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1x3_spinful_current.py
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# Prerequisites
from __future__ import division, print_function
#get_ipython().magic(u'matplotlib inline')
import matplotlib.pyplot as plt
import numpy as np
import qmeq
import random
from sympy import *
# Quantum dot parameters
cL1 = 1.6
cR1 = 0.3
cm12 = 1.0
cm13 = 0.5
cU1 = 0.4
cD1 = 0.4
cL2 = 0.8
cR2 = 0.8
cm21 = 1.0
cm23 = 1.0
cU2 = 0.4
cD2 = 0.4
cL3 = 0.3
cR3 = 1.6
cm31 = 0.5
cm32 = 1.0
cU3 = 0.4
cD3 = 0.4
q10 = 0.0
q20 = 0.0
q30 = 0.0
c11 = cL1 + cR1 + cm12 + cm13 + cU1 + cD1
c22 = cL2 + cR2 + cm23 + cm21 + cU2 + cD2
c33 = cL3 + cR3 + cm32 + cm31 + cU3 + cD3
vGU, vGD, vL, vR = symbols("vGU, vGD, vL, vR")
n1, n2, n3 = symbols("n1, n2, n3")
q1 = (cL1*vL+cR1*vR+cU1*vGU+cD1*vGD)*6.24/1000+q10
q2 = (cL2*vL+cR2*vR+cU2*vGU+cD2*vGD)*6.24/1000+q20
q3 = (cL3*vL+cR3*vR+cU3*vGU+cD3*vGD)*6.24/1000+q30
C = Matrix([[c11, -cm12, -cm13],
[-cm21, c22, -cm23],
[-cm31, -cm32, c33]])*6.24
Q = Matrix([[q1 - n1],
[q2 - n2],
[q3 - n3]])
U=0.5*((C.inv()*Q).T*Q)*1000
U0x0 = U.subs([(n1,0),(n2,0),(n3,0)])[0]
U1x1 = U.subs([(n1,1),(n2,0),(n3,0)])[0]
U2x1 = U.subs([(n1,0),(n2,1),(n3,0)])[0]
U3x1 = U.subs([(n1,0),(n2,0),(n3,1)])[0]
U1x2 = U.subs([(n1,2),(n2,0),(n3,0)])[0]
U2x2 = U.subs([(n1,0),(n2,2),(n3,0)])[0]
U3x2 = U.subs([(n1,0),(n2,0),(n3,2)])[0]
mu1 = U1x1 - U0x0
mu2 = U2x1 - U0x0
mu3 = U3x1 - U0x0
U12x1x1 = U.subs([(n1,1),(n2,1),(n3,0)])[0]
U23x1x1 = U.subs([(n1,0),(n2,1),(n3,1)])[0]
U13x1x1 = U.subs([(n1,1),(n2,0),(n3,1)])[0]
Um12 = U12x1x1 - U0x0 - mu1 - mu2
Um23 = U23x1x1 - U0x0 - mu2 - mu3
Um13 = U13x1x1 - U0x0 - mu1 - mu3
U11 = U1x2 - U1x1 - mu1
U22 = U2x2 - U2x1 - mu2
U33 = U3x2 - U3x1 - mu3
omegapres, omegaflip = 1.0, 0.0
vgateup, vgatedown, vbiasL, vbiasR = 0.0, 0.0, 8.0, -8.0
V0=20 # unit meV*nm
a=11 # unit: nm
Ev1=-V0*(1/a+0.5/a)
Ev2=-V0*(1/a+1/a)
Ev3=-V0*(1/a+0.5/a)
Je = 0.00
Jp = 0.00
Jt1 = 0.00
Jt2 = 0.00
Eq1 = 7.0
Eq2 = -4.0
Eq3 = 3.0
# Lead parameters
temp = 0.5
dband = 1200
# Tunneling amplitudes
gam = 0.005
t0 = np.sqrt(gam/(2*np.pi))
t00 = 0.0*t0
nsingle = 6
nstate = 2**nsingle
hsingle = {(0,0): Eq1+Ev1+mu1.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(1,1): Eq1+Ev1+mu1.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(2,2): Eq2+Ev2+mu2.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(3,3): Eq2+Ev2+mu2.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(4,4): Eq3+Ev3+mu3.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(5,5): Eq3+Ev3+mu3.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(0,2): -omegapres,
(0,4): -omegapres/5,
(2,4): -omegapres,
(1,3): -omegapres,
(1,5): -omegapres/5,
(3,5): -omegapres
}
# 0 is up, 1 is down
coulomb = {(0,1,1,0):U11.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(2,3,3,2):U22.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(4,5,5,4):U33.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(0,2,2,0):Um12.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(0,3,3,0):Um12.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(1,2,2,1):Um12.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(1,3,3,1):Um12.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(0,4,4,0):Um13.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(0,5,5,0):Um13.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(1,4,4,1):Um13.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(1,5,5,1):Um13.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(2,4,4,2):Um23.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(2,5,5,2):Um23.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(3,4,4,3):Um23.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(3,5,5,3):Um23.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)])
