NANO LETTERS
Predicting the Thermal Resistance of Nanosized Constrictions
2005 Vol. 5, No. 11 2155-2159
Ravi Prasher* Intel Corporation, CH5-157, 5000 West Chandler BouleVard, Chandler, Arizona 85226 Received August 26, 2005
ABSTRACT Various devices and technologies using nanowires and nanoparticles are under intense investigation because of their promise. In these devices, nanowires or nanoparticles are typically in contact with another surface. The contact between a nanowire and a nanoparticle with a substrate forms a constriction of the order of a few nanometers. A continuum description of heat transport at these nanosized constrictions will break down. In this paper, an analytical model is presented in which the relevant length scales have been taken into consideration. The results show that the constriction resistance of nanoconstrictions is much higher than those predicted using macroscopic approaches. The Knudsen number is the key parameter for constriction formed between the same materials, whereas the microscopic Biot number based on phonon thermal boundary resistance is the key parameter for constriction formed between dissimilar materials. Finally, the model is applied to calculate the thermal resistance of the nanowire/planar interface.
Thermal transport in nanosctructures and their interfaces is very important for a variety of technologies.1 Such examples include thermal transport in nanoscale microelectronic devices, thin-film theromoelectrics,2 and nanoparticle-based thermal interface materials.3,4 Other applications include the development of accurate metrologies for the measurement of thermal properties such as the thermal conductivity of nanowires.5 In most of the applications and metrologies, the nanostructures, such as nanowires or nanoparticles, are typically in contact with a substrate. Thermal transport in these nanostructures will be dominated by the thermal resistance at the interface formed between the nanostructure and the substrate.1 Considerable progress, both in modeling and experiments, has been made in understanding the thermal transport across planar interfaces such as thin-film supperlattices.2 Theoretical modeling of interfacial thermal transport between a planar surface and a curved nanostructure such as a nanowire is still in its infancy.6 The technological potential of this geometrical configuration is immense. Therefore, it is very important to develop models for the thermal resistance of nanosized curved surfaces such as spherical nanoparticles and cylindrical nanowires with a planar surface. Recently, Bahadur et al.6 modeled the interfacial thermal resistance of a nanowire/plane interface using macroscopic constriction resistance models. They developed a model to calculate the interfacial area between a nanowire and a substrate by using the van der Waals force between the nanowire and the substrate. They used the size-dependent thermal conductivity of the nanowire in their constriction * Corresponding author. E-mail:
[email protected]. 10.1021/nl051710b CCC: $30.25 Published on Web 10/19/2005
© 2005 American Chemical Society
model. The thermal model by Bahadur et al.6 is not very accurate for the modeling of the constriction resistance of nanosized constrictions because it ignores the ballistic nature of thermal transport in the vicinity of the constrictions. Ballistic transport of phonons will be very important at the constriction if the constriction size is comparable to the mean free path (mfp) of carriers such as phonons or electrons. Their calculation showed that the width of the constriction formed between the silicon (Si) nanowire and the Si substrate was in the range of 1-3 nm. The mfp (l) of phonons in Si at room temperature is approximately 250 nm,2,7 therefore assuming that the bulk model for such small constrictions is not justified. Recently, Yu et al.8 used the model by Bahadur et al.6 to estimate the thermal resistance of the carbon nanofiber/platinum interface. They found that the model prediction was approximately an order of magnitude smaller than the experimental data. Considering the discussions in the previous paragraphs, it is very important to develop a model for the thermal resistance of these nanosized constrictions that can account for the microscale effects. This paper develops an approximate analytical model that accounts for the ballistic transport near a constriction. The focus is on phononmediated transport. The model is first developed for two planar surfaces in contact where the thermal transport is possible only through a small area at the interface. Figure 1 shows two surfaces in contact where thermal transport at the interface is possible only in the circular aperture. The surface outside the aperture is completely insulated. Circular constriction will be formed between a spherical nanoparticle and a planar substrate. The radius of the circular aperture is
method2 calculates an effective Vg that includes the contribution from all three modes. Only acoustic phonons participate in thermal transport because of the very small group velocity of the optical phonons.2 The heat flux (q) from side 1 at the aperture shown in Figures 1 and 2 is given by11 q1 )
Vg
ω π/2 2π ∫ ∫ ∫ ∑ 0 0 0 4π 3
pω
m
exp
( ) pω
kbT
D(ω) dω dθ dφ ) -1 U(T1)Vg 4
Figure 1. Phonon transport through the circular constriction. The area surrounding the constriction is insulated.
