International Journal of Heat and Mass Transfer 147 (2020) 118941

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Abnormal thermal boundary resistance of thin films with heat source

https://doi.org/10.1016/j.ijheatmasstransfer.2019.1189410017-9310/� 2019 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: mrwang@tsinghua.edu.cn (M. Wang).

Xin Ran, Moran Wang ⇑Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics and CNMM, Tsinghua University, Beijing 100084, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 5 September 2019Received in revised form 16 October 2019Accepted 22 October 2019

Keywords:Interfacial heat transportHeat sourceMonte Carlo simulationMicro- and nano- scale heat transport

Interfacial phonon transport through thin films with heat source exists widely in devices, such aselectronics and thermoelectrics, yet the mechanism is still unclear. In this work, thin films made ofbi-material pairs with heat sources are simulated by a developed Monte Carlo method, namelykinetic-type Monte Carlo method. The size effects of the thermal boundary resistance are found to bemuch different at different interfaces in double-layer and three-layer thin films when heat source is con-sidered, even for a same material pair. Meanwhile they are much different from those with only theisothermal boundary considered. The strength of the heat souce is proved to have little impact on thethermal boundary resistance. The results show that the thermal boundary resistance depends stronglyon the external temperature difference, the position of heat source, and the layout of materials. Thesefindings and corresponding explanations unlock some of the physical mechanism for interfacial phonontransport with heat source. Moreover they are very valuable for optimization and design of nanodeviceswith the interface and heat source.

� 2019 Elsevier Ltd. All rights reserved.

1. Introduction

In recent years, rapid developments of micro- and nano-technology have made micro- and nano-scale heat transport moreand more attractive and important [1–4]. Electrons and phononsplay the major roles in heat conductions in semiconductors andthe present work only focuses on the contributions of phonons.At such a small scale, interfaces will cause non-ignorable resis-tances when phonons transport across them, which are mostlynegligible at large scales [5–13]. This influence makes interfacialphonon transport popular in applications in the thermoelectrics[14–16] and the thermal managements [17,18]. However, rarelymature theories can describe phonons transport across the inter-face successfully, which are necessary for further engineering andindustrial developments.

Basically, the phonons will transmit or be reflected back whenthey scatter with an interface. There are mainly two interface mod-els to quantify the probability for a phonon transmitting across theinterface also named the transmissivity. The first model is theaccoustic mismatch model (AMM) based on assuming completelyspecular scattering at the interface for phonons [6]. Its transmissiv-ity is a constant calculated from the acoustic impedance of thematerial pair at both sides of an interface [6]. This model can pre-dict the thermal boundary conductance at very low temperatures

since most of scattered phonons with the interface are low-frequency with wave-lengths much larger than the scale of theinterfacial asperity [6,10]. The second model is the diffuse mis-match model (DMM) developed from the assumption of com-pletely diffuse scattering for phonons interacting with aninterface [10]. Its transmissivity is a constant obtained from thephonon dispersions for both materials forming the interface [10].The DMM works better at higher temperatures such as aroundthe ordinary temperature, where a majority of phonons are popu-lated at high frequency with wave-lengths comparable to or smal-ler than the size of the interfacial asperity [10]. Many extendedinterface models are improvements based on DMM to betterdescribe interfacial phonon transport at ordinary temperatures[19–22]. The spectral diffuse mismatch model (SDMM) is currentlythe most appropriately and intuitively analytical interface modelaccounting for spectral property of transmissivity, which has beenobserved by experiments [23] and microscropic calculations [24].Therefore, the SDMM will be adopted in the present work to studythe phonon scattering at interfaces.

There are many methods to study phonon transport at micro-and nano-scale, such as the microscopic method (e.g. moleculardynamics method [25,26] and Green’s function method [5,27])and the mesoscopic method (e.g. discrete-ordiante method[28,29], lattice Boltzmann method [30,31], and Monte Carlomethod [8,32–36]). The former one is hardly applicable for arelatively large system compared with the latter one adopted inthe present work. Among the mesoscopic methods, the

2 X. Ran, M. Wang / International Journal of Heat and Mass Transfer 147 (2020) 118941

discrete-ordinate method solves the phonon Boltzmann equationdirectly and very complicated in treating interfaces, esspeciallyfor complex interfaces. The lattice Boltzmann method is not welldeveloped in accounting for spectral properties of phonons. TheMonte Carlo method has its special advantages in micro- andnano-scale phonon transport due to its clear physical figure andeasy implement in treating interfaces and considering spectrumof phonons. The present study uses the Monte Carlo method forthe interfacial phonon transport.

