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# 物理代写|固体物理代写Solid Physics代考|KYA322

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## 物理代写|固体物理代写Solid Physics代考|Spin–OrbitEffects on thek-Dependence of Bands

We now move on to the next order of the tight-binding approximation (1.9.20), namely terms for nearest neighbors. For simplicity, we will use the example of a simple cubic lattice, in which each atom has six nearest neighbors, as illustrated in Figure 1.42. For nearest neighbors in the $x$-direction, we must compute terms of the following type:
\begin{aligned} U_{x x} & =\frac{-i \hbar^2}{m^2 c^2} \int d^3 r \Phi_x^(\vec{r})\left[\left(\nabla_x U\right) \nabla_y-\left(\nabla_y U\right) \nabla_x\right] \Phi_x(\vec{r}-a \hat{x}), \ U_{x y} & =\frac{-i \hbar^2}{m^2 c^2} \int d^3 r \Phi_x^(\vec{r})\left[\left(\nabla_x U\right) \nabla_y-\left(\nabla_y U\right) \nabla_x\right] \Phi_y(\vec{r}-a \hat{x}) . \end{aligned}

Because the two orbitals $\Phi_n(\vec{r})$ and $\Phi(\vec{r}-a \hat{x})$ are not centered on the same location, parity in the $x$-direction is no longer a concern. However, parity in the $y$ – and $z$-directions still matters, and $U_{x x}$ vanishes due to negative parity in the $y$-direction. However, $U_{x y}$ is nonzero.
If we change the variable $x$ in (1.13.13) to $-x$, then the $x$-integral becomes
\begin{aligned} I(a) & =\int_{-\infty}^{\infty} d x \Phi_x^(x, y, z)\left[\left(\nabla_x U\right) \nabla_y-\left(\nabla_y U\right) \nabla_x\right] \Phi_y(x-a, y, z) . \ & =\int_{-\infty}^{\infty} d x \Phi_x^(-x, y, z)\left[\left(-\nabla_x U\right) \nabla_y+\left(\nabla_y U\right) \nabla_x\right] \Phi_y(-x-a, y, z) . \end{aligned}
Since $\Phi^*(-x, y, z)=-\Phi(x, y, z)$ and $\Phi_y(-x, y, z)=\Phi_y(x, y, z)$, we therefore have $I(a)=$ $I(-a)$. Therefore, the sum of the two nearest-neighbor terms in the $x$-direction in the tightbinding formula (1.9.20) is
$$i \sigma_z U_{x y}\left(e^{i k_x a}+e^{-i k_x a}\right)=2 i \sigma_z U_{x y} \cos k_x a .$$

## 物理代写|固体物理代写Solid Physics代考|Quasiparticles

We saw in Section 1.11 that it is very common to have a solid with one or more entirely full bands and some entirely empty bands. Suppose that we have two bands with minima and maxima at the center of the Brillouin zone, as shown in Figure 2.1, which is a fairly typical band structure for a semiconductor. In the ground state of the crystal, the lower band, known as the valence band, is completely full, and the upper band, known as the conduction band, is completely empty. None of the electrons in the lower band can change its state because all the nearby states are filled. If we put energy into the system, however, we can promote an electron from a state in the lower band to a state in the upper band. In this case, the electron in the upper band can move freely into other states in the same band with very little energy change. We call this a free electron

At the same time, an empty state is left in the valence band. If another electron from the same band moves into this state, it will leave an empty state in a new place, as illustrated in Figure 2.1.

Rather than keeping track of all the electrons in the band that move to fill the empty state, we can simply keep track of where the empty state goes. We call this empty state a hole. One can think of it in the same way as a bubble in a glass of water. When you look at a bubble rising in a glass of water, you don’t think “the water fell, to fill the empty spot, leaving an empty spot further up,” although this of course is what happens. It is much easier simply to think, “the bubble moved up.”

This leads to a new way of thinking about the electrons in a material. Instead of worrying about all the electrons frozen by Pauli exclusion in the lower band, we can define a new “vacuum” state which is equal to the ground state of the crystal. If we put energy in, we “create” two new particles, which are really excitations of the original ground state. These new particles can be called quasiparticles. The created quasiparticles can also annihilate each other if an electron in the upper band falls back down into a hole in the lower band. This process is called recombination.

This picture of the ground state of a system being a vacuum (that is, a renormalized vacuum) and the excitations being new particles that are created and destroyed is a very general and important concept in condensed matter physics. It is used not only for semiconductors but for many other many-body systems, as we will see in later chapters.
Calling them quasiparticles makes them sound as if they are not real. They are real, however, in the sense that they transport energy, charge, and mass. Actually, one can argue that every normal particle is also a quasiparticle. In relativistic quantum mechanics, the field equation for the electron is the Dirac equation, which implies the existence of negativeenergy states of electrons as well as positive-energy states (see Appendix F). In order to avoid a collapse of all particles to infinite negative energy, Dirac hypothesized that the negative-energy states are all filled with electrons in the same way as the lower band in Figure 2.1, and this prevents positive-energy particles from falling into the negative-energy states. This is called the Dirac sea ${ }^1$ In this case, just as in the bands of Figure 2.1, an electron can be promoted from the lower band to the upper band with the input of sufficient energy. This leads to the existence of a free, positive-energy electron, as well as an empty state in the lower band. This empty state, or hole, is called a positron. Not only the electrons but every fermion has an antiparticle; these antiparticles are the antimatter of high-energy physics. The infinite number of electrons at negative energy doesn’t matter, because these are just part of the ground state of the system, which we define as our vacuum.

## 物理代写|固体物理代写Solid Physics代考|Spin–OrbitEffects on thek-Dependence of Bands

\begin{aligned} U_{x x} & =\frac{-i \hbar^2}{m^2 c^2} \int d^3 r \Phi_x^(\vec{r})\left[\left(\nabla_x U\right) \nabla_y-\left(\nabla_y U\right) \nabla_x\right] \Phi_x(\vec{r}-a \hat{x}), \ U_{x y} & =\frac{-i \hbar^2}{m^2 c^2} \int d^3 r \Phi_x^(\vec{r})\left[\left(\nabla_x U\right) \nabla_y-\left(\nabla_y U\right) \nabla_x\right] \Phi_y(\vec{r}-a \hat{x}) . \end{aligned}

\begin{aligned} I(a) & =\int_{-\infty}^{\infty} d x \Phi_x^(x, y, z)\left[\left(\nabla_x U\right) \nabla_y-\left(\nabla_y U\right) \nabla_x\right] \Phi_y(x-a, y, z) . \ & =\int_{-\infty}^{\infty} d x \Phi_x^(-x, y, z)\left[\left(-\nabla_x U\right) \nabla_y+\left(\nabla_y U\right) \nabla_x\right] \Phi_y(-x-a, y, z) . \end{aligned}

## Matlab代写

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