∎

11institutetext: Rahul Goswami 22institutetext: Department of Mathematics,
IIT Guwahati,
Guwahati, India
22email<EMAIL_ADDRESS>
33institutetext: Arabin Kumar Dey 44institutetext: Department of Mathematics,
IIT Guwahati,
Guwahati, India
44email<EMAIL_ADDRESS>

# Concordance based Survival Cobra with regression type weak learners

Rahul Goswami and Arabin Kumar Dey

###### Abstract

In this paper, we predict conditional survival functions through a combined
regression strategy. We take weak learners as different random survival trees.
We propose to maximize concordance in the right-censored set up to find the
optimal parameters. We explore two approaches, a usual survival cobra and a
novel weighted predictor based on the concordance index. Our proposed
formulations use two different norms, say, Max-norm and Frobenius norm, to
find a proximity set of predictions from query points in the test dataset. We
illustrate our algorithms through three different real-life dataset
implementations.

###### Keywords:

Survival Tree; COBRA; Concordance Index; Right Censored

††journal:

##  1 Introduction

Combined regression strategy (COBRA) [biau2013cobra , biau2016cobra ] is a
powerful ensemble technique which has its own orientation. Majority of the
work available in prediction of conditional survival function so far uses
Adaboost type of ensemble and its variations [bellot2018boosted ,
solomatine2004adaboost , djebbari2008ensemble ]. No work is available for
predicting conditional survival function using Combined regression strategy
(COBRA).

In this paper, we explore the COBRA ensemble setup by maximizing the
concordance index. We choose different survival tree-based models as weak
learners. The Concordance index (C-index) extends the usage for right-censored
data too. This approach is a combination of several pure regressor-type
models. We use two norms (Frobenius and max) to check similarities between
observations and compare the results. Existing COBRA implementation in
regression setup uses a one-dimensional response variable. In survival
regression, the response variable is the survival function. Therefore it is
represented by a vector of values. COBRA structure implementation changes
mainly due to this functional structure of the response variable.

There is a wide variety of ensemble algorithms [dietterich2000ensemble ,
giraud2014introduction , shalev2014understanding ], with a crushing majority
devoted to linear or convex combinations. Boosting techniques have other
several variants used in different context, like Gradient Boosting
[friedman2002stochastic ], SurvivalBoost.R, SurvivalBoost.T [bellot2018boosted
], XGboost [chen2015xgboost ] which is similar but different than COBRA. A
non-linear way of combining estimators is available by
[mojirsheibani1999combining ]. The adaboost [solomatine2004adaboost ] also
uses threshold parameters. However, Adaboost uses an exponential error
function, whereas original COBRA for normal classification and regression
builds its integration through square error loss. In Survival COBRA, we use
Concordance loss instead of square error loss.

Since the weak learners are Survival Trees we compare COBRA with similar
ensemble structure like Random survival forest. We study the amount of
improvement obtained by combining all the weak-learners. The current study
does not compare among the different existing variations of the adaboost with
current proposed structure of COBRA, since adaboost orientation is totally
different and does not show better result than Random Survival Forest when
C-index is taken as metric [see bellot2018boosted ].

We organize the paper in the following way. In section 2, we provide the
conventional structure of COBRA. We propose a Survival COBRA structure to
predict conditional survival in section 3. An illustration of the real dataset
is available in section 4. We conclude the paper in section 5.

##  2 Usual Cobra

Combined regression strategy (COBRA) combines predictions from different weak
learners to make the prediction. Suppose we have a data set of the following
form : $(X_{i},y_{i})\in R^{d+1};i=1,\ldots,N$. We divide the data set into
training and test set, where test set observations are, $(X_{i},y_{i})\in
R^{d+1};i=1,\ldots,l$. Let’s denote the M weak learners that COBRA uses are,
say
$r_{1}(\cdot),r_{2}(\cdot),r_{3}(\cdot),\ldots,r_{M}(\cdot):R^{d}\rightarrow
R.$ COBRA comes with an objective to find an ensembled strategy to combine all
the regressor to predict a new observation with better accuracy. It uses its
predictor at particular instances ($x$) as a weighted sum of those test
samples $Y_{i}$ where weights are calculated based on some test sample
instances, which are in the neighborhood of predictions made by each predictor
at the particular instances.

Mathematically, the COBRA strategy expresses a predictive estimator $T_{n}$,
which we can write as

$T_{n}(r_{k}(\textbf{x}))=\sum_{i=1}^{l}W_{n,i}(\textbf{x})Y_{i},~{}~{}~{}~{}~{}~{}x\in
R^{d}$

where the random weights $W_{n,i}(\textbf{x})$ take the form :

$W_{n,i}(\textbf{x})=\frac{1_{\cap_{m=1}^{M}{|r_{k,m}(x)-r_{k,m}(X_{i})|\leq\epsilon_{l}}}}{\sum_{j=1}^{l}1_{\cap_{m=1}^{M}{|r_{k,m}(x)-r_{k,m}(X_{i})|\leq\epsilon_{l}}}}.$

Here $\epsilon_{l}$ is a positive number and $\frac{0}{0}=0$ (by convention).
The parameter $\epsilon_{l}$ plays the role of smoothing parameters. In
addition, we can provide an alternative way to express COBRA when all original
observations have the same, equally valued opinion on the importance of the
observation $X_{i}$ (within the range of $\epsilon_{l}$). Generally, the
corresponding $Y_{i}$ forms $T_{n}$ in such cases. However the unanimity
constraint may be relaxed by imposing, for example, a fixed fraction
$\alpha\in\\{\frac{1}{M},\frac{2}{M},\cdots,1\\}.$ of the machines agrees on
the importance of $X_{i}$. In that case, the weights take a more sophisticated
form :

$W_{n,i}(x)=\frac{1_{\sum_{m=1}^{M}1_{|r_{k,m}(x)-r_{k,m}(X_{j})|\leq\epsilon_{l}}\geq
M\alpha}}{\sum_{j=1}^{l}1_{\sum_{m=1}^{M}1_{|r_{k,m}(x)-r_{k,m}(X_{j})|\leq\epsilon_{l}}\geq
M\alpha}}$

Instead of counting the number of observations similar to the query point and
expressing the same in terms of an indicator function, researchers used kernel
approximation of the above expression as a smooth representation so that we
can choose the optimal direction of all parameters.

Key Implementation Issues in above Context :

Since the algorithm requires extraction of similar observations depending on a
particular choice of threshold, it may not always be capable of selecting a
non-zero number of similar samples. This situation will arise as the threshold
value comes closer to zero or exactly equals to zero. A solution to the
problem is to increase the test set size. Alternatively, the algorithm
suggests using a proportion ($\alpha$) of machine predictions instead of
considering all the machine predictions.

##  3 Proposed Survival COBRA

The usual general regression setup uses the response variable/output variable
with only one dimension. The conditional survival prediction predicts a
function, and thus output variable/prediction space is always a higher
dimension (since a set of points represents the function). Therefore usual
COBRA structure does not work in a conditional survival prediction setup. We
propose two Survival COBRAs in this context.

###  3.1 Vanilla Survival Cobra

We perform the experimental procedure in the following way. In the first step,
we focus on the calculation of the parameters. In the second step, we find the
curve estimates and benchmark scores based on the optimized parameters. We
divide the whole set as train and test sets and start optimizing the
parameters based on the train set for an arbitrary split division. The
algorithm will further subdivide the whole dataset into two parts, (say,
$D_{l}$ and $D_{k}$), where all the weak-learners training use $D_{k}$ only.
In this process test set would be a query set or validation set. The
prediction uses $D_{l}$. However, a small original dataset would create
$D_{l}$ much smaller due to such division, which could be a bottleneck for
getting the desired optimal parameters. Therefore, we choose a slightly
different way to perform the split division. First, we use the whole dataset
to find the optimal parameters and then split the dataset into training and
test sets to get the survival estimate at optimized COBRA parameters.

Our proposed algorithm uses weak learners as different survival trees. We take
eight weak-learners and follow up on the corresponding results. Since each
weak-learners are different regressions producing survival functions, the
output dimension is high (depends on the number of time points considered).
While aggregating similar observations in the test set, we first choose a
suitable norm to measure the similarity. We take two different norms in this
case : (1) Frobenius norm and (2) Sup norm, which calculates the maximum of
all prediction distances. In general, the choice of the value of the threshold
parameter ($\epsilon_{l}$) and a proportion of a total number of predictors
($\alpha$) depend on the minimization of the cross-validated test error. Our
current exploration uses the concordance index for right-censored data. In a
different note, we propose to choose this $\epsilon_{l}$ and $\alpha$ by
maximizing concordance index measured for right censored data. We study the
other benchmarks in a separate work.

Concordance Index  The C-index or concordance index $C(t)$ for two patients
$i$ and $j$ can be given by the following expression :
$C(t)=P(\hat{S}_{i}(t)>\hat{S}_{j}(t)|\delta_{i}=1,t\leq T_{j},T_{i}>T_{j})$
where, $T_{i},T_{j}$ are time to event data for patients $i$ and $j$
respectively and $\hat{S}_{i}(t)$, $\hat{S}_{j}(t)$ are the conditional
survival functions for patient $i$ and patient $j$ respectively.
Alternatively, we can choose to use
$C(t)=P(\hat{r}_{i}<\hat{r}_{j}|\delta_{i}=1,t\leq T_{j},T_{i}>T_{j})$ where,
$\hat{r}_{i}$, $\hat{r}_{j}$ are the risk scores corresponding to the $i$th
and $j$-th patient. We define a risk score as total sum of cumulative hazard
function calculated across all predecided time points.

If the data is censored we choose $\delta_{i}=1$. Since, we propose to
maximize C-index to train Survival COBRA, We can call the algorithm as
CindexSurvCobraBoosting.

