Thermal conductance of graphene and dimerite
aa r X i v : . [ c ond - m a t . m t r l - s c i ] M a y Thermal conductance of graphene and dimerite
Jin-Wu Jiang, Jian-Sheng Wang, and Baowen Li
1, 2, ∗ Department of Physics and Centre for Computational Science and Engineering,National University of Singapore, Singapore 117542, Republic of Singapore NUS Graduate School for Integrative Sciences and Engineering, Singapore 117597, Republic of Singapore (Dated: October 28, 2018)We investigate the phonon thermal conductance of graphene regarding the graphene sheet as thelarge-width limit of graphene strips in the ballistic limit. We find that the thermal conductancedepends weakly on the direction angle θ of the thermal flux periodically with period π/
3. It isfurther shown that the nature of this directional dependence is the directional dependence of groupvelocities of the phonon modes in the graphene, originating from the D h symmetry in the honey-comb structure. By breaking the D h symmetry in graphene, we see more obvious anisotropic effectin the thermal conductance as demonstrated by dimerite. PACS numbers: 81.05.Uw, 65.80.+n
I. INTRODUCTION
As a promising candidate material for nanoelectronicdevice, graphene has been attracted intensive atten-tion in research in past years (for review, see e.g.Ref. 1). It demonstrates not only peculiar electronicproperties , but also very high (as high as 5000Wm − K − ) thermal conductivity, which is beneficialfor the possible electronic and thermal device applica-tions of graphene. A recent work has studied thestructure of an interesting new allotrope of graphene bya first principle calculation. The ground state energyin this allotrope is about 0.28 eV/atom above graphene,which is 0.11 eV/atom lower than C .In this paper, we calculate the ballistic phonon ther-mal conductance for the graphene sheet by treating thegraphene as the large width limit of graphene strips,which can be described by a lattice vector ~R = n ~a + n ~a . The phonon dispersion of the graphene is ob-tained in the valence force field model (VFFM), wherethe out-of-plane acoustic phonon mode is a flexure mode,i.e., it has the quadratic dispersion around Γ point inthe Brillouin zone. Our result shows that the thermalconductance has a T . dependence at low temperature,which is due to the contribution of the flexure mode. At room temperature, our result are comparable with therecent experimental measured thermal conductivity. We find that the thermal conductance in graphene de-pends on the direction angle θ of the thermal flux pe-riodically with π/ × Wm − K − , which is about 1% varia-tion. Our study shows that this directional dependencefor the graphene is attributed to the directional depen-dence of the velocities of the phonon modes, which originsfrom the D h symmetry of the honeycomb structure.For the dimerite, where the D h symmetry is broken,the thermal conductance shows more obvious anisotropyof 10% and the value is about 40 % smaller than that ofthe graphene at room temperature. The present paper is organized as follows. In Sec.II,we describe the graphene strip by a lattice vector. Theformulas we used in the calculation of the thermal con-ductance are derived in Sec. III. Calculation results forgraphene and dimerite are discussed in Sec. IV A andSec. IV B, respectively. Sec. V is the conclusion. II. CONFIGURATION
In graphene, the primitive lattice vectors are ~a and ~a , with | ~a | = | ~a | = √ b . b = 1 .
