A presentation about "How electrons move in a periodic one-dimensional potential." Here is an introduction to Bloch theorem.

1. Electrons in periodic potentials
Basavaraja G
DOS in Physics
January 30, 2019
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 1 / 26

2. Table of Content
1 Recalling some quantum facts!
2 Periodic potentials
3 Translate (by L) operator
4 Bloch wave
5 Dirac comb
6 Graphical solution and interpretation.
7 Effect of force
8 Effective mass
9 Summary
10 References
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3. Quantum facts!
Any arbitrary state can be expressed as a linear combination of
eigenstates of any observable.
For a free particle eigenstates of momentum are given by eikx with
some normalisation constant.
Noether’s theorem: Every symmetry is associated with a conserved
quantity.
Translation in position is given by a unitary operator e
i ˆpˆx
.
Commutators: [ ˆA, ˆB]
[ ˆA, ˆB] = 0 =⇒ existance of common eigenstates.
[ ˆH, ˆA] = 0 =⇒ < ˆA > is time invariant.
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4. Periodic potential
Model of a lattice
↑ V (x)
L x →
· · ·· · ·
It is of the form V (x + L) = V (x), where L is the periodicity.
examples
Kronig-Penney potentials : A series of finite potential wells with regular
arrangement.
Dirac comb : A series of delta potentials.
These can be considered as models for periodic lattice.
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5. Translation operator ˆTL
ˆTL is an operator, which on any function f (x) acts like
ˆTL f (x) = f (x + L)
Potential is periodic, i.e., ˆTL V (x) = V (x + L) = V (x).
And
ˆTL = e
i ˆpL
So, ˆTL
ˆ
T†
L = I =⇒ Unitary.
[ ˆH, ˆTL] = 0 =⇒ There exists simultaneous eigenstates of ˆH and ˆTL.
Let ψE,α be these states such that
ˆH ψE,α = E ψE,α
and
ˆTL ψE,α = λα ψE,α
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6. And we can write λα as eiα so that ˆTL ψE,α = eiα ψE,α, since it is a
pure phase.
Let us define a function u(x) as u(x) = e−iqx ψE,α
Operating ˆTL on u(x).
ˆTLu(x) = ˆTL(e−iqx
ψE,α) = e−iq(x+L) ˆTLψE,α
ˆTLu(x) = e−iq(x+L−α)
ψE,α = e−i(qL−α)
e−iqx
ψE,α
∴ ˆTLu(x) = e−i(qL−α)
u(x)
u(x) will be periodic if α = qL.Gives
ˆTLu(x) = u(x)
∴ ψE,q = eiqx
u(x)
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7. Bloch waves ψE,q = eiqx
u(x)
These Bloch waves are not periodic by L (unless q = 0)
But the probability density of the Bloch waves are periodic i.e.,
|ψE,q(x)|2
= |ψE,q(x + L)|2
This suggests that electron wavefunction is not localised anywhere.
This means electron wavefunction has equal amplitude overall the
wells.They are extended.
Therefore we have to consider wavepackets to have normalised states.
These ψE,q are not momentum eigenstates and q is not the
momentum.
q is called ”Crystal momentum”. And we can see that q + 2π
L = q.
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8. Dirac comb
x → L
· · ·· · ·
↑ V (x)
Let us consider our periodic potentials to be delta functios, such that
V (x) =
∞
n=−∞
2
2mL
goδ(x − nL)
where go is a dimensionless strength.
Using this potential in Schrodinger wave equation we get solution
(for 0 < x < L) as
ψE,q(x) = Aeikx
+ Be−ikx
where k2 = 2mE
2 and A, B are constants.
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9. This is for the range 0 < x < L. But our potential is periodic and we
have ˆTL.
i.e., for the range L < x < 2L wavefunction is
ψE,q(x + L) = ˆTLψE,q(x) = eiqL
ψE,q(x) = eiqL
(Aeikx
+ Be−ikx
)
In general the solution is
ψE,q(x + nL) = einqL
(Aeikx
+ Be−ikx
)
where n ∈ Z
Now if we apply boundary conditions at x = nL which are
continuity of ψE,q at x = nL.
discontinuity of derivative of ψE,q at x = nL.
periodicity of the lattice.
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 9 / 26