C-index calculation takes the following steps:

  1. 1.

After forming all possible pair cases, omit those pairs whose shorter survival
time is censored. Omit pairs i and j if $T_{i}=T_{j}$ unless at least one is a
death. Let Permissible denote the total number of permissible pairs.

  2. 2.

For each permissible pair where $T_{i}\neq T_{j}$ , count 1 if the shorter
survival time has worse risk score as defined above; count 0.5 if predicted
outcomes are tied. For each permissible pair, where $T_{i}=T_{j}$ and both are
deaths, count 1 if predicted outcomes are tied; otherwise, count 0.5. For each
permissible pair where $T_{i}=T_{j}$ , but not both are deaths, count 1 if the
death has worse predicted outcome; otherwise, count 0.5. Let Concordance
denote the sum over all permissible pairs.

  3. 3.

he C-index, C, is defined by C = Concordance/Permissible.

Concordance intuitively means that two samples were ordered correctly by the
model. More specifically, two samples are concordant, if the one with a higher
estimated risk score has a shorter actual survival time.

###  3.2 Proposed Algorithm- Weighted Survival Cobra

One of the problems in the above Vanilla COBRA is the approach does not allow
evaluating the survival curve properly for a given patient if a sufficient
number of observations from the test set comes in proximity to that patient
observation. We suggest an alternative procedure to find the survival curve
rectifying the problem of an insufficient number of observations in the
proximity of the query of the point. The proposed methodology works as long as
there is a non-zero observation available in the proximity of the query point.

Key algorithmic steps of the weighted version COBRA are as follows :

  1. 1.

Find optimized parameters from the data set with five-fold cross-validation.

  2. 2.

Divide the data set into train and test. Use test dataset as query point and
further divide train dataset into two subdivisions where the we train the weak
learners on one such subdivision and prediction is made based on the other.

  3. 3.

Collect all similar points in the proximity of the query point based on the
chosen optimized parameters, (there should be at least one).

  4. 4.

Use machine prediction (weak learners) as an average of all the predictions on
the set of similar points.

  5. 5.

Final prediction is the weighted average of all machine predictions where
weights are taken as normalized concordance index by each machine.

The above approach does not hurt the original notion of COBRA as it uses the
contribution of the observed response variables in the weight calculation. The
final prediction ensures that the algorithm generates a survival curve or
cumulative hazard function even when only one observation is available in the
proximity setting.

Key issues/Limitations with Weighted Survival Cobra Although Weighted Survival
COBRA is capable of generating the survival curve in both the suggested norms,
its performance depends on the choice of the optimal parameters. The algorithm
requires at least one observation in the proximity of the query point. The
computational run time increases with more inclusion of machines.

##  4 Data Analysis

###  4.1 Data Set Descriptions

We have taken three datasets for experimentation

  * •

Dataset I : Worcester Heart Attack Study dataset , it has 500 samples and 14
features. Total number of deaths recorded are for 215 patients.

  * •

Dataset II : German Breast Cancer Study Group 2 dataset : The dataset has 686
samples and 8 features. The endpoint is recurrence free survival, which
occurred for 299 patients (43.6%). Response variables are : $cens$: Boolean
indicating whether the endpoint has been reached or the event time is right
censored. $time$: Total length of follow-up.

  * •

Dataset III : Veteran Lung Cancer Data , it has 137 samples and 6 features out
of which number of deaths recorded are 128.

All datasets are publicly available and extracted directly from python module
sksurv.

###  4.2 Numerical Results

All codes are run at IIT Guwahati computers with the following configurations
: (i) Intel(R) Core(TM) i5-6200U CPU 2.30 GHz. (ii) Ubuntu 20.04 LTS OS, and
(iii) 12 GB Memory. All codes in written in python 3.9 in Jupyter Notebook.
All codes can be made available on request to authors.

The conditional survival function obtained for different datasets are
available in Figure-1, Figure-2, Figure-3 and Figure-4. The following 12
Figures show conditional survival curves for different patients in two
different benchmarks and with two different COBRA implementations. We consider
eight survival trees as weak learners to construct the survival functions. In
this experiment, we choose 50% initial division of the whole dataset to form
$D_{l}$ and $D_{k}$ and try to optimize the parameters. We choose 80% and 20%
division ratios for the train and test set for the final prediction during the
final prediction. Two parameters take the optimal value at the cross-validated
test set concordance score. We use five-fold cross-validation to calculate the
above score.

Table-1 provides loss-score calculations for different datasets and for
different norms. We denote COBRA-1 when it uses the Frobenius norm and COBRA-2
when it uses Sup-norm. We randomly select the first consecutive weak learners
and show the loss-score results. We verify whether all score calculation
matches their intuition. On a different note, they should be higher in value
than what COBRA should provide. The calculation of mean and standard deviation
of concordance loss (1 - concordance Index) presented in the table uses 1000
iterations. Each iteration consists of a random split of 80% and 20% division
of the train and test dataset. We compare the results with similar ensemble
techniques like Random Survival Forest and observe that all different COBRAs
perform better or at the same accuracy level. Weighted COBRA beats all
algorithms not only in terms of accuracy but also due to its ability to
produce survival curves.

Table 1: Concordance Index for different Dataset and different COBRA algorithms | DATASET I | DATASET II | DATASET III
---|---|---|---
| mean |  | mean |  | mean |
| sd. |  | sd. |  | sd |
Weighted-Survival COBRA 1 | 0.2813 |  | 0.3690 |  | 0.3832 |
0.0391 |  | 0.040388 |  | 0.0634 |
Weighted-Survival COBRA 2 | 0.2705 |  | 0.350436 |  | 0.3702 |
0.039 |  | 0.033199 |  | 0.0614 |
Vanilla-Survival COBRA 1 | 0.274064 |  | 0.370564 |  | 0.423134 |
0.033473 |  | 0.039722 |  | 0.078164 |
Vanilla-Survival COBRA 2 | 0.310436 |  | 0.388382 |  | 0.395362 |
0.052027 |  | 0.044932 |  | 0.079831 |
Random Survival Forest | 0.2807 |  | 0.3714 |  | 0.4044 |
0.0404 |  | 0.0405 |  | 0.0763 |

##  5 Conclusion

Usual vanilla Survival CoBRA does not provide the desired result in specific
norms and datasets. However, Weighted Survival Cobra works with almost all
datasets and norms. Weighted Survival COBRA provides a better loss score too.
The information provided in this paper can be beneficial to the doctors and
the insurance companies. Since the data uses patient-specific information as
covariates, real-life implementation requires data privacy protection or
consent to access patient data from hospital authorities or patients, which
may not allow real-life integration many times in practice. However, we can
produce the result without taking too much time. It requires storing the
training information separately before running on Mobile or other devices.
There are multiple different ways COBRA architecture can solve the same
problem. Also, the paper does not consider a cause-specific analysis in such a
setup. We explore those problems in separate research work.

##  6 Declarations

We keep all relevant declarations in the following subsections.

### Datasets

All datasets are publicly available in the python module called sksurv.

### Funding and Financial Affiliation of the work

The work is not associated with funding agency.

### Conflicts of interest/Competing interests

The content of this article contains unique contribution. None of the author
has any kind of conflict of interest in contributing for the article. All
authors have consent in contribution of the work.

### Availability of Code

Authors can also provide all relevant codes on request in case of any
difficulty in accessing the above repository.

## References

  * (1) Bellot, A., van der Schaar, M.: Boosted trees for risk prognosis.  In: Machine Learning for Healthcare Conference, pp. 2–16 (2018)
  * (2) Biau, G., Fischer, A., Guedj, B., Malley, J.D.: Cobra: a nonlinear aggregation strategy.  arXiv preprint arXiv:1303.2236 (2013)
  * (3) Biau, G., Fischer, A., Guedj, B., Malley, J.D.: Cobra: A combined regression strategy.  Journal of Multivariate Analysis 146, 18–28 (2016)
  * (4) Chen, T., He, T., Benesty, M., Khotilovich, V., Tang, Y.: Xgboost: extreme gradient boosting.  R package version 0.4-2 pp. 1–4 (2015)
  * (5) Dietterich, T.G.: Ensemble methods in machine learning.  In: International workshop on multiple classifier systems, pp. 1–15. Springer (2000)
  * (6) Djebbari, A., Liu, Z., Phan, S., Famili, F.: An ensemble machine learning approach to predict survival in breast cancer.  International journal of computational biology and drug design 1(3), 275–294 (2008)
  * (7) Friedman, J.H.: Stochastic gradient boosting.  Computational statistics & data analysis 38(4), 367–378 (2002)
  * (8) Giraud, C.: Introduction to high-dimensional statistics, vol. 138.  CRC Press (2014)
  * (9) Mojirsheibani, M.: Combining classifiers via discretization.  Journal of the American Statistical Association 94(446), 600–609 (1999)
  * (10) Shalev-Shwartz, S., Ben-David, S.: Understanding machine learning: From theory to algorithms.  Cambridge university press (2014)
  * (11) Solomatine, D.P., Shrestha, D.L.: Adaboost. rt: a boosting algorithm for regression problems.  In: 2004 IEEE International Joint Conference on Neural Networks (IEEE Cat. No. 04CH37541), vol. 2, pp. 1163–1168. IEEE (2004)

(a) $\xi_{1}$ (b) $\xi_{2}$ (c) $\xi_{3}$

Figure 1: Survival Curve for Vanilla Survival Cobra for three datasets with
Frobenius Norm

(a) $\xi_{1}$ (b) $\xi_{2}$ (c) $\xi_{3}$

Figure 2: Survival Curve for Vanilla Survival Cobra for three datasets with
Max Norm

(a) $\xi_{1}$ (b) $\xi_{2}$ (c) $\xi_{3}$

Figure 3: Survival Curve for Weighted Survival Cobra for three datasets with
Frobenius Norm

(a) $\xi_{1}$ (b) $\xi_{2}$ (c) $\xi_{3}$

Figure 4: Survival Curve for Weighted Survival Cobra for three datasets with
Max Norm


The following article has been submitted to/accepted by Applied Physics
Letters.

