def central_dos(hetero,energies=[0.0],num_rep=100,
mixing=0.7,eps=0.0001,green_guess=None,max_error=0.0001):
""" Calculates the density of states
of a heterostructure by a
green function approach, input is a heterostructure class"""
from green import dyson
dos = [] # list with the density of states
intra = hetero.central_intra # central intraterm
iden = np.matrix(np.identity(len(intra),dtype=complex)) # create idntity
for energy in energies: # loop over energies
# right green function
intra = hetero.right_intra
inter = hetero.right_inter
gr = dyson(intra,inter,energy=energy,num_rep=num_rep,mixing=mixing,
eps=eps,green_guess=green_guess,max_error=max_error)
# left green function
intra = hetero.left_intra
inter = hetero.left_inter
gl = dyson(intra,inter,energy=energy,num_rep=num_rep,mixing=mixing,
eps=eps,green_guess=green_guess,max_error=max_error)
# left selfenergy
inter = hetero.left_coupling
selfl = inter*gl*inter.H # left selfenergy
# right selfenergy
inter = hetero.right_coupling
selfr = inter*gr*inter.H # right selfenergy
# central green function
intra = hetero.central_intra
gc = energy*iden -intra -selfl -selfr # dyson equation for the center
gc = gc.I # calculate inverse
dos.append(-gc.trace()[0,0].imag) # calculate the trace of the Green function
return dos
def central_dos(hetero,energies=[0.0],num_rep=100,
mixing=0.7,eps=0.0001,green_guess=None,max_error=0.0001):
""" Calculates the density of states
of a heterostructure by a
green function approach, input is a heterostructure class"""
from green import dyson
dos = [] # list with the density of states
intra = hetero.central_intra # central intraterm
iden = np.matrix(np.identity(len(intra),dtype=complex)) # create idntity
for energy in energies: # loop over energies
# right green function
intra = hetero.right_intra
inter = hetero.right_inter
gr = dyson(intra,inter,energy=energy,num_rep=num_rep,mixing=mixing,
eps=eps,green_guess=green_guess,max_error=max_error)
# left green function
intra = hetero.left_intra
inter = hetero.left_inter
gl = dyson(intra,inter,energy=energy,num_rep=num_rep,mixing=mixing,
eps=eps,green_guess=green_guess,max_error=max_error)
# left selfenergy
inter = hetero.left_coupling
selfl = inter*gl*inter.H # left selfenergy
# right selfenergy
inter = hetero.right_coupling
selfr = inter*gr*inter.H # right selfenergy
# central green function
intra = hetero.central_intra
gc = energy*iden -intra -selfl -selfr # dyson equation for the center
gc = gc.I # calculate inverse
dos.append(-gc.trace()[0,0].imag) # calculate the trace of the Green function
return dos
def effective_central_hamiltonian(hetero, energy=0.0, write=False, gf=None):
""" Plots the local density of states in the central part"""
from green import dyson
from hamiltonians import is_number
if not hetero.block_diagonal:
intra = hetero.central_intra # central intraterm
if hetero.block_diagonal:
intra = hetero.central_intra[0][0] # when it is diagonal
# perform dyson calculation
intra = hetero.right_intra
inter = hetero.right_inter
gr = dyson(intra, inter, is_sparse=hetero.is_sparse, gf=gf)
hetero.right_green = gr # save green function
# left green function
intra = hetero.left_intra
inter = hetero.left_inter
gl = dyson(intra, inter, is_sparse=hetero.is_sparse, gf=gf)
hetero.left_green = gl # save green function
# save green functions
hetero.write_green()
print "Saved green functions"
# left selfenergy
inter = hetero.left_coupling
selfl = inter * gl * inter.H # left selfenergy
# right selfenergy
inter = hetero.right_coupling
selfr = inter * gr * inter.H # right selfenergy
# central green function
intra = hetero.central_intra
# dyson equation for the center
# full matrix
if not hetero.block_diagonal:
heff = intra + selfl + selfr
hetero.heff = heff
if save_heff:
print "Saving effective hamiltonian in ", hetero.file_heff
if write: # save hamiltonian if desired
hetero.write_heff()
# reduced matrix
if hetero.block_diagonal:
from copy import deepcopy
heff = deepcopy(intra)
