OUTLINE

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Lecture 5

• OUTLINE
• Intrinsic Fermi level
• Determination of EF
• Degenerately doped semiconductor
• Carrier properties
• Carrier drift
• Read Sections 2.5, 3.1

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Intrinsic Fermi Level, Ei

• To find EF for an intrinsic semiconductor, use
  the fact that n p

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n(ni, Ei) and p(ni, Ei)

• In an intrinsic semiconductor, n p ni and EF
  Ei

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Example Energy-band diagram

• Question Where is EF for n 1017 cm-3 ?

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Dopant Ionization

• Consider a phosphorus-doped Si sample at 300K
  with
• ND 1017 cm-3. What fraction of the donors are
  not ionized?
• Answer Suppose all of the donor atoms are
  ionized.
• Then
• Probability of non-ionization ?

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Nondegenerately Doped Semiconductor

• Recall that the expressions for n and p were
  derived using the Boltzmann approximation, i.e.
  we assumed
• The semiconductor is said to be nondegenerately
  doped in this case.

Ec
3kT
EF in this range
3kT
Ev
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Degenerately Doped Semiconductor

• If a semiconductor is very heavily doped, the
  Boltzmann approximation is not valid.
• In Si at T300K Ec-EF lt 3kT if ND gt 1.6x1018
  cm-3
• EF-Ev lt 3kT if NA gt 9.1x1017 cm-3
• The semiconductor is said to be degenerately
  doped in this case.
• Terminology
• n ? degenerately n-type doped. EF ? Ec
• p ? degenerately p-type doped. EF ? Ev

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Band Gap Narrowing

• If the dopant concentration is a significant
  fraction of the silicon atomic density, the
  energy-band structure is perturbed ? the band gap
  is reduced by DEG

N 1018 cm-3 DEG 35 meV N 1019 cm-3 DEG
75 meV
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Mobile Charge Carriers in Semiconductors

• Three primary types of carrier action occur
  inside a semiconductor
• Drift charged particle motion under the
  influence of an electric field.
• Diffusion particle motion due to concentration
  gradient or temperature gradient.
• Recombination-generation (R-G)

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Electrons as Moving Particles
In vacuum
In semiconductor
F (-q)E mna where mn is the electron
effective mass
F (-q)E moa
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Carrier Effective Mass

• In an electric field, E, an electron or a hole
  accelerates
• Electron and hole conductivity effective masses

electrons
holes


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Thermal Velocity
Average electron kinetic energy
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Carrier Scattering

• Mobile electrons and atoms in the Si lattice are
  always in random thermal motion.
• Electrons make frequent collisions with the
  vibrating atoms
• lattice scattering or phonon scattering
• increases with increasing temperature
• Average velocity of thermal motion for electrons
  107 cm/s _at_ 300K
• Other scattering mechanisms
• deflection by ionized impurity atoms
• deflection due to Coulombic force between
  carriers
• carrier-carrier scattering
• only significant at high carrier concentrations
• The net current in any direction is zero, if no
  electric field is applied.

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Carrier Drift

• When an electric field (e.g. due to an externally
  applied voltage) is applied to a semiconductor,
  mobile charge-carriers will be accelerated by the
  electrostatic force. This force superimposes on
  the random motion of electrons

• Electrons drift in the direction opposite to the
  electric field
• ? current flows
• Because of scattering, electrons in a
  semiconductor do not achieve constant
  acceleration. However, they can be viewed as
  quasi-classical particles moving at a constant
  average drift velocity vd

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Electron Momentum

• With every collision, the electron loses momentum
• Between collisions, the electron gains momentum
• (-q)Etmn
• tmn is the average time between electron
  scattering events

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Carrier Mobility
mnvd (-q)Etmn
vd qEtmn / mn mn E

• ?n ? qtmn / mn is the electron mobility

vd qEtmp / mp ? mp E
Similarly, for holes

• ?p ? qtmp / mp is the hole mobility

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Electron and Hole Mobilities
? has the dimensions of v/E
Electron and hole mobilities of selected
intrinsic semiconductors (T300K)
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Example Drift Velocity Calculation
a) Find the hole drift velocity in an intrinsic
Si sample for E 103 V/cm. b) What is the
average hole scattering time? Solution a) b)
vd mn E
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Mean Free Path

• Average distance traveled between collisions

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Summary

• The intrinsic Fermi level, Ei, is located near
  midgap
• Carrier concentrations can be expressed as
  functions of Ei and intrinsic carrier
  concentration, ni
• In a degenerately doped semiconductor, EF is
  located very near to the band edge
• Electrons and holes can be considered as
  quasi-classical particles with effective mass m
• In the presence of an electric field e, carriers
  move with average drift velocity ,
  where m is the carrier mobility