基于维格纳方程的电子的古斯-汉欣位移

doi:10.19596/j.cnki.1001-246x.8310

摘要/Abstract

摘要：

通过求解电子的维格纳方程研究二维电子气中电子的输运性质。我们发现电子在倾斜入射到势垒界面并反射时, 出现与光波类似的古斯-汉欣位移。通过维格纳方程可以得到电子的瞬态演化, 不仅可以计算古斯-汉欣位移还能研究电子在势垒内部的运动轨迹以及出现稳定古斯-汉欣位移的时间。与稳定相位法得到的古斯-汉欣位移对比发现, 考虑古斯-汉欣位移的界面反射较几何光学反射在时间上有一定迟缓, 这种迟缓与入射角无关, 但会随着势垒宽度的增加而增加; 电子的古斯-汉欣位移与势垒厚度无关, 随着入射角或入射能量的增大而增大。基于此, 我们提出一种电子分束器模型, 向输入端注入初始动能不同的高斯波包, 当电子能量低于0.01 eV时, 约85%的电子运动至第二输出端; 而当电子能量高于0.07 eV时, 约85%的电子运动至第一输出端。

关键词: 古斯-汉欣位移, 维格纳方程, 电子分束器, 平面纳米器件

Abstract:

We studied transport properties of electrons in two-dimensional electron gas with Wigner equation. It was found that as electrons enter the barrier interface obliquely and are reflected, Goos-Hanchen shift similar to light waves presents. With Wigner equation we get transient evolution of electrons, with which we can calculate Goos-Hanchen shift and study trajectory of electrons inside the potential barrier as well. Compared with Goos-Hanchen shift obtained with the stable phase method, it is found that the interface reflection considering Goos-Hanchen shift has a certain retardation in time compared with geometric optical reflection, which is independent of incident angle, but increases with the increase of width. However, Goos-Hanchen shift of electrons is independent of barrier width and increases with the incident angle or incident energy. According to this, we propose an electron beam splitter model in which Gaussian wave packets with different initial kinetic energies inject into the input. As electron energy is below 0. 01 eV, about 85% of the electrons move to the second output, while as electron energy is above 0. 07 eV, about 85% of the electrons move to the first output.

Key words: Goos-Hanchen shift, Wigner formalism, electron beam splitter, planar nanometer device

中图分类号:

O463

程受广, 尹云倩, 钟菊莲, 许坤远. 基于维格纳方程的电子的古斯-汉欣位移[J]. 计算物理, 2021, 38(4): 431-440.

Shouguang CHENG, Yunqian YIN, Julian ZHONG, Kunyuan XU. Goos-Hanchen Shift of Electrons Based on Wigner Equation[J]. Chinese Journal of Computational Physics, 2021, 38(4): 431-440.

图/表 12

图1 基于符号粒子的维格纳蒙特卡罗方法流程图

Fig.1 Flowchart of the Wigner Monte Carlo methodbased on signed particles

表1 模拟参数

Table 1 Simulation parameters

L_x/nm	L_y/nm	L_c/nm	Δk/nm^-1	Δl/fs	Δx/nm	Δy/nm	k_x0/nm^-1	σ/nm	势垒宽度a/nm	势垒高度/eV
150	150	100	π/L_c	1.0	1.0	1.5	0.25	10	10	0.05

图2 优化再分配方案对比(a), (b), (c)为原程序密度分布；(d), (e), (f)为优化后密度分布

Fig.2 Comparison of optimizing redistribution scheme (a), (b), (c) are density distributions by the original program; (d), (e), (f) represent density distributions after optimization

表2 方势垒演化的线程-加速比

Table 2 Thread-acceleration ratios of thesquare barrier evolution

线程数	计算时间/s	加速比
1	94 443	1
2	54 627	1.73
3	36 064	2.62
4	30 710	3.08
5	28 684	3.29
6	23 071	4.09
7	21 187	4.45
8	19 285	4.9

图3 光在界面处的全反射及古斯-汉欣位移

Fig.3 Total reflection of light at interface andGoos-Hanchen shift

图4 波包的中心轨迹与几何光学反射

Fig.4 Total reflection of light and Goos-Hanchen shift

图5 势垒宽度对古斯-汉欣位移的影响

Fig.5 Influence of barrier width on Goos-Hanchen shift

图6 入射角度对古斯-汉欣位移的影响

Fig.6 Influence of incident angle on Goos-Hanchen shift

图7 入射能量对古斯-汉欣位移的影响

Fig.7 Influence of incident energy on Goos-Hanchen shift

图8 器件结构

Fig.8 Device structure

图9 不同入射能量的波包在势垒界面的运动(a)初始时刻波包位置；(b)入射动能E=0.01 eV波包在t=270 fs时的位置；(c)入射动能E=0.028 eV波包在t=180 fs时的位置；(d)入射动能E=0.07 eV波包在t=140 fs时的位置

Fig.9 Movement of wave packets with different incident energy at the barrier interface (a) Initial position of the wave packet; (b) Position of a incident kinetic energy E=0.01 eV wave packet at t=270 fs; (c) Position of a incident kinetic energy E=0.028 eV wave packet at t=180 fs; (d) Position of a incident kinetic energy E=0.0 7 eV wave packet at t=140 fs

图10 不同入射动能的波包到达两个输出端的比率

Fig.10 The ratio of wave packets to two output terminalswith different incident kinetic energy

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