heff[0][0] = intra[0][0] + selfl
heff[-1][-1] = intra[-1][-1] + selfr
# save the green function
from scipy.sparse import bmat
hetero.heff = bmat(heff)
if write: # save hamiltonian
print "Saving effective hamiltonian in ", hetero.file_heff
hetero.write_heff()
def effective_central_hamiltonian(hetero,energy=0.0,write=False,gf=None):
""" Plots the local density of states in the central part"""
from green import dyson
from hamiltonians import is_number
if not hetero.block_diagonal:
intra = hetero.central_intra # central intraterm
if hetero.block_diagonal:
intra = hetero.central_intra[0][0] # when it is diagonal
# perform dyson calculation
intra = hetero.right_intra
inter = hetero.right_inter
gr = dyson(intra,inter,is_sparse=hetero.is_sparse,gf=gf)
hetero.right_green = gr # save green function
# left green function
intra = hetero.left_intra
inter = hetero.left_inter
gl = dyson(intra,inter,is_sparse=hetero.is_sparse,gf=gf)
hetero.left_green = gl # save green function
# save green functions
hetero.write_green()
print "Saved green functions"
# left selfenergy
inter = hetero.left_coupling
selfl = inter*gl*inter.H # left selfenergy
# right selfenergy
inter = hetero.right_coupling
selfr = inter*gr*inter.H # right selfenergy
# central green function
intra = hetero.central_intra
# dyson equation for the center
# full matrix
if not hetero.block_diagonal:
heff = intra + selfl + selfr
hetero.heff = heff
if save_heff:
print "Saving effective hamiltonian in ",hetero.file_heff
if write: # save hamiltonian if desired
hetero.write_heff()
# reduced matrix
if hetero.block_diagonal:
from copy import deepcopy
heff = deepcopy(intra)
heff[0][0] = intra[0][0] + selfl
heff[-1][-1] = intra[-1][-1] + selfr
# save the green function
from scipy.sparse import bmat
hetero.heff = bmat(heff)
if write: # save hamiltonian
print "Saving effective hamiltonian in ",hetero.file_heff
hetero.write_heff()
def plot_local_central_dos(hetero, energies=0.0, gf=None):
""" Plots the local density of states in the central part"""
from green import dyson
from green import gf_convergence
from hamiltonians import is_number
if gf == None: gf = gf_convergence("lead")
if not hetero.block_diagonal:
intra = hetero.central_intra # central intraterm
dimhc = len(intra) # dimension of the central part
if hetero.block_diagonal:
intra = hetero.central_intra[0][0] # when it is diagonal
# dimension of the central part
dimhc = len(hetero.central_intra) * len(intra)
iden = np.matrix(np.identity(len(intra), dtype=complex)) # create idntity
ldos = np.array([0.0 for i in range(dimhc)]) # initialice ldos
# initialize ldos
for energy in energies: # loop over energies
# right green function
gr = None
gl = None
# perform dyson calculation
intra = hetero.right_intra
inter = hetero.right_inter
gr = dyson(intra, inter, energy=energy, gf=gf)
hetero.right_green = gr # save green function
# left green function
intra = hetero.left_intra
inter = hetero.left_inter
gl = dyson(intra, inter, energy=energy, gf=gf)
hetero.left_green = gl # save green function
# save green functions
# hetero.write_green()
# left selfenergy
inter = hetero.left_coupling
selfl = inter * gl * inter.H # left selfenergy
# right selfenergy
inter = hetero.right_coupling
selfr = inter * gr * inter.H # right selfenergy
# central green function
intra = hetero.central_intra
# dyson equation for the center
# full matrix
if not hetero.block_diagonal:
heff = intra + selfl + selfr
hetero.heff = heff
gc = (energy + 1j * eps) * iden - heff
gc = gc.I # calculate inverse
# get the local density of states
ldos += np.array([-gc[i, i].imag for i in range(len(gc))])
# if save_heff:
# hetero.write_heff()
# reduced matrix
if hetero.block_diagonal:
from copy import deepcopy
heff = deepcopy(intra)
heff[0][0] = intra[0][0] + selfl
heff[-1][-1] = intra[-1][-1] + selfr
dd = (energy + 1j * gf.eps) * iden
for i in range(len(intra)): # add the diagonal energy part
heff[i][i] = heff[i][i] - dd # this has the wrong sign!!
