#  Dual Teaching: A Practical Semi-supervised Wrapper Method

Fuqaing LIU1 Chenwei DENG1 Fukun BI2 Yiding YANG1
1Department of Information and Electronic, Beijing Institute of Technology,
Beijing
2College of Information Engineering, North China University of Technology,
Beijing
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Semi-supervised wrapper methods are concerned with building effective
supervised classifiers from partially labeled data. Though previous works have
succeeded in some fields, it is still difficult to apply semi-supervised
wrapper methods to practice because the assumptions those methods rely on tend
to be unrealistic in practice. For practical use, this paper proposes a novel
semi-supervised wrapper method, Dual Teaching, whose assumptions are easy to
set up. Dual Teaching adopts two external classifiers to estimate the false
positives and false negatives of the base learner. Only if the recall of every
external classifier is greater than zero and the sum of the precision is
greater than one, Dual Teaching will train a base learner from partially
labeled data as effectively as the fully-labeled-data-trained classifier. The
effectiveness of Dual Teaching is proved in both theory and practice.

## Intruduction

Semi-supervised learning (SSL) is concerned with utilizing some labeled data
as well as amounts of unlabeled data to train better classifiers and
regressors (?; ?). Some SSL methods are called ”Semi-supervised wrapper
methods” (?) and they procedure external ”wrappers” around the base learner, a
supervised learning model. The purpose of semi-supervised wrapper methods is
to train a supervised model from partially labeled data as effectively as the
same model trained from fully labeled data. Compared with other semi-
supervised methods, semi-supervised wrapper methods can use almost any user-
specified supervised learner as a subroutine, which allows us to take
advantage of advances in supervised learning without deriving new algorithms.

However, it is difficult to apply semi-supervised wrapper methods in practical
learning cases (?; ?; ?). Though most of previous wrapper methods propose
theoretical analysis to prove their effectiveness, there are still many gaps
between realities and theories (?; ?; ?). In real-world learning scenarios,
assumptions those semi-supervised wrapper methods rely on might be untenable,
which makes those methods ineffective. For instance, self-training (?) assumes
that predictions of the base learner tend to be correct in early generation
otherwise it would grow worse when using the unlabeled data. However for a
base learner initialized from a few labeled data, making good predictions on
amounts of unlabeled data is a tough job. More related semi-supervised wrapper
methods including Co-Training (?) and Cluster-Then-Label-Methods (?) as well
as the reason why they are hard to set up in practice are discussed in Related
Work Related Work.

For practical use, this paper proposes a novel semi-supervised wrapper method
named ”Dual Teaching” whose assumptions are easy to set up in practical
learning tasks. Dual Teaching only adopts two external classifiers outside the
base learner. One assumption of Dual Teaching is that the recall of each
auxiliary classifier is greater than zero. The other is that the sum of
precisions is greater than one. Compared with assumptions of previous works,
these assumptions are much easier to set up in practice. The two assumptions
themselves are so simple that they can be matched even by two weak
classifiers. In addition, the assumptions have no relationship with the base
learner so that Dual Teaching can choose a proper supervised learner as the
base learner according to the practical data.

The above easy assumptions originate from a novel framework. Different from
previous works, auxiliary wrappers in Dual Teaching estimate the error of the
base learner rather than directly estimate the label of unlabeled data. One
external classifier named ”FP Teacher” picks out false positives from the
positive outputs of the base learner. The other one named ”FN Teacher” picks
out false negatives from the negative outputs. To begin with, the base learner
is zero-initialized. Then an effective learner would be trained by the
following recursive processing. In every generation, the unlabeled data are
classified by the base learner firstly; secondly the two teachers are tested
on a validation set composed by part of labeled data and if the teachers do
not match the assumptions, they will be tuned based on the rest labeled data
until their performance on validation set matches the assumptions; thirdly the
two teachers point out the unlabeled examples mis-classified by the base
learner and then the base learner is re-trained using all previous mis-
classified examples with their corrected labels. The base learner would grow
better in every generation if the assumptions are valid.

The rest of the paper is organized as follows. We first review related semi-
supervised wrapper methods. Next, the proposed method is detailed. Then we
analyzes the effectiveness of Dual Teaching in theory. This is followed by a
number of experiments to show the effectiveness in practice. This paper
finishes with conclusions and suggestions for future research.

## Related Work

This section reviews a series of semi-supervised wrapper methods including
Self-Training, Co-Training and Cluster-then-label-methods. Differences between
Dual Teaching and the above methods are also discussed.

Self-Training (?) is characterized by the strategy that the learner uses its
own predictions to teach itself. It starts by training a base learner from
some labeled data and then evaluates the learner on the unlabeled data.
Examples along with the labels predicted by the base learner are added to the
training set and then the classifier is re-trained. Because the early mistakes
made by the base learner would be reinforced by inputting the false labels to
the early training set, self-training assumes that predictions of the base
learner tend to be correct. However, it is unrealistic in some practical
scenarios that a supervised model trained form a few labeled could classify
amounts of unlabeled data successfully.

Co-Training (?) is a wrapper method that works with two classifiers. The
learning is initialized by training two classifiers from two separate feature
sets of the labeled data. Then, one classifier make predictions on unlabeled
data and then the data with their predicted labels are fed back for re-
training another classifier. The two classifiers work as the above step
alternately. The high performance of Co-Training relies on two independent
feature-spaces. In addition, each classifier should make good predictions on
unlabeled data in early generation. The above two assumptions tend to be
violated in practice. First, two independent feature-spaces are difficult to
be available because the diffusion of the real-world data is complicated.
Second, it is difficult for the two classifiers to make good predictions on
unlabeled data at the very start when they are trained from a few labeled
data.

Cluster-then-label-Methods (?) start by identifying clusters by unsupervised
clustering algorithms and unlabeled data. Then supervised classifiers are
learned from the labeled examples in each cluster. Each cluster should learn a
classifier and if there is no labeled example falls into the cluster, the
classifier would be trained from all labeled data. After that, the unlabeled
examples in each cluster are labeled by the corresponding classifier. Cluster-
then-label-methods will work effectively only if one cluster contains examples
belong to a single class only. But in practice, one cluster usually contains
examples from more than one class.

Dual Teaching is very different from previous works. Dual Teaching makes
assumptions on its wrappers rather than the base learner or the data.
Accordingly, the wrappers can be adjusted to match the assumptions in the
process of Dual Teaching. The special assumptions are derived from a novel
framework. Dual Teaching tries to estimate the error of the base learner,
while previous works try to directly estimate the labels of unlabeled data.
Thanks to this strategy, Dual Teaching can provide theoretical analysis for
general cases with arbitrary inner base learners.

## Description of Dual Teaching

We focus on binary classification. Suppose $x$ is an example from a feature-
space $X$ and $y$ is its label from a label space $Y=\\{1,-1\\}$. Suppose
$X_{l}$ is a labeled data set, $L_{l}$ is its corresponding label set and
$X_{u}$ is an unlabeled data set, where $l$ and $u$ represent the number of
labeled examples and that of unlabeled examples respectively. Assuming that
$L_{u}$ is an in-existent label set that contains true labels of the unlabeled
data, suppose $F:X\rightarrow Y$ be an supervised classifