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single_island_coulomb.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
#from IPython.display import Image
import qmeq
Cg = 2.6
Cs = 2.8
Cd = 2.8
Csigma = 2 * Cg + Cs + Cd
E_c = (1/Csigma)/6.24*1000
Q_0 = 0
# Quantum dot parameters
vgate = 0.0
bfield = 0.0
omega = 0.0
# In[5]:
# Lead parameters
vbias = 2
temp = 0.5
dband = 1000.0
# Tunneling amplitudes
gam = 0.5
t0 = np.sqrt(gam/(2*np.pi))
nsingle = 6
# 0 is up, 1 is down
hsingle = {(0, 0): 0.5*E_c - 3 * E_c,
(1, 1): 0.5*E_c - 2 * E_c,
(2, 2): 0.5*E_c - 2 * E_c,
(3, 3): 0.5*E_c + 0 * E_c,
(4, 4): 0.5*E_c + 1 * E_c,
(5, 5): 0.5*E_c + 2 * E_c}
#
coulomb = {(0,1,1,0): E_c,
(0,2,2,0): E_c,
(0,3,3,0): E_c,
(0,4,4,0): E_c,
(0,5,5,0): E_c,
(1,2,2,1): E_c,
(1,3,3,1): E_c,
(1,4,4,1): E_c,
(1,5,5,1): E_c,
(2,3,3,2): E_c,
(2,4,4,2): E_c,
(2,5,5,2): E_c,
(3,4,4,3): E_c,
(3,5,5,3): E_c,
(4,5,5,4): E_c}
tleads = {(0, 0):t0, # L, up <-- up
(1, 0):t0, # R, up <-- up
(0, 1):t0, # L, down <-- down
(1, 1):t0,
(0, 2):t0,
(1, 2):t0,
(0, 3):t0,
(1, 3):t0,
(0, 4):t0, # L, up <-- up
(1, 4):t0, # R, up <-- up
(0, 5):t0, # L, down <-- down
(1, 5):t0} # R, down <-- down
# lead label, lead spin <-- level spin
nleads = 2
# L,up R,up L,down R,down
mulst = {0: vbias/2, 1: -vbias/2}
tlst = {0: temp, 1: temp}
system = qmeq.Builder(nsingle, hsingle, coulomb,
nleads, tleads, mulst, tlst, dband,
kerntype='Pauli')
system.solve()
print('Pauli 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))
# If we want to change the approach we could redefine the system with the **qmeq.Builder** by specifying the new *kerntype*. It is also possible just to change the value of *system.kerntype*:
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))
#
for j1 in range(vgpnt):
print(j1)
for j2 in range(vpnt):
system.change(hsingle={(0,0):-2.5*E_c + Cs/Csigma * vlst[j2]/2 - Cd/Csigma * vlst[j2]/2 - Cg/Csigma * vglst[j1],
(1,1):-2.5*E_c + Cs/Csigma * vlst[j2]/2 - Cd/Csigma * vlst[j2]/2 - Cg/Csigma * vglst[j1],
(2,2):-2.5*E_c + Cs/Csigma * vlst[j2]/2 - Cd/Csigma * vlst[j2]/2 - Cg/Csigma * vglst[j1],
(3,3):-2.5*E_c + Cs/Csigma * vlst[j2]/2 - Cd/Csigma * vlst[j2]/2 - Cg/Csigma * vglst[j1],
(4, 4): -2.5*E_c + Cs/Csigma * vlst[j2]/2 - Cd/Csigma * vlst[j2]/2 - Cg/Csigma * vglst[j1],
(5, 5): -2.5*E_c + Cs/Csigma * vlst[j2]/2 - Cd/Csigma * vlst[j2]/2 - Cg/Csigma * vglst[j1]})
system.solve(masterq=False)
system.change(mulst={0: vlst[j2]/2, 1: -vlst[j2]/2})
system.solve(qdq=False)
stab[j1, j2] = abs(system.current[0])
#
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]:
system.kerntype = 'Pauli'
vpnt, vgpnt = 201, 201
vlst = np.linspace(-40, 40, vpnt)
vglst = np.linspace(-300, 300, 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=(6,4.2))
#
p1 = plt.subplot(1, 1, 1)
p1.set_xlabel('$V_{g}(meV)$', fontsize=20)
p1.set_ylabel('$V_{b}(meV)$', fontsize=20)
p1_im = plt.imshow(stab.T/gam, extent=[xmin, xmax, ymin, ymax], aspect='auto', origin='lower', cmap = plt.get_cmap('Spectral'))
cbar1 = plt.colorbar(p1_im)
cbar1.set_label('Current [$\Gamma$]', fontsize=20)
tt = " levels = " + str(nsingle) + " Single island"
plt.title(tt, loc='center')
#
#p2 = plt.subplot(1, 2, 2)
#p2.set_xlabel('$V_{g}$', fontsize=20);
#p2.set_ylabel('$V_{b}$', fontsize=20);
#p2_im = plt.imshow(stab_cond.T, extent=[xmin, xmax, ymin, ymax], aspect='auto', origin='lower')
#cbar2 = plt.colorbar(p2_im)
#cbar2.set_label('Conductance $\mathrm{d}I/\mathrm{d}V$', fontsize=20)
#
plt.tight_layout()
plt.show()
stab_plot(stab, stab_cond, vlst, vglst, gam)