*2.1. Categories*

The results section is categorized, as follows.


#### *2.2. Coupling Currents of Entropy and Charge in Thermoelectric Materials*

When a thermoelectric material is simultaneously placed in a gradient of the electrochemical potential ∇*μ*˜ and a gradient of the temperature ∇ *T*, electrical flux density **j***q*, and entropy flux density **j***S* are observed [34,46].

$$
\begin{pmatrix} \mathbf{j}\_q \\ \mathbf{j}\_S \end{pmatrix} = \begin{pmatrix} \sigma & \sigma \cdot \mathbf{a} \\ \sigma \cdot a & \sigma \cdot a^2 + \Lambda\_{\text{CC}} \end{pmatrix} \cdot \begin{pmatrix} -\nabla \vec{\mu} / q \\ -\nabla T \end{pmatrix} \tag{1}
$$

With the classical thermodynamic potential gradients ∇*μ*˜ (per electric charge *q*) and ∇ *T* being employed, the basic transport Equation (1) has the following structure.

$$\text{flux densities} \quad = \begin{array}{c} \text{material tensor} \quad \text{potential gradients} \end{array} \tag{2}$$

The thermoelectric material tensor in Equation (1) is composed of only three quantities, which are the isothermal electrical conductivity *σ*, the Seebeck coefficient *α*, and the entropy conductivity at electrical open circuit ΛOC (i.e., at vanishing electrical current). In principle, all three quantities are tensors themselves, but, for homogenous materials, they are often treated as scalars.

The entropy conductivity Λ is related to the traditional "heat" conductivity *λ* by the absolute temperature *T* [32,34,37]. This, in principle, indicates that the traditional "heat" conduction is based on a more fundamental entropy conduction. The author proposes using the generic term thermal conductivity to address either the "heat" conductivity or the entropy conductivity [47,48].

$$
\lambda = T \cdot \Lambda \tag{3}
$$

It is emphasized that Equation (1) refers to a steady-state non-equilibrium situation. Instead of the quantities electric charge *q* and entropy *S*, their local flux densities appear. According to Falk [35], considering local flux densities allows addressing local energy conversion or better to say local power conversion. Because flowing quantities are involved, preference should be given to local power density. Remember, power is the flux of energy. Equation (1) allows for locally varying quantities to be considered, which can be expressed with the positional vector **r**: **j***q* = **j***q* (**r**), **j***S* = **j***S* (**r**), *σ* = *σ* (**r**), *α* = *α* (**r**), ΛOC = ΛOC (**r**), ∇*μ*˜ = ∇*μ*˜ (**r**), ∇*T* = ∇*T* (**r**). Of course, the thermodynamic potentials are locally varying when gradients are present: *μ*˜ = *μ*˜ (**r**), *T* = *T* (**r**).

However, if the local variation of all quantities in Equation (1) is neglected, a simplified formulation of the transport equation can be observed [34,49,50]. If a further weak temperature dependence is assumed for the electron chemical potential *μ* (i.e., *∂μ∂T* ≈ 0), the temperature dependence of the electrochemical potential *μ*˜ = *μ* + *q* · *ϕ* is only in the electrical potential *ϕ*. With ∇*μ*/*q* ≈ 0 follows ∇*μ*˜/*q* = ∇*μ*/*q* + ∇*ϕ* ≈ ∇*ϕ*. The assumption of constant gradients (i.e., linear potential curves) allows for them to be substituted by the difference of the respective potential along the thermoelectric material of length *L*: ∇*ϕ* → <sup>−</sup>Δ*ϕ*/*L*, ∇*T* → <sup>−</sup>Δ*T*/*L*. Furthermore, for a thermoelectric material of cross-sectional area *A*, the local flux densities can be replaced by the integrative currents of electrical charge and entropy, respectively: **j***q* → *Iq*/*A*, **j***S* → *IS*/*A*. Subsequently, the transport equation follows as:

$$
\begin{pmatrix} I\_q \\ I\_S \end{pmatrix} = \frac{A}{L} \cdot \begin{pmatrix} \sigma & \sigma \cdot a \\ \sigma \cdot a & \sigma \cdot a^2 + \Lambda\_{\text{OC}} \end{pmatrix} \cdot \begin{pmatrix} \Delta \varrho \\ \Delta T \end{pmatrix} \tag{4}
$$

Equation (4) describes the coupling of currents of electrical charge *Iq* and entropy *IS* in the thermoelectric material, which causes the occurence of either an electrically-induced entropy current [51] (Peltier effect) or a thermally-induced electrical current [52,53] (Seebeck effect). Note that Equation (4) describes these effects in a thermoelectric material, which is schematically shown in Figure 1, apart from a device.

#### *2.3. Material's Voltage—Electrical Current and Electrical Power—Electrical Current Characteristics*

Different working conditions of the thermoelectric material in this article are discussed with reference to the voltage–electrical current curve, which is derived from Equation (4) as Equation (5). Remember that the voltage Δ*ϕ* is the electrical potential difference along the thermoelectric material.

$$
\Delta \varphi = -\mathfrak{a} \cdot \Delta T + \frac{I\_q}{\frac{A}{L} \cdot \sigma} \tag{5}
$$

**Figure 1.** This paper discusses characteristics of a thermoelectric material of cross-sectional area *A* and length *L* when exposed to a temperature difference Δ*T* = *T*hot − *T*cold between a hot reservoir at *T*hot and a cold reservoir at *T*cold.

According to Equation (5), the voltage–electrical current characteristics is a line, which has the material's electrical resistance *R* = 1*A L* ·*σ* as its slope. This line is only determined by the voltage Δ*ϕ*OC under electrically open-circuited conditions (i.e., at zero electrical current) and the electrical current *I*SC at electrically short-circuited conditions (i.e., at zero voltage). The OC is of practical importance for the measurement of temperature using thermocouples.

$$
\Delta \mathfrak{q}\_{\rm OC} = -\mathfrak{a} \cdot \Delta T \tag{6}
$$

$$I\_{q\text{SC}} = \frac{A}{L} \cdot a \cdot \sigma \cdot \Delta T \tag{7}$$

Obviously, the sign of the Seebeck coefficient *α* determines the sign of both the voltage Δ*ϕ*OC under electrically short-circuited conditions and the electrical current *Iq*,SC under electrically short-circuited conditions. Thus, the voltage–electrical current characteristics of *p*-type (*α* > 0) or *n*-type (*α* < 0) conductors differ from each other by principle (cf. Appendix A).

