# Aestimo Tutorial #4¶

In this tutorial, we are going to model optical intersubband transitions between the conduction subbands of a quantum well. In the conduction band, intersubband transitions only couple to the electric field component across the structure (along the growth direction/perpendicular to the plane of the quantum well etc)[ref]. This means that the medium is anisotropic which leads to interesting properties like a shift in the absorption frequency due to a 'depolarisation shift'[ref] as well as standing wave effects[ref]. I could go on about this for a while [my ref] but instead, let's cut straight to the action.

First we will model a 'normal' quantum well structure

In [1]:
import aestimo.aestimo as solver
import aestimo.config as ac
ac.messagesoff = True # turn off logging in order to keep notebook from being flooded with messages.
import numpy as np
import matplotlib.pyplot as plt
import copy
from pprint import pprint
%matplotlib notebook

WARNING aestimo logs automatically to aestimo.log in the current working directory.
INFO using psi_at_inf_cython module

In [2]:
class Structure(object): pass
s1 = Structure() # this will be our datastructure

# TEMPERATURE
s1.T = 15.0 #Kelvin

# COMPUTATIONAL SCHEME
# 0: Schrodinger
# 1: Schrodinger + nonparabolicity
# 2: Schrodinger-Poisson
# 3: Schrodinger-Poisson with nonparabolicity
# 4: Schrodinger-Exchange interaction
# 5: Schrodinger-Poisson + Exchange interaction
# 6: Schrodinger-Poisson + Exchange interaction with nonparabolicity
s1.computation_scheme = 6

# Non-parabolic effective mass function
# 0: no energy dependence
# 1: Nelson's effective 2-band model
# 2: k.p model from Vurgaftman's 2001 paper
s1.meff_method = 2

# Non-parabolic Dispersion Calculations for Fermi-Dirac
s1.fermi_np_scheme = True #needed only for aestimo_numpy2.py

# QUANTUM
# Total subband number to be calculated for electrons
s1.subnumber_e = 3

# APPLIED ELECTRIC FIELD
s1.Fapplied = 0.00 # (V/m)

# GRID
# For 1D, z-axis is choosen
s1.gridfactor = 0.1 #nm
s1.maxgridpoints = 200000 #for controlling the size

# REGIONS
# Region input is a two-dimensional list input.
#         | Thickness (nm) | Material | Alloy fraction | Doping(cm^-3) | n or p type |
s1.material =[
[ 20.0, 'AlGaAs', 0.3, 0.0, 'n'],
[ 11.0, 'GaAs', 0, 2e18, 'n'],
[ 20.0, 'AlGaAs', 0.3, 0.0, 'n'],
]

structure1 = s1

In [3]:
# Initialise structure class
model1 = solver.StructureFrom(structure1,adatabase) # structure could also be a dictionary.

#calculate QW states
result1 = solver.Poisson_Schrodinger(model1)

#solver.save_and_plot(result,model) # Write the simulation results in files
solver.QWplot(result1,figno=None) # Plot QW diagram
solver.logger.info("Simulation is finished.")

INFO Total layer number: 3
INFO Total number of materials in database: 23
WARNING The calculation of Vxc depends upon m*, however when non-parabolicity is also
considered m* becomes energy dependent which would make Vxc energy dependent.
Currently this effect is ignored and Vxc uses the effective masses from the
bottom of the conduction bands even when non-parabolicity is considered
elsewhere.
INFO calculation time  3.39166 s

INFO Simulation is finished.


Now we will calculate the intersubband transitions using the appropriately named 'intersubband_optical_transitions' module. The module contain several different models starting with semiclassical models and finishing with a perturbative quantum mechanical model that accounts for collective interactions between all of the transitions. The theory behind the intersubband transitions is a little more complicated than might be expected, see [?] for details.

In [4]:
import aestimo.intersubband_optical_transitions as isbt

# set dielectric constants
case = 2

if case==1: #scalar dielectric constants
eps_b = 12.90
eps_z = 12.90

elif case==2: #z-dependent dielectric constants
eps_b = 10.364
eps_gaas = 10.364 # @ 16um
eps_algaas = 8.2067
#eps_z = isbt.eps_background_GaAs(model,eps_gaas,eps_algaas)

eps_z = np.zeros(model1.n_max)
position = 0.0 # keeping in nanometres (to minimise errors)
for layer in model1.material:
startindex = solver.round2int(position*1e-9/model1.dx)
position += layer[0] # update position to end of the layer
finishindex = solver.round2int(position*1e-9/model1.dx)
#
matType = layer[1]
if matType == 'GaAs':
eps_z[startindex:finishindex] = eps_gaas
elif matType == 'AlGaAs':
eps_z[startindex:finishindex] = eps_algaas

elif case==3: #w-dependent dielectric constants
#because the zeroth axis is assumed to be the z-axis, our eps_z array must be 2d
pass
#currently the matrix model doesn't cope with frequency dependent dielectric constants
#therefore the classical model would be the best approach (model2) although it seems
#to over-estimate the coupling between the transitions.
#Alternatively, we could resolve the matrix at each frequency (for each value of the
#background dielectric constant which would be accurate but may be quite computationally
#intensive.

elif case==4: #z-dependent and w-dependent dielectric constants
pass
#currently the matrix model doesn't cope with frequency dependent dielectric constants
#therefore the classical model is the best approach (model2) although it seems
#to over-estimate the coupling between the transitions.
#Alternatively, we could resolve the matrix at each frequency (for each value of the
#background dielectric constant which would be accurate but may be quite computationally
#intensive.

#Set thickness of effective medium
#Lperiod = sum([layer[0] for layer in model.material])*1e-9 #m
Lperiod = model1.x_max #m

# Define Linewidth function (or constant)
def linewidth(freq): return 0.1*freq #define linewidth in THz

# Optical Intersubband Transitions
transitions_table,(hdr,units) = isbt.transitions(result1,Lperiod,eps_z,linewidth)

isbt.print_levels(result1)
isbt.print_transitions(transitions_table,hdr,units)


INFO the energy levels\population are (meV)	(m**-2):
INFO 1019.9	2.1643e+16
INFO 1092.7	3.5687e+14
INFO 1197.5	2.2983e-21
INFO T = 15K
INFO the energy levels gaps are
INFO (meV)	(THz)	(um)	(wavno)
INFO 72.8	17.6	17	587
INFO 178	42.9	6.98	1.43e+03
INFO 105	25.3	11.8	845
INFO Summary of Intersubband Transitions
INFO        j                        0          1          2
INFO   ilevel                        0          0          1
INFO   flevel                        1          2          2
INFO       dE           meV       72.8        178        105
INFO     freq           THz       17.6       42.9       25.3
INFO   lambda            um         17       6.98       11.8
INFO    wavno          cm-1        587   1.43e+03        845
INFO       dN 1e11cm-2 @15K       21.3       21.6      0.357
INFO     z_if            nm      -2.65      0.321         -3
INFO        f                    0.921     0.0328       1.69
INFO     Leff            nm       9.29       15.4       6.83
INFO     S_if            nm      0.821      0.203      0.776
INFO   S_if_b            nm      0.822      0.203      0.777
INFO       wp           THz       16.2       12.7       2.44
INFO        R           THz       2.06      0.393      0.362
INFO  Lperiod            nm         51         51         51
INFO     y_if           THz       1.76       4.29       2.53
INFO    eps_w                     10.3       10.3       10.3

In [5]:
# Save default rcParams so we can reset them later
# WARNING: Do not run this cell after changing rcParams, as it will overwrite the
# defaults that we are trying to preserve.
rcdef = plt.rcParams.copy()

In [6]:
newparams = {'savefig.dpi': 120, 'legend.fontsize': 'smaller'}
#newparams = {'axes.labelsize': 14, 'axes.linewidth': 1, 'savefig.dpi': 150,
#             'lines.linewidth': 1.5, #'figure.figsize': (8, 3),
#             'figure.subplot.wspace': 0.4,
#             'ytick.labelsize': 12, 'xtick.labelsize': 12,
#             'legend.fontsize': 12, 'legend.frameon': False,
#             'legend.handlelength': 1.5}

# Update the global rcParams dictionary with the new parameter choices
# Before doing this, we reset rcParams to its default again, just in case
plt.rcParams.update(rcdef)
plt.rcParams.update(newparams)
f1=isbt.plotting_absorption(model1,result1,transitions_table,eps_b,eps_z,linewidth)

INFO calc_wR_multiplasmon: using eigh() solver for Hermitian matrix
INFO calc_wR_multiplasmon2: using eigh() solver for Hermitian matrix
INFO calc_wR_multiplasmon3: using eigh() solver for Hermitian matrix


In the plot above, the vertical blue lines mark the energy gaps between the quantum well's energy levels. We can also see that apart from the 'naive' model, the three different models are almost in agreement for this case. The result shows that the absorption strength of the lower transition has been shifted to a higher frequency which is an example of the 'depolarisation shift'. It also appears that the first and second transitions have almost merged, in fact we often find that when transitions are close, the higher energy transition will 'steal' the transition strength of the lower one due to the coulombic interactions between the different transitions.

Let's model an even wider quantum well to show what happens when there are many transtions interacting.

In [7]:
structure2 = copy.copy(structure1)

# DOPING
Nd = 1e18

structure2.material =[
[ 10.0, 'AlGaAs', 0.3, 0.0, 'n'],
[ 30.0, 'GaAs', 0, Nd, 'n'],
[ 10.0, 'AlGaAs', 0.3, 0.0, 'n']]

structure2.subnumber_e = 7

# Initialise structure class
model2 = solver.StructureFrom(structure2,adatabase) # structure could also be a dictionary.

INFO Total layer number: 3
INFO Total number of materials in database: 23


This example also allows me to show that sometimes the Poisson-Schrodinger algorithm needs to be optimised. Using the standard parameters, we get weird results. If we run aestimo on the command-line we can see how the energy-state solutions evolve as the algorithm iterates; if we looked at the output for this simulation, we would see that the solutions are drastically changing between each iteration, the algorithm has gone into oscillations rather than tending towards the correct solution. There will probably be a warning that aestimo has reached the maximum number of iterations.

In [8]:
ac.damping = 0.5 # standard value of this parameter

#calculate QW states
result2 = solver.Poisson_Schrodinger(model2)

#solver.save_and_plot(result,model) # Write the simulation results in files
solver.QWplot(result2,figno=None) # Plot QW diagram
solver.logger.info("Simulation is finished.")

WARNING The calculation of Vxc depends upon m*, however when non-parabolicity is also
considered m* becomes energy dependent which would make Vxc energy dependent.
Currently this effect is ignored and Vxc uses the effective masses from the
bottom of the conduction bands even when non-parabolicity is considered
elsewhere.
WARNING Have reached maximum number of iterations
INFO calculation time  27.6719 s

INFO Simulation is finished.


Now, we change one of the parameters in the configuration module so that the algorithm is more 'damped'. This allows the algorithm to find the correct solution.

In [9]:
# Optimise stability of Poisson-Schrodinger Algorithm
ac.damping = 0.3

#calculate QW states
result2 = solver.Poisson_Schrodinger(model2)

#solver.save_and_plot(result,model) # Write the simulation results in files
solver.QWplot(result2,figno=None) # Plot QW diagram
solver.logger.info("Simulation is finished.")

WARNING The calculation of Vxc depends upon m*, however when non-parabolicity is also
considered m* becomes energy dependent which would make Vxc energy dependent.
Currently this effect is ignored and Vxc uses the effective masses from the
bottom of the conduction bands even when non-parabolicity is considered
elsewhere.
INFO calculation time  12.5607 s

INFO Simulation is finished.

In [10]:
# set dielectric constants
case = 2

if case==1: #scalar dielectric constants
eps_b2 = 12.90
eps_z2 = 12.90

elif case==2: #z-dependent dielectric constants
eps_b2 = 10.364
eps_gaas = 10.364 # @ 16um
eps_algaas = 8.2067
eps_z2 = np.zeros(model2.n_max)

position = 0.0 # keeping in nanometres (to minimise errors)
for layer in model1.material:
startindex = solver.round2int(position*1e-9/model2.dx)
position += layer[0] # update position to end of the layer
finishindex = solver.round2int(position*1e-9/model2.dx)
#
matType = layer[1]
if matType == 'GaAs':
eps_z2[startindex:finishindex] = eps_gaas
elif matType == 'AlGaAs':
eps_z2[startindex:finishindex] = eps_algaas

#Set thickness of effective medium
#Lperiod = sum([layer[0] for layer in model.material])*1e-9 #m
Lperiod2 = model2.x_max #m

# Optical Intersubband Transitions
transitions_table2,(hdr,units) = isbt.transitions(result2,Lperiod2,eps_z2,linewidth)

isbt.print_levels(result2)
isbt.print_transitions(transitions_table2,hdr,units)

#plot output
isbt.plotting_absorption(model2,result2,transitions_table2,eps_b2,eps_z2,linewidth)

plt.gca().set_xlim(xmin=0,xmax=30)
plt.gca().set_ylim(bottom=0,top=0.09)

INFO the energy levels\population are (meV)	(m**-2):
INFO 1002.7	1.4672e+16
INFO 1015.4	1.1073e+16
INFO 1039	4.2554e+15
INFO 1071.5	3.8099e+08
INFO 1111.1	1.881e-05
INFO 1156.3	1.2895e-20
INFO 1205	3.0188e-37
INFO T = 15K
INFO the energy levels gaps are
INFO (meV)	(THz)	(um)	(wavno)
INFO 12.7	3.07	97.8	102
INFO 36.3	8.78	34.1	293
INFO 68.8	16.6	18	555
INFO 108	26.2	11.4	874
INFO 154	37.1	8.07	1.24e+03
INFO 202	48.9	6.13	1.63e+03
INFO 23.6	5.72	52.5	191
INFO 56.1	13.6	22.1	452
INFO 95.7	23.2	12.9	772
INFO 141	34.1	8.8	1.14e+03
INFO 190	45.9	6.54	1.53e+03
INFO 32.5	7.85	38.2	262
INFO 72.1	17.4	17.2	582
INFO 117	28.4	10.6	946
INFO 166	40.1	7.47	1.34e+03
INFO 39.6	9.59	31.3	320
INFO 84.8	20.5	14.6	684
INFO 134	32.3	9.28	1.08e+03
INFO 45.2	10.9	27.4	364
INFO 93.9	22.7	13.2	758
INFO 48.7	11.8	25.4	393
INFO Summary of Intersubband Transitions
INFO        j                        0          1          2          3
INFO   ilevel                        0          0          0          0
INFO   flevel                        1          2          3          4
INFO       dE           meV       12.7       36.3       68.8        108
INFO     freq           THz       3.07       8.78       16.6       26.2
INFO   lambda            um       97.8       34.1         18       11.4
INFO    wavno          cm-1        102        293        555        874
INFO       dN 1e11cm-2 @15K        3.6       10.4       14.7       14.7
INFO     z_if            nm      -6.43   -0.00719     -0.627     0.0432
INFO        f                    0.927   3.32e-06     0.0478   0.000358
INFO     Leff            nm       21.6       27.2       30.6       33.5
INFO     S_if            nm       2.07      0.573      0.269      0.156
INFO   S_if_b            nm       2.07      0.574      0.269      0.156
INFO       wp           THz       4.64       7.03       7.87       7.52
INFO        R           THz      0.964     0.0031      0.442     0.0382
INFO  Lperiod            nm         50         50         50         50
INFO     y_if           THz      0.307      0.878       1.66       2.62
INFO    eps_w                     9.28       9.28       9.28       9.28
INFO        j                        4          5          6          7
INFO   ilevel                        0          0          1          1
INFO   flevel                        5          6          2          3
INFO       dE           meV        154        202       23.6       56.1
INFO     freq           THz       37.1       48.9       5.72       13.6
INFO   lambda            um       8.07       6.13       52.5       22.1
INFO    wavno          cm-1   1.24e+03   1.63e+03        191        452
INFO       dN 1e11cm-2 @15K       14.7       14.7       6.82       11.1
INFO     z_if            nm     -0.214     0.0812      -6.48     0.0331
INFO        f                   0.0124    0.00236       1.75   0.000109
INFO     Leff            nm       36.2       40.3         14       21.5
INFO     S_if            nm      0.102     0.0694       1.72      0.469
INFO   S_if_b            nm      0.102     0.0695       1.72      0.469
INFO       wp           THz       7.24       6.85       7.94       8.15
INFO        R           THz      0.225     0.0981       1.82     0.0183
INFO  Lperiod            nm         50         50         50         50
INFO     y_if           THz       3.71       4.89      0.572       1.36
INFO    eps_w                     9.28       9.28       9.28       9.28
INFO        j                        8          9         10         11
INFO   ilevel                        1          1          1          2
INFO   flevel                        4          5          6          3
INFO       dE           meV       95.7        141        190       32.5
INFO     freq           THz       23.2       34.1       45.9       7.85
INFO   lambda            um       12.9        8.8       6.54       38.2
INFO    wavno          cm-1        772   1.14e+03   1.53e+03        262
INFO       dN 1e11cm-2 @15K       11.1       11.1       11.1       4.26
INFO     z_if            nm     -0.747      0.112     -0.313      -6.57
INFO        f                   0.0943     0.0031     0.0328       2.48
INFO     Leff            nm       26.5       31.2       35.4       10.3
INFO     S_if            nm      0.223      0.129     0.0844       1.69
INFO   S_if_b            nm      0.223      0.129     0.0846       1.69
INFO       wp           THz       7.34       6.77       6.36        7.3
