single_island_ggmap.py

# 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



Cg1 = 2.6
Cg2 = 2.4
Cs = 2.8
Cd = 2.8
Csigma = Cg1 + Cg2 + 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 = 1600.0
# Tunneling amplitudes
gam = 0.5
t0 = np.sqrt(gam/(2*np.pi))




nsingle = 8

# 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,
           (6, 6): 0.5*E_c + 3 * E_c,
           (7, 7): 0.5*E_c + 4 * E_c}


#

coulomb = {}



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,
          (0, 6):t0,
          (1, 6):t0,
          (0, 7):t0,
          (1, 7):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):
            xxx = - Cg2/Csigma * vlst[j2] - Cg1/Csigma * vglst[j1]
            system.change(hsingle={(0,0):0.5*E_c - 3 * E_c + xxx,
                          (1,1):0.5*E_c - 2 * E_c + xxx,
                          (2,2):0.5*E_c - 1 * E_c + xxx,
                          (3,3):0.5*E_c + 0 * E_c + xxx,
                          (4, 4): 0.5*E_c + 1 * E_c + xxx,
                          (5, 5): 0.5*E_c + 2 * E_c + xxx,
                          (6, 6): 0.5*E_c + 3 * E_c + xxx,
                          (7, 7): 0.5*E_c + 4 * E_c + xxx})

            system.solve(masterq=False)
            system.solve(qdq=False)
            stab[j1, j2] = abs(system.current[0])
            #
            #system.add(mulst={0: dV/2, 1: 0})
            #system.solve(qdq=False)
            #stab_cond[j1, j2] = (system.current[0] - 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]:

system.kerntype = 'Pauli'
vpnt, vgpnt = 201, 201
vlst = np.linspace(-200, 200, vpnt)
vglst = np.linspace(-200, 200, 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 = "Vbias = " + str(vbias) + " meV" + " 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)