NANO LETTERS
Thermal Conductivity Modeling of Core−Shell and Tubular Nanowires
2005 Vol. 5, No. 6 1111-1115
Ronggui Yang* and Gang Chen Mechanical Engineering Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Mildred S. Dresselhaus Department of Physics, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received April 7, 2005; Revised Manuscript Received April 22, 2005
ABSTRACT The heteroepitaxial growth of crystalline core−shell nanostructures of a variety of materials has become possible in recent years, allowing the realization of various novel nanoscale electronic and optoelectronic devices. The increased surface or interface area will decrease the thermal conductivity of such nanostructures and impose challenges for the thermal management of such devices. In the meantime, the decreased thermal conductivity might benefit the thermoelectric conversion efficiency. In this paper, we present modeling results on the lattice thermal conductivity of core−shell and tubular nanowires along the wire axis direction using the phonon Boltzmann equation. We report the dependence of the thermal conductivity on the surface conditions and the core−shell geometry for silicon core−germanium shell and tubular silicon nanowires at room temperature. The results show that the effective thermal conductivity changes not only with the composition of the constituents but also with the radius of the nanowires and nanopores due to the nature of the ballistic phonon transport. The results in this work have implications for the design and operation of a variety of nanoelectronic devices, optoelectronic devices, and thermoelectric materials and devices.
Core-shell heterostructure nanowires1 and tubular nanowires2,3 have recently been synthesized. Thermal properties of such nanostructures are of interest for many applications, for example, in the thermal management of nanoelectronics and nanophotonics, energy conversion, and sensor development.4-10 When the characteristic size of the nanostructures is comparable to the wavelength and mean free path of the heat carriers, the size effect on the effective thermal conductivity is expected, which often appears as a decrease in the thermal conductivity of nanostructures compared to their bulk counterparts. The reduced thermal conductivity not only creates hurdles for the thermal management of electronics and optoelectronic devices but also provides new opportunities for data storage devices and thermoelectric energy conversion. The thermal conductivity of nanowires and superlattice nanowires (axial heterostructure nanowires) has been studied both theoretically based on molecular dynamics simulation11-13 as well as several approaches using the Boltzmann transport equation,14-19 and also experimentally.20-23 In contrast, studies of the thermal conductivity of core-shell nanowires have been rare. Zeng and Liu presented a model on the thermal conductivity in the radial direction of core-shell structures.24 To our knowledge, there has been neither 10.1021/nl0506498 CCC: $30.25 Published on Web 04/30/2005
© 2005 American Chemical Society
theoretical nor experimental study of the thermal conductivity of core-shell nanostructure along the axial direction, including core-shell and tubular nanowires. This work presents a model calculation of thermal transport in core-shell and tubular semiconductor nanowires along the axial direction. In the past, the modeling of the thermal conductivity of nanowires has taken several directions.14-19 One approach is based on modifying the phonon dispersion relation, calculated either from continuum mechanics or lattice dynamics perspective.14-16 However, it is found that to explain the experimental data, diffuse interface scattering must be added, and the major contribution to the thermal conductivity reduction comes from the diffuse interface scattering rather than from the modification to the phonon dispersion relations. This conclusion is consistent with previous studies on thin films and superlattices.25,26 The diffuse boundary scattering, due to the short wavelength of the dominant phonon heat carriers in comparison to interface roughness, can not only reduce the phonon mean free path but can also destroy the coherence of the phonons.26 The other approach is to neglect the modification to the phonon dispersion relation and to consider interface scattering only.17-19,24 This approach is consistent with the picture that diffuse interface scattering destroys the phonon coherence,
Figure 1. (a) Generic model for phonon transport in core-shell nanostructures; (b) silicon core-germanium shell nanowires; (c) tubular silicon nanowires.
