Electrons riding the wave of atoms
Table of Contents
1. Bloch Electrons
- Electrons moving in the periodic potentials can be represented through the wave equation: \[\Psi(r)=\mathrm{e}^{\imath k r} u(r).\] Here, \(\Psi(r)\) is the wave function of the electrons at position \(r\), and \(u(r)\) is the periodic function with some period \(R\). The periodicity of the function takes into account the periodicity of the atoms and the other two and three electrons processes. In short the periodic function is some self consistent field felt by single electron generated by the background atomic structures and other electrons. Another way of writing the Bloch wave function is: \[\Psi(r+R)= \mathrm{e}^{\imath k R} \Psi(r).\] These two representation is known as the Bloch theorem.
- As the wave function \(\Psi(r)\) depends on the momentum \(k\), for different \(k\) we will get different wave function, e.g. \(\Psi(r,k_{1})\), \(\Psi(r,k_{2})\), \(\Psi(r,k_{3})\), e.t.c. All these individual wave function will satisfy the Schrodinger equation. Hence, definition of the Bloch electrons: All the individual electrons which satisfy the Schrodinger equation (with potential) are called the Bloch electrons.
2. Effect of Born-Karmann (BK) boundary conditions
- In the Sommerfeld model of the free electrons the effect of the effect of BK boundary condition is that the electrons can take the quantized momentum \(k=2n\pi/L\). Physically it means that the \(k\) is chosen such that the total phase change inside the bounded box is multiple of \(2\pi\) for every single electrons.
- When atoms are placed inside this box then the above picture is applicable for the whole box, however, a new picture in which the periodicity increases. For electrons inside the box without any atomic the lattice periodicity was unity, however when atoms are placed in the box at some fixed distance the peridicity increases; it is equal to the number of atoms inside the box. One can think this in terms of necklace. When there was no atoms inside the box, the neclace was made up of two atoms, however, when the atom is placed inside the box, the necklace is made of \(N\) no. of atoms. Now every portion in between two beads are same, as in a necklace due to translational symmetry all the portions between two beads looks same.
- The allowed value of momentum is \[k=\frac{m}{N} 2 \pi.\]
3. Properties of the Bloch function
- The momentum in the Bloch wave function \(\Psi(r)=\mathrm{e}^{\imath k r} u(r)\) is known as quasi momentum. As it does not give the momentum of the wave function when the momentum operator \(\hat{p}= i \nabla_{r}\) is applied on the \(\Psi(r)\).
- Only the momentum in the first Brillouin zone is important as \(\mathrm{e}^{i (k+K)r} = \mathrm{e}^{i k r}\).
- For a given \(k\) there are \(n\) no. of solutions to the wave function. It comes from the fact that the wave function for give \(k\) can be written as \(\Psi(r)= \sum\limits_{K} c_{k-K} \mathrm{e}^{\imath (k-K)r}\). Physically it means to construct the wave function at \(k\), the contribution for all \(k-K\) comes. As the system is invariant with BK boundary condition (necklace), hence, no matter which BZ one chooses, there will be a given solution. This gives the band structure.
- The velocity of the Bloch electrons is given as \[v_{n}(k) = -\frac{1}{\hbar} \nabla_{k} E_{n}(k)\]. Physically it means that the electron velocity depends on the slope of the electronic dispersion. Hence if there is very large change in energy then the velocity is high, however, if the slope is almost horizontal then the electrons travels very slowly. It also means that, electron velocity depends only on the quasi-momentum \(k\); the time dependence of the velocity is absent. It means the electrons don't loose their energy with time, hence travel with constant velocity. It is not correct, because in that case the superconductivity is present in the system.
4. Fermi surface and different types of materials
- Dielectrics (\(\Delta \gg k_{B}T\)) - Materials whose Fermi surface is filled fully, and the energy gap \(\Delta \gg k_{B}T\) then they are known as the dielectric. Physically it means the valence band can not be acquired by electrons only through thermally excited process, external electric field is needed to fill the valence band.
- Semiconductors (\(\Delta < k_{B}T\)) - Materials whose Fermi surface is filled fully and the energy gap \(\Delta < k_{B}T\) then they are known as semiconductors. Physically, it means that the electron can jump to the valence band by heating also.
- Metals - Materials where the energy zone is filled partially, are called metals.
5. Density of states
- All the properties of the materials are related to the energy levels of the materials. Usually one sum (integrate) the property in question over the available energy levels: \(q = \sum\limits_{k} Q(k)\). The limit of the \(k \in \left( 0, k_{f} \right)\). However, one can think it from another point of view. To every \(k\) there is a corresponding energy level. Hence for a given energy difference \(\Delta E\) there will be some number of \(\Delta k\). Hence one can take the above summation of the properties of the system as, summation over \(\Delta E\). To do this one need to multiply the Δ E with a function \(g(E)\) which gives number of \(k\) states are present in the given energy \(E\) for an energy difference of \(\Delta E\). It should be remembered that the all the states (\(\Delta k\)) corresponding to \(E\) contain in the energy gap \(\Delta E\) centered around \(E\) has the same energy. This is called the density of states.
- The continium defination of the density of states is: \[g(E)=\int dk \delta(E-E(k)).\] Physically it gives all the number of states withe energy \(E\) in whole Brillouion zone. In the ideal case it will be an positive integer.
- Density of states can also be thought of as isoenergy surface. One can find the number of states in the energy gape \(\Delta E\) by integrating the area on the \(k\) space corresponding to two isoenergy surfaces.