Energy Bands and Charge Carriers in Semiconductors

1
Topic III

• Energy Bands and Charge Carriers in Semiconductors

2
Energy Bands

• Energy Bands can be thought of in two different
  ways
• (1) Two atoms share two electrons and the Pauli
  exclusion principles requires the energy level of
  the individual atoms become two levels. Many
  atoms form bands of allowed energy with gaps of
  energy the electrons may not have.
• (2) Solution of the Shrodinger equation for a
  periodic potential gives a range or band of
  energy the electron may have and gaps it may not
  have. Kronig Penny uses delta function V(xa)
  V(x) ?

3
Schrodinger in Crystal

• Periodic potential solution of Shrodinger is a
  Bloch wave
• ?k(x)U(k,x)exp(ikx)
• where U(k,x) is a periodic function with period
  of the lattice, and the wave length associated
  with k has a period much longer than the lattice.
• Resulting energy as a function of monmentum E(k)
  has gaps Eg in allowed levels and can have
  minimum above the gap having k not being zero.
  Such a case is called a indirect bandgap
  semiconductor whereas if k 0 in the allowed
  energy level above Eg the semiconductor is
  referred to as direct band gap. Silicon is a
  indirect bandgap while GaAs is an direct bandgap.

4
Satellite Bands

• E(k) can have more than one minimum
• E(k) is a function of the direction of k relative
  to the lattice
• E(k) is discontinuous when k?/a the lattice
  constant and that k marks the Brillouin zone in k
  space.
• E(k) is graphed along symmetry points, ?(k?0
  point all directions low energy, L point k
  k?/a 111 direction, X point k k?/a 100
  direction Figure 3-10 page 78
• For GaAs the minimum at the L point is called the
  L satellite band.

5
Effective mass

• The quantum E(k) for a free electron and for a
  electron in a crystal with effective mass m is
• m is found from the E(k) found from the
  Shrodinger equation

6
Multiple effective masses

• Silicon has two effective mass ratios associated
  with the indirect bandgap minimum about 0.89
  fraction of the way to the X 100 point not
  being symmetric with respect to energy but a
  paraboloidal with the 100 direction
  longitudinal mass ratio ml/m0 0.98 and the two
  transverse mass ratios mt/m0 0.19 Figure 3-10
  (b) page 78
• GaAs has three effective mass ratios that are
  symmetric and associated with the symmetry points
  of E(k), Gamma minimum m?/m0 0.067, L minimum
  m?/m0 0.55, Gamma minimum m?/m0 0.85 Figure
  3-10 (a) page 78

7
AlxGa1-xAs Band Structure

• AlAs is an indirect bandgap semiconductor with
  the minimum at the X symmetry point and Eg 2.16
  eV
• GaAs is a direct bandgap semiconductor with a
  minimum at the ? point and Eg 1.43 eV
• In AlxGa1-xAs as the composition x varies from 0
  to 1.0 the material goes from a direct bandgap
  material to a indirect bandgap material at x ?
  0.4
• The gap voltages at the ?, L, X symmetry points
  all vary with x as seen in Figure 3-6 c page 72

8
Semiconductor Holes

• Holes are missing valence electrons in a
  semiconductor crystal lattice
• With energy at least Egap a valance electron goes
  into a conduction band state leaving behind a
  missing valence electron called a hole.
• With an applied electric field the adjacent
  valence electrons can jump to the hole leaving
  behind a hole so it appears as if the hole moves
  in the direction opposite.
• In the area of the hole the missing electron and
  made the area charge neutral but with the missing
  electron there is now a unbalanced positive
  electric charge q equal in magnitude to the
  electron charge.

9
Hole effective mass

• The hole effective mass is given by the E(k)
  relationship from the Shrodinger equation for the
  energies less than Egap. The hole energy is
  positive in the downward direction and the
  effective mass is given the same as for the
  electron

10
Multiple Hole Effective Masses

• The valence band E(k) maximum is always at the k
  0 point and is the E(k) minimum for the hole
  (positive energy downward).
• There are multiple bands for the holes, the heavy
  hole band with effective mass ratio mhh/m0,
  light hole band with effective mass ration
  mll/m0 , and split off or Spin Orbit mass
  mso/m0. The heavy and light hole masses are
  given for some semiconductors in Appendix III.

11
Electron effective masses in semiconductor
equations

• E(k) masses have already been developed
• Mobility mass for ml and mt masses averages the
  three diections and they act in parallel
• Density of states mass uses states that are
  products of the three directions and the density
  of states is the 3/2 power of mass and sometimes
  includes a directions factor.
• Hole density of states mass assumes equal heavy
  and light hole populations.

12
Electron effective masses in semiconductor
processes

• In materials with satellite bands like GaAs as
  the electrons gain energy in an Electric Field
  the population of m?, mL, and mX electrons
  change as electrons jump or tunnel to satellite
  bands. The result is a mobility effective mass
  that increases with electric field decreasing
  mobility, electron velocity, and electric
  current. The mobility and mass populations are
  usually found by Monte Carlo methods.

13
Intrinsic Semiconductors

• In Intrinsic (Undoped) semiconductors
• n p ni the intrinsic conduction electron and
  hole concentrations (/cm3).
• ni 1.5x1010/cm3 for Silicon at 23 deg C and
  1.48x106/cm3 for GaAs.
• pn ni2 because the ni value depends on the
  probability the electrons and holes find each
  other to achieve equilibrium of generation and
  recombination.

14
N Type Doped Semiconductors

• In Silicon dopant Group V atoms have an extra
  electron that does not participate in the lattice
  bonding that is easily made a mobile conduction
  electron. The atom has donated an electron.
• n ND where ND is the concentration /cm3 of
  donor atoms.
• p ni2/n ni2/ND and is a small number
• n type semiconductors are donor doped and the
  conductivity is due to mobile electrons.

15
P Type Doped Semiconductors

• In Silicon dopant Group III atoms need an extra
  electron to participate in the lattice bonding so
  it captures a nearby valence electron creating a
  hole. The atom has accepted an electron.
• p NA where NA is the concentration /cm3 of
  acceptor atoms.
• n ni2/p ni2/NA and is a small number
• p type semiconductors are acceptor doped and the
  conductivity is due to mobile holes.

16
Fermi Level in Semiconductors

• Electrons in semiconductors obey Fermi Dirac
  Statistics
• The electron concentration is given by
• Fermi function x Density of States

17
Intrinsic Fermi Level and Reference

• For undoped semiconductors the intrinsic Fermi
  level is Ei Ef Eg/2 to get equal electron and
  hole populations (for holes the probability of
  occupation is 1-f(E,Ef))
• Using Ei as a reference in general for donor
  doped n type semiconductors
• Using Ei as a reference in general for acceptor
  doped p type semiconductors