Open access peer-reviewed chapter

# Introduction to (p × n)-Type Transverse Thermoelectrics

Written By

Matthew Grayson, Qing Shao, Boya Cui, Yang Tang, Xueting Yan and Chuanle Zhou

Submitted: October 30th, 2017 Reviewed: May 15th, 2018 Published: November 5th, 2018

DOI: 10.5772/intechopen.78718

From the Edited Volume

## Bringing Thermoelectricity into Reality

Edited by Patricia Aranguren

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## Abstract

This chapter will review (p × n)-type transverse thermoelectrics (TTE). Starting with the device advantages of single-leg (p × n)-type TTE’s over other thermoelectric paradigms, the theory of (p × n)-type TTE materials is given. Then, the figure of merit, transport equations, and thermoelectric tensors are derived for an anisotropic effective-mass model in bulk three-dimensional materials (3D), quasi-two-dimensional (2D), and quasi-one-dimensional (1D) materials. This chapter concludes with a discussion of the cooling power for transverse thermoelectrics in terms of universal heat flux and electric field scales. The importance of anisotropic ambipolar conductivity for (p × n)-type TTEs highlights the need to explore noncubic, narrow-gap semiconductor or semimetallic candidate materials.

### Keywords

• transverse thermoelectrics
• Seebeck tensor
• transport equations
• transverse thermoelectric figure of merit
• transverse cooling power

## 1. Introduction

The paradigm of (p× n)-type transverse thermoelectrics (TTE) [1, 2, 3] occurs in an anisotropic semiconductor or semimetal with p-type Seebeck response along one axis and n-type Seebeck orthogonal, whereby an appropriately applied electric current at an angle with respect to these axes can induce a purely orthogonalheat flow (the transverse Peltier effect), or conversely, an applied temperature gradient can generate a purely orthogonalelectric field (the transverse Seebeck effect).

Transverse thermoelectric effects in general [4, 5, 6] require a broken symmetry to generate the necessary off-diagonalcomponent in the Seebeck tensor. An off-diagonal component of the Seebeck tensor Sxydescribes, for example, an x-direction gradient in the electrochemical potential xVthat results from a transverse y-direction temperature gradient yT, expressed as

Sxy=xVyTE1

within the full tensor expression

iV=jSijjTE2

where i,jxyz. The Kelvin relations show how the corresponding transverse Peltier coefficient is related to the transverse Seebeck coefficient:

πxy=SxyTE3

For standard thermoelectrics (called “longitudinal thermoelectrics” in the present context), the Seebeck tensor is typically assumed to be diagonal and isotropic, and intentionally doped to be unipolar (entirely n-type or p-type), with no off-diagonal terms. Although many materials found in standard thermoelectrics are anisotropic in their crystalline form, for optimal thermoelectric performance these crystals are typically crushed into a powder and sintered into a dense randomly oriented polycrystal with many grain boundaries to reduce thermal conductivity, resulting in a macroscopically isotropic Seebeck tensor. For example, even if the original crystals were to have a Seebeck tensor S0that was anisotropic,

S0=Sxx000Syy000SzzE4

the resulting Seebeck tensor Ssntof the sintered polycrystal of randomly oriented grains becomes isotropic,

Ssnt=Str000Str000Str.E5

If the thermal conductivity of the original crystal is approximately isotropic, or if the subsequent sintered material is dominated by grain boundary scattering, the diagonal components of the sintered Seebeck tensor can be approximated by the trace of the original anisotropic crystal, Str=13trS0.

To generate off-diagonal terms, a transport symmetry must be broken. One way to do this is with a magnetic field, which breaks time-reversal symmetry. According to the Nernst-Ettingshausen effect (N-E) when a magnetic field is applied to a medium with ambipolar electron and hole conduction, an antisymmetric off-diagonal Seebeck response Sij=Sji=αBis induced proportional to the field strength Bwith N-E coefficient αand ij. For a review of the Nernst-Ettingshausen effect, please see [4, 7] and references therein. A magnetic field-induced transverse thermoelectric effect has also been observed in unipolar materials, where a “spin voltage” is generated orthogonal to a temperature gradient, named spin Seebeck effect [8, 9, 10, 11].

