模型和能带, 学术

二维离散ky能带的叠加和宽度为Ny的条带能带的关系(附Python代码)

在物理上,宽度为Ny的条带能带应该要接近于二维离散ky能带的叠加。本篇想通过代码例子,看它们之间是完全相等,还是仅仅在某种程度上的接近。

首先给出自己的猜想(大概率是对的):

  • 在周期性边界条件下,两者是完全等价的。只是要选对离散的ky值,但找到对应的离散ky值是个难点。
  • 在开放边界条件下,两者不完全等价。当Ny较大时,也会接近二维离散ky能带的叠加,但需要额外加上边缘态。

需要说明的是:

  • 本篇没有给出正确的离散ky值,即:黑色实线(二维的多个ky截面能带的叠加)和黄色点线(周期边界条件下的条带能带)没有完全重合,因此无法直接在数值上进行验证。
  • 如果找到正确的离散ky值,黑色实线应该会和黄色点线完全重合。
  • 大概思路是:要找到Ny值和ky数组之间的关系,但还不清楚在理论上是否可行,即Ny值是否已经包含了ky数组的全部信息。如果有知道怎么找到正确的离散ky值,可留言或联系我讨论。

之前已经有给出一些能带图:

本篇使用Guan软件包(http://py.guanjihuan.com,版本>=0.0.109)来写代码,调用一些常用的函数,而不是重新写或者来回复制了,这样会简洁一些。函数的具体内容可通过Guan软件包的源代码来查看。

一、方格子

Python代码:

"""
This code is supported by the website: https://www.guanjihuan.com
The newest version of this code is on the web page: https://www.guanjihuan.com/archives/22691
"""


import guan
import numpy as np
import functools


# 2D square lattice
k1_array = np.linspace(-np.pi, np.pi, 100)
k2_array = np.linspace(-np.pi, np.pi, 100)
eigenvalue_array = guan.calculate_eigenvalue_with_two_parameters(k1_array, k2_array, guan.hamiltonian_of_square_lattice)
guan.plot_3d_surface(k1_array, k2_array, eigenvalue_array, xlabel='kx', ylabel='ky', zlabel='E')


# 2D square lattice for discrete ky array
Ny = 10
ky_array = np.linspace(-np.pi, np.pi, Ny) # important
print(ky_array)

kx_array = np.linspace(-np.pi, np.pi, 100)
i0 = 0
for ky in ky_array:
    hamiltonian_function = functools.partial(guan.hamiltonian_of_square_lattice, k2=ky)
    eigenvalue_array = guan.calculate_eigenvalue_with_one_parameter(kx_array, hamiltonian_function)
    if i0 == 0:
        eigenvalue_array_for_discrete_ky = eigenvalue_array
    else:
        eigenvalue_array_for_discrete_ky = np.append(eigenvalue_array_for_discrete_ky, eigenvalue_array, axis=1)
    i0 += 1


# 1D square ribbon
hamiltonian_function = functools.partial(guan.hamiltonian_of_square_lattice_in_quasi_one_dimension, N=Ny)
eigenvalue_array_2 = guan.calculate_eigenvalue_with_one_parameter(kx_array, hamiltonian_function)


# 1D square ribbon with periodic boundary condition in y direction
hamiltonian_function = functools.partial(guan.hamiltonian_of_square_lattice_in_quasi_one_dimension, N=Ny, period=1)
eigenvalue_array_3 = guan.calculate_eigenvalue_with_one_parameter(kx_array, hamiltonian_function)


# Plot figures
guan.plot_three_array(kx_array, eigenvalue_array_for_discrete_ky, eigenvalue_array_2, eigenvalue_array_3, xlabel='kx', ylabel='E', style_1='-k', style_2='--r', style_3='.y', linewidth_1=3, markersize_2=1, markersize_3=1)

运行结果:

二、石墨烯

Python代码:

"""
This code is supported by the website: https://www.guanjihuan.com
The newest version of this code is on the web page: https://www.guanjihuan.com/archives/22691
"""


import guan
import numpy as np
import functools


# 2D graphene lattice
k1_array = np.linspace(-2*np.pi, 2*np.pi, 300)
k2_array = np.linspace(-2*np.pi, 2*np.pi, 300)
eigenvalue_array = guan.calculate_eigenvalue_with_two_parameters(k1_array, k2_array, guan.hamiltonian_of_graphene)
guan.plot_3d_surface(k1_array, k2_array, eigenvalue_array, xlabel='kx', ylabel='ky', zlabel='E', rcount=300, ccount=300)


