kp theory

Introduction of $k\cdot p$ theory

Pertubation theory

$k\cdot p$ is a straightforward way to understand the essential physics in soild state physics. It can combine the band structure in reciprocal space with the physcial properties in experiments by pertubation theory.
Here, we will start from the Schr$\ddot o$dinger equation with a periodic potential V to introduce the $k\cdot p$ theory.

$$H \psi = \left(-\frac{\hbar^2}{2m}\nabla^2 + V(r)\right)\psi = E\psi$$

Here, the $V(r)=V(r+R)$ which means the periodic potential obeys the crystal’s symmetry(translation symmetry). So the wavefunction $\psi$ have the Bloch’s formula.

$$\psi_{n,\vec{k}}(\vec{r}) = e^{i\vec{k}\vec{r}}u_{n,\vec{k}}(\vec{r})$$

Here, $\vec{k}$ is the wavevector and $n$ is the band index. $u_{n,\vec{k}}(\mathbf{r})=u_{n,\vec{k}}(\vec{r}+\vec{R})$ shares the same periodicity with the crystal.
Take this wavefunction into Schr$\ddot o$dinger equantion,
we can obtain:

$$\begin{align*} H\psi_{n,\vec{k}}(\vec{r}) &= \left(-\frac{\hbar^2}{2m}\nabla^2 + V(\vec{r})\right) \psi_{n,\vec{k}}(\vec{r})\\ &=\left(-\frac{\hbar^2}{2m}\nabla^2 + V(\vec{r})\right)\left(e^{i\vec{k}\vec{r}}u_{n,\vec{k}}(\vec{r})\right)\\ &=-\frac{\hbar^2}{2m}\nabla\left(i\vec{k}e^{i\vec{k}\vec{r}}u_{n,\vec{k}}(\vec{r})+e^{i\vec{k}\vec{r}}\nabla u_{n,\vec{k}}(\vec{r})\right)+V(\vec{r})e^{i\vec{k}\vec{r}}u_{n,\vec{k}}(\vec{r})\\ &=-\frac{\hbar^2}{2m}\left( -k^2e^{i\vec{k}\vec{r}}u_{n,\vec{k}}(\vec{r}) +2i\vec{k}\nabla u_{n,\vec{k}}(\vec{r})e^{i\vec{k}\vec{r}} +e^{i\vec{k}\vec{r}}\nabla^2u_{n,\vec{k}}(\vec{r}) \right)+V(\vec{r})e^{i\vec{k}\vec{r}}u_{n,\vec{k}}(\vec{r})=E_{n,\vec{k}}e^{i\vec{k}\vec{r}}u_{n,\vec{k}}(\vec{r}) \end{align*}$$

Here, we can use $\hat{p}=-i\hbar\nabla$ and $\hat{p}^2=-\hbar^2\nabla^2$ to modify this equation. The expression can be simplified and rewrited as:

$$\begin{align*} \left(\frac{\hbar^2}{2m}\left(k^2-2ik\nabla-\nabla^2\right)+V(\vec{r})\right)u_{n,\vec{k}}(\vec{r}) &= E_{n,\vec{k}}u_{n,\vec{k}}(\vec{r})\\ \left(\frac{\hbar^2}{2m}\left(k + \frac{1}{i}\nabla \right)^2 + V(\vec{r})\right) u_{n,\vec{k}}(\vec{r})&= E_{n,\vec{k}}u_{n,\vec{k}}(\vec{r}) \end{align*}$$ $$\left(\frac{\hat{p}^2}{2m}u_{n,\vec{k}}(\vec{r})+ \frac{\hbar}{m}(k\cdot \hat{p})+ \frac{\hbar^2k^2}{2m}+ V(\vec{r}) \right)u_{n,\vec{k}}(\vec{r})=E_{n,\vec{k}}u_{n,\vec{k}}(\vec{r})$$

We can divide them into two separate terms:

$$\begin{align*} H_0 &= \frac{\hat{p}^2}{2m} + V(\vec{r})\\ H_k &= \frac{\hbar}{m}(k\cdot \hat{p})+ \frac{\hbar^2k^2}{2m} \end{align*}$$

$H_0$ is the Hamiltonian of free electron and $H_k$ is the k-dependent Hamiltonian from the crystal symmetry, which can be related to the perturbation theory.

$k\cdot p$ Effective Mass

Here, we extend the $k\rightarrow k+q$ and $q$ is very small.

$$\begin{align*} H_{k+q} &= H_k + \frac{\hbar}{m}q\cdot(\hbar k + \hat{p}) +\frac{\hbar q^2}{2m}\\ &=H_k+\underbrace{\frac{\hbar^2}{m}q\cdot(k+\frac{1}{i}\nabla) +\frac{\hbar^2 q^2}{2m}}_{perturbation} \end{align*}$$

Here, we use the second-order perturbation theory

$$E_n^{2} \approx E_n^{0} + H^{\prime}_{n,n} + \sum_{m \neq n}{\frac{|H_{n,m}^{\prime}|^2}{E_n-E_m}}$$

At the same time, $E(k+q)$ can be extended by Taylor Series.

$$E_n(k+q)=E_n(k)+\sum_i \frac{\partial E_n}{\partial k_i}q_i+\frac{1}{2}\sum_{ij}\frac{\partial^2 E_n}{\partial k_i\partial k_j}q_iq_j+\mathcal{O}(q^3)$$

Based on above two equations:

$$\frac{1}{2}\sum_{ij}\frac{\partial^2 E_n}{\partial k_i\partial k_j}q_iq_j= \frac{\hbar^2q^2}{2m} + \sum_{m\neq n}\frac{\bra{u_n}\frac{\hbar^2}{m}q\cdot(k+\frac{1}{i}\nabla)\ket{u_m}\bra{u_m}\frac{\hbar^2}{m}q\cdot(k+\frac{1}{i}\nabla)\ket{u_n}}{E_n-E_m}$$

We know that:

$$\hat{p}\psi_{nk}=e^{ikr}\hbar(k+\frac{1}{i}\nabla)u_{n,k}$$

So the second derivate can be simplified to

$$\frac{1}{2}\sum_{ij}\frac{\partial^2 E_n}{\partial k_i\partial k_j}q_iq_j= \frac{\hbar^2q^2}{2m} + \sum_{m\neq n}\frac{\hbar^2}{m^2}\frac{\bra{\psi_n}q\cdot\hat{p}\ket{\psi_m}\bra{\psi_m}q\cdot\hat{p}\ket{\psi_n}}{E_n-E_m}$$

It means that:

$$\frac{1}{\hbar^2}\frac{\partial^2 E_n}{\partial k_i\partial k_j}= \frac{1}{m}\delta_{i,j} + \sum_{m\neq n}\frac{2}{m^2}\frac{\bra{\psi_n}\hat{p}_i\ket{\psi_m}\bra{\psi_m}\hat{p}_j\ket{\psi_n}}{E_n-E_m}$$

As we all know, the effective mass can be expressed as:

$$\begin{align*} \frac{1}{m^\star} &= \frac{1}{\hbar^2}\left(\frac{\partial^2 E}{\partial^2 k}\right)\\ &= \frac{1}{m}\delta_{i,j} + \sum_{m\neq n}\frac{2}{m^2}\frac{\bra{\psi_n}\hat{p}_i\ket{\psi_m}\bra{\psi_m}\hat{p}_j\ket{\psi_n}}{E_n-E_m} \end{align*}$$

From this we can conclude that the level repulsion causes bands to curve as bandgap is reduceed.