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Interfacial thermal transport in atomic junctions

We study ballistic interfacial thermal transport across atomic junctions. Exact expressions for phonon transmission coefficients are derived for thermal transport in one-junction and two-junction chains, and verified by numerical calculation based on a nonequilibrium Green's function method. For a single-junction case, we find that the phonon transmission coefficient typically decreases monotonically with increasing freqency. However, in the range between equal frequency spectrum and equal acoustic impedance, it increases first then decreases, which explains why the Kapitza resistance calculated from the acoustic mismatch model is far larger than the experimental values at low temperatures. The junction thermal conductance reaches a maximum when the interfacial coupling equals the harmonic average of the spring constants of the two semi-infinite chains. For three-dimensional junctions, in the weak coupling limit, we find that the conductance is proportional to the square of the interfacial coupling, while for intermediate coupling strength the conductance is approximately proportional to the interfacial coupling strength. For two-junction chains, the transmission coefficient oscillates with the frequency due to interference effects. The oscillations between the two envelop lines can be understood analytically, thus providing guidelines in designing phonon frequency filters.

We study ballistic interfacial thermal transport across atomic junctions. Exact expressions for phonon transmission coefficients are derived for thermal transport in one-junction and two-junction chains, and verified by numerical calculation based on a nonequilibrium Green's function method. For a single-junction case, we find that the phonon transmission coefficient typically decreases monotonically with increasing freqency. However, in the range between equal frequency spectrum and equal acoustic impedance, it increases first then decreases, which explains why the Kapitza resistance calculated from the acoustic mismatch model is far larger than the experimental values at low temperatures. The junction thermal conductance reaches a maximum when the interfacial coupling equals the harmonic average of the spring constants of the two semi-infinite chains. For three-dimensional junctions, in the weak coupling limit, we find that the conductance is proportional to the square of the interfacial coupling, while for intermediate coupling strength the conductance is approximately proportional to the interfacial coupling strength. For two-junction chains, the transmission coefficient oscillates with the frequency due to interference effects. The oscillations between the two envelop lines can be understood analytically, thus providing guidelines in designing phonon frequency filters.

I. INTRODUCTION
In the past decade there has been a significant research focus on thermal transport in micro scale 1 . Several conceptual thermal devices, such as thermal rectifiers/diodes, thermal transistors, thermal logical gates, and thermal memory [2][3][4][5] , have been proposed, which, in principle, make it possible to control heat due to phonons and process information with phonons. The issue of quantum thermal transport in nanostructures was also addressed 6 . In this context, the critical information is in phonon transmission coefficients that in quasi-onedimensional atomic models can be calculated by transfer matrix method [7][8][9][10] . However, the evaluation of the transfer matrix may be numerically unstable, particularly when the system size becomes large. Alternatively, nonequilibrium Green's function (NEGF) method is an efficient way to calculate the transmission coefficient 11 . Unfortunately, both of these two methods are numerical in nature and do not give analytical expressions.
For thermal transport and control, the interfacial thermal scattering process is becoming increasingly important, especially in practical devices. Two theories, acoustic mismatch model 22 and the diffuse mismatch model 13 , have been proposed to study the mechanism of the thermal interfacial resistance. However, both models offer limited accuracy in nanoscale interfacial resistance predictions 14 because they neglect atomic details of actual interfaces. A scattering boundary method within the lattice dynamic approach was first proposed by Lumpkin and Saslow to study the Kapitza conductance in a one-dimensional (1D) lattice 15 , and was then applied to calculate the Kapitza resistance in two-and threedimensional (3D) lattices 16,17 . This method can predict thermal interfacial conductance between heterogeneous materials with full consideration of the atomic structures in the interface. Recently, this method was applied to study the ballistic thermal transport in nanotube junctions 18 , spin chains 19 , and honeycomb lattice ribbons 20 .
In this paper we give an explicit analytical expression of transmission coefficient obtained through the scattering boundary method, and use it to study the interfacial thermal transport across atomic junctions. First, in Sec. II, we introduce a model in which two semi-infinite 1D atomic chains are coupled either via a point junction or an extended junction region. By using the boundary scattering method we derive the exact expressions for phonon transmission coefficients for thermal transport in one-junction and two-junction chains in Sec. III. The role of various parameters on the junction conductance is analyzed and discussed in Sec. IV. In section IV we also estimate the interfacial conductance between two 3D solids. In Sec. V, we introduce briefly the NEGF method, and use it to verify the results from analytical formulae for the thermal transport in our model. A short summary is presented in Sec. VI.

