def cal_sstates(self, tp, energies, s, q):
#To see how much the transmision from one orbital(local basis) in lead a
# to the orbital in the lead b, the transmission is decomposited to
# Bloch wave in leads
MinErr = 1e-8
MaxLambda = 1 + MinErr
energies = np.real(energies)
ne = len(energies)
t_all = np.zeros([ne, tp.lead_num, tp.lead_num])
t_lead_all = []
for i in range(tp.lead_num):
t_lead_all.append([])
for j in range(tp.lead_num):
t_lead_all[i].append(np.zeros([tp.nblead[i], tp.nblead[j],
ne]))
total_t = []
total_vc = []
total_k = []
total_vl = []
sk1 = []
sk2 = []
for n, energy in enumerate(energies):
t, vc, k, vl = self.central_scattering_states(tp, energy, s, q)
total_t.append(t)
total_vc.append(vc)
total_k.append(k)
total_vl.append(vl)
for i in range(tp.lead_num):
for j in range(tp.lead_num):
t0 = t[i][j]
t_all[n, i, j] = np.sum(np.dot(t0.T.conj(), t0))
for m1 in range(t0.shape[0]):
sk1.append(
self.lead_k_matrix(tp, i, s, q, k[i][m1], 'S'))
for m2 in range(t0.shape[1]):
sk2.append(
self.lead_k_matrix(tp, i, s, q, k[j][m2], 'S'))
for m1 in range(t0.shape[0]):
for m2 in range(t0.shape[1]):
w1 = (vl[j][:, m1].conj() *
np.dot(sk1[m1], vl[j][:, m1])).real
w1[find(w1 < 0)] = 0
w1 /= np.sum(w1)
w2 = (vl[i][:, m2].conj() *
np.dot(sk2[m2], vl[i][:, m2])).real
w2[find(w2 < 0)] = 0
w2 /= np.sum(w2)
t_lead_all[i][j][:, :,
n] += abs(t0[m1, m2])**2 * np.dot(
w2, w1.T.conj())
return np.array(t_lead_all), np.array(t_all)
def cal_sstates(self, tp, energies, s, q):
#To see how much the transmision from one orbital(local basis) in lead a
# to the orbital in the lead b, the transmission is decomposited to
# Bloch wave in leads
MinErr = 1e-8
MaxLambda = 1 + MinErr
energies = np.real(energies)
ne = len(energies)
t_all = np.zeros([ne, tp.lead_num, tp.lead_num])
t_lead_all = []
for i in range(tp.lead_num):
t_lead_all.append([])
for j in range(tp.lead_num):
t_lead_all[i].append(np.zeros([tp.nblead[i],
tp.nblead[j], ne]))
total_t = []
total_vc = []
total_k = []
total_vl = []
sk1 = []
sk2 = []
for n, energy in enumerate(energies):
t, vc, k, vl = self.central_scattering_states(tp, energy, s, q)
total_t.append(t)
total_vc.append(vc)
total_k.append(k)
total_vl.append(vl)
for i in range(tp.lead_num):
for j in range(tp.lead_num):
t0 = t[i][j]
t_all[n, i, j] = np.sum(np.dot(t0.T.conj(), t0))
for m1 in range(t0.shape[0]):
sk1.append(self.lead_k_matrix(tp, i, s, q, k[i][m1], 'S'))
for m2 in range(t0.shape[1]):
sk2.append(self.lead_k_matrix(tp, i, s, q, k[j][m2], 'S'))
for m1 in range(t0.shape[0]):
for m2 in range(t0.shape[1]):
w1 = (vl[j][:, m1].conj() * np.dot(sk1[m1],
vl[j][:, m1])).real
w1[find(w1 < 0)] = 0
w1 /= np.sum(w1)
w2 = (vl[i][:, m2].conj() * np.dot(sk2[m2],
vl[i][:, m2])).real
w2[find(w2 < 0)] = 0
w2 /= np.sum(w2)
t_lead_all[i][j][:, :, n] += abs(t0[m1,
m2]) ** 2 * np.dot(w2, w1.T.conj())
return np.array(t_lead_all), np.array(t_all)
def central_scattering_states(self, tp, energy, s, q):
#To get the scattering states corresponding to a Bloch vector in lead
