5649.ppt

Chapter 3. Carrier Action
Carrier Action
Sung June Kim
kimsj@snu.ac.kr
http://nanobio.snu.ac.kr
Chapter 3.
1

Subject
1. What relation do mobility of electron and resistivity have?
2. Contribution of diffusion of excess carrier in current
3. Carrier injection and carrier action
1. direct gap semiconductor and indirect semiconductor
2. recombination lifetime
3. quasi-Fermi level
4. How does current change while considering diffusion and
recombination ?
2

Contents
 Drift
 Diffusion
 Generation-Recombination
 Equations of State
3

4
 Drift
• Definition-Visualization
 Charged-particle motion in response to electric field
 An electric field tends to accelerate the +q charged holes in the
direction of the electric field and the –q charged electrons in the
opposite direction
Collisions with impurity atoms and thermally agitated lattice atoms
 Repeated periods of acceleration and subsequent decelerating
collisions
 Measurable quantities are macroscopic observables that reflect the
average or overall motion of the carriers
The drifting motion is actually superimposed upon the always-
present thermal motion.
 The thermal motion of the carriers is completely random and
therefore averages out to zero, does not contribute to current

5
• Drift Current
 I (current) = the charge per unit time crossing a plane oriented
normal to the direction of flow
(a) Motion of carriers within a biased semiconductor bar; (b) drifting hole
on a microscopic or atomic scale; (c) carrier drift on a macroscopic scale
Thermal motion of carrier

6
Expanded view of a biased p-type
semiconductor bar of cross-sectional
area A
....... All holes this distance back from the normal plane will
cross the plane in a time
....... All holes in this volume will cross the plane in a time
d d
d
d
v t v
t
v tA t
pv tA

....... Holes crossing the plane in a time
..... Charge crossing the plane in a time
...... Charge crossing the plane per unit time
d
d
t
qpv tA t
qpv A t

7
P drift d hole drift current
I qpv A
  
P drift d
qpv
 
J
The vd is proportional to E at low electric fields, while at high electric
fields vd saturates and becomes independent of E

8
 0
sat
. . . 0
0
d . . .
1/
0
sat
1
v
v
v





 
 
 
 
 
 
 
 
 
 
E E
E
E
E
where for holes and for electrons, is the constant of
proportionality between vd and E at low to moderate electric fields, and
is the limiting or saturation velocity
sat
v
 In the low field limit d 0
v 
 E
P drift p
q p

 
J E
N drift n
q n

 
J E |
, ,
d n N drift d
q v E qnv

    
J
1

 2

 0


9
• Mobility
 Mobility is very important parameter in characterizing transport due
to drift
 Unit: cm2/Vs
 Varies inversely with the amount of scattering
(i) Lattice scattering
(ii) Ionized impurity scattering
Displacement of atoms leads to lattice scattering
The internal field associated with the stationary array of atoms is
already taken into account in
 , where is the mean free time and is the
conductivity effective mass
 The number of collisions decreases   varies inversely with
the amount of scattering
*
/
q m
 
   
  *
m
*
m

 

10
Room temperature carrier mobilities as a function of the dopant
concentration in Si

11
 Temperature dependence
 Low doping limit: Decreasing temperature causes an ever-
decreasing thermal agitation of the atoms, which decreases the
lattice scattering
 Higher doping: Ionized impurities become more effective in
deflecting the charged carriers as the temperature and hence the
speed of the carriers decreases

12
Temperature dependence of electron mobility in Si for dopings ranging
from < 1014 cm-3 to 1018 cm-3.

13
• Resistivity
 Resistivity () is defined as the proportionality constant between
the electric field and the total current per unit area

 J
E
1


 
J E E =1/: conductivity
drift N drift P drift n p
( )
q n p
 
 
   
J J J E
n p
1
( )
q n p

 


n D
1
... type semiconductor
n
q N


  
p A
1
... type semiconductor
p
q N


  

