Blochtheorem

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 (byL) operator
4
Bloch wave
5
Dirac comb
6
Graphical solution and interpretation.
7
Eect of force
8
Eective mass
9
Summary
10
References
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 2 / 26

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 bye
ikx
with
some normalisation constant.
Noether's theorem: Every symmetry is associated with a conserved
quantity.
Translation in position is given by a unitary operatore
i^p^x
~.
Commutators: [^A;^B]
[
^
A;
^
B] = 0 =)existance of common eigenstates.
[
^
H;
^
A] = 0 =)<
^
A>is time invariant.
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 3 / 26

4. Periodic potential
Model of a lattice
"V(x)Lx!
It is of the formV(x+L) =V(x), whereLis the periodicity.
examples
Kronig-Penney potentials : A series of nite potential wells with regular
arrangement.
Dirac comb : A series of delta potentials.
These can be considered as models for periodic lattice.
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 4 / 26

5. Translation operator
^
TL
^TLis an operator, which on any functionf(x) acts like
^TLf(x) =f(x+L)
Potential is periodic, i.e.,^TLV(x) =V(x+L) =V(x).
And
^
TL=e
i^pL
~
So,
^
TL
^
T
y
L
=I=)Unitary.
[^H;^TL] = 0 =)There exists simultaneous eigenstates of^Hand^TL.
Let E;be these states such that
^H E;=E E;
and
^
TL E;= E;
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 5 / 26

6. And we can writease
i
so that
^
TL E;=e
i
E;, since it is a
pure phase.
Let us dene a functionu(x) asu(x) =e
iqx
E;
Operating^TLonu(x).
^TLu(x) =^TL(e
iqx
E;) =e
iq(x+L)^TL E;
^TLu(x) =e
iq(x+L)
E;=e
i(qL)
e
iqx
E;
)^TLu(x) =e
i(qL)
u(x)
u(x) will be periodic if=qL.Gives
^TLu(x) =u(x)
) E;q=e
iqx
u(x)
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 6 / 26

7. Bloch waves E;q=e
iqx
u(x)
These Bloch waves are not periodic byL(unlessq= 0)
But the probability density of the Bloch waves are periodic i.e.,
j E;q(x)j
2
=j E;q(x+L)j
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 E;qare not momentum eigenstates andqis not the
momentum.
qis called "Crystal momentum". And we can see thatq+
2
L
=q.
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 7 / 26

8. Dirac comb
x!L "V(x)
Let us consider our periodic potentials to be delta functios, such that
V(x) =
1
X
n=1
~
2
2mL
go(xnL)
wheregois a dimensionless strength.
Using this potential in Schrodinger wave equation we get solution
(for 0<x<L) as
E;q(x) =Ae
ikx
+Be
ikx
wherek
2
=
2mE
~
2andA;Bare constants.
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 8 / 26

9. This is for the range 0<x<L. But our potential is periodic and we
have^TL.
i.e., for the rangeL<x<2Lwavefunction is
E;q(x+L) =
^
TL E;q(x) =e
iqL
E;q(x) =e
iqL
(Ae
ikx
+Be
ikx
)
In general the solution is
E;q(x+nL) =e
inqL
(Ae
ikx
+Be
ikx
)
wheren2Z
Now if we apply boundary conditions atx=nLwhich are
continuity of E;qatx=nL.
discontinuity of derivative of E;qatx=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)
wherek
2
=
2mE
~
2
Hereqis a free parameter such that

2
q

2
And the equation gives a relation betweenqandE.
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=
~
2
q
2
2m
The plot of cos(qL) vsElooks like
E!cos(qL)"0+11
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 11 / 26

12. "Eq!0

L
+

L
Graph of E vs q
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 12 / 26

13. If we consider one well and plot everything in one period width,
"Ek!0

L
+

L
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 13 / 26

14. Now let's apply potential i.e.,go6= 0.Then we get our graph as
E!cos(qL)"0+11
The graph deviates from the free particle case.
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 14 / 26

15. As we increase the value ofgothe curve deviates more.
E!cos(qL)"0+11
For larger and largergothe curve approach a sequence of vertical
lines.
For any value ofq,jcosqLj 6>1.
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 15 / 26

16. Such values of energy are forbidden.
So we get alternating Allowed and Forbidden values for energy.
E!cos(qL)"0+11
As the value ofEincreases the width of the allowed energy increases.
In these allowed region the energy is continuous and are called Bands.
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 16 / 26

17. TheEvsqgraph looks like,
It is clear that Energy eigenvalues are restricted to lie within the band
and bands are separated by gaps.
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 17 / 26

18. The original unfolded graph of this is
The range of allowed regions are called Brillouin zones.
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 18 / 26

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 byvg=
d!
dq
Which is equivalent to the slope of the graphEvsq(*E=~!).
vgq

L
+

L
0
As we increseqfrom zero the velocity increases becomes maximum
and then becomes zero again atq=

L
.
As we increase further it attains negative velocity.
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 19 / 26

20. We know that<F>=
d
dt
~q.
=)qvaries linearly withF.
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 eective transfer results in conduction.
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 21 / 26

22. Eective mass
From position time graph we see that the electron oscillates.For this it
should havenegative masssince it is moving against the force applied.
Let's nd the mass!
From group velocity we havevg=
<q>
m

When we substitute forvgand<q>we get
1
m

=
1
~
2
@
2
E
@q
2
If we plot
1
m
vsq, we get,
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 22 / 26

23. 1
m
q0

L
+

L
from graph we can see that mass takes innite value and also
negative values. This is not actual mass of the electron, it is called
theeective massof the electron.
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 23 / 26

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 eld applied the electron undergoes Bloch oscillations.
The electron has an eective mass.
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 24 / 26

25. References
Quantum physics of atoms, molecules, solids, nuclei and particles-
Robert Eisberg.
Introduction to quantum mechanics - D J Griths.
Quantum physics- Stephen Gasiorowicz
Principles of quantum mechanics - R Shankar
Solid state physics A J Dekker.
Basavaraja G ( DOS in Physics ) Electrons in periodic potentials January 30, 2019 25 / 26

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