3-1 Simulation Conditions of one source
In order to understand the transport phenomena, we simulate a simple case first, one source in the chamber. Taking computational efficiency and simulation accuracy into account, we select the quarter half sphere for simulation region. The geometry is shown in Fig. 3.1. The source is in the center of the simulation region with 200 cm in diameter.
The metal put on the source is titanium.
Because we focus on evaporation behavior above the source, the unstructured mesh in the source is denser than other region. The mesh size observe the rule that it should double or triple larger than mean free path, which defined 2
2
σ is collision diameter. Then, we apply DSMC code to model the flow condition. To sample the flow properties, we take an appropriate interval of time step that flow has already reach steady state and then average the flow properties by statistic concept. During each time step, motion and collisions are assumed to be decoupled such that particles are generated at the source and moved according to their velocities and boundary interactions. Collisions are calculated and particles are given new velocity.
A DSMC simulation predicts the trajectories and collisions of representative particles, each of which represents large numbers of atoms. Usually, we decide how many particles to simulate, based on how many cells in the flow condition. Around 10 or 20 particles in each cell are acceptable. Particles are introduced into the cell from the source surface at a rate given by the Langmuir equation. These particles are given a
random velocity whose magnitude follows a Maxwellian distribution (i.e. kinetic energy follows a Boltzmann distribution) such that average velocity is a function of source temperature, and whose direction follows a cosine distribution. Pairs of particles within a cell are chosen randomly, their collision probability calculated based on their relative velocities, and if a random variable is chosen which satisfies that probability, they are collided and given new velocities, which are given by various collision models. For the purpose of determining collisions and resulting velocities, particles are modeled as hard sphere. Other collision models include the variable hard sphere model, in which collision diameter depends on relative velocity, and soft sphere models involving energy transfer between kinetic energy and the internal vibrational, rotational and electronic energies. In this study, the model we select is hard sphere.
3-1-1 Verification of Flux Distribution
To verify if the flux distribution will obey cosine distribution, we select five different source diameters for simulation. The five diameters are 0.2 cm, 2.0 cm, 4.0 cm, 8.0 cm and 16.0 cm, which corresponding the values of d/λ0 are 0.26, 2.61, 5.22, 10.44 and 20.89. The source temperature and vapor pressure are 2050 K and 4.5 Pa. As saying above, in the ideal condition when source size becomes smaller, the flux distribution will tend to approximate consine distribution. As a result, Fig. 3.2 shows the normalized vapor flux distributions from titanium disk sources at 2050 K with four different diameters, and cos(θ) for reference. It has been speculated that the extent of focusing should depend only on the ratio of source diameter to equivalent mean free path
/λ0
d [Powell, 1997], as discussed in chapter 1. By using the DSMC method to calculate
vapor flux profiles in titanium evaporation, the mechanism of vapor plume focusing toward the normal has been confirmed to be based on the collisions between evaporant atoms in the dense region immediately above the source.
3-1-2 Variations of Inflow Field
Fig. 3.3 and Fig. 3.4 show the contour about density, temperature. The simulated results correspond to general expectation. Compared with simulation condition, the percentage of error is small. The value of density and temperature close to the source is higher than others. As far away the source, the value of density and temperature decreases gradually. From Fig. 3.5, when it comes to velocity distribution, z-directional velocity is needed to pay attention. Because of great pressure difference between source and chamber, the atoms evaporated from the source move rapidly. According to compressible fluid theorem, if there is too much pressure difference, the flow condition will become supersonic flow. As a result, the value of velocity can be calculated. Fig. 3.6 shows the velocity vector in the chamber. Obviously, the greater parts of atoms leave the source at a strongly off-normal angle.
The calculations presented here were performed under relatively clean conditions, that is, with uniform temperature distribution on the source, effects of beam scan rate ignored, and the background gas neglected. In order to make this study more closely simulate the conditions in an actual evaporation system, however, we should take these factors into consideration. For this reason, we will add the background gas effect into simulation in the next section.
