Chapter 9: Semiconductors

Question 1

Semiconductors Basics

(3 points)

Answer 1

• Conductivity highly dependent on T,E,B
• Metallic or insulator like properties
• Filled valence band, so technically an insulator

Question 2

Semiconductor Classification

(4 points)

Answer 2

• Intrinsic semiconductors are “naturally” occurring and have low conductivity
• Extrinsic semiconductors are doped intrinsic semiconductors
• A direct semiconductor has the minimum of the conduction and the maximum of the valence band at same k
• An indirect semiconductor has the minimum of the conduction band and the maximum of the valence band at different k

Question 3

Temperature Dependence of Band Gap

(5 points)

Answer 3

• As temperature increases, band gap decreases
  
  • Because interatomic spacing increases with temperature, so potential felt by electron decreases
• Described by Varhsni Equation (see below)
  
  • Parabolic at low T
  • Linear at high T

Question 4

Temperature Dependence of Band Gap

(Graph)

Question 5

Optical Absorption in Semiconductors

(4 points)

Answer 5

• Photon delivers only energy
  
  • ​Optically transparent for (h-bar)ω < E_g
• Phonon delivers momentum
  
  • Required for indirect band gap

Question 6

Doped Semiconductors

(7 points)

Answer 6

• Types:
  
  • Donor: Impurity that adds electron to conduction band
  • Acceptor: Impurity that adds hole to valence band
• Binding energy can be estimated with Bohr model (meV
• Classification:
  
  • Doped with more donors → n-type
  • Doped with more acceptors → p-type

Question 7

Temperature Dependence of Chemical Potential

(Regimes: 4 points)

Answer 7

1. Compensation Regime
2. Impurity Reserve Regime
3. Saturtion of Impurities
4. Intrinsic Regime

Question 8

Compensation Regime

(5 points)

Answer 8

• k_BT <<< E_d
  
  • ​Donors not ionized
• For T → 0, then µ = E_d
• As T increases, electrons excited into conduction band, because acceptors are saturated
• µ shifts towards E_c and n_C increases exponentially (−E_d/k_BT)

Question 9

Impurity Reserve Regime

(3 points)

Answer 9

• k_BT ≪ E_d
• Donors start to contribute to conduction band, but not all
• µ approximately in the middle of E_d and E_c

Question 10

Saturation of Impurities

(3 points)

Answer 10

• k_BT ≥ E_d
• Temperature ionizes all donors, but still too small to excited a large enough number of VB electrons
• µ moves with increasing T towards middle of band gap

Question 11

Intrinsic Regime

(3 points)

Answer 11

• k_BT >> E_d
• VB excitations become important
• µ moves to middle of band gap

Question 12

pn-Junction in Thermal Equilibrium

(Before Contact: 2 points)

Answer 12

• µ_n < E_d (n-type)
• µ_p > E_a (p-type)

Question 13

pn-Junction in Thermal Equilibrium

(After Contact: 5 points)

Answer 13

• µ_n = µ_p
• Diffusion of electrons in p-side (or holes into n-side)
• Creates space charge separation
• Creates electric field
• Induces drift current in opposition to diffusion current

Question 14

Schottky Approximation

(3 points)

Answer 14

• Space charge approximated as step-functions
• Electric field determined from E ∝ \int ρ dx
• Potential determined from V ∝ − \int E dl

Question 15

pn-Junction Under Forward Bias

(4 points)

Answer 15

• Bands “squished” together (∆E = e(V_o − V_app))
• Electron Fermi energy raises
• Induced electric field diminishes → Drift current weaker → diffusion current dominates
• IV characteristic: forward bias region (below)

Question 16

pn-Junction Under Reverse Bias

(4 points)

Answer 16

• Band separated (∆E = e(V_o − (−V_app)))
• Electron Fermi energy lowered
• Potential barrier becomes thin → Tunneling occurs → ”leakage current” (Reverse bias region below)
• If reverse bias becomes to high, breakdown occurs (Reverse breadown region below)