14
Resistivity versus
impurity
concentration at
300 K in Si

15
• Band Bending
When exists the band
energies become a function of
position
 If an energy of precisely EG is
added to break an atom-atom bond,
the created electron and hole
energies would be Ec and Ev,
respectively, and the created carriers
would be effectively motionless
motionless


16
 E-Ec= K.E. of the electrons
 Ev-E= K.E. of the holes
The potential energy of a –q charged
particle is
P.E. qV
 
c ref
P.E. E E
 
c ref
P.E. E E
 
c ref
1
( )
V E E
q
   
dV
dx
 
E

17
 By definition,
V
 
E
dV
dx
 
E
c v i
1 1 1
dE dE dE
q dx q dx q dx
  
E
 In one dimension,

18
 Diffusion
 Diffusion is a process whereby
particles tend to spread out or redistribute
as a result of their random thermal motion,
migrating on a macroscopic scale from
regions of high concentration into region
of low concentration

19
• Diffusion and Total Currents
 Diffusion Currents: The greater the concentration gradient, the
larger the flux
 Using Fick’s law,
]
/
[# 2
s
cm
D
F 


 
Diffusion constant

20
P diff P
N diff N
qD p
qD n


  
  
J
J
 Total currents
P P drift P diff p P
N N drift N diff n N
drift diffusion
q p qD p
q n qD n


 
 
    
 
    
J J J
J J J
E
E
 Total particle currents
N P
 
J J J
s
cm
D /
]
[ 2

+

21
• Relating Diffusion coefficients/Mobilities
 Einstein relationship
Consider a nonuniformly doped semiconductor under equilibrium
 Constancy of the Fermi Level: nonuniformly doped n-type
semiconductor as an example / / / 0
F F F
dE dx dE dy dE dz
  
= 0

22
To prove that (Under equilibrium, the EF should be continuous.)
Under thermal equilibrium, there are no charge movements between
two materials,
Also, the rate of electron movement from material 2 to material 1 is in
proportion to
Under equilibrium condition, these terms are equal. So,
f1(E) = f2(E) and EF1 = EF2
That is, dEF /dx = 0.
The rate of electron movement from material 1 to material 2 is in
proportion to
Fig. Junction of two materials in equilibrium
state. Actually, there is no electron movement,
Fermi level should be constant hrough the
entire material.

23
 Under equilibrium conditions
N P N P
0 0
    
J J J J = J
p P
n N
for nonuniform doping
0 and
or for uniform doping
0
q p qD p
q n qD n
p n


 



 





   

E
E
E
 Einstein relationship:
 Nondegenerate, nonuniformly doped semiconductor,Under
equilibrium conditions, and focusing on the electrons,
N drift N diff n N 0
dn
J J q n qD
dx

 
   
E
i
1 dE
q dx

E F i
( )/
i
E E kT
n n e 

&
= -
Electron diffusion current flowing in the +x direction “Built-in”
electric field in the –x direction  drift current in the –x
direction

24
 With dEF/dx=0,
F i
( )/
i i
E E kT
n dE
dn q
e n
dx kT dx kT

    E
 Substituting
n N
( ) ( ) 0
q
qn qn D
kT
  
E E
N
n
Einstein relationship for electrons
D kT
q


P
p
Einstein relationship for holes
D kT
q


 Einstein relationship is valid even under nonequilibrium
 Slightly modified forms result for degenerate materials

25
 Recombination-Generation
 When a semiconductor is perturbed  an excess or deficit in the
carrier concentrations  Recombination-generation
 Recombination: a process whereby electrons and holes (carriers)
are annihilated or destroyed
 Generation: a process whereby electrons and holes are created
• Band-to-Band Recombination
 The direct annihilation of an electron and a hole  the production of
a photon (light)

26
• R-G Center Recombination
 R-G centers are lattice defects or impurity atoms (Au)
 The most important property of the R-G centers is the introduction
of allowed electronic levels near the center of the band gap (ET)
Two-step process
 R-G center recombination (or indirect recombination) typically
releases thermal energy (heat) or, equivalently, produces lattice
vibration

