Thermal conductivity and thermal boundary resistance of nanostructures
© Termentzidis et al; licensee Springer. 2011
Received: 9 December 2010
Accepted: 4 April 2011
Published: 4 April 2011
We present a fabrication process of low-cost superlattices and simulations related with the heat dissipation on them. The influence of the interfacial roughness on the thermal conductivity of semiconductor/semiconductor superlattices was studied by equilibrium and non-equilibrium molecular dynamics and on the Kapitza resistance of superlattice's interfaces by equilibrium molecular dynamics. The non-equilibrium method was the tool used for the prediction of the Kapitza resistance for a binary semiconductor/metal system. Physical explanations are provided for rationalizing the simulation results.
68.65.Cd, 66.70.Df, 81.16.-c, 65.80.-g, 31.12.xv
Understanding and controlling the thermal properties of nanostructures and nanostructured materials are of great interest in a broad scope of contexts and applications. Indeed, nanostructures and nanomaterials are getting more and more commonly used in various industrial sectors like cosmetics, aerospace, communication and computer electronics. In addition to the associated technological problems, there are plenty of unresolved scientific issues that need to be properly addressed. As a matter of fact, the behaviour and reliability of these devices strongly depend on the way the system evacuates heat, as excessive temperatures or temperature gradients result in the failure of the system. This issue is crucial for thermoelectric energy-harvesting devices. Energy transport in micro and nanostructures generally differs significantly from the one in macro-structures, because the energy carriers are subjected to ballistic heat transfer instead of the classical Fourier's law, and quantum effects have to be taken into account. In particular, the correlation between grain boundaries, interfaces and surfaces and the thermal transport properties is a key point to design materials with preferred thermal properties and systems with a controlled behaviour.
Fabrication process of superlattices
We aim to fabricate optimized vertical nano-superlattices (with layers ranging <100 nm each) with high thermoelectric efficiency. High thermoelectric efficiency occurs for high electrical conductivity and low thermal conductivity. The electronic conductivity will be controlled though the Si doping and the use of metal to fill in the trenches. The film thickness needs to be decreased, to decrease the individual layer thermal conductivity and increase the influence of the interfacial thermal resistance.
To obtain such dimension on a large area at low cost, we are developing a process based on the transfer by DRIE of 30-nm line patterns made of di-block copolymers . For this purpose, it is required to characterize them to the best possible degree of accuracy. Measurements at this scale will possibly be plagued by quantum effects [5, 6]. That is the reason why we fabricated first micro-scale superlattices, to make thermoelectric measurements free from quantum effects and then applied the method to characterize the final nano-superlattice thermoelectric devices.
Simulations: thermal conductivity of superlattices
When the layer thickness of the superlattices is comparable to the phonon mean free path (PMFP), the heat transport remains no longer diffusive, but ballistic within the layers. Furthermore, decreasing the dimensions of a structure increases the effects of strong inhomogeneity of the interfaces. Interfaces, atomically flat or rough, impact the selection rules, the phonon density of states and consequently the hierarchy or relative strengths of their interactions with phonons and electrons. Thus, it is important to study and predict the heat transfer and especially the influence of the height of superlattice's interfaces on the cross and in-plane thermal conductivities. This is a formidable task, from a theoretical point of view, as one needs to account for the ballistic motion of the phonons and their scattering at interfaces. Molecular dynamics is a relatively simple tool which accounts for these phenomena, and it has been applied successfully to predict heat-transfer properties of superlattices.
Two routes can be adopted to compute the thermal conductivity, namely, the non-equilibrium (NEMD)  and the equilibrium molecular dynamics (EMD) . In this article, we have considered both methods to characterize the thermal anisotropy of the superlattices. In the widely used direct method (NEMD), the structure is coupled to a heat source and a heat sink, and the resulting heat flux is measured to obtain the thermal conductivity of the material [9, 10]. Simulations are held for several systems of increasing size and finally thermal conductivity is extrapolated for a system of infinite size [11, 12]. The NEMD method is often the method of choice for studies of nanomaterials, while for bulk thermal conductivity, particularly that of high conductivity materials, the equilibrium method is typically preferred because of less severe size effects. Comparisons between the two methods have been done previously, concluding that the two methods can give consistent results [13, 14]. Green-Kubo method for nanostructures is proven to have greater uncertainties than those of NEMD, but a correct description of thermal conductivity with EMD is achieved by establishing statistics from several results, starting from different initial conditions.
