Transport Properties

From Thermal-FluidsPedia

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
$μ$ ^a (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
k^a(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
c_p^a (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
D₁₂^b (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
k₁₂^Tc	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 $μ, k, c p = c 0 + c 1 T + c 2 T 2$ ; 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

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

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

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

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

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

where $T c$ and $T b$ 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 $R g$ is the gas constant.

Table 2 Dependence of collision integral

Ω μ

on dimensional temperature ${{k}_{b}}T/\varepsilon$

$k b T / σ$	$Ω μ$	$Ω D,12$	$k b T / σ$	$Ω μ$	$Ω D,12$	$k b T / σ$	$Ω μ$	$Ω D,12$
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

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

$Ω D,12$ 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, 2^nd 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.

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Transport Properties

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