## Abstract

We explore non-local effects in radially symmetric heat transport in silicon thin layers and in graphene sheets. In contrast to one-dimensional perturbations, which may be well described by means of the Fourier law with a suitable effective thermal conductivity, two-dimensional radial situations may exhibit a more complicated behaviour, not reducible to an effective Fourier law. In particular, a hump in the temperature profile is predicted for radial distances shorter than the mean-free path of heat carriers. This hump is forbidden by the local-equilibrium theory, but it is allowed in more general thermodynamic theories, and therefore it may have a special interest regarding the formulation of the second law in ballistic heat transport.

## 1. Introduction

Systems with a long mean-free path (mfp) for heat carriers are expected to show some different features in heat transport than those obeying the classical Fourier law. Up to the first order, a long mfp implies a high thermal conductivity, consistent with the Fourier law. However, in a more general setting, it also implies a long collision time and, therefore, memory effects, which are not included in the Fourier law, but which will be relevant in high-frequency perturbations and thermal wave propagation, as well as effects of second or higher order in the mfp. In the present paper, we will consider a steady situation, so that memory effects will not be relevant, and will focus on non-local effects beyond the Fourier law. Non-local effects will show if heat-flux (or temperature) inhomogeneities are sufficiently steep. One-dimensional temperature perturbations have been much studied in such systems, but two-dimensional perturbations may provide some new perspectives on non-local effects, related to the influence of the mfp on the steady-state temperature profile, which are not directly accessible in one-dimensional problems. These effects could be exhibited by radial heat transport in silicon thin layers, or in graphene sheets, which are systems with relatively long mfps (Ju & Goodson 1999; Balandin *et al.* 2008; Ghosh *et al.* 2008; Balandin 2011).

Here, we consider radial heat transport from a point source in a flat system (see figure 1 for a qualitative sketch), and we explore the possible consequences of non-local effects. In more detail, we suppose that, through a very thin wire of radius *r*_{0}, the layer is supplied with a constant amount of heat *Q*_{0} per unit time that propagates radially away from that source, and we look for the influence of non-local effects on the radial temperature profile.

The layout of the paper is as follows. In §2, the radial temperature profile is analysed to explore the consequences of non-local terms arising from a well-known phenomenological equation for heat transport. In §3, we numerically illustrate the temperature profile in two physical systems of special interest in nanoscience: silicon thin layers (§3*a*) and graphene sheets (§3*b*). In §4, we discuss the thermodynamic implications of a surprising temperature hump appearing when the radial distance from the source is smaller than (or equal to) the phonon mfp, whose consistency with the second law of thermodynamics requires more general formulations than the local-equilibrium one. In §5, we comment on the essential results of the paper.

## 2. Non-local heat transport and the steady-state temperature profile

Current theoretical developments of transport equations devote a special interest to memory, non-local and nonlinear effects (Tzou 1997; Müller & Ruggeri 1998; Luzzi *et al.* 2002; Chen 2005; Zhang 2007; Lebon *et al.* 2008; Jou *et al.* 2010) because of the interest in fast perturbations (memory effects), miniaturized systems (non-local effects) and high-power situations (nonlinear effects). In the present section, we focus our attention on non-local effects in heat transport, which are especially relevant in systems whose sizes are comparable to (or smaller than) the mfp of heat carriers, i.e. a situation that is often encountered at nanoscale. In particular, we analyse the consequences on the radial temperature profile of a flat system (represented in figure 1), arising from non-local effects in the evolution equation of the heat flux. The choice of this geometry takes us beyond the usual one-dimensional situations, in which some of the non-local effects vanish identically. In fact, in one-dimensional heat transport in nanowires and thin layers, the non-local effects are also present through the variation of the heat flux inside the system, but their effects may be embedded in size-dependent thermal conductivity (Alvarez *et al.* 2009, 2011; Sellitto *et al.* 2010*b*). Although this approach is very useful for practical applications, it slightly obscures some essential points of non-local transport.

As a starting point for our analysis, we take the following evolution equation of Guyer–Krumhansl type (Guyer & Krumhansl 1966*a*,*b*; Jou *et al.* 2010) for the heat flux **q**:
2.1where *τ*_{R} is the relaxation time related to the resistive interaction mode between the heat carriers (i.e. the phonons; Balandin *et al.* 2008), *λ*_{0} is the Ziman limit (Ziman 2001) for bulk thermal conductivity (namely, , where *ϱ* is the mass density, *c*_{v} the specific heat capacity at constant volume and the modulus of the average phonon speed), *T* is the temperature and ℓ is the mfp of phonons. Note that the first term on the right-hand side of equation (2.1) corresponds to the classical Fourier law, whereas the non-local terms are introduced by the product of ℓ^{2} and the second-order spatial derivatives of **q**. These non-local terms arise in the kinetic theory, in the so-called Burnett approximation (Burnett 1936; Chapman & Cowling 1970). They are expected to be relevant when the terms ℓ^{2}(∇^{2}**q**+2∇∇⋅**q**) are comparable to (or bigger than) **q**, i.e. when the inhomogeneities in **q** in ℓ-size regions are relevant enough.

