ARTICULOS

Thermal transport in a 2D stressed nanostructure with mass gradient

R. Barreto,^1,2 M. F. Carusela,^{1, 2} A. Mancardo Viotti,¹ A. G. Monastra^1,2*

*E-mail: amonast@ungs.edu.ar
1 Instituto de Ciencias, Universidad Nacional de Gral. Sarmiento, J. M. Gutiérrez 1150, 1613 Los Polvorines, Argentina.
2 Consejo Nacional de Investigaciones Científicas y Técnicas, Argentina.

---

Received: 20 November 2015, Accepted: 17 April 2015
Edited by: C. A. Condat, G. J. Sibona
Licence: Creative Commons Attribution 3.0
DOI: http://dx.doi.org/10.4279/PIP.070008

---

Inspired by some recent molecular dynamics (MD) simulations and experiments on suspended graphene nanoribbons, we study a simplified model where the atoms are disposed in a rectangular lattice coupled by nearest neighbor interactions which are quadratic in the interatomic distance. The system has a mechanical strain, and the border atoms are coupled to Langevin thermal baths. Atom masses vary linearly in the longitudinal direction, modeling an isotope or doping distribution. This asymmetry and tension modify thermal properties. Although the atomic interaction is quadratic, the potential is anharmonic in the coordinates. By direct MD simulations and solving Fokker-Planck equations at low temperatures, we can better understand the role of anharmonicities in thermal rectification. We observe an increasing thermal current with an increasing applied mechanical tension. The temperatures and thermal currents vary along the transverse direction. This effect can be useful to establish which parts of the system are more sensitive to thermal damage. We also study thermal rectification as a function of strain and system size.

I. Introduction and motivation

Efficient energy consumption is one of the biggest challenges for modern societies. Moreover, miniaturization of electronic devices together with their increasing computing capabilities make heat production per surface/volume unit tend to increase constantly. New technologies are necessary for efficient heat transport. At nanometric scales and for low-dimensional systems, Fouriers law is not fullfilled and new phenomena arise. Among them, thermal rectification would make possible to build thermal diodes, being low dimensional systems the best candidates (e.g., atomic chains, graphene) [1-3]. In particular, graphene nanoribbons (GNR) are promising candidates in nanoelectronics. However, at the chiplevel integrated circuits, the power density highly increased, making the thermal managment vital to ensure a stable operation of any practical graphene-based device [4, 5]

Thermal conductance can be modified by defects, impurities, shapes, geometries, mechanical strains, asymmetries, etc. It is known that it is necessary to have an asymmetry on the system to achieve heat rectification [6]. Thus, systems with mass gradients due to dopping concentrations, deposition of heavy molecules or a variable width are candidates to present this phenomena [1,7-9].

On the other hand, thermal conductance of 2D systems, as GNRs, is remarkably affected by tensile strain. Moreover, mechanical tension is relatively easy to control experimentally at nanoscale, being a good candidate to tune the phononic heat transport in a system [10-13].

From this motivation, we present a simple model for a 2D system to better understand heat transport and the possibility of thermal rectification in a layered-device with variable widths subject to a mechanical longitudinal tension.

II. The model

Figure 1: Schematic of the system. Particles bordered by dash-dot lines are coupled to Langevin thermal baths.

that only depends on the positions.

III. Fokker Planck equations - Harmonic approximation

due to the square root. In the last equality, we have used Taking into account that the proposed model is valid for small displacements, we can make an expansion considering Collecting terms of the same order in , we arrive to

We see explicitly in this expansion that the potential energy is not quadratic on the coordinates. Nevertheless, for very low temperatures, where displacements are small with respect to the lattice constants, we can neglect the cubic and higher order terms. Making this approximation for all bonds in x and y direction, after collecting terms, we arrive to a quadratic approximation for the total potential energy of the system

where and or 1 otherwhise. In this harmonic approximation, the directions x and y are completely decoupled. The dependence on the tension comes through the transversal spring constant k₁.