Phonon transport across interfaces has been studied using theMonte Carlo method [8,32,33,36,37]. In these works, the spectraltransmissivity between homogenous or heterogeneous materialpairs were considered [8,32,33,36,37]. Some of them exploredthe relation between thermal conductivity and mean free path ofphonons [32,33,37]. The effects from the interface roughness andthe film size on thermal boundary conductance were studied in adouble-layer thin film [8]. The size effects on the thermal boundaryconductance have been clarified within the multilayer thin film[36]. On the other hand, several studies focused on exploring theimpact of heat source on heat transport within a single material[38–40]. However, the mechanism of interfacial phonon transportinfluenced by heat source has still not been explored. Actually, it iscommon that heat will be generated inside devices when they areworking [41–43]. Therefore, the present work will study the effectof heat source on phonon transport across interfaces with an inter-face model, SDMM, using Monte Carlo method. The following con-tents will be organized as four sections: the numerical method andthe physical models adotped in the present work will be intro-duced in Section 2. Section 3 will provide the verifications for thepresent numerical algorithm by modeling cross-plane phonontransport through both single-layer and double-layer thin filmswith isothermal boundaries. The results and analyses will beshown and discussed in Section 4: including the effects of sizeand heat source strength on the thermal boundary resistance, thereverse of the thermal boundary conductance, and the abnormalthermal boundary resistances in some special cases. Conclusionswill be made in Section 5 finally.

Fig. 1. Physical models for cross-plane interfacial phonon transport with the heatsource: (a) and (b) are double-layer thin films; (c) and (d) are three-layer thin films.1 and 2 represent material labels, and orange parts denote locations of heat sources.

2. Numerical method & physical models

The Monte Carlo method is a statistical approach to solve theBoltzmann equation with a physical figure of pseudo-particletransport, which is very convenient to treat complex interfaces[8,34–36,44,45]. A novel phonon Monte Carlo scheme, namelythe kinetic-type Monte Carlo method (KMC), is adopted to studyinterfacial phonon transport with heat source whose physicalmodels will be introduced shortly. KMC improves the previousMonte Carlo scheme by solving the linearized version of energy-based deviational phonon Boltzmann equation, which is developedfrom the conventional phonon Boltzmann equation [35,44]:

@ed

@tþ vg x;pð Þ � red ¼ �

ed � eeqloc � eeqTeq� �

s x;p; Tð Þ ; ð1Þ

where ed ¼ h�x f � f eqTeq� �

represents the deviational distribution

with the reduced Plank constant h�, the phonon angular frequencyx, the phonon distribution f and Bose-Einstein distributionf eqTeq ¼ 1

exp h�x=kBTeqð Þ�1at the referenced equilibrium temperature Teq

with the Boltzmann constant kB. vg x;pð Þ and s x;p; Tð Þ denote thephonon group velocity and relaxation time, respectively, with thepolarization p at the thermodynamic temperature T. eeqloc ¼ h�xf eqlocand eeqTeq ¼ h�xf eqTeq are the local pseudo-equilibrium and equilibrium

energy distributions at the local pseudo-equilibrium and referencedequilibrium temperatures, Tloc and Teq, respectively.

KMC consumes less memory and is more efficient than the con-ventional phonon Monte Carlo methods by tracking energy packetsone by one [35,44]. It starts with the initialization for each energypacket through determining phonon properties (frequency, polar-ization, group velocity, sign, initial position, and initial time). Theadvections and scatterings happen alternately until the trackedpacket is absorbed by the isothermal boundaries or the simulationends. Finally, the macroscopic information (temperature distribu-tion and heat flux) will be obtained through averaging over allenergy packets at a certain time. For initializations, each packetwill be emitted from the boundary, initial, or source term respec-tively. Two kinds of scattering are considered, namely Umklappand interface scatterings. In the present work, SDMM is adoptedto describe the interface scattering with the requirement of diffusescattering and the neglect of inelastic scattering and polarizationconversion [8,19,36]. During advections of a tracked packet, theinterface scattering will happen if the interface scattering time isshortest among all scattering times. One random number will begenerated to be compared with the spectral transmissivity. Whenthis number is smaller than the tranmissivity, the packet willtransmit through the interface with the update of the travel direc-tion and group velocity based on the frequency and new disper-sions. Otherwise, the packet will be reflected back without anychanges of phonon properties except the travel direction [8].

Cross-plane interfacial phonon transport through double-layerand three-layer thin films with heat sources are considered withthe equal thickness of each layer in the present work, as shownin Fig. 1 where 1 and 2 represent the labels of materials for twokinds of pairs, namely Al/Si and Ge/Si. The detailed correspondingrelations between materials and labels will be introduced in Sec-tion 4. The heat source will be located at material 1 or 2 to studythe influence of the position of heat source on heat transportthrough the same material pair. Equal and non-equal temperaturesof isothermal boundaries are considered in the present work toinvestigate the effect of the boundary temperature on phonontransport across interfaces. Finally, the dispersions for Si and Geare fitted at [1 0 0] direction by fourth degree polynomial equa-tions to Ref. [46] and Ref. [47], respectively. The optical phonons

Fig. 2. Physical models for cross-plane phonon transport: (a) is the single-layer thinfilm; (b) is the double-layer thin film. 1 and 2 represent Al or Ge and Si for Al/Si orGe/Si respectively, whose volume ratios of two materials are always fixed at one.

