update

2020.03.25_Hamiltonian_and_bands_of_bilayer_graphene/bands_of_bilayer_graphene_with_label_sequence_1.py

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+"""
+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/4322
+"""
+
+import numpy as np
+import matplotlib.pyplot as plt
+from math import * 
+import cmath 
+import functools
+
+
+def hamiltonian(k, N):  
+    # 初始化为零矩阵
+    h = np.zeros((4*N, 4*N), dtype=complex)
+
+    t=1
+    a=1
+    t0=0.2   # 层间跃迁
+    V=0.2    # 层间的势能差为2V
+
+    for i in range(N):
+        h[i*4+0, i*4+0] = V
+        h[i*4+1, i*4+1] = V
+        h[i*4+2, i*4+2] = -V
+        h[i*4+3, i*4+3] = -V
+
+        h[i*4+0, i*4+1] = -t*(1+cmath.exp(1j*k*a))
+        h[i*4+1, i*4+0] = -t*(1+cmath.exp(-1j*k*a))
+        h[i*4+2, i*4+3] = -t*(1+cmath.exp(1j*k*a))
+        h[i*4+3, i*4+2] = -t*(1+cmath.exp(-1j*k*a))
+
+        h[i*4+0, i*4+3] = -t0
+        h[i*4+3, i*4+0] = -t0
+
+    for i in range(N-1):
+        # 最近邻
+        h[i*4+1, (i+1)*4+0] = -t
+        h[(i+1)*4+0, i*4+1] = -t
+
+        h[i*4+3, (i+1)*4+2] = -t
+        h[(i+1)*4+2, i*4+3] = -t
+    return h
+
+
+def main():
+    hamiltonian0 = functools.partial(hamiltonian, N=100)
+    k = np.linspace(-pi, pi, 400)
+    plot_bands_one_dimension(k, hamiltonian0)
+
+
+def plot_bands_one_dimension(k, hamiltonian):
+    dim = hamiltonian(0).shape[0]
+    dim_k = k.shape[0]
+    eigenvalue_k = np.zeros((dim_k, dim))
+    i0 = 0
+    for k0 in k:
+        matrix0 = hamiltonian(k0)
+        eigenvalue, eigenvector = np.linalg.eig(matrix0)
+        eigenvalue_k[i0, :] = np.sort(np.real(eigenvalue[:]))
+        i0 += 1
+        print(k0)
+    for dim0 in range(dim):
+        plt.plot(k, eigenvalue_k[:, dim0], '-k')
+    plt.show()
+
+
+if __name__ == '__main__':
+    main()

2020.03.25_Hamiltonian_and_bands_of_bilayer_graphene/bands_of_bilayer_graphene_with_label_sequence_2.py

@@ -0,0 +1,83 @@
+"""
+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/4322
+"""
+
+import numpy as np
+import matplotlib.pyplot as plt
+from math import * 
+import cmath 
+import functools 
+
+
+def hamiltonian(k, N):  
+    # 初始化为零矩阵
+    h = np.zeros((4*N, 4*N), dtype=complex)
+    h11 = np.zeros((4*N, 4*N), dtype=complex)  # 元胞内
+    h12 = np.zeros((4*N, 4*N), dtype=complex)  # 元胞间
+
+    t=1
+    a=1
+    t0=0.2   # 层间跃迁
+    V=0.2    # 层间的势能差为2V
+
+    for i in range(N):
+        h11[i*2+0, i*2+0] = V
+        h11[i*2+1, i*2+1] = V
+
+
+        h11[N*2+i*2+0, N*2+i*2+0] = -V
+        h11[N*2+i*2+1, N*2+i*2+1] = -V
+
+
+        h11[i*2+0, i*2+1] = -t 
+        h11[i*2+1, i*2+0] = -t
+
+
+        h11[N*2+i*2+0, N*2+i*2+1] = -t
+        h11[N*2+i*2+1, N*2+i*2+0] = -t
+
+        h11[i*2+0, N*2+i*2+1] = -t0
+        h11[N*2+i*2+1, i*2+0] = -t0
+
+
+    for i in range(N-1):
+        h11[i*2+1, (i+1)*2+0] = -t 
+        h11[(i+1)*2+0, i*2+1] = -t
+
+        h11[N*2+i*2+1, N*2+(i+1)*2+0] = -t
+        h11[N*2+(i+1)*2+0, N*2+i*2+1] = -t
+
+
+    for i in range(N):
+        h12[i*2+0, i*2+1] = -t
+        h12[N*2+i*2+0, N*2+i*2+1] = -t
+
+    h= h11 + h12*cmath.exp(-1j*k*a) + h12.transpose().conj()*cmath.exp(1j*k*a)    
+    return h
+
+
+def main():
+    hamiltonian0 = functools.partial(hamiltonian, N=100) 
+    k = np.linspace(-pi, pi, 400)
+    plot_bands_one_dimension(k, hamiltonian0)
+
+
+def plot_bands_one_dimension(k, hamiltonian):
+    dim = hamiltonian(0).shape[0]
+    dim_k = k.shape[0]
+    eigenvalue_k = np.zeros((dim_k, dim))
+    i0 = 0
+    for k0 in k:
+        matrix0 = hamiltonian(k0)
+        eigenvalue, eigenvector = np.linalg.eig(matrix0)
+        eigenvalue_k[i0, :] = np.sort(np.real(eigenvalue[:]))
+        i0 += 1
+        print(k0)
+    for dim0 in range(dim):
+        plt.plot(k, eigenvalue_k[:, dim0], '-k') 
+    plt.show()
+
+
+if __name__ == '__main__':
+    main()