MICROSCALE HEAT TRANSFER IN MULTILAYER STRUCTURE

Some microscale heat transfer problems are associated with cylindrical geometry.

For example, heat transport in silicon and germanium multishell nanowire heterostructures. Lauhon et al. [91] synthesized core-shell nanowires by chemical vapor deposition. Then, they developed a high-performance coaxially gated field-effect transistor of core-shell structures, as illustrated in Fig. 3.1. In this chapter, EPRT and DMM are employed to analyze microscale heat transfer in multilayer structures.

3.1 Analysis

3.1.1 Mathematical Formulation

Phonon heat transfer in dielectric thin films can be modeled by the Boltzmann transport equation [31]. Under the first-order relaxation time approximation, the BTE is reduced to

R

f f f

t v f

τ ^ω

ω ω

ω + ⋅∇ = −

∂

∂ v ⁰

, (3-1)

where f denotes the phonon distribution function as a function of frequency _ω ω , v

substituting Eq. (3-2) into Eq. (3-1) yields

The second term on the left-hand-side in the above equation is described in the form [90]

I was employed in the above equation due to axisymmetric assumption.

Substituting Eq. (3-4) into Eq. (3-3) and letting _µ =sin_ζ cos_ϕ, _η=sin_ζsin_ϕ, Eq.

(3-3) becomes

v

The equilibrium phonon intensity I_ω⁰ can be approximated by assuming equilibrium at every frequency, then

=

∫

Substituting Eq. (3-6) into Eq. (3-5) leads to I

This study considers a two- layer concentric cylinder with inner radius r and _i outer radius r , as illustrated in Fig. 3.2. The medium temperature initially is _o T . At _o time t =0 , the temperature at r =r_i rises to T (_i T_i >T_o ). Meanwhile, the temperature at r =r_o remains T . For convenience, the subsequent analysis assumes _o the medium to be gray, i.e., frequency independent, while the EPRT can be expressed as

The subscript indices k =1 and 2 represent layer 1 (inner layer) and layer 2 (outer layer), respectively. The initial conditions of this system can be written as

)

Moreover, the boundary conditions are )

∫

Many theories have been developed to describe the interfacial condition between two dissimilar materials, and various α_ij have been calculated using different models. Chen [92] derived the energy transmission coefficient for diffuse interface, as follows:

where C is the volumetric specific heat and v is the phonon group velocity. The diffuse mismatch model (DMM) is based on the assumption that phonons arriving at the interface totally lose their memory on the side which they come from. Swartz and Pohl [35] stated that the phonon reflectivity from layer 1 to layer 2 equals the phonon transmissivity from layer 2 to layer 1. It can be expressed as

ij ji = R

α , i, j=1,2. (3-15)

Once the transmissivity is obtained from Eq. (3-14), the reflectivity can be calculated according to the requirement to conserve energy, and thus

ij

Rij =1−α , i, j=1,2. (3-16)

∫ ∑

₌

where w are the weighting factors for the Gaussian quadrature. The governing _i equation then is transformed to

k

The third term on the left- hand-side in Eq. (3-19) can be approximated as [90]

( )

Substituting Eq. (3-20) into Eq. (3-19) leads to

( )

In this study, the S scheme, which means ₂ m=2, is selected to deal with the governing equations. The finite difference method is employed to approximate the differential terms in the governing equations. Backward difference is used in space when 0<µi<1 and forward difference is used in space when −1<µi<0 . Addit ionally, forward difference is used in time in both cases. Thus, Eq. (3.24) becomes where n and j represent the time index and the space index in the radial direction, respectively. To solve the simultaneous governing equations, an iterative procedure is performed in this study. The convergence criterion is that the relative error for temperature is less than 10⁻⁴.

cases, GaAs/AlAs superlatices [40] and diamond thin film deposited on silicon substrate are selected as examples for demonstration.