To discuss the materials independently of the sign of the Seebeck coefficient, the absolute of the voltage | Δ*ϕ* | is plotted in Figure 2 versus the absolute value of the electrical current | *Iq* |. In order to diminish Ohmic losses, the electrical resistance *R* = 1*A L* ·*σ* must be reduced, which, for the given geometry, requires the electrical conductivity *σ* to be increased.

To make the discussion independent from even the material parameters and temperature difference Δ*T*, the normalized electrical current *i* and normalized voltage *u*, as normalized to electrically short-circuited and open-circuited conditions, respectively, are considered in subsequent sections.

$$i = \frac{I\_q}{I\_{q, \text{SC}}} = \frac{I\_q}{\frac{A}{L} \cdot \alpha \cdot \sigma \cdot \Delta T} \tag{8}$$

$$u = \frac{\Delta \varphi}{\Delta \varphi\_{\rm OC}} = \frac{\Delta \varphi}{-\kappa \cdot \Delta T} = 1 - i \tag{9}$$

The electrical power *P*el is determined by the product of voltage and electrical current as given by Equation (10). It increases linearly with the electrical current, but it is parabolically damped at high electrical currents due to the limited electrical conductivity (Ohmic dissipation [54]).

$$\begin{aligned} P\_{\text{el}} &= \Delta \boldsymbol{\varrho} \cdot I\_{\text{q}} &= \left( -\boldsymbol{\kappa} \cdot \Delta T + \frac{l\_{\text{q}}}{\frac{\Delta}{\Gamma} \cdot \boldsymbol{\sigma}} \right) \cdot I\_{\text{q}} \\ &= -\boldsymbol{\kappa} \cdot \Delta T \cdot I\_{\text{q}} + \frac{l\_{\text{q}}^{2}}{\frac{\Delta}{\Gamma} \cdot \boldsymbol{\sigma}} \\ &= -\frac{A}{L} \cdot \boldsymbol{\sigma} \cdot \boldsymbol{\kappa}^{2} \cdot (\Delta T)^{2} \cdot \left( \boldsymbol{i} - \boldsymbol{i}^{2} \right) \end{aligned} \tag{10}$$

**Figure 2.** Absolute voltage | Δ*ϕ* | – electrical current | *Iq* | curve (green), with slope given by the electrical resistance *R* = 1*A L* ·*σ* , and the absolute electrical power | *P*el | – electrical current | *Iq* | curve (red) for a thermoelectric material. Here, Δ*T* = *T*hot−*T*cold *T*hot is the temperature difference along the thermoelectric material of cross-sectional area *A* and length *L*. These quantities, together with the (isothermal) electrical conductivity *σ* and the Seebeck coefficient *α*, determine the electrical current *I*SC under electrically short-circuited conditions. The voltage Δ*ϕ*OC under electrically open-circuited conditions is determined by the Seebeck coefficient and the temperature difference. Generator mode refers to a positive sign and entropy pump mode to a negative sign of the electrical power (cf. Appendix A).

The absolute of the electrical power | *P*el | is plotted in Figure 2 versus the absolute value of the electrical current | *Iq* | to discuss the thermoelectric materials independent of the sign of the Seebeck coefficient.

It is obvious from Figure 2 that the electrical power to be put into the material in entropy pump mode may distinctly exceed the electrical power that can be gained in generator mode if the material is applied to the same temperature difference.

#### *2.4. Material's Thermal Conductivity—Electrical Current Characteristics*

From Equation (4), the entropy current *IS* flowing through the material is obtained. It depends on not only the temperature difference Δ*T* but also the Peltier effect that is associated with the thermally induced electrical current *Iq*, which can be expressed by the normalized electrical current *i* as given in Equation (8).

$$\begin{array}{ll} I\_S & = \frac{A}{L} \cdot \Lambda\_{\text{OC}} \cdot \Delta T + \alpha \cdot I\_q \\ & = \frac{A}{L} \cdot \Lambda\_{\text{OC}} \cdot \Delta T + \frac{A}{L} \cdot \sigma a^2 \cdot i \cdot \Delta T \\ & = \frac{A}{L} \cdot \left(\Lambda\_{\text{OC}} + \sigma a^2 \cdot i\right) \Delta T \\ & = \frac{A}{L} \cdot \Lambda \cdot \Delta T \end{array} \tag{11}$$

From Equation (11), it follows that the thermal conductivity, expressed here by the entropy conductivity Λ, is dependent on the electrical current *i*.

$$
\Lambda = \Lambda \left( i \right) = \Lambda\_{\rm OC} + \sigma \mathfrak{a}^2 \cdot i \tag{12}
$$

When compared to electrically open-circuited conditions, the power factor *σα*˙ 2 gives an additional contribution to the entropy conductivity, which increases linearly with the electrical current. Under electrically short-circuited conditions (SC, i.e., *i* = 1), the entropy conductivity reaches its maximum value.

$$
\Lambda\_{\text{SC}} = \Lambda\_{\text{OC}} + \sigma a^2 \tag{13}
$$

Under electrically short-circuited conditions, the electrical potential is spatially constant (i.e., its gradient vanishes: ∇*ϕ* = 0). Note that the entropy conductivity at electrical short circuit ΛSC, as given by Equation (13), is identical to tensor element *M*22 of the thermoelectric material tensor in the transport Equation (4).