# now change the sign
for i in range(len(intra)):
for j in range(len(intra)):
try:
heff[i][j] = -heff[i][j]
except:
heff[i][j] = heff[i][j]
# save the green function
hetero.heff = heff
# if save_heff:
# from scipy.sparse import bmat
# hetero.heff = bmat(heff)
# hetero.write_heff()
# calculate the inverse
from green import gauss_inverse # routine to invert the matrix
# list with the diagonal matrices
ldos_e = ldos * 0.0 # initialice ldos at this energy
ii = 0 # counter for the element
for i in range(len(heff)): # loop over blocks
gci = gauss_inverse(heff, i, i) # calculate each block element
for j in range(len(heff[0][0])): # loop over each block
ldos_e[ii] = -gci[j, j].imag
ii += 1 # increase counter
if not ii == dimhc:
print("Wrong dimensions", ii, dimhc)
raise
ldos += ldos_e # add to the total ldos
# save the effective hamiltonian
if hetero.has_spin: # resum ldos if there is spin degree of freedom
ldos = [ldos[2 * i] + ldos[2 * i + 1] for i in range(len(ldos) / 2)]
if hetero.has_eh: # resum ldos if there is eh
ldos = [ldos[2 * i] + ldos[2 * i + 1] for i in range(len(ldos) / 2)]
# ne = len(ldos)/2
# ldos = [ldos[i]+ldos[ne+i] for i in range(ne)]
ldos = np.array(ldos) # transform into an array
if min(ldos) < 0.0:
print("Negative density of states")
raise
g = hetero.central_geometry # geometry of the central part
fldos = open("LDOS.OUT", "w") # open file for ldos
fldos.write("# X Y LDOS\n")
for (ix, iy, il) in zip(g.x, g.y, ldos):
fldos.write(str(ix) + " " + str(iy) + " " + str(il) + "\n")
fldos.close()
# scale the ldos
# save the LDOS in a file
if True:
# if True>0.001:
# ldos = np.sqrt(ldos)
# ldos = np.arctan(7.*ldos/max(ldos))
print("Sum of the LDOS =", sum(ldos))
ldos = ldos * 300 / max(ldos)
else:
ldos = ldos * 0.0
# now create the figure
fig = py.figure() # create figure
fig.subplots_adjust(0.2, 0.2)
fig.set_facecolor("white") # face in white
sldos = fig.add_subplot(111) # create subplot for the DOS
# plot the lattice
if not len(g.x) == len(ldos):
raise
sldos.scatter(g.x, g.y, color="red", s=ldos) # plot the lattice
sldos.scatter(g.x, g.y, color="black", s=4) # plot the lattice
sldos.set_xlabel("X")
sldos.set_xlabel("Y")
sldos.axis("equal") # same scale in axes
return fig
def plot_local_central_dos(hetero,energies=0.0,gf=None):
""" Plots the local density of states in the central part"""
from green import dyson
from green import gf_convergence
from hamiltonians import is_number
if gf==None: gf = gf_convergence("lead")
if not hetero.block_diagonal:
intra = hetero.central_intra # central intraterm
dimhc = len(intra) # dimension of the central part
if hetero.block_diagonal:
intra = hetero.central_intra[0][0] # when it is diagonal
# dimension of the central part
dimhc = len(hetero.central_intra)*len(intra)
iden = np.matrix(np.identity(len(intra),dtype=complex)) # create idntity
ldos = np.array([0.0 for i in range(dimhc)]) # initialice ldos
# initialize ldos
for energy in energies: # loop over energies
# right green function
gr = None
gl = None
# perform dyson calculation
intra = hetero.right_intra
inter = hetero.right_inter
