In analyzing *zT* improvements due to lattice thermal conductivity (*κ _{L}*) reduction, electrical conductivity (

*σ*) and total thermal conductivity (

*κ*) are often used to estimate the electronic component of the thermal conductivity (

_{Total}*κ*) and in turn

_{E}*κ*from

_{L}*κ*= ∼

_{L}*κ*−

_{Total}*LσT*. The Wiedemann-Franz law,

*κ*=

_{E}*LσT*, where

*L*is Lorenz number, is widely used to estimate

*κ*from

_{E}*σ*measurements. It is a common practice to treat

*L*as a universal factor with 2.44 × 10

^{−8}WΩK

^{−2}(degenerate limit). However, significant deviations from the degenerate limit (approximately 40% or more for Kane bands) are known to occur for non-degenerate semiconductors where

*L*converges to 1.5 × 10

^{−8}WΩK

^{−2}for acoustic phonon scattering. The decrease in

*L*is correlated with an increase in thermopower (absolute value of Seebeck coefficient (

*S*)). Thus, a first order correction to the degenerate limit of

*L*can be based on the measured thermopower, |

*S*|, independent of temperature or doping. We propose the equation: $L=1.5+exp \u2212 | S | 116 $ (where

*L*is in 10

^{−8}WΩK

^{−2}and

*S*in μV/K) as a satisfactory approximation for

*L*. This equation is accurate within 5% for single parabolic band/acoustic phonon scattering assumption and within 20% for PbSe, PbS, PbTe, Si

_{0.8}Ge

_{0.2}where more complexity is introduced, such as non-parabolic Kane bands, multiple bands, and/or alternate scattering mechanisms. The use of this equation for

*L*rather than a constant value (when detailed band structure and scattering mechanism is not known) will significantly improve the estimation of lattice thermal conductivity.

A semiconductor with large Seebeck coefficient, high electrical conductivity, and low thermal conductivity is a good candidate for a thermoelectric material. The thermoelectric material’s maximum efficiency is determined by its figure of merit $zT= S 2 \sigma T \kappa E + \kappa L $, where *T*, *S*, *σ*, *κ _{E}*, and

*κ*are the temperature, Seebeck coefficient, electrical conductivity, and the electronic and lattice contributions to the thermal conductivity, respectively. Because the charge carriers (electrons in

_{L}*n*-type or holes in

*p*-type semiconductors) transport both heat and charge,

*κ*is commonly estimated using the measured

_{E}*σ*using the Wiedemann-Franz law:

*κ*=

_{E}*LσT*, where

*L*is the Lorenz number. Once

*κ*is known,

_{E}*κ*is computed by subtracting the

_{L}*κ*from the total thermal conductivity,

_{E}*κ*=

_{Total}*κ*+

_{E}*κ*. For this method, the bipolar thermal conductivity (

_{L}*κ*) will also be included which can be written

_{B}*κ*+

_{B}*κ*=

_{L}*κ*−

_{Total}*LσT*.

Since a high *zT* requires low *κ _{Total}* but high

*σ*simultaneously, one of the more popular routes towards improving

*zT*has been to reduce

*κ*.

_{L}^{1}However, depending on the value of

*L*, which maps from

*σ*to

*κ*, the resulting

_{E}*κ*can often be misleading. For instance, in the case of lanthanum telluride, incautious determination of

_{L}*L*can even cause

*κ*to be negative, which is not physical.

_{L}^{2}Therefore, careful evaluation of

*L*is critical in characterizing enhancements in

*zT*due to

*κ*reduction.

_{L}For most metals, where charge carriers behave like free-electrons, *L* converges to $ \pi 2 3 k B e 2 =2.44\xd71 0 \u2212 8 W\Omega K \u2212 2 $ (degenerate limit). Although some heavily doped semiconductor thermoelectric materials have an *L* very close to the degenerate limit, properly optimized materials often have charge carrier concentrations between the lightly doped (non-degenerate) and heavily doped (degenerate) regions^{3} (*ξ _{optimum}* is near the band edge where

*ξ*is the electronic chemical potential) which can result in errors of up to ∼40%.

^{4}

Direct measurement of *L*^{5} requires high mobility—typically beyond that attainable at the temperatures of interest (>300 K). Thus, *L* is typically estimated either as a constant (2.44 × 10^{−8} WΩK^{−2}) or by applying a transport model—such as the single parabolic band (SPB) model obtained by solving the Boltzmann transport equations—to experimental data.