To discuss the characteristics of the entropy conductivity in a general manner, it is normalized to its value under electrically open-circuited conditions:

$$
\tilde{\Lambda} = \tilde{\Lambda}\left(i\right) = \frac{\Lambda}{\Lambda\_{\rm OC}} = 1 + \frac{\sigma a^2}{\Lambda\_{\rm OC}} \cdot i = 1 + zT \cdot i \tag{14}
$$

In Equation (14), a figure-of-merit *zT* has been identified, which only depends on the three material parameters *σ*, *α* and ΛOC, which make up the material tensor of Equation (4).

$$zT = \frac{\sigma \cdot a^2}{\Lambda\_{\text{OC}}} \tag{15}$$

Equation (14) is visualized in Figure 3 for some hypothetical thermoelectric materials with *zT* = 0.1, 0.5, 1, 2, 4 and 8. Working points for electrically open-circuited (OC) conditions, maximum electrical power point (MEPP), and electrical short-circuited (SC) conditions are indicated on the voltage–electrical current curve. Note that the entropy conductivity inversion point (ECIP) is given by the negative reciprocal of the figure-of-merit <sup>−</sup>1/*zT*. Only for electrical currents being below the ECIP, effective entropy pump mode is reached with a negative entropy conductivity of the thermoelectric material. Only then, more entropy is pumped against the temperature difference than flows down it. Obviously, the measurements of the thermal conductivity of a thermoelectric material must refer to the working point on the voltage–electrical current curve.

**Figure 3.** Normalized entropy conductivity Λ ˜ as function of normalized electrical current *i* for some hypothetical thermoelectric materials. Depending on the figure-of-merit *zT*, the curves pivot through the working point for electrically open-circuited (OC) conditions. The figure-of-merit *zT* gives the slope of the curve and its negative reciprocal −1/*zT* indicates the entropy conductivity inversion point (ECIP). For some thermoelectric materials, the respective ECIP is indicated as working point on the normalized voltage *u*–normalized electrical current *i* curve. Note that the ECIP for materials with *zT* = 0.1. and *zT* = 0.5 is out of the applied scale. The term entropy pump mode is put into brackets because a net entropy current against the temperature difference will only occur if the magnitude of the electrical current is beyond the respective ECIP. For generator mode, the working points MEPP and SC are indicated.

#### *2.5. Thermoelectric Material in Generator Mode*

#### 2.5.1. Working Point for Maximum Electrical Power

Remember, the characteristics of the thermoelectric material are all discussed for Δ*T* being different from zero, which implies non-isothermal conditions. It can be easily seen from Equation (10) that maximum electrical power output is obtained for half of the electrically short-circuited electrical current (*i*MEPP = 12, cf. Appendix B.1):

$$|P\_{\rm el, max} = |\; P\_{\rm el} \left( i\_{\rm MEPP} = 0.5 \right)| = \frac{1}{4} \cdot \frac{A}{L} \cdot \sigma \cdot a^2 \cdot \left( \Delta T \right)^2 \tag{16}$$

To make the discussion independent from material parameters and temperature difference, the normalized electrical power *p*el, as normalized to the maximum electrical power in generator mode, is plotted in Figure 4.

$$p\_{\rm el} = \frac{|P\_{\rm el}|}{P\_{\rm el, max}} = 4 \cdot |\ i - i^2 \mid \tag{17}$$

The maximum electrical power point (MEPP) is indicated on the normalized voltage–electrical current curve in Figure 4. It is clearly seen that the MEPP (*i*MEPP = 0.5, *u*MEPP = 0.5) is at half of the open-circuited voltage as well as at half of the electrically short-circuited electrical current, which also follows from Equation (9).

**Figure 4.** Normalized curves for both voltage *u* – electrical current *i* characteristics and electrical power *p*el–electrical current *i* characteristics of a thermoelectric material when it is operated in generator mode. The working points open-circuited (OC), maximum electrical power point (MEPP), and short-circuited (SC) are indicated.

#### 2.5.2. Thermal Conductivity

For the thermoelectric material being operated in generator mode, Equation (12) is graphically expressed in Figure 5. The electrically open-circuited entropy conductivity ΛOC is purely dissipative, while the part of the entropy conductivity depending on the power factor *σ* · *α*2 couples to the electrical current, and it fully contributes to the thermal-to-electric power conversion. Obviously, to maximize

the electrical power at a given temperature difference, the power *σ* · *α*2 must be maximized, which is in accordance with Equation (10).

**Figure 5.** Entropy conductivity Λ as function of the normalized electrical current *i* for a thermoelectric material with *zT* = 2 in generator mode. The working points OC, MEPP, and SC are indicated on the normalized voltage–electrical current curve.

The thermally induced electrical current carries electrical energy, which, however, with increasing electrical current, is diminished by Ohmic losses due to the limited (isothermal) electrical conductivity *σ* as discussed above. At maximum electrical power, the entropy conductivity is increased by half of the power factor as compared to electrically open-circuited conditions. Under electrically short-circuited conditions, the entropy conductivity reaches its maximum (see Equation (13)).

#### 2.5.3. Thermal Power

The thermal input power and the thermal output power depend on the electrical current *i*. According to Fuchs [33], the available thermal power *P*th is determined by the fall of entropy down the temperature difference Δ*T* along the material.

$$P\_{\rm th} = I\_{\rm S} \cdot \Delta T = \Lambda \cdot \left(\Delta T\right)^2 = \frac{A}{L} \cdot \left(\Lambda\_{\rm OC} + \sigma \mathbf{a}^2 \cdot i\right) \cdot \left(\Delta T\right)^2 \tag{18}$$

Thus, the available thermal power, as given by Equation (18), depends on the electrical current in the same manner as the entropy conductivity in Figures 3 and 5.