Generally, to obtain the behaviour of the temperature profile, the evolution equation of the heat flux is coupled both with the local balance of the specific internal energy *e* per unit volume, i.e.
2.2and with an equation of state relating *e* and *T*, as for instance d*e*=*c*_{v}d*T*.

In steady-state situations, from equation (2.2) we have ∇⋅**q**=0. Since in one-dimensional situations, the heat flux has only one component (e.g. the component along the *x*-axis), this equation implies that *q*_{x,x}=*q*_{x,xx}=0, where the symbols ,_{x} and ,_{xx} mean the first- and second-order spatial derivatives of the corresponding quantity, respectively. Therefore, such situations are not the most suitable to analyse the role played by non-local effects.

In contrast, in two-dimensional situations wherein the heat diffuses radially from a source, instead of propagating longitudinally, the role played by non-local effects can be shown. Owing to the radial symmetry, in fact, in these cases, the heat still has only one component (i.e. the radial component), but the steady-state condition implies that the corresponding heat-flux radial profile *q* is
2.3with *r* being the radial distance from the centre of the hot spot, *Γ*=*Q*_{0}/(2*πh*) and *h* being the thickness of the layer (figure 1). It is important to observe that, since we are approaching the problem from a macroscopic point of view, we assume that the layer behaves isotropically, namely, we suppose that the heat propagates in the same way along all directions in the plane.

Introducing equation (2.3) in the steady-state version of equation (2.1), when *r*>*r*_{0}, we have
2.4

Finally, by integration of equation (2.4), we obtain the radial behaviour of the temperature profile, which reads
2.5once the condition that the temperature is *T*_{0} in the circular spot of radius *r*_{0} has been used. In the absence of non-local terms, instead, one would have
2.6

The previously mentioned results point out that in a range of radial distances, the orders of which are several times greater than those of the mfps ℓ, the radial behaviour of the temperature *T*(*r*) is strongly modified by the non-local terms. It seems important to note that, according to equation (2.4), in an annular region with *r*<ℓ (i.e. the dotted circular area in the zoomed-in area of figure 1), the temperature increases with radius, instead of decreasing, as could have been expected both on intuitive grounds and from the Fourier law. This behaviour will be explicitly illustrated in §3. Although this phenomenon is against the local-equilibrium formulation of the second law of thermodynamics, it is allowed by more general formulations of the second law. We refer the readers to §4, where we deal with this topic deeper.

## 3. Consequences of non-local terms in silicon and graphene

The temperature behaviour in equation (2.5) could be explored, for instance, in two physical situations, which are popular topics nowadays: silicon thin layers and graphene sheets. These systems have been much studied theoretically (Alvarez *et al.* 2009, 2011) and experimentally (Asheghi *et al.* 1997; Ju & Goodson 1999; Liu & Asheghi 2004; Saito *et al.* 2007; Balandin *et al.* 2008; Ghosh *et al.* 2008; Balandin 2011). However, the thermodynamical behaviour in two-dimensional radial transport problems remains largely unexplored because the interest of researchers has been focused principally on one-dimensional perturbations.

### (a) Silicon thin layers

Thin layers are widely applied in the nanoelectronics industry for the fabrication of transistors, solid-state lasers, sensors and actuators. Since the thermal-transport properties affect the performance and reliability of these devices, a better understanding of their thermal behaviour is useful for practical applications.

In order to investigate the consequences of equation (2.5), we assume that heat is supplied to a point (or a small circular spot) of a thin layer of silicon. The heat flux will be inhomogeneous, both in the radial direction (i.e. along the plane of the layer) and in the transverse direction (i.e. across the layer, as shown in figure 2).

Indeed, one-dimensional perturbations across the transverse section of the layer are well described by an effective thermal conductivity, which decreases for decreasing thickness of the layer (Alvarez *et al.* 2009, 2011; Sellitto *et al.* 2010*a*,*b*), both because of the increased role of collisions with the walls and because of phonon confinement effects (Balandin & Wang 1998). Therefore, in such a situation, the evolution equation of the heat flux is still given by equation (2.1), but with the bulk value of the thermal conductivity *λ*_{0} replaced by an effective value of the thermal conductivity *λ*_{eff}, dependent on the ratio between ℓ and *h*, i.e. the so-called Knudsen number Kn=ℓ/*h*. This effective thermal conductivity is being explored from both microscopic grounds and mesoscopic perspectives.

