In analyzing zT improvements due to lattice thermal conductivity (κL) reduction, electrical conductivity (σ) and total thermal conductivity (κTotal) are often used to estimate the electronic component of the thermal conductivity (κE) and in turn κL from κL = ∼ κTotalLσT. The Wiedemann-Franz law, κE = LσT, where L is Lorenz number, is widely used to estimate κE from 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 | 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, Si0.8Ge0.2 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 z T = S 2 σ T κ E + κ L , where T, S, σ, κE, and κL are the temperature, Seebeck coefficient, electrical conductivity, and the electronic and lattice contributions to the thermal conductivity, respectively. Because the charge carriers (electrons in n-type or holes in p-type semiconductors) transport both heat and charge, κE is commonly estimated using the measured σ using the Wiedemann-Franz law: κE = LσT, where L is the Lorenz number. Once κE is known, κL is computed by subtracting the κE from the total thermal conductivity, κTotal = κE + κL. For this method, the bipolar thermal conductivity (κB) will also be included which can be written κB + κL = κTotalLσT.

Since a high zT requires low κTotal but high σ simultaneously, one of the more popular routes towards improving zT has been to reduce κL.1 However, depending on the value of L, which maps from σ to κE, the resulting κL can often be misleading. For instance, in the case of lanthanum telluride, incautious determination of L can even cause κL to be negative, which is not physical.2 Therefore, careful evaluation of L is critical in characterizing enhancements in zT due to κL reduction.

For most metals, where charge carriers behave like free-electrons, L converges to π 2 3 k B e 2 = 2 . 44 × 1 0 8 W Ω K 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) regions3 (ξoptimum 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 L5 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, nH, 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/kBT, where kB is Boltzmann constant); thus, no explicit knowledge of temperature (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 κL for a few known thermoelectric materials including: PbTe,8–10 Zintl materials,11–13 co-doped FeV 0.6Nb0.4Sb Half Heusler,14 La3−xTe4,2 resulting in much more satisfactory values for κL than the degenerate limit result (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 κTotal is 10% and that SPB-APS is only an approximation, a much simpler equation would supply sufficient accuracy. Here, we propose the equation

L = 1 . 5 + exp | S | 116
(1)

(where L is in 10−8 WΩK−2 and S in μV/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 (Sr3GaSb3),11 Half Heusler (ZrNiSn),19 and Si0.8Ge0.220).

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

L = k B e 2 1 + λ 3 + λ F λ η F λ + 2 η 2 + λ 2 F λ + 1 η 2 1 + λ 2 F λ η 2 ,
(2)
S = k B e 2 + λ F λ + 1 η 1 + λ F λ η η .
(3)

Where F j η represents the Fermi integral,

F j η = 0 ϵ j d ϵ 1 + Exp ϵ η .
(4)

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 temperature17,21), 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 μV/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 μV/K. For |S| < 10 μV/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: α = k B T E g (Eg 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, κL estimates will maintain the order: κL,deg < κL,SPBAPS < κL,SKBAPS with the largest errors being for the degenerate limit when applied in the non-degenerate case.22 

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 (Si0.8Ge0.2 in Table I)–although limited data is available. For ZrNiSn0.99Sb0.01 (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 mechanism7 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 PbTe0.85Se0.15 and n-type Si0.8Ge0.2, 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. κL calculated from the difference between κTotal and κE 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 temperatures27).

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.

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Supplementary Material