MaxLambda = 1e2
MinErr = 1e-8
bc = tp.inner_mol_index
nc = len(bc)
molhes = tp.hsd.H[s][q].recover(True) - \
tp.hsd.S[q].recover(True) * energy
blead = []
lead_hes = []
lead_couple_hes = []
for i in range(tp.lead_num):
blead.append(tp.lead_layer_index[i][-1])
lead_hes.append(tp.lead_hsd[i].H[s][q].recover() -
tp.lead_hsd[i].S[q].recover() * energy)
lead_couple_hes.append(tp.lead_couple_hsd[i].H[s][q].recover() -
tp.lead_couple_hsd[i].S[q].recover() *
energy)
ex_ll_index = tp.hsd.S[0].ex_ll_index
total_k = []
total_vk = []
total_lam = []
total_v = []
total_pro_right_index = []
total_pro_left_index = []
total_left_index = []
total_right_index = []
total_kr = []
total_kt = []
total_lambdar = []
total_lambdat = []
total_vkr = []
total_vkt = []
total_vr = []
total_vt = []
total_len_k = []
total_nblead = []
total_bA1 = []
total_bA2 = []
total_bproin = []
total_nbB2 = []
total_bB2 = []
for i in range(tp.lead_num):
k, vk = self.lead_scattering_states(tp, energy, i, s, q)
total_k.append(k)
total_vk.append(vk)
lam = np.exp(2 * np.pi * k * 1.j)
total_lam.append(lam)
#calculating v = dE/dk
de2 = 1e-8
k2, vk2 = self.lead_scattering_states(tp, energy + de2, i, s, q)
v = de2 / (k2 - k) / 2 / np.pi
total_v.append(v)
#seperating left scaterring states and right scattering states
#left scattering: lead->mol, right scattering: mol->lead
proindex = find(abs(k.imag) < MinErr)
pro_left_index = proindex[find(v[proindex].real < 0)]
pro_left_index = pro_left_index[-np.arange(len(pro_left_index)) -
1]
pro_right_index = proindex[find(v[proindex].real > 0)]
left_index = find(k.imag > MinErr)
left_index = np.append(left_index, pro_left_index)
right_index = pro_right_index.copy()
right_index = np.append(right_index, find(k.imag < -MinErr))
total_pro_left_index.append(pro_left_index)
total_pro_right_index.append(pro_right_index)
total_left_index.append(left_index)
total_right_index.append(right_index)
kr = k[left_index]
kt = k[right_index]
total_kr.append(kr)
total_kt.append(kt)
lambdar = np.diag(np.exp(2 * np.pi * kr * 1.j))
lambdat = np.diag(np.exp(2 * np.pi * kt * 1.j))
total_lambdar.append(lambdar)
total_lambdat.append(lambdat)
vkr = np.take(vk, left_index, axis=1)
vkt = np.take(vk, right_index, axis=1)
vr = v[pro_left_index]
vt = v[pro_right_index]
total_vkr.append(vkr)
total_vkt.append(vkt)
total_vr.append(vr)
total_vt.append(vt)
#abstract basis information
len_k = len(right_index)
total_len_k.append(len_k)
#lead i basis seqeunce in whole matrix
nblead = len(ex_ll_index[i][-1])
total_nblead.append(nblead)
bA1 = nc + int(np.sum(total_nblead[:i])) + np.arange(nblead)
#sequence of the incident wave
bA2 = nc + int(np.sum(total_len_k[:i])) + np.arange(len_k)
# the first n in vkt are scattering waves, the rest are the decaying ones
# the first n in vkr are decaying waves, the rest are the scattering ones
total_bA1.append(bA1)
total_bA2.append(bA2)
bproin = np.arange(len(pro_right_index)) + \
len(kr) - len(pro_right_index)
### this line need to check it...
total_bproin.append(bproin)
nbB2 = len(bproin)
total_nbB2.append(nbB2)
bB2 = int(np.sum(total_nbB2[:i])) + np.arange(nbB2)
total_bB2.append(bB2)
ind = get_matrix_index(bc)
Acc = molhes[ind.T, ind]
total_Alc = []
total_hcl = []
total_All = []
total_Acl = []
total_Bc = []
total_Bl = []
for i in range(tp.lead_num):
ind1, ind2 = get_matrix_index(blead[i], bc)
Alc = molhes[ind1, ind2]
ind1, ind2 = get_matrix_index(bc, blead[i])
hcl = molhes[ind1, ind2]
vkt = total_vkt[i]
All = np.dot(tp.lead_hsd[i].H[s][q].recover(), vkt) + \
np.dot(np.dot(tp.lead_couple_hsd[i].H[s][q].recover(), \
vkt), total_lambdat[i])
Acl = np.dot(hcl, vkt)
bpi = total_bproin[i]
vkr = total_vkr[i]
vkr_bpi = np.take(vkr, bpi, axis=1)
Bc = -np.dot(hcl, vkr_bpi)
ind = get_matrix_index(bpi)
Bl = -np.dot(tp.lead_hsd[i].H[s][q].recover(), vkr_bpi) + \
np.dot(np.dot(tp.lead_couple_hsd[i].H[s][q].recover(),
vkr_bpi) ,total_lambdar[i][ind.T, ind])
total_Alc.append(Alc)
total_Acl.append(Acl)
total_hcl.append(hcl)
total_All.append(All)
total_Bc.append(Bc)
total_Bl.append(Bl)
total_bc = molhes.shape[-1]
MatA = np.zeros([total_bc, nc + np.sum(total_len_k)], complex)
MatB = np.zeros([total_bc, np.sum(total_nbB2)], complex)
ind = get_matrix_index(np.arange(nc))
MatA[ind.T, ind] = Acc
for i in range(tp.lead_num):
ind1, ind2 = get_matrix_index(total_bA1[i], total_bA2[i])
MatA[ind1, ind2] = total_All[i]
ind1, ind2 = get_matrix_index(np.arange(nc), total_bA2[i])
MatA[ind1, ind2] = total_Acl[i]
ind1, ind2 = get_matrix_index(total_bA1[i], np.arange(nc))
MatA[ind1, ind2] = total_Alc[i]
ind1, ind2 = get_matrix_index(np.arange(nc), total_bB2[i])
MatB[ind1, ind2] = total_Bc[i]
ind1, ind2 = get_matrix_index(total_bA1[i], total_bB2[i])
MatB[ind1, ind2] = total_Bl[i]
Vx, residues, rank, singular = np.linalg.lstsq(MatA, MatB)
total_vl = []
for i in range(tp.lead_num):
total_k[i] = total_k[i][total_pro_right_index[i]]
total_vl.append(
np.take(total_vk[i], total_pro_right_index[i], axis=1))
total_vc = []
t = []
for i in range(tp.lead_num):
t.append([])
for j in range(tp.lead_num):
t[i].append([])
for i in range(tp.lead_num):
ind1, ind2 = get_matrix_index(np.arange(nc), total_bB2[i])
total_vc.append(Vx[ind1, ind2])
for j in range(tp.lead_num):
bx = total_bA2[j]
bx = bx[:len(total_pro_right_index[j])]
ind1, ind2 = get_matrix_index(bx, total_bB2[i])
#t[j].append(Vx[ind1, ind2])
t[j][i] = Vx[ind1, ind2]
for m in range(len(total_pro_left_index[i])):
for n in range(len(total_pro_right_index[j])):
t[j][i][n, m] *= np.sqrt(
abs(total_vt[i][n] / total_vr[j][m]))
return np.array(t), np.array(total_vc), \
np.array(total_k), np.array(total_vl)
def lead_scattering_states(self, tp, energy, l, s, q):
#Calculating the scattering states in electrodes
#l ---- index of electrode
#s ---- index of spin
#q ---- index of local k point
#if it is multi-terminal system, should add a part that can rotate
# the lead hamiltonian
MaxLambda = 1e2
