CROSS-REFERENCE TO RELATED APPLICATION

This application claims the benefit of U.S. Provisional Application No. 63/218192, filed July 2, 2021 , the disclosure of which is expressly incorporated herein by reference in its entirety.

BACKGROUND

Charge pumping in microelectronic devices has been extensively studied over the last several decades. Quantum pumping of charge in Coulomb blockade devices has resulted in metrological applications; the modern realization of the capacitance standard relies on quantization of pumped charge in Coulomb blockade devices. Another known conventional technology is based on a Peltier effect. With the Pel tier-effect device, a voltage is applied across joined conductors to create an electric current. When the current flows through the junctions of the two conductors, heat is transported from one junction to another. Depending on a polarity of the device, cooling or heating of a target object can occur as the heat is deposited from one junction to another. Some common applications of Peltier effect are electronic cooling and small-scale refrigeration.

However, Peltier cooling is inefficient due to ohmic heating associated with the current as well as to parasitic thermal conduction, and Peltier cooling has not been achieved below about 200 K. Because of the energy losses, Peltier effect devices are characterized by a relatively low coeffici ent of effi ciency in transporting the heat from one side of the device to another. Accordingly, systems and methods for improved heat transfer are still needed. SUMMARY

This summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. This summary is not intended to identify key features of the claimed subject matter, nor is it intended to be used as an aid in determining the scope of the claimed subject matter.

In one embodiment, electron flow heat pump assembly includes a channel configured for pumping heat by a flow of electrical charges, and a source of excitation of the electrical charges. The channel is on average charge-neutral, and the electrical charges are distributed along the channel as wave-like density variations.

In one aspect, a characteristic frequency of the flow of electrical charges is more than one order of magnitude higher than a characteristic frequency of the source of excitation.

In another aspect, the characteristic frequency of the flow of electrical charges is in a GHz range, and the characteristic frequency of the source of excitation is in a kHz or MHz range.

In one aspect, the source of excitation is an alternating current (AC) source.

In another aspect, the AC source is configured for generating phased voltages that are coupled to the channel by an array of gate electrodes.

In one aspect, the source of excitation is a piezoelectric vibrator.

In another aspect, the source of excitation is configured for generating surface acoustic waves (SAW).

In one aspect, the density variations are distributed as travelling waves.

In one aspect, the flow of electrical charges comprises electrons and holes.

In one aspect, the channel is made of a graphene material. In another aspect, the graphene material is a monolayer graphene or a natural bilayer graphene.

In one embodiment, a method for pumping heat using electron flow includes: exciting electrical charges in a channel; in response to exciting the electrical charges, arranging the electrical charges in wave-like density variations along the channel; and pumping heat along the channel by wave-like density variations along the channel. The channel is on average charge-neutral.

In one aspect, a characteristic frequency of a flow of electrical charges is more than one order of magnitude higher than a characteristic frequency of a source of excitation.

In another aspect, the characteristic frequency of a flow of electrical charges is in a GHz range, and the characteristic frequency of a source of excitation is in a kHz or MHz range.

In one aspect, the electrical charges in the channel are excited by an alternating current (AC) source.

In another aspect, the electrical charges in the channel are excited by a piezoelectric vibrator.

In yet another aspect, the piezoelectric vibrator is configured for generating surface acoustic waves (SAW).

In one aspect, the density variations are distributed as travelling waves.

In another aspect, the flow of electrical charges comprises electrons and holes.

In one aspect, the channel is made of a graphene material.

In another aspect, the graphene material is a monolayer graphene or a natural bilayer graphene. DESCRIPTION OF THE DRAWINGS

The foregoing aspects and many of the attendant advantages of this invention will become more readily appreciated as the same become better understood by reference to the following detailed description, when taken in conjunction with the accompanying drawings, wherein:

FIGURES 1A and IB are schematic representation of different pumping modes for liquids;

FIGURES 1C and I D are schematic representation of different heat pumping modes for electrical charges;

FIGURE 2A illustrates heat flow in a heat pump in accordance with an embodiment of the present technology;

FIGURE 2B illustrates time-dependent pumping potential of a heat pump in accordance with an embodiment of the present technology,

FIGURE 3 illustrates operation of a heat pump in accordance with an embodiment of the present technology;

FIGURE 4 illustrates operation of a heat pump in accordance with an embodiment of the present technology; and

FIGURE 5 is a flowchart illustrating a method of operating a heat pump in accordance with embodiments of the present technology.

