On the concept of an effective temperature Seebeck ratchet

Thermoelectric generators (TEGs) are a promising concept for the production of clean power. However, modest efficiencies and high costs per watt prevent widespread applications, irrespective of the type of the active material that is used.¹ The thermodynamic limit of an energy harvesting thermoelectric generator (TEG) is given by

where the first term on the right-hand side is the Carnot efficiency and the second term approaches unity when the figure of merit $ZT ,$

approaches infinity. Here, S is the Seebeck coefficient, $σe=neμ$ is the electrical conductivity that equals the product of the volume charge density $ne$ and mobility $μ$⁠, and $Th−Tc$ is the temperature difference between hot and cold sides. Since $σe$ is, via the Wiedemann–Franz law, typically linearly proportional to the electronic contribution $κel$ of the thermal conductivity $κ$⁠, optimizing $ZT$ by minimizing the lattice contribution $κlat$ to $κ$ while trying to maintain or even increase $σe$ and S is a common strategy for inorganic thermoelectrics.^2,3

Here, we propose an alternative approach, by exploiting the fact that in disordered semiconductors, thermalization of “hot” electrons to the lattice temperature $Tlat$ occurs relatively slowly.⁴ This is very prominent in organic semiconductors, which usually show a high degree of energetic disorder, i.e., exceeding the thermal energy $kBT$ by at least a factor 2–3. The slow thermalization of excited charge carriers in the density of localized states allows us to heat up the charge carrier distribution without heating up the lattice and, thus, to decouple the two contributions to the thermal conductivity. Specifically, in disordered systems, the application of an electric field $F$ can significantly alter the conductivity of the sample, which can be understood in terms of an effective temperature $Teff,$⁵ 

Here, $αl$ is the localization length, e is the elementary charge, and $β=2$ and $γ=0.89$ are empirical coefficients.⁶ As can be seen in Fig. 1, the effective temperature becomes relevant at fields above 10⁷ V/m and is well described by Eq. (3). The kinetic Monte Carlo (KMC) model that is used to generate the reference data (symbols) in Fig. 1 has been described in Ref. 7. In short, it describes the stochastic hopping of charge carriers between localized sites with random energies on a random 3D lattice using Miller–Abrahams rates. The effective temperature is determined by fitting a Fermi–Dirac distribution $f=1/1+expE−EF/kBTeff$ to the density of occupied states.

Figure 1 shows that the field-induced effective temperature can become more than twice the lattice temperature at high, but experimentally achievable fields around 10⁸ V/m. In principle, having such a high $Teff$ on the hot side of a TEG allows us to have a highly efficient electronic Seebeck effect. Since the lattice can remain at a constant temperature, the lattice contribution to the loss channel (thermal conductivity) is zero.

To utilize this idea for power generation, we draw inspiration from the conventional TEG, see Fig. 2(a), in which a temperature modulation along the current path (chosen to be the $±z$ direction) drives the device. A corresponding modulation in effective temperature can be realized by a spatial modulation (along the current path) of a static electric field that is oriented perpendicular to the current path, e.g., $Fyz$⁠. However, applying an electric field through the connecting metallic electrodes of an otherwise conventional TEG is problematic, and we, therefore, propose an alternative geometry that has the additional advantage of eliminating the need to have both n- and p-type semiconductors.

We integrate the effective temperature concept into the Seebeck ratchet scheme that was developed by Büttiker, see Fig. 2(b).^8,9 It was shown that particles that are subjected to (spatially) periodic modulations of both the potential and the diffusivity or temperature will show a net directional motion, depending on the phase difference $ϕ$ between the two. In the original concept, this motion is maximal at a phase difference of $ϕ=π/2$ and vanishes for zero phase. In the example of Fig. 2(b), this can be understood as the potential barriers are more likely to be overcome by diffusing charges from the hot side than from the cold side. This leads to a directional difference in the carrier transition rates from potential minimum to minimum. The phase of $π/2$ leads to the maximal temperature difference between hot and cold slopes while at a phase of 0 or $π$ the slopes would not differ in temperature.

Following Büttiker, the transition rates r for a potential modulation $Vq=V0sinq$ can be expressed as⁸ 

The term $Δ$ is the slope of the generalized potential, defined as

Here, $α$ is the amplitude of the diffusion modulation D along the normalized spatial coordinate $q=z/P$ with $P=nzαNN2π$ (typically $30×1.8−9 m2π$⁠); the modulation period equals the entire simulation region of our KMC simulations,

This gives rise to a net current of the following form:⁸ 

With the charge density $n=N0cfreee$ derived from the site density $N0$ and the relative free charge concentration $cfree$ (typically 0.01). The above equations hold for the overdamped regime, in which inertial effects can be ignored, which is a reasonable assumption for electronic charges hopping in a disordered medium.

