SEMICONDUCTORS

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SEMICONDUCTORS

• Semiconductors
• Semiconductor devices

Electronic Properties Robert M Rose, Lawrence A
Shepart, John Wulff Wiley Eastern Limited, New
Delhi (1987)
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Energy gap in solids

• In the free electron theory a constant potential
  was assumed inside the solid
• In reality the presence of the positive ion
  cores gives rise to a varying potential field
• The travelling electron wave interacts with this
  periodic potential (for a crystalline solid)
• The electron wave can be Bragg diffracted

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Bragg diffraction from a 1D solid
1D ? ?90o
n? 2d
n? 2d Sin?

• The Velocity of electrons for the above values
  of k are zero
• These values of k and the corresponding E are
  forbidden in the solid
• The waveform of the electron wave is two
  standing waves
• The standing waves have a periodic variation in
  amplitude and hence the electron probability
  density in the crystal
• The potential energy of the electron becomes a
  function of its position (cannot be assumed to
  be constant (and zero) as was done in the free
  electron model)

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k ?
E ?
Band gap
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• The magnitude of the Energy gap between two
  bands is the difference in the potential energy
  of two electron locations

K.E of the electron increasingDecreasing
velocity of the electron?ve effective mass (m)
of the electron
Within a band
E ?
k ?
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Effective energy gap ? Forbidden gap ? Band gap
110
100
Effective gap
E ?
E ?
k ?
k ?
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• The effective gap for all directions of motion
  is called the forbidden gap
• There is no forbidden gap if the maximum of a
  band for one direction of motion is higher
  than the minimum for the higher band for another
  direction of motion ? this happens if the
  potential energy of the electron is not a
  strong function of the position in the crystal

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Energy band diagram METALS
Divalent metals
Monovalent metals

• Monovalent metals Ag, Cu, Au ? 1 e? in the
  outermost orbital ? outermost energy band is
  only half filled
• Divalent metals Mg, Be ? overlapping conduction
  and valence bands ? they conduct even if the
  valence band is full
• Trivalent metals Al ? similar to monovalent
  metals!!! ? outermost energy band is only half
  filled !!!

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Energy band diagram SEMICONDUCTORS
2-3 eV

• Elements of the 4th column (C, Si, Ge, Sn, Pb) ?
  valence band full but no overlap of valence and
  conduction bands
• Diamond ? PE as strong function of the position
  in the crystal ? Band gap is 5.4 eV
• Down the 4th column the outermost orbital is
  farther away from the nucleus and less bound ?
  the electron is less strong a function of the
  position in the crystal ? reducing band gap down
  the column

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Energy band diagram INSULATORS
gt 3 eV
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Intrinsic semiconductors

• At zero K very high field strengths ( 1010 V/m)
  are required to move an electron from the top of
  the valence band to the bottom of the
  conduction band
• ? Thermal excitation is an easier route

P(E) ?
E ?
Eg
Eg/2
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T gt 0 K
? Unity in denominator can be ignored

• ne ? Number of electrons promoted across the
  gap ( no. of holes in the valence band)
• N ? Number of electrons available
  at the top of the valance band for excitation

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Conduction in an intrinsic semiconductor

• Under applied field the electrons (thermally
  excited into the conduction band) can move using
  the vacant sites in the conduction band
• Holes move in the opposite direction in the
  valence band
• The conductivity of a semiconductor depends on
  the concentration of these charge carriers (ne
  nh)
• Similar to drift velocity of electrons under an
  applied field in metals in semiconductors the
  concept of mobility is used to calculate
  conductivity

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Conductivity as a function of temperature
Ln(?)?
1/T (/K) ?
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Extrinsic semiconductors

• The addition of doping elements significantly
  increases the conductivity of a semiconductor

• Doping of Si ? V column element (P, As, Sb) ?
  the extra unbonded electron is
  practically free (with a radius of motion of
  80 Å) ? Energy level near the conduction
  band ? n- type semiconductor ? III column
  element (Al, Ga, In) ? the extra electron for
  bonding supplied by a neighbouring Si atom ?
  leaves a hole in Si. ? Energy level near
  the valence band ? p- type semiconductor

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• Ionization Energy? Energy required to
  promote an electron from the Donor level to
  conduction band
• EIonization lt Eg ? even at RT large fraction
  of the donor electrons are exited into
  the conduction band

n-type
EIonization
Eg
Donor level

• Electrons in the conduction band are the
  majority charge carriers
• The fraction of the donor level electrons
  excited into the conduction band is much
  larger than the number of electrons excited from
  the valence band
• Law of mass action (ne)conduction band x
  (nh)valence band Constant
• The number of holes is very small in an n-type
  semiconductor
• ? Number of electrons ? Number of holes

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p-type
Acceptor level
Eg
EIonization

• At zero K the holes are bound to the dopant atom
• As T? the holes gain thermal energy and break
  away from the dopant atom ? available for
  conduction
• The level of the bound holes are called the
  acceptor level (which can accept and electron)
  and acceptor level is close to the valance band
• Holes are the majority charge carriers
• Intrinsically excited electrons are small in
  number
• ? Number of electrons ? Number of holes

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Intrinsic
slope
All dopant atoms have been excited
Exhaustion
Exponentialfunction
? (/ Ohm / K)?
Extrinsic
ve slope due to Temperature dependentmobility
term
Slope can be usedfor the calculationof
EIonization
0.1
0.06
0.02
0.04
0.08
50 K
10 K
1/T (/K) ?
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• Semiconductor device ? chose the flat region
  where the conductivity does not change much
  with temperature
• Thermistor (for measuring temperature) ? maximum
  sensitivity is required