Electrons in a Weak Periodic Potential
Electrons in a Weak Periodic Potential
Motivation
We start from the Sommerfeld free-electron model and then add a weak periodic potential.
The crystal contains ions arranged periodically, so the electron feels a potential
\[ U(\mathbf{r}) \]
that has the same periodicity as the crystal lattice.
The central question is:
How does a weak periodic potential change the free-electron energy levels?
For a free electron,
\[ \varepsilon_{\mathbf{k}}^0 = \frac{\hbar^2 k^2}{2m} \]
The weak periodic potential modifies this dispersion, especially near points where free-electron states become degenerate.
Why Can the Periodic Potential Be Treated as Weak?
At first, the electron-ion Coulomb interaction seems strong.
However, the effective potential felt by conduction electrons can be treated as weak for two main reasons.
Core-electron exclusion: Conduction electrons cannot easily enter the immediate neighborhood of the ion cores because these regions are already occupied by core electrons. Therefore, conduction electrons mostly move in the space between ions.
Screening: Positive ions attract conduction electrons. The mobile conduction electrons accumulate around the ions and partially cancel the bare ionic electric field. This effect is called screening.
Therefore, the potential felt by conduction electrons is not the full bare Coulomb potential, but a weaker screened potential.
3. Schrödinger Equation in a Periodic Potential
We solve the one-electron Schrödinger equation
\[ \left[ -\frac{\hbar^2}{2m}\nabla^2 + U(\mathbf{r}) \right] \psi(\mathbf{r}) = E\psi(\mathbf{r}) \]
where the periodic potential satisfies
\[ U(\mathbf{r}+\mathbf{R}) = U(\mathbf{r}) \]
for every Bravais lattice vector \(\mathbf{R}\).
Because \(U(\mathbf{r})\) is periodic, it can be expanded in reciprocal lattice vectors:
\[ U(\mathbf{r}) = \sum_{\mathbf{K}} U_{\mathbf{K}} e^{i\mathbf{K}\cdot\mathbf{r}} \]
where \(\mathbf{K}\) are reciprocal lattice vectors.
The Fourier coefficients are
\[ U_{\mathbf{K}} = \frac{1}{v} \int_{\text{cell}} d^3r\, e^{-i\mathbf{K}\cdot\mathbf{r}} U(\mathbf{r}) \]
where \(v\) is the primitive cell volume.
If \(U(\mathbf{r})\) is real, then
\[ U_{-\mathbf{K}} = U_{\mathbf{K}}^* \]
If the crystal has inversion symmetry,
\[ U(\mathbf{r}) = U(-\mathbf{r}) \]
then
\[ U_{-\mathbf{K}} = U_{\mathbf{K}} \]
so the Fourier coefficients can be chosen real.
Because the potential is periodic, a natural basis is a sum of plane waves whose wave vectors differ by reciprocal lattice vectors.
We write
\[ \psi(\mathbf{r}) = \sum_{\mathbf{K}} c_{\mathbf{k}-\mathbf{K}} e^{i(\mathbf{k}-\mathbf{K})\cdot\mathbf{r}} \]
Here:
- \(\mathbf{k}\) is a wave vector in the first Brillouin zone,
- \(\mathbf{K}\) is a reciprocal lattice vector,
- \(c_{\mathbf{k}-\mathbf{K}}\) is the coefficient of the plane wave,
- the plane wave momentum is \(\hbar(\mathbf{k}-\mathbf{K})\).