Similarly, structural asymmetrycan cause nonzero off-diagonal Seebeck terms, provided the asymmetry is maintained at a macroscopic scale. For example, sintering of randomly oriented polycrystals is disallowed, but in certain cases, twinned polycrystals with common c-axis alignment might still manifest the necessary anisotropy. If the diagonal Seebeck tensor elements Siiand Sjjin Eq. (4) are not equal, a simple rotation Rcan induce symmetric nonzero off-diagonal terms Sij=+Sjifor ij. With appropriate choice of the rotation tensor R, the off-diagonal terms become

Sij=kRik1SkkRkjE6

Note that a purely p- or n-type unipolarmaterial can never generate purelyorthogonal Seebeck response—any transverse Seebeck will always be accompanied by a finite longitudinal Seebeck response. Such unipolar transverse Seebeck effects have been observed and studied, for example, in Refs. [12, 13, 14, 15, 16].

A more interesting structural asymmetry is one that includes an ambipolar Seebeck tensor, whereby the diagonalized Seebeck tensor elements Sxx, Syy, and Szzin Eq. (3) have at least one p-type S>0along one direction and at least one n-type S<0along another direction, as illustrated in Figure 1.

In such a material, there doesexists a rotation Rsuch that the Seebeck response is purely orthogonal, such that an applied thermal gradient can induce a purely orthogonal electric field, or equivalently,

TV=0E7

where Vis determined from Tvia Eq. (2). A well-documented example of such a transverse thermoelectric is the composite multilayered transverse thermoelectric [4, 5, 6, 17, 21, 22] (see Figure 3), whereby a sequence of macroscopic alternating n- (typically metallic or semimetallic) and p-type (semiconducting) layers create the necessary structural asymmetry. Orthogonal to the layers, electrical and thermal resistance is in series from adjacent p- to n-layers, whereas parallel to the layers, the electrical and thermal conduction is in parallel.

The most recent addition to the lexicon of transverse thermoelectric phenomena is single crystals that themselves possess orthogonal p- and n-type Seebeck components, and which the authors have dubbed “(p × n)-type” [1, 2]. The cause of such ambipolar behavior is fundamentally different from that of the composite multilayered materials, since the (p × n)-type materials are bulk crystals, and thus both the electrical and thermal conductions are in parallel in alldirections. Although this may seem like a trivial distinction, the consequences are profound. Once a bulk crystal is solely responsible for the ambipolar Seebeck tensor, this material can be scaled to arbitrary size—large or small—allowing one to envision both sheets of active cooling layers as well as microscale cooling applications for integrated thermal management. Furthermore, because such ambipolar materials operate close to the intrinsic limit with minimal doping, there is no danger of dopant freeze-out; thus, these materials can be expected to achieve transverse thermoelectric performance at arbitrarily low temperatures, provided that the band gap is of order the operation temperature. Finally, the (p × n)-type materials continue to have the same structural advantage of all transverse phenomena, namely that they can be implemented as single-leg devices, allowing for improved cooling differentials in tapered structures [19] as well as geometric implementation in other unconventional geometries that standard thermoelectrics cannot achieve.

Because the underlying phenomenology of these (p × n)-type transverse thermoelectrics materials has only recently been introduced, their band characteristics are just now being explored theoretically and experimentally. Section 2 reviews intuition behind how (p × n)-type materials function in simple devices, while Section 3 reviews the key band-theoretical equations for generating the necessary ambipolar Seebeck tensor in bulk materials from a simple effective mass model for 3D bulk semiconductors. Because quasi-2D and quasi-1D materials represent extreme limits of anisotropic band structure, the equations for calculating Seebeck tensors in such limits are also provided. Section 4 reviews how the transverse figure of merit is optimized for transverse materials in general, and Section 5 identifies the cooling power for devices made of such transverse materials.

