def test_atomic_chain_two_kinds_of_atoms():
""" """
site_energy1 = -1.0
site_energy2 = -2.0
coupling = -1.0
l_const = 2.0
a = tb.Orbitals('A')
a.add_orbital(
title='s',
energy=site_energy1,
)
b = tb.Orbitals('B')
b.add_orbital(
title='s',
energy=site_energy2,
)
xyz_file = """2
H cell
A 0.0000000000 0.0000000000 0.0000000000
B 0.0000000000 0.0000000000 1.0000000000
"""
tb.set_tb_params(PARAMS_A_B={'ss_sigma': coupling})
h = tb.HamiltonianSp(xyz=xyz_file, nn_distance=1.1)
h.initialize()
PRIMITIVE_CELL = [[0, 0, l_const]]
h.set_periodic_bc(PRIMITIVE_CELL)
num_points = 10
kk = np.linspace(0, 3.14 / 2, num_points, endpoint=True)
band_structure = []
for jj in range(num_points):
vals, _ = h.diagonalize_periodic_bc([0.0, 0.0, kk[jj]])
band_structure.append(vals)
band_structure = np.array(band_structure)
desired_value = np.zeros(band_structure.shape)
b = site_energy1 + site_energy2
c = site_energy1 * site_energy2 - (2.0 * coupling *
np.cos(0.5 * kk * l_const))**2
desired_value[:, 0] = 0.5 * (b - np.sqrt(b**2 - 4.0 * c))
desired_value[:, 1] = 0.5 * (b + np.sqrt(b**2 - 4.0 * c))
np.testing.assert_allclose(band_structure, desired_value, atol=1e-9)
def graphene_nanotube():
from ase.build.tube import nanotube
from ase.visualize.plot import plot_atoms
n = 10
m = 10
atoms = nanotube(n, m)
atoms.wrap()
period = np.array([list(atoms.get_cell()[2])])
period[:, [1, 2]] = period[:, [2, 1]]
coord = atoms.get_positions()
coord[:, [1, 2]] = coord[:, [2, 1]]
coords = []
coords.append(str(len(coord)))
coords.append('Nanoribbon')
for j, item in enumerate(coord):
coords.append('C' + str(j + 1) + ' ' + str(item[0]) + ' ' + str(item[1]) + ' ' + str(item[2]))
coords = '\n'.join(coords)
# --------------------------- Basis set --------------------------
s_orb = tb.Orbitals('C')
s_orb.add_orbital("pz", energy=0, orbital=1, magnetic=0, spin=0)
# s_orb.add_orbital("py", energy=0, orbital=1, magnetic=1, spin=0)
# s_orb.add_orbital("px", energy=0, orbital=1, magnetic=-1, spin=0)
# ------------------------ set TB parameters----------------------
t = 2.8
tb.set_tb_params(PARAMS_C_C={'pp_pi': t})
# --------------------------- Hamiltonian -------------------------
h = tb.Hamiltonian(xyz=coords, nn_distance=1.7, comp_angular_dep=False)
h.initialize()
h.set_periodic_bc(period)
k_points = np.linspace(0.0, np.pi/period[0][1], 20)
band_structure = np.zeros((len(k_points), h.h_matrix.shape[0]))
for jj, item in enumerate(k_points):
band_structure[jj, :], _ = h.diagonalize_periodic_bc([0.0, item, 0.0])
# visualize
ax = plt.axes()
ax.set_title('Band structure of carbon nanotube, ({0}, {1}) \n 1st nearest neighbour approximation'.format(n, m))
ax.set_ylabel('Energy (eV)')
ax.set_xlabel(r'Wave vector ($\frac{\pi}{a}$)')
ax.plot(k_points, np.sort(band_structure), 'k')
ax.xaxis.grid()
plt.show()
ax1 = plot_atoms(atoms, show_unit_cell=2, rotation='10x,50y,30z')
ax1.axis('off')
plt.show()
def graphene_third_nearest_neighbour_with_overlaps():
"""
All parameters are taken from Reich et al, Phys. Rev. B 66, 035412 (2002)
