# Power Conversion and Its Efficiency in Thermoelectric Materials

## Abstract

**:**

## 1. Introduction

#### 1.1. Controversial Points of View

#### 1.2. Implications of Natural Philosophy

#### 1.3. Evolution of Thermodynamics

#### 1.4. Modern Thermodynamics

#### 1.5. Entropy in Thermoelectrics

#### 1.6. Aim of This Work

## 2. Results

#### 2.1. Categories

- Section 2.2: Coupling currents of entropy and charge in thermoelectric materials
- Section 2.3: Material’s voltage–electrical current and electrical power–electrical current characteristics
- Section 2.4: Material’s thermal conductivity–electrical current characteristics
- Section 2.5: Thermoelectric material in generator mode
- Section 2.5.1: Working point for maximum electrical power
- Section 2.5.2: Thermal conductivity
- Section 2.5.3: Thermal power
- Section 2.5.4: Power conversion efficiency (thermal to electrical)
- Section 2.5.5: Working points for maximum conversion efficiency and maximum electrical power
- Section 2.6: Thermoelectric material in entropy pump mode
- Section 2.6.1: Power conversion efficiency (electrical to thermal)
- Section 2.6.2: Electrical and thermal power
- Section 2.7: Complete picture

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

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

#### 2.4. Material’s Thermal Conductivity—Electrical Current Characteristics

#### 2.5. Thermoelectric Material in Generator Mode

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

#### 2.5.2. Thermal Conductivity

#### 2.5.3. Thermal Power

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

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

^{nd}law power conversion efficiency at the MEPP is obtained as follows (cf. Appendix B.1).

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

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

^{nd}-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):

#### 2.6.2. Electrical and Thermal Power

#### 2.7. Complete Picture

^{nd}-law conversion efficiency for both modes are identical. Some values are given in Table 2. In addition, values of the 2

^{nd}-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

## 4. Discussion

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

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

^{nd}-law efficiency at the MCEP (see Appendix B.4 and Appendix B.5 for a device in generator mode; see Appendix C.4 and Appendix 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 $\Delta T$, both of the models give nearly the same results.

#### 4.3. Remarks on Narducci’s Model

#### 4.4. Remarks on ${\mathsf{\Lambda}}_{\mathrm{OC}}$

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

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

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

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

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

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ECIP | Entropy Conductivity Inversion Point |

MCEP | Maximum Conversion Efficiency Point (either in generator mode or entropy pump mode) |

MEPP | Maximum Electrical Power Point (in generator mode) |

OC | (Electrical) Open Circuit |

SC | (Electrical) Short Circuit |

Symbols | |

The following symbols are used in this manuscript: | |

Geometry | |

A | cross-sectional area of thermoelectric material |

L | length of thermoelectric material |

Material properties | |

$\alpha $ | Seebeck coefficient |

f | figure-of-merit (as proposed by Zener [67]) |

$\lambda $ | “heat” conductivity |

${\lambda}_{\mathrm{OC}}$ | “heat” conductivity under electrically open-circuited (OC) conditions |

$\mathsf{\Lambda}$ | entropy conductivity |

${\mathsf{\Lambda}}_{\mathrm{OC}}$ | entropy conductivity under electrically open-circuited (OC) conditions |

${\mathsf{\Lambda}}_{\mathrm{SC}}$ | entropy conductivity under electrically open-circuited (SC) conditions |

$\tilde{\mathsf{\Lambda}}$ | normalized entropy conductivity |

${M}_{22}$ | tensor element (of the thermoelectric material tensor) |

R | electrical resistance (of thermoelectric material) |

$\sigma $ | isothermal electrical conductivity |

z | thermoelectric factor (as introduced by Ioffe [56]) |

$zT$ | figure-of-merit (as introduced by Ioffe [56]) |

$z{T}_{\mathrm{max}}$ | maximum figure-of-merit |

Thermodynamic potentials | |

$\mu $ | chemical potential |

$\tilde{\mu}$ | electrochemical potential ($\tilde{\mu}=\mu +q\xb7\phi $) |

$\nabla \tilde{\mu}$ | gradient of the electrochemical potential |

$\nabla \tilde{\mu}/q$ | gradient of the electrochemical potential per electric charge ($\nabla \tilde{\mu}/q=\nabla \mu /q+\nabla \phi $) |

$\phi $ | electrical potential |

$\nabla \phi $ | gradient of the electrical potential |

$\Delta \phi $ | difference of electrical potential (along the thermoelectric material) |

$\Delta {\phi}_{\mathrm{OC}}$ | voltage under electrically open-circuited (OC) conditions |