INFO        R           THz      0.539     0.0977      0.318       1.71
INFO  Lperiod            nm         50         50         50         50
INFO     y_if           THz       2.32       3.41       4.59      0.785
INFO    eps_w                     9.28       9.28       9.28       9.28
INFO        j                       12         13         14         15
INFO   ilevel                        2          2          2          3
INFO   flevel                        4          5          6          4
INFO       dE           meV       72.1        117        166       39.6
INFO     freq           THz       17.4       28.4       40.1       9.59
INFO   lambda            um       17.2       10.6       7.47       31.3
INFO    wavno          cm-1        582        946   1.34e+03        320
INFO       dN 1e11cm-2 @15K       4.26       4.26       4.26   3.81e-07
INFO     z_if            nm     0.0962     -0.811      0.201       -6.7
INFO        f                  0.00118      0.136     0.0119       3.14
INFO     Leff            nm       17.8       23.2       29.7       8.38
INFO     S_if            nm      0.442      0.208      0.115        1.7
INFO   S_if_b            nm      0.442      0.208      0.115        1.7
INFO       wp           THz       5.56       4.87        4.3    0.00242
INFO        R           THz     0.0374      0.402      0.119   0.000577
INFO  Lperiod            nm         50         50         50         50
INFO     y_if           THz       1.74       2.84       4.01      0.959
INFO    eps_w                     9.28       9.28       9.28       9.28
INFO        j                       16         17         18         19
INFO   ilevel                        3          3          4          4
INFO   flevel                        5          6          5          6
INFO       dE           meV       84.8        134       45.2       93.9
INFO     freq           THz       20.5       32.3       10.9       22.7
INFO   lambda            um       14.6       9.28       27.4       13.2
INFO    wavno          cm-1        684   1.08e+03        364        758
INFO       dN 1e11cm-2 @15K   3.81e-07   3.81e-07   1.88e-20   1.88e-20
INFO     z_if            nm      0.182     -0.891      -6.86      0.307
INFO        f                  0.00499      0.187       3.75     0.0157
INFO     Leff            nm       15.5       21.1       7.21       14.3
INFO     S_if            nm      0.431      0.201       1.74      0.422
INFO   S_if_b            nm      0.431      0.201       1.74      0.422
INFO       wp           THz    0.00178    0.00153    5.8e-10   4.12e-10
INFO        R           THz    2.3e-05   0.000141    1.4e-10   9.06e-12
INFO  Lperiod            nm         50         50         50         50
INFO     y_if           THz       2.05       3.23       1.09       2.27
INFO    eps_w                     9.28       9.28       9.28       9.28
INFO        j                       20
INFO   ilevel                        5
INFO   flevel                        6
INFO       dE           meV       48.7
INFO     freq           THz       11.8
INFO   lambda            um       25.4
INFO    wavno          cm-1        393
INFO       dN 1e11cm-2 @15K   1.29e-35
INFO     z_if            nm      -7.12
INFO        f                     4.36
INFO     Leff            nm       6.47
INFO     S_if            nm        1.8
INFO   S_if_b            nm        1.8
INFO       wp           THz    1.6e-17
INFO        R           THz   3.96e-18
INFO  Lperiod            nm         50
INFO     y_if           THz       1.18
INFO    eps_w                     9.28

INFO calc_wR_multiplasmon: using eigh() solver for Hermitian matrix
INFO calc_wR_multiplasmon2: using eigh() solver for Hermitian matrix
INFO calc_wR_multiplasmon3: using eigh() solver for Hermitian matrix

Out[10]:
(0, 0.09)

So when there are multiple transitions, we can see that the matrix model gives a very different result compared to treating the transitions independently. In general, we find that the frequency of the absorption will be at a higher frequency than expected and most of the transition strength will be concentrated into a single transition.

1. T. Ando, "Intersubband optical absorption in space-charge layers on semiconductor surfaces", Z. PHysik B, 26, 263-272 (1977)
2. G. Pegolotti, A. Vasanelli, Y. Todorov and C. Sirtori, "Quantum model of coupled intersubband plasmons Phys. Rev. B 90, 035305 (2014)