Thermoelectric Cooling and Power Generation based on the Quantum Hall Effect

Abstract

A quantum Hall system can be used for extremely efficient thermoelectric cooling and power generation. Such a quantum Hall system can be implemented as a two-dimensional (2D) material that is subject to a quantizing magnetic field and whose opposite ends are electrically and thermally coupled to a heat sink and heat source, respectively. The edges of the 2D material connecting those opposite ends are coupled to respective ohmic contacts. The massive degeneracy and the metallicity of a partially-filled Landau level in the quantum Hall system enable thermoelectric energy conversion with unprecedented efficiency at low temperature. This efficiency occurs because the thermoelectric figure of merit is constant for a transverse thermoelectric device using the =0 quantum Hall state of Dirac materials at charge neutrality. Under these conditions, electron-hole symmetry causes the electrical Hall effect to vanish and the thermoelectric Hall effect to peak.

Claims

1. An apparatus for thermoelectric cooling and/or power generation, the apparatus comprising: a first heat bath; a second heat bath; a two-dimensional (2D) material in thermal and electrical communication with the first heat bath and the second heat bath; and a magnetic field source, in electromagnetic communication with the 2D material, to apply a magnetic field to the 2D material, the magnetic field causing electrons and holes to flow along edges of the 2D material between the first heat bath and the second heat bath, the electrons and holes carrying heat from the first heat bath to the second heat bath.

2. The apparatus of claim 1, wherein the apparatus has a finite thermoelectric figure of merit that is independent of temperature over a temperature range of about 0.1 K to about 200 K.

3. The apparatus of claim 1, wherein the 2D material comprises at least one of graphene or a topological insulator thin film.

4. The apparatus of claim 1, wherein the magnetic field source is configured to apply the magnetic field at an amplitude of about 0.1 Tesla to about 1.0 Tesla in a direction orthogonal to a plane of the 2D material.

5. The apparatus of claim 1, wherein the 2D material is configured to conduct an electrical current in a direction perpendicular to magnetic field and to a flow of the heat.

6. The apparatus of claim 5, further comprising: a first electrode and a second electrode, in electrical communication with the 2D material, to conduct electrical current generated by the flow of the electrons and holes out of the 2D material.

7. The apparatus of claim 6, further comprising: a voltage source, in electrical communication with the first electrode and the second electrode, to apply a potential difference across the first electrode and the second electrode, the potential difference causing the heat to flow against a thermal gradient between the first heat bath and the second heat bath.

8. The apparatus of claim 6, further comprising: a resistive load, in electrical communication with the first electrode and the second electrode, to convert the electrical current into electrical power.

9. The apparatus of claim 1, wherein the magnetic field applied by the magnetic field source quantizes Landau energy levels of the 2D material.

10. The apparatus of claim 9, wherein the 2D material has a peak thermoelectric Hall conductivity over the range of temperatures T satisfying <<k.sub.BT<<h.sub.c when the Landau energy levels are partially filled, where is disorder-induced Landau level broadening, k.sub.B is Boltzmann's constant, and h.sub.c is cyclotron energy.

11. A method comprising: applying a magnetic field to the 2D material connecting a first heat bath with a second heat bath, the magnetic field causing electrons and holes to flow along edges of the 2D material between the first heat bath and the second heat bath, the electrons and holes carrying heat from the first heat bath to the second heat bath.

12. The method of claim 11, wherein the 2D material has a finite thermoelectric figure of merit that is independent of temperature over a temperature range of about 0.1 K to about 200 K.

13. The method of claim 11, wherein the magnetic field is at an amplitude of about 0.1 Tesla to about 1.0 Tesla and in a direction orthogonal to a plane of the 2D material.

14. The method of claim 11, further comprising: conducting an electrical current via the 2D material in a direction perpendicular to magnetic field and to a flow of the heat.

15. The method of claim 14, further comprising: converting the electrical current into electrical power with a resistive load coupled to the 2D material.

16. The method of claim 11, further comprising: applying a potential difference across the 2D material in a direction perpendicular to magnetic field and to a flow of the heat, the potential difference causing the heat to flow against a thermal gradient between the first heat bath and the second heat bath.

