Lecture 3
Semiconductor Physics (II)
Carrier Transport
Outline
• Thermal Motion
• Carrier Drift
• Carrier Diffusion
Reading Assignment:
Howe and Sodini; Chapter 2, Sect. 2.4-2.6
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1. Thermal Motion
In thermal equilibrium, carriers are not sitting still:
• Undergo collisions with vibrating Si atoms
(Brownian motion)
• Electrostatically interact with each other and with
ionized (charged) dopants
Characteristic time constant of thermal motion:
⇒ mean free time between collisions
τ c ≡ collison time [s]
In between collisions, carriers acquire high velocity:
−1
v th ≡ thermal velocity [cms ]
…. but get nowhere!
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Characteristic length of thermal motion:
λ ≡ mean free path [cm]
λ = v thτ c
Put numbers for Si at room temperature:
τc
−13
≈ 10
s
7
vth ≈ 10 cms
−1
⇒ λ ≈ 0.01 µm
For reference, state-of-the-art production MOSFET:
Lg ≈ 0.1 µm
⇒ Carriers undergo many collisions as they travel
through devices
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2. Carrier Drift
Apply electric field to semiconductor:
E ≡ electric field [V cm-1]
⇒ net force on carrier
F = ±qE
E
Between collisions, carriers accelerate in the direction of
the electrostatic field:
qE
v(t) = a • t = ±
t
mn,p
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But there is (on the average) a collision every τc and the
velocity is randomized:
net velocity
in direction
of field
τc
time
The average net velocity in direction of the field:
qE
qτ c
v = vd = ±
τc = ±
E
2mn,p
2mn,p
This is called drift velocity [cm s-1]
Define:
qτ c
2 −1 −1
µn, p =
≡ mobility [cm V s ]
2mn,p
Then, for electrons:
and for holes:
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v dn = − µn E
v dp = µp E
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Mobility - is a measure of ease
of carrier drift
• If τc ↑, longer time between collisions ⇒ µ ↑
• If m ↓, “lighter” particle ⇒ µ ↑
At room temperature, mobility in Si depends on doping:
1400
1200
electrons
mobility (cm2/Vs)
1000
800
600
holes
400
200
0
1013
1014
1015
1016
1017
1018
Nd + Na total dopant concentration (cm−3)
1019
1020
• For low doping level, µ is limited by collisions with
lattice. As Temp ->INCREASES; µ-> DECREASES
• For medium doping and high doping level, µ limited by
collisions with ionized impurities
• Holes “ heavier” than electrons
– For same doping level, µn > µp
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Drift Current
Net velocity of charged particles ⇒ electric current:
∝ carrier drift velocity
∝ carrier concentration
∝ carrier charge
Drift current density
Drift current densities:
drift
Jn
= −qnv dn = qnµ n E
drift
J p = qpv dp = qpµ p E
Check signs:
E
vdn
E
vdp
-
+
Jpdrift
Jndrift
x
x
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Total Drift Current Density :
(
)
drift
J drift = J drift
+
J
= q nµ n + pµ p E
n
p
Has the form of Ohm’s Law
J = σE =
E
ρ
Where:
σ ≡ conductivity [Ω-1 • cm-1]
ρ ≡ resistivity [Ω • cm]
Then:
σ=
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ρ
(
= q nµ n + pµ p
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Resistivity is commonly used to specify the doping level
• In n-type semiconductor:
• In p-type semiconductor:
1
ρn ≈
qN d µn
1
ρp ≈
qN aµ p
1E+4
Resistivity (ohm.cm)
1E+3
1E+2
1E+1
p-Si
1E+0
n-Si
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 (cm-3)
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Numerical Example:
Si with Nd = 3 x 1016 cm-3 at room temperature
µ n ≈ 1000 cm 2 / V • s
ρn ≈ 0.21Ω • cm
n ≈ 3X1016 cm −3
Apply E = 1 kV/cm
vdn ≈ 10 cm / s << vth
6
J
drift
n
≈ qnvdn = qnµ n E = σ E =
J
drift
n
≈ 4.8 × 10 A / cm
3
E
ρ
2
Time to drift through L = 0.1 µm
fast!
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L
td =
= 10 ps
vdn
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3. Carrier Diffusion
Diffusion = particle movement (flux) in response to
concentration gradient
n
x
Elements of diffusion:
• A medium (Si Crystal)
• A gradient of particles (electrons and holes) inside
the medium
• Collisions between particles and medium send
particles off in random directions
– Overall result is to erase gradient
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Fick’s first lawKey diffusion relationship
Diffusion flux ∝ - concentration gradient
Flux ≡ number of particles crossing a unit area per unit
time [cm-2 • s-1]
For Electrons:
Fn = −Dn
For Holes:
dn
dx
dp
Fp = −Dp
dx
Dn ≡ electron diffusion coefficient [cm2 s-1]
Dp ≡ hole diffusion coefficient [cm2 s-1]
D measures the ease of carrier diffusion in response to a
concentration gradient: D ↑ ⇒ Fdiff ↑
D limited by vibration of lattice atoms and ionized dopants.
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Diffusion Current
Diffusion current density =charge × carrier flux
dn
diff
Jn = qDn
dx
dp
diff
J p = −qD p
dx
Check signs:
n
p
Fn
Fp
Jpdiff
Jndiff
x
x
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Einstein relation
At the core of drift and diffusion is same physics:
collisions among particles and medium atoms
⇒ there should be a relationship between D and µ
Einstein relation [will not derive in 6.012]
D kT
=
q
µ
In semiconductors:
Dn
µn
=
kT
q
Dp
=
µp
kT/q ≡ thermal voltage
At room temperature:
kT
≈ 25mV
q
For example: for Nd = 3 x 1016 cm-3
µn ≈ 1000 cm2 / V • s ⇒ Dn ≈ 25cm 2 / s
µ p ≈ 400 cm 2 / V • s ⇒ D p ≈ 10 cm 2 / s
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Total Current Density
In general, total current can flow by drift and diffusion
separately. Total current density:
Jn =
J drift
n
+
J diff
n
Jp =
drift
Jp
+
diff
Jp
dn
= qnµ n E + qD n
dx
dp
= qpµ pE − qD p
dx
J total = J n + J p
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What did we learn today?
Summary of Key Concepts
• Electrons and holes in semiconductors are mobile
and charged
– ⇒ Carriers of electrical current!
• Drift current: produced by electric field
J
drift
∝E J
drift
dφ
∝
dx
• Diffusion current: produced by concentration
gradient
J
diffusion
dn dp
∝
,
dx dx
• Diffusion and drift currents are sizeable in modern
devices
• Carriers move fast in response to fields and
gradients
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