3-1-3 Comparison of Simulations with and without Background Gas
we investigate the following cases based on dimensions that company SYSKEY provided. The chamber is shown in Fig. 3.7, with 32 cm in length, 32.5 cm in width. The height of the substrate can be adjusted upward or downward. In this study, the height of the substrate is 27.5 cm. The diameter of the substrate and source are 7.62 cm and 1.5 cm, respectively. The chamber temperature is 300 K, and source temperature 2050 K. The metal put on the source is titanium. The chamber temperature and pressure are 300 K and vacuum condition. A parameter named sticking coefficient is fixed for 1 such that the particles vaporized from the source can stick on the chamber wall or substrate completely.
For the purpose of obtaining preliminary reorganization of behavior in the chamber, I take the same way of simplifying the geometry of the chamber. The simulation domain contains one source and one substrate. All of them are quarter of real body. Under the condition of vacuum in the chamber, the distribution of density, temperature and z-directional velocity are shown in Fig. 3.8. At this time we are curious that if we fill the chamber with the background gas, what will happen. From Fig. 3.9, it seems obvious to see the differences from the case which background gas is air (80% N2 + 20% O2), and background pressure torr. The reason why the distribution of temperature and velocity differ from the vacuum condition is that when atoms of evaporant leave the source, the background gas will be compressed. As a result, background gas will give resistance to keep evaporated atoms from going forward. Once this occurs, it will appear the phenomena of shock wave. According to compressible fluid theorem, if there is shock wave in the flow field, the temperature and pressure will increase rapidly at this shock region. Contrarily, velocity will decrease rapidly. Therefore, we can explain the sharp
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change on such region above the source. Fig. 3.10 shows us the change of pressure will cause different results. As the background pressure is lower, the effect of shock wave becomes not obvious.
Fig. 3.11 shows the number of deposited atoms comparison with and without background gas. We can see that the number of deposited atoms on the same position is less when the background gas is considered. That is because the atoms in the vacuum condition will spray under faster speed and have more opportunities to deposit on the substrate.
The influence with and without background gas is studied, and the properties of distribution of energy and incident angle are now investigated. The specific cells of the substrate using for sampling unit are labeled in Fig. 3.12. As we discuss the influences on distribution of energy and incident angle due to background gas, the direct thought is that the evaporated atoms will collide with background gas unceasingly. Consequently, as evaporated atoms strike on the substrate, it is expected that energy will become smaller when velocity becomes slower. We can verify this statement from Fig. 3.13. The incident angle distribution is shown in the Fig. 3.14 as the background pressure torr is considered. The differences between comparing with vacuum condition are not obvious just because the pressure is too rare.
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11 .
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3-2 Simulation Conditions of Multiple Sources
The thesis objective is to model the multiple sources of EBPVD. There is a titanium alloy applied widely for semiconductor and aerospace industry based on its excellent properties. This common alloy is Ti6Al4V composed of Ti, Al and V, which the weight
ratio is about 0.9, 0.06 and 0.04, respectively. The properties of individual atom are shown in Table I. The composition of Ti6Al4V is shown in Table II [35]. Its typical physical properties and mechanical properties are shown in Table III [35] and Table IV [35].
Because we want to simulate the forming of alloy, the accurate composition deposited on the substrate is needed to control well. In order to fully mixing the evaporated atoms, the position relation between three sources is designed to triangle. The distance is 1.732 cm between two sources. Fig. 3.15 shows the relative position of three sources and substrate in the simulation domain. Once sources are heated, evaporated atoms will diffuse everywhere. Fig. 3.16 shows the position of evaporated atoms at a specific steady time step.
3-2-1 Uniformity of Composition
From Table II, the content of Ti6Al4V is mostly composed of Ti, Al and V. The reasonable range of weight ratio of aluminum is from 0.055 to 0.0676, and vanadium is from 0.035 to 0.045. If we want to deposit the alloy which corresponds to composition like that, the number ratio of individual atoms evaporated from the source should also correspond to the ratio described above. From the Fig. 3.17[Powell, 1997], we can calculate the number of atom of each evaporant according to evaporation rate by assigning source temperature. In this study, as background pressure is torr, the source temperature of titanium, aluminum and vanadium is set 2000℃, 1000℃ and 1880℃. The deposited distributions of individual evaporated atom are shown in Fig. 3.18.
We can see the value of distributions falling in the acceptable range. More strictly, we set 10 5
11 .
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some value as a benchmark to show the distribution of error percentage, as displayed in the Fig. 3.19. These values for titanium, aluminum and vanadium are 0.9, 0.06 and 0.04, respectively. As a result, we can use the same way to show the uniformity of composition in the following case.
3-2-2 Uniformity of Thickness
In addition to discuss the uniformity of composition, another subject we need to focus on is the uniformity of thickness. The uniform thickness on the substrate is a significant factor in the semiconductor. Nowadays, for the purpose of reducing cost, the semiconductor manufacturers are making efforts in developing larger wafer’s dimension.
Therefore, the uniformity of thickness will relate to the yield of product directly.
The Ti6Al4V alloy is mostly composed of titanium. Based on this reason, we suppose that the crystal lattice structure of Ti6Al4V will be the same with titanium. And the crystal lattice structure of titanium is Hexagonal Closest Packing (HCP). The method to calculate the thickness is based on the equation:
i crystal lattice structure, is the lattice points per crystal lattice structure. The value of
Wi
i Acls c
ncls
ac for titanium is 1.586, where is the distance between adjacent lattice points in the basal plane.
a
The simulation condition is the same described in the last section. The sample starts from steady condition. It takes 36000 time step in total. The distribution of thickness on the substrate is shown in the Fig. 3.20.
3-2-3 Growth Rate
The growth rate is governed by other process parameters such as evaporation rate, background pressure, substrate temperature, and so on. The deposited thickness is calculated based on crystal lattice structure. We gather statistics at time step 5000, 15000, 25000 and 35000. The performance coefficients are identical to the condition described above. Fig. 3.21 shows the distributions of growth rate in the period that we assign. The unit of growth rate here is m/s. We can see that the growth rate does not change obviously because it has already reached the steady condition in the chamber when gathering statistics.
3-3 Uniformity Discussion of Different Parameters
In this section, we try to change some parameters to discuss the uniformity problem.
The uniformity of composition and thickness is the subjects that we need to compare. The parameters are inclusive of the altitude of the substrate, background pressure and distance between the sources. For each case, we will show the table listed maximum value, minimum value and average value of composition of Ti6Al4V. We also select different error percentages to obtain the ratio of acceptable value to total value. The acceptable value is defined that if it is in the interval of the error percentage we assign. The calculated results are also listed in the same table. The error percentage of titanium,
aluminum, vanadium and thickness is assigned as ±0.2%, ±1.5%, ±2% and ±10%, respectively.
3-3-1 Altitude of the Substrate
Contrasted with the original simulated case that the substrate is located at a height of 27.5cm, we try to adjust the altitude of the substrate for the two cases of 10 cm and 40 cm.
In the Fig. 3.22 we show the composition and thickness to compare with each other. All of the performance coefficients are the same. Table V shows the relevant value on the substrate for each case.
We can see that the distribution of composition will be better when the substrate is located at appropriate altitude. In other words, the altitude of the substrate can not be neither too high nor too low. The appropriate altitude should be decided according to experimental results. On the other side, the growth rate becomes slower because of the increasing altitude.
3-3-2 Background Pressure
As discussed if the background pressure will make influences on composition and thickness, we select three different background pressures to check it. The simulated background pressure is torr, torr and vacuum. The composition and thickness are shown in the Fig. 3.23.
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It seems that the distribution of composition will be more uniform when the background pressure is rare. On the other hand, the background pressure effect doesn’t make more influence on the distribution of thickness. Table VI shows the relevant value
on the substrate for each case.
3-3-3 Distance between the Sources
Another parameter we need to discuss is the distance between the sources. There are three cases which the distance is 1.732 cm, 4.330 cm and 6.928 cm, respectively. The results are shown in the Fig. 3.24.
We observe that the composition of aluminum and vanadium exceeds the expected value of Ti6Al4V gradually when the distance between sources is farther. The thickness is also thinner due to evaporated characteristic. Table VII shows the relevant value on the substrate for each case.