17. The method of claim 11, wherein the flow of heat cools the first heat bath to a temperature less than about 200 K.

18. The method of claim 11, wherein the flow of heat cools the first heat bath to a temperature less than about 10 K.

Description

BRIEF DESCRIPTIONS OF THE DRAWINGS

[0010] The skilled artisan will understand that the drawings primarily are for illustrative purposes and are not intended to limit the scope of the inventive subject matter described herein. The drawings are not necessarily to scale; in some instances, various aspects of the inventive subject matter disclosed herein may be shown exaggerated or enlarged in the drawings to facilitate an understanding of different features. In the drawings, like reference characters generally refer to like features (e.g., elements that are functionally and/or structurally similar).

[0011] FIG. 1 shows a conventional thermoelectric generator.

[0012] FIG. 2A illustrates thermoelectric cooling based on the quantum Hall effect.

[0013] FIG. 2B illustrates thermoelectric generation based on the quantum Hall effect.

DETAILED DESCRIPTION

[0014] Thermoelectric cooling and power generation can be accomplished using Landau levels in a two-dimensional (2D) material, such as graphene, under a quantizing magnetic field. A partially filled Landau level in a 2D material exhibits a massive ground state degeneracy in the clean noninteracting limit. Provided that disorder and electron interaction are weak, the entropy per charge carrier remains finite down to very low temperature. Because of its non-vanishing entropy and metallicity, a partially filled Landau level enables thermoelectric cooling and power generation with unprecedented efficiency at low temperature. In fact, a thermoelectric cooler with quantized Landau levels in a 2D material could cool to temperatures of less than 100 K, which is colder than the temperatures achievable with conventional thermoelectric coolers.

[0015] Increasing Thermoelectric Efficiency in 2D and 3D Materials with Magnetic Fields

[0016] An inventive thermoelectric cooler/power generator uses a magnetic field to quantize Landau levels in the 2D material. Magnetic fields have been used to improve thermoelectric efficiency before, although not with 2D materials. Continuous cooling from room temperature to around 100 K was demonstrated using the giant Nernst effect in bismuth-antimony alloy under a modest magnetic field. (In the Nernst effect, applying a magnetic field and a temperature gradient perpendicular to each other in an electrically conductive sample yields an electric field that is perpendicular to both the magnetic field and the temperature gradient.) And a recent theoretical work proposed thermoelectric applications using the large, non-saturating Seebeck effect of Dirac/Weyl semimetals in the high-field quantum limit when electrical resistivity is dominantly transverse. In these works, the use of three-dimensional (3D) materials with a finite density of states at the Fermi level sets a fundamental limit that the Seebeck and Nernst responses are proportional to the temperature T, making the figure of merit zTT.sup.2 degrade rapidly as T.fwdarw.0. In contrast, Landau levels in a 2D material provide a flat band with a singularly large density of states for the transport of carriers, while maintaining the metallicity.

[0017] To see why Landau levels in a 2D material provide a flat band, consider coupled electrical and heat transport under a magnetic field B. The coupling between electricity and heat is often described in terms of Seebeck and Nernst signals S.sub.ij, which measure the voltage generated by a temperature difference under open-circuit conditions. As explained below, it can be preferable to consider thermoelectric conductivity .sub.ij. This thermoelectric conductivity .sub.ij represents the electrical current I generated by a temperature gradient VT in the absence of any voltage (short-circuit condition): I.sub.i=.sub.ij.sub.jT. By the Onsager relation, the thermoelectric conductivity also measures the heat current Q generated by an electric field under isothermal conditions: Q.sub.i=T.sub.ijE.sub.j. Since heat current is carried by thermal excitations, thermoelectric conductivity is purely a Fermi surface property, and the contributions from different Fermi pockets simply add up. Neither is the case for thermopower or Nernst signal, which are equal to S.sub.ij=.sub.ik.sub.kj, where is resistivity.

[0018] In the presence of a magnetic field B, thermoelectric conductivity generally has a component that is odd in magnetic field B. This component is called the thermoelectric Hall conductivity. For an isotropic system, it appears as the transverse component satisfying .sub.xy(B)=.sub.yx(B)=.sub.xy(B), where the xy plane is perpendicular to the magnetic field B.

[0019] Throughout this specification, consider a quantizing fieldor equivalently weak disorderthat satisfies B>>1, where is the carrier mobility of the 2D material. Under this condition, Landau levels are well separated by the cyclotron energy .sub.c>>, where is disorder-induced Landau level broadening. Both mobility and broadening arise from disorder. A higher mobility equates to a smaller broadening and means that Landau levels are formed at small fields. At temperatures on the order of a few Kelvin, for example, Landau levels appear in graphene at magnetic fields as small as 0.1 Tesla.