## 2. (p × n)-type transverse thermoelectric devices

(p × n)-Type transverse thermoelectrics have potential device advantages over other thermo-electric solutions when considering microscale devices or cryogenic operation. Conventional longitudinal thermoelectric devices (Figure 2) [6] or multilayer composite transverse thermoelectrics (Figure 3) [4, 5, 17] require at least one component with extrinsic p- or n-type doping, which limits their use at cryogenic temperatures since the dopants freeze out. A typical minimum operation temperature is T = 150 K. Similarly, the minimum device size is limited for the multileg structure of conventional longitudinal thermoelectric in Figure 2. And for multilayer composite transverse thermoelectrics of Figure 3, the macroscopic stacked sublayers set a minimum device size on the order of centimeters. For this reason, submillimeter scale devices are not feasible with either of the above thermoelectric paradigms.

On the other hand, (p × n)-type transverse thermoelectric bulk materials have distinct advantages in the cryogenic and size-scaling regimes since they operate as nominally undoped, single-leg devices. Transverse thermoelectric bulk materials have optimal performance near intrinsic doping with ambipolar electron and hole transport. As a consequence, narrow gap (p × n)-type materials should be able to work at arbitrarily low temperatures down to the cryogenic limit. The single-leg geometry also makes it straightforward to scale up to unconventional sheet-like geometries or to scale down to microscale devices since the full thermoelectric function is contained within a single material.

There are additional device advantages to single-leg thermoelectrics that result from the reduced fabrication complexity. For conventional two-leg thermoelectric devices, it is known that by stacking thermoelectric units one on top of the other with ever smaller areas, the resulting thermoelectric cascade can achieve a lower base temperature than a single stage, alone. When longitudinal thermoelectrics require multiple devices and multiple stages [18] to create such a cascade structure, transverse thermoelectrics can achieve the same “cascade” function by simply tapering a single thermoelectric leg [19]. The result acts as an “infinite-stage” Peltier refrigerator, which achieves superior cooling efficiency compared to the multiple discrete-element cascade stages by simply tapering a piece of transverse thermoelectric as a trapezoid or exponential taper. The tapering strategy allows one to achieve enhanced temperature differences even with a somewhat smaller transverse figure of merit zxyT[1, 19].

A typical longitudinal thermoelectric device structure is shown in Figure 2. As can be observed from the schematic diagram, each thermocouple unit has two legs, one p-type leg and one n-type leg. For Peltier refrigeration, the common side of both legs on the top is connected to the object to be cooled while the other side is connected to the heat sink. Following the flow of heat Qpand Qnin each leg, the top junction is cooled and the heat is transferred to the bottom heat sink.

The TTE unit in Figure 4, on the other hand, is made of one single material. Depending on the direction of current flow, only one kind of charge carrier, holes or electrons, will dominate conduction within each leg. For instance, we can observe electron current Jnin the right branch and hole current Jpin the left counterpart. Moreover, the heat current of both legs is flowing downward, just like the heat flow of the conventional device.

As demonstrated in Figure 5, a simpler single-leg geometry is possible with transverse thermoelectrics. With the electrons and holes, transportation directions of the p×n-type transverse thermoelectric are indicated with the crossed-arrow symbol on the upper right. The macroscopic transport of charge and heat is a vector sum of the net electron-hole electrical and heat currents, respectively. This picture depicts net charge current Jxto the right and net heat current Qyup.

An important quantity in comparing different transverse thermoelectric materials is the transverse figure of merit zxyT, which is used in the expressions of device efficiency and performance (see Section 5). For transverse materials, the dimensionless figure of merit zxyTis given as:

zxyT=Sxy2ρxxκyyTE8

In the transverse figure of merit expression above, the off-diagonal Seebeck element Sxyin the numerator is clearly the relevant component for generating a transverse thermoelectric response. In the denominator, to minimize Joule heating along the x-direction of current flow, a small resistivity component ρxxis needed; and to minimize passive return of Fourier heat in the y-direction of the temperature differential, a small thermal conductivity κyyis needed. Note that for longitudinal thermoelectrics, the expression for zTtypically includes the conductivity in the numerator since the scalar equation σ=1/ρis valid. However, when solving for transverse thermoelectric tensors, which by necessity have anisotropic conductivities, one must take care to calculate the resistivity component and place it in the denominator of the expression above, since, in general σxx1/ρxx.