II. MODEL
The one-dimensional atomic chain consists of three parts: two semi-infinite leads and an center region (see   1). The two leads are in equilibrium at different temperatures T L and T R . The three parts are coupled by harmonic springs with constant strength k 12 and k 23 ; all of which are harmonic chains with mass and spring constants m 1 , k 1 , m 2 , k 2 and m 3 , k 3 , respectively. So the total Hamiltonian can be written as (1) here, Where x α,i is the relative displacement of i-th atom in α-th part. If there is no center part, that is, the two semi-infinite leads connected directly by k 12 , then by setting α = 1, 2 and k 23 = 0 in Eq. (1), we can obtain the corresponding Hamiltonian. For the semi-infinite leads, N α = ∞.

III. ANALYTICAL SOLUTION FROM THE SCATTERING BOUNDARY METHOD
Heat current flowing from left to right through a junction connecting two leads kept at different equilibrium heat-bath temperatures T L and T R is given by the Landauer formula 6 which allows us to develop the junction conductance formula here, f L,R = {exp[hω/(k B T L,R )] − 1} −1 is the Bose-Einstein distribution for phonons, and T [ω] is the frequency dependent transmission coefficient. Therefore, the key step for the thermal transport characterization is to calculate the transmission coefficients.
We first consider a point-junction case, that is, two semi-infinite harmonic chains connected by a spring with constant strength k 12 . We assume a wave solution transmitting from the left lead to the right lead. We label the atoms as −∞, · · · , −1, 0, 1, 2, · · · , +∞. Atoms 0 and 1 are connected by k 12 spring. An incident wave from left is assumed as x I = λ j 1 e −iωt . When it arrives at the interface, it will be partially reflected and partially transmitted. The reflected wave amplitude is x R = r 12 λ −j 1 e −iωt and the transmission wave can be written as x T = t 12 λ j−1 2 e −iωt . So at each atom we have · · · , x −1 = (λ −1 1 + r 12 λ 1 )e −iωt , x 0 = (1 + r 12 )e −iωt , x 1 = t 12 e −iωt , x 2 = t 12 λ 2 e −iωt , · · ·. Here, λ j = e iqj aj , q j is the wave vector, a j is the interatomic spacing. For the atom in the j − th part, we can have the equation of motion as each wave transport separately and satisfies such equation. Thus λ j satisfies the dispersion relation of the corresponding lead as The quadratic equation has two roots. Which one should we choose? Replacing ω with ω + iη, η = 0 + , none of the eigenvalues λ will have modulus exactly 1. We find for the traveling waves 21 thus the forward moving waves with group velocity v > 0 have |λ| < 1. Therefore we should take the one with |λ| < 1 of the two roots which are given as From the scattering boundary method, the coefficients r 12 , t 12 can be obtained from the continuity condition at the interface as: Finally we can get the transmission coefficient as here, Of course, we can also use t 12 to express T [ω] as m2v2/a2 m1v1/a1 |t 12 | 2 , here the group velocity v i = dω dqi = ai 2 4ki mi − ω 2 , which is derived from the dispersion relation given by Eq. (6). Thus, the transmission coefficient can also be expressed as here For the long-wave limit, that is, ; and the transmission is This result is consistent with the one obtained for the acoustic mismatch model, i.e., We note that in acoustic mismatch model the transmission coefficient is frequency independent, and in reality it only applies in the limit of low frequency/long wavelengths. In this case the phonon sees the interface only as a discontinuity between two semiinfinite media and the transmission does not depend on the coupling spring strength k ij . If the two leads have the same acoustic impedance for long wave limit, then For a two-junction case, which is shown in Fig. 1, the transmission wave will be reflected and transmitted by the second boundary, leading to multiple reflections. Finally the total transmitted wave function is obtained as a superposition of multiple reflections and transmissions, resulting in the transmission coefficient through the center part here r ij and λ i are determined by Eq. (12) and Eq. (8); N C is the number of atoms in the center atomic chain. From this expression, we can find that the transmission coefficient oscillates with frequency, and is between the envelope lines of maximum and minimum transmis-