[0020] (The cyclotron energy .sub.c is the energy gap between Landau levels. At temperatures below the cyclotron gap, transport is dominated by Landau levels. In graphene, the cyclotron energy at a magnetic field of 1 Tesla corresponds to a temperature of about 400 K. At a magnetic field of 0.1 Tesla, the cyclotron energy corresponds to a temperature of about 130 K.)

[0021] At temperatures k.sub.BT>>.sub.c, the Landau levels are thermally smeared, and semiclassical transport theory is applicable. Under an applied electric field E perpendicular to the magnetic field B, charge carriers acquire a drift velocity v.sub.d, which in the clean limit is simply determined by the balance of electric force and Lorenz force: v.sub.d=EB/B.sup.2. This creates, in addition to a transverse electrical current, a transverse heat current Q=Tsv.sub.d, where s is the entropy density. Therefore, the thermoelectric Hall conductivity is given by

[00001]${}_{x.Math.y}=\frac{s}{B}.$

[0022] Since entropy is associated with the number of thermal excitations within the energy k.sub.BT from the Fermi level, in the temperature range .sub.c<<k.sub.BT<<E.sub.F (E.sub.F is Fermi energy; in ordinary metals, it is on the order of electron volts), the thermoelectric Hall conductivity follows the proportionality .sub.xy sk.sub.BT for metals and degenerate semiconductors.

[0023] The entropy-based formula for the thermoelectric Hall conductivity .sub.xy above continues to hold at temperatures low enough that k.sub.BT<.sub.c in the limit of weak disorder /.sub.c.fwdarw.0. However, the entropy is now strongly modified by Landau quantization of density of states, which in two dimensions is a set of sharp peaks at discrete energies. When the Fermi energy is at the center of a Landau level, each Landau orbital has probability 1/2 of being occupied and of being empty (assuming <<k.sub.BT), resulting in a maximum entropy density s=(log 2)k.sub.B (B/.sub.0), where .sub.0=h/e is the flux quantum. Therefore, in the temperature range <<k.sub.BT<<.sub.c, thermoelectric Hall conductivity is peaked whenever a Landau level is half-filled, and the peak value is universal:

[00002]${}_{x.Math.y}=\frac{\left(\log .Math.2\right).Math.k_B.Math.e}{h}.$

[0024] In the dissipation-less limit, the Seebeck coefficient S.sub.xx=.sub.xy.sub.yx also depends on the number of completely filled Landau levels via .sub.yx and hence is less universal. For example, in the quantum Hall regime of graphene at charge neutrality, S.sub.xx=0 while .sub.xy=s/B still holds.

[0025] Thanks to the finite thermoelectric Hall conductivity .sub.xy, at low temperature, quantum Hall systems are advantageous over traditional thermoelectric materials that employ .sub.xx. The latter decreases linearly with temperature when k.sub.BT<<E.sub.F, as seen from the generalized Mott formula .sub.xx=(.sup.2k.sub.B.sup.2T/3e)d(E)/dE|.sub.E.sub.F, where (E) is energy-dependent conductivity. For 3D systems under a magnetic field, the continuous energy spectrum of one-dimensional Landau band dispersing along the field direction leads to .sub.xy T, which decreases at low temperature, in contrast with the 2D case.

[0026] Thermoelectric Cooling and Power Generation with a Quantum Hall System

[0027] FIGS. 2A and 2B illustrate a device 200 for thermoelectric cooling and power generation, respectively, motivated by the consideration of thermoelectric Hall conductivity .sub.xy in 2D materials. As shown in FIGS. 2A and 2B, the system 200 includes a 2D material that is in thermal contact with two heat baths or thermal reservoirs 201 and 202 at different temperatures. Each bath/reservoir 201, 202 exchanges energy the 2D material 212 without transferring any net charge. The 2D material 212 is also connected via electrical leads 203 and 204 to an external circuita battery 206 or other voltage source in the case of cooling (FIG. 2A) and a resistive load R.sub.L in the case of power generation (FIG. 2B).

