NANO LETTERS
Predicting the Thermal Conductivity of Si and Ge Nanowires
2003 Vol. 3, No. 12 1713-1716
Natalio Mingo* and Liu Yang Eloret Corp., NASA Ames Research Center, Moffett Field, California 94035
Deyu Li and Arun Majumdar Department of Mechanical Engineering, UniVersity of California, Berkeley, California Received August 29, 2003
ABSTRACT We theoretically predict the thermal conductivity versus temperature dependence of Si and Ge nanowires. Three methods are compared: the traditional Callaway and Holland approaches, and our “real dispersions” approach. Calculations with the former two show large disagreements with experimental data. On the contrary, the real dispersions approach yields good agreement with experiments for Si nanowires between 37 and 115 nm wide, approximately. In all cases, only bulk data are used as inputs for the calculation. Predictions for Ge nanowires of varying diameters are given, enabling future experimental verification.
Crystalline semiconductor nanowires are very promising materials in the present miniaturization of devices toward the nanoscale.1,2 Recently, measurements of the thermal conductivity of individual Si nanowires have been reported in ref 3. Thermal conductivity of individual GaAs nanowires4 and carbon nanotubes5 have also been recently measured. Understanding and predicting the thermal conductivity of nanowires plays a central role in two of their most fundamental potential applications: (i) heat dissipation, essential in the design of future nanochips, where nanowires may be used as heat drains,6 and (ii) new thermoelectric materials, in which a small thermal conductivity combined with high electronic conductivity and high thermopower can result in nanowires having a much better performance than any known materials presently used for thermoelectric refrigeration.7,8 It is then clear that theoretical prediction of the thermal conductivity of the nanowires prior to their fabrication is a desirable goal, both from the applied and basic research points of view. Our aim in this paper is to show a way to achieve this, using only data from bulk measurements, which are readily available. Alternative methods to compute thermal transport in materials exist, including molecular dynamics,9-11 the equation of phonon radiative transfer,12,13 solutions to the Boltzmann transport equation under different approximations,14-19 and others.20 Investigation of some of these alternative approaches will be presented elsewhere. Over the past four decades, two models have proven successful in parametrizing the thermal conductivity versus temperature curve for bulk Ge and Si: Callaway’s model21-24 * Corresponding author. E-mail:
[email protected]. 10.1021/nl034721i CCC: $25.00 Published on Web 10/24/2003
© 2003 American Chemical Society
and Holland’s model.22 Therefore, as a first attempt, one would try to use these two approaches to model the nanowire case. In this paper we will show how those two approaches fail to provide reasonable predictions of even the qualitative behavior. After briefly explaining the fundamentals of those models and the results obtained with them, we will introduce our “real dispersions” theoretical approach and compare our calculated curves with experimental results. We will then discuss the case of narrow (less than 30 nm in diameter) nanowires, for which disagreement arises. Finally, we will show predicted curves for Ge nanowires, for which no experiments are available to date, and we will summarize the conclusions. In Callaway’s model, the general expression for the thermal conductivity can be derived as21,22 κ)
1 1 2π2c kT2
e τ(ω) dω ∫0ω (pω)2ω2(epω/kT - 1)2 D
pω/kT
(1)
where c is the speed of sound averaged between the three acoustic branches, two transverse and one longitudinal, τ(ω) is the phonon relaxation time, and ωD is the Debye frequency cutoff. It has been shown that a correction resulting from normal scattering processes can be disregarded in the Si and Ge cases.22 The above considers all branches as equivalent. If one considers different lifetimes and cutoff frequencies for the three branches, then eq 1 is used for each of the branches (divided by 3). The inverse lifetime is given as a sum of the contributions from different sources: boundary, impurity, and Umklapp. The boundary scattering rate is the only one that directly depends on the diameter of the wire,
Figure 1. Si experimental nanowire thermal conductivities and theoretical results using Callaway’s, Holland’s, and our real dispersions approach. Every thick curve corresponds to a wire twice as wide as the previous one (see text). The same set of parameters also matches the bulk thermal conductivity limit.
