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Optimized couplers for interfacial thermal transport
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 125401 (http://iopscience.iop.org/0953-8984/27/12/125401) View the table of contents for this issue, or go to the journal homepage for more
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 125401 (8pp)
doi:10.1088/0953-8984/27/12/125401
Optimized couplers for interfacial thermal transport Bo Chen1 and Lifa Zhang2,3 1 Tongda College, Nanjing University of Posts and Telecommunications, Nanjing Jiangsu 210003, People’s Republic of China 2 Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, 210023, People’s Republic of China
E-mail: [email protected] Received 8 December 2014, revised 30 January 2015 Accepted for publication 16 February 2015 Published 5 March 2015 Abstract
To find optimized interfacial thermal couplers we study thermal transport though a one dimensional atomic chain which includes three parts—two semi-infinite leads and a center region acting as a thermal coupler. It is found that the spring constant and atomic mass of an interfacial coupler can be selected to maximize interfacial thermal conductance. For the case of two lead materials with equal cutoff phonon frequencies, the interfacial system will reach a maximum thermal conductance if the coupler has the same cutoff frequency as those of the leads and its spring constant is equal to the geometric mean of those of leads. If two leads have equal acoustic impedances, one can find optimized interfacial coupler around the point where its acoustic impedance is the same as those of the leads and its spring constant is equal to the harmonic mean of those of leads. For general lead materials the interfacial thermal conductance can be optimized near the cross point of the geometric mean of impedances and harmonic mean of cutoff frequencies of the leads. Our findings can have potential applications in high performance interfacial thermal transport. Keywords: interfacial thermal transport, atomic chain, interfacial coupler (Some figures may appear in colour only in the online journal)
atomic details of actual interface structures, both models offer limited accuracy in nanoscale interfacial thermal resistance predictions [13]. Another method, the scattering boundary method which is a lattice dynamic approach, was first proposed by Lumpkin and Saslow to study the Kapitza conductance in a one-dimensional lattice [14]. Then is was applied to calculate the Kapitza resistance in two- and three-dimensional lattices [15, 16]. With full consideration of the atomic structures at interfaces, the scattering boundary method can predict interfacial thermal conductance with a high accuracy, which has been applied to study ballistic thermal transport in nanotube junctions [17], spin chains [18] and honeycomb lattice ribbons [19]. By using this method, Shafranjuk has been successfully observed a conversion of thermal energy into electricity in the electrically polarized graphene [20]. Recently, an analytic formula was obtained for interfacial thermal transport in 1D atomic chains [21], where the authors found that when the interfacial coupling equals to the harmonic
1. Introduction
In modern information techniques, accumulation of heat becomes a bottleneck for the progress of microelectronic devices due to a rapid increasing power density. Thus to efficiently dissipate waste heat or manipulate heat flow has been recognized to be a crucial issue in information and energy technologies [1, 2]. The past decade has witnessed significant research which focuses on thermal transport in nano structures [3, 4]. As the dimensions of materials shrink into the nanoscale, interfaces dramatically affect thermal transport [5–10], which makes it a lucrative field to be explored. An interfacial scattering process is a most important mechanism for thermal resistance especially in practical devices. To study thermal interfacial resistance, the acoustic mismatch model [11] and diffuse mismatch model [12] have been proposed for a long time. However, since they neglect 3
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J. Phys.: Condens. Matter 27 (2015) 125401
B Chen and L Zhang
mean of the spring constants of leads the interfacial thermal conductance reaches its maximum, which was reproduced and detailed explained by Saltonstall et al in [22] by a non-equilibrium Greens functions approach. In these studies the interface is simply modeled by a bond coupling. Very recently Kuar et al [23] reported a sixfold enhancement in thermal interface conductance between metal surfaces and vertically aligned multiwall-carbon-nanotube arrays through bridging the interface with short, covalently bonded organic molecules. This finding is significant since it solves the long lasting problem of limited performance of nanotubes in practical systems, where the nanostructured systems have low thermal interfacial conductances due to weak adhesion at interfaces [24] while carbon nanotubes have very high thermal conductivities [25]. Therefore one cannot obtain optimized interfacial thermal transport through coupling the two materials directly, that is, the maximized conductance found in [21] can not be applied here. However, a possible way is to find a thermal coupler to bridge them. To help device engineers select thermal interface couplers, we need know not only the coupling strength but also the atomic mass. However, the method of obtaining and optimized interface coupler with certain spring constant and atomic mass is still an open question to date. In this paper we apply the scattering boundary method to study an optimized thermal coupler bridging two dissimilar materials to get a optimized performance for interfacial thermal transport. We investigate the optimized spring constant and atomic mass for interface thermal couplers, where different cases are discussed, such as two leads with equal cut-off frequencies, two leads with equal acoustic impedance and a general case in the end.
which can be obtained though the scattering boundary method. Assuming a wave solution transmitting from one region i to the other region j , by the continuity condition at the interface kij , one can obtain the reflection coefficient and transmission coefficient. For the model with three regions, the total transmitted wave function is obtained as a superposition of multiple reflections and transmissions, resulting in a total transmission coefficient. In the end, the transmission probability can be written as [21] Tr[ω] =
(1 − |r12 |2 )(1 − |r23 |2 ) C −1) 2 |1 − r23 r21 λ2(N | 2
.
(3)
Here NC is the number of atoms in the center atomic chain and the reflection coefficient rij is given by rij =
ki (λi − 1/λi )(kj − kij − kj /λj ) − 1. (4) (ki − kij − ki /λi )(kj − kij − kj /λj ) − kij2
Here, i = 1, 2, or 3, ki , kj is the spring constant of the i, j region and kij is the spring constant between i and j regions. λi is the one with |λ| < 1 of the two roots which are given as λj =
−hj ±
2
h2j − 4
,
hj =
mj (ω + iη)2 − 2, kj
(5)
Here iη is a tiny pure imaginary number. In later calculations, we will use dimensionless unit, that is, we set h ¯ = 1, kB = 1 for simplicity. For conversion from the dimensionless units to physical units, if one takes the energy unit [E] = 1 meV, length unit [L] = 1 Å, then temperature unit is [T ] = 11.6 K, thermal conductance unit is [σ ] = 20.9 nW mK−1 and the spring constant unit is [k] = 1 meV Å−2 [10]. As shown in figure 1, the transmission coefficient oscillates with changing of frequency due to the interference effect in the center region [6], where the number of peaks increases with enlarging system size; however the inset shows that the thermal conductance is independent of system size because of ballistic transport. Thus without loss of generality, in the following we set a constant of NC = 1000 to calculate thermal conductance.