X. Ran, M. Wang / International Journal of Heat and Mass Transfer 147 (2020) 118941 3

of both materials are ignored due to their tiny contributions to heattransport at the steady state. The dispersions of Al are fitted at[1 0 0] direction by fourth degree polynomial equations to Ref.[48].

For simplifications, the same formulas for relaxation times of Siand Ge are adopted referred to Ref. [49]. Impurity and Umclappscattering rates are all obtained by formulas s�1

I ¼ AIx4 ands�1U ¼ BUTx2exp �C=Tð Þ respectively for both Transverse Acoustic

(TA) and Longitudinal Acoustic (LA) branches [49]. Normal scatter-ing rates are expressed as s�1

T ¼ BTxT4 and s�1L ¼ BLx2T3 for TA

and LA respectively [49]. Most of parameters are fitted by the bulkthermal conductivities in Ref. [50] by ourselves except impurityscattering rates, whose parameters are based on analytical analy-ses from Ref. [51,52]. Table 1 gives all of parameters for the previ-ous formulas additionally with a boundary scattering rate s�1

B . Thereferred thermal conductivities for both Si and Ge agree well withthe predictions by these formulas with the present parameters atthe entire temperature range. For Al, electrons carry much heatand also couple with phonons during heat transport. However, pre-sent simulations neglect both pure electron transport and elec-tron–phonon coupling and only consider thermal conduction dueto the lattice to simplify transport process. Therefore, a constantrelaxation time 8.95 � 10�12 s is chosen to derive the lattice ther-mal conductivity of Al about 30 W/(m K) [21].

3. Verifications

To verify the present KMC algorithm, two cases are considered:the cross-plane phonon transport through single-layer and double-layer thin films with isothermal boundaries at both sides, as shownin Fig. 2. The single-layer thin films are made of Si, Al, and Ge andthe double-layer thin films are made of Al/Si and Ge/Si with thevolume ratio of two materials as one respectively, of which the dis-persions are introduced in Section 2.

All films are initially maintained at 299 K, and then the left andthe right sides change to Tl = 301 K and Tr = 299 K, respectively,with a referenced equilibrium temperature as Teq = 300 K. Thenumerical parameters in the simulations are listed in Tables 2and 3 for single-layer and double-layer thin films, respectively.The temperature and the heat flux distributions at a spatial unitcell are computed after the systems approach to the steady stateby [35,44]:

T ¼ Teq þXi

sigdeff =CVV ; ð2Þ

q ¼Xi

sigdeffv i;x=V ; ð3Þ

where si represents the sign of the tracked deviational energypacket, gd

eff the effective deviational energy of an energy packet,CV the volumetric heat capacity, V the volume of one unit cell, andvi,x being the velocity along the direction of heat flux. All of theresults are compared with the calculations of the discrete-ordinate method (DOM) from the Ref. [8].

Table 2 gives the parameters for the cross-plane phonon trans-port through single-layer thin films made of Si, Al, and Ge at threetotal thicknesses, 4 nm, 70 nm, and 700 nm. Fig. 3 shows agreeablecomparisons of temperature distributions along films at the steady

Table 1Parameters for each formula of various scattering rates.

Material AI BU BT

Si 1.32 � 10�45 s3 1 � 10�19 s/K3 1 � 10�

Ge 2.4 � 10�44 s3 8.5 � 10�20 s/K3 6.8 � 10

state and the effective thermal conductivities at various thick-nesses for the single-layer thin film. The effective cross-plane ther-mal conductivity is calculated by keff ¼ qaveL= T l � Trð Þ, whoseaverage heat flux qave is obtained by averaging the heat flux overthe whole film at the steady state, and L denotes the thickness ofthe thin film. And an average Knudsen number is introduced tobetter express the relation between the effective thermal conduc-tivity and the film thickness as [53]:

Knh i ¼ K�

L; ð4Þ

where an average MFP is defined as: K�¼ P

p

Rx D

deeqTeqdT vgsdx=

Pp

Rx D

deeqTeqdT dx, D being the density of phonon states. Results indi-

cate that larger temperature jump occurs at the boundary whenthe film thickness is smaller. This means that the non-equilibriumeffect at the boundary is larger due to more phonons interact withboundaries with a smaller film thickness [54].

Table 3 gives some of the parameters for cross-plane interfacialphonon transport through double-layer thin films made of Al/Siand Ge/Si at three total thicknesses, 20 nm, 120 nm, and 220 nm.Figs. 4 and 5 also give good comparisons of temperature distribu-tions along the whole film at the steady state and thermal bound-ary conductance at various thicknesses for the double-layer thinfilms of Al/Si and Ge/Si, respectively. The thermal boundary con-ductance is based on equivalent equilibrium temperature [8,10]and calculated by the formula, G ¼ qave=DT , where DT is the equiv-alent equilibrium temperature jump at the interface. As the resultsshow, the temperature jump at the interface is larger for the filmwith the smaller thickness. This indicates that more phonons inter-act with the interface for a smaller film thickness, which causes astronger non-equilibrium effect. Furthermore, the temperaturejump at the interface is larger than that at the boundary for a fixedfilm thickness. It means that the interface can cause a strongernon-equilibrium effect than the boundary due to reflections ofphonons at the interface [36]. Thus, the present KMC algorithmhas been verified by all of these comparisons and can be appliedto studying physical problems introduced in Section 2.

