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能带结构

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在固体物理学里，固体的能带结构 (又称电子能带结构) 描述了"禁止"或"允许"电子带有的能量，这是周期性晶格中的量子动力学电子波衍射引起的。材料的能带结构决定了多种特性，特别是它的电子学和光学性质。

目录

[编辑] 为何有能带

单个自由原子的电子占据了原子轨道，形成一个分立的能级结构。如果几个原子集合成分子，他们的原子轨道发生类似于耦合振荡的分离。这会产生与原子数量成比例的分子轨道。当大量（数量级为1020或更多）的原子集合成固体时，轨道数量急剧增多，轨道相互间的能量的差别变的非常小。但是，无论多少原子聚集在一起，轨道的能量都不是连续的。

The electrons of a single free-standing atom occupy atomic orbitals, which form a discrete set of 能 levels. If several atoms are brought together into a molecule, their atomic orbitals split like in a coupled oscillation. This produces a number of 分子轨道s proportional to the number of atoms. When a large number of atoms (of order 1020 or more) are brought together to form a solid, the number of orbitals becomes exceedingly large, and the difference in energy between them becomes very small. However, some intervals of energy contain no orbitals, no matter how many atoms are aggregated.

这些能级如此之多甚至无法区分。首先，固体中能级的分离与电子和声原子振动持续的交换能相比拟。其次，由于相当长的时间间隔，它接近于由于海森伯格的测不准原理引起的能量的不确定度。

These energy levels are so numerous as to be indistinct. First, the separation between energy levels in a solid is comparable with the energy that electrons constantly exchange with phonons (atomic vibrations). Second, it is comparable with the energy uncertainty due to the Heisenberg uncertainty principle, for reasonably long intervals of time.

物理学中流行的方法是从不带电的电子和原子核出发，因为它们是自由的平面波，可以具有任意能量，并在带电后衰减。这导致了布拉格反射和带结构。

A view popular in physics is to start with uncharged electrons and cores, which are therefore both free and plane waves and can have any energy, and then fade in the charge. This leads to Bragg reflection and therefore bands.

[编辑] 基本概念

Simplified diagram of the electronic band structure of an insulator or semiconductor.
Simplified diagram of the electronic band structure of an insulator or semiconductor.

Any solid has a large number of bands. In theory, it can be said to have infinitely many bands (just as an atom has infinitely many energy levels). However, all but a few lie at energies so high that any electron that reaches those energies escapes from the solid. These bands are usually disregarded.

Bands have different widths, based upon the properties of the atomic orbitals from which they arise. Also, allowed bands may overlap, producing (for practical purposes) a single large band.

金属s contain a band that is partly empty and partly filled regardless of temperature. Therefore they have very high conductivity.

The uppermost occupied band in an 绝缘体 or 半导体 is called the 价带 by analogy to the valence electrons of individual atoms. The lowermost unoccupied band is called the conduction band because only when electrons are excited to the conduction band can current flow in these materials. The difference between insulators and semiconductors is only that the forbidden band gap between the valence band and conduction band is larger in an insulator, so that fewer electrons are found there and the electrical conductivity is less. Because one of the main mechanisms for electrons to be excited to the conduction band is due to thermal energy, the conductivity of semiconductors is strongly dependent on the temperature of the material.

This band gap is one of the most useful aspects of the band structure, as it strongly influences the electrical and optical properties of the material. Electrons can transfer from one band to the other by means of carrier generation and recombination processes. The band gap and defect states created in the band gap by 掺杂 can be used to create 半导体器件 such as 太阳能电池、二极管、晶体管、激光二极管以及其它。

Anderson's rule is used to create band diagrams between two semi-conductors.

[编辑] 不同固体的能带结构

Although electronic band structures are usually associated with 晶体line materials, quasi-crystalline and amorphous solids may also exhibit band structures. However, the periodic nature and symmetrical properties of crystalline materials makes it much easier to examine the band structures of these materials theoretically. In addition, the well-defined symmetry axes of crystalline materials makes it possible to determine the dispersion relationship between the momentum (a 3-dimension vector quantity) and energy of a material. As a result, virtually all of the existing theoretical work on the electronic band structure of solids has focused on crystalline materials.

[编辑] 态密度

While the 能态密度 in a band is very great, it is not uniform. It approaches zero at the band boundaries, and is generally greatest near the middle of a band. The density of states for the 自由电子模型 is given by,

D(\epsilon)= \frac{V}{2\pi^2}\left(\frac {2m}{\hbar^2}\right)^{3/2} \epsilon^{1/2}

[编辑] 能带填充

Although the number of states in all of the bands is effectively infinite, in an uncharged material the number of electrons is equal only to the number of protons in the atoms of the material. Therefore not all of the states are occupied by electrons ("filled") at any time. The likelihood of any particular state being filled at any temperature is given by the Fermi-Dirac statistics. The probability is given by the following:

f(E) = \frac{1}{1 + e^{\frac{E-E_F}{kT}}}

where:

The Fermi level naturally is the level at which the electrons and protons are balanced.