2. Method
To study interfacial thermal transport, we use a simple model as follows: a one-dimensional atomic chain consisting of three parts—two semi-infinite leads and a center region (interfacial coupler). The left and right leads are in equilibrium at different temperatures TL and TR , respectively. The left lead, center region and right lead are harmonic chains with mass and spring constants m1 , k1 , m2 , k2 and m3 , k3 . The three parts are connected by harmonic springs with constant strength k12 and k23 . Heat current flowing through the system is given by the Landauer formula [4] ∞ 1 h ¯ ω fL (ω) − fR (ω) Tr[ω]dω, (1) I= 2π 0
3. Results
From equation (3), we can calculate the transmission coefficient of the interfacial system, then input it to equation (2) and obtain the thermal conductance. The reference [21] focused on interfacial coupling and reported that the interfacial thermal conductance reaches its maximum when the interfacial coupling strength equals the harmonic mean of the spring constants of the leads, that is, k12m = 2k1 k2 /(k1 + k2 ). In this work we study the optimized materials for an interfacial coupler and will focus on k2 and m2 to obtain an optimized thermal interfacial coupler. In the following we will consider two kinds of cases: one is a simplified case where k12 = k23 = k2 . Another is the maximum-coupling case where k12 = k12m and k23 = k23m = 2k2 k3 /(k2 + k3 ).
which allows us to develop the junction conductance formula as ∞ 1 ∂f (ω) σ = dω h ¯ ω Tr[ω] , (2) 2π 0 ∂T here, fL,R = {exp[¯hω/(kB TL,R )] − 1}−1 is the Bose-Einstein distribution for phonons, kB is the Boltzman constant and Tr[ω] is the transmission probability for phonon with frequency ω. Therefore the key step for characterization of interfacial thermal transport is to calculate the transmission probability, 2
J. Phys.: Condens. Matter 27 (2015) 125401
B Chen and L Zhang
Figure 1. The transmission probability Tr[ω] as a function of frequency for different system size. The inset shows the thermal conductance for 1D lead materials as a function of size. Here k1 = 2, m1 = 3, k1 = 9, m1 = 3.5 and k3 = 7, m3 = 4, k12 = 1, k23 = 3.
Figure 2. Thermal conductance for lead materials with equal cutoff frequencies in large ranges of k2 and m2 of the interfacial material. Here the solid line corresponds to interfacial material with cutoff frequency being equal to that of leads; the dashed and dotted lines corresponds to interfacial material with acoustic impedance being equal to that of left and right leads, respectively. k1 = m1 = 5 and k3 = m3 = 6. For (a) k12 = k23 = k2 , for (b) k12 = k12m = 2k1 k2 /(k1 + k2 ), k23 = k23m = 2k2 k3 /(k2 + k3 ).
panels) have large conductances due to the less scattering at the boundaries of the center part. The solid line in each panel of figure 2 divides the contour into two regions. For all four plots, the right-bottom region has a larger conductance than that of the left-top one since in the left-top region the cutoff frequency of the interface coupler is lower than that of the leads which will block phonons with high frequencies. However, if the atomic mass of the center is very large, the thermal conductance will be very small although the cutoff frequency is larger than that of the leads. We know that
3.1. Two lead materials with equal cutoff phonon frequencies
We first focus on the interface √ with two leads of equal cutoff phonon frequency ωC = 2 k/m that is k1 /m1 = k3 /m3 . The contour plots of interfacial thermal conductance as functions of k2 and m2 are shown in figure 2. It is obviously shown that the interfacial conductance is dependent on both spring constant k2 and atomic mass m2 of the center coupler material. Therefore using only interfacial coupling (a spring constant) in [21] to study interfacial thermal transport is not a proper model. As shown in figure 2, if k2 is very small or m2 is too large, the conductance will be very small. Only both m2 and k2 are comparable to those of leads, the interfacial conductance can be quite large. The maximum-coupling cases (bottom two
ω 2 m2
the phonon impedance equals to Zi = ki mi − 4 i [22] and at low√frequencies ω → 0, we obtain the acoustic impedance Zi (0) ki mi . Thus in figure 2 we plot two lines where the 3
J. Phys.: Condens. Matter 27 (2015) 125401
B Chen and L Zhang
Figure 3. The zoomed-in plots of figure 2 between the two lines of acoustic impedances of the leads. The insets show the thermal
conductance along the impedance equal to left lead (black line Z22 = k2 m2 = 25) and right lead (red line Z22 = k2 m2 = 36). The parameters are the same as those in figure 2.
