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two_orbital.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 time
import qmeq
import random
# Quantum dot parameters
vgate = 0.0
bfield = 0.0
omega = 0.0
vbias = 20
cL1,cG1,cm = 1.6, 0.6, 1.0
cL2,c12,c21,cG2 = 0.3, 0.2, 0.2, 0.5
cR2,cR1 = 1.6, 0.3
q10 = 0.0
q20 = 0.0
csigma1 = cL1 + cG1 + cm + c12 + cR1
csigma2 = cR2 + cG2 + cm + c21 + cL2
U1 = (csigma2/(csigma1*csigma2-cm**2))/6.24*1000
U2 = (csigma1/(csigma1*csigma2-cm**2))/6.24*1000
Um = (cm/(csigma1*csigma2 - cm**2))/6.24*1000
print(U1)
print(U2)
print(Um)
vgate1, vgate2, vbiasL, vbiasR = 0.0, 0.0, 10.0, 0.0
q1 = (c12*vgate2 + cG1*vgate1 + cL1*vbiasL + cR1*vbiasR)*6.24/1000 + q10
q2 = (c21*vgate1 + cG2*vgate2 + cL2*vbiasL + cR2*vbiasR)*6.24/1000 + q10
mu1 = Um*q2 + U1*q1 - U1/2
mu2 = Um*q1 + U2*q2 - U2/2
omegapres, omegaflip = 0.01*U1, 0.00*U1
Jev = 0.01 * random.random()
Jt1v = 0.01 * random.random()
Jt2v = 0.01 * random.random()
Je = 0.00*U1
Jp = 0.00*U1
Jt1 = 0.00*U1
Jt2 = 0.00*U1
print(Je)
print(Jt1)
print(Jt2)
Eq1 = 0.5 * U1
Eq2 = 0.0 * U1
Eq1h = 0.5 * U1
Eq2h = 0.0 * U1
# Lead parameters
temp = 0.5
dband = 2000
# Tunneling amplitudes
gam = 0.005
t0 = np.sqrt(gam/(2*np.pi))
t00 = 0.0*t0
offset1 = 6
offset2 = 4
nsingle = 8
hsingle = {(0,0): Eq1-mu1+bfield/2,
(1,1): Eq1-mu1-bfield/2,
(2,2): Eq2-mu2+bfield/2,
(3,3): Eq2-mu2-bfield/2,
(4,4): Eq1-mu1 + offset1,
(5,5): Eq1-mu1 + offset1,
(6,6): Eq2-mu2 + offset2,
(7,7): Eq2-mu2 + offset2,
(0,2): -omegapres,
(1,3): -omegapres,
(0,5): -omegapres,
(2,4): -omegapres,
(0,3): -omegaflip,
(1,2): -omegaflip
}
# 0 is up, 1 is down
"""
coulomb = {(0,1,1,0):U1,
(1,2,2,1):Um,
(0,2,2,0):Um,
(1,3,3,1):Um,
(0,3,3,0):Um,
(2,3,3,2):U2}
"""
coulomb = {(0,1,1,0):U1,
(0,4,4,0):U1,
(0,5,5,0):U1,
(1,4,4,1):U1,
(1,5,5,1):U1,
(4,5,5,4):U1,
(2,3,3,2):U2,
(2,6,6,2):U2,
(2,7,7,2):U2,
(3,6,6,3):U2,
(3,7,7,3):U2,
(6,7,7,6):U2,
(1,2,2,1):Um,
(0,2,2,0):Um,
(4,2,2,4):Um,
(5,2,2,5):Um,
(1,3,3,1):Um,
(0,3,3,0):Um,
(4,3,3,4):Um,
(5,3,3,5):Um,
(1,6,6,1):Um,
(0,6,6,0):Um,
(4,6,6,4):Um,
(5,6,6,5):Um,
(1,7,7,1):Um,
(0,7,7,0):Um,
(4,7,7,4):Um,
(5,7,7,5):Um
}
nleads = 12
tleads = {(0, 0):-t0, # L, up <-- up
(1, 0):-t00, # R, up <-- up
(2, 1):-t0, # L, down <-- down
(3, 1):-t00,
(4, 2):-t00,
(5, 2):-t0,
(6, 3):-t00,
(7, 3):-t0,
(8, 4):-t0,
(9, 4):-t00,
(8, 5): -t0,
(9, 5): -t00,
(10, 6): -t00,
(11, 6): -t0,
(10, 7): -t00,
(11, 7): -t0} # R, down <-- down
# lead label, lead spin <-- level spin
vbiasL = vbias/2
vblasR = -vbias/2
# L,up R,up L,down R,down
mulst = {0: -vbiasL, 1: -vbiasR, 2: -vbiasL, 3: -vbiasR,
4: -vbiasL, 5: -vbiasR, 6: -vbiasL, 7: -vbiasR,
8: -vbiasL, 9: -vbiasR, 10: -vbiasL, 11: -vbiasR}
tlst = {0: temp, 1: temp, 2: temp, 3: temp,
4: temp, 5: temp, 6: temp, 7: temp,
8: temp, 9: temp, 10: temp, 11: temp}
system = qmeq.Builder(nsingle, hsingle, coulomb,
nleads, tleads, mulst, tlst, dband,
