Universal scaling formalism and analytical optimization criterion for multiscale geometric design of thermoelectric metamaterials

Abstract

Thermoelectric (TE) generators directly convert heat into electricity, yet their performance is often limited by small temperature gradients. Width-modulated metamaterials with constrictions and expansions (constricted geometries) sustain larger temperature differences ΔT than constant-width counterparts due to reduced Transmissivity (Tr) – the geometric ratio of constriction to expansion cross-sections. A scaling behavior of transport and key TE performance metrics with Transmissivity is demonstrated from the nanoscale to the macroscale using analytical formalism and simulations across single- and multiple-constriction profiles. It is shown that ΔT, electrical and thermal resistances, efficiency, and output power are governed by a single scaling function g(Tr) – the conductance of a constricted geometry relative to a uniform-width counterpart–independent of carrier type, material, or operating conditions. Universal scaling formalism leads to Performance Design Maps and an analytical optimization criterion: maximum TE performance occurs at an optimal Transmissivity Tr_opt, where g(Tr_opt) = Bi, with Bi = hL/k denoting the Biot number and h, L, and k the convection coefficient, length, and thermal conductivity, respectively. Compared with the uniform geometry, the optimal constricted geometry produces a maximum output power reduced under fixed ΔT, by a factor of Bi/4, and enhanced under identical convective operating conditions by a factor of (1 + Bi)²/(4Bi). This work provides a theoretical framework for multiscale design and optimization of constricted geometries, thereby enabling systematic exploration of design strategies for next-generation TE modules based on advanced thermoelectric metamaterials analogous to nature's hierarchical structures for optimized functionality.

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1. Introduction

Thermoelectric (TE) energy conversion directly transforms heat into electricity, offering a solid-state solution to energy and environmental challenges.^1–3 Thermoelectric devices, with their compact and scalable design, offer a sustainable response to the growing power demands of microelectronics, AI, autonomous sensors, and next-generation technologies by directly converting heat into electricity and recycling waste heat.^3–10

The performance of TE devices is governed by a combination of properties of the material constituting the TE leg – primarily the figure of merit (ZT), which depends on the Seebeck coefficient, the electrical conductivity, and the thermal conductivity – as well as device-level factors such as maintenance of high temperature gradients with the use of efficient heat exchangers,^2,11,12 low contact/interface resistances to decrease output power losses^1,2,13–16 and alternatives to the traditional π-module geometry to meet the needs of various applications.^12,17–19 Although materials with high ZT values have been discovered, their integration into efficient modules remains limited due to unresolved technological challenges.¹⁶ Today, a revolutionary shift in TE research recognizes material geometry as an independent factor capable of maximizing the thermal resistance of the leg and maintaining a high temperature difference, ΔT, under various operating conditions. Optimizing conventional TE leg geometry involves adjusting the height and cross-sectional area of a cuboid-shaped material to balance the requirements for high thermal resistance and low electrical resistance.^2,14,20 Non-cuboid shapes with variable cross-section – particularly pyramidal geometries – have drawn research interest in TEs since the early years of research,^21–23 but remained of limited focus until recent advances in metamaterials science and technology renewed attention. Width-modulated metamaterials featuring constrictions and expansions (constricted geometries) were proposed to increase the TE efficiency by geometrical tuning the electrical and thermal transport.^24–28 Research interest in pyramidal^29–32 and constricted geometries^33–35 has been reignited, theoretically demonstrating enhanced TE performance. Furthermore, significant progress in advanced manufacturing techniques – such as additive manufacturing and Direct Ink Writing – enabled fabrication of non-conventional TE legs and initial experimental validation of predictions.^36,37 These findings sparked a surge of research on non-cuboid TE leg configurations.^38–53 Future advances will depend on multi-factor optimization of TE devices based on non-cuboid legs.^16,54,55 Optimizing metamaterial geometry with respect to TE performance metrics is a central prerequisite for advancing thermoelectric devices. Numerical studies have examined a wide range of non-cuboid shapes, demonstrating that variable cross-sections can improve performance under diverse boundary conditions and constraints (e.g., constant volume, constant surface area).^{33,35,37,43,46,56–58} Among these, constricted geometries consistently outperform cuboid and other variable cross-section designs.^34,35,37,49 Recent proposals for novel TE leg architectures – enabled by advanced manufacturing techniques – underscore the growing emphasis on enhancing TEG efficiency through geometry optimization, informed by high-fidelity simulations.¹⁶ Despite this progress, three key challenges persist:⁵⁴ (i) an incomplete understanding of how shape governs the thermal and electrical resistances of metamaterials, (ii) the lack of validated optimization criteria applicable across different geometries, and (iii) the absence of standardized descriptors required for AI-driven design. The present work addresses these challenges introducing a physically grounded analysis, global optimization criteria, and performance metrics, supported by universal scaling formalism validated through finite element calculations across a range of geometry modulation profiles.

