ECSE-6230 Semiconductor Devices and Models I Lecture 8

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ECSE-6230Semiconductor Devices and Models
ILecture 8

• Prof. Shayla Sawyer
• Bldg. CII, Rooms 8225
• Rensselaer Polytechnic Institute
• Troy, NY 12180-3590
• Tel. [phone]
• Fax. [phone]
• e-mail sawyes_at_rpi.edu

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sawyes_at_rpi.edu www.rpi.edu/sawyes/courses.html

December 2, 2013
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Lecture Outline

• PN Junctions
• Built in Potential
• Depletion Approximation
• Depletion Width
• Depletion Capacitance
• Linearly Graded Junction

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PN Junction Background

• The pn junction theory serves as the foundation
  of the physics of semiconductor physics
• The basic equations are used to develop the ideal
  static and dynamic characteristics of pn
  junctions
• Departures from ideal
• Generation and recombination in the depletion
  layer
• High injection
• Series resistance
• Junction breakdown

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Built-in Potential Abrupt pn junction

• Joining the two regions of p and n type doping
  causes diffusion
• Large carrier concentration gradient at the
  junction
• Space charge region diffusion carriers leave
  behind uncompensated donor ions in the n region
  (Nd) and uncompensated acceptor ions in the p
  region (Na-)
• Drift current opposes diffusion current

n
p
n
p
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Built-In Potential Abrupt pn junction

• Constant Fermi level throughout the sample
• Band bending and built in potential occurs
• Electric field is proportional to the slope of
  the bands, charge density proportional to the
  curvature

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Built-In Potential Derivation

• Finding potential
• Solving for the electric field from JN equation
  and with Einsteins relationship, integrate the
  electric field to find Vbi
• Vbi for a non degenerate semiconductor is

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Built-In Potential Derivation

• Built-in potential is the internal potential
  difference between the p-side and the n-side of
  the junction

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Depletion Approximation

• Electrostatic Solution of an abrupt, uniformly
  doped pn junction at thermal equilibrium ( J 0
  )
• Poissons Equation
• Depletion approximation allows Poissons equation
  to be solved easily
• It assumes that the mobile carriers (n and p) are
  small in number compared to the donor and
  acceptor ion concentrations in the depletion
  region
• The device is charge neutral elsewhere

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Depletion Width, Vbi, Carrier Conc.

• Poissons Equation
• Integrate above For -xp lt x lt 0 (
  p-side),
• For 0 lt x lt xn (
  n-side),
• Maximum electric field is given by

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Depletion Width, Vbi, Carrier Conc.

• Integrate electric field equation to get
  potential distribution

Potential across different regions
?bi
-?n
?p
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Depletion Width, Vbi, Carrier Conc.

• Depletion widths are calculated to be
• Penetration of the transition region into the n
  and p materials

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Depletion Capacitance

• Variation of charge within a pn junction with an
  applied voltage variation yields a capacitance
• Capacitance in non-linear derive small signal
  capacitance associated with the depletion layer
  of a pn junction.
• Define change of charge
• DC signal VA has a small signal superimposed onto
  it va, W increases or decreseas by an increment
  of ?W, for small signals valtltVA, ?WltltW
• Charge is only added and removed at the edge of
  the depletion region
• When vagt0, W decreases (neutralizing ions)
• When va lt 0, W increases (more depletion)

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Depletion Capacitance

• Two observations
• Charges are majority carriers which respond to
  the voltage change at roughly the dielectric
  relaxation time of the material
• At normal doping levels the majority carrier
  response time is from 10-10 to 10-12 sec.
  Therefore, the phenomena will be independent of
  the frequency of va up to very high frequencies
• The incremental charge diagrams are similar to
  charge fluctuations in parallel plates of a
  capacitor with area A and width separation W

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Depletion Capacitance

• Depletion layer capacitance per unit area pn
  junction
• Depletion layer capacitance per unit area for a
  one sided abrupt junction p n where NagtgtgtNd and
  xnoW and xpo is negligible

Parameter extraction Rearrange and solve for 1/C2
plot vs Voltage
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Depletion Layer Widths in PN Junctions
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Debye Length

• Capacitance voltage data are insensitive to
  changes in doping profiles that occur in a
  distance less than a Debye Length
• Limit of a potential change in response to an
  abrupt change in the doping profile
• If the depletion width is smaller than the Debye
  Length, the analysis using the Poissons equation
  is no longer valid
• Also defined as the width of the transition
  region in which the carrier depletion goes from
  100 to 0 and can written as
• Typically, W 8LD in Si 6 LD in Ge
  10 LD in GaAs

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Extrinsic Debye Length
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Linearly Graded PN Junction

• In practical devices the doping profiles are not
  abrupt
• Near the metallurgical junction the two types
  compensate each other
• When depletion widths terminate within this
  transition region, the doping profile can be
  approximated as a linear function

Poissons equation
a is the doping gradient in cm-4
Electric field (integrate above)
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Linearly Graded PN Junction

• Built in potential related to the depletion width
• or
• Depletion layer capacitance

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Example