Fig. 3.4 shows the transient temperature distributions on GaAs/AlAs superlattices with inner radius ri =10⁻⁷ m under different film thicknesses:

(a)L₁ = L₂ =5×10⁻⁹ m, (b)L₁ = L₂ =5×10⁻⁷ m. Due to the ballistic transport of phonons, temperature discontinuities (dropping at high-temperature boundary and jumping at low-temperature boundary) occur at two boundaries. Both temperatures at

ri

r = and r =r_o increase with time and approach to the steady state. It is noticed that the dimensionless temperature of layer 2 is zero in the early time (t<10⁻¹⁰ s) for case (b) because it takes time to make phonons thermal equilibrium. Moreover, the tube thickness significantly influences phonon heat transport. Comparing Figs. (3.4a) and (3.4b) reveals that the temperature discontinuities at boundaries decrease with increasing tube thickness. With increasing tube thickness, the temperature profile presents the diffuse-like behavior – that is diffusive transport dominates. Otherwise, ballistic transport dominates, since the tube is thin. Ballistic transport dominates for small length scale which is comparable to phonon mean free path.

The length scale considered here is the micro/nano- meter. In practical engineering applications, steady states are reached within a micro-second. The steady state thus is assumed in the following demonstrating cases. Fig. 3.5 illustrates the effect of tube thickness on the temperature profiles of GaAs/AlAs superlattices at

difference divided by the heat flux at the interface is defined as interface thermal resistance (ITR), thus

q

ITR= ∆T^int , (3-26)

where ∆T_int is the temperature difference at the interface. Numerous models are proposed for describing the interface thermal resistance. This study employs the DMM model. Fig. 3.5 clearly shows that the interface temperature discontinuity increases with decreasing tube thickness. Additionally, the temperature discontinuity also increases at boundaries with decreasing tube thickness. The temperature drop inside each layer is small compared to that at the interface when the layers are very thin. In other words, the thermal resistance at the surfaces, including the boundary and interface, is greater than that inside the materials. Surface properties thus dominate heat transfer in a thin tube.

Fig. 3.6 displays the effect of curvature on the temperature profiles of GaAs/AlAs superlattices at L₁ = L₂ =10⁻⁷ m. Notably, the temperature profile approaches that for slab when the inner radius (r ) is large. In this case, no obvious _i difference exists between slab and r_i =10⁻⁵ m. This phenomenon can be explained

by Eq. (3-9). Once the term

η ϕ^ω

∂

∂I r

1 equals zero, Eq. (3-9) is reduced to the

∂I

calculating the heat transfer rate. According to the Fourier law, the effective thermal conductivity k_eff is defined as [93]

) (

2

) ln(

o i

i o T

eff T T

r r k Q

= −

π , (3-27)

where Q is the total net radial heat transfer. To show the effect of film thickness on _T thermal conductivity, the plane-parallel GaAs/AlAs superlattices are chosen for a demonstration. Furthermore, the numerical calculations are compared to the experimental data. Fig. 3.7 reveals that the thermal conductivity of GaAs/AlAs superlattices is smaller than its bulk value. Additionally, the thermal conductivity decreases with decreasing film thickness. The simulation results agree closely with the experimental data. The miniaturization of microelectronic devices reduces the heat transfer ability and causes device failure if the size effect is ignored. The size effect thus must be considered when designing a microelectronic device.

Fig. 3.8 illustrates the effect of curvature on the thermal conductivity of GaAs/AlAs superlattices. Three inner radii: ri =10⁻⁵ m, ri =10⁻⁶ m, and ri =10⁻⁷ m are chosen as examples. The thermal conductivity increases with the increasing curvature. Moreover, this phenomenon is amplified by large tube thickness. However, no obvious difference exists among these three cases when the tube is getting thinner.

When the film thickness is small compared to its inner radius, the phonon transport in the ultra-thin tube approaches that in the slab. Consequently, the effect of curvature on

resistance of GaAs/AlAs superlattices. The figure reveals that no visible discrepancy exists under three different inner radii, indicating that the effect of curvature on the interface thermal resistance of GaAs/AlAs superlattices is insignificant. Additionally, the size effect on the interface thermal resistance is also unimportant. The size and curvature effects on the interface thermal resistance are not significant if the diffuse mismatch model is employed to describe the interface condition.