acoustic impedance Z2 of the interfacial material is equal to those of left and right leads. It can be seen that the peaks of interfacial thermal conductance located in the zones which are between the two impedance lines and below the frequency line. Therefore we zoom into the peak area in figure 3. √ If the cut-off frequency of the interfacial material k2 /m2 is less than those of the leads the thermal conductance will be small due to the blockade of phonons with high frequencies. Thus we focus on the contour area below the frequency line and between the two impedance lines, as shown in figure 3. It is obviously shown that the conductance will maximize at a point where the cut-off frequency equals to those of the leads. And from the two contour plots, the peaks of interfacial thermal conductance are on the left-up edge where the center part has the same cut-off frequency as that of the leads. There is not much difference for these two contour plots. To further investigate the maximum interfacial thermal conductance we study the interfacial conductance along the left-up edge and plot the curves in figure 4. From calculations of the interfacial thermal conductance along the line where the interfacial coupler has the same cutoff frequency as that of the leads, we find that the conductance will reach its maximum when the acoustic impedance Z2 (0) √ equals to the geometric mean of the of leads Z1 (0)Z3 (0), that is, k2 m 2 = k 1 m 1 k 3 m3 . (6)
frequencies. In figure 4, we investigate the conductance ef σ (k2 ) in the range between k2 = 0.75 k2m and k2 = 1.25 ef k2m , where the conductance has minimum at k2 = 0.75 ef ef k2m and maximum at k2 = k2m , thus we show the ratio of ef ef ef [σ (k2 ) − σ (0.75 k2m )]/[σ (k2m ) − σ (0.75 k2m )] in order to get values in the range between 0 and 1. As shown in figure 4, for the simplified-coupling case k12 = k23 , although the interfacial conductances are quite different for different parameters as show in the inset, all the cases have a maximum conductance ef at k2m , where all the curves almost scale to one line with a tiny deviation. If we change the cut-off frequency for all three parts, the interfacial thermal conductance will change a lot, but the scaling behavior remains fixed and all cases ef maximize at k2m , as shown in the right panel of figure 4. If we change from the simplified case to maximum-coupling case (k12 = k12m = 2k1 k2 /(k1 +k2 ), k23 = k23m = 2k2 k3 /(k2 +k3 )), all the curves of interfacial thermal conductance will scale to ef exact one line in the vicinity of k12m while values of interfacial conductances are still quite different. For different cut-off frequency cases, the scaling behaviors are exactly the same. Therefore, for the interface structure with leads of equal cutoff-frequency, if the interface coupler has the same cut-off frequency and its spring constant equals the geometric mean of those of the leads the interfacial thermal conductance will reach its maximum.
Combing the cut-off frequency condition
3.2. Two lead materials with equal acoustic impedance
k2 /m2 = k1 /m1 = k3 /m3 , we obtain k2 =
ef k1 k3 ≡ k2m ,
We have found an exact point of maximum interfacial thermal conductance for a simple case where the leads have an equal cut-off frequency. As a further step we study interfacial structures where the leads have an equal acoustic impedance, that is, k1 m1 = k3 m3 . If the cut-off frequency of the coupler is less than any one of the leads, the thermal conductance will be very small. The peak area of the conductance is distributed along the solid line of equal acoustic impedance and in the area
(7)
(8)
when the interfacial conductance will reach the maximum. ef Here k2m denotes the spring constant of k2 which makes the conductance maximize for the case of two leads with equal 4
J. Phys.: Condens. Matter 27 (2015) 125401
B Chen and L Zhang
ef
ef
Figure 4. The thermal conductance versus the ratio of k2 /k2m . Here k2m is the geometric mean of the spring constants of the two semi-infinite leads which have equal cutoff frequencies. √ k12 = k23 = k2 . Left panel: the cut-off frequency for all three parts are 2; right panel: the cut-off frequency for all three parts are 2/ 3.
Figure 5. Thermal conductance for lead materials with equal acoustic impedance in the area between the two lines of cutoff frequency of the leads. The sold line corresponds to acoustic impedance being equal to that of leads and the dotted line corresponds to the cutoff frequency being equal to the harmonic mean of those of left and right leads. The stars show the maximum conductances. The white line in each plot labels a contour of a certain value of thermal conductance. k1 = 4, = m1 = 6 and k3 = 5, m3 = 4.8. For (a) k12 = k23 = k2 , for (b) k12 = k12m = 2k1 k2 /(k1 + k2 ), k23 = k23m = 2k2 k3 /(k2 + k3 ).
(colorful region) between the two lines of cut-off frequencies, which is shown in figure 5. The star marks the point where the interfacial thermal conductance reaches its maximum. We plot a dotted line where the frequency equals to the harmonic mean of those of the leads, that is √ √ 2 k1 /m1 k3 /m3 k2 /m2 = √ . (9) √ k1 /m1 + k3 /m3
It crosses with the equal impedance line k2 m2 = k1 m1 = k3 m3
(10)
at the point of k2 = 5
2k1 k3 ei ≡ k2m , k1 + k 3
(11)
J. Phys.: Condens. Matter 27 (2015) 125401
B Chen and L Zhang
Figure 6. (a) The thermal conductance normalized by maximum conductance versus spring constant k2 of interfacial materials for different ei acoustic impedance. The k2m = 2k1 k3 /(k1 + k3 ) is marked for guidance. (b) The thermal conductance normalized by maximum conductance versus spring constant k2 of interfacial materials at fixed acoustic impedance for different spring constant k1 of the left lead. The stars are ei marked for points of k2m = 2k1 k3 /(k1 + k3 ). ei here k2m denotes the spring constant of k2 to make the conductance maximize for the case of two leads with equal impedance. It is not the same point with the maximum conductance as shown figure 7. Nevertheless, according to ei has a value more than contour lines in figure 5, the point of k2m 99% of the maximum. Therefore although we cannot obtain an ei exact formula for the maximum interfacial conductance, k2m can act as an alternative to find the optimized material for an interface thermal coupler. We change the impedance for the interfacial systems and plot the interfacial thermal conductance as a function of k2 in figure 6(a). If k1 and k3 are fixed, for different acoustic impedance cases, the interfacial thermal ei conductance maximizes at the same point which is near to k2m ei and the interfacial conductance at k2m is always larger than 99% of its maximum. If we change the spring constant of one of leads and keep the three parts a fixed acoustic impedance, the normalized interfacial thermal conductances are plotted in ei , which are near figure 6(b). The stars mark the points of k2m to the corresponding maximums; furthermore the interfacial ei have thermal conductances more than 99% couplers with k2m of their maxima. Therefore for the case of two leads with an equal acoustic impedance, we can choose an interfacial coupler with the same acoustic impedance (Z(ω = 0+)) as those of leads and the spring constant equal to the harmonic mean of the leads to obtain an almost maximum of the thermal conductance.