[0028] The thermal reservoirs 201 and 202 can be made of 2D or bulk materials in Ohmic contact with the quantum Hall system (the 2D material 212). Suitable 2D materials 212 include but are not limited to graphene and topological insulator (e.g., HgTe and Bi.sub.2Se.sub.3) thin films. Layered Dirac materials can also be used as the 2D materials 212. For instance, the 2D material 212 may be a single layer of graphene or multiple layers of graphene. Multi-layer graphene may conduct higher fluxes of heat and current than single-layer graphene. The electrical leads 203 and 204 can be formed as ohmic contacts on the 2D material 212.

[0029] A magnetic field source 220, such as a permanent magnet or an electromagnet, applies a magnetic field B to the 2D material 212. This magnetic field B is orthogonal to the plane of the 2D material (out of the plane in FIG. 2A and into the plane in FIG. 2B) and causes the 2D material 212 to behave as a quantum Hall system. The amplitude of this magnitude field B is large enough to satisfy the condition B>>1, where is the carrier mobility of the 2D material 212. For suitable 2D materials, the magnitude field amplitude may range from about 0.1 T to about 1.0 T or higher.

[0030] Power generation is achieved by natural heat flux from the hot bath 201 to the cold bath 202 as shown in FIG. 2B. This heat flux produces a voltage between the leads 203 and 204 and thus supplies electrical power to the resistive load R.sub.L. On the other hand, passing a sufficiently large electrical current between the leads 203 and 204 cools the cold bath 201 by directing heat from the cold bath 201 into the hot bath 202 against the opposing temperature difference as shown in FIG. 2A. In this device 200, electrical current and heat current run in orthogonal directions (electrical current runs from electrical contact 204 to electrical contact 203 in both the cooling and power generation configurations). For such transverse thermoelectric geometry, there is no need to employ both n and p-type materials as in traditional Peltier coolers and Seebeck generators (e.g., as in FIG. 1).

[0031] In thermal equilibrium, the device's terminals (cold bath 201, hot bath 202, and electrical contacts 203 and 204) are at the same temperature T.sub.j=T and chemical potential .sub.j=. (The subscripts j=1,2,3,4 correspond to the cold bath 201, hot bath 202, first electrical contact 203, and second electrical contact 204, respectively.) While the device 200 is operating, T.sub.j and .sub.j are generally different from their equilibrium values, so T.sub.j=T+T.sub.j, .sub.j=+eV.sub.j and V.sub.j are called generalized forces. There may also be net charge and heat currents, denoted as I.sub.j and Q.sub.j, that flow within the 2D material 212 into (defined as positive) or out of (defined as negative) the terminals 203 and 204.

[0032] Assume for simplicity that the device 200 has a twofold rotation symmetry that exchanges baths 201.Math.202 and 203.Math.204 at opposite ends. Then, while the device 200 is in a working state, the currents and forces at terminal 201 (203) are opposite to those at 202 (204) and can be written as J.sub.1=J.sub.2J.sub.y, J.sub.4=J.sub.3J.sub.x, and F.sub.1=F.sub.2 F.sub.y/2, F.sub.4=F.sub.3 F.sub.x/2, where J stands for I or Q, and F for V or T. By definition, there is no net charge current flowing into a heat bath, so I.sub.y=0. Assume also that the two electrical leads 203 and 204 are at the same temperature (as is the case when the external circuit is a perfect thermal conductor), so T.sub.x=0.

[0033] For a given temperature difference T.sub.y between the cold bath 201 and the hot bath 202, solving the coupled electrical/thermal transport equation under the condition I.sub.y=0 yields the electrical current I.sub.x and the heat current Q.sub.y in the device 200 used as a cooler for a given voltage V.sub.x, which set by the external battery 206. Likewise, when the device 200 is used as a generator with an external resistance R.sub.L, one can obtain I.sub.x and Q.sub.y by further using Ohm's law I.sub.x=V.sub.x/R.sub.L.

[0034] To simplify calculations further, consider electron-hole balanced quantum Hall systems (2D material 212) in the quantum Hall regime, such as 2D Dirac materials like graphene and topological insulator (e.g., HgTe and Bi.sub.2Se.sub.3) thin films. Under a magnetic field, the massless Dirac fermion exhibits a special n=0 Landau level at zero energy, which is exactly half-filled at charge neutrality and composed equally of electrons and holes.

