HW#1_Solution

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Apr 3, 2024

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Step (1) I am provide a brief overview of the energy band diagrams for p-MOSFETs under the specified conditions: (a) **Threshold (Vertical Direction):** At threshold, the energy band diagram shows a flat band condition from the gate to the gate dielectric to the substrate. The Fermi level aligns with the mid-gap energy, indicating the minimum energy required for carriers to move from the source to the drain. (b) **On-State (Lateral Direction):** In the on-state, the energy band diagram between the source and drain shows a slope from the source to the drain, representing the potential energy gradient that facilitates the flow of holes (in the case of p-MOSFET) from source to drain. (c) **Off-State (Lateral Direction):** In the off-state, the energy band diagram between source and drain is relatively flat, indicating a barrier that restricts the flow of carriers between source and drain. The Fermi level remains aligned with the mid-gap energy. For detailed and accurate diagrams, you may refer to your class materials, textbooks, or consult relevant resources on semiconductor physics. Explanation: (2):(a) **Threshold Voltage (Vth):** The threshold voltage for a long-channel MOSFET can be calculated using the equation: \[V_{th} = \phi_f + 2\phi_{ms} + \frac{Q_{\text{dep}}}{C_{\text{ox}}} \] where: - \(\phi_f\) is the metal work function, - \(\phi_{ms}\) is the semiconductor-metal work function difference, - \(Q_{\text{dep}}\) is the depletion charge per unit area, and - \(C_{\text{ox}}\) is the oxide capacitance per unit area. Given that the gate material is n++ poly-Si, \(\phi_f\) is approximately 4.1 eV, and \(\phi_{ms}\) is the difference between the metal and semiconductor work functions. \[ \phi_{ms} = \phi_{\text{poly-Si}} - \phi_{\text{Si}} \] Assuming \(\phi_{\text{Si}}\) is around 4.19 eV, you can calculate \(\phi_{ms}\) and then determine \(Q_{\text{dep}}\) and \(C_{\text{ox}}\) using the provided parameters. (b) **Drain Saturation Current (\(I_{\text{dsat}})\):** The drain saturation current can be calculated using the saturation current equation: \[ I_{\text{dsat}} = \frac{1}{2} \mu_n C_{ox} \frac{W}{L} (V_{gs} - V_{th})^2 \] where: - \( \mu_n \) is the electron mobility, - \( W \) is the channel width, - \( L \) is the channel length, and - \( V_{gs} - V_{th} \) is the overdrive voltage . Given \(V_{gs}\) and \(V_{th}\) from part (a), you can use this equation to find \(I_{\text{dsat}}\). Ensure that \(V_{gs} - V_{th}\) is in the saturation region (typically \(V_{gs} > V_{th}\)). 3:(a) **Effect of Parameter Variations on Channel Current:** 1. **Threshold Voltage (\(V_{th}\)):** An increase in \(V_{th}\) would result in a decrease in the channel current (\(I_{ds}\)), as a higher threshold voltage makes it

