半导体器件原理 1.4 PN Junction Formation

About formation of the PN junction, band diagram, depletion region width calculation, one-sided PN junction, and whether we can measure the built-in potential.

# Doping of the PN Junction

• Diode: Putting P-type and N-type silicon together, forming a PN junction.
• Recall recombination:
  • When electrons coming from donors meet holes from acceptor, recombination happens.
  • After recombination, location of dopants carries a small positive charge, and location of acceptors carries a small negative charge.
• When we put P-type silicon on the left, and N-type silicon on the right:

     P   |   N
  . . . .|. . . .

  we assume that recombination happens in an orderly manner, starting from the interface of the two types of silicon (the junction).

     P   |   N
  . . . -|+ . . .

     P   |   N
  . . - -|+ + . .
      ^^^^^^^
     depletion region

  areas near the junction are depleted of free carriers, forming a depletion region, where generally assumed to have no carriers.
  • Covalence bounds are similar to intrinsic silicon.
  • Assumption is not accurate, as there are still carriers in intrinsic silicon
  • The number is much smaller than the number of those added by doping, so we can ignore them.
• As recombination happens, charges accumulates, which tend to oppose further diffusion of carriers across the junction.
• An equilibrium will be reached eventually, with a particular depletion width.
• Simplifications we made:
  • Abrupt junction approximation: Assumes there is a clear boundary between P-type and N-type silicon, where a transition region (both donors and acceptors exist) exists in reality.
  • Depletion approximation: Assumes the depletion region is fully devoid of charge carriers and there is a clear boundary between the depletion region and the neutral region.

# Band Diagram of the PN Junction

• When PN junction is formed, the Fermi levels of P-type and N-type silicon must align at equilibrium.
• In P side, $E_i>E_F$
• In N side, $E_i<E_F$
• The built-in potential $V_{bi}$ satisfies:$$q{V}_{bi}=q{V}_p+q{V}_n$$where $q{V}_p=E_{i_P}-E_F$, $q{V}_n=E_F-E_{i_N}$
• $$\{\begin{array}{l}{E}_{i_P}-E_F=kT\ln \frac{N_A}{n_i}\\ E_F-E_{i_N}=kT\ln \frac{N_D}{n_i}\end{array}$$
• Therefore,$$V_{bi}=\frac{kT}{q}\ln \frac{N_A{N}_D}{n_i^2}$$

# Calculating the Depletion Region Width

• Depletion region width $x_d=x_p+x_n$
• Charge neutrality:$$N_A{x}_p=N_D{x}_n$$
• Charge density:$$\rho (x)=\{\begin{array}{ll}-q{N}_A,& -x_p<x<0\\ +q{N}_D,& 0<x<x_n\\ 0,& \text{elsewhere}\end{array}$$
• Using Poisson’s equation:$${\mathrm{\nabla }}^{2}V=-\frac{\rho }{\epsilon }$$
• In 1D:$$\frac{d^{2}V}{d{x}^2}=-\frac{\rho }{\epsilon }$$
• Integrate twice:$$V_{bi}=\frac{q{N}_A{x}_p^2}{2{\epsilon }_{\text{Si}}}+\frac{q{N}_D{x}_n^2}{2{\epsilon }_{\text{Si}}}$$
• Solve with equation (1) and (2):$$x_p=\sqrt{\frac{2{\epsilon }_{\text{Si}}{V}_{bi}}{q}\cdot \frac{N_D}{N_A(N_A+N_D)}}$$$$x_n=\sqrt{\frac{2{\epsilon }_{\text{Si}}{V}_{bi}}{q}\cdot \frac{N_A}{N_D(N_A+N_D)}}$$
• Finally:$$x_d=\sqrt{\frac{2{\epsilon }_{\text{Si}}{V}_{bi}}{q}(\frac{1}{N_A}+\frac{1}{N_D})}$$where$$V_{bi}=\frac{kT}{q}\ln \frac{N_A{N}_D}{n_i^2}$$

# One-Sided PN Junction

• Doping on two sides are asymmetrical
  • Usually, a PN junction is formed by counter doping to convert part of a material to the opposite type
  • Counter doping concentration is much higher to minimize the background dopants
• In the depletion region expression, higher doping concentration term can be removed, and the depletion region width is mainly controlled by the lightly doped side
  • Graphically, the depletion region extends much more into the lightly doped side (PN junction is one-sided)

# Measuring the Built-in Potential

• The built-in potential cannot be measured directly with a voltmeter
• The potential difference will be canceled out by the contact potential when the voltmeter is connected

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