For every particle, we have two degrees of freedom, so the system has M = 2N total number of degrees of freedom. we call this coordinates as , we can rewrite the equations of motion as

where p_n = m_nq_n are the momenta, and

are elements of a force matrix. This is a set of stochastic linear equations, the so called Fokker Planck (FP) equations, that can be exactly integrated for a given realization of the random forces . We explain in more detail the general solution following [14].

with K the force matrix, and the diagonal matrices . This matrix A can be diagonalized, obtaining 2M complex eigenvalues that come in complex conjugate pairs. Their real part is always positive. The eigenvectors are also complex, and arranging them as columns, we obtain the diagonalizing unitary matrix U, which fuls AU = UA0, where A0 is a diagonal matrix with the eigenvalues as elements. Transforming and defning X0 = U- 1X, and F0(t) = U- 1F(t), the matrix Eq. (9) transforms to

Taking the solutions for each x'_i, and using that X(t) = UX0(t), we fnally obtain the solutions for positions qn and momenta p_n. For suciently long times, the term proportional to the initial condition will vanish, provided that

Now we are interested in the statistical behavior of the system, particularly in the stationary regime. With these solutions, one can compute the mean values doing averages over the ensemble of random forces, which are necessary to estimate time and spatial correlations, currents per bond and site temperatures.

Taking into account the correlations of the random forces that depend mostly on bath temperatures, and after some calculations, the time and spatial correlations of all dynamical variables can be computed. First, we defne the diagonal matrix , where Tk corresponds to the temperature for the moment component which are coupled to a thermal bath, or zero otherwise. Then, transforming this matrix as D0 = U1D (U-1)T , and defning a new matrix

We plot on Figs. 2 and 3 the time correlation functions computed by FP equations. We observe that particles coupled to a heat bath, or near to it, decorrelate in shorter times, as expected due to the random forces. Particles in the middle of the system have longer correlation times. The frequencies on these functions are related to normal modes weakly coupled to the heat baths. We observe that for times of the order of 200, most of these functions decay signifcantly, except for the velocity in y direction. Anyway, the rapid and non periodic uctuations make it difficult to predict on average the behavior for long times. From these time correlation functions, we conclude that the system can attain a stationary regime at times of the order of 500, and measurements of dynamical variables separated by this time scale can be considered independent.

We observed thermal rectification in a model with an interatomic potential quadratic on the distance, although as a two dimensional model, the potential is non-linear in the coordinates. Also the mass gradient, which makes the system asymmetric, is essential for the thermal rectification. The effect could be even stronger incorporating cubic and quartic terms in the interatomic potentials (Fermi- Pasta-Ulam models), directional terms (strongly present in carbon-carbon interaction), and flexural modes.

1 N A Roberts, D G Walker, A review of thermal rectification ohservations and models in solid materials, Int. J. Thermal Sci. 50, 648 (2011).         [ Links ]

2 A Dhar, Heat transport in low-dimensional systems, Adv. in Phys. 57, 457 (2008).         [ Links ]

3 N Li, J Ren, L Wang, G Zhang, P Hánggi, B. Li, Colloquium: Phononics: Manipulating heat flovj with electronic analogs and beyond, Rev. Mod. Phys. 84, 1045 (2012).         [ Links ]

4 Z Li, H Qian, J Wu, B-L Gu, W Duan, Role of symmetry in the transport properties of graphene nanoribbons under bias, Phys. Rev. Lett. 100, 206802 (2008).         [ Links ]

5 E Pop, S Sinha, K E Goodson, Heat generation and transport in nanometer-scale transistors, Proc. IEEE 94, 1587 (2006).         [ Links ]

6 S Das, A Dhar, O Narayan, Heat conduction in the Fermi-Pasta-Ulam chain, J. Stat. Phys. 154, 204 (2014).

7 J Hu, X Rúan, Y P Chen, Thermal conductivity and thermal rectification in graphene nanoribbons: A molecular dynamics study, Nano Lett. 9, 2730 (2009).

8 E Pereira, Graded anharmonic crystals as genuine thermal diodes: Analytical description of rectification and negative differential thermal resistance, Phys. Rev. E 82, 040101(R) (2010).         [ Links ]

9 C W Chang, D Okawa, A Majumdar, A Zettl, Solid-state thermal rectifier, Science 314, 1121

10 N Wei, L Xu, H-Q Wang, J-C Zheng, Strain engineering of thermal conductivity in graphene sheets and nanoribbons: a demon-stration of magic flexibility, Nanotechnology 22, 105705 (2011).

11 K G S H Gunawardana, K Mullen, J Hu, Y P Chen, X Ruan, Tunable thermal transport and thermal rectification in strained graphene nanoribbons, Phys. Rev. B 85, 245417 (2012).         [ Links ]

12 G I Giannopoulos, I A Liosatos, A K Moukanidis, Parametric study of elastic mechanical properties of graphene nanoribbons by a new structural mechanics approach, Physica E 44, 124 (2011).         [ Links ]

13 K Kim et al., High-temperature stability of suspended singlelayer graphene, Phys. Status Solidi 4, 302 (2010).         [ Links ]

14 H Risken, The Fokker-Planck equation, Springer, New York (1989).         [ Links ]