## 3. Seebeck tensor of (p×n)-type transverse thermoelectrics

### 3.1. Thermoelectric tensors definition

Below, we derive how parallel anisotropic electron and hole conductivity give rise to the observed transverse thermoelectric behavior in (p× n)-type thermoelectrics. For an intrinsic semiconductor with anisotropic conductivity, we describe the electrical conductivity of the separate electron and hole bands with tensors σnand σpand the Seebeck response with tensors as snand sp. Considering the conduction along the two principal axes of interest labeled aand b, which manifest the transport anisotropy, we obtain the following diagonal matrices:

σn=σn,aa00σn,bb,    σp=σp,aa00σp,bbsn=sn00sn,    sp=sp00sp,E9

where the diagonal elements satisfy sn<0, sp>0. Note that single-band Seebeck tensors snand spare typically isotropic, but conductivity tensors σnand σpcan be strongly anisotropic (see Sections 3.3–3.5). The total conductivity tensor Σand total resistivity tensor Pare related by Σ=P1=σn+σp.

The total Seebeck tensor for the two-band system is defined as the weighted sum of the single-band Seebeck tensors by the conductivity tensors:

S=σp+σn1σpsp+σnsnE10

We remark again that this parallel conductionof bands within the same material is fundamentally different from stacked synthetic multilayer transverse thermoelectrics of Figure 3 in which the out-of-plane Seebeck arises from serieselectrical and thermal resistancesof two different materials. From Eq. (10), if oppositely charged carriers dominate conduction along aand b, respectively, the total Seebeck coefficients in the two orthogonal directions will have opposite signs. If we assume that p-type conduction dominates along aand n-type, conduction dominates along b, then the total Seebeck tensor is

S=Sp,aa00Sn,bb,E11

with elements

Sp,aa=Spσp,aa+Snσn,aaσp,aa+σn,aa>0,Sn,bb=Spσp,bb+Snσn,bbσp,bb+σn,bb<0,E12

where the first inequality is valid provided that p-type conduction in the a-direction is sufficiently dominant σp,aa/σn,aa>snsp,and the second valid provided n-type conduction in the b-direction is sufficiently dominant σn,bb/σp,bb>sp/sn. The result is the desired ambipolar Seebeck tensor where one of the diagonal elements has opposite sign.

When transverse thermoelectric materials are cut into a shape such that the transport directions x,yare at an angle θto the principal axes aand b, as shown in Figure 1, the Seebeck tensor in the x-ytransport basis can yield the necessary off-diagonal terms:

S=cosθsinθsinθcosθSp,aa00Sn,bbcosθsinθsinθcosθ=cos2θSp,aa+sin2θSn,bbsinθcosθSp,aaSn,bbsinθcosθSp,aaSn,bbcos2θSn,bb+sin2θSp,aa = SxxSxySyxSyyE13

This nonzero off-diagonal component of the Seebeck tensor Sxyin the transport basis is the essential prerequisite for any transverse thermoelectric effect.

### 3.2. Thermoelectric tensor calculation

In the following, we will demonstrate how the anisotropic electrical transport tensors of each separate band can be calculated. Standard longitudinal thermoelectric devices have both heat and electrical current flowing along the same axis, so their electrical resistivity, thermal conductivity, and Seebeck coefficient can be treated as scalars. In contrast, the thermoelectric properties in an anisotropic thermoelectric material must be described by tensors for the electrical conductivity σ, the Seebeck coefficient Sand the thermal conductivity κ. The anisotropic transport tensors of each electron or hole band can be calculated according to the material’s band structure, and then the equations of the previous section can be used to determine the total Seebeck and resistivity tensors of the two-band system. Because the compounds of interest tend to be highly anisotropic, in addition to the 3D effective mass model, we will also consider the case of quasi-2D and quasi-1D materials, which host effective mass band-conduction along two axes or one axis, respectively, and hopping transport along the remaining orthogonal axes.

