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Annealing, an important step in semiconductor materials processing, can be accomplished by rapidly heating the silicon wafer to a high temperature for a short period of time. The schematic shows a method involving the use of a hot plate operating at an elevated temperature \(T_{h}\). The wafer, initially at a temperature of \(T_{w, i}\), is suddenly positioned at a gap separation distance \(L\) from the hot plate. The purpose of the analysis is to compare the heat fluxes by conduction through the gas within the gap and by radiation exchange between the hot plate and the cool wafer. The initial time rate of change in the temperature of the wafer, \(\left(d T_{w} / d t\right)_{i}\), is also of interest. Approximating the surfaces of the hot plate and the wafer as blackbodies and assuming their diameter \(D\) to be much larger than the spacing \(L\), the radiative heat flux may be expressed as \(q_{\text {rad }}^{\prime \prime}=\sigma\left(T_{h}^{4}-T_{w}^{4}\right)\). The silicon wafer has a thickness of \(d=0.78 \mathrm{~mm}\), a density of \(2700 \mathrm{~kg} / \mathrm{m}^{3}\), and a specific heat of 875 \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\). The thermal conductivity of the gas in the gap is \(0.0436 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). (a) For \(T_{h}=600^{\circ} \mathrm{C}\) and \(T_{w, i}=20^{\circ} \mathrm{C}\), calculate the radiative heat flux and the heat flux by conduction across a gap distance of \(L=0.2 \mathrm{~mm}\). Also determine the value of \(\left(d T_{\mathrm{w}} / d t\right)_{i}\), resulting from each of the heating modes. (b) For gap distances of \(0.2,0.5\), and \(1.0 \mathrm{~mm}\), determine the heat fluxes and temperature-time change as a function of the hot plate temperature for \(300 \leq\) \(T_{h} \leq 1300^{\circ} \mathrm{C}\). Display your results graphically. Comment on the relative importance of the two heat

Step by step solution

01

a) Calculation of radiative heat flux

Given the temperatures of the hot plate and wafer, we can calculate the radiative heat flux using the provided formula: \(q_{rad}'' = \sigma(T_h^4 - T_w^4)\), where: \(q_{rad}''\) = radiative heat flux (\(W/m^2\)) \(\sigma\) = Stefan-Boltzmann constant = \(5.67 \times 10^{-8} W/m^2K^4\) \(T_h\) = hot plate temperature = \(600 + 273.15 = 873.15 K\) \(T_w\) = wafer temperature = \(20 + 273.15 = 293.15 K\) Plugging in the values into the formula, we obtain: \(q_{rad}'' = 5.67 \times 10^{-8}(873.15^4 - 293.15^4) \approx 10055.9 W/m^2\)

02

a) Calculation of heat flux by conduction

The heat flux by conduction, \(q_{cond}''\), can be calculated using Fourier's Law of conduction: \(q_{cond}'' = \frac{-k(T_h - T_w)}{L}\), where: \(q_{cond}''\) = heat flux by conduction (\(W/m^2\)) \(k\) = thermal conductivity of the gap gas = \(0.0436 W/m K\) \(L\) = gap distance = \(0.2 mm = 0.0002 m\) Plugging in the values, we obtain: \(q_{cond}'' = \frac{-0.0436 (873.15 - 293.15)}{0.0002} \approx 603665.93 W/m^2\)

03

a) Calculation of the initial rate of temperature change for radiative heating

To calculate the initial rate of temperature change (\(\frac{d T_w}{d t}_i\)) due to radiative heating, we will use the following formula: \(\frac{d T_w}{d t}_i = \frac{q_{rad}''}{\rho C_p d}\), where: \(\rho\) = density of the silicon wafer = \(2700 kg/m^3\) \(C_p\) = specific heat of the silicon wafer = \(875 J/kg K\) \(d\) = thickness of the wafer = \(0.78 mm = 0.00078 m\) Plugging in the values, we obtain: \(\frac{d T_w}{d t}_i = \frac{10055.9}{2700 \times 875 \times 0.00078} \approx 5.398 K/s\)

04

a) Calculation of the initial rate of temperature change for conductive heating

For the heat flux by conduction, we need to use the same formula as above with the only difference being substituting \(q_{cond}''\) instead of \(q_{rad}''\): \(\frac{d T_w}{d t}_i = \frac{q_{cond}''}{\rho C_p d}\), Using the values obtained earlier, we get: \(\frac{d T_w}{d t}_i = \frac{603665.93}{2700 \times 875 \times 0.00078} \approx 326.174 K/s\)

05

b) Heat fluxes and temperature-time change for varying gap distances and hot plate temperatures