[0035] In general ground, electrical Hall conductivity, thermal Hall conductivity (denoted by .sub.xy) as well as diagonal thermoelectric conductivity and Seebeck coefficient are odd under charge conjugation. In contrast, the thermoelectric Hall conductivity is invariant under charge conjugation. As a result, the =0 quantum Hall state at charge neutrality has .sub.xy=.sub.xy=0 and .sub.xx=S.sub.xx=0 due to electron-hole cancellation, but a nonzero thermoelectric Hall conductivity .sub.xy that takes a universal value under the specified conditions. Thus, the thermoelectric Hall effect is the only remaining Hall response at =0!

[0036] For such an electron-hole-balanced system, the simplified transport equation with .sub.xy=.sub.xy=.sub.xx=0 takes the general form

I.sub.x=GV.sub.x+L.sup.ehT.sub.y

Q.sub.y=TL.sup.heV.sub.x+{tilde over (K)}T.sub.y,

where G is longitudinal electrical conductance, {tilde over (K)} is longitudinal thermal conductance in the absence of any voltage, and L.sup.eh, L.sup.he are transverse thermoelectric conductance.

[0037] To understand how the device 200 in FIGS. 2A and 2B enables thermoelectric cooling and power generation, consider the short-circuit and open-circuit limits. In the presence of a temperature difference T.sub.y, when R.sub.L=0 to short-circuit the generator, a transverse electrical current is produced by thermoelectric Hall effect, denoted as I.sub.x.sup.0. Alternatively, in an open circuit with R.sub.L=, an opposing voltage difference V.sub.x.sup.0 arises to cancel the electrical current that would otherwise be present and enforces I.sub.x=0. For finite R.sub.L, I.sub.x is nonzero and smaller than I.sub.x.sup.0. The ratio of output electrical power and the heat current |Q.sub.y| defines the coefficient of performance, .sub.p=I.sub.x.sup.2R.sub.L/|Q.sub.y|. .sub.p is at a maximum at a certain load resistance.

[0038] In cooling mode (FIG. 2A), the external battery 206 sets a forward voltage V.sub.x (opposite to) V.sub.x.sup.0) to increase the electrical current I.sub.x above the short-circuit current I.sub.x.sup.0. When V.sub.x is sufficiently large, the thermoelectric Hall effect generates a heat current Q.sub.y going from the cold batch 201 to the hot bath 202 against the opposing temperature difference. The cooling power q is given by Q.sub.y minus Joule heating in the cold bath 201, q=Q.sub.yI.sub.xV.sub.x/2 (half of I.sub.x goes through the cold bath 201). The coefficient of performance is defined as the ratio of q and the rate of electrical power input, .sub.c=(Q.sub.yI.sub.xV.sub.x/2)/(I.sub.xV.sub.x). Since Q.sub.y V.sub.x and Joule heating is proportional to V.sub.x.sup.2, .sub.c should be at a maximum at a certain applied voltage V.sub.x.

[0039] The maximum coefficient of performance .sub.c or .sub.p of the device 200 increases with a dimensionless quantity ZT known as the thermoelectric figure of merit,

[00003]$Z.Math.T=\frac{L^{e.Math.h}.Math.L^{h.Math.e}.Math.T}{G.Math.K}$

where T=(T.sub.1+T.sub.2)/2 is the mean temperature of heat baths; K={tilde over (K)}+TL.sup.ehL.sup.he IG is thermal conductance in the absence of any electrical current, i.e., in an open-circuit condition. For a cooler, the maximum temperature difference attainable is T.sub.max=ZT.sub.1.sup.2/2. For a generator, the efficiency approaches the Carnot limit of (T.sub.1-T.sub.2)/T.sub.1 as ZT.fwdarw..

[0040] This thermoelectric figure of merit ZT depends on a combination of electrical, thermal, and thermoelectric conductance for the 2D material 212 in the four-terminal geometry shown in FIGS. 2A and 2B. If the 2D material 212 is large enough, the electrical, thermal, and thermoelectric conductance are each proportional to the corresponding local conductivity. A half-filled Landau level at charge neutrality shows a finite, T-independent longitudinal conductivity at low temperature, which is on the order of e.sup.2/h in graphene and topological insulator thin films. Assuming lattice thermal conductivity is negligible at low temperature, it follows from the Wiedemann-Franz law that is on the order of (k.sub.B/e).sup.2Ta. Using these values for and along with the universal value of .sub.xy=s/B, the thermoelectric figure of merit ZT is of order unity throughout the temperature range <<k.sub.B T<<.sub.c (practically, over a temperature range of about 0.1 K to about 200 K).