The (effective) temperature enters these equations via the mobility $μ$ that is connected to the diffusion $D$ via the Einstein relation,

with the Boltzmann constant $kB$⁠. The charge carrier mobility in a disordered organic semiconductor is modeled after Upreti et al.⁷ The typical hop is assumed to take place from a site at the Fermi energy $EF$ to a final site with energy $ϵ$ and has a probability that is calculated as a Miller–Abrahams rate,

which enters the mobility as^7,10

where $Et$ is the (temperature dependent) transport energy that can be calculated by maximizing Eq. (9) while adhering to the percolation criterion that connects the characteristic hopping distance $R*$ to $Et,$

In the above, we use a critical number of Bonds $Bc$ of 2.7. The empirical model parameter B is  3.⁷ 

The temperature dependent Fermi energy can be calculated from the particle concentration using $N0cfree=∫DOSEfE,EFdE,$ where $f=1/1+expE−EF/kBT$ is the Fermi–Dirac distribution and the density of states (DOS), which we will assume to be a Gaussian of width $σDOS$ (0.07 eV),

Using Eqs. (8)–(11), the diffusion variation amplitude $α$ in Eq. (6) can be calculated from the (effective) temperature variation Eq. (3), and from that the DC ratchet current can be calculated.

Specifically, we divide the z-space in a discrete set of $Ns$ slices in the x,y-plane. For each slice $i$⁠, the (effective) temperature, Fermi energy, and mobility $μi$ are calculated. We then calculate the average mobility in the z-direction as

With the mobility values $μi,$ we use Eq. (8) to determine the local diffusion coefficient $Di$ for each slice and from those we calculate the base diffusion and the diffusion modulation amplitude that enters Eq. (6) as

and

Here, $Dmax/min$ stands for the maximal and minimal diffusion among the slices. The total current density in the z-direction is calculated as a sum over the ratchet current Eq. (7) and a field-induced (Ohmic) component $jz,Ohm=cμz,avFz,app$⁠. Because of the many simplifications made in the analytical model, the calculated currents are multiplied by a calibration factor $κcal=O1$⁠.

At this point, we also considered the option to exchange the temperature modulation with a modulation of an electrical field in the y-direction, i.e., the field direction is perpendicular to the modulation direction of the potential, along which the ratchet current is generated, viz.,

this again results in a ratchet current in the z-direction, despite the absence of an applied field in that direction. In this case, the total current density along y is calculated by averaging over the current density in each slice as

The entire python code used to perform these calculations can be found in the supplementary material.

To calibrate the analytical model, we ran kinetic Monte Carlo (KMC) simulations of Seebeck ratchets, applying sinusoidal potential and temperature modulations with a phase difference of $π/2$ along the z-axis, as shown in Fig. 3. Here, both classical Seebeck ratchets driven by sinusoidal temperature variation and a field driven ratchet are shown. When comparing the shape of the curves, one can see that field and temperature driven ratchets behave very similarly toward applied potential modulation $V0$⁠. The main differences lie in the absolute current values. One should note here that the effective temperatures resulting from the field modulation in the black curve lie between 300 and 395 K, making the modulation amplitude and base temperature values comparable to the blue curve. The analytical model reproduces the functional shape well, independent of calibration factors. One possible reason for the nonunity calibration factor is that we considered a homogeneous carrier concentration in the analytical model. While the KMC simulation naturally accounts for it, the self-consistency problem is difficult if possible to analytically solve. The need for different calibration factors for temperature and field driven ratchets further indicates that subtle differences between both cases exist, which are not yet fully accounted for. This can be attributed to the fact that Büttiker formulated his theory for sinusoidal modulations of $D$⁠, which does not fully hold in our case due to the field to effective temperature relation being nonlinear. While the diffusion modulation amplitude in the analytical theory is unaffected by this, the mean diffusion can deviate significantly between temperature and field driven ratchets. All further simulations where done at a potential modulation $V0$ = 0.05 V to maintain consistency across simulations. A further remarkable consequence of the increased complexity of the current system is the dependence of the device current on the phase angle $ϕ$⁠. As shown in the supplementary material, Sec. 1, the output current is still periodic in $ϕ$⁠, but with a nonsinusoidal functional form.

To illustrate that useful power can be extracted from such a device and to find the corresponding maximum-power point, we applied a bias field $Fz$ against this current, as shown in Fig. 4 for different $Fmod$⁠.

The effective temperature Seebeck ratchet shows essentially linear current-field curves in which the open circuit field needed to compensate the driving force of the ratchet is about two orders of magnitude below the driving field, cf. $Fz,0≈2.5×106$ for $Fmod=1×108$ V/m. This allows us to make a rough estimate of the device efficiency as $η ∼Fz,0/Fmod2∼6×10−4$⁠. The short circuit ratchet current (at $Fz$ = 0) as expected matches the open circuit field induced current, justifying the use of the linear expression for $jz,Ohm$⁠.

