Supporting Information for Phonon Dominated Heat Conduction ...
Supporting Information for Phonon Dominated Heat Conduction Normal to Mo/Si Multilayers with Period below 10 nm Zijian Li1, Si Tan1, Elah Bozorg-Grayeli1, Takashi Kodama1, Mehdi Asheghi1, Gil Delgado2, Matthew Panzer2, Alexander Pokrovsky2, Daniel Wack2, and Kenneth E. Goodson1,* 1
Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA 2
KLA-Tencor Corporation, Milpitas, CA95035, USA
* Email: [email protected], Phone: (650) 725-2086
I. Additional TEM images of the Mo/Si multilayers
Figure S-1. Cross sectional transmission electron microscope (TEM) image of an extreme ultraviolet (EUV) mask made of Mo/Si multilayers. A total of 40 Mo/Si bilayers are deposited on 1
silicon substrate. EUV masks require a TaN absorber and a Ru capping layer on top of the Mo/Si multilayers. These two layers are removed before fabricating the nanoheaters.
poly-Mo a-Si 5 nm
c-Si
Figure S-2. Cross sectional TEM image near the bottom of the Mo/Si multilayers. The Si substrate is single crystalline (c-Si), buffer Si layer is amorphous (a-Si), and the metal region is polycrystalline (poly-Mo).
II. Diffuse mismatch model for phonon-phonon interface conductance The diffuse mismatch model (DMM) evaluates the thermal conductance of phonon-phonon coupling across the Mo/Si interaface, hpp.
The hypothesis of diffusive phonon reflection at interfaces is
appropriate for the room temperature at which the phonon wavelength is below the interface roughness observed using TEM. Assuming the temperature drop across the interface is small, we can write the thermal interface conductance as1
3 2 hpp , Mo Si Mo Si vMo , D vMo , j CMo , v T j 12
(S-1)
2
hpp , Si Mo Si Mo vSi3 , D vSi2, j CSi ,v T j 12
(S-2)
where Cv(T) is volumetric heat capacity at T, and v is the speed of sound with its subscript j denoting the polarization (l for longitudinal and t for transverse). The phonon transmission coefficient1, 2 at Mo/Si interface under the DMM can be written as
Mo Si
Si Mo
j
j
vSi2, j
(S-3)
v 2 j vSi2, j j Mo , j
j
2 vMo ,j
(S-4)
2 2 vMo , j j vSi , j
and vD is the speed of sound calculated using the Debye model:
vD vl3 2vt3
1/3
(S-5)
where the longitudinal and transverse components, vl and vt, are taken from Table S-1. The DMM estimate used here can lead to substantial deviations in the thermal interface conductance near room temperature because the Debye approximation does not properly account for the complexity of the phonon dispersion relationships at the edge of the Brillouin zone.3,
4
Depending on the material
combination and temperature, Reddy et al reported as large as 100 percent difference between the Debye model and a numerical computation considering the full dispersion relationship over the entire Brillouin zone.4 There is some evidence that the presence of at least one amorphous material can mitigate the error due to the strong scattering of short wavelength phonons – for which dispersion is most important – in the amorphous material.5, 6 The determination of the error is further complicated by the polycrystalline structure of molybdenum on the opposite side, which may render the use of an average, isotropic dispersion relationship (as in the Debye model) more appropriate. Further work on crystalline-amorphous interfaces considering the full dispersion relationship on one side and strong 3
scattering on the other would be useful in refining the modeling developed in the present manuscript. For ultrathin metal/dielectric multilayers at room temperature, the Debye approximation shows ~35% overestimate for ~3 nm thick W/Al2O3,3 and achieved very good accuracy for ~5 nm Ta/TaOx.7 Based on these facts, we use the Debye approximation to estimate the phonon conduction across the interface. It is worth noting that the density of an amorphous Si (a-Si) thin film affects the calculation of sound velocity and eventually the DMM result. The density of crystalline Si is well characterized as 2.33 g/cm3. However, the density of a-Si depends on the concentrations of material defects, local disorder, impurity, and stress in the film.8 Our sample is prepared by a sputtering process which is refined to produce excellent quality films for EUV mirrors (see TEM image in Fig. 1, and in Supporting Information I). The density of high-quality a-Si has been reported as ~95% of the crystalline value.9 Although up to a ~12% decrease in a-Si density has been reported, it is due to a different sputtering process that leads to void formation in the a-Si films.10 Furthermore, a-Si thin films with thickness of a few nanometers have also been reported to have 98% of the crystalline density.11 Based on this evidence from multiple sources, we do not expect the density of our 2-5 nm a-Si thin films to deviate more than 5% from the crystalline value. The speed of sound calculation here based on Debye model, 4.1 km/s, is also consistent with the measurement data of 4.4 ± 0.5 km/s.12
Table S-1. Material properties of silicon and molybdenum used in the calculation.