[0041] Next, consider the thermoelectric figure of merit ZT in the phase-coherent transport regime, where the electrical, thermal, and thermoelectric conductance are determined by the transmission probability of an electron or a hole going from one terminal to another terminal. In the weak disorder limit, the conductance is dominated by edge-state transport. This allows the calculation of G, K, L.sup.eh, L.sup.he explicitly using a scattering approach and thus the determination of ZT.

[0042] The zero-energy n=0 Landau levels in graphene, HgTe quantum wells, and topological insulator thin films each have twofold degeneracy. These twofold degeneracies are associated with valley, spin, and top/bottom surface layer degrees of freedom in graphene, HgTe quantum wells, and topological insulator thin films, respectively. The applied magnetic field splits each degeneracy at the edge of the sample, giving rise to two branches of edge modes: a counter-clockwise branch at E>0 and a clockwise branch at E<0. As a result, electrons and holes go in opposite directions at the edge, as illustrated in FIGS. 2A and 2B, in tandem with the sign change of the quantized Hall conductance. Without being bound by any particular theory, the existence of these ambipolar edge states within the energy gap between n=0 and n=1 Landau levels is guaranteed by topology: it is required by the first quantized Hall plateau on the electron and hole side. At a given energy, the edge state is chiral and elastic backscattering from impurity is forbidden.

[0043] Due to its chirality, the transmission probability of an electron that occupies an E>0 edge mode going from terminal i to j is 1 if j is a downstream neighbor of i, and 0 otherwise. In contrast, due to its opposite chirality, the transmission probability of a holethe state of having an unoccupied E<0 edge modegoing from terminal i to j is 1 if j is an upstream neighbor of i, and 0 otherwise.

[0044] Therefore, applying a voltage or temperature change at a given terminal can only produce nonzero electrical and/or heat currents at its downstream neighbor, at its upstream neighbor, and at itselfthe last being a sum of the first two by the law of current conservation. The downstream (upstream) current is solely carried by electrons (holes) and thus depends only on the change of occupation of the E>0 (E<0) edge modes due to =eV or T.

[0045] To obtain electrical conductance G, calculate the electrical current I.sub.4 produced by V.sub.4. For eV.sub.4>0 as shown in FIG. 2A, the increase in the chemical potential at lead 204 sends more electrons to bath 201 and fewer holes to bath 202, which add to yield the electrical current:

[00004]$I_4={}_0^{}.Math.d.Math.E\left(\frac{e}{h}\right).Math.\left(-\frac{f}{E}.Math.e.Math.V_4\right)+{}_{-}^0.Math.d.Math.E\left(\frac{-e}{h}\right).Math.\left(\frac{f}{E}.Math.e.Math.V_4\right)$

where f (E)=1/(exp(E)+1) is the Fermi-Dirac distribution at charge neutrality. The change of electron occupation is f, while the change of hole occupation is f. This current-voltage relation yields an electrical conductance:

G=I.sub.4/(2V.sub.4)=(1/2)e.sup.2/h

[0046] The prefactor 1/2 is due to two resistors in series in the four-terminal geometry of the device 200 in FIGS. 2A and 2B.

[0047] Similarly, calculate the heat current Q.sub.1 produced by the temperature change T.sub.1. As shown in FIG. 2B, the increase of temperature at bath 201 sends more electrons to lead 203 and more holes to lead 204, which add to yield the heat current

[00005]$Q_1={}_0^{}.Math.d.Math.E\left(\frac{E}{h}\right).Math.\left(-\frac{f}{E}.Math.\frac{E.Math..Math..Math..Math.T_1}{T}\right)+{}_{-}^0.Math.d.Math.E\left(\frac{-E}{h}\right).Math.\left(\frac{f}{E}.Math.\frac{E.Math..Math.T_1}{T}\right)$

using the identity f/T=f/E.Math.(E/T). A hole that corresponds to an unoccupied E<0 mode is an excitation that costs energy E>0. The thermal conductance is then given by