The free-electron energy of a plane wave with wave vector \(\mathbf{q}\) is
\[ \varepsilon_{\mathbf{q}}^0 = \frac{\hbar^2 q^2}{2m} \]
so for the component \(\mathbf{q}=\mathbf{k}-\mathbf{K}\),
\[ \varepsilon_{\mathbf{k}-\mathbf{K}}^0 = \frac{\hbar^2|\mathbf{k}-\mathbf{K}|^2}{2m} \]
Substituting
\[ \psi(\mathbf{r}) = \sum_{\mathbf{K}} c_{\mathbf{k}-\mathbf{K}} e^{i(\mathbf{k}-\mathbf{K})\cdot\mathbf{r}} \]
and
\[ U(\mathbf{r}) = \sum_{\mathbf{K}} U_{\mathbf{K}} e^{i\mathbf{K}\cdot\mathbf{r}} \]
into Schrödinger’s equation gives
\[ \left[ \varepsilon_{\mathbf{k}-\mathbf{K}}^0 - E \right] c_{\mathbf{k}-\mathbf{K}} + \sum_{\mathbf{K}'} U_{\mathbf{K}'-\mathbf{K}} c_{\mathbf{k}-\mathbf{K}'} = 0 \]
or equivalently,
\[ \left[ E - \varepsilon_{\mathbf{k}-\mathbf{K}}^0 \right] c_{\mathbf{k}-\mathbf{K}} = \sum_{\mathbf{K}'} U_{\mathbf{K}'-\mathbf{K}} c_{\mathbf{k}-\mathbf{K}'} \]
This is the main equation of the nearly-free-electron model.
It says that the periodic potential couples plane waves whose wave vectors differ by reciprocal lattice vectors.
The coefficient
\[ U_{\mathbf{K}'-\mathbf{K}} \]
couples the plane wave
\[ e^{i(\mathbf{k}-\mathbf{K}')\cdot\mathbf{r}} \]
to the plane wave
\[ e^{i(\mathbf{k}-\mathbf{K})\cdot\mathbf{r}} \]
The potential can scatter an electron from one wave vector to another if the difference is a reciprocal lattice vector.
This is why Bragg reflection appears naturally in the band theory.
8. Case 1: Non-Degenerate Free-Electron Levels
Suppose we choose one plane wave
\[ \mathbf{k}-\mathbf{K}_\ell \]
whose free-electron energy is
\[ \varepsilon_{\mathbf{k}-\mathbf{K}_\ell}^0 \]
and assume that it is far away from all other free-electron energies:
\[ \left| \varepsilon_{\mathbf{k}-\mathbf{K}_\ell}^0 - \varepsilon_{\mathbf{k}-\mathbf{K}}^0 \right| \gg |U| \]
for
\[ \mathbf{K} \neq \mathbf{K}_\ell \]
Then the weak periodic potential does not strongly mix this state with the others.
For \(\mathbf{K}\neq \mathbf{K}_\ell\), the coefficient is small:
\[ c_{\mathbf{k}-\mathbf{K}} \simeq \frac{ U_{\mathbf{K}_\ell-\mathbf{K}} c_{\mathbf{k}-\mathbf{K}_\ell} } { \varepsilon_{\mathbf{k}-\mathbf{K}_\ell}^0 - \varepsilon_{\mathbf{k}-\mathbf{K}}^0 } + O(U^2) \]
So distant states only enter weakly.
Energy shift
Substituting this back into the main equation gives
\[ E = \varepsilon_{\mathbf{k}-\mathbf{K}_\ell}^0 + \sum_{\mathbf{K}\neq \mathbf{K}_\ell} \frac{ \left| U_{\mathbf{K}-\mathbf{K}_\ell} \right|^2 } { \varepsilon_{\mathbf{k}-\mathbf{K}_\ell}^0 - \varepsilon_{\mathbf{k}-\mathbf{K}}^0 } + O(U^3) \]
Therefore, away from degeneracies, the weak periodic potential shifts the free-electron energy only in second order.
So away from Bragg planes, the energy bands remain almost free-electron-like.
9. Case 2: Nearly Degenerate Free-Electron Levels
Now suppose several free-electron levels are close together:
\[ \varepsilon_{\mathbf{k}-\mathbf{K}_1}^0, \varepsilon_{\mathbf{k}-\mathbf{K}_2}^0, \dots, \varepsilon_{\mathbf{k}-\mathbf{K}_m}^0 \]
These energies are close on the scale of the potential:
\[ \left| \varepsilon_{\mathbf{k}-\mathbf{K}_i}^0 - \varepsilon_{\mathbf{k}-\mathbf{K}_j}^0 \right| \sim |U| \]
but far from all other free-electron levels.
Then the weak periodic potential strongly mixes these nearly degenerate states.