After it is published, it will be found at http://apl.aip.org/.

# Spin lifetime measurements in GaAsBi thin films

Brennan Pursley<EMAIL_ADDRESS>Applied Physics Program, University if
Michigan, Ann Arbor, MI 48109  M. Luengo-Kovac  Department of Physics,
University of Michigan, Ann Arbor, MI 48109  G. Vardar  Department of
Materials Science and Engineering, University of Michigan, Ann Arbor, MI 48109
R. S. Goldman  Applied Physics Program, University if Michigan, Ann Arbor, MI
48109  Department of Physics, University of Michigan, Ann Arbor, MI 48109
Department of Materials Science and Engineering, University of Michigan, Ann
Arbor, MI 48109  V. Sih  Applied Physics Program, University if Michigan, Ann
Arbor, MI 48109  Department of Physics, University of Michigan, Ann Arbor, MI
48109

(November 8, 2012)

###### Abstract

Photoluminescence spectroscopy and Hanle effect measurements are used to
investigate carrier spin dephasing and recombination times in the
semiconductor alloy GaAsBi as a function of temperature and excitation energy.
Hanle effect measurements reveal the product of g-factor and effective spin
dephasing time ($gT_{s}$) ranges from 0.8 ns at 40 K to 0.1 ns at 120 K. The
temperature dependence of $gT_{s}$ provides evidence for a thermally activated
effect, which is attributed to hole localization at single Bi or Bi cluster
sites below 40 K.

The field of spintronics, firmly entrenched in giant magnetoresistive (GMR)
read heads for hard-drive technology, is rapidly progressing to encompass
other elements of computer memory such as magnetic random access memory
(MRAM).Keatley  _et al._ (2011) However, a superior spintronic analogue to
silicon compimentary metal-oxide semiconductor (CMOS) logic systems has yet to
be realized. The dilute bismuthides GaAs(1-x)Bix, also referred to as
bismides, are a potentially promising family of semiconductor alloys for both
spintronic and electronic applications. These alloys can be grown on common
GaAs substrates, Bi incorporation has minimal effect on electron mobility,Kini
_et al._ (2009); Cooke  _et al._ (2006) and by incorporating nitrogen, the
significant band gap tunability of both bismuthide and nitride alloys might be
achieved while remaining lattice matched to GaAs.Tiedje, Young, and
Mascarenhas (2008) Furthermore, giant spin-orbit bowing in films with low Bi
incorporation has been reportedFluegel  _et al._ (2006) which allows tuning of
the spin-orbit splitting. Although bismuthide alloys have intriguing
properties that would be useful for scalable spintronic devices, the carrier
spin dynamics in GaAsBi have not yet been reported.

In this work, we investigate spin dephasing and carrier recombination times in
dilute bismuthide thin films using photoluminescence spectroscopy and Hanle
effect measurements. Photoluminescence (PL) below 1.4 eV is attributed to
carrier recombination in the GaAsBi epilayer and we observe a power dependent
blue shift of the emission peak in agreement with existing
literature.Kudrawiec  _et al._ (2009); Imhof  _et al._ (2010); Francoeur  _et
al._ (2008) In addition, we report excitation dependent broadening of the
spectra between the main bismuthide and GaAs emission. Hanle effect
measurements reveal the product of g-factor and effective spin dephasing time
($gT_{s}$) ranges from 0.8 ns at 40 K to 0.1 ns at 120 K with $gT_{s}$ nearly
constant below and decreasing above 40 K. All observed phenomena are
attributed to hole localization that occurs below 40 K at single Bi or Bi
cluster sites.

GaAsBi epilayers were grown by molecular-beam epitaxy on GaAs substrates at
350º C with a growth rate of 0.1 $\mu$m/hr and As2/Ga and Bi/As2 beam
equivalent pressure ratios of 6 and 0.01, respectively. A 100 nm thick
GaAs0.992Bi0.008 film was grown on a 520 nm GaAs buffer layer on a semi-
insulating (001) GaAs substrate. The GaAsBi film composition was confirmed
using Rutherford back-scattering spectroscopy.

Samples were mounted on the cold-finger of a continuous flow liquid helium
cryostat. PL was performed using a tunable-wavelength mode-locked Ti:Sapphire
laser with pulse duration $\sim$3 ps and a repetition rate of 76 MHz as the
excitation source. Two excitation energies, ExA=1.59eV and ExB=1.45eV, were
used to selectively excite carriers either above (ExA) or below (ExB) the
observed GaAs band gap at all temperatures. The excitation and collection
paths were both normal to the sample surface. Incident light was focused to a
75 $\mu$m diameter spot with intensities ranging from 1 W/cm2 to 250 W/cm2. A
grating spectrometer and liquid nitrogen cooled CCD camera with 0.2 nm
resolution were used for analysis. Using both excitation energies over the
entire intensity range, PL spectra were recorded at temperatures between 10 K
and 200 K. Laser scatter was reduced by orthogonally oriented linear
polarizers and long pass filters.

We measured the Hanle effect using either ExA or ExB excitation, with an
irradiance of 23 W/cm2. A variable retarder, placed in the incident path,
allowed for right ($\sigma_{+}$) or left ($\sigma_{-}$) circularly polarized
excitation. A quarter-wave plate, combined with a linear polarizer and placed
in the collection path, selectively passed $\sigma_{+}$ PL. We recorded PL
within the range from 1.30 eV to 1.38 eV, and in our analysis, we present
Hanle data averaged within a $\pm 2$ meV range of selected energies. An
electromagnet was used to generate a field with maximum magnitude of 250 mT in
the sample plane. For each measurement, the applied magnetic field was varied
from -250 mT to +250 mT. At each field step, the $\sigma_{+}$ PL intensity was
recorded for both $\sigma_{+}$ and $\sigma_{-}$ excitation. The entire process
was repeated several times to average out noise due to fluctuations in the
laser power. The relative polarization
$P_{rel}=(I_{\sigma_{+}}-I_{\sigma_{-}})/(I_{\sigma_{+}}+I_{\sigma_{-}})$,
where $I_{\sigma_{j}}$ is the intensity of $\sigma_{+}$ PL under $\sigma_{j}$
excitation, was calculated using the average intensity of a $\pm$2 meV range
about selected PL energies.

The polarization of PL in direct-gap III-V semiconductors is proportional to
the spin polarization of the photoexcited carriers. Equation 1 describes the
temporal evolution of electron spin polarization where $\Omega$ is the Larmor
precession frequency, $\tau_{D}$ is the ensemble dephasing time, $\tau_{R}$ is
the carrier recombination time, and $S_{0}$ is the initial spin
polarization.Meier and Zakharchenya (1984)

$\frac{d\mathbf{S}}{dt}=\mathbf{\Omega}\times\mathbf{S}-\frac{\mathbf{S}}{\tau_{D}}-\frac{\mathbf{S}-\mathbf{S}_{0}}{\tau_{R}}$
(1)

In bulk GaAs, the degeneracy of the light and heavy hole transitions lead to
$S_{0}$=0.5 for transitions involving circularly-polarized light near k =
0.Žutić and Das Sarma (2004) However, $S_{0}$ could have a different value in
bismuthide samples due to carrier trappingFrancoeur  _et al._ (2008) and
strain-induced splitting of the valence bands.Francoeur  _et al._ (2003)
Therefore, our only assumption is that $S_{0}$ is a constant.
$\mathbf{\Omega}$, a function of the applied field, is defined by Eq. 2 where
$g$ is the effective electron g-factor, $\mu_{B}$ is the Bohr magneton, and
$\hbar$ is the reduced Planck’s constant.

$\mathbf{\Omega}=\frac{g\mu_{B}\mathbf{B}}{\hbar}$ (2)

It is convenient to define an effective spin lifetime, $T_{S}$, by Eq. 3.

$\frac{1}{T_{S}}=\frac{1}{\tau_{D}}+\frac{1}{\tau_{R}}$ (3)

In steady state, the combination of Eqs. 1 through 3 yields 4.

$0=\frac{g\mu_{B}}{\hbar}\mathbf{B}\times\mathbf{S}-\frac{\mathbf{S}}{T_{S}}+\frac{\mathbf{S_{0}}}{\tau_{R}}$
(4)

The spin polarization, $S_{0}$, of the photo-generated carriers is defined to
be along $+\mathbf{\hat{z}}$, the same direction as the incident light. By
applying a magnetic field $\mathbf{B}$ in the sample plane, which is
perpendicular to $\mathbf{\hat{z}}$, we obtain

$\displaystyle S_{z}(B)$ $\displaystyle=$
$\displaystyle\frac{S_{z}(0)}{1+(\frac{gT_{s}\mu_{B}}{\hbar}B)^{2}}$ (5)
$\displaystyle S_{z}(0)$ $\displaystyle=$
$\displaystyle\frac{S_{0}}{1+\frac{\tau_{R}}{\tau_{D}}}$ (6)

where $S_{z}(0)$ is the zero-field maximum and $S_{z}(B)$ has a Lorentzian
lineshape. Note that $gT_{s}$, the product of the effective spin lifetime and
g-factor, cannot be uncoupled through Hanle measurements. The corresponding
equations for the emitted PL polarization are

$\displaystyle P(B)$ $\displaystyle=$
$\displaystyle\frac{P(0)}{1+(\frac{gT_{s}\mu_{B}}{\hbar}(B-B_{0}))^{2}}+P_{bkg}$
(7) $\displaystyle P(0)$ $\displaystyle=$
$\displaystyle\frac{S_{0}^{2}}{1+\frac{\tau_{R}}{\tau_{D}}}$ (8)

The $S_{0}^{2}$ in Eq. 8 accounts for both the optical generation and
recombination pathways of spin polarized carriers. $B_{0}$ and $P_{bkg}$ are,
respectively, horizontal and vertical offsets added for better fitting.