Returns
-------
"""
coords = """2
Graphene
C1 0.00 0.00 0.00
C2 {} 0.00 0.00
""".format(lat_const)
# --------------------------- Basis set --------------------------
s_orb = tb.Orbitals('C')
s_orb.add_orbital("pz", energy=-0.28, orbital=1, magnetic=0, spin=0)
# ------------------------ set TB parameters----------------------
gamma0 = -2.97
gamma1 = -0.073
gamma2 = -0.33
s0 = 0.073
s1 = 0.018
s2 = 0.026
tb.set_tb_params(PARAMS_C_C1={'pp_pi': gamma0},
PARAMS_C_C2={'pp_pi': gamma1},
PARAMS_C_C3={'pp_pi': gamma2},
OV_C_C1={'pp_pi': s0},
OV_C_C2={'pp_pi': s1},
OV_C_C3={'pp_pi': s2})
# --------------------------- Hamiltonian -------------------------
h = tb.Hamiltonian(xyz=coords, nn_distance=3.1, comp_overlap=True)
h.initialize(radial_dep)
h.set_periodic_bc(period)
band_structure = np.zeros((sum(num_points), h.h_matrix.shape[0]))
for jj, item in enumerate(k_points):
band_structure[jj, :], _ = h.diagonalize_periodic_bc(item)
# visualize
global fig_counter
plt.figure(fig_counter)
fig_counter += 1
ax = plt.axes()
ax.set_title('Band structure of graphene, 3d NN \n after Reich et al, Phys. Rev. B 66, 035412 (2002)')
ax.set_ylabel('Energy (eV)')
ax.plot(np.sort(band_structure), 'k')
ax.plot([0, band_structure.shape[0]], [0, 0], '--', color='k', linewidth=0.5)
plt.xticks(np.insert(np.cumsum(num_points) - 1, 0, 0), labels=sym_points)
ax.xaxis.grid()
plt.show()
def single_atom_chain():
"""Test set for a single-atom chain.
:return:
Parameters
----------
Returns
-------
"""
sys.path.insert(0, '/home/mk/TB_project/tb')
a = tb.Orbitals('A')
a.add_orbital('s', 0.7)
tb.set_tb_params(PARAMS_A_A={'ss_sigma': 0.5})
xyz_file = """1
H cell
A1 0.0000000000 0.0000000000 0.0000000000
"""
h = tb.Hamiltonian(xyz=xyz_file, nn_distance=1.1)
h.initialize()
h.set_periodic_bc([[0, 0, 1.0]])
h_l, h_0, h_r = h.get_hamiltonians()
num_sites = h_0.shape[0]
energy = np.linspace(-3.0, 3.0, 300)
tr = np.zeros((energy.shape[0]))
dos = np.zeros((energy.shape[0]))
sgf_l = []
sgf_r = []
for j, E in enumerate(energy):
se_l, se_r = negf.surface_greens_function(E,
h_l,
h_0,
h_r,
iterate=False)
sgf_l.append(se_l)
sgf_r.append(se_r)
gf = np.linalg.pinv(E * np.identity(num_sites) - h_0 - se_l - se_r)
gamma_l = 1j * (se_l - se_l.conj().T)
gamma_r = 1j * (se_r - se_r.conj().T)
tr[j] = np.real(np.trace(
gamma_l.dot(gf).dot(gamma_r).dot(gf.conj().T)))
dos[j] = np.real(np.trace(1j * (gf - gf.conj().T)))
sgf_l = np.array(sgf_l)
sgf_r = np.array(sgf_r)
return energy, dos, tr, h, sgf_l, sgf_r
def main(energy):
# define the basis set - one s-type orbital
orb = tb.Orbitals('A')
orb.add_orbital('s', energy=-1.0)
# set TB parameters
tb.set_tb_params(PARAMS_A_A={"ss_sigma": 1.0})
# define atomic coordinates for the unit cell
input_file = """4
Nanostrip
A1 0.0 0.0 0.0
A2 0.0 1.0 0.0
A3 0.0 2.0 0.0
A4 0.0 3.0 0.0
"""
# compute Hamiltonian matrices
h = tb.Hamiltonian(xyz=input_file, nn_distance=1.4)
h.initialize()
period = [0, 0, 1.0]
h.set_periodic_bc([period])
