# Drift–Diffusion Balance in Non-Uniformly Doped n-type Semiconductor ## Key Equations ### Electron Current Density \[ J_n = q n \mu_n E(x) + q D_n \frac{dn(x)}{dx} \] At **equilibrium**: \[ J_n = \] Therefore, \[ q n \mu_n E(x) + q D_n \frac{dn}{dx} = \] Divide both sides by \( q \): \[ n\mu_nE(x) + D_n\frac{dn}{dx} = \] Solve for electric field: \[ E(x) = -\frac{D_n}{\mu_n}\frac{1}{n}\frac{dn}{dx} \] Using the **Einstein relation**: \[ \frac{D_n}{\mu_n} = \frac{kT}{q} \] Thus, \[ E(x) = -\frac{kT}{q}\frac{1}{n}\frac{dn}{dx} \] At equilibrium, \[ n(x) \approx N_D(x) \] --- ## (a) Band Diagram Sketch Use the relation: \[ E(x) = \frac{1}{q}\frac{dE_i}{dx} \] **Interpretation:** The slope of the band diagram is proportional to the electric field. From the doping profile: - \( N_D \) decreases from \( x = 1\,\mu\text{m} \rightarrow 4\,\mu\text{m} \) - Reaches a minimum near \( B \) - Increases again from \( 4\,\mu\text{m} \rightarrow 7\,\mu\text{m} \) Since \[ n(x) = N_D(x) \] the electron concentration follows the same shape. ### Band Bending Rule When **doping decreases with \( x \)**: \[ \frac{dn}{dx} < \] \[ E(x) > \] Bands **slope upward** with \( x \). When **doping increases**: \[ \frac{dn}{dx} > \] \[ E(x) < \] Bands **slope downward** with \( x \). ### Final Band Diagram Behavior | Region | Doping trend | Band bending | |----------|--------------|--------------------------| | 1–4 µm | decreasing | bands slope **upward** | | ~4–5 µm | constant | **flat bands** | | 5–7 µm | increasing | bands slope **downward** | | 7–8 µm | constant | **flat bands** | - All bands (\( E_c, E_i, E_v \)) move **together**. - Fermi level (\( E_F \)) remains **flat** at equilibrium. --- ## (b) Built-in Electric Field \( E(x) \) Using \[ E(x) = -\frac{kT}{q}\frac{1}{n}\frac{dn}{dx} \] Electric field depends on the **gradient of doping**. ### Region 1–4 µm Doping **decreases**: \[ \frac{dn}{dx} < \] \[ E(x) > \] Field points in the **\(+x\) direction**. --- ### Around B (\( \approx 4\text{–}5\,\mu\text{m} \)) Doping **constant**: \[ \frac{dn}{dx} = \] \[ E(x) = \] --- ### Region 5–7 µm Doping **increases**: \[ \frac{dn}{dx} > \] \[ E(x) < \] Field points in the **\(-x\) direction**. --- ### Region 7–8 µm Doping is **constant**: \[ E(x) = \] --- #### Shape of \( E(x) \): - **Positive** from \( 1\text{–}4\,\mu\text{m} \) - **Zero** near \( 4\text{–}5\,\mu\text{m} \) - **Negative** from \( 5\text{–}7\,\mu\text{m} \) - **Zero** after \( 7\,\mu\text{m} \) --- ## (c) Direction of Diffusion Current Diffusion current: \[ J_\text{diff} = q D_n \frac{dn}{dx} \] Electrons diffuse **from high concentration to low concentration**. ### Point A Doping **decreases with \( x \)**. Electrons move **left → right**: \[ J_\text{diff} \rightarrow \] --- ### Point B Minimum doping: \[ \frac{dn}{dx} = \] \[ J_\text{diff} = \] --- ### Point C Doping **increases**: Electrons diffuse **right → left**: \[ J_\text{diff} \leftarrow \] --- ## (d) Direction of Drift Current Drift current: \[ J_\text{drift} = q n \mu_n E(x) \] - Electrons drift **opposite** to the electric field, but **current follows field direction**. - At equilibrium: \[ J_\text{drift} = -J_\text{diff} \] --- ### Point A Diffusion current → right, so drift current must be: \[ J_\text{drift} \leftarrow \] --- ### Point B \[ J_\text{diff} = \] \[ J_\text{drift} = \] --- ### Point C Diffusion current → left, so drift current: \[ J_\text{drift} \rightarrow \] --- ## Final Answers ### (a) Band Diagram - \( 1\text{–}4\,\mu\text{m} \): bands slope **upward** - \( 4\text{–}5\,\mu\text{m} \): **flat** - \( 5\text{–}7\,\mu\text{m} \): bands slope **downward** - \( E_F \): **flat** --- ### (b) Electric Field - **Positive** (\( \rightarrow \)) from \( 1\text{–}4\,\mu\text{m} \) - **** near \( 4\text{–}5\,\mu\text{m} \) - **Negative** (\( \leftarrow \)) from \( 5\text{–}7\,\mu\text{m} \) - **** after \( 7\,\mu\text{m} \) --- ### (c) Diffusion Current | Point | Direction | |-------|-----------| | A | \( \rightarrow \) | | B | | | C | \( \leftarrow \) | --- ### (d) Drift Current | Point | Direction | |-------|-----------| | A | \( \leftarrow \) | | B | | | C | \( \rightarrow \) | ---