IV. RESULTS AND DISCUSSIONS
A. Thermal transport in 1D one-junction chains In Sec. III, we have derived the analytical expressions for the phonon transmission coefficient for pointjunction and extended-junction (two point junction) cases Eq. (11), Eq. (12) and Eq. (16) by using the scattering boundary method. Using these analytical expressions, we analyze the role of various parameters on the thermal transport in one-and two-point junctions. Figure 2 shows the transmission coefficient as a function of frequency for a different interface spring constant k 12 for the point-junction model. The maximum frequency at which the transmission coefficient is above zero is equal to the minimum of 2 k 1 /m 1 and 2 k 2 /m 2 . In Fig. 2(a), the two semi-infinite atomic chains have the same mass and spring constant. When the inter- face coupling k 12 equals to that of the chains, the transmission is equal to one in the whole frequency domain, because of the homogeneity of the chain structure. If k 12 increases or decreases, the transmission coefficient decreases. If we set k 1 /m 1 = k 2 /m 2 , the transmission coefficient exhibits similar behavior, the only difference is that the transmission coefficient changes to the value obtained by Eq. (15). In Fig. 2(b), the two semi-infinite atomic chains have different masses and spring constants. The transmission decreases with increased frequency for all the coupling values k 12 . Also, it appears that for a given frequency the transmission is maximized for a k 12 value residing between k 1 and k 2 . From Eq. (11) and Eq. (12), T [ω] = 0, if k 12 = 0; and T [ω] has definite The maximum transmission concept results in the maximum junction conductance as shown in Fig. 3. With the increasing of k 12 , we find that the conductance will first increase, then arrive at maximum value, and then slightly decrease and at last it will tend to a constant. We find that the maximum transmission or conductance occurs at k 12 given by i.e., when the coupling spring stiffness is equal to the harmonic average of spring connecting atoms in the two semi-infinite chains. In Fig. 4, we show the thermal conductance vs the ratio of k 12 and k 12m . For the two semiinfinite chains with the same mass m 1 = m 2 , the maximum conductance occurs exactly at k 12m . If the two leads have different masses m 1 = m 2 , the maximum con- ductance is almost exactly at the k 12m point, for mass ratios ranging from 0.01 to 100. In Fig. 5, we show the curves of the transmission as a function of frequency for interface coupling equal to k 12m . If k 1 /m 1 = k 2 /m 2 , that is, when both chains have the same frequency spectrum of [0, 2 k 1 /m 1 ], the transmission equals to a constant T [ω] = T [0 + ], which can be seen from the solid line in Fig. 5, and which is consistent with Fig. 2(a). Thus for chains with matched spectra the transmission is frequency independent. Let us now fix k 1 , k 2 and k 2 , and decrease m 2 . In the range between the point of equal-spectrum (ω m = k 1 /m 1 = k 2 /m 2 ) and the one of equal-impedance (Z(ω = 0 + ) = k 1 m 1 = k 2 m 2 ), the transmission will first increase with frequency and then decrease. Otherwise, there is a monotonic decrease. The former behavior is quite interesting, as one  expects that the transmission should be the largest in the long wavelength limit. For highly dissimilar materials, the transmission coefficient in the whole frequency range is much larger than that in the long wave limit T [ω = 0 + ] = 4Z1Z2 (Z1+Z2) 2 , thus the real conductance is far larger than that calculated from the acoustic mismatch model. This result explain why the interfacial resistance calculated from the acoustic mismatch model is far lager than the experimental value measured at low temperatures, where the phonon transport can be regarded as ballistic transport.
In many real interfaces, interface coupling is very weak, that is, the k 12 is less than k 12m . So it is desirable to study the thermal transport in atomic chains in the weak coupling limit. Figure 6 shows the transmission coefficient as function of interface coupling. In the weak coupling limit, with the frequency increasing, the transmission decreases rapidly to zero, so the frequency region where phonons are effectively transmitted is very narrow. With interface strength increasing, more and more modes contribute to the transmission and the phonon transmission window widens. If the interface coupling increases further, that is k 12 /k 12m > 0.1, out of the weak interface coupling limit, all the phonons contribute to the transmission. The only further change with increasing k 12 is the actual values of the transmission coefficients increase. In Fig. 7(a), we show the transmission cutoff frequency as function of the interface coupling. Here, we define the cutoff frequency ω cutoff at which the transmission T (ω cutoff ) = 0.1T (0 + ). We find that the cutoff frequency shows linear dependance on interface coupling in the weak coupling limit k 12 < 0.1k 12m . If the interface strength increase further, the cutoff frequency is saturated. In Fig. 7(b), we show the transmission as function of interface coupling for several different phonons. We find that in the weak interface coupling region, the transmission is proportional to the square of the interface coupling, which is consistent with the formulas Eq. (13)  and Eq. (14). In the weak interface coupling region, for the 1D atomic one-junction chains, it is shown that the thermal conductance is linear with the interface coupling (see Fig. 8). If we strengthen the interface coupling between the two chains, the conductance will be linearly enhanced. For different mismatched chains, the absolute values of the conductance are different, but dependence on the coupling strength is the same.

