1. The Conductivity of Doped Semiconductors
Ian Steegmayer

2. Conductivity of pure semiconductors
𝜎=𝑒 𝑛 𝜇 𝑛 +𝑝 𝜇 𝑝
𝑛= 𝐸 𝐿 ∞ 𝐷 𝐿 𝐸 𝑓 𝐸,𝑇 𝑑𝐸 , 𝑝= −∞ 𝐸 𝑉 𝐷 𝑉 𝐸 1−𝑓 𝐸,𝑇 𝑑𝐸
𝐷 𝐿 𝐸 = 2 𝑚 𝑛 ∗ 3/2 2 𝜋 2 ℏ 3 𝐸− 𝐸 𝐿 , 𝐸> 𝐸 𝐿
𝐷 𝑉 𝐸 = 2 𝑚 𝑝 ∗ 3/2 2 𝜋 2 ℏ 𝐸 𝑉 −𝐸 , 𝐸< 𝐸 𝑉 , band gap in between
𝑓 𝐸,𝑇 = 𝑒 𝐸− 𝐸 𝐹 𝑘𝑇 +1 −1 ≈ 𝑒 − 𝐸− 𝐸 𝐹 𝑘𝑇 , 1−𝑓 𝐸,𝑇 ≈ 𝑒 − 𝐸 𝐹 −𝐸 𝑘𝑇
⇒ 𝑛=𝐶⋅ 𝑒 𝐸 𝐹 𝑘𝑇 𝐸 𝐿 ∞ 𝐸− 𝐸 𝐿 𝑒 − 𝐸 𝑘𝑇 𝑑𝐸 = 𝑁 𝐿 𝑒 − 𝐸 𝐿 − 𝐸 𝐹 𝑘𝑇

3. Intrinsic charge carrier density
𝑛⋅𝑝= 𝑁 𝐿 𝑁 𝑉 exp − 𝐸 𝑔 𝑘𝑇 characteristic for semiconductor
Intrinsic semiconductor:
𝑛 𝑖 = 𝑝 𝑖 = 𝑁 𝐿 𝑁 𝑉 exp − 𝐸 𝑔 𝑘𝑇
𝐸 𝐹 = 𝐸 𝐿 + 𝐸 𝑉 𝑘𝑇 ln 𝑚 𝑝 ∗ 𝑚 𝑛 ∗
Energy
Density of states
Fermi function
Energy

4. Why doped Semiconductor
𝜎 𝑖 𝑆𝑖 = 𝑛 𝑖 𝑒 𝜇 𝑛 ≈4⋅ 10 −4 1 Ω𝑚
Without dopants carrier density and thus conductivity too small for most applications

5. Hydrogen approximation
Photon energy
Absorption coefficient
Eigenenergy of donor
𝐸 𝜈 =− 𝑚 ∗ 𝑒 𝜋 𝜖 𝑟 𝜖 ℏ 2 ⋅ 1 𝜈 2
times smaller than
for H atom
Large effective Bohr radius
Similar for acceptors
Unlike bands, impurities energy changes significantly when occupied with two electrons

6. Hydrogen Approximation
Energy
Donor niveau
Acceptor niveau
Spatial coordinate
semiconductor
Hydrogen Approximation
Ionization energy is small (≈meV)
Thermic excitation ionizes dopants
Free charge carriers from dopants contribute to conduction
Compensation
Impurity states are localized
Significant increase in energy with higher occupation
Energy
Density of states
Spatial coordinate

7. Charge carrier density of doped semiconductor
𝑛 𝐷 = 𝑛 𝐷 0 + 𝑛 𝐷 + and 𝑛 𝐴 = 𝑛 𝐴 0 + 𝑛 𝐴 −
Occupation of ground state (not ionized): 𝑛 𝐷 0 𝑛 𝐷 = 1 𝑔 𝐷 ⋅ 𝑒 𝐸 𝐷 −𝐸 𝐹 𝑘𝑇 +1
intrinsic el. concentration: 𝑛= 𝑁 𝐿 𝑒 − 𝐸 𝐿 − 𝐸 𝐹 𝑘𝑇
Semiconductor shall be neutral: 𝑛+ 𝑛 𝐴 − =𝑝+ 𝑛 𝐷 +
Assumptions:
One type dominant, (here donors, 𝑛 𝐷 ≫ 𝑛 𝐴 )
𝐸 𝐿 − 𝐸 𝐹 ≫𝑘𝑇
Doping provides the majority of free charge carriers, (here 𝑛 𝐷 + ≫𝑝)

8. Charge carrier density of doped semiconductor
All acceptors neutralized for n-type:
𝑛= 𝑛 𝐷 + − 𝑛 𝐴 = 𝑛 𝐷 − 𝑛 𝐷 0 − 𝑛 𝐴 = 𝑛 𝐷 1− 1 𝑒 (𝐸 𝐷 − 𝐸 𝐹 )/𝑘𝑇 +1 − 𝑛 𝐴
Result: 𝑛 𝑛 𝐴 +𝑛 𝑛 𝐷 − 𝑛 𝐴 −𝑛 = 𝑁 𝐿 𝑒 𝐸 𝑑 𝑘𝑇
𝛿- compensation, 𝑘𝑇≪ 𝐸 𝑑
𝛾- ionization regime, 𝑛 𝐴 ≪𝑛≪ 𝑛 𝐷
𝛽- saturation regime
𝛼- intrinsic ( 𝑒 − from valence band)
Reciprocal temperature 1 𝑇
Electron density

9. Electron density - saturation regime
Reciprocal temperature
Saturation regime:
𝑘𝑇≈ 𝐸 𝑑 ⇒ exp − 𝐸 𝑑 𝑘𝑇 ≈1
𝑛 2 ≈ 𝑁 𝐿 𝑛 𝐷 −𝑛 and 𝑛≪ 𝑁 𝐿
𝑛 𝐷 −𝑛≈0
Reciprocal temperature 1 𝑇
Electron density

10. Mobility 𝜇= 𝑒𝜏 𝑚 ∗ Main contribution to 𝜏 Low temperatures
𝜇= 𝑒𝜏 𝑚 ∗
Main contribution to 𝜏
Scattering on impurities
Scattering on phonons
Low temperatures
(Ionized) impurity scattering
High temperatures
Electron-phonon scattering

11. Conductivity
Reciprocal temperature
Conductivity
𝜎=𝑒 𝑛 𝜇 𝑛 +𝑝 𝜇 𝑝

12. Conclusion Doping is necessary for most devices
Knowledge of free charge carrier density is important
Important impact on functionality of devices in certain environment
e.g. in space
Good models are needed for prediction of new materials
Interesting topic in research

13. All figures taken from:
S. Hunklinger, Festkörperphysik, 3rd edition, 2011, Oldenburgverlag