能带理论
提出原因：尽管量子自由电子理论已经说明了金属热容 等传统理论无法解释的现象，但对金属许多重要性质， 如金属、半金属、半导体和绝缘体间的本质区别，二价 镁比一价铜导电性差固体电阻差异大，等无能为力。 问题的关键：电子所处势场并非一个常数而是一个周期 性势场。 解决办法：写出存在相互作用的所有离子和电子系统的 薛定谔方程，并严格求解。实际根本办不到。 近似办法－能带理论 1、假设原子实固定不动，并按周期排列，从而分开电子 运动和晶格振动，多体问题多电子问题 2、假设电子间交互作用可用某种平均作用代替，则作用 到每个电子上的势场只与该电子的位置相关，而与其它 电子的位置和状态无关。多电子问题单电子问题
(+) =|(+)|2cos2 (x/a),
(-) =|(-)|2sin2 (x/a),
The upper function (+) piles up electrons (negative charge) on the positive ions centered at x=0, a, 2a, …., where the potential energy is lowest
出发点 固体中的电子可以在整个固体中运动 电子在运动过程中要受晶格原子势场的作用 两个基本假设：
Born－Oppenheimer绝热近似：所有原子核都周期性地静 止排列在其格点位置上，因而忽略了电子与声子的碰撞 Hatree－Fock平均场近似：忽略电子与电子间的相互作用， 用平均场代替电子与电子间的相互作用 能带论是单电子近似理论。用这种方法求出的电子能量状 态将不再是分立的能级，而是由能量的允带和禁带相间组 成的能带，故称为能带论。
Magnitude of the energy gap
The first-order energy difference between the two standing wave states is
The gap is equal to the Fourier component of the crystal potential.
Bloch Functions
Proof: we assume that k is nondegenerate and consider N identical lattice points on a ring of length Na. The potential energy is periodic in a, with U(x)=U(x+sa), where s is an integer.
Bloch Functions
Bloch theorem: The eigenfunctions of the wave equation for a periodic potential are the product of a plane wave exp(ik•r) times a function uk(r) with the periodicity of the crystal lattice.
(x)=uk(x) exp(i2 sx/Na)
satisfies (x+a) =C(x), provided that uk(x) has the periodicity a, so that uk(x) = uk(x+a). With k=2s/Na, we have the Bloch result.
Since the symmetry of the ring, we look for solutions of the wave equation such that
(x+a)=C(x).
Where C is a constant. Then, on going once around the ring,
(+)= exp(i x/a)+ exp(-i x/a) =2cos(x/a) (-)= exp(i x/a)- exp(-i x/a) =2isin(x/a) 均由向左和向右行波的相等部分组成
Nearly Free Electron Model
The probability density (+) and (-) for the standing wave (+) are respectively
The existence of a band gap is the most important and new property of crystals.
Nearly Free Electron Model
We must extend the free electron model to consider the periodic lattice of the solid.
Kronig-Penney Model
square-well array. The wave equation is
Kronig-Penney Model
The solution in the region a<x<a+b must be related to the above solution in the region b<x< 0 by the Bloch theorem:
n
a
Nearly Free Electron Model
The wavefunctions at k=±/a are not the traveling waves, such as exp(ix/a) or exp(-ix/a) of free electrons. The time-independent state is represented by standing waves:
The solutions of the Schrodinger equation for a periodic potential must be of a special form:
Bloch Functions
where uk(r) has the period of the crystal with uk(r)= uk(r+T) .
Magnitude of the energy gap
The potential energy of an electron in the crystal at point x is
U(x)=Ucos(2x/a)
The wavefunctions at the Brillouin zone boundary k= /a are 2cos (x/a) and 2sin(x/a), normalized over unit length of line. The first-order energy difference between the two standing wave states is
自由电子模型＋周期性点阵
Nearly Free Electron Model: the band electrons are treated as perturbed only weakly by the periodic potential of the ion cores. This model answers almost all the qualitative questions about the behavior of electrons in metals.
Energy Bands－Core ideas
Electrons in crystals are arranged in energy bands separated by regions in energy for which no wavelike electron orbitals exist; Band gaps result from the interaction of the conduction electron waves with the ion cores of the crystal.
基本出发点和假设
Energy Bands
Energy band theory tells us: electrons in crystals are arranged in energy bands.
It can illustrate the huge difference (32 orders) between a good conductor and a good insulator.
Chapter 7: Energy Bands
The free electron model does work in heat capacity, thermal conductivity, electrical conductivity, and other electrodynamics of metals. 问题的提出：fails to help us with other large questions: 1. fails to explain The distinction between metals, semiconductors, and insulators; 2. fails to explain Positive values of the Hall coefficient; 结果：New theory is proposed and developed to understand such questions －Energy Bands Theory
A simple problem: the origin of energy gaps in a linear solid of lattice constant a.
自由电子模型