[00006]$\tilde{K}=\frac{1}{2}.Math.\frac{{}^2}{3}.Math.\frac{k_B^2.Math.T}{h}$

[0048] Finally, calculate the electrical current I.sub.3 produced by opposite temperature changes at the two baths 201 and 202: T.sub.1=T.sub.2 T. As shown in FIG. 2B, the increase of temperature at bath 201 sends more electrons to lead 203, while the decrease of temperature at bath 202 sends fewer holes to lead 203. The two contributions add to yield the electrical current:

[00007]$I_3={}_0^{}.Math.d.Math.E\left(\frac{e}{h}\right).Math.\left(-\frac{f}{E}.Math.\frac{E.Math..Math..Math..Math.T}{2.Math..Math.T}\right)+{}_{-}^0.Math.d.Math.E\left(\frac{-e}{h}\right)\left[\frac{f}{E}.Math.\frac{E\left(-.Math.T\right)}{2.Math.T}\right].$

[0049] This gives the thermoelectric Hall conductance

L.sup.eh=log(2)k.sub.Be/h.

[0050] Likewise, from the heat current Q.sub.3 produced by concurrent voltages V.sub.1=V.sub.2, L.sup.he=L.sup.eh, in accordance with the general symmetry property in scattering theory of thermoelectric transport.

[0051] Putting these conductance values together shows that the thermoelectric figure of merit ZT is a temperature-independent constant over a temperature range of about 0.1 K to about 200 K:

[00008]$Z.Math.T=\frac{{\log }_2.Math.2}{\frac{1}{2}\left[\frac{{}^2}{6}+2.Math.{\log }_2.Math.2\right]}0.3.Math.7.$

[0052] This result is independent of any additional degeneracy of the n=0 Landau level that may be present (e.g., spin degeneracy in addition to the aforementioned valley degeneracy in graphene), because such a degeneracy would increase electrical conductance, thermal conductance, and thermoelectric Hall conductance by the same factor without affecting ZT. Since a large electronic thermal conductance is desirable in order to outweigh the phonon contribution, a large Landau level degeneracy is advantageous.

[0053] Such a record-high thermoelectric figure of merit can be achieved with existing 2D materials, including graphene, HgTe, and Bi.sub.2Se.sub.3 thin films, which each have a Dirac velocity on the order of 10.sup.6 m/s. A 1 T magnetic field creates an energy gap of about 400K between the n=0 and n=1 Landau levels. In high-mobility samples, the Landau level width is about 10 K. Thus, thermoelectric cooling and power generation should be efficient in over the range of temperatures T spanning <k.sub.BT<.sub.c.

[0054] It is encouraging that ample evidence of electrical transport mediated by ambipolar edge states has been observed in the =0 quantum Hall state in graphene, HgTe, and Bi.sub.2Se.sub.3 thin films. Moreover, a peak of the thermoelectric Hall conductivity .sub.xy approaching the universal value (.sub.xy=s/B) has been observed at =0 in graphene and bilayer graphene in the range of temperatures T spanning <k.sub.BT<.sub.c.

[0055] In practice, a 2D quantum Hall system could be used for thermoelectric cooling of a small quantum device. In order to cool a 3D bath at low temperature, it may be useful to employ bulk crystals formed with weakly coupled layers, such as graphite, ZrTe.sub.5, or organic molecular crystals in which 3D quantum Hall states have recently been observed. For instance, a layered material where each layer hosts Dirac bands can also be placed in a three-dimensional quantum Hall regime by applying a magnetic field and thus can be used for thermoelectric cooling. In such a layered material, each layer acts in parallel, thus enhancing the device's overall cooling power.

[0056] While the detailed analysis above is focused on the electron-hole-symmetric =0 quantum Hall state, the conclusion that the thermoelectric figure of merit ZT remains finite at low temperature follows from two features of partially filled Landau levels (more generally flat bands with Chern number): (1) a finite thermoelectric Hall conductivity .sub.xy due to massive degeneracy and (2) a finite electrical conductivity .sub.xx due to its metallicity, together with Wiedemann-Franz law.

[0057] Last but not the least, the role of Coulomb interaction in lifting Landau level degeneracy has been neglected in the preceding analysis. The characteristic energy scale associated with Coulomb interaction is e.sup.2/l.sub.B, where E is the dielectric constant and l.sub.B is the magnetic length. For thermoelectric cooling and power generation, this energy scale can be sufficiently suppressed by strong dielectric screening (e.g., by placing the 2D material near a metal) or by working with a small magnetic field. For example, at a magnetic field of 0.1 Tesla to 1.0 Tesla, the Coulomb interaction effect is not important.

