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

#### 3.1. Thermoelectric tensors definition

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 trans-

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

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

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

verse figure of merit zxyT [1, 19].

86 Bringing Thermoelectricity into Reality

transferred to the bottom heat sink.

rectangles represent metal contacts.

downward, just like the heat flow of the conventional device.

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 labeled a and b, which manifest the transport anisotropy, we obtain the following diagonal matrices:

$$\begin{aligned} \sigma\_n &= \begin{bmatrix} \sigma\_{n,a} & & 0 \\ & 0 & \sigma\_{n,bb} \end{bmatrix}, & \sigma\_p &= \begin{bmatrix} \sigma\_{p,a} & & 0 \\ & 0 & \sigma\_{p,bb} \end{bmatrix} \\ \mathbf{s}\_n &= \begin{bmatrix} s\_n & 0 \\ 0 & s\_n \end{bmatrix}, & \mathbf{s}\_p &= \begin{bmatrix} s\_p & 0 \\ & 0 & s\_p \end{bmatrix} \end{aligned} \tag{9}$$

S<sup>0</sup> ¼

¼

¼

3.2. Thermoelectric tensor calculation

current J and heat flow Q are given by

to the Peltier tensor <sup>Π</sup> and <sup>σ</sup> by <sup>Π</sup> <sup>¼</sup> <sup>D</sup>σ�<sup>1</sup>

<sup>S</sup> <sup>¼</sup> <sup>C</sup>σ�<sup>1</sup>

2 4 cosθ �sinθ sinθ cosθ

Sxx Sxy Syx Syy " #

essential prerequisite for any transverse thermoelectric effect.

respectively, and hopping transport along the remaining orthogonal axes.

that instead of scalar coefficients, here, we show the complete tensor derivations.

, and <sup>S</sup> follows the Kelvin relation <sup>S</sup> <sup>¼</sup> <sup>Π</sup>=<sup>T</sup> <sup>¼</sup> <sup>D</sup>σ�<sup>1</sup>

cally transformed so that J and G are the independent variables

" # Sp, aa 0

<sup>0</sup> Sn, bb " # cos<sup>θ</sup> sin<sup>θ</sup>

cos<sup>2</sup>θSp, aa <sup>þ</sup> sin<sup>2</sup>θSn, bb sinθcos<sup>θ</sup> Sp, aa � Sn, bb � �

sinθcos<sup>θ</sup> Sp, aa � Sn, bb � � cos<sup>2</sup>θSn, bb <sup>þ</sup> sin<sup>2</sup>θSp, aa

This nonzero off-diagonal component of the Seebeck tensor Sxy in the transport basis is the

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 S and 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,

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 κ is ideally obtained from experimental measurements of the material of interest, whereas transport tensors of the Seebeck S and electrical resistivity r can 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

When both an electric field E and a temperature gradient G ¼ �∇T are present, the electrical

J ¼ σE þ CG

where σ is the electrical conductivity tensor, κ is the thermal conductivity tensor , D is related

<sup>Q</sup> <sup>¼</sup> DE <sup>þ</sup> <sup>κ</sup><sup>G</sup> (14)

, C is related to the Seebeck tensor S and σ by

=T. These equations are typi-

�sinθ cosθ

3 5

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

(13)

89

" #

where the diagonal elements satisfy sn < 0, sp > 0. Note that single-band Seebeck tensors s<sup>n</sup> and s<sup>p</sup> are typically isotropic, but conductivity tensors σ<sup>n</sup> and σ<sup>p</sup> can be strongly anisotropic (see Sections 3.3–3.5). The total conductivity tensor Σ and total resistivity tensor P are related by <sup>Σ</sup> <sup>¼</sup> <sup>P</sup>�<sup>1</sup> <sup>¼</sup> <sup>σ</sup><sup>n</sup> <sup>þ</sup> <sup>σ</sup>p.

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

$$\mathbf{S} = \left(\mathbf{\sigma}\_p + \mathbf{\sigma}\_n\right)^{-1} \left(\mathbf{\sigma}\_p s\_p + \mathbf{\sigma}\_n s\_n\right) \tag{10}$$

