The document discusses the band structure of electrons in solids. It explains that when electrons are placed in a periodic potential, as in metals and semiconductors, the allowed energy levels split and form bands separated by band gaps. The nearly-free electron model is introduced to account for this band structure by treating electrons as interacting with a periodic lattice potential rather than being completely free. The key outcomes are that the energy-momentum relationship becomes a series of bands rather than continuous, and band gaps open up where electron states are forbidden. This distinguishes conductors, semiconductors and insulators.

1. Nanotechnology & Quantum
Phenomena
Lecture 6 (29th Oct 2008)
The Nearly-Free electron model and band structure
In this lecture, we are going to look at what happens when we have electrons in a periodic
potential, as is the case in metals and semiconductors. The form of the electrical potential in a
crystal is such that there are potential wells centred on the atomic cores. Each of these wells
will have discrete allowed energy levels of a form similar to those of Hydrogen. Due to the
proximity of the atoms to each other, the tails of the potentials overlap and modify the overall
potential. This coupling causes the energy levels to shift and split, and for a number N of
atoms, we will have N energy states, corresponding to 2N possible electron states (2 comes
from the 2 spin states of an electron).
Our aim is to find the form of the E-k relationship for electrons in such a potential, and see how
it compares to the free electron case. Then we will understand the origin of the differences
between metals, semiconductors and insulators.

2. Electrons in solids
First of all, consider a finite square well separated by two barriers, as we
encountered earlier in the course. For example, imagine if the dimensions are such that
there are two bound states.
If the wells are close enough, the tail of the wave-functions within
each well can extend appreciably into neighbouring wells. This
gives rise to a coupling between the wells, and generates new
E1 wave-functions which are combinations of the original ones. For
E2
two wells, those combinations are the sum and difference between
the original functions. Hence, there will be two states instead of
x one. For N wells, each state will become N closely spaced states.
The closer the wells are to each other, the larger will be the splitting
in energy. This is similar to beats in the addition of waves: the
Now add in another well: closer the frequency, the more pronounced are the beats.
In a crystal, the potential is close to that above: energy
bands and a band gap. To consider the behaviour of
electrons in solids, people generally use the so-called
“Free electron model”, where electrons are treated as
completely free. This can explain the origin of the Fermi
energy and simple conduction phenomena, however it
x does not explain why some materials are conductors and
others insulators. We will see that if we treat the electrons
in a solid as being in a periodic potential (which they are),
Now for a very large number of wells: then we can see the origin of the band-gap.
Energy levels become bands.

3. Free-electron model:
This simple model assumes we can consider electrons in a solid as being like a gas. It is
generally assumed that each atom contributes one electron to the gas, and the electrons obey
Fermi-Dirac statistics (the first models of metals assumed electrons followed a Maxwell-Boltzmann
velocity distribution, and they gave reasonable estimates for electrical and thermal conductivity, and
the ratio of the two, but Fermi-Dirac showed better agreement). However, it cannot account for
band gaps.
For free electrons, E = ħ2k2/2m
E-k relationship is parabolic (remember, it is linear for
Ef electromagnetic radiation), and electronic wave-functions
are pure, travelling sine-waves, i.e.
How about we now see the effects of putting the electron in a periodic potential like that
in a crystal?

4. The Nearly Free Electron Model
• Coupling of electronic states from neighbouring atomic cores causes
creation of extra levels (one per site at each site), and the formation of an
energy gap
• What happens for a periodic potential?
a
Ion cores
Interatomic
potential
Steps:
1. Define potential as seen by an electron
2. Expand as a Fourier series (must be possible, as the potential is inherently periodic)
3. Solution to Schrödinger’s equation contains information about periodic potential super-
imposed on free-electron wave-functions (which are simple plane waves)
4. This periodic part to the solution can also be expressed as a Fourier series
5. Insert all of the above into Schrödinger’s equation, try to find a solution!

5. The reciprocal lattice vectors are given by G, and the lattice spacing is a. As we are
dealing with a periodic system, it is useful to use the Fourier expansion of the crystal
potential, i.e.
(1)
where the Fourier coefficients are given by
(2)
and p = 0, ±1, ±2,….. and Gp = 2πp/a
The general solution of the Schrödinger equation with a periodic potential is ψ(x) =
eikxu(x). This is a plane wave modulated by the function u(x), where u(x) is a periodic
function with the periodicity of the lattice, i.e. u(x) represents the influence of the crystal
potential. This is known as Bloch’s theorem, and u(x) as a Bloch function. In the figure
below, we show the typical form of the wave-functions for the free and nearly-free electron
models, and we include the approximate lattice potential for reference.
Free electrons
Nearly-free electrons
Lattice potential

6. In the same way as we expanded the potential as a Fourier series,
we can now do the same for u(x), to obtain:
(3)
where n = 0, ±1, ±2,….. and Gn = 2πn/a
That gives for the total expansion of the wave function:
ψ (x) = ∑ Cn e( (
i k +G n ) x )
n
(4)
We now insert the Fourier expansions of both ψ(x) and u(x) into Schrodinger’s equation,
(-ħ2/2md2/dx2 + V) Ψ(x) = EΨ(x)
(5)
We end up with a set of simultaneous equations in the unknown Cn. Note that the Vp are known,
as the form of the crystal potential is assumed initially. There are an infinite number of terms, so
to make the problem manageable, we artificially truncate the series and consider only the
leading-order terms given by n = 0, ±1. This is justified for weak potentials such as those found in
metals.
i.e., we write V(x) as:
V(x) = V0 + V1eiG1x + V-1eiG-1x (6)
= V0 + 2V1cos(G1x)

