# Impact of the Regularization Parameter in the Mean Free Path Reconstruction Method: Nanoscale Heat Transport and Beyond

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{nano}and that of the bulk α

_{bulk}. The suppression has been derived for different experimental geometries from the Boltzmann transport equation [7,8,9].

_{nano}(d) and the suppression function was originally derived for thermal conductivity [4,10], and more recently has been used to determine the MFP of magnons and the spin diffusion length distribution [11]. This relation can be expressed by means of a cumulative MFP distribution as

_{bulk}/d, Λ

_{bulk}is the bulk MFP and d the characteristic length of the sample and F

_{acc}is the accumulation function given by

_{c}, to the total transport property, and is the object that will be recovered by applying the MFP reconstruction technique. From this definition it is easy to see that the accumulated function is subject to some physical restrictions: it cannot take values lower than zero for Λ

_{c}= 0 or higher than one for Λ

_{c}→ ∞, and it must be monotonous [4]. We can recognize that Equation (2) is a Fredholm integral equation of the first kind that transforms the accumulation function F

_{acc}(Λ

_{bulk}) into α with $K=\frac{dS(\chi )}{d\chi}\frac{d\chi}{d{\Lambda}_{bulk}}$ acting as a kernel. As the inverse problem of reconstructing the accumulation function F

_{acc}is an ill-posed problem with infinite solutions [10]. Minnich demonstrated some restriction can be imposed on F

_{acc}to obtain a unique solution [4]. Furthermore, it is reasonable to require the smoothness conditions mentioned before on F

_{acc}, since it is unlikely to have abrupt behavior in all its domain. These requirements can be applied through the Tikhonov regularization method, where the criterion to obtain the best solution F

_{acc}is to find the following minimum:

_{i}is the normalized i-th measurement, $A=K({\chi}_{i,j})\times {\beta}_{i,j}$ is an m × n matrix, where m is the number of measurements and n is the number of discretization points, ${K}_{i,j}$ is the value of the kernel at ${\chi}_{i,j}={\Lambda}_{i,j}/{d}_{i}$, and ${\beta}_{i,j}$ the weight of this point for the quadrature. The operators $\parallel \text{}{\parallel}_{2}$ and ${\Delta}^{2}$ are the 2-norm and the (n − 2) × n trigonal Toeplitz matrix which represent a second order derivative operator, respectively. The first term of Equation (3) is related to how well our result fits to experimental data (residual norm), while the second term represents the smoothness of the accumulation function (solution norm). The balance between both is controlled by the regularization parameter μ. In other words, μ sets the equilibrium between how good the experimental data is fitted and how smooth is the fitting function. The choice of μ will thus have a huge impact on the final result of the accumulation function, and a criterion to obtain the optimal value must be established. The selection of the most adequate μ is still an open question in mathematics. Several heuristic methods are frequently used, such as the Morozov’s discrepancy principle, the Quasi-Optimally criterion, the generalized cross validation, L-curve criterion, the Gfrerer/Raun method, to name a few [5,6]. Among these methods, the L-curve criterion is one of the most popular methods due to its robustness, convergence speed and efficiency. This method establishes a balance between the size of the discrepancy of the fitting function and the experimental data (residual norm) with the size of the regularized solution (solution norm) for different values of μ. The curve has an L-like shape composed by a steep part where the solutions are dominated by perturbation errors and a flat part where the solution is dominated by regularization errors. The corner represents a compromise between a good fit of the experimental data and the smoothness of the solution. It has been found that the corresponding point lies in the corner of the L-curve in the residual norm-solution norm plane, which can be defined as the point of maximum curvature. The optimal value of µ can be found by locating the peak position of the curvature as a function of μ [5,6].

_{acc}using a convex optimization package for MATLAB called CVX [12,13].

## 3. Results and Discussions

#### 3.1. Phonons in Out-of-Plane Thermal Transport in Graphite from Molecular Dynamic Simulations

_{acc}as a function of the MFP for different μ-values. For this particular example, we can observe that depending on the μ-values, the span of MFP can fluctuate from a very wide (5 nm–6.88 μm for μ = 10) to very narrow (60–100 nm for μ = 0.1) distribution. It is also important to notice that μ-values between 0.1–4.0 lead to quite good agreement between the simulated κ (“experimental data”) and A·F

_{acc}product (see Figure 3b), although these values yield to a completely different span of the MFP. The MFP distribution varies from very narrow (low μ-values) to very wide (high μ-values). This difference in the MFP distribution is a direct consequence of the weight given to each component in the Equation (3). The low μ-values give a large weight to the residual norm and, as a consequence, it leads to an overfitting of the reconstructed function. At the same time, large μ-values give higher importance to the solution norm, leading to and over smoothing of the reconstructed function.

_{acc}will be strongly affected by the “shape” of the selected SF. Therefore the choice of a different SF will lead to a different MFP distribution [11].