By using a phonon-hydrodynamic approach (Jou *et al.* 2010, 2011), in the steady-state situation and when Kn>1 (i.e. when *h*<ℓ) in Alvarez *et al.* (2009) (see eqn (16) therein), the following effective thermal conductivity has been obtained:
3.1where *C* is a coefficient that describes the effects of the interactions between the phonons and walls (Alvarez *et al.* 2009, 2011), namely, the specular and diffusive reflections of phonons. This parameter depends both on the wall characteristics and on the temperature (Sellitto *et al.* 2010*a*,*b*). Equation (3.1) arises from the observation that the longitudinal heat flux *q* has two different contributions (Alvarez *et al.* 2009, 2011): the bulk contribution (*q*_{b}), obtained as the solution of the steady-state version of equation (2.1), and the wall contribution (*q*_{w}), obtained as the spatial derivative of *q*_{b} along the transverse direction, valued on the wall. Both contributions can be seen in the zoomed-in area of figure 2.

However, *λ*_{eff} takes into account the non-local effects in the transverse, but not in the radial direction. To do this, we again use equation (2.5). Thus, in a thin layer, the radial behaviour of the temperature profile becomes
3.2with *λ*_{eff} given by equation (3.1), instead of *λ*_{0} appearing in equation (2.5). Indeed, one may observe that though in one-dimensional situations the Fourier law may provide a useful description of heat transport in nanosystems whenever a suitable effective thermal conductivity *λ*_{eff} is used (namely, if one supposes **q**=−*λ*_{eff}∇*T*), it would lead to the following radial behaviour of the temperature profile:
3.3outside the heated spot. Equation (3.3) is functionally different from the behaviour predicted by equation (3.2), where the non-local terms yield a further contribution, proportional to *r*^{−2}, which is lacking in equation (3.3) and cannot be described by means of an effective thermal conductivity.

The different behaviours may also be clearly seen in figure 3, wherein the consequences of non-local effects on the radial behaviour of the temperature have been shown for a silicon thin layer with *h*=50×10^{−9} m. In the computation, we have supposed that the layer is heated with *Q*_{0}=10^{−8} W, and the spot radius is *r*_{0}=50×10^{−9} m. Note that we have chosen such a value for *r*_{0} because, here, we are interested in a small value of the radius, smaller than the phonon mfp, and we have taken as an indicative size that of the layer thickness. However, the effect of the radius will be further considered later, in figure 4. We have taken two different values for *T*_{0}, namely, *T*_{0}=150 K and *T*_{0}=100 K. The corresponding wall-accommodation parameters *C*=0.36 and *C*=0.46 have been taken from Sellitto *et al.* (2010*b*) and refer to a smooth wall, so that the effective thermal conductivity at *T*_{0}=150 K and *T*_{0}=100 K are, respectively, *λ*_{eff}=23.2 Wm^{−1}K^{−1} and *λ*_{eff}=18.8 Wm^{−1}K^{−1}, according to equation (3.1).

As it is possible to observe, the temperature behaviour arising from equation (3.2) (solid lines in figure 3) shows a hump; namely, when *r*<ℓ, the temperature increases with radius. This does not appear in the temperature behaviour given by equation (3.3) (dashed lines in figure 3), which, instead, is always decreasing. Moreover, figure 3 shows that the difference in temperature Δ*T*,
3.4between the predictions of equations (3.2) and (3.3) increases for decreasing *T*_{0}. This is because, in silicon, for decreasing temperature, the phonon mfp increases. Although the Δ*T* in figure 3 is small (of the order of 10^{−2} K in the peak at 150 K, and of the order of 10^{−1} K in the peak at 100 K), it could be measured in principle. However, since Δ*T* is inversely proportional to *λ*_{eff}, it would be still higher in the case of rough-walled thin layers (Alvarez *et al.* 2009; Sellitto *et al.* 2010*a*,*b*), where *λ*_{eff} is an order of magnitude smaller than in smooth-walled thin layers.