10. We get a transcendental equation for Energy as
cos(qL) = cos(kL) +
go
2kL
sin(kL)
where k2 = 2mE
2
Here q is a free parameter such that −π
2 ≤ q ≤ π
2
And the equation gives a relation between q and E.
We can solve that transcedental equation from Graphical Method.
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 10 / 26

11. Graphical solutions for energy
Let us consider a free particle situation i.e., go = 0. Which gives
cos(qL) = cos(kL)
or
q = k
gives energy
E =
2q2
2m
The plot of cos(qL) vs E looks like
E →
cos(qL)
↑
0
+1
−1
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 11 / 26

12. ↑ E
q →0−π
L +π
L
Graph of E vs q
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13. If we consider one well and plot everything in one period width,
↑ E
k →
0−π
L +π
L
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14. Now let’s apply potential i.e., go = 0.Then we get our graph as
E →
cos(qL)
↑
0
+1
−1
The graph deviates from the free particle case.
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15. As we increase the value of go the curve deviates more.
E →
cos(qL)
↑
0
+1
−1
For larger and larger go the curve approach a sequence of vertical
lines.
For any value of q , | cos qL| > 1.
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16. Such values of energy are forbidden.
So we get alternating Allowed and Forbidden values for energy.
E →
cos(qL)
↑
0
+1
−1
As the value of E increases the width of the allowed energy increases.
In these allowed region the energy is continuous and are called Bands.
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17. The E vs q graph looks like,
It is clear that Energy eigenvalues are restricted to lie within the band
and bands are separated by gaps.
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18. The original unfolded graph of this is
The range of allowed regions are called Brillouin zones.
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19. When force applied
We have seen that the state of the electron is extended and we have
to have wavepacket construction.
The group velocity of the particle is given by vg = dω
dq
Which is equivalent to the slope of the graph E vs q ( E = ω).
vg
q
−π
L
+π
L0
As we increse q from zero the velocity increases becomes maximum
and then becomes zero again at q = π
L .
As we increase further it attains negative velocity.
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20. We know that < F >= d
dt q.
=⇒ q varies linearly with F.
The corresponding position time graph looks like,
< x >
T0
The electron oscillates! The oscillations are called
Bloch Oscillations.
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 20 / 26

21. Reason for Conduction
Electrons collide with the ions and transfer momentum which sets it
to vibrate, that momentum is transfered again to another electron.
The process continues and the effective transfer results in conduction.
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 21 / 26

22. Effective mass
From position time graph we see that the electron oscillates.For this it
should have negative mass since it is moving against the force applied.
Let’s find the mass!
From group velocity we have vg = <q>
m∗
When we substitute for vg and < q > we get
1
m∗
=
1
2
∂2E
∂q2
If we plot 1
m∗ vs q, we get,
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 22 / 26

23. 1
m∗
q
0
−π
L +π
L
from graph we can see that mass takes infinite value and also
negative values. This is not actual mass of the electron, it is called
the effective mass of the electron.
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24. Summary
The lattice is modelled as periodic potentials.
The energy eigenstates of the electrons are extended over the lattice
with cetain allowed energies forming bands.
When electric field applied the electron undergoes Bloch oscillations.
The electron has an effective mass.
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25. References
Quantum physics of atoms, molecules, solids, nuclei and particles-
Robert Eisberg.
Introduction to quantum mechanics - D J Griffiths.
Quantum physics- Stephen Gasiorowicz
Principles of quantum mechanics - R Shankar
Solid state physics A J Dekker.
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26. Thank You
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