DETAILED DESCRIPTION

While illustrative embodiments have been illustrated and described, it will be appreciated that various changes can be made therein without departing from the spirit and scope of the invention. The inventive technology is directed to electron-mediated pumping of heat without needing a charge current through the channel. The inventive technology may be realized in charge-neutral electron materials, for example graphene, where the conductive path is coupled to an external pumping potential, for example voltage potential or a source of vibration. Flow of heat in this pumping cycle is not accompanied by voltage buildup along the system, which offers advantages over traditional thermoelectric cooling setups. In a system, the heat flux becomes independent of the strength of the pumping potential. In particular, for a potential moving with a velocity c the pumping is perfect, the entire heat content of the electron liquid is advected with velocity c. For a general pumping cycle, the heat flux is determined by the cycle geometry and disorder strength. Efficiency of heat pumping and magnitude of heat flux may be described through the hydrodynamic regime for weak disorder.

In some embodiments, heat pumping based on the electron flow can be initiated by the application of a driving potential at one side of the heat pump, e.g., at one side of a graphene bar. The driving potential drives the electron flow from one side of the heat pump to another. Examples of suitable driving potential are an alternative current (AC) potential or mechanical vibration that may be generated by, for example, a piezoelectric vibrator. It is important to recognize that the frequency of oscillation of the electron flow wave is generally higher by several orders of magnitude than the frequency of the driving potential. Therefore, the driving potential serves to excite the electron flow, which then proceeds to behave as an electron wave at a frequency that is decoupled from the driving potential frequency.

FIGS. 1A and 1B are schematic representation of different pumping modes for liquids. In particular, FIG. 1A illustrates pumping of a liquid by peristaltic constrictions moving along a tube. FIG. 1B corresponds to pumping a liquid by wavelike undulations of a canal bottom. In peristaltic pumping, packets of water (liquid) are pushed along a flexible tube at speed c by travelling constrictions (FIG. 1A). Notice that if the tube is only partially squeezed and not closed off there will still be some flow, albeit less, provided the tube is not so sloping that gravity prevents the flow. FIG. 1B il lustrates a scenario where the “top” of the tube is removed to make an open canal with an undulating bottom. In this case, the flow may be larger, because now the bulk of water moves uniformly at speed c in equilibrium within its rest frame (with just some of the fluid near the bottom slowed by viscous friction). Pumping of the electron flow may be explained with the above analogy in mind with respect to FIGS. 1C and 1D below.

FIGS. 1C and 1D are schematic representation of different pumping modes for electrical charges. In particular, FIG. 1C illustrates that the electrons in graphene, acting as an electron fluid, can analogously be pumped by a travelling wave of electric potential. The shading in FIG. 1C represents thermal broadening of the Fermi distribution. FIG. 1D illustrates that with the introduction of a band gap and by making the amplitude large, equal numbers of electrons (above the gap) and holes (below' the gap) are pushed along, causing a flow' of heat, but w'ithout the flow of electrical charge.

Here, electrons in a conductor collide primarily with each other rather than with material defects and lattice vibrations. This hydrodynamic electron regime is not common in bulk crystalline materials, but it may be achievable over a wide temperature range in monolayer or bilayer graphene, for example. Graphene as a heat pump material may also have additional advantageous property. For example, graphene is relatively easily penetrated by electric fields, providing a way to pump the electron fluid. Furthermore, the graphene naturally contains equal numbers of electrons and holes, which allows the net electric current accompanying the pumping to vanish.

With the electron flow illustrated in FIGS 1C and 1D, the electrons are pushed away from the peaks of the potential wave, rather as the water in FIG. IB is pushed away from the high points on the undulating canal bottom. The holes, or empty electron states, are pushed away from the dips. In a normal conductor, the scattering from defects and phonons tends to randomize the motion of the carriers, but in the hydrodynamic regime the flow of electrons and holes scatter mostly from each other resulting in a nearly uniform average flow at the speed of the wave, similar to the water flow in FIG. 1B.

FIG. 2A illustrates heat flow in a heat pump in accordance with an embodiment of the present technology. In some embodiments, heat flow 20 can be directed from the cold side (C) of the heat pump 10 to the hot side (H) of the heat pump. Different materials may be used for the heat pump 10 (also referred to as a ‘channel’), for example, graphene, copper or other metal, with a trade-off being that increased lattice defects will reduce the efficiency of the electron flow pumping. In different embodiments, graphene material may be a monolayer or natural bilayer graphene. In other embodiments, different 2D materials may be used, for example a 2D topological insulator, monolayer WTe ₂ , which possess one-dimensional topological edge states that act as 1D quantum wires. Another heat pump material is a twisted- magic angle graphene in its superconducting state.