and this physical approach will be adapted in the present work. In this article, we apply the phonon Boltzmann equation to study the classical size effect on the thermal conductivity of core-shell nanostructures, including both core-shell and tubular nanowires, at room temperature. Figure 1a shows the geometry of the nanostructure considered in the generic phonon transport model that we developed for core-shell nanostructures, consisting of a tubular core layer and a shell layer. Here we use the following notations: r0 is the inner radius of the core layer, r1 is the outer radius of the core layer, and r2 is the outer radius of the shell. We assume partially diffuse and partially specular surface scattering at the inner surface of the core layer and the outer surface of the shell layer where the nanostructures interface with the external environment. The specularity parameter at the inner surface of the core layer and the outer surface of the shell layer is represented by p1 and p2 respectively, where p () p1 or p2) ) 0 corresponds to diffuse scattering and p ) 1 corresponds to specular scattering at the surface. This generic model can be used to simulate a variety of nanostructures, including nanowires and nanocomposites, by changing some of the input parameters. The results of nanocomposite modeling will be reported elsewhere.27 In this paper we focus on the thermal conductivity of core-shell and tubular nanowires. For example, when the inner radius of the tubular core layer r0 ) 0 and the interface specularity p1 ) 1 at r0 ) 0, the structure represents a solid core-shell nanowire structure (Figure 1b). When we use the same material for both the core and the shell layer, the interface between the core and shell layers disappears and the scattering at r1 dies out; the structure becomes a tubular nanowire with r0 as inner pore radius and r2-r0 as the shell thickness (Figure 1c). 1112
Table 1. Room-Temperature Parameters Used in the Simulation
silicon germanium
Cp (J/m3K)
v (m/s)
Λ (nm)
k (W/m K)
0.93 × 106 0.87 × 106
1804 1042
268.2 198.6
150 60
The phonon Boltzmann transport model is based on the following assumptions: (1) the phonon wave effect can be excluded; (2) the frequency-dependent scattering rate in the bulk medium can be approximated by using an average phonon mean free path (MFP). Assumption (1) can be well justified since the wavelength for the dominant phonons responsible for thermal transport is around 1 nm. The justification for these assumptions can be found in refs 25 and 28. In our simulation, we used the phonon MFP and the group velocity, which are obtained by approximating the dispersion of the transverse and the longitudinal-acoustic phonons with simple sine functions and neglecting the optical phonon contribution to the thermal conductivity k.25 This estimation leads to a longer mean free path than using the simple kinetic theory expression k ) 1/3CVΛ, and this finding is consistent with experiments from Goodson’s group.29 The phonon properties at room temperature used in our simulation are taken from ref 25 and listed in Table 1 for convenience. In terms of the total phonon intensity I,30,31 the 2-D phonon Boltzmann equation under the single mode relaxation time approximation in cylindrically axisymmetric coordinates can be written as ∂Ii Ii - Ioi 1 ∂ µ ∂ (rIi) (ηIi) + ζ ) r ∂r r ∂φ ∂z Λi
(1)
where the subscript i ()1,2) denotes the properties of the Nano Lett., Vol. 5, No. 6, 2005
Figure 2. Phonon transport in cylindrical coordinates.
core, and shell material Λi is the average phonon MFP, while µ, η, and ζ are the directional cosines defined as µ ) sinθcosφ, η ) sinθsinφ, ζ ) cosθ
(2)
where θ and φ are polar and azimuthal angles, respectively, as shown in Figure 2. In the phonon Boltzmann transport equation (eq 1), there are two sets of coordinate systems (shown in Figure 2): spatial coordinates (r, φc, and z) and directional coordinates (θ and φ) for the transport, which correspond to the movement of carriers in spatial and momentum space. As a phonon travels through a curved geometry such as in the cylindrical coordinates, the propagating direction is constantly varying, even though the phonon does not physically change its direction. This is why an additional term, the second term of the left-hand side in eq 1 is introduced. The difficulty of carrying out numerical simulation is also due to this term. There exist many procedures to address this term in the neutron transport literature.32 The local intensity Ioi of eq 1 is determined by the Bose-Einstein distribution of phonons and depends on the local equilibrium temperature. In nanostructures, local equilibrium cannot be established, and thus the temperature obtained should not be treated in the same way as in the case of local thermal equilibrium. An energy balance shows that Ioi can be calculated from33 Ioi (r,z) )
