2. Electrons in Solids
• Considering electrons in metals as free
particles, electron gas in a box explains
many experimental results which?
• Still, why are some solids metals and others
insulators?
– For metals we assumed that electrons are free
– We know, however, that there are no free
electrons in insulators
• QM give us the answer!
– Need a more realistic potential for electrons
reflecting periodic ionic structure of solids
4. “Realistic” Potential in Solids
• For one dimensional case where atoms
(ions) are separated by distance d, we can
write condition of periodicity as
5. “Realistic” Potential in Solids
• Multi-electron atomic potentials are complex
• Even for hydrogen atom with a “simple”
Coulomb potential solutions are quite
complex
• So we use a model one-dimensional periodic
potential to get insight into the problem
6. Bloch’s Theorem
• Bloch’s Theorem states that for a particle
moving in the periodic potential, the
wavefunctions ψ(x) are of the form
• uk(x) is a periodic function with the
periodicity of the potential
– The exact form depends on the potential
associated with atoms (ions) that form the solid
7. Bloch’s Theorem
• Bloch’s Theorem imposes very special
conditions on any solution of the
Schrödinger equation, independent of the
form of the periodic potential
• The wave vector k has a two-fold role:
1. It is still a wave vector in the plane wave part of
the solution
2. It is also an index to uk(x) because it contains all
the quantum numbers, which enumerate the
wavefunction
8. Bloch’s Theorem
• What is probability density of finding
particle at coordinate x?
• But |uk(x)|2
is periodic, so P(x) is as well
9. Bloch’s Theorem
The probability of finding an electron at
any atom in the solid is the same!!!
• Each electron in a crystalline solid
“belongs” to each and every atom
forming the solid
10. Covalent Bonding Revisited
• When atoms are covalently bonded
electrons supplied by atoms are
shared by these atoms since pull of
each atom is the same or nearly so
– H2, F2, CO,
• Example: the ground state of the
hydrogen atoms forming a molecule
– If the atoms are far apart there is very
li le overlap between their wavefunctions
– If atoms are brought together the
wavefunctions overlap and form the
compound wavefunction, ψ1(r)+ψ2(r),
increasing the probability for electrons to
exist between the atoms
12. Schrödinger Equation Revisited
• If a wavefunctions ψ1(x) and ψ2(x) are
solutions for the Schrödinger equation for
energy E, then functions
– -ψ1(x), -ψ2(x), and ψ1(x)±ψ2(x) are also solutions
of this equations
– the probability density of -ψ1(x) is the same as
for ψ1(x)
13. • Consider an atom with only one electron in s-state
outside of a closed shell
• Both of the wavefunctions below are valid and the
choice of each is equivalent
• If the atoms are far apart, as before, the
wavefunctions are the same as for the isolated
atoms
Band Theory of Solids
14. Band Theory of Solids
• The sum of them is shown in the
figure
• These two possible
combinations represent two
possible states of two atoms
system with different energies
• Once the atoms are brought together the
wavefunctions begin to overlap
– There are two possibilities
1. Overlapping wavefunctions are the same (e.g., ψs
+
(r))
2. Overlapping wavefunctions are different
22. Atoms and Band Structure
• Consider multi-electron atoms:
1. The outer electrons (large n and
l) are “closer” to each other than
the inner electrons
• Thus, the overlap of the wave-
functions of the outer electrons is
stronger than overlap of those of
inner electrons
• Therefore, the bands formed from
outer electrons are wider than the
bands formed from inner electrons
• Bands with higher energies
are therefore wider!
28. Insulators, Semiconductors,
Metals
• The last completely filled (at least at T = 0 K)
band is called the Valence Band
• The next band with higher energy is the
Conduction Band
– The Conduction Band can be empty or partially
filed
• The energy difference between the bo om of
the CB and the top of the VB is called the
Band Gap (or Forbidden Gap)
29. • Can be found using computer
• In 1D computer simulation of light in a
periodic structure, we found the
frequencies and wave functions
• Allowed modes fall into quasi-
continuous bands separated by
forbidden bands just as would be
expected from the tight binding model
Computer simulation can give
exact solution in simple cases
30. Insulators, Semiconductors,
Metals
• Consider a solid with the empty
Conduction Band
• If apply electric field
to this solid, the
electrons in the
valence band (VB)
cannot participate in
transport (no current)
31. Insulators, Semiconductors,
Metals
• The electrons in the VB do not
participate in the current, since
– Classically, electrons in the
electric field accelerate, so they
acquire [kinetic] energy
– In QM this means they must
acquire slightly higher energy
and jump to another quantum
state
– Such states must be available, i.
e. empty allowed states
– But no such state are available
in the VB!