27
Near-midgap energy levels
introduced by some common
impurities in Si

28
• Generation Processes
thermal energy > EG  direct thermal generation
light with an energy > EG  photogeneration
 The thermally assisted generation of carriers with R-G centers

29
Impact ionization
 e-h is produced as a results of energy released when a highly
energetic carrier collide with the lattice
High -field regions


30
• Momentum Considerations
One need be concerned only with the dorminat process
Crystal momentum in addition to energy must be conserved.
The momentum of an electron in an energy band can assume only
certain quantized values.
where k is a parameter proportional to the electron momentum

31
Si, Ge
GaAs

32
Photons, being massless entities,
carry very little momentum, and a
photon-assisted transition is
essentially vertical on the E-k plot
The thermal energy associated with lattice
vibrations (phonons) is very small (in the 10-
50 meV range), whereas the phonon
momentum is comparatively large. A phonon-
assisted transition is essentially horizontal on
the E-k plot. The emission of a photon must
be accompanied by the emission or
absorption of a phonon.
Photons: light
Phonons: lattice vibration quanta

33
• R-G Statistics
 It is the time rate of change in the carrier concentrations (n/t, p
t) that must be specified
 B-to-B recombination is totally negligible compared to R-G center
recombination in Si
Even in direct materials, the R-G center mechanism is often the
dominant process.

34
• Indirect Thermal Recombination-Generation
n0, p0 …. carrier concentrations when equilibrium conditions
prevail
n, p ……. carrier concentrations under arbitrary conditions
n=n-n0 … deviations in the carrier concentrations from their
equilibrium values
p=p-p0 …
NT ………. number of R-G centers/cm3
material
type
p
a
in
p
p
p
n
material
type
n
an
in
n
n
n
p
implies
injection
level
low









0
0
0
0
,
,

35
A specific example of ND=1014 cm-3 Si subject to a perturbation
where n= p=109 cm-3
Although the majority carrier concentration remains essentially
unperturbed under low-level injection, the minority carrier
concentration can increase by many orders of magnitude.
excess
carrier
p
p
p
p
cm
N
n
p
n
n
n
n
cm
N
n
D
i
D

















0
3
6
2
0
0
0
3
14
0
10
/
10

36
p T
R
p
c N p
t

 

 The greater the number of filled R-G centers, the greater the
probability of a hole annihilating transition and the faster the rate of
recombination
Under equilibrium, essentially all of the R-G centers are filled with
electrons because EF>>ET
With , electrons always vastly outnumber holes and
rapidly fill R-G levels that become vacant  # of filled centers during
the relaxation NT
 The number of hole-annihilating transitions should increase almost
linearly with the number of holes
 Introducing a proportionality constant, cp,

0
n
p 

T
R
N
t
p




37
Generateion rate: p/t|G depends only on # of empty R-G centers
The recombination and generation rates must precisely balance under
equilibrium conditions, or p/t|G = p/t|G-equilibrium=- p/t|R-equilibrium , so
p T 0
G
p
c N p
t



p T 0
i thermal R G
R G
( )
p p p
c N p p
t t t

 
  
    
  
p T
i thermal
R G
or for holes in an type material
p
c N p n
t 
 

    

n T
i thermal
R G
for electrons in a type material
n
c N n p
t 
 

    

 The net rate
 An analogous set of arguments yields
um
equiliburi
G
G t
p
t
p







38
 cn and cp are referred to as the capture coefficients
cpNT and cnNT must have units of 1/time
p n
p T n T
1 1
c N c N
  
 
i thermal
p
R G
for holes in an type material
p p
n
t 

 
 
   

i thermal
n
R G
for electrons in a type material
n n
p
t 

 
 
   

These are the excess minority carrier life times.