#### 3.2. In-Plane Thermal Transport in 400 nm Thick Si Membrane

_{2}is Fuchs-Sondheimer SF for in-plane thermal transport given by [10]

_{1}function is unity in the limit qΛ’ << 1, in the diffusive limit, and goes like (qΛ’)

^{−2}for qΛ’ >> 1, in the ballistic regime, thus describing the transition between both regimes necessary to adequately interpret the measured quantities in the experiment [17].

_{opt}= 1.05 at room temperature. In Figure 4 we can observe the effect of the different values of μ. Note that, in this case, the reduction of µ affects mainly the smoothness of the reconstructed function with large changes on the concavity and convexity of the accumulation function. However, there was not a significant change on the A·F

_{acc}product for small μ-values. The oscillations observed for low μ-values are a direct consequence of the overfitting of the reconstructed function, due to the large weight imposed to the minimization problem. The increase of the regularization parameter beyond the optimal value results in an increase of the span of the MFP of the carriers, as shown in Figure 4a, and a poor agreement with the experimental data, as can be seen in Figure 4b.

#### 3.3. In-Plane Thermal Transport in Si: Reconstruction by Changing the Thickness of the Membrane

## 4. Conclusions

## Supplementary Materials

_{opt}. (b) Normalized Longitudinal spin-Hall torque coefficient for different thickness of the Pt filmvalues of μ for the Magnon-MFP reconstruction at different temperatures for LSSE experiments.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Flow-chart of the reconstruction method used to obtain the accumulation function from experimental data.

**Figure 2.**(

**a**) Three-dimensional visualization of the relation between the L-Curve (blue) and the different values of μ for a cross-plane Fuchs-Sondheimer suppression function at T = 300 K for graphite simulations. (

**b**) L-curve or xz projection of 3D curve, the maximum curvature is displayed as heat-like color bar.

**Figure 3.**(

**a**) Phonon mean free path distribution of the accumulation function reconstructed for different values of μ (0.1–10) using the FS cross-plane suppression function and the simulated κ made by Wei et al. [14]. The black-dashed and black-dotted lines represent reconstructions obtained by using an optimum (0.95) and large (4.0) μ-value, respectively. The cyan solid and grey dotted line represent the MFP reconstruction obtained by Zhang et al. [16] and Wei et al. [14] (

**b**) Thermal conductivity normalized to the bulk value and/or A·F

_{acc}product as function of graphite thickness. The open dots represent the simulated thermal conductivity (“experimental data”). Heat-like lines correspond to different A·F

_{acc}for μ-values in a range of 0.1 < μ < 10.

**Figure 4.**(

**a**) Phonon mean free path distribution of the accumulation function for a 400 nm Si film reconstructed for μ

_{opt}(green line). The blue and red lines are the result obtained using a low and high value of μ, respectively. (

**b**) Thermal conductivity of 400 nm Si film corresponding to the different accumulation functions (red, green, and blue lines) for different transient grating periods in the experimental technique [17].

**Figure 5.**Three-dimensional visualization of the relation between the L-Curve (blue) and the different values of μ for an in-plane Fuchs-Sondheimer suppression function at T = 300 K for silicon measurements.

**Figure 6.**(

**a**) Phonon mean free path distribution of bulk Silicon reconstructed from the thickness dependence of thermal conductivity. The dark green, red, dotted grey and dotted dark cyan lines represent reconstructions for different μ-values. The black stared and blue dotted lines represent the MFP reconstruction and molecular dynamic calculation performed by Cuffe et al. [10] and Henry et al. [18], respectively. The orange-solid dots represent the experimental phonon-MFP distribution of bulk silicon measured by Regner et al. [19]. (

**b**) Thermal conductivity corresponding to the different accumulation functions for different samples with different thickness [10].

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**MDPI and ACS Style**

Sanchez-Martinez, M.-Á.; Alzina, F.; Oyarzo, J.; Sotomayor Torres, C.M.; Chavez-Angel, E.
Impact of the Regularization Parameter in the Mean Free Path Reconstruction Method: Nanoscale Heat Transport and Beyond. *Nanomaterials* **2019**, *9*, 414.
https://doi.org/10.3390/nano9030414

**AMA Style**

Sanchez-Martinez M-Á, Alzina F, Oyarzo J, Sotomayor Torres CM, Chavez-Angel E.
Impact of the Regularization Parameter in the Mean Free Path Reconstruction Method: Nanoscale Heat Transport and Beyond. *Nanomaterials*. 2019; 9(3):414.
https://doi.org/10.3390/nano9030414

**Chicago/Turabian Style**

Sanchez-Martinez, Miguel-Ángel, Francesc Alzina, Juan Oyarzo, Clivia M. Sotomayor Torres, and Emigdio Chavez-Angel.
2019. "Impact of the Regularization Parameter in the Mean Free Path Reconstruction Method: Nanoscale Heat Transport and Beyond" *Nanomaterials* 9, no. 3: 414.
https://doi.org/10.3390/nano9030414