Geometry modulation affects transport on two distinct levels: material and structural. At the material level, it influences electric and thermal transport by modifying the energy states of electrons and phonons and altering their scattering.^24–26,59 At the structural level, transport is constrained by reduced Transmissivity – a mechanism arising purely from geometry, and fundamentally different from conventional boundary or interface scattering mechanisms.²⁷ The concept of Transmissivity, fully defined by the geometry-modulation profile, was first introduced as an intuitive framework for analyzing nanoscale thermal transport in constricted metamaterials.²⁷ Follow-up studies employing phonon Monte Carlo simulations validated these initial insights and uncovered distinct features associated with this mechanism.²⁸ A central finding was the characteristic scaling between thermal conduction and Transmissivity.^27,28 In this work, this concept is extended to the macroscale by demonstrating universal scaling dependence of electrical transport, thermal transport, and TE performance metrics on Transmissivity in constricted geometries with single- and multiple-constriction profiles at millimetric dimensions (Fig. 1). A universal scaling formalism is developed, together with and an analytical performance optimization criterion in terms of Transmissivity, which is thereby established as a fundamental geometric descriptor enabling general multiscale design rules and global optimization criteria for enhancing TE performance.

Fig. 1 Multiscale scaling with Transmissivity. Thermal conductance of width-modulated metamaterials versus Transmissivity (a) at the nanoscale,²⁸ and (b) at the macroscale, for dimensions detailed in Fig. 2. Thermal conductance is expressed relative to that of the constant-width material. (c) Definition of Transmissivity Tr as the ratio of the cross-sectional area of the constriction A_C to that of the expansion A. (d) Artistic representation of the concept of multiscale design of constricted metamaterials for energy convertors, reminiscent of nature-inspired designs that employ hierarchical scaling for optimized functionality.

Section 2 introduces the theoretical model and methodology. Section 3 presents and discusses the simulation results along with the analytical formalism, establishing universal scaling relations for electrical resistance, thermal resistance, temperature difference, efficiency and maximum output power with Transmissivity. It also introduces Performance Design Maps, formulates a global optimization criterion for maximizing the TE performance of metamaterials with constricted geometry and discusses the application of the geometry optimization formalism in connection with device-level factors and practical device design.

2. Method and validation

Calculations were carried out using the finite element method where the structure is discretized into individual cells. At two opposite sides of the material are imposed electrical potential difference ΔV and temperature difference ΔT. The current densities, J, and heat flux, q, are calculated using the following definitions:^2,14

J = −σ(∇V + S∇T) (1)

q = −k∇T + STJ (2)

where σ is the electrical conductivity, S is the Seebeck coefficient and k is the thermal conductivity of the material. The electrical potential, V, and the absolute temperature, T, profiles are determined applying electric current continuity and energy conservation conditions:

∇·J = 0 (3)

∇·q = −∇V·J. (4)

Calculations performed across different materials, dimensions, and temperature gradients consistently revealed the same trends. For clarity, we present representative results for n-type Bi₂Te_2.7Se_0.3 constricted geometries with single- and multiple-constriction modulation profiles. In the calculation, the experimentally measured⁶⁰ temperature dependence of the electrical conductivity, Seebeck coefficient, and thermal conductivity were used for the corresponding material parameters. The discretization of the structures varied depending on the complexity of the modulation and the length of the structure. For each geometry, the mesh was refined until further reduction of the cell size produced no change in the computed temperature and bias profiles. Convergence was assumed when additional refinement loops left these profiles unchanged and the continuity conditions were satisfied to within a few percent (typically 1–5%). A constant electrical bias was applied at the two opposite ends of the TE leg. Currents were then calculated as a function of the applied bias, and independent simulations were performed for a series of bias values to determine the current–voltage characteristics. The temperature at the hot side was fixed (eqn (38)). At the cold side, the boundary condition depended on the operating mode: the temperature was set equal to the ambient temperature T_a under fixed-temperature conditions, whereas eqn (39) was applied under convective cooling. Calculations were tested over a range of boundary-condition parameters. Representative results are presented for a hot-side temperature of T_h = 400 K, ambient temperature at T_a = 300 K and convection coefficients h = 200 Wm⁻² K⁻¹ (efficient convection) and h = 20 Wm⁻² K⁻¹ (weak convection).

Simulation results are interpreted by analytical formalism; their mutual comparison provides validation for both approaches. The finite-element simulations were also compared with recent experimental data, and very good agreement was obtained.⁶¹ These experimental results were originally reported to be consistent with COMSOL simulations. Our calculations show quantitative agreement with those simulations as well as with other standard finite-element solvers widely used in the community, and can be readily reproduced using such tools. In addition, the Monte Carlo calculations were performed using an advanced simulator that is now publicly available.⁶²

3. Results and discussion

3.1. Scaling of transport with Transmissivity

Evaluation of a TE device performance typically relies on the figure of merit ZT, the conversion efficiency, η, η_max and the maximum output power, P_max, of the material constituting the TE leg:²where P_el is the electrical output power and Q_in is the incoming thermal power. V_OC is the open circuit voltage, T_h(c) is the temperature of the hot (cold) side of the material, ΔT is the temperature difference across the structure and R_el is the material electrical resistance.

To improve TE efficiency, it is essential to enhance the intrinsic material's figure of merit ZT and maintain a large temperature difference ΔT across the material [eqn (6b)]. The value of ZT can be increased by engineering the material microscale morphology to optimize the transport properties: σ, S, and k. Preserving a high ΔT requires minimizing the thermal conductance of the material G_th (or equivalently, maximizing the material's thermal resistance R_th), thereby reducing parasitic heat flow. This reduction can be achieved by lowering the material's intrinsic thermal conductivity at the microscale or by geometrically tuning G_th (R_th). Among various geometry design strategies, constricted geometry is a suitable approach because width-modulated materials by constrictions exhibit lower G_th than the corresponding G_th⁰ of constant-width cuboids. This was initially demonstrated in width-modulated nanostructures.^27,28 In particular, it was shown that the thermal conductance G_th of width-modulated nanoslabs (wires and films) decreases monotonically with decreasing the constriction width, relative to the conductance G_th⁰ of the corresponding uniform structure with constant width. For multiple-constriction modulation profiles (Fig. 1a), this monotonic reduction follows a simple analytical relation:²⁸

where A and A_C are the cross-sectional areas of the expansions and the constrictions respectively (Fig. 1). Tr defined as the ratio of the two characteristic cross-sections of the modulated material, expresses the Transmissivity of the actual constricted geometry.^27,28 This definition should be appropriately modified for other metamaterial geometries.

Eqn (8) makes it explicit that in constricted geometries G_th is smaller than G_th⁰ because the constriction area A_C is smaller than the expansion area A(Tr < 1). Moreover, this equation shows that the ratio G_th/G_th⁰ scales directly with Tr, i.e. the decrease in G_th relative to G_th⁰ is governed by the ratio A_C/A. Consequently, the relative decrease in thermal conductance induced by geometric modulation will be identical across structures with different absolute values of A (or A_C) as long as they have the same Tr.

The scaling of thermal conductance with Transmissivity holds irrespective of the modulation profile, although the exact functional dependence is dictated by the specific form of that profile.^28,63,64 Eqn (8) can therefore be recast in a more general form to account for this dependence:

where g(Tr) denotes the functional dependence of G_th/G_th⁰ on Tr.

The reduction of G_th in constricted metamaterials, relative to G_th⁰, can be fully attributed to the decreased Transmsissivity imposed by the modulation geometry. The scaling dependence of the relative conductance G_th/G_th⁰ on Tr – a quantity determined solely by geometry – reflects the geometric origin of the reduced thermal conductance in this class of metamaterials. This scaling behavior has been consistently observed across different nanoscale modulation profiles.^28,63,64

Here, this behavior is demonstrated in constricted materials with millimetre-scale dimensions, as in typical TE legs. Finite element calculations were performed varying cross-sectional areas A and A_C. Representative results are shown for single-constriction (Fig. 2) and multiple-constriction modulation profiles (Fig. 3), under both one- (1D) and two- (2D) dimensional width-modulation schemes (Table 1): (i) varying A while keeping A_C fixed, and (ii) varying A_C with A held constant. In all cases, the simulations confirm that G_th/G_th⁰ scales with Tr.

Table 1 Classification of geometry types and modulation schemes

Geometry class	Modulation scheme	Figure	Color coding
Single constriction	1D and 2D	Fig. 2	1D: purple; 2D: blue and cyan
Multiple constrictions	1D and 2D	Fig. 3	1D: purple; 2D: blue and cyan
Modulation strategy A	Vary AC with A fixed	All figures	Blue and purple symbols
Modulation strategy B	Vary A with AC fixed	All figures	Cyan symbols

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Fig. 2 Scaling behavior for single-constriction width-modulation. (a) Calculations for structures with 1D (purple) and 2D (blue and cyan) width-modulation geometries. (b) Relative (electrical and thermal) conductance/resistance. (c) Temperature difference ΔT. (d) Open-circuit voltage V_OC. (e) Maximum output power density P_max/A, under three operating conditions with decreasing convection coefficient: h = ∞ (fixed ΔT), 200 Wm⁻² K⁻¹, and 20 Wm⁻² K⁻¹. The inset in (e) corresponds to h = 20 Wm⁻² K⁻¹. Two modulation schemes are compared: fixed A = 4 mm × 4 mm with variable A_C (blue and purple symbols), and fixed A_C = 2 mm × 2 mm with variable A (cyan symbols). The structure length L is fixed at 8 mm.

Fig. 3 Scaling behavior for multiple-constriction width-modulation. (a) Calculations for structures with 1D (purple) and 2D (blue and cyan) width-modulation geometries. (b) Relative electrical and thermal conductance/resistance. (c) Temperature difference ΔT and efficiency n_max. (d) Open-circuit voltage V_OC. (e) Maximum output power density P_max/A, under three operating conditions with decreasing convection coefficient: h = ∞ (fixed ΔT), 200 Wm⁻² K⁻¹, and 20 Wm⁻² K⁻¹. The inset in (e) corresponds to h = 20 Wm⁻² K⁻¹. Symbol colours and structures dimensions are identical to those presented in Fig. 2.