42 is the C-C bondlength in graphene. The corresponding reciprocal unitvectors are ~b = ( π b , − π √ b ), ~b = ( π b , π √ b ).As shown in Fig. 1 (a), a strip in the graphene sheet canbe described by a lattice vector ~R = n ~a + n ~a . The reallattice vector ~H = p ~a + p ~a is introduced through : n p − n p = N ( N is the greatest common divisor of n and n ). The strip is denoted by N H ~H × N R ~R , where N H and N R are numbers of the periods in the directionsalong ~H and ~R , respectively. Instead of ~a and ~a , we use( ~H , ~R/N ) as the basic vectors in the following, and ~b H and ~b R are their corresponding reciprocal unit vectors: ~b H = 1 N ( − n ~b + n ~b ) ,~b R = p ~b − p ~b . Any wave vector in the reciprocal space can be writtenas: ~k = k H ~b H + k R ~b R . (1)Using the periodic boundary conditions, this strip has N H translational periods in the ~H direction and N R × N translational periods in the ~R direction. As shown inFig. 1 (b), the Brillouin zone for the graphene strip is N R × N discrete segments, which are parallel or co-incide with ~b H . The coordinates for the wave vectorson these lines are ( k H , k R )=( i/N H , j/N R N ), with i = 0 , , , ..., N H − j = 0 , , , ..., N R N − FIG. 1: Graphene strip is described by a lattice vector ~R = n ~a + n ~a . (a). Strip with ( n , n ) = (4 ,
2) and ( N H , N R )=(12, 1); (b). The Brillouin zone for this special strip istwo discrete segments (solid) in the reciprocal space. The graphene sheet is actually a strip in the limit of N H −→ ∞ and N R −→ ∞ . In this case, the Brillouinzone for the strip, i.e., N R × N discrete lines, turns tothe two-dimensional Brillouin zone for the graphene. III. CONDUCTANCE FORMULAS
The contribution of the phonon to the thermal con-ductance in the ballistic region is: σ ( T ) = 12 π Z ∞ T ( ω )¯ hω dfdT dω, where f ( T, ω ) is the Bose-Einstein distribution function. T ( ω ) is transmission function. In the ballistic region, T ( ω ) is simply the number of phonon branches at fre-quency ω .From the above expression, the thermal conductancein the graphene strip can be written as: σ ( T ) = NN R − X j =0 6 X n =1 X ~v θn > π Z b H dk H × ¯ hω n ( ~k ) dfdT v θn ( ~k ) T n ( ~k ) , (2)where θ determines the direction of the thermal flux: ~e θ = (cos θ, sin θ ). ~k = k H ~b H + jNN R ~b R is the wave vectorin the Brillouin zone of the strip, i.e., on the N R × N dis-crete lines. The transmission function for a phonon mode T n ( ~k ) is assumed to be one. v θn ( ~k ) = ∂ω n ( ~k ) ∂k θ is the groupvelocity of mode ( ~k, n ) in ~e θ direction. The value of thegroup velocity can be accurately calculated through thefrequency and the eigen vector of this phonon mode: v θn ( ~k ) = ∂ω n ( ~k ) ∂k θ = ~u † n ( ~k ) · ∂D∂k θ · ~u n ( ~k )2 ω n ( ~k ) , (3)where D is the dynamical matrix and ~u n ( ~k ) is the eigenvector. Only those phonon modes with ~v θn > ~e θ direction.In the two-dimensional graphene strip system, it isconvenient to use conductance reduced by cross section:˜ σ = σ/s , where s = W × h is the cross section. Thethickness of the strip, h = 3 .
35 ˚A, is chosen arbitrarilyto be the same as the space between two adjacent layersin the graphite. The width for the strip is W = N R | ~R | ,where, the thermal flux in the strip is set to be in thedirection perpendicular to ~R , i.e., ~e θ = ~b H /b H . We ad-dress a fact that the integral parameter ( k H ) in Eq. (2)is the quantum number along the thermal flux direction. The thermal conductance in ~e θ direction of thegraphene can be obtained by:˜ σ ( T ) = lim N R −→∞ W h σ ( T ) . (4) IV. CALCULATION RESULTS ANDDISCUSSIONA. graphene results
The phonon spectrum of graphene is calculated inthe VFFM, which has been successfully applied tostudy the phonon spectrum in the single-walled carbonnanotubes and multi-layered graphene systems. Inpresent calculation, we utilize three vibrational poten-tial energy terms. They are the in-plane bond stretch-ing ( V l ) and bond bending ( V BB ), and the out-of-planebond bending ( V rc ) vibrational potential energy. Thethree force constants are taken from Ref. 21 as: k l =305.0Nm − , k BB =65.3 Nm − and k BB =14.8 Nm − .
1. temperature dependence for thermal conductance
In Fig. 2, the temperature is 100 K and the directionangle for the thermal flux is θ = π/
3. It is shown that thethermal conductance for a strip decreases with increasingwidth. At about W =100 ˚A, the thermal conductance TABLE I: The dependence of the thermal conductance on group velocities of phonon modes. ‘b’ in the 2nd line is a fittingparameter (see text). In the 4th line, the down (up) arrow indicates the decreasing (increasing) of ∆˜ σ when the correspondingphonon mode is excited more.velocity α v v v v v sign( b ) − − + − + − ˜ σ ∝ √ α v v v v v ∆˜ σ ↓ ↓ ↑ ↑ ↓ ↑ s / s ( W m − K − ) W (Å)T=100 K
FIG. 2: Convergence for the thermal conductance of thegraphene strip with increasing width at temperature 100 K.In the large width limit (
W > s / s ( W m − K − ) T (K)
FIG. 3: The thermal conductance of the graphene sheet v.s.temperature. Inset is log ˜ σ v.s. log T in extremely low tem-perature region. The calculated results (filled squares) can befitted by function f ( x ) = 13 .