The coefficients outside this nearly degenerate subspace are small, so we keep only the coupled equations inside the subspace:
\[ \left[ E - \varepsilon_{\mathbf{k}-\mathbf{K}_i}^0 \right] c_{\mathbf{k}-\mathbf{K}_i} = \sum_{j=1}^{m} U_{\mathbf{K}_j-\mathbf{K}_i} c_{\mathbf{k}-\mathbf{K}_j} \]
This gives a first-order energy splitting.
Thus, the weak periodic potential has its strongest effect near degeneracies.
Two-Level Mixing
The most important case is two nearly degenerate free-electron levels.
Consider the two plane waves
\[ e^{i\mathbf{q}\cdot\mathbf{r}} \]
and
\[ e^{i(\mathbf{q}-\mathbf{K})\cdot\mathbf{r}} \]
Their free-electron energies are
\[ \varepsilon_{\mathbf{q}}^0 = \frac{\hbar^2 q^2}{2m} \]
and
\[ \varepsilon_{\mathbf{q}-\mathbf{K}}^0 = \frac{\hbar^2|\mathbf{q}-\mathbf{K}|^2}{2m} \]
The coupled equations are
\[ \left(E-\varepsilon_{\mathbf{q}}^0\right)c_{\mathbf{q}} = U_{\mathbf{K}}c_{\mathbf{q}-\mathbf{K}} \]
and
\[ \left(E-\varepsilon_{\mathbf{q}-\mathbf{K}}^0\right)c_{\mathbf{q}-\mathbf{K}} = U_{-\mathbf{K}}c_{\mathbf{q}} \]
Since
\[ U_{-\mathbf{K}} = U_{\mathbf{K}}^* \]
the matrix equation is
\[ \begin{pmatrix} E-\varepsilon_{\mathbf{q}}^0 & -U_{\mathbf{K}} \\ -U_{\mathbf{K}}^* & E-\varepsilon_{\mathbf{q}-\mathbf{K}}^0 \end{pmatrix} \begin{pmatrix} c_{\mathbf{q}} \\ c_{\mathbf{q}-\mathbf{K}} \end{pmatrix} = 0 \]
For a nonzero solution, the determinant must vanish:
\[ \left(E-\varepsilon_{\mathbf{q}}^0\right) \left(E-\varepsilon_{\mathbf{q}-\mathbf{K}}^0\right) - |U_{\mathbf{K}}|^2 = 0 \]
Solving gives
\[ E_{\pm} = \frac{1}{2} \left( \varepsilon_{\mathbf{q}}^0 + \varepsilon_{\mathbf{q}-\mathbf{K}}^0 \right) \pm \sqrt{ \frac{1}{4} \left( \varepsilon_{\mathbf{q}}^0 - \varepsilon_{\mathbf{q}-\mathbf{K}}^0 \right)^2 + |U_{\mathbf{K}}|^2 } \]
This formula describes the avoided crossing of two free-electron parabolas.
11. Bragg Planes
The two free-electron levels become degenerate when
\[ \varepsilon_{\mathbf{q}}^0 = \varepsilon_{\mathbf{q}-\mathbf{K}}^0 \]
Using
\[ \varepsilon_{\mathbf{q}}^0 = \frac{\hbar^2 q^2}{2m} \]
we get
\[ q^2 = |\mathbf{q}-\mathbf{K}|^2 \]
Expanding the right-hand side,
\[ |\mathbf{q}-\mathbf{K}|^2 = q^2 - 2\mathbf{q}\cdot\mathbf{K} + K^2 \]
Therefore,
\[ q^2 = q^2 - 2\mathbf{q}\cdot\mathbf{K} + K^2 \]
so
\[ 2\mathbf{q}\cdot\mathbf{K} = K^2 \]
or
\[ \mathbf{q}\cdot\mathbf{K} = \frac{K^2}{2} \]
This can be rewritten as
\[ \left( \mathbf{q} - \frac{\mathbf{K}}{2} \right) \cdot \mathbf{K} = 0 \]
This means that
\[ \mathbf{q} - \frac{\mathbf{K}}{2} \]
is perpendicular to \(\mathbf{K}\).