Figure 1: a) Power dependent photoluminescence (PL) generated by a Ti:Sapphire
laser tuned to 1.45 eV (ExB) for a sample temperature of 10 K. b) Comparison
of PL, normalized to the dilute bismuthide emission peak, for excitations of
1.45 eV and 1.59 eV (ExA) at 10 K and various powers, as labeled. Red lines
are for ExB and blue lines for ExA excitations, respectively. c) Power
dependence of the bismuthide emission peak location at 10 K for both ExB (red
down triangles) and ExA (blue up triangles).

Figure 1(a) shows PL attributed to carrier recombination in the GaAsBi
epilayer measured at 10 K with ExB at intensities of 2.3 W/cm2 to 207 W/cm2.
Increasing power blue shifts the bismuthide emission peak from 1.32 eV to 1.36
eV and increases the low energy tail. PL measured at 10 K, 40 K, 80 K, and 120
K all exhibit similar power dependence even though PL intensity diminishes
with increasing temperature. Figure 1(b) shows a comparison of ExA and ExB
generated PL collected at 10 K and normalized to the bismuthide signal
maximum, which is below 1.40 eV. The three peaks observed above 1.40 eV are
attributed to emission from the underlying GaAs: 1.46 eV, neutral acceptor
exciton; 1.48 eV, valence to conduction band transition; 1.51 eV, free and
neutral donor excitons.Kang, Woo, and Kim (1996); Zemon  _et al._ (1986) ExA
at 23 W/cm2 yielded roughly the same bismuthide emission peak location as ExB
at 207 W/cm2, but with ExA there is a broadening of the bismuthide emission at
higher energies. At all temperatures, excitation with ExA compared to ExB
generated a broader high energy tail to the bismuthide signal. Furthermore,
low powers of ExA generated PL with the same bismuthide peak locations as high
powers of ExB. The bismuthide peak locations obtained at 10 K for both
excitations as a function of power is shown in Figure 1(c).

We attribute the differences in PL linewidth and bismuthide emission peak
location between excitation energies to carrier transfer from the underlying
GaAs to the dilute bismuthide layer as opposed to sample heating. Migrating
carriers generated in GaAs by ExA would increase the carrier density of the
bismuthide layer and fill higher energy states, continuing the power dependent
trends for ExB, as observed in 1(b-c). PL measurements show that the
bismuthide peak location exhibits minimal shift when the sample temperature is
varied from 10 K to 120 K. Carrier transfer is supported by the theoretical
prediction,Broderick  _et al._ (2011); Usman  _et al._ (2011) and likely
observation,Riordan  _et al._ (2012) of a type I heterojunction at the
GaAsBi/GaAs interface which energetically favors both electron and hole
migration from GaAs to the bismuthide layer. The power dependence of the
bismuthide signal shift is likely due to the filling of first the defect,
single-Bi, and Bi-cluster bound exciton states, followed by the band edges.
Several recent studies support the interpretation of hole localization at Bi
and Bi-cluster sites through Bi atom distributions,Sales  _et al._ (2011);
Ciatto  _et al._ (2008) bandgap transition behavior,Kudrawiec  _et al._
(2009); Imhof  _et al._ (2010); Francoeur  _et al._ (2008) hole mobility,Kini
_et al._ (2011) and hole diffusion.Sales  _et al._ (2011) The measured spin
behavior, discussed below, provides further support for the effects of carrier
transfer and localization.

Figure 2: Temperature dependent Hanle data for excitation energies of a) 1.59
eV (ExA) and b) 1.45 eV (ExB) evaluated at their respective 10 K bismuthide
photoluminescence peak locations of 1.35 eV and 1.33 eV. Polarization and
linewidth monotonically decrease as temperature decreases.

Figure 2 shows the temperature dependent evolution of Hanle data for PL
energies of 1.35 eV with ExA (Fig. 2a) and 1.33 eV with ExB (Fig. 2b), the
respective PL peak energies for each excitation at 10 K. The data were shifted
to remove the magnetic field independent vertical offset $P_{bkg}$ attributed
to laser scatter. Nonlinear least-squares fits were performed to all data with
Eq. 7 and are shown as black lines in Fig. 2. The fitting parameters were
$P(0)$, $gT_{S}$, $P_{bkg}$, and $B_{0}$. For both excitation energies, the
amplitude and linewidth of the Hanle curves increases with increasing
temperatures. Equation 7 shows that the value of $gT_{S}$ has a direct impact
on the observed Hanle linewidth–large $gT_{S}$ leads to small widths and vice
versa. From Eq. 8, and assuming that $S_{0}$ is constant, an increase in
polarization corresponds to a decrease in the ratio $\tau_{R}/\tau_{D}$. Since
$P(0)$ can obtain at least 7$\%$ at 120 K, this implies that at low
temperature, where the polarization has decreased by an order of magnitude,
$\tau_{R}$ is at least an order of magnitude larger than $\tau_{D}$. According
to Eq. 3, a very large $\tau_{R}/\tau_{D}$ value implies $gT_{S}\simeq
g\tau_{D}$. Therefore $g\tau_{D}$ dominates the observed low temperature Hanle
linewidth.

Figure 3:  $P(0)$ as a function of temperature with excitation energies of a)
1.59 eV (ExA) and b) 1.45 eV (ExB) at selected photoluminescence (PL)
energies. $gT_{s}$ as a function of temperature with excitation c) ExA and d)
ExB at selected PL energies and fits (solid black lines) as described in the
text. The blue filled up-triangles and red filled down-triangles respectively
correspond to the ExA and ExB generated Hanle curves shown in Fig. 2. The size
of all data points includes the standard error of the fit values.

Plots of $P(0)$ and $gT_{s}$ as functions of temperature and various PL energy
for both excitations are shown in Fig. 3. Evaluation of ExA generated PL at
1.35 eV reveals an increase in $P(0)$ from 0.1$\%$ to 4$\%$ polarization and a
decrease in $gT_{s}$ from 0.4 ns to 0.1 ns as temperature increases from 10 K
to 120 K. For ExB generated PL, evaluation at 1.33 eV also reveals an increase
of $P(0)$ but from 0.6$\%$ to 7$\%$ and a decrease in $gT_{S}$ from 0.7 ns to
0.2 ns as temperature increases from 10 K to 120 K. The temperature dependence
of $P(0)$ and $gT_{S}$, shown in Fig. 3, appears to show a threshold for
behavior change around 40 K. The change in $gT_{S}$ is more apparent than for
polarization but both occur around 40 K regardless of excitation. As PL energy
increases, P(0) increases in value below 40 K while $gT_{S}$ temperature
dependence remains unchanged. This behavior is attributed to hole localization
which can be achieved by impurities, Bi clusters, or possibly single Bi
atoms.Francoeur  _et al._ (2008) The $P(0)$ dependence on PL energy might be
explained by a correlation of 1.37 eV with free carriers and 1.31 eV to bound
carriers. Recently, a strong nonlinear recombination dependence on Bi
composition was reportedNargelas  _et al._ (2011) indicating that AsGa defects
play a lesser role in carrier lifetime. Also observed was thermal activation
of hole diffusion which was attributed to strong hole localization.
Furthermore, a report of bismuthide energy gap broadeningsKudrawiec  _et al._
(2012) shows that valence to conduction band transitions are much broader than
split-off to conduction band transitions, which is uncommon in III-V alloys
and implies unusually strong perturbations. Bismuth incorporation appears to
strongly perturb the valence band and localize holes at isolated Bi atoms or
clusters.

The role of the D’yakonov-PerelD’yakonov and Perel (1972) and Elliot-
YafetElliott (1954); Yafet (1963) mechanisms in limiting the measured spin
dephasing time should be distinguishable using a temperature dependent power
law model Song and Kim (2002) and an appropriate assumption for the
temperature dependence of the momentum scattering time. However, the sharp
transition in behavior observed at 40 K cannot easily be explained by those
mechanisms, and we have insufficient data at higher temperatures to fit the
measured spin dephasing time to a temperature dependent power law. We believe
that the dominant temperature dependent effect on $gT_{s}$ is due to changes
in carrier localization. Below 40 K, holes are localized and therefore undergo
minimal scattering; above 40 K, the holes are no longer bound and experience
some combination of the above mechanisms. An Arrhenius function (Eq. 9),
analogous to one used in Ref. 20, was used to fit the temperature dependence
of $gT_{s}$ for both excitation energies, and can be seen as black lines in
Fig. 3.

$\frac{1}{gT_{S}}=\alpha e^{-\frac{\Delta E}{kT}}+\frac{1}{g\tau_{0}}$ (9)

$\alpha$ is the pre-exponential factor, T is the sample temperature, k is
Boltzmann’s constant, $\tau_{0}$ is the 0 K limit of the dephasing time, and
both $g$ and $T_{S}$ are the same as previously defined. $\Delta E$ is
interpreted as the thermal activation energy for holes with average fit values
of 33$\pm$8 meV for ExA and 40$\pm$6 meV for ExB. These values are similar to
the value of 46 meV in Ref. 20 when a similar model was applied to the
temperature dependence of hole diffusion in GaAsBi. Future work must isolate
the contribution of hole localization in order to determine the underlying
dephasing mechanisms.

