Electronic States in Perturbed Periodic Systems
Electronic States in Perturbed Periodic Systems
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扰动周期系统中的电子态
DOI：
10.1103/physrev.76.1611
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发表时间：
1949
期刊：
影响因子：
--
通讯作者：
H. James
中科院分区：
文献类型：
--
作者：
H. James
从定性和定量的角度讨论了具有离散能量的局域态在扰动周期系统（例如不纯晶体）中的出现。半经典的考虑，以通常的方式通过波概念进行修改，清楚地表明杂质将根据杂质的离子电荷小于或大于它所取代的离子的离子电荷而产生高于或低于相应允许能带的杂质态。晶体界面和自由表面的局域态可以用同样的方式讨论。对波包行为的考虑导致了 Peckar 有效质量波动方程的公式化。然后，通过将对于未扰动势的单个周期有效的解连接在一起，构造扰动周期波动方程的完整解。当扰动缓慢变化时（尽管其总效应不一定很小），这种方法会导致问题的解析解，该问题涉及单个细胞上扰动势的变化与粒子总动能之比的量级误差。有效质量方程与该解的近似形式一起出现，但其解 rp(x) 与正确波函数 p(x) 的关系比以前认识到的更为复杂。要在任何小区域中构造 p(x)，应将 q(x) 局部解析为两个指数 C expI +(i/Ii) pgxI 之和，将每个指数乘以适当的周期函数，然后将结果相加。二次可积 q(x) 对应于具有相同能量的二次可积 P(x)；因此，通过求解有效质量波动方程确定的稳态能量被发现非常可靠。
The occurrence in perturbed periodic systems, such as impure crystals, of localized states with discrete energies is discussed from both qualitative and quantitative points of view. Semiclassical considerations, modi6ed in the usual way by wave concepts, make it clear that impurities will give rise to impurity states above or below corresponding permitted bands of energy, according as the ionic charge of the impurity is less than or greater than that of the ion it replaces. Localized states at crystal interfaces and free surfaces can be discussed in the same way. Consideration of the behavior of wave packets leads to formulation of the effectivemass wave equation of Peckar. Complete solutions of the perturbed-periodic wave equation are then constructed by joining together solutions valid for a single period of the unperturbed potential. XVhen the perturbation is slowly varying (though not necessarily small in its total efkct) this approach leads to an analytic solution of the problem involving errors of the order of the ratio of the change in the perturbation potential across a single cell to the total kinetic energy of the particle. The effectivemass equation appears in connection with an approximate form of this solution, but the relation of its solution rp{x) to the correct wave function p(x) is more complex than has previously been realized. To construct p(x) in any small region one should resolve q(x) locally into the sum of two exponentials C expI +(i/Ii) pgxI, multiply each by the appropriate periodic function, and add the results. A quadradically integrable q(x) corresponds to a quadratically integrable P(x) with the same energy; thus stationarystate energies determined by solving the effective-mass wave equation are found to be surprisingly reliable.