3.3. A general interface
After investigating the above two special cases we move to a general case, where two leads have different cut-off frequencies and different acoustic impedances. Similarly to the above two cases, the maximum thermal conductance is in the range bounded by the two cut-off frequencies and two acoustic impedances (Z(ω = 0+ )) of the leads. The contour plots of this region for the interfacial thermal conductance are shown in figure 7. Similarly to the second case, one still cannot get an analytic formula for the maximum point of interfacial conductance, but with the hint from the above two cases, we plot the lines of geometric mean of impedances (equation (6), the solid line) and the harmonic mean of frequencies (equation (9), the dotted line) in figure 7 for guidance. Comparing the maximum point of conductance (marked as stars), we find that cross points of the two lines are located in the contour with values larger than 99.00%) of their maxima. Therefore the cross point of the two lines can be a good choice to obtain optimized interfacial conductance for a general interface. 4. Discussion
For interfacial thermal transport, the scattering due to the mismatch at interfaces causes thermal resistance. Without mismatch, that is, in a homogeneous system, for each 6
J. Phys.: Condens. Matter 27 (2015) 125401
B Chen and L Zhang
Figure 7. Thermal conductance for lead materials with general spring constants and masses as a function of the spring constant k2 and atomic mass m2 of the interfacial material. Here the contours are plotted in the regions bounded by the two lines of cutoff frequency of the leads and the two lines of acoustic impedance of the leads. The sold line corresponds to acoustic impedances equal to the geometric mean of those of the leads and the dotted line corresponds the cutoff frequency being equal to the harmonic mean of those of leads. The stars show the maximum conductances. The white line in each plot labels a contour of a certain value of thermal conductance. Here k1 = 2, m1 = 3 and k3 = 7, m3 = 4. For (a) k12 = k23 = k2 , for (b) k12 = k12m = 2k1 k2 /(k1 + k2 ), k23 = k23m = 2k2 k3 /(k2 + k3 ).
phonon mode the transmission probability will be 1 when no phonon scattering exists and the system is a perfect thermal conductor. Without considering nonlinear phonon-phonon interaction, defects and rough surfaces, all scattering comes from interfaces—the mismatch between different materials. The goal to find optimized materials is to minimize scattering from interfacial mismatch. In [21], if interfacial coupling equals to the harmonic mean of those of two coupling leads, the conductance will be maximized. However, the direct coupling between materials sometimes is very weak and one needs find a thermal coupler to bridge them. In many realistic materials, the direct coupling strength between them is small, which can be increasing by shortening distance between them. If the coupling strength between the thermal coupler and two lead materials is still very small, the interfacial thermal conductance of the whole system will be small. This is because of the large scattering at interfaces, which is similar to the weak direct coupling between two lead materials. Therefore, we need to find thermal coupler which has relatively larger coupling to leads. If the coupling strength between a thermal coupler and leads is in the range around maximized couplings k2m , k3m , one could obtain optimized thermal conductance when a thermal coupler has acoustic impedance equal to that of the leads and spring constant equal to the harmonic mean of the leads. If coupling strength is around the spring constant of materials themselves (k1 , k2 , k3 ), thermal conductance of whole systems can also be optimized near the two means. We use a simplified one-dimensional atomic model to investigate optimized thermal couplers for interfacial thermal transport. For most quasi-one-dimensional thermal transport
our findings can be applied. The Landauer formula (2) can be applied to any-dimension systems. For different dimensions, the transmission probability Tr(ω) will be quite different. Using the same transmission as that of one-dimensional case and replacing ω with ω3 , one can calculate thermal conductance to mimic thermal transport in three-dimensional systems avoiding introducing more complicated structures. Our calculations show that the optimized interfacial thermal coupler with properties near the geometric mean of impedances and harmonic mean of cutoff frequencies of leads can still be applied to three-dimensional interfaces. Thus our finding on optimized interface coupler can be qualitatively applied to real systems. Therefore to find an optimized interfacial thermal coupler in real systems, one needs to find materials with relativey larger couplings to the leads and with properties near the geometric mean of impedances and harmonic mean of cutoff frequencies of lead materials. As reported in [23], using a coupler of covalently bonded organic molecules, thermal interface conductance between metal and multiwall-carbon-nanotube arrays has a sixfold enhancement. Such organic molecules can have better adhesion with metal and carbon nanotube, if the organic molecules have properties near the geometric mean of impedances and harmonic mean of cutoff frequencies of metal and carbon nanotubes. The thermal conductance can be much more enhanced. 5. Conclusion
Using the scattering boundary method, we study optimized couplers for interfacial thermal transport where couplers 7
J. Phys.: Condens. Matter 27 (2015) 125401
B Chen and L Zhang
cannot be simplified as only spring constants. The spring constant and atomic mass of the coupler can be selected to optimize interfacial thermal conductance. For an interface with two leads having equal cutoff phonon frequencies, to get the maximum interfacial conductance, the interfacial coupler must have a spring constant equal to the geometric mean of the leads. It must also have the same cutoff frequency as the leads. For two lead materials with equal acoustic impedances, one can find optimized interfacial materials around the point where the acoustic impedance is equal to those of leads and the spring constant is equal to the harmonic mean of those of leads. For general materials one can find optimized interfacial couplers with spring constant and atomic mass near the cross point of the geometric mean of impedances and harmonic mean of cutoff frequencies of leads.
