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-y transport basis can be determined. The transverse thermoelectric figure of merit zxyð Þ θ T is defined as:

$$z\_{xy}(\theta)T = \frac{S\_{xy}^2}{\kappa\_{yy}\rho\_{xx}} = \frac{\sin^2\theta\cos^2\theta \left(S\_{p,aa} - S\_{n,bb}\right)^2}{\left(\sin^2\theta\kappa\_{aa} + \cos^2\theta\kappa\_{bb}\right)\left(\cos^2\theta\rho\_{aa} + \sin^2\theta\rho\_{bb}\right)}.\tag{35}$$

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

$$\cos^2 \theta\_\perp = \frac{1}{1 + \sqrt{\frac{\kappa\_{bb}/\kappa\_{aa}}{\rho\_{bb}/\rho\_{aa}}}} \tag{36}$$

and the maximum value z⊥T becomes

$$z\_{\perp}T = z\_{xy}(\theta\_{\perp})T = \frac{\left(\mathcal{S}\_{p,aa} - \mathcal{S}\_{n,bb}\right)^2 T}{\left(\sqrt{\rho\_{aa}\kappa\_{aa}} + \sqrt{\rho\_{bb}\kappa\_{bb}}\right)^2} \tag{37}$$

Eq. (36) shows that θ<sup>⊥</sup> is independent of the Seebeck anisotropy, and it approaches <sup>π</sup> <sup>4</sup> when the thermal conductivity anisotropy matches the resistivity anisotropy <sup>κ</sup>bb <sup>κ</sup>aa <sup>¼</sup> <sup>r</sup>bb raa .

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 PF<sup>⊥</sup> as

$$PF\_{\perp} = \frac{\left(\mathcal{S}\_{p,aa} - \mathcal{S}\_{n,bb}\right)^2}{\left(\sqrt{\rho\_{aa}} + \sqrt{\rho\_{bb}}\right)^2} \,' \tag{38}$$

dQy

<sup>2</sup> � Sxy <sup>þ</sup> Syx SxySyx

dlnT þ

dT " # dT

Note for constant thermoelectric coefficients, the derivatives with respect to temperature are

þ

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

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

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

� � and <sup>E</sup><sup>0</sup> <sup>¼</sup> SxyTh=<sup>L</sup> � �, respectively, are introduced, generating normalized ver-

<sup>T</sup><sup>∗</sup> � <sup>1</sup> <sup>þ</sup> zxyTh

1 þ zxyTh � �T<sup>∗</sup> zxyTh

� �T<sup>∗</sup> � � dT<sup>∗</sup>

d2 T∗

dln SyxSxy=rxx � �

zero, and for transverse thermoelectrics, Sxy ¼ Syx. Eq. (44) thus becomes:

� <sup>d</sup> Tdy � �<sup>2</sup>

Ex Sxy

5. Cooling power for transverse thermoelectrics

<sup>0</sup> <sup>¼</sup> <sup>1</sup> SxySyx Ex

does have an analytical solution.

coolers with identical figure of merit.

Q∗

<sup>y</sup> ¼ � zxyTh � �E<sup>∗</sup>

<sup>0</sup> <sup>¼</sup> <sup>E</sup><sup>∗</sup> � dT<sup>∗</sup>

dy<sup>∗</sup> � �<sup>2</sup>

þ

<sup>Q</sup><sup>0</sup> <sup>¼</sup> <sup>κ</sup><sup>c</sup>

yyTh=L

sions of the Eqs. (42) and (45):

þ 1 þ

ficients:

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

þ

1 zxy

1 þ zxyT zxy

dln Syx=rxx � � dlnT

Sxy � �Ex

dlnκyc y

> d2 T

dy <sup>¼</sup> ExJx: (43)

dT dy

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

> 1 þ zxyT zxy

d2 T dy<sup>2</sup>

dy<sup>2</sup> <sup>¼</sup> <sup>0</sup>: (45)

dy<sup>∗</sup> , (46)

d y<sup>∗</sup> ð Þ<sup>2</sup> , (47)

(44)

95

1

dy � �<sup>2</sup> þ

where S and r tensors can be calculated by the use of semiclassical Boltzmann transport theory for the corresponding scattering mechanisms [2]. Thus, for a given band structure, PF<sup>⊥</sup> can be theoretically estimated to evaluate the performance of transverse thermoelectrics.

