ECE 230B: Winter 2003
Solid-State Electronic Devices
Professor Yuan Taur
Electrical & Computer Engineering
University of California, San Diego
1/8/2003
Yuan Taur
1
ECE 230B: Winter 2003
Solid-State Electronic Devices
Prerequist: ECE 135A, B, or equivalent and ECE 230A
Instructor: Yuan Taur
Office: EBU1-4807
Phone: 534-3816
taur@ece.ucsd.edu
Office hour: Open
This course covers the physics of solid-state electronic devices, including
p-n junctions, MOS devices, field-effect transistors, bipolar transistors, etc.
Principles of CMOS and bipolar scaling to nanometer dimensions and their high
frequency performance in digital and analog circuits will be taught.
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Topics to be Covered
1) Band diagram, Fermi level, Poisson’s
eq., Carrier transport (1.5 wks)
2) P-n junction (1 wk)
3) MOS device (1 wk)
4) MOSFETs (1.5 wks)
5) CMOS scaling and design (1.5 wks)
6) CMOS performance factors (1 wk)
7) Bipolar transistors (1.5 wks)
8) SiGe bipolar device (1 wk)
Textbook: “Fundamentals of Modern VLSI Devices”
Y. Taur and T. H. Ning, Cambridge Univ. Press (1998)
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Homework and Grading Policy
•
•
•
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Five open book quizzes (bring
your calculator)
Five homework assignments
Equal proportion of credits
among above
4
Electron Energy Levels and Bands
Discrete electron energy
levels in an atom:
Broadening into electron
energy bands in a solid:
There are forbidden energy gaps between
allowed electron energy bands.
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The Origin of Energy Gap in a
Crystalline Solid
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The Origin of Energy Gap in a
Crystalline Solid
(From Kittel)
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Crystalline Lattice
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Energy Band Structures
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Metals, Insulators, and Semiconductors
Insulators: Eg > 4-5 eV
Semiconductors: Eg < 4-5 eV
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Metals, Insulators, and Semiconductors
(From Muller and Kamins)
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Energy Band Diagram of Silicon
Hole
energy
Conduction band
Ec
Eg
E
i
Ev
Valence band
Electron
energy
Eg = 1.12 eV
Free electron (− )
kT/q = 0.026 V
@ 300 K
Free hole (+)
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Fermi-Dirac Statistical Distribution
f ( E) =
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1
1+ e
( E − E f )/ kT
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Intrinsic Silicon
E
E
E
E
Conduction
band
n
Ec
Eg
E
f
Ev
p
Valence
band
1/2
Free electron (−)
Free hole (+)
N(E)
N (E) =
∞
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f (E)
8πg 2 m x m y m z
n = ∫ N ( E ) f ( E )dE
Ec
1
n and p
E − Ec (Appendix C)
h3
p = ∫ N ( E )[1 − f ( E )]dE
Ev
−∞
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Fermi Level in Intrinsic Silicon
1
Fermi-Dirac distribution:
f ( E) =
For (E - Ef)/kT >> 1,
f ( E) ≈ e
For (E - Ef)/kT << -1,
f ( E) ≈ 1 − e
1+ e
( E− E f )/ kT
− ( E − E f )/ kT
− ( E f − E )/ kT
Carrying out the integrations,
n = Nc e
− ( Ec − E f )/ kT
p = Nv e
Nc, Nv: Effective density of states, N c =
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2 g mx m y mz
h3
( 2πkT ) 3 / 2
Ec + Ev kT  Nc 
Ei = E f =
−
ln 
2
2  Nv 
Intrinsic silicon:
charge neutrality
requires n = p = ni,
− ( E f − Ev )/ kT
ni =
N c N v e − ( E c − E v )/ 2 kT =
NcNv e
− E g / 2 kT
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Physical Properties
Si
SiO2
Atomic/molecular weight
28.09
60.08
Atoms or molecules/cm3
5.0×1022
2.3×1022
Density (g/cm3)
2.33
2.27
Crystal structure
Diamond
Amorphous
Lattice constant (Å)
5.43
---
Energy gap (eV)
1.12
8-9
Dielectric constant
11.7
3.9
1.4×1010
---
Electron: 1430
---
Hole: 470
---
Conduction band Nc: 3.2×1019
---
Valence band Nv: 1.8×1019
---
Breakdown field (V/cm)
3×105
>107
Melting point (°C)
1415
1600-1700
Thermal conductivity (W/cm-°C)
1.5
0.014
Specific heat (J/g-°C)
0.7
1.0
Thermal diffusivity (cm2/s)
0.9
0.006
Thermal expansion coefficient (°C−1)
2.5×10−6
Intrinsic carrier concentration (cm−3)
Carrier mobility (cm2/V-s)
Effective density of states (cm−3)
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0.5×10−6
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Extrinsic (n-type and p-type) Silicon
Si
Si
Si
Si
Si
Si
Si
Si
Si
P+
Si
Si
B−
Si
−q
Si
Si
Si
Si
+q
Si
Si
Si
Si
Si
Si
Si
Si
(a)
(b)
(c)
Intrinsic
n-type
p-type
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Si
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Band Diagrams of n-type and
p-type Silicon
Conduction band
Conduction band
Ec
E
d
E
f
Ei
Eg
Ec
E
i
E
f
E
Ev a
Eg
Ev
Valence band
Free electron (−)
(a) n-type
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Valence band
Free hole (+)
(b) p-type
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Fermi Level in Extrinsic Silicon
n = Nc e
− ( Ec − E f )/ kT
p = Nv e
− ( E f − Ev )/ kT
For n-type silicon with donor concentration Nd, charge neutrality
requires n = p + Nd+ ≈ Nd (assuming complete ionization).
