```Haynes-Shockley Experiment
J.R. Haynes and W. Shockley, Bell Telephone Laboratories, “The Mobility
and Life of Injected Holes and Electrons in Germanium”, Physical Review,
81, 835 (1951).
n-type semiconductor (n0 >>p0 )
•
Shine short pulse of laser light on sample
•
Create localized pulse of excess minority carriers
(holes)
Three effects:
-pulse spreads out in time (diffusion)
-pulse travels with drift velocity vD=µE
-area of pulse decreases with time because of
recombination
Phys 365 2002-1: Non equilibrium effects
130
Case (a): No external E field, no recombination
•
Diffusion only (no drift)
•
After light pulse generation rate gp is zero.
•
rp=0
∂δp ( x, t )
1 ∂J p
=−
∂t
q ∂x
Continuity equation:
(assume minority carrier lifetime is long compared to
observation time)
∂δp
Assuming only diffusion current: J p = − qD p ∂x
∂δp ( x , t )
∂ 2δp( x , t )
= Dp
∂t
∂x 2
Solution:
δp( x, t ) =
1
4πD p t
e
 x2 
−

 4 D pt 
• Hole pulse width increases with t
Width of the Gaussian pulse: ∆x = 4 D p t
Correction in presence of recombination:
δp( x, t ) =
1
4πD p t
e
 x2 
−

 4 D p t 
e
−t /τ p
Case (b) Include drift (no recombination)
•
Center of Gaussian pulse moves with vd= µpE
•
Observe pulse as it passes a fixed contact point
•
If hole pulse width is ∆x, this is observed as a pulse of
width ∆t ~ ∆x / vd on the oscilloscope screen.
•
Use the relation ∆x = 4 D p t to determine Dp.
Phys 365 2002-1: Non equilibrium effects
131
Summary of Haynes-Shockley Experiment:
• Diffusivity Dp which is related to the width of the pulse
as a function of time
• Drift velocity is given by the delay between the launch
of the pulse and its detection at some distance along the
sample.
• Minority carrier lifetime is given by the decay of the
integral of the pulse shape
Phys 365 2002-1: Non equilibrium effects
132
Junctions (Mostly p-n junctions)
Equilibrium Properties
•
diffusion current of electrons to the p-type side.
•
Similarly, net flow of holes toward the n-type side.
•
Near the junction, we get electron-hole recombination,
which would result in "depletion" of holes from p-type side
and "depletion" of electrons from n-type side.
•
n-doped region near the junction becomes positively
charged due to depletion of electrons
•
p-doped region becomes negatively charged due to
depletion of holes.
Built in field:
As the electrons and holes move across the junction, the
exposed charges that they leave behind set up a built-in
electric field which opposes further charge separation.
Gauss’ law:
In one dimension:
Electric potential:
∇ ⋅E =
ρ
ε rε0
dEx
ρ
=
dx ε r ε 0
Ex = −
Phys 365 2002-1: Non equilibrium effects
∂V
∂x
133
∂ 2V
ρ
=
−
Poisson’s equation:
εr ε0
∂x 2
Consider the solution of Gauss’s law for two finite charged
slabs of opposite charge
Energy bands:
Note: electrostatic potential is repulsive to holes on the left
and repulsive to electrons on the right.
Phys 365 2002-1: Non equilibrium effects
134
Location of Fermi level for electrostatic and thermal
equilibrium
• In a system in electrostatic and thermal equilibrium,
the Fermi level remains at a constant value
independent of x.
qV
Calculation of the Built-in Potential V0
In neutral p-type region:
p = ni e ( Ei ( p )− E F ) / kT and p=Na
 NA 


E
(
p
)
−
E
=
kT
ln
i
F
Therefore:
 ni 
In neutral n-type region:
Phys 365 2002-1: Non equilibrium effects
135
n = ni e ( E F − Ei ( n )) / kT
Therefore:
and n=Nd
N
EF − Ei ( n) = kT ln D
 ni




By inspection of above figure: qV0= Ei(p)-Ei(n)
Therefore:
 N 
N
qV 0 = E i ( p ) − E i ( n ) = kT  ln  A  + ln  D
 ni
  ni 
Therefore:
V0 =
kT  N A N D
ln
q  ni 2





 

Built-in voltage
Example:
For a Si p-n junction with NA=1018 cm-3 and
ND=1015 cm-3 we have
kT  N A N D
V0 =
ln
q  ni 2

 (1018 )(10 15
 = (0.0259 ) ln 
 (1.45 × 10 10 ) 2




 = 0.755V

Higher doping levels => higher barrier height
For example, for NA=ND = 1×1018 cm-3 we get: V0=0.93.
Comparing this with the band gap of Si of EG= 1.1 eV, we
can see that V0 approaches EG in the limit of high doping
Phys 365 2002-1: Non equilibrium effects
136
Analysis of the Depletion Region: Abrupt Junction
Abrupt junction:
•
Uniform carrier concentrations in the p and n regions
respectively
•
Transition between n and p is perfectly abrupt.
Energy bands:
Phys 365 2002-1: Non equilibrium effects
137
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