}
tleads = {(0, 0):-t0, # L, up <-- up
(1, 4):-t0, # R, up <-- up
(2, 1):-t0, # L, down <-- down
(3, 5):-t0}
nleads = 4
# L,up R,up L,down R,down
mulst = {0: -vbiasL, 1: -vbiasR, 2: -vbiasL, 3: -vbiasR}
tlst = {0: temp, 1: temp, 2: temp, 3: temp}
system = qmeq.Builder(nsingle, hsingle, coulomb,
nleads, tleads, mulst, tlst, dband,
kerntype='Lindblad')
# Here we have chosen to use **Pauli master equation** (*kerntype='Pauli'*) to describe the stationary state. Let's calculate the current through the system:
system.solve()
print("System indexing: ")
print(system.indexing)
print("Dots Eigenenergies: ")
print(system.Ea)
################################## density matrix in eigenstate basis #######################################
dm_eigen = np.zeros((nstate,nstate),dtype=np.complex_)
for i in range(0, nstate):
for j in range(0, nstate):
dm_eigen[i,j] = system.get_phi0(i,j)
print("Reduced density matrix: ")
print(dm_eigen)
################################## eigenstates in fock basis full 1x16 array #################################
def fock_coeff():
result = np.zeros((nstate,nstate),dtype=np.complex_)
for i in range(0, nstate):
temp = system.coeff_state(i)
for j in range(0, len(temp)):
result[i,j] = temp[j]
return result
eigenstate_in_fock = fock_coeff()
eig_T = np.transpose(eigenstate_in_fock)
eig_C = np.conjugate(eigenstate_in_fock)
print("Eigenstates in fock basis: ")
print(eigenstate_in_fock)
################################# density matrix in fock basis #########################################
dm_fock = np.matmul(eig_T, np.matmul(dm_eigen, eig_C)) # transform from eigenstate basis to fock basis
print("Dot density matrix in fock basis: ")
print(dm_fock)
for i in range(0,nstate):
system.print_state(i)
################################# construct the current operater here |1d> to |2d> ############################
current_1dx2d = np.zeros((nstate,nstate),dtype=np.complex_)
current_1dx2d[1,2] = 1
current_1dx2d[2,1] = -1
current_1dx2d[5,6] = 1
current_1dx2d[6,5] = -1
current_1dx3d = np.zeros((nstate,nstate),dtype=np.complex_)
current_1dx3d[1,4] = 1
current_1dx3d[4,1] = -1
current_1dx3d[3,6] = 1
current_1dx3d[6,3] = -1
current_2dx3d = np.zeros((nstate,nstate),dtype=np.complex_)
current_2dx3d[2,4] = 1
current_2dx3d[4,2] = -1
current_2dx3d[3,5] = 1
current_2dx3d[5,3] = -1
#print(current_1dx2d)
################################ The current is the trace of product dm_fock and current_1dx2d ##############################
#product = np.matmul(dm_fock,current_1dx2d)
#print("Current 1d->2d is: ")
#print(np.trace(product))
for i in range(0,4):
print("#####################")
def stab_calc(system, bfield, vlst, vglst, dV=0.0001):
vpnt, vgpnt = vlst.shape[0], vglst.shape[0]
stab = np.zeros((vpnt, vgpnt))
stab_cond = np.zeros((vpnt, vgpnt))
print(vpnt)
for j1 in range(vgpnt):
vgateup = vglst[j1]
print(j1)
for j2 in range(vpnt):
vgatedown = vlst[j2]
system.change(hsingle = {(0,0): Eq1+Ev1+mu1.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(1,1): Eq1+Ev1+mu1.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(2,2): Eq2+Ev2+mu2.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(3,3): Eq2+Ev2+mu2.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(4,4): Eq3+Ev3+mu3.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(5,5): Eq3+Ev3+mu3.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(0,2): -omegapres,
(0,4): -omegapres/5,
(2,4): -omegapres,
(1,3): -omegapres,
(1,5): -omegapres/5,
(3,5): -omegapres})
