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# 物理代写|凝聚态物理代写Condensed Matter Physics代考|CRN9072 Nearly-free electron model

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## 物理代写|凝聚态物理代写Condensed Matter Physics代考|Nearly-free electron model

The model of electrons interacting with a weak potential is an excellent one for representing the electronic structure of many solids. In many metals, the electrons can be considered as almost free, so their energies and wavefunctions are not expected to differ much from the results obtained for a free electron model (FEM). The standard approach is to use perturbation theory on the FEM to obtain results for the nearly-free electron model (NFEM) rather than using the Kronig-Penney method. Because of the appropriateness of the model for real systems, the model also illustrates many of the general features of band structures.
Since the extension of the FEM and the NFEM to the three-dimensional case is straightforward with no fundamental complication, we proceed directly to three dimensions.

Letting $V(\mathbf{r}) \neq 0$ (but weak), we can obtain $E(\mathbf{k})$ via standard perturbation theory. This gives to second order
$$E(\mathbf{k})=E_{0}(\mathbf{k})+V_{\mathbf{k k}}+\sum_{\mathbf{k} \neq \mathbf{k}^{\prime}} \frac{\left|V_{\mathbf{k}^{\prime} \mathbf{k}}\right|^{2}}{E_{0}(\mathbf{k})-E_{0}\left(\mathbf{k}^{\prime}\right)},$$
with
$$V_{\mathbf{k}^{\prime} \mathbf{k}}=\frac{1}{\Omega_{x}} \int e^{i\left(\mathbf{k}-\mathbf{k}^{\prime}\right) \cdot \mathbf{r}} V(\mathbf{r}) d \mathbf{r}$$
and
$$V_{\mathbf{k k}}=\frac{1}{\Omega_{x}} \int V(\mathbf{r}) d \mathbf{r}=V_{0}$$
where $E_{0}(\mathbf{k})$ is the FEM energy, $\Omega_{x}$ is the crystal volume, which for convenience can be set to unity, $V_{0}$ is the average potential, and the $V_{\mathbf{k k}^{\prime}}$ are matrix elements of the potential between FEM states $\Psi_{\mathbf{k}}^{0}$ and $\Psi_{\mathbf{k}^{\prime}}^{0}$, where $\Psi_{\mathbf{k}}^{0}(\mathbf{r})=\frac{e^{i \mathbf{k} \cdot \mathbf{r}}}{\sqrt{\Omega_{x}}}$. The constant $V_{0}$ shifts all the energies irrespective of their k-values; hence, it is used just to set the zero of energy.

## 物理代写|凝聚态物理代写Condensed Matter Physics代考|Tight-binding model

At the other limit relative to the NFEM is the tight-binding model (TBM). This approximation is most appropriate for solids in which the constituent atoms resemble slightly perturbed free atoms. Whereas the NFEM starts with completely free electrons perturbed by a weak potential, the TBM begins with atomic-like states and perturbs them with a potential due to interaction with nearby atoms. The NFEM is appropriate for close-packed solids in which the overlap between electrons from adjacent atoms is large. Hence, metals are particularly suited to this type of analysis. The TBM is more appropriate for wide gap insulators where the overlap between adjacent atoms is small and atomic separations are large. Semiconductors tend to fall in between, and for these cases, both methods have been used.

It is possible to begin with the Kronig-Penney model to gain insight into the TBM. For example, consider a situation in which the above Kronig-Penney potential is appropriate for atomic-like systems where the starting point is a set of bound states. For the repulsive potential case considered earlier, it is not possible to get a bound state. This can be seen by examining the Kronig-Penney equation, assuming a bound state with $E<0$. The condition $E<0$ implies that $\alpha$ is imaginary. Assuming $\alpha=i \phi$, we have
$$\cos K=\cosh \phi+\frac{P \sinh \phi}{\phi} .$$
This can be solved only if we consider an attractive potential.
Now $P$ (Eq. (3.30)) is negative, and one has the possibility that $E$ can be negative. Equation (3.52) can be written as
$$\cos K=\cosh \phi-|P| \frac{\sinh \phi}{\phi} .$$
This is easily solvable if $|\phi| \gg 1$. Expanding, we find
$$\cos K=\frac{e^{\phi}}{2}\left(1-\frac{|P|}{\phi}\right)$$

## 物理代写儗聚态物理代写Condensed Matter Physics代考|Nearly-free electron model

$$E(\mathbf{k})=E_{0}(\mathbf{k})+V_{\mathbf{k} \mathbf{k}}+\sum_{\mathbf{k} \neq \mathbf{k}^{\prime}} \frac{\left|V_{\mathbf{k}^{\prime} \mathbf{k}}\right|^{2}}{E_{0}(\mathbf{k})-E_{0}\left(\mathbf{k}^{\prime}\right)}$$

$$V_{\mathbf{k}^{\prime} \mathbf{k}}=\frac{1}{\Omega_{x}} \int e^{i\left(\mathbf{k}-\mathbf{k}^{\prime}\right) \cdot \mathbf{r}} V(\mathbf{r}) d \mathbf{r}$$

$$V_{\mathbf{k k}}=\frac{1}{\Omega_{x}} \int V(\mathbf{r}) d \mathbf{r}=V_{0}$$

## 物理代写|凝聚态物理代与写Condensed Matter Physics代考|Tight-binding model

$E<0$. 条件 $E<0$ 暗示 $\alpha$ 是虚构的。假设 $\alpha=i \phi$ ，我们有
$$\cos K=\cosh \phi+\frac{P \sinh \phi}{\phi}$$

$$\cos K=\cosh \phi-|P| \frac{\sinh \phi}{\phi} .$$

$$\cos K=\frac{e^{\phi}}{2}\left(1-\frac{|P|}{\phi}\right)$$

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