#### 2.5.4. Power Conversion Efficiency (Thermal to Electrical)

From Equations (10) and (18), the second-law power conversion efficiency for the thermoelectric material in generator mode is obtained:

$$\begin{array}{lcl} \eta\_{\text{II,gen}} & = | \frac{P\_{\text{el}}}{P\_{\text{th,val}}} | & = \frac{\frac{\text{A}}{\text{L}} \cdot r \cdot a^2 \cdot (\Delta T)^2 \cdot \left(i - i^2\right)}{\frac{\text{A}}{\text{L}} \cdot \left(\Lambda\_{\text{OC}} + r \sigma^2 \cdot i\right) \cdot (\Delta T)^2} \\ & = \frac{\frac{\text{i} \cdot r^2}{\text{i} - i^2}}{\frac{\text{A} \cdot \text{s}}{\text{s} + \text{s}^2}} \\ & = \frac{\text{i} - \text{s}^2}{\text{i} + \frac{1}{\text{s}^2}} \end{array} \tag{19}$$

Equation (19) is plotted in Figure 6 as solid blue curves for some hypothetical thermoelectric materials with different values of the figure-of-merit *zT*. Obviously, the figure-of-merit *zT* must be maximized in order to maximize the thermal-to-electrical power conversion efficiency at a given (thermally induced) electrical current.

Equation (19) can be read as the coupled thermal power being converted into electrical power with the constraint; however, with increasing electrical current, Ohmic dissipation gains overhead. As a result, the optimum power conversion efficiency is obtained at lower electrical current than the optimum electrical power output, and the working points for one or other task differ from each other, which can be seen in Figure 6.

According to Fuchs [33], the second-law efficiency *η*II,gen is related to the first-law efficiency *η*I,gen by Carnot's efficiency *η*C.

$$
\eta\_{\text{l,gen}} = \eta\_{\text{C}} \cdot \eta\_{\text{II,gen}} = \frac{T\_{\text{hot}} - T\_{\text{cold}}}{T\_{\text{hot}}} \cdot \eta\_{\text{II,gen}} \tag{20}
$$

Carnot's efficiency *η*C places a theoretical limit for the case in which the second-law efficiency *η*II,gen = 1, which refers to the unrealistic case of vanishing dissipation. Nevertheless, the second-law efficiency *η*II,gen is the only material-dependent factor and has been used by Altenkirch [55] and Ioffe [56] in order to estimate the performance of thermoelectric materials by treating thermogenerators. It is worth noting that the entropy-based approach presented here allows for power conversion and its efficiency for a single thermoelectric material apart from a device to be discussed.

#### 2.5.5. Working Points for Maximum Conversion Efficiency and Maximum Electrical Power

From the maximum of Equation (19), the maximum conversion efficiency point (MCEP) is obtained with the normalized electrical current *<sup>i</sup>*MCEP,gen being, as follows (cf. Appendix B.2):

$$\begin{array}{rcl} \text{ $i$ }\_{\text{MCEP}, \text{gcm}} & = \frac{1}{\sqrt{1+zT+1}} \\\\ \end{array} \tag{21}$$

At the MCEP, the maximum power conversion efficiency of the thermoelectric material in generator mode is then obtained, as follows (cf. Appendix B.2):

$$\eta\_{\text{Il,geen,max}} = \eta\_{\text{Il,geen}} \left( i\_{\text{MCE,geen}} \right)\_{\text{ }} = \frac{\sqrt{1+z}\overline{1}-1}{\sqrt{1+z}\overline{1}+1} \tag{22}$$

Equation (23), which shows the variation of the MCEP with varying *<sup>i</sup>*MCEP,gen due to varying *zT*, is plotted in Figure 6 as dotted blue line.

$$\left(\eta\_{\text{II,gen,max}}\left(i\_{\text{MCE,gen}}\right)\right) \quad = 1 - 2 \cdot i\_{\text{MCE,gen}}\tag{23}$$

Note that with increasing figure-of-merit *zT*, not only does the MCEP drift apart from the MEPP, but the electrical power output also decreases with respect to the MEPP (see Equation (16)), both of which can be seen in Figure 6 (cf. Appendix B.2).

$$P\_{\rm el,MCEP} = \frac{4 \cdot \sqrt{1 + zT}}{\left(\sqrt{1 + zT} + 1\right)^2} \cdot P\_{\rm el,max} \tag{24}$$

**Figure 6.** Thermal to electrical power conversion efficiency for some hypothetic materials with figure-of-merit *zT* varying from 0.5 to 100. Respective working points MCEP (blue) are indicated on the voltage–electrical current curve as well as the MEPP (red). Vertical lines indicate the electrical power output at the MCEP for the example materials. Note that the MCEP drifts apart from the MEPP with increasing figure-of-merit *zT*. The dashed line indicates the dependence of the MCEP with varying *zT*.

Obviously, with increasing figure-of-merit *zT*, the electrical power at the MCEP converges to zero. Figure 7 shows that a notable difference in electrical power output between MCEP and MEPP can be expected for thermoelectric materials with *zT* > 0.3 only (red curves). A notable difference in the power conversion efficiency of the thermoelectric material being operated in the MCEP or the MEPP can only be expected when *zT* > 2. This is also obvious from Table 2, which, for some hypothetical values of the material's figure-of-merit *zT*, gives values of the second-law power conversion efficiency at the working points under discussion. The 2n<sup>d</sup> law power conversion efficiency at the MEPP is obtained as follows (cf. Appendix B.1).

$$
\eta\_{\text{II,gen,MEPP}} = \eta\_{\text{II,gen}}(i\_{\text{MEPP}} = 0.5) \quad = \frac{1}{2} \cdot \frac{\bar{z}\_{\text{II}}^T}{\bar{z}\_{\text{II}}^T + 2} \tag{25}
$$

**Figure 7.** Electrical power output (red lines) and thermal-to-electrical power conversion efficiency (blue lines) for some hypothetic materials with figure-of-merit *zT* varying from 0.01 to 1000 when operated in two distinct working points, respectively. Solid lines refer to the MCEP and dashed lines refer to the MEPP.

It is worth noting that, for a thermoelectric material with *zT* < 2, there is no benefit from operating it apart from the MEPP.


**Table 2.** Second-law power conversion efficiency of a thermoelectric material at the MCEP in either entropy pump mode or generator mode and at the MEPP in generator mode for some hypothetical values of the figure-of-merit *zT*.

