Superlattices and Microstructures 45 (2009) 60–64

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Superlattices and Microstructures

journal homepage: www.elsevier.com/locate/superlattices

Graphite C-axis thermal conductivityKe Sun, Michael A. Stroscio ∗, Mitra DuttaDepartment of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, IL 60607, United States

a r t i c l e i n f o

Article history:Received 12 September 2008Received in revised form17 November 2008Accepted 18 November 2008Available online 10 January 2009

Keywords:Graphitec-axisThermal conductivity

a b s t r a c t

This paper models the c-axis thermal conductivity of thin graphitelayers taking into account phonon confinement. A Debye modelis used to calculate graphite c-axis thermal conductivity, which isfound to be 4 orders of magnitude smaller than in the graphitebasal plane. This reduced thermal conductivity is promising fordevices with improved thermoelectric figure of merit, ZT , andthermal conduction along graphite c-axis. Results of graphitethermal conductivity in the basal plane are also presented anddiscussed. These calculations have been done for ideal graphitestructures that are a few monolayers thick, free of defects,and free of boundary scattering processes. To achieve the lowcalculated values of thermal conductivity, it will be necessary tofabricate high-quality graphite structures; this will pose significantfabrication challenges.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The thermoelectric figure of merit ZT is given by

ZT = S2σT/λ,

where S is the Seebeck coefficient, σ is the electric conductivity, T is the absolute temperature andλ is the thermal conductivity. In order to achieve high ZT , it is important to design thermoelectricsystems withmaterials of high electric conductivity and low thermal conductivity. Graphite is knownfor its high electric conductivity. It also has large thermal conductivity in the basal plane, that is, inthe plane of the graphene sheets normal to the c-axis of graphite. In this paper, a model of graphitethermal conductivity along the c-axis, perpendicular to the basal plane, is presented to investigategraphite as a material for improved high-ZT thermoelectrics. Theoretical expressions for graphite

∗ Corresponding author.E-mail address: stroscio@uic.edu (M.A. Stroscio).

0749-6036/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.spmi.2008.11.018

K. Sun et al. / Superlattices and Microstructures 45 (2009) 60–64 61

thermal conductivity along the c-axis are derived fromaDebyemodel for roomandhigh temperatureswhere equipartition holds [1]. The intrinsic thermal conductivity, which arises from three-phononinteractions, is calculated for thermal conduction along the c-axis.While point defects reduce thermalconductivity [2], their effects need not be considered for the relatively low defect concentrationsrealizable in practice. Since graphite displays low thermal conductivity along the c-axis, it is expectedthat thermoelectric devices with graphite engineered accordingly will obtain high ZT values.

2. Model and application

Previous models of thermal transport in graphite provide the essential features for the presentanalysis of c-axis thermal transport in graphite. From graphite dispersion relations [3], it is observedthat above 4 THz, latticewaves propagate in the basal plane and a two-dimensional phonon gasmodelis appropriate. In two dimensions, the phonon spectral specific heat is

C2 (f ) =4kfa3f 2d

,

where a3 is the volume of one molecular group of the solid, k is Boltzmann’s constant, f is the phononfrequency, and fd is the Debye frequency; fd = 4.6×1013 Hz. The expression for the two-dimensionalintrinsic mean free path is [2]

li (f , T ) =Mv3fd4πγ 2kTf 2

,

where M is the mass of a carbon atom, v is the phonon velocity, and γ is the Grüneisen parameter.Here γ 2 = 4 is adopted, following Klemens’ treatment [1,2]. Thermal conductivity in the basal planeis thus given by

λi(T ) =12

∫ fd

fcC2 (f ) vli (f , T ) df ,

where fc = 4 THz, the low-frequency limit below which lattice waves propagate along the c-axis inthree dimensions. The expression for the 2-D thermal conductivity is given by,

λi (T ) =v4M

2πa3γ 2Tfd(ln fd − ln fc).