Therefore,
 Nc 
Ec − E f = kT ln 
 Nd 
Or, in terms of ni and Ei,
 Nd 
E f − Ei = kT ln 
 ni 
For Nd > Nc, Ef > Ec Æ degenerate n+ silicon. Need full F-D eq.
Note that np = ni2, independent of Ef (n- or p-type).
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Fermi Level vs. Temperature
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Fermi Level in Extrinsic Silicon
If incomplete ionization, Nd+ < Nd where


1
N d = N d [1 − f ( E d )] = N d 1 −
( E d − E f )/ kT 
 1 + (1 / 2 ) e

Ef is then solved from
+
Nce
− ( E c − E f ) / kT
=
Nd
1 + 2e
− ( E d − E f ) / kT
+ Nv e
− ( E f − E v ) / kT
Æ Condition for complete ionization:
N d ( E c − E d ) / kT
<< 1
e
Nc
Requires the donor energy level to be within a few kT of
the bottom of the conduction band.
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Donor and Acceptor Levels in Silicon
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Dopant Solubility in Silicon
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Carrier Concentration vs. Temperature
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Carrier Transport: Drift
v d = − qEτ / m
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Mobility
vd = µE
Mobility (cm2/V-s)
1,400
T = 300 K
Electrons
1,200
40
30
1,000
800
600
20
Holes
400
10
Diffusion coefficient (cm2/s)
1,600
200
0
1E+14
1E+15
1E+16
1E+17
1E+18
1E+19
0
1E+20
Doping concentration (cm-3)
1
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µ
=
1
µL
+
1
µI
+L
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Resistivity of Silicon
J n,drift = qnvd = qnµ n E
ρn =
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1
qnµ n
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Velocity Saturation
Carrier velocity (cm/s)
1E+7
1E+6
E
1E+5
1E+2
n
ro
t
lec
s
H
es
l
o
T=300 K
1E+3
1E+4
1E+5
Electric field (V/cm)
vsat ≈ 107 cm/s
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Electron concentration, n
Carrier Transport: Diffusion
dn
Jn, diff = qDn
dx
n(x+l)
n(x)
n(x-l)
kT
Dn =
µn
q
kT
Dp =
µp
q
x-l x x+l
Distance
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Poisson’s Equation
Define intrinsic potential:
Ei
ψi = −
q
Electric field:
E=-
Poisson’s eq.:
Or,
ρ ( x)
d 2ψ
dE
=
−
=
−
ε si
dx 2
dx
d 2ψ
dE
q
[ p ( x) − n( x ) + N d ( x ) − N a ( x)]
=
−
=
−
2
dx
dx
ε si
Gauss’s law:
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dψ
dx
E=
1
ε si
∫ ρ ( x)dx =
Qs
ε si
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Carrier Concentration in terms
of Electrostatic Potential
Silicon
surface
Ec
qψ s
(> 0)
qψ ( x )
qψ
Eg
B
E
i
E
f
Ev
kT  N b 
ln 
ψB ≡ ψ f − ψi =
q  ni 
p-type silicon
Oxide
( E f− Ei )/ kT
= nie
q ( ψ i −ψ f )/ kT
( Ei − E f )/ kT
= nie
q ( ψ f −ψ i )/ kT
n = nie
p = nie
x
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Debye Length
d 2ψ i
q
q ( ψ i − ψ f ) / kT
=
−
N
(
x
)
−
n
e
[
]
i
ε si d
dx 2
LD
Ef
Ei
Nd
Nd +∆Nd
d 2 ( ∆ψi ) q 2 N d
q
(
∆
ψ
)
∆N d ( x )
−
=
−
i
ε si kT
ε si
dx 2
Solution: ∆ψi ~ exp(-x/LD)
where
LD ≡
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ε si kT
q2 Nd
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Current Density Equations
Combining drift and diffusion components,
Or,
J n = qnµ n E + qDn
dn
dx
J p = qpµ p E − qD p
dp
dx
 dψ i kT dn 
dφ n
Jn = −qnµ n 
−
 = −qnµ n
dx
 dx qn dx 
 dψ i kT dp 
dφ p
J p = −qpµ p
+
 = −qpµ p
dx
 dx qp dx 
where
φ n ≡ψ i −
kT  n 
ln
q  ni 
φ p ≡ψ i +
kT  p 
ln
q  ni 
are the quasi-Fermi potentials.
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Current Continuity Equations
From conservation of mobile charge,
∂ n 1 ∂ Jn
=
− Rn + Gn
∂t q ∂x
1 ∂ Jp
∂p
=−
− Rp + G p
q ∂x
∂t
where Gn, Gp, Rn, Rp are electron-hole generation
and recombination rates.
In the steady state with negligible electron-hole
generation and recombination rates,
dJn/dx = 0
dJp/dx = 0
Simply continuity of electron and hole currents.
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Dielectric Relaxation Time
Neglect Rn, Gn in the current continuity equation,
∂ n 1 ∂ Jn
=
∂t q ∂x
From Ohm’s law, Jn = E/ρn
And from Poisson’s equation with majority carrier
density n, ∂E/∂x = −qn/εsi
Substituting into current continuity equation:
∂n
n
=−
∂t
ρ nε si
The solution is of the form n(t) ∝ exp(−t/ρnεsi)
where ρnεsi ≈ 10-12 s is the majority carrier
response time.
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