MinErr = 1e-8
energy += tp.bias[l]
hes00 = tp.lead_hsd[l].H[s][q].recover() - \
tp.lead_hsd[l].S[q].recover() * energy
hes01 = tp.lead_couple_hsd[l].H[s][q].recover() - \
tp.lead_couple_hsd[l].S[q].recover() * energy
nb = hes00.shape[-1]
dtype = hes00.dtype
A = np.zeros([2 * nb, 2 * nb], dtype)
B = np.zeros([2 * nb, 2 * nb], dtype)
A[:nb, nb:2 * nb] = np.eye(nb)
A[nb:2 * nb, :nb] = hes01.T.conj()
A[nb:2 * nb, nb:2 * nb] = hes00
B[:nb, :nb] = np.eye(nb)
B[nb:2 * nb, nb:2 * nb] = -hes01
from scipy.linalg import eig
D, V = eig(A, B)
index = np.argsort(abs(D))
D = D[index]
V = V[:, index]
#delete NaN
index = find(np.abs(D) >= 0)
D = D[index]
V = V[:, index]
#delete some unreasonable solutions
index = find(abs(D) > MaxLambda)
cutlen = len(D) - index[0]
index = np.arange(cutlen, len(D) - cutlen)
D = D[index]
V = V[:, index]
k = np.log(D) * (-1.j) / 2 / np.pi
Vk = V[:nb]
#sort scattering states
proindex = find(abs(k.imag) < MinErr)
if len(proindex) > 0:
k_sort = np.sort(k[proindex].real)
index = np.argsort(k[proindex].real)
k[proindex] = k[proindex[index]].real
Vk[:, proindex] = Vk[:, proindex[index]]
#normalization the scattering states
j = proindex[0]
while j <= proindex[-1]:
same_k_index = find(abs((k - k[j]).real) < MinErr)
sk = self.lead_k_matrix(tp, l, s, q, k[j])
Vk[:, same_k_index] = eig_states_norm(Vk[:, same_k_index], sk)
j += len(same_k_index)
return np.array(k[proindex]), np.array(Vk[:, proindex])
def central_scattering_states(self, tp, energy, s, q):
#To get the scattering states corresponding to a Bloch vector in lead
MaxLambda = 1e2
MinErr = 1e-8
bc = tp.inner_mol_index
nc = len(bc)
molhes = tp.hsd.H[s][q].recover(True) - \
tp.hsd.S[q].recover(True) * energy
blead = []
lead_hes = []
lead_couple_hes = []
for i in range(tp.lead_num):
blead.append(tp.lead_layer_index[i][-1])
lead_hes.append(tp.lead_hsd[i].H[s][q].recover() -
tp.lead_hsd[i].S[q].recover() * energy)
lead_couple_hes.append(tp.lead_couple_hsd[i].H[s][q].recover() -
tp.lead_couple_hsd[i].S[q].recover() * energy)
ex_ll_index = tp.hsd.S[0].ex_ll_index
total_k = []
total_vk = []
total_lam = []
total_v = []
total_pro_right_index = []
total_pro_left_index = []
total_left_index = []
total_right_index = []
total_kr = []
total_kt = []
total_lambdar = []
total_lambdat = []
total_vkr = []
total_vkt = []
total_vr = []
total_vt = []
total_len_k = []
total_nblead = []
total_bA1 = []
total_bA2 = []
total_bproin = []
total_nbB2 = []
total_bB2 = []
for i in range(tp.lead_num):
k, vk = self.lead_scattering_states(tp, energy, i, s, q)
total_k.append(k)
total_vk.append(vk)
lam = np.exp(2 * np.pi * k * 1.j)
total_lam.append(lam)
#calculating v = dE/dk
de2 = 1e-8
k2, vk2 = self.lead_scattering_states(tp, energy + de2, i, s, q)
v = de2 / (k2 - k) / 2 / np.pi
total_v.append(v)
#seperating left scaterring states and right scattering states
#left scattering: lead->mol, right scattering: mol->lead
proindex = find(abs(k.imag) < MinErr)