44 + 1 . x (blue line). It indicatesthat the thermal conductance has a T . dependence in thisregion. reaches a saturate value, which is actually the thermalconductance for the graphene. In the calculation, thewidth we used is about 300 ˚A, which ensures that thestrip is wide enough to be considered as a graphene sheet.In Fig. 3, the thermal conductance versus the temper-ature is displayed. In the low temperature region, thethermal conductance has a T . dependence. This is theresult of the flexure mode in the graphene sheet, whichhas the dispersion ω = αk . In the very low tempera-ture region, this mode makes the largest contribution tothe thermal conductance. Its contribution to the thermalconductance is ˜ σ ∝ T . / √ α , which can be seen fromthe figure in the low temperature region. At room tem-perature T =300 K, the value for the thermal conductanceis about 4 . × Wm − K − . This result agrees with therecent experimental value for the thermal conductance inthe graphene. In the experiment, the thermal conduc-tivity is measured to be about 5 . × Wm − K − atroom temperature. The distance for the thermal fluxto transport in the experiment is L =11.5 µ m. So thereduced thermal conductance can be deduced from thisexperiment as ˜ σ = σs = κL = 0 . × Wm − K − . Ourtheoretical result is much larger than this experimentalvalue. Because our calculation is in the ballistic region,while in the experiment, there is scattering on defects,edges or impurities and thus the transport is partiallydiffusive. At the high temperature limit T = 1000 K , ourcalculation gives the value 8.9 × Wm − K − , which isin consistency with the previous theoretical result.
2. directional dependence for thermal conductance
As shown in Fig. 4, at T =100 K, the thermal conduc-tance varies periodically with the direction angle θ . Thecalculated results can be fitted very well by the function f ( θ ) = a + b cos(6 θ ) + c cos(12 θ ). The difference betweenthe thermal conductance in the two directions with an-gle θ = 0 and π /2 is about 1.2 × Wm − K − . Thisdifference is very stable for graphene strips with differ-ent width (see Fig. 5). At T =100 K, the lattice thermalconductance is about two orders larger than the elec-tron thermal conductance. So the experimental mea-sured thermal conductance at T =100 K is mainly due tothe contribution of the phonons. As a result, our calcu-lated directional dependence of the lattice thermal con-ductance in the graphene can be carefully investigated in s / s ( W m − K − ) q ( p ) calculated resultsfitted curve FIG. 4: The direction dependence of thermal conductance. θ is the direction angle for the thermal flux. The calculatedresults (filled squares) are fitted by the function f ( x ) = a + b cos(6 θ ) + c cos(12 θ ) with a = 1 . × , b = − . × and c = − . × . s / s ( W m − K − ) Width (Å) q =0.5 q =0.0 FIG. 5: Thermal conductance in θ = 0 and π/ the experiment. In the following, we say that two quan-tities Q = a + b cos 6 θ and Q = a + b cos 6 θ havethe same (opposite) dependence on θ , if the signs of b and b are the same (opposite).To find the underlying mechanism for this directionaldependence for the thermal conductance, firstly we showin Fig. 6 the coefficient α for the flexure mode and thevelocities for the other five phonon modes at the Γ point.Interestingly, this coefficient and velocities are also direc-tional dependent with the period π/
3. Obviously, theycan be fitted by function f ( θ ) = a + b cos(6 θ ). In Table I,the sign of the fitting parameter b for this coefficient andfive velocities are listed, which can be read from Fig. 6.In the third line of Table I, we list the contribution ofthe six phonon modes to the thermal conductance. Ifthe three low frequency modes are excited, the thermal FIG. 6: The coefficient and velocity of the six phonon spec-trum around Γ point in the Brillouin zone: (a). the coefficientof the out-of-plane acoustic mode with ω = αk in the unitof 10 − m s − ; (b)-(f). velocities of the other five phononspectrum (from low frequency to high frequency), in the unitof ms − . The horizontal axises in all figures are the directionangle θ in the unit of π . −4 0 4 8 12 16 0 200 400 600 800 1000 Ds / s ( W m − K − ) T (K) −0.4 0 0.4 0 40 80