BL C s�1B

16 K�4 5 � 10�25 s/K 120 K 1.16 � 106 s�1

�14 K�4 2.5 � 10�25 s/K 32 K 0.27 � 106 s�1

Table 2Numerical parameters in Monte Carlo simulations for cross-plane phonon transport through single-layer thin films made of Si, Al, and Ge.

Materials Si Al Ge

Film Thickness (nm) 4 70 700 4 70 700 4 70 700Sample Number (million) 20 50 250 56 80 320 56 200 190

Table 3Numerical parameters in Monte Carlo simulations for cross-plane interfacial phonon transport through double-layer thin films made of Al/Si and Ge/Si.

Material Pairs Al/Si Ge/Si

Total Thickness of film (nm) 20 120 220 20 120 220Sample Number (million) 50 80 100 80 80 100

(a) (b)

(b) (d)Fig. 3. The non-dimensional temperature distributions and the effective thermal conductivities of cross-plane phonon transport through single-layer thin films, made of Si,Al, and Ge: (a), (b), and (c) are the non-dimensional temperature distributions for Si, Al, and Ge, respectively; (d) the effective thermal conductivities for Si, Al, and Ge thinfilms at various Knudsen numbers.

4 X. Ran, M. Wang / International Journal of Heat and Mass Transfer 147 (2020) 118941

4. Results & discussions

In this section, the present verified KMC is adopted to investi-gate interfacial phonon transport with heat source within systemsintroduced in Section 2, with a treatment for heat source in Ref.[44]. The following discussions are organized by three parts: sizeeffect, effect of heat source strength, and other special cases.

4.1. Size effect

Thermal properties of materials are strongly dependent on thesystem size in nanoscale heat transport. The size effects on the

thermal boundary resistance within the double-layer and multi-layer thin film with isothermal boundaries have been well studiedin Ref. [8,36,55,56]. However, the effect in these system with heatsource is still needed to be explored. The present work studies thesize effects on the thermal boundary resistance in double-layer andthree-layer thin films made of Al/Si and Ge/Si with heat source. Allthin films are occupied by two isothermal boundaries at 300 K toexclude the influence from the non-equilibrium effects at bound-aries and with a referenced equilibrium temperature Teq = 300 K.The stengths of heat source are fixed at 5 � 1016 W/m3 and2 � 1016 W/m3 for Al/Si and Ge/Si, respectively. And the heatsource is located in the left layer, in order to only focus on the

(b)(a)

Fig. 4. The non-dimensional temperature distribution and the thermal boundary conductance in cross-plane interfacial phonon transport through the Al/Si double-layer thinfilm: (a) non-dimensional temperature distribution at three total thicknesses, L, 20 nm, 120 nm, 220 nm; (b) thermal boundary conductance at the reverses of various totalthicknesses.

(a) (b)

Fig. 5. The non-dimensional temperature distribution and the thermal boundary conductance in cross-plane interfacial phonon transport through the Ge/Si double-layer thinfilm: (a) non-dimensional temperature distribution at three total thicknesses, L, 20 nm, 120 nm, 220 nm; (b) thermal boundary conductance at the reverses of various totalthicknesses.

X. Ran, M. Wang / International Journal of Heat and Mass Transfer 147 (2020) 118941 5

influence from the system size. The physical models have beenshown in Fig. 1(a) and (c), whose labels 1 and 2 represent Al orGe and Si for Al/Si or Ge/Si.

The thermal boundary resistances are derived from the temper-ature and the heat flux distributions after systems approach to thesteady state by the formula, R ¼ DT=q0

ave, with the average heatflux, q0

ave, along a whole layer nearest to the interface without heatsource. We have also given the thermal boundary resistances ofdouble-layer and three-layer thin films only with two isothermalboundaries Tl = 305 K and Tr = 300 K. This can reveal the differencebetween two conditions of only considering the isothermal bound-ary or heat source. As Fig. 6 shows, the thermal boundary resis-tance at the second interface in the three-layer thin film is muchsmaller than that at the first interface for both Al/Si and Ge/Si withheat source. The latter one is close to that in the double-layer thinfilm for both material pairs, both of which are also slightly differentfrom that only considering the isothermal boundary. All these ther-mal boundary resistances will almost approach to two constantvalues at the large enough size for two material pairs. Furthermore,for Al/Si, the thermal boundary resistance at the second interface

decreases with the increasing system size more smoothly withheat source. While those of the interface in double-layer thin filmand the first interface decrease with the increasing system sizesharply. Different from Al/Si, the thermal boundary resistance atthe second interface even increases with the increasing system sizesmoothly for Ge/Si with heat source. But those of interface indouble-layer thin film and the first interface for Ge/Si decreasewith the increasing system size sharply. It means that the exis-tences of heat sources influence the thermal boundary resistancesof different material pairs at different interfaces very differently.Moreover, their influences are also much different from those onlyconsidering the isothermal boundary due to different distances foremitted phonons to reach interfaces and scatter, whose moredetailed descriptions will be shown in Subsection 4.3. The trans-missivity between the material pair will be introduced to explainthis abnormal behavior of the thermal boundary resistance whenconsidering the heat source.