#### *2.6. Thermoelectric Material in Entropy Pump Mode*

#### 2.6.1. Power Conversion Efficiency (Electrical to Thermal)

Traditional approaches consider a coefficient of performance when addressing the performance of a thermoelectric cooling or heating device [56,57]. Analogously, a coefficient of performance *COP* of the thermoelectric material, when used in a cooler, can be considered. It is the thermal power removed from the cold side *T*cold · *IS* related to the electrical power (cf. Appendix C.1).

$$\begin{array}{rcl} \text{COP}\_{\text{cooker}} & = \begin{vmatrix} \frac{T\_{\text{coalg}} \cdot I\_{\text{S}}}{P\_{\text{ol}}} \end{vmatrix} & = \begin{array}{rcl} \frac{T\_{\text{coalg}}}{\Delta T} \cdot | \; \frac{P\_{\text{th}}}{P\_{\text{sl}}} \; | \; \\\\ & = \begin{array}{rcl} \frac{T\_{\text{copl}}}{\Delta T} \cdot \eta \text{II}\_{\text{cSP}} \end{array} \end{array} \tag{26}$$

If instead of a cooler, the thermoelectric material is used in a heater (see Fuchs [32], p. 135ff), the thermal power released to the hot side *T*hot · *IS* becomes the reference parameter, and the *COP* is then (cf. Appendix C.1):

 $\text{COP}\_{\text{heater}} = |\begin{array}{c} T\_{\text{bat}} \cdot I\_{\text{S}} \ | \ \ \ \end{array}| \ \ = \frac{T\_{\text{bat}}}{\Delta T} \cdot | \ \ | \ \frac{P\_{\text{th}}}{P\_{\text{el}}} \ |$ 
$$= \frac{T\_{\text{bat}}}{\Delta T} \cdot \eta\_{\text{II,cp}} \tag{27}$$

$$= \frac{1}{\eta\_{\text{C}}} \cdot \eta\_{\text{II,cp}}$$

In both cases, Equations (26) and (27), the *COP* can be factorized into a temperature factor and the second-law efficiency for the thermoelectric material in entropy pump mode *η*II,ep (see Fuchs [32], p. 135ff). When the thermoelectric material is used in a heater (Equation (27)), the temperature factor is the inverse of Carnot's efficiency *η*C [32]. The second-law efficiency for the thermoelectric material in entropy pump mode *η*II,ep relates the thermal power *P*th that is needed to pump a certain entropy current from the cold side to the hot side to the electrical power *P*el (cf. Appendix C.1).

$$|\eta\_{\rm II,cp}| = |\begin{array}{c} \frac{p\_{\rm fin}}{p\_{\rm el}} \end{array}| \;= \begin{array}{c} \frac{i + \frac{1}{\pi I}}{-i^2 + i} \end{array} \tag{28}$$

The second-law efficiency for the thermoelectric material in entropy pump mode *η*II,ep only depends on the normalized electrical current *i* (i.e., working point on the voltage–electrical current curve) and the material's figure-of-merit *zT*. It can be used to assess the performance of the thermoelectric material when it is used to pump entropy, regardless of whether the purpose is cooling or heating.

Note that the second-law efficiency for the thermoelectric material in entropy pump mode *η*II,ep (Equation (28)) is the inverse of the second-law efficiency for the thermoelectric material in generator mode (Equation (19)). Because a net entropy current from the cold side to the hot side will only be obtained for negative entropy conductivity (see Equation (14) and Figure 3), here *η*II,ep will make sense only for the normalized electrical current being *i* ≤ 1*zT* . For this parameter range it is plotted in Figure 8 for some hypothetic thermoelectric materials with figure-of-merit *zT* between 0.5 and 100.

The maximum 2nd-law power conversion efficiency for a thermoelectric material operated in entropy pump mode is dependent on the material's figure-of-merit *zT* (cf. Appendix C.2):

$$
\eta\_{\text{II,eq,max}} \qquad = \frac{\sqrt{1+zT}-1}{\sqrt{1+zT}+1} \tag{29}
$$

It is obtained at a normalized electrical current *<sup>i</sup>*MCEP,ep, which corresponds to the thermoelectric material's maximum conversion efficiency point (MCEP) in entropy pump mode (cf. Appendix C.2). Respective working points for some hypothetic thermoelectric materials are indicated on the voltage–electrical current curve presented in Figure 8.

$$\begin{array}{rcl} \text{ $i$ } \text{ $M$ +} \text{CEP}\_{\text{d} \text{°P}} & = & -\frac{1}{\sqrt{1+z}T-1} \end{array} \tag{30}$$

The dependence of the maximum second-law efficiency on the electrical current is shown in Figure 8 as a hyperbolic line (cf. Appendix C.2).

$$\left(\eta\_{\text{II,ep,max}}\left(i\_{\text{MCE,ep}}\right)\right)\_{\text{=}} = \frac{1}{1 - 2 \cdot i\_{\text{MCE,ep}}}\tag{31}$$

Obviously, an ideal thermoelectric material would have an infinite *zT* , but the MCEP converges then to the OC working point at vanishing electrical current and, thus, zero electrical power. On the contrary, for the limit of vanishing *zT*, the maximum second-law efficiency converges to zero at infinite magnitude of the electrical current.

**Figure 8.** Electrical-to-thermal power conversion efficiency as a function of the reduced electrical current for some hypothetic materials with figure-of-merit *zT* varying from 0.5 to 100. Respective working points MCEP (blue) are indicated on the voltage–electrical current curve for *zT* = 100, 32, 18, 8 and 4. Further vertical lines (blue) indicate the MCEP for *zT* = 2, 1. The MCEP for *zT* = 0.5 is out of display. The hyperbolic curve indicates the dependence of the MCEP with varying *zT*. The red curve indicates electrical power–electrical current characteristics. The set of inclined parallel lines (magenta) indicate the thermal power–electrical current characteristics for the respective *zT*. All of the power curves are normalized to the MEPP in generator mode.