Notably, electrical conductance exhibits the same dependence on Tr as thermal conductance. Regardless of the absolute values of thermal or electrical resistance, the normalized ratios G_th(el)/G_th(el)⁰ follow the same functional form g(Tr) (Fig. 4). This universality observed across all modulation profiles (Fig. 2–4) indicates that the impact of geometry modulation on transport is inherently geometric and independent of the nature of carriers. Hence, it holds:

Fig. 4 Scaling of resistance with Transmissivity. Electrical and thermal resistances – shown both in absolute terms and normalized to the constant cross-section structure R_th(el)/R_th(el)⁰ – as a function of Transmissivity for width-modulated structures with: (a) single-constriction and (b) multiple-constriction profiles. Blue and cyan symbols correspond to the two modulation schemes described in the caption of Fig. 2.

Eqn (10) and (11) make explicit that, in constricted metamaterials electrical and thermal conduction remain coupled similarly as in uniform materials, meaning that geometry modulation by itself does not offer a way around the persistent trade-off between them.

The functional form g(Tr) is determined by the specific modulation profile. Both nanoscale^28,63,64 and macroscale calculations, indicate the following approximate relation:

g(Tr) ∼ Trⁿ (12)

with n ≈ 1 for multiple-constriction modulation and n ≈ 0.5 for single-constriction modulation.

These scaling exponents are consistent with the underlying physics. For a single constriction, the dominance of constriction (spreading) resistance in the 3D solution of Laplace's equation leads to an effective conductance scaling as , implying g(Tr) ∝ Tr^0.5. For a series of constrictions along the current path, the current is repeatedly ‘pinched’ and never fully spreads into the volume of the intervening expansions. The effect depends on the modulation degree (constriction density).^28,64 When the constriction modulation is sufficiently dense, the structure behaves effectively as a resistor of cross section A_C giving G ∝ A_C and therefore g(Tr) ∝ Tr.

This behaviour holds at both the macroscale and the nanoscale within the diffusive transport regime that dominates in good TE materials. At the macroscale, this has been demonstrated through finite-element analysis, which is well suited for macroscopic transport and which we used to understand and optimize width-modulated TE legs with millimetric features. At the nanoscale, additional mechanisms – such as boundary scattering, size-dependent thermal conductivity, and non-diffusive (ballistic or quasi-ballistic) transport – become important. These effects have been examined in our previous work using Monte Carlo simulations, which are appropriate for nanoscale transport and for analysing width-modulated nanostructures. These studies showed that the conductance-scaling behaviour holds at the nanoscale and that size-dependent effects are captured by the uniform-width nanomaterial conductivity.²⁸ Deviations from this scaling behavior are expected in the single-constriction regime in nanostructures whose dimensions fall well below the carriers mean free paths. Electron mean free paths in typical TE materials are on the order of 1 nm. For example, in crystalline Bi₂Te₃ the electron mean free path – depending on doping – lies in the range of 2–5 nm at 500 K. In nanostructured TE materials, this mean free path may be further reduced due to distorted crystallinity, allowing electron transport to be safely treated as diffusive. Large phonon mean free paths contribute significantly to thermal transport in materials with high thermal conductivity. In good TE materials, which have low thermal conductivity, phonons with short mean free paths dominate heat transport. For instance, crystalline Bi₂Te₃ has a thermal conductivity of order 1 Wm⁻¹ K⁻¹, corresponding to phonon mean free path of roughly 1 nm. In nanostructures, contributions from phonons with long paths are suppressed by scattering at defects, boundaries, and interfaces – one of the key reasons nanostructuring is an effective strategy for reducing thermal conductivity and enhancing TE performance. Consequently, ballistic phonon effects are not expected to dominate in nanostructured TE materials, and thermal transport can be treated as diffusive when optimizing constricted TE legs with nanoscale features.

The relative conductance G_th(el)/G_th(el)⁰ decreases monotonically with decreasing Tr (Fig. 2b and 3b). In contrast, the absolute values of conductance (or resistance) may not vary monotonically with Tr, as the outcome depends on which dimension is held constant. This effect is illustrated in Fig. 4 where resistance is shown in both absolute and relative terms. Specifically, resistance increases with decreasing Tr when A_C decreases at fixed A, whereas it decreases with decreasing Tr when A increases at fixed A_C. These opposite trends arise because, in the first case, R_th(el)⁰ remains constant, and resistance is governed solely by the variation in Tr due to varying A_C. In the second case, however, R_th(el)⁰ itself varies with A, so resistance is primarily determined by the increase in R_th(el)⁰ as A decreases.

The demonstrated scaling dependence has been validated across different materials and temperature gradients, confirming that varying Transmissivity serves as a purely geometric mechanism for tuning conduction. This mechanism is thus distinct from conventional mechanisms such as intrinsic scattering or boundary/interface roughness scattering, which depend on carrier type and intrinsic material properties. The consistent scaling behavior of transport with Tr – across length scales from the nanoscale to the macroscale–demonstrates that Transmissivity reflects a geometric property of the constricted metamaterial rather than a simple geometric ratio. For example, assigning the same numerical ratio A_C/A to pyramidal geometries would not capture the physical essence of Transmissivity. Owing to this scaling relationship, the reduction (increase) in conductance (resistance) in a constricted metamaterial relative to the corresponding uniform material can be fully determined by this geometric property, positioning Transmissivity as a robust descriptor of the effect of geometry modulation on transport across multiple scales.

These results establish that conduction in constricted metamaterials exhibits a universal scaling dependence on Transmissivity – independent of transport carrier (electrons or phonons), intrinsic material properties, or specific modulation profile.