vl [km/s]
vt [km/s]
Cv [MJ/m3K]
ne [1028 m3]
τe [10-15 s]
a-Si
8.30
5.33
1.654
--
--
Mo
6.25
3.35
2.580
7.53
3.80
4
III. BTE verification of the thermal resistor model
The approximate thermal resistor model shown in Fig. 3(a) lends insight into the physical mechanisms in the electron-phonon coupled thermal conduction in metal/dielectric multilayers. In order to verify this model, we use the equation of electron-phonon transport (EEPT) based on the Boltzmann transport equation (BTE) to calculate the thermal properties in the multilayer. Although it is much more computational intensive and might obscure the physical intuition, this BTE-based approach more rigorously describes the electron-phonon thermalization process in thin metal layer bounded by dielectrics in a multilayer structure. The governing equations for electrons and phonons are given as 1 1 1 1 Ie d Ie 1I p d g I e I 1 I e 1 e 2 2 x v f t vfe v f ep
(S-6)
1 1 1 1 I p 2 1I p d I p g 2 1I e d I p 1 I p x vs t vs p vs pe
(S-7)
where g (C p vs ) / (Ce v f ) is for the gray case which matches the temperature between electron system and phonon system during the thermalization process. The phonon and electron intensity are respectively defined as
I p , , x, t
D
v , f x, t D d p
0
p
p
(S-8)
p
I e , , x, t v f f e x, t e De d e
(S-9)
e
where vf is the Fermi velocity, ω is phonon or electron frequency, and θ and ϕ are the polar and azimuthal angles, respectively. The integral calculates the transient intensity at spatial position x and time t. The summation in Eq. (S-8) is over the phonon polarizations p. Under the gray approximation, the phonon and electron intensity reduce to 5
I p Cs vsTp / 4
(S-10)
I e Ce v f Te / 4
(S-11)
where Cs and Ce are the heat capacity of phonons and electrons, respectively, and the transverse and longitudinal phonon velocities are simplified into an average velocity vs. The corresponding heat flux under gray approximation can be calculated as
qe 2 i I ei wi
(S-12)
q p 2 i I pi wi
(S-13)
i
i
where µ = cosθ, and w is the weighing factor obtained by the discrete ordinate method. This calculation uses a totally diffusive boundary condition as illustrated in Fig. S-3. Energy conservation at the interface between Layer 1 and Layer 2 would dictate that the forward propagation of phonon in Layer 1 equals the sum of the portion that is scattered backwards to Layer 1 plus the portion that gets transmitted to Layer 2. Explicitly, it is given as
I x , dΩ p1
0
1
1
1
I x , dΩ
2
0
2
2
(S-14)
Rd 21 I p2 x0 , 2 2 dΩ1 Td 21 I p1 x0 , 1 1dΩ 2
(S-15)
2
2
p2
Rd 12 I p1 x0 , 1 1dΩ1 Td 12 I p2 x0 , 2 2 dΩ 2
1
2
2
2
where Rdij and Tdij indicate diffusive reflectivity and transmissivity from layer i to layer j, respectively, and dΩi = 2πsinθi is the differential solid angle.
6
Heat Flux
I p1
I p2
I p1
I p2
Layer 1
Layer 2
Figure S-3. Schematic of the totally diffusive boundary condition for the phonon system. The
phonon intensity (Wm-2sr-1) is denoted as Ip. A similar boundary condition is applied to the electron system.
The boundary condition for electrons is achieved by simply setting I e / x 0 because electrons are isolated at metal-dielectric interface. By solving Eq. (S-1) – (S-2) subject to boundary conditions (S-14) – (S-15) under different film thickness, the heat flux of electrons and phonons can be obtained by Eq. (S-12) and (S-13). The effective thermal resistances associated with electrons and phonons are given by definition as
Re
ΔT ΔT , Rq qe qp
(S-16)
The BTE solutions are plotted as the dashed curves in Fig. S-4, which agree reasonably well with the thermal resistor model presented in Fig. 3. The deviation at very small thickness is partly due to the breakdown of the TTM for length scales close to the atomic spacing. A full numerical computation on the multilayer system based on BTE will provide a more detailed view of the multiple conduction mechanisms and their relative strengths.
