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A furnace for processing semiconductor materials is formed by a silicon carbide chamber that is zone heated on the top section and cooled on the lower section. With the elevator in the lowest position, a robot arm inserts the silicon wafer on the mounting pins. In a production operation, the wafer is rapidly moved toward the hot zone to achieve the temperature-time history required for the process recipe. In this position the top and boltom surfaces of the wafer exchange radiation with the hot and cool rones, respectively, of the chamber. The zone temperatures are \(T_{\mathrm{a}}=1500 \mathrm{~K}\) and \(T_{c}=\) \(330 \mathrm{~K}\), and the emissivity and thickness of the wafer are \(e=0.65\) and \(d=0.78 \mathrm{~mm}\), respectively. With the ambient gas at \(T_{w}=700 \mathrm{~K}\), convection coefficients at the upper and lower surfaces of the wafer are 8 and \(4 \mathrm{~W} / \mathrm{m}^{2}-\mathrm{K}\). respectively. The silicon wafer has a den. sity of \(2700 \mathrm{~kg} / \mathrm{m}^{3}\) and a specific heat of \(875 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). (a) For an initial condition corresponding fo a wafer tempersture of \(T_{w i}=300 \mathrm{~K}\) and the position of the wafer shawn schematically. determine the corresponding time rate of change of the wafer tempereture, \(\left(d T_{w} / d r\right)_{0}\). (b) Determine the steady state temperature reached by the wafer if it remains in this position. How significant is convection heat transfer for this situation? Sketch how you would expect the wafer temperature to vary as a function of vertical distance.

Step by step solution

01

Understand the Heat Transfer Mechanisms

We have radiation exchange between the wafer and the hot and cool zones, as well as convection heat transfer from the ambient gas to the wafer. The objective is to determine the time rate of change of temperature initially and the steady-state temperature.

02

Scenario Analysis for Initial Rate of Temperature Change

Consider the wafer's initial temperature, initial rate of temperature change, heat transfer through convection, and radiation. The net heat transfer will be calculated by considering the energy gained through radiation and convection from the ambient gas.

03

Apply the Law of Energy Conservation

The rate of energy change in the wafer equals the difference in energy absorbed due to radiation minus the energy lost due to convection.Use:\[\dot{Q}_{net} = \rho c V \frac{dT_w}{dt}\]where \(\rho = 2700\, \text{kg/m}^3\), \(c = 875\, \text{J/kg.K}\), and \(V\) is the volume of the wafer.

04

Calculate Radiation Heat Transfer

For radiation:\[\dot{Q}_{rad} = \varepsilon \sigma A \left( T_a^4 - T_w^4 \right) - \varepsilon \sigma A \left( T_c^4 - T_w^4 \right)\]Set emissivity \(\varepsilon = 0.65\), Stefan-Boltzmann constant \(\sigma = 5.67 \times 10^{-8} \text{W/m}^2\text{.K}^4\), area \(A\), and substitute given temperatures.

05

Calculate Convection Heat Transfer

For convection, we have:\[\dot{Q}_{conv} = h_u A (T_a - T_w) + h_l A (T_c - T_w)\]where the upper and lower convection coefficients are \(h_u = 8\, \text{W/m}^2\text{-K}\) and \(h_l = 4\, \text{W/m}^2\text{-K}\) respectively.

06

Solve the Energy Balance Equation

Substitute \(\dot{Q}_{rad}\) and \(\dot{Q}_{conv}\) into our energy conservation equation and solve for \(\frac{dT_w}{dt}\). Use the wafer's initial conditions to calculate the rate of temperature change.

07

Determine Steady State Temperature

For steady state, the net heat exchange is zero:\[\dot{Q}_{net} = \dot{Q}_{rad} + \dot{Q}_{conv} = 0\]Solve the resulting equation to find the steady-state temperature of the wafer.

08

Analyze the Role of Convection

Evaluate the heat transfer contributions from both convection and radiation to determine the significance of convection compared to radiation. Discuss its effect on reaching the steady-state temperature.

09

Sketch Temperature Variation with Distance

With dominant radiation heat transfer at the wafer's surfaces, temperature distribution may not be uniform. Sketch a curve indicating higher temperatures near the hot zone and lower near the cool zone.

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Key Concepts

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

Radiation Heat Transfer

In the semiconductor processing scenario, radiation heat transfer plays a significant role. Radiation is the method by which energy transfers as electromagnetic waves. When the silicon wafer is positioned between the hot and cool zones, it exchanges heat primarily through radiation with these zones.

The hot zone at a temperature of 1500 K radiates energy to the wafer, while the wafer itself emits radiation towards the cooler zone and its surroundings. The net radiation heat transfer \( \dot{Q}_{rad} \) can be calculated using the formula:

• \( \dot{Q}_{rad} = \varepsilon \sigma A \left( T_a^4 - T_w^4 \right) - \varepsilon \sigma A \left( T_c^4 - T_w^4 \right) \)

Here, \( \varepsilon \) represents emissivity (0.65 for silicon), \( \sigma \) is the Stefan-Boltzmann constant (5.67 x 10^-8 W/m虏路K鈦�), and \( A \) is the area of the wafer. This formula incorporates both the radiation received from the hot zone and emitted to the cool zone.

Radiation is especially impactful due to the high temperature differences ( between 1500 K and 330 K) and the nature of electromagnetic wave transfer, which does not require a medium. This makes it essential to consider radiation when analyzing the heat exchange in wafer processing.

Convection Heat Transfer

Convection heat transfer also contributes to the temperature change of the wafer, though its significance may vary. Convection involves the transfer of heat through a fluid, in this case, the ambient gas around the silicon wafer.

The upper surface of the wafer faces the hot zone and is subject to a convection coefficient of 8 W/m虏-K, while the lower surface, facing the cool zone, has a convection coefficient of 4 W/m虏-K. The heat transfer through convection can be represented by:

• \( \dot{Q}_{conv} = h_u A (T_a - T_w) + h_l A (T_c - T_w) \)

where \( h_u \) and \( h_l \) are the convection coefficients for the upper and lower surfaces, respectively.

Convection is generally a weaker form of heat transfer compared to radiation, given the larger scale of energy exchanged through electromagnetic waves in radiation. However, it still plays a role in reaching equilibrium, particularly influencing the wafer's temperature towards the ambient air's influence.

Understanding the balance between radiation and convection is crucial for effectively controlling the temperature of the wafer during semiconductor manufacturing.

Energy Conservation

Energy conservation is a fundamental principle in determining the rate of temperature change and the steady-state condition of the wafer. The concept revolves around the energy balance in which the net energy into the wafer is equal to the rate of change of its internal energy.

By applying energy conservation, one can determine the initial temperature change rate using:

• \( \dot{Q}_{net} = \rho c V \frac{dT_w}{dt} \)

where \( \rho \) is the density, \( c \) the specific heat, and \( V \) the volume of the wafer. \( \dot{Q}_{net} \) is the net heat transfer, comprising both radiation and convection contributions.

For steady-state, the principle implies that the total heat gained and lost by the wafer equals to zero. This balance allows us to solve for the wafer's constant temperature in the steady state:

• \( \dot{Q}_{rad} + \dot{Q}_{conv} = 0 \)

This equation helps us find the final steady-state temperature, ensuring that all absorbed and lost energy equilibrates.

Through energy conservation, we quantify how efficiently a semiconductor wafer reaches its desired temperature profile during processing. Proper adherence to these principles is key to effective semiconductor manufacturing.