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 series electrical and thermal resistances of two different materials. From Eq. (10), if oppositely charged carriers dominate conduction along a and b, respectively, the total Seebeck coefficients in the two orthogonal directions will have opposite signs. If we assume that p-type conduction dominates along a� and n-type, conduction dominates along b, then the total Seebeck tensor is

$$\mathbf{S} = \begin{bmatrix} \mathbf{S}\_{p,aa} & \mathbf{0} \\ \mathbf{0} & \mathbf{S}\_{n,bb} \end{bmatrix}' \tag{11}$$

with elements

$$\begin{split} \mathcal{S}\_{p,aa} &= \frac{\mathcal{S}\_p \sigma\_{p,aa} + \mathcal{S}\_n \sigma\_{n,aa}}{\sigma\_{p,aa} + \sigma\_{n,aa}} > 0, \\ \mathcal{S}\_{n,bb} &= \frac{\mathcal{S}\_p \sigma\_{p,bb} + \mathcal{S}\_n \sigma\_{n,bb}}{\sigma\_{p,bb} + \sigma\_{n,bb}} < 0, \end{split} \tag{12}$$

where the first inequality is valid provided that p-type conduction in the a-direction is sufficiently dominant σp, aa=σn, aa > sn sp � � � � � �, and the second valid provided <sup>n</sup>-type conduction in the <sup>b</sup>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, y are at an angle θ to the principal axes a and b, as shown in Figure 1, the Seebeck tensor in the x-y transport basis can yield the necessary off-diagonal terms:

$$\begin{aligned} \mathbf{S}' &= \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} S\_{p,aa} & 0 \\ 0 & S\_{n,bb} \end{bmatrix} \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} \\ &= \begin{bmatrix} \cos^2\theta S\_{p,aa} + \sin^2\theta S\_{n,bb} & \sin\theta\cos\theta (S\_{p,aa} - S\_{n,bb}) \\\\ \sin\theta\cos\theta (S\_{p,aa} - S\_{n,bb}) & \cos^2\theta S\_{n,bb} + \sin^2\theta S\_{p,aa} \end{bmatrix} \\ &= \begin{bmatrix} \mathbf{S}\_{\mathbf{x}\mathbf{y}} & \mathbf{S}\_{\mathbf{x}\mathbf{y}} \\ \mathbf{S}\_{\mathbf{y}\mathbf{x}} & \mathbf{S}\_{\mathbf{y}\mathbf{y}} \end{bmatrix} \end{aligned} \tag{13}$$

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

#### 3.2. Thermoelectric tensor calculation

labeled a and b, which manifest the transport anisotropy, we obtain the following diagonal

, sp ¼

where the diagonal elements satisfy sn < 0, sp > 0. Note that single-band Seebeck tensors s<sup>n</sup> and s<sup>p</sup> are typically isotropic, but conductivity tensors σ<sup>n</sup> and σ<sup>p</sup> can be strongly anisotropic (see Sections 3.3–3.5). The total conductivity tensor Σ and total resistivity tensor P are related

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

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 series electrical and thermal resistances of two different materials. From Eq. (10), if oppositely charged carriers dominate conduction along a and b, respectively, the total Seebeck coefficients in the two orthogonal directions will have opposite signs. If we assume that p-type conduction dominates along a� and n-type, con-

<sup>S</sup> <sup>¼</sup> Sp, aa <sup>0</sup>

Sp, aa <sup>¼</sup> Spσp, aa <sup>þ</sup> Snσn, aa σp, aa þ σn, aa

Sn, bb <sup>¼</sup> Spσp, bb <sup>þ</sup> Snσn, bb σp, bb þ σn, bb

where the first inequality is valid provided that p-type conduction in the a-direction is suffi-

direction is sufficiently dominant σn, bb=σp, bb > ∣sp=sn∣. The result is the desired ambipolar

When transverse thermoelectric materials are cut into a shape such that the transport directions x, y are at an angle θ to the principal axes a and b, as shown in Figure 1, the Seebeck

0 Sn, bb � �

> 0,

< 0,

�, and the second valid provided <sup>n</sup>-type conduction in the <sup>b</sup>-

σpsp þ σnsn

, σ<sup>p</sup> ¼

σp, aa 0

sp 0

" #

0 sp

" #

0 σp, bb

,

� � (10)

, (11)

(9)

(12)

σ<sup>n</sup> ¼

s<sup>n</sup> ¼

band Seebeck tensors by the conductivity tensors:

duction dominates along b, then the total Seebeck tensor is

sp � � � � �

Seebeck tensor where one of the diagonal elements has opposite sign.

tensor in the x-y transport basis can yield the necessary off-diagonal terms:

σn, aa 0

sn 0

" #

0 sn

S ¼ σ<sup>p</sup> þ σ<sup>n</sup> � ��<sup>1</sup>

" #

0 σn, bb

matrices:

88 Bringing Thermoelectricity into Reality

by <sup>Σ</sup> <sup>¼</sup> <sup>P</sup>�<sup>1</sup> <sup>¼</sup> <sup>σ</sup><sup>n</sup> <sup>þ</sup> <sup>σ</sup>p.

with elements

ciently dominant σp, aa=σn, aa > sn

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 S and 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 κ is ideally obtained from experimental measurements of the material of interest, whereas transport tensors of the Seebeck S and electrical resistivity r can 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 E and a temperature gradient G ¼ �∇T are present, the electrical current J and heat flow Q are given by

$$\begin{aligned} \mathbf{J} &= \mathbf{\sigma} \mathbf{E} + \mathbf{\mathbf{C}} \mathbf{G} \\ \mathbf{Q} &= \mathbf{D} \mathbf{E} + \mathbf{\kappa} \mathbf{G} \end{aligned} \tag{14}$$

where σ is the electrical conductivity tensor, κ is the thermal conductivity tensor , D is related to the Peltier tensor <sup>Π</sup> and <sup>σ</sup> by <sup>Π</sup> <sup>¼</sup> <sup>D</sup>σ�<sup>1</sup> , C is related to the Seebeck tensor S and σ by <sup>S</sup> <sup>¼</sup> <sup>C</sup>σ�<sup>1</sup> , and <sup>S</sup> follows the Kelvin relation <sup>S</sup> <sup>¼</sup> <sup>Π</sup>=<sup>T</sup> <sup>¼</sup> <sup>D</sup>σ�<sup>1</sup> =T. These equations are typically transformed so that J and G are the independent variables

$$\mathbf{E} = \mathbf{pJ} - \mathbf{S}\mathbf{G}$$

$$\mathbf{Q} = \Pi \mathbf{J} + \mathbf{\kappa}^{\mathbf{c}} \mathbf{G}.\tag{15}$$

ka ¼

kb ¼

sinαcosϕa^ þ

vkiτkdSk

2E ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mimjml p ℏ

ffiffiffiffi

ffiffiffiffiffiffiffiffiffiffi mjml mi

ð Þ kBT <sup>s</sup>þ<sup>3</sup>

where EF ¼ 0 is defined at the valence band edge, E<sup>g</sup> is the bandgap, the chemical potential μ<sup>o</sup> is defined relative to the band edge μ<sup>o</sup> ¼ EF � E<sup>g</sup> for the conduction band and μ<sup>o</sup> ¼ �EF for

e

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

ii ð ÞE

ii <sup>¼</sup> kB

<sup>2</sup>Γ s þ

5 2 � �F<sup>3</sup>

<sup>2</sup>þs

<sup>0</sup> <sup>t</sup> n <sup>1</sup>þet�<sup>ξ</sup> <sup>d</sup>t.

ð Þ 5=2 þ s Fsþ3=<sup>2</sup>

ð Þ 3=2 þ s Fsþ1=<sup>2</sup>

μo kBT

> μo kBT � �

> > μo kBT

� � � <sup>μ</sup><sup>o</sup>

kBT

(27)

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

r

s

ν<sup>k</sup> ¼

become as

σ<sup>3</sup><sup>D</sup> ii ð Þ¼ E

e2 4π<sup>3</sup>ℏ

<sup>¼</sup> <sup>e</sup><sup>2</sup>γE<sup>s</sup> 4π<sup>3</sup>ℏ

<sup>¼</sup> <sup>2</sup> ffiffiffi 2 <sup>p</sup> <sup>e</sup><sup>2</sup><sup>γ</sup> 3π<sup>2</sup>ℏ<sup>3</sup>

S<sup>3</sup><sup>D</sup> ii <sup>¼</sup> <sup>1</sup> eT Ð dE ∂f 0 <sup>∂</sup><sup>E</sup> E � μ<sup>o</sup> � �σ<sup>3</sup><sup>D</sup>

ð<sup>2</sup><sup>π</sup> 0 dϕ ðπ 0 dα vki j j vk

ð<sup>2</sup><sup>π</sup> 0 dϕ ðπ 0 dα v2 ki

> ffiffiffiffiffiffiffiffiffiffi mjml mi

E sþ3=2

the valence band, and Fnð Þ <sup>ξ</sup> is the Fermi integral Fnð Þ¼ <sup>ξ</sup> <sup>Ð</sup> <sup>∞</sup>

σ<sup>3</sup><sup>D</sup>

r

σ<sup>3</sup><sup>D</sup> ii <sup>¼</sup> <sup>2</sup> ffiffiffi 2 <sup>p</sup> <sup>e</sup><sup>2</sup><sup>γ</sup> 3π<sup>2</sup>ℏ<sup>3</sup>

ffiffiffiffiffiffi 2e ma r

where α is the polar angle and ϕ is the azimuthal angle.

