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Electronic Properties
of Materials - Classical Theory of Electron Gas
Here we discuss the motion of an electron in an energy band.
Group Velocity:
$$ v_g = \frac{d\omega}{dk} $$
$$ \omega = \frac{\hbar}{k}$$
By putting them together:
$$v_g = \frac{1}{\hbar} \frac{dE}{dk}$$
$$v_g = \frac{1}{\hbar} \nabla _k E(k) $$
The work done by the elctric field on an electron:
$$\delta E = -eE v_g \delta t $$
$$\delta E = \frac{dE}{dk} dk - \hbar v_g dk $$
This gives :
$$ \habr \frac{dk}{dt} $$
This is an important relation. In a periodic potential, \hbar dk/dt equals the external force acting on the electron. When an electron is moving in the crystal, is is subjected to forces from the external forces as well as the forces from lattice potential. The external force can come from electric or magnetic field.
Effective Mass:
$$ \frac{d v_g}{dt}=\frac{1}{\hbar} \frac{d^2E}{dk dt} $$
$$\frac{d v_g}{dt}= \frac{1}{\hbar} \frac{d^2 E}{dk^2} \frac{dk}{dt} $$
$$\frac{d v_g}{dt} = \left( \frac{1}{\hbar}\right)^2 \frac{d^2 E}{dk^2} \hbar \frac{dk}{dt} $$
By putting the expression for external force:
$$\frac{d v_g}{dt}=\left(\frac{1}{\hbar^2} \frac{d^2E}{dk^2} \right) F $$
This equation resembles $$F=m^* a$$
with $$m^* = \frac{\hbar^2}{d^2K/dk^2}$$
FOr anisotropic crystals, since the E-k relationship depends on the direction, effective mass will also be anisotropic.
Effective mass of electron is confirmed through the cyclotron resonance in semiconducting materials which is given by $$\omega _c = \frac{eB}{m^* c}$$
In the next lesson, we will focus on the electronic properties of semiconducting Materials.
Learning Goals:
Solving Schrodinger Equation in the presence of a Periodic Potential:
We have been working on understanding the electron dynamics in the presence of a periodic potential.
In the last class, we discussed how to include the periodic potential as a perturbation. We found with the degenerate perturbation theory that the degeneracy is lifted in the presence of the perturbation, which introduces forbidden regions or electron energy gaps.
However, we have not yet solved the Schrodinger equation for the periodic potential. F. Bloch (through his PhD. work) introduced a way to solve the Schrodinger equation for electrons in the periodic potential. For this reason, electrons in a periodic potential are some times referred to as Bloch electrons.
Bloch theorem says, solutions for the Schrodinger equation for electrons in a periodic potential is given by:
$$\Psi_k(\vec{r})=U_k(\vec{r} e^{i \vec{k}.\vec{r}} $$
where Uk® has the periodicity of the lattice.
$$U_k(r+T)=U_k (r) $$
T is the translation vector of the periodic system.
In otherwords,
the eggenfunctions of the wave function for a periodic potential are the product of a plane wave e^{irk} times a function Uk® , which has the periodicity of the crystal
It basically says the wave function is delocalized through out the crystal.
Graphically, this can be represented as follows.
Physical Proof:
Bloch Theorem basically says that the physical properties are invariant under crystal translation. We can prove that statement as follows.
$\Psi_k®=U_k®e^{ii \vec{k}.\vec{r}}$$
$$\Psi_k(r+T)=U_k(r+T)e^{ii \vec{k}.\vec{r}+\vec{T}}$$
We know that $$U_k(r+T)=U_k® $$
$$\Psi_k(r+T)=U_k®e^{ikr}e^{ikT}$$
which can be written as
$$\Psi_k(r+T)=\Psi_k®e^{ikT} $$
Thant tells under the crystal translation, the wave function only picks a phase factor.
That means, the physical properties will not change under the crystal translation.
Proof of Bloch Theorem:
Let's prove the Bloch theorem for a 1D problem. Genralization for 3D will be straight forward. We are going to discuss the important consequences of this proof.
The envelope function U(x) has the same periodicty as the lattice.
$$U(x+a)=U(x) $$
So the potential energy can be expanded as a Fourier series of the reciprocal lattice vectors G.
$$U(x) = \sum_G U_G e^{iGx} $$
We do not know the exact form of the wave function. In fact, that's what we are looking for. So now what we are going to do is to expand the wave function interns of plane waves.
$$\Psi_k(x)=\sum_k C_k e^{ikx}$$
What are these Ck's? That's what we are looking for.