The superlattice system under study is made of superposition of Lennard-Jones crystals and fcc structures, oriented along the  direction. The molecular dynamics code LAMMPS [15–17] is used in all the NEMD and EMD simulations. The mass ratio of the two materials of the superlattice is taken as equal to 2, and this ratio reproduces approximately the same acoustic impedance difference as that between Si and Ge. Periodic boundary conditions are used in all the three directions. Superlattices with period of 40a 0 are discussed, where a 0 is the lattice constant. The shape of the roughness is chosen as a right isosceles triangle. The roughness height was varied from one atomic layer (1 ML = 1/2a 0) to 24a 0. For each roughness, heat transfer simulations with NEMD were performed for several system sizes in the heat flux direction to extrapolate the thermal conductivity for a system of infinite size . For EMD simulations, the size of the system is smaller than with NEMD simulations and only one size is considered 20a 0 × 10a 0 × 40a 0, where the last dimension is perpendicular to interfaces.
In the diffuse-mismatch model, on the other hand, phonons are diffusively scattered at interfaces, and their energy is redistributed in all the directions . In practice, the acoustic model describes the physics of interfacial heat transfer at low temperatures, for phonons having large wavelengths, while the diffuse model is relevant for small wavelengths phonons. At the considered temperature in the current study, we are most probably in an intermediate situation where the physics is not captured by one single model. Nevertheless, both models predict that a moderate amount of interfacial roughness will tend to decrease the in-plane TC, because rough interfaces will increase specular reflection and diffusive scattering of phonons travelling in the in plane direction. However, if the roughness is large enough, then locally, the phonons encounter smooth-like interfaces, and the partial group of phonons that are diffusely scattered in all space direction decreases. This might explain the further increase of the thermal conductivity when the roughness is large enough.
The behaviour of the cross-plane thermal conductivity is different: it increases monotonously with the interfacial roughness. For smooth interfaces, the cross-plane thermal conductivity is 50% lower than the in-plane thermal conductivity. This anisotropy has to be taken into account for thermal behaviour of systems made of sub-micronic solid layers. Invoking again the acoustic mismatch model, we conclude that the transmission coefficient of the solid/solid interface is smaller than the reflection coefficient, which is not surprising if we consider the acoustic impedance ratio of the two materials. Roughness increases the transmission coefficient as it increases the diffused scattering at the interface .
The same qualitative trend regarding the influence of the roughness on the thermal conductivity of superlattices has been reported previously for materials with diffusive behaviour, without thermal contact resistance . In this case, the variation of the in-plane and cross-plane conductivities with the interfacial roughness is due to the heat flux line deviation that minimizes the heat flux path in the material that has the lower thermal conductivity. This tends to increase the cross-plane thermal conductivity. On the other hand, the increase of the roughness leads to the heat flux constrictions that decrease the in-plane thermal conductivity. The qualitative interpretation of the results shows that the thermal contact resistance of the interface has a strong influence on the superlattice thermal conductivity.
Simulations: Kapitza resistance
Superlattices with rough interfaces
where S is the interface area. The latter formula expresses the fact that the resistance is controlled by the transmission of all the phonons travelling across the interface.
In the situation of interest to us here, the transmission of phonons is expected to be strongly anisotropic, and thus the resistance developed by an interface should depend on the main direction of the heat flux. To measure this anisotropy, we have generalised the previous equation and introduced the concept of directional resistance, by considering the heat flux q θ(t) in the direction θ in (0,π/2) with the normal of the interface.
This angular Kapitza resistance quantifies the transmission of the heat flux in the direction making an angle θ with the normal of the interface.
The Cu and Ag films on Si-oriented substrates are the principal combinations in large-scale integrate circuits. Furthermore, with the fabrication process of vertical-built superlattices described in previous section, we are interested in the heat transfer phenomena related to the metal/semiconductor interfaces. The prediction of heat transfer in these systems becomes challenging when the thickness of the layers reaches the same order of magnitude as the PMFP. For heat transfer studies, MD is well suited for dielectrics since only phonons carry heat. For metals, coupling between phonons and electrons can be modelled with the two-temperature model . For the above systems, it has been proven that the Kapitza resistance is mainly due to phonon energy transmission through the interfaces [23, 24]. The interfacial thermal resistance, known as the Kapitza resistance [25, 26] is important to be studied as it might become of the same order of magnitude than the film thermal resistance. In this section, interatomic potentials for Ag and Si are discussed. Using NEMD simulations, for an average temperature of 300 K, the Kapitza resistance of Si/Ag systems is determined.