#### 2.6.2. Electrical and Thermal Power

All of the power curves in Figure 8, for the thermoelectric material in entropy pump mode, are normalized to the MEPP in generator mode (see Figures 2 and 4) when the material is exposed to the same temperature difference Δ*T*. According to Equations (16) and (18), the normalized thermal power *p*th in Figure 8 is given by a straight line that intersects the horizontal axis at − 1*zT* and it has a slope of −4 (cf. Appendix C.3).

$$p\_{\rm th} \quad = \frac{|P\_{\rm th}|}{P\_{\rm cl,max}} \quad = 4 \cdot |\,\,\frac{1}{\overline{z}T} + i\,\, |\tag{32}$$

For different values of the figure-of-merit *zT*, a set of inclined parallel lines results. Only the lines for *zT* = 0.5, 1 and 2 are labelled in Figure 8. With increasing figure-of-merit *zT*, the normalized thermal power curve approaches the normalized electrical power curve, which is in accordance with the increasing power conversion efficiency. However, when the thermoelectric material is operated in its MCEP, the thermal power will decrease with increasing figure-of-merit *zT*, which becomes obvious when Equation (30) is combined with Equation (32) (cf. Appendix C.2).

$$\begin{array}{rcl}p\_{\text{th,MCEP}} & = p\_{\text{th}} \left(i\_{\text{MCEP}; \text{cp}}\right) & = \mathbf{4} \cdot \frac{\sqrt{1 + zT}}{zT} \end{array} \tag{33}$$

The normalized thermal power at MCEP would be steeply curved in Figure 8, with the data point out of scale for *zT* < 8, but has been skipped for clarity. Instead, relevant values for the MCEP are listed in Table 3, together with the normalized electrical power and the normalized electrical current.


**Table 3.** Values of normalized electrical current *<sup>i</sup>*MCEP,ep, normalized thermal power *p*th,MCEP, and normalized electrical power *p*el,MCEP at the MCEP in entropy pump mode for some hypothetical values of the figure-of-merit *zT*. Values of the second law power conversion efficiency can be read from Table 2

### *2.7. Complete Picture*

With the approach chosen here, working points on the voltage–electrical current curve relate the power conversion properties of the thermoelectric material in generator mode and entropy pump mode to each other. Figure 9 illustrates the concise result for a hypothetical thermoelectric material with figure-of-merit *zT* = 3.5.

**Figure 9.** Related characteristics of a hypothetic thermoelectric material with figure-of-merit *zT* = 3.5 in entropy pump mode and generator mode: normalized voltage, normalized electrical power, normalized thermal power, and 2nd-law conversion efficiency as a function of the normalized electrical current. Different working points are indicated on the voltage–electrical current curve. Note that, for current state-of-the-art materials, the MCEP in entropy pump mode would be out of display (see Table 3).

For a given figure-of-merit *zT*, according to Equations (22) and (29), the values of the maximum 2nd-law conversion efficiency for both modes are identical. Some values are given in Table 2. In addition, values of the 2nd-law conversion efficiency at the MEPP in generator mode are given (see Equation (25)). Remember, the obtained power requires consideration of the absolute value of the electrical power, as determined by the power factor (see Equation (16)).

#### **3. Materials and Methods**

Detailed calculations, as given in Appendixs B and C, were made using pencil and paper. The manuscript was prepared using Latex in MikTex distribution. Figures were drawn with the aid of Microcal's Origin and Microsoft's PowerPoint.

## **4. Discussion**

#### *4.1. Remarks on the Use of Working Points*

Traditionally, a thermoelectric device is considered and, in generator mode, the operational conditions are set by an external load resistance. The approach of this work, which uses working points on the material's voltage–electrical voltage curve, gives consistent results, which is explicitly shown in Appendix B.3. However, consideration of working points comes with the advantage that the contribution of individual thermoelectric materials in a device can be easily understood [58]. Moreover, the material's voltage–electrical voltage curve directly relates generator mode and entropy pump mode.

#### *4.2. Remarks on the Altenkirch-Ioffe Model*

Due to the prominence of the Altenkirch-Ioffe model [55,56], it is worth comparing it to the model, which has been introduced in this work. A comparison of important quantities described by the model of this work and the Altenkirch-Ioffe model is shown in Figure 10.

Remember, Equation (4) has been derived for a thermoelectric material apart from a device. Furthermore, a constant temperature gradient has been assumed, which means a constant slope of the temperature profile, which then connects the hot side at *T*hot and the cold side at *T*cold by a straight line (solid line in Figure 10a). The further assumption of a temperature-independent entropy conductivity ΛOC at electrical open-circuit is plotted in Figure 10b as a solid line. As a consequence of these assumptions, at a given electrical current (including electrically open-circuited conditions), the entropy current will carry the highest energy current at the hot side of the thermoelectric material. When advancing through the thermoelectric material to lower temperatures, the entropy current cannot further carry all thermal energy ("heat"), which then needs to be dissipated. Following Walstrom's approach [59], thermal energy is assumed to be dissipated transversally together with instantaneously produced excess entropy as its carrier. It is important to emphasize that excess entropy leaves the thermoelectric material in directions transversal to the flow of the entropy inserted at the hot side. The ability to conduct thermal energy is decreased with decreasing temperature, which is reflected in a decreasing "heat" conductivity, as plotted in Figure 10c as a solid line.

Traced back to Altenkirch [55] and Ioffe [56], often a model is discussed that considers a two-leg thermogenerator and assumes constant "heat" conductivity. Concerning the thermoelectric material, the model is purely one-dimensional and does not allow for transversal dissipation of entropy and energy. All dissipation has to be considered parallel or antiparallel to the flow of entropy and thermal energy along the thermoelectric material. In fact, only the parallel option (i.e., down the temperature gradient) remains physically meaningful. Under electrically open-circuited conditions (i.e., vanishing electrical current), the temperature profile can still be linear. However, Heikes and Ure [60] have shown that, in the presence of a thermally-induced electrical current, the temperature profile is flattened at the hot side and steeply sloping at the cold side, which is shown in Figure 10a as a dashed line. As a consequence of the curved temperature profile and the constant "heat" conductivity (see dashed line in Figure 10c), the "heat" flux is diminished at the hot side (thermal energy input) and increased at the cold side (thermal energy output). The change in the temperature profile is such that, as compared to the zero electrical current situation, the thermal energy input is diminished by half of the Joule "heat" and the thermal energy release at the cold side is increased by half of the Joule "heat", as shown by Heikes and Ure [60,61]. This is to account for the dissipation of thermal energy being parallel to the flow of entropy and thermal energy. As a consequence, when compared to electrically open-circuited conditions, the thermoelectric material would be thermally less transparent when an electrical current flows.