3.2. Scaling of ΔT and TE efficiency with Transmissivity

The efficiency of a TE module is governed by the temperature difference ΔT across the leg material [eqn (6b)]. This difference reaches its maximum under fixed contact temperature conditions. In practical operating environments, however, contact temperatures often fluctuate due to convective heat exchange. Under such convective conditions, the contact temperature is determined by both the thermal resistance R_th of the leg and the strength of convective flow – quantified by the convection heat transfer coefficient, h. For uniform materials, it can be shown (Appendix A) that:where ΔT_max ≡ T_h − T_a. L denotes the length of the material along the temperature gradient.

A higher value of h corresponds to stronger heat exchange, causing the contact temperature to approach that of the ambient. In the theoretical limit h → ∞, the contact temperature equals that of the environment yielding ΔT = ΔT_max. As h decreases, the attainable ΔT is reduced, a well-known challenge for TE devices operating under weak convection. This relationship is captured by eqn (13) and is confirmed by the finite element calculations for constant-width geometries (A_C = A and Tr = 1) (Fig. 2c and 3c). The thermal conductivity is weakly temperature dependent.⁶⁰ Simulations show exact quantitative agreement with eqn (13) when using the average thermal conductivity k = 1.8 Wm⁻¹ K⁻¹ and the length L = 8 mm of the simulated structures.

Finite element calculations indicate that, under convective conditions ΔT is higher in constricted materials (A_C < A and Tr < 1) than in the corresponding uniform material with constant cross-section A (Fig. 2c and 3c). This demonstrates that the temperature difference ΔT diminishes less (is preserved better) in constricted geometries compared to their cuboid counterparts under the same convective operating conditions. Importantly, ΔT increases monotonically as Tr decreases, reflecting the concurrent increase of the relative thermal resistance (R_th/R_th⁰) and exhibits a direct scaling relationship with Tr. These simulation results are further validated by the following analytical formalism obtained by extending eqn (13) to non-uniform materials (Appendix A):

Eqn (14) and finite element calculations show quantitative agreement across all simulated structures (Tr ≤ 1). The simulation results in Fig. 2c and 3c match exactly the predictions of eqn (14) when using k = 1.8 Wm⁻¹ K⁻¹, L = 8 mm and the calculated values of G_th/G_th⁰ in Fig. 2b and 3b.

Eqn (14) makes clear that the enhancement of ΔT arises from the reduced relative thermal conductance G_th/G_th⁰, which is directly governed by the smaller Transmissivity of the constricted geometry. It also interprets the scaling of ΔT with Tr observed in the simulations, showing that it originates from the scaling dependence of G_th/G_th⁰ on Tr. A key implication of this scaling is that structures with identical modulation ratios (A_C/A) – and thus the same Tr – will exhibit the same ΔT, regardless of their absolute cross-sectional areas. This leads to the important conclusion that, for fixed h, L and k, ΔT is determined by the Transmissivity of the material constricted geometry.

A comparison of the results for a given h (Fig. 2c and 3c) shows that ΔT rises more sharply with decreasing Tr in multiple-constriction modulations than in single-constriction cases. This behavior stems from the fact that G_th/G_th⁰ is lower in structures with multiple constrictions than in those with a single constriction at the same Tr (Fig. 2b and 3b). As the number of constrictions increases, conductance decreases (and resistance increases) progressively with the degree of modulation.⁶² Consequently, geometries with more extended modulation yield a more pronounced increase in ΔT as Tr decreases.

At the macroscale, the figure of merit ZT remains constant since the intrinsic material transport properties (σ, k and S) are independent of the material dimensions. As a result, the TE efficiency directly follows the behavior of ΔT [eqn (6b)]. The maximum efficiency η_max increases monotonically as Tr increases (Fig. 3c). Like ΔT, η_max also scales with Tr, owing to its dependence on the ratio G_th/G_th⁰, as captured by the following analytical expression derived from eqn (6b) and (14):

From eqn (10), (14) and (15), universal scaling relations for ΔT and η_max can be derived in terms of the same function g(Tr):

The optimization of the constricted geometry is strongly dictated by its Transmissivity, which serves as a key design parameter. Enhancement in TE efficiency [eqn (6b)] arises primarily from the elevated thermal resistance R_th, compared with R_th⁰. Eqn (14) explicitly shows that the increase (decrease) in R_th (G_th) directly enhances the temperature difference ΔT under convective operation. To explore the role of the constriction, the data of Fig. 2c for ΔT due to a single-constriction modulation are re-plotted against the inverse Transmissivity (Tr⁻¹) (Fig. 5b). Then, a distinct progression is observed regardless of the specific constriction profile – whether abrupt or gradual (Fig. 5a): an abrupt growth in ΔT is followed by saturation, where the plateau asymptotically approaches the ambient temperature. This saturation limit is prescribed by eqn (14), which identifies the convection coefficient h, material conductivity k, and length L as the key determinants of the plateau height. The emergence and extent of the plateau are governed by the functional g(Tr) [eqn (16)], thereby linking the phenomenon directly to geometric Transmissivity.