All measurements in this study exhibit an excitation energy dependence which
can be explained by changes in carrier concentration. Spin polarized carriers
optically injected in the underlying GaAs could migrate to the bismuthide
layer, fill the trap states, and substantially increase the carrier population
at the bismuthide band edges. These spins could partially dephase while in
GaAs before crossing the GaAs/GaAsBi interface. Furthermore, spin polarization
of the carriers could scatter as they cross the GaAs/GaAsBi interface, as
reported for GaAs/GaNAs Puttisong  _et al._ (2011) and GaAs/ZnSe Malajovich
_et al._ (2000) heterointerfaces. The additional spin dephasing of transferred
carriers would explain the lower measured polarizations and effective
dephasing times for ExA compared to ExB (see Fig. 3c,d). Carrier transfer
could also account for the difference in $\Delta E$ values since a higher
carrier concentration would lead to a larger number of free carriers, and
hence, lower extracted thermal activation energies.

In conclusion, we present measurements of spin dynamics in a dilute bismuthide
semiconductor alloy using excitation energies both above and below the GaAs
band gap and at temperatures ranging from 10 K to 120 K. The values of
$gT_{s}$, extracted from Hanle effect measurements, range from 0.8 ns at 40 K
to 0.1 ns at 120 K, and are modeled by an Arrhenius function with thermal
activation energies of 33$\pm$8 meV for ExA and 40$\pm$6 meV for ExB.
Observations of broad PL spectra below the GaAs bandgap and thermally
activated behavior are consistent with existing literature and both phenomena
are likely due to hole localization caused by single bismuth or bismuth
cluster sites.

This material is based upon work supported by the National Science Foundation
under Grant No. ECCS-0844908, and the Materials Research Science and
Engineering Center program DMR-1120923. BP was supported in part by the
Graduate Student Research Fellowship under Grant No. DGE 1256260. VS
acknowledges support from the Air Force Office of Scientific Research (Award
No. FA9550-12-1-0258) and the Office of Naval Research (Award No.
N00014-12-1-0519). GV was supported in part by a Fulbright Foreign Student
Fellowship. GV and RSG were supported in part by NSF DMR Grant No. 1006835.

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# Sub-Landau levels in two-dimensional electron system in magnetic field

G.-Q. Hai  email<EMAIL_ADDRESS>Instituto de Física de São Carlos,
Universidade de São Paulo, 13560-970, São Carlos, SP, Brazil

###### Abstract

Considering the role of electron correlation in a two-dimensional (2D) system
in strong magnetic field, we investigate the quantum states of two interacting
electrons in the quantum Hall effect regime. We introduce sub-Landau levels of
the two-electron states by determining their energies and degeneracies. The
effects of electron correlation, Zeeman splitting and energy-level broadening
on the stability of the electron-pair states are discussed. It is shown that
the two electrons in the triplet state with spins aligned parallel to the
magnetic field can form a stable electron pair in the 2D system in GaAs/AlGaAs
heterojunction. A wavefunction of many electron pairs are presented and
compared with the Laughlin wavefunction.

Introduction.$-$ The discovery of quantum Hall effects has significantly
enriched our knowledge of quantum mechanics and condensed matter physics[1, 2,
3]. Although the integer quantum Hall effect (IQHE) is directly related to the
degeneracy of Landau levels in the 2D electron system subjected to a strong
magnetic field, the fractional quantum Hall effect (FQHE) is a manifestation
of strong electronic correlation.[1] The Laughlin wavefunction[4] and Jain’s
composite fermions[5] are important concepts for explaining the complex
phenomena observed in the FQHE, offering complementary perspectives on the
behavior of strongly correlated electrons in the 2D system. However, a simpler
physical picture on the mechanism of electron interaction and correlation is
still highly desired for the quantum Hall effects. In this work we will
present a study on electron pair states in 2D electron system and discuss
their possible relation with the quantum Hall states. Our calculation is
applied to the electron system in GaAs/AlGaAs heterojunction. The obtained
results may provide useful clues for understanding recent experimental
observations showing electron pairing in the quantum Hall regime[6, 7, 8].

For a single electron with charge -$e$ and effective mass $m^{*}_{e}$ confined
in the two-dimensional $(x,y)$ plane and subjected to an external magnetic
field ${\bm{B}}=B\hat{\bm{z}}$ in the $z$ direction, using the symmetric gauge
for vector potential ${\bm{A}}={\bm{B}}\times{\bm{r}}/2=(-By/2,Bx/2,0)$, the
Hamiltonian can be written as

$\widehat{H}_{\rm
s}({\bm{r}})=\frac{1}{2}\left(-i{\bm{\nabla}}+\frac{1}{2}\hat{\bm{z}}\times{\bm{r}}\right)^{2},$
(1)

where the magnetic length $l_{\rm B}=\sqrt{\hbar c/eB}$ and cyclotron energy
$\hbar\omega_{c}=\hbar eB/m^{*}_{e}c$ are used for the length and energy
units, respectively. In the polar coordinates ($r,\varphi$), the corresponding
Schrödinger equation is written as

$\widehat{H}_{\rm s}({\bm{r}})\psi^{\rm s}_{nm}({r,\varphi})=E^{\rm
s}_{nm}\psi^{\rm s}_{nm}({r,\varphi}).$ (2)

The solutions yield the Landau levels $E^{\rm s}_{nm}=E_{\rm L}(n,m)$ with

$E_{\rm L}(n,m)=n+\frac{m+|m|+1}{2},$ (3)

for $m=0,\pm 1,\pm 2,\cdots$, and $n=0,1,2,\cdots$. The corresponding
wavefunctions are given by

$\psi^{\rm s}_{nm}({r,\varphi})=\frac{1}{\sqrt{2\pi}}e^{im\varphi}R^{\rm
s}_{nm}(r),$ (4)

with

$R^{\rm
s}_{nm}(r)=\sqrt{\frac{n!}{(n+|m|)!}}\left(\frac{r^{2}}{2}\right)^{\frac{|m|}{2}}{\rm
L}^{|m|}_{n}(\frac{{}_{r^{2}}}{{}^{2}})e^{-\frac{r^{2}}{4}},$ (5)

where ${\rm L}^{|m|}_{n}(x)$ is the generalized Laguerre polynomial. The
angular momentum of the electron is given by ${\bm{L}}=m\hbar\hat{\bm{z}}$.
The Landau levels are highly degenerate for $m\leq 0$. The degeneracy (i.e.,
the number of single-electron states per unit area) of the lowest Landau level
is given by $n_{\phi_{0}}=1/(2\pi l_{B}^{2})=B/{\phi_{0}}$, where
$\phi_{0}={2\pi\hbar c/e}$ is the quantum of flux.

When there are $N_{e}$ electrons confined in the 2D plane in the magnetic
field, the Hamiltonian of the many-electron system can be written as,

$\displaystyle\widehat{H}=\sum_{i=1}^{N_{e}}\widehat{H}_{s}({\bm{r}}_{i})+\sum_{i=1}^{N_{e}}\sum_{j<i}\frac{\gamma_{B}}{|{\bm{r}}_{i}-{\bm{r}}_{j}|},$
(6)

where ${\bm{r}}_{i}$ denotes the position of electron $i$. The electron-
electron Coulomb interaction strength is measured by parameter
$\gamma_{B}=l_{B}/a^{*}_{B}$ with the effective Bohr radius
$a^{*}_{B}=\epsilon_{0}\hbar^{2}/m^{*}_{e}e^{2}$. The challenge in dealing
with such a system is the electron correlation, which is also the most
important and interesting part of the problem.

Two-electron states.$-$ We will first study the solutions for a two-electron
system in magnetic field by determining their energy levels, wavefunctions,
angular momenta and the degeneracies of the quantum states. The importance of
the electron correlation in the electron pairing mechanism and the stability
of the electron pairs will be discussed. The Hamiltonian of a two-electron
system is given by

$\widehat{H}_{p}({\bm{r}}_{1},{\bm{r}}_{2})=\widehat{H}_{\rm
s}({\bm{r}}_{1})+\widehat{H}_{\rm
s}({\bm{r}}_{2})+\frac{\gamma_{B}}{|{\bm{r}}_{2}-{\bm{r}}_{1}|}.$ (7)

It is known from both classical mechanics and quantum mechanics that two
electrons in a 2D system orbit each other in a strong magnetic field and can
bind together.[9, 10, 11] The problem of two interacting spinless electrons in
a 2D plane in magnetic field has been studied by different authors in the last
decades.[11, 12, 13, 14] Most investigations focused on the so-called quasi-
exact solutions, i.e, for specific values of $\gamma_{B}$, the eigenfunctions
are in closed form given by a product of a polynomial of finite degree and an
exponential function. The problem has also been studied from different point
of views, i.e., 2D electrons or anyons. In the following, we will present the
numerical solutions of the two-electron system in combination with the quasi-
exact solutions. We use the center-of-mass (CM) and relative coordinates
defined as

${\bm{R}}=\frac{1}{\sqrt{2}}({\bm{r}}_{1}+{\bm{r}}_{2})\;\;\;{\rm
and}\;\;\;{\bm{r}}=\frac{1}{\sqrt{2}}({\bm{r}}_{2}-{\bm{r}}_{1}),$ (8)

respectively. In the new coordinates, the two-electron Hamiltonian becomes

$\displaystyle\widehat{H}_{p}({\bm{r}}_{1},{\bm{r}}_{2})=H_{\rm
cm}({\bm{R}})+H_{\rm rel}({\bm{r}}),$ (9)

with

$\displaystyle\widehat{H}_{\rm
cm}({\bm{R}})=\frac{1}{2}\left(-i{\bm{\nabla}}_{\bm{R}}+\frac{1}{2}\hat{\bm{z}}\times{\bm{R}}\right)^{2}$
(10)

and

$\displaystyle\widehat{H}_{\rm
rel}({\bm{r}})=\frac{1}{2}\left(-i{\bm{\nabla}}_{\bm{r}}+\frac{1}{2}\hat{\bm{z}}\times{\bm{r}}\right)^{2}+\frac{\gamma_{B}}{|\bm{r}|}.$
(11)

It is seen that $\widehat{H}_{\rm cm}({\bm{R}})$ and the first part of
$\widehat{H}_{\rm rel}({\bm{r}})$ have the same form as Eq. (1).