**Figure 10.** Comparison of the model of this work (constant entropy conductivity) to the Altenkirch-Ioffe model [33,55,56,60] (constant "heat" conductivity) with the schematic profiles of the following quantities over the thermoelectric material when the material is carrying a (thermally induced) electrical current: (**a**) temperature *T*; (**b**) electrically open-circuited entropy conductivity ΛOC; and, (**c**) electrically open-circuited "heat" conductivity *λ*OC. Note that profiles are not drawn to scale.

In contrast, the model of this work predicts the thermoelectric material to become thermally more transparent with increasing electrical current, which is reflected in the then reversible increased entropy conductivity Λ(*i*) (see Equations (12) and (14)). In the author's opinion, this is an important characteristic of thermoelectric materials, which is fully embezzled in the traditional model.

In the Altenkirch-Ioffe model, all the excess entropy and excess thermal energy are dissipated to the cold side, which is reflected in an irreversible increase of the entropy conductivity along the thermoelectric material, as visualized in Figure 10b. The aforementioned assumption introduces a ratio of *T*hot/*T*cold into the formula for the 2nd-law efficiency at the MCEP (see Appendixs B.4 and B.5 for a device in generator mode; see Appendixs C.4 and C.5 for a device in entropy pump mode). Ioffe [56] has shown that the deviation from Equation (22) (generator mode) or Equation (29) (entropy pump mode), however, is only a few per cent when the efficiency itself is small. In other words, for a small temperature difference Δ*T*, both of the models give nearly the same results.

It must be emphasized that both of the models rely on very special assumptions and, thus, cannot claim general validity [62]. In this sense, all of the results have to be considered semi-quantitatively when it comes to real thermoelectric materials and devices. More general considerations, as provided by Equation (1), need to consider the local variation of thermoelectric parameters but are beyond the scope of this work. Heikes and Ure [60] and Gryasnov et al. [63] have considered the local variation of thermoelectric parameters to some extent. However, the advantage of the model of this work is not only to consider the thermoelectric material apart from a device, but also to clearly separate the dissipation of entropy and thermal energy from the reversible thermoelectric coupling. The simplicity of thermoelectrics is manifested.

#### *4.3. Remarks on Narducci's Model*

Narducci has put the question "Do we really need high thermoelectric figures of merit?" and found in his calculations that, when considering constant Δ*T*, the electrical power output of a two-leg thermogenerator device at the MEPP increases with increased thermal conductivity (see Narducci [64], Figure 2). The situation that is discussed by Narducci corresponds to a decreasing figure-of-merit (i.e., *zT* → 0 limit) with the electrical power converging to what we have obtained here as *<sup>P</sup>*el, max (see Equation (16)). In light of this work, it becomes obvious that the MCEP and the MEPP of the thermoelectric material(s) then merge (see Figures 6 and 7).

#### *4.4. Remarks on* Λ*OC*

In the model applied in this work, the electrically open-circuited entropy conductivity ΛOC originates only from non-charge transporting excitations of the solid (mostly phonons). Here, the contribution from electrons to the entropy conductivity solely originates from the power factor (see Figures 3 and 5). Subsequently, distinguishing contributions from electrons and phonons to the thermal conductivity is straightforward (see Ioffe [56], p. 44) and has been coined the "phonon glass–electron crystal" (PGEC) concept by Slack [65]. In this case, ΛOC is identical to the phonon contribution to the entropy conductivity.

However, as mentioned by Ioffe (see Ioffe [56], p. 46), in materials with charge carriers of both signs (electrons and holes from multiple bands), the situation is more intricate. Subsequently, important electronic contribution to the thermal conductivity can be expected for vanishing net flux of charge. In other words, the electrically open-circuited entropy conductivity ΛOC has contributions from both phonons and electrons. The application of the empirical Wiedemann-Franz law to describe the relationship between thermal and electrical conductivity is questionable for these materials [48,56]. In practice, this is the case for many semiconductors and metals. To improve the thermoelectric properties of these materials, it is not sufficient to reduce the phonon contribution by the PGEC concept. In addition, electronic band engineering is required in order to diminish the electron contribution to ΛOC. The theory in this work can be easily extended to treat this case by introducing a second type of charge carrier into Equations (1) and (4).

#### *4.5. Remarks on Figure-of-Merit zT*

In this work, the figure-of-merit has been introduced in context with the entropy conductivity (cf. Equation (14)) to underline that it is the dimensionless ratio of two entropy conductivities. Initially, the thermoelectric figure-of-merit was introduced by Ioffe [56] as a parameter *z* = *<sup>σ</sup>*·*α*<sup>2</sup> *λ*OC in the treatment of a thermogenerator referring to the "heat" conductivity. In subsequent treatment, Ioffe has taken into account the medium temperature *T* of the device and elucidated the thermoelectric material's figure-of-merit to be *zT* = *<sup>σ</sup>*·*α*<sup>2</sup> *λ*OC · *T*, which has subsequently been widely used as *zT*. With this formulation of the figure-of-merit, researchers often have been confused by the intensive variable temperature *T* showing up explicitly besides material parameters [66]. It is seen as a persistent residual outcome of the historical dispute between Ostwald and Boltzmann (see Section 1.2) that it has not been realized that the use of entropy conductivity Λ instead of the "heat" conductivity *λ* makes the figure-of merit depend on three material parameters only, which all implicitly depend on temperature (see Equations (3) and (15)).