B. Thermal transport in 3D single-interface structures
The thermal conductance Eq. (4) can also be written as 23 : because of v(ω) = ∂ω/∂k and phonon density of states in 1D structure, D(ω) = 1/(2πv), we can obtain Eq. (4). In order to estimate the behavior of the interfacial thermal transport across interfaces in 3D structures, we only need to change the phonon density of states in the above equation. Because the density of states for 3D structure within the Debye approximation is D(ω) ∼ ω 2 , therefore we can replace ω with ω 3 in Eq. (4); the thermal conductance as a function of the coupling strength is shown in Fig. 9. From Fig. 9(a), we find that in the weak interface limit, conductance is proportional to the square of interface coupling, which is consistent with the results from other models [24][25][26] , while it is linear dependent on the interface coupling in 1D junctions. This is due to the fact that in 3D low frequency region contributes relatively little to the conductance as the density of states is low there. If the interface coupling increases further, that is k 12 /k 12m > 0.1, out of the weak interface coupling limit, all the modes contribute to the transmittance, the conductance is no longer proportional to the square of the interface coupling, and the slope continuously decreases. In some intermediate ranges the conductance is approximately proportional to the interfacial coupling (see Fig. 9(b)), which is consistent with the results from molecular simulation approach 27 . For stronger coupling the conductances for the 1D case and 3D one have similar behaviors, the slope of both cases will decrease continuously to be zero at point k 12m , where the conductance will be maximized and then decrease slightly to a limiting value.

C. Thermal transport in extended junctions
Now we focus on a case where the junction is extended and involves a center part. The overall behavior of the transmission is the combination of the transmission behavior in single point-junction case and the oscillatory behavior due to phonon interferences arising form multiple scattering. We show the transmission coefficient as a function of frequency of an arbitrary case in Fig. 10(a).
Here, the three chain parts have different masses and spring constants, and the interface coupling is not special. From the analytical expression of Eq. (16), we plot curves of the maximum transmission and minimum transmission, N c = 4 and N c = 9. The transmission oscillates between the envelop lines of maximum and minimum transmission. The maximum transmission line will increase first, and the minimum transmission line will monotonically decrease with frequency. However for interface coupling that is the same with the leads, the two envelop lines will monotonically decrease, which can be seen in Fig. 10(b). For some special cases, the transmission coefficient in the frequency domain has interesting phenomena, which are shown in Fig. 11. In Fig. 11(a), the transmission for the case of two identical leads is shown. In this case, the maximum transmission is equal to one, the infinite-long wavelength phonon and the resonance mode can transmit fully through the center part. The minimum transmission is very low, indicating efficient destructive interference. Figure 11(b) shows the transmission when all three parts are different and connected by interface couplings k 12m and k 23m . We find that overall trend for the maximum and minimum transmission lines is increasing first, then decreasing. If, in addition, the ratios of k i /m i are the same for three parts, then the maximum and minimum transmission are constants in the whole frequency range, and the transmission coefficient through finite-size center part oscillate between the two constants, which can be clearly seen in Fig. 11(c). Therefore, we can use the above properties of transmission to design the frequency filters. Figure 12 shows the maximum and minimum transmission coefficient for the filter. If the spring constant of the center part is very different from the ones of the the two leads, the oscillatory peak is sharp, and transmission for most of the frequency will tend to zero, only few resonant frequency can be transmitted. This finding provides guidelines for the design of selective frequency filters.