Fig. 7 shows that the transmissivity from Al or Ge to Si is smallerthan that from Si to Al or Ge for both branches and the differencebetween Ge and Si is much larger than that between Al and Si.

(a) (b)

Fig. 6. The thermal boundary resistances at different thicknesses of each layer for Al/Si and Ge/Si: (a) material pair Al/Si; (b) material pair Ge/Si. The blue lines with squares,the cyan lines with diamonds, the red lines with left triangles are for the interface in the double-layer thin film, and the first and the second interfaces from left side for thethree-layer thin film with heat source. And the green lines with circles, the magenta lines with right triangles, the black lines with stars are for the interface in the double-layer thin film, and the first and the second interfaces from left side for the three-layer thin film only with two isothermal boundaries Tl = 305 K and Tr = 300 K. (Forinterpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

(a) (b)

Fig. 7. The transmissivity of each branch for Al/Si and Ge/Si: (a) material pair Al/Si; (b) material pair Ge/Si.

6 X. Ran, M. Wang / International Journal of Heat and Mass Transfer 147 (2020) 118941

When phonons are generated by the left layer made of Al or Ge, theinterface of the double-layer thin film or the first interface of thethree-layer thin film play a role as filters. It means that some ofphonons will be reflected back and less phonons can transmitacross the interface to scatter with the boundary or the secondinterface. The major scattering event occurs at the interface ofthe double-layer thin film or the first interface of the three-layerthin film. This causes much larger non-equilibrium effects at theseinterfaces then much larger thermal boundary resistances. Com-pared with Al/Si, the reduction of transmitting phonons is rela-tively larger for Ge/Si due to its smaller transmissivity. Therefore,the increase of the non-equilibrium effect from the second inter-face to the first interface for Ge/Si is larger than that for Al/Si.When the system size decreases, the constraint effect from the firstinterface to the second interface for Ge/Si is stronger than that forAl/Si. This kind of effect has been introduced in Ref. [36] to illus-trate the influence among interfaces with the varying size. Finally,the non-equilibrium effect ratio at the second interface evendecreases with the decreasing system size thus the decreasingthermal boundary resistance for Ge/Si. However, those constraint

effects in the three-layer thin film made of Al/Si are not strongenough to lead to a decreasing non-equilibrium effect ratio. Thusthe thermal boundary resistances at the second interface inthree-layer thin films of Al/Si increase with the decreasing systemsize. The non-equilibrium effect ratios at the interface in thedouble-layer thin film or the first interface in three-layer thin filmsincrease with the decreasing system size. Thus their thermalboundary resistances will increase for both material pairs withthe decreasing system size.

4.2. Effect of heat source strength

In the real applications, the strengths of heat sources are usuallydifferent due to different working conditions, whose effect on ther-mal boundary resistance has never been reported before. In thispart, this effect will be studied in the same system in Subsection4.1 with the same heat souce location and the thickness of eachlayer as 10 nm. The systems are also occupied by two isothermalboundaries with the temperature Tl = Tr = 300 K and a referencedequilibrium temperature Teq = 300 K. The various strengths of heat

X. Ran, M. Wang / International Journal of Heat and Mass Transfer 147 (2020) 118941 7

source are chosen to avoid too high temperature increases afterevolutions of systems, which will break down the assumption ofthe tiny temperature difference for the present KMC scheme. Thethermal boundary resistances are calculated based on the temper-ature and the heat flux distributions at the steady state by the for-mula, R ¼ DT=q0

ave. As the Fig. 8 shows, almost constant thermalboundary resistances are obtained for different strengths of heatsource at different interfaces. This observation indicates that thestrength of heat source will not change the non-equilibrium effectratio under two isothermal boundaries with two equal tempera-tures. In detail, a larger strength of heat source will cause a largernon-equilibrium effect of different parts of the system. But thenon-equilibrium effect ratio of each part will keep constantbecause of the same level of increases.

4.3. Other special cases

Previous two parts have revealed that the effects of the systemsize and the strength of heat source on phonon transport throughthin films with heat source. All these effects are studied in systemswith an equal temperature at isothermal boundaries, and fixedlocations of heat source, and a fixed material layout. However,devices may work at some special conditions, such as non-equaltemperatures at two sides. The heat source may be also locatedat the right side for the present double-layer thin film or insidethe multilayer thin film rather than only in the layer near theboundary. Furthermore, different material layouts for the three-layer thin film should be also considered in real applications. Theyare valuable to explore the effect of heat source on the thermalboundary resistance. While for a double-layer thin film, differentmaterial layouts are the equivalent cases with different locationsof heat source. Therefore, the systems with different temperaturesat isothermal boundaries, different locations of heat source, anddifferent material layouts are considered in the present part.