The author has used *zT* to be consistent with the conventional nomenclature of the thermoelectric community. All of the formulas in this article, which contain the figure-of-merit, however, would look more straightforward if *zT* were to be substituted by a single letter, for instance, *f* as used by Zener [67].

$$f = \frac{\sigma \cdot a^2}{\Lambda\_{\text{OC}}} = \frac{\sigma \cdot a^2}{\lambda\_{\text{OC}}} \cdot T = zT \tag{34}$$

#### *4.6. Remarks on State-of-the-Art and Emerging Thermoelectric Materials*

It is worth noting that, for a thermoelectric material with *zT* < 2, there is no benefit from operating it apart from the MEPP (see Figure 6, Figure 7 and Table 2). In this context, it is important to perceive that current state-of-the-art materials hardly exceed a *zT* value of 2. The values listed in Table 4 are peak values. Among the materials of Table 4, PbTe0.7S0.3-2.5%K has a peak *zT* of 2.2 at 923 K and a record high average *zT* of 1.56 in the temperature interval of 300–900 K [68]. Conclusively, the tracking of the MEPP [69], but not of the MCEP, is reported for thermogenerators. However, for the application of emerging thermoelectric materials with further improved figure-of-merit, and thus more distant working points, tracking of the MCEP might become relevant.

**Table 4.** Maximum figure-of-merit *zT*max and corresponding power factor *σ* · *α*2 of some state-of-the-art and emerging thermoelectric materials at temperature *T* with indication of conduction type.


The benefit of an increased figure-of-merit *zT* will be an increased power conversion efficiency at the MEPP anyway. Figure 6, Figure 7, and Table 2 indicate that the material's second-law power conversion efficiency at the MEPP will not exceed the value of 0.5 (see also Equation (25)). Interestingly, this value corresponds to the lower limit of the Curzon-Ahlborn efficiency of a Carnot engine operated at its MEPP [91,92]. At the MEPP, a real thermoelectric material will always be operated at less than half of the Carnot efficiency.

#### *4.7. Remarks on the Importance of the Power Factor and Choice of Materials for Thermogenerators*

Because normalized curves are discussed in this work, one might lose sight of the fact that the power factor *σ* · *α*2 is at least as important as the figure-of-merit *zT*. According to Equation (16), it rules over the maximum achievable absolute electrical power when the thermoelectric material is operated in generator mode at MCEP. For a material with high *zT* (e.g., 100), the electrical power is much lower at the MCEP compared to the MEPP (Figures 6 and 7). This is because, at the low electrical current of the MCEP, the thermoelectric material is less permeable to entropy when compared to the MEPP (see Figure 5). Thus, less thermal power is available to be thermoelectrically converted into electrical power. The amount of useful thermal power depends on the power factor and the electrical current (see the second summand in Equation (18)).

The open-circuited entropy conductivity ΛOC causes a thermoelectrically-inactive bypass, which eventually leads the temperature difference Δ*T*, which squared determines the maximum electrical power in Equation (16), to drop. To provide large Δ*T*, the open-circuited entropy conductivity ΛOC should be kept small. Here, in addition to a high power factor *σα*˙ 2, the figure-of-merit *zT* comes into play, which relates the aforementioned contributions to the entropy conductivity (see Equation (12) and Equation (15)). The materials that are listed in Table 4 represent those with the highest values of the figure-of-merit reported thus far. In the author's opinion, the most interesting materials are those that also have a high power factor of at least 30 μWcm−1K−2.

A high electrical conductivity *σ* is also advantageous, as already mentioned in Section 2.3. The choice of materials can easily be made with the help of type-1 Ioffe plots [56] (*σα*<sup>2</sup> − *σ*) and type-2 Ioffe plots (ΛOC − *σ*) [56,93], which have been recently revitalized on the example of current thermoelectric materials [47,48,94]. The reader is referred to Fuchs [32] (p. 135ff) for further details.

#### *4.8. Remarks on the Second-Law Power Conversion Efficiency vs. Coefficient of Performance for Entropy Pumps*

While the upper limit of the coefficient of performance will depend on temperature conditions, as involved in the Carnot efficiency *η*C (Equation (27)) or the temperature factor *T*cold Δ*T* (Equation (26)), the upper limit for the second-law efficiency is fixed to unity (i.e., *η*II,ep ≤ 1). The unity value of the second-law efficiency refers to an ideal material. While the coefficient of performance is related to a floating scale, the second-law efficiency allows for the estimation of how far from ideal a thermoelectric material is. Another advantage is that the second-law efficiency in Equation (28) only depends on the figure-of-merit and the electrical current and, thus, allows for evaluation of the performance of the thermoelectric material apart from specific temperature conditions, as well as independent from use in a cooler or a heater.

Note that, according to Equations (29) and (22), the maximum second-law efficiency of a thermoelectric material is identical in entropy pump mode and generator mode:

$$\eta\_{\text{II,ep,max}} = \eta\_{\text{II,ep}} \left( i\_{\text{MCE,ep}} \right) \\ \quad = \eta\_{\text{II,gen}} \left( i\_{\text{MCE,gen}} \right) \\ \quad = \eta\_{\text{II,gen,max}} \tag{35}$$

This is also apparent from Figure 9.

#### *4.9. Remarks on the Choice of Materials for Entropy Pumps*

Remember, electrical and thermal power in Figure 8 are normalized to the MEPP in generator mode (see Equations (17) and (32)). Thus, the absolute thermal power in entropy pump mode is

determined by the material's power factor *σ* · *α*2 (see Equation (16)). A low open-circuited entropy conductivity Λ*OC* is desired to prevent the thermoelectrically inactive fall of entropy along the temperature difference Δ *T*, which would make it difficult to maintain the Δ *T*. Thus, in addition to a high power factor *σ* · *α*2, a high figure-of-merit *zT* is favourable, which relates the aforementioned quantities (see Equation (15)).

Operating the thermoelectric material in entropy pump mode requires good performance at ambient temperature and below (e.g., for cooling 150–300 K) or above (e.g., for heating 300–400 K). Among the materials listed in Table 4, only bismuth telluride-based materials fulfil all requirements; and, they are the current materials of choice for the mentioned applications and are conclusively found in commercial devices.

According to Figure 8, emerging materials with improved figure-of-merit at a power factor comparable to bismuth telluride-based materials would have the benefit that comparable thermal power could be pumped from the cold to hot side at a lower electrical current and electrical power.
