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guanjihuan 2022-04-21 22:03:21 +08:00
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academic_codes/2021.01.10_Berry_curvature_distribution_of_graphene_under_broken_inversion_symmery/Berry_curvature_distribution_of_graphene_under_broken_inversion_symmery.py
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@ -10,7 +10,7 @@ import cmath
import time import time
def hamiltonian(k1, k2, t1=2.82*sqrt(3)/2, a=1/sqrt(3)): # 石墨烯哈密顿量(a为原子间距，不赋值的话默认为1/sqrt(3)） def hamiltonian(k1, k2, t1=2.82*sqrt(3)/2, a=1/sqrt(3)): # 石墨烯哈密顿量（a为原子间距，不赋值的话默认为1/sqrt(3)）
h = np.zeros((2, 2))*(1+0j) h = np.zeros((2, 2))*(1+0j)
h[0, 0] = 0.28/2 h[0, 0] = 0.28/2
h[1, 1] = -0.28/2 h[1, 1] = -0.28/2

226
academic_codes/2021.01.10_Berry_curvature_distribution_of_graphene_under_broken_inversion_symmery/Berry_curvature_distribution_of_graphene_under_broken_inversion_symmery_with_ky=0.py → academic_codes/2021.01.10_Berry_curvature_distribution_of_graphene_under_broken_inversion_symmery/Berry_curvature_distribution_with_ky=0.py Executable file → Normal file
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@ -1,114 +1,114 @@
""" """
This code is supported by the website: https://www.guanjihuan.com 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/8536 The newest version of this code is on the web page: https://www.guanjihuan.com/archives/8536
""" """
import numpy as np import numpy as np
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
from math import * from math import *
import cmath import cmath
import time import time
def hamiltonian(k1, k2, t1=2.82*sqrt(3)/2, a=1/sqrt(3)): # 石墨烯哈密顿量(a为原子间距，不赋值的话默认为1/sqrt(3)） def hamiltonian(k1, k2, t1=2.82*sqrt(3)/2, a=1/sqrt(3)): # 石墨烯哈密顿量（a为原子间距，不赋值的话默认为1/sqrt(3)）
h = np.zeros((2, 2))*(1+0j) h = np.zeros((2, 2))*(1+0j)
h[0, 0] = 0.28/2 h[0, 0] = 0.28/2
h[1, 1] = -0.28/2 h[1, 1] = -0.28/2
h[1, 0] = t1*(cmath.exp(1j*k2*a)+cmath.exp(1j*sqrt(3)/2*k1*a-1j/2*k2*a)+cmath.exp(-1j*sqrt(3)/2*k1*a-1j/2*k2*a)) h[1, 0] = t1*(cmath.exp(1j*k2*a)+cmath.exp(1j*sqrt(3)/2*k1*a-1j/2*k2*a)+cmath.exp(-1j*sqrt(3)/2*k1*a-1j/2*k2*a))
h[0, 1] = h[1, 0].conj() h[0, 1] = h[1, 0].conj()
return h return h
def main(): def main():
start_time = time.time() start_time = time.time()
n = 2000 # 取点密度 n = 2000 # 取点密度
delta = 1e-9 # 求导的偏离量 delta = 1e-9 # 求导的偏离量
for band in range(2): for band in range(2):
F_all = [] # 贝里曲率 F_all = [] # 贝里曲率
for kx in np.linspace(-2*pi, 2*pi, n): for kx in np.linspace(-2*pi, 2*pi, n):
for ky in [0]: # 这里只考虑ky=0对称轴上的情况 # np.linspace(-pi, pi, n): for ky in [0]: # 这里只考虑ky=0对称轴上的情况 # np.linspace(-pi, pi, n):
H = hamiltonian(kx, ky) H = hamiltonian(kx, ky)
eigenvalue, eigenvector = np.linalg.eig(H) eigenvalue, eigenvector = np.linalg.eig(H)
vector = eigenvector[:, np.argsort(np.real(eigenvalue))[band]] # 价带波函数 vector = eigenvector[:, np.argsort(np.real(eigenvalue))[band]] # 价带波函数
# print(np.argsort(np.real(eigenvalue))[0]) # 排序索引（从小到大） # print(np.argsort(np.real(eigenvalue))[0]) # 排序索引（从小到大）
# print(eigenvalue) # 排序前的本征值 # print(eigenvalue) # 排序前的本征值
# print(np.sort(np.real(eigenvalue))) # 排序后的本征值（从小到大） # print(np.sort(np.real(eigenvalue))) # 排序后的本征值（从小到大）
H_delta_kx = hamiltonian(kx+delta, ky) H_delta_kx = hamiltonian(kx+delta, ky)
