Mid-temperature thermoelectric conversion efficiencies of the IV-VI materials were calculated under the Boltzmann transport theory of carriers, taking the Seebeck, Peltier, and Thomson effects into account. The conversion efficiency was discussed with respect to the lattice thermal conductivity, keeping other parameters such as Seebeck coefficient and electrical conductivity to the same values. If room temperature lattice thermal conductivity is decreased up to 0.5W/mK, the conversion efficiency of a PbS based material becomes as high as 15% with the temperature difference of 500K between 800K and 300K.

## I. INTRODUCTION

Development of materials with high thermoelectric conversion efficiency is important for energy and environmental issues. Dimensionless figure of merit *ZT* = *S*^{2}*σT*/*κ*, where *S* is the Seebeck coefficient, *σ* the electrical conductivity, and *κ* the thermal conductivity, is used as an indicator of the conversion efficiency. Among the parameters, thermal conductivity is an important parameter because the reduction of thermal conductivity increases the *ZT* and also reduces the necessary amount of the material for device fabrication. Seebeck coefficient is another important factor because it has the effect of second power on the figure of merit. Narrow gap IV-VI semiconductors such as PbTe-, PbS-, and GeTe-based materials have been studied as high efficiency thermoelectric materials, because they have small lattice thermal conductivity, and relatively high Seebeck coefficient and electrical conductivity.^{1–7} Introduction of a superlattice or a nanostructure into the materials can decrease the thermal conductivity keeping a high electrical conductivity.^{8–10} Thus many nanostructure thermoelectric materials have been reported recently. In this paper we describe the calculation of thermoelectric conversion efficiency, and discuss how high conversion efficiency is possible by decreasing the lattice thermal conductivity. A calculation of conversion efficiency had been discussed by Snyder et al.^{11} and a conventional relation to estimate conversion efficiency from the thermoelectric figure of merit had been derived. However, there are some difficulties to apply the equation in general cases. The thermoelectric conversion is the result of Peltier effect and Seebeck effect, and the conversion efficiency can be exactly calculated from the thermal conductivity, Joule heating and Thomson effect inside the materials with the former two thermoelectric effects. In this paper, relations among these effects will be clearly described, and thermoelectric-conversion efficiencies of IV-VI materials are discussed with the temperature dependent Seebeck coefficient, electrical conductivity, and lattice thermal conductivity.

## II. CALCULATION OF THERMOELECTRIC CONVERSION EFFICIENCY

Under Boltzmann transport theory, electrical conductivity *σ* and Seebeck coefficient *S* are given by^{12,13}

where *k _{B}* is the Boltzmann constant,

*T*the absolute temperature,

*f*

_{0}the Fermi-Dirac distribution,

*τ*the electron scattering time,

*v*the electron velocity along temperature gradient,

_{x}*E*the Fermi level,

_{F}*E*(

**k**) the electron energy in wavevector space, and $ E \u0304 $ is an average electron energy related to the electronic conduction. The integral is performed in wavevector space

**k**and summation is performed for all valleys related to the electrical conduction. Using Eq. (1) as electrical conductivity and considering that each electron carries the energy

*E*-

*E*, energy flow inside the material by current

_{F}*I*is just given by

*TSI*. Thus Peltier coefficient

*Π*is given by

*Π*=

*ST*. Thomson effect is a phenomenon of heating or cooling inside a material by electric current under the temperature gradient, and the Thomson coefficient

*μ*is related to the Seebeck coefficient as

_{T}*μ*=

_{T}*TdS*/

*dT*.

In thermoelectric power generation, these three effects affect the efficiency although other two coefficients are obtained from the Seebeck coefficient. Figure 1 shows energy band diagram of a Seebeck device, and also carrier and energy flows. In the lead-salt IV-VI material such as PbS or PbTe, the band gap increases with the temperature. At high temperature contact with temperature *T _{H}*, electrons absorb energy and electron flow from valence band in p-type semiconductor to conduction band in n-type semiconductor occurs causing a Peltier cooling at the high temperature contact. In the conduction band of the

*n*-type region and valence band of the

*p*-type region, electrons lose some of the average energy $ E \u0304 $, resulting in Thomson heating in the two regions. At low temperature contact, Peltier heating proportional to the current flow occurs. The difference of Fermi level between the two contacts at the low temperature sides corresponds to the output energy per one electron flow. Figure 2 shows graphical energy relations of the Peltier, Thomson, and Seebeck effects. The rectangular area formed by areas A, B, and C corresponds to Peltier cooling at the high temperature contact. The areas B and C correspond to Peltier heating at the low temperature contact and Thomson heating inside the semiconductor, respectively. The summation of Seebeck output power and Joule heating inside the semiconductor corresponds to the area A. If thermal conductivity of the material is zero, the ratio of areas A/(A+ B+ C) gives the maximum conversion efficiency under infinitesimally small output current.