# 2D graphene lattice for discrete ky array
Ny = 5
ky_array = np.linspace(-np.pi, np.pi, Ny*4) # important
print(ky_array)

kx_array = np.linspace(-np.pi, np.pi, 100)
i0 = 0
for ky in ky_array:
    hamiltonian_function = functools.partial(guan.hamiltonian_of_graphene, k2=ky)
    eigenvalue_array = guan.calculate_eigenvalue_with_one_parameter(kx_array, hamiltonian_function)
    if i0 == 0:
        eigenvalue_array_for_discrete_ky = eigenvalue_array
    else:
        eigenvalue_array_for_discrete_ky = np.append(eigenvalue_array_for_discrete_ky, eigenvalue_array, axis=1)
    i0 += 1


# 1D graphene ribbon
hamiltonian_function = functools.partial(guan.hamiltonian_of_graphene_with_zigzag_in_quasi_one_dimension, N=Ny)
eigenvalue_array_2 = guan.calculate_eigenvalue_with_one_parameter(kx_array, hamiltonian_function)


# 1D graphene ribbon with periodic boundary condition in y direction
hamiltonian_function = functools.partial(guan.hamiltonian_of_graphene_with_zigzag_in_quasi_one_dimension, N=Ny, period=1)
eigenvalue_array_3 = guan.calculate_eigenvalue_with_one_parameter(kx_array, hamiltonian_function)


# Plot figures
guan.plot_three_array(kx_array, eigenvalue_array_for_discrete_ky, eigenvalue_array_2, eigenvalue_array_3, xlabel='kx', ylabel='E', style_1='-k', style_2='--r', style_3='.y', linewidth_1=3, markersize_2=3, markersize_3=3)

运行结果:

三、Haldane模型

Python代码:

"""
This code is supported by the website: https://www.guanjihuan.com
The newest version of this code is on the web page: https://www.guanjihuan.com/archives/22691
"""


import guan
import numpy as np
import functools


# 2D Haldane lattice
k1_array = np.linspace(-2*np.pi, 2*np.pi, 100)
k2_array = np.linspace(-2*np.pi, 2*np.pi, 100)
eigenvalue_array = guan.calculate_eigenvalue_with_two_parameters(k1_array, k2_array, guan.hamiltonian_of_haldane_model)
guan.plot_3d_surface(k1_array, k2_array, eigenvalue_array, xlabel='kx', ylabel='ky', zlabel='E')


# 2D Haldane lattice for discrete ky array
Ny = 5
ky_array = np.linspace(-np.pi, np.pi, Ny*4) # important
print(ky_array)

kx_array = np.linspace(-np.pi, np.pi, 100)
i0 = 0
for ky in ky_array:
    hamiltonian_function = functools.partial(guan.hamiltonian_of_haldane_model, k2=ky)
    eigenvalue_array = guan.calculate_eigenvalue_with_one_parameter(-kx_array, hamiltonian_function)
    if i0 == 0:
        eigenvalue_array_for_discrete_ky = eigenvalue_array
    else:
        eigenvalue_array_for_discrete_ky = np.append(eigenvalue_array_for_discrete_ky, eigenvalue_array, axis=1)
    i0 += 1


# 1D Haldane ribbon
hamiltonian_function = functools.partial(guan.hamiltonian_of_haldane_model_in_quasi_one_dimension, N=Ny)
eigenvalue_array_2 = guan.calculate_eigenvalue_with_one_parameter(kx_array, hamiltonian_function)


# 1D Haldane ribbon with periodic boundary condition in y direction
hamiltonian_function = functools.partial(guan.hamiltonian_of_haldane_model_in_quasi_one_dimension, N=Ny, period=1)
eigenvalue_array_3 = guan.calculate_eigenvalue_with_one_parameter(kx_array, hamiltonian_function)


# Plot figures
guan.plot_three_array(kx_array, eigenvalue_array_for_discrete_ky, eigenvalue_array_2, eigenvalue_array_3, xlabel='kx', ylabel='E', style_1='-k', style_2='--r', style_3='.y', linewidth_1=3, markersize_2=3, markersize_3=3)

运行结果:

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