h_l, h_0, h_r = h.get_hamiltonians()
# compute DOS and transmission using Green's functions
dos = np.zeros((energy.shape[0]))
tr = np.zeros((energy.shape[0]))
for j, E in enumerate(energy):
# compute surface Green's functions
L, R = surface_greens_function(E, h_l, h_0, h_r)
# recursive Green's functions
g_trans, grd, grl, gru, gr_left = recursive_gf(E, [h_l], [h_0 + L + R],
[h_r])
# compute DOS
dos[j] = np.real(np.trace(1j * (grd[0] - grd[0].conj().T)))
# compute left-lead coupling
gamma_l = 1j * (L - L.conj().T)
# compute right-lead coupling
gamma_r = 1j * (R - R.conj().T)
# compute transmission
tr[j] = np.real(
np.trace(gamma_l.dot(g_trans).dot(gamma_r).dot(g_trans.conj().T)))
print("{} of {}: energy is {}".format(j + 1, energy.shape[0], E))
tr = np.array(tr)
dos = np.array(dos)
return dos, tr
def graphene_first_nearest_neighbour():
coords = """2
Graphene
C1 0.00 0.00 0.00
C2 {} 0.00 0.00
""".format(lat_const)
# --------------------------- Basis set --------------------------
s_orb = tb.Orbitals('C')
s_orb.add_orbital("pz", energy=0, orbital=1, magnetic=0, spin=0)
# ------------------------ set TB parameters----------------------
t = 2.8
tb.set_tb_params(PARAMS_C_C={'pp_pi': t})
# --------------------------- Hamiltonian -------------------------
h = tb.Hamiltonian(xyz=coords, nn_distance=1.5)
h.initialize()
h.set_periodic_bc(period)
band_structure = np.zeros((sum(num_points), h.h_matrix.shape[0]))
for jj, item in enumerate(k_points):
band_structure[jj, :], _ = h.diagonalize_periodic_bc(item)
# visualize
global fig_counter
plt.figure(fig_counter)
fig_counter += 1
ax = plt.axes()
ax.set_title(r'Band structure of graphene, 1st NN')
ax.set_ylabel('Energy (eV)')
ax.plot(np.sort(band_structure), 'k')
ax.plot([0, band_structure.shape[0]], [0, 0], '--', color='k', linewidth=0.5)
plt.xticks(np.insert(np.cumsum(num_points) - 1, 0, 0), labels=sym_points)
ax.xaxis.grid()
plt.show()
def test_simple_atomic_chain():
""" """
site_energy = -1.0
coupling = -1.0
l_const = 1.0
a = tb.Orbitals('A')
a.add_orbital(
title='s',
energy=-1,
)
xyz_file = """1
H cell
A 0.0000000000 0.0000000000 0.0000000000
"""
tb.set_tb_params(PARAMS_A_A={'ss_sigma': -1.0})
h = tb.HamiltonianSp(xyz=xyz_file, nn_distance=1.1)
h.initialize()
PRIMITIVE_CELL = [[0, 0, l_const]]
h.set_periodic_bc(PRIMITIVE_CELL)
num_points = 10
kk = np.linspace(0, 3.14 / l_const, num_points, endpoint=True)
band_structure = []
for jj in range(num_points):
vals, _ = h.diagonalize_periodic_bc([0.0, 0.0, kk[jj]])
band_structure.append(vals)
band_structure = np.array(band_structure)
desired_value = site_energy + 2 * coupling * np.cos(l_const * kk)
np.testing.assert_allclose(band_structure,
desired_value[:, np.newaxis],
atol=1e-9)
def complex_chain():
""" """
sys.path.insert(0, '/home/mk/TB_project/tb')
a = tb.Orbitals('A')
a.add_orbital('s', -0.7)
b = tb.Orbitals('B')
b.add_orbital('s', -0.5)
c = tb.Orbitals('C')
c.add_orbital('s', -0.3)
tb.set_tb_params(PARAMS_A_A={'ss_sigma': -0.5},