[7] Hsieh W P, Lyons A S, Pop E, Keblinski P and Cahill D G 2011 Phys. Rev. B 84 184107 [8] Li X and Yang R 2012 Phys. Rev. B 86 054305 [9] Wilson R B and Cahill D G 2012 Phys. Rev. Lett. 108 255901 [10] Zhang L, Thingna J, He D, Wang J-S and Li B 2013 Europhys. Lett. 103 64002 Zhang L, Lu J-T, Wang J-S and Li B 2013 J. Phys.: Condens. Matter 25 445801 [11] Little W A 1959 Can. J. Phys. 37 334 [12] Swartz E and Pohl R 1989 Rev. Mod. Phys. 61 605 [13] Stevens R, Smith A and Norris P 2005 J. Heat Transfer 127 315 [14] Lumpkin M E and Saslow W M 1997 Phys. Rev. B 17 4295 [15] Paranjape B V, Arimitsu N and Krebes E S 1987 J. Appl. Phys. 61 888 [16] Young D A and Maris H J 1989 Phys. Rev. B 40 3685 [17] Wang J and Wang J-S 2006 Phys. Rev. B 74 054303 [18] Zhang L, Wang J-S and Li B 2008 Phys. Rev. B 78 144416 [19] Cuansing E and Wang J-S 2009 Eur. Phys. J. B 69 505 [20] Shafranjuk S E 2013 arXiv:1208.4112 [21] Zhang L, Keblinski P, Wang J-S and Li B 2011 Phys. Rev. B 83 064303 [22] Saltonstall C B, Polanco C A, Duda J C, Ghosh A W, Norris P M and Hopkins P E 2013 J. Appl. Phys. 113 013516 [23] Kaur S, Raravikar N, Helms B A, Prasher R and Ogletree D F 2014 Nat. Commun. 5 3082 [24] Marconnet A M, Panzer M A and Goodson K E 2013 Rev. Mod. Phys. 85 1295 [25] Prasher R 2008 Phys. Rev. B 77 075424
References [1] Li N, Ren J, Wang L, Zhang G, H¨anggi P and Li B 2012 Rev. Mod. Phys. 84 1045 [2] Cahill D G, Ford W K, Goodson K E, Mahan G D, Majumdar A, Maris H J, Merlin R and Phillpot S R 2003 J. Appl. Phys. 93 793 [3] Dhar A 2008 Adv. Phys. 57 457 [4] Wang J-S, Wang J and L¨u J T 2008 Eur. Phys. J. B 62 381 [5] Costescu R M, Cahill D G, Fabreguette F H, Sechrist Z A and George S M 2004 Science 303 989 [6] Hu L, Zhang L, Hu M, Wang J-S, Li B and Keblinski P 2010 Phys. Rev. B 81 235427
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Optimized couplers for interfacial thermal transport
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 125401 (http://iopscience.iop.org/0953-8984/27/12/125401) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 131.91.169.193 This content was downloaded on 09/09/2015 at 01:08
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 125401 (8pp)
doi:10.1088/0953-8984/27/12/125401
Optimized couplers for interfacial thermal transport Bo Chen1 and Lifa Zhang2,3 1 Tongda College, Nanjing University of Posts and Telecommunications, Nanjing Jiangsu 210003, People’s Republic of China 2 Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, 210023, People’s Republic of China
E-mail: [email protected] Received 8 December 2014, revised 30 January 2015 Accepted for publication 16 February 2015 Published 5 March 2015 Abstract
To find optimized interfacial thermal couplers we study thermal transport though a one dimensional atomic chain which includes three parts—two semi-infinite leads and a center region acting as a thermal coupler. It is found that the spring constant and atomic mass of an interfacial coupler can be selected to maximize interfacial thermal conductance. For the case of two lead materials with equal cutoff phonon frequencies, the interfacial system will reach a maximum thermal conductance if the coupler has the same cutoff frequency as those of the leads and its spring constant is equal to the geometric mean of those of leads. If two leads have equal acoustic impedances, one can find optimized interfacial coupler around the point where its acoustic impedance is the same as those of the leads and its spring constant is equal to the harmonic mean of those of leads. For general lead materials the interfacial thermal conductance can be optimized near the cross point of the geometric mean of impedances and harmonic mean of cutoff frequencies of the leads. Our findings can have potential applications in high performance interfacial thermal transport. Keywords: interfacial thermal transport, atomic chain, interfacial coupler (Some figures may appear in colour only in the online journal)
atomic details of actual interface structures, both models offer limited accuracy in nanoscale interfacial thermal resistance predictions [13]. Another method, the scattering boundary method which is a lattice dynamic approach, was first proposed by Lumpkin and Saslow to study the Kapitza conductance in a one-dimensional lattice [14]. Then is was applied to calculate the Kapitza resistance in two- and three-dimensional lattices [15, 16]. With full consideration of the atomic structures at interfaces, the scattering boundary method can predict interfacial thermal conductance with a high accuracy, which has been applied to study ballistic thermal transport in nanotube junctions [17], spin chains [18] and honeycomb lattice ribbons [19]. By using this method, Shafranjuk has been successfully observed a conversion of thermal energy into electricity in the electrically polarized graphene [20]. Recently, an analytic formula was obtained for interfacial thermal transport in 1D atomic chains [21], where the authors found that when the interfacial coupling equals to the harmonic
1. Introduction
In modern information techniques, accumulation of heat becomes a bottleneck for the progress of microelectronic devices due to a rapid increasing power density. Thus to efficiently dissipate waste heat or manipulate heat flow has been recognized to be a crucial issue in information and energy technologies [1, 2]. The past decade has witnessed significant research which focuses on thermal transport in nano structures [3, 4]. As the dimensions of materials shrink into the nanoscale, interfaces dramatically affect thermal transport [5–10], which makes it a lucrative field to be explored. An interfacial scattering process is a most important mechanism for thermal resistance especially in practical devices. To study thermal interfacial resistance, the acoustic mismatch model [11] and diffuse mismatch model [12] have been proposed for a long time. However, since they neglect 3
Author to whom any correspondence should be addressed.
0953-8984/15/125401+08$33.00
1
© 2015 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 27 (2015) 125401
B Chen and L Zhang
mean of the spring constants of leads the interfacial thermal conductance reaches its maximum, which was reproduced and detailed explained by Saltonstall et al in [22] by a non-equilibrium Greens functions approach. In these studies the interface is simply modeled by a bond coupling. Very recently Kuar et al [23] reported a sixfold enhancement in thermal interface conductance between metal surfaces and vertically aligned multiwall-carbon-nanotube arrays through bridging the interface with short, covalently bonded organic molecules. This finding is significant since it solves the long lasting problem of limited performance of nanotubes in practical systems, where the nanostructured systems have low thermal interfacial conductances due to weak adhesion at interfaces [24] while carbon nanotubes have very high thermal conductivities [25]. Therefore one cannot obtain optimized interfacial thermal transport through coupling the two materials directly, that is, the maximized conductance found in [21] can not be applied here. However, a possible way is to find a thermal coupler to bridge them. To help device engineers select thermal interface couplers, we need know not only the coupling strength but also the atomic mass. However, the method of obtaining and optimized interface coupler with certain spring constant and atomic mass is still an open question to date. In this paper we apply the scattering boundary method to study an optimized thermal coupler bridging two dissimilar materials to get a optimized performance for interfacial thermal transport. We investigate the optimized spring constant and atomic mass for interface thermal couplers, where different cases are discussed, such as two leads with equal cut-off frequencies, two leads with equal acoustic impedance and a general case in the end.
which can be obtained though the scattering boundary method. Assuming a wave solution transmitting from one region i to the other region j , by the continuity condition at the interface kij , one can obtain the reflection coefficient and transmission coefficient. For the model with three regions, the total transmitted wave function is obtained as a superposition of multiple reflections and transmissions, resulting in a total transmission coefficient. In the end, the transmission probability can be written as [21] Tr[ω] =
(1 − |r12 |2 )(1 − |r23 |2 ) C −1) 2 |1 − r23 r21 λ2(N | 2
.