7. If we continue along the same lines, we can assume that the wave-function also only
contains leading-order terms, i.e.
ψ(x) = [C0 + C1eiG1x + C-1eiG-1x]eikx (7)
Substituting equations (6) and (7) into (5), we get:
(-ħ2/2md2/dx2 + V0 + V1eiG1x + V-1eiG-1x)[C0 + C1eiG1x + C-1eiG-1x]eikx =
E[C0 + C1eiG1x + C-1eiG-1x]eikx
If we just consider a region where C0 and C-1 dominate, we are left with the relationships
(noting that G-1 = -G1 etc.):
(- ħ2k2C0/2m + V0C0 + C0V1eiG1x + C0V-1eiG-1x - ħ2(k + G-1)2C-1 eiG-1x /2m +
V0C-1eiG-1x + V1C-1 + V-1C-1e2iG-1x) = EC0 + EC-1eiG-1x
Collecting terms in eiG-1x, we find that:
C0V-1 = [ħ2(k + G-1)2/2m + E - V0]C-1
Terms without any exponent give: (8)
C-1V1 = [(ħ2k2/2m) + E - V0]C0

8. For a non-trivial solution, both ratios for C-1/C0 must be equal, i.e.
C-1/C0 = [(ħ2k2/2m) + E - V0]/V1
= V-1/[ħ2(k + G-1)2/2m + E - V0]
or, [E – V0 + (ħ2k2/2m)] [E – V0 + ħ2(k + G-1)2/2m]
= V1V-1 = |V1|2 (Everything is symmetric)
Solving for E in terms of k, and plotting, we get a relationship of the form:
30
25
20
E
G
= 2V1
15
10 E
V
c
o
5
-1.5 -1 -0.5 0 0.5 1 1.5
k

9. Points of interest to note relative to free-electron case:
The energy is shifted up by the amount V0, where V0 is the spatial average of V(x)
At certain values of k, the terms in brackets in Equation (8) are equal, i.e. at k = -G-1/2
= π/a. The gaps appear at these points, which define the Brillouin Zone boundary,
and this corresponds to Bragg reflection. At this value of k, the energy can have two
values:
E = V0 + ħ2(π/a)2 ± |V1|
The separation in energy is 2|V1|, which is just twice the first term in the Fourier
series expansion of the crystal potential, and the Schrödinger equation has no wave-
like solutions in this gap
The wavefunction at the energy gap has two solutions corresponding to:
ψ±(x) = C0[1 ± e(-2iπx/a)]e(iπx/a)
or,
ψ-(x) = i2C0sin(πx/a) and ψ+(x) = 2C0cos(πx/a)
These are standing waves! i.e., electrons at these energies cannot travel through the
crystal, and hence do not contribute to conduction.

10. Now, ψ+(x) is the lower energy state, and has maxima at the atomic cores, whereas ψ-(x) is the
higher energy state, with maxima at the mid-points between the atomic cores. These correspond
to the valence and conduction bands, respectively.
How do we draw band diagrams?
There are two methods: the extended and
reduced-zone schemes.
valence band
Conduction band
Extended zone scheme Reduced zone scheme
In the reduced zone scheme, we just shift everything in order to show all the bands as lying
within the first Brillouin zone.
The first band is always full, and often the second band is more than half-full.

11. Interesting point:
The electronic wave-functions outside the energy gap are described by travelling
waves superimposed on a (Bloch) function with the lattice periodicity. On average
then, the electrons are represented as travelling waves, with equal probability of being
anywhere within the conductor; i.e. from this model, there is no reason for electrical
resistance - electrons should move unhindered throughout a metal! We will see later
in the course that electrical resistance is caused by deviations from the lattice
periodicity.
More about band structure
Metals: electrons carry current
Semiconductors: electrons & holes. Both types of carrier have distinct band structure,
i.e. the conduction & valence bands respectively.
Real materials have significantly more complex band structure, as they are 3-
Dimensional! Also, for most materials, the atomic spacings are not the same in all
directions, so the apparent band gap can seem different depending on which direction
an electron is travelling in!

12. Electrons
(conduction
Si Band band)
structure
holes
(valence
Depending on the relative position of the conduction band)
band minimum to the valence band maximum, we
have either direct or indirect gap semiconductors.
Direct gap Indirect gap
In a direct-gap semiconductor (e.g. GaAs, Ge), to cause a transition of an electron from the
Valence band to the Conduction band only requires energy (Eg), whereas in an indirect-gap
semiconductor (e.g. Si), momentum (q) is also required. Therefore, only direct-gap
semiconductors can be used in optical devices (e.g. photodiodes). The momentum of a photon
is generally extremely small, and not enough to induce a transition, so the transition must
involve something else (known as a phonon, which we will see later).

13. Surface states…..
Interatomic spacing, a Surface i.e. the break in
atom
Vacuum level symmetry induces the
Work formation of new
function
states within the
conduction band band gap. These are
Bulk band gap Band of surface called “Shockley
states
Valence band states”
Also, atoms on surfaces are unstable, may re-arrange their positions: “surface
reconstruction” . This also induces new energy states, called “Tamm states”
Native surface Reconstructed surface
Top atomic layer

14. Native surface reconstructed surface