#### 4.1. Transport equations

The current flow J ¼ Jxx^ defines 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 Q<sup>Π</sup> ¼ ΠJ ¼ ð Þ TS J with longitudinal and transverse components:

$$Q\_{II,x} = \ \ Q\_{II} \cdot \hat{\mathbf{x}} = \ \ \left(\mathcal{S}\_{p,\text{at}} \cos^2 \theta + \mathcal{S}\_{n,bb} \sin^2 \theta\right) T\!\!/\_{x} \tag{39}$$

$$Q\_{\Pi,y} = \ \mathcal{Q}\_{\Pi} \cdot \hat{y} = \ \left(\mathcal{S}\_{p, \text{at}} - \mathcal{S}\_{n, lb}\right) \cos\Theta \sin\Theta \ T\mathcal{I}\_x\tag{40}$$

The total heat flux density <sup>Q</sup> <sup>¼</sup> <sup>Q</sup><sup>Π</sup> � <sup>κ</sup><sup>c</sup> ∇T includes both Peltier and thermal conduction effects, <sup>κ</sup><sup>c</sup> as notated in Ref. [23] defines the open-circuit thermal conductivity tensor at <sup>J</sup> <sup>¼</sup> 0. The thermal gradient is orthogonal to the current density <sup>∇</sup><sup>T</sup> <sup>¼</sup> dT dy y^; the longitudinal electric field component Ex is constant everywhere [23]; and the heat flux component Qy depends only on y. Therefore, the longitudinal current and transverse heat flow are

$$J\_{\mathbf{x}} = \frac{1}{\rho\_{\mathbf{x}\mathbf{x}}} E\_{\mathbf{x}} - \frac{S\_{\mathbf{xy}}}{\rho\_{\mathbf{x}\mathbf{x}}} \frac{dT}{dy} \tag{41}$$

$$Q\_y = T \frac{S\_{yx}}{\rho\_{xx}} E\_x - \left(1 + z\_{xy} T\right) \kappa\_{yy}^c \frac{dT}{dy} \tag{42}$$

with transverse figure of merit zxyT <sup>¼</sup> SxySyxT rxxκ<sup>c</sup> yy . Steady state requires <sup>∇</sup> � <sup>J</sup> <sup>¼</sup> 0 and <sup>∇</sup>� <sup>Q</sup> <sup>þ</sup> <sup>μ</sup><sup>J</sup> � � <sup>¼</sup> 0, where the scalar μ is the electrochemical potential and �∇μ ¼ E is the electric field. Longitudinal Joule heating ExJx sources a divergence in the transverse heat flux density Qy:

Introduction to (*p* × *n*)-Type Transverse Thermoelectrics http://dx.doi.org/10.5772/intechopen.78718 95

$$\frac{dQ\_y}{dy} = E\_x I\_x.\tag{43}$$

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

$$\begin{split} 0 &= \frac{1}{S\_{\text{xy}}S\_{\text{yx}}}E\_{\text{x}}^{-2} - \left[\frac{S\_{\text{xy}} + S\_{\text{yx}}}{S\_{\text{xy}}S\_{\text{yx}}} + \frac{d\ln\left(S\_{\text{yx}}/\rho\_{\text{xx}}\right)}{d\ln T}\frac{1}{S\_{\text{xy}}}\right]E\_{\text{x}}\frac{dT}{dy} \\ &+ \left[1 + \frac{d\ln\left(S\_{\text{yx}}S\_{\text{xy}}/\rho\_{\text{xx}}\right)}{d\ln T} + \frac{1}{z\_{\text{xy}}}\frac{d\ln\kappa\_{\text{y}}^{\text{yc}}}{dT}\right]\left(\frac{dT}{dy}\right)^{2} + \frac{1 + z\_{\text{xy}}T}{z\_{\text{xy}}}\frac{d^{2}T}{dy^{2}} \end{split} \tag{44}$$

Note for constant thermoelectric coefficients, the derivatives with respect to temperature are zero, and for transverse thermoelectrics, Sxy ¼ Syx. Eq. (44) thus becomes:

$$
\left(\frac{E\_x}{S\_{xy}} - \frac{d}{Tdy}\right)^2 + \frac{1 + z\_{xy}T}{z\_{xy}}\frac{d^2T}{dy^2} = 0.\tag{45}
$$