Generally, if the graphene is undoped, the electron-hole fluid is neutral and there is no net electrical current, but the electrons and holes still have thermal energy and all move in the same direction, so their heat (i.e., thermal energy or, more precisely, entropy) is carried along with the wave. An example of such wave is discussed with respect to FIG. 2B below.

FIG. 2B illustrates time-dependent pumping potential in a heat pump in accordance with an embodiment of the present technology. Here, the thermal energy (entropy) is carried in the direction 20 by waves 22 that represent density-variation of the charges (e.g., electrons, or electrons and holes). The vertical axis indicates energy potential of the oscillating electrons in the electron flow'. The decoupling of heat from the electron (charge) flow' means that there need not be any ohmic heating in the process, and the pump (channel) 10 can at least in principle achieve the ideal maximum thermodynamic efficiency. It should be noted that in some embodiments the production of the travelling potential wave is likely to be inefficient, but the associated heat generation can be kept remote from the region from which the heat is to be removed. For example, heat generated due to inefficiencies can be kept from the cold side of the heat pump 20 (e.g., the side attached to the electronic device that is cooled), which is unlike with the Peltier device that produces heat precisely at the junction of the device.

In some embodiments, the wave amplitude is large compared with k _B T (T is temperature, k _B Boltzmann’s constant) and assuming a small band gap that separates the electrons and holes (as in Fig. 1D) the electron flow pumping scenario can be understood as the alternating pockets of hole fluid and electron fluid moving along together at speed c, conveying their heat, but with no net (electrical) charge flow. In some embodiments, performance of the heat pump may be improved by adjusting geometry, scaling up or parallelizing the devices, using different piezoelectric substrates and films, and replacing the graphene with other materials.

FIG. 3 illustrates operation of a heat pump assembly 100 in accordance with an embodiment of the present technology . Here, the electron flow of the heat pump (channel) 10 is initiated by the AC source 15, having several phases 15-i . In the illustrated embodiment, the AC source includes three phases 15-1, 15-2 and 15-3, but in other embodiment different number of AC phases may be used. In operation, the heat pump 10 pumps heat from a cold side 30-1 (C) to a hot side 30-2 (H). The heat pump 10 may be made of graphene or other material. In some embodiments, a pumping potential may be generated by an array of gate electrodes with phased voltages on them, potentially reducing the power losses and allowing local cooling on a silicon substrate, therefore making it easier to integrate with other electrical components.

The operation of the heat pump assembly may be controlled by a controller 40. In the illustrated embodiment, no moving parts are required for the operation of the pump.

FIG. 4 illustrates operation of a heat pump in accordance with an embodiment of the present technology. In the illustrated embodiment, a vibrator 16 is a source of driving potential . In turn, the vibrator 16 may be driven by an oscillating electrical potential 15 (e.g., an AC source). In some embodiments, the vibrator 16 is a piezoelectric vibrator having a lithium niobate substrate, lithium niobate being a strongly piezoelectric material in which strain is accompanied by large electrical polarization. Surface acoustic waves (SAWs) in the heat pump 10 may be launched by applying an AC voltage from the source 15 to the interdigitated transducer 16. When an AC voltage from the source 15 is applied between the electrodes of the interdigitated vibrator (transducer) 16 at the appropriate frequency, a SAW is generated travelling away from the transducer at the surface sound speed. This carries a wavelike variation of polarization, and hence electric field, which is experienced by a piece of graphene, or other material (i .e., the heat pump material) placed on the surface of the vibrator. Therefore, the SAWs cause heat pumping toward the right-hand side of the heat pump 10, in the process cooling the upstream end of the heat pump (cold (C) at the left hand side of the heat pump) and warming the downstream end (hot (H) at the right hand side of the heat pump) When tuned precisely to the charge-neutral point, a net charge current wil l be generated between the upstream and downstream ends of the heat pump 10.

