MSE 310-ECE 340
MSE 310-ECE 340
Elec Props of Matls
Elec Props of Matls
Extrinsic S/C Prop.: Free Carrier Statistics
Extrinsic S/C Prop.: Free Carrier Statistics
 Let’s define our extrinsic S/C system:
 Free Carrier Statistics:
 Previously, we determined the eqns:
 One type of donor (when I say “donor” or
“acceptor”, I mean shallow-level donor or
acceptor):
o n = f(T)
o p = f(T)
 Carrier concentrations as a function of
temperature in intrinsic S/C’s.
 We found that:
ni2  N c N v e
o The donor has energy, ED
o The donor concentration, ND
 One type of acceptor
 Eg
kT
o The acceptor has energy, EA
o The acceptor concentration, NA
 Since we have been discussing impurities in S/Cs
(extrinsic S/Cs), we would like to find n and p for
the extrinsic case.
 That is, we want to determine how many free e-’s
and h+’s exist near their respective band edges a a
function of:
 The position of the Fermi energy level, Ef , will
be determined self-consistently by the:
o Temperature, T
o Impurity concentrations, ND & NA
o Band gap energy, Eg
o Impurity concentration
o Temperature
 One additional factor to introduce: Degeneracy,
g, in k-space:
 Called “Free Carrier Statistics”, this is again
performed using:
o Donors:
o Statistical Mechanics
o Quantum Mechanics
• g = 2, (spin ½) mj = ±½ (2 spin states)
o Acceptors:
• g = 4, (spin 3/2) mj = ±3/2 (4 spin states)
» mj is the total angular momentum
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MSE 310-ECE 340
MSE 310-ECE 340
Elec Props of Matls
Elec Props of Matls
Extrinsic S/C Prop.: Free Carrier Statistics
Extrinsic S/C Prop.: Free Carrier Statistics
 The Fermi-Dirac distribution function is
(w/degeneracy g):
 We now can ask: “What is the probability that
a donor level at energy ED with degeneracy g
is occupied?”
1
F (E) 
1  ge
 In order to answer this question, we need to
use a distribution statistics of e-’s & h+’s
(EE f )
kT
 The probability for occupancy of a donor level
at energy ED is:
 What distribution function would you suggest?
F ( ED ) 
1
1  gDe
( ED  E f )
kT
 The fraction of donor levels, N*, which are
occupied by e-s is:
N *  N D  F ( ED ) 
ND
1  gDe
( ED  E f )
kT
 Let’s choose a specific case in which:
ND > NA
 What “type” of extrinsic S/C do we have?
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MSE 310-ECE 340
MSE 310-ECE 340
Elec Props of Matls
Elec Props of Matls
Extrinsic S/C Prop.: Free Carrier Statistics
Extrinsic S/C Prop.: Free Carrier Statistics
 Since:
 The maximum number of e-’s that can be
promoted to the CB is:
N *  N D  F ( ED )
 Substituting N* in the previous equation:
N  ND  N A
ND
N
 There are NA
e-’s
compensated by acceptors.
1  gDe
 Thus:
 Let: n = number or concentration of e-’s in CB
 Thus:
1  gDe
N n

ND
 Therefore, they can be safely neglected at low
temperatures: T < room temperature
kT
ND

 Since the acceptor states lie very close to the
VB, the compensated e-’s in the acceptor states
need nearly the Eg to be promoted to the CB.
n
 ED  E f 
ED
kT
n
Ef
kT
e
1
1  gDe
ED
Ef
kT
e
kT
 Let’s draw the energy band diagram when the
S/C is compensated and define certain energy
levels.
N  ND  N A  N *  n
Ee Where N* is concentration of donor states
occupied by e-’s
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ED
EA
ECB
EVB
ED
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MSE 310-ECE 340
MSE 310-ECE 340
Elec Props of Matls
Elec Props of Matls
Extrinsic S/C Prop.: Free Carrier Statistics
Extrinsic S/C Prop.: Free Carrier Statistics
 Continuing with our previous equation and taking
the reciprocal of both sides, we have:
Ef
ED
ND
 1  g D e kT e kT
N n
Ef
ED
ND
 1  g D e kT e kT
N n
Ef
ED
ND  N  n
 g D e kT e kT
N n
 Solving for Ef, we have:
 n 
E f  Ec  kT ln 

 NC 
 Substituting Ef into equation. (i), we obtain:
 ( EC  ED )
NC
ND  N  n
kT
 gDe
n
N n
(i)
Let : ED  EC  ED &   N C g D e
So :
 From our earlier determination of the number of e-’s
in the CB, we found that:
n  NC e

