VLSI homework 2

7th-Semester-Fall-2024/VLSI/homework/homework-2/homework-2.md

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+# VSLI Homework 2 - Aidan Sharpe
+
+## Problem 1
+A 90[nm] long transistor has a gate oxide thickness $t_\text{ox}$ of 16[$\text{\r{A}}$]. What is its gate capcaitance per micrion of width?
+
+```python
+eps_0 = 8.85E-12
+k_ox = 3.9
+
+L = 90E-9       # 90nm expressed in meters
+t_ox = 16E-10   # 16A expressed in meters
+
+C_permeter = k_ox * eps_0 * L / t_ox
+C_permicron = C_permeter * 1E-6
+
+print(C_permicron)
+```
+$$\boxed{C_\text{permicron} = 1.94\text{[fF/$\mu$m]}}$$
+
+## Problem 2
+Consider the nMOS transistor in a 0.6[$\mu$m] process with gate oxide thickness of 100[$\text{\r{A}}$]. The doping level is $N_A = 2 \times 10^{17}$[cm$^{-3}$] and the nominal threshold voltage is 0.7[V]. The body is tied to ground with a substrate contact. How much does the threshold change at room temperature if the source is at 4[V] instead of 0[V]?
+
+```python
+from math import log, sqrt
+
+V_t0 = 0.7              # The nominal threshold voltage
+t_ox = 100E-8           # The gate threshold voltage in angstrom with CGS units
+N_A = 2E17              # The doping level in cm^-3
+
+k_ox = 3.9
+k_si = 11.7
+eps_0 = 8.85E-14        # Vacuum permittivity with CGS units
+k = 1.380E-23           # Boltzmann's constant
+q = 1.602E-19           # The charge of an electron
+
+T = 300                 # Room temperature in Kelvin
+
+v_T = k*T/q
+n_i = 1.45E10           # The intrinsic carrier concentration of undoped Si
+
+eps_ox = k_ox * eps_0
+eps_si = k_si * eps_0
+
+V_b = 0
+V_s0 = 0
+V_s1 = 4
+
+gamma = (t_ox / eps_ox) * sqrt(2*q*eps_si*N_A)
+phi_s = 2 * v_T * log(N_A / n_i)
+
+def V_t(V_t0, V_s, V_b, gamma, phi_s):
+    V_sb = V_s - V_b
+    return V_t0 + gamma*(sqrt(phi_s + V_sb) - sqrt(phi_s))
+
+Delta_V_t = V_t(V_t0, V_s1, V_b, gamma, phi_s) \
+          - V_t(V_t0, V_s0, V_b, gamma, phi_s)
+
+print(Delta_V_t)
+```
+
+$$\boxed{\Delta V_t = 0.955583\text{[V]}}$$

7th-Semester-Fall-2024/VLSI/homework/homework-2/problem_1.py

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+eps_0 = 8.85E-12
+k_ox = 3.9
+
+L = 90E-9       # 90nm expressed in meters
+t_ox = 16E-10   # 16A expressed in meters
+
+C_permeter = k_ox * eps_0 * L / t_ox
+C_permicron = C_permeter * 1E-6
+
+print(C_permicron)

7th-Semester-Fall-2024/VLSI/homework/homework-2/problem_2.py

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+from math import log, sqrt
+
+V_t0 = 0.7              # The nominal threshold voltage
+t_ox = 100E-8           # The gate threshold voltage in angstrom with CGS units
+N_A = 2E17              # The doping level in cm^-3
+
+k_ox = 3.9
+k_si = 11.7
+eps_0 = 8.85E-14        # Vacuum permittivity with CGS units
+k = 1.380E-23           # Boltzmann's constant
+q = 1.602E-19           # The charge of an electron
+
+T = 300                 # Room temperature in Kelvin
+
+v_T = k*T/q
+n_i = 1.45E10           # The intrinsic carrier concentration of undoped Si
+
+eps_ox = k_ox * eps_0
+eps_si = k_si * eps_0
+
+V_b = 0
+V_s0 = 0
+V_s1 = 4
+
+gamma = (t_ox / eps_ox) * sqrt(2*q*eps_si*N_A)
+phi_s = 2 * v_T * log(N_A / n_i)
+
+def V_t(V_t0, V_s, V_b, gamma, phi_s):
+    V_sb = V_s - V_b
+    return V_t0 + gamma*(sqrt(phi_s + V_sb) - sqrt(phi_s))
+
+Delta_V_t = V_t(V_t0, V_s1, V_b, gamma, phi_s) \
+          - V_t(V_t0, V_s0, V_b, gamma, phi_s)
+
+print(Delta_V_t)