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

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

PF<sup>⊥</sup> <sup>¼</sup> Sp, aa � Sn, bb

ffiffiffiffiffiffi raa <sup>p</sup> <sup>þ</sup> ffiffiffiffiffiffi rbb

where S and r tensors can be calculated by the use of semiclassical Boltzmann transport theory for the corresponding scattering mechanisms [2]. Thus, for a given band structure, PF<sup>⊥</sup> can be theoretically estimated to evaluate the performance of transverse

The current flow J ¼ Jxx^ defines 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,

With Peltier tensor Π, the total Peltier heat flux density becomes Q<sup>Π</sup> ¼ ΠJ ¼ ð Þ TS J with

The total heat flux density <sup>Q</sup> <sup>¼</sup> <sup>Q</sup><sup>Π</sup> � <sup>κ</sup><sup>c</sup>∇<sup>T</sup> includes both Peltier and thermal conduction effects, <sup>κ</sup><sup>c</sup> as notated in Ref. [23] defines the open-circuit thermal conductivity tensor at <sup>J</sup> <sup>¼</sup> 0.

field component Ex is constant everywhere [23]; and the heat flux component Qy depends only

Ex � Sxy rxx dT

Ex � <sup>1</sup> <sup>þ</sup> zxyT � �κ<sup>c</sup>

where the scalar μ is the electrochemical potential and �∇μ ¼ E is the electric field. Longitudinal

yy dT

QΠ, <sup>y</sup> ¼ Q<sup>Π</sup> � y^ ¼ Sp, aa � Sn, bb

Jx <sup>¼</sup> <sup>1</sup> rxx

> Syx rxx

rxxκ<sup>c</sup> yy

Joule heating ExJx sources a divergence in the transverse heat flux density Qy:

The thermal gradient is orthogonal to the current density <sup>∇</sup><sup>T</sup> <sup>¼</sup> dT

on y. Therefore, the longitudinal current and transverse heat flow are

Qy ¼ T

<sup>Q</sup>Π,x <sup>¼</sup> <sup>Q</sup><sup>Π</sup> � <sup>x</sup>^ <sup>¼</sup> Sp, aacos<sup>2</sup><sup>θ</sup> <sup>þ</sup> Sn, bbsin<sup>2</sup><sup>θ</sup> � �TJx (39)

(40)

dy y^; the longitudinal electric

dy (41)

. Steady state requires <sup>∇</sup> � <sup>J</sup> <sup>¼</sup> 0 and <sup>∇</sup>� <sup>Q</sup> <sup>þ</sup> <sup>μ</sup><sup>J</sup> � � <sup>¼</sup> 0,

dy (42)

� � cosθsinθTJx

� �<sup>2</sup>

� � <sup>p</sup> <sup>2</sup> , (38)

factor PF<sup>⊥</sup> as

94 Bringing Thermoelectricity into Reality

thermoelectrics.

4.1. Transport equations

but the errors in that reference are corrected below.

longitudinal and transverse components:

with transverse figure of merit zxyT <sup>¼</sup> SxySyxT

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 <sup>Q</sup><sup>0</sup> <sup>¼</sup> <sup>κ</sup><sup>c</sup> yyTh=L � � and <sup>E</sup><sup>0</sup> <sup>¼</sup> SxyTh=<sup>L</sup> � �, respectively, are introduced, generating normalized versions of the Eqs. (42) and (45):

$$Q\_y^\* = -\left(z\_{xy}T\_h\right)E^\*T^\* - \left(1 + \left(z\_{xy}T\_h\right)T^\*\right)\frac{dT^\*}{dy^{\*\prime}}\tag{46}$$

$$0 = \left(E^\* - \frac{dT^\*}{dy^\*}\right)^2 + \frac{1 + \left(z\_{xy}T\_h\right)T^\*}{z\_{xy}T\_h} \frac{d^2T^\*}{d\left(y^\*\right)^2},\tag{47}$$

