Feasibility Study of a New Model for the Thermal Boundary Resistance at an Interface of Solid Thin Films

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1 Feasibility Study of a New Model for the Theral Boundary Resistance at an Interface of Solid Thin Fils Soon Ho Choi* and Shigeo Maruyaa** Departent of Mechanical Engineering, The University of Tokyo, Tokyo, Japan The theral boundary resistance (TBR) is still a challengeable subject since its echanis to explain quantitatively is not clear in spite of its iportance. We perfored non-equilibriu olecular dynaic (NEMD) siulation to evaluate TBR at an epitaxial solid interface coposed of two different aterials. Solid argon is selected as a siulation aterial since it is a non-conductor electrically, so that energy transportation is caused only by lattice vibration. The history to grasp the echanis on TBR is fairly long and the various theories were developed such as an acoustic isatch odel (AMM), an acoustic ipedance isatch odel (AIMM) and a diffuse isatch odel (DMM), however any odel can t ake a success of analyzing TBR quantitatively. The echanis of TBR can be explained as the energy reflection at an interface caused by the discontinuity of acoustic ipedance even in a icroscale syste fro this study. However it should be corrected with the consideration of a icroscale characteristic. The corrected icroscale acoustic ipedance isatch odel (CM-AIMM), which is developed in this study, fairly well predicts TBR copared with any other existing odels. I. INTRODUCTION The nano- and icro-technologies have been ore critical to the related industries such as a sei-conductor, a icro-electroechanical syste (MEMS) and a nano-electroechanical syste (NEMS). These newly growing technologies ake the various devices perfor the advanced functions with a ore reduced size and change even the huan life []. However these new technologies require at the sae tie that we should thoroughly understand, analyze the new phenoena resulted fro the extreely sall size and the application liits of the existing theories based on a acro syste [-3]. In MEMS and NEMS the theral boundary resistance (TBR) at a solid interface coposed of two different aterials or a grain boundary is focused one s attention [3-4] because it plays key role in the heat dissipation capacity of those devices. For exaple the heat generation fro CPU or an electronic chip degrades the perforance of the, so that a proper heat reoval is very iportant to assure their designed functions. However the existence of an interface between a substrate and a ounted chip on it prevents fro flowing out a heat current, which causes a device to fail an intended perforance or a proper operation. Recent thin fil deposition technology, which is able to control to the range of one ato thickness nowadays, akes supperlattices be possible as shown in Fig.. This aterial is an artificial fil not existed in nature since it is possible to be anufactured fro E-ail : choi@photon.t.u-tokyo.ac.jp ** E-ail: aruyaa@photon.t.u-tokyo.ac.jp arbitrary selections of any eleent [5]. Many studies report that there are two anifest features in a theral phenoenological sense. One is that there is a large teperature jup at an interface in supperlattices and the other is that the theral conductivity of each fil is lower than that in a bulk state [6-]. The latter is relatively well explained and analyzed by the concept of a phonon ean free path (MFP) in Ref. subitted with this paper []. However it can be said within our knowledge at least that there is no theory to explain quantitatively the agnitude of TBR at a solid interface, so that it reains still a challengeable subject [3-] since Kapitza found the existence of TBR at an interface between the etal surface and liquid Heliu [-]. After Kapitza s observation of TBR, any researchers have focused all their attention on TBR between solid-liquid or solid-solid interface. The first attept to explain the echanis of TBR was done by Little [5] who applied the concept of a phonon reflection and transission at a solid-solid interface, which is now called as an acoustic isatch odel (AMM). Hereafter the alost experiental or coputational results were copared with the predictions based on AMM, however the coparisons were far fro the quantitative agreeent except in the case that the teperature of a syste is lower than about 0.0 K [3-4, 6]. To overcoe this discrepancy Swartz proposed a diffuse isatch odel that is called as DMM [3]. Although it shows a little iproved prediction of a theral conductance in the region of a high teperature, there is still a large gap between an experiental result and a prediction by DMM. In contrast to a icroscale situation, TBR sees to be considerably reduced fro the results of a