Since the curvature effect on the interface is insignificant in this study (DMM is employed), a plane-parallel diamond thin film deposited on the silicon substrate is used to illustrate the behavior of the interface thermal resistance. Fig. 3.10 displays the comparison of interface thermal resistance for diamond/silicon with experimental data. The calculated interface thermal resistance is smaller than the experimental value. Since DMM is a simplified model, it can not completely describe the interface condition. The interface roughness, inelastic scattering resulting from the anharmonic interatomic force interaction, and the phonon mode conversion at the interface may cause diffuse scattering at the interface. Additionally, measurement errors may also contribute the discrepancy between the numerically predicted and experimental values.

All of them make the calculated interface thermal resistance lower than the experimental value.

Fig. 3.1 Schematic diagram of coaxially- gated nanowire transistors [91].

p-Si i-Ge SiOx

p-Ge

r _o

r _b r _i

I

₁ ^–

I

₁ ⁺

I

₂ ⁺

I

₂ ^–

T

₁

T

2

L

₁

L

₂

0 0.2 0.4 0.6 0.8 1 ( r - r

_i

) / ( r

_o

- r

_i

)

0 0.2 0.4 0.6 0.8 1

( T - T

o

) / ( T

i

- T

o

)

21 Grids 31 Grids 51 Grids

Fig. 3.3 Grid-refinement test for the numerical scheme.

0 0.4 0.8 1.2

( T - To ) / ( Ti - To )

t = 10^-13 (sec) t = 10^-12 (sec) t = 5x10^-12 (sec) steady state

0 0.2 0.4 0.6 0.8 1

0 0.4 0.8

( T - To ) / ( Ti - To )

t = 10^-11 (sec) t = 10^-10 (sec) t = 10^-9 (sec) steady state

(a)

(b)

0 0.2 0.4 0.6 0.8 1 ( r - r

_i

) / ( r

_o

- r

_i

)

0 0.4 0.8 1.2

( T - T

o

) / ( T

i

- T

o

)

L₁=L₂=10^-8 m L₁=L₂=10^-7 m L₁=L₂=10^-6 m

0 0.2 0.4 0.6 0.8 1 ( r - r

_i

) / ( r

_o

- r

_i

)

0 0.2 0.4 0.6 0.8 1

( T - T

o

) / ( T

i

- T

o

)

slab r_i=10^-5 m r_i=10^-6 m r_i=10^-7 m

1x10^-9 1x10^-8 1x10^-7 1x10^-6 FILM THICKNESS (m)

1 10 100

THERMAL CONDUCTIVITY (W/mK)

present results bulk (Chen, 1998) experiments

(Capinski and Maris, 1996)

Fig. 3.7 Effect of film thickness on the thermal conductivity of GaAs/AlAs superlattices.

present results bulk [52]

experiments [53]

1x10^-9 1x10^-8 1x10^-7 1x10^-6 FILM THICKNESS (m)

1 10 100

THERMAL CONDUCTIVITY (W/mK)

r_i = 10^-5 m r_i = 10^-6 m r_i = 10^-7 m

1x10^-9 1x10^-8 1x10^-7 1x10^-6 FILM THICKNESS (m)

1x10^-10 1x10^-9 1x10^-8

THERMAL RESISTANCE (m2K/W)

r_i = 10^-3 m r_i = 10^-5 m r_i = 10^-7 m

Fig. 3.9 Effect of curvature on the interface thermal resistance of GaAs/AlAs

1x10

^-7

1x10

^-6

FILM THICKNESS (m)

1x10

^-10

1x10

^-9

1x10

^-8

1x10

^-7

1x10

^-6

1x10

^-5

T H E R M A L R E S IS T A N C E ( m

2

K /W )

experiments

(Goodson et al., 1995) present results

experiments [94]

4. MICROSCALE HEAT TRANSFER IN

在文檔中 固態層狀結構中微觀熱傳現象之分析 (頁 70-90)