Multiple control of thermoelectric dual‐function metamaterials

Thermal metamaterials based on transformation theory offer a practical design for controlling heat flow by engineering spatial distributions of material parameters, implementing interesting functions such as cloaking, concentrating, and rotating. However, most existing designs are limited to serving a single target function within a given physical domain. Here, we analytically prove the form invariance of thermoelectric (TE) governing equations, ensuring precise controls of the thermal flux and electric current. Then, we propose a dual‐function metamaterial that can concentrate (or cloak) and rotate the TE field simultaneously. In addition, we introduce two practical control methods to realize corresponding functions: one is a temperature‐switching TE rotating concentrator cloak that can switch between cloaking and concentrating; the other is an electrically controlled TE rotating concentrator that can handle the temperature field precisely by adjusting external voltages. The theoretical predictions and finite‐element simulations agree well with each other. This work provides a unified framework for manipulating the direction and density of the TE field simultaneously and may contribute to the study of thermal management, such as thermal rectification and thermal diodes.

On the other hand, it is important to address the problem that traditional metamaterials only serve single-target applications.
Recently, multifunctional metamaterials [23][24][25][26][27][28] under the thermoelectric (TE) coupling field have been proposed, which is expected to pave the way for new approaches. However, most of the investigations neglect the regulating effect of the electric field itself on the thermal field. It is well known that the Peltier effect (one of the TE effects) [29][30][31][32][33][34] can convert electrical energy into heat due to the electron transfer of heat energy between two different materials. The electron releases (absorbs) heat to the outside world as it moves from a region of high (low) energy level to a region of low (high) energy level. Inspired by this concept, the electric field can be treated as a tunable parameter 35 to manipulate the thermal field.
In this work, we prove the form invariance of the TE coupling and derive the transformation relationship of constitutive parameters. Then, we establish a general framework to regulate the direction and density of heat flux (electric current) in a given region.
Based on this framework, we design a dual-function metamaterial that can concentrate (or cloak) and rotate the TE field at the same time. To further control the TE field, we first introduce a temperature-dependent conductivity [36][37][38][39][40][41][42][43][44][45] with a step function, with the ability to switch between a rotating concentrator and cloak. In addition, we can not only manipulate the distribution of electric potential but also use the external electric potential as a means of regulating the thermal field. By adjusting the difference in external voltage, we can precisely control the temperature in the core−shell structure. These methods are demonstrated by finite-element simulation and achieve the desired effect. From an experimental point of view, the methods can be implemented by layer structure and shape memory alloys. 46,47 This work provides a broad platform for the manipulation of multiple physical fields and may inspire the research of thermal management devices such as thermal rectification 48,49 and diodes. 36 where indexes u l v k i v , , , , , take 1, 2, 3 and g is the determinant of the Note that we here take the second-order tensor in this form M v u on the basis of g u and g v , which can be transformed into M M g = uv i u iv on the basis of g u and g v .
For the sake of explanation, we start from a known shape (the red line in Figure 1A) of the isotherm T x x x ( , , ) 1′ 2′ 3′ (the same as the isopotential line μ x x x ( , , ) 1′ 2′ 3′ ) in the Cartesian coordinates with Assume that the isotherm shape presented in Figure 1B where the domain equation can be described by Equation (4).
Because there is only coordinate transformation from Figure 1A to B, the constitutive parameters in the curvilinear coordinates can be expressed as where the indices u and v can be replaced by any other indices.
Substituting Equation (7) into Equation ( It is worth mentioning that these two coordinates describe the the isotherm shape in the curvilinear coordinates is different from that in the Cartesian coordinates. In other words, the shape plotted in the curvilinear coordinates is virtual, requiring us to retain the equivalent shape in the Cartesian coordinates ( Figure 1C). As we know, two factors determining the field distributions are boundary conditions and the governing equations. Therefore, we look for a new set of parameters κ σ ,ṽ u v ũ˜, and S v ũ in the Cartesian coordinates to match 3 can be easily expressed as by taking g δ = ij ij˜˜˜a nd g = 1. As long as the forms of Equations (8) and (9) Figure 1C). Due to the equivalence between x u and x ũ , one can obtain transformation relations of constitutive parameters, As we usually discuss these tensors on the contravariant basis of g ũ and g ṽ , they should be transformed by the metric of the Cartesian coordinates, for example, κ κ δ κ = = uv i u iv v ũ˜˜˜˜˜˜. Then, we rewrite the transformation relation as can enhance the area of heat flux in a given region, and thermal rotators can change the direction of heat flux. Here, we take these two functions into account in a device. As shown in Figure 1D, the space in Cartesian coordinates is artificially divided into four regions with the same thermal conductivity κ 0 , electrical conductivity σ 0 , 3 ); and Region III′ (r R ′ > 3 ). Then, we perform the following coordinate transformation for these regions, Here, R 2 ranges from 0 to R 3 , yielding three different devices: rotating the imperfect (perfect) cloak for R R < Substituting Equation (14) into Equation (10), we can obtain the spatial distribution of material parameters ( Figure 1F), The transformation matrix is presented as follows:

| Temperature-switching TE rotating concentrator cloak
The above discussion is based on linear materials, for which thermal conductivity is independent of temperature. However, nonlinearity phenomena are common, and their underlying mechanisms are significant for understanding and designing complex systems.
Generally, the thermal conductivities of natural materials are basically dependent on temperature (nonlinear), which provides a hint for function switching at different temperatures. Inspired by this concept, we introduce temperature into geometrical transformation relations, where T c is a critical point and α is a scaling coefficient for ensuring the step change around T c . By combining Equations (15) and (18) Similarly, polar diagrams in Figure 3A3,B3 show that this device can rotate the heat flow and electric current by a predetermined angle.

| Electrically controlled TE rotating concentrator
In addition, the external voltage is another manipulating method for temperature distribution. We first solve the temperature field before transformation by introducing a generalized auxiliary potential 33 and then, Equation (3) reduces to   The first equation is the Laplace equation with respect to U, which can be easily calculated from The latter is a Poisson equation, which has the particular solution, where the first term is generated by external heat sources and the other term is generated by the TE effect. So far, we have derived the temperature distribution before the coordinate transformation.
Then, we substitute the transformation relation (equivalent to Equations 11 and 13), into Equation (21), and the temperature distributions of the core and background can be written as Clearly, the correlation between temperature and the external electric voltage leads to an approach to regulate the thermal field further. To demonstrate the effect of the external electric voltage, we keep the external thermal field (T L and T R ) constant. When the difference in external voltage is low, the TE effect can be ignored, so the temperature distribution is regulated only by the transformation theory ( Figure 4A1). When the difference increases, the TE effect increases gradually, heating all regions ( Figure 4B1). When the difference is relatively high, the TE effect dominates, even leading to a maximum temperature higher than that of external sources ( Figure 4C1). The temperature field excited by the TE effect is a quadratic function, so the core region (x = 0) is heated up the most.
In addition, this effect can be preserved in the process of coordinate transformation, that is, concentration and rotation. To further validate the calculation, we compare the simulated temperature distribution along different boundaries with theoretical predictions ( Figure 4A2-C2). Obviously, the simulated curves coincide well with the prediction (scatter points) of Equation (23).

| DISCUSSION AND CONCLUSION
In this work, we investigate the TE field by two methods: coordinate transformation and solving the TE coupling equation directly. Both methods yield analytical solutions for physical fields, which guarantees the consistency of the simulation results (Figures 2-4) and theoretical predictions. Nevertheless, these metamaterials based on the transformation theory usually require extreme constitutive parameters, such as anisotropy, inhomogeneity, or even singularity, thus limiting the practical applications. Fortunately, the effective medium theory can provide the possibility of complicated parameters. For example, we can design the spatial distribution of two natural materials to achieve the desired spatial distribution of anisotropic thermal conductivity. In addition, the temperature-switching TE rotating concentrator cloak requires thermal (electric) conductivity varying with temperature as a step function, which can be solved by shape memory alloys. 46