1 4π
∫
I (b, r Ω B ) dΩ ) 4π i 1 4π
∫02π ∫0π Ii(r,z,θ,φ) sinθ dθ dφ
(3)
and the corresponding temperature obtained from energy balance is a measure of the local energy density. For this work, we are interested only in the size effect due to the constraints in the radial direction, not in the wire axial direction. However, because heat flows along the axial direction, the simulation requires choosing a proper length and the corresponding boundary conditions at the two ends. We use a periodic boundary condition proposed previously for phonon transport.25,28 The periodic boundary condition requires that the deviation of the phonon intensity in each direction at each point in one boundary is the same as the deviation of the corresponding point and direction in the other boundary of the unit cell along the heat transport direction. For transport along the wire axis direction, the periodicity Nano Lett., Vol. 5, No. 6, 2005
is arbitrary. So we can choose an arbitrary length L and use the periodic boundary condition to obtain thermal conductivity values that are independent of the simulation domain length L. We also note that the nature of the phonon Boltzmann equation requires iterations to obtain convergent results with the boundary conditions we have defined. Equation 1 is similar to the photon radiative transport equation (RTE).33 The key is to solve for the intensity distribution I(r b,θ,φ). A variety of solution methods are available in the thermal radiative transfer literature.32,34 For phonon transport in nanostructures, the challenge is to reduce the ray effect, which often happens similarly to thermal radiation in the optical thin limit. In our previous work,35 double Gauss-Legendre quadratures have been used to replace the conventional SN quadratures for the discrete ordinate method, and this approach was shown to successfully resolve the ray effect problem in our phonon transport simulation in nanostructures. In the present work, we extend the previous work in Cartesian coordinates to cylindrical coordinates. The method used there separately discretizes the integrating points in ζ ) cosθ (the angle θ) and in the angle φ using the Gauss-Legender quadrature. To obtain high accuracy, ζ is discretized into 120 points from -1 to 1 and φ is discretized into 24 points for 0 to π (not 0 to 2π due to symmetry). Then eq 3 can be written in discrete form as I0i(r,z) )
2 4π
∑ ∑Ii(r,z,ζn,φm,)wnw′m m n
(4)
where the weights satisfy ∑m∑nwnw′m ) 2π. Following the conventional artifice of Carlson and Lathrop32 and Lewis and Miller,36 which maintains phonon radiative energy conservation and permits minimal directional coupling, the angular derivative term can be written as follows: Rn,m+1/2Ii ∂ nm nm (η Ii ) ) ∂φ
- Rn,m-1/2In,m-1/2 i wnw′m
n,m+1/2
(5)
where wnw′m is a weight and Rn,m(1/2 is the coefficient for the angular derivative term determined from the nondivergent flow condition as the following recursive equation Rn,m+1/2 - Rn,m-1/2 ) (wnw′m)µn,m
(6)
A non-uniform spatial grid system and the step scheme are used for spatial discretization to accurately capture the physics of the transport phenomena and to minimize the calculation time. Once the intensity is found by solving eq 1, the heat flux qr(r,z), qz(r,z) and the effective temperature T(r,z) can be determined through numerical integration over the whole solid angle 4π. The effective thermal conductivity along the wire direction can thus be calculated. Figure 3 shows results for the thermal conductivity of silicon core-germanium shell nanowires along the wire axis 1113
Figure 4. Thermal conductivity of tubular silicon nanowires along the wire axis direction as a function of the core radius and the specularity p at the inner (p1) and outer (p2) surfaces for shell thickness (r2 - r0) ) 50 nm and (r2 - r0) ) 20 nm tubular nanowires.
Figure 3. Thermal conductivity of silicon core-germanium shell nanowires along the wire axis direction: (a) as a function of silicon core radius r1 and (r2 - r1)/r1, assuming totally diffusive surface scattering; (b) for silicon core radius r1 ) 25 nm and 50 nm as a function of (r2 - r1)/r1 and the specularity parameter p2 at the outer surface of the shell.