# Drift–Diffusion Balance in Non-Uniformly Doped n-type Semiconductor ## Key Equations ### Electron Current Density \[ J_n = q n \mu_n E(x) + q D_n \frac{dn(x)}{dx} \] At **equilibrium**: \[ J_n = 0 \] Therefore, \[ q n \mu_n E(x) + q D_n \frac{dn}{dx} = 0 \] Divide by \( q \): \[ n\mu_nE(x) + D_n\frac{dn}{dx}=0 \] Solve for electric field: \[ E(x)=-\frac{D_n}{\mu_n}\frac{1}{n}\frac{dn}{dx} \] Using the **Einstein relation**: \[ \frac{D_n}{\mu_n}=\frac{kT}{q} \] Thus, \[ E(x)=-\frac{kT}{q}\frac{1}{n}\frac{dn}{dx} \] At equilibrium: \[ n(x) \approx N_D(x) \] --- ## (a) Band Diagram Sketch Use the relation: \[ E(x)=\frac{1}{q}\frac{dE_i}{dx} \] **Interpretation:** The slope of the band diagram is proportional to the electric field. From the doping profile: - \( N_D \) decreases from \( x=1\,\mu\text{m} \) to \( 4\,\mu\text{m} \) - Reaches a minimum near \( B \) - Increases again from \( 4\,\mu\text{m} \) to \( 7\,\mu\text{m} \) Since: \[ n(x)=N_D(x) \] the electron concentration follows the same shape. ### Band Bending Rule When **doping decreases with \( x \)**: \[ \frac{dn}{dx} < 0 \] \[ E(x) > 0 \] Bands **slope upward** with \( x \). When **doping increases**: \[ \frac{dn}{dx} > 0 \] \[ E(x) < 0 \] Bands **slope downward** with \( x \). ### Final Band Diagram Behavior | Region | Doping trend | Band bending | |----------|--------------|--------------------------| | 1–4 µm | decreasing | bands slope **upward** | | ~4–5 µm | constant | **flat bands** | | 5–7 µm | increasing | bands slope **downward** | | 7–8 µm | constant | **flat bands** | - All bands (\( E_c, E_i, E_v \)) move **together**. - Fermi level (\( E_F \)) remains **flat** at equilibrium. --- ## (b) Built-in Electric Field \( E(x) \) Using: \[ E(x)=-\frac{kT}{q}\frac{1}{n}\frac{dn}{dx} \] Electric field depends on the **gradient of doping**. ### Region 1–4 µm Doping **decreases**: \[ \frac{dn}{dx} < 0 \] \[ E(x) > 0 \] Field points in the **\(+x\) direction**. --- ### Around B (\( \approx 4\text{–}5\,\mu\text{m} \)) Doping **constant**: \[ \frac{dn}{dx} = 0 \] \[ E(x) = 0 \] --- ### Region 5–7 µm Doping **increases**: \[ \frac{dn}{dx} > 0 \] \[ E(x) < 0 \] Field points in the **\(-x\) direction**. --- ### Region 7–8 µm Doping is **constant**: \[ E(x) = 0 \] --- #### Shape of \( E(x) \): - **Positive** from \( 1\text{–}4\,\mu\text{m} \) - **Zero** near \( 4\text{–}5\,\mu\text{m} \) - **Negative** from \( 5\text{–}7\,\mu\text{m} \) - **Zero** after \( 7\,\mu\text{m} \) --- ## (c) Direction of Diffusion Current Diffusion current: \[ J_\text{diff}=qD_n\frac{dn}{dx} \] Electrons diffuse **from high concentration to low concentration**. ### Point A Doping **decreasing with \( x \)**. Electrons move **left → right**: \[ J_\text{diff} \rightarrow \] --- ### Point B Minimum doping: \[ \frac{dn}{dx}=0 \] \[ J_\text{diff}=0 \] --- ### Point C Doping **increasing**: Electrons diffuse **right → left**: \[ J_\text{diff} \leftarrow \] --- ## (d) Direction of Drift Current Drift current: \[ J_\text{drift}=qn\mu_nE(x) \] - Electrons drift **opposite** to the electric field, but **current follows field direction**. - At equilibrium: \[ J_\text{drift} = - J_\text{diff} \] --- ### Point A Diffusion current → right, so drift current must be: \[ J_\text{drift} \leftarrow \] --- ### Point B \[ J_\text{diff}=0 \] \[ J_\text{drift}=0 \] --- ### Point C Diffusion current → left, so drift current: \[ J_\text{drift} \rightarrow \] --- ## Final Answers ### (a) Band Diagram - \( 1\text{–}4\,\mu\text{m} \): bands slope **upward** - \( 4\text{–}5\,\mu\text{m} \): **flat** - \( 5\text{–}7\,\mu\text{m} \): bands slope **downward** - \( E_F \): **flat** --- ### (b) Electric Field - **Positive** (\( \rightarrow \)) from \( 1\text{–}4\,\mu\text{m} \) - **0** near \( 4\text{–}5\,\mu\text{m} \) - **Negative** (\( \leftarrow \)) from \( 5\text{–}7\,\mu\text{m} \) - **0** after \( 7\,\mu\text{m} \) --- ### (c) Diffusion Current | Point | Direction | |-------|-----------| | A | \( \rightarrow \) | | B | 0 | | C | \( \leftarrow \) | --- ### (d) Drift Current | Point | Direction | |-------|-----------| | A | \( \leftarrow \) | | B | 0 | | C | \( \rightarrow \) | ---