presented in Fig. 2, A and B. The evolution of both ∆

*N*and ∆

*T*is faithfully described by the theoretical model [eq. S14 in (

*18*)], which does not involve any adjustable parameter. The dynamics of both temperature and atom-number evolution (Fig. 2C) is slowing down with increasing ν

*, as expected because the number of conducting modes is reduced. The experimental values for τ*

_{z}_{0}are extracted from fitting the experimental data with the theoretical model with τ

_{0}as the only free parameter (

*18*). They agree well with the ab initio theoretical predictions and with independent measurements performed with a pure particle number imbalance (

*18*).

The amplitude of the transient imbalance also increases with stronger confinement. We define the thermoelectric response , where ∆*T*_{0} is the initial temperature difference. In Fig. 2D, we present the maximum thermoelectric response *, which we fitted using the theoretical model where τ*_{0} was left as a free parameter. It displays an approximately linear increase with ν* _{z}*, well reproduced by the theory (

*18*).

We now investigate the effects of disorder on thermoelectricity. We project a blue detuned laser speckle pattern on the channel, which creates a random potential of average value and standard deviation (*18*). This leads to a random, isotropic distribution of hills and wells in the plane of the channel. Figure 3A presents the time evolution of ∆*N*/*N*_{tot} for increasing disorder, for fixed ν* _{z}* = 3.5 kHz. First, we observe that the time scale increases: The resistance increases as the channel crosses over from ballistic to diffusive. In addition to this slowdown, increases from 0.17(8) without disorder to 0.55(16) for a strong disorder of 1.1 μK [still below the percolation threshold of the disorder of ∼1.8 μK (

*23*)].

For the strongest disorder, we observe (Fig. 3B) that the thermoelectric response saturates, while the time scale τ_{0} keeps increasing, indicating the continuous increase of resistance with disorder. To further investigate this point, we performed experiments for a fixed confinement in the channel of 4.95 kHz, and several large disorder strengths ranging from 542 to 1220 nK. In this regime, the full data set collapses to a single curve (Fig. 3C), provided the time axis is rescaled by the time scale τ_{0} extracted from an independent atomic conduction experiment (*13*, *18*).

This scaling property comes from the thermopower being a ratio of two linear response coefficients. Thus, it does not depend on the actual value of transparency, as the resistance does, but only on the way the transparency varies with energy. The variation is fixed by the diffusive nature of the transport process, leading to a universal thermoelectric response independent of the scattering time. This confirms that thermopower is less sensitive than resistance to the details of the conductor, a fact widely used in condensed-matter physics (*24*).

The effect of disorder can be described by extending our theoretical model, introducing an energy-dependent transparency of the constriction [eq. S10 in (*18*)]. This transparency involves the energy-dependent mean free path in the channel, which is the product of the particle velocity and scattering time. At strong disorder, the scattering time is assumed to be energy independent. When expressed as a function of *t*/τ_{0}, a unique theoretical scaling curve independent of is predicted for the thermoelectric response, which agrees well with the experimental data (Fig. 3C), with τ_{0} as the only adjustable parameter. The dependence of τ_{0} on disorder strength, fitted to the strongest disorder data only, together with the requirement that resistance interpolates between Ohm’s law and ballistic transport from strong to weak disorder, yields a full model for the ballistic-to-diffusive crossover (*18*). The resulting theoretical curves for the transient particle imbalance accurately describe the data over the entire crossover (Fig. 3A) without extra fitting parameters.

Confinement and disorder are two independent ways of influencing the thermoelectric properties. To compare their respective merits, we display as a function of τ_{0} in Fig. 3D. We find that disorder is more efficient than confinement to increase . For the largest time scales, we observe an approximately threefold increase in in the diffusive case compared to the ballistic one. This is because thermoelectricity is due to the asymmetry between the motion of low- and high-energy particles. In the ballistic case, it originates from an increase in particle velocity and density of states with energy. For diffusive motion with a given scattering time, an extra dependence emerges: The rate at which particles with a given energy will cross the disordered region is itself an increasing function of energy, as it is proportional to the mean free path.

In the experiment, a controlled exchange of heat between a hot and a cold reservoir is used to produce a directed current, i.e., work. This motivates an analysis in terms of heat engines. To do so, we evaluate the work, efficiency, and power of the process. The area enclosed in the μ − *N* plane (Fig. 4A) represents the work produced during the evolution, which we evaluate from *N*_{c,h} and *T*_{c,h}. Similarly, we evaluate the irreversible heat associated to the transport process [eq. S20 in (*18*)].

We introduce the relative efficiency η = *W*/*Q* (*2*, *18*). For a reversible (Carnot) process, η = 1. We find that η is largest in the strongly disordered regime, where the thermoelectric response is largest (Fig. 4, B and C). However, the efficiency of any heat engine is expected to increase as the dynamics slows down, because the thermodynamic processes become closer to reversibility. Therefore, a complementary criterion to evaluate the merits of the various thermoelectric configurations is the cycle-averaged power (*25*), estimated by *W*/τ_{0}. Whereas the power is constant for the ballistic case (Fig. 4D), it shows a maximum in the diffusive case (Fig. 4E) and decreases for the largest disorder, because the increase in work is overcompensated by the slowdown. This is in qualitative agreement with the theory [eq. S23 in (*18*)]. We attribute the discrepancy with experiments to a small, finite imbalance offset not taken into account in the calculation.

We now focus on the channel, independently of the reservoirs. We use the transport coefficients extracted with our model to estimate the dimensionless figure of merit (*2*) of the channel (Fig. 4, F and G). This number is related to the efficiency achievable with a channel operating at maximum power and is used as a criterion for engineering thermoelectric devices (*26*). For the largest thermoelectric response observed, we infer *ZT* = 2.4, which is as large as the best values observed in any solid-state material (*3*). This is partly because our setup also allows one to explore thermoelectricity in the regime where phonons are absent and the ratio of temperature to Fermi temperature is large. This is not the case in conventional metals, which have a small *ZT*. However, in materials of current interest for thermoelectricity, such as strongly correlated “bad metals” (*27*), the relevant scale is the effective degeneracy temperature of quasiparticles , so that can be large.

We have demonstrated thermoelectric effects in quantum gases and shown that thermopower is a sensitive observable in this context. Our technique can be straightforwardly generalized to interacting systems where thermoelectric properties are of fundamental interest (*27*–*30*). The reversed operation of our device leads in principle to cooling by the Peltier effect. This may be useful to cool quantum gases to low entropy, which is needed to explore strongly correlated fermions in lattices.

## Supplementary Materials

## References and Notes

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**Acknowledgments:**We acknowledge fruitful discussions with J. Blatter, M. Büttiker, D. Papoular, and S. Stringari. We acknowledge funding from the Swiss National Science Foundation, National Centres of Competence in Research Materials with Novel Electronic Properties (MaNEP) and Quantum Science and Technology (QSIT), the European Research Council Project SQMS, the FP7 Project Nanodesigning of Atomic and Molecular Quantum Matter (NAME-QUAM), the ETH Zurich Schrödinger chair, the Defense Advanced Research Projects Agency–OLE program, Agence Nationale de la Recherche (Far From Equilibrium Quantum Systems), and ETH Zurich. J.-P.B. is partially supported by the European Union through a Marie Curie Fellowship. The data presented in this paper are available upon request to T.E.