effective mass
TRANSCRIPT
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The Quasi-Free Electron and
Electron Effective Mass, m*
ECE G201(Partly adapted from Prof. Hopwood)
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Quasi-Free Electron:
This is the BIG approximation in looking at electrons in
crystals.
What is a Quasi-Free Electron?
Under some conditions (often found in devices)electrons behave like free particles with an effective mass
that is different than the mass in vacuum.
We want to understand this approximation.
We also want to understand when behavior beyond this
approximation occurs in devices.
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Goal: show that an electron
behaves like a particle with mass
m* = 2(d2E/dK2)-1
Recall that the electron energy is related to
the frequency of the electron waveE =
and the group velocity of the wave is the
velocity of the electronvg = d/dK = 1/ dE/dK (as in text)
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The acceleration of a particle is given by
the time-derivative of its velocity:
a = dvg/dt =d/dt(
d/dK)
=
d
/dK(
d
/dK)
dK
/dt= (1/
2) d/dK(d/dK)(
d (K)/dt)
= (1/2)(d2E/dK2)(d (K)/dt)
This is the term we are looking to show is:
(1/2)(d2E/dK2) = 1/m*
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What is d (K)/dt?
If we apply an external force on the electron, forexample an electric field (Fext=qE), then we will do
work on the electron:
dWe = Fextdx = Fext(vgdt) since vg = dx/dt = d/dK
= Fext(d/dK)dt
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Doing work on the electron
increases its energydWe = Fext(
d/dK)dt = dE
= (dE/dK)dK
= [d()/dK]dK
= (d/dK)dK
therefore: Fextdt = dK
or Fext = d (K)/dtnote: since F=d(mv)/dt,
K is called the crystal momentum
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Finally
a = (1/2)(d2E/dK2)(d (K)/dt)
and
Fext = d(K)/dt
gives us
a = (1/m*)Fext or Fext = m*a
Where m* = [(1/2)(d2E/dK2)]-1 = 2 (d2E/dK2)-1
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Without the crystal lattice, thehole cannot exist. It is an artifact
of the periodic potential (Ep)created by the crystal.
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E(K) and E(x)
p/a
E(K)
K
conduction band
valence band
EC
EV +
-
x
E(x)
Eg
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Generation and Recombination
of electron-hole pairs
conduction band
valence band
EC
EV +
-
x
E(x)
+
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Non-cubic lattices:
(FCC, BCC, diamond, etc.)
a
b
x
y
p/a
E(Kx)
Kx
p/b
E(Ky)
Ky
Different lattice spacings lead to different curvatures for E(K)
and effective masses that depend on the direction of motion.
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(from S.M. Sze, 1981)
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Memory Aid
a hairpin is lighter than a frying pan
light m*
(larger d2E/dK2)
heavy m*
(smaller d2E/dK2)
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Notes on E(K)
The extrema for the conduction and valence
bands are at different values of K for silicon
and germaniumthese are called indirectbandgap semiconductors
The conduction band minimum and valence
band maximum both occur at K=0 for GaAsthis is called a directbandgap semiconductor
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