6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 18-1
Lecture 18 - Metal-Semiconductor Junction
(cont.)
March 16, 2007
Contents:
1. Metal-semiconductor junction outside equilibrium (cont.)
Reading assignment:
del Alamo, Ch. 7, §7.2.3
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6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 18-2
Key questions
• In a metal-semiconductor junction under bias, is there current
flow? If so, how exactly does it happen?
• What are the key dependences of the current in a metal-semiconductor
junction?
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6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 18-3
1. Metal-semiconductor junction outside TE (cont.)
� I-V Characteristics
Few minority carriers anywhere → majority carrier device
Bottleneck: transport through SCR
Je=0
EF
qjBn
qjBn
Ec
EF
a) equilibrium
Ev
Je
Ef
q(jBn-V)
qjBn
Ec
Efe
b) forward bias
Ev
Je
Ef
qjBn
c) reverse bias
q(jBn-V)
Ec
Efe
Ev
• in forward bias, J ∝ eqV/kT
• in reverse bias, J saturates with V
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6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 18-4
Balance between electron drift and diffusion in SCR:
• TE: perfectly balanced
• forward bias: E ↓ ⇒ diffusion > drift
• reverse bias: E ↑ ⇒ diffusion < drift
Net current due to imbalance of drift and diffusion ⇒
� Drift-diffusion model
Start with electron current equation:
Je = qµe nE + qDe
dn
qn dφ dn
= qDe (−
+ )
dx
kT dx dx
qφ
Multiply by exp(− kT
):
Je exp(−
qφ
qn dφ
qφ
dn
qφ
) = qDe[−
exp(− ) +
exp(− )]
kT
kT dx
kT
dx
kT
= qDe
d
qφ
[n exp(− )]
dx
kT
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6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 18-5
Integrate along the depletion region:
• left-hand side: Je � J (negligible hole contribution), and J
independent of x:
� x
d
0
� x
qφ
qφ
Je exp(− )dx = J 0 d exp(− )dx
kT
kT
• right-hand side
� x
d
0
qDe
d
qφ
qφ x
[n exp(− )]dx = qDen exp(− )|0 d
dx
kT
kT
For left-hand side, use φ(x) obtained earlier:
x2 2x
φ(x) = −(φbi − V )( 2 −
+ 1)
xd xd
for 0 ≤ x ≤ xd
For right-hand side, use boundary conditions:
• At x = 0:
φ(0) = −(φbi − V )
Bn exp qV
n(0) = ND exp −q(φkTbi−V ) = Nc exp −qϕ
kT
kT
• At x = xd :
φ(xd) = 0
n(xd ) = ND
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6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 18-6
Do integral, substitute boundary conditions and get:
�
�
�
c�
�
2
J=
q De N
kT
2q(φbi − V )ND
−qϕBn
qV
exp
(exp
− 1)
�
kT
kT
Total current, multiply J by area Aj :
I = IS (exp
qV
− 1)
kT
IS ≡ saturation current (A)
I
log |I|
0.43 q
kT
IS
0
0
IS
linear scale
V
0
V
semilogarithmic scale
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6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 18-7
Key dependencies of drift-diffusion model:
qV
• I ∝ exp kT
−1
Bn
• IS ∝ exp −qϕ
kT
• IS weakly dependent on V
Experiments (Schottky diode from Analog Devices):
−4
10
data at T = 299 K
−6
Magnitude of current (A)
10
ideal Schottky diode equation with I0 = 1.4*10−13 A
−8
10
−10
10
−12
10
−14
10
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
Voltage (V)
0.2
0.3
0.4
0.5
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6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 18-8
Courtesy of the American Institute of Physics. Used with permission. Figure 5 on page 1598 in Akiya, Masahiro, and Hiroaki
Nakamura. "Low Ohmic Contact to Silicon with a Magnesium/Aluminum Layered Metallization." Journal of Applied Physics 59, no.
5 (1986): 1596-1598.
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6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 18-9
Temperature dependence of IS :
IS ∝ T 1/2 exp
−qϕBn
kT
not seen in practice!
What one finds experimentally is:
IS ∝ T 2 exp
−qϕBn
kT
Made implicit assumption: quasi-equilibrium across SCR ⇒
drift/diffusion balance only slightly broken.