The Schrödinger equation for the relative motion of the two electrons is
written as

$\widehat{H}_{\rm rel}({\bm{r}})\psi^{\rm rel}({r,\theta})=E^{\rm
rel}\psi^{\rm rel}({r,\theta}),$ (12)

where ${\bm{r}}=({r,\theta})$. For $\gamma_{B}=0$, the solution of the above
equation reduces to the same expressions for the single-electron state given
by Eqs. (3) and (4). For $\gamma_{B}>0$, the relative motion maintains the
angular symmetry. The orbital angular momentum (${\widehat{L}}_{\rm
rel}=-i\hbar\partial/\partial\theta$) is conserved. The eigenfunction has the
following form,

$\psi^{\rm rel}_{nm}({r,\theta})=\frac{e^{im\theta}}{\sqrt{2\pi}}R^{\rm
rel}_{nm}(r),$ (13)

for $m=0,\pm 1,\pm 2,\cdots$. The angular momentum of the relative motion is
given by ${L}^{\rm rel}_{m}=m\hbar$. And the radial wavefunction can be
written as,

$\displaystyle R_{nm}^{\rm
rel}(r)=\sqrt{\frac{n!}{(n+|m|)!}}\left(\frac{r^{2}}{2}\right)^{\frac{|m|}{2}}h^{|m|}_{n}(\frac{{}_{r^{2}}}{{}^{2}})e^{-\frac{r^{2}}{4}},$
(14)

for $n=0,1,2,\cdots$, and the function $h^{|m|}_{n}(x)$ is defined as

$h^{|m|}_{n}(x)=\sum_{n^{\prime}}a^{(m)}_{nn^{\prime}}\sqrt{\frac{n^{\prime}!(n+|m|)!}{n!(n^{\prime}+|m|)!}}\;{\rm
L}^{|m|}_{n^{\prime}}(x),$ (15)

where the coefficients $a^{(m)}_{nn^{\prime}}$ are determined numerically by
the following linear equations for each $m$,

$\sum_{n^{\prime}}\left[\left(E_{\rm L}(n,m)-E_{nm}^{\rm
rel}\right)\delta_{nn^{\prime}}+\gamma_{B}M^{|m|}_{nn^{\prime}}\right]a^{(m)}_{nn^{\prime}}=0,$
(16)

with the matrix element for the e-e interaction
$M^{|m|}_{nn^{\prime}}=\int_{0}^{\infty}drR^{s}_{n^{\prime}m}(r)R^{s}_{nm}(r)$.

Figure 1: The eigenenergy $E_{nm}^{\rm rel}$ as a function of $\gamma_{B}$
indicated by ($n,m$) for $n=0$ (the black curves), $n=1$ (the red), $n=2$ (the
green), $n=3$ (the blue), and $m\leq 0$ ($m=0,-1,-2,...,-9$) (the solid
curves), $m$=+1 (the dashed), $m$=+2 (the dotted), and $m$=+3 (the dash-
dotted). The symbols show the quasi-exact solutions indicated by $n_{p}$.

Since electrons are fermions, the total wavefunction of the two electrons
(including spins) must by antisymmetric. Therefore, to interchange two
electrons in the relative coordinates $(r,\theta)\to(r,\theta+\pi)$, the
wavefunction in Eq. (13) must have the following property,

$\psi^{\rm rel}_{nm}(r,\theta+\pi)=e^{i{\sigma_{s}\pi}}\psi^{\rm
rel}_{nm}(r,\theta),\;\;\;{\rm for}\;\;\sigma_{s}=0\;{\rm or}\;1.$ (17)

Here $\sigma_{s}=0$ corresponds to the spin-singlet state and $\sigma_{s}=1$
to spin-triplet state. The condition $\sigma_{s}=0$ ($\sigma_{s}=1$) is
satisfied when taking $m$ as an even (odd) number in Eq. (13). Therefore, we
obtain the wavefunctions with $m=0,\pm 2,\pm 4,\cdots$ for singlet and with
$m=\pm 1,\pm 3,\pm 5,\cdots$ for triplet states.

The eigenenergies $E^{\rm rel}_{nm}$ obtained from the numerical solutions of
Eq. (16) are given in Fig. 1 indicated by two quantum numbers ($n,m$). The so-
called quasi-exact solutions at specific $\gamma_{B}$ are also indicated by
the symbols in the figure. The energy $E^{\rm rel}_{nm}$ at $\gamma_{B}=0$
reduces to $E^{\rm rel}_{nm}=n+(m+|m|+1)/2$. For $\gamma_{B}>0$, the Coulomb
interaction lifts the degeneracy of the states with different $m$. The energy
levels with $m\leq-10$ are not shown in the figure. For $m\to-\infty$, $E^{\rm
rel}_{nm}\to n+1/2$ at any $\gamma_{B}$.

The function $h^{|m|}_{n}(x)$ in Eq. (15) is obtained from the numerical
calculations. At specific $\gamma_{B}$ where there exists quasi-exact
solution, $h^{|m|}_{n}(x)$ recovers analytic expression in the form of a
polynomial $h^{|m|}_{n}(r^{2}/2)=\sum_{j=0}^{n_{p}}c_{n,j}^{|m|}r^{j}$
truncated at the $n_{p}^{\rm th}$ power. For instance, at $\gamma_{B}=0$ and
$E^{\rm rel}_{0m}=(1+|m|+m)/2$ (the open red dots in Fig. 1),
$h^{|m|}_{0}(r^{2}/2)$ is a constant with $n_{p}=0$. At
$\gamma_{B}=\sqrt{(2|m|+1)/2}$ and $E^{\rm rel}_{nm}=(2+|m|+m)/2$ (the solid
red dots), $h^{|m|}_{n}(r^{2}/2)$ is a linear function of $r$ with $n_{p}=1$.

The Schrödinger equation for the CM motion is written as

$\widehat{H}_{\rm cm}({\bm{R}})\psi^{\rm cm}_{NM}({R,\Theta})=E^{\rm
cm}_{NM}\psi^{\rm cm}_{NM}({R,\Theta}),$ (18)

where ${\bm{R}}=({R,\Theta})$. The eigenvalue $E^{\rm cm}_{NM}$ and the
eigenfunction $\psi^{\rm cm}_{NM}({R,\Theta})$ of the above equation have the
same expressions as given by Eqs. (3) to (5) with $E^{\rm
cm}_{NM}=E_{L}(N,M)=N+(M+|M|+1)/2$ for $N=0,1,2,...$ and $M$ being an integer.
Since Eqs. (18) and (12) describe the motion of the same electrons, the value
of $M$ depends on $m$ given in Eq. (13). From the periodic conditions in the
$\theta$ and $\Theta$ coordinates for the total wavefunction of the two
electrons, $M$ must be equal to $m$ multiplied by a positive integer, that is,
$M=m\cdot k$ for $k=1,2,3,\cdots$.

Therefore, the total eigenenergy of the two electrons is given by $E^{\rm
pair}_{NM,nm}=E^{\rm cm}_{NM}+E^{\rm rel}_{nm}$ with the corresponding
wavefunction

$\displaystyle\Psi_{NM,nm}({\bm{R}},{\bm{r}})=\psi^{\rm
cm}_{NM}({R,\Theta})\psi_{nm}^{\rm rel}(r,\theta)$
$\displaystyle=\sqrt{\frac{N!n!}{(N+|M|)!(n+|m|)!}}\frac{e^{iM\Theta}}{\sqrt{2\pi}}\left(\frac{R^{2}}{2}\right)^{|M|\over
2}{\rm L}^{|M|}_{N}(\frac{{}_{R^{2}}}{{}^{2}})$
$\displaystyle\;\;\times\frac{e^{im\theta}}{\sqrt{2\pi}}\left(\frac{r^{2}}{2}\right)^{\frac{|m|}{2}}h^{|m|}_{n}(\frac{{}_{r^{2}}}{{}^{2}})\;e^{-\frac{R^{2}+r^{2}}{4}},$
(19)

for $n=0,1,2,\cdots$, $m=0,\pm 1,\pm 2,\cdots$, $N=0,1,2,\cdots$, and $M=mk$
with $k=1,2,3,\cdots$. In which, even $m$ for spin-singlet and odd $m$ for
spin-triplet states. We will call these eigenenergies $E^{\rm pair}_{NM,nm}$
as sub-Landau levels. For fixed quantum numbers $n,m$, and $N$, the sub-Landau
level $E^{\rm pair}_{NM,nm}$ is degenerate for all negative values of $M$. But
it happens only for $m<0$ since $M=mk$. Notice that, in the above solution,
the correlation energy between the two electrons is fully included. The
Coulomb interaction between the electrons does not affect the total angular
momentum $\widehat{L}_{p}=\widehat{L}_{\rm cm}+\widehat{L}_{\rm rel}$, where
$\widehat{L}_{\rm cm}$ and $\widehat{L}_{\rm rel}$ are the CM and relative
angular momentum, respectively. They are gauge invariant and conserved quantum
quantities with the eigenvalues given by ${\bm{L}}^{\rm
cm}_{M}=M\hbar\hat{\bm{z}}$ and ${\bm{L}}^{\rm rel}_{m}=m\hbar\hat{\bm{z}}$.
In fact, $\widehat{H}_{\rm cm}$, $\widehat{H}_{\rm rel}$, $\widehat{L}_{\rm
cm}$ and $\widehat{L}_{\rm rel}$ are mutually commuting operators.