Real semiconductor has a thermal conductivity which decreases the efficiency of the thermoelectric conversion. The thermal conductivity has mainly two components; lattice thermal conductivity *κ _{L}* and carrier thermal conductivity

*κ*. Infrared radiation also affects the thermal conduction, but this effect is ignored in this discussion. The carrier thermal conductivity is calculated under the Boltzmann transport framework and given in the form

_{E}^{13,14}:

where σ is the electrical conductivity given by Eq. (1). The carrier thermal conductivity includes the thermal conductivity due to majority carriers and that due to bipolar transport which increases significantly at high temperature. For carrier mobility of bulk or epitaxial IV-VI materials proportional to T^{−α} (α =2-2.5), the thermal conductivity due to majority carriers is approximately proportional to T^{−β } and the value of β is smaller than α (β = α-1 in Wiedemann-Franz law) because average energy carried by one electron increases with temperature. In bipolar transport, electron and hole pair carriers an energy as high as E_{g} and the current density of the minority carriers is approximately proportional to diffusion current ignoring the effect of weak electric field. Thus thermal flow *J*_{E,bipolar} by the bipolar transport is given by

where *N _{C}* and

*N*are effective densities of states in conduction and valence bands, respectively, and

_{V}*N*is majority donor or acceptor concentration,

_{I}*D*is the diffusion coefficient of the minority carrier. Thus thermal conductivity may be written in the form:

where *κ _{L0}* and

*κ*are room-temperature lattice and majority-carrier thermal conductivities, respectively, and

_{E0}*κ*is a constant for the bipolar thermal conductivity. The index

_{E1}*γ*is a fitting parameter comes from the temperature dependence of effective densities of states

*N*,

_{C}*N*, and diffusion coefficient

_{V}*D*, etc. Thermoelectric-power efficiency

*η*is calculated by the equation:

where *P* is the output electric power, and *Q* is the heat lost at the high temperature side. The value *Q* is calculated from the sum of Peltier cooling and thermal conduction at high temperature side or the sum of Peltier heating and thermal conduction at low temperature side and Seebeck output *P*.

Figure 3(a) shows theoretical Seebeck coefficient calculated for n-type PbS (blue solid circle and blue solid triangle), n-type PbTe (red solid square and red solid triangle) and p-type PbTe (red open circle and red open triangle) with carrier concentrations 5 × 10^{18} cm^{−3} and 2 × 10^{19} cm^{−3}, respectively. Detail calculations of the Seebeck coefficient and carrier thermal conductivity are described in previous papers.^{12–14} The direct bandgaps of PbS and PbTe are 410 and 320 meV at 300K, respectively, and the bandgaps increase by 0.5meV/K with the temperature. In the calculation, we assumed a density of states effective mass 0.125m_{0} for each L-point valleys of both conduction and valence bands in PbS, and 0.089m_{0} for each L-point valleys in PbTe at 300K. The effective masses increases with temperature proportional to the band gap.^{15} In the PbTe, we assumed additional twelve indirect valence band maxima with temperature independent band gap of 370meV along Σ axes in the Brillouin zone.^{16} The density of states effective mass of each indirect valley was assumed to be 0.069m_{0}, which corresponds to a total density of states effective mass of 0.36m_{0} for the twelve indirect valleys. Seebeck coefficient of p-type PbTe is higher than that of n-type PbTe owing to the existence of the indirect valleys near the L-point valence band maxima.