eigenvalue, eigenvector = np.linalg.eig(H_delta_kx) eigenvalue, eigenvector = np.linalg.eig(H_delta_kx)
vector_delta_kx = eigenvector[:, np.argsort(np.real(eigenvalue))[band]] # 略偏离kx的波函数 vector_delta_kx = eigenvector[:, np.argsort(np.real(eigenvalue))[band]] # 略偏离kx的波函数
# vector_delta_kx = find_vector_with_the_same_gauge(vector_delta_kx, vector) # 如果波函数不连续需要使用这个 # vector_delta_kx = find_vector_with_the_same_gauge(vector_delta_kx, vector) # 如果波函数不连续需要使用这个
H_delta_ky = hamiltonian(kx, ky+delta) H_delta_ky = hamiltonian(kx, ky+delta)
eigenvalue, eigenvector = np.linalg.eig(H_delta_ky) eigenvalue, eigenvector = np.linalg.eig(H_delta_ky)
vector_delta_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[band]] # 略偏离ky的波函数 vector_delta_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[band]] # 略偏离ky的波函数
# vector_delta_ky = find_vector_with_the_same_gauge(vector_delta_ky, vector) # 如果波函数不连续需要使用这个 # vector_delta_ky = find_vector_with_the_same_gauge(vector_delta_ky, vector) # 如果波函数不连续需要使用这个
H_delta_kx_ky = hamiltonian(kx+delta, ky+delta) H_delta_kx_ky = hamiltonian(kx+delta, ky+delta)
eigenvalue, eigenvector = np.linalg.eig(H_delta_kx_ky) eigenvalue, eigenvector = np.linalg.eig(H_delta_kx_ky)
vector_delta_kx_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[band]] # 略偏离kx和ky的波函数 vector_delta_kx_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[band]] # 略偏离kx和ky的波函数
# vector_delta_kx_ky = find_vector_with_the_same_gauge(vector_delta_kx_ky, vector) # 如果波函数不连续需要使用这个 # vector_delta_kx_ky = find_vector_with_the_same_gauge(vector_delta_kx_ky, vector) # 如果波函数不连续需要使用这个
# 价带的波函数的贝里联络(berry connection) # 求导后内积 # 价带的波函数的贝里联络(berry connection) # 求导后内积
A_x = np.dot(vector.transpose().conj(), (vector_delta_kx-vector)/delta) # 贝里联络Ax（x分量） A_x = np.dot(vector.transpose().conj(), (vector_delta_kx-vector)/delta) # 贝里联络Ax（x分量）
A_y = np.dot(vector.transpose().conj(), (vector_delta_ky-vector)/delta) # 贝里联络Ay（y分量） A_y = np.dot(vector.transpose().conj(), (vector_delta_ky-vector)/delta) # 贝里联络Ay（y分量）
A_x_delta_ky = np.dot(vector_delta_ky.transpose().conj(), (vector_delta_kx_ky-vector_delta_ky)/delta) # 略偏离ky的贝里联络Ax A_x_delta_ky = np.dot(vector_delta_ky.transpose().conj(), (vector_delta_kx_ky-vector_delta_ky)/delta) # 略偏离ky的贝里联络Ax
A_y_delta_kx = np.dot(vector_delta_kx.transpose().conj(), (vector_delta_kx_ky-vector_delta_kx)/delta) # 略偏离kx的贝里联络Ay A_y_delta_kx = np.dot(vector_delta_kx.transpose().conj(), (vector_delta_kx_ky-vector_delta_kx)/delta) # 略偏离kx的贝里联络Ay
# 贝里曲率(berry curvature) # 贝里曲率(berry curvature)
F = ((A_y_delta_kx-A_y)/delta-(A_x_delta_ky-A_x)/delta)*1j F = ((A_y_delta_kx-A_y)/delta-(A_x_delta_ky-A_x)/delta)*1j
# print(F) # print(F)
F_all = np.append(F_all,[F], axis=0) F_all = np.append(F_all,[F], axis=0)
plt.plot(np.linspace(-2*pi, 2*pi, n)/pi, np.real(F_all)) plt.plot(np.linspace(-2*pi, 2*pi, n)/pi, np.real(F_all))
plt.xlabel('k_x (pi)') plt.xlabel('k_x (pi)')
plt.ylabel('Berry curvature') plt.ylabel('Berry curvature')
if band==0: if band==0:
plt.title('Valence Band') plt.title('Valence Band')
else: else:
plt.title('Conductance Band') plt.title('Conductance Band')
plt.show() plt.show()
end_time = time.time() end_time = time.time()