To determine the ratchets efficiency more accurately, we divided the output power (inner product of current and bias field) by the power dissipated in the driving y-direction. In KMC simulations, we calculate the latter by summing over all hopping movements, each multiplied by the (local) field in the y-direction. In the analytical model, we multiply the y-current in each slice by the respective y-field and average over all slices. For our base configuration, we found an efficiency maximum around $1.5×10−4$⁠, as shown in Fig. 5. When it comes to efficiency, the analytical model is able to predict the general trends but struggles with accurate predictions of the exact values. The main reason for this can probably be found in the analytical model treating motion in each spatial direction independently, apart from a coupling through $Teff$⁠.

As our system has two current directions, one driving direction and one output direction, we expected the device efficiency to be influenced by anisotropy in the material. Indeed, using different localization lengths along y and z leads to an increased or decreased efficiency, as shown in Fig. 5. Experimentally, such anisotropies are realistically obtainable using stretching or rubbing.¹¹ 

To put the seemingly low efficiencies around $10−4$ for the Seebeck ratchet into perspective, we used Eqs. (1) and (2) to calculate the efficiency of a conventional TEG composed of n- and p-type materials with the same materials parameters as used above. The required conductivity, Seebeck coefficient, and electronic thermal conductivity were determined from the separate KMC simulation as 2.4 $×10−4$ S/m, 0.33 V/K, and 1.2 $×10−6$ W/(m K). We assume a base temperature $Tc$ of 300 K and set the hot temperature to the effective temperature corresponding to a 10⁸ V/m field applied on top of the base temperature. This approximately corresponds to $2Tc$⁠. The lattice thermal conductivity is assumed to be 0.2 W/(m K), which is an optimistic lower limit.^11,12 Under these conditions, we arrive at an efficiency of $∼2×10−5$⁠, which is roughly one order of magnitude below the Seebeck ratchet efficiency. It should be noted here that a conventional TEG would theoretically be able to reach efficiencies of up to 35% under the same conditions when the lattice contribution to the thermal conductivity is removed, further demonstrating that the use of effective temperature in such devices might yield huge improvements.

Finally, Fig. 6 gives an idea of how a proof-of-concept Seebeck ratchet might be realized; we do not expect this particular incarnation to become economically relevant. The potential variation inside a sheet of doped organic semiconductor can be accomplished by buried, interdigitated stripe electrodes as used in previous ratchet designs.^13,14 The high electric field strengths that are needed to create the phase-shifted effective temperature profile could be accomplished at the sharp ends of plasmonic nano-antennas, which would sit on top of the organic layer. The device can be driven by illumination with radiation matching the resonance wavelength of the nano-antennas. As effective temperature is independent of the field direction, even high frequencies should be accessible so that by adjusting the resonator size, the device can be tuned to a wide range of radiation wavelengths from GHz to the infrared and even visible. For infrared radiation, for which so far no efficient light-to-electricity conversion device exists, the required resonator size would be in the micrometer range. To manufacture the proposed device, two lithography and one spin coating step might be sufficient; no heat exchangers or similar parts are necessary. The main disadvantage of the setup is its reliance on high fields as effective temperature is barely relevant below 10⁷ V/m, which necessitates the use of focused light, in addition to the field amplification by the plasmonic resonators.

A 10⁷ V/m field corresponds roughly to an intensity $I$ of 1 $×$ 10⁸ sun (1 $×$ 10¹¹ W/m²) without field amplification. This was calculated from $I=cε0E2/2$ with the electric field $E$⁠, the light velocity $c,$ and the vacuum permittivity $ϵ0$⁠. As the dependence of intensity on field is quadratic, a field amplification of a factor of 100 would reduce the necessary incoming intensity to 1 $×$ 10⁴ sun, which is challenging but should be achievable with light concentration. At these intensities, heating effects will have to be mitigated, for instance by cooling and choice of materials.

In summary, we have demonstrated that field induced effective temperatures can theoretically be used to drive thermoelectric generators. In particular, we argue that this can greatly increase efficiency as the system becomes independent of lattice thermal conductivity, which constitutes the major loss channel in conventional TEGs. As a specific example, we introduced the Seebeck ratchet and showed that it can have very good efficiencies compared to classical TEGs made from the same materials. While further work is necessary for a full analytical description, we are able to predict the general trends, which might prove useful in device optimization. The concept of effective temperature has not seen much experimental attention in the thermoelectric community, and we hope that the proposed device inspires others to further explore the concept, specifically in the rectification of electromagnetic radiation with wavelengths that currently are inaccessible for power generation.

See the supplementary material that provides additional simulation results on the role of the relative phase difference between potential and field modulation and lists the Python code for the numerical implementation of the analytical model.

This research is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy–2082/1-390761711. M.K. thanks the Carl Zeiss Foundation for financial support.