as τ-1 b (ω) ) c/(Fl), where l is the diameter of the nanowire. The phonon mean free path approaches ∼Fl if the boundary scattering dominates over other mechanisms. Thus F is related to the degree of specularity25 of the boundary scattering, which is not known a priori. For this reason, we will show all the calculation results in terms of Fl, rather than l. The particular choice of functional dependences for the lifetimes with frequency crucially determines the quality of the calculation as compared with experimental results. Callaway’s original paper used analytical forms obtained from first-order perturbation theory.26 With these, the thermal conductivity of bulk Ge and Si were fitted rather successfully, except in the higher temperature range. The lack of agreement in the high temperatures led Holland to modify the model, considering the transverse and longitudinal branches separately and using several different functional forms for the Umklapp scattering in different regions of the spectrum.22 This improved model yielded an excellent fitting of the bulk thermal conductivity in the whole temperature range. Nevertheless, the increased number of parameters and features deprive Holland’s calculation from the appealing simplicity of Callaway’s original model. The results from using Callaway and Holland’s models for Si nanowires are shown in Figure 1 (two upper graphs). In each case, we show a series of conductivity versus temperature curves, each of them corresponding to a different nanowire diameter. Together with them, we show the 1714
experimental results for four different nanowires.3 The theoretical curves correspond to nanowires of increasing diameter, according to Fln ) 20.25n nm, where n is an integer. In other words, every four curves, the corresponding diameter is doubled, considering F to be diameter independent. The model’s predictive capability is readily assesed by direct observation of the graphs: the smaller the number of theoretical curves crossed by one experimental set, the better the model. Ideally, each experimental set of points should lay on just a single theoretical curve. Our calculation using Callaway’s model clearly does not agree with the experimental results. Each of the experimental sets for the four individual nanowires crosses a dozen or more of the theoretically predicted curves. This means that the prediction error is of ∼300% or more, for every set. Using Holland’s model we get slightly better predictions, but still markedly inaccurate. The sets cross through about five different theoretical curves, which corresponds to deviations of more than 40%. By looking at the higher temperature region we can see that the predicted slope is also incorrect, so that larger errors will still be incurred if a larger temperature range is measured. Two main factors seem responsible for the poor predictions obtained with the two models above: the use of modelistic phonon dispersion relations and an inadequate form of the anharmonic scattering rates. To overcome them, our method of calculation differs essentially from the two models above. Instead of employing linearized dispersion relations, we calculate the real dispersion relations of the nanowires, using the interatomic potentials described in ref 27. The thermal conductivity is calculated as28 κ(T) )
∫0∞ Lu (ω)
1 s
dfBose pω dω dT 2π
(2)
where u(ω) is the transmission function for phonons in the wire, s is the nanowire cross section, fBose is the BoseEinstein distribution, and L is the nanowire length. For a translationally periodic system, if the phonon lifetime τ(ω) does not depend on the branch, the transmission function can be calculated as Lu ) Nb(ω)Vavg z (ω)τ(ω)
(3)