pro_left_index = proindex[find(v[proindex].real < 0)]
pro_left_index = pro_left_index[-np.arange(len(pro_left_index))
- 1]
pro_right_index = proindex[find(v[proindex].real > 0)]
left_index = find(k.imag > MinErr)
left_index = np.append(left_index, pro_left_index)
right_index = pro_right_index.copy()
right_index = np.append(right_index, find(k.imag < -MinErr))
total_pro_left_index.append(pro_left_index)
total_pro_right_index.append(pro_right_index)
total_left_index.append(left_index)
total_right_index.append(right_index)
kr = k[left_index]
kt = k[right_index]
total_kr.append(kr)
total_kt.append(kt)
lambdar = np.diag(np.exp(2 * np.pi * kr * 1.j))
lambdat = np.diag(np.exp(2 * np.pi * kt * 1.j))
total_lambdar.append(lambdar)
total_lambdat.append(lambdat)
vkr = np.take(vk, left_index, axis=1)
vkt = np.take(vk, right_index, axis=1)
vr = v[pro_left_index]
vt = v[pro_right_index]
total_vkr.append(vkr)
total_vkt.append(vkt)
total_vr.append(vr)
total_vt.append(vt)
#abstract basis information
len_k = len(right_index)
total_len_k.append(len_k)
#lead i basis seqeunce in whole matrix
nblead = len(ex_ll_index[i][-1])
total_nblead.append(nblead)
bA1 = nc + int(np.sum(total_nblead[:i])) + np.arange(nblead)
#sequence of the incident wave
bA2 = nc + int(np.sum(total_len_k[:i])) + np.arange(len_k)
# the first n in vkt are scattering waves, the rest are the decaying ones
# the first n in vkr are decaying waves, the rest are the scattering ones
total_bA1.append(bA1)
total_bA2.append(bA2)
bproin = np.arange(len(pro_right_index)) + \
len(kr) - len(pro_right_index)
### this line need to check it...
total_bproin.append(bproin)
nbB2 = len(bproin)
total_nbB2.append(nbB2)
bB2 = int(np.sum(total_nbB2[:i])) + np.arange(nbB2)
total_bB2.append(bB2)
ind = get_matrix_index(bc)
Acc = molhes[ind.T, ind]
total_Alc = []
total_hcl = []
total_All = []
total_Acl = []
total_Bc = []
total_Bl = []
for i in range(tp.lead_num):
ind1, ind2 = get_matrix_index(blead[i], bc)
Alc = molhes[ind1, ind2]
ind1, ind2 = get_matrix_index(bc, blead[i])
hcl = molhes[ind1, ind2]
vkt = total_vkt[i]
All = np.dot(tp.lead_hsd[i].H[s][q].recover(), vkt) + \
np.dot(np.dot(tp.lead_couple_hsd[i].H[s][q].recover(), \
vkt), total_lambdat[i])
Acl = np.dot(hcl, vkt)
bpi = total_bproin[i]
vkr = total_vkr[i]
vkr_bpi = np.take(vkr, bpi, axis=1)
Bc = -np.dot(hcl, vkr_bpi)
ind = get_matrix_index(bpi)
Bl = -np.dot(tp.lead_hsd[i].H[s][q].recover(), vkr_bpi) + \
np.dot(np.dot(tp.lead_couple_hsd[i].H[s][q].recover(),
vkr_bpi) ,total_lambdar[i][ind.T, ind])
total_Alc.append(Alc)
total_Acl.append(Acl)
total_hcl.append(hcl)
total_All.append(All)
total_Bc.append(Bc)
total_Bl.append(Bl)
total_bc = molhes.shape[-1]
MatA = np.zeros([total_bc, nc + np.sum(total_len_k)], complex)
MatB = np.zeros([total_bc, np.sum(total_nbB2)], complex)
ind = get_matrix_index(np.arange(nc))
MatA[ind.T, ind] = Acc
for i in range(tp.lead_num):
ind1, ind2 = get_matrix_index(total_bA1[i], total_bA2[i])
MatA[ind1, ind2] = total_All[i]
