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# 物理代写|固体物理代写Solid Physics代考|k · p Theory

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## 物理代写|固体物理代写SOLID PHYSICS代考|k · p Theory

Another approximation method, which is very useful for understanding interactions between bands, uses a perturbation expansion of a different type. This method takes note of the fact that the critical points of the Brillouin zone have well-defined properties. If the energies at these critical points are known, then we can treat the band energy at a nearby point in the Brillouin zone as the sum of the energy at the critical point plus a small perturbation.
We begin by writing the Schrödinger equation in terms of the Bloch functions,
$$\left(\frac{p^2}{2 m}+U(\vec{r})\right) u_{n \vec{k}}(\vec{r}) e^{i \vec{k} \cdot \vec{r}}=E_n(\vec{k}) u_{n \vec{k}}(\vec{r}) e^{i \vec{k} \cdot \vec{r}},$$

where $\vec{p}=-i \hbar \nabla$. Since the derivative of $e^{i \vec{k} \cdot \vec{r}}$ is known, we can rewrite this as
$$\left(\frac{1}{2 m}|\vec{p}+\hbar \vec{k}|^2+U(\vec{r})\right) u_{n \vec{k}}(\vec{r})=E_n(\vec{k}) u_{n \vec{k}}(\vec{r}) .$$
We can then write this as the sum of three terms,
$$\left(H_0+H_1+H_2\right) u_{n \vec{k}}(\vec{r})=E_n(\vec{k}) u_{n \vec{k}}(\vec{r})$$
where
\begin{aligned} H_0 & =\frac{p^2}{2 m}+U(\vec{r}) \ H_1 & =\frac{\hbar}{m} \vec{k} \cdot \vec{p} \ H_2 & =\frac{\hbar^2 k^2}{2 m} . \end{aligned}

## 物理代写|固体物理代写SOLID PHYSICS代考|Other Methods ofCalculating Band Structure

We have already seen in Section 1.6 that the Bloch states of different bands are orthogonal. Since the core electrons are nearly the same as the atomic states, which have slow variation near the atomic nucleus, this means that the electron wave functions for higher levels will tend to have strong spatial oscillations near a nucleus, so that the overlap integral $\int \psi_n^* \psi_m d^3 r$ will vanish. This leads to problems for numerical calculations.

One way to solve for the higher band states without using rapidly oscillating wave functions is the pseudopotential method. In this method, instead of using just the potential $U(\vec{r})$ of the bare nucleus, a new $U(\vec{r})$ is used which includes the effects of the Coulomb repulsion and Pauli exclusion of the core electrons, to repel the electrons in higher states from the core region.

Using this new $U(\vec{r})$, the upper electron states can be calculated using the nearly free electron approximation; the inner, core electron states are assumed to remain nearly the same as the atomic core states. This strong distinction between the two types of states is one of the major assumptions of this method.

There is no exact way of calculating the potential $U(\vec{r})$; in this method one simply starts with a guess and then improves $U(\vec{r})$ by iteration. This can be done either by comparing the calculated band structure to experimental data or by adjusting $U(\vec{r})$ to give self-consistency. Once the valence electron states are calculated, the local charge density due to these electrons can be calculated, which is proportional to $\rho(\vec{r})=\psi^*(\vec{r}) \psi(\vec{r})$. The Coulomb repulsion from this charge density then gives an adjustment to $U(\vec{r})$. Eventually, the adjusted $U(\vec{r})$ will not change upon iteration, when it is consistent with the charge density of the valence states.

The band structure of silicon in Figure 1.26(a) was calculated using a pseudopotential method. Notice how the bands have the character of nearly free electrons – for example, the lowest energy band is nearly parabolic and the next energy band has a maximum at zone center, as in Figure 1.30. In general, pseudopotential methods give reasonable predictions of many band structure parameters, but still require some experimental input for realistic calculations.

## 物理代写|固体物理代写SOLID PHYSICS代考|k · p Theory

$$\left(\frac{p^2}{2 m}+U(\vec{r})\right) u_{n \vec{k}}(\vec{r}) e^{i \vec{k} \cdot \vec{r}}=E_n(\vec{k}) u_{n \vec{k}}(\vec{r}) e^{i \vec{k} \cdot \vec{r}},$$

$$\left(\frac{1}{2 m}|\vec{p}+\hbar \vec{k}|^2+U(\vec{r})\right) u_{n \vec{k}}(\vec{r})=E_n(\vec{k}) u_{n \vec{k}}(\vec{r}) .$$

$$\left(H_0+H_1+H_2\right) u_{n \vec{k}}(\vec{r})=E_n(\vec{k}) u_{n \vec{k}}(\vec{r})$$

\begin{aligned} H_0 & =\frac{p^2}{2 m}+U(\vec{r}) \ H_1 & =\frac{\hbar}{m} \vec{k} \cdot \vec{p} \ H_2 & =\frac{\hbar^2 k^2}{2 m} . \end{aligned}

## MATLAB代写

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