gr = dyson(intra,inter,energy=energy,gf=gf)
hetero.right_green = gr # save green function
# left green function
intra = hetero.left_intra
inter = hetero.left_inter
gl = dyson(intra,inter,energy=energy,gf=gf)
hetero.left_green = gl # save green function
# save green functions
# hetero.write_green()
# print "Saved green functions"
# left selfenergy
inter = hetero.left_coupling
selfl = inter*gl*inter.H # left selfenergy
# right selfenergy
inter = hetero.right_coupling
selfr = inter*gr*inter.H # right selfenergy
# central green function
intra = hetero.central_intra
# dyson equation for the center
# full matrix
if not hetero.block_diagonal:
heff = intra + selfl + selfr
hetero.heff = heff
gc = (energy+1j*eps)*iden - heff
gc = gc.I # calculate inverse
# get the local density of states
ldos += np.array([-gc[i,i].imag for i in range(len(gc))])
# if save_heff:
# print "Saving effective hamiltonian in ",hetero.file_heff
# hetero.write_heff()
# reduced matrix
if hetero.block_diagonal:
from copy import deepcopy
heff = deepcopy(intra)
heff[0][0] = intra[0][0] + selfl
heff[-1][-1] = intra[-1][-1] + selfr
dd = (energy+1j*gf.eps)*iden
for i in range(len(intra)): # add the diagonal energy part
heff[i][i] = heff[i][i] - dd # this has the wrong sign!!
# now change the sign
for i in range(len(intra)):
for j in range(len(intra)):
try:
heff[i][j] = -heff[i][j]
except:
heff[i][j] = heff[i][j]
# save the green function
hetero.heff = heff
# if save_heff:
# print "Saving effective hamiltonian in ",hetero.file_heff
# from scipy.sparse import bmat
# hetero.heff = bmat(heff)
# hetero.write_heff()
# calculate the inverse
from green import gauss_inverse # routine to invert the matrix
# list with the diagonal matrices
ldos_e = ldos*0.0 # initialice ldos at this energy
ii = 0 # counter for the element
for i in range(len(heff)): # loop over blocks
gci = gauss_inverse(heff,i,i) # calculate each block element
for j in range(len(heff[0][0])): # loop over each block
ldos_e[ii] = -gci[j,j].imag
ii += 1 # increase counter
if not ii==dimhc:
print "Wrong dimensions",ii,dimhc
raise
ldos += ldos_e # add to the total ldos
# save the effective hamiltonian
if hetero.has_spin: # resum ldos if there is spin degree of freedom
ldos = [ldos[2*i]+ldos[2*i+1] for i in range(len(ldos)/2)]
if hetero.has_eh: # resum ldos if there is eh
ldos = [ldos[2*i]+ldos[2*i+1] for i in range(len(ldos)/2)]
# ne = len(ldos)/2
# ldos = [ldos[i]+ldos[ne+i] for i in range(ne)]
ldos = np.array(ldos) # transform into an array
if min(ldos)<0.0:
print "Negative density of states"
print ldos
raise
g = hetero.central_geometry # geometry of the central part
fldos = open("LDOS.OUT","w") # open file for ldos
fldos.write("# X Y LDOS\n")
for (ix,iy,il) in zip(g.x,g.y,ldos):
fldos.write(str(ix)+" "+str(iy)+" "+str(il)+"\n")
fldos.close()
# scale the ldos
# save the LDOS in a file
if True:
# if True>0.001:
# ldos = np.sqrt(ldos)
# ldos = np.arctan(7.*ldos/max(ldos))
print "Sum of the LDOS =",sum(ldos)
ldos = ldos*300/max(ldos)
else:
ldos = ldos*0.0
# now create the figure
fig = py.figure() # create figure
fig.subplots_adjust(0.2,0.2)