PARAMS_B_B={'ss_sigma': -0.5},
PARAMS_A_B={'ss_sigma': -0.5},
PARAMS_B_C={'ss_sigma': -0.5},
PARAMS_A_C={'ss_sigma': -0.5})
xyz_file = """4
H cell
A1 0.0000000000 0.0000000000 0.0000000000
B2 0.0000000000 0.0000000000 1.0000000000
A2 0.0000000000 1.0000000000 0.0000000000
B3 0.0000000000 1.0000000000 1.0000000000
"""
h = tb.Hamiltonian(xyz=xyz_file, nn_distance=1.1)
h.initialize()
h.set_periodic_bc([[0, 0, 2.0]])
h_l, h_0, h_r = h.get_hamiltonians()
energy = np.linspace(-3.0, 1.5, 700)
sgf_l = []
sgf_r = []
for E in energy:
left_se, right_se = negf.surface_greens_function(E,
h_l,
h_0,
h_r,
iterate=5)
sgf_l.append(left_se)
sgf_r.append(right_se)
sgf_l = np.array(sgf_l)
sgf_r = np.array(sgf_r)
num_sites = h_0.shape[0]
gf = np.linalg.pinv(
np.multiply.outer(energy, np.identity(num_sites)) - h_0 - sgf_l -
sgf_r)
tr = np.zeros((energy.shape[0]))
dos = np.zeros((energy.shape[0]))
for j, E in enumerate(energy):
gf0 = gf[j, :, :]
gamma_l = 1j * (sgf_l[j, :, :] - sgf_l[j, :, :].conj().T)
gamma_r = 1j * (sgf_r[j, :, :] - sgf_r[j, :, :].conj().T)
tr[j] = np.real(
np.trace(gamma_l.dot(gf0).dot(gamma_r).dot(gf0.conj().T)))
dos[j] = np.real(np.trace(1j * (gf0 - gf0.conj().T)))
return energy, dos, tr, h, sgf_l, sgf_r
def graphene_nanoribbons_armchair():
from ase.build.ribbon import graphene_nanoribbon
from ase.visualize.plot import plot_atoms
atoms = graphene_nanoribbon(11, 1, type='armchair')
period = np.array([list(atoms.get_cell()[2])])
period[:, [1, 2]] = period[:, [2, 1]]
coord = atoms.get_positions()
coord[:, [1, 2]] = coord[:, [2, 1]]
coords = []
coords.append(str(len(coord)))
coords.append('Nanoribbon')
for j, item in enumerate(coord):
coords.append('C' + str(j+1) + ' ' + str(item[0]) + ' ' + str(item[1]) + ' ' + str(item[2]))
coords = '\n'.join(coords)
s_orb = tb.Orbitals('C')
s_orb.add_orbital("pz", energy=-0.28, orbital=1, magnetic=0, spin=0)
# ------------------------ set TB parameters----------------------
gamma0 = -2.97
gamma1 = -0.073
gamma2 = -0.33
s0 = 0.073
s1 = 0.018
s2 = 0.026
tb.set_tb_params(PARAMS_C_C1={'pp_pi': gamma0},
PARAMS_C_C2={'pp_pi': gamma1},
PARAMS_C_C3={'pp_pi': gamma2},
OV_C_C1={'pp_pi': s0},
OV_C_C2={'pp_pi': s1},
OV_C_C3={'pp_pi': s2})
# --------------------------- Hamiltonian -------------------------
h = tb.Hamiltonian(xyz=coords, nn_distance=3.1, comp_overlap=True)
h.initialize(radial_dep)
h.set_periodic_bc(period)
k_points = np.linspace(0.0, np.pi/period[0][1], 20)
band_structure = np.zeros((len(k_points), h.h_matrix.shape[0]))
for jj, item in enumerate(k_points):
band_structure[jj, :], _ = h.diagonalize_periodic_bc([0.0, item, 0.0])
# visualize
ax = plt.axes()
ax.set_title('Graphene nanoribbon, armchair 11')
ax.set_ylabel('Energy (eV)')
ax.set_xlabel(r'Wave vector ($\frac{\pi}{a}$)')
ax.plot(k_points, np.sort(band_structure), 'k')
ax.xaxis.grid()
plt.show()
ax1 = plot_atoms(atoms, show_unit_cell=2, rotation='90x,0y,00z')
ax1.axis('off')
plt.show()
summation method should be used over numerical integration.