7
2
Thermal Resistance [m2K/GW]
10
Rp, approx. resistor model Re, approx. resistor model Rp, BTE
1
Re, BTE
10
0
10
Dominant Carrier Transition Thickness
-1
10
0
10
1
2
10 10 Film Thickness [nm]
3
10
Figure S-4. Comparison between the theoretical solutions from the approximate thermal resistor
model in Fig. 3 (solid lines) and the Boltzmann transport equation (dashed lines). A single period of 2.8 nm /4.1 nm Mo/Si multilayer is used in the computation. The thermal resistance of the phonon direct transmission (Rp) becomes smaller than the electron-phonon coupling path (Re) for a film thickness less than the dominant carrier transition thickness, which is approximately 14 nm in this Mo/Si sample.
References
1.
De Bellis, L.; Phelan, P. E.; Prasher, R. S., J Thermophys Heat Tr 2000, 14 (2), 144-150.
2.
Swartz, E.; Pohl, R., Reviews of Modern Physics 1989, 61 (3), 605-668.
3.
Costescu, R. M.; Cahill, D. G.; Fabreguette, F. H.; Sechrist, Z. A.; George, S. M., Science 2004,
303 (5660), 989-990. 4.
Reddy, P.; Castelino, K.; Majumdar, A., Appl. Phys. Lett. 2005, 87 (21).
8
5.
Stevens, R. J.; Zhigilei, L. V.; Norris, P. M., International Journal of Heat and Mass Transfer
2007, 50 (19-20), 3977-3989.
6.
Beechem, T.; Hopkins, P. E., J. Appl. Phys. 2009, 106 (12).
7.
Ju, Y. S.; Hung, M.-T.; Usui, T., Journal of Heat Transfer 2006, 128 (9), 919.
8.
Coxsmith, I. R.; Liang, H. C.; Dillon, R. O., J Vac Sci Technol A 1985, 3 (3), 674-677.
9.
Renner, O.; Zemek, J., Czech J Phys 1973, B 23 (11), 1273-1276.
10.
Hauser, J. J., Phys Rev B 1973, 8 (8), 3817-3823.
11.
Ruppert, A. F.; Persans, P. D.; Hughes, G. J.; Liang, K. S.; Abeles, B.; Lanford, W., Phys Rev B
1991, 44 (20), 11381-11385.
12.
Testardi, L. R.; Hauser, J. J., Solid State Commun 1977, 21 (11), 1039-1041.
9
Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA 2
KLA-Tencor Corporation, Milpitas, CA95035, USA
* Email: [email protected], Phone: (650) 725-2086
I. Additional TEM images of the Mo/Si multilayers
Figure S-1. Cross sectional transmission electron microscope (TEM) image of an extreme ultraviolet (EUV) mask made of Mo/Si multilayers. A total of 40 Mo/Si bilayers are deposited on 1
silicon substrate. EUV masks require a TaN absorber and a Ru capping layer on top of the Mo/Si multilayers. These two layers are removed before fabricating the nanoheaters.
poly-Mo a-Si 5 nm
c-Si
Figure S-2. Cross sectional TEM image near the bottom of the Mo/Si multilayers. The Si substrate is single crystalline (c-Si), buffer Si layer is amorphous (a-Si), and the metal region is polycrystalline (poly-Mo).
II. Diffuse mismatch model for phonon-phonon interface conductance The diffuse mismatch model (DMM) evaluates the thermal conductance of phonon-phonon coupling across the Mo/Si interaface, hpp.