print('运行时间(min)=', (end_time-start_time)/60) print('运行时间(min)=', (end_time-start_time)/60)
def find_vector_with_the_same_gauge(vector_1, vector_0): def find_vector_with_the_same_gauge(vector_1, vector_0):
# 寻找近似的同一的规范 # 寻找近似的同一的规范
phase_1_pre = 0 phase_1_pre = 0
phase_2_pre = pi phase_2_pre = pi
n_test = 10001 n_test = 10001
for i0 in range(n_test): for i0 in range(n_test):
test_1 = np.sum(np.abs(vector_1*cmath.exp(1j*phase_1_pre) - vector_0)) test_1 = np.sum(np.abs(vector_1*cmath.exp(1j*phase_1_pre) - vector_0))
test_2 = np.sum(np.abs(vector_1*cmath.exp(1j*phase_2_pre) - vector_0)) test_2 = np.sum(np.abs(vector_1*cmath.exp(1j*phase_2_pre) - vector_0))
if test_1 < 1e-9: if test_1 < 1e-9:
phase = phase_1_pre phase = phase_1_pre
# print('Done with i0=', i0) # print('Done with i0=', i0)
break break
if i0 == n_test-1: if i0 == n_test-1:
phase = phase_1_pre phase = phase_1_pre
print('Gauge Not Found with i0=', i0) print('Gauge Not Found with i0=', i0)
if test_1 < test_2: if test_1 < test_2:
if i0 == 0: if i0 == 0:
phase_1 = phase_1_pre-(phase_2_pre-phase_1_pre)/2 phase_1 = phase_1_pre-(phase_2_pre-phase_1_pre)/2
phase_2 = phase_1_pre+(phase_2_pre-phase_1_pre)/2 phase_2 = phase_1_pre+(phase_2_pre-phase_1_pre)/2
else: else:
phase_1 = phase_1_pre phase_1 = phase_1_pre
phase_2 = phase_1_pre+(phase_2_pre-phase_1_pre)/2 phase_2 = phase_1_pre+(phase_2_pre-phase_1_pre)/2
else: else:
if i0 == 0: if i0 == 0:
phase_1 = phase_2_pre-(phase_2_pre-phase_1_pre)/2 phase_1 = phase_2_pre-(phase_2_pre-phase_1_pre)/2
phase_2 = phase_2_pre+(phase_2_pre-phase_1_pre)/2 phase_2 = phase_2_pre+(phase_2_pre-phase_1_pre)/2
else: else:
phase_1 = phase_2_pre-(phase_2_pre-phase_1_pre)/2 phase_1 = phase_2_pre-(phase_2_pre-phase_1_pre)/2
phase_2 = phase_2_pre phase_2 = phase_2_pre
phase_1_pre = phase_1 phase_1_pre = phase_1
phase_2_pre = phase_2 phase_2_pre = phase_2
vector_1 = vector_1*cmath.exp(1j*phase) vector_1 = vector_1*cmath.exp(1j*phase)
# print('二分查找找到的规范=', phase) # print('二分查找找到的规范=', phase)
return vector_1 return vector_1
if __name__ == '__main__': if __name__ == '__main__':
main() main()

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academic_codes/2022.04_21_Berry_curvature/Berry_curvature_distribution_of_graphene_under_broken_inversion_symmery_with_Kubo_formula.py Normal file
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@ -0,0 +1,55 @@
"""
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/20869
"""
import numpy as np
import matplotlib.pyplot as plt
from math import *
import cmath
def hamiltonian(k1, k2, t1=2.82*sqrt(3)/2, a=1/sqrt(3)): # 石墨烯哈密顿量（a为原子间距，不赋值的话默认为1/sqrt(3)）
h = np.zeros((2, 2))*(1+0j)
h[0, 0] = 0.28/2
h[1, 1] = -0.28/2
h[1, 0] = t1*(cmath.exp(1j*k2*a)+cmath.exp(1j*sqrt(3)/2*k1*a-1j/2*k2*a)+cmath.exp(-1j*sqrt(3)/2*k1*a-1j/2*k2*a))
h[0, 1] = h[1, 0].conj()
return h
def main():
n = 2000 # 取点密度
delta = 1e-9 # 求导的偏离量
for band in range(2):
F_all = [] # 贝里曲率
for kx in np.linspace(-2*pi, 2*pi, n):
for ky in [0]: # 这里只考虑ky=0对称轴上的情况
H = hamiltonian(kx, ky)
eigenvalue, eigenvector = np.linalg.eig(H)
if band==0:
vector_0 = eigenvector[:, np.argsort(np.real(eigenvalue))[0]]
vector_1 = eigenvector[:, np.argsort(np.real(eigenvalue))[1]]
elif band==1:
vector_0 = eigenvector[:, np.argsort(np.real(eigenvalue))[1]]
vector_1 = eigenvector[:, np.argsort(np.real(eigenvalue))[0]]
eigenvalue = np.sort(np.real(eigenvalue))
H_delta_kx = hamiltonian(kx+delta, ky)-hamiltonian(kx, ky)
H_delta_ky = hamiltonian(kx, ky+delta)-hamiltonian(kx, ky)
berry_curvature = 1j*(np.dot(np.dot(np.dot(np.dot(np.dot(vector_0.transpose().conj(), H_delta_kx/delta), vector_1), vector_1.transpose().conj()), H_delta_ky/delta), vector_0)- np.dot(np.dot(np.dot(np.dot(np.dot(vector_0.transpose().conj(), H_delta_ky/delta), vector_1), vector_1.transpose().conj()), H_delta_kx/delta), vector_0))/(eigenvalue[0]-eigenvalue[1])**2
F_all = np.append(F_all,[berry_curvature], axis=0)
plt.plot(np.linspace(-2*pi, 2*pi, n)/pi, np.real(F_all))
plt.xlabel('k_x (pi)')
plt.ylabel('Berry curvature')
if band==0:
plt.title('Valence Band')
else:
plt.title('Conductance Band')
plt.show()
if __name__ == '__main__':
main()

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academic_codes/2022.04_21_Berry_curvature/Berry_curvature_distribution_of_graphene_under_broken_inversion_symmery_with_the_efficient_method.py Normal file
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@ -0,0 +1,63 @@
"""
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/20869
"""
import numpy as np
import matplotlib.pyplot as plt
from math import *
import cmath
def hamiltonian(k1, k2, t1=2.82*sqrt(3)/2, a=1/sqrt(3)): # 石墨烯哈密顿量（a为原子间距，不赋值的话默认为1/sqrt(3)）
h = np.zeros((2, 2))*(1+0j)
h[0, 0] = 0.28/2
h[1, 1] = -0.28/2
h[1, 0] = t1*(cmath.exp(1j*k2*a)+cmath.exp(1j*sqrt(3)/2*k1*a-1j/2*k2*a)+cmath.exp(-1j*sqrt(3)/2*k1*a-1j/2*k2*a))
h[0, 1] = h[1, 0].conj()
return h
def main():
n = 2000 # 取点密度
delta = 4*pi/n # 求导的偏离量
for band in range(2):
F_all = [] # 贝里曲率
for kx in np.linspace(-2*pi, 2*pi, n):
for ky in [0]: # 这里只考虑ky=0对称轴上的情况
H = hamiltonian(kx, ky)
eigenvalue, eigenvector = np.linalg.eig(H)
vector = eigenvector[:, np.argsort(np.real(eigenvalue))[band]] # 价带波函数
H_delta_kx = hamiltonian(kx+delta, ky)
eigenvalue, eigenvector = np.linalg.eig(H_delta_kx)
vector_delta_kx = eigenvector[:, np.argsort(np.real(eigenvalue))[band]] # 略偏离kx的波函数
H_delta_ky = hamiltonian(kx, ky+delta)
eigenvalue, eigenvector = np.linalg.eig(H_delta_ky)
vector_delta_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[band]] # 略偏离ky的波函数
H_delta_kx_ky = hamiltonian(kx+delta, ky+delta)
eigenvalue, eigenvector = np.linalg.eig(H_delta_kx_ky)
vector_delta_kx_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[band]] # 略偏离kx和ky的波函数
Ux = np.dot(np.conj(vector), vector_delta_kx)/abs(np.dot(np.conj(vector), vector_delta_kx))
Uy = np.dot(np.conj(vector), vector_delta_ky)/abs(np.dot(np.conj(vector), vector_delta_ky))
Ux_y = np.dot(np.conj(vector_delta_ky), vector_delta_kx_ky)/abs(np.dot(np.conj(vector_delta_ky), vector_delta_kx_ky))
Uy_x = np.dot(np.conj(vector_delta_kx), vector_delta_kx_ky)/abs(np.dot(np.conj(vector_delta_kx), vector_delta_kx_ky))
F = cmath.log(Ux*Uy_x*(1/Ux_y)*(1/Uy))/delta/delta*1j
F_all = np.append(F_all,[F], axis=0)
plt.plot(np.linspace(-2*pi, 2*pi, n)/pi, np.real(F_all))
plt.xlabel('k_x (pi)')
plt.ylabel('Berry curvature')
if band==0:
plt.title('Valence Band')
else:
plt.title('Conductance Band')
plt.show()
if __name__ == '__main__':
main()