Fig. 5 Constriction Thermal Resistance across scales. (a) Abrupt (blue) and smooth (violet) constriction geometries. (b) Temperature difference ΔT vs. inverse Transmissivity (Tr⁻¹) for single-constriction structures: fixed A with variable A_C (blue, purple and violet), and fixed A_C and variable A (cyan), for dimensions as in Fig. 2. (c) Formation of Constriction Thermal Resistance (CTR) at the macroscale. (d) Formation of CTR at the nanoscale (adapted from ref. 28).

Analysis of the temperature distribution (T-profile) across the constricted material provides further insight into the underlying mechanism for the occurrence of the plateau (Fig. 5b). For large Tr, the T-profile is nearly linear. As Tr decreases, however, it progressively distorts and eventually develops into a broad, stable window centered at the constriction. By Fourier's law, thermal conductivity is inversely related to the local temperature gradient. Thus, the sharp gradient appearing at the constriction at the onset of the plateau signals the formation of a dominant thermal resistance. This localized resistance, referred to as Constriction Thermal Resistance (CTR), first identified at the nanoscale (Fig. 5d)²⁸ and demonstrated here at the macroscale (Fig. 5c) – remains nearly constant within the plateau regime, as indicated by the weak variation in T-profiles across different Tr values. The formation of CTR accounts for the abrupt rise in ΔT below a critical Tr. As a result, maximum TE efficiency is achieved at the onset of this plateau, marking the transition into the CTR-dominated regime.

3.3. Scaling of the output power density with Transmissivity

The maximum output power P_max is governed by the trade-off between the open-circuit voltage V_OC and the electrical resistance R_el [eqn (7)]. V_OC is directly proportional to ΔT following the relation: V_OC = SΔT. The calculated values of V_OC (Fig. 2d and 3d) are fully consistent with this relation. Moreover, the spatial distribution of voltage across the constricted material mirrors the corresponding temperature distribution (Fig. 6).

Fig. 6 Temperature and voltage distribution. (a) Temperature and (b) Voltage profiles across different constricted geometries under the same convective operating conditions.

Under fixed ΔT, P_max of a constricted material is lower than that of the corresponding uniform material P_max⁰. This occurs because V_OC remains unchanged, while R_el is reduced relative to R_el⁰. This relationship is made explicit by the following expression derived from eqn (7):

Eqn (18) shows that the ratio P_max/P_max⁰ is directly proportional to G_el/G_el⁰, and therefore decreases as Tr is reduced. Furthermore, P_max/P_max⁰ follows the same scaling dependence on Tr, governed by the function g(Tr).

From eqn (19) it is obtained:

Eqn (18) and (20), give:

Eqn (21) demonstrates that the output power density P_max/A scales with the same functional dependence as the relative conductance G_el/G_el⁰, consistent with the simulation results (Fig. 2e and 3e).

Both P_max/P_max⁰ and P_max/A decrease monotonically with Tr according to g(Tr). However, the absolute value of P_max does not necessarily vary monotonically with Tr, depending on which structural dimension is held constant – analogous to the case of R_el. As shown in Fig. 7, P_max decreases with decreasing Tr when A_C decreases at fixed A, whereas it increases when A increases at fixed A_C. These trends arise directly from the corresponding dependencies of R_el (Fig. 4) and their previously discussed interpretation.

Fig. 7 Scaling of output power with Transmissivity. Absolute output power P_max and output power density P_max/A for (a) single-constriction modulation, and (b) multiple-constriction modulation, as in Fig. 4. Results are presented under three operating conditions for decreasing convection coefficient h: fixed ΔT (h = ∞), h = 200 Wm⁻² K⁻¹ and h = 20 Wm⁻² K⁻¹.

Under convective operating conditions, the combination of eqn (7) and (14) yields:

or equivalently, using eqn (10) and (11):

Then, the output power density P_max/A can be expressed:

or equivalently:

For uniform materials (Tr = 1), P_max decreases with decreasing convection coefficient h, due to the corresponding decrease in ΔT as expected by eqn (7) and (14) and illustrated by finite element calculations in Fig. 7. P_max/P_max⁰ and P_max/A also decrease accordingly [eqn (22) and (24)]. Moreover, they are functions of g(Tr) and thus scale with Transmissivity [eqn (23) and (25)]. Corresponding finite element calculations are shown in Fig. 2e and 3e for P_max/A.

For constricted materials (Tr < 1), P_max may not vary monotonically with Transmissivity depending on which dimension is kept fixed, as explained in the case of R_el. Finite element calculations show that, contrary to the fixed ΔT case, under convective boundary conditions, P_max/P_max⁰ and P_max/A may show non-monotonic variation with Tr. Such a non-monotonic dependence is illustrated in Fig. 3e for weak convection with h = 20 Wm⁻² K⁻¹. In this case, P_max/A shows a maximum at an optimal Transmissivity Tr_opt ≈ 0.09. The non-monotonic dependence is depicted by the analytical formalism, eqn (23) and (25). These equations have global maximum when:

At Tr_opt, P_max/P_max⁰ and P_max/A reach their maxima. This result is confirmed by the finite element calculations. For example, applying this criterion to the dataset of Fig. 2b – where g(Tr) ≈ Tr^0.5 – yields Tr_opt ≈ 0.8 for h = 200 Wm⁻² K⁻¹ and Tr_opt ≈ 0.008 for h = 20 Wm⁻² K⁻¹, consistent with the simulation results (Fig. 2e). Similarly, applying it to the dataset of Fig. 3b – where g(Tr) ≈ Tr – yields Tr_opt ≈ 0.9 for h = 200 Wm⁻² K⁻¹ and Tr_opt ≈ 0.09 for h = 20 Wm⁻² K⁻¹, again consistent with the simulation results (Fig. 3e).