39
 Equations of State
• Continuity Equations
 Carrier action – whether it be drift, diffusion, indirect or direct
thermal recombination, indirect or direct generation, or some other
type of carrier action – gives rise to a change in the carrier
concentrations with time
Let’s combine all the mechanisms
thermal other processes
drift diff
R-G (light, etc.)
thermal other processes
drift diff
R-G (light, etc.)
n n n n n
t t t t t
p p p p p
t t t t t
    
   
    
    
   
    

40
processes
other
G
R
thermal
P
processes
other
G
R
thermal
N
P
Pz
Py
Px
diff
drift
N
Nz
Ny
Nx
diff
drift
t
p
t
p
J
q
t
p
t
n
t
n
J
q
t
n
J
q
z
J
y
J
x
J
q
t
p
t
p
J
q
z
J
y
J
x
J
q
t
n
t
n





























































1
1
1
)
(
1
1
)
(
1

x
x
x
J
x
J
q
t
p p
p
x
x
x
n










)
(
)
(
1
t
p
x
t
x
p
x
q
q
v
p
x
J
q 

















1
t
x
v




42
• Minority Carrier Diffusion Equations
 Simplifying assumptions:
(1) One-dimensional
(2) The analysis is limited or restricted to minority carriers
(3) .—no drift current term- so it is called Diffusion Equations.
(4) The equilibrium minority carrier concentrations are not a
function of position
(5) Low level injection
(6) Indirect thermal R-G is the dominant thermal R-G mechanism
(7) There are no “other processes”, except possibly
photogeneration
0
E
0 0 0 0
( ), ( )
n n x p p x
 
N
N
1 1 J
q q x

 

J
N n N N
n n
J q n qD qD
x x

 
 
 
E



43
0
n n n
  

0
n
n n n
x x x x

  
  
   
2
N N 2
1 n
D
q x
 
 

J
 With low level injection,
thermal
n
R-G
n n
t 
 
 

L
other
processes
n
G
t



 GL=0 without illumination

44
 The equilibrium electron concentration is never a function of time
0
n
n n n
t t t t

  
  
   
2
p p p
N L
2
n
2
n n n
P L
2
p
n n n
D G
t x
p p p
D G
t x


   
  
 
   
  
 
Minority carrier
diffusion equations

45
 No concentration gradient or no diffusion current
2 2
2 2
0 0
p n
N P
n p
D D
x x
   
 
 
 
 
 
 No drift current or  no further simplification
0
E
 No thermal R-G
0 0
p n
n p
n p
 
 
 
 
 
 
 
 No light
L 0
G 

• Simplifications and Solutions
0 0
p n
n p
t t
 
 
 
 
 
 
 Steady state

is set equal to zero.
L
G
L
P
n
n
P
n
G
p
x
p
D
t
p











2
2
L
P
n
n
G
p
dt
p
d





P
t
n
n e
p
t
p 
/
)
0
(
)
( 



Therefore, the solution is

2
p p p
N L
2
n
2
n n n
P L
2
p
n n n
D G
t x
p p p
D G
t x


   
  
 
   
  
 

Sample Problem No.1 : A uniformly donor-doped silicon wafer
maintained at room temperature is suddenly illuminated with light at
time t=0. Assuming , , and a light-
induced creation of electrons and holes per
throughout the semiconductor, determine for .
3
15
/
10 cm
ND  sec
10 6


P

17
10 sec
3

cm
)
(t
pn
 0

t
Solution :
Step1 – Si, T=300K, , and
at all points inside the semiconductor.
Step2 -
with uniform doping, the
equilibrium and values are the same
everywhere throughout the semiconductor.
3
15
/
10 cm
ND  sec
/
10 3
17

 cm
GL
3
15
0
3
10
/
10
,
,
/
10 cm
N
n
n
N
cm
n D
i
D
i 



3
5
2
0 /
10
/ cm
N
n
p D
i 


Step3 – Prior to t=0, equilibrium conditions prevail and
illumination -> increase.
excess carrier number ↑ -> thermal recombination rate ↑
∴ light generation and thermal recombination rates must
balance under steady state conditions, we can even state
or
if low-level injection prevails.
Step4 –
the boundary condition
0

 n
p
n
p

P
n
L t
p
G 
/
)
( 


 P
L
n
n G
p
t
p 





 max
|
)
(
3
15
0
3
11
max
| /
10
/
10 cm
n
cm
G
p P
L
n 



 
L
P
n
n
P
n
G
p
x
p
D
t
p











2
2
0
|
)
( 0 
 
t
n t
p

is not function of position, the diffusion equation becomes an
ordinary differential equation and simplifies to
The general solution is
Applying the boundary condition yields
and
n
p