We, therefore, perform a complete derivation of the thermoelectric tensor components from first principles corresponding to 3D, 2D, and 1D anisotropic transport scenarios. The thermal conductivity tensor κis ideally obtained from experimental measurements of the material of interest, whereas transport tensors of the Seebeck Sand electrical resistivity ρcan be calculated with simple assumptions outlined below if the band structure is known. The derivation of thermoelectric equations uses the intuitive notations borrowed from Chambers [20] except that instead of scalar coefficients, here, we show the complete tensor derivations.

When both an electric field Eand a temperature gradient G=Tare present, the electrical current Jand heat flow Qare given by

J=σE+CGQ=DE+κGE14

where σis the electrical conductivity tensor, κis the thermal conductivity tensor ,Dis related to the Peltier tensor Πand σby Π=1, Cis related to the Seebeck tensor Sand σby S=1, and Sfollows the Kelvin relation S=Π/T=1/T. These equations are typically transformed so that Jand Gare the independent variables

E=ρJSGQ=ΠJ+κcG.E15

where ρ=σ1is the electrical resistivity tensor, and the normally measured open circuit thermal conductivity κcis defined as κc=κσSΠ.

σand Dtensor components σijand Dijcan be calculated from the band structure

σij=0f0ϵσijϵE16
σijϵ=e24π3vkivkvkjτkdSkE17
Dij=f0ϵDijϵE18
Dijϵ=1eϵμσijϵE19

where ϵis the carrier energy relative to the edge of the energy band,f0ϵ=1/1+eϵ/kBTis the Fermi-Dirac distribution function, eis the electron charge, and EFis the chemical potential. We assume that the scattering time τ=γϵsobeys a power law in the energy of the carrier relative to the band minimum with exponent s. vkis the carrier velocity vector for wave vector k, which is defined as vk=vkaâ+vkbb̂+vkcĉwith each velocity component vki=ℏdki. Skis the equienergy k-space surface area at energy ϵand wave vector k. The indices i,j,lin the subscripts represent three orthogonal crystal axes a,b,andc.

In the next subsections, we will analyze 3D, 2D, and 1D transport and deduce their thermoelectric tensors. The 3D anisotropic case is for anisotropic effective mass in bulk materials, e.g., an ellipsoidal effective mass such as in noncubic lattices. The 2D anisotropic case is relevant for quasi-2D materials and can be found in parallel quantum wells or weakly coupled superlattice layers with approximately infinite cross-plane effective mass. The 1D anisotropic case can be applied to quasi-1D materials or arrays of nanowires or nanotubes, which have weak tunnel coupling in two directions.

### 3.3. Three-dimensional transport

For a general orthorhombic lattice, the carrier energy relative to the band edge in a given energy band can be expressed with a three-dimensional (3D) effective mass approximation:

ϵ=2ka22ma+2kb22mb+2kc22mcE20

where miis the effective mass in the idirection. In spherical coordinates, the wave vectors and the velocity are as follows:

ka=2maϵsinαcosϕE21
kb=2mbϵsinαsinϕE22
kc=2mcϵcosαE23
νk=2ϵmasinαcosϕâ+2ϵmbsinαsinϕb̂+2ϵmccosαĉE24

where αis the polar angle and ϕis the azimuthal angle.

If the principle axes of mass anisotropy are chosen as the coordinate, then the transport tensors are all diagonal, and the diagonal components of the energy-dependent 3D conductivity become as

σii3Dϵ=e24π302π0πvkivkvkiτkdSk=e2γϵs4π302π0πvki22ϵmimjmlsin2αcos2ϕmi+sin2αsin2ϕmj+cos2αml2ϵsin2αcos2ϕmi+sin2αsin2ϕmj+cos2αml=22e2γ3π23mjmlmiϵs+3/2E25

Integrating this expression in Eq. (16) yields the final conductivity tensor:

σii3D=22e2γ3π23mjmlmikBTs+32Γs+52F32+sμokBT,E26

where EF=0is defined at the valence band edge,Egis the bandgap, the chemical potential μois defined relative to the band edge μo=EFEgfor the conduction band and μo=EFfor the valence band, and Fnξis the Fermi integral Fnξ=0tn1+etξdt.

