2. (p n)-type transverse thermoelectric devices

(p n)-Type transverse thermoelectrics have potential device advantages over other thermoelectric 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.

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 geom-

figure, the rotation angle between x-y transport coordinate system and a-b crystal axis coordinate system is θ.

Figure 1. The (p n)-type thermoelectrics have p-type dominant conduction and Seebeck coefficient along the a-axis, where holes are the main charge carriers to transport (blue dashed lines), and n-type-dominant conduction and Seebeck coefficient along the orthogonal b-axis, where electrons are the main charge carriers to transport (orange dashed lines). The (p n)-type character is notated by the blue and orange crossed arrow symbol at the upper right. The movement of electrons (orange arrows) and holes (blue arrows) in orthogonal directions results in net charge current J<sup>x</sup> to the right and net particle or heat flow Q<sup>y</sup> up. The carrier transport shown in this figure can be driven either by drift due to an electric field along the +x direction or by diffusion due to a temperature gradient along the y direction. As shown at the bottom left corner of this

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

(p n)-Type transverse thermoelectrics have potential device advantages over other thermoelectric solutions when considering microscale devices or cryogenic operation. Conventional

etries that standard thermoelectrics cannot achieve.

2. (p n)-type transverse thermoelectric devices

of such transverse materials.

84 Bringing Thermoelectricity into Reality

Figure 2. The conventional thermoelectric heat pump or Peltier refrigerator with one p-type leg and the other n-type leg will drive heat flow Q parallel or antiparallel, respectively, to the conventional electrical current J. Solid gray rectangles represent metal contacts.

Figure 3. Stacked multilayer synthetic transverse thermoelectric. Alternating layers of p-type semiconductor and n-type (semi)metal create a stacked composite whose behavior in a tilted electric field results in skewed electron and hole currents within each layer, giving a net transverse Seebeck effect. Solid gray rectangles represent metal contacts.

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

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 zxyT

Figure 5. Microscopic electron-hole picture of the p � n-type transverse thermoelectric depicts net charge current Jtotal to the right, and net particle or heat current Qtotal up, based on the vector sum of charge current of holes Qp and charge current of electrons Qn, heat current of holes Jp and heat current of electrons Jn. Solid gray rectangles represent metal

zxyT <sup>¼</sup> Sxy

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

In the transverse figure of merit expression above, the off-diagonal Seebeck element Sxy in 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 rxx is needed; and to minimize passive return of Fourier heat in the y-direction of the temperature differential, a small thermal conductivity κyy is needed. Note that for longitudinal thermoelectrics, the expression for zT typically includes the conductivity in the numerator since the scalar equation σ ¼ 1=r is 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

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 σ<sup>n</sup> and σ<sup>p</sup> and the Seebeck response with tensors as s<sup>n</sup> and sp. Considering the conduction along the two principal axes of interest

2 rxxκyy

T (8)

Introduction to (*p* × *n*)-Type Transverse Thermoelectrics http://dx.doi.org/10.5772/intechopen.78718 87

is given as:

contacts.

σxx 6¼ 1=rxx.

3.1. Thermoelectric tensors definition

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 Jn in the right branch and hole current Jp in 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 Jx to the right and net heat current Qy up.

Figure 4. Sketch of p n-type transverse thermoelectrics in a device structure mimicking that of the standard double-leg thermoelectric device in Figure 2. Here, the same material can be used for both legs, as long as the crystal axis is oriented parallel to the p-type direction for the p-leg current Jp and parallel to the n-type direction for the n-leg current Jn. Solid gray rectangles represent metal contacts.

Figure 5. Microscopic electron-hole picture of the p � n-type transverse thermoelectric depicts net charge current Jtotal to the right, and net particle or heat current Qtotal up, based on the vector sum of charge current of holes Qp and charge current of electrons Qn, heat current of holes Jp and heat current of electrons Jn. Solid gray rectangles represent metal contacts.

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 zxyT is given as:

$$z\_{xy}T = \frac{{\bf S\_{xy}}^2}{\rho\_{xx}\kappa\_{yy}}T\tag{8}$$

In the transverse figure of merit expression above, the off-diagonal Seebeck element Sxy in 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 rxx is needed; and to minimize passive return of Fourier heat in the y-direction of the temperature differential, a small thermal conductivity κyy is needed. Note that for longitudinal thermoelectrics, the expression for zT typically includes the conductivity in the numerator since the scalar equation σ ¼ 1=r is 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 σxx 6¼ 1=rxx.
