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Semiconductor Physics
Carrier Transport

Contents:
1
...
Carrier drift
3
...
Thermal Motion
In thermal equilibrium, carriers are not sitting still:

• undergo collisions with vibrating Si atoms (Brownian
motion)
• electrostatically interact with charged dopants and
with each other

Characteristic time constant of thermal motion - mean
free time between collisions:
τc ≡ collision time [s]
In between collisions, carriers acquire high velocity:
vth ≡ thermal velocity [cm/s]

...
1 µm

⇒ carriers undergo many collisions in modern devices


2
...


Define:
µn,p =

qτc
2mn,p

≡ mobility [cm2 /V · s]

Then, for electrons:
vdn = −µnE
for holes:

vdp = µp E

Mobility is measure of ease of carrier drift:
• if τc ↑, longer time between collisions → µ ↑
• if m ↓, ”lighter” particle → µ ↑
Mobility depends on doping
...


• In n-type semiconductor:
ρn

1
qNdµn

• In p-type semiconductor:
ρp

1
qNaµp

For Si at 300K:
1E+4
1E+3

Resistivity (ohm
...
21 Ω · cm

• apply |E| = 1 kV /cm
|vdn|
drif
|Jn t |

106 cm/s

vth

4
...
1 µm:
td =
fast!


L
= 10 ps
vdn

3
...


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, particle movement down the gradient

Key diffusion relationship (Fick’s first law):


Diffusion flux ∝ - concentration gradient
Flux ≡ number of particles crossing unit area per unit
time [cm−2 · s−1]
For electrons:
dn
Fn = −Dn
dx
For holes:
Fp = −Dp

dp
dx

Dn ≡ electron diffusion coefficient [cm2 /s]
Dp ≡ hole diffusion coefficient [cm2 /s]
D measures the ease of carrier diffusion in response to a
concentration gradient: D ↑ ⇒ F dif f ↑
...
012]:
D kT
=
µ
q
In semiconductors:
Dn Dp kT
=
=
µn
µp
q
kT
q

≡ thermal voltage [V ]

At 300 K:
kT
q

25 mV

For example: for Nd = 3 × 1016 cm−3 :
µn
µp

1000 cm2/V · s → Dn 25 cm2/s
400 cm2/V · s → Dp 10 cm2 /s

Total current

In general, current can flow by drift and diffusion separately
...


Key conclusions

• Electrons and holes in semiconductors are mobile and
charged ⇒ carriers of electrical current!
• Drift current: produced by electric field

J drif t ∝ E
• Diffusion current: produced by concentration gradient
J dif f ∝

dn dp
,
dx dx

• Carriers move fast in response to fields and gradients

• Diffusion and drift currents are sizable in modern devices



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