T | absolute temperature |

${T}_{\mathrm{cold}}$ | temperature of the thermoelectric material at its cold side |

${T}_{\mathrm{hot}}$ | temperature of the thermoelectric material at its hot side |

$\nabla T$ | gradient of the temperature |

$\Delta T$ | difference of temperature (along the thermoelectric material) |

u | normalized voltage |

${u}_{\mathrm{MEPP}}$ | normalized voltage at the maximum electrical power point (MEPP) |

Fluxes | |

A | cross-sectional area of thermoelectric material |

L | length of thermoelectric material |

i | normalized electrical current |

${i}_{\mathrm{MCEP},\mathrm{ep}}$ | normalized electrical current at the maximum conversion efficiency point (MCEP) in entropy pump mode |

${i}_{\mathrm{MCEP},\mathrm{gen}}$ | normalized electrical current at the maximum conversion efficiency point (MCEP) in generator mode |

${i}_{\mathrm{MEPP}}$ | normalized electrical current at the maximum electrical power point (MEPP) |

${I}_{q}$ | electrical current |

${I}_{q,SC}$ | electrical current at electrically short-circuited (SC) conditions |

${I}_{S}$ | entropy current |

${\mathbf{j}}_{q}$ | electrical flux density |

${\mathbf{j}}_{S}$ | entropy flux density |

q | electric charge |

S | entropy |

Performance | |

${COP}_{\mathrm{cooler}}$ | coefficient of performance of the thermoelectric material when used in a cooler |

${COP}_{\mathrm{heater}}$ | coefficient of performance of the thermoelectric material when used in a heater |

${\eta}_{\mathrm{I},\mathrm{gen}}$ | first-law power conversion efficiency of the thermoelectric material in generator mode |

${\eta}_{\mathrm{II},\mathrm{gen}}$ | second-law power conversion efficiency of the thermoelectric material in generator mode |

${\eta}_{\mathrm{II},\mathrm{gen},\mathrm{max}}$ | maximum second-law power conversion efficiency of the thermoelectric material in generator mode |

${\eta}_{\mathrm{II},\mathrm{ep}}$ | second-law power conversion efficiency of the thermoelectric material in entropy pump mode |

${\eta}_{\mathrm{II},\mathrm{ep},\mathrm{max}}$ | maximum second-law power conversion efficiency of the thermoelectric material in entropy pump mode |

${\eta}_{\mathrm{C}}$ | Carnot’s efficiency |

${p}_{\mathrm{el}}$ | normalized electrical power |

${P}_{\mathrm{el}}$ | electrical power, needed for lifting electrical charge (generator mode) |

or made available by the fall of electric charge (entropy pump mode); | |

simplified called output (generator mode) or input (entropy pump mode), | |

when the electrical potential on one side of the thermoelectric material is set to zero | |

${P}_{\mathrm{el},\phantom{\rule{4.pt}{0ex}}\mathrm{max}}$ | maximum electrical power output of the thermoelectric material in generator mode (at the MEPP) |

${P}_{\mathrm{el},\mathrm{MCEP}}$ | electrical power output, of the thermoelectric material in generator mode, at the MCEP |

${P}_{\mathrm{th}}$ | thermal power, made available by the fall of entropy (generator mode) |

or needed for lifting entropy (entropy pump mode) |

## Appendix A. Voltage–Electrical Current and Electrical Power–Electrical Current Characteristics: p- and n-Type Materials

**Figure A1.**Voltage $\Delta \phi $ – electrical current ${I}_{q}$ characteristics (green curves) and electrical power ${P}_{\mathrm{el}}$ – electrical current characteristics ${I}_{q}$ (red curves) for materials with: (

**a**) Seebeck coefficient $\alpha $ being positive, which refers to p-type conduction and (

**b**) Seebeck coefficient $\alpha $ being negative, which refers to n-type conduction. Here, $\Delta T=\frac{{T}_{\mathrm{hot}}-{T}_{\mathrm{cold}}}{{T}_{\mathrm{hot}}}$ is the temperature difference along a thermoelectric material of length L and cross-sectional area A. These quantities, together with the (isothermal) electrical conductivity $\sigma $ and the Seebeck coefficient, determine the electrical current ${I}_{\mathrm{SC}}$ under electrical short-circuited conditions. The voltage $\Delta {\phi}_{\mathrm{OC}}$ under electrical short-circuited conditions is determined by the Seebeck coefficient and the temperature difference. When the electrical power ${P}_{\mathrm{el}}$ is negative (electrical power output), the material is in generator mode (thermal-to-electrical power conversion). When the electrical power ${P}_{\mathrm{el}}$ is positive (electrical power input), the material is in entropy pump mode (electrical-to-thermal power conversion).