Another way to increase the difference in temperature Δ*T* would be either to increase the amount of heat per unit time *Q*_{0}, or to reduce the heated-spot radius *r*_{0}. In figure 4, the variation of Δ*T* in terms of *r*_{0} is examined, at a distance *r*=ℓ from the centre of the heated spot, for fixed values of *h*, *Q*_{0} and *T*_{0} (they are the same as those used in figure 3). Figure 4 points out that the smaller the *r*_{0}, the greater the Δ*T*. We see that reducing the radius by a factor 2 (i.e. from 50 to 25 nm) raises Δ*T* by a factor 4. Thus, from the operational point of view, it seems easier to go from *T*_{0}=150 K to *T*_{0}=100 K, which increases Δ*T* by a factor 10, without a need to reduce the radius of the heat-supplying nanowire.

### (b) Graphene sheets

Graphene is a flat monolayer of carbon atoms tightly packed into a two-dimensional honeycomb lattice (Geim & Novoselov 2007; Balandin *et al.* 2008; Ghosh *et al.* 2008), which can be wrapped up into zero-dimensional fullerenes, rolled into one-dimensional nanotubes, or stacked into three-dimensional graphite. This means that here, we could practically speak about a sheet, or a very thin layer, the thickness of which is *h*≈5×10^{−10} m, namely, it is of the order of the atomic diameter of carbon. At room temperature, the bulk thermal conductivity of graphene is in the range *λ*_{0}∼ 3080–5150 Wm^{−1}K^{−1}, and the phonon mfp is of the order of ℓ=7.75×10^{−7} m (Ghosh *et al.* 2008; Nika *et al.* 2009*a*,*b*; Balandin 2011). By using these data, assuming the same values for the amount of heat per unit time *Q*_{0} and for the spot radius *r*_{0} we used in §3*a*, for graphene sheets, the difference in temperature Δ*T* predicted by equation (3.4) is in the range Δ*T*∼ 0.12–0.07 K, which is of the same order of magnitude as the silicon thin layers considered earlier.

The thermal properties of graphene have been recently reviewed in detail by Balandin (2011), with further information about analogous properties in nanostructured carbon materials. It is pointed out how graphene (or other materials that are able to conduct heat well) could be essential for the refrigeration of small devices, as in the next generation of highly miniaturized integrated circuits, or optoelectronic and photonic devices, by removing the heat dissipated during functioning at very high frequencies. Thus, the hot spot considered in our paper could be imagined as a device connected to a graphene sheet, instead of a wire supplying heat from outside the system. In this case, the radius of the spot would be related to the size of the device. This more realistic and practically interesting situation enhances the possible interest in the problem examined here, namely, radial propagation of heat in physical conditions in which the Fourier equation is not strictly valid because of the comparable size of the mfp (some 700 nm) to the size of some miniaturized devices.

## 4. Thermodynamic aspects

In §2, we obtained, in equation (2.5), the radially dependent temperature profile and we noted a surprising behaviour in the annular region between *r*_{0} and ℓ. In this region, the temperature increases with radius, instead of decreasing, as would have been expected on classical grounds and predicted by the classical Fourier law. The decrease is found when *r*>ℓ. This has been explicitly illustrated in §3. The temperature-hump behaviour is forbidden by the classical local-equilibrium formulation of the second law, stating that
4.1with *σ*_{le} as the entropy production per unit time and volume in the local-equilibrium approximation, which must be non-negative along any thermodynamic processes at any time and any point. In fact, introducing in the previously mentioned equation (2.3) for *q*(*r*) and equation (2.4) for d*T*/d*r*, we are led to
4.2which is negative for *r*<ℓ, i.e. in the region where the temperature increases with radius. Thus, if equation (4.2) is used, then equation (2.1) would be forbidden. It is known, however, that equation (2.1) follows from the Boltzmann equation for phonons and that it describes some important features of heat transport in dielectric systems (Guyer & Krumhansl 1966*a*,*b*; Jou *et al.* 2010), so that it is worthy of physical consideration, instead of being forbidden.

However, it is known that the transport equation (2.1) is compatible with the second law of thermodynamics in the framework of extended irreversible thermodynamics (Lebon *et al.* 2008; Jou *et al.* 2010; Mendez *et al.* 2010). In this theory, the heat flux **q** and the flux of heat flux **Q** (which is a second-order tensor) are considered as independent variables in the state space, on the same grounds as the internal energy (Jou *et al.* 1999, 2010; Lebon *et al.* 2008). Therefore, the extended entropy has the functional dependence *s*_{eit}=*s*_{eit}(*e*;**q**;**Q**). If the relaxation time of **Q** is sufficiently smaller than that of **q**, then the flux of heat flux reduces to **Q**=ℓ^{2}∇**q** (Jou *et al.* 1999; Sellitto *et al.* 2010*a*), and the extended entropy becomes *s*_{eit}=*s*_{eit}(*e*;**q**;∇**q**), exhibiting, in a more explicit way, the influence of non-local terms. In this formalism, the entropy production is given by Jou *et al.* (1999) and Sellitto *et al.* (2010*a*),
4.3which reduces to equation (4.1) whenever ℓ=0. Thus, it is seen that the formulation of the second law in ballistic situations (i.e. for distances shorter than ℓ) is especially demanding.

