ENE 311
Lecture VI
We have learned how to find new position of Fermi level for extrinsic
semiconductors. Now let us consider the new electron density in case of both donors
ND and acceptors NA are present simultaneously. The Fermi level will adjust itself to
preserve overall charge neutrality as
n  N A  p  N D
(1)
By solving (1) with n. p  ni2 , the equilibrium electron and hole concentrations
in an n-type semiconductors yield
nn 
1 
N D  N A 

2
N

D
2

 N A   4ni2 

ni2
pn 
nn
Similarly to p-type semiconductors, the electron and hole concentrations are
expressed as
pp 
1 
N A  N D 

2
N

D
2

 N A   4ni2 

ni2
np 
pp
Generally, in case of all impurities are ionized, the net impurity concentration
ND – NA is larger than the intrinsic carrier concentration ni; therefore, we may simply
rewrite the above relationship as
nn  N D  N A if N D  N A
p p  N A  N D if N A  N D
The below figure shows electron density in Si as a function of temperature for
a donor concentration of ND = 1015 cm-3. At low temperature, not all donor impurities
could be ionized and this is called “Freeze-out region” since some electrons are frozen
at the donor level. As the temperature increased, all donor impurities are ionized and
this remains the same for a wide range of temperature. This region is called “Extrinsic
region”. Until the temperature is increased even higher and it reaches a point where
electrons are excited from valence band. This makes the intrinsic carrier concentration
2
becomes comparable to the donor concentration. At this region, the semiconductors
act like an intrinsic one.
Degenerate semiconductor
If the semiconductors are heavily doped for both n- or p-type, EF will be
higher than EC or below EV, respectively. The semiconductor is referred to as
degenerate semiconductor. This also results in the reduction of the bandgap. The
bandgap reduction Eg for Si at room temperature is expressed by
Eg  22
N
meV
1018
where the doping is in the unit of cm-3.
Ex. Si is doped with 1016 arsenic atoms/cm3. Find the carrier concentration and the
Fermi level at room temperature (300K).
Soln At room temperature, complete ionization of impurity atoms is highly possible,
then we have n = ND = 1016 cm-3.
ni2  9.65 10
p

ND
1016

9 2
 9.3 103 cm-3
The Fermi level measured from the bottom of the conduction band is
3
The Fermi level measured from the intrinsic Fermi level is
Direct Recombination
When a bond between neighboring atoms is broken, an electron-hole pair is
generated. The valence electron moves upward to the conduction band due to getting
thermal energy. This results in a hole being left in the valence band. This process is
called carrier generation with the generation rate Gth (number of electron-hole pair
generation per unit volume per time). When an electron moves downward from the
conduction band to the valence band to recombine with the hole, this reverse process
is called recombination. The recombination rate represents by Rth. Under thermal
equilibrium, the generation rate Gth equals to the recombination rate Rth to preserve
the condition of pn  ni2 . The direct recombination rate R can be expressed as
R   np
where  is the proportionality constant.
(2)
4
Therefore, for an n-type semiconductor, we have
Gth  Rth   nn 0 pn 0
(3)
where nn0 and pn0 represent electron and hole densities at thermal
equilibrium.
If the light is applied on the semiconductor, it produces electron-hole pairs at a
rate GL, the carrier concentrations are above their equilibrium values. The generation
and recombination rates become
G  GL  Gth
(4)
R   nn pn    nn0  n  pn0  p 
(5)
where n and p are the excess carrier concentrations
n  nn  nn 0
p  pn  pn 0
(6)
and n = p to maintain the overall charge neutrality.
The net rate of change of hole concentration is expressed as
dpn
 G  R  GL  Gth  R
dt
(7)
In steady-state, dpn/dt = 0. From (7) we have
GL  R  Gth  U
(8)
where U is the net recombination rate.
Substituting (3) and (5) into (9), this yields
U    nn0  pn0  p  p
(9)
For low-level injection p, pn0 << nn0, (9) becomes
U   nn 0 p 
pn  pn 0 pn  pn 0

1/  nn 0
p
(10)
where p is called excess minority carrier lifetime .
pn  pn 0   p GL
(11)
We may write pn in the function of t as
pn (t )  pn 0   p GL exp  t /  p 
(12)
5
Ex. A Si sample with nn0 = 1014 cm-3 is illuminated with light and 1013 electron-hole
pairs/cm3 are created every microsecond. If n = p = 2 s, find the change in the
minority carrier concentration.
Continuity Equation
We shall now consider the overall effect when drift, diffusion, and
recombination occur at the same time in a semiconductor material. Consider the
infinitesimal slice with a thickness dx located at x shown in the figure below. The
number of electrons in the slice may increase because of the net current flow and the
net carrier generation in the slice. Therefore, the overall rate of electron increase is the
sum of four components: the number of electrons flowing into the slice at x, the
number of electrons flowing out at x+dx, the rate of generated electrons, and the rate
of recombination. This can be expressed as
6
n
 J ( x) A J e ( x  dx) A 
Adx   e