The physical models adopted here have been shown in Fig. 1,where 1 and 2 denote Al and Si or Si and Al for Al/Si while denoteGe and Si or Si and Ge for Ge/Si. The first or the second case consid-ered in this part is for Tl = 305 K and Tr = 300 K or Tl = 300 K andTr = 305 K, respectively. The heat sources are all located in themateiral 1 as Fig. 1(a) and (c) show for the first and the secondcases. The heat source will be changed into the material 2 withan equal temperature 300 K at boundaries as Fig. 1(b) and (d)show, as the third case. The material pair 1 and 2 represent Al

(a)

Fig. 8. The thermal boundary resistances at different strengths of heat source for Al/Si anthe cyan lines with diamonds, the red lines with left triangles are for the interface in thethree-layer thin film. (For interpretation of the references to colour in this figure legend

and Si or Ge and Si for previous three cases. At last, different mate-rial layouts for the three-layer thin film are simulated also with anequal temperature 300 K as the fourth case, namely 1 and 2 denot-ing Si and Al for Al/Si while Si and Ge for Ge/Si. All cases are sim-ulated at the strength of heat source as 5 � 1016 W/m3 or 2 � 1016

W/m3 for material pair Al/Si or Ge/Si, respectively, the thickness ofeach layer as 10 nm, and a referenced equilibrium temperatureTeq = 300 K. The thermal boundary resistances are obtained basedon a formula introduced before at the steady state, R ¼ DT=q0

ave,with the average heat flux, q0

ave, along a whole layer without heatsource nearest to the interface. The results with the thickness ofeach layer as 10 nm and only considering non-equal temperaturesat boundaries in Subsection 4.1 are also shown here. Finally, allprevious results are compared with films with the same size andonly considering heat sources in Subsection 4.1, named the stan-dard case.

Tables 4 and 5 give the thermal boundary resistances of variouscases for two material pairs Al/Si and Ge/Si. As the results show, allof the thermal boundary resistances for each case are much differ-ent from those for the standard case for both material pairs. Indetail, the thermal boundary resistances of cases with an externaltemperature difference as 5 K are all smaller than those of the stan-dard case for Al/Si, whose external temperature difference is 0 K.And mostly, the thermal boundary resistances of the case with ahigher temperature at the left boundary is larger than those witha higher temperature at the other boundary. For Ge/Si, the compar-isons for the thermal boundary resistance at the interface of thedouble-layer thin film and the first interface of the three-layer thinfilm are the same as those for Al/Si. But they are different for theanother interface, as both thermal boundary resistances of thecases with the external temperature difference are larger thanthose of the standard case. The thermal boundary resistance atthe second interface of the case with a higher temperature at theright boundary is larger than that with a higher temperature atthe other side. Then for the third case, all of the thermal boundaryresistances of each interface are larger than those for standardcases for both material pairs. In addition, for the different materiallayout from the standard case, the thermal boundary resistances ofthe first and the second interfaces of the fourth case are larger andsmaller than those of the standard case for Al/Si, respectively.While all of the thermal boundary resistances of two interfacesin the fourth case are much larger than those of the standard casefor Ge/Si. Finally, the thermal boundary resistances only

(b)

d Ge/Si: (a) material pair Al/Si; (b) material pair Ge/Si. The blue lines with squares,double-layer thin film, and the first and the second interfaces from left side for the, the reader is referred to the web version of this article.)

Table 5The thermal boundary resistances for five cases of the material pair Ge/Si. The 1st caserepresents Tl = 305 K and Tr = 300 K. The 2nd case represents Tl = 300 K and Tr = 305 K.The 3rd case means that the heat source is changed into material 2. And the 4th caserepresents different material layout as 1(Si)2(Ge). Finally, the 5th case denotes thinfilms without heat sources and only with two non-equal temperature at boundariesTl = 305 K and Tr = 300 K, while the 6th case is the standard case.

Cases 2-Layer(m2 K/GW)

3-Layer 1st Interface(m2 K/GW)

3-Layer 2nd Interface(m2 K/GW)

1st Case 13.18 9.86 6.242nd Case 9.13 3.36 7.533rd Case 83.21 109.40 109.254th Case 133.71 11.765th Case 11.30 6.89 6.846th Case 39.75 44.47 4.99

Table 4The thermal boundary resistances for five cases of the material pair Al/Si. The 1st caserepresents Tl = 305 K and Tr = 300 K. The 2nd case represents Tl = 300 K and Tr = 305 K.The 3rd case means that the heat source is changed into material 2. And the 4th caserepresents different material layout as 1(Si)2(Al). Finally, the 5th case denotes thinfilms without heat sources and only with two non-equal temperature at boundariesTl = 305 K and Tr = 300 K, while the 6th case is the standard case.