system.change(coulomb = {(0,1,1,0):U11.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(2,3,3,2):U22.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(4,5,5,4):U33.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(0,2,2,0):Um12.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(0,3,3,0):Um12.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(1,2,2,1):Um12.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(1,3,3,1):Um12.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(0,4,4,0):Um13.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(0,5,5,0):Um13.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(1,4,4,1):Um13.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(1,5,5,1):Um13.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(2,4,4,2):Um23.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(2,5,5,2):Um23.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(3,4,4,3):Um23.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)]),
(3,5,5,3):Um23.subs([(vL, vbiasL),(vR, vbiasR),(vGU, vgateup),(vGD, vgatedown)])})
system.solve(masterq=False)
################# ploting the expectation value of 1d->2d current in gate-gate map ########################
system.solve(qdq=False)
#for i in range(0, nstate):
# for j in range(0, nstate):
# dm_eigen[i,j] = system.get_phi0(i,j)
#eigenstate_in_fock = fock_coeff()
#eig_T = np.transpose(eigenstate_in_fock)
#eig_C = np.conjugate(eigenstate_in_fock)
#dm_fock = np.matmul(eig_T, np.matmul(dm_eigen, eig_C))
#product = np.matmul(dm_fock,current_1dx2d)
#stab[j1, j2] = abs(np.trace(product))
#temp = 0
#mini = 0
#for j in range(0,nstate):
# if (system.Ea[j] < mini):
# mini = system.Ea[j]
# temp = j
#result = 0
#if temp == 0:
# result = 0
#elif temp > 0 and temp < 4:
# result = 1
#elif temp > 3 and temp < 7:
# result = 2
#else:
# result = 3
#stab[j1, j2] = result
#for i in range(0, nstate):
# for j in range(0, nstate):
# dm_eigen[i,j] = system.get_phi0(i,j)
#eigenstate_in_fock = fock_coeff()
#eig_T = np.transpose(eigenstate_in_fock)
#eig_C = np.conjugate(eigenstate_in_fock)
#dm_fock = np.matmul(eig_T, np.matmul(dm_eigen, eig_C))
#product = np.matmul(dm_fock,(current_2dx3d + current_1dx3d))
#stab[j1, j2] = abs(np.trace(product))
stab[j1, j2] = system.current[0] + system.current[2]
stab[j1, j2] = abs(stab[j1, j2])
return stab, stab_cond
# We changed the single particle Hamiltonian by calling the function **system.change** and specifying which matrix elements to change. The function **system.add** adds a value to a specified parameter. Also the option *masterq=False* in **system.solve** indicates just to diagonalise the quantum dot Hamiltonian and the master equation is not solved. Similarly, the option *qdq=False* means that the quantum dot Hamiltonian is not diagonalized (it was already diagonalized previously) and just master equation is solved.
#system.kerntype = 'Lindblad'
vpnt, vgpnt = 101, 101
vlst = np.linspace(-200, 600, vpnt)
vglst = np.linspace(-200, 600, vgpnt)
stab, stab_cond = stab_calc(system, 0, vlst, vglst)
# The stability diagram has been produced. Let's see how it looks like:
def stab_plot(stab, stab_cond, vlst, vglst, gam):
(xmin, xmax, ymin, ymax) = np.array([vglst[0], vglst[-1], vlst[0], vlst[-1]])
fig = plt.figure(figsize=(8,6))
#
p1 = plt.subplot(1, 1, 1)
p1.set_xlabel('$V_{g1}(mV)$', fontsize=20)
p1.set_ylabel('$V_{g2}(mV)$', fontsize=20)
p1_im = plt.imshow(stab.T, extent=[xmin, xmax, ymin, ymax], aspect='auto', origin='lower', cmap = plt.get_cmap('Spectral'))
cbar1 = plt.colorbar(p1_im)
cbar1.set_label('Current [unit]', fontsize=20)
plt.tight_layout()
plt.show()
stab_plot(stab, stab_cond, vlst, vglst, gam)