   Gn  Rn  Adx
t
e
 e
(13)
where A is the cross-section area and Adx is the volume of the slice.
By expanding the expression for the current at x+dx in Taylor series yields
J e ( x  dx)  J e ( x) 
J e
dx  ...
x
Thus, we have the basic continuity equation for electrons and holes as
n 1 J e

  Gn  Rn 
t e x
p
1 J h

  G p  Rp 
t
e x
(14)
We can substitute the total current density for holes and electrons
dn
dx
dn
J h  e h nE  eD p
dx
J e  ee nE  eDn
and (10) into (14). For low-injection condition, we will have the continuity equation
for minority carriers as
n p
n p
 2n p
n  n po
E
 n p e
 e E
 Dn 2  Gn  p
t
x
x
x
n
pn
E
pn
 2 pn
p  pno
  pn h
 h E
 Dp
 Gp  n
2
t
x
x
x
p
(15)
7
The Haynes-Shockley Experiment
This experiment can be used to measure the carrier mobility μ. The voltage
source establishes an electric field in the n-type semiconductor bar. Excess carriers
are produced and effectively injected into the semiconductor bar at contact (1). Then
contact (2) will collect a fraction of the excess carriers drifting through the
semiconductor bar. After the pulse, the transport equation given by equation (15) can
be rewritten as
pn
p
 2 pn pn  pno
 h E n  Dp

t
x
x 2
p
If there is no applied electric field along the bar, the solution is given by
 x2
N
t 
pn ( x, t ) 
exp  
   pno
 4D t  
4 Dpt
p
p 

(16)
8
where N is the number of electrons or holes generated per unit area. If an electric field
is applied along the sample, an equation (16) will becomes
  x   Et 2 t 
N
p
pn ( x, t ) 
exp  
   pno

4 D pt
p 
4 D pt


Ex. In Haynes-Shockley experiment on n-type Ge semiconductor, given the bar is 1
cm long, L = 0.95 cm, V1 = 2 V, and time for pulse arrival = 0.25 ns. Find mobility μ.
Ex. In a Haynes-Shockley experiment, the maximum amplitudes of the minority
carriers at t1 = 100 μs and t2 = 200 μs differ by a factor of 5. Calculate the minority
carrier lifetime.
Soln
  x   Et 2 t 
N
p
pn ( x, t ) 
exp  
   pno

4 D pt
p 
4 D pt


  x   Et 2 t 
N
p
p  pn  pno 
exp  
 

4 D pt
p 
4 D pt


The maximum amplitude
p
 t 
N
exp   
  
4 D pt
 p
Therefore,
t2 exp  t1 /  p 
 200  100 
p (t1 )
200


exp 
  5

p (t2 )
100

t1 exp  t2 /  p 
p


 p  79 s
9
Thermionic emission process
It is the phenomenon that carriers having high energy thermionically emitted
into the vacuum. In other words, electrons escapes from the hot or high temperature
surface of the material. This is called “thermionic emission process”.
There are new interesting terms we should know from the above figure.
 Electron affinity qχ is the energy difference between the conduction band
edge and the vacuum.
 Work function q is the energy between the Fermi level and the vacuum level
in the semiconductor.
It is clearly seen that an electron can thermionically escape from the
semiconductor surface into the vacuum if its energy is above qχ. The electron density
with energies above qχ can be found by
 q    Vn  
n
(
E
)
dE

N
exp
C



kT


q

nth 
10
where Vn is the difference between the bottom of the conduction band and the
Fermi level.
If escaping electrons with velocity normal to the surface and having energy
greater than EF + q, the thermionic current density is equal to
J   qvN ( E ) F ( E )dE
 4 (2m)3/ 2   ( E  EF ) / kT
  qv 
dE
e
h3


h2k 2
Then we use p = mv and E 
, and after integration, it yields
8 2 m
 q 
J  A*T 2 exp 

 kT 
4 qmk 2
*
A 
 Richardson constant
h3
 1.2  106 A/(m 2 .K 2 )  120 A/(cm 2 .K 2 )
Ex. Calculate the thermionically emitted electron density nth at room temperature for
an n-type silicon sample with an electron affinity qχ = 4.05 eV and qVn = 0.2 eV. If
we reduce the effective qχ to 0.6 eV, what is nth?
As we clearly see that at the room temperature, there is no thermionic
emission of electrons into the vacuum. This thermionic emission process is important
for metal-semiconductor contacts.
Tunneling Process
The figure shows the energy band when two semiconductor samples are
brought close to each other. The distance between them (d) is sufficient small, so that
the electrons in the left-side semiconductor may transport across the barrier and move
11
to the right-side semiconductor even the electron energy is much less than the barrier
height. This process is called “quantum tunneling process”.
The transmission coefficient can be expressed as

2me* (qV0  E ) 
T  2  exp 2d

2
A


C
2
This process is used in tunnel diodes by having a small tunneling distance d, a low
potential barrier qV0, and a small effective mass.
12