Therefore, the degenerate points form a plane perpendicular to \(\mathbf{K}\), halfway between \(0\) and \(\mathbf{K}\).
This plane is called a Bragg plane.
12. Energy Gap at a Bragg Plane
At a Bragg plane,
\[ \varepsilon_{\mathbf{q}}^0 = \varepsilon_{\mathbf{q}-\mathbf{K}}^0 = \varepsilon^0 \]
Then the two energies become
\[ E_+ = \varepsilon^0 + |U_{\mathbf{K}}| \]
and
\[ E_- = \varepsilon^0 - |U_{\mathbf{K}}| \]
Therefore, the energy gap is
\[ \Delta E = E_+ - E_- \]
so
\[ \Delta E = 2|U_{\mathbf{K}}| \]
Thus, the Fourier component \(U_{\mathbf{K}}\) of the periodic potential opens a band gap at the Bragg plane associated with \(\mathbf{K}\).
If
\[ U_{\mathbf{K}}=0 \]
then no first-order gap opens at that Bragg plane.
13. Velocity at the Bragg Plane
Near a Bragg plane, the energy can be written in terms of
\[ \mathbf{q} - \frac{\mathbf{K}}{2} \]
The group velocity is related to
\[ \frac{\partial E}{\partial \mathbf{q}} \]
At the Bragg plane,
\[ \left( \mathbf{q} - \frac{\mathbf{K}}{2} \right) \cdot \mathbf{K} = 0 \]
This means that the group velocity is parallel to the Bragg plane.
Therefore, near the Bragg plane, the constant-energy surfaces become pressed against the plane.
14. Wavefunctions at the Gap
At the Bragg plane,
\[ \varepsilon_{\mathbf{q}}^0 = \varepsilon_{\mathbf{q}-\mathbf{K}}^0 = \varepsilon^0 \]
The two coupled equations become
\[ (E-\varepsilon^0)c_{\mathbf{q}} = U_{\mathbf{K}}c_{\mathbf{q}-\mathbf{K}} \]
and
\[ (E-\varepsilon^0)c_{\mathbf{q}-\mathbf{K}} = U_{\mathbf{K}}^*c_{\mathbf{q}} \]
Assume \(U_{\mathbf{K}}\) is real and positive.
For the upper level,
\[ E_+ = \varepsilon^0 + |U_{\mathbf{K}}| \]
so
\[ c_{\mathbf{q}} = c_{\mathbf{q}-\mathbf{K}} \]
The wavefunction is approximately
\[ \psi_+(\mathbf{r}) \propto e^{i\mathbf{q}\cdot\mathbf{r}} + e^{i(\mathbf{q}-\mathbf{K})\cdot\mathbf{r}} \]
At the Bragg plane, write
\[ \mathbf{q} = \frac{\mathbf{K}}{2} + \mathbf{p} \]
Then
\[ \mathbf{q}-\mathbf{K} = -\frac{\mathbf{K}}{2} + \mathbf{p} \]
Substituting,
\[ \psi_+(\mathbf{r}) \propto e^{i\mathbf{p}\cdot\mathbf{r}} \left[ e^{i\mathbf{K}\cdot\mathbf{r}/2} + e^{-i\mathbf{K}\cdot\mathbf{r}/2} \right] \]
Using
\[ e^{ix}+e^{-ix}=2\cos x \]
we get
\[ \psi_+(\mathbf{r}) \propto 2e^{i\mathbf{p}\cdot\mathbf{r}} \cos\left( \frac{\mathbf{K}\cdot\mathbf{r}}{2} \right) \]
Therefore,
\[ |\psi_+(\mathbf{r})|^2 \propto \cos^2\left( \frac{\mathbf{K}\cdot\mathbf{r}}{2} \right) \]
For the lower level,
\[ E_- = \varepsilon^0 - |U_{\mathbf{K}}| \]
the coefficients have opposite sign:
\[ c_{\mathbf{q}} = - c_{\mathbf{q}-\mathbf{K}} \]
so
\[ \psi_-(\mathbf{r}) \propto e^{i\mathbf{q}\cdot\mathbf{r}} - e^{i(\mathbf{q}-\mathbf{K})\cdot\mathbf{r}} \]
and
\[ |\psi_-(\mathbf{r})|^2 \propto \sin^2\left( \frac{\mathbf{K}\cdot\mathbf{r}}{2} \right) \]
Thus, at the Bragg plane, the two states become standing waves.
