EE40
Lecture 32
Prof. Chang-Hasnain
11/21/07
Reading: Supplementary Reader
EE40 Fall 2007
Slide 1
Prof. Chang-Hasnain
Electron and Hole Densities in Doped Si
• Instrinsic (undoped) Si
n  p  ni
np  ni
2
• N-doped Si
– Assume each dopant contribute to one electron
n  Nd  Nce
( E f  Ec ) kT
p  ni 2 N d
• p-doped Si
– Assume each dopant contribute to one hole
p  Na  Nve
( Ev  E f ) kT
p  ni 2 N a
EE40 Fall 2007
Slide 2
Prof. Chang-Hasnain
Summary of n- and p-type silicon
Pure silicon is an insulator. At high temperatures it conducts
weakly.
If we add an impurity with extra electrons (e.g. arsenic,
phosphorus) these extra electrons are set free and we have a
pretty good conductor (n-type silicon).
If we add an impurity with a deficit of electrons (e.g. boron) then
bonding electrons are missing (holes), and the resulting holes
can move around … again a pretty good conductor (p-type
silicon)
Now what is really interesting is when we join n-type and p-type
silicon, that is make a pn junction. It has interesting electrical
properties.
EE40 Fall 2007
Slide 3
Prof. Chang-Hasnain
Junctions of n- and p-type Regions
p-n junctions form the essential basis of all semiconductor devices.
A silicon chip may have 108 to 109 p-n junctions today.
How do they behave*? What happens to the electrons and holes?
What is the electrical circuit model for such junctions?
n and p regions are brought into contact :
aluminum
aluminum
?
wire
n
p
*Note that the textbook has a very good explanation.
EE40 Fall 2007
Slide 4
Prof. Chang-Hasnain
The pn Junction Diode
Schematic diagram
p-type
net acceptor
concentration NA
Circuit symbol
ID
n-type
net donor
concentration ND
cross-sectional area AD
Physical structure:
(an example)
+
+ ID
VD
metal
SiO2
For simplicity, assume that
the doping profile changes
abruptly at the junction.
EE40 Fall 2007
VD
–
SiO2
p-type Si
n-type Si
–
metal
Slide 5
Prof. Chang-Hasnain
Depletion Region Approximation
• When the junction is first formed, mobile carriers diffuse
across the junction (due to the concentration gradients)
– Holes diffuse from the p side to the n side,
leaving behind negatively charged immobile acceptor
ions
– Electrons diffuse from the n side to the p side,
leaving behind positively charged immobile donor ions
acceptor ions
p
donor ions
–
–
–
–
–
+
+
+
+
+
n
A region depleted of mobile carriers is formed at the junction.
• The space charge due to immobile ions in the depletion region
establishes an electric field that opposes carrier diffusion.
EE40 Fall 2007
Slide 6
Prof. Chang-Hasnain
Summary: pn-Junction Diode I-V
• Under forward bias, the potential barrier is reduced, so
that carriers flow (by diffusion) across the junction
– Current increases exponentially with increasing forward bias
– The carriers become minority carriers once they cross the
junction; as they diffuse in the quasi-neutral regions, they
recombine with majority carriers (supplied by the metal contacts)
“injection” of minority carriers
• Under reverse bias, the potential barrier is increased, so
that negligible carriers flow across the junction
– If a minority carrier enters the depletion region (by thermal
generation or diffusion from the quasi-neutral regions), it will be
swept across the junction by the built-in electric field
ID (A)
“collection” of minority carriers  reverse current
VD (V)
EE40 Fall 2007
Slide 7
Prof. Chang-Hasnain
Charge Density Distribution
Charge is stored in the depletion region.
acceptor ions
p
quasi-neutral p region
donor ions
–
–
–
–
–
+
+
+
+
+
n
depletion region
quasi-neutral n region
charge density (C/cm3)
distance
EE40 Fall 2007
Slide 8
Prof. Chang-Hasnain
Two Governing Laws
Gauss’s Law describes the relationship of charge (density) and
electric field.
1
 E  dA      dV 
S
dE 

dx 
V
Qencl

E ( x)  E ( x0 ) 
1

x
x0
 ( x)dx
Poisson’s Equation describes the relationship between electric
field distribution and electric potential
d 2 ( x)
dE ( x)
 ( x)


2
dx
dx

x
 ( x)   ( x0 )    E ( x)dx
x0
EE40 Fall 2007
Slide 9
Prof. Chang-Hasnain
Depletion Approximation 1
 qN a
 0 x   
 qN d
 x
p0
 x  0
0  x  xn0 
and  0 x   0 x   x p 0 , x  xn 0 
ρo(x)
p
qNd
n
x
E0 ( x) 
( x  x po )
s
E0 ( x)  
x
E0 ( x) 
-xpo
qN a
xno
x
( x po  x  0)
-qNa
0 ( x)
qN d
dx  E0 ( xno ) 
( xno  x)  0
s
s
qN d
s
xno
( x  xno )
Gauss’s Law
E0(x)
p
n
-xpo
(0  x  xno )
xno
x
x
E0 (0) 
EE40 Fall 2007
Slide 10
 qN a x po
s

 qN d xno
s
Prof. Chang-Hasnain
Depletion Approximation 2
p
-xpo
d
2s
xno2 
qN
a
2s
n
xno
E0 (0) 
Poisson’s Equation
qN
E0(x)
 qNa x po
s