ind1, ind2 = get_matrix_index(np.arange(nc), total_bA2[i])
MatA[ind1, ind2] = total_Acl[i]
ind1, ind2 = get_matrix_index(total_bA1[i], np.arange(nc))
MatA[ind1, ind2] = total_Alc[i]
ind1, ind2 = get_matrix_index(np.arange(nc), total_bB2[i])
MatB[ind1, ind2] = total_Bc[i]
ind1, ind2 = get_matrix_index(total_bA1[i], total_bB2[i])
MatB[ind1, ind2] = total_Bl[i]
Vx, residues, rank, singular = np.linalg.lstsq(MatA, MatB)
total_vl = []
for i in range(tp.lead_num):
total_k[i] = total_k[i][total_pro_right_index[i]]
total_vl.append(np.take(total_vk[i],
total_pro_right_index[i], axis=1))
total_vc = []
t = []
for i in range(tp.lead_num):
t.append([])
for j in range(tp.lead_num):
t[i].append([])
for i in range(tp.lead_num):
ind1, ind2 = get_matrix_index(np.arange(nc), total_bB2[i])
total_vc.append(Vx[ind1, ind2])
for j in range(tp.lead_num):
bx = total_bA2[j]
bx = bx[:len(total_pro_right_index[j])]
ind1, ind2 = get_matrix_index(bx, total_bB2[i])
#t[j].append(Vx[ind1, ind2])
t[j][i] = Vx[ind1, ind2]
for m in range(len(total_pro_left_index[i])):
for n in range(len(total_pro_right_index[j])):
t[j][i][n, m] *= np.sqrt(abs(total_vt[i][n]/
total_vr[j][m]))
return np.array(t), np.array(total_vc), \
np.array(total_k), np.array(total_vl)
def lead_scattering_states(self, tp, energy, l, s, q):
#Calculating the scattering states in electrodes
#l ---- index of electrode
#s ---- index of spin
#q ---- index of local k point
#if it is multi-terminal system, should add a part that can rotate
# the lead hamiltonian
MaxLambda = 1e2
MinErr = 1e-8
energy += tp.bias[l]
hes00 = tp.lead_hsd[l].H[s][q].recover() - \
tp.lead_hsd[l].S[q].recover() * energy
hes01 = tp.lead_couple_hsd[l].H[s][q].recover() - \
tp.lead_couple_hsd[l].S[q].recover() * energy
nb = hes00.shape[-1]
dtype = hes00.dtype
A = np.zeros([2*nb, 2*nb], dtype)
B = np.zeros([2*nb, 2*nb], dtype)
A[:nb, nb:2*nb] = np.eye(nb)
A[nb:2*nb, :nb] = hes01.T.conj()
A[nb:2*nb, nb:2*nb] = hes00
B[:nb, :nb] = np.eye(nb)
B[nb:2*nb, nb:2*nb] = -hes01
from scipy.linalg import eig
D, V = eig(A, B)
index = np.argsort(abs(D))
D = D[index]
V = V[:, index]
#delete NaN
index = find(np.abs(D) >= 0)
D = D[index]
V = V[:, index]
#delete some unreasonable solutions
index = find(abs(D) > MaxLambda)
cutlen = len(D) - index[0]
index = np.arange(cutlen, len(D) - cutlen)
D = D[index]
V = V[:, index]
k = np.log(D) * (-1.j) / 2 / np.pi
Vk = V[:nb]
#sort scattering states
proindex = find(abs(k.imag) < MinErr)
if len(proindex) > 0:
k_sort = np.sort(k[proindex].real)
index = np.argsort(k[proindex].real)
k[proindex] = k[proindex[index]].real
Vk[:, proindex] = Vk[:, proindex[index]]
#normalization the scattering states
j = proindex[0]
while j <= proindex[-1]:
same_k_index = find(abs((k - k[j]).real) < MinErr)
sk = self.lead_k_matrix(tp, l, s, q, k[j])
Vk[:, same_k_index] = eig_states_norm(Vk[:, same_k_index],
sk)
j += len(same_k_index)
return np.array(k[proindex]), np.array(Vk[:, proindex])