kc ¼

ffiffiffiffiffiffiffiffiffiffi <sup>2</sup>ma<sup>E</sup> <sup>p</sup>

ffiffiffiffiffiffiffiffiffiffi <sup>2</sup>mb<sup>E</sup> <sup>p</sup>

> ffiffiffiffiffiffiffiffiffiffi <sup>2</sup>mc<sup>E</sup> <sup>p</sup>

> > ffiffiffiffiffiffi 2e mb

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

sin<sup>2</sup>

sin<sup>2</sup>αcos<sup>2</sup>ϕ mi

s

αcos<sup>2</sup>ϕ mi

<sup>2</sup><sup>E</sup> <sup>p</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

þ sin<sup>2</sup>

sinαsinϕ^

b þ

s

<sup>ℏ</sup> sinαcos<sup>ϕ</sup> (21)

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

<sup>ℏ</sup> sinαsin<sup>ϕ</sup> (22)

<sup>ℏ</sup> cos<sup>α</sup> (23)

cosα^c (24)

ffiffiffiffiffiffi 2e mc r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

αsin<sup>2</sup>ϕ mj

mj

þ cos<sup>2</sup>α ml

þ cos<sup>2</sup>α ml

� �, (26)

(25)

91

þ sin<sup>2</sup> αsin<sup>2</sup> ϕ

where <sup>r</sup>¼σ�<sup>1</sup> is the electrical resistivity tensor, and the normally measured open circuit thermal conductivity κ<sup>c</sup> is defined as κ<sup>c</sup> ¼κ�σSΠ.

σ and D tensor components σij and Dij can be calculated from the band structure

$$
\sigma\_{i\dot{\jmath}} = \int\_0^\infty d\epsilon \frac{\partial f\_0}{\partial \epsilon} \sigma\_{i\dot{\jmath}}(\epsilon) \tag{16}
$$

$$
\sigma\_{\vec{\eta}}(\epsilon) = \frac{e^2}{4\pi^3 \hbar} \left[ \frac{v\_{\vec{k}}}{|v\_{\vec{k}}|} v\_{\vec{k}} \tau\_k dS\_k \right. \tag{17}
$$

$$D\_{\vec{\eta}} = \int d\epsilon \frac{\partial f\_0}{\partial \epsilon} D\_{\vec{\eta}}(\epsilon) \tag{18}$$

$$D\_{\vec{\eta}}(\epsilon) = \frac{1}{\mathcal{e}} \left( \epsilon - \mu \right) \sigma\_{\vec{\eta}}(\epsilon) \tag{19}$$

where <sup>E</sup> is the carrier energy relative to the edge of the energy band, <sup>f</sup> <sup>0</sup>ð Þ¼ <sup>E</sup> <sup>1</sup><sup>=</sup> <sup>1</sup> <sup>þ</sup> <sup>e</sup>�E=kBT � � is the Fermi-Dirac distribution function, e is the electron charge, and EF is the chemical potential. We assume that the scattering time <sup>τ</sup> <sup>¼</sup> <sup>γ</sup>E<sup>s</sup> obeys a power law in the energy of the carrier relative to the band minimum with exponent s. v<sup>k</sup> is the carrier velocity vector for wave vector k, which is defined as vk¼v<sup>k</sup>aa^þv<sup>k</sup><sup>b</sup> ^ <sup>b</sup>þv<sup>k</sup>c^<sup>c</sup> with each velocity component vki <sup>¼</sup> <sup>d</sup><sup>E</sup> ℏdki . Sk is the equienergy k-space surface area at energy E and wave vector k. The indices i, j, l in the subscripts represent three orthogonal crystal axes a, b, and c.

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:

$$
\epsilon = \frac{\hbar^2 k\_a^2}{2m\_a} + \frac{\hbar^2 k\_b^2}{2m\_b} + \frac{\hbar^2 k\_c^2}{2m\_c} \tag{20}
$$

where mi is the effective mass in the i direction. In spherical coordinates, the wave vectors and the velocity are as follows:

Introduction to (*p* × *n*)-Type Transverse Thermoelectrics http://dx.doi.org/10.5772/intechopen.78718 91

$$k\_a = \frac{\sqrt{2m\_d}\epsilon}{\hbar} \sin \alpha \cos \phi \tag{21}$$

$$k\_b = \frac{\sqrt{2m\_b\epsilon}}{\hbar} \text{sinəsinə}$$

$$k\_{\epsilon} = \frac{\sqrt{2m\_{\epsilon}\epsilon}}{\hbar} \cos \alpha \tag{23}$$

$$\mathbf{v\_k} = \sqrt{\frac{2\mathbf{\hat{e}}}{\mathbf{m\_a}}} \mathbf{\hat{s}} \mathbf{n} \mathbf{a} \cos \phi \mathbf{\hat{a}} + \sqrt{\frac{2\mathbf{\hat{e}}}{\mathbf{m\_b}}} \mathbf{\hat{s}} \mathbf{n} \mathbf{a} \sin \phi \mathbf{\hat{b}} + \sqrt{\frac{2\mathbf{\hat{e}}}{\mathbf{m\_c}}} \mathbf{cos} \mathbf{a} \mathbf{\hat{c}} \tag{24}$$

where α is the polar angle and ϕ is the azimuthal angle.

E ¼ rJ � SG <sup>Q</sup> <sup>¼</sup> <sup>Π</sup><sup>J</sup> <sup>þ</sup> <sup>κ</sup><sup>c</sup>

where <sup>r</sup>¼σ�<sup>1</sup> is the electrical resistivity tensor, and the normally measured open circuit

¼κ�σSΠ.

e2 4π<sup>3</sup>ℏ

> 1 e

where <sup>E</sup> is the carrier energy relative to the edge of the energy band, <sup>f</sup> <sup>0</sup>ð Þ¼ <sup>E</sup> <sup>1</sup><sup>=</sup> <sup>1</sup> <sup>þ</sup> <sup>e</sup>�E=kBT � � is the Fermi-Dirac distribution function, e is the electron charge, and EF is the chemical potential. We assume that the scattering time <sup>τ</sup> <sup>¼</sup> <sup>γ</sup>E<sup>s</sup> obeys a power law in the energy of the carrier relative to the band minimum with exponent s. v<sup>k</sup> is the carrier velocity vector for wave vector

equienergy k-space surface area at energy E and wave vector k. The indices i, j, l in the sub-

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

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 mi is the effective mass in the i direction. In spherical coordinates, the wave vectors and

<sup>E</sup> <sup>¼</sup> <sup>ℏ</sup><sup>2</sup> k 2 a 2ma þ ℏ2 k2 b 2mb þ ℏ2 k2 c 2mc

ð vki ∣vk∣

σ and D tensor components σij and Dij can be calculated from the band structure

σij ¼ ð∞ 0 dE ∂f 0 ∂E

Dij ¼ ð dE ∂f 0 ∂E

Dijð Þ¼ E

^

scripts represent three orthogonal crystal axes a, b, and c.

σijð Þ¼ E

thermal conductivity κ<sup>c</sup> is defined as κ<sup>c</sup>

90 Bringing Thermoelectricity into Reality

k, which is defined as vk¼v<sup>k</sup>aa^þv<sup>k</sup><sup>b</sup>

coupling in two directions.

the velocity are as follows:

3.3. Three-dimensional transport

<sup>G</sup>: (15)

σijð ÞE (16)

vkjτkdSk (17)

Dijð ÞE (18)

ℏdki

. Sk is the

(20)

<sup>E</sup> � <sup>μ</sup> � �σijð Þ<sup>E</sup> (19)

<sup>b</sup>þv<sup>k</sup>c^<sup>c</sup> with each velocity component vki <sup>¼</sup> <sup>d</sup><sup>E</sup>

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

$$\begin{split} \sigma\_{\mu}^{20}(\epsilon) &= \frac{e^{2}}{4\pi^{3}\hbar} \Big|\_{0}^{2\pi} d\phi \int\_{0}^{\pi} d\alpha \frac{\eta\_{li}}{|v\_{k}|} v\_{k} \tau\_{k} dS\_{k} \\ &= \frac{e^{2} \gamma \epsilon^{s}}{4\pi^{3}\hbar} \Big|\_{0}^{2\pi} d\phi \Big|\_{0}^{2\pi} d\alpha \frac{\tau\_{ki}^{2} \frac{2\epsilon\sqrt{m\_{i}m\_{j}m\_{l}}}{\hbar} \sqrt{\frac{\sin^{2}\alpha \cos^{2}\phi}{m\_{i}} + \frac{\sin^{2}\alpha \sin^{2}\phi}{m\_{j}} + \frac{\cos^{2}\alpha}{m\_{l}}}{\sqrt{2}\epsilon\sqrt{\frac{\sin^{2}\alpha \cos^{2}\phi}{m\_{i}} + \frac{\sin^{2}\alpha \sin^{2}\phi}{m\_{j}} + \frac{\cos^{2}\alpha}{m\_{l}}}} \\ &= \frac{2\sqrt{2}\epsilon^{2}\gamma}{3\pi^{2}\hbar^{3}} \sqrt{\frac{m\_{i}m\_{l}}{m\_{i}}} \epsilon^{s+3/2} \end{split} \tag{25}$$

