Lecture 5
OUTLINE
• Semiconductor Fundamentals (cont’d)
– Carrier diffusion
• Diffusion current
• Einstein relationship
– Generation and recombination
• Excess carrier concentrations
• Minority carrier recombination lifetime
Reading: Pierret 3.2-3.3; Hu 2.3, 2.5-2.6
Diffusion
Particles diffuse from regions of higher concentration to regions
of lower concentration region, due to random thermal motion.
C. C. Hu, Modern Semiconductor Devices for Integrated Circuits, Figure 2-9
EE130/230A Fall 2013
Lecture 5, Slide 2
1-D Diffusion Example
• Thermal motion causes particles to
move into an adjacent compartment
every t seconds
– Each particle has an equal
probability of jumping to the left
or jumping to the right.
EE130/230A Fall 2013
Lecture 5, Slide 3
R.F. Pierret, Semiconductor Fundamentals, Figure 3.11
Diffusion Current
J n,diff
dn
 qDn
dx
J p,diff
dp
  qD p
dx
D is the diffusion constant, or diffusivity.
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Lecture 5, Slide 4
Total Current
J  Jn  J p
J n  J n ,drift  J n ,diff
dn
 qn n ε  qDn
dx
J p  J p ,drift  J p ,diff
dp
 qp p ε  qD p
dx
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Lecture 5, Slide 5
Non-Uniformly-Doped Semiconductor
• The position of EF relative to the band edges is determined by
the carrier concentrations, which is determined by the net
dopant concentration.
• In equilibrium EF is constant; therefore, the band-edge energies
vary with position in a non-uniformly doped semiconductor:
Ec(x)
EF
Ev(x)
EE130/230A Fall 2013
Lecture 5, Slide 6
Potential Difference due to n(x), p(x)
• The ratio of carrier densities at two points depends exponentially
on the potential difference between these points:
 n1 
 n1 
EF  Ei1  kT ln    Ei1  EF  kT ln  
 ni 
 ni 
 n2 
Similarly, Ei2  EF  kT ln  
 ni 
Therefore
  n2 
 n1 
 n2 
Ei1  Ei2  kT ln    ln    kT ln  
 ni 
 n1 
  ni 
1
kT  n2 
V2  V1  Ei1  Ei2  
ln  
q
q  n1 
EE130/230A Fall 2013
Lecture 5, Slide 7
Ev(x)
Built-In Electric Field due to n(x), p(x)
Ef
Ec(x)
Consider a piece of a non-uniformly doped semiconductor:
n  Nce
n-type semiconductor
dn
N
dEc
  c e ( Ec  EF ) / kT
dx
kT
dx
Decreasing donor concentration
Ec(x)
EE130/230A Fall 2013
 ( Ec  E F ) / kT
EF
n dEc

kT dx
Ev(x)
n

qε
kT
Lecture 5, Slide 8
Einstein Relationship between D, 
• In equilibrium there is no net flow of electrons or holes
Jn = 0 and Jp = 0
 The drift and diffusion current components must balance
each other exactly. (A built-in electric field exists, such that
the drift current exactly cancels out the diffusion current due
to the concentration gradient.)
dn
J n  qn n ε  qDn
0
dx
dp
J p  qp p ε  qD p
0
dx
The Einstein relationship is valid for a non-degenerate semiconductor, even under non-equilibrium conditions.
EE130/230A Fall 2013
Lecture 5, Slide 9
Example: Diffusion Constant
What is the hole diffusion constant in a sample of silicon with
p = 410 cm2 / V s ?
Answer:
Remember: kT/q = 26 mV at room temperature.
EE130/230A Fall 2013
Lecture 5, Slide 10
Quasi-Neutrality Approximation
• If the dopant concentration profile varies gradually with position,
then the majority-carrier concentration distribution does not
differ much from the dopant concentration distribution.
ND ( x)  p( x)  NA ( x)  n( x)
– n-type material: n( x)  ND ( x)  NA ( x)
– p-type material: p( x)  N A ( x)  N D ( x)