Figure 3(b) shows theoretical carrier thermal conductivity calculated for n-type PbS (blue solid circle, and blue solid triangle), n-type PbTe (red solid square, and red solid triangle) and p-type PbTe (red open circle, and red open triangle) with the carrier concentration of 5 × 10^{18} cm^{−3} and 2 × 10^{19} cm^{−3}, respectively . The solid lines are fitted ones using Eq. (5) for n-type PbS and PbTe with carrier concentration 5 × 10^{18} cm^{−3}. Table I shows mobility parameters of majority and minority carriers to calculate the carrier thermal conductivity and parameters used for the fitting curves: The μ_{0} and μ_{1} represent majority- and minority-carrier mobilities at 300K, respectively, and α the index for the temperature dependence. These values were determined from experimental values for the PbS and PbTe films. For n-type PbTe and PbS, theoretical carrier thermal conductivities calculated by Eq. (3) are well reproduced by Eq. (5), and the index *β* = 1.7 which represents temperature dependence of the majority-carrier thermal conductivity was slightly higher than the value expected by Wiedemann-Franz law (*β* = *α*-1=1.5). The enhancement of carrier thermal conductivity of n- or p-type PbTe at high temperature is due to bipolar transport and the enhancement is much higher than that of PbS owing to narrow indirect band gap of the PbTe.

. | n
. | μ_{0}
. | μ_{1}
. | . | κ_{E0}
. | . | κ_{E1}
. | . |
---|---|---|---|---|---|---|---|---|

Material . | [cm^{−3}]
. | [cm^{2}/Vs]
. | [cm^{2}/Vs]
. | α
. | [W/mK] . | β
. | [W/mK] . | γ
. |

n-PbS | 5 × 10^{18} | 500 | 500 | 2.5 | 0.193 | 1.7 | 105 | -1.5 |

n-PbS | 2 × 10^{19} | 300 | 300 | 2.5 | - | - | - | - |

n-PbTe | 5 × 10^{18} | 1,500 | 750 | 2.5 | 0.604 | 1.7 | 80 | 0 |

n-PbTe | 2 × 10^{19} | 800 | 400 | 2.5 | - | - | - | - |

p-PbTe | 5 × 10^{18} | 750 | 1500 | 2.5 | - | - | - | - |

p-PbTe | 2 × 10^{19} | 400 | 800 | 2.5 | - | - | - | - |

. | n
. | μ_{0}
. | μ_{1}
. | . | κ_{E0}
. | . | κ_{E1}
. | . |
---|---|---|---|---|---|---|---|---|

Material . | [cm^{−3}]
. | [cm^{2}/Vs]
. | [cm^{2}/Vs]
. | α
. | [W/mK] . | β
. | [W/mK] . | γ
. |

n-PbS | 5 × 10^{18} | 500 | 500 | 2.5 | 0.193 | 1.7 | 105 | -1.5 |

n-PbS | 2 × 10^{19} | 300 | 300 | 2.5 | - | - | - | - |

n-PbTe | 5 × 10^{18} | 1,500 | 750 | 2.5 | 0.604 | 1.7 | 80 | 0 |

n-PbTe | 2 × 10^{19} | 800 | 400 | 2.5 | - | - | - | - |

p-PbTe | 5 × 10^{18} | 750 | 1500 | 2.5 | - | - | - | - |

p-PbTe | 2 × 10^{19} | 400 | 800 | 2.5 | - | - | - | - |

In p-type PbTe, the carrier thermal conductivity at low temperature region cannot be fitted by the Eq. (5) because the majority-carrier thermal conductivity has a multiple-valley effect similar to the bipolar transport. Since there is certain energy separation ΔE between the L-point valence-band edge and the indirect valley, the majority carrier can bring the energy proportional to the separation ΔE by the multiple-valley transport. In bipolar transport, a small number of minority carriers can bring a large energy corresponding to the bandgap E_{g}, and the carrier thermal conductivity enhances significantly at high temperature. On the other hand, relatively small energy separation between the valence-band valleys causes the multiple-valley transport at low temperature. Thus the slope of thermal conductivity in the logarithmic scale increases below the crossing temperature (β>1.7 below 400K), and the slope decreases above 400K, corresponding to β <1.7 as shown in Fig. 3(b).