Any negative change ΔT in the temperature of the upstream "cold end" of the bi layer graphene indicates heat pumping. On the other hand, a temperature rise at the "hot end" could have a trivial origin. Generally, ΔT is no larger than -Q/K, where K is the total thermal conductance from the cold end to the surroundings which is dominated by the thermal conductance of the graphene (or other material of the heat pump) itself between cold and hot ends. A rough upper estimate gives K ~ Kd ~ 3 x 10 ^-8 WK ^-1 where K » 100Wm ^-1 K ^-1 is the thermal conductivity of graphene and d ~ 0.3 nm is its thickness. For Q ~ 1μW, the result is ΔT ~ 30 K. In some embodiments, ΔT may be detected by measuring the resistance R of the bilayer graphene itself, which is highly sensitive to electron temperature. For example, a small current can be passed between two electrical contacts to the cold end between to measure R. To improve measurement precision, /? can be separately calibrated against the cryostat temperature T in equilibrium with no SAW. In many embodiments, a decrease in temperature by 30 K from 100 K becomes easily detectable this way.

In some embodiments, the heat pump 10 is on a mm or larger physical scale, while being capable of pumping heat in a mW or μW range, which may be useful for surface cooling applications. The wave frequency (also referred to as a characteristic frequency) of the electron flow wave may be in a GHz scale (e.g., about 3 GHz) for a graphene heat pump. For perspective, the frequency of the excitation SAW may be in a kHz range.

The excitation frequency is generally many times smaller than the frequency of the electron flow. For example, the excitation frequency may be in a kHz or a MHz range, while the characteristic frequency of the electron flow is in a GHz range. Analogously, the amplitude of the excitation frequency is generally many times larger than the frequency of the electron flow.

FIGURE 5 is a flowchart illustrating a method of operating a heat pump in accordance with embodiments of the present technology. In different embodiments, illustrated method may include additional steps or may include other steps not shown in the flowchart. The method may start in block 605. In block 610, electrical charges in the channel 10 are excited by a source of excitation (e.g., an AC source or a source of vibration). In response, in block 615, variations in the density of charge are generated along the channel. In response, in block 620, heat is pumped from one side of the channel to another. In block 625, heat is removed from the cold side of the heat pump (e.g., from the electronics component to be cooled) to the hot side of the heat pump. The method may end in block 630.

Device fabrication

In some embodiments, the heat pump is made as a 2D device from stacks of graphene and other exfoliatable layered materials. A monolayer or bilayer graphene may be patterned into the desired shape by direct electrochemical atomic force microscope (AFM) lithography , or AFM-cleaned, and then encapsulated between hBN flakes, using a "dry transfer" polymer stamp process, so as to improve cleanness and homogeneity of the material as much as possible. A graphite top gate can provide the ability to tune the overall charge density. The top hBN may be 40 nm thick to reduce screening of the SAW potential by this gate, while not having bottom metallic gate underneath the graphene channel as that could screen out the SAW potential. In some embodiments, bottom gates are placed under the end zones, so that they can be independently tuned for optimal sensitivity to temperature changes and characterization. The vibrators (metal transducers and contacts) can be tested on LiNbO ₃ .

Heat pump model

Generally, the flow of heat through substances is described by the Fourier law, q = which corresponds to a relatively slow diffusive spreading of heat through the substance. Because of this, heat pumping cycles are typically based on convective heat transfer, in which the heat content (entropy) moves together with the matter transporting it (the working body). For convective heat transfer the ratio of heat flux to the pumped electric current is given by the Peltier coefficient. However, the phenomenological Fourier law' and diffusive character of heat spreading through matter are not universal; under some conditions propagation of heat through physical substances may become ballistic. Some examples are second sound waves in superfluid helium (He) and pure crystals in the regime of phonon hydrodynamics. In contrast to the usual adiabatic sound, in which entropy and matter move together, the temperature oscillations in the second sound wave are practically decoupled from the density oscillations (density variations). They correspond to temperature/entropy waves that move ballistically in stationary substances. Crystallization waves in He represent another example of decoupling between the flows of order and matter, In particular, a fall of a He crystallite in superfluid He proceeds partly via melting at the top and crystallization at the bottom of the crystallite. As a result the crystallite boundary moves at a faster speed than the atoms inside it.

Decoupling of flows of heat and matter can also be realized in electronic systems in the regime of electron hydrodynamics. In systems with equal densities of electron- and holelike carriers (e.g., graphene at charge neutrality) the hydrodynamic flow of the electron liquid is decoupled from charge flow and corresponds to the flow of heat. Hydrodynamic regime of charge-neutral electron liquid can be realized in single and bilayer graphene, and may in principle be realized in clean semimetals.

