PN junctions and diodes

1
PN junctions and diodes

• Week 7
• Reference Chapter 6 class text

2
pn junction in equilibrium

• Far from a pn junction of p type and n type
  material we would expect the normal p type and n
  type Fermi level diagram. In equilibrium with no
  applied voltage dEf/dx is constant so the Fermi
  levels of the two materials are referenced to a
  consant Fermi level. In the region of the
  junction there is a built in Electric Field
  indicated by band bending of Ec, Ev, and Ei.

3
PN junction in equilibrium
4
PN junction in equilibrium built in potential Vbi
from Fermi level diagram

• For the diagram on the left for the p material
• For the diagram on the right for the n material
• The total change in Ei is the built in potential
  Vbi

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PN junction equilibrium built in potential from
drift diffusion

• No current flows in equilibrium so the current
  equation is
• Taking the integral across the junction and using
  the Einstein mobility relationship D/uKT/q
  results in the equation

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PN junction equilibrium built in potential from
drift diffusion 2

• Using the n(0) ni2/NA and n(L) ND results in
• From either the graph or the integral of the
  drift diffusion equation then the built in
  potential of a pn junction with no external
  aplied voltage is

7
PN junction with applied voltage

• I f we apply a voltage vdiode to the p side and
  assume the current is zero and that the n side
  concentration remains ND we get
• Solving for n(0) the n concnetration at the edge
  of the p region gives

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PN junction with applied voltage 2

• By a simular argument for holes
• If we now consider outside the junction region
  the carriers are diffusing with recombination in
  a charge neutral region
• The excess holes diffusing and recombining in the
  n region is given by

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PN junction with applied voltage 3

• The hole diffusion current density at x 0 is
  then
• The electron distribution on th p side for xlt0 is
  (note the junction is assumed to be zero width
  compared with the diffusion lengths)

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PN junction with applied voltage 4

• The electron current density is then
• The diode current is then given by
• The diode magnitude constant I0 is then

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pn junction solution assumptions

• (1) the depletion region is much smaller than the
  diffusion lengths Lp and Ln.
• (2) beyond the depletion region the electric
  field is negligible so diffusion with
  recombination applies
• (3) the recombining majority carriers and the
  majority carrier flow to and across the junction
  can be supplied by a minute ohmic voltage drop
  disreguared for the diffusion solution.

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Depletion region approximation

• Electrons diffusing into the p region recombine
  with holes leaving just acceptor atoms with
  captured electrons representing fixed immobile
  negative charge.
• Holes diffusing into the n region recombine with
  electrons leaving just ionized donor atoms that
  have given up an electron and have net positive
  electric charge

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Poissons equation for the depletion width of the
junction

• In equilibrium with no applied voltage where now
  we assume the depletion region and electric field
  begin at x -xp and end at xn and that the
  doping abruptly changes from NA to ND at the
  junction at x 0.
• Poissons equation in the two regions is then
• -xp lt x lt0 0 lt x lt xn

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Poissons equation continued 2

• Taking the integrals we get for xlt0 andn xgt0
• The resulting ?(x) for x lt0 and xgt0 are
• For x 0 equating the two ?(0) results in the
  relationship between depletion region width and
  doping

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Poissons equation continued 3

• Taking the integral of ?(x) over the depletion
  region for xlt0 and xgt 0 results in
• The total voltage V(xn) must be Vbi
• The total width of the junction W can be
  expressed as

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Poissons equation continued 4

• Using the expressions to eleminate xp and xn
• Further algebraic reduction results in
• Finally solving for W results in
• If a forward voltage Vdiode is applied it works
  to reduce the built in voltage resulting in

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Equilibrium depletion width of the junction

• The region width doping equation and the
  depletion width equations are
• The p and n doping ratio determines the ratio of
  xn and xp. If NAgtND then xngtxp and if NAltND then
  xpgtxn. The depletion width extends mostly into
  the side opposite the heavily doped side.
• The total width W is inversely proportional to
  the doping. The higher the doping the smaller the
  junction width.

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Nonequilibrium depletion width of the junction

• With applied voltage the junction width W is
  given by
• With forward bias the junction width decreases,
  and with reverse bias increases.
• With AC varying voltage the boundaries between
  the depletion region and the charge neutral
  regions move making an effective AC junction
  depletion capacitance per unit area