Nb(ω) is the number of phonon subbranches crossing frequency ω and Vavg z (ω) is the averaged sound velocity at that frequency in the nanowire’s axial direction. The calculation method and derivation of eqs 2 and 3 are explained in ref 28. The Callaway formula can be derived from eqs 2 and 3 if one inserts the Nb and Vavg z corresponding to a linear dispersion approximation.29 We use Klemens’ expressions for the impurity and boundary scattering rates. For the Umklapp scattering rate, we use30 τu-1 ) BTω2e-C/T
(4)
Nano Lett., Vol. 3, No. 12, 2003
Here, B and C are parameters to be adjusted from the bulk data. This form adequately accounts for the high and lowtemperature limits.31 Therefore, the calculation method involves no fitting to any nanowire measurements, and it is thus a fully predictive approach. The only fitting is that of the Umklapp scattering parameters B and C, which involves only the bulk thermal conductivity measurements. For Si, we obtain B ) 1.73 × 10-19 s/K and C ) 137.39 K. The value of A is given by an analytical expression and it should not be adjusted, so we keep the values given by ref 22. (The cited reference contains an errata in the exponent. The proper value is A ) 1.32 × 10-45 s3.) We label each theoretical curve by the value of Fl. SEM measurements of the nanowire diameters can be then used to derive the value of F by comparing the experimental and theoretical sets. As Figure 1 (lower graph) shows, now the agreement between the theoretical predictions and the measured sets is good for the three wider nanowires. Each set crosses over no more than two theoretical curves, which corresponds to errors of less than 15%. Only the narrowest nanowire clearly disagrees in shape with the prediction. This will be discussed in the next paragraph. The SEM-measured nanowire diameters are 115, 56, 37, and 22 nm, implying F ∼ 1-1.3 for the three wider nanowires. The narrowest of the wires measured presents a curve that increases almost linearly from 50 K up. Its second derivative is, furthermore, positive in the range from 50 to 320 K. This implies that this wire contains vibrational modes with frequencies higher than the highest frequency available in bulk Si. This can be seen by considering the thermal conductivity as calculated by eq 2. Let the integrand in eq 2 be named f (T,ω). For any given ω, f (T) has only one inflection point, Tinfl, changing from concave to convex. The higher the frequency, the higher this inflection temperature becomes. For ω ∼100 THz, corresponding to the highest frequency available in Si, the inflection temperature is still only ∼200 K. Therefore, if only frequencies lower than 100 THz were present, the thermal conductivity should have a negative second derivative for T above 200 K, which is not the case in this wire. To have Tinfl > 320 K the frequency must be at least ∼160 THz. This suggests the presence of higher frequency modes in the wire that do not exist in pure Si. These modes might be associated with the oxidation of the nanowire’s surface. The highest frequency available in silica is ∼200 THz, which makes this a reasonable conjecture. In addition, phonon confinement and nanowire low speed modes can cause important effects in sub 30 nm wires or superlattices.18 The results in Figure 1 (lower graph) indicate that, at least down to nanowires ∼37 nm in diameter or wider, the set of two parameters B and C, obtained exclusively from bulk measurements, is sufficient to predict the sets of curves for nanowires with good accuracy. Since we have all the necessary experimental data for bulk Ge, we have followed the same approach as for Si, to obtain the thermal conductivity sets of Ge nanowires. Fitting the bulk thermal conductivity of Ge yields the values B ) 8.8 × 10-20 s/K and C ) 57.6 Nano Lett., Vol. 3, No. 12, 2003
Figure 2. Theoretical results for Ge nanowires. Wire widths, l, range from Fl ) 24.5 = 22.6 to Fl ) 27.5 = 181.0 nm.