In some embodiments, the decoupling of heat and charge flows in electron liquids at charge neutrality results in heat pumps that do not involve mechanically moving parts and transfer heat in the absence of net flow of matter or charge. As an example, a pumping setup, in which heat transfer is mediated by an electron liquid in a charge-neutral system, such as graphene monolayer or bilayer, may be subjected to an external electric pumping potential U (r, t). The latter may be generated by applying time-dependent voltage to a series of gates to achieve quantized charge pumping in carbon nanotubes or by placing the system on a piezoelectric substrate driven by a surface acoustic wave (SAW).

The essential features of the pumping mechanism can be understood based on ciean systems in the regime of electron hydrodynamics. Considering a unidirectional geometry with a periodic pumping potential of the form U(x — ct), at zero temperature difference between the reservoirs, and for slow pumping velocities c, the electron liquid will remain in local thermal equilibrium corresponding to the instantaneous realization of the pumping potential U (% - ct). Therefore the densities of electrons, n(x, t) and entropy, s(x, t), are given by the equilibrium values corresponding to the local value of U. Since the latter moves with velocity c, the electron liquid will also move with the hydrodynamic velocity u ~ c. The local densities of charge and entropy of the electron liquid will propagate with the same velocity. As a result, the entire heat content of the electron liquid will be entrained by this flow; producing a net heat flux density T(s)c where (... ) denotes spatial average. In contrast, the net charge pumping current vanishes, because at charge neutrality because the vanishing average electron density, (n) ~ 0.

However. in the presence of disorder and temperature gradient the entrainment of the electron liquid by the pumping potential is no longer prefect; the pressure gradient proportional to the temperature gradient and the disorder-induced friction force causes the electron liquid to lag behind the pumping potential. Evaluating the heat flux and pumping cycle efficiency in this case requires a quantitative model.

Below, a heat pumping model is developed in the regime of electron hydrodynamics in the creeping flow approximation. The hydrodynamic description applies provided the rate of momentum-conserving electron-electron collisions exceeds the momentum relaxation rate and the pumping frequency ω. The underlying assumption is that the correlation radius ξ of the disorder potential satisfies the condition ξ,q « 1. In this case pumping may be described by averaging the flow of the electron liquid over length scales of order 0 For slow pumping the corresponding macroscopic hydrodynamic equations may be written in the form

Here P is the pressure, u is the hydrodynamic velocity, and denotes the viscous stress tensor, and k denotes the disorder-induced “friction” coefficient. The densities of particles, entropy, and momentum are denoted by n, s, and p, respectively. The electric potential Φ is related to the electron charge density en by the Poisson equation. The current densities of particles, j, and entropy, may be expressed as where the first term in the right hand side (r.h.s.) represents the equilibrium components of the currents, while the second represents the dissipative components. The latter are linear in the temperature gradient VT and the electromotive force (with μ being the chemical potential). The elements of the Onsager matrix of the intrinsic kinetic coefficients of the electron liquid are the electrical and thermal conductivities σ, and K, and thermoelectric coefficient γ. Finally, is entropy production rate per unit area, caused by dissipative processes in the electron liquid and loss of heat to the lattice.

Adiabatic pumping

Here the regime of slow pumping is considered, where the heat flux is linear in the rate of change of the pumping potential. Further assumption is that the temperature difference ΔT between the hot and cold reservoirs is small, and work in linear order accuracy in and ΔT. In this approximation in Eq. (1) we may neglect and thereby reducing Eq. (3) to a continuity equations for the entropy current. Furthermore, we may replace the densities of entropy and particles by their equilibrium values in the presence of pumping potential U.

Considering a unidirectional geometry, in which a two-dimensional system of length L (in the x-direction) and width w (in the y-direction) is subjected to a periodic in space and time pumping potential of the form U (x, t) = U(x + λ, t) = U(x, t + τ). Being interested in the bulk effects we can evaluate the pumping heat flux per unit width of the system for L » A. In this case, adn without loss of generality, we can set L/k to be an integer, and impose at the reservoirs periodic boundary conditions on the system variables, eε, and u.

Furthermore, assuming that both the pumping potential U and disorder are small in comparison to T, in this regime the hydrodynamic velocity u becomes independent of U/T and is controlled by the disorder strength and time-dependence of the pumping potential.