## Appendix B. Thermal-to-Electrical Power Conversion: Calculations and Established Models

#### Appendix B.1. Maximum Electrical Power Point (MEPP): Material in Generator Mode

^{nd}-law power conversion efficiency at the MEPP is then obtained as follows.

#### Appendix B.2. Maximum Conversion Efficiency Point (MCEP): Material in Generator Mode

^{nd}-law power conversion efficiency for a thermoelectric material operated in generator mode is obtained as follows.

^{nd}law power conversion efficiency can be written as follows.

#### Appendix B.3. Comparison to Power Conversion Efficiency after Fuchs: Thermogenerator Device

^{nd}-law efficiency by the ratio of useful to available power and expressed the efficiency with respect to the internal resistance of the device ${R}_{\mathrm{TEG}}$ and an external load resistance ${R}_{\mathrm{ext}}$.

^{nd}-law efficiency of the device has its maximum at.

^{nd}-law power conversion efficiency is as follows.

#### Appendix B.4. Comparison to Power Conversion Efficiency after Altenkirch: Thermogenerator Device

^{st}-law power conversion efficiency. Altenkirch [55] has factorized the 1

^{st}law power conversion efficiency into the Carnot efficiency and what we call here the 2

^{nd}-law power conversion efficiency ${\eta}_{\mathrm{II}}$. The latter has been of the following form.

^{nd}-law power conversion efficiency ${\eta}_{\mathrm{II},\mathrm{TEG},\mathrm{max}}$ to be (see Altenkirch [55], Equation (5)) as follows.

^{nd}law power conversion efficiency ${\eta}_{\mathrm{II},\mathrm{TEG},\mathrm{max}}$ as a function of $x=\frac{{R}_{\mathrm{ext}}}{{R}_{\mathrm{TEG}}}$ for different values of his “${\eta}^{\prime}$”, which despite a dimensionless factor has been identified with $zT$. In the plot, he indicated the shift of the MCEP with varied figure-of-merit.

#### Appendix B.5. Comparison to Power Conversion Efficiency after Ioffe: Thermogenerator Device

## Appendix C. Electrical-to-Thermal Power Conversion: Calculations and Established Models

#### Appendix C.1. Power Conversion Efficiency

**Figure A2.**When the thermoelectric material is operated in entropy pump mode, electrical power ${P}_{\mathrm{el}}$, which is available by the fall of electric charge along $\Delta \phi $, drives the pumping of entropy from the cold side to hot side. The thermal power ${P}_{\mathrm{th}}=\Delta T\xb7{I}_{S}={T}_{\mathrm{hot}}\xb7{I}_{S}-{T}_{\mathrm{cold}}\xb7{I}_{S}$ for lifting entropy along the temperature difference $\Delta T$ adds to the thermal power removed from the cold side ${T}_{\mathrm{cold}}\xb7{I}_{S}$ to give the thermal power released to the hot side ${T}_{\mathrm{hot}}\xb7{I}_{S}$. Different width of arrows refers to different magnitudes of thermal power at the opposite sides of the material, which is due to thermoelectric power conversion.

#### Appendix C.2. Maximum Conversion Efficiency Point (MCEP): Material in Entropy Pump Mode

^{nd}-law power conversion efficiency, as given by Equation (A26), vanishes.

^{nd}-law power conversion efficiency for a thermoelectric material operated in entropy pump mode is then as follows.

#### Appendix C.3. Normalized Thermal Power

#### Appendix C.4. Comparison to Power Conversion Efficiency after Altenkirch: Thermoelectric Cooler Device

^{nd}-law power conversion efficiency for a thermoelectric material operated in entropy pump mode ${\eta}_{\mathrm{II},\mathrm{ep},\mathrm{max}}$, becomes as follows.

^{nd}-law power conversion efficiency for thermoelectric cooler ${\eta}_{\mathrm{II},\mathrm{TEC},\mathrm{max}}$ becomes the following.

#### Appendix C.5. Comparison to Power Conversion Efficiency after Ioffe: Thermoelectric Cooler Device

^{nd}-law efficiency ${\eta}_{\mathrm{II},\mathrm{ep},\mathrm{max}}$. After Ioffe [56], the device-related analogue of the latter has been as follows.

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**Figure 1.**This paper discusses characteristics of a thermoelectric material of cross-sectional area A and length L when exposed to a temperature difference $\Delta T={T}_{\mathrm{hot}}-{T}_{\mathrm{cold}}$ between a hot reservoir at ${T}_{\mathrm{hot}}$ and a cold reservoir at ${T}_{\mathrm{cold}}$.