fig.set_facecolor("white") # face in white
sldos = fig.add_subplot(111) # create subplot for the DOS
# plot the lattice
if not len(g.x)==len(ldos):
raise
sldos.scatter(g.x,g.y,color="red",s=ldos) # plot the lattice
sldos.scatter(g.x,g.y,color="black",s=4) # plot the lattice
sldos.set_xlabel("X")
sldos.set_xlabel("Y")
sldos.axis("equal") # same scale in axes
return fig
from time import clock
es = np.linspace(-3.0,3.0,500)
told = clock()
if True:
bulk = []
surf = []
for e in es:
dos_bulk,dos_surf = green.green_renormalization(h.intra,
h.inter,energy=e)
bulk.append(-dos_bulk.trace()[0,0].imag)
surf.append(-dos_surf.trace()[0,0].imag)
py.plot(es,surf,color="blue",marker="o")
py.plot(es,bulk,color="red",marker="x")
print told-clock()
told = clock()
if False:
for e in es:
dos = green.dyson(h.intra,h.inter,energy = e)
dos = -dos.trace()[0,0].imag
py.scatter(e,dos,color="red",marker="x")
print told-clock()
py.show()
def landauer(hetero, energies=0.0, gf=None, delta=0.001):
""" Calculates transmission using Landauer formula"""
from green import dyson
from green import gf_convergence
import green
if gf == None: gf = gf_convergence("lead")
from hamiltonians import is_number
if not hetero.block_diagonal:
intra = hetero.central_intra # central intraterm
dimhc = len(intra) # dimension of the central part
if hetero.block_diagonal:
intra = hetero.central_intra[0][0] # when it is diagonal
# dimension of the central part
dimhc = len(hetero.central_intra) * len(intra)
iden = np.matrix(np.identity(len(intra), dtype=complex)) # create idntity
trans = [] # list with the transmission
for energy in energies: # loop over energies
intra = hetero.right_intra
inter = hetero.right_inter
grb, gr = green.green_renormalization(intra,
inter,
energy=energy,
delta=delta)
gr = dyson(intra, inter, energy=energy, gf=gf)
hetero.right_green = gr # save green function
# left green function
intra = hetero.left_intra
inter = hetero.left_inter
glb, gl = green.green_renormalization(intra,
inter,
energy=energy,
delta=delta)
hetero.left_green = gl # save green function
# left selfenergy
inter = hetero.left_coupling
selfl = inter * gl * inter.H # left selfenergy
# right selfenergy
inter = hetero.right_coupling
selfr = inter * gr * inter.H # right selfenergy
#################################
# calculate spectral functions
#################################
gammar = selfr - selfr.H
gammal = selfl - selfl.H
#################################
# dyson equation for the center
#################################
# central green function
intra = hetero.central_intra
# full matrix
if not hetero.block_diagonal:
heff = intra + selfl + selfr
hetero.heff = heff
gc = (energy + 1j * gf.eps) * iden - heff
gc = gc.I # calculate inverse
G = (gammar * gc * gammal.H * gc.H).trace()[0, 0].real
trans.append(G)
# reduced matrix
if hetero.block_diagonal:
from copy import deepcopy
heff = deepcopy(intra)
heff[0][0] = intra[0][0] + selfl
heff[-1][-1] = intra[-1][-1] + selfr
dd = (energy + 1j * gf.eps) * iden
for i in range(len(intra)): # add the diagonal energy part
heff[i][i] = heff[i][i] - dd # this has the wrong sign!!
