论文标题
在典型的一维正弦电位中的非相互作用电子
Noninteracting Electrons in a Prototypical One-Dimensional Sinusoidal Potential
论文作者
论文摘要
研究了一个含有非互动电子的一维金属元素固体的原型模型，其中余弦势能周期性的论点包含第一个相互晶格载体g1 = 2pi/a，其中a是晶格常数。时间无关的schrodinger方程可以用简化的变量写为MATHIEU方程，以获得数值的频带结构和波函数的溶液。条带结构的带隙随着余弦电位的振幅Q的增加而增加。在扩展区域方案中，能量差距随着brillouin-Zone边界KA = N PI的增加而减小，其中K是电子的晶体动量。发现频带底部和顶部的波函数分别是真实的或虚构的，与这些能量处的站立波相对应。无论在第一个布里渊区内的波矢量k不管，都发现电子概率密度与晶格周期性。波函数的傅立叶成分是衍生而成的Q，除非q = 0，否则揭示了多个在波函数中具有可变振幅的相互关系矢量成分。发现傅立叶组件的幅度可在n〜3到45的功率上呈n〜3至45的功率，以ka = pi/2和q = 2和q = 2和qusise and fit n〜3至45且for n〜3至45。还讨论了从波函数获得的概率密度和概率电流。发现在能带的顶部和底部的晶体动量的概率电流为零，因为这些晶体动量的波函数是站立波。最后，从中心方程计算带结构，并将其与数值精确的频带结构进行比较。
A prototypical model of a one-dimensional metallic monatomic solid containing noninteracting electrons is studied, where the argument of the cosine potential energy periodic with the lattice contains the first reciprocal lattice vector G1 = 2pi/a, where a is the lattice constant. The time-independent Schrodinger equation can be written in reduced variables as a Mathieu equation for which numerically-exact solutions for the band structure and wave functions are obtained. The band structure has band gaps that increase with increasing amplitude q of the cosine potential. In the extended-zone scheme, the energy gaps decrease with increasing index n of the Brillouin-zone boundary ka = n pi where k is the crystal momentum of the electron. The wave functions at the bottoms and tops of the bands are found to be real or imaginary, respectively, corresponding to standing waves at these energies. Irrespective of the wave vector k within the first Brillouin zone, the electron probability density is found to be periodic with the lattice. The Fourier components of the wave functions are derived versus q, which reveal multiple reciprocal-lattice-vector components with variable amplitudes in the wave functions unless q = 0. The magnitudes of the Fourier components are found to decrease exponentially as a power of n for n ~ 3 to 45 for ka = pi/2 and q = 2 and a precise fit is obtained to the data. The probability densities and probability currents obtained from the wave functions are also discussed. The probability currents are found to be zero for crystal momenta at the tops and bottoms of the energy bands, because the wave functions for these crystal momenta are standing waves. Finally, the band structure is calculated from the central equation and compared to the numerically-exact band structure.