The hypothesis of diffusive phonon reflection at interfaces is
appropriate for the room temperature at which the phonon wavelength is below the interface roughness observed using TEM. Assuming the temperature drop across the interface is small, we can write the thermal interface conductance as1
3 2 hpp , Mo Si Mo Si vMo , D vMo , j CMo , v T j 12
(S-1)
2
hpp , Si Mo Si Mo vSi3 , D vSi2, j CSi ,v T j 12
(S-2)
where Cv(T) is volumetric heat capacity at T, and v is the speed of sound with its subscript j denoting the polarization (l for longitudinal and t for transverse). The phonon transmission coefficient1, 2 at Mo/Si interface under the DMM can be written as
Mo Si
Si Mo
j
j
vSi2, j
(S-3)
v 2 j vSi2, j j Mo , j
j
2 vMo ,j
(S-4)
2 2 vMo , j j vSi , j
and vD is the speed of sound calculated using the Debye model:
vD vl3 2vt3
1/3
(S-5)
where the longitudinal and transverse components, vl and vt, are taken from Table S-1. The DMM estimate used here can lead to substantial deviations in the thermal interface conductance near room temperature because the Debye approximation does not properly account for the complexity of the phonon dispersion relationships at the edge of the Brillouin zone.3,
4
Depending on the material
combination and temperature, Reddy et al reported as large as 100 percent difference between the Debye model and a numerical computation considering the full dispersion relationship over the entire Brillouin zone.4 There is some evidence that the presence of at least one amorphous material can mitigate the error due to the strong scattering of short wavelength phonons – for which dispersion is most important – in the amorphous material.5, 6 The determination of the error is further complicated by the polycrystalline structure of molybdenum on the opposite side, which may render the use of an average, isotropic dispersion relationship (as in the Debye model) more appropriate. Further work on crystalline-amorphous interfaces considering the full dispersion relationship on one side and strong 3
scattering on the other would be useful in refining the modeling developed in the present manuscript. For ultrathin metal/dielectric multilayers at room temperature, the Debye approximation shows ~35% overestimate for ~3 nm thick W/Al2O3,3 and achieved very good accuracy for ~5 nm Ta/TaOx.7 Based on these facts, we use the Debye approximation to estimate the phonon conduction across the interface. It is worth noting that the density of an amorphous Si (a-Si) thin film affects the calculation of sound velocity and eventually the DMM result. The density of crystalline Si is well characterized as 2.33 g/cm3. However, the density of a-Si depends on the concentrations of material defects, local disorder, impurity, and stress in the film.8 Our sample is prepared by a sputtering process which is refined to produce excellent quality films for EUV mirrors (see TEM image in Fig. 1, and in Supporting Information I). The density of high-quality a-Si has been reported as ~95% of the crystalline value.9 Although up to a ~12% decrease in a-Si density has been reported, it is due to a different sputtering process that leads to void formation in the a-Si films.10 Furthermore, a-Si thin films with thickness of a few nanometers have also been reported to have 98% of the crystalline density.11 Based on this evidence from multiple sources, we do not expect the density of our 2-5 nm a-Si thin films to deviate more than 5% from the crystalline value. The speed of sound calculation here based on Debye model, 4.1 km/s, is also consistent with the measurement data of 4.4 ± 0.5 km/s.12
Table S-1. Material properties of silicon and molybdenum used in the calculation.
vl [km/s]
vt [km/s]
Cv [MJ/m3K]
ne [1028 m3]
τe [10-15 s]
a-Si
8.30
5.33
1.654
--
--
Mo
6.25
3.35
2.580
7.53
3.80
4
III. BTE verification of the thermal resistor model
The approximate thermal resistor model shown in Fig. 3(a) lends insight into the physical mechanisms in the electron-phonon coupled thermal conduction in metal/dielectric multilayers. In order to verify this model, we use the equation of electron-phonon transport (EEPT) based on the Boltzmann transport equation (BTE) to calculate the thermal properties in the multilayer. Although it is much more computational intensive and might obscure the physical intuition, this BTE-based approach more rigorously describes the electron-phonon thermalization process in thin metal layer bounded by dielectrics in a multilayer structure. The governing equations for electrons and phonons are given as 1 1 1 1 Ie d Ie 1I p d g I e I 1 I e 1 e 2 2 x v f t vfe v f ep
(S-6)
1 1 1 1 I p 2 1I p d I p g 2 1I e d I p 1 I p x vs t vs p vs pe
(S-7)
where g (C p vs ) / (Ce v f ) is for the gray case which matches the temperature between electron system and phonon system during the thermalization process. The phonon and electron intensity are respectively defined as
I p , , x, t
D
v , f x, t D d p
0
p
p
(S-8)
p
I e , , x, t v f f e x, t e De d e
(S-9)
e
where vf is the Fermi velocity, ω is phonon or electron frequency, and θ and ϕ are the polar and azimuthal angles, respectively. The integral calculates the transient intensity at spatial position x and time t. The summation in Eq. (S-8) is over the phonon polarizations p. Under the gray approximation, the phonon and electron intensity reduce to 5
I p Cs vsTp / 4
(S-10)
I e Ce v f Te / 4
(S-11)
where Cs and Ce are the heat capacity of phonons and electrons, respectively, and the transverse and longitudinal phonon velocities are simplified into an average velocity vs. The corresponding heat flux under gray approximation can be calculated as
qe 2 i I ei wi
(S-12)
q p 2 i I pi wi
(S-13)
i
i
where µ = cosθ, and w is the weighing factor obtained by the discrete ordinate method. This calculation uses a totally diffusive boundary condition as illustrated in Fig. S-3. Energy conservation at the interface between Layer 1 and Layer 2 would dictate that the forward propagation of phonon in Layer 1 equals the sum of the portion that is scattered backwards to Layer 1 plus the portion that gets transmitted to Layer 2. Explicitly, it is given as
I x , dΩ p1
0
1
1
1
I x , dΩ
2
0
2
2
(S-14)
Rd 21 I p2 x0 , 2 2 dΩ1 Td 21 I p1 x0 , 1 1dΩ 2
(S-15)
2
2
p2
Rd 12 I p1 x0 , 1 1dΩ1 Td 12 I p2 x0 , 2 2 dΩ 2
1
2
2
2
where Rdij and Tdij indicate diffusive reflectivity and transmissivity from layer i to layer j, respectively, and dΩi = 2πsinθi is the differential solid angle.
