*1.1. State of the Art*

Nowadays, a heat transfer problem in the case of modern electronic structure is one of the most important research areas in the high-tech industry. The main reason for this fact is a significant downsizing of integrated circuits that is implemented in all modern electronic devices. Moreover, it also connects with a meaningful increase of an operation frequency of the mentioned devices. Both of these facts cause a rapid growth of a heat density generated inside such small structures. Consequently, the increased internal heat generation results in a huge increase of a temperature in critical parts of a device during its operation. It should also be highlighted that assurance of a proper cooling condition in the case of nanosized electronic structures is a non-trivial issue. Thus, all of these factors may influence an unstable operation and shorten the life cycle of an entire structure. Therefore, a thermal analysis seems to be one of the most important steps in the process of designing and developing modern electronic structures.

For a long time, heat conduction problems have been solved based on Fourier's theory [1,2]. This approach uses the Fourier's law and resulting Fourier–Kirchhoff (F–K) model for solids. They can be described by the following system of equations:

$$\begin{cases} \mathbf{q}(\mathbf{x},t) = -k \cdot \nabla T(\mathbf{x},t) \\ \ c\_v \frac{\partial T(\mathbf{x},t)}{\partial t} + \nabla \circ \mathbf{q}(\mathbf{x},t) = q\_V(\mathbf{x},t) \end{cases} \quad \mathbf{x} \in R^n, \ n \in N, \ t \in \mathbb{R}\_+ \cup \{0\} \tag{1}$$

where **q** is a heat flux density vector; *k* means a thermal conductivity of investigated material; *T* is a function regarding temperature rise above the ambient temperature; *cv* means a volumetric heat capacity being a product of a specific heat of a material for a constant pressure (*cp*) and its density (ρ); and *qV* reflects the value of internally generated heat.

The classical F–K equation can also be directly derived from the classical thermodynamics for a positive-definite entropy-production rate, for which the thermodynamic state transition in a heat transfer process is extremely slow (quasi-stationary process). Therefore, the process time (*t*) should be longer than a system's relaxation times.

Despite the numerous advantages of the Fourier–Kirchho ff approach, the research has shown that the application of the mentioned model may not reflect real phenomena [3–5]. One of its biggest disadvantages is an assumption considering the infinite speed of heat propagation. Apart from that, an instantaneous change of a temperature gradient or a heat flux should also be emphasized as a non-physical behavior postulated by the investigated theory. None of them, especially in the case of electronic structure with physical dimensions in nanometer-scale, has been empirically confirmed [6,7]. Thus, a thermal model including improvements of the F–K approach should be used instead of the classical theory.

The thermodynamic state is nonequilibrium in a fast transient process where time is comparable with the system relaxation times (e.g., an Umklapp phonon–phonon scattering process relaxation time in semiconductors [8,9]). The extended irreversible thermodynamic (EIT) can be applied to describe this kind of heat transfer system, assuming the second order the Taylor series expansion of entropy in EIT [10,11]. As a consequence, in mid-1990s, the new Dual-Phase-Lag (DPL) model was introduced by Tzou [12,13].

This model is an advanced version of an approach based on the Fourier–Kirchho ff theory and includes so-called time lags describing the time needed to change the temperature gradient, as well as the heat flux density. Each change is reflected by a separate lag value: a temperature time lag (<sup>τ</sup>*T*) and a heat flux time lag (<sup>τ</sup>*q*), respectively. The mathematical description of analyzed model can be presented in the form of the following system of equations:

$$\begin{cases} \nabla \circ \mathbf{q}(\mathbf{x},t) = -c\_{\upsilon} \frac{\partial T(\mathbf{x},t)}{\partial t} + q\nu(\mathbf{x},t) \\\ k\tau\_{T}\frac{\partial \nabla T(\mathbf{x},t)}{\partial t} + \tau\_{q}\frac{\partial \mathbf{q}(\mathbf{x},t)}{\partial t} = -k \cdot \nabla T(\mathbf{x},t) - \mathbf{q}(\mathbf{x},t) \end{cases} \quad \mathbf{x} \in \mathbb{R}^{n}, \ n \in \mathcal{N}, \ t \in \mathbb{R}\_{+} \cup \{0\} \tag{2}$$

The other general approach can be derived by using the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) equation proposed in 1997 by M. Grmela and H. C. Öttinger [14–16]. This approach can be useful to obtain a model for the Monte Carlo simulation of modern nanostructures [17] and also for deterministic approaches [18]. Apart from that, an approach known as Guyer-Krumhansl-type heat conduction [19,20] can be also used. Moreover, the research presented by Pop et al. [21], related to heat generation and transportation problems in nanosized transistors, is also worth considering.