Cases 2-Layer(m2 K/GW)

3-Layer 1st Interface(m2 K/GW)

3-Layer 2nd Interface(m2 K/GW)

1st Case 2.39 2.16 2.012nd Case 2.15 1.81 2.003rd Case 8.36 6.84 6.854th Case 12.09 1.595th Case 2.27 1.98 2.026th Case 4.28 4.77 2.16

8 X. Ran, M. Wang / International Journal of Heat and Mass Transfer 147 (2020) 118941

considering non-equal temperatures at boundaries at the left aregenerally smaller than those only considering heat source, exceptfor the second interface for Ge/Si.

Here the influence of the external temperature difference willbe explained firstly, namely the first and the second cases. Whenapplying the external temperature at the boundary, the non-equilibrium effect ratio of each part will be reduced due to theexternal non-equilibrium effect at the boundary. Viewing from thispoint, the thermal boundary resistance of each part will be reducedas well. However, when applying the external temperature differ-ence at the boundary, extra phonons will be injected into systemsfrom the boundaries at the direction of the drop of this externaltemperature difference. Therefore, the scattering rate with inter-faces will be promoted with a higher temperature at the leftboundary. It is due to moving directions of generated phonons bythe heat source are the same with those by the external boundarytemperature. While the influence on the scattering rate fromboundaries with a higher temperature at the right is partially orcompletely offset by the heat source, due to opposite directionsof their generated phonon fluxes. Finally, the thermal boundaryresistances of cases with the external temperature difference aresmaller than those without the external temperature differencewhen considering the non-equilibrium effect of the boundarymostly. Moreover, the thermal boundary resistance with a highertemperature at the left side is larger than that at the right side,by the mutual promotion of phonon fluxes motivated by the heatsource and the external temperature difference. Particularly, ifthe phonon fluxs motivated by boundaries are too large, the scat-tering rate with interface will increase so much that the non-equilibrium effect ratio of each part will increase as well. As aresult, the thermal boundary resistance will increase comparedwith that without the external temperature difference, such asthe second interface within three-layer thin films for Ge/Si in thepresent simulations. To conclude, the external temperature differ-

ence can readujst the non-equilibruim effect ratio of each part of asystem with heat source and change the thermal boundaryresistance.

The another case is that the location of heat source is changedinto material 2, namely the third case. To explain the abnormalthermal boundary resistance of this case, the comparisons for thetransmissivity and the mean free path of each material pair willbe introduced in Figs. 7 and 9. It should be noted that Fig. 9 onlygives comparions of the mean free path of partial frequencydomains for two material pairs. This is to avoid giving too largemean free path for phonons with very small frequencies to makeit be easily identified in a single figure. While the conclusions forthe comparisons of whole domains are the same with those for thispartial comparions. From comparisons of the mean free path, it isnoticed that the mean free paths of Si are much larger than thoseof Al or Ge for both branches, and the differences for Al/Si are evenlarger. And also it is mentioned again that the transmissivity fromAl or Ge to Si is smaller than that from Si to Al or Ge, and moreover,the difference for Ge/Si is larger, shown in Fig. 7. The smaller trans-missivity from Al to Si means more phonons will be stored in Alwhen the heat source is located in Al. This will cause more frequentinterface scattering within the double-layer thin film or at the firstinterface of the three-layer thin film. While extremely larger meanfree paths of Si make much more frequent interface scatteringwhen phonons are generated in Si as the heat source. For Al/Si,the influence from difference of the mean free path is larger thanthat from the transmissivity due to the former is more obvious.Even, phonons will be reflected back and more likely rescatter withthe other interface when locating heat source in the middle layermade of Si within the three-layer thin film. Then the interface scat-tering rate when locating the heat source in Si is larger than thatwhen locating the heat source in Al, since it causes a larger non-equilibrium effect ratio of each interface. Therefore, the thermalboundary resistance of each interface of the third case is largerthan that of the standard case. The analyses of the comparions ofthermal boundary resistances for Ge/Si and conclusions are thesame as well.

Then change the material layout for the three-layer thin film,namely the fourth case. It is stated that the non-equilibrium effectis mainly caused by interface scattering rather than the intrinsicand the boundary scattering when the thickness of each layer issmall. Due to much larger mean free path of Si than Al, more pho-nons scatter with the first interface while less phonons scatter withthe second interface. This makes a smaller thermal boundary resis-tance of the second interface for the fourth case of Al/Si than that ofthe standard case. These factors make a much larger thermalboundary resistance at the first interface of the fourth case thanthat of the standard case. For the material pair Ge/Si, the transmis-sivity from Si to Ge is much larger than that at the reverse direc-tion, so more phonons will transmit across the first interface andscatter with the second interface. This cause a larger non-equilibrium effect at the second interfaces then larger thermalboundary resistances for the fourth case. On the other hand, pho-nons with larger mean free paths generated in Si will more likelyscatter with the first interface, which causes a larger non-equilibrium effect at the first interface. Therefore, the thermalboundary resistance of the first interface for the fourth caseincreases.