where ω is the frequency of the phonons, ωm is the maximum frequency of phonons, D(ω) is density of states, U is the internal energy per unit volume of the acoustic phonons, T is the temperature, θ is the polar angle, and φ is the azimuthal angle. Therefore, the net heat flux assuming that T1 is larger than T2 is given by q)
Figure 2. Side of view of Figure 1 showing that phonon transport at the interface is allowed only though the aperture (constriction).
a as shown in Figure 1. Figure 2 shows the side of view of Figure 1. In the first part of the paper, the model is developed for circular constriction and the material on both sides is considered to be same. The extension to rectangular constriction and different materials on each side is a simple extension and is presented in the second part of the paper. In the completely diffusive limit where a is much larger than the mfp (a . l), the diffusive constriction resistance (Rcd) is given by the Maxell constriction resistance9 Rcd )
1 1 1 + ) 4ka 4ka 2ka
(1)
In the completely ballistic limit (a , l), eq 1 is not valid. In the completely ballistic limit, Rc (constriction resistance) can be calculated using the method to calculate the flow rate of gas molecules through an orifice in the free molecular flow regime.10 Free molecular flow of gases is analogous to the ballistic transport of phonons because both correspond to the case where the mfp is much larger than the characteristic dimension of the system under consideration. In calculating Rc in the ballistic regime, frequency-independent group velocity (Vg) is assumed (grey medium approximation). The extension to frequency-dependent Vg is simple. Although a frequency-independent Vg is assumed, Vg is calculated based on realistic dispersion relations using Chen’s method.2 Vg is also assumed to be the same for all three modes. Chen’s 2156
(2)
Vg [U(T1) - U(T2)] 4
(3)
If the temperature difference across the interface is not large, then the specific heat per unit volume (C) on both the sides can be assumed to be the same (the material is same on both sides). Therefore, eq 3 can be written as q)
VgC [T1 - T2] 4
(4)
Therefore, the ballistic constriction resistance (Rc) can be written as Rcb )
T1 - T2 4 ) qA AVgC
(5)
where A is the area of the constriction. Using k ) 1/3CVgl, eq 5 can be written as Rcb )
4l 3kA
(6)
Equation 6 is same as that derived by Sharvin9,12 for electron transport across constriction formed between the same materials in the ballistic regime. Equation 6 shows that Rcb does not depend on the shape of the constriction. Therefore, eq 6 will be same for circular or rectangular constriction. For nanoconstrictions, the transport across the constriction will typically fall in the ballistic regime because the mfp of phonons in crystalline solids is ∼100 nm, whereas the contact radius or width between the nanowire and a substrate will be ∼1-5 nm.6 Therefore, in most cases eq 6 can be used to calculate the Rc of nanoconstirctions. The question that needs to be answered is what is the constriction (Rc) for a∼l. This regime requires a rigorous solution of the Boltzmann Nano Lett., Vol. 5, No. 11, 2005
Figure 3. Nanowire-planar substrate system. Elastic deformation due to van der Waals forces leads6 to a line contact of width 2a.
transport equation (BTE). Various authors have solved the BTE for electron transport across constrictions9,13,14 and have shown that adding Rcd and Rcb is a very good approximation for Rc. The Rc given by adding Rcd and Rcb was in very good agreement with the rigorous solution obtained from BTE for electron transport.9,13,14 This conclusion will be equally applicable for phonon transport. Therefore, Rc can be calculated accurately using Rc ) Rcd + Rcb. For a circular constriction, Rc can be written as Rc )
1 8 1 + Kn 2ka 3π
[
]
(7)
where Kn is the Knudsen number (l/a). Equation 7 shows that if Kn f 0, then Rc reduces to the diffusive resistance and if Kn f ∞, then Rc reduces to the ballistic resistance. The model can be extended easily for a cylindrical surface in contact with a planar surface. This geometry is shown in Figure 3. The constriction between a cylinder and planar surface is rectangular. The diffusive resistance for this geometry is given by6 Rcd )
[ ( )
( )]
2D D 1 1 1 1 ln + ln L πkw a 2kw πks πa
(8)
where D is the diameter of the cylindrical wire, L is the length of the wire, kw is the thermal conductivity of the wire, and ks is the thermal conductivity of the substrate. The first two terms in eq 8 are the constriction resistance from the cylinder side, and the last term is the constriction resistance from the substrate side. Therefore, the total thermal resistance including the ballistic component is given by adding eq 8 and eq 5. Bahadur et al.6 computed the constriction width ()2a) between the nanowire and the substrate by calculating the deformation of the nanowire at the interface as shown in Figure 3 due to van der Waals forces. They used eq 8 to calculate Rc using the measured value of k of various Si nanowires.15 Figure 4 shows the comparison between Rcd and Rcb for a 1-µm-long Si nanowire on a Si substrate where the gap between the nanowire and the substrate is filled with air. The value of C and Vg for Si has been taken from Chen.2 The calculation has been performed for an air environment in Figure 4. Figure 4 shows that the ballistic resistance is approximately 1 order of magnitude higher than the diffusive Nano Lett., Vol. 5, No. 11, 2005
Figure 4. Various interfacial thermal resistance of a 1-µm Si nanowire on an Si substrate with air as the fluid in the gap.