FIG. 7: The difference of the thermal conductance betweendirections of θ = 0 and π/ conductance is in inverse proportion to their velocities. While as can be seen from Eq. (4), the thermal conduc-tance is proportional to the velocities for the three highfrequency optically modes when they are excited. In eachtemperature region, there will be a key mode which is themost important contributor to the thermal conductance.The direction dependence of the velocity of this key modedetermines the direction dependence of the thermal con-ductance.We then further study the difference between the ther-mal conductance in two directions with θ = 0 and π :∆˜ σ = ˜ σ ( π ) − ˜ σ (0). In the fourth line of Table I, we dis-play the effect of different modes on ∆˜ σ . It shows that∆˜ σ will decrease, if the first, second and fifth phononmodes are excited sufficiently with increasing tempera-ture. The other three phonon modes have the oppositeeffect on the thermal conductance. The dependence of∆˜ σ on the temperature is shown in Fig. 7, where fivedifferent temperature regions are exhibited.(1) [0, 4]K: In this extremely low temperature region,only the flexure mode is excited. This mode results in∆˜ σ <
0. Because the coefficient α depends on the di-rection angle θ very slightly, the absolute value of ∆˜ σ ispretty small (see inset of Fig. 7).(2) [4, 10]K: The second acoustic mode is excited inthis temperature region. In respect that this mode hasmore sensitive direction dependence and favors to de-crease ∆˜ σ , ∆˜ σ decreases much faster than region (1).(3) [10, 70]K: In this temperature region, the thirdacoustic mode begins to have an effect on the thermalconductance. This mode’s directional dependence is op-posite of the previous two acoustic modes and it willincrease ∆˜ σ . The competition between this mode andthe other two acoustic modes slow down the decrease ofthe value ∆˜ σ at temperature below T =40K. The thirdacoustic mode becomes more and more important withtemperature increasing, and ∆˜ σ begins to increase after T =40K as can be seen from the inset of Fig. 7.(4) [70, 500]K: The third acoustic mode becomes thekey mode in this temperature region. As a result, ∆˜ σ changes into a positive value and keeps increasing.(5) [500, 1000]K: In this high temperature region, theoptical mode will also be excited one by one in the fre-quency order with increasing temperature. Since thereare two optical modes (1st and 3rd optical modes) favorsto increase ∆˜ σ , while only one optical mode (2nd opti-cal mode) try to decrease ∆˜ σ , the competition result isincreasing of ∆˜ σ in the high temperature region. T =100K is in region (4), where the direction depen-dence of the thermal conductance is controlled by thevelocity of the third mode, so the dependence of ˜ σ on θ in Fig. 4 is opposite to the dependence of velocity v ( θ )in Fig. 6 (c). B. dimerite results
The adatom defect is used as a basic block to man-ufacture a new carbon allotrope of graphene, named dimerite , and the relaxed configuration of this new ma-terial is investigated by a first principle calculation. Theunit cell for the dimerite is shown in Fig. 8 (a), where thetwo basic unit vectors are ~a and ~a , with | ~a | = 7 .
60 ˚A,and | ~a | = 7 .
05 ˚A. The angle between these two vectors is0.7 π . In each unit cell, there is a (7-5-5-7) defect, whichleads to two anisotropic directions: the (7-7) directionand the (5-5) direction. (7-7) direction is from the centerof a heptagon to the opposite heptagon, and the (5-5)direction is from the center of a pentagon to the neigh-boring pentagon. These two directions are perpendicular FIG. 8: (Color online) Configuration for the dimerite. (a) isone unit cell for dimerite. There are 16 atoms in each unit cell.The length for the two unit vectors are 7.60 ˚Aand 7.05 ˚A. Theangle between them is 0.7 π . (b) shows 49 unit cells together. w ( c m − ) G M K M G K FIG. 9: (Color online) Phonon spectrum for the dimeritealong high symmetry lines in the Brillouin zone. Around Γpoint, there are low frequency optical modes with frequencyabout 200 cm − , which do not exist in pure graphene. to each other, and corresponding to θ = 0 . π , 0.9 π re-spectively in Fig. 8 (a), where θ is the direction anglewith respect to x axis. From graphene to dimerite, thesymmetry is reduced from D h to D h .
