## 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*×

*)-type transverse thermoelectrics (TTE) [1, 2, 3] occurs in an anisotropic semiconductor or semimetal with*n

*-type Seebeck response along one axis and*p

*-type Seebeck orthogonal, whereby an appropriately applied electric current at an angle with respect to these axes can induce a*n

*heat flow (the transverse Peltier effect), or conversely, an applied temperature gradient can generate a*purely orthogonal

*electric field (the transverse Seebeck effect).*purely orthogonal

Transverse thermoelectric effects in general [4, 5, 6] require a broken symmetry to generate the necessary * off-diagonal*component in the Seebeck tensor. An off-diagonal component of the Seebeck tensor

*-direction gradient in the electrochemical potential*x

*-direction temperature gradient*y

within the full tensor expression

where

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

*-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*p

the resulting Seebeck tensor

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,

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

*with N-E coefficient*B

*and*α

Similarly, * structural asymmetry*can 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

*-axis alignment might still manifest the necessary anisotropy. If the diagonal Seebeck tensor elements*c

Note that a * purely p*- or

*-type*n

*material can never generate*unipolar

*orthogonal 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].*purely

A more interesting structural asymmetry is one that includes an ambipolar Seebeck tensor, whereby the diagonalized Seebeck tensor elements * p*-type

*-type*n

In such a material, there * does*exists a rotation

*, such that an applied thermal gradient can induce a purely orthogonal electric field, or equivalently,*purely orthogonal

where * n*- (typically metallic or semimetallic) and

*-type (semiconducting) layers create the necessary structural asymmetry. Orthogonal to the layers, electrical and thermal resistance is in series from adjacent*p

*- to*p

*-layers, whereas parallel to the layers, the electrical and thermal conduction is in parallel.*n

The most recent addition to the lexicon of transverse thermoelectric phenomena is single crystals that themselves possess orthogonal * p*- and

*-type Seebeck components, and which the authors have dubbed “(*n

*×*p

*)-type” [1, 2]. The cause of such ambipolar behavior is fundamentally different from that of the composite multilayered materials, since the (*n

*×*p

*)-type materials are bulk crystals, and thus both the electrical and thermal conductions are in parallel in*n

*directions. 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 (*all

*×*p

*)-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.*n

Because the underlying phenomenology of these (* p* ×

*)-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 (*n

*×*p

*)-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.*n

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

(* p* ×

*)-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*n

*- or*p

*-type doping, which limits their use at cryogenic temperatures since the dopants freeze out. A typical minimum operation temperature is*n

*= 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.*T

On the other hand, (* p* ×

*)-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 (*n

*×*p

*)-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.*n

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 * T*[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

*-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*n

*and*Q

_{p}

*in each leg, the top junction is cooled and the heat is transferred to the bottom heat sink.*Q

_{n}

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 * J*in the right branch and hole current

_{n}

*in the left counterpart. Moreover, the heat current of both legs is flowing downward, just like the heat flow of the conventional device.*J

_{p}

As demonstrated in Figure 5, a simpler single-leg geometry is possible with transverse thermoelectrics. With the electrons and holes, transportation directions of the

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

*is given as:*T

In the transverse figure of merit expression above, the off-diagonal Seebeck element * x*-direction of current flow, a small resistivity component

*-direction of the temperature differential, a small thermal conductivity*y

*typically includes the conductivity in the numerator since the scalar equation*zT

## 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*×

*)-type thermoelectrics. For an intrinsic semiconductor with anisotropic conductivity, we describe the electrical conductivity of the separate electron and hole bands with tensors*n

where the diagonal elements satisfy

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:

We remark again that this * parallel conduction*of 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

*electrical and thermal*series

*of two different materials. From Eq. (10), if oppositely charged carriers dominate conduction along*resistances

with elements

where the first inequality is valid provided that * p*-type conduction in the

*-direction is sufficiently dominant*a

*-type conduction in the*n

*-direction is sufficiently dominant*b

When transverse thermoelectric materials are cut into a shape such that the transport directions * x-y*transport basis can yield the necessary off-diagonal terms:

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

_{xy}

### 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

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

When both an electric field

where

where

where

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:

where

where

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

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

where

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

### 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

where

Assuming that the in-plane mass

The Seebeck tensor remains diagonal with components:

### 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

where

The diagonal Seebeck tensor components are

## 4. Transverse thermoelectric figure of merit z xy T

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-y*transport basis can be determined. The transverse thermoelectric figure of merit

We define the angle

and the maximum value

Eq. (36) shows that

In semiconductors, the thermal conductivity is usually dominated by the lattice thermal conductivity [2]. Therefore, under the assumption of isotropic

where

### 4.1. Transport equations

The current flow

With Peltier tensor

The total heat flux density

with transverse figure of merit

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

Note for constant thermoelectric coefficients, the derivatives with respect to temperature are zero, and for transverse thermoelectrics,

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

where

Thus,

and

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 (

Figure 7, right axis, plots the normalized maximum cooling power

Figure 8 shows the cooling power _{0} and _{0} are known.

## 6. Conclusion

This review of (* p* ×

*)-transverse thermoelectrics explains the origin of materials with*n

*-type Seebeck along one axis and*p

*-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*n

Q

_{0}and

E

_{0}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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