constriction resistance. The reason for this behavior is that a is approximately 1-3 nm,6 whereas the mfp of Si is approximately 250 nm. Therefore, the constriction resistance is dominated completely by the ballistic resistance. Bahadur et al.6 also calculated the thermal resistance of the surrounding fluid in the gap assuming bulk thermal conductivity (kf) of the fluid. The thermal resistance due to the fluid gap (Rf) derived by them is 1 ) 2Lkf Rf
∫aD/2
dx
(9)
D/2 - xD2/4 - x2
The mfp of air molecules at room temperature and atmospheric pressure is around 64 nm.16 The maximum gap thickness between the substrate and the nanowire is D/2. Therefore, for nanowires gap thickness is comparable to the mfp of air molecules. The average gap thickness (δav) is given by
δav )
∫aD/2 (D/2 - xD2/4 - x2)dx D 2
D/2 - a πD2 a 16 2
[
)
x(D2 ) - a - D8 sin (2aD)] (10) 2
2
2
-1
D/2 - a
Because a is much smaller6 than D, eq 10 can be simplified to yield δav ≈ D/9. That means that even for a 100-nm nanowire, δav is approximately 11 nm, which is much smaller than the mfp of air molecules. Because δav is comparable to the mfp of air molecules, the air conductivity will also be smaller than its bulk value. The effective conductivity of air is given by17 kfef (x) )
kfδ(x) δ(x) + 2gλf
(11) 2157
where λf is the mean free path of air molecules and g ) 2[(2 - β)/β][2/(γ + 1)][(kf /(µCV))]
(12)
where β is the accommodation coefficient, γ is the ratio of specific heats, µ is the viscosity of the gas, and CV is the specific heat of air at constant volume. β for air16 is typically 0.9. Substituting the properties of air, g ) 2 at room temperature. Following the procedure of Bahadur et al.,6 the effective thermal resistance (Rfef) of the air gap can be written as 1 ) 2Lkf Rfef
∫aD/2
dx D/2 - xD /4 - x2 + 2gλf
(13)
2
Figure 4 shows the comparison between Rf calculated using eq 9 and Rfef using eq 13. Figure 4 shows that thermal resistance of the air in the gap is much higher than the predictions made assuming diffusive behavior. The total thermal resistance (Rtot) for the nanowire/substrate interface is given by parallel combination of Rc and Rfef, leading to Rtot )
(Rcd + Rcb)Rfef Rcd + Rcb + Rfef
(14)
In the previous paragraphs, the two adjoining surfaces were assumed to be of the same material while calculating the ballistic constriction resistance. In this part, the ballistic constriction resistance is calculated if the two adjoining solids are different. In this case, reflection of phonons will take place at the boundary.11 If the transmission coefficient (R) and reflection coefficient (r) are assumed to be independent of phonon frequency, then using eq 3 q can be shown to be q ) R1f2
U1(T1)Vg1 U2(T2)Vg2 - R2f1 ) 4 4 C1Vg1T1 C2Vg2T2 - R2f1 (15) R1f2 4 4
For the calculation of R, if the diffuse mismatch model is assumed then11 R1f2 ) 1 - R2f1. Using this relation and the principle of detailed balance,2,11 it can be shown easily that q)
T1 - T2 R1f2C1Vg1 (T1 - T2) ) 4 Rb
(16)
where Rb ) 4/(R1f2C1Vg1) is the well-known thermal boundary resistance.2,11 Rb has the units of thermal impedance. Therefore, for dissimilar materials the total constriction resistance for circular constriction is given by Rc ) 2158
Rb 1 1 + 2+ 4k1a πa 4k2a
(17)
Bahadur et al.6 ignored the contribution of Rb in the constriction resistance of dissimilar materials. Equation 17 can be written as Rc )
[
]
[
1 2 R bk 1 2 1+ ) 1 + Bi 2ka π a 2ka π
]
(18)