Regardless of the temperature, f(EF) = 1 / 2. At T=0, the distribution is a simple step function:

f(E) = \begin{cases} 1 & \mbox{if}\ 0 < E \le E_F \\ 0 & \mbox{if}\ E_F < E \end{cases}

At nonzero temperatures, the step "smooths out", so that an appreciable number of states below the Fermi level are empty, and some states above the Fermi level are filled.

[编辑] 晶体的能带结构

[编辑] 布里渊区

Because electron momentum is the reciprocal of space, the dispersion relation between the energy and momentum of electrons can best be described in reciprocal space. It turns out that for crystalline structures, the dispersion relation of the electrons is periodic, and that the 布里渊区 is the smallest repeating space within this periodic structure. For an infinitely large crystal, if the dispersion relation for an electron is defined throughout the Brillouin zone, then it is defined throughout the entire reciprocal space.

[编辑] 晶体能带结构理论

The ansatz is the special case of electron waves in a periodic crystal lattice using Bloch waves as treated generally in the dynamical theory of diffraction. Every crystal is a periodic structure which can be characterized by a Bravais lattice, and for each Bravais lattice we can determine the reciprocal lattice, which encapsulates the periodicity in a set of three reciprocal lattice vectors (\mathbf{b_1}, \mathbf{b_2}, \mathbf{b_3}). Now, any periodic potential V(\mathbf{r}) which shares the same periodicity as the direct lattice can be expanded out as a Fourier series whose only non-vanishing components are those associated with the reciprocal lattice vectors. So the expansion can be written as:

V(\mathbf{r}) = \sum_{\mathbf{K}}{V_{\mathbf{K}}e^{i \mathbf{K}\cdot\mathbf{r}}}

where \mathbf{K} = m_1 \mathbf{b}_1 + m_2 \mathbf{b}_2 + m_3 \mathbf{b}_3 for any set of integers (m1,m2,m3).

From this theory, an attempt can be made to predict the band structure of a particular material, however most ab initio methods for electronic structure calculations fail to predict the observed band gap.

[编辑] 近自由电子近似

The nearly-free electron approximation in solid state physics is similar in some respects to the Hydrogen-like atom of quantum mechanics in that interactions between electrons are completely ignored. This allows us to use Bloch's Theorem which states that electrons in a periodic potential have 波函数s and energies which are periodic in wavevector up to a constant phase shift between neighboring reciprocal lattice vectors. This can be described mathematically by:

\Psi(\mathbf{r}) = e^{i \mathbf{k}\cdot\mathbf{r}} u(\mathbf{r})

where the function u(\mathbf{r}) is periodic over the 晶格。

(详情见近自由电子模型)

[编辑] Mott绝缘体

Although the nearly-free electron approximation is able to describe many properties of electron band structures, one consequence of this theory is that it predicts the same number of electrons in each unit cell. If the number of electrons is odd, we would then expect that there is an unpaired electron in each unit cell, and thus that the valence band is not fully occupied, making the material a conductor. However, materials such as CoO that have an odd number of electrons per unit cell are insulators, in direct conflict with this result. This kind of material is known as a Mott insulator, and requires new theories, such as the Hubbard model, to explain the discrepancy.

[编辑] 其它

Calculating band structures is an important topic in theoretical solid state physics. In addition to the models mentioned above, other models include the following:

  • The 紧束缚模型, which assumes that each electron is usually associated with only one atom at a time, and treats the other atoms in the solid as perturbations.
  • The Kronig-Penney model, which depicts the atoms as barriers to electron motion, while the electrons are otherwise free and independent. While simple, it predicts many important phenomena, but is not quantitatively accurate.
  • Bands may also be viewed as the large-scale limit of 分子轨道理论. A solid creates a large number of closely spaced molecular orbitals, which appear as a band.
  • Methods involving 格林函数
  • Hubbard model
  • 密度函数理论

The band structure has been generalised to wavevectors that are 复数，resulting in what is called a 复杂能带结构, which is of interest at surfaces and interfaces.

Each model describes some types of solids very well, and others poorly. The nearly-free electron model works well for metals, but poorly for non-metals. The tight binding model is extremely accurate for ionic insulators, such as metal halide salts (e.g. 氯化钠NaCl).

[编辑] 参考资料

  1. Kotai no denshiron (The theory of electrons in solids), by Hiroyuki Shiba, ISBN 4-621-04135-5
  2. Microelectronics, by Jacob Millman and Arvin Gabriel, ISBN 0-07-463736-3, Tata McGraw-Hill Edition.
  3. Solid State Physics, by Neil Ashcroft and N. David Mermin, ISBN 0-03-083993-9,
  4. Introduction to Solid State Physics by Charles Kittel, ISBN 0-471-41526-X
  5. Electronic and Optoelectronic Properties of Semiconductor Structures - Chapter 2 and 3 by Jasprit Singh, ISBN 0-521-82379-X

[编辑] 参见

来自“http://www.wikilib.net/wiki/%E8%83%BD%E5%B8%A6%E7%BB%93%E6%9E%84”

4个分类: 正在翻譯的條目 | 凝聚态物理学 | 半导体物理学 | 来自中文维基百科