# Transport Properties

The thermophysical properties of various gaseous precursors are necessary to utilize the transport models outlined above. Since the temperature varies significantly throughout a CVD system, the transport phenomena must be modeled using variable thermal physical properties. This requires knowledge of the dependence of the thermophysical properties on temperature. The thermophysical properties of the commonly used gas(es) in CVD are tabulated in Table 1.

The viscosity and thermal conductivity of some precursors that are not readily available can be estimated using the method recommended by Bird et al. (2002). The viscosity is

$\mu =2.6693\times {{10}^{-6}}\frac{\sqrt{MT}}{{{\sigma }^{2}}{{\Omega }_{\mu }}} \qquad \qquad(1)$

Table 1 Transport properties of the common gas(es) for CVD

 Properties Gas(es) c0 c1 c2 μa (N-s/m2) TMGa -1.15 × 10-6 3.35 × 10-8 -6.68 × 10-12 AsH3 -4.32 × 10-7 5.94 × 10-8 -1.46 × 10-11 H2 2.63 × 10-6 2.22 × 10-8 -5.19 × 10-12 N2 4.93 × 10-6 4.55 × 10-8 -1.08 × 10-11 SiH4 1.47 × 10-6 3.66 × 10-8 -6.81 × 10-12 ka(W/m-K) TMGa -3.52 × 10-3 3.85 × 10-5 -3.84 × 10-8 AsH3 -7.16 × 10-3 6.53 × 10-5 -3.47 × 10-9 H2 5.77 × 10-2 4.43 × 10-4 -7.54 × 10-8 N2 8.15 × 10-3 6.24 × 10-5 -4.48 × 10-9 SiH4 -2.12 × 10-2 1.45 × 10-4 -1.31 × 10-8 cpa (kJ/Kg-K) TMGa 5.40 × 102 1.60 0 AsH3 2.45 × 102 1.08 × 100 -4.24 × 10-4 H2 1.44 × 104 -2.61 × 10-1 8.67 × 10-4 N2 1.03 × 103 4.58 × 10-3 1.34 × 10-4 SiH4 4.74 × 102 3.26 -1.08 × 10-3 D12b (m2/s) SiH4, N2 -9.64 × 10-1 6.25 × 10-3 8.50 × 10-6 SiH4, H2 -2.90 2.06 × 10-2 2.81 × 10-5 N2, H2 -3.20 2.44 × 10-2 3.37 × 10-5 TMGa, H2 -1.87 1.64 × 10-2 3.13 × 10-5 TMGa, N2 -4.17 × 10-1 2.89 × 10-3 4.93 × 10-6 AsH3, H2 -2.26 1.73 × 10-2 2.80 × 10-5 AsH3, N2 -6.15 × 10-1 4.57 × 10-3 7.49 × 10-6 TMGa, AsH3 -2.26 × 10-1 1.27 × 10-3 3.18 × 10-6 k12Tc H2,SiH4 -2.74 × 10-1 -1.70 -6.35 × 10-3 N2,SiH4 -5.15 × 10-2 -1.69 -4.94 × 10-3 H2,N2 -2.71 × 10-1 -1.61 -9.15 × 10-3 TMGa, H2 1.32 -1.54 -3.57 × 10-3 TMGa, N2 6.36 × 10-1 -1.58 -3.36 × 10-3 AsH3,H2 8.86 × 10-1 -1.57 -4.35 × 10-3 AsH3, N2 3.09 × 10-1 -1.55 -4.06 × 10-3 TMGa, AsH3 1.94 × 10-1 -1.79 -1.91 × 10-3

a μ,k,cp = c0 + c1T + c2T2; b ${{D}_{12}}={{D}_{21}}=\left( {{c}_{0}}+{{c}_{1}}T+{{c}_{2}}{{T}^{2}} \right)/p$; c $k_{12}^{T}=-k_{12}^{T}={{c}_{0}}{{x}_{1}}{{x}_{2}}\left[ 1+{{c}_{1}}\exp ({{c}_{2}}T) \right],\text{ for }{{\text{x}}_{\text{1}}}\to 0.$ T is absolute temperature (K);

where M is the molecular mass and σ is the collision diameter (&Aring= 10-10 m) of the molecule that can be estimated by

$\sigma =0.841\bar{v}_{c}^{1/3} \qquad \qquad(2)$

or

$\sigma =1.166\bar{v}_{b,liq}^{1/3} \qquad \qquad(3)$

where ${{\bar{v}}_{c}}$ and ${{\bar{v}}_{b,liq}}$ are the specific volumes (cm3/mol) of the precursor at critical point, and of the saturated liquid at normal boiling point, respectively.

The collision integral Ωμ in eq. (1) is a slowly-varying function of dimensionless temperature, ${{k}_{b}}T/\varepsilon$, and is tabulated in Table 2. kb is the Boltzmann constant and $\varepsilon$ is a characteristic energy of interaction between molecules which can be estimated by

$\frac{\varepsilon }{{{k}_{b}}}=0.77{{T}_{c}} \qquad \qquad(4)$

or

$\frac{\varepsilon }{{{k}_{b}}}=1.15{{T}_{b}} \qquad \qquad(5)$

where Tc and Tb are critical temperature and normal boiling point, respectively. The thermal conductivity of the polyatomic gas is related to its viscosity by

$k=\left( {{c}_{p}}+\frac{5}{4}{{R}_{g}} \right)\mu \qquad \qquad(6)$

where Rg is the gas constant.