 Ec  E f

n  ND  N  n
N n
ED
kT
=  (ii)
 With some algebra, we obtain a quadratic
equation in n:
kT
n2   N D    N  n  N   0
 Where we assumed the Boltzmann Approximation.
 Thus, we are assuming that the S/C is
nondegenerate.
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MSE 310-ECE 340
MSE 310-ECE 340
Elec Props of Matls
Elec Props of Matls
Extrinsic S/C Prop.: Free Carrier Statistics
Extrinsic S/C Prop.: Free Carrier Statistics
 To determine the number of e-’s in CB, solve
for n.
 Using the Quadratic equation, solve for n:

 Case 1: NA = 0 therefore N = ND.
 @ low T, n << ND.
 n = concentration of CB e-’s, a.k.a. the free carrier
concentration

1
2
n    N D    N    N D    N   4N
2
1 
4N
   N D    N    N D    N  1 
2
2
 ND    N 


We can use equation (ii) to examine several Cases:



n  g D N D NC e


1
4N
1

 N D    N   1 

2
2
 N D    N  

 Since n ≥ 0, we take only the positive solution.
ED
kT
  g D N D NC  e
12
ED
2 kT
Plot of ln(n) vs 1/T
Freeze-out Curve
lnn


1
4N
n   ND    N   1
 1
2
2
 N D    N  

Where :   N C g D e
ED
1/T
kT
 This last expression is good to within several
kT of the CB b/c the Boltzmann Approximation
was used.
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MSE 310-ECE 340
MSE 310-ECE 340
Elec Props of Matls
Elec Props of Matls
Extrinsic S/C Prop.: Free Carrier Statistics
Extrinsic S/C Prop.: Free Carrier Statistics
 Case 2: NA  0. T~low, thus n<<NA<ND
n  g D NC
 Case 3: Increase T such that NA<< n <ND
N D  N A ED kT
e
NA
n   g D NC N D  e
12
2 kT
 Similar to case 1, we find that the slope is ED/2k
or “half-slope”.
 Hence, the region in blue below is known as the
“half-slope regime”.
 Plot of ln(n) vs 1/T
Freeze-out Curve
lnn
ED
Freeze-out Curve
E
slope   D
kb
ED  kb  slope
slope  
1/T
lnn
 This region is also known as the “Full-slope
regime”.
E D
2 kb
ED  2kb  slope
NA
1/T
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MSE 310-ECE 340
MSE 310-ECE 340
Elec Props of Matls
Elec Props of Matls
Extrinsic S/C Prop.: Free Carrier Statistics
Extrinsic S/C Prop.: Free Carrier Statistics
 Case 4:
 Fermi Energy Level of a Doped S/C as a Function
of Temperature
 As T continues to increase, a region is attained in
which the free e- concentration, n, remains
constant.
 This is known as the
 Assume equilibrium
 At temperatures sufficiently high for the S/C to be
intrinsic, Ef lies near mid-gap.
 As T decreases Ef moves either toward the:
o Exhaustion regime
o Saturation regime
o CB for n-type
o VB for p-type
 In this condition, n is given by:
n  ND  N A
 Ef coincides with the majority dopant energy level
at the T for which 50% of the majority dopants are
frozen out (i.e., neutral).
Freeze-out Curve
 N D  N Do  N D
lnn
N Do  N D
saturation/exhaustion regime
 For uncompensated S/C (ideal case), Ef would
approach a position close to the mid-point between
the dopant level and a band edge (CB or VB).
 For compensated S/Cs, a fraction of the majority
dopants remain ionized at the lowest T and Ef will
be very close to the majority dopant energy level.
ND-NA
NA
1/T
 Question: What happens at higher temps beyond
the saturation regime?
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MSE 310-ECE 340
Elec Props of Matls
MSE 310-ECE 340
Extrinsic S/C Prop.: Free Carrier Statistics
Elec Props of Matls
Extrinsic S/C Prop.: Free Carrier Statistics
 In-class Exercise:







Determine if n- or p-type?
|ND-NA| ?
Minority dopant concentration?
Full slope or half slope?
Eg? What is the S/C material?
ED or EA? What is the dopant?
Show where intrinsic region and extrinsic region and saturation regimes are.
 Freezeout Curve
Case 4
15
Case 1
Half-slope
thus,
uncompensated
Singh, Semiconductor Devices: Basic Principles (2000)
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MSE 310-ECE 340
Elec Props of Matls
Extrinsic S/C Prop.: Free Carrier Statistics
 Freezeout Curve
McCluskey & Haller, Dopants & Defects in Semiconductors (CRC Press, 2012) fig. 4.13, p. 120
After: Haller, Hansen, & Goulding, Adv. Phys. (1981)
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