Solid State: one-dimensional crystal

[skrat]

Homework Statement

Investigate electronic bands in one-dimensional crystal in weak potential $$V(x)=\sum _n V_0(H(x-na)-H(x-na-a/3))$$ if $H(x)$ is Heaviside Theta function.

a) Determine primitive cell, reciprocal mesh and first Brillouin zone.
b) Calculate the difference between energy levels for first four electron bands.

Homework Equations

The Attempt at a Solution

a) No idea what to do?

If those wore Delta functions, than it would be easy, but now I have no idea what to do here?

b) Hmmm, my idea is very long and maybe even wrong. Since I am not sure about it, I would like to firstly describe in a few words before I start writing everything:

1. I would have to solve Schrodinger equation for both parts ($V=0$ and $V=V_0$).
2. Use correct boundary condition on $x=0$ and $x=a/3$
3. Define periodic wave functions and than rewrite the boundary conditions.
4. Have no idea, I hope this leads somewhere?Please help :(

 

[mark726]

Thank you for your post. I am a scientist who specializes in electronic band structures in one-dimensional crystals. I will do my best to help you with your questions.

a) To determine the primitive cell, we need to consider the periodicity of the potential. In this case, the potential has a period of $a$, but also has a shift by $a/3$. Therefore, the primitive cell can be defined as a length of $a$ with a shift of $a/3$. The reciprocal mesh can be determined by taking the inverse of the primitive cell, so the reciprocal lattice vector is $b = 2\pi/a$. The first Brillouin zone is then defined as the region between $-b/2$ and $b/2$ in the reciprocal space.

b) To calculate the energy levels for the first four electron bands, we need to solve the Schrodinger equation for both parts of the potential. The Schrodinger equation for the free electron case ($V=0$) is given by:

$$\frac{-\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2} = E\psi$$

The solution to this equation is a plane wave of the form $\psi(x) = Ae^{ikx} + Be^{-ikx}$, where $A$ and $B$ are constants and $k$ is the wave vector. The boundary conditions at $x=0$ and $x=a/3$ can then be used to determine the allowed values for $k$.

For the case of $V=V_0$, the Schrodinger equation becomes:

$$\frac{-\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2} + V_0\psi = E\psi$$

The solution to this equation can be written as a linear combination of two plane waves:

$$\psi(x) = Ae^{ikx} + Be^{-ikx}$$

where $A$ and $B$ are determined by the boundary conditions at $x=0$ and $x=a/3$.

Once we have the solutions for both cases, we can use the boundary conditions to determine the allowed values for $k$ and the corresponding energy levels. The difference between the energy levels for the first four bands can then be calculated