For example, Larsen *et al.* proposed an approximate analytical solution of *L* based on the SPB model as a function of carrier concentration (*n*) and (*m*^{*}*T*)^{−3/2} (where *m*^{*} is the effective mass) along with various sets of parameters for distinct carrier scattering mechanisms.^{6} However, when the Hall carrier concentration, *n _{H}*, of a material is not available, the use of the approximate solution by Larsen is not possible. It can be shown that for the SPB model with acoustic phonon scattering (SPB-APS), both

*L*and

*S*are parametric functions of only the reduced chemical potential (

*η*=

*ξ*/

*k*, where

_{B}T*k*is Boltzmann constant); thus, no explicit knowledge of temperature (

_{B}*T*), carrier concentration (

*n*), or effective mass (

*m*

^{*}) is required to relate them.

^{7}We have utilized this correlation between

*L*and measured

*S*to estimate

*κ*for a few known thermoelectric materials including: PbTe,

_{L}^{8–10}Zintl materials,

^{11–13}co-doped FeV

_{0.6}Nb

_{0.4}Sb Half Heusler,

^{14}La

_{3−x}Te

_{4},

^{2}resulting in much more satisfactory values for

*κ*than the degenerate limit result (

_{L}*L*= 2.44 × 10

^{−8}WΩK

^{−2}) would have.

While the SPB model works well to estimate *L*, a transcendental set of equations is needed to solve for *L* in terms of *S*—requiring a numerical solution. Considering that the typical measurement uncertainty for *κ _{Total}* is 10% and that SPB-APS is only an approximation, a much simpler equation would supply sufficient accuracy. Here, we propose the equation

(where *L* is in 10^{−8} WΩK^{−2} and *S* in *μ*V/K) as a satisfactory approximation for *L*.

Equation (1) allows for a facile estimation of *L* from an experimental *S* only without requiring a numerical solution. We characterize the effectiveness of this estimate for *L* using some experimental data from relevant thermoelectric materials (PbSe,^{15} PbS,^{16} PbTe,^{17,18} Zintl material (Sr_{3}GaSb_{3}),^{11} Half Heusler (ZrNiSn),^{19} and Si_{0.8}Ge_{0.2}^{20}).

For a single parabolic band, *L* and *S* are both functions of reduced chemical potential (*η*) and carrier scattering factor (λ) only

Where $ F j \eta $ represents the Fermi integral,

By assuming that the carrier relaxation time is limited by acoustic phonon scattering (one of the most relevant scattering mechanisms for thermoelectric materials above room temperature^{17,21}), Eqs. (2) and (3) can be solved numerically for *L* and the corresponding *S* as shown in Fig. 1 along with the proposed approximation (Eq. (1)).

According to the Fig. 1, the degenerate limit of *L* (2.44 × 10^{−8} WΩK^{−2}) is valid with errors less than 10% for materials whose thermopower is smaller than 50 *μ*V/K (highly degenerate). In contrast, if the thermopower is large, the discrepancy with the degenerate limit can be up to 40%.

To decide an appropriate value of *L* with a known *S* easily, rather than graphically extracting it from Fig. 1, Eq. (1) can be used to quickly estimate *L*, given a measured thermopower. Equation (1) is accurate within 5% for single parabolic band where acoustic phonon scattering is dominant scattering mechanism when |*S*| > ∼ 10 *μ*V/K. For |*S*| < 10 *μ*V/K, while the SPB model converges to the degenerate limit, Eq. (1) increases exponentially, thus reducing the accuracy of the Eq. (1). Although estimation of *L* with an accuracy within 0.5% for SPB-APS is possible, this requires an approximate equation more complex than Eq. (1).^{22}

Exceptions are known where *L* has been found to be outside the uncertainty described above for SPB-APS which are presented in Fig. 2 and Table I.^{22} These exceptions typically involve either non-parabolic band structures (PbTe, PbSe, and PbS) or alternative scattering mechanisms (other than acoustic phonons). Narrow-gap semiconductors (lead chalcogenides, for example) are often better described by the non-parabolic Kane model which yields a different *η* dependence of *L* and *S* which depends on the non-parabolicity parameter: $\alpha = k B T E g $ (*E _{g}* is the gap between conduction and valence band).

^{23,24}For well-studied lead chalcogenides (PbTe, PbSe, and PbS), a reasonable range of

*α*is from 0.08 (300 K) to 0.16 (850 K).

^{25}Figure 2 shows that

*L*is at most ∼26% lower than that of the SPB-APS and Eq. (1) results over the entire range of temperatures. In other words,

*κ*estimates will maintain the order:

_{L}*κ*

_{L,deg}<

*κ*

_{L,SPB−APS}<

*κ*

_{L,SKB−APS}with the largest errors being for the degenerate limit when applied in the non-degenerate case.