Figure 2: (a) The sub-Landau levels $E^{\rm pair}_{NM,nm}$ indicated by
$(n,m)$ as a function of $B$ for $N$=0. Only the odd $|m|\leq 9$ are shown for
$n=0$ (the black curves), $n=1$ (the red curves), etc. The Landau levels
$n_{s}=0$, 1 and 2 are shown by the thick horizontal lines. (b) The density of
states of the sub-Landau levels for $N$=$n$=0 and $m$=-1,-3,…, -9 at $B$=10 T.

Fig. 2(a) shows the sub-Landau levels $E^{\rm pair}_{NM,nm}$ for $N=0$ (with
$E^{\rm cm}_{0M}=\hbar\omega_{c}/2$) and $m=+3,+1,-1,-3,-5,-7$, and $-9$
(i.e., only the odd $m$ for triplet states are plotted). The energy in the
figure is given by $E^{\rm pair}_{NM,nm}$ divided by 2 (i.e., measured by
energy per electron) in order to compare with the Landau levels indicated by
$n_{s}$. Notice that, the energy levels with $m\leq-10$ are not shown in the
figure. For each set of the sub-Landau levels with fixed $N$, $n$ and
different $m$ (for $m\leq 0$ ), the lowest level is found at
$[(N+n+1)/2]\hbar\omega_{c}$ for $m\to-\infty$. Therefore, the transition from
a Landau level of single-electron states to the sub-Landau levels of electron-
pair states is quasi-continue. It means that there is no energy gap between
them and consequently, any weak e-e repulsion may push electrons in the
single-electron states to the electron-pair states.

To determine the degeneracy of the sub-Landau levels, we will focus on the set
of the lowest energy levels $E^{\rm pair}_{0M,0m}$ = $E^{\rm cm}_{0M}$ \+
$E^{\rm rel}_{0m}$ with $N=n=0$, and $m$=$-1,-2,-3,\cdots$. At $\gamma_{B}=0$
($B\to\infty$), $E^{\rm pair}_{0M,0m}=(M+|M|+1)/2+(m+|m|+1)/2$ which are
degenerate for both negative $M$ and $m$. We can compare Eqs. (7) and (9) of
the same Hamiltonian of the two-electron system in different coordinates. The
above degeneracy is equivalent to two non-interacting electron systems of each
one with degeneracy $n_{\phi_{0}}=B/{\phi_{0}}$. However, for $\gamma_{B}>0$,
the Coulomb interaction lifts the degeneracy in $m$. The sub-Landau level
$E^{\rm pair}_{0M,0m}$ is only degenerate for different $M$ with the
degeneracy being half of the non-interacting systems, but with each state
($0M,0m$) containing two correlated electrons. Notice that, the e-e
interaction does not alter the total angular momentum ${L}^{\rm
cm}_{M}+{L}^{\rm rel}_{m}=M\hbar+m\hbar$. For $m=-1$, the sub-Landau level
$E^{\rm pair}_{0M,0-1}$ is degenerate for $M=-1,-2,-3,\cdots$, with the total
number of the electron-pair states $n_{\phi_{0}}/2$. Thus, the total
degeneracy, namely, the “states for single electrons per unit area” is the
same as that of the lowest Landau level $n_{\phi_{0}}$. For $m=-2$, the sub-
Landau level $E^{\rm pair}_{0M,0-2}$ is degenerate for $M=-2,-4,-6,\cdots$,
and, therefore, the total degeneracy for single electrons is $n_{\phi_{0}}/2$.
In general, for a fixed negative $m$ value, possible values for $M$ are given
by $M=m,2m,3m,\cdots$. Consequently, the total degeneracy $n^{(m)}_{\phi}$ of
the sub-Landau level $E^{\rm pair}_{0M,0m}$ is given by

$n^{(m)}_{\phi}=\frac{n_{\phi_{0}}}{|m|}=\frac{B}{|m|\phi_{0}},\;\;{\rm
for}\;m<0.$ (20)

The density of states of the sub-Landau levels ($0M,0m$) for $m$=-1, -3, …, -9
at $B$=10 T with a small broadening is shown in Fig. 2(b) together with the
lowest Landau level.

Electron pairing.$-$ To better understand the mechanism of electron pairing,
we may “visualize” the two-electron system in Fig. 3. Each electron is
subjected to the magnetic field ${\bm{B}}$ and two electrons interact to each
other according to Coulomb’s law. The e-e interaction includes direct Coulomb
repulsion, exchange interaction and electron correlation. The exchange
interaction determines the symmetry of the total wavefunction. The correlation
attraction due to the duality of the electrons is incorporated in the solution
of the Schrödinger equation. The two electrons circulate around their center
of mass with the angular momentum ${\bm{L}}^{\rm rel}$ (corresponding to a
velocity ${\bm{v}}$). Each electron experiences a Lorentz force
${\bm{F}}_{B}=-e{\bm{v}}\times{\bm{B}}$ from the magnetic field and a direct
Coulomb repulsion force ${\bm{F}}_{C}$ of the other electron. A necessary
condition for such a circulation being stable is a negative relative angular
momentum $L^{\rm rel}_{m}=m\hbar$ and thus the force ${\bm{F}}_{B}$ pointing
to the center of the circle overcoming (partially) the direct Coulomb
repulsion. This means that the potentially stable sub-Landau state of an
electron pair must be diamagnetic with $m<0$. In addition, the electronic
correlation creates a Coulomb hole.[15, 16] From the symmetry of the two-
electron wavefunction (and the charge density distribution), we know that the
center of the Coulomb hole is located at ${\bm{r}}=0$, the exact position of
the CM. The attractive force ${\bm{F}}_{\rm corr}$ exerted on each electron
due to the Coulomb hole is centripetal, pointing always to the CM and ensuring
the stability of the rotation of the two electrons. We understand this intra-
orbital correlation is at the heart of the formation of such a correlated
rotating electron pair (CREP) as shown in Fig. 3. Moreover, the rotating
motion of the electron pair with relative orbital angular momentum $m\hbar$
creates a vortex with vortex charge $m$.

Electron correlation depends on the orbitals of the electrons and can in turn
modify the electron orbitals. The dependence of the correlation energy on the
orbitals leads to different intra- and inter-orbital correlations. For
instance, in atomic systems, the correlation of an electron with the other
occupying the same orbital is much stronger than its correlation with another
electron in a different orbital.[17] In a 2D system of many electrons in
strong magnetic field, the intra-orbital electron correlations of the CREPs in
highly degenerated sub-Landau levels can be essential for the quantum Hall
effects.

Figure 3: Diagram of the correlated rotating electron pair with ${\bm{L}}^{\rm
rel}=m\hbar\hat{\bm{z}}$ ($m<0$).

We will not determine now the exact value of the electron pair correlation
energy. However, from our study on the electron correlation energies in many-
electron atoms and molecules, we know that the intra-orbital electron pair
correlation energy is related to the size of the orbitals. In atoms and
molecules, such a correlation energy is about 20 to 40 millihartree.[17, 16,
18, 19] Although the absolute value of the correlation energy in GaAs
semiconductor is orders of magnitude smaller than that in atoms and molecules,
the nature of the electron correlation is the same but the energy is re-scaled
to the effective Hartree $E^{*}_{\rm
h}=\hbar^{2}/m^{*}_{e}{a^{*}_{B}}^{2}=m^{*}_{e}e^{4}/\epsilon_{0}^{2}\hbar^{2}$.
We can assume that the correlation energy $\varepsilon_{c}$ between two
electrons of a CREP in the sub-Landau level $E^{\rm pair}_{0M,0-1}$ (with
$n=0$ and $m=-1$) is of the order of $10^{-2}E^{*}_{\rm h}$ =10 m$E^{*}_{\rm
h}$ at $B=10$ T. This is consistent with the correlation energy between two
electrons in a quantum ring of the similar orbital size.[20, 21] In
GaAs/AlGaAs heterostructure, $\varepsilon_{c}$ = 10 m$E^{*}_{\rm h}\simeq$
0.12 meV, corresponding to a temperature of 1.4 K. Moreover, this correlation
energy decreases as $|m|$ increases. Therefore, the intra-orbital correlation
energy $\varepsilon_{c}$ can hold the electron pair at low temperature
($\lesssim 1.4$ K) in a very clean GaAs/AlGaAs sample.

Zeeman splitting and stability of the electron pair.$-$ Although the Coulomb
interaction is fully considered in the above two-electron system including the
exchange and correlation effects, the Zeeman splitting and spin-orbital
coupling are not included. The spin of two electrons leads to the additional
Zeeman energy, given by

$E_{S_{z}}=2\Delta_{Z}S_{z}$ (21)

with $\Delta_{Z}=g^{*}\mu_{B}B/2=g^{*}(m_{e}^{*}/4m_{e})\hbar\omega_{c}$,
where $\mu_{B}$ is the Bohr magneton, $g^{*}$ the effective Landé factor, and
$S_{z}$ the spin component of the two electrons. $S_{z}=0$ for the singlet and
$S_{z}=0,\pm 1$ for the triplet states. For GaAs, $m_{e}^{*}=0.067m_{e}$ and
$g^{*}=-0.44$, we obtain $\Delta_{Z}=-0.00737\hbar\omega_{c}$.