Modified embedded-atom method (MEAM) is the only appropriate potential that can be used for metal/semiconductor systems. The first nearest-neighbour MEAM (1NN MEAM) potential by Baskes et al.  and the second nearest-neighbour MEAM (2NN MEAM) by Lee  are examined in the current study. The general MEAM potential is a good candidate for simulating the dynamics of a binary system with a single type of potential. For example, it can be applied for both fcc and bcc structures. Furthermore, this potential includes directional bonding, and thus can be applied for Si systems.
The 2NN MEAM potential allows recovering the expansion coefficient for Ag quite accurately while the 1NN MEAM potential significantly underestimates it. For Ag, the two potentials provide a good description for the more basic properties, such as cohesive energy, lattice parameters and bulk modulus . Even if the 1NN MEAM potential gives results closer to the experimental values for dispersion curves, the values obtained for the linear thermal expansion are not reasonable. Therefore, the 1NN MEAM potential cannot be considered appropriate for simulating heat transfer for silver. Regarding the investigation of heat-transfer temperature, the 2NN MEAM gives the best results for harmonic and anharmonic properties for silver and for silicon using the previous results of the literature . Kapitza resistance is predicted for the 2NN MEAM Si/Ag potential. The interface thermal resistance, also known as Kapitza resistance, R K, creates a barrier to heat flux and leads to a discontinuous temperature, ΔT, drop across the interfaces.
The Kapitza resistance obtained with NEMD is 4.9 × 10-9 m2K/W. The temperature profile for Si is almost flat due its high thermal conductivity. With MD simulations, it is not possible to simulate heat transfer due to the electrons, and thus the steep slope of Ag is due to its low lattice thermal conductivity. The value R KT is in the range 1.4-125 × 10-9 m2K/W which also includes the Kapitza conductance for dielectric/metal systems [33, 34].
Conclusions - Discussion
A new fabrication method for superlattices is used, reducing the time and fabrication costs. With the fabrication of vertical superlattices, several questions a rose for the influence of the roughness' height of the superlattices and the quality of interface on the thermal transport. When the length of the superlattice's period is comparable to the phonon-free mean path, the heat transfer becomes ballistic.
The cross-plane and in-plane thermal conductivities of a dielectric/dielectric (representing Si-Ge systems) superlattice are predicted using EMD and NEMD simulations. Both methods give the same tendencies for the anisotropic heat transfer at superlattices with rough interfaces. The in-plane thermal conductivity exhibits a minimum for a certain interfacial width, while the cross-plane thermal conductivity increases modestly in increasing the width of the interfaces. The Kapitza resistance of these interfaces is also studied, with a proposed methodology in this article, introducing the concept of directional thermal resistance. Values presented here are coherent with the difference between the in-plane and cross-plane thermal conductivities.
Molecular dynamics simulations are also used to study the metal/semiconductor interfaces. Among all the interatomic potentials that are available, the MEAM potential is a good alternative to work with since it can be used for different materials. At 300 K, the 2NN MEAM potential gives the best results for the fundamental properties associated with the heat transfer of silicon and silver. Previous results [23, 24, 32] suggest that interfacial thermal conductance depends predominantly on the phonon coupling between silicon and metal lattices so that Si/Ag can be simulated without considering the contribution of electron heat transfer. The value of magnitude of the Kapitza resistance for a Si/Ag system is within the range of Kapitza resistance proposed in the literature.
This study proves that making rough instead of smooth interfaces in superlattices is a useful way to decrease the thermal conductivity and finally to design materials with desired thermal properties. Furthermore, when more interfaces are added (rough or smooth), i.e. when the superlattice's period decreases, the interfacial thermal resistance becomes comparable to the superlattice's layers thermal conductivity. With these two parameters, namely, the introduction of rough interfaces and the decrease of the superlattice's period, we can create systems with controlled values of the thermal conductivity.
deep reactive ion etching
scanning electron microscope
phonon mean free path
non-equilibrium molecular dynamics
equilibrium molecular dynamics
This study has been conducted within the framework of the projects ANR-COFISIS (ANR-07-NANO-047-03). COFISIS (Collective Fabrication of Inexpensive Superlattices in Silicon) is a project with collaboration between theoretical and experimental groups in ESIEE Paris, CETHIL and MATEIS at INSa of Lyon. The project COFISIS intends to develop integrated silicon-based and low-cost superlattices.
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