## III. CALCULATED RESULTS AND DISCUSSIONS

Thermoelectric figure of merit, thermoelectric conversion efficiencies, and output powers were calculated for the n-type PbS, n-type PbTe, and p-type PbTe. Figure 4 shows the temperature dependence of theoretical figure of merit *ZT* for (a) n-type PbS, (b) n-type PbTe, and (c) p-type PbTe with carrier concentration 5 × 10^{18} cm^{−3} with a parameter of lattice thermal conductivity. PbS and PbTe have lattice thermal conductivity of 3 W/mK and 2 W/mK at room temperature, and the lattice thermal conductivities are inversely proportional to the absolute temperature. Therefore, room-temperature lattice thermal conductivities used for the calculation were 3, 1, 0.5, and 0 W/mK for the PbS, and 2, 1, 0.5, and 0 W/mK for the PbTe, and lattice thermal conductivities at higher temperature were assumed to be inversely proportional to the absolute temperature. If we can decrease lattice thermal conductivity keeping electrical conductivity and Seebeck coefficient, the thermoelectric figure of merit increases significantly. However, the enhancement of *ZT* for PbTe with decreasing the lattice thermal conductivity is smaller than that for PbS at high temperature because of higher carrier thermal conductivity in PbTe due to bipolar transport.

Figure 5 shows theoretical conversion efficiency of (a) n-type PbS, (b) n-type PbTe, and (c) p-type PbTe with carrier concentration 5 × 10^{18} cm^{−3} with a parameter of lattice thermal conductivity and thermoelectric output power. In the calculation, the temperatures at high and low temperature sides were *T _{H}* =800K and

*T*=300K, respectively, with a temperature difference of 500K, and thickness of the device was 5 mm. The device thickness does not affect the maximum efficiency but it affects maximum output power because the resistivity of the device depends on the thickness. The maximum conversion efficiency calculated for the n-type PbS, n- and p-type PbTe were 4.7%, 10.3% and 5.7%, respectively, and the ultimate efficiencies assumed zero lattice thermal conductivity were 29.4%, 22.7%, and 14.5%, respectively. Although it is impossible to decrease the lattice thermal conductivity to zero, the ultimate efficiency is a useful indicator to give an upper limit of conversion efficiency. If room-temperature lattice thermal conductivity is decreased to 0.5W/mK, a large conversion efficiency as high as 15% is expected even for PbS. The output current at maximum efficiency decreases with decreasing the lattice thermal conductivity owing to the temperature and resultant resistivity increases inside the material because of the weak dissipation of the Joule and Thomson heats.

_{L}Figure 6(a) shows the dependence of maximum efficiency on carrier concentration for n-type PbS and n- and p-type PbTe, and (b) shows output power at the maximum efficiency operations. We assumed the carrier mobility listed in TABLE I, which decreases with carrier concentration and temperature. If the mobilities are kept at high values in the carrier concentration region above 1 × 10^{19} cm^{−3}, higher conversion efficiencies are expected for all the materials. The conversion efficiency significantly increases in the p-type PbTe at the high carrier concentration region as shown in the Fig. 6(a). It is due to the large density of states in the valence band comes from indirect valleys, and also due to the decrease of carrier thermal conductivity by the reduction of bipolar-transport effect with increasing the carrier concentration.

## IV. CONCLUSIONS

Thermoelectric conversion efficiencies of IV-VI compounds were calculated from the theoretical Seebeck coefficient and carrier thermal conductivity under the Boltzmann transport theory, and effects of lattice and carrier thermal conductivities on the conversion efficiency were discussed. The carrier thermal conductivity of p-type PbTe deviates strongly from Wiedemann-Franz law owing to the existence of two kinds of majority carriers at low temperature side. An enhancement of the carrier thermal conductivity by bipolar transport also appears significantly in PbTe at high temperature side, owing to the narrow indirect band gap. IV-VI compounds such as PbTe and PbS have small lattice thermal conductivity as low as 2∼3 [W/mK], and thermoelectric conversion efficiency of 4∼10% with mid-temperature region. Farther decrease of the lattice thermal conductivity is expected by the introduction of quantum wells or nanostructures. The conversion efficiency increases significantly by decreasing the lattice thermal conductivity. If room temperature lattice thermal conductivity is decreased up to 0.5W/mK, the conversion efficiency of PbS and PbTe based material becomes as high as 15% with the temperature difference of 500K between 800K and 300K.