(3)
Here NC is the number of atoms in the center atomic chain and the reflection coefficient rij is given by rij =
ki (λi − 1/λi )(kj − kij − kj /λj ) − 1. (4) (ki − kij − ki /λi )(kj − kij − kj /λj ) − kij2
Here, i = 1, 2, or 3, ki , kj is the spring constant of the i, j region and kij is the spring constant between i and j regions. λi is the one with |λ| < 1 of the two roots which are given as λj =
−hj ±
2
h2j − 4
,
hj =
mj (ω + iη)2 − 2, kj
(5)
Here iη is a tiny pure imaginary number. In later calculations, we will use dimensionless unit, that is, we set h ¯ = 1, kB = 1 for simplicity. For conversion from the dimensionless units to physical units, if one takes the energy unit [E] = 1 meV, length unit [L] = 1 Å, then temperature unit is [T ] = 11.6 K, thermal conductance unit is [σ ] = 20.9 nW mK−1 and the spring constant unit is [k] = 1 meV Å−2 [10]. As shown in figure 1, the transmission coefficient oscillates with changing of frequency due to the interference effect in the center region [6], where the number of peaks increases with enlarging system size; however the inset shows that the thermal conductance is independent of system size because of ballistic transport. Thus without loss of generality, in the following we set a constant of NC = 1000 to calculate thermal conductance.
2. Method
To study interfacial thermal transport, we use a simple model as follows: a one-dimensional atomic chain consisting of three parts—two semi-infinite leads and a center region (interfacial coupler). The left and right leads are in equilibrium at different temperatures TL and TR , respectively. The left lead, center region and right lead are harmonic chains with mass and spring constants m1 , k1 , m2 , k2 and m3 , k3 . The three parts are connected by harmonic springs with constant strength k12 and k23 . Heat current flowing through the system is given by the Landauer formula [4] ∞ 1 h ¯ ω fL (ω) − fR (ω) Tr[ω]dω, (1) I= 2π 0
3. Results
From equation (3), we can calculate the transmission coefficient of the interfacial system, then input it to equation (2) and obtain the thermal conductance. The reference [21] focused on interfacial coupling and reported that the interfacial thermal conductance reaches its maximum when the interfacial coupling strength equals the harmonic mean of the spring constants of the leads, that is, k12m = 2k1 k2 /(k1 + k2 ). In this work we study the optimized materials for an interfacial coupler and will focus on k2 and m2 to obtain an optimized thermal interfacial coupler. In the following we will consider two kinds of cases: one is a simplified case where k12 = k23 = k2 . Another is the maximum-coupling case where k12 = k12m and k23 = k23m = 2k2 k3 /(k2 + k3 ).
which allows us to develop the junction conductance formula as ∞ 1 ∂f (ω) σ = dω h ¯ ω Tr[ω] , (2) 2π 0 ∂T here, fL,R = {exp[¯hω/(kB TL,R )] − 1}−1 is the Bose-Einstein distribution for phonons, kB is the Boltzman constant and Tr[ω] is the transmission probability for phonon with frequency ω. Therefore the key step for characterization of interfacial thermal transport is to calculate the transmission probability, 2
J. Phys.: Condens. Matter 27 (2015) 125401
B Chen and L Zhang
Figure 1. The transmission probability Tr[ω] as a function of frequency for different system size. The inset shows the thermal conductance for 1D lead materials as a function of size. Here k1 = 2, m1 = 3, k1 = 9, m1 = 3.5 and k3 = 7, m3 = 4, k12 = 1, k23 = 3.
Figure 2. Thermal conductance for lead materials with equal cutoff frequencies in large ranges of k2 and m2 of the interfacial material. Here the solid line corresponds to interfacial material with cutoff frequency being equal to that of leads; the dashed and dotted lines corresponds to interfacial material with acoustic impedance being equal to that of left and right leads, respectively. k1 = m1 = 5 and k3 = m3 = 6. For (a) k12 = k23 = k2 , for (b) k12 = k12m = 2k1 k2 /(k1 + k2 ), k23 = k23m = 2k2 k3 /(k2 + k3 ).
panels) have large conductances due to the less scattering at the boundaries of the center part. The solid line in each panel of figure 2 divides the contour into two regions. For all four plots, the right-bottom region has a larger conductance than that of the left-top one since in the left-top region the cutoff frequency of the interface coupler is lower than that of the leads which will block phonons with high frequencies. However, if the atomic mass of the center is very large, the thermal conductance will be very small although the cutoff frequency is larger than that of the leads. We know that
3.1. Two lead materials with equal cutoff phonon frequencies
We first focus on the interface √ with two leads of equal cutoff phonon frequency ωC = 2 k/m that is k1 /m1 = k3 /m3 . The contour plots of interfacial thermal conductance as functions of k2 and m2 are shown in figure 2. It is obviously shown that the interfacial conductance is dependent on both spring constant k2 and atomic mass m2 of the center coupler material. Therefore using only interfacial coupling (a spring constant) in [21] to study interfacial thermal transport is not a proper model. As shown in figure 2, if k2 is very small or m2 is too large, the conductance will be very small. Only both m2 and k2 are comparable to those of leads, the interfacial conductance can be quite large. The maximum-coupling cases (bottom two
ω 2 m2
the phonon impedance equals to Zi = ki mi − 4 i [22] and at low√frequencies ω → 0, we obtain the acoustic impedance Zi (0) ki mi . Thus in figure 2 we plot two lines where the 3
J. Phys.: Condens. Matter 27 (2015) 125401
B Chen and L Zhang
Figure 3. The zoomed-in plots of figure 2 between the two lines of acoustic impedances of the leads. The insets show the thermal
conductance along the impedance equal to left lead (black line Z22 = k2 m2 = 25) and right lead (red line Z22 = k2 m2 = 36). The parameters are the same as those in figure 2.