The Seebeck tensor is isotropic for a single band, and the diagonal Seebeck component is

Sii3D=1eTf0∂ϵϵμoσii3Dϵσii3D=kBe5/2+sFs+3/2μokBT3/2+sFs+1/2μokBTμokBTE27

### 3.4. Quasi-two-dimensional transport

If carriers propagate in one direction via weak tunnel coupling, then the lattice behaves as a quasi-2D lattice or as a superlattice with weak tunneling between layers. If, for example, carriers follow the effective mass approximation in the acplane and obey a weak-coupling model in the bdirection, the following energy dispersion can be assumed as follows:

ϵ=2ka22ma+2kc22mc+2tb1coskbdE28

where dis the superlattice period or quantum well width and tbis the nearest neighbor hopping matrix element between the weakly coupled layers. In-plane momenta are

ka=2maϵ2tb1coskbdcosϕkc=2mcϵ2tb1coskbdsinϕ.E29

Assuming that the in-plane mass ma=mcand that kris the wave vector in acplane, we obtain the conductivity tensor components:

σaa2D=σcc2D=e2γπ2dkBTs+1Γs+2F1+sμokBTσbb2D=2e2γmatb2dπ4kBTsΓs+1FsμokBTE30

The Seebeck tensor remains diagonal with components:

Sii2D=1eTf0∂ϵϵμoσii2Dϵσii2D=kBe2+sFs+1μokBT1+sFsμokBTμokBTE31

### 3.5. Quasi-one-dimensional transport

If carriers propagate in two orthogonal directions via weak tunnel coupling, the lattice is a quasi-1D lattice with weak coupling between chains. Hence, if carriers obey the effective mass approximation in the a-direction only and tunnel perpendicularly in the b- and c-directions, the following energy dispersion can be assumed:

ϵ=2ka22ma+2tb1coskbdb+2tc1coskcdcE32

where tband tcare the nearest neighbor hopping matrix elements between the weakly coupled nanowires, and dband dcare the distances between nanowires in the b- and c-directions. We can arrive at an analytical solution in the second line of σii1Dϵderivations only under the assumption that the carrier velocity in the tunnel directions is much smaller than that in the wire direction, νkaνkb,νkc.So, the conductivity components are as follows:

σaa1D=2e2γπℏdbdcmakBTs+1/2Γs+32F1/2+sμokBTσbb1D=2e2γtb2dbmaπ3dckBTs1/2Γs+12Fs1/2μokBTE33

The diagonal Seebeck tensor components are

Sii1D=1eTf0∂ϵϵμoσii1Dϵσii1D=kBe3/2+sFs+1/2μokBT1/2+sFs1/2μokBTμokBTE34

## 4. Transverse thermoelectric figure of merit zxyT

Inserting the conductivity and Seebeck tensors for the individual bands from Sections 3.3–3.5 into Eq. (10), and then rotating according to Eq. (13), the tensor components of all transport quantities in the x-ytransport basis can be determined. The transverse thermoelectric figure of merit zxyθTis defined as:

zxyθT=Sxy2κyyρxx=sin2θcos2θSp,aaSn,bb2sin2θκaa+cos2θκbbcos2θρaa+sin2θρbb.E35

We define the angle θas that which maximizes zxyθT:

cos2θ=11+κbb/κaaρbb/ρaaE36

and the maximum value zTbecomes

zT=zxyθT=Sp,aaSn,bb2Tρaaκaa+ρbbκbb2E37

Eq. (36) shows that θis independent of the Seebeck anisotropy, and it approaches π4when the thermal conductivity anisotropy matches the resistivity anisotropy κbbκaa=ρbbρaa.

In semiconductors, the thermal conductivity is usually dominated by the lattice thermal conductivity [2]. Therefore, under the assumption of isotropic κ, we define a transverse power factor PFas

PF=Sp,aaSn,bb2ρaa+ρbb2,E38

where Sand ρtensors can be calculated by the use of semiclassical Boltzmann transport theory for the corresponding scattering mechanisms [2]. Thus, for a given band structure, PFcan be theoretically estimated to evaluate the performance of transverse thermoelectrics.