To determine the heat fluxes and temperature-time change as a function of gap distance and hot plate temperature, you should create two matrices (one for radiative heat flux and one for conductive heat flux). The rows of these matrices should correspond to the different gap distances (0.2 mm, 0.5 mm, and 1.0 mm), and the columns should correspond to the different hot plate temperatures from 300°C to 1300°C (step size 100°C or any suitable interval). Then, fill the matrices by calculating the heat fluxes using the formulas from parts (a) for each combination of gap distance and hot plate temperature. After that, create another set of matrices for temperature-time change calculations using the formula from part (a) with corresponding heat flux values from the first matrices. Once you've completed the matrices, plot the resultant heat flux and temperature-time change values as separate graphs, with hot plate temperature on the x-axis and heat flux/temperature-time change on the y-axis. Display separate lines/curves for each gap distance. Then, analyze the graphs to understand the relative importance of radiation and conduction heat transfer mechanisms and their effects on the heating of the silicon wafer. Remember to label the axes and provide a legend to indicate the gap distances in each plot. You should observe patterns that help explain how the two heat transfer mechanisms contribute differently to the heating process for varying gap distances and hot plate temperatures.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radiative Heat Flux Calculation

When we talk about the annealing of semiconductor materials, one key element that comes into play is radiative heat transfer. This process involves the exchange of thermal energy through electromagnetic waves, and it's particularly significant in scenarios where the objects are not in direct contact or when the intervening medium has a low thermal conductivity coefficient. In the case of semiconductor annealing where a silicon wafer is placed near a hot plate, radiative heat transfer is crucial.

The radiative heat flux can be calculated using the following relationship:
\[\begin{equation}q_{rad}'' = \text{Stefan-Boltzmann constant} \times (T_{\text{hot}}^4 - T_{\text{wafer}}^4)\end{equation}\]The Stefan-Boltzmann constant (\(\sigma\)) is a fundamental physical constant that quantifies how much radiant energy is emitted by a blackbody per unit area based on its temperature. In this specific exercise, we assume the surfaces of the hot plate and the wafer act as blackbodies. By substituting the given temperatures (in Kelvin) and the Stefan-Boltzmann constant into the equation, one can determine the radiative heat flux, a critical value for understanding the heat transfer experienced by the wafer.

Conductive Heat Flux Calculation

While radiative heat transfer is important in a vacuum or where the medium's thermal properties are negligible, conductive heat transfer becomes significant when there's a medium between the two objects. In the context of semiconductor annealing, the conduction aspect of heat transfer occurs through the gas that fills the gap between the wafer and the hot plate.

Fourier's Law of conduction is the guiding principle for this type of heat transfer, which can be mathematically expressed as:
\[\begin{equation}q_{\text{cond}}'' = -k \frac{T_h - T_w}{L}\end{equation}\]Where:

• \(q_{\text{cond}}''\) is the conductive heat flux,
• \(k\) is the thermal conductivity of the gas,
• \(T_h\) and \(T_w\) are the temperatures of the hot plate and the wafer, respectively,
• \(L\) is the gap distance.

To determine the conductive heat flux, we should input the corresponding temperatures, the thermal conductivity of the intervening gas, and the separation distance between the wafer and the hot plate into the above formula. This allows us to measure the rate at which heat is transferred by conduction through the gas.

Temperature Rate of Change

In semiconductor manufacturing, understanding how rapidly a wafer's temperature changes is critical to control the annealing process's quality. The temperature rate of change is indicative of how fast the wafer is heating up or cooling down.

To determine this rate due to heat transfer—whether by radiation or conduction—we can use the following formula based on a balance of energy:
\[\begin{equation}\left(\frac{d T_w}{d t}\right)_i = \frac{q''}{\rho C_p d}\end{equation}\]Here, \(q''\) represents either the radiative or conductive heat flux calculated earlier, \(\rho\) is the material's density, \(C_p\) is the specific heat capacity, and \(d\) is the thickness of the wafer. By dividing the heat flux by the product of these three properties, we're essentially quantifying how effectively the wafer's temperature can change given its material properties and the external heat input.

Fourier's Law of Conduction

Fourier's Law is a cornerstone in the study of heat transfer, particularly when it comes to conduction—the process by which thermal energy is transmitted through collisions between adjacent atoms or molecules. When we analyze the rate of heat flow across a medium, this law comes handy. It postulates that the heat flux is proportional to the negative gradient of temperature, implying that heat transfers from regions of high to low temperature.

The law is mathematically expressed as:
\[\begin{equation}q_{\text{cond}}'' = -k abla T\end{equation}\]Where \(q_{\text{cond}}''\) is the heat flux due to conduction, \(k\) is the thermal conductivity of the medium, and \(abla T\) is the temperature gradient. In the context of our semiconductor annealing problem, this law allows us to calculate the conductive heat flux by considering the temperature difference between the hot plate and the wafer and the gap medium's thermal conductivity.

Stefan-Boltzmann Constant

The Stefan-Boltzmann constant is a fundamental parameter in the realm of thermodynamics and heat transfer, symbolized as \(\sigma\). It is essential when dealing with radiative heat transfer processes and is a key component of the equation used to determine the radiative heat flux emitted by black bodies.

Value-wise, the Stefan-Boltzmann constant is \(5.67 \times 10^{-8} W/m^2K^4\). It enables us to relate the temperature of an object to the total energy radiated per unit surface area. Essentially, it tells us that the power radiated per unit area of a black body is directly proportional to the fourth power of the black body's thermodynamic temperature. So, in our exercise, where the hot plate and wafer approximate blackbody conditions, this constant is utilised in conjunction with the temperatures to calculate the radiative heat flux.