2 acroscale syste if the actual contacting area at an interface is increased by any ethod [-6]. Those results ake one understand that TBR is disappeared if the interface is perfect contact as shown in Fig., which is considered as fairly reasonable. However, against our naive expectation, the recent study by Matuoto et al. reported that TBR still exists at an epitaxial interface between dissiilar aterials even though an interface is perfectly contacted in view of an atoic level [7]. Matuoto s study was perfored with changing the olecular ass of two species and a potential well depth ε, but failed to predict TBR quantitatively. This study is carried out to develop a new odel to predict TBR accurately using NEMD and investigated only the effect caused by a olecular ass ratio. Solid argon is selected as a siulation aterial because of the sae reason described in another paper subitted with this []. It has been found fro this study that TBR at an interface is explicitly dependent on the ass ratio between two different aterials and the higher a ass ratio is the higher TBR. The echanis of TBR can be explained as the energy reflection at an interface caused by the discontinuity of acoustic ipedance. AIMM is corrected for the icroscale syste and it is confired to evaluate TBR qualitatively and quantitatively copared with any other odel such as AMM, AIMM and DMM. The detailed description will be given in the next sections. condition (PBC), which behaviors just as an actual thin fil. The velocity scaling ethod is used for the teperature control of both TCLs and the equation of otion is integrated by the Velocity Verlet ethod in the sae as Ref.. An interolecular distance is deterined to aintain a syste to be under a freestanding state, which eans a syste to be in the zero stress state internally during a siulation [8-3]. The selected interolecular length in this study is the sae as Ref.. Eighteen argon olecules are arranged on x and y direction respectively in one layer and total layer of a syste is thirty in z direction. A teperature gradient is easured in the ediu of the eighteen layers excluding the teperature control layers and the adiabatic wall at each side. Tie interval for an iteration calculation is selected as Δt.0x0-5 s ( fs) and the cut-off length an interolecular interaction 3.5 σ AR. The properties of argon are σ AR 3.4 Å, AR 6.634x0-6 kg and ε AR.67x0 - J. (a) View fro the Top (a) Epitaxial Supperlattice (b) Non-epitaxial Supperlattice Fig.. Scheatic Cross Sections of Superlattices II. SIMULATION METHOD A siulation syste is arranged with fcc <> as shown in Fig. and the siulation ethod is actually the sae as described in Ref. except that the upper half of a syste has the different olecular ass fro the lower half. The adiabatic wall of 3 layers and the teperature control layers (TCLs) are placed at both ends of a syste. The forer are for isolating the syste fro circustances and the latter for controlling both TCLs to each desired teperature. The botto is controlled to a high teperature and the top to a low teperature, so that a heat current flows up (z direction). The x-y plane perpendicular to the heat flow direction is set as a periodic boundary (b)overall View (c)view fro the Front (d)view fro the Side (8 x 8 olecules in a plane and 30 layers in length) Fig.. The Siulation Syste of fcc<> with an Adiabatic Wall The lower half of Fig. is coposed of the argon olecules to have the earlier specified properties, however the upper half the iaginary argon olecules that have a different ass. The latter has the sae properties except for the ass, which ake one evaluate TBR especially caused by the ass difference. TBR is investigated in case that the ass ratio is :, :3, :5 and :7 respectively. The first siulation akes a syste aintain an initial equilibriu state at any setting teperature and then controlling both TCLs respectively develops a teperature gradient in a syste. These siulations are perfored six ties each of which is 400,000 iterations (400 ps), however the first one is excluded in the evaluation of a theral conductivity since it ust be a transient period to develop a teperature gradient in a syste. Therefore the theral conductivity is the averaged value over the rest five