 qNd xno
s
0(x)
xpo2
n=1017
p=105
P=1018
n=104
-xpo
EE40 Fall 2007
x
Slide 11
xno
x
Prof. Chang-Hasnain
EE40
Lecture 33
Prof. Chang-Hasnain
11/26/07
Reading: Supplementary Reader
EE40 Fall 2007
Slide 12
Prof. Chang-Hasnain
Depletion Approximation 3
0 ( x)  
x
 x po
 E0 ( x)dx  0 ( x po )  
qN a
x
s
 x po
( x  x po )dx  0
x
qN a  x


x
dx

x
dx

 xpo po 
 s   x po
qN a
0 ( x) 
( x  x po ) 2
2 s
(  x po  x  0)
x
x
0
0
0 ( x)    E0 ( x)dx  0 (0)   

qN d
s
 
x
0
x dx  
x
0

qN d
s
qN a
xno dx 
x po 2
2 s
qN d
qN a
2
0 ( x) 
x(2 xno  x) 
x po 2
2 s
2 s
EE40 Fall 2007
qN a
( x  xno )dx 
(0  x po ) 2
2 s
Slide 13
(0  x  xno )
Prof. Chang-Hasnain
Effect of Applied Voltage
VD
p
–
–
–
–
–
+
+
+
+
+
n
• The quasi-neutral p and n regions have low resistivity,
whereas the depletion region has high resistivity. Thus,
when an external voltage VD is applied across the
diode, almost all of this voltage is dropped across
the depletion region. (Think of a voltage divider
circuit.)
• If VD > 0 (forward bias), the potential barrier to carrier
diffusion is reduced by the applied voltage.
• If VD < 0 (reverse bias), the potential barrier to carrier
diffusion is increased by the applied voltage.
EE40 Fall 2007
Slide 14
Prof. Chang-Hasnain
Depletion Approx. – with VD<0 reverse bias
p
-xp -xpo
E0(x)
n
xno x
n
E0 (0) 
 qNa x po
s
x

 qNd xno
s
Higher barrier and few holes in np=105
type lead to little current!
0(x)
qN
qN
bi-qVD
a
d
Built-in potential bi=
xno2 
xpo2
2s
2s
n=1017
P=1018
bi
n=104
x
-xp -xpo
xnoxn
EE40 Fall 2007
Slide 15
Prof. Chang-Hasnain
Depletion Approx. – with VD>0 forward bias
p
-xpo
-x
p
E0(x)
n
xnxno
E0 (0) 
 qNa x po
x

 qNd xno
s
s
Poisson’s Equation
Lower barrier and large hole (electron) density
at the right places lead to large current!
0(x)
qN
qN
a
d
Built-in potential bi=
xno2 
xpo2
P=1018
2s
2s
bi
bi-qVD
n=104
-xp
-xpo
EE40 Fall 2007
Slide 16
xnxno
n=1017
p=105
x
Prof. Chang-Hasnain
Forward Bias
• As VD increases, the potential barrier to carrier
diffusion across the junction decreases*, and
current increases exponentially.
VD > 0
p
–
–
–
–
–
+
+
+
+
+
I D  I S (e
The carriers that diffuse across the
junction become minority carriers in
the quasi-neutral regions; they then
recombine with majority carriers,
“dying out” with distance.
ID (Amperes)
n
qVD kT
 1)
VD (Volts)
* Hence, the width of the depletion region decreases.
EE40 Fall 2007
Slide 17
Prof. Chang-Hasnain
Reverse Bias
• As |VD| increases, the potential barrier to carrier
diffusion across the junction increases*; thus, no
carriers diffuse across the junction.
VD < 0
p
–
–
–
–
–
+
+
+
+
+
A very small amount of reverse
current (ID < 0) does flow, due to
minority carriers diffusing from the
quasi-neutral regions into the depletion
region and drifting across the junction.
ID (Amperes)
n
VD (Volts)
* Hence, the width of the depletion region increases.
EE40 Fall 2007
Slide 18
Prof. Chang-Hasnain
Optoelectronic Diodes
• Light incident on a pn junction generates electron-hole pairs
• Carriers are generated in the depletion region as well as ndoped and p-doped quasi-neutral regions.
• The carriers that are generated in the quasi-neutral regions
diffuse into the depletion region, together with the carriers
generated in the depletion region, are swept across the
junction by the electric field
• This results in an additional component of current flowing in
the diode:
qVD kT
I D  I S (e
1)  I optical
where Ioptical is proportional to the intensity of the light
EE40 Fall 2007
Slide 19
Prof. Chang-Hasnain
Example: Photodiode
• An intrinsic region is placed
between the p-type and n-type
regions
 Wj  Wi-region, so that most of the
electron-hole pairs are generated
in the depletion region
 faster response time
(~10 GHz operation)
ID (A)
in the dark
VD (V)
operating point
with incident light
EE40 Fall 2007
Slide 20
Prof. Chang-Hasnain
Planck Constant
•
•
•
•
Planck’s constant h = 6.625·10-34 J·s
E=hnhc/l1.24 eV-mm/lmm)
C is speed of light and hn is photon energy
The first type of quantum effect is the quantization of
certain physical quantities.
• Quantization first arose in the mathematical formulae of
Max Planck in 1900. Max Planck was analyzing how the
radiation emitted from a body was related to its
temperature, in other words, he was analyzing the
energy of a wave.
• The energy of a wave could not be infinite, so Planck
used the property of the wave we designate as the
frequency to define energy. Max Planck discovered a
constant that when multiplied by the frequency of any
wave gives the energy of the wave. This constant is
referred to by the letter h in mathematical formulae. It is
a cornerstone of physics.
EE40 Fall 2007
Slide 21
Prof. Chang-Hasnain
Bandgap Versus Lattice Constant
Si
EE40 Fall 2007
Slide 22
Prof. Chang-Hasnain