L
P
n
n
G
p
dt
p
d





P
t
P
L
n Ae
G
t
p 
 /
)
( 



P
L
G
A 


solution
)
1
(
)
( /



  P
t
P
L
n e
G
t
p 


Step5 – A plot of the solution equation is shown below.
Note that, starts from zero at t=0 and eventually
saturates at after a few .
)
(t
pn

P
L
G  P


Epilogue – Light output from the stroboscope can be modeled to first
order by the pulse train pictured below.

)
1
(
)
( / P
t
P
L
n e
G
t
p 
 



The upper equation approximately describes the carrier build-up
during a light pulse.
However, the stroboscope light pulses have a duration of
compared to a minority carrier lifetime of .
With , one sees only the very first portion of the Fig.
3.25 transient before the light is turned off and the semiconductor is
allowed to decay back to equilibrium.
The light-off expression derivation closely parallels the light-
on sample problem solution.
Exceptions -> at the beginning of the light-off = at the end
of the light-on.
sec
1

on
t
sec
150
 
P
1
/ 
P
on
t 
)
(t
pn

n
p
 n
p


55
 Sample problem 2: As shown below, the x=0 end of a uniformly
doped semi-infinite bar of silicon with ND=1015 cm-3 is illuminated so
as to create pn0=1010 cm-3 excess holes at x=0. The wavelength of
the illumination is such that no light penetrates into the interior (x>0)
of the bar. Determine pn(x).

56
 Solution: at x=0, pn(0)= pn0= 1010 cm-3 , and pn  0 as x  
 The light first creates excess carriers right at x=0
GL=0 for x>0
Diffusion and recombination
 As the diffusing holes move into the bar their numbers are reduced
by recombination
 Under steady state conditions it is reasonable to expect an excess
distribution of holes near x=0, with pn(x) monotonically decreasing
from pn0 at x=0 to pn0= 0 as x  
 Can we ignore the drift current? E 0 ? Yes
because 1>Excess hole pile-up is very small , here
2> Also the majority carriers redistribute in such a way to partly
cancel the minority carrier charge. Thus use of diffusion eq. is justified.

)
( max i
n n
p 


57
 The general solution
P P
/ /
n ( ) x L x L
p x Ae Be

   p p p
L D 

where
exp(x/Lp)   as x   0
B
 
 With x=0,
n0
A p
 
P
/
n n0
( ) solution
x L
p x p e
   
2
n n
p 2
p
0 for 0
d p p
D x
dx 
 
   
n
0
x
p  
 
n0
n 0 n 0
x x
p p p

   
    
 So---Under steady state conditions with GL=0 for x > 0

58
N N n
L D 

LP and LN represent the average distance minority carriers can
diffuse into a sea of majority carriers before being annihilated
 The average position of the excess minority carriers inside the
semiconductor bar is
n n P
0 0
( ) ( )
x x p x dx p x dx L
 
     
 
 Supplemental Concepts
• Diffusion Lengths
 minority carrier diffusion lengths
P P p
L D 


59
• Quasi-Fermi Levels
 Quasi-Fermi levels are energy levels used to specify the carrier
concentrations under nonequilibrium conditions
 The convenience of being able to deduce the carrier
concentrations by inspection from the energy band diagram is
extended to nonequilibrium conditions through the use of quasi-Fermi
levels by introducing FN and FP

60
Equilibrium conditions Nonequilibrium conditions
N i
i P
( )/
N i
( )/
P i
or ln
or ln
F E kT
i
i
E F kT
i
i
n
n n e F E kT
n
p
p n e F E kT
n


 
      
 
 
      
 

11 -3 11 -3 15 -3
0
10 cm , 10 cm , 10 cm
n L p
p G p p p n

       3
15
0 10 

 cm
n
n

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