3 (a) Mass Ratio of : (b) Mass Ratio of :3 (T ave ; 40.3 K) (T ave ; 40. K) Fig. 5. Teperature Gradient Ratio of the Syste with a Mass Ratio (c) Mass Ratio of :5 (d) Mass Ratio of :7 (T ave ; 40.3 K) (T ave ; 40. K) Fig. 3. Teperature Jup at Interface by Different Mass Ratio (a) Mass Ratio of : (b) Mass Ratio of :3 (T ave ; 40.3 K) (T ave ; 40. K) (c) Mass Ratio of :5 (d) Mass Ratio of :7 (T ave ; 40.3 K) (T ave ; 40. K) Fig. 4. Heat Flux Reductions by Different Mass Ratio siulations. The initial equilibriu teperature is 40 K. During all siulations for developing a teperature gradient the high TCLs is controlled to 4 K and the low TCLs to 38 K always (4 K). It is confired that the averaged teperature of a syste is aintained to 40 K during all siulations although each TCLs is in the different setting teperature, that is 38 K and 40 K. III. RESULTS AND DISCUSSION III-I. TEMPERATURE JUMP AT AN INTERFACE AND HEAT FLUX There is no need to describe the ethod of the calculation of a heat flux, a theral conductivity and the deterination of an interolecular length here repeatedly since the details of the are explained in Ref.. Fro the siulation results it is observed that a heat flux is reduced and a teperature jup is increased when a ass ratio gets ore increased as our expectation. Fig. 3 and Fig. 4 shows respectively the exaples of the teperature jup at an interface and the heat flux of each corresponding ass ratio. It is clear that the heat current is prevented fro flowing through an interface owe to the increased TBR as the ass ratio is increased. The thick dotted line on Fig. 4 is the heat flux of a syste in which all olecules are argon, that is no interface. Heat inflow and outflow by the velocity scaling of both TCLs are alost the sae (axiu 3 % deviation), which justifies that the syste is fully under a non-equilibriu steady state during a siulation. The theoretical ratio of two teperature gradients of a syste as shown in Fig. 3 can be calculated fro the definition of a diensionless theral conductivity [9-30]. Δ * L σ σ λ q λ k B ε k B ε () In Eq. () q is a heat flux, ΔL a syste length, the teperature difference between both ends and kb is the Boltzann constant,.38x0-3 J/K. Considering that the upper half and the lower half of a syste have the sae properties except for their ass, it is easily derived fro Eq. () that the 3

4 theoretical ratio of two teperature gradients is proportional to the square root of a ass ratio. ΔL ΔL () Fig. 5 is the easured ratios of the teperature gradients fro all siulation and each data point are averaged over five siulations. It shows that the MD results are fairly well agreed with the theoretical value of Eq. () and the standard deviation ( σ STD ) is about 0 % of the average at ost. III-II. PREDICTION OF THERMAL BOUNDARY RESISTANCE BY AIMM The history to grasp the echanis of TBR is fairly long as described previously and the typical odels are AMM and DMM, which are concerned with phonon transportation phenoena [3-5]. However it is well known that neither of the agrees with experiental data except for an extreely low teperature region although both odels give rather siilar predictions for any cases [6]. Moreover these odels were too coplicate to handle easily for engineering use. We don t ention the details of AMM and DMM again since other references provide with the full description on these odels [3-4]. Recently Matuoto et al. tried to evaluate TBR by coparing with the heat fluxes between the reference syste that consists of all the sae olecules of argon (hereafter called as SYS REF ) and the syste that has a ass ratio (hereafter called as SYS MR ) [7]. The concept by Matyoto is very siple. They defined the energy reflection coefficient (ERC) fro the above heat fluxes and then copared with ERC fro an acoustic ipedance isatch odel (AIMM). As well known, AIMM is the odel for calculating ERC occurred at an interface of a acroscale syste considered as a continuu. If we assue that an incident wave to an interface is A exp{i(ω t-k x)}, a reflection wave fro it B exp{i(ω t+k x)} and a transitted wave through it A exp{i(ω t-k x)}, then ERC is given as follows [9-30, 33]. ρ c ρ c ERC ρ c + ρ c c c c + c (3) q MR ERC (4) q REF In Eq. (4) q is the heat flux of SYSREF and REF q the heat flux of SYS MR MR. The acoustic velocity of a solid aterial, c is theoretically the square root of a ratio of Young s odulus to a density. Since Young s odulus is related with a potential well, ε and two species are assued to have the sae potential well in this study, Eq.