To show this, we can approximate all quantities to leading order in U/T. Since the local electron density n is linear in the pumping potential, n/s α U/T « 1, it follows from Eq. (1) that the particle current, and thus the EMF e£ in Eq. (2), are linear in U/T. In contrast, deviations of the entropy density from that at charge neutrality, s ₀ , are quadratic in U/T and may be neglected. Then it follows from Eq. (3) that to within second order accuracy in U /T the entropy flux is uniform. Since the latter is given by and y α U/T, we see that to leading order in U/T the hydrodynamic velocity u is uniform, while the inhomogeneous components of w and temperature gradients _x induced by the pumping are quadratic in U/T. Thus, to leading order in U/T we may neglect the thermoelectric contribution to the particle current in (4), and write the continuity equation (1) in the form 0. This yields the following expression for EMF

The uniform components of u and d _x T can be obtained by averaging the momentum evolution Eq. (2) over space. Using the definition of EMF eε, and the thermodynamic relation dP = ndμ + sdT, we get where denotes spatial averaging. This equation expresses force balance. The second term in the r.h.s. describes the force density caused by thermally induced pressure gradient. This force is balanced by the disorder-induced friction force in the left hand side (l.h.s.) and the force exerted by the pumping potential (first term in the r.h.s.). The magnitude of this force as can be seen from Eq. (5).

Substituting Eq. (7) into Eq. (6) we obtain the hydrodynamic velocity in the form where we expressed the friction coefficient in terms of the variance of disorder-induced density modulations,

Equation (8) determines the hydrodynamic velocity, and thus the heat flux Tj _s = Ts _o μ, in the presence of pumping and temperature difference between the reservoirs. At ΔT ™ 0 the velocity u is determined only by disorder and the pumping cycle parameters. This pumping contribution is given by

In particular, for a potential in the form of a traveling wave generated by SAW,

U (x, t) = U ₀ (x- ct) we get

At vanishing disorder this expression reproduces the expected result of perfect pumping, l

Note that for a general periodic pumping cycle at weak disorder, the pumping velocity in Eq. (9) reaches the value which independent of the pumping potential amplitude. Using the Fourier series representation, and may be interpreted as a connection in the space of periodic functions with zero mean.

Averaging the hydrodynamic velocity in Eq. (8) over time we can write the pumping heat flux per unit width of the system in the form where represents the effective thermal conductivity of the system,

It is inversely proportional to the variance of deviations of electron density from charge neutrality, which are caused by both disorder and the pumping potential. Note while disorder-induced density variations, ((δn) ² ), are statistically isotropic, the density variations induced by the pumping potential have a uniaxial character. Because of that their variance in the denominator is not multiplied by a factor 1/2.

Equations (12), (9), and (13) describe the average heat flux across the system. At ΔT = 0 the pumping heat current is given by the first term in the r.h.s of Eq. (12). The pumping velocity of the electron liquid depends only on the pumping potential and disorder strength. Moreover, it follows from Eq. (9) that if the pumping potential exceeds the disorder strength, the pumping velocity becomes independent of the amplitude of pumping potential and is determined only by the pumping cycle geometry, see Eq. (11). In particular, for a potential U in the form of a wave traveling with velocity c heat pumping becomes perfect; the entire heat content of the electron liquid is transported with velocity c. The backflow of heat at ΔT 0 is described by the second term in the r.h.s. of (12) and is proportional to the effective thermal conductivity of the system in Eq. (13). The latter also depends on the pumping potential.

Some noteworthy features of heat pumping cycle considered here are: i) Similar to peristaltic pumps the pumping potential i s di stributed throughout the system. Therefore, pumped heat flux given by the first term in the r.h.s. of Eq. (12) does not decrease with the system length L. In contrast, the backflow of heat (second term in the r.h.s. of Eq. (12)) at a fixed temperature difference between the reservoirs is inversely proportional to L. Because of this, in longer systems the pumping heat flux can significantly exceed the backflow flow even at relatively low pumping strengths. ii) Although the mechanism of heat transfer considered here is thermoelectric by nature, in contrast to traditional thermoelectric cooling schemes, the pumping of heat is not accompanied by a voltage buildup between the reservoirs as it proceeds in a charge-neutral (on average) system. This may prove advantageous for certain applications. iii) For sufficiently small temperature differences, A , the backflow of heat may be neglected, and the pumping heat flux is well approximated by the first term in Eq. (12), which depends on the cycle geometry and disorder strength. For weak disorder, the heat moves with the velocity (9) that is determined only by the cycle geometry.