OK. Now let's put the above two forms of the wave function and the potential in the Schrodinger equation. Goal is to find the wave function (i.e. Cks) and the corresponding eigenvalues.
$$H \Psi =\epsilon \Psi $$
$$\left[\frac{-\hbar^2 }{2m} + U \right] \Psi = \epsilon \Psi $$
$$ \left[ \frac{-\hbar^2}{2m} + \sum_G U_G e^{iGx} \right] \sum_k C_k e^{ikx} = \epsilon \sum_k C_k e^{ikx} $$
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OK, Now we know the form of the solution for electrons in a periodic potential. There are different ways of solving the Schrodinger equation to find out the energy eigen values and form of the wave function for specific materials.
In this lesson, we will only brielfy talk about the Tight binding Approach.
Tight Binding Approach:
Tight binding is probably one of the simple, and transparent methods of solving the electron band structure of Bloch electrons. Even though it is simple, it is being widely used in advanced research studies for materials. Also, its extended versions of Hubbard Model , Anderson Model etc. exist for further studies.
We can construct the wave function as a Linear Combination of Atomic Orbitals (LCAO).
$$ \Psi_k(r) = \sum_j a_j \phi_j(r-r_j)$$
This form of the wave function can be applied to any system. As long as we have an infinite summation, we can expand the wave function for any system.
For the specific case of electrons in a periodic solid, we can truncate the expansion by using a specific form of the wave function within LCAO:
$$\Psi_k(r)=\sum_{T} e^{i\vec{k}.\vec{T}} \phi(r-T) $$
The summation goes through all possible nearest neighbor unit cells.
Why do we use this specific form of the wave function? Just because it satisfies the Bloch form of the wave function:
In the homework problem, you will prove this for a 1D problem.
Application of Tight Binding Problem - 1D- 1Atomic wire:
1 D atomic wire figure here
Ok, let's consider the Schrodinger equation:
$$H |\Psi> = E |\Psi> $$
$$ H \sum_T e^{i k_x ja}|\phi (r-ja)> = E \sum j e^{i k_x ja} |\phi(r-ja)> $$
Sum goes through nearest neighbor cells.
$$ H \sum_T e^{i k_x ja}|\phi (r-ja)> = E \sum_ j e^{i k_x ja} |\phi(r-ja)> $$
By only considering the nearest neighbor interactions, we only get j=-1,0,+1 terms. Which gives:
$$E(k)=t e^{-ika}+\epsilon + t e^{ika} $$
$$ E(k) = \epsilon + 2 t Cos ka $$
Here we have just given a very fundamental introduction. Still, this band structure is based on the hopping parameter t. The model can be extended to consider more basis functions. Also it is important to note that these can be empirical parameters or evaluated based on the theory. Most of the time empirical parameters are used. The predictive power is lost when empirical parameters are used. However, further device modeling can be done using existing experimental observations. It is tough to predict novel materials with this approach.
Electronic structure theory is a complete subject by itself. There are various approaches to calculate electron band structure of periodic solids. Hartree Approximation, Hartree Fock Approximation, Density Functional Theory are few to name. There are various commercial and open-source packages which can be used to calculate the electron band structure of materials. However, we are not going to further discuss how to calculate electronic properties of solids.
Important Information from Electronic Band Structure:
What is k?
Bloch function introduced a wave vector k in the
In the travelling wave function in the free electron model,
Let's apply the momentum operator on Bloch functions:
$$\frac{\hbar}{i}\nabla \Psi(r) =\frac{\hbar}{i}\nabla (u_k(r) e^{ikr}) =\hbar k \Psi_k(r) + e^{ikr} \frac{\hbar}{i} \nabla u_k(r)$$
$$ \left(u_k(r)e^{ikr} \right) $$
$$=\hbar k \Psi(r)+e^{ikr}\frac{\hbar}{i} \nabla u_k(r) $$
That is
$$\frac{\hbar}{i} \nabla \Psi_k(r) \ne \hbar k \Psi_k(r) $$
In other words, Bloch functions are not momentum eigenstates. is called the crystal momentum of Bloch electrons.
Dynamics of Bloch Electrons