1. phonon dispersions in dimerite
The phonon dispersion for the dimerite from the aboveVFFM is shown in Fig. 9. Similar with pure graphene,there are three modes with zero frequency. Two of themare the acoustic modes in the xy plane. The other zero- s / s ( W m − K − ) T (K) dimeritegraphene
FIG. 10: (Color online) The thermal conductance for thedimerite (blue solid line) and graphene (red dotted line) arecompared in a large temperature region. The value of thedimerite is about 40% smaller than the graphene at roomtemperature. s / s ( W m − K − ) q ( p ) calculated resultsfitted curve FIG. 11: (Color online) The directional dependence for thethermal conductance of dimerite. The calculated results(filled squares) are fitted by function f ( x ) = a + b cos(2( θ + c ))with a =1.9 × , b =1.1 × , and c = 0.1 π (blue line). frequency mode is a flexure mode with parabolic dis-persion of ω = βk . This flexure mode corresponds tothe vibration in the z direction. Different from the puregraphene, there are a lot of optical phonon modes withfrequency around 150 cm − in the dimerite.
2. thermal conductance in dimerite
In Fig. 10, we compare the thermal conductance of thedimerite with the graphene. At room temperature, thethermal conductance of the dimerite is about 40% smallerthan the graphene. As can be seen from Fig. 11, thethermal conductance in the dimerite is more anisotropic than graphene. The calculated results can be best fittedwith function f ( x ) = a + b cos(2( θ + c )) with a=1.9 × ,b=1.1 × , and c= 0.1 π . The thermal conductance hasa minimum value at θ = 0 . π , and maximum value at θ = 0 . π . As mentioned previously, the directions with θ = 0 . π , 0.9 π are the (7-7) and the (5-5) directions,respectively. So, the thermal conductance in (5-5) direc-tion is 12% larger than (7-7) direction, which is aboutone order larger than that of the pristine graphene. Weexpect this anisotropic effect detectable experimentally ifthe precision in the current experiment can be improved. V. CONCLUSION
In conclusion, we have calculated the phonon thermalconductance for graphene in the ballistic region, by con-sidering the graphene as the large width limit of graphenestrips. The calculated value for the thermal conductanceat room temperature is comparable with the recent ex-perimental results, while at high temperature region ourresults are consistent with the previous theoretical cal-culations. We have found that the thermal conductanceis directionally dependent and the reason is the direc-tional dependence of the velocities of different phononmodes, which can be excited in the frequency order withincreasing temperature. By breaking the D h symmetryin graphene, we can see more obvious anisotropic effectof the thermal conductance as demonstrated by dimerite.We have following two further remarks:(1). Since the anisotropic effect of thermal conduc-tance in graphene is small (1%), it requires high accu-racy in the calculation of phonon mode’s group velocityto see this anisotropic effect. Thanks to the superiorityof the VFFM, we can derive an analytic expression forthe dynamical matrix and calculate accurately the valueof the group velocity following Eq. (3). Thus we can ob-tain the 1% anisotropic effect in graphene as discussed inthis manuscript. We also used the Brenner empirical po-tential implemented in the “General Utility Lattice Pro-gram” (GULP) to calculate the phonon dispersion andgroup velocity in graphene. For lack of analytic expres-sion for the dynamical matrix, we find that the accuracyis not high enough for the group velocity and it is diffi-culty to see this anisotropic effect.(2). From symmetry analysis , when thermal trans-port is in the diffusive region where the Fourier’s lawexists, the D h symmetry of graphene constrains thethermal conductivity to be a constant value. So theanisotropic effect in graphene can not be seen if thethermal transport is in the diffusive region, yet it canonly be seen in the ballistic region as discussed in thismanuscript. But the situation changes in dimerite, wherethe D h symmetry is broken into D h . The D h symme-try does not constrain the thermal conductivity to be aconstant value even the Fourier’s law is valid. So we canexpect to see the isotropic effect in the dimerite both inthe ballistic and diffusive region. Acknowledgements
We thank Lifa Zhang for helpful discussions. Thework is supported by a Faculty Research Grant of R-144-000-173-112/101 of NUS, and Grant R-144-000-203-112from Ministry of Education of Republic of Singapore, andGrant R-144-000-222-646 from NUS.
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