Dimensionless Figure of Merit

• Dimensionless figures of merit are scale-independent ratios that combine measurable observables to benchmark the performance and quality of materials and devices.
• They facilitate direct comparisons across systems by canceling units, with applications in thermoelectricity, acousto-optics, optoelectronics, and cosmology.
• Optimization strategies using these metrics guide experimental design and theoretical analysis, balancing key parameters to enhance device efficiency.

A dimensionless figure of merit is a scalar quantity—constructed by combining experimentally accessible or theoretically computable observables—that quantifies the performance or quality of a process, material, or device relative to a particular function. Its lack of units facilitates direct comparison between disparate systems, across scales and disciplines. In physical sciences and engineering, dimensionless figures of merit, including ZT for thermoelectricity, FoM for cosmological constraints, and device metrics for acousto-optic or optoelectronic systems, play a central role in assessing and optimizing performance within a well-defined theoretical framework.

1. Formulation and Physical Meaning

Dimensionless figures of merit are always ratios, products, or combinations of physical parameters with their units canceling, so that the resulting number provides an intrinsic, scale-independent indication of performance. In thermoelectricity, the canonical figure is

$ZT = \frac{S^2 \sigma T}{\kappa}$

where $S$ is the Seebeck coefficient, $\sigma$ the electrical conductivity, $T$ the absolute temperature, and $\kappa$ the total (electronic + lattice) thermal conductivity. This metric quantifies the efficiency of thermal-to-electrical energy conversion—the higher the $ZT$, the closer a device approaches Carnot efficiency. Similar logic appears for the acoustooptic figure as in $M^2 = n^6 p_\text{eff}^2 / (\rho v^3)$ (Mys et al., 2014), or for optoelectronic phase modulators with $$\xi = (\ln 10/10d)\, \text{Re}[{\chi^{(2)}/n_0}] / \text{Im}[n_0]$$ (Mossman et al., 2016). Cosmological analyses define a constraint-based figure such as

$\text{FoM} = [\det C(w_0, w_1)]^{-1/2}$

where $C$ is the covariance matrix of dark energy parameters $w_0$ and $w_1$ (Su et al., 2011). These definitions share the property of being dimensionless, facilitating optimization and cross-comparison.

2. Thermoelectric ZT and Advanced Formulations

The dimensionless thermoelectric figure of merit $ZT$ is central to energy conversion science. It measures the interplay of charge and heat flow and identifies the optimal balance between high electrical conductivity, large thermopower, and low thermal conductivity.

• The definition generalizes to incorporate both electronic and phononic contributions: $$ZT = G S^2 T/(G^\text{el}_\text{th} + G^\text{ph}_\text{th})$$ (0805.3374).
• Engineering strategies affecting $ZT$ include nanostructuring, impurity doping, and band engineering, as in Te-doped FeSb$_2$ (Pokharel et al., 2013), Si-Ge-Fe-P nanocomposites (Ghodke et al., 2019), or organic $\tau$-type conductors (Yoshino et al., 2010).
• The role of the Lorenz number and the Wiedemann-Franz law is critical: in molecular junctions, $ZT$ can diverge due to the suppression of $G^\text{el}_\text{th}$, violating the Wiedemann-Franz relation (0805.3374). In universal analyses, $ZT$ at fixed quality factor $b_L$ follows an almost material-independent curve (Witkoske et al., 2018).

Alternatively, superlattice and low-dimensional systems require modified similarity criteria and "dimensionless temperature" $t = kT/E_F$ in their theoretical evaluation (Gorskyi, 2015). Fermi integral methods, polylogarithmic formalism, and exact band calculations further support the calculation and optimization of $ZT$ (Kakemoto, 2018, Yadav et al., 2019).