### 4.1. Transport equations

The current flow J=Jxx̂defines the x-axis. Eqs. (35)(37) in general apply to all transverse thermoelectrics [4, 5, 17], but they are rederived above for completeness. The dependence of temperature on position within the transverse thermoelectric must now be carefully derived. The derivation below most closely follows that of Ref. [23] for the Nernst-Ettingshausen effect, but the errors in that reference are corrected below.

With Peltier tensor Π, the total Peltier heat flux density becomes QΠ=ΠJ=TSJwith longitudinal and transverse components:

QΠ,x=QΠ·x̂=Sp,aacos2θ+Sn,bbsin2θTJxE39
QΠ,y=QΠ·ŷ=Sp,aaSn,bbcosθsinθTJxE40

The total heat flux density Q=QΠκcTincludes both Peltier and thermal conduction effects, κcas notated in Ref. [23] defines the open-circuit thermal conductivity tensor at J=0. The thermal gradient is orthogonal to the current density T=dTdyŷ; the longitudinal electric field component Exis constant everywhere [23]; and the heat flux component Qydepends only on y. Therefore, the longitudinal current and transverse heat flow are

Jx=1ρxxExSxyρxxdTdyE41
Qy=TSyxρxxEx1+zxyTκyycdTdyE42

with transverse figure of merit zxyT=SxySyxTρxxκyyc. Steady state requires ·J=0and ·Q+μ¯J=0, where the scalar μ¯is the electrochemical potential and μ¯=Eis the electric field. Longitudinal Joule heating ExJxsources a divergence in the transverse heat flux density Qy:

dQydy=ExJx.E43

Eqs. (41)(43) define the differential equation for temperature-dependent thermoelectric coefficients:

0=1SxySyxEx2Sxy+SyxSxySyx+dlnSyx/ρxxdlnT1SxyExdTdy+1+dlnSyxSxy/ρxxdlnT+1zxydlnκyycdTdTdy2+1+zxyTzxyd2Tdy2E44

Note for constant thermoelectric coefficients, the derivatives with respect to temperature are zero, and for transverse thermoelectrics, Sxy=Syx. Eq. (44) thus becomes:

ExSxydTdy2+1+zxyTzxyd2Tdy2=0.E45

This equation can be integrated to determine the temperature profile inside a rectangular solid of transverse thermoelectric material under constant current density. Note again, that unlike for the Nernst-Ettingshausen effect, the above Eq. (45) must be integrated numerically and does have an analytical solution.

## 5. Cooling power for transverse thermoelectrics

The cooling power for transverse Peltier refrigeration has recently been studied in detail [3]. The transport equations have no analytical solution, so the graphical results are presented here to allow simple estimations of cooling power for generic transverse thermoelectric scenarios. Here, we identify the characteristic heat flux scale and electric field scale for a transverse thermoelectric to define a normalized expression for thermoelectric transport [3]. The resulting study demonstrates the superiority of transverse thermoelectric coolers over longitudinal coolers with identical figure of merit.

One starts with the expression in Eq. (45) to identify the temperature distribution in a transverse cooler. To generalize this expression, the following heat flux and electric field scales Q0=κyycTh/Land E0=SxyTh/L, respectively, are introduced, generating normalized versions of the Eqs. (42) and (45):

Qy=zxyThET1+zxyThTdTdy,E46
0=EdTdy2+1+zxyThTzxyThd2Tdy2,E47

where T=T/Th, E=E/E0, y=yLy,and Q=Qy/Q0are normalized temperature, electric field, ycoordinate, and heat flux density, respectively. Eqs. (46) and (47) indicate that the normalized heat flux density Qyand the normalized temperature profile Tyonly depend on the normalized electrical field Eand transverse figure of merit zxyTh. To determine the maximum normalized temperature difference, one sets the cooling power at the cold side Qcto zero, to achieve ΔT=1Tc=1Ty=1whereby the optimal Eis determined by

QcEE=Eopt=0E48

Thus, ΔTmaxzxyThis only a function of zxyTh. Similarly, the maximum of the cooling power at the cold side Qc=Qyy=1for a given Tccan be obtained when Esatisfies:

TEE=Eopt=0E49

and Qc,maxdepends only on Tcand zTh.