For the spin-singlet pair with $S_{z}=0$, two electrons have different spins.
The spin-up and spin-down electrons in magnetic field gain extra energies
$+\Delta_{Z}$ and $-\Delta_{Z}$, respectively. Although the total Zeeman
energy $E_{S_{z}}=0$, the energy splitting between the two electrons is
2$|\Delta_{Z}|=0.0147\hbar\omega_{c}$. At $B=10$ T, this splitting is $0.254$
meV, and it increases with increasing magnetic field. Because 2$|\Delta_{Z}|$
is much larger than the correlation energy $\varepsilon_{c}$ between the
electrons in typical magnetic field for quantum Hall effects, the Zeeman
splitting destabilizes the spin-singlet CREP. For the three spin-triplet
states, they have energy difference $2\Delta_{Z}$ with the state $S_{z}=0$ in
the middle. The state with $S_{z}=-1$ is of highest energy due to $g^{*}<0$ in
GaAs. Both the states with $S_{z}=0$ and $S_{z}=-1$ are unlikely stable mostly
because of the instability of $S_{z}=0$ state with two electrons of opposite
spins. Therefore, the only possible stable state of the CREP in the condition
of quantum Hall effects in GaAs is the triplet state with $S_{z}=+1$ and
energy $E^{\rm pair}_{NM,nm}+2\Delta_{Z}$. This is consistent with Laughlin’s
assumption for a spin polarized Landau level in FQHE[4]. The corresponding
energy levels are the same as given in Fig. 2 but with a downward shift
$\Delta_{Z}=-0.00737\hbar\omega_{c}$.

On the other hand, the so-called disorder effects due to presence of defects
such as impurities and interface roughness in experimental samples play an
important role in the quantum Hall effects. The defects scatter the electrons
in crystal leading to an energy-level broadening. In GaAs, a broadening of
0.12 meV of the single-electron level corresponds to a quantum mobility of
$2\times 10^{5}$ cm2/Vs.[22] Therefore, the primary condition for the
existence of a stable CREP at low temperature is the intra-orbital correlation
energy $\varepsilon_{c}$ being larger than the level broadening. In other
words, the above triplet CREPs can exist only in a very clean sample (with
higher electron mobility). Since $\varepsilon_{c}$ in the CREP decreases with
increasing $|m|$, the sub-Landau levels with very large $|m|$ cannot be formed
in a real sample even with small level broadening.

Many electron-pair wavefunction.$-$ For a system with $N_{p}$ electron pairs
formed by $N_{e}=2N_{p}$ electrons, the Hamiltonian given in Eq. (6) becomes,

$\widehat{H}=\sum_{\alpha=1}^{N_{p}}\widehat{H}_{p}({\bm{r}}_{\alpha,1},{\bm{r}}_{\alpha,2})+\sum_{\beta\
<\alpha}^{N_{p}}\sum_{i,j=1}^{2}\frac{\gamma_{B}}{|{\bm{r}}_{\alpha,i}-{\bm{r}}_{\beta,j}|},$
(22)

where ${\bm{r}}_{\alpha,i}$ represents the pair $\alpha$ with two electrons
$i$=1 and 2. We may denote an electron pair $\alpha$ at
${\bm{\xi}}_{\alpha}=({\bm{r}}_{\alpha,1},{\bm{r}}_{\alpha,2})=({\bm{R}}_{\alpha},{\bm{r}}_{\alpha})$
in the state $\mu=(NM,nm)$ with wavefunction
$\psi_{\mu}({\bm{\xi}}_{\alpha})=\Psi_{NM,nm}({\bm{r}}_{\alpha,1},{\bm{r}}_{\alpha,2})$
given by Eq. (Sub-Landau levels in two-dimensional electron system in magnetic
field). There are the following relations $\int
d{\bm{\xi}}\psi^{*}_{\mu}({\bm{\xi}})\psi_{\mu^{\prime}}({\bm{\xi}})=\delta_{N,N^{\prime}}\delta_{M,M^{\prime}}\delta_{n,n^{\prime}}\delta_{m,m^{\prime}}=\delta_{\mu,\mu^{\prime}}$,
and
$\sum_{\mu}\psi^{*}_{\mu}({\bm{\xi}^{\prime}})\psi_{\mu}({\bm{\xi}})=\delta(\bm{\xi}-\bm{\xi}^{\prime})$
. The interaction potential between the CREPs is given by the second part in
Eq. (22).

For the set of lowest sub-Landau levels ($0M,0m$) with $N=0$, $n=0$, $m<0$ and
$M=m,2m,3m,\cdots$, the wavefunction can written as

$\displaystyle\psi_{\mu}({\bm{\xi}}_{\alpha})=\Psi_{0M,0m}({\bm{R}}_{\alpha},{\bm{r}}_{\alpha})$
$\displaystyle=\frac{(z_{\alpha,2}+z_{\alpha,1})^{|M|}}{\pi
2^{|M|+|m|+1}\sqrt{|M|!|m|!}}(z_{\alpha,2}-z_{\alpha,1})^{|m|}h^{|m|}_{0}(d_{\alpha}^{2}/4)$
$\displaystyle\;\;\;\times
e^{-\frac{|z_{\alpha,1}|^{2}+|z_{\alpha,2}|^{2}}{4}},$ (23)

where we have used the notations $z=re^{-i\theta}$ = $x-iy$,
$z_{\alpha,j}$=$x_{\alpha,j}-iy_{\alpha,j}$ ($j=1,2$),
${\bm{r}}_{\alpha}=(x_{\alpha},y_{\alpha})$, and
$d_{\alpha}$=$|z_{\alpha,2}-z_{\alpha,1}|$=
$|{\bm{r}}_{\alpha,2}-{\bm{r}}_{\alpha,1}|$. In a moderate or strong magnetic
field, $h^{|m|}_{0}(d_{\alpha}^{2}/4)$ is sub-linear function. Remember that
the stable CREPs are spin triplet with $S_{z}=1$.

Because the CREPs are effectively bosons, within the Hartree-Fock theory for
bosons[23], the total wavefunction of $N_{p}$ CREPs in one of the lowest sub-
Landau level (with $N=n=0$ and fixed $m$) can be approximately written as the
following permanent,

$\displaystyle\Phi({\bm{\xi}}_{1},{\bm{\xi}}_{2},...,{\bm{\xi}}_{N_{p}})=$
$\displaystyle\frac{1}{\sqrt{N_{p}!}}\;{\rm
perm}\left(\begin{array}[]{cccc}\psi_{1}({\bm{\xi}}_{1})&\psi_{1}({\bm{\xi}}_{2})&\ldots&\psi_{1}({\bm{\xi}}_{N_{p}})\\\
\psi_{2}({\bm{\xi}}_{1})&\psi_{2}({\bm{\xi}}_{2})&\ldots&\psi_{2}({\bm{\xi}}_{N_{p}})\\\
\vdots&\vdots&\ddots&\vdots\\\
\psi_{N_{p}}({\bm{\xi}}_{1})&\psi_{N_{p}}({\bm{\xi}}_{2})&\ldots&\psi_{N_{p}}({\bm{\xi}}_{N_{p}})\\\
\end{array}\right)$ (28) $\displaystyle=$ $\displaystyle{\rm
perm}\left(\begin{array}[]{cccc}(z_{1,1}+z_{1,2})^{|m|}&(z_{2,1}+z_{2,2})^{|m|}&\ldots&(z_{{}_{N_{p},1}}+z_{{}_{N_{p},2}})^{|m|}\\\
(z_{1,1}+z_{1,2})^{2|m|}&(z_{2,1}+z_{2,2})^{2|m|}&\ldots&(z_{{}_{N_{p},1}}+z_{{}_{N_{p},2}})^{2|m|}\\\
\vdots&\vdots&\ddots&\vdots\\\
(z_{1,1}+z_{1,2})^{N_{p}|m|}&(z_{2,1}+z_{2,2})^{N_{p}|m|}&\ldots&(z_{{}_{N_{p},1}}+z_{{}_{N_{p},2}})^{N_{p}|m|}\\\
\end{array}\right)$ (34)
$\displaystyle\;\;\;\;\;\times\left\\{\frac{1}{\sqrt{N_{p}!}}\prod_{\alpha=1}^{N_{p}}\frac{(z_{\alpha,2}-z_{\alpha,1})^{|m|}h^{|m|}_{0}(d_{\alpha}^{2}/4)}{{\pi
2^{(\alpha+1)|m|+1}\sqrt{|m|!(\alpha|m|)!}}}\right\\}\times\exp\left(-\frac{1}{4}\sum_{j=1}^{N_{e}}|z_{j}|^{2}\right).$

We can compare the above wavefunction with the Laughlin wavefunction for the
fractional quantum Hall states given by, for $m=-3,-5,\cdots$,

$\Phi(z_{1},...,z_{N_{e}})\propto\prod_{j<k}^{N_{e}}(z_{j}-z_{k})^{|m|}\exp\left(-\frac{1}{4}\sum_{j=1}^{N_{e}}|z_{j}|^{2}\right).$
(35)

It is seen that our wavefunction has the basic characteristics of the Laughlin
wavefunction. Considering that the Laughlin wavefunction is exact, the
wavefunction in Eq. (28) includes only the intra-pair e-e interaction. The
inter-pair interaction has to be considered through the pair-pair interaction
potential in Eq. (22) for $\alpha\neq\beta$. The inter-pair correlation energy
should be small because the overlap between different pairs is small and the
intra-pair and inter-pair correlations are not entangled.[17] On the other
hand, our wavefunction is presented for both the IQHE ($m=-1$) and FQHE
($m\leq-3$) states. Each state has well defined energy level and degeneracy
$n_{\phi_{0}}/|m|$. There are energy gaps between different states. The study
in this work may provide clues for explanation of the experimental
observations showing electron pairing in the quantum Hall effect regime[6, 7,
8]. Furthermore, we have constructed a system of many bosonic quasiparticles
of CREPs for the quantum Hall states. It opens a possibility of studying Bose-
Einstein condensation in the quantum Hall effects.

###### Acknowledgements.

This work was supported by FAPESP (São Paulo Research Foundation, under the
grant 2024/00484-2) and CNPq (Brazil).

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