3. Device Figures: Acoustooptic, Optoelectronic, SBS, and Photodetection

Device figures of merit capture the optimization tradeoffs in specialized contexts:

• Acoustooptic: $M^2 = (n^6 p_\text{eff}^2)/(\rho v^3)$, quantifying AO diffraction efficiency and reflecting anisotropy in crystalline or amorphous solids (Mys et al., 2014).
• SBS waveguide amplification: $\mathcal{F} = -\beta/(\sqrt{\alpha\gamma})$ describes the competition among linear loss, two-photon absorption, and carrier-induced loss; $\mathcal{F} > 1$ is necessary for net SBS gain (Wolff et al., 2015).
• Electro-optic phase modulator: Figure of merit $\xi$ balances low half-wave voltage and optical loss (Mossman et al., 2016).
• Photodetection: All classical and quantum metrics—bandwidth, efficiency, photon-number resolution, timing—are constructed from positive-operator-valued measure (POVM) elements, e.g., $\Omega_k^{(1)} = \mathrm{Tr}(\Pi_k^{(1)})$, purity $$\mathrm{Pur}(\Pi_k) = \mathrm{Tr}(\Pi_k^2)/[\mathrm{Tr}(\Pi_k)]^2$$ (Enk, 2017).

4. Data-Driven and Comparative Figures of Merit

Figures of merit also serve as diagnostic tools in experimental and simulation contexts:

• Cosmological parameter constraints: FoM is a function of the error ellipse area, with larger FoM indicating tighter constraints on dark energy (Su et al., 2011).
• Comparative benchmarking: Gaussian process regression and dynamic time warping generate normalized, dimensionless metrics for fuel cycle comparison across time series, independent of underlying units or sampling (Scopatz, 2015).
• Concentrated solar power: FOM, receiver effectiveness, and the normalized metric $$\eta_\text{receiver} = \text{FOM}_\text{actual}/\text{FOM}_\text{max}$$ provide consistent assessments; maximum system efficiency is bounded by $$\eta_\text{max} = \text{FOM}_\text{max} \times \eta_\text{Carnot}$$ (Elahi et al., 2020).

5. Optimization Principles, Anisotropy, and Frequency Effects

Dimensionless figures of merit enable optimization by parameter tuning:

• For $ZT$, maximization protocols target specific regimes: weak molecule–lead coupling $\gamma\to 0$, optimal level offset $\delta\sim2.4$, or nanostructuring to suppress phonon transfer (0805.3374).
• AO device optimization leverages anisotropy in EEC and slow acoustic velocities, seeking optimal propagation/polarization directions (Mys et al., 2014).
• Harman method metrology: Frequency-dependent phase delay must be corrected in four-probe zT measurement; analytical solutions based on the heat equation allow precise extraction (error $<$ 20%) regardless of voltage terminal placement (Okawa et al., 2 Apr 2024).

6. Interdisciplinary and Universal Properties

Several cross-cutting themes emerge:

• Universal behavior: In thermoelectricity, the dependence of $zT$ on the quality factor $b_L$ is robust against electronic complexity (Witkoske et al., 2018).
• Optimization of figures of merit is usually more fruitful than optimization of constituent parameters in isolation; for instance, maximizing the figure of merit in optoelectronic devices may require balancing strong nonlinearity and controllable loss, not only maximizing $\chi^{(2)}$ (Mossman et al., 2016).
• Some figures (e.g., those constructed from POVMs) offer platform-independent assessments and can rigorously define trade-offs (or lack thereof) among classical and quantum photodetection metrics (Enk, 2017).

7. Practical Impact and Future Directions

Dimensionless figures of merit play a central role in guiding experimental design, theoretical exploration, and device engineering. Their universality and scale independence facilitate:

• Direct optimization and material/device selection,
• System-level performance comparisons,
• Rigorous benchmarking and metrology.

Their use extends across condensed matter physics, optical engineering, quantum information, cosmology, and energy conversion science. Current trends include the development of figures for novel transport regimes, integration of multidimensional data-driven comparisons, and the search for fundamental bounds and trade-offs in coupling-limited or topological systems. Identifying exceptions to universal behavior (e.g., ways to break cancellation in $zT$ vs. $b_L$) remains a frontier for enhancing device functionality and materials performance.