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PHY 752 Solid State Physics
11-11:50 AM MWF Olin 107
Plan for Lecture 26:
 Chap. 19 in Marder
 Properties of semiconductors
 Diodes and other electronic
devices
Some of the lecture materials are from slides prepared
by Marder
3/30/2015
PHY 752 Spring 2015 -- Lecture 26
1
3/30/2015
PHY 752 Spring 2015 -- Lecture 26
2
Charge and voltage profile near a semiconductor p-n junction
Equilibrium distributions
3/30/2015
PHY 752 Spring 2015 -- Lecture 26
3
Charge and voltage profile near a semiconductor p-n junction
Internal voltage:
3/30/2015
PHY 752 Spring 2015 -- Lecture 26
4
Charge and voltage profile near a semiconductor p-n junction
 0
for x   x p

Simplified model:
eN a for  x p  x  0
 ( x)  
 eN d for 0  x  xn

for x  xn
 0
3/30/2015
PHY 752 Spring 2015 -- Lecture 26
5
Charge and voltage profile near a semiconductor p-n junction
Simplified model – continued:
2 e
Vbi  V ()  V () 
N a x 2p  N d xn2 

ò
3/30/2015
PHY 752 Spring 2015 -- Lecture 26
6
Charge and voltage profile near a semiconductor p-n junction
Effects of adding an applied voltage to junction
Voltage across depletion zone:
V  Vbi  VA
Depletion size in presence of applied voltage
1/2
 VA 
xn (VA )  xn (0) 1  
 Vbi 
1/2
 VA 
x p (VA )  x p (0) 1  
 Vbi 
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PHY 752 Spring 2015 -- Lecture 26
7
Charge and voltage profile near a semiconductor p-n junction
VA=0
VA<0
VA>0
V(x)
(x)
3/30/2015
PHY 752 Spring 2015 -- Lecture 26
8
Boltzmann equation treatment of current response to diode
(relaxation time approximation)
3/30/2015
PHY 752 Spring 2015 -- Lecture 26
9
Boltzmann equation treatment of current response to diode
(relaxation time approximation)
electron mobility
Electron current:
electron diffusion coefficient;
Einstein relation
jn  en r  en E  eDnn
3/30/2015
PHY 752 Spring 2015 -- Lecture 26
10
Boltzmann equation treatment of current response to diode
(relaxation time approximation)
Summary of equations:
3/30/2015
PHY 752 Spring 2015 -- Lecture 26
11
Boltzmann equation treatment of current response to diode
Approximate solution ignoring “minority” carriers and
assuming spatially constant currents jn and jp
For HW, you will show that the above solutions are
consistent with the equations:
3/30/2015
PHY 752 Spring 2015 -- Lecture 26
12
-VA
When VA  0 :
p  Na n 
p majority
n minority
n majority
p minority
2
i
n
Na
Estimation of minority carriers
3/30/2015
PHY 752 Spring 2015 -- Lecture 26
When VA  0 :
n  Nd
ni2
p
Nd
13
Boltzmann equation treatment of current response to diode
Approximate steady state solutions in presence of
applied voltage VA
=0
≈0

3/30/2015
PHY 752 Spring 2015 -- Lecture 26
14
Boltzmann equation treatment of current response to diode
Approximate steady state solutions in presence of
applied voltage VA
Solution to diffusion equations:
for x>xn
for x<xp
ni2 
ni2   ( x  xn )/ Lp
p( x) 
  p ( xn ) 
for x  xn
e
Nd 
Nd 
ni2 
ni2   ( x  x p )/ Ln
n( x ) 
  n( x p ) 
for x  x p
e
Na 
Na 
3/30/2015
PHY 752 Spring 2015 -- Lecture 26
15
Boltzmann equation treatment of current response to diode
Approximate steady state solutions in presence of
applied voltage VA
Current through device:
j =  ejn ( x p )  ej p ( xn )
eD
= n
Ln

ni2  eD p
 n( x p ) 

N
Lp
a 


ni2 
 p ( xn ) 

N
d 


ni2 
Dn  ni2  eVA ni2 
jn ( x p )   Dn

 n( x p ) 

 e

dx
N
L
N
N
a 
n 
a
a 

dp ( xn ) D p 
ni2  D p  ni2  eVA ni2 
j p ( xn )  D p

e

 p ( xn ) 



dx
Lp 
N d  Lp  N d
Nd 
dn( x p )

 j  en e
2
i
3/30/2015
 eVA
D
 n
Ln
 Dn
Dp 
1 


L
N
L
N
 n a
p d 


PHY 752 Spring 2015 -- Lecture 26
16
Behavior of diode:
j
VA<0
3/30/2015
PHY 752 Spring 2015 -- Lecture 26
VA>0
17