(4) can be rewritten as Eq. (5). ERC + (5) Matuoto et al. supposed that ERC by the heat flux ratio of Eq. (4) is equal to ERC by AIMM of Eq. (5), however their MD results were largely different fro the prediction by Eq. (5). Although the attept by the showed not to predict TBR quantitatively it is worthwhile noting their concept because it is very siple and shows a little iproved estiation of TBR copared with AMM or DMM. We investigated their concept thoroughly fro the starting point of AIMM theory and found that it can be an excellent odel to anticipate TBR if it is corrected for a icroscale syste. This will be described in the next section. III-III. CORRECTED MICROSCALE ACOUSTIC IMPEDANCE MISMATCH MODEL (CM-AIMM) As entioned previously the ethod of Matuoto s et al. is very siple to understand and easy to handle while it fails to predict TBR quantitatively. It ay be considered that the discrepancy between MD results and AIMM prediction results fro the direct application of AIMM to a icroscale syste without any consideration. If we consider the fact that AIMM was originally developed for a acro syste, it is reasonable that soe assuptions should be corrected for applying AIMM to a icroscale syste. AIMM was developed under two boundary conditions (BCs) of () the displaceents of both sides at an interface are the sae iediately for all tie and () the force acting to both sides at an interface are also the sae. These conditions assure that there is no discontinuity of the displaceent and the force at an interface. However it is questionable that BC () is applicable to a icroscale syste though BC () is considered to be reasonable still. Under our assuptions the displaceents of olecules are not collective but rather rando fro a icroscopic viewpoint. In a acroscale syste such as a string, a slender bar or a slinky the iparted energy for a wave otion is extreely large copared with the energy of a olecular otion, so that the energy by the olecular otions can be ignored copletely. Therefore ERC can be calculated sufficiently considering only the collective otion of a body. However in a icroscale syste like this study the energy transport depends only on the theral otions fro the olecules and it ust be 4

5 considered when evaluating TBR. Supposing two olecules with a different ass respectively under the Lennard-Jones (L-J) potential and their oveents, it can be easily iagined that the aplitudes of two olecules are different fro each other and expected to be proportional to each ass. Fig. 6 shows the respective aplitudes of each with the different ass, which are easured fro each equilibriu position during a otion. It is concluded fro this figure that our assuption can be accepted for a correction of AIMM. Consequently BC () should be corrected for a icroscale syste as () the displaceents of both sides at an interface has the ratio proportional to their ass ratio for all tie. Under these corrected BCs, ERC by AIMM is derived as Eq. (6). ERC Z + α Z Z α α / Z (6) Z ρ c case that the ass ratio is : and (b) the siplified teperature profile of (a). If there is no teperature jup at an interface of a siulation syste, T ust A be decreased and T B increased as shown in (b). However the corrected teperature gradient should satisfy the following relationships. Δ T + (7) GAP A B A GAP + GAP + B (8) (9) Fig. 6. Aplitude of Two olecules with Different Mass under the L-J potential In Eq. (6) Z is acoustic ipedance that is the product of the density and the acoustic velocity of a aterial. Eq. (6) is naed as CM-AIMM in this study and it is identical with Eq. (3) if α is replaced as.0. Before applying Eq.(6) to a icroscale syste, it should be considered what syste can be regarded as a reference. Matuoto et al. considered that a reference syste is SYS REF under the sae siulation conditions. However the syste with a ass ratio is certainly different fro one without a ass ratio even though the sae siulation conditions are applied such as the sae physical diensions and the teperature difference at both ends. Rather the reference of a siulation syste ay well be own itself to be assued without the teperature jup at an interface (hereafter called as SYS REF ), which is easily created fro Eq.(). Figure 7 (a) is the exaple of MD siulation in 5 (a) Measured Teperature Profile by MD (b) Siplified Teperature Profile Fig. 7. Teperature Profile of the Reference Syste