V. VERIFICATION BY NONEQUILIBRIUM GREEN'S FUNCTION METHOD
The NEGF method is an exact approach to study the ballistic thermal transport through junctions. Following the discussion in Sec. II, if we use a transformation for the coordinates, u j = √ m j x j , which is called the massnormalized displacement, then the Hamiltonian can be written as where H α = 1 2 P T α P α + U T α K α U α . K α is the massnormalized spring constant matrix, and V 12 = (V 21 ) T is the coupling matrix of the left lead to the central region and similarly for V 23 is the coupling matrix of the right lead to the central region. As stated in Ref. ? , the element of the coupling matrix V ij α,β is equal to −k ij /sqrtm i m j which corresponding to the coupling between the i th atom in region α and the j th atom in region β.
We can use the nonequilibrium Green's function method 6 to study the thermal transport in the atomic chain. We define the contour-ordered Green's function as where α and β refer to the region that the coordinates belong to and T is the contour-ordering operator. Then the equations of motion of the Green's function can be derived. In particular, the retarded Green's function for the central region in frequency domain is Here, Σ r = α=1,3 Σ r α , and Σ α = V 2,α g α V α,2 is the selfenergy due to interaction with the heat bath, g r α = [(ω + iη) 2 −K α ] −1 . And in the advanced Green's function G a = (G r ) † , the transmission coefficient can be calculated by the so-called Caroli formula as . For single-junction atomic chains, if we regard the two atoms in the interface (atom 0 and atom 1) as the center part, then we can still use the formulae above to study the phonon transmission leading to the exact formula yielding the same result with the one obtained from the scattering boundary method. In Appendix A, We give the analytical proof of this fact.
For two-junction atomic chains, according to the NEGF formulas, we do the numerical calculation and plot the curves of the transmission coefficient as a function of frequency and compare them to the results obtained the scattering boundary method (see Fig. 13). We find that for any arbitrary case, the results from the NEGF method and the scattering boundary method are exactly the same. If there is no many-body interaction, that is, for the ballistic thermal transport the scattering matrix approach and the Green's function method give the same results. These two methods are equivalent, which has been proved from other points of view in Refs. 28,29 .

VI. CONCLUSION
In this paper, we study the ballistic interfacial thermal transport in atomic junctions, we give the analytical simple formulae Eq. (11), Eq. (12) and Eq. (16) for the transmission of one-junction and two-junction cases, which are consistent with the results from the NEGF method.
For one-junction case, we find the transmission and conductance are maximized when the interface spring constant equals to the harmonic average of the two spring constants of the leads. At the point near k 12 = k 12m , the transmission T [ω] is a constant if k 2 /m 2 = k 1 /m 1 ; if not equal, in the range between k 1 /m 1 = k 2 /m 2 and k 1 m 1 = k 2 m 2 , the transmission coefficient increases first then decreases with the increasing of frequency, otherwise the transmission monotonically decreases as the frequency increasing. For weak interface coupling, the cutoff frequency and the interface conductance for 1D chain is linear dependent with the interface coupling strength.
Because of different density of states, we change the formula of conductance to mimic the thermal transport in 3D junctions. In weak interface coupling limit, we find that the conductance is proportional to the square of the interface coupling, which is consistent with the results from other models. The slope of the conductance as function of interfacial coupling strength decreases continuously from two to zero, in certain range of which, the conductance is linear proportional to the interface coupling, which are consistent with the results of other molecular simulations.
For two-junction case, the transmission will oscillate with frequency in the envelop lines of maximum and minimum transmission which are determined by the onejunction picture. The transmission sometimes oscillates between two decreasing envelop lines, sometimes between two increasing envelop curves, or between two constants, etc.