Introducing equations (2.3) and (2.4) into equation (4.3), we are now led to 4.4which is positive everywhere. Thus, if the second law is expressed as the requirement of the extended entropy, equation (2.1) (and the subsequent temperature hump considered here) is compatible with the second law of thermodynamics. Note that the first term in equation (4.4) corresponds to the usual Fourier law combined with local-equilibrium theory, as it should be, because when ℓ=0, equation (2.1) reduces to the Fourier law for heat transport.

From this point of view, the hump in temperature profile we observed between *r*_{0} and ℓ would have much interest from a basic thermodynamic perspective because it would explicitly show both the influence of non-local effects in heat transport (not describable by the Fourier law) and the need of generalizing the formulation of the second law. For the sake of completeness, let us observe that a similar hump, in the travelling front solution of a reaction–diffusion equation with a non-local term has been obtained, on theoretically grounds, as a consequence of non-localities in the equation for the diffusion flux (Gourley 2000) but this behaviour has not been addressed from a thermodynamic perspective.

Let us remark that in Sellitto *et al.* (2010*a*), we already pointed out the need of considering equation (4.3) in the thermodynamic description of phonon backscattering in rough-walled nanowires. Indeed, in that situation, there was a very thin layer, near the wall of the system, the characteristic size of which was of the order of the roughness size, wherein the heat flux pointed in the same direction as the temperature gradient, instead of pointing in the opposite direction, as required by the Fourier law. However, the situation shown here seems more easily accessible to experiments, as the size of the region involved (of the order of 180–750 nm in the situations considered in §3) is considerably bigger than that of the wall roughness in nanowires (of the order of 2–8 nm).

## 5. Concluding remarks

In this paper, we have shown that non-local effects described by equation (2.1), which have been much studied in longitudinal heat flux along nanowires and thin layers (Alvarez *et al.* 2009, 2011; Sellitto *et al.* 2010*a*,*b*), could be pointed out in a more direct way in situations where the heat flux propagates radially in an axial-symmetric geometry. In particular, in the steady-state situations we have considered here, equation (2.1) clearly exhibits its difference with respect to a constitutive equation for the heat flux of the Fourier-law type, namely,
5.1where *λ*_{eff}(Kn) is a size-dependent effective thermal conductivity, Kn=ℓ/*r* being the Knudsen number, which seems to be sufficient in the study of longitudinal heat flux along nanosystems.

The most surprising aspect is the appearance of a hump in the temperature profile, whenever the radial distance *r* from the heated-spot is smaller than the mfp ℓ. The hump is at odds with the local-equilibrium formulation of the second law, but not with the extended thermodynamic formulation of the second law. A physical interpretation of this hump could be that in the regions characterized by *r*<ℓ, the incoming phonons interact very rarely with the particles of the system. The maximum interactions will happen when *r*≈ℓ, where the ordered energy of non-scattered phonons will become the disordered internal energy of the system, thus raising the temperature, understood here as a measure of the disordered vibrational energy of the particles (Si atoms or C atoms in the system studied here).

Note that the difference between the temperature profile following from equations (2.5) and (2.6) can be found not only when *r*<ℓ, but also at distances of the order of several times ℓ. This difference could be checked in all that region. In this way, nanosystems also become an exciting ground for the exploration of the range of validity of thermodynamic concepts as, for instance, the more suitable form of the transport equation and of the formulation of the second law of thermodynamics. Furthermore, this suggests that axially symmetric geometries (as, for instance, concentric superlattices) could be worth studying, as they could show features not expected from current analysis based on the effective Fourier law.

## Acknowledgements

A.S. acknowledges the financial support from the Departament de Física of Universitat Autònoma de Barcelona for his stay in the Autonomous University of Barcelona. D.J. and J.B. acknowledge the financial support from the Dirección General de Investigación of the Spanish Ministry of Science and Innovation under grant FIS no. 2009-13370-C02-01, the Consolider Project NanoTherm (grant CSD-2010-00044) and the Direcció General de Recerca of the Generalitat of Catalonia under grant no. 2009-SGR-00164.

- Received September 25, 2011.
- Accepted November 1, 2011.

- This journal is © 2011 The Royal Society