6
Heat Flux
I p1
I p2
I p1
I p2
Layer 1
Layer 2
Figure S-3. Schematic of the totally diffusive boundary condition for the phonon system. The
phonon intensity (Wm-2sr-1) is denoted as Ip. A similar boundary condition is applied to the electron system.
The boundary condition for electrons is achieved by simply setting I e / x 0 because electrons are isolated at metal-dielectric interface. By solving Eq. (S-1) – (S-2) subject to boundary conditions (S-14) – (S-15) under different film thickness, the heat flux of electrons and phonons can be obtained by Eq. (S-12) and (S-13). The effective thermal resistances associated with electrons and phonons are given by definition as
Re
ΔT ΔT , Rq qe qp
(S-16)
The BTE solutions are plotted as the dashed curves in Fig. S-4, which agree reasonably well with the thermal resistor model presented in Fig. 3. The deviation at very small thickness is partly due to the breakdown of the TTM for length scales close to the atomic spacing. A full numerical computation on the multilayer system based on BTE will provide a more detailed view of the multiple conduction mechanisms and their relative strengths.
7
2
Thermal Resistance [m2K/GW]
10
Rp, approx. resistor model Re, approx. resistor model Rp, BTE
1
Re, BTE
10
0
10
Dominant Carrier Transition Thickness
-1
10
0
10
1
2
10 10 Film Thickness [nm]
3
10
Figure S-4. Comparison between the theoretical solutions from the approximate thermal resistor
model in Fig. 3 (solid lines) and the Boltzmann transport equation (dashed lines). A single period of 2.8 nm /4.1 nm Mo/Si multilayer is used in the computation. The thermal resistance of the phonon direct transmission (Rp) becomes smaller than the electron-phonon coupling path (Re) for a film thickness less than the dominant carrier transition thickness, which is approximately 14 nm in this Mo/Si sample.
References
1.
De Bellis, L.; Phelan, P. E.; Prasher, R. S., J Thermophys Heat Tr 2000, 14 (2), 144-150.
2.
Swartz, E.; Pohl, R., Reviews of Modern Physics 1989, 61 (3), 605-668.
3.
Costescu, R. M.; Cahill, D. G.; Fabreguette, F. H.; Sechrist, Z. A.; George, S. M., Science 2004,
303 (5660), 989-990. 4.
Reddy, P.; Castelino, K.; Majumdar, A., Appl. Phys. Lett. 2005, 87 (21).
8
5.
Stevens, R. J.; Zhigilei, L. V.; Norris, P. M., International Journal of Heat and Mass Transfer
2007, 50 (19-20), 3977-3989.
6.
Beechem, T.; Hopkins, P. E., J. Appl. Phys. 2009, 106 (12).
7.
Ju, Y. S.; Hung, M.-T.; Usui, T., Journal of Heat Transfer 2006, 128 (9), 919.
8.
Coxsmith, I. R.; Liang, H. C.; Dillon, R. O., J Vac Sci Technol A 1985, 3 (3), 674-677.
9.
Renner, O.; Zemek, J., Czech J Phys 1973, B 23 (11), 1273-1276.
10.
Hauser, J. J., Phys Rev B 1973, 8 (8), 3817-3823.
11.
Ruppert, A. F.; Persans, P. D.; Hughes, G. J.; Liang, K. S.; Abeles, B.; Lanford, W., Phys Rev B
1991, 44 (20), 11381-11385.
12.
Testardi, L. R.; Hauser, J. J., Solid State Commun 1977, 21 (11), 1039-1041.
9