Another indication for these comparions for Al/Si and Ge/Si is:after changing material layout, the relative increase of the differ-ence of the thermal boundary resistance between two interfacesof Al/Si is much larger than Ge/Si. It can be explained by the trans-missivity and the mean free path for Al/Si and Ge/Si. At first, thedifference of the transmissivity at two directions for Al/Si is smallerthan that for Ge/Si. After changing material layout, more phononswill transmit across the first interface and scatter with the second

(a) (b)

Fig. 9. The mean free path of each branch for Al/Si and Ge/Si: (a) material pair Al/Si; (b) material pair Ge/Si.

X. Ran, M. Wang / International Journal of Heat and Mass Transfer 147 (2020) 118941 9

interface for Ge/Si while this increase for Al/Si is smaller. Viewingfrom this point, the increase of the non-equilibrium effect at thesecond interface for Ge/Si is larger. While the difference for themean free path of Al/Si is much larger than that of Ge/Si, the scat-tering at the first interface of Al/Si is thus more frequent than thatof Ge/Si after changing material layout. This cause the increase ofthe non-equilibrium effect at the first interface for Al/Si is largerthan that for Ge/Si. In summary, the relative increase of the differ-ence of the non-equilibrium effect between two interfaces of Al/Siis much larger than Ge/Si thus a same conclusion for the thermalboundary resistance is obtained.

Finally, we just give a brief illustration for the differencebetween two conditions only considering non-equal temperaturesat boundaries or the heat source, due to the main mechanismshave been explored before. Emitted phonons will travel longer dis-tances to scatter with the interface when only considering non-equal boundary temperatures, resulting in smaller non-equilibrium effects at the interface. So thermal boundary resis-tances only considering non-equal boundary temperatures aremostly smaller than those only considering the heat source. Spe-cially, for Ge/Si, due to a extremely large non-equilibrium effectat the first interface only with the heat source, the non-equilibrium effect ratio for the second interface is relatively small.Therefore, its thermal boundary resistance is smaller than that atthe second interface for the case only with non-equal temperaturesat boundaries.

5. Conclusions

In summary, the cross-plane interfacial phonon transportthrough double-layer and three-layer thin films with heat sourcesare studied in this work. With the verified Monte Carlo algorithm,the thermal boundary resistances between Al/Si and Ge/Si at vari-ous cases are calculated. Firstly, the thermal boundary resistancesat different interfaces with heat sources located in the first layerand two isothermal boundaries with an equal temperature are dif-ferent even with the same thickness of each layer, and vary withsize differently as well. The thermal boundary resistances of inter-faces farther from heat sources are much smaller than those nearerheat sources. While those for the double-layer thin film and forinterfaces nearer heat sources within the three-layer thin film forboth material pairs are almost the same. The thermal boundaryresistances decrease sharply with the increasing size for the

double-layer thin film or interfaces nearer heat sources withinthe three-layer thin film for Al/Si and Ge/Si. Moreover, the thermalboundary resistances for another kind of interfaces decreaseslightly or even increase with increasing system size for Al/Si orGe/Si, respectively. This can be explained by the different non-equilibrium effect caused by different transmissivity at differentdirections for Al/Si and Ge/Si and influences among each interfacefor three-layer thin film. In addition, when only the isothermalboundaries with non-equal temperatures are considered, the ther-mal boundary resistance is mostly smaller than that with only theheat source considered. It is mainly caused by different distancesfor emitted phonons to reach the interfaces and scatter.

Secondly, the thermal boundary resistances at differentstrengths of heat sources are studied and almost constant thermalboundary resistances are found. This means that the strength ofheat source hardly changes the non-equilibrium effect ratio of eachpart. Some special cases are considered for both material pairsfinally, namely applying external temperature differences, andchanging locations of heat sources into the other material for bothkinds of films, and changing the material layout for three-layer thinfilms. All of thermal boundary resistances are much different fromthose of the standard case, as in the size effect part of thickness10 nm for each layer. Particularly, the thermal boundary resis-tances are much larger than those of the standard cases when heatsources are located in middle layers for both materials. The differ-ences among these special cases and the standard cases are allattributed to the different transmissivities at different directions,different mean free paths of materials, and rescattering whenlocating heat souces in the middle layer. It is still needed moreworks to build up the quantitative relations between these factorsand the thermal boundary resistance in the future. It should beemphasized that the complete neglect of optical phonons will leadto a lower thermal boundary conductance or higher thermalboundary resistance. The reason is that optical phonons canexchange energy with acoustic phonons transmitting the interfaceeven optical phonons can not across the interface thus enhancethermal boundary conductance. The conclusions of the presentwork can be guides for the design for nanodevices with heatsource, esspecially for the multilayer systems.

Declaration of Competing Interest

We declare that there is no conflict of interests for this work.

10 X. Ran, M. Wang / International Journal of Heat and Mass Transfer 147 (2020) 118941

Acknowledgements

This work is financially supported by NSF of China (No.51621062, 91634107).

Appendix A. Supplementary material

Supplementary data to this article can be found online athttps://doi.org/10.1016/j.ijheatmasstransfer.2019.118941.

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