kT  1  dn
kT  1  dN D




 
q  n  dx
q  N D  dx
EE130/230A Fall 2013
Lecture 5, Slide 11
in n-type material
Generation and Recombination
• Generation:
– A process by which
electrons & holes are
created in pairs.
• Recombination:
– A process by which
electrons and holes
are annihilated in pairs.
• Generation and recombination processes act to change the
carrier concentrations, and thereby indirectly affect current flow
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Lecture 5, Slide 12
Generation Processes
R.F. Pierret, Semiconductor Fundamentals, Figure 3.15
Band-to-Band
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R-G Center
Lecture 5, Slide 13
Impact Ionization
Recombination Processes
R.F. Pierret, Semiconductor Fundamentals, Figure 3.15
Direct
R-G Center
Auger
Recombination in Si is primarily via R-G centers
EE130/230A Fall 2013
Lecture 5, Slide 14
Direct vs. Indirect Band Gap Materials
Energy (E) vs. momentum (p=ħk) Diagrams
Direct:
Indirect:
Little change in momentum
is required for recombination
Large change in momentum
is required for recombination
 momentum is conserved by
photon emission
 momentum is conserved by
phonon + photon emission
EE130/230A Fall 2013
Lecture 5, Slide 15
Excess Carrier Concentrations
equilibrium values
n  n  n0
p  p  p0
Charge neutrality condition:
n  p
EE130/230A Fall 2013
Lecture 5, Slide 16
“Low-Level Injection”
• Often the disturbance from equilibrium is small, such that the
majority-carrier concentration is not affected significantly:
– For an n-type material:
| n || p | n0 so n  n0
– For a p-type material:
| n || p | p0 so p  p0
However, the minority carrier concentration can be
significantly affected.
EE130/230A Fall 2013
Lecture 5, Slide 17
Indirect Recombination Rate
Suppose excess carriers are introduced into an n-type Si sample
(e.g. by temporarily shining light onto it) at time t = 0.
How does p vary with time t > 0?
1.Consider the rate of hole recombination via traps:
p
t R
 c p NT p
2.Under low-level injection conditions, the hole generation rate is
not significantly affected:
p
t G

EE130/230A Fall 2013
p
t G  equilibrium

p
t R  equilibrium
Lecture 5, Slide 18
 c p NT p0
3.
The net rate of change in p is therefore
p
t R G
p
t R G

p
t R
 c p NT p  c p NT p0
p
  c p N T ( p  p0 )    p
where  p 
EE130/230A Fall 2013

p
t G
1
c p NT
Lecture 5, Slide 19
Minority Carrier (Recombination) Lifetime
 p  c 1N
p
T
 n  c 1N
n
T
The minority carrier lifetime  is the average time an excess
minority carrier “survives” in a sea of majority carriers
 ranges from 1 ns to 1 ms in Si and depends on the density of
metallic impurities (contaminants) such as Au and Pt, and the
density of crystalline defects. These impurities/defects give rise
to localized energy states deep within the band gap. Such deep
traps capture electrons or holes to facilitate recombination and
are called recombination-generation centers.
EE130/230A Fall 2013
Lecture 5, Slide 20
Relaxation to Equilibrium State
Consider a semiconductor with no current flow in which
thermal equilibrium is disturbed by the sudden creation of
excess holes and electrons. The system will relax back to the
equilibrium state via the R-G mechanism:
n
n

t
n
for electrons in p-type material
p
p

t
p
for holes in n-type material
EE130/230A Fall 2013
Lecture 5, Slide 21
Example: Photoconductor
Consider a sample of Si doped with 1016 cm-3 boron,
with recombination lifetime 1 s. It is exposed continuously
to light, such that electron-hole pairs are generated
throughout the sample at the rate of 1020 per cm3 per second,
i.e. the generation rate GL = 1020/cm3/s
What are p0 and n0 ?
What are n and p ?
(Hint: In steady-state, generation rate equals recombination rate.)
EE130/230A Fall 2013
Lecture 5, Slide 22
What are p and n ?
What is the np product ?
Note: The np product can be very different from ni2.
EE130/230A Fall 2013
Lecture 5, Slide 23
Net Recombination Rate (General Case)
For arbitrary injection levels, the net rate of carrier recombination is:
pn  ni2
n
p



t
t
 p (n  n1 )   n ( p  p1 )
where n1  ni e ( ET  Ei ) / kT and p1  ni e ( Ei  ET ) / kT
EE130/230A Fall 2013
Lecture 5, Slide 24
Summary
• Electron/hole concentration gradient  diffusion
J n,diff  qDn
dn
dx
J p,diff   qD p
dp
dx
• Current flowing in a semiconductor is comprised of drift and
diffusion components for electrons and holes
J = Jn,drift + Jn,diff + Jp,drift + Jp,diff
In equilibrium Jn = Jn,drift + Jn,diff = 0 and Jp = Jp,drift + Jp,diff = 0
• The characteristic constants of drift and diffusion are related:
D
kT


q
EE130/230A Fall 2013
Lecture 5, Slide 25
Summary (cont’d)
• Generation and recombination (R-G) processes affect carrier
concentrations as a function of time, and thereby current flow
– Generation rate is enhanced by deep (near midgap) states
due to defects or impurities, and also by high electric field
– Recombination in Si is primarily via R-G centers
• The characteristic constant for (indirect) R-G is the minority
carrier lifetime:
 p  c 1N
p
(n - typematerial)
T
 n  c 1N
n
(p - typematerial)
T
• Generally, the net recombination rate is proportional to
EE130/230A Fall 2013
Lecture 5, Slide 26
np  n
2
i