One standing wave has charge density concentrated near potential minima, while the other has charge density concentrated near potential maxima.
This is why their energies split.
15. Energy Bands in 1D
For a free electron in one dimension,
\[ \varepsilon_k^0 = \frac{\hbar^2 k^2}{2m} \]
The reciprocal lattice vectors are
\[ K = \frac{2\pi n}{a} \]
where \(a\) is the lattice spacing.
The Bragg points occur at
\[ k = \frac{K}{2} = \frac{\pi n}{a} \]
At these points, the free-electron parabola intersects a shifted parabola:
\[ \frac{\hbar^2}{2m}|k-K|^2 \]
Without a periodic potential, the parabolas cross.
With a weak periodic potential, a gap opens at each crossing.
The gap size is
\[ 2|U_K| \]
16. Extended-Zone Scheme
In the extended-zone scheme, the bands are plotted continuously over many Brillouin zones.
The free-electron parabolas are modified near Bragg points.
At the Bragg points,
\[ k = \pm\frac{\pi}{a}, \pm\frac{2\pi}{a}, \pm\frac{3\pi}{a}, \dots \]
the periodic potential opens gaps.
So the crossings become avoided crossings.
17. Reduced-Zone Scheme
In the reduced-zone scheme, all wave vectors are folded back into the first Brillouin zone.
For a one-dimensional lattice, the first Brillouin zone is
\[ -\frac{\pi}{a} \leq k \leq \frac{\pi}{a} \]
The same energy states are now represented as multiple bands inside the first Brillouin zone.
18. Reduced-Zone Scheme with Gaps
When the weak periodic potential is included, gaps appear at the Brillouin zone boundaries.
At
\[ k = \pm\frac{\pi}{a} \]
the first and second bands are separated by a gap
\[ \Delta E = 2|U_K| \]
The free-electron crossing becomes an avoided crossing.
19. Energy Bands and Fermi Surfaces in 3D
In three dimensions, the free-electron constant-energy surfaces are spheres:
\[ \varepsilon_{\mathbf{k}}^0 = \frac{\hbar^2 k^2}{2m} \]
At the Fermi energy, the free-electron Fermi surface is a sphere.
However, in a periodic potential, this sphere is modified near Bragg planes.
The strongest changes occur where the free-electron Fermi sphere intersects Bragg planes.
At these intersections, gaps open and the Fermi surface becomes distorted.
A useful procedure is:
- Draw the free-electron Fermi sphere centered at \(\mathbf{k}=0\).
- Draw the Bragg planes of the reciprocal lattice.
- Identify where the Fermi sphere crosses the Bragg planes.
- At these crossings, the weak periodic potential opens gaps.
- The Fermi surface is deformed near these regions.
Therefore, the Fermi surface of a real metal is not always a perfect sphere.
It is most strongly changed near Bragg planes.
20. Brillouin Zones
Brillouin zones are constructed from Bragg planes.
The first Brillouin zone is the set of points in reciprocal space closer to the origin than to any other reciprocal lattice point.
Equivalently, it is the Wigner-Seitz cell of the reciprocal lattice.
The first Brillouin zone is bounded by the nearest Bragg planes.
The second Brillouin zone is the set of points reached from the origin by crossing one Bragg plane.
The third Brillouin zone is the set of points reached by crossing two Bragg planes.
In general, the \(n\)-th Brillouin zone is the set of points that can be reached from the origin by crossing \(n-1\) Bragg planes, but no fewer.
Each Brillouin zone has the same volume as a primitive cell of the reciprocal lattice.
21. Structure Factor and Band Gaps
Now consider the Fourier coefficient \(U_{\mathbf{K}}\) of the periodic potential.