harder for the MOSFET to turn on. 2. **Oxide Thickness ((t_{\text{ox}})\):* An increase in \(t_{\text{ox}}\) would also lead to a decrease in \(I_{ds}\). This is because a thicker oxide increases the capacitance, making it harder to control the flow of charge carriers. 3. **Effective Charge in the Oxide (\(Q_{\text{ox}})\):** An increase in \(Q_{\text{ox}}\) would result in an increase in \(I_{ds}\). This is because a higher charge in the oxide facilitates the flow of charge carriers in the channel. (b) **Maximum 'Worst Case' Increase and Decrease:** The worst-case scenarios for current increase and decrease can be calculated by considering the extremes of the specified 5% variation for each parameter. 1. **Maximum Increase in Current:** Consider the simultaneous increase in \(V_{th}\), \(t_{\text{ox}}\), and \(Q_{\text{ox}}\) by 5%. Calculate the corresponding increase in \(I_{ds}\) using the MOSFET current equation. 2. **Maximum Decrease in Current:** Consider the simultaneous decrease in \(V_{th}\), \(t_{\text{ox}}\), and \(Q_{\text{ox}}\) by 5%. Calculate the corresponding decrease in \(I_{ds}\) using the MOSFET current equation. These worst-case scenarios provide an estimate of the maximum potential variation in MOSFET current due to fabrication parameter fluctuations. Step 4:(a) **Threshold Voltage (\(V_{th}\)):** The threshold voltage for a short-channel MOSFET can be calculated using the following equation: \[ V_{th} = \phi_f + 2\phi_{ms} + \frac{Q_{\text{dep}}}{C_{\text{ox}}} \] Given that the gate material is n++ poly-Si, \(\phi_f\) is approximately 4.1 eV, and \(\phi_{ms}\) is the difference between the metal and semiconductor work functions. You'll need to calculate \(Q_{\text{dep}}\) and \(C_{\text{ox}}\) using the provided parameters. (b) **Drain Saturation Current (\(I_{\text{dsat}})\) at \(V_{ds} = 3.3 V\):** The drain saturation current can be calculated using the MOSFET saturation current equation: \[ I_{\text{dsat}} = \frac{1}{2} \mu_n C_{ox} \frac{W}{L} (V_{gs} - V_{th})^2 \] Given \(V_{gs}\) as the gate-source voltage and \(V_{th}\) from part (a), you can use this equation to find \(I_{\text{dsat}}\). Ensure that \(V_{gs} - V_{th}\) is in the saturation region (typically \(V_{gs} > V_{th}\)). Remember to use the appropriate mobility (\(\mu_n\)) for the short-channel MOSFET and the intrinsic carrier concentration (\(n_i\)) in your calculations. Follow-up Questions You10 hours ago Could you improve answer's formatting? It's almost impossible to read the equations and numbers vijaykumarchinthapoola1 hour ago (4) To calculate the threshold voltage (Vth) for the MOSFET, you can use the equation V th= vfbfb- fb Where: - \(V_{FB}\) is the flat-band voltage,

- \(\phi_F\) is the bulk Fermi potential, - \(Q_{\text{oxide}}\) is the charge in the oxide, - \(C_{\text{ox}}\) is the oxide capacitance. Given that \(V_{FB} = \phi_{MS} + \frac{Q_{\text{ss}}}{C_{\text{ox}}}\), where \(\phi_{MS}\) is the metal-semiconductor work function difference and \(Q_{\text{ss}}\) is the surface state charge density, you can calculate each term based on the provided parameters. For part (b), the drain saturation current (\(I_{\text{dsat}}\)) can be estimated using the traditional MOSFET saturation current equation: Idsat=1/2un C ox W/L( V gs - V th) 2 Given that the drain voltage (\(V_{\text{ds}}\)) is equal to the supply voltage (\(V\)) in this case, you can use \(V_{\text{gs}} = V - V_{\text{th}}\) in the equation. Remember to use consistent units and convert any parameters as necessary. Let me know if you'd like me to perform the calculations with the provided information. (5)(a) Let's analyze how the variation of each parameter affects the channel current: 1. **\(t_{\text{ox}}\) (Oxide Thickness):** - Increase: Decreases the capacitance (\(C_{\text{ox}}\)), resulting in an increase in threshold voltage (\(V_{\text{th}}\)), which lowers the channel current. - Decrease: Increases \(C_{\text{ox}}\), leading to a decrease in \(V_{\text{th}}\) and an increase in channel current. 2. **\(N_{\text{a}}\) (Substrate Doping Concentration):** - Increase: Enhances charge carrier concentration, leading to a decrease in \(V_{\text{th}}\) and an increase in channel current. - Decrease: Reduces carrier concentration, resulting in an increase in \(V_{\text{th}}\) and a decrease in channel current. 3. **\(Q_{\text{ox}}\) (Charge in the Oxide):** - Increase: Increases \(V_{\text{th}}\), leading to a decrease in channel current. - Decrease: Decreases \(V_{\text{th}}\), resulting in an increase in channel current. (b) To calculate the maximum 'worst case' increase and decrease of the current, you would need to consider the combined effect of these variations. This involves evaluating the extremes of each parameter's variation simultaneously to obtain the maximum and minimum values for channel current. The final result will depend on the specific relationships between these parameters and the device characteristics. If you have specific equations or relationships between these parameters and channel current, I can provide more detailed calculations. Otherwise, you may need to consult device models or specifications for your specific MOSFET to determine the impact of parameter variations on current.

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