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

$$
\sigma\_{\vec{u}}^{\rm 3D} = \frac{2\sqrt{2}e^2\gamma}{3\pi^2\hbar^3} \sqrt{\frac{m\_l m\_l}{m\_i}} (k\_B T)^{s + \frac{3}{2}} \Gamma \left( s + \frac{5}{2} \right) F\_{\frac{3}{2} + s} \left( \frac{\mu\_o}{k\_B T} \right) \tag{26}
$$

where EF ¼ 0 is defined at the valence band edge, E<sup>g</sup> is the bandgap, the chemical potential μ<sup>o</sup> is defined relative to the band edge μ<sup>o</sup> ¼ EF � E<sup>g</sup> for the conduction band and μ<sup>o</sup> ¼ �EF for the valence band, and Fnð Þ <sup>ξ</sup> is the Fermi integral Fnð Þ¼ <sup>ξ</sup> <sup>Ð</sup> <sup>∞</sup> <sup>0</sup> <sup>t</sup> n <sup>1</sup>þet�<sup>ξ</sup> <sup>d</sup>t.

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

$$\mathcal{S}\_{\vec{n}}^{\text{3D}} = \frac{1}{eT} \frac{\int d\epsilon \frac{\delta \ell\_0}{\delta \epsilon} (\epsilon - \mu\_o) \sigma\_{\vec{n}}^{\text{3D}}(\epsilon)}{\sigma\_{\vec{n}}^{\text{3D}}} \quad = \frac{k\_{\text{B}}}{e} \left[ \frac{(5/2 + s) F\_{s+3/2} \left(\frac{\mu\_o}{k\_{\text{B}}T}\right)}{(3/2 + s) F\_{s+1/2} \left(\frac{\mu\_o}{k\_{\text{B}}T}\right)} - \frac{\mu\_o}{k\_{\text{B}}T} \right] \tag{27}$$

#### 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 a � c plane and obey a weak-coupling model in the b direction, the following energy dispersion can be assumed as follows:

$$
\epsilon = \frac{\hbar^2 k\_a^2}{2m\_a} + \frac{\hbar^2 k\_c^2}{2m\_c} + 2t\_b(1 - \cos k\_b d) \tag{28}
$$

where tb and tc are the nearest neighbor hopping matrix elements between the weakly coupled nanowires, and db and dc are the distances between nanowires in the b- and c-directions. We

assumption that the carrier velocity in the tunnel directions is much smaller than that in the

ð Þ kBT <sup>s</sup>�1=<sup>2</sup>

E � μ<sup>o</sup> � �σ<sup>i</sup>1<sup>D</sup>

> σ<sup>i</sup>1<sup>D</sup> i

ð Þ 3=2 þ s Fsþ1=<sup>2</sup>

ð Þ 1=2 þ s Fs�1=<sup>2</sup>

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

> <sup>θ</sup><sup>⊥</sup> <sup>¼</sup> <sup>1</sup> 1 þ

<sup>z</sup>⊥<sup>T</sup> <sup>¼</sup> zxyð Þ <sup>θ</sup><sup>⊥</sup> <sup>T</sup> <sup>¼</sup> Sp, aa � Sn, bb � �<sup>2</sup>

Eq. (36) shows that θ<sup>⊥</sup> is independent of the Seebeck anisotropy, and it approaches <sup>π</sup>

thermal conductivity anisotropy matches the resistivity anisotropy <sup>κ</sup>bb

<sup>¼</sup> sin<sup>2</sup>

sin<sup>2</sup>

cos<sup>2</sup>

Γ s þ 3 2 � �F<sup>1</sup>=2þ<sup>s</sup>

> Γ s þ 1 2 � �Fs�1=<sup>2</sup>

> > <sup>i</sup> ð ÞE

μo kBT � �

> μo kBT

� � � <sup>μ</sup><sup>o</sup>

<sup>θ</sup>cos<sup>2</sup><sup>θ</sup> Sp, aa � Sn, bb � �<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffi κbb=κaa

ffiffiffiffiffiffiffiffiffiffiffiffi <sup>r</sup>aaκaa <sup>p</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffi

θκaa <sup>þ</sup> cos2θκbb � � cos2θraa <sup>þ</sup> sin<sup>2</sup>θrbb � � : (35)

T

<sup>r</sup>bb=raa <sup>q</sup> (36)

<sup>r</sup>bbκbb � � <sup>p</sup> <sup>2</sup> (37)

<sup>κ</sup>aa <sup>¼</sup> <sup>r</sup>bb raa .

kBT

wire direction, j j νka ≫ j j νkb , νkc j j: So, the conductivity components are as follows:

ffiffiffiffiffiffi ma <sup>p</sup> ð Þ kBT <sup>s</sup>þ1=<sup>2</sup>

> Ð dE ∂f 0 ∂E

πℏ<sup>3</sup>dc

¼ kB e

4. Transverse thermoelectric figure of merit zxyT

xy κyyrxx

We define the angle θ<sup>⊥</sup> as that which maximizes zxyð Þ θ T:

ii ð ÞE derivations only under the

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

(33)

93

(34)

<sup>4</sup> when the

μo kBT � �

> μo kBT � �

can arrive at an analytical solution in the second line of σ<sup>1</sup><sup>D</sup>

ffiffiffi 2 <sup>p</sup> <sup>e</sup><sup>2</sup><sup>γ</sup> πℏdbdc

ffiffiffi 2 <sup>p</sup> <sup>e</sup><sup>2</sup>γ<sup>t</sup> 2 bdb ffiffiffiffiffiffi ma p

S<sup>i</sup>1<sup>D</sup> <sup>i</sup> <sup>¼</sup> <sup>1</sup> eT

σ<sup>1</sup><sup>D</sup> aa ¼

σ<sup>1</sup><sup>D</sup> bb ¼

The diagonal Seebeck tensor components are

merit zxyð Þ θ T is defined as:

zxyð Þ <sup>θ</sup> <sup>T</sup> <sup>¼</sup> <sup>S</sup><sup>2</sup>

and the maximum value z⊥T becomes

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

$$\begin{split}k\_{a} &= \frac{\sqrt{2m\_{a}(\epsilon - 2t\_{b}(1 - \cos(k\_{b}d)))}}{\hbar} \cos\phi\\k\_{c} &= \frac{\sqrt{2m\_{c}(\epsilon - 2t\_{b}(1 - \cos(k\_{b}d)))}}{\hbar} \sin\phi.\end{split} \tag{29}$$

Assuming that the in-plane mass ma ¼ mc and that kr is the wave vector in a � c plane, we obtain the conductivity tensor components:

$$\begin{aligned} \sigma\_{\text{au}}^{2D} &= \sigma\_{\text{cc}}^{2D} = \frac{e^2 \mathcal{V}}{\pi \hbar^2 d} (k\_B T)^{s+1} \Gamma(s+2) F\_{1+s} \left( \frac{\mu\_o}{k\_B T} \right) \\\\ \sigma\_{bb}^{2D} &= \frac{2e^2 \mathcal{V} m\_t t\_b^2 d}{\pi \hbar^4} (k\_B T)^s \Gamma(s+1) F\_s \left( \frac{\mu\_o}{k\_B T} \right) \end{aligned} \tag{30}$$

The Seebeck tensor remains diagonal with components:

$$\begin{split} S\_{ii}^{2D} &= \frac{1}{eT} \frac{\int d\epsilon \,\frac{\partial f\_0}{\partial \epsilon} \left( \epsilon - \mu\_o \right) \sigma\_{ii}^{2D}(\epsilon)}{\sigma\_{ii}^{2D}} \\ &= \frac{k\_B}{e} \left[ \frac{(2+s)F\_{s+1}\left(\frac{\mu\_o}{k\_B T}\right)}{(1+s)F\_s\left(\frac{\mu\_o}{k\_B T}\right)} - \frac{\mu\_o}{k\_B T} \right] \end{split} \tag{31}$$