kerntype='Pauli')
system.solve(masterq=False)
system.solve(qdq=False)
print(system.current[0] + system.current[2] + system.current[4] + system.current[6])
# Here we have chosen to use **Pauli master equation** (*kerntype='Pauli'*) to describe the stationary state. Let's calculate the current through the system:
# In[11]:
#system.solve()
#print('Current:')
#print(system.current)
#print(system.energy_current)
# The four entries correspond to current in $L\uparrow$, $R\uparrow$, $L\downarrow$, $R\downarrow$ lead channels. We see that the current through left lead and right lead channels is conserved up to numerical errors:
# In[12]:
#print('Current continuity:')
#print(np.sum(system.current))
#print(system.indexing)
for i in range(0,4):
print("#####################")
#system.print_state(i)
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):
vgate1 = vglst[j1]
print(j1)
#print(vgate1)
#print(vgate2)
for j2 in range(vpnt):
vgate2 = vlst[j2]
q1 = (c12*vgate2 + cG1*vgate1 + cL1*vbiasL + cR1*vbiasR)*6.24/1000 + q10
q2 = (c21*vgate1 + cG2*vgate2 + cL2*vbiasL + cR2*vbiasR)*6.24/1000 + q20
#print(q1)
#print(q2)
mu1 = Um*q2 + U1*q1 - U1/2
mu2 = Um*q1 + U2*q2 - U2/2
gg = 0.01*U1
system.change(hsingle={(0,0): Eq1-mu1+bfield/2,
(1,1): Eq1-mu1-bfield/2,
(2,2): Eq2-mu2+bfield/2,
(3,3): Eq2-mu2-bfield/2,
(4,4): Eq1-mu1 + offset1,
(5,5): Eq1-mu1 + offset1,
(6,6): Eq2-mu2 + offset2,
(7,7): Eq2-mu2 + offset2,
(0,2): -gg,
(1,3): -gg,
(0,6): -gg,
(1,7): -gg,
(2,4): -gg,
(3,5): -gg,
(4,6): -gg,
(5,7): -gg,
(0,3): -omegaflip,
(1,2): -omegaflip})
system.solve(masterq=False)
system.solve(qdq=False)
stab[j1, j2] = system.current[0] + system.current[2] + system.current[4] + system.current[6] + system.current[8] + system.current[10]
stab[j1, j2] = abs(stab[j1, j2])
#0.000000001 good
#stab[j1, j2] = np.log(stab[j1, j2])
#stab[j1,j2] = mu1-mu2
#
#system.add(mulst={0: dV/2, 1: -dV/2,
#2: dV/2, 3: -dV/2})
#system.solve(qdq=False)
#stab_cond[j1, j2] = (system.current[0] + system.current[2] - stab[j1, j2])/dV
#
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.
# In[19]:
start_time = time.time()
system.kerntype = 'Pauli'
vpnt, vgpnt = 101, 101
vlst = np.linspace(-100, 500, vpnt)
vglst = np.linspace(0, 600, vgpnt)
stab, stab_cond = stab_calc(system, bfield, vlst, vglst)
# The stability diagram has been produced. Let's see how it looks like:
# In[20]:
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)
#
#p2 = plt.subplot(1, 2, 2)
#p2.set_xlabel('$V_{g}/U$', fontsize=20);
#p2.set_ylabel('$V/U$', fontsize=20);
#p2_im = plt.imshow(stab_cond.T, extent=[xmin, xmax, ymin, ymax], aspect='auto', origin='lower', cmap = plt.get_cmap('Spectral'))
#cbar2 = plt.colorbar(p2_im)
#cbar2.set_label('Conductance $\mathrm{d}I/\mathrm{d}V$', fontsize=20)
#
plt.tight_layout()
plt.show()
print("--- %s seconds ---" % (time.time() - start_time))
stab_plot(stab, stab_cond, vlst, vglst, gam)