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Slide1
Semiconductor Device Modeling and Characterization – EE5342 Lecture 7 – Spring 2011
Professor Ronald L. Carter
ronc@uta.edu
http://www.uta.edu/ronc/
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First Assignment
e-mail to listserv@listserv.uta.edu
In the body of the message include subscribe EE5342
This will subscribe you to the EE5342 list. Will receive all EE5342 messages
If you have any questions, send to ronc@uta.edu, with EE5342 in subject line.
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Second Assignment
Submit a signed copy of the document that is posted at
www.uta.edu/ee/COE%20Ethics%20Statement%20Fall%2007.pdf
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Schedule Changes Due to the University Closures last week
Plan to meet until noon some days in the next few weeks. This way we will make up the lost time. The first extended class will be Wednesday, February 9.
The MT will be postponed until Wednesday, February 16. All other due dates and tests will remain the same.
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Equipartition
theorem
The thermodynamic energy per degree of freedom is kT/2
Consequently,
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Carrier velocity
saturation
1
The mobility relationship v =
m
E is limited to “low” fields
v < v
th
= (3kT/m*)
1/2
defines “low”
v =
m
o
E[1+(E/E
c
)
b
]
-1/
b
,
m
o
= v
1
/E
c
for Si
parameter electrons holes
v
1
(cm/s) 1.53E9 T
-0.87
1.62E8 T
-0.52
E
c
(V/cm) 1.01 T
1.55
1.24 T
1.68
b
2.57E-2 T
0.66
0.46 T
0.17
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v
drift
[cm/s]
vs.
E
[V/cm]
(Sze
2
, fig. 29a)
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Carrier velocity
saturation (cont.)
At 300K, for electrons,
m
o
= v
1
/E
c
= 1.53E9(300)
-0.87
/1.01(300)
1.55
= 1504 cm
2
/V-s, the low-field mobility
The maximum velocity (300K) is v
sat
=
m
o
E
c
= v
1
=
1.53E9 (300)
-0.87
= 1.07E7 cm/s
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Diffusion of
carriers
In a gradient of electrons or holes,
p and
n are not zero
Diffusion current,
`
J
=
`
J
p
+
`
J
n
(note D
p
and D
n
are diffusion coefficients)
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Diffusion of
carriers (cont.)
Note (
p)
x
has the magnitude of dp/dx and points in the direction of increasing p (uphill)
The diffusion current points in the direction of decreasing p or n (downhill) and hence the - sign in the definition of
`
J
p
and the + sign in the definition of
`
J
n
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Diffusion of
Carriers (cont.)
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Current density
components
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Total current
density
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Doping gradient
induced E-field
If N = N
d
-N
a
= N(x), then so is E
f
-E
fi
Define
f
= (E
f
-E
fi
)/q = (kT/q)ln(n
o
/n
i
)
For equilibrium, E
fi
= constant, but
for dN/dx not equal to zero,
E
x
= -d
f
/dx =- [d(E
f
-E
fi
)/dx](kT/q) = -(kT/q) d[ln(n
o
/n
i
)]/dx = -(kT/q) (1/n
o
)[dn
o
/dx] = -(kT/q) (1/N)[dN/dx], N > 0
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Induced E-field
(continued)
Let V
t
= kT/q, then since
n
o
p
o
= n
i
2
gives n
o
/n
i
= n
i
/p
o
E
x
= - V
t
d[ln(n
o
/n
i
)]/dx = - V
t
d[ln(n
i
/p
o
)]/dx = - V
t
d[ln(n
i
/|N|)]/dx, N = -N
a
< 0
E
x
= - V
t
(-1/p
o
)dp
o
/dx = V
t
(1/p
o
)dp
o
/dx = V
t
(1/N
a
)dN
a
/dx
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The Einstein
relationship
For E
x
= - V
t
(1/n
o
)dn
o
/dx, and
J
n,x
= nq
m
n
E
x
+ qD
n
(dn/dx)
= 0
This requires that nq
m
n
[V
t
(1/n)dn/dx] = qD
n
(dn/dx)
Which is satisfied if
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Direct carrier
gen/recomb
gen
rec
-
+
+
-
E
v
E
c
E
f
E
fi
E
k
E
c
E
v
(Excitation can be by light)
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Direct gen/rec
of excess carriers
Generation rates, G
n0
= Gp0Recombination rates, Rn0 = Rp0In equilibrium: Gn0 = Gp0 = Rn0 = Rp0In non-equilibrium condition:n = no +
dn and p = po + dp, where nopo=ni2and for dn and dp > 0, the recombination rates increase to R’n and R’p
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Direct rec for
low-level injection
Define low-level injection as
d
n = dp < no, for n-type, and dn = dp < po, for p-typeThe recombination rates then are R’n = R’p = d
n(t)/tn0, for p-type, and R’n = R’p = dp(t)/tp0, for n-typeWhere tn0 and tp0 are the minority-carrier lifetimes
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Shockley-Read-
Hall Recomb
E
v
E
cEf
Efi
E
k
E
c
E
v
E
T
Indirect, like Si, so intermediate state
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S-R-H trap
characteristics
1
The Shockley-Read-Hall Theory requires an intermediate “trap” site in order to conserve both E and p
If trap neutral when orbited (filled) by an excess electron - “donor-like” Gives up electron with energy Ec - ET“Donor-like” trap which has given up the extra electron is +q and “empty”
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S-R-H trap
char. (cont.)
If trap neutral when orbited (filled) by an excess hole - “acceptor-like”
Gives up hole with energy E
T - Ev“Acceptor-like” trap which has given up the extra hole is -q and “empty”Balance of 4 processes of electron capture/emission and hole capture/ emission gives the recomb rates
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References
*Fundamentals of Semiconductor Theory and Device Physics
, by
Shyh
Wang, Prentice Hall, 1989.
**
Semiconductor Physics & Devices
, by Donald A.
Neamen
, 2nd ed., Irwin, Chicago.
M&K =
Device Electronics for Integrated Circuits
, 3rd ed., by Richard S. Muller, Theodore I.
Kamins
, and
Mansun
Chan, John Wiley and Sons, New York, 2003.
1
Device Electronics for Integrated Circuits
, 2 ed., by Muller and
Kamins
, Wiley, New York, 1986.
2
Physics of Semiconductor Devices
, by S. M.
Sze
, Wiley, New York, 1981.
3
Physics of Semiconductor Devices
,
Shur
, Prentice-Hall, 1990.