#### 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:

$$\epsilon = \frac{\hbar^2 k\_\text{\tiny{k}}^2}{2m\_\text{\tiny{m}}} + 2t\_b(1 - \cos k\_b d\_b) + 2t\_c(1 - \cos k\_c d\_c) \tag{32}$$

where tb and tc are the nearest neighbor hopping matrix elements between the weakly coupled nanowires, and db and dc are the distances between nanowires in the b- and c-directions. We can arrive at an analytical solution in the second line of σ<sup>1</sup><sup>D</sup> ii ð ÞE derivations only under the assumption that the carrier velocity in the tunnel directions is much smaller than that in the wire direction, j j νka ≫ j j νkb , νkc j j: So, the conductivity components are as follows:

$$\begin{split} \sigma\_{\rm ad}^{1D} &= \frac{\sqrt{2}c^2\gamma}{\pi\hbar d\_b d\_c\sqrt{m\_d}} (k\_B T)^{s+1/2} \Gamma \left( s + \frac{3}{2} \right) F\_{1/2+s} \left( \frac{\mu\_o}{k\_B T} \right) \\ \sigma\_{b\psi}^{1D} &= \frac{\sqrt{2}c^2\gamma t\_b^2 d\_b \sqrt{m\_d}}{\pi\hbar^3 d\_c} (k\_B T)^{s-1/2} \Gamma \left( s + \frac{1}{2} \right) F\_{s-1/2} \left( \frac{\mu\_o}{k\_B T} \right) \end{split} \tag{33}$$

The diagonal Seebeck tensor components are

3.4. Quasi-two-dimensional transport

92 Bringing Thermoelectricity into Reality

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 a � c plane and obey a weak-coupling

where d is the superlattice period or quantum well width and tb is the nearest neighbor

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>mað Þ <sup>E</sup> � <sup>2</sup>tbð Þ <sup>1</sup> � cosð Þ kbd <sup>p</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>mcð Þ <sup>E</sup> � <sup>2</sup>tbð Þ <sup>1</sup> � cosð Þ kbd <sup>p</sup>

Assuming that the in-plane mass ma ¼ mc and that kr is the wave vector in a � c plane, we

ð Þ kBT <sup>s</sup>þ<sup>1</sup>

E � μ<sup>o</sup> � �σ<sup>2</sup><sup>D</sup>

> σ<sup>2</sup><sup>D</sup> ii

ð Þ 2 þ s Fsþ<sup>1</sup>

ð Þ 1 þ s Fs

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

2 bd <sup>π</sup>ℏ<sup>4</sup> ð Þ kBT <sup>s</sup>

<sup>ℏ</sup> cos<sup>ϕ</sup>

<sup>ℏ</sup> sinϕ:

Γð Þ s þ 2 F<sup>1</sup>þ<sup>s</sup>

Γð Þ s þ 1 Fs

ii ð ÞE

μo kBT � �

� � � <sup>μ</sup><sup>o</sup>

kBT

þ 2tbð Þþ 1 � coskbdb 2tcð Þ 1 � coskcdc (32)

μo kBT

μo kBT � �

> μo kBT � �

þ 2tbð Þ 1 � coskbd (28)

(29)

(30)

(31)

model in the b direction, the following energy dispersion can be assumed as follows:

hopping matrix element between the weakly coupled layers. In-plane momenta are

<sup>E</sup> <sup>¼</sup> <sup>ℏ</sup><sup>2</sup>k<sup>2</sup> a 2ma þ ℏ2 k2 c 2mc

ka ¼

kc ¼

cc <sup>¼</sup> <sup>e</sup><sup>2</sup><sup>γ</sup> πℏ<sup>2</sup>d

bb <sup>¼</sup> <sup>2</sup>e<sup>2</sup>γmat

Ð dE ∂f 0 ∂E

¼ kB e

σ<sup>2</sup><sup>D</sup>

obtain the conductivity tensor components:

3.5. Quasi-one-dimensional transport

following energy dispersion can be assumed:

<sup>E</sup> <sup>¼</sup> <sup>ℏ</sup><sup>2</sup>k<sup>2</sup> a 2ma

σ<sup>2</sup><sup>D</sup> aa <sup>¼</sup> <sup>σ</sup><sup>2</sup><sup>D</sup>

The Seebeck tensor remains diagonal with components:

S<sup>2</sup><sup>D</sup> ii <sup>¼</sup> <sup>1</sup> eT

$$\begin{split} S\_i^{1D} &= \frac{1}{\varepsilon T} \frac{\int d\epsilon \,\frac{\partial f\_0}{\partial \epsilon} \left( \epsilon - \mu\_o \right) \sigma\_i^{1D}(\epsilon)}{\sigma\_i^{1D}} \\ &= \frac{k\_B}{\varepsilon} \left[ \frac{(3/2 + s) F\_{s + 1/2} \left( \frac{\mu\_o}{k\_B T} \right)}{(1/2 + s) F\_{s - 1/2} \left( \frac{\mu\_o}{k\_B T} \right)} - \frac{\mu\_o}{k\_B T} \right] \end{split} \tag{34}$$