^{22}

Band^{a}
. | Scattering^{b}
. | Examples . | Maximum error (%) . |
---|---|---|---|

P | AP | Sr_{3}Ga_{0.93}Zn_{0.07}Sb_{3}^{11} | 4.4 |

2P | AP + II | Si_{0.8}Ge_{0.2}^{20} | 7.5 |

K | AP | PbTe_{0.9988}I_{0.0012}^{18} | 19.7 |

K | AP + PO | Pb_{1.002}Se_{0.998}Br_{0.002}^{15} | 19.5 |

PbS_{0.9978}Cl_{0.0022}^{16} | 19.4 | ||

K | AP + PO + AL | ZrNiSn_{0.99}Sb_{0.01}^{19} | 25.6 |

2K + P | AP | PbTe_{0.85}Se_{0.15}^{17} | 14.9 |

Band^{a}
. | Scattering^{b}
. | Examples . | Maximum error (%) . |
---|---|---|---|

P | AP | Sr_{3}Ga_{0.93}Zn_{0.07}Sb_{3}^{11} | 4.4 |

2P | AP + II | Si_{0.8}Ge_{0.2}^{20} | 7.5 |

K | AP | PbTe_{0.9988}I_{0.0012}^{18} | 19.7 |

K | AP + PO | Pb_{1.002}Se_{0.998}Br_{0.002}^{15} | 19.5 |

PbS_{0.9978}Cl_{0.0022}^{16} | 19.4 | ||

K | AP + PO + AL | ZrNiSn_{0.99}Sb_{0.01}^{19} | 25.6 |

2K + P | AP | PbTe_{0.85}Se_{0.15}^{17} | 14.9 |

^{a}

Band is the type and number of bands involved in evaluating *L*. For instance, “2K + P” means two non-parabolic Kane bands (K) and a parabolic band (P).

^{b}

Scattering is the type of scattering mechanism assumed in estimating *L*. AP, II, PO, and AL are acoustic phonon, ionized impurities, polar, and alloy scattering, respectively. For example, “AP + PO” means that both acoustic phonon and polar scatterings are assumed in calculating *L*.

Alternative scattering mechanisms can also yield deviations from the SPB-APS. For example, when ionized impurity scattering dominates (λ = 2), the *L* actually increases with increasing *S*; however, this example is not particularly prevalent in materials which have high dielectric constants (including the lead chalcogenides)^{26} or at high temperatures. However, when the ionized impurity scattering and acoustic phonon scattering are both considered, the deviation from the SPB-APS is not significant (Si_{0.8}Ge_{0.2} in Table I)–although limited data is available. For ZrNiSn_{0.99}Sb_{0.01} (Table I), acoustic phonon scattering and two other scattering mechanisms (polar and alloy scatterings) are taken into account; these result in a larger deviation as the Seebeck becomes larger. At low temperatures (<100 K), as *S* approaches zero, it is expected that *L* converges to the degenerate limit regardless of carrier scattering mechanism^{7} and parabolicity of bands involved in transport.^{22} However, a pronounced inelastic electron-electron scattering due to high mobility of carriers decreases *L* from the degenerate limit, even for strongly degenerate materials. In case of *n*-type PbTe, *L* at 100 K is approximately 40% lower than its value at 300 K.^{24}

Multiple band behavior (present in *p*-type PbTe_{0.85}Se_{0.15} and *n*-type Si_{0.8}Ge_{0.2}, Fig. 2) can also lead to deviations in the thermopower-dependence of the Lorenz number. In the case of PbTe, hole population of both the light and heavy bands yields a more complicated relationship between *L* and *S*; it is not simply a parametric function of *η* and depends on the specific effective mass and mobility contributions from each band.

One last, prevalent source of error occurs because the Wiedemann-Franz law does not take the bipolar thermal conductivity into consideration. *κ _{L}* calculated from the difference between

*κ*and

_{Total}*κ*does include varying portion of bipolar conduction with respect to temperature and band structure of materials (which can become important for lightly doped materials with narrow gaps at high temperatures

_{E}^{27}).

An equation for *L* entirely in terms of the experimentally determined *S* is proposed and found to be accurate (within 20%) for most common band structures/scattering mechanisms found for thermoelectric materials. Use of this equation would make estimates of lattice thermal conductivity much more accurate without requiring additional measurement. Therefore, *zT* improvement due to lattice thermal conductivity reduction can be calculated with much improved accuracy and access.

The authors would like to acknowledge funding from The Materials Project: supported by Department of Energy’s Basic Energy Sciences program under Grant No. EDCBEE, DOE Contract No. DE-AC02-05CH11231 and as part of the Solid-State Solar-Thermal Energy Conversion Center (S3TEC), an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, and Basic Energy Sciences under Award No. DE-SC0001299.