This solid
would behave
as an
insulator
32. Insulators, Semiconductors,
Metals
• Consider a solid with the half filled
Conduction Band (T = 0K)
• If an electric field is applied
to this solid, electrons in the
CB do participate in
transport, since there are
plenty of empty allowed
states with energies just
above the Fermi energy
• This solid would behave as
a conductor (metal)
33. Band Overlap
• Many materials are
conductors (metals) due to
the “band overlap”
phenomenon
• Often the higher energy
bands become so wide that
they overlap with the lower
bands
– additional electron energy
levels are then available
34. Band Overlap
• Example: Magnesium (Mg; Z =12): 1s2
2s2
2p6
3s2
– Might expect to be insulator; however, it is a metal
– 3s-band overlaps the 3p-band, so now the
conduction band contains 8N energy levels, while
only have 2N electrons
– Other examples: Zn, Be, Ca, Bi
35. Band Hybridization
• In some cases the opposite occurs
– Due to the overlap, electrons from different
shells form hybrid bands, which can be
separated in energy
– Depending on the magnitude of the gap, solids
can be insulators (Diamond); semiconductors (Si,
Ge, Sn; metals (Pb)
36. Insulators, Semiconductors, Metals
• There is a qualitative difference between
metals and insulators (semiconductors)
– the highest energy band “containing” electrons
is only partially filled for Metals (sometimes due
to the overlap)
• Thus they are good conductors even at very low
temperatures
• The resisitvity arises from the electron sca ering from
la ice vibrations and la ice defects
• Vibrations increases with temperature ⇒ higher
resistivity
• The concentration of carriers does not change
appreciably with temperature
37. Insulators, Semiconductors, Metals
• The difference between Insulators and
Semiconductors is “quantitative”
– The difference in the magnitude of the band gap
• Semiconductors are “Insulators” with a
relatively small band gap
– At high enough temperatures a fraction of electrons
can be found in the conduction band and therefore
participate in transport
38. Insulators vs Semiconductors
• There is no difference between Insulators and
Semiconductors at very low temperatures
• In neither material are there any electrons in the
conduction band – and so conductivity vanishes in
the low temperature limit
39. Insulators vs Semiconductors
• Differences arises at high temperatures
– A small fraction of the electrons is thermally
excited into the conduction band. These
electrons carry current just as in metals
– The smaller the gap the more electrons in the
conduction band at a given temperature
– Resistivity decreases with temperature due to
higher concentration of electrons in the
conduction band
40. Holes
• Consider an insulator (or semiconductor)
with a few electrons excited from the valence
band into the conduction band
• Apply an electric field
– Now electrons in the valence band have some
energy sates into which they can move
– The movement is complicated since it involves
~ 1023
electrons
41. Concept of Holes
• Consider a semiconductor with a small number of
electrons excited from the valence band into the
conduction band
• If an electric field is applied,
– the conduction band electrons will participate in the
electrical current
– the valence band electrons can “move into” the empty
states, and thus can also contribute to the current
42. Holes from the Band Structure
Point of View
• If we describe such changes via
“movement” of the “empty” states – the
picture can be significantly simplified
• This “empty space” is a Hole
– “Deficiency” of negative charge – holes are
positively charged
– Holes often have a larger effective mass
(heavier) than electrons since they represent
collective behavior of many electrons
43. Holes
• We can “replace” electrons at the top of
eth band which have “negative” mass (and
travel in opposite to the “normal”
direction) by positively charged particles
with a positive mass, and consider all
phenomena using such particles
• Such particles are called Holes
• Holes are positively charged and move in
the same direction as electrons “they
replace”
44. Hole Conduction
• To understand hole motion, one requires
another view of the holes, which represent
them as electrons with negative effective
mass
• To imagine the movement of the hole think
of a row of chairs occupied by people with
one chair empty
• To move all people rise all together and
move in one direction, so the empty spot
moves in the same direction
45. Concept of Holes
• If we describe such changes via
“movement” of the “empty” states – the
picture will be significantly simplified
• This “empty space” is called a Hole
– “Deficiency” of negative charge can be treated
as a positive charge
– Holes act as charge carriers in the sense that
electrons from nearby sites can “move” into the
hole
– Holes are usually heavier than electrons since
they depict collective behavior of many electrons