Eqn (26) states that P_max/P_max⁰ and P_max/A are maximized when the functional g(Tr) equals the Biot number Bi (= hL/k). The Biot number quantifies the balance between external convection and internal conduction. For strong convection, Bi ≈ 1 indicating that internal conduction is comparable to surface convection. In this regime, a significant temperature gradient is established across the material, and uniform-width geometry with high electrical conductance are optimal for maximizing TE output power. For weak convection, Bi << 1 indicating that internal heat conduction is much faster than external convection establishing nearly uniform temperature in the material. In this regime, a constricted geometry with low Transmissivity is preferred because it has decreased thermal conduction and thus preserves higher temperature gradient and enhanced output power.

The relevance of this condition for optimizing constricted geometries for maximum TE performance becomes even clearer when considering that: (i) constricted geometries consistently yield higher TE efficiency than uniform-width geometries due to enhanced ΔT under convective operating conditions, but (ii) the corresponding output power density does not necessarily increase, because while the reduced Tr boosts ΔT, it also increases the relative electrical resistance, partially offsetting the efficiency gains in efficiency. Therefore, maximizing TE performance ultimately requires maximizing output power density. In this context, eqn (26) provides the optimization criterion for designing constricted geometries for maximum output power and maximum TE performance under convective operating conditions. By applying eqn (26) to eqn (23) and (25) yields the optimal maximum output power P^opt_max, which can be expressed as follows:

Eqn (23), (25), and (26), along with their graphical representations (such as in Fig. 2e and 3e) constitute analytical TE Performance Desing Maps. These maps provide practical, predictive tools for evaluating constricted metamaterial geometries. They enable designers to determine, in advance, whether a certain constricted geometry will yield a net increase in output power, and if so, to quantify its magnitude for a given choice of geometric parameters and cross-sectional areas (A and A_C). Eqn (27) and (28) provide a means to calculate the maximum output power – and its density – attainable with the optimized constricted geometry. Eqn (27) can be rewritten in terms of the Biot number as:

Eqn (29) explicitly shows that the maximum output power of the optimal constricted geometry is inferior to that of the uniform geometry under fixed temperature difference by a factor of Bi/4.

The maximum power for the uniform, cuboid structure P^cuboid_max is obtained from eqn (23) by setting g(Tr) = 1:

Eqn (29) and (30) give:

Eqn (31) analytically shows that, under identical operating conditions, the optimal constricted geometry produces a maximum output power greater that that of the uniform (cuboid) geometry by a factor (1 + Bi)²/(4Bi). It consistently yields that P^opt_max = P^cuboid_max, for Bi = g(Tr) = 1 reflecting that the cuboid structure is preferable to the constricted geometry for efficient convective cooling.

3.4. Geometry optimization formalism, device-level factors and device design

The derived scaling formalism and the analytical optimization condition provide a rigorous and systematic scheme for optimizing the constricted geometry of a TE leg with cross-sectional area A and length L. The optimal constriction geometry – whether single or multiple-characterized by A^opt_C, can be obtained either graphically from the TE Performance Design Maps or numerically from the analytical condition g(Tr) = Bi under the relevant operating conditions. This optimization scheme can be seamlessly integrated into standard TE device-optimization frameworks that account for additional device-level factors such as contact thermal resistance, electrode losses, and mechanical stability. In practice, the optimal TE leg dimensions A_opt and L_opt can be first determined using established device-level optimization methods, after which the optimal constricted geometry can be derived using our analytical criterion. This enables a truly global optimization of TE devices – something that has not been possible previously due to the absence of a validated optimization framework for constricted material geometries.

The importance of optimizing the material geometry as a prerequisite for optimizing composite TE legs – comprising both the constricted material and the contact resistances – was demonstrated in our recent work,⁶¹ where the scaling formalism was used to interpret experimental observations on hourglass-shaped TE legs. In that study, we developed a unified optimization framework that incorporates both the constricted material and the contact electrodes. This analysis revealed that performance enhancements previously attributed to geometry modulation may, in fact, arise predominantly from contact resistance rather than from geometry alone. Moreover, the commonly used geometric parameter, which was employed to interpret the experiment, was shown to be an inappropriate metric for guiding the design of constricted legs: it violates current-continuity requirements and fails to capture the full influence of geometry modulation on electrical and thermal resistances. In contrast, this influence is fully governed by Transmissivity through its scaling relationship with transport properties, establishing it as a robust and physically sound descriptor of the impact of geometry modulation on TE performance.

