Recitation 3
Carrier Action
6.012 Spring 2009
Recitation 3: Carrier Action
Yesterday we talked about the movement of the carriers inside a semiconductor. There
is a direct relationship between the velocity of carriers and the electrical current that is
generated.
Current Density = |Jn | = |n · q · vn |
= |Jp | = |p · q · vp |
This is because:
|I | =
# of charges across cross-section area Q q · # of charges across cross-section area = = time
t
t
q · # density · volume q · n · L · A = = |q · n · vn · A| ∵ Lt = velocity
t
t
=
I n
= |Jn | = |q · n · vn |
A
Table 1: Drift vs. Diffusion
Drift
Diffusion
Mechanism
Due to electric field E
Carrier Velocity
vn = μn E vp = μp E
Current Density
Jn = −q · n · vn = q · n · μn · E
Jp = q · p · vp = q · p · μp · E
Important Parameter
Mobility μn , μp
q · τc
q · τc
μn =
μp =
2 · mn
2 · mp
1
Due to concentration gradient
dn dp
dx dx
dn
dp
Fp = −Dp
Flux Fn = −Dn
dx
dx
Fn
Fp
vn =
vp =
n
p
dn
Jn = −q · Fn = q · Dn ·
dx
dp
Jp = q · Fp = −q · Dp ·
dx
Diffusion Coefficient Dn , Dp
Dp
kT
=
μ
q
Recitation 3
Carrier Action
6.012 Spring 2009
Note: physical intuition rather than remembering the equation
• Current (can usually measure) always related to charge velocity (can back calculate)
• τc (collision time) is related to the imperfection of the lattice
• Mobility depends on collision time and temperature:
1. μ ∝ τc : doping(impurity) increases =⇒ more collisions =⇒ μ ↓
2. temperature (lattice vibration): higher T =⇒ more collision =⇒ μ ↓
• same semiconductor, the difference between μn & μp are due to mn & mp
• high mobility is extremely important for high performance devices
Si: μn = 1400 cm2 /V · sec μp = 500 cm2 /V · sec for doping 1013 cm−3
GaAs: μn = 8000 cm2 /V · sec μp = 400 cm2 /V · sec
1400
1200
Electrons
mobility (cm2 / V S)
Holes
1000
800
600
400
200
0
1013
1014
1015
1016
1017
1018
1019
1020
Nd + Nα total dopant concentration (cm-3)
Mobility limited by (scattering with)
lattice vibrations
Mobility limited by (scattering with)
impurities (dopants)
Carrier Mobility vs. Doping Concentration in Si
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Figure by MIT OpenCourseWare.
Recitation 3
Carrier Action
6.012 Spring 2009
Example 1: Integrated Resistor
Our first IC device:
Area: w x t
n-type slab of Si
Fabricated by ion implantation from top
As long as doping is uniform along L, no diffusion along L
Therefore, only drift current along L
Note: doping need not be uniform along t (as you will see)
J
E
I
I
But I
∴R
= Jn + Jp = q(nμn + pμp )E
V
, A=w·t
=
L
V
= J · A = q(nμn + pμp )
· (w · t)
L
w·t
= q(nμn + pμp )
·V
L
V
=
Ohm’s Law
R
1
1
L
L
= w·t = q(nμ + pμ ) · w · t = ρ · w · t
q(nμn + pμp ) L
n
p
3
Recitation 3
Carrier Action
Resistivity = ρ =
6.012 Spring 2009
1
or σ = q(nμn + pμp )
q(nμn + pμp )
1
, and vice versa).
q · n · μn
Since ρ (or σ) can be measured easily, it can be used to derive doping of a semiconductor
(n or p). If we take a Si wafer, it will be hard to know the doping a priori unless someone
specifies the doping level, but we can use resistivity to find out.
Usually majority dominates resistivity (n-type majority =⇒ ρ ≈
Example 2: Resistivity of Si
What is the resistivity of (1) intrinsic Si, (2) Si with Nd = 1013 and (3) Si with Na = 1020 ?
1. no = po = 1010 cm−3 . Therefore, ρ is:
=
=
1
1.6 ×
10−19 C(1450 cm2 /V
1.6 ×
10−19
· sec ×
1010 cm−3
+ 500 cm2 /V · sec × 1010 cm−3 )
1
= 3.2 × 105 Ω · cm (make sure the units are correct)
× 1.95 × 1013
Poor conductivity, quite insulating
2. Nd = 1013 cm−3 ni = 1010 =⇒ no ≈ Nd = 1013 , po =
n2i
= 107
no
1
1
=
= 430 Ω · cm
q(n · μn + p · μp )
1.6 × 10−19 × (1450 × 1013 + 500 × 107 )
(check on the curve)
ρ=
3. Na = 1020 ni = 1010 , =⇒ po ≈ Na = 1020 , no =
ρ≈
n2i
=1
po
1
1
=
= 1.25 × 10−3 Ω · cm like metal
−19
1.6 × 10
× 50 × 1020
q · p · μp
From this example, we can see that Si resistivity can be tuned several orders of magnitude
by doping, from insulator-like to metal-like.
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Recitation 3
Carrier Action
6.012 Spring 2009
1E+4
p-Si
n-Si
1E+3
Resistivity (ohm/cm)
1E+2
1E+1
1E+0
1E-1
1E-2
1E-3
1E-4
1E+12 1E+13 1E+14 1E+15 1E+16
1E+17 1E+18 1E+19 1E+20 1E+21
Doping (cm3)
Figure by MIT OpenCourseWare.
Sheet Resistance
R=
ρ L
t
w
The unit of ρ is Ω · cm, the unit of t is cm meaning that the unit of ρt is Ω - that of
ρ
resistance. We call t the sheet resistance Rs . This is a convenient metric for IC design
as:
• ρ, t: process and material parameters
•
L
: # of squares with dimensions w - layout design parameter
w
Sheet resistance is also a very useful parameter to characterize (thin) film resistivity.
Fabricating an IC Resistor
How to fabricate an IC resistor?
Make an n-type region in a p-type substrate. We will see why this isolation can work soon.
5
MIT OpenCourseWare
http://ocw.mit.edu
6.012 Microelectronic Devices and Circuits
Spring 2009
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