where Bi is the microscopic Biot number18 based on the radius of the nanoconstrictions and k is the harmonic mean thermal conductivity given by k ) 2k1k2/(k1 + k2). The Biot number signifies the importance of surface-dominated resistance. Equation 18 shows that if Bi is large then Rc is dominated by surface resistance, that is, by Rb. Bi will be large for smaller values of a, larger values of Rb, or larger values of k. Rb between solids in the 150-300 K temperature range typically ranges between1 1 × 10-8 and 5 × 10-8 K m2 W-1. The constriction radius between a nanoparticle and a planar surface due to the deformation of the nonplanar because of surface forces can range between 1 and 5 nm.6,8 Assuming that the thermal conductivity of a crystalline nanoparticle ranges between 1 and 10 W m-1 K-1, Bi can range between 2 and 500. This order of magnitude calculation shows that the Rc of a nanoconstriction will be dominated by Rb rather than the diffuse constriction resistance. For the nanowire/planar surface interface, Rc is given by Rc )
[
]
Rb 2D D 1 1 1 1 ln + ln + L πkw a 2kw πks πa 2a
( )
( )
(19)
Recently, Yu et al.8 used the model by Bahadur et al.6 to estimate the constriction resistance of a carbon nanowire and planar platinum surface. They estimated the value of Rc (per unit length basis) to be ∼0.15 K m W-1. From their experiments, Yu et al. estimated that Rc in their study was ∼3.6 K m W-1, which is approximately 1 order of magnitude larger than the theoretical predictions of Bahadur et al.6 Yu et al.8 estimated the contact width ()2a) for the nanofiber to be around 10 nm, assuming van der Waals force. Rb between solids in the temperature range of 150-300 K typically ranges between1 1 × 10-8 and 5 × 10-8 K m2 W-1, as mentioned earlier. Using this value of Rb, a ) 5 nm, and eq 19, Rc can estimated to be 1.15-5.15 K m W-1, which is much closer to the experimental data by Yu et al.8 Another set of calculations performed by Bahadur et al.6 was with water as the surrounding fluid in the gap between the nanowire and the substrate. Because the average gap thickness is small, the thermal boundary resistance between water and the nanowire and water and the substrate will play an important role in thermal transport across the water gap. Recent measurements19,20 of Rb between water and various solids have shown that Rb is ∼1 × 10-8 K m2 W-1. In the presence of Rb between the fluid/solid interfaces, the fluid gap resistance can be written as 1 ) 2Lkf Rfef
∫aD/2
dx D/2 - xD /4 - x2 + 2kf Rbf
(20)
2
Nano Lett., Vol. 5, No. 11, 2005
calculate the thermal resistance of the nanowire/planar surface. The nanowire/planar interface as shown in Figure 3 is now used routinely to estimate the thermal conductivity of nanowires.5,8 The knowledge of the thermal resistance between the nanowire and the planar surface such as the heaters is very important in estimating the intrinsic conductivity of the nanowire. The model developed in this paper can be used to estimate the impact of interfacial resistance on the thermal conductivity of the nanowire. References
Figure 5. Various interfacial thermal resistance of a 1-µm Si nanowire on an Si substrate with water as the fluid in the gap.
where Rbf is the thermal boundary resistance of the fluid. Figure 5 shows the results with water as the fluid in the gap. Figure 5 also shows that Rcb is much higher than Rcd. Figure 5 also shows that thermal boundary resistance between the liquid and the surrounding solid can substantially increase the thermal resistance of the fluid. For nanosized geometries, thermal boundary resistance between the liquid and the solid must be considered. In conclusion, an analytical model for thermal resistance of nanosized constriction was developed. The model captures the effects of appropriate length scales. The model shows that for constriction formed by the same materials, the Knudsen number is the appropriate dimensionless parameter, whereas for constriction formed by dissimilar materials the microscopic Biot number is the appropriate parameter. The macroscopic constriction resistance is not very important for nanosized constrictions. Finally, the model was applied to
Nano Lett., Vol. 5, No. 11, 2005
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