To examine the effect of the contact thermal resistance R^c_th on the optimization condition – and on the resulting optimal Transmissivity – the Biot number of the composite leg, consisting of the leg and the two contact resistances, is expressed as:

The thermal resistance of the composite leg is:

R^comp_th = 2R^c_th + R_th. (33)

Assuming that the contact thermal resistance per interface is a fraction f of the leg thermal resistance R_th, yields that:

R^comp_th = (1 + 2f)R_th. (34)

It is thereby obtained that:

k_comp = k/(1 + 2f) (35)

and

Bi_comp = (1 + 2f)Bi. (36)

For well-designed TE modules, the contact thermal resistance per interface is a small fraction of the leg thermal resistance and neither the Biot number nor the optimal Transmissivity is expected to be significantly affected by contact thermal resistance. For instance, for contacts with thermal resistance equal to 5% of the leg thermal resistance, f = 0.05, Bi_comp = 1.1Bi, Tr^comp_opt = 1.12Tr_opt for single constriction and Tr^comp_opt = 1.1Tr_opt for multiple constrictions.

4. Conclusions

A universal scaling behavior of transport and key TE performance metrics with Transmissivity is demonstrated in width-modulated metamaterials with constrictions and expansions, spanning from the nanoscale to the macroscale, using analytical formalism and simulations across a range of modulation profiles with single and multiple constrictions.

Analytical formalism quantitatively interprets finite element calculations and shows explicitly that the effect of geometric modulation on temperature difference ΔT, electrical and thermal resistances, TE efficiency, and output power is governed by a single scaling function, g(Tr) of the Transmissivity Tr. This function, defined by the ratio G_th(el)/G_th(el)⁰, represents the (thermal/electrical) conductance of a constricted geometry relative to a uniform-width counterpart. Crucially, this relationship is independent of carrier type, material choice, exact constricted geometry profile and operating conditions. Transmissivity is thereby established as a reliable, multiscale design parameter for engineering transport properties – analogous to nature's use of hierarchical structures to achieve optimized functionality.

Constricted geometries consistently enhance TE efficiency compared to uniform-width structures – primarily through increased ΔT under convective operating conditions. However, this enhancement does not always translate into higher output power. The analytical scaling formalism for the output power density together with its graphical representation, constitute TE Performance Design Maps – practical, predictive tools that identify the conditions under which constricted geometries can enhance TE performance. Furthermore, an analytical optimization criterion is established: maximum TE performance is achieved at an optimal Transmissivity, Tr_opt, where g(Tr_opt) = Bi, with Bi = hL/k the Biot number. The optimization condition leads to the generic conclusion that compared with the uniform geometry, the optimal constricted geometry produces a maximum output power reduced under fixed ΔT, by a factor of Bi/4, and enhanced under identical convective operating conditions by a factor of (1 + Bi)²/(4Bi).

Although the physical meaning of the Biot number is well established in thermal science, its connection to a geometry-dependent scaling function and its use as an analytical optimization condition for constricted TE legs have not been previously reported. The condition g(Tr_opt) = Bi emerges uniquely from the common scaling of electrical and thermal conductance with Transmissivity – an insight introduced here for the first time. Unlike previous models based on equivalent resistances, which can inadvertently neglect current continuity requirements, our formalism provides a mathematically rigorous design rule. The criterion g(Tr_opt) = Bi therefore provides the first explicit, closed-form optimization condition for geometry-modulated TE legs.

The universal scaling formalism provides a rigorous framework for evaluating TE performance in constricted metamaterials and establishes the foundation for a unified optimization strategy for composite TE legs that incorporate both metamaterial and electrode contacts. Beyond establishing this universal framework, our results provide the missing analytical tools required to integrate geometry-modulated legs into full device-level optimization workflows. By enabling the systematic determination of optimal constricted geometries and their coupling with contact and module-level constraints, the present formalism offers a practical pathway toward globally optimized thermoelectric devices – an advance that was previously not achievable due to the absence of a validated optimization strategy for geometry-modulated materials.

This work offers a theoretical platform for multiscale design and optimization of constricted geometries, enabling systematic exploration of design strategies for next-generation TE modules based on advanced thermoelectric metamaterials.

Author contributions

X. Z.: conceptualization; methodology, investigation, formal analysis, visualization, writing.

Conflicts of interest

There are no conflicts to declare.

Data availability

The data that support the findings of this study are included in the article.

Appendix A: Analytical formalism for the temperature difference ΔT in uniform and non-uniform materials under convective operation conditions

Let us consider a uniform cuboid material of cross-sectional area A and length L. For a hot-side temperature T_h, the cold-side temperature T_c under convective conditions is determined by the solution of the 1D steady-state thermal conduction equation:where constants C₁ and C₂ are determined applying the boundary conditions:

T(0) = T_h (38)

where k is the material thermal conductivity, h is the convection coefficient and T_a is the ambient temperature. This gives:

For corresponding non-uniform materials, eqn (40) can be extended as follows:

in terms of the effective thermal conductivity k_eff, defined from the non-uniform material's thermal conductance G_th through the relation:

The intrinsic thermal conductivity k and the thermal conductance of the uniform cuboid material G_th⁰ are related by the expression:

Combining eqn (42) and (43) yields:

Then, from eqn (41) and (44), it is obtained:

or equivalently expressed, in terms of ΔT_max ≡ T_h − T_a:

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