direction. The phonon scattering at the silicon-germanium interface is assumed to be diffuse, and the phonon transmissivity Td12 at the interface can be calculated as19 Td12(T) )
U2(T)V2 U1(T)V1 + U2(T)V2
(7)
where U is the volumetric internal energy, V is the velocity of phonon, and the subscript i ()1,2) denotes the materials at different sides of the interface. Figure 3a shows thermal conductivity as a function of the core-shell radius ratio (r2 - r1)/r1 for different values of the core radius r1, assuming totally diffusive surface scattering at outer shell surface p2 ) 0. Also shown in Figure 3a is the thermal conductivity of the core-shell wire in the Fourier limit (bulk). Apparently, the thermal conductivity of a bulk core-shell wire would decrease as the composition of germanium, i.e., (r2 - r1)/r1, increases, since Si has much higher thermal conductivity than Ge. However, the thermal conductivity of core-shell nanowires shows a very different trend. First of all, due to the interface and surface scattering, the effective thermal conductivity of core-shell nanowires decreases as the silicon core dimension decreases. For a very small germanium shell, i.e., (r2 - r1)/r1 is small, the effective thermal conductivity has the same trend as that of the bulk structure. For larger shell thicknesses, the effective thermal conductivity increases 1114
as the composition of the lower thermal conductivity component, Ge, i.e., (r2 - r1)/r1, increases. In between, there exists a minimum. This trend can be understood as follows. Due to interface scattering, the effective thermal conductivity of the Si wire keSi is decreased to well below the bulk Ge thermal conductivity. The effective thermal conductivity of the Ge matrix keGe increases as (r2 - r1)/r1 increases, since the scattering surface per unit volume of Ge decreases. The effective thermal conductivity of the core-shell structure can be written as [keSi × r12 + keGe × (r2 - r1)2]/r22, thus accounting for the existence of a minimum value. When the silicon core radius r1 is approximately twice that of the phonon MFP in silicon, the minimum disappears and the thermal conductivity asymptotically approaches the bulk value. Figure 3b shows the change of the effective thermal conductivity with specularity for 25 nm and 50 nm silicon core-shell nanowires, showing that thermal conductivity increases when the surface of the outer shell becomes more specular (p2 increases). The same code can be used to study the thermal conductivity of tubular nanowires. Figure 4 shows results for the thermal conductivity of tubular silicon wires as a function of pore radius r0 for various values of the tube thickness and the specularity at the inner and outer surfaces. The objective of the study is to see how the transport in a tubular nanowire asymptotically approaches the 1-D simple nanowire and 2-D thin film limits. Comparing the thermal conductivity value for (r2 - r0) ) 50 nm and (r2-r0) ) 20 nm with p1 ) p2 ) 0, Figure 4 clearly shows that the thermal conductivity decreases as the shell thickness (r2 - r0) decreases, which is consistent with the thermal conductivity results for both thin film and simple nanowires. The difference of the thermal conductivity of p1 ) 0.8 (p2 ) 0) and p2 ) 0.8 (p1 ) 0) curves for the r2 - r0 ) 50 nm silicon tubular nanowire shows the asymmetric effect of the boundary scattering in tubular nanowires. When r0 is very large, the transport along the wire asymptotically approaches the transport along a thin film, i.e., the asymmetric effect disappears. When p2 is large and p1 is small (p1 ) 0, p2 ) 0.8), the nanostructure resembles more as a periodic stack of tubular nanowires Nano Lett., Vol. 5, No. 6, 2005
cylindrical nanoporous material. For fixed shell thickness (r2 - r0), increasing r0 means increasing surface scattering area and the porosity of nanoporous material, thus decreasing the effective thermal conductivity. When p1 is large and p2 is small (p1 ) 0.8, p2 ) 0), the nanostructure resembles more a simple wire. Increasing r0 for fixed shell thickness (r2 - r0) means effectively increasing the simple wire radius, thus increasing the effective thermal conductivity since the outer shell is totally diffuse. In summary, we developed a generic phonon transport model in cylindrical coordinates to study the thermal conductivity of core-shell nanostructures, i.e., core-shell silicon-germanium nanowires and tubular silicon nanowires. The results show that the effective thermal conductivity changes not only with the composition of the constituents but also with the radius of both the nanowires and nanopores, due to the nature of ballistic phonon transport. The results in this work have implications for the design and operation of a variety of nanoelectronic devices, optoelectronic devices, and thermoelectric materials and devices. Acknowledgment. The work is financially supported by DOE (DE-FG02-02ER45977) and NASA Contract #NAS303108. References (1) Lauhon, L. J.; Gudiksen, M. S.; Wang, D.; Lieber, C. M. Nature 2002, 420, 57. (2) Yin, L. W.; Bando, Y.; Golberg, D.; Lee, M. S. Appl. Phys. Lett. 2004, 85, 3869. (3) Remskar, M. AdV. Mater. 2004, 16, 1497. (4) Xia, Y. N.; Yang, P. D.; Sun, Y. G.; Wu, Y. Y.; Mayers, B. AdV. Mater. 2003, 15, 353. (5) Law, M.; Goldberger, J.; Yang, P. D. Annu. ReV. Mater. Res. 2004, 34, 83. (6) Dresselhaus, M. S.; Lin, Y. M.; Cronin, S. B. Semicond. Semimetals 2003, 71, 1. (7) Tien, C. L.; Majumdar, A.; Gerner, F. M. Microscale Energy Transport; Taylor & Francis: Washington, DC, 1998. (8) Chen, G. Nanoscale Energy Transfer and ConVersion, Oxford Press: New York, 2005.
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