Chapter 2: Graphene
2.3. Preparation of graphene
2.3.6. Basic principle of CVD process
CVD is one of the effective way for depositing thin films, which can be also fabricated by evaporation, sputtering, electroplating, etc. The advantage of CVD method in the film deposition is the superior conformity and large area availability, although the cost of CVD is not the lowest. One typical CVD system is mainly composed of quartz tube with good leakproofness, gas flow controllers which inject the precursor of deposition to the chamber accurately by the set value, a temperature controller that can precisely control the temperature, and a vacuum pump that can eject the residual gas or air remaining inside the tube or provides a low-pressure reaction environment.
a
b c
d e f
31
We often use “sccm” for the gas measurement. “sccm” is a volume unit which means “standard cubic centimeters per minute”. The standard conditions here are 1 standard atmosphere pressure and 0 ℃.According to the ideal gas law: PV=nRT, P is the pressure, V the volume, n the number of moles, R the universal gas constant (~8.314 J mol-1 K-1) and T the absolute temperature. Then I can calculate that 1 sccm contains ~7.4×10-7 moles per second. As the actual operation condition of CVD in ambient is close to the standard condition, the value still can be used by ignoring the thermal expansion.
The velocities of molecules in an ideal gas obey a Maxwell distribution and the mean velocity is:
𝐶
𝑚𝑒𝑎𝑛= √
8𝑘𝑇𝜋𝑚 (2.10)
Bear in mind that the direction of the velocity is random, thus I can assume half of gas molecules will not heat the wall of tube: nCmean/2. For the gas molecules strike to the wall can also be divided into the direct movement and perpendicular to the wall. Therefore, I can describe the flux of molecules to the wall as:
𝐽 =
𝑛𝐶𝑚𝑒𝑎𝑛4
= 3.51 × 10
22 𝑃√𝑀𝑇 (2.11)
where n is N/V, V is the volume and N is the molecules. The unit of J is in mole cm-2 sec, P in Torr and M in g mole-1.
The mean free path (λ) is the average distance traveled by a moving particle without striking, which plays an important role in the determination of how fast transport mass, energy and momentum happen in the gas.
It can be calculated by:
λ =
1√2𝜋𝑎2𝑛 (2.12) where πa2 is the cross-section area.
The flux can be converted to an equivalent “deposition rate”:
𝑅[A/min] =
𝐾𝑠1.7×10−10𝜌𝑚
[𝑝𝑟𝑒]
𝑐ℎ (2.13)32
where Ks is the rate constant (=exp(-Ea/RT), cm s-1), [pre]ch is the species concentration, 𝜌𝑚 is the molar density of the deposited film.
Next let’s have an investigation of the CVD reactor. To simplify, here all gas concentrations are taken to be constant at all locations in the reactor, corresponding parameters are shown in Figure 2.15. According to the law of conservation of mass:
F
in[pre]
in=K
sS[pre]
ch+F
in[pre]
ch (2.14)[𝑝𝑟𝑒]
𝑐ℎ=
𝐹𝑖𝑛[𝑝𝑟𝑒]𝑖𝑛𝐹𝑖𝑛+𝐾𝑠𝑆
(2.15)
The time of a gas molecule stays in the chamber are be expressed as residence time
t
res: 𝑡
𝑟𝑒𝑠=
𝑉𝐹𝑖𝑛 (2.16)
The consumption file time
t
con shows the average time of a molecule survived before reacted with the substrate:𝑡
𝑐𝑜𝑛=
𝑉𝐾𝑠𝑆 (2.17) Therefore, formula 2.15 can be rewritten as
[𝑝𝑟𝑒]
𝑐ℎ=
[𝑝𝑟𝑒]𝑖𝑛1+𝑡𝑟𝑒𝑠/𝑡𝑐𝑜𝑛
(2.18)
Figure 2.15 CVD reactor and terminology.91
33
In the case of very low surface reaction (t
res <<t
con), according to 2.18,[pre]
ch≈[pre]
in.
Thus the deposition reaction is only limited by the rate of the surface reaction.On the other hand, very fast surface reaction (
t
res >>t
con)
, eq. 2.18 can be rewritten as:[𝑝𝑟𝑒]
𝑐ℎ= [𝑝𝑟𝑒]
𝑖𝑛𝑡𝑐𝑜𝑛𝑡𝑟𝑒𝑠
(2.19)
This means the rate determined by the supply of reactant to the surface, which is transport limitation. In this case, the rate of surface reaction can be:
𝑅 = 𝐾
𝑠[𝑝𝑟𝑒]
𝑖𝑛𝑡𝑐𝑜𝑛𝑡𝑟𝑒𝑠
= [𝑝𝑟𝑒]
𝑖𝑛𝐹𝑖𝑛𝑆 (2.20)
Thus it can be seen that the reaction rate is independent of temperature but relies on the inlet flow rate.
For a typical CVD condition of graphene growth on Cu foil in this thesis, the volume of graphene heating zone (V) is π× 22×30=376.8 cm3. The input volume flow is 300 sccm=5 cm3 s-1 at 760 Torr. The temperature is 1075 ℃=1348 K. Thus the volume expand to 5× (1348/273) =24.7 cm3 s-1 . Therefore, the residence time can be estimated to
t
res=V/F
in=376.8/24.7=15.2 seconds. In this thesis, CH4 is used as a carbon source and the typical mole fraction is 15 ppm (seen as an ideal gas).The surface area S can be estimated by the area of quartz holder, where Cu foil is put on. The length is 5 cm and worth is 3.5 cm. Thus S is 17.5 cm2 and [pre]in=4.5×10-5 moles/cm3×760× (273/1348) ×15×10
-6=1.04×10-7 moles cm-3. For the reaction rate constant Ks= exp(-Ea/RT). Ea can be estimated to ~2.6 eV from the previous reported.86 Thus Ks≈1 cm/s and the consumption lifetime
t
on=V/K
sS
=376.8/17.5=21.5 seconds, which is close tot
res.
This means the as-designed graphene growth condition here is high efficient for over half of the precursor will be utilized to form graphene film on Cu.34
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