# now change the sign
for i in range(len(intra)):
for j in range(len(intra)):
try:
heff[i][j] = -heff[i][j]
except:
heff[i][j] = heff[i][j]
# calculate green function
from green import gauss_inverse # routine to invert the matrix
# calculate only some elements of the central green function
gc1n = gauss_inverse(heff, 0,
len(heff) - 1) # calculate element 1,n
# and apply Landauer formula
G = (gammal * gc1n * gammar.H * gc1n.H).trace()[0, 0].real
trans.append(G)
print "Landauer transmission E=", energy, "G=", G
return trans # return transmission
def landauer(hetero,energies=0.0,gf=None,delta = 0.001):
""" Calculates transmission using Landauer formula"""
from green import dyson
from green import gf_convergence
import green
from hamiltonians import is_number
if not hetero.block_diagonal:
intra = hetero.central_intra # central intraterm
dimhc = len(intra) # dimension of the central part
if hetero.block_diagonal:
intra = hetero.central_intra[0][0] # when it is diagonal
# dimension of the central part
dimhc = len(hetero.central_intra)*len(intra)
iden = np.matrix(np.identity(len(intra),dtype=complex)) # create idntity
trans = [] # list with the transmission
for energy in energies: # loop over energies
intra = hetero.right_intra
inter = hetero.right_inter
grb,gr = green.green_renormalization(intra,inter,
energy=energy,delta=delta)
gr = dyson(intra,inter,energy=energy,gf=gf)
hetero.right_green = gr # save green function
# left green function
intra = hetero.left_intra
inter = hetero.left_inter
glb,gl = green.green_renormalization(intra,inter,
energy=energy,delta=delta)
hetero.left_green = gl # save green function
# left selfenergy
inter = hetero.left_coupling
selfl = inter*gl*inter.H # left selfenergy
# right selfenergy
inter = hetero.right_coupling
selfr = inter*gr*inter.H # right selfenergy
#################################
# calculate spectral functions
#################################
gammar = selfr-selfr.H
gammal = selfl-selfl.H
#################################
# dyson equation for the center
#################################
# central green function
intra = hetero.central_intra
# full matrix
if not hetero.block_diagonal:
heff = intra + selfl + selfr
hetero.heff = heff
gc = (energy+1j*gf.eps)*iden - heff
gc = gc.I # calculate inverse
G = (gammar*gc*gammal.H*gc.H).trace()[0,0].real
trans.append(G)
# reduced matrix
if hetero.block_diagonal:
from copy import deepcopy
heff = deepcopy(intra)
heff[0][0] = intra[0][0] + selfl
heff[-1][-1] = intra[-1][-1] + selfr
dd = (energy+1j*gf.eps)*iden
for i in range(len(intra)): # add the diagonal energy part
heff[i][i] = heff[i][i] - dd # this has the wrong sign!!
# now change the sign
for i in range(len(intra)):
for j in range(len(intra)):
try:
heff[i][j] = -heff[i][j]
except:
heff[i][j] = heff[i][j]
# calculate green function
from green import gauss_inverse # routine to invert the matrix
# calculate only some elements of the central green function
gc1n = gauss_inverse(heff,0,len(heff)-1) # calculate element 1,n
# and apply Landauer formula
G = (gammal*gc1n*gammar.H*gc1n.H).trace()[0,0].real
trans.append(G)
print "Landauer transmission E=",energy,"G=",G
return trans # return transmission
py.figure()
from time import clock
es = np.linspace(-3.0, 3.0, 500)
told = clock()
if True:
bulk = []
surf = []
for e in es:
dos_bulk, dos_surf = green.green_renormalization(h.intra,
h.inter,
energy=e)
bulk.append(-dos_bulk.trace()[0, 0].imag)
surf.append(-dos_surf.trace()[0, 0].imag)
py.plot(es, surf, color="blue", marker="o")
py.plot(es, bulk, color="red", marker="x")
print told - clock()
told = clock()
if False:
for e in es:
dos = green.dyson(h.intra, h.inter, energy=e)
dos = -dos.trace()[0, 0].imag
py.scatter(e, dos, color="red", marker="x")
print told - clock()
py.show()