Transcript Chemistry 140a - California Institute of Technology

Chemistry 140a
Lecture #5
Jan, 29 2002
Fermi-Level Equilibration
• When placing two surfaces in contact, they will equilibrate; just
like the water level in a canal lock.
• The EF of the semi-conductor will always lower to the EF of the
metal or the solution. This can be understood by looking at the
density of states for each material/soln.
Semi-Conductor
Metal/Soln.
Initial EF
Eq. EF
Eq. E
Initial
EFF
Fermi-Level Equilibration
• Charge comes from the easiest thing to ionize, the dopant
atoms. This leads to a large region of (+) charges within the
semi-conductor.
• In the metal all of the charge goes to the surface. (Gauss’s Law)
• The more charge transferred the more band bending.
E
E
E
E
ECB
ECB
EF
EF
EVB
EVB
x
x
V
Vbi
bi
EF
EF
ECB
ECB
EVB
xx
EVB
VbiVbi
EF
EF
Depletion Approximation
• All donors are fully ionized to a certain
distance, W, from the interface.
• W=W(ND,Vbi)
+++++ -+++++ -W
ND
W
Vbi
W
X
Final Picture
E
E
EVac
EVac
ECB
EF
EVB
x
m
sc
Vbi
EF
ECB
Vbi
Eg
EVB
x
+
++ -
EF
Useful Equations
E(x) = Electric Field (V/cm)
(x) = Electric Potential (V)
(x) = Electric Potential Energy (J)
d(x)
E(x)  
dx
E x  q(x)
x  qpx  nx  NA x  ND x

Poisson’s Eqn:
d 2 (x)

dx2
x 

K0

Electric Potential (V)
Integrate Poisson’s Eqn.
(x)x)
d 2 (x) x 


2
dx
K0
B.C.’s
 d (x)
0
dx
 x   0
Result:

+++++++
Q=qNDW
x W
W
x W
quadratic
qND
2
 (x) 
x  W 
2K0
qN
D
X
- x
-qNDW2
V-bi=
(2K0)
-
Depletion Width
• Rearranging for W:
2K0Vbi
W
qND
• As expected, W increases w/ Vbi and decreases w/
ND

• If one accounts for the free carrier distribution’s tail
around x=W

kT
2K0 Vbi  
q 

W 
qND
Typical Values
Vbimax (V)
ND (cm-3)
W (m)
Q (C/cm2)
1
1013
11
1010
1
1016
0.36
3*1011
Electric Potential Energy
• E(x) = -q(x)
E
Net = 0 @ Eq.
 (0) = -Vbi
ECB
• qVbi = (EF,SC-EF,M)
 B = Vbi + Vn
Vbi
(x)
B
e-
Vn
EF
x
EVB
– Barrier height
x
– Independent of doping
– Vbi and Vn are doping dependent
h+
Electric Field (V/cm)
qN
d

(x)
D
E(x)  

x  W 
K0
dx
Ex)

W

x
Emax=-qNDW/(K
I-V Curve
I
No Band Bending
Low Band Bending
High Band Bending
V
Review
• N-type
P-type
E
E
EVac
EVac
m
sc
ECB
Vbi
Eg
EVB
x
sc
-
+
++ -
EF
Vbi
EVB
x
m
__
-
ECB
Eg
+
+
+
EF
Solution Contact
• 10^17 atoms in 1mL of 1mM solution
• D.O.S. argument holds
• Difference in exchange current across the
interface
++++
++++
++++
++++
AAAAAA-
Li+
Li+
Li+
Li+
Li+
Li+
*Significantly less than typical W ~ 10nm
5-10 Angstroms
Semiconductor Contacting
Phase
• No longer 1-Sided Abrupt Jxn. as the semi-conductor
doesn’t have infinite capacity to accept charge
• Assume ND(n-type)=NA(p-type), then Wn=Wp
n-type
p-type
e-
h+
Diode
directionalized current
Degenerate Doping
• Dope p-type degenerately
• NA>>ND --> 1-sided Abrupt Jxn.
P-N Homojunction
W n Wp
N-type
B
B
B
N-type
P+-type
Heterojunctions
• 2 different semiconductors grown w/ the
same cyrstal structure (difficult)
– Ge/GaAs
Normal
ao~5.65 angstroms
Staggered
Broken
LASERs
• 3 Pieces --> 2 Heterjunctions
– p-(Al,Ga)As | GaAs | n-(Al, Ga) As
e-
h
h+
Traps electrons and holes
Fermi-Level Pinning

•Ideal Case
1
(only works for very ionic
semiconductors like TiO2 and SnO2)
EF,M
Never works for Si
Fermi-Level Pinning
Sze p. 278
Slope
1
TiO2 SnO2
CdS
Si GaAs
A-B
What’s Missing?
E
E
• Fermi-Level pinning hurts
– Hinders our ability to fine tune Vbi
EVac
EVac Vbi/Ni~Vbi/Pt~Vbi/Au
vs.
• Why does this happen?
ECB
B
EF
ECB
B
EF
EVB
EVB
*Solution contact for GaAs sees Fermi-level
x pinning, while the barrier
x
height correlates well with the electro-chemical potential for solution
contact to Si
Devious Experimenter
• Given a Si sample with a
magic type of metal on the
surface X
• Thus the Fermi-level will
alwaysequilibrate to the
Fermi-level of X
• Thin interface --> e-’s tunnel
through it and no additional
potential drop is observed
E
E
ECB
EVB
x
EF,X-EF,M
EF,X
EF,X
What is X?
• Any source or sink for charge at the
interface
– Dangling bonds
– Surface states
– etc.
Questions
• Questions
– Abrupt 1-sided junction
(What is it?)
– Sign of Electric P.E. and Electric Potential
(Are they correct? I put them as they were in
the notes, but this doesn’t seem to agree
with the algebra to me)