Fig. 1 shows the relationship between the two-dimensional intrinsic thermal conductivity andtemperature; as temperature increases, the intrinsic thermal conductivity decreases. Here, the groupvelocity is taken as v = 1.86 × 104 m/s, by averaging the longitudinal

(2.36× 104 m/s

)and fast

transverse(1.59× 104 m/s

)velocities from the phonon dispersion relation for the basal plane as

calculated by [4]

2〈v〉2=

1〈vLA〉

2 +1〈vTA〉

2 .

In the c-axis, the LA and TA branches have velocities of 1960 m/s and 700 m/s, respectively, ascalculated from the phonon dispersion relation. Following the same equation to calculate the averagevelocity [4], we obtained v = 932 m/s for the c-axis.In one dimension, the phonon spectral specific heat is

C1(f ) =kN2π fd

=k2π fd

G21G3,

in which G3 = 1/a3 is the number of layers in a crystal of unit thickness and unit width, each layerhaving G21 = 1/a

22 atoms per unit area, and the number of phonon states N = G

21G3. Note here that

62 K. Sun et al. / Superlattices and Microstructures 45 (2009) 60–64

Fig. 1. Two-dimensional intrinsic thermal conductivity as a function of temperature.

the 1-D specific heat is not a function of the phonon frequency and rather it is constant for the c-axis transport over the frequency range of interest. Accordingly, one-dimensional intrinsic thermalconductivity is given by

λi =

∫ fupper

flowerC1vlidf ,

where flower and fupper are determined by the c-axis phonon dispersion relation. Our expression forthe mean free path is that derived by Klemens [4] for the anharmonic relaxation process; it has beenevaluated for the velocity along the c-axis consistent with its use in our calculation. The Grüneisenconstant for the 1D mean free path, is to the best of our knowledge, not known with precisionand, accordingly, we used the same Grüneisen constant as for the basal plane; there is, however,some evidence that the c-axis and basal plane Grüneisen constants are within a factor of 2 of eachother [5]. When the film thickness decreases below 10 nm [6], phonon quantization occurs thatresults in discrete dispersion relations and it is likely that these quantization effects will be important.fupper = 2.75 THz and flower = fupper/L, where L is the number of atoms in the one dimensional chainof carbon atoms. This frequency range over which the thermal conductivity is integrated is about 20times smaller than that used for the 2-D case. Therefore, the expression for one dimensional thermalconductivity is

λi (T , L) =ρv4

8π2γ 2T

(1

fupper/L−

1fupper

).

Here, instead of a logarithmic function, as in the 2-D case, an inverse relation is obtained for c-axisthermal conductivity and frequency, as shown in Fig. 2.The intrinsic thermal conductivity is about 4 orders of magnitude smaller in one dimension than

in two dimensions (Fig. 3). This result is consistent with the following simple scaling considerations.Given the velocity in one dimension, which is about 20 times smaller than that in two dimensions,and the mean free path which varies with the third power of the velocity, it is not surprising to obtainthe thermal conductivity 4 orders of magnitude smaller in one dimension than in two dimensions.Considering that the 1-D specific heat is not a function of frequency and thus the resulting thermalconductivity is inversely related to frequency, while the 2-D specific heat is a linear function offrequency, which results in a logarithmic relation between the thermal conductivity and frequency;accordingly, the predicted 4-order-of-magnitude reduction is consistent with these simple scaling

K. Sun et al. / Superlattices and Microstructures 45 (2009) 60–64 63

Fig. 2. One dimensional thermal conductivity of c-axis graphite. The solid line is for a chain of 3 graphene planes, the dottedline is for 6 graphene planes, and the dashed line is for 10 graphene planes.