Table 2 Dependence of collision integral Ωμ on dimensional temperature ${{k}_{b}}T/\varepsilon$

 kbT / σ Ωμ ΩD,12 kbT / σ Ωμ ΩD,12 kbT / σ Ωμ ΩD,12 0.30 2.840 2.649 1.70 1.249 1.141 4.2 0.9598 0.8748 0.35 2.676 2.468 1.75 1.235 1.128 4.3 0.9551 0.8703 0.40 2.531 2.314 1.80 1.222 1.117 4.4 0.9506 0.8659 0.45 2.401 2.182 1.85 1.209 1.105 4.5 0.9462 0.8617 0.50 2.284 2.066 1.90 1.198 1.095 4.6 0.9420 0.8576 0.55 2.178. 1.965 1.95 1.186 1.085 4.7 0.9380 0.8537 0.60 2.084 1.877 2.00 1.176 1.075 4.8 0.9341 0.8499 0.65 1.999 1.799 2.10 1.156 1.058 4.9 0.9304 0.8463 0.70 1.922 1.729 220 1.138 1.042 5.0 0.9268 0.8428 0.75 1.853 1.667 2.30 1.122 1.027 6.0 0.8962 0.8129 0.80 1.790 1.612 2.40 1.107 1.013 7.0 0.8727 0.7898 0.85 1.734 1.562 2.50 1.0933 1.0006 8.0 0.8538 0.7711 0.90 1.682 1.517 2.60 1.0807 0.9890 9.0 0.8380 0.7555 0.95 1.636 1.477 2.7 1.0691 0.9782 10.0 0.8244 0.7422 1.00 1.593 1.440 2.8 1.0583 0.9682 12.0 0.8018 0.7202 1.05 1.554 1.406 2.9 1.0482 0.9588 14.0 0.7836 0.7025 1.10 1.518 1.375 3.0 1.0388 0.9500 16.0 0.7683 0.6878 1.15 1.485 1.347 3.1 1.0300 0.9418 18.0 0.7552 0.6751 1.20 1.455 1.320 3.2 1.0217 0.9340 20.0 0.7436 0.6640 1.25 1.427 1.296 3.3 1.0139 0.9267 25.0 0.7198 0.6414 1.30 1.401 1.274 3.4 1.0066 0.9197 30.0 0.7010 0.6235 1.35 1.377 1.253 3.5 0.9996 0.9131 35.0 0.6854 0.6088 1.40 1.355 1.234 3.6 0.9931 0.9068 40.0 0.6723 0.5964 1.45 1.334 1.216 3.7 0.9868 0.9008 50.0 0.6510 0.5763 1.50 1.315 1.199 3.8 0.9809 0.8952 75.0 0.6140 0.5415 1.55 1.297 1.183 3.9 0.9753 0.8897 100.0 0.5887 0.5180 1.60 1.280 1.168 4.0 0.9699 0.8845 1.65 1.264 1.154 4.1 0.9647 0.8796

For a mixture of different gases, as is usually the case in CVD processes, the viscosity and the thermal conductivity of the mixture are related to those of the individual components by

$\mu =\sum\limits_{i=1}^{N}{\frac{{{x}_{i}}{{\mu }_{i}}}{\sum\nolimits_{j=1}^{N}{{{x}_{i}}{{\phi }_{ij}}}}} \qquad \qquad(7)$

$k=\sum\limits_{i=1}^{N}{\frac{{{x}_{i}}{{k}_{i}}}{\sum\nolimits_{j=1}^{N}{{{x}_{i}}{{\phi }_{ij}}}}} \qquad \qquad(8)$

where

${{\phi }_{ij}}=\frac{1}{\sqrt{8}}{{\left( 1+\frac{{{M}_{i}}}{{{M}_{j}}} \right)}^{-1/2}}{{\left[ 1+{{\left( \frac{{{\mu }_{i}}}{{{\mu }_{j}}} \right)}^{1/2}}{{\left( \frac{{{M}_{j}}}{{{M}_{i}}} \right)}^{1/4}} \right]}^{2}} \qquad \qquad(9)$

The specific heat of the gaseous mixture is related to those of the individual components by

${{c}_{p}}=\sum\limits_{i=1}^{N}{{{x}_{i}}{{c}_{p,i}}} \qquad \qquad(10)$

For applications that involve unknown mass diffusivity, it can be estimated by

${{D}_{12}}=1.8583\times {{10}^{-7}}\frac{\sqrt{{{T}^{3}}\left( M_{1}^{-1}+M_{2}^{-1} \right)}}{p\sigma _{12}^{2}{{\Omega }_{D,12}}} \qquad \qquad(11)$

where the unit for pressure is atm and

${{\sigma }_{12}}=\frac{1}{2}\left( {{\sigma }_{1}}+{{\sigma }_{2}} \right) \qquad \qquad(12)$

ΩD,12 is a function of ${{k}_{b}}T/{{\varepsilon }_{12}}$ that can be obtained from Table 2 using

${{\varepsilon }_{12}}=\sqrt{{{\varepsilon }_{1}}{{\varepsilon }_{2}}} \qquad \qquad(13)$

The concentration of the reactant is usually much lower than that of the carrier gas(es). When the reactant is a single gas diluted by the carrier gas, the diffusivity of the reactant to the carrier gaseous mixture is of interest. If the reactant is defined as component 1 in the precursor, and the carrier gases are components 2 through N, the diffusivity of the reactant – 1 – to the carrier gas mixture – m – can be obtained by

$\frac{1-{{x}_{1}}}{{{D}_{1m}}}=\sum\limits_{j=2}^{N}{\frac{{{x}_{j}}}{{{D}_{1j}}}} \qquad \qquad(14)$

## References

Bird, R.B., Stewart, W.E., and Lightfoot, E.N., 2002, Transport Phenomena, 2nd ed., John Wiley and Sons, New York.

Mahajan, R.L., 1996, “Transport Phenomena in Chemical Vapor-Deposition Systems,” Advances in Heat Transfer, Academic Press, San Diego, CA.