Transcript Chap 4. Semiconductor in Equilibrium

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
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
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.
n( E )  g ( E ) f ( E )
c
The thermal equilibrium conc. of electrons no is given by

no   g c ( E ) f ( E )
Ec
 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.
p( E )  g v ( E )[1  f ( E )]
 And the thermal equilibrium conc. Of holes po is given by
Ev
po   gv ( E )[1  f ( E )]

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Equilibrium Distribution of Electrons and Holes
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The no and po eqs.
 Recall the thermal equilibrium conc. of electrons

no   g c ( E ) f ( E )
Ec
 Assume that the Fermi energy is within the bandgap. For electrons in
the conduction band, if Ec-EF >>kT, then E-EF>>kT, so the Fermi
probability function reduces to the Boltzmann approximation,
f ( E )  exp
 Then
no  

Ec

4 2mn*
h3

32
[( E  EF )]
kT
 2mn* kT 
  ( E  EF ) 

E  Ec exp 
dE  2
2

kT
h




 2mn* kT 
 We may define N c  2 h 2 


32
  ( Ec  E F ) 
exp 

kT


32
, (at T =300K, Nc ~1019 cm-3), which
is called the effective density of states function in the conduction band
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The no and po eqs.
 The thermal equilibrium conc. of holes in the valence band is given by
Ev
po   gv ( E )[1  f ( E )]

 For energy states in the valence band, E<Ev. If (EF-Ev)>>kT,
 Then,
po  
Ev

1  f ( E )  exp

4 2m

* 32
p
h3
[( EF  E )]
kT
 2m*p kT 
  ( EF  E ) 

Ev  E exp 
dE  2
2



kT


 h

 2m*p kT 
 We may define N v  2 h 2 


32
  ( E F  Ev ) 
exp 

kT


32
, (at T =300K, Nv ~1019 cm-3), which
is called the effective density of states function in the valence band
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nopo product
 The product of the general expressions for no and po are given by
  Eg 
  ( Ec  EF ) 
  ( EF  Ev ) 
no po  N c exp 

N
exp

N
N
exp
c v


 v


kT
kT




 kT 
 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
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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,
32
 2mn* kT 
  ( Ec  EFi ) 
  ( Ec  EFi ) 

no  ni  2
exp

N
exp
c
2





kT
kT




 h

32
 2m*p kT 
  ( EFi  Ev ) 
  ( EFi  Ev ) 

po  pi  2
exp

N
exp
v




 h2 
kT
kT






  Eg 
 n  N c N v exp 
, where Eg is the bandgap energy
kT


2
i
 For an given semiconductor at a constant temperature, the value of ni is
constant, and independent of the Fermi energy.
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Intrinsic Carrier Conc.
 Commonly accepted values
of ni at T = 300 K
Silicon
GaAs
Germanium
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Chap. 4
ni = 1.5x1010 cm-3
ni = 1.8x106 cm-3
ni = 1.4x1013 cm-3
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Intrinsic Fermi-Level Position
 For an intrinsic semiconductor, ni = pi,
 ( Ec  EFi )
 ( E Fi  Ev )
 N c exp[
]  N v exp[
]
kT
kT
m*p
Nv
1
3
1
3
 EFi  ( Ec  Ev )  kT ln( )  ( Ec  Ev )  kT ln( * )
2
4
Nc
2
4
mn




Emidgap =(Ec+Ev)/2: is called the midgap energy.
If mp* = mn*, then EFi = Emidgap (exactly in the center of the bandgap)
If mp* > mn*, then EFi > Emidgap (above the center of the bandgap)
If mp* < mn*, then EFi < Emidgap (below the center of the bandgap)
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Dopant and Energy Levels
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Dept. of E. E. NCU
Acceptors and Energy Levels
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Ionization Energy
 Ionization energy is the energy required to elevate the donor electron
into the conduction band.
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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
  (E  E ) 
  (E  E ) 
ni  N c exp 

c
kT
Fi
  N v exp 


Fi
kT
v


  ( Ec  EF ) 
  ( Ec  EFi )  ( EF  EFi ) 
no  N c exp 

N
exp
c



kT
kT




  ( E F  Ev ) 
  ( EF  EFi )  ( Ev  EFi ) 
po  N v exp 

N
exp
v



kT
kT




 E  EFi 
  ( EF  EFi ) 
 no  ni exp  F
and
p

n
exp
o
i



kT
 kT 


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Extrinsic Semiconductor
 When the donor impurity atoms are added, the density of electrons is
greater than the density of holes, (no > po)  n-type; EF > EFi
 When the acceptor impurity atoms are added, the density of electrons
is less than the density of holes, (no < po)  p-type; EF < EFi
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Dept. of E. E. NCU
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
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Degenerate and Nondegenerate
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Dept. of E. E. NCU
Statistics of Donors and Acceptors
 The probability of electrons occupying the donor energy state was
given by
N
nd 
d
E  EF
1
1  exp( d
)
g
kT
, g : degeneracy factor
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
pa 
Na
, g  4 for the acceptor level in Si and GaAs
E F  Ea
1
1  exp(
)
g
kT
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Complete Ionization
 If we assume Ed-EF>> kT or EF-Ea >> kT (e.g. T= 300 K), then
  ( Ed  E F ) 

nd  2 N d exp 

N
 N d  nd  N d
d

kT


  ( E F  Ea ) 

pa  4 N a exp 

N
 N a  pa  N a
a

kT


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.
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Dept. of E. E. NCU
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.
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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 > Na and a p-type semiconductor
occurs when Na > Nd.
 The charge neutrality condition is expressed by
no  N a  po  N d
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.
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Compensated Semiconductor
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Compensated Semiconductor
 If we assume complete ionization, Nd+ = Nd and Na- = Na, then
ni2
no  N a  po  N d , recall po 
no
ni2
2
no  N a 
 N d  no  ( N d  N a )no  ni2  0
nn
 no
N  N a  
 d
2
 Nd  Na 
2

  ni
2


2
 If Na = Nd = 0, (for the intrinsic case), no = po
 If Nd >> Na, no = Nd
 If Na > Nd,  po
N  N d  
 a
2
 Na  Nd 
2

  ni
2


2
is used to
calculate the conc. of holes in valence band
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Compensated Semiconductor
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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
no  N c exp 

 Ec  EF 
 and p  N exp  EF  Ev  
o
v
kT 
kT 

N 
 Ec  EF  kT ln  c 
 no 
N 
and EF  Ev  kT ln  v 
 po 
n 
or EF  EFi  kT ln  o 
 ni 
p 
and EFi  Ev  kT ln  o 
 ni 
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EF(n, p, T)
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EF(n, p, T)
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Homework
 4.18
 4.20
 4.24
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Dept. of E. E. NCU