Thermoelectric Figure of Merit

• The thermoelectric figure of merit (ZT) is a dimensionless metric that quantifies a material's ability to convert heat to electricity by balancing power factor and thermal conductivity.
• This concept is central to advancing thermoelectric devices for power generation and cooling, with innovative strategies leveraging multivalley degeneracy and band complexity.
• Combining ab initio transport modeling with defect engineering, recent studies show that optimized doping in materials like LaSO can push ZT values beyond conventional limits.

The thermoelectric figure of merit, denoted $ZT$, is a dimensionless parameter that quantifies the efficiency of materials for direct thermal-to-electrical energy conversion. The search for high-$ZT$ materials remains central to the development of advanced thermoelectric devices for power generation and solid-state cooling. $ZT$ encapsulates the interplay of electronic structure, scattering mechanisms, and phonon transport, and its enhancement relies on simultaneously optimizing the power factor and minimizing total thermal conductivity. Recent work highlights multivalley degeneracy and band complexity—exemplified by the n-doped lanthanum oxysulphate LaSO system—as potent design strategies, pushing device-relevant $ZT$ well beyond conventionally accepted limits (Farris et al., 2021).

1. Fundamental Definition and Physical Meaning

The figure of merit is defined as

$ZT = \frac{S^2 \sigma T}{\kappa_e + \kappa_L}$

where:

• $S$: Seebeck coefficient (thermopower), in V/K
• $\sigma$: electrical conductivity, in S/m
• $T$: absolute temperature (K)
• $\kappa_e$: electronic contribution to thermal conductivity, W/(m·K)
• $\kappa_L$: lattice (phonon) contribution to thermal conductivity, W/(m·K)

A large $ZT$ signifies efficient thermoelectric conversion, requiring maximization of the numerator (the electrical power output, i.e., the power factor $S^2 \sigma$) and minimization of the denominator (thermal leakage). The Marxwell–Kelvin relation shows that the highest conversion efficiency is reached when both the magnitude of the Seebeck effect and the electrical conductivity are optimized, provided the overall heat conductivity is kept low.

2. Power Factor and the Role of Band Structure

The power factor,

$$\mathrm{PF} = S^2 \sigma \quad [\text{W}/\text{m}\,\text{K}^2]$$

is the direct product of a material’s ability to produce voltage from a thermal gradient ($S$) and to sustain electrical current ($\sigma$). At fixed $T$ and total $\kappa$, $ZT$ scales with the power factor.

Recent ab initio studies on LaSO attribute extraordinarily high $ZT$ values to a large geometric band complexity factor ($C_b$), arising from a multivalleyed band structure. Specifically,

$C_b = N_v K_v$

with $N_v$ the valley degeneracy and $K_v$ a measure of mass-tensor anisotropy. In LaSO, the four lowest conduction bands yield $N_v = 16$ active valleys within ~150 meV above the band edge, and $K_v \simeq 1$ (in-plane isotropy). The empirical scaling

$\mathrm{PF} \sim C^{0.6}$

indicates that materials with high valley multiplicity naturally exhibit enhanced power factors, crucial to achieving large $ZT$ (Farris et al., 2021).

3. Ab Initio Transport Modelling: Bloch–Boltzmann Approach

For reliable prediction of thermoelectric coefficients, electronic structure is computed via DFT-GGA and interpolated on dense $k$-grids using Fourier–Wannier methods. The semiclassical Boltzmann equation is solved in the relaxation-time approximation (RTA), with relaxation time $\tau(E, T, \mu)$ parameterized to account for:

• Acoustic phonon scattering (deformation-potential theory)
• Charged-impurity scattering (Brooks–Herring)
• Polar optical phonon scattering (Fröhlich interaction)

The tensor transport coefficients are then evaluated as: $$\mathcal{L}^{(n)}_{ij} = \frac{1}{V} \int dE\, \left(-\frac{\partial f}{\partial E}\right) (E-\mu)^n\, \Sigma_{ij}(E)\, \tau(E)$$ where $\Sigma_{ij}(E)$ is the velocity-squared DOS. Explicitly,

$$\sigma_{ij} = e^2 \mathcal{L}^{(0)}_{ij}, \quad S_{ij} = -\frac{1}{eT} \frac{\mathcal{L}^{(1)}_{ij}}{\mathcal{L}^{(0)}_{ij}}, \quad \kappa_{e,ij} = \frac{1}{T} \left[\mathcal{L}^{(2)}_{ij} - \frac{(\mathcal{L}^{(1)}_{ij})^2}{\mathcal{L}^{(0)}_{ij}}\right]$$

This workflow allows energy-, chemical potential-, and temperature-dependent relaxation times, capturing the full intricacies of multi-band/multivalley transport in complex materials.