6 directional displaceents of the olecules were exaggerated for the visualization. Fig. 9 is the kinetic energy behaviors of each layer with the tie and corresponds to Fig. 8. Fig. 8. Wave Behavior in the Syste with an Interface Assuing that there is no teperature jup at an interface, the heat flux of SYS REF ust be increased copared with SYS MR and it can be calculated fro the ratio of the corrected teperature difference to the originally easured one by MD siulation. q REF q MR REF REF (0) In Eq. (0) q is the heat flux of SYSREF and REF q the heat flux of SYSMR. As a result, ERC by the MR concept of a reference syste is calculated by subtracting the reverse of Eq. (0) fro unit. ERC () REF REF In this study Eq. (), which is a result by MD siulation and corresponds to an actual experient, is copared with the theoretical ERC by CM-AIMM of Eq. (6). However it is necessary that the acoustic ipedance of a aterial should be deterined before calculating ERC. For this purpose we prepared the long syste of fcc <> arrangeent and iparted a weak pulse to the botto after achieving an initial equilibriu state. Fro the observation of the kinetic energy of a syste we easured the acoustic velocity. Fig. 8 is the snapshots that show the reflection and transission of a wave occurred at an interface. The z Fig. 9. Kinetic Energy Behavior of a Layer in the Syste with an Interface Fro the previous results of Fig. 8 and 9 we calculated the acoustic velocity of each aterial and 6

7 confired that the echanis of TBR is reasonable, which is caused by a wave reflection at an interface. The easured acoustic velocity is different fro the theoretical prediction that is proportional to the square root of the ratio of Young s odulus to a density. Fig. 0 is the observed value of the acoustic velocity with the change of a ass and resulted in the power of 0.6 approxiately. Fig. shows ERC results of this study, in which the dotted line is ERC by AIMM calculated fro Eq. (3) and the solid line is that by the CM-AIMM fro Eq. (6). Both theoretical values are based on the easured acoustic velocity of Fig. 0 and tabulated in Table. The opened circle is the ERC by Matuoto et al., which was on the syste arranged with fcc<00> and the heat flux ratio between SYS MR Fig. 0. Measureent of Acoustic Velocity Mass Ratio TABLE : Coparison of ERC aong the Theoretical Values and MD Results ERC fro Matuoto et al. ERC fro this Study ERC by AIM M ERC by CM-AIMM : 53 % 0 % 5 % 0 % : % 7 % 4 % :4 6 % --- % 55 % : % 5 % 65 % : % 0 % 76 % :9 77 % % 83 % and SYS REF. The opened square is the ERC by this study and also based on the heat flux ratio as sae as the above. The solid square is ERC by this study however they are based on the heat flux ratio between SYS MR and SYS REF, which is assued that there is no teperature jup at an interface. As seen in Figure, it is clear that the developed CM-AIMM, which is based on the concept of the reflection of an incident wave or energy at an interface, predicts well TBR copared with any other odel. Moreover this new odel can evaluate the teperature profile through Eq.(7) to (9). Considering the results of this study, it is supposed that AIMM based on a acroscal syste is still applicable to a icroscale syste if a proper coefficient is introduced. Indeed a correction coefficient α of Eq. (6) plays such a role. Therefore it can be explained that the reported discrepancy between the experients and the theoretical odels such as AMM, DMM and AIMM results fro ignoring the characteristic of a olecular otion when easuring TBR at an interface between thin solid fil. IV. CONCLUSION Fig.. Coparison of ERC by Each Model The olecular dynaics siulation is perfored to analyze the TBR at an interface of a syste with different aterials and it is confired that the echanis of TBR can be explained by the concept of a classical wave theory in which an incident wave is partly reflected and partly transitted on an interface. Soe correction is ade to a conventional AIMM for a acro syste and the new concept, which is the heat flux ratio between a siulation syste and a reference syste, is introduced to evaluate TBR. The CM-AIMM developed in this study predicts TBR accurately copared with other existing odels such as AMM, DMM and AIMM. The axiu deviation between the ERCs by CM-AIMM and MD results within %, which is the case of the ass ratio of :7. This is quite satisfactory result for engineering use. Moreover this new odel 7

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