But because Eqs. (46)(49) cannot be exactly solved with analytical methods, it is illustrative to plot the numerically calculated temperature profile and heat flux, and thereby investigate the cooling power of the transverse coolers.

Figure 6 shows the normalized temperature profile under the condition of maximum temperature difference (Qc=0) for various transverse figures of merit zxyTh. The temperature gradient at the hot side (y=0) is zero, indicating that there is no net heat diffusion from the heat sink into, or out of, the device. Thus, 100% of the Peltier cooling power compensates the Joule heating in the device. The temperature gradient from the hot to the cold side becomes steeper as zxyThincreases, indicating that the higher figure of merit can compensate a larger thermal diffusive flux under larger ohmic heating. Figure 7, left axis, shows the dependence of the maximum temperature difference ΔT=1Tcon zxyTh, left axis. Transverse coolers (solid line) show a larger ΔTthan the analytically solved ΔT=1+11+2zxyThzxyThfor longitudinal coolers (dashed line) [24]. For zxyTh=1, a 30 %temperature reduction is observed for the transverse cooler, which is slightly larger than the 27 %reduction of the conventional longitudinal cooler with the same zxyTh. The trend becomes more obvious when an unphysically large zxyThof 4 results in a 60 %temperature reduction with the transverse cooler, whereas the longitudinal cooler achieves only 50%. Note that ΔTmaxcan be further increased with geometric tapering of the transverse cooler [1, 19], allowing for additional advantage over longitudinal cooling for achieving large temperature differences.

Figure 7, right axis, plots the normalized maximum cooling power Qc,maxof the transverse cooler when Tc=Thas a function of zxyTh. Unlike the linear dependence in Qc,max=1/2zxyThfor longitudinal coolers (dashed line) when Tc=1, Qc,maxshows a superlinear dependence on zxyThfor transverse coolers (solid line), which exceeds the longitudinal limit for all zxyTh, approaching the longitudinal behavior only in the limit of small zxyTh. The cooling power enhancement in Qc,maxfor transverse coolers over longitudinal coolers with the same zxyThis 28 %when zxyTh=1and rapidly increases to 220 %when zxyTh=4.

Figure 8 shows the cooling power Qc,maxof transverse coolers as a function of Tcfor various zxyThvalues. For a given zxyTh, the Fourier diffusion heat flow increases when Tcdecreases; thus, a larger portion of the Peltier cooling power is used to compensate the diffusive heat flow, and the remaining cooling power at the cold side Qc,maxwill decrease. The intersection of the curves with the horizontal axis and vertical axis corresponds to the maximum normalized temperature difference case and maximum cooling power case in Figure 7, respectively. The performance of a transverse cooler can be readily predicted from Figure 8 for any given heat load or cold side temperature, once the scales Q0 and E0 are known.

## 6. Conclusion

This review of (p × n)-transverse thermoelectrics explains the origin of materials with p-type Seebeck along one axis and n-type Seebeck orthogonal. The rigorous derivation of all thermoelectric transport tensors for anisotropic thermoelectric phenomena is given, as well as the transport equations from which one can derive all essential material performance parameters. The necessarily anisotropic band structure is expected to arise via anisotropic band or hopping conduction, whose transport tensors are derived for 3D, 2D and 1D effective mass approximations. The cooling power is expressed in a normalized notation relative to heat flux and electric field scales Q0 and E0 that are a function of the thermoelectric transport parameters. Numerical calculation of the maximum temperature difference and cooling power shows enhanced performance compared with longitudinal coolers with the same figure of merit. This work motivates the search for novel transverse thermoelectric materials with high figure of merit.

## Acknowledgments

This work is supported by the Air Force Office of Scientific Research (AFOSR) Grant FA9550-15-1-0377 and the Institute for Sustainability and Energy at Northwestern (ISEN) Booster Award, and the work of M. Ma is supported by the China Scholarship Council program (No. 201406280070).

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Written By

Matthew Grayson, Qing Shao, Boya Cui, Yang Tang, Xueting Yan and Chuanle Zhou

Submitted: October 30th, 2017 Reviewed: May 15th, 2018 Published: November 5th, 2018