'''
import matplotlib.pyplot as plt
import numpy as np
import nanonet.tb as tb
from nanonet.negf.greens_functions import simple_iterative_greens_function, surface_greens_function
from nanonet.negf import pole_summation_method
from nanonet.negf.pole_summation_method import fermi_fun
# First we design a tight-binding model. We choose a 15 site model
# so that it is symmetric and that features may be clearly ob-
# served, one or two site models would look to similar to clearly
# differentiate whether something erroneously similar was happening
a = tb.Orbitals('A')
a.add_orbital('s', 0)
tb.Orbitals.orbital_sets = {'A': a}
tb.set_tb_params(PARAMS_A_A={'ss_sigma': -1})
xyz_file = """15
A cell
A1 0.0000000000 0.0000000000 0.0000000000
A2 1.0000000000 0.0000000000 0.0000000000
A3 2.0000000000 0.0000000000 0.0000000000
A4 3.0000000000 0.0000000000 0.0000000000
A5 4.0000000000 0.0000000000 0.0000000000
A6 5.0000000000 0.0000000000 0.0000000000
A7 6.0000000000 0.0000000000 0.0000000000
A8 7.0000000000 0.0000000000 0.0000000000
def main(surf_greens_fun):
""" An example for the Green's function usage"""
a = tb.Orbitals('A')
a.add_orbital('s', -0.7)
tb.Orbitals.orbital_sets = {'A': a}
tb.set_tb_params(PARAMS_A_A={'ss_sigma': -0.5})
xyz_file = """1
A cell
A1 0.0000000000 0.0000000000 0.0000000000
"""
h = tb.Hamiltonian(xyz=xyz_file, nn_distance=1.1)
h.initialize()
h.set_periodic_bc([[0, 0, 1.0]])
h_l, h_0, h_r = h.get_hamiltonians()
energy = np.linspace(-3.0, 1.5, 700)
sgf_l = []
sgf_r = []
for E in energy:
sf = surf_greens_fun(E, h_l, h_0, h_r, damp=0.001j)
if isinstance(sf, tuple):
L = sf[0]
R = sf[1]
else:
L = sf
R = surf_greens_fun(E, h_r, h_0, h_l, damp=0.001j)
sgf_l.append(L)
sgf_r.append(R)
sgf_l = np.array(sgf_l)
sgf_r = np.array(sgf_r)
num_sites = h_0.shape[0]
gf = np.linalg.pinv(
np.multiply.outer(energy, np.identity(num_sites)) - h_0 - sgf_l -
sgf_r)
dos = -np.trace(np.imag(gf), axis1=1, axis2=2)
tr = np.zeros((energy.shape[0]), dtype=complex)
for j, E in enumerate(energy):
gf0 = gf[j, :, :]
gamma_l = 1j * (sgf_l[j, :, :] - sgf_l[j, :, :].conj().T)
gamma_r = 1j * (sgf_r[j, :, :] - sgf_r[j, :, :].conj().T)
tr[j] = np.real(
np.trace(np.linalg.multi_dot([gamma_l, gf0, gamma_r,
gf0.conj().T])))
dos[j] = np.real(np.trace(1j * (gf0 - gf0.conj().T)))
fig, axs = plt.subplots(2, figsize=(5, 7))
fig.suptitle('Green\'s function technique')
axs[0].plot(energy, dos, 'k')
# axs[0].title.set_text('Density of states')
axs[0].set_xlabel('Energy (eV)')
axs[0].set_ylabel('DOS')
axs[1].plot(energy, tr, 'k')
# axs[1].title.set_text('Transmission function')
axs[1].set_xlabel('Energy (eV)')
axs[1].set_ylabel('Transmission probability')
plt.show(block=False)
