Minority Carriers, Recombination, Generation, Drift, Diffusion

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Minority Carriers, Recombination, Generation,
Drift, Diffusion

• Week 6

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Photon Absorption

• Photons with energy h? gt Eg (direct or indirect)
  are absorbed by semiconductors.
• The absorbed photon frees a valance electron
  creating a conduction electron and hole pair.
• The absorption is described by the equation
• ? the optical absorption coefficient is a
  function of frequency ? increasing for h? gt Eg
• I0 has units Watts/cm2 or Joules/sec/cm2

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Optical Generation

• For monochromatic light (single frequency) the
  decrease in intensity is also the number of
  photons absorbed per cm3
• If the efficiency for absorption is 100 the
  expression also gives the electron and hole
  generation rate.

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Equilibrium electron hole concentrations for
intrinsic material

• Equilibrium electron hole concentrations for
  intrinsic (undoped) semiconductors is given by
• The ni value is determined by equilibrium between
  electron hole pairs being thermally generated and
  electron hole pairs recombining.
• The probability the electrons and holes find each
  other is proportional to the p and n product

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Equilibrium electron hole concentrations in doped
material

• The pn product based on the probability the
  conduction electrons and holes find each other is
  applicable to all semiconductors
• n type doped ND n ND p minority carrier
• p type doped NA p NA n minority carrier

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Nonequilibrium electron hole concentrations

• If pn gt ni2 the electrons and holes easily find
  each other and recombine restoring equilibrium.
• The rate they the recombine R has a
  characteristic time constant called the
  recombination rate or lifetime tn or tp
• If they recombine directly tn tp but if
  recombination centers are involved tn and tp may
  not be equal
• The rate of recombination is proportional to the
  minority carrier concentration.
• The equations are

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Nonequilibrium and quasi Fermi level

• When electron and hole concentrations are not in
  equilibrium a nonequilibrium Fermi level F is
  said to exist.
• In non-equilibrium the conduction electron quasi
  Fermi level Fn and hole quasi Fermi level Fp will
  not be equal.
• The quasi Fermi levels if p and n are known can
  be found from the equations (Fn in eV)

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Steady state optical generation with recombination

• For steady state optical generation with
  recombination
• The excess electrons and holes are then
• For a photo resistor the illuminated resistivity
  will then be given by

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Recombination processes

• Three primary methods of recombination
• (1) SRH Shockley Read Hall recombination through
  defects characterized by defect density and
  energy level in the band gap for dopant levels
  lt1017
• (2) Radiative recombination in direct band gap
  semiconductors GaAs B 10-10 Silicon B10-14
• (3) Auger recombination requiring two electrons
  and a hole or two holes and an electron for
  doping levels gt 1017

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SRH Recombination

• SRH net recombination rate R-G is
• SRH recombination determined by a trap density
  Nt, trap energy level Et, and trap capture cross
  sections ?n and ?p all determined from
  measurement
• The SRH lifetimes are given in terms of thermal
  velocity, capture cross section cm2, and trap
  density

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Radiative recombination

• Radiative recombination lifetime is given by a
  measured constant B and majority carrier
  concentration as (if ?ngtND use ?n not ND)
• n type p type
• Values for B are
• Silicon B 1.0x10-14 cm3/sec
• GaAs B 1.2x10-10 cm3/sec

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Auger Recombination

• Auger recombination requires two holes and an
  electron or two electrons and a hole
• The Auger lifetime is given by
• Values for C (units are cm6/sec )are
• Silicon Cn 2.8x10-31 Cp 1.0x10-31
• GaAs Cn 8.0x10-32 Cp 2.8x10-31

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Silicon recombination rate

• Silicon measured recombination rate as a function
  of doping has been modeled with the equations
• n type
• p type electron recombination lifetime

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Surface recombination rate

• Surface recombination rate due to defects (SRH)
  tend to be higher than bulk rates due to surface
  defects with Et then being an interface trap
  level.
• The region of applicability of surface
  recombination would be the depth of the depletion
  region associated with the surface.
• Surface recombination is significant in
  heterojunction bipolar transistors

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Diffussion in Semiconductors

• Ficks law applied to semiconductors is
• D is the diffusion constant
• Diffusion is a statistical process and occurs
  anywhere there is a concentration gradient
• The Einstein mobility relationship relates the
  diffusion constant to the mobility as D/? KT/q
  (note that it rhymes and also it is Dn/un and
  Dp/up)

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Drift Diffusion Equations

• If both ohmic electric field conduction (drift)
  and concentration dependant diffusion are present
  the current density equations are
• It can be shown that the above expressions can be
  written( note that Ij0 if dEf/dx0)

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Energy band bending due to Electric Field

• In the presence of an electric field ? the
  intrinsic energy level varies as the electric
  field
• If n(x) is constant
• The bracket expression must be zero so
  dFn/dxdEidx and by the same reasoning dEc/dx
  dEv/dxdEi/dx-dV/dx

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The continuity equation

• The continuity equations for electrons and holes
  are

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Diffusion with recombination

• Steady state diffusion with recombination for
  minority carrier electrons in a p region is
  described by the continuity equation for
  electrons
• Assuming Dn is constant results in the
  differential equation

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Diffusion with recombination

• The solution is
• Where Ln is a characteristic length for the
  excess carriers to extend into the p region and
  is called the diffusion length.
• A similar equation exists for holes ?p(x)
  diffusing with recombination in the n region
• Diffusion in the charge neutral regions (no
  electric field) is responsible for diode current

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Poissons Equation

• Poissons equation is used to find the electric
  field and the electric potential
• For a n type uniformly doped semiconductor
  depleted of electrons and holes
• Taking the first integral results in

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Poissons Equation 2

• Using the electric field in terms of potential
  ?-dV/dx and the boudary condition ?(0) 0
  results in the equation for the electric field as
• Taking the integral of the electric field gives
  the potential (and taking V(0) 0V)