Effective mass in terms of electron states

In summary, on page 94 of the paper, the second order perturbation expansion for the effective mass matrix is solved using the Taylor expansion of the exact result. The perturbation sum is then obtained by canceling terms with the same sign. The missing factor of ##m^2## is obtained by taking the inverse of the expression in post #3.
  • #1
ergospherical
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I'm trying to figure out the second order extension of the "trick" used on page 92 (https://www.damtp.cam.ac.uk/user/tong/aqm/solid3.pdf) for the calculation of the effective mass matrix ##m^{\star}_{ij} = \hbar^2 (\partial^2 E/ \partial k_i \partial k_j)^{-1}## on page 94. I think for this one would need to consider the following perturbation:\begin{align*}
\delta H &= \frac{\partial H_{\mathbf{k}}}{\partial \mathbf{k}} \cdot \mathbf{q} + \frac{1}{2}\frac{\partial^2 H_{\mathbf{k}}}{\partial k_i \partial k_j} q_i q_j \\
&= \frac{\hbar^2}{m}\mathbf{q} \cdot (-i\nabla + \mathbf{k}) + \frac{\hbar^2}{m} \delta_{ij} q_i q_j
\end{align*}Then I can equate the second order perturbation expansion to the Taylor expansion of the exact result ##E(\mathbf{k} + \mathbf{q})##,\begin{align*}
\frac{\partial E}{\partial \mathbf{k}} \cdot \mathbf{q} + \frac{1}{2}\frac{\partial^2 E}{\partial k_i \partial k_j} q_i q_j &= \langle \psi_{n,\mathbf{k}}| \delta H | \psi_{n,\mathbf{k}} \rangle + \sum_{n \neq n'} \frac{|\langle \psi_{n,\mathbf{k}} | \delta H | \psi_{n', \mathbf{k}} \rangle|^2}{E_{n}(\mathbf{k}) - E_{n'}(\mathbf{k})} \\
\frac{1}{2}\frac{\partial^2 E}{\partial k_i \partial k_j} q_i q_j &= \frac{\hbar^2}{m} \delta_{ij} q_i q_j + \frac{\hbar^2}{m^2} \sum_{n \neq n'} \frac{|\langle \psi_{n,\mathbf{k}} | \mathbf{q} \cdot -i \hbar \nabla | \psi_{n', \mathbf{k}} \rangle|^2}{E_{n}(\mathbf{k}) - E_{n'}(\mathbf{k})}
\end{align*}where I canceled the first order terms from both sides, and also dropped the fourth order terms in ##q_i## from the second term. Then\begin{align*}
\hbar^2 \left(\frac{\partial^2 E}{\partial k_i \partial k_j} \right)^{-1} &= \frac{m}{2}\left[ \delta_{ij} - \frac{1}{m} \sum_{n\neq n'} \frac{\langle \psi_{n,\mathbf{k}} | p_i | \psi_{n',\mathbf{k}} \rangle \langle \psi_{n',\mathbf{k}} |p_j | \psi_{n,\mathbf{k}} \rangle}{E_n(\mathbf{k}) - E_{n'}(\mathbf{k})} \right]
\end{align*}What's wrong?
 
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  • #2
Just as a bewildered observer, what do you think is wrong with this result? Seems like a complicated second order perturbation matrix answer to a complicated problem. I assume once the perturbation sums are computed the off diagonal terms go away because of symmetry perhaps?
 
  • #3
I'm trying to get to this result:

1648734934772.png
 
  • #4
I see now, you made a mistake somewhere :rolleyes:. I’m still trying to see how ##p_i## comes about. Then all one needs is the missing factor of ##m^2## and the sign error…..
 
  • #5
It's because ##\mathbf{q} \cdot -i\hbar \nabla = q_i p_i##. The expression in post #3 is also the inverse matrix and I don't know where the Hermitian conjugate (##\mathrm{h.c.}##) terms come from.
 
  • #6
Okay, I see ##k## and ##q## In the development. What’s ##p##?
 
  • #7
I have ##p_i## as the crystal momentum eigenvalues.
 

FAQ: Effective mass in terms of electron states

1. What is effective mass in terms of electron states?

Effective mass in terms of electron states refers to the measure of the mass of an electron in a solid material, taking into account its interactions with other particles in the material. It is an important concept in condensed matter physics and is used to describe the behavior of electrons in a material.

2. How is effective mass related to band structure?

Effective mass is closely related to the band structure of a material. In a solid, electrons are restricted to certain energy levels, or bands, due to the periodic arrangement of atoms. The curvature of these bands determines the effective mass of the electrons within them.

3. What is the significance of effective mass in electronic devices?

Effective mass plays a crucial role in the design and functioning of electronic devices. It affects the mobility of electrons, which determines the speed at which they can move through a material. This, in turn, affects the performance of devices such as transistors and diodes.

4. Can effective mass be measured experimentally?

Yes, effective mass can be measured experimentally using various techniques such as cyclotron resonance, Hall effect, and optical spectroscopy. These methods involve applying a magnetic field or an electric field to the material and measuring the resulting changes in the electron's motion.

5. How does effective mass vary in different materials?

The effective mass of electrons can vary greatly between different materials. It depends on the band structure of the material, which is determined by factors such as the type of atoms, their arrangement, and the strength of electron-electron interactions. Materials with a lower effective mass tend to have better electrical conductivity.

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