acoustic impedance Z2 of the interfacial material is equal to those of left and right leads. It can be seen that the peaks of interfacial thermal conductance located in the zones which are between the two impedance lines and below the frequency line. Therefore we zoom into the peak area in figure 3. √ If the cut-off frequency of the interfacial material k2 /m2 is less than those of the leads the thermal conductance will be small due to the blockade of phonons with high frequencies. Thus we focus on the contour area below the frequency line and between the two impedance lines, as shown in figure 3. It is obviously shown that the conductance will maximize at a point where the cut-off frequency equals to those of the leads. And from the two contour plots, the peaks of interfacial thermal conductance are on the left-up edge where the center part has the same cut-off frequency as that of the leads. There is not much difference for these two contour plots. To further investigate the maximum interfacial thermal conductance we study the interfacial conductance along the left-up edge and plot the curves in figure 4. From calculations of the interfacial thermal conductance along the line where the interfacial coupler has the same cutoff frequency as that of the leads, we find that the conductance will reach its maximum when the acoustic impedance Z2 (0) √ equals to the geometric mean of the of leads Z1 (0)Z3 (0), that is, k2 m 2 = k 1 m 1 k 3 m3 . (6)
frequencies. In figure 4, we investigate the conductance ef σ (k2 ) in the range between k2 = 0.75 k2m and k2 = 1.25 ef k2m , where the conductance has minimum at k2 = 0.75 ef ef k2m and maximum at k2 = k2m , thus we show the ratio of ef ef ef [σ (k2 ) − σ (0.75 k2m )]/[σ (k2m ) − σ (0.75 k2m )] in order to get values in the range between 0 and 1. As shown in figure 4, for the simplified-coupling case k12 = k23 , although the interfacial conductances are quite different for different parameters as show in the inset, all the cases have a maximum conductance ef at k2m , where all the curves almost scale to one line with a tiny deviation. If we change the cut-off frequency for all three parts, the interfacial thermal conductance will change a lot, but the scaling behavior remains fixed and all cases ef maximize at k2m , as shown in the right panel of figure 4. If we change from the simplified case to maximum-coupling case (k12 = k12m = 2k1 k2 /(k1 +k2 ), k23 = k23m = 2k2 k3 /(k2 +k3 )), all the curves of interfacial thermal conductance will scale to ef exact one line in the vicinity of k12m while values of interfacial conductances are still quite different. For different cut-off frequency cases, the scaling behaviors are exactly the same. Therefore, for the interface structure with leads of equal cutoff-frequency, if the interface coupler has the same cut-off frequency and its spring constant equals the geometric mean of those of the leads the interfacial thermal conductance will reach its maximum.
Combing the cut-off frequency condition
3.2. Two lead materials with equal acoustic impedance
k2 /m2 = k1 /m1 = k3 /m3 , we obtain k2 =
ef k1 k3 ≡ k2m ,
We have found an exact point of maximum interfacial thermal conductance for a simple case where the leads have an equal cut-off frequency. As a further step we study interfacial structures where the leads have an equal acoustic impedance, that is, k1 m1 = k3 m3 . If the cut-off frequency of the coupler is less than any one of the leads, the thermal conductance will be very small. The peak area of the conductance is distributed along the solid line of equal acoustic impedance and in the area
(7)
(8)
when the interfacial conductance will reach the maximum. ef Here k2m denotes the spring constant of k2 which makes the conductance maximize for the case of two leads with equal 4
J. Phys.: Condens. Matter 27 (2015) 125401
B Chen and L Zhang
ef
ef
Figure 4. The thermal conductance versus the ratio of k2 /k2m . Here k2m is the geometric mean of the spring constants of the two semi-infinite leads which have equal cutoff frequencies. √ k12 = k23 = k2 . Left panel: the cut-off frequency for all three parts are 2; right panel: the cut-off frequency for all three parts are 2/ 3.
Figure 5. Thermal conductance for lead materials with equal acoustic impedance in the area between the two lines of cutoff frequency of the leads. The sold line corresponds to acoustic impedance being equal to that of leads and the dotted line corresponds to the cutoff frequency being equal to the harmonic mean of those of left and right leads. The stars show the maximum conductances. The white line in each plot labels a contour of a certain value of thermal conductance. k1 = 4, = m1 = 6 and k3 = 5, m3 = 4.8. For (a) k12 = k23 = k2 , for (b) k12 = k12m = 2k1 k2 /(k1 + k2 ), k23 = k23m = 2k2 k3 /(k2 + k3 ).
(colorful region) between the two lines of cut-off frequencies, which is shown in figure 5. The star marks the point where the interfacial thermal conductance reaches its maximum. We plot a dotted line where the frequency equals to the harmonic mean of those of the leads, that is √ √ 2 k1 /m1 k3 /m3 k2 /m2 = √ . (9) √ k1 /m1 + k3 /m3
It crosses with the equal impedance line k2 m2 = k1 m1 = k3 m3
(10)
at the point of k2 = 5
2k1 k3 ei ≡ k2m , k1 + k 3
(11)
J. Phys.: Condens. Matter 27 (2015) 125401
B Chen and L Zhang
Figure 6. (a) The thermal conductance normalized by maximum conductance versus spring constant k2 of interfacial materials for different ei acoustic impedance. The k2m = 2k1 k3 /(k1 + k3 ) is marked for guidance. (b) The thermal conductance normalized by maximum conductance versus spring constant k2 of interfacial materials at fixed acoustic impedance for different spring constant k1 of the left lead. The stars are ei marked for points of k2m = 2k1 k3 /(k1 + k3 ). ei here k2m denotes the spring constant of k2 to make the conductance maximize for the case of two leads with equal impedance. It is not the same point with the maximum conductance as shown figure 7. Nevertheless, according to ei has a value more than contour lines in figure 5, the point of k2m 99% of the maximum. Therefore although we cannot obtain an ei exact formula for the maximum interfacial conductance, k2m can act as an alternative to find the optimized material for an interface thermal coupler. We change the impedance for the interfacial systems and plot the interfacial thermal conductance as a function of k2 in figure 6(a). If k1 and k3 are fixed, for different acoustic impedance cases, the interfacial thermal ei conductance maximizes at the same point which is near to k2m ei and the interfacial conductance at k2m is always larger than 99% of its maximum. If we change the spring constant of one of leads and keep the three parts a fixed acoustic impedance, the normalized interfacial thermal conductances are plotted in ei , which are near figure 6(b). The stars mark the points of k2m to the corresponding maximums; furthermore the interfacial ei have thermal conductances more than 99% couplers with k2m of their maxima. Therefore for the case of two leads with an equal acoustic impedance, we can choose an interfacial coupler with the same acoustic impedance (Z(ω = 0+)) as those of leads and the spring constant equal to the harmonic mean of the leads to obtain an almost maximum of the thermal conductance.
