1. Atilla Ozgur Cakmak, PhD
Nanophotonics
Atilla Ozgur Cakmak, PhD

2. Lecture 16: Electrons in Solids
Unit 2
Lecture 16: Electrons in Solids

3. Outline Bloch Waves Reciprocal Space Effective Mass
Band Structures in Solids
Quasiparticles: Holes, Excitons
Defect States

4. A couple of words…
We have seen the basics of understanding the electron wave propagation in periodic lattices. Now, we will try to look into bulk solid materials and will try to adapt what we have learned from previous lectures. Rather than looking into the wave propagation, this time we will consider the fundamentals of semiconductor physics.
Suggested reading: Sergey V. Gaponenko, “Introduction to Nanophotonics”, 4th Chapter.

5. Bloch Waves
So far we have seen periodic potentials and also investigated the Bloch waves without really labelling them as Bloch waves. Let us write Schrodinger’s equation again.
Let us say that the potential is periodic like we discussed in the previous lectures
This will dictate that the wave function will also be periodic.
This condition was derived by Floquet in Accordingly, the wave functions become

6. Bloch Waves
Graphically, the wave function and the periodic potential are shown below.
If we consider the 3D Schrodinger equation while U(r) is again periodic, Bloch theorem can be generalized:

7. Bloch Waves
If we plot the dispersion graphs, like we did for Kronig Penney model,
We will end up with the band gaps as we had previously discussed. For this 1D problem,
-π/a < k < π/a is called the reduced zone. This is a special zone because if we return back to the Bloch’s theorem,
Reciprocal wave vector

8. Reciprocal Space
The lattices are converted to their k-space (reciprocal space) counterparts below for the square and triangular lattices. The reduced zone or the 1st Brillouin zone is the locus of points that are closer to the origin than to any other reciprocal lattice origins.
Any k value can be expressed with the help of G: reciprocal wave vector. We name the Brillouin zone corners with Greek letters.
π/a
-π/a
-π/a
π/a

9. Reciprocal Space
In 3D elementary cells are shown below where every node represents an atom.
Accordingly, the reciprocal lattice vectors are defined as follows:
where is the volume of the elementary cell of the original lattice. Translational symmetry of the original lattice gives rise to translational symmetry of the wave vectors in the reciprocal lattice space.

10. Effective mass
One important consequence that we can observe using the dispersion bands is that
near the center and edges of the Brillouin zone. In other words, the group velocity goes to zero. And near one of these points, if we try to expand for the energy:
by putting E0 and k0 are both zero, which give a parabola. Remember that for the free particle
Depending on the second derivative, the particle can be lighter in a periodic potential or heavier compared to the case in free space.

11. Band structures in solids
The simple lattice formations are for elementary solids like Si, C, Ge, etc. For compound semiconductors like CdS and CdSe a much more complex lattice can be formed. A c-axis is a result of an anisotropic displacement of atoms. For such cases, dispersion bands can be plotted like the following.
The highest occupied band by the electrons is referred as the valance band and the lowest unoccupied is called the conduction band. The bandgap is given as:

12. Band structures in solids
The semiconductors with coinciding valance max. and conduction min. are called direct bandgap semiconductors, others are indirect bandgap semiconductors. Some of the semiconductors and their bandgaps are shown:

13. Quasiparticles: Holes, Excitons
If we get back to our quantum mechanics notes, we have the electron with charge –e, spin ½, mass me (will become me* in a crystal), a momentum ћk. Only charge and spin remain the same for an electron in vacuum and a crystal. Electron in the conduction band is a particle in a many-body system consisting of large number of positive nuclei and negative electrons.
Hole is a quasiparticle with +e, ½ spin, effective mass mh* and also momentum.
Vacuum state: No electron in the conduction band.
First excited state: One electron in the conduction band+one hole in the valance band.
An e-h pair is created.
Photon absorption is required to create this e-h pair. Examine the figure in the next slide!
Energy and momentum conservation requires that:
Photon momentum is very small such that the transition is vertical.
Many body systems
Quasiparticles (consisting real particles)

14. Quasiparticles: Holes, Excitons
The reverse process of emitting a photon with the aid of an e-h pair could have also happened.
Now, let us go back to the Hydrogen Atom’s Schrodinger equation.

15. Quasiparticles: Holes, Excitons
Here, we had to switch from the mere masses of electrons and holes to the effective masses, since what the particles feel in the crystal is critical. We also had to include ε, the dielectric function of the material, since it is not in vacuum. Moreover, we had to assign a specific distance, since none of the particles is fixed and 1/r became 1/|re–rh|.
Let us assume that we take the Laplacians under the same umbrella by having:
Hence, the exciton Bohr radius becomes:
Meanwhile, let us remember Rydberg constant for the Hydrogen Atom:

16. Quasiparticles: Holes, Excitons
If we try to estimate the same constants for the excitons:
Then, the dispersion relation of the excitons can be described while writing the total energy:
Here M is the total mass of the exciton, Eg is the band gap of the semiconductor and K is the wave vector of the exciton. In case, it is created with photon absorption K≈0.

17. Quasiparticles: Holes, Excitons
As a result, the dispersion curves for the exciton can be determined like shown below (left):
And when the incoming photon with negligible momentum is absorbed to create an exciton, we see a huge peak for the corresponding 1st energy level of the excitons in direct-gap semiconductor monocrystals (right). Notice that y-axis is 1/T (Transmission). ZnSe Eg at 88K (the measurement temperature) is eV and Ry*=18 meV. The exciton absorption peak corresponds to n=1 state.

18. Defect States
Ions are introduced as impurity atoms and they result in substitution of the corresponding intrinsic states of electrons by impurity states. This sort of doping addition is outlined with pictures below.
Let us assume that (again) the impurity atom is several magnitudes heavier than the electron and is immobile => we go back to Hydrogen electron model with electron under Coulomb potential. 1 2 1- 2-

19. Defect States
Similar to the excitons, let us write the Hamiltonian again for this case:

20. Defect States
As we are loading the impurity ions with concentrations N (1/cm3), we get to see the following modification of the defect states:
While D(E), density of states can be described in terms of the Gaussian distribution: