Instructor: Pei-Wen Li

1
Chap 4. Semiconductor in Equilibrium

• Carriers in Semiconductors
• Dopant Atoms and Energy Levels
• Extrinsic Semiconductor
• Statistics of Donors and Acceptors
• Charge Neutrality
• Position of Fermi Energy

2
Equilibrium Distribution of Electrons and Holes

• The distribution of electrons in the conduction
  band is given by the density of allowed quantum
  states times the probability that a state will be
  occupied.
• The thermal equilibrium conc. of electrons no is
  given by
• Similarly, the distribution of holes in the
  valence band is given by the density of allowed
  quantum states times the probability that a state
  will not be occupied by an electron.
• And the thermal equilibrium conc. Of holes po is
  given by

3
Equilibrium Distribution of Electrons and Holes
4
The no and po eqs.

• Recall the thermal equilibrium conc. of electrons
  
• Assume that the Fermi energy is within the
  bandgap. For electrons in the conduction band, if
  Ec-EF gtgtkT, then E-EFgtgtkT, so the Fermi
  probability function reduces to the Boltzmann
  approximation,
• Then
• We may define , (at
  T 300K, Nc 1019 cm-3), which
• is called the effective density of states
  function in the conduction band

5
The no and po eqs.

• The thermal equilibrium conc. of holes in the
  valence band is given by
• For energy states in the valence band, EltEv. If
  (EF-Ev)gtgtkT,
• Then,
• We may define , (at
  T 300K, Nv 1019 cm-3), which
• is called the effective density of states
  function in the valence band

6
nopo product

• The product of the general expressions for no and
  po are given by
• ? for a semiconductor in thermal equilibrium, the
  product of no and po is always a constant for a
  given material and at a given temp.
• Effective Density of States Function

7
Intrinsic Carrier Concentration

• For an intrinsic semiconductor, the conc. of
  electrons in the conduction band, ni, is equal to
  the conc. of holes in the valence band, pi.
• The Fermi energy level for the intrinsic
  semiconductor is called the intrinsic Fermi
  energy, EFi.
• For an intrinsic semiconductor,
• For an given semiconductor at a constant
  temperature, the value of ni is constant, and
  independent of the Fermi energy.

8
Intrinsic Carrier Conc.

• Commonly accepted values
• of ni at T 300 K
• Silicon ni 1.5x1010 cm-3
• GaAs ni 1.8x106 cm-3
• Germanium ni 1.4x1013 cm-3

9
Intrinsic Fermi-Level Position

• For an intrinsic semiconductor, ni pi,
• Emidgap (EcEv)/2 is called the midgap energy.
• If mp mn, then EFi Emidgap (exactly in the
  center of the bandgap)
• If mp gt mn, then EFi gt Emidgap (above the
  center of the bandgap)
• If mp lt mn, then EFi lt Emidgap (below the
  center of the bandgap)

10
Dopant and Energy Levels
11
Acceptors and Energy Levels
12
Ionization Energy

• Ionization energy is the energy required to
  elevate the donor electron into the conduction
  band.

13
Extrinsic Semiconductor

• Adding donor or acceptor impurity atoms to a
  semiconductor will change the distribution of
  electrons and holes in the material, and
  therefore, the Fermi energy position will change
  correspondingly.
• Recall

14
Extrinsic Semiconductor

• When the donor impurity atoms are added, the
  density of electrons is greater than the density
  of holes, (no gt po) ? n-type EF gt EFi
• When the acceptor impurity atoms are added, the
  density of electrons is less than the density of
  holes, (no lt po) ? p-type EF lt EFi

15
Degenerate and Nondegenerate

• If the conc. of dopant atoms added is small
  compared to the density of the host atoms, then
  the impurity are far apart so that there is no
  interaction between donor electrons, for example,
  in an n-material.
• ?nondegenerate semiconductor
• If the conc. of dopant atoms added increases such
  that the distance between the impurity atoms
  decreases and the donor electrons begin to
  interact with each other, then the single
  discrete donor energy will split into a band of
  energies. ?EF move toward Ec
• The widen of the band of donor states may overlap
  the bottom of the conduction band. This occurs
  when the donor conc. becomes comparable with the
  effective density of states, EF ? Ec
• ?degenerate semiconductor

16
Degenerate and Nondegenerate
17
Statistics of Donors and Acceptors

• The probability of electrons occupying the donor
  energy state was given by
• where Nd is the conc. of donor atoms, nd is the
  density of electrons occupying the donor level
  and Ed is the energy of the donor level. g 2
  since each donor level has two spin orientation,
  thus each donor level has two quantum states.
• Therefore the conc. of ionized donors Nd Nd
  nd
• Similarly, the conc. of ionized acceptors Na-
  Na pa, where

18
Complete Ionization

• If we assume Ed-EFgtgt kT or EF-Ea gtgt kT (e.g. T
  300 K), then
• that is, the donor/acceptor states are almost
• completely ionized and all the
  donor/acceptor
• impurity atoms have donated an electron/hole
• to the conduction/valence band.

19
Freeze-out

• At T 0K, no electrons from the donor state are
  thermally elevated into the conduction band this
  effect is called freeze-out.
• At T 0K, all electrons are in their lowest
  possible energy state that is for an n-type
  semiconductor, each donor state must contain an
  electron, therefore, nd Nd or Nd 0, which
  means that the Fermi level must be above the
  donor level.

20
Charge Neutrality

• In thermal equilibrium, the semiconductor is
  electrically neutral. The electrons distributing
  among the various energy states creating negative
  and positive charges, but the net charge density
  is zero.
• Compensated Semiconductors is one that contains
  both donor and acceptor impurity atoms in the
  same region. A n-type compensated semiconductor
  occurs when Nd gt Na and a p-type semiconductor
  occurs when Na gt Nd.
• The charge neutrality condition is expressed by
• where no and po are the thermal equilibrium
  conc. of e- and h in the conduction band and
  valence band, respectively. Nd is the conc. Of
  positively charged donor states and Na- is the
  conc. of negatively charged acceptor states.

21
Compensated Semiconductor
22
Compensated Semiconductor

• If we assume complete ionization, Nd Nd and
  Na- Na, then
• If Na Nd 0, (for the intrinsic case), ?no
  po
• If Nd gtgt Na, ?no Nd
• If Na gt Nd, is used to
• calculate the conc. of holes in valence band

23
Compensated Semiconductor
24
Position of Fermi Level

• The position of Fermi level is a function of the
  doping concentration and a function of
  temperature, EF(n, p, T).
• Assume Boltzmann approximation is valid, we have

25
EF(n, p, T)
26
EF(n, p, T)
27
Homework

• 4.18
• 4.20
• 4.24