CONCLUSION

[0058] While various inventive embodiments have been described and illustrated herein, those of ordinary skill in the art will readily envision a variety of other means and/or structures for performing the function and/or obtaining the results and/or one or more of the advantages described herein, and each of such variations and/or modifications is deemed to be within the scope of the inventive embodiments described herein. More generally, those skilled in the art will readily appreciate that all parameters, dimensions, materials, and configurations described herein are meant to be exemplary and that the actual parameters, dimensions, materials, and/or configurations will depend upon the specific application or applications for which the inventive teachings is/are used. Those skilled in the art will recognize or be able to ascertain, using no more than routine experimentation, many equivalents to the specific inventive embodiments described herein. It is, therefore, to be understood that the foregoing embodiments are presented by way of example only and that, within the scope of the appended claims and equivalents thereto, inventive embodiments may be practiced otherwise than as specifically described and claimed. Inventive embodiments of the present disclosure are directed to each individual feature, system, article, material, kit, and/or method described herein. In addition, any combination of two or more such features, systems, articles, materials, kits, and/or methods, if such features, systems, articles, materials, kits, and/or methods are not mutually inconsistent, is included within the inventive scope of the present disclosure.

[0059] Also, various inventive concepts may be embodied as one or more methods, of which an example has been provided. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments.

[0060] All definitions, as defined and used herein, should be understood to control over dictionary definitions, definitions in documents incorporated by reference, and/or ordinary meanings of the defined terms.

[0061] The indefinite articles a and an, as used herein in the specification and in the claims, unless clearly indicated to the contrary, should be understood to mean at least one.

[0062] The phrase and/or, as used herein in the specification and in the claims, should be understood to mean either or both of the elements so conjoined, i.e., elements that are conjunctively present in some cases and disjunctively present in other cases. Multiple elements listed with and/or should be construed in the same fashion, i.e., one or more of the elements so conjoined. Other elements may optionally be present other than the elements specifically identified by the and/or clause, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, a reference to A and/or B, when used in conjunction with open-ended language such as comprising can refer, in one embodiment, to A only (optionally including elements other than B); in another embodiment, to B only (optionally including elements other than A); in yet another embodiment, to both A and B (optionally including other elements); etc.

[0063] As used herein in the specification and in the claims, or should be understood to have the same meaning as and/or as defined above. For example, when separating items in a list, or or and/or shall be interpreted as being inclusive, i.e., the inclusion of at least one, but also including more than one, of a number or list of elements, and, optionally, additional unlisted items. Only terms clearly indicated to the contrary, such as only one of or exactly one of, or, when used in the claims, consisting of, will refer to the inclusion of exactly one element of a number or list of elements. In general, the term or as used herein shall only be interpreted as indicating exclusive alternatives (i.e. one or the other but not both) when preceded by terms of exclusivity, such as either, one of only one of, or exactly one of. Consisting essentially of when used in the claims, shall have its ordinary meaning as used in the field of patent law.

[0064] As used herein in the specification and in the claims, the phrase at least one, in reference to a list of one or more elements, should be understood to mean at least one element selected from any one or more of the elements in the list of elements, but not necessarily including at least one of each and every element specifically listed within the list of elements and not excluding any combinations of elements in the list of elements. This definition also allows that elements may optionally be present other than the elements specifically identified within the list of elements to which the phrase at least one refers, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, at least one of A and B (or, equivalently, at least one of A or B, or, equivalently at least one of A and/or B) can refer, in one embodiment, to at least one, optionally including more than one, A, with no B present (and optionally including elements other than B); in another embodiment, to at least one, optionally including more than one, B, with no A present (and optionally including elements other than A); in yet another embodiment, to at least one, optionally including more than one, A, and at least one, optionally including more than one, B (and optionally including other elements); etc.

[0065] In the claims, as well as in the specification above, all transitional phrases such as comprising, including, carrying, having, containing, involving, holding, composed of, and the like are to be understood to be open-ended, i.e., to mean including but not limited to. Only the transitional phrases consisting of and consisting essentially of shall be closed or semi-closed transitional phrases, respectively, as set forth in the United States Patent Office Manual of Patent Examining Procedures, Section 2111.03.