where <sup>T</sup><sup>∗</sup> <sup>¼</sup> ð Þ <sup>T</sup>=Th , <sup>E</sup><sup>∗</sup> <sup>¼</sup> ð Þ <sup>E</sup>=E<sup>0</sup> , <sup>y</sup><sup>∗</sup> <sup>¼</sup> <sup>y</sup> Ly , and <sup>Q</sup><sup>∗</sup> <sup>¼</sup> Qy=Q<sup>0</sup> are normalized temperature, electric field, y coordinate, and heat flux density, respectively. Eqs. (46) and (47) indicate that the normalized heat flux density Q<sup>∗</sup> <sup>y</sup> and the normalized temperature profile <sup>T</sup><sup>∗</sup> <sup>y</sup><sup>∗</sup> ð Þ only depend on the normalized electrical field E<sup>∗</sup> and transverse figure of merit zxyTh. To determine the maximum normalized temperature difference, one sets the cooling power at the cold side Q<sup>∗</sup> <sup>c</sup> to zero, to achieve <sup>Δ</sup>T<sup>∗</sup> <sup>¼</sup> <sup>1</sup> � <sup>T</sup><sup>∗</sup> <sup>c</sup> <sup>¼</sup> <sup>1</sup> � <sup>T</sup><sup>∗</sup> <sup>y</sup>ð Þ <sup>∗</sup> <sup>¼</sup> <sup>1</sup> whereby the optimal <sup>E</sup><sup>∗</sup> is determined by

$$\frac{\partial \Delta Q\_c^\*}{\partial E^\*} \mid\_{E^\* = E\_{opt}^\*} = 0 \tag{48}$$

as zxyTh increases, 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

coolers (dashed line) [24]. For zxyTh ¼ 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 zxyTh of 4 results in a 60 % temperature reduction with the transverse cooler, whereas the

ric tapering of the transverse cooler [1, 19], allowing for additional advantage over longitudi-

<sup>c</sup> <sup>¼</sup> 1, <sup>Q</sup><sup>∗</sup>

c,max for transverse coolers over longitudinal coolers with the same zxyTh is

c,max of transverse coolers as a function of T<sup>∗</sup>

<sup>c</sup> <sup>¼</sup> <sup>1</sup> � � are numerically calculated for the transverse coolers in this study

zxyTh for 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

various zxyTh values. For a given zxyTh, the Fourier diffusion heat flow increases when T<sup>∗</sup>

decreases; thus, a larger portion of the Peltier cooling power is used to compensate the

Figure 7. The dependence on zxyT of maximum normalized temperature difference (left axis) whereby <sup>Δ</sup>T<sup>∗</sup> <sup>¼</sup> ð Þ Th � Tc =Th and maximum cooling power when Tc ¼ Th (right axis) for transverse thermoelectric coolers in comparison

and analytically solved for the longitudinal coolers according to standard equations in the literature [1, 22, 23].

<sup>c</sup> on zxyTh, left axis. Transverse coolers (solid

<sup>1</sup>� ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ2zxyTh p

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

max can be further increased with geomet-

c,max shows a superlinear dependence on

zxyTh for longitudinal

c,max of the transverse

c,max ¼ 1=2zxyTh

c,max will decrease.

<sup>c</sup> for

c

97

maximum temperature difference <sup>Δ</sup>T<sup>∗</sup> <sup>¼</sup> <sup>1</sup> � <sup>T</sup><sup>∗</sup>

longitudinal cooler achieves only 50%. Note that ΔT<sup>∗</sup>

nal cooling for achieving large temperature differences.

for longitudinal coolers (dashed line) when T<sup>∗</sup>

Figure 8 shows the cooling power Q<sup>∗</sup>

enhancement in Q<sup>∗</sup>

with longitudinal coolers. ΔT<sup>∗</sup>

max and Q<sup>∗</sup>

c,max T<sup>∗</sup>

line) show a larger <sup>Δ</sup>T<sup>∗</sup> than the analytically solved <sup>Δ</sup>T<sup>∗</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup>

Figure 7, right axis, plots the normalized maximum cooling power Q<sup>∗</sup>

28 % when zxyTh ¼ 1 and rapidly increases to 220 % when zxyTh ¼ 4.

diffusive heat flow, and the remaining cooling power at the cold side Q<sup>∗</sup>

cooler when Tc <sup>¼</sup> Th as a function of zxyTh. Unlike the linear dependence in <sup>Q</sup><sup>∗</sup>