Fig. 3. Ratio of the 2D thermal conductivity to the 1D thermal conductivity. The solid line is for 3 atoms (representing 3graphene planes) in a 1D chain, the dotted line is for 6 atoms and the dashed line is for 10 atoms.

estimates. Also, this phenomenon is stable over the temperature range we are interested in. Ouranalysis only takes into consideration the isolated c-axis thermal conductivity. The low value ofthermal conductivity along the c-axis is thus promising for improving thermoelectric figure of merit.Notice that Ref. [7] has a graphite c-axis thermal conductivity of 80 W/mK at 273 K, which is about1236 times greater than the value obtained in this work. This is not surprising since there is a largevariation in reported experimental values of thermal conductivity, and frequently the highest valuesare reported; moreover, the calculations in the present paper consider confinement effects, and aresulting lower frequency cutoff, in contrast to the reported experimental c-axis thermal conductivityfor bulk graphite.

64 K. Sun et al. / Superlattices and Microstructures 45 (2009) 60–64

3. Discussion

Based on previously-determined values of c-axis electrical conductivity [7], the electrical tothermal conductivity ratio may be estimated using our calculated values of thermal conductivity.Refs. [7,8] give consistent values for graphite c-axis electrical conductivity and here Ref. [7] is used.In the basal plane, the electrical conductivity is about 105 �−1 m−1 and thermal conductivity is1900W/mK at 300 K, which yields a ratio of 52.6 K/�W. Along the c-axis, the electrical conductivityis 2.44 × 104 �−1 m−1 and thermal conductivity is 0.0131 W/mK, which gives a ratio of 1.86 ×106 K/�W. Based on these estimates, the electrical to thermal conductivity ratio is found to be largeralong the c-axis than in the basal plane. The implication on the system level is the possible abilityto realize high-ZT thermoelectrics utilizing the property of low thermal conductivity along c-axis ofgraphite. This conclusion depends critically on our assumption that ideal graphite structures that area few monolayers thick and free of defects may be fabricated. Not only will the thermal conductivitybe affected by the lack of such ideal structures, but the electrical conductivity will also be affected.Additionally, the c-axis thermal conductivity decreases as temperature increases.

4. Conclusion

A Debye model is used to calculate graphite c-axis thermal conductivity, which is found to be 4orders of magnitude smaller than in the graphite basal plane. This reduced thermal conductivity ispromising for deviceswith improved thermoelectric figure ofmerit, ZT , and thermal conduction alonggraphite c-axis. Results of graphite thermal conductivity in the basal plane are also presented anddiscussed. These calculations have been done for ideal graphite structures that are a few monolayersthick, free of defects, and free of boundary scattering processes. To achieve the low calculated valuesof thermal conductivity, it will be necessary to fabricate high-quality graphite structures; this willpose significant fabrication challenges.

Acknowledgement

One of the authors,MAS, thanks Prof. P. Klemens formany enlightening conversations on his theoryof thermal conductivity.

References

[1] P.G. Klemens, Theory of thermal conduction in thin ceramic films’, International Journal of Thermophysics 22 (1) (2001).[2] P.G. Klemens, Graphite, graphene and carbon nanotubes, in: Ralph B. Dinwiddie (Ed.), Proceedings of the Twenty-SixthInternational Thermal Conductivity Conference, in: Thermal Conductivity, vol. 26, DEStech Publ., Lancaster, PA, ISBN: 1-932078-36-3, 2004, pp. 48–57.

[3] R. Nicklow, N. Wakabayashi, H.G. Smith, Lattice dynamics of pyrolytic graphite, Physical Review B 5 (12) (1972).[4] P.G. Klemens, D.F. Pedraza, Thermal conductivity of graphite in the basal plane, Carbon 32 (4) (1994) 735–741.[5] C.N. Hooker, A.R. Ubbelohde, D.A. Young, Anisotropy of thermal conductance in near-ideal graphite, in: Proceedings of theRoyal Society of London, Series A, Mathematical and Physical Sciences, vol. 284, 1964, p. 17.

[6] M. Stroscio, M. Dutta, Phonons in Nanostructures, Cambridge University Press, Cambridge, 2001.[7] R.L. Powell, G.E. Childs, American Institute of Physics Handbook, 1972, pp. 4-142–4-160.[8] W. Primak, C-axis electrical conductivity of graphite, Physical Review 103 (3) (1956).