Quasi-equilibrium assumption good if:
|J | � |Je (drif t)|, |Je (dif f )|
Test at x = 0:
|J |
�1!
|Je(drif t)|
Assumption fails at x = 0. Need to look at situation closely around
x = 0.
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6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 18-10
� Thermionic-emission theory
Closer than a mean free path from the interface, arguments of
drift and diffusion do not work!
In the last mean free path,
• electrons do not suffer any collisions,
• only those with enough EK get over the barrier
• actually, only half of those with enough EK do
• this is bottleneck: thermionic emission theory
�
Ef
0
lce
x
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6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 18-11
�
Ef
0
lce
x
In steady state:
Jt � Je = −qn(x)ve (x)
Focus on bottleneck at x = 0:
J = −qn(0)ve (0)
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6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 18-12
� First compute n(0):
n(0) is only a fraction of the carriers present at n(lce ):
n(0) =
n(lce )
q[φ(0) − φ(lce )]
exp
2
kT
If rest of SCR is in quasi-equilibrium:
n(lce ) � ND exp
qφ(lce )
kT
Also:
φ(0) = −(φbi − V )
Then:
n(0) =
ND
−q(φbi − V ) Nc
−q(ϕBn − V )
exp
=
exp
2
kT
2
kT
• n(0) is exactly half of what one would obtain if it was in TE
with bulk.
• All electrons at x = 0 are injected into metal.
• Note ∝ eqV/kT dependence on n(0).
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6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 18-13
� Then compute ve (0):
Over the last mean free path, carriers basically travel at vth
But, velocity pointing at different angles. After taking care of statis­
tics:
ve (0) = −
�
�
�
�
�
�
vth
2kT
=−
2
πm∗ce
(minus sign indicates carriers traveling against x)
� Finally, electron current:
J = A∗T 2 exp
−qϕBn
qV
exp
kT
kT
with:
�
�
� m∗
�
de 3
o�
� mo
�
� m∗ce
mo
4πqk 2m
A =
h3
∗
(
)
A∗ ≡ Richardson’s constant
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6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 18-14
Still must subtract electron injection from metal to semiconductor in
TE, so that when V → 0, J → 0:
J = A∗T 2 exp
−qϕBn
qV
(exp
− 1)
kT
kT
Valid in forward and reverse bias.
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6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
IS = Aj A∗T 2 exp
Lecture 18-15
−qϕBn
kT
IS /T 2 is thermally activated with Ea = qϕBn
−2
10
−4
10
−6
Current (A)
10
T = 344 K
−8
10
−10
10
T = 209 K
−12
ΔT = 15 K
10
−14
10
0
0.05
0.1
0.15
0.2
0.25
0.3
Voltage (V)
0.35
0.4
0.45
0.5
−15
10
−16
10
−17
10
−18
I0/T2 (A/K2)
10
−19
10
−20
10
−21
10
−22
10
−23
10
measurements
ideal Schottky diode equation with
qφB = 0.96 eV
−24
10
2.5
3
3.5
1000/T (1/K)
4
4.5
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6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 18-16
� Thermionic emission theory valid if:
thermionic current � drift current
for lce ≤ x ≤ xd.
Bethe condition:
lce |Emax| > 1.5
kT
q
Easily satisfied in Si at around room temperature (mean free paths
are rather long).
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6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 18-17
� If thermionic emission theory applies:
• Ef e flat throughout SCR up to x = lce.
• Beyond x = lce , Ef e has no physical meaning (electron distribu­
tion is not Maxwellian!)
Ef
Ec
Ef
forward bias
Ev
Ef
Ec
Ef
reverse bias
Ev
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6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 18-18
Key conclusions
• Minority carriers play no role in I-V characteristics of MS junc­
tion.
• Energy barrier preventing carrier flow from S to M modulated
by V , barrier to carrier flow from M to S unchanged by V ⇒
rectifying behavior:
I = IS (exp
qV
− 1)
kT
• Drift-diffusion theory of current: small perturbation of balance
of drift and diffusion inside SCR.
• Drift-diffusion theory of current exhibits several dependences
observed in devices, but fails temperature dependence.
• Thermionic emission theory of current: bottleneck is flow of
carriers over energy barrier at M-S interface. Transport at this
bottleneck is of a ballistic nature.
• IS /T 2 is thermally activated; activation energy is qϕBn.
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6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 18-19
Self study
• Thermionic emission theory
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