**Figure 2.**Absolute voltage $\mid \Delta \phi \mid $ – electrical current $\mid {I}_{q}\mid $ curve (green), with slope given by the electrical resistance $R=\frac{1}{\frac{A}{L}\xb7\sigma}$, and the absolute electrical power $\mid {P}_{\mathrm{el}}\mid $ – electrical current $\mid {I}_{q}\mid $ curve (red) for a thermoelectric material. Here, $\Delta T=\frac{{T}_{\mathrm{hot}}-{T}_{\mathrm{cold}}}{{T}_{\mathrm{hot}}}$ is the temperature difference along the thermoelectric material of cross-sectional area A and length L. These quantities, together with the (isothermal) electrical conductivity $\sigma $ and the Seebeck coefficient $\alpha $, determine the electrical current ${I}_{\mathrm{SC}}$ under electrically short-circuited conditions. The voltage $\Delta {\phi}_{\mathrm{OC}}$ 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).

**Figure 3.**Normalized entropy conductivity $\tilde{\mathsf{\Lambda}}$ 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.

**Figure 4.**Normalized curves for both voltage u – electrical current i characteristics and electrical power ${p}_{\mathrm{el}}$–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.

**Figure 5.**Entropy conductivity $\mathsf{\Lambda}$ 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.

**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 $zT$.

**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.

**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 $zT=100,32,18,8\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}4$. 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.

**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 2

^{nd}-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).

**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 ${\mathsf{\Lambda}}_{\mathrm{OC}}$; and, (

**c**) electrically open-circuited “heat” conductivity ${\lambda}_{\mathrm{OC}}$. Note that profiles are not drawn to scale.

**Table 1.**Working points on the voltage–electrical current curve of a thermoelectric material in both operational modes, as addressed in this work.

Abbreviation | Working Point | Operational Mode |
---|---|---|

MCEP | Maximum (power) conversion efficiency point | entropy pump mode |

EICP | Entropy conductivity inversion point | entropy pump mode |

OC | (electrical) open circuit | generator mode |

MCEP | (see above) | generator mode |

MEPP | Maximum (electrical) power point | generator mode |

SC | (electrical) short circuit | generator mode |

**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 $zT$.

$\mathbf{zT}$ | Maximum 2^{nd} Law Efficiency | 2^{nd} Law Efficiency at MEPP |
---|---|---|

0.1 | 0.02 | 0.02 |

0.5 | 0.1 | 0.1 |

1 | 0.17 | 0.17 |

1.5 | 0.23 | 0.21 |

2 | 0.27 | 0.25 |

2.5 | 0.30 | 0.28 |

3 | 0.33 | 0.3 |

3.5 | 0.36 | 0.32 |

4 | 0.38 | 0.33 |

8 | 0.5 | 0.4 |

16 | 0.61 | 0.44 |

32 | 0.70 | 0.47 |

100 | 0.82 | 0.49 |

**Table 3.**Values of normalized electrical current ${i}_{\mathrm{MCEP},\mathrm{ep}}$, normalized thermal power ${p}_{\mathrm{th},\mathrm{MCEP}}$, and normalized electrical power ${p}_{\mathrm{el},\mathrm{MCEP}}$ 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

$\mathbf{zT}$ | ${\mathit{i}}_{\mathbf{MCEP},\mathbf{ep}}$ | ${\mathit{p}}_{\mathbf{th},\mathbf{MCEP}}$ | ${\mathit{p}}_{\mathbf{el},\mathbf{MCEP}}$ |
---|---|---|---|

0.1 | $-20.49$ | 41.95 | 1761.32 |

0.5 | $-4.45$ | 9.80 | 97.01 |

1 | $-2.41$ | 5.66 | 32.87 |

1.5 | $-1.72$ | 4.22 | 19.67 |

2 | $-1.36$ | 3.46 | 12.83 |

2.5 | $-1.48$ | 2.99 | 10.77 |

3.0 | $-1$ | 2.68 | 8.93 |

3.5 | $-0.89$ | 2.42 | 7.56 |

4 | $-0.80$ | 2.2 | 5.76 |

8 | $-0.50$ | 1.5 | 3.00 |

16 | $-0.32$ | 1.03 | 1.69 |

32 | $-0.21$ | 0.71 | 1.02 |

100 | $-0.11$ | 0.40 | 0.49 |

**Table 4.**Maximum figure-of-merit $z{T}_{\mathrm{max}}$ and corresponding power factor $\sigma \xb7{\alpha}^{2}$ of some state-of-the-art and emerging thermoelectric materials at temperature T with indication of conduction type.

Material | Type | ${\mathbf{zT}}_{\mathbf{max}}$ | $\mathit{\sigma}\xb7{\mathit{\alpha}}^{2}$ | T | Ref. |
---|---|---|---|---|---|

[$\mathit{\mu}$Wcm${}^{-1}$K${}^{-2}$] | [K] | ||||

(Bi${}_{0.25}$Sb${}_{0.75}$)${}_{2}$Te${}_{3}$ | p | 1.05 | 43 | 323 | [70] |

FeNb${}_{0.8}$Ti${}_{0.2}$Sb | p | 1.10 | 53 | 973 | [48,71] |

Hf${}_{0.6}$Zr${}_{0.4}$Hf${}_{0.25}$NiSn${}_{0.995}$Sb${}_{0.005}$ | n | 1.20 | 47 | 900 | [48,72] |

Bi${}_{2}$(Te${}_{0.94}$Se${}_{0.06}$)${}_{3}$ (0.017 wt.% Te, 0.068 wt.% I) | n | 1.25 | 57 | 298 | [73] |

(Bi${}_{0.25}$Sb${}_{0.75}$)${}_{2}$Te${}_{3}$ (8wt.% Te) | p | 1.27 | 58 | 298 | [73] |

nano (Bi${}_{0.25}$Sb${}_{0.75}$)${}_{2}$Te${}_{3}$ | p | 1.4 | 38 | 373 | [70] |

ZrCoBi${}_{0.65}$Sb${}_{0.15}$Sn${}_{0.20}$ | p | 1.42 | 38 | 973 | [48,74] |

FeNb${}_{0.88}$Hf${}_{0.12}$Sb | p | 1.45 | 51 | 1200 | [48,75] |

Bi${}_{0.88}$Ca${}_{0.06}$Pb${}_{0.06}$CuSeO | p | 1.5 | 8 | 873 | [48,76] |

$\beta $-Cu${}_{2-x}$Se | p | 1.5 | 12 | 1000 | [77] |

Ti${}_{0.5}$Zr${}_{0.25}$Hf${}_{0.25}$NiSn${}_{0.998}$Sb${}_{0.002}$Se | n | 1.5 | 62 | 700 | [48,78] |

Mg${}_{3}$Sb${}_{1.48}$Bi${}_{0.4}$Te${}_{0.04}$ | n | 1.65 | 13 | 725 | [79] |

Ba${}_{0.08}$La${}_{0.05}$Yb${}_{0.04}$Co${}_{4}$Sb${}_{12}$ | n | 1.7 | 51 | 850 | [80] |

Mg${}_{3.175}$Mn${}_{0.025}$Sb${}_{1.5}$Bi${}_{0.49}$Te${}_{0.01}$ | n | 1.71 | 20 | 700 | [48,81] |

B-doped Si${}_{80}$Ge${}_{20}$ + YSi${}_{2}$ | p | 1.81 | 39 | 1073 | [48,82] |

Cu${}_{2-y}$S${}_{1/3}$Se${}_{1/3}$Te${}_{1/3}$ | p | 1.9 | 8 | 1000 | [83] |

AgPb${}_{m}$SbTe${}_{2+m}$ | n | 2.2 | 11 | 800 | [84] |

PbTe${}_{0.7}$S${}_{0.3}$-2.5%K | p | 2.2 | 14 | 923 | [68] |

PbTe-4%SrTe-2%Na | p | 2.2 | 24 | 915 | [85] |

Ge${}_{0.89}$Sb${}_{0.1}$In${}_{0.01}$Te | p | 2.3 | 37 | 650 | [86] |

PbTe-8%SrTe | p | 2.5 | 30 | 923 | [87] |

SnSe single crystal’s b-axis | p | 2.6 | 10 | 923 | [88] |

$\beta $-Cu${}_{2}$Se/CuInSe${}_{2}$ (1% In) | p | 2.6 | 12.5 | 850 | [89] |

SnSe${}_{0.97}$Br${}_{0.03}$ single crystal’s a-axis | n | 2.8 | 9 | 773 | [90] |

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Feldhoff, A.
Power Conversion and Its Efficiency in Thermoelectric Materials. *Entropy* **2020**, *22*, 803.
https://doi.org/10.3390/e22080803

**AMA Style**

Feldhoff A.
Power Conversion and Its Efficiency in Thermoelectric Materials. *Entropy*. 2020; 22(8):803.
https://doi.org/10.3390/e22080803

**Chicago/Turabian Style**

Feldhoff, Armin.
2020. "Power Conversion and Its Efficiency in Thermoelectric Materials" *Entropy* 22, no. 8: 803.
https://doi.org/10.3390/e22080803