Heat pump efficiency

To keep the expressions simpler we may consider the traveling wave potential of the form U ₀ (X — ct). Efficiency of heat pumps is characterized by the coefficient of performance (COP), defined as the ratio of useful power to the power IE consumed by the pump. For cooling/heating cycle the useful power (per unit width of the system) is given by T j _s ~ Ts _o u, where T is the temperature of cold/hot reservoir. The power consumed by the pump may be obtained from a mechanical consideration. For a traveling wave potential, U _o (x - ct), the force density exerted by the pumping potential on the electron liquid in Eq. (7) has the form of a friction force, Multiplying it by the velocity c and integrating over the system length one finds

This yields Where is the Carnot efficiency. Expressing ΔT in terms of u using Eq. (8) we get where we introduced the dimensionless disorder strength

At a fixed temperature difference, the maximal COP in Eq. (15) is achieved at

From Eq. (8) it is easy to see that this occurs at

The maxima! value of COP is given by

For weak disorder, a « 1, it nearly reaches the Carnot limit. In this case u/c 1 in Eq. (17), and heat pumping is almost perfect.

Equation (14) and subsequent results for the heat pump efficiency may be also obtained by considering energy dissipation in the pump. The considerations above Eq. (5) show that within second order accuracy in U/T the contributions of temperature gradients and viscous stresses to energy dissipation may be neglected. Thus the rate of energy dissipation per unit area is given by Then using Eq. (5) we obtain

As demonstrated above, the energy dissipation rate in the system is characterized by the intrinsic conductivity. This reflects the fact that at small deviations from charge neutrality dissipation is caused by the electric fields arising in the electron liquid. The power consumed by the pump is given by sum of dissipated energy, and heat flux TJ _S . This reproduces Eq. (14).

Model predictions

To summarize the outcome of the above model, a mechanism of electronic pumping of heat at charge neutrality is modeled. The heat transfer proceeds at zero net charge current because the system is on average charge-neutral. As a result pumping of heat is not accompanied by voltage buildup along the system. This may prove advantageous for potential cooling applications (e.g. of microelectronic devices) where voltage buildup or presence of mechanically moving parts is undesirable. Consideration focused on slow pumping in the regime of electron hydrodynamics. An important parameter of the pumping cycle is the dimensionless ratio of disorder to pumping strength (α in Eq. (16) in a traveling wave setup). At α « 1 the heat flux is determined by the geometry of the pumping potential. For potentials in the form of a traveling wave the optimal efficiency of the pump, Eq. (19), may come close to the Carnot limit. In this case the pumping is nearly perfect; the entire heat content of the electron liquid is entrained by the pumping potential. Equation (18) shows that optimal efficiency may be reached for a wide range of temperature differences by adjusting the pumping parameters. Many embodiments of the technology described above may take the form of computer- or controller-executable instructions, including routines executed by a programmable computer or controller. Those skilled in the relevant art will appreciate that the technology cars be practiced on computer/controller systems other than those shown and described above. The technology can be embodied in a special -purpose computer, controller or data processor that is specifically programmed, configured or constructed to perform one or more of the computer-executable instructions described above. Such computers, controllers and data processors may include a non-transitory computer-readable medium with executable instructions. Accordingly, the terms "computer" and "controller" as generally used herein refer to any data processor and can include Internet appliances and hand-held devices (including palm-top computers, wearable computers, cellular or mobile phones, multi -processor systems, processor-based or programmable consumer electronics, network computers, mini computers and the like).

From the foregoing, it will be appreciated that specific embodiments of the technology have been described herein for purposes of illustration, but that various modifications may be made without deviating from the disclosure. Moreover, while various advantages and features associated with certain embodiments have been described above in the context of those embodiments, other embodiments may also exhibit such advantages and/or features, and not all embodiments need necessarily exhibit such advantages and/or features to fall within the scope of the technology. Where methods are described, the methods may include more, fewer, or other steps. Additionally, steps may be performed in any suitable order. Accordingly, the disclosure can encompass other embodiments not expressly shown or described herein. In the context of this disclosure, the term "about" means +/- 5% of the stated value. For the purposes of the present disclosure, lists of two or more elements of the form, for example, "at least one of A, B, and C," is intended to mean (A), (B), (C), (A and B), (A and C), (B and C), or (A, B, and C), and further includes all similar permutations when any other quantity of elements is listed.