半导体器件原理 1.3 Effects of Doping

About doping, dopant states, and how to calculate carrier density and locate the Fermi level in doped semiconductors.

# Doping of Silicon

• Common group V elements used for doping: P and As
  • Others may not be suitable due to size mismatch
• Group V elements give an extra electron which is forced into the conduction band
  • Donor dopant
  • N-type silicon, N for Negative carriers
• Common group III elements used for doping: B
• Group III elements have one less electron, creating a hole in the valence band
  • Acceptor dopant
  • P-type silicon, P for Positive carriers

# Dopant States

• Donor dopants have an extra positive charge in the nucleus
  • Stronger attraction to electrons
  • Some electrons can stay in the bandgap where they are not allowed in intrinsic silicon
  • Introduces a donor energy level $E_D$
• Acceptor dopants have one less positive charge in the nucleus
  • Weaker attraction to electrons
  • Introduces a loosely bound hole, electrons can stay a bit further away from the nucleus at the acceptor energy level $E_A$
• ${\text{Density}}_{\text{Si}}\gg {\text{Density}}_{\text{dopants}}$ due to solid solubility limit in the amount of dopants that can be taken by silicon before it loses intrinsic properties
  • Locations of donor sites are very sparse
  • $E_C-E_D$ is very small, electrons can be easily excited and move around in the conduction band
  • Once the electron comes out of the dip, it may not be distinguished among others
    • We can assume all electrons coming from donor dopants will be delocalized and become electrons in the conduction band
• In N-type silicon, electrons significantly outnumber holes because of donor dopants
  • Electrons are the majority carriers
  • Holes are the minority carriers
• P-type silicon is similar, but with holes as majority carriers and electrons as minority carriers

# Carrier Density

• When doped with both donor and acceptor dopants:
  • Recombination: electrons from donor dopants can recombine with holes from acceptor dopants
  • Cancel each other out instead of adding up the number of carriers
• When dopants are added to the system originally at thermal equilibrium (take donor dopants as an example):
  • Holes can more easily recombine with electrons
  • Newly added electrons $\gg$ number of holes
    • Number of electrons does not change much
    • Number of holes decreases significantly
  • At new thermal equilibrium:
    • $pn=n_i^2$ (always the case at thermal equilibrium)
    • $n=p+N_D$ (every hole must come from an electron)
    • $N_D\gg p$ (Majority carriers come from dopants)
    • Thus, $p=n_i^2/N_D$

# Locating the Fermi Level

• For N-type silicon, with additional electrons in the conduction band, the probability of finding an electron in the conduction band increases
  • Temperature is the same, the shape of the Fermi-Dirac distribution does not change
  • The Fermi level must have moved up to increase the probability of finding an electron in the conduction band
  • The Fermi level of intrinsic silicon is marked as $E_i$
  • $$\begin{array}{rl}n& =N_C{e}^{-\frac{E_C-E_F}{kT}}\\ n& =N_D\\ \Rightarrow E_C-E_F& =kT\ln \frac{N_C}{N_D}\end{array}$$
  • Equivalent density of states $N_C$ does not change with doping$$\{\begin{array}{ll}n=N_C{e}^{-\frac{E_C-E_F}{kT}}& \text{in N-type silicon}\\ n_i=N_C{e}^{-\frac{E_C-E_i}{kT}}& \text{in intrinsic silicon}\end{array}$$
  • Dividing the two equations:$$n=n_i{e}^{\frac{E_F-E_i}{kT}}$$or$$E_F-E_i=kT\ln \frac{n}{n_i}$$
• For P-type silicon, the Fermi level moves down
  • $$\begin{array}{rl}p& =N_V{e}^{-\frac{E_F-E_V}{kT}}\\ p& =N_A\\ \Rightarrow E_F-E_V& =-kT\ln \frac{N_V}{N_A}\end{array}$$
  • $$\begin{array}{rl}& \{\begin{array}{ll}p=N_V{e}^{-\frac{E_F-E_V}{kT}}& \text{in P-type silicon}\\ p_i=n_i=N_V{e}^{-\frac{E_i-E_V}{kT}}& \text{in intrinsic silicon}\end{array}\\ \Rightarrow & p=n_i{e}^{\frac{E_i-E_F}{kT}}\\ \Rightarrow & E_i-E_F=kT\ln \frac{p}{n_i}\end{array}$$

# Water Analogy for Fermi Level

• Water level represents the Fermi level
• When dopants are added, the water level rises or falls
• For N-type silicon, the weight of the box representing the semiconductor increases, pushing the box down and the water level up
• For P-type silicon, the weight of the box decreases, causing the box to rise and the water level to fall

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