3.3. A general interface
After investigating the above two special cases we move to a general case, where two leads have different cut-off frequencies and different acoustic impedances. Similarly to the above two cases, the maximum thermal conductance is in the range bounded by the two cut-off frequencies and two acoustic impedances (Z(ω = 0+ )) of the leads. The contour plots of this region for the interfacial thermal conductance are shown in figure 7. Similarly to the second case, one still cannot get an analytic formula for the maximum point of interfacial conductance, but with the hint from the above two cases, we plot the lines of geometric mean of impedances (equation (6), the solid line) and the harmonic mean of frequencies (equation (9), the dotted line) in figure 7 for guidance. Comparing the maximum point of conductance (marked as stars), we find that cross points of the two lines are located in the contour with values larger than 99.00%) of their maxima. Therefore the cross point of the two lines can be a good choice to obtain optimized interfacial conductance for a general interface. 4. Discussion
For interfacial thermal transport, the scattering due to the mismatch at interfaces causes thermal resistance. Without mismatch, that is, in a homogeneous system, for each 6
J. Phys.: Condens. Matter 27 (2015) 125401
B Chen and L Zhang
Figure 7. Thermal conductance for lead materials with general spring constants and masses as a function of the spring constant k2 and atomic mass m2 of the interfacial material. Here the contours are plotted in the regions bounded by the two lines of cutoff frequency of the leads and the two lines of acoustic impedance of the leads. The sold line corresponds to acoustic impedances equal to the geometric mean of those of the leads and the dotted line corresponds the cutoff frequency being equal to the harmonic mean of those of leads. The stars show the maximum conductances. The white line in each plot labels a contour of a certain value of thermal conductance. Here k1 = 2, m1 = 3 and k3 = 7, m3 = 4. For (a) k12 = k23 = k2 , for (b) k12 = k12m = 2k1 k2 /(k1 + k2 ), k23 = k23m = 2k2 k3 /(k2 + k3 ).
phonon mode the transmission probability will be 1 when no phonon scattering exists and the system is a perfect thermal conductor. Without considering nonlinear phonon-phonon interaction, defects and rough surfaces, all scattering comes from interfaces—the mismatch between different materials. The goal to find optimized materials is to minimize scattering from interfacial mismatch. In [21], if interfacial coupling equals to the harmonic mean of those of two coupling leads, the conductance will be maximized. However, the direct coupling between materials sometimes is very weak and one needs find a thermal coupler to bridge them. In many realistic materials, the direct coupling strength between them is small, which can be increasing by shortening distance between them. If the coupling strength between the thermal coupler and two lead materials is still very small, the interfacial thermal conductance of the whole system will be small. This is because of the large scattering at interfaces, which is similar to the weak direct coupling between two lead materials. Therefore, we need to find thermal coupler which has relatively larger coupling to leads. If the coupling strength between a thermal coupler and leads is in the range around maximized couplings k2m , k3m , one could obtain optimized thermal conductance when a thermal coupler has acoustic impedance equal to that of the leads and spring constant equal to the harmonic mean of the leads. If coupling strength is around the spring constant of materials themselves (k1 , k2 , k3 ), thermal conductance of whole systems can also be optimized near the two means. We use a simplified one-dimensional atomic model to investigate optimized thermal couplers for interfacial thermal transport. For most quasi-one-dimensional thermal transport
our findings can be applied. The Landauer formula (2) can be applied to any-dimension systems. For different dimensions, the transmission probability Tr(ω) will be quite different. Using the same transmission as that of one-dimensional case and replacing ω with ω3 , one can calculate thermal conductance to mimic thermal transport in three-dimensional systems avoiding introducing more complicated structures. Our calculations show that the optimized interfacial thermal coupler with properties near the geometric mean of impedances and harmonic mean of cutoff frequencies of leads can still be applied to three-dimensional interfaces. Thus our finding on optimized interface coupler can be qualitatively applied to real systems. Therefore to find an optimized interfacial thermal coupler in real systems, one needs to find materials with relativey larger couplings to the leads and with properties near the geometric mean of impedances and harmonic mean of cutoff frequencies of lead materials. As reported in [23], using a coupler of covalently bonded organic molecules, thermal interface conductance between metal and multiwall-carbon-nanotube arrays has a sixfold enhancement. Such organic molecules can have better adhesion with metal and carbon nanotube, if the organic molecules have properties near the geometric mean of impedances and harmonic mean of cutoff frequencies of metal and carbon nanotubes. The thermal conductance can be much more enhanced. 5. Conclusion
Using the scattering boundary method, we study optimized couplers for interfacial thermal transport where couplers 7
J. Phys.: Condens. Matter 27 (2015) 125401
B Chen and L Zhang
cannot be simplified as only spring constants. The spring constant and atomic mass of the coupler can be selected to optimize interfacial thermal conductance. For an interface with two leads having equal cutoff phonon frequencies, to get the maximum interfacial conductance, the interfacial coupler must have a spring constant equal to the geometric mean of the leads. It must also have the same cutoff frequency as the leads. For two lead materials with equal acoustic impedances, one can find optimized interfacial materials around the point where the acoustic impedance is equal to those of leads and the spring constant is equal to the harmonic mean of those of leads. For general materials one can find optimized interfacial couplers with spring constant and atomic mass near the cross point of the geometric mean of impedances and harmonic mean of cutoff frequencies of leads.
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