K. The predicted thermal conductivity curves for Ge nanowires (Figure 2) present their maxima at lower temperatures than those for Si nanowires. Also, for the same values of Fl, Ge nanowires have a smaller thermal conductivity than their Si counterparts. It still remains to verify experimentally these predicted curves. In conclusion, we have presented calculations of the thermal conductivity of Si and Ge nanowires, using a method that employs the full phonon dispersion relations of the material. Good agreement with experimental results is obtained for Si nanowires. On the other hand, we have shown that the traditional Callaway and Holland models for Si do not predict the thermal conductivity of Si nanowires correctly. The phonon dispersion relations were computed using interatomic potentials, and the only two parameters involved were obtained exclusively from fitting bulk data. Thus, in principle, this method can be used to predictively calculate thermal conductivity of nanowires made of other materials. We have presented theoretical curves for Ge, which await experimental verification. Acknowledgment. We acknowledge Philip Kim and Deepak Srivastava for helpful discussions. Eloret employees were supported under NASA contract. References (1) Gudiksen, M. S.; Lauhon, L. J.; Wang, J.; Smith, D. C.; Lieber, C. M. Nature 2002, 415(6872), 617-20. (2) Xia, Y.; Yang, P.; Sun, Y.; Wu, Y.; Mayers, B.; Gates, B.; Yin, Y.; Kim, F.; Yan, H. AdV. Mater. 2003, 15, 353. (3) Li, D.; Wu, Y.; Kim, P.; Shi, L.; Yang, P.; Majumdar, A. Appl. Phys. Lett. 2003, 83, 2934. (4) Fon, W.; Schwab, K. C.; Worlock, J. M.; Roukes, M. L. Phys. ReV. B 2002, 66, 045302. (5) Kim, P.; Shi, L.; Majumdar, A.; McEuen, P. L. Phys. ReV. Lett. 2001, 87, 215502. (6) Zou, J.; Balandin, A. Proc. Electrochem. Soc. 2001, 2001-19, 7080. (7) Mahan, G.; Sales, B.; Sharp, J. Phys. Today 1997, March, 42. (8) Hicks, L. D.; Dresselhaus, M. S. Phys. ReV. B 1993, 47, 12727. (9) Osman, M. A.; Srivastava, D. Nanotechnology 2001, 12, 21. (10) Schelling, P. K.; Phillpot, S. R.; Keblinski, P. Phys. ReV. B 2002, 65, 144306. (11) Volz, S. G.; Chen, G. Appl. Phys. Lett. 1999, 75, 2056. (12) Lu, X.; Chu, J. H.; Shen, W. Z. J. Appl. Phys. 2003, 93, 1. (13) Volz, S.; Lemonnier, D. Phys. Low-Dimens. Semicond. Struct. 2000, 5-6, 91. (14) Walkauskas, S. G.; Broido, D. A.; Kempa, K.; Reinecke, T. L. J. Appl. Phys. 1999, 85, 2579. (15) Hyldgaard, P.; Mahan, G. D. Phys. ReV. B 1997, 56, 10754. (16) Simkin, M. V.; Mahan, G. D. Phys. ReV. Lett. 2000, 84, 927. 1715
(17) (18) (19) (20)
(21) (22) (23) (24) (25)
1716
Chen, G.; Neagu, M. Appl. Phys. Lett. 1997, 71, 2761. Balandin, A.; Wang, K. Phys. ReV. B 1998, 58, 1544. Sparavigna, A. Phys. ReV. B 2002, 66, 174301. Cahill, D.; Ford, W. K.; Goodson, K. E.; Mahan, G. D.; Majumdar, A.; Maris, H. J.; Merlin, R.; Phillpot, S. R. Appl. Phys. ReV. 2003, 93, 793. Callaway, J. Phys. ReV. 1959, 113, 1046. Holland, M. G. Phys. ReV. 1963, 132, 2461. Parrott, J. E.; Stuckes, A. D. Thermal conductiVity of solids; Pion Ltd.: London, 1975. Berman, R. Thermal conductiVity of solids; Clarendon Press: Oxford, 1976. Ziman, J. M. Electrons and Phonons; Oxford University Press: Oxford, 1963.
(26) Klemens, P. G. Solid State Physics; Seitz, F., Turnbull, D., Eds.; Academic: New York, 1958; Vol. 7, p 1. (27) Harrison, W. A. Electronic Structure and the Properties of Solids; Dover: Mineola, NY, 1989. (28) Mingo, N. Phys. ReV. B 2003, 68, 113308. Also, cond-mat/0308587. (29) In a linear dispersion approximation Nlinear ) (3ω2/4πc2)l2, and Vavg b z ) (2/3)c. (30) Asen-Palmer, M.; Bartkowski, K.; Gmelin, E.; Cardona, M.; Zhernov, A. P.; Inyushkin, A. V.; Taldenkov, A.; Ozhogin, V. I.; Itoh, K. M.; Haller, E. E. Phys. ReV. B 1997, 56, 9431. (31) As stated in ref 23, pp 95-96, the low-temperature behavior of τu is determined by the exponential factor, so the power of T does not matter much at low T.
NL034721I
Nano Lett., Vol. 3, No. 12, 2003