Let us consider, more precisely, one of the most common heat transfer models for electronic structures—the DPL model (Table 1a—left column), the ballistic-conductive heat transfer model (Table 1b—left column), and the model proposed in this paper, the DPL model with fractional order of the temperature function space derivative based on the Grünwald–Letnikov theory (Table 1c—left column). Suppose also an isotropic medium parameter properties and idempotent equations parameters *k* > 0, *cv* > 0, *k*21, *k*12, τ*T*, <sup>τ</sup>*q*, <sup>τ</sup>*Q*, etc. To simplify the analysis, the Taylor-series expansions mapping into a function with time delay is considered, as presented in Equation (3). The terms of higher orders are neglected.

$$f(\mathbf{x}, t) + \tau \frac{\partial f(\mathbf{x}, t + \tau)}{\partial t} \to f(\mathbf{x}, t + \tau) \tag{3}$$

Then, time delayed PDEs are obtained and presented in the right column of Table 1. In the first case (Table 1a—right column), the state is appointed by using a gradient of a heat flux (−*k*·Δ *T*) delayed by <sup>τ</sup>*T*-τ*q*. It should be emphasized that results produced by the DPL model are consistent with many experiments and measurements at nanoscale (for more, see [9]).

In the case of the ballistic-conductive heat transfer model (Table 1b), the rate of value temperature *T*(**<sup>x</sup>**, *t* + <sup>τ</sup>*q*) change at *t*+ τ*q* and **x** is dependent on a gradient of heat flux (−*k*·Δ*T*) at the current time (*t*), increased by a difference of growth rate of the averaged value of temperature (*T*) over the infinitesimal neighborhood of point **x** and the value of temperature (*T*) in this point, both from the past time (*t* − <sup>τ</sup>*Q*):

$$\Delta DT \left(\mathbf{x}, t + \tau\_{\neq}\right) \propto k \cdot \Delta T \left(\mathbf{x}, t\right) \cdot Dt + |\mathbf{c}\_{\upsilon} k\_{12} k\_{21}| \cdot D \Delta T \left(\mathbf{x}, t - \tau\_{\neq}\right) \tag{4}$$

where *D*(·) is the difference operator ("change in") corresponding to changes for *Dt*→0 and also for *k*12·*k*21 ≤ 0. Therefore, the temperature *T*(**<sup>x</sup>**, *t* + <sup>τ</sup>*q*) dynamic is additionally intensified by the dynamics of the rate of heat flux gradient (∂Δ*T*(**<sup>x</sup>**, *t* − <sup>τ</sup>*Q*)/∂*t*) in the past time (*t* − <sup>τ</sup>*Q*).

The approach proposed in this work is based on the temperature *T*(**<sup>x</sup>**, *t* + <sup>τ</sup>*q*) dynamic control, using the fractional order of Laplace operator (GLΔα*x*) in the DPL model (Table 1c—right column). The integral or differential behavior of this operator and time–space discretization scheme is obtained by the value of parameter α*x*. The application of the fractional derivative can be interpreted as an inhomogeneous space for the heat transfer in relation to the local energy distribution (interpreted as temperature at nanoscale) in infinitesimal neighborhood of considered point. Therefore, the infinitesimal distance in Laplacian is modulated in relation to the temperature distribution around the considered point at nanoscale. The physical interpretation of the fractional derivatives is also presented in [22–25].


**Table 1.** Selected Heat Transfer Model transformed into time delayed Partial Differential Equation.

Hence, the proposed model (and proposed time–space scheme) is a link between experimentally confirmed DPL mesoscopic model with the ballistic heat transport model with dynamic temperature changes' intensification useful for quasi 1-D nanostructures and for radiative heat transport without phonon collisions (e.g., metal nanowires and ultra-thin metal–oxide films).

One of the biggest advantages of the DPL approach is a universality of its application. It can be used in the case of parabolic partial differential equations, as well as for hyperbolic ones. It means that the Dual-Phase-Lag model can successfully replace the classical model based on parabolic Fourier–Kirchho ff di fferential equation. Of course, the DPL approach also has some disadvantages. For example, it is characterized by a bigger complexity than F–K model, what results in longer time needed for computational simulation results, especially in the case of complex structures. However, this issue will not be investigated in this paper.

The crucial achievement described in this paper is the application of the Grünwald–Letnikov fractional space-derivative in Dual-Phase-Lag heat transfer model to more accurate modeling of delays and modulation of Laplacian in relation to the temperature distribution around the considered point. This approach seems to be beneficial to represent real heat transfer behavior at the nanoscale. The proposed solution is more accurate than the F–K heat transfer model. The computation time reduction is also obtained in comparison to the DPL model. Mentioned features will be demonstrated by using a special test nanostructure developed in microelectromechanical system (MEMS) technology.

The structure of the paper is as follows. Primarily, a description of investigated real test structure is prepared. Then, analyses related to mathematical modeling of the temperature distribution inside the test structure are included. The structure's discretization method is considered, as well. Key components of authors' approach, i.e., DPL model approximation scheme and heat transfer enhancement (including Grünwald–Letnikov fractional derivative and its application), are described in Sections 2.3 and 2.4. Next, simulation results are demonstrated and compared to real measurements described in [32,33]. Finally, results are discussed, and the research is summarized.