4. Lattice Thermal Conductivity: Phonon Contributions and Modeling

The lattice thermal conductivity $\kappa_L$ is computed using the single-mode relaxation-time approximation (SMA) for phonons, with phonon frequencies and mode Grüneisen parameters provided by density-functional perturbation theory (DFPT). Phonon–phonon scattering rates follow the Bjerg et al. model: $$\tau_{pp}^{-1}(\omega) \sim \gamma^2(\omega)\, \omega^2\, T\, \exp(-\Theta_D/T)\, /\, v_s^2$$ where $\gamma(\omega)$ is the mode Grüneisen parameter, $v_s$ sound velocity, $\Theta_D$ Debye temperature. The analytic Debye-model for phonon DOS is used, optionally incorporating Casimir (boundary) and point-defect scattering.

Uncertainty due to Debye temperature choice is bracketed by two reference points: $\Theta_D = 215\,\text{K}$ (higher $\kappa_L$) and $\Theta_D = 335\,\text{K}$ (lower $\kappa_L$), bounding the predicted $ZT$ (Farris et al., 2021).

5. Numerical Results: LaSO and the Dependence of $ZT$ on Material Parameters

For n-doped LaSO, the following summary characterizes the interplay between doping, temperature, and $ZT$:

Temperature (K)	Optimal Carrier Concentration ($10^{20}$ cm$^{-3}$)	$ZT$
400	0.5	~1.5
600	1.0	~3
800	2.0	~4.5
1100	3.5	3.5–6.5

• Maximal in-plane power factor $\sim 15$ mW/(m K$^2$) at 400 K; decreases moderately at higher temperatures.
• At 1100 K, $ZT$ reaches 6.5 for low $\kappa_L$, 3.5 for high $\kappa_L$, superior to most known oxide/sulfide thermoelectrics.
• $ZT > 2$ persists even at modest doping (~$2\times10^{19}$ cm$^{-3}$) at 800 K.

A high geometric band complexity ($C_b \sim 16$) is the principal driver of electronic transport superiority; the moderately low phonon conductivity ensures that gains in power factor are fully expressed in $ZT$.

6. Doping Strategies and Defect Engineering

Efficient $n$-type doping is imperative to reach the optimal carrier concentrations. Among substitutional candidates on La sites, Hf shows shallow donor characteristics (+0.05 eV above the CBM), superior to Zr and decisively better than Ce (which forms deep, inactive $f$-state levels). Ab initio simulation suggests that under O-lean conditions ($p_{{\rm O}_2} \lesssim 10^{-6}$ atm at 1000 K), Hf can deliver carrier densities $n \sim 10^{20}$ cm$^{-3}$ at 700 K—fully sufficient to reach the high-$ZT$ regime. An important nuanced mechanism is that, due to the spatial separation of Hf donors (La–O layers) and S-derived conduction-band states (S–S layers), impurity scattering is potentially mitigated, preserving electron mobility (Farris et al., 2021).

7. Significance, Outlook, and Materials Design Implications

The demonstration of $ZT \gtrsim 6$ at high temperature in a bulk oxide/sulfide marks a paradigm shift in thermoelectric materials design. The key controlling parameter is now recognized as the “band complexity factor,” a combination of multivalley degeneracy and controlled effective mass anisotropy. This insight, validated by ab initio transport theory, directs the focus of materials discovery to compounds exhibiting extensively degenerate conduction-band minima and favorable mass-tensor characteristics.

This strategy—amplifying electronic structure-driven power factors while managing but not necessarily minimizing phonon conductivity—contrasts with previous singular emphasis on extreme phonon glass formation or one-dimensional nanostructuring. The identification of optimal dopants (e.g., Hf for LaSO) is a necessary step for realizing predicted device performance.

These principles are directly generalizable to other families where multivalley degeneracy is accessible (e.g., via chemical engineering, strain, or low-dimensional structure) and have motivated renewed investigation of oxyselenides, layered chalcogenides, and 2D semiconductors as high-$ZT$ candidates.

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References

• Farris, D. F., Kim, B., Barbier, L., et al., "Giant thermoelectric figure of merit in multivalley high-complexity-factor LaSO" (Farris et al., 2021).