Thus, ΔT<sup>∗</sup> max zxyTh is only a function of zxyTh. Similarly, the maximum of the cooling power at the cold side Q<sup>∗</sup> <sup>c</sup> <sup>¼</sup> <sup>Q</sup><sup>∗</sup> <sup>y</sup> <sup>y</sup>ð Þ <sup>∗</sup> <sup>¼</sup> <sup>1</sup> for a given <sup>T</sup><sup>∗</sup> <sup>c</sup> can be obtained when <sup>E</sup><sup>∗</sup> satisfies:

$$\frac{\partial \Delta T^\*}{\partial E^\*} \mid\_{E^\* = E\_{\text{opt}}^\*} = 0 \tag{49}$$

and Q<sup>∗</sup> c,max depends only on T<sup>∗</sup> <sup>c</sup> and 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 (Q<sup>∗</sup> <sup>c</sup> ¼ 0) for various transverse figures of merit zxyTh. The temperature gradient at the hot side (y<sup>∗</sup> <sup>¼</sup> 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

Figure 6. Normalized temperature profile of transverse thermoelectric coolers operating at maximum temperature difference for various zxyTh values. At <sup>y</sup><sup>∗</sup> <sup>¼</sup> <sup>y</sup>=Ly <sup>¼</sup> 0, the heat sink temperature <sup>T</sup><sup>∗</sup> <sup>¼</sup> <sup>T</sup>=Th <sup>¼</sup> 1 and at the <sup>y</sup><sup>∗</sup> <sup>¼</sup> 1, the cold side heat flow Qc <sup>¼</sup> Qy <sup>y</sup>ð Þ¼ <sup>∗</sup> <sup>¼</sup> <sup>1</sup> 0.

as zxyTh increases, 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 <sup>Δ</sup>T<sup>∗</sup> <sup>¼</sup> <sup>1</sup> � <sup>T</sup><sup>∗</sup> <sup>c</sup> on zxyTh, left axis. Transverse coolers (solid line) show a larger <sup>Δ</sup>T<sup>∗</sup> than the analytically solved <sup>Δ</sup>T<sup>∗</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>1</sup>� ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ2zxyTh p zxyTh for longitudinal coolers (dashed line) [24]. For zxyTh ¼ 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 zxyTh of 4 results in a 60 % temperature reduction with the transverse cooler, whereas the longitudinal cooler achieves only 50%. Note that ΔT<sup>∗</sup> max can be further increased with geometric tapering of the transverse cooler [1, 19], allowing for additional advantage over longitudinal cooling for achieving large temperature differences.

where <sup>T</sup><sup>∗</sup> <sup>¼</sup> ð Þ <sup>T</sup>=Th , <sup>E</sup><sup>∗</sup> <sup>¼</sup> ð Þ <sup>E</sup>=E<sup>0</sup> , <sup>y</sup><sup>∗</sup> <sup>¼</sup> <sup>y</sup>

normalized heat flux density Q<sup>∗</sup>

96 Bringing Thermoelectricity into Reality

to achieve <sup>Δ</sup>T<sup>∗</sup> <sup>¼</sup> <sup>1</sup> � <sup>T</sup><sup>∗</sup>

max zxyTh

<sup>c</sup> <sup>¼</sup> <sup>Q</sup><sup>∗</sup>

c,max depends only on T<sup>∗</sup>

cooling power of the transverse coolers.

Thus, ΔT<sup>∗</sup>

and Q<sup>∗</sup>

the cold side Q<sup>∗</sup>

ature difference (Q<sup>∗</sup>

side heat flow Qc <sup>¼</sup> Qy <sup>y</sup>ð Þ¼ <sup>∗</sup> <sup>¼</sup> <sup>1</sup> 0.

Ly 

mum normalized temperature difference, one sets the cooling power at the cold side Q<sup>∗</sup>

∂ΔQ<sup>∗</sup> c ∂E<sup>∗</sup> j <sup>E</sup>∗¼E<sup>∗</sup>

∂ΔT<sup>∗</sup> ∂E<sup>∗</sup> j <sup>E</sup>∗¼E<sup>∗</sup>

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

Figure 6 shows the normalized temperature profile under the condition of maximum temper-

ent at the hot side (y<sup>∗</sup> <sup>¼</sup> 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

Figure 6. Normalized temperature profile of transverse thermoelectric coolers operating at maximum temperature difference for various zxyTh values. At <sup>y</sup><sup>∗</sup> <sup>¼</sup> <sup>y</sup>=Ly <sup>¼</sup> 0, the heat sink temperature <sup>T</sup><sup>∗</sup> <sup>¼</sup> <sup>T</sup>=Th <sup>¼</sup> 1 and at the <sup>y</sup><sup>∗</sup> <sup>¼</sup> 1, the cold

<sup>c</sup> ¼ 0) for various transverse figures of merit zxyTh. The temperature gradi-

<sup>y</sup> <sup>y</sup>ð Þ <sup>∗</sup> <sup>¼</sup> <sup>1</sup> for a given <sup>T</sup><sup>∗</sup>

<sup>c</sup> and zTh.

electric field, y coordinate, and heat flux density, respectively. Eqs. (46) and (47) indicate that the

the normalized electrical field E<sup>∗</sup> and transverse figure of merit zxyTh. To determine the maxi-

, and <sup>Q</sup><sup>∗</sup> <sup>¼</sup> Qy=Q<sup>0</sup>

<sup>c</sup> <sup>¼</sup> <sup>1</sup> � <sup>T</sup><sup>∗</sup> <sup>y</sup>ð Þ <sup>∗</sup> <sup>¼</sup> <sup>1</sup> whereby the optimal <sup>E</sup><sup>∗</sup> is determined by

is only a function of zxyTh. Similarly, the maximum of the cooling power at

<sup>y</sup> and the normalized temperature profile <sup>T</sup><sup>∗</sup> <sup>y</sup><sup>∗</sup> ð Þ only depend on

<sup>c</sup> can be obtained when <sup>E</sup><sup>∗</sup> satisfies:

opt ¼ 0 (48)

opt ¼ 0 (49)

are normalized temperature,

<sup>c</sup> to zero,

Figure 7, right axis, plots the normalized maximum cooling power Q<sup>∗</sup> c,max of the transverse cooler when Tc <sup>¼</sup> Th as a function of zxyTh. Unlike the linear dependence in <sup>Q</sup><sup>∗</sup> c,max ¼ 1=2zxyTh for longitudinal coolers (dashed line) when T<sup>∗</sup> <sup>c</sup> <sup>¼</sup> 1, <sup>Q</sup><sup>∗</sup> c,max shows a superlinear dependence on zxyTh for 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 Q<sup>∗</sup> c,max for transverse coolers over longitudinal coolers with the same zxyTh is 28 % when zxyTh ¼ 1 and rapidly increases to 220 % when zxyTh ¼ 4.

Figure 8 shows the cooling power Q<sup>∗</sup> c,max of transverse coolers as a function of T<sup>∗</sup> <sup>c</sup> for various zxyTh values. For a given zxyTh, the Fourier diffusion heat flow increases when T<sup>∗</sup> c decreases; 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 Q<sup>∗</sup> c,max will decrease.

Figure 7. The dependence on zxyT of maximum normalized temperature difference (left axis) whereby <sup>Δ</sup>T<sup>∗</sup> <sup>¼</sup> ð Þ Th � Tc =Th and maximum cooling power when Tc ¼ Th (right axis) for transverse thermoelectric coolers in comparison with longitudinal coolers. ΔT<sup>∗</sup> max and Q<sup>∗</sup> c,max T<sup>∗</sup> <sup>c</sup> <sup>¼</sup> <sup>1</sup> � � are numerically calculated for the transverse coolers in this study and analytically solved for the longitudinal coolers according to standard equations in the literature [1, 22, 23].

Acknowledgements

201406280070).

Author details

References

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.

Introduction to (*p* × *n*)-Type Transverse Thermoelectrics http://dx.doi.org/10.5772/intechopen.78718 99

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

Electrical Engineering and Computer Science, Northwestern University, Evanston, USA

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\*Address all correspondence to: m-grayson@northwestern.edu

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Figure 8. The maximum normalized cooling power Q<sup>∗</sup> c,max for transverse thermoelectric cooling as a function of the normalized cold side temperature T<sup>∗</sup> <sup>c</sup> for various zxyTh values.

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 Q<sup>0</sup> and E<sup>0</sup> 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 Q<sup>0</sup> and E<sup>0</sup> 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.