Suppose the crystal potential is built from atomic potentials centered at
\[ \mathbf{R}+\mathbf{d}_j \]
where:
- \(\mathbf{R}\) is a Bravais lattice vector,
- \(\mathbf{d}_j\) is the position of atom \(j\) inside the basis.
Then the total potential is
\[ U(\mathbf{r}) = \sum_{\mathbf{R}} \sum_j \phi(\mathbf{r}-\mathbf{R}-\mathbf{d}_j) \]
where \(\phi\) is the potential of one ion.
The Fourier coefficient is
\[ U_{\mathbf{K}} = \frac{1}{v} \int_{\text{cell}} d^3r\, e^{-i\mathbf{K}\cdot\mathbf{r}} U(\mathbf{r}) \]
Substitute the full potential:
\[ U_{\mathbf{K}} = \frac{1}{v} \int_{\text{cell}} d^3r\, e^{-i\mathbf{K}\cdot\mathbf{r}} \sum_{\mathbf{R}} \sum_j \phi(\mathbf{r}-\mathbf{R}-\mathbf{d}_j) \]
Summing over all Bravais lattice vectors and integrating over one cell is equivalent to integrating over all space.
So,
\[ U_{\mathbf{K}} = \frac{1}{v} \sum_j e^{-i\mathbf{K}\cdot\mathbf{d}_j} \int d^3r\, e^{-i\mathbf{K}\cdot\mathbf{r}} \phi(\mathbf{r}) \]
Define the Fourier transform of the atomic potential:
\[ \phi_{\mathbf{K}} = \int d^3r\, e^{-i\mathbf{K}\cdot\mathbf{r}} \phi(\mathbf{r}) \]
Then
\[ U_{\mathbf{K}} = \frac{1}{v} \phi_{\mathbf{K}} \sum_j e^{-i\mathbf{K}\cdot\mathbf{d}_j} \]
Define the structure factor
\[ S_{\mathbf{K}} = \sum_j e^{-i\mathbf{K}\cdot\mathbf{d}_j} \]
Therefore,
\[ U_{\mathbf{K}} = \frac{1}{v} \phi_{\mathbf{K}}S_{\mathbf{K}} \]
This shows that the Fourier component of the lattice potential depends on the same kind of structure factor that appears in diffraction.
Since the energy gap is
\[ \Delta E = 2|U_{\mathbf{K}}| \]
we get
\[ \Delta E = \frac{2}{v} |\phi_{\mathbf{K}}S_{\mathbf{K}}| \]
Thus, the structure factor controls which Bragg-plane gaps appear in the nearly-free-electron band structure.
If
\[ S_{\mathbf{K}} = 0 \]
then
\[ U_{\mathbf{K}} = 0 \]
and no first-order gap opens at that Bragg plane.
22. Main Results
The nearly-free-electron model starts with free-electron energies
\[ \varepsilon_{\mathbf{k}}^0 = \frac{\hbar^2 k^2}{2m} \]
and adds a weak periodic potential.
Away from degeneracies, the energy shift is second order:
\[ E = \varepsilon_{\mathbf{k}-\mathbf{K}_\ell}^0 + \sum_{\mathbf{K}\neq \mathbf{K}_\ell} \frac{ \left| U_{\mathbf{K}-\mathbf{K}_\ell} \right|^2 } { \varepsilon_{\mathbf{k}-\mathbf{K}_\ell}^0 - \varepsilon_{\mathbf{k}-\mathbf{K}}^0 } + O(U^3) \]
Near degeneracies, the periodic potential produces first-order splitting.
For two degenerate levels,
\[ E_{\pm} = \varepsilon^0 \pm |U_{\mathbf{K}}| \]
so the band gap is
\[ \Delta E = 2|U_{\mathbf{K}}| \]
Degeneracies occur at Bragg planes:
\[ \left( \mathbf{q} - \frac{\mathbf{K}}{2} \right) \cdot \mathbf{K} = 0 \]
The nearly-free-electron model explains:
- why free-electron parabolas become energy bands,
- why gaps open at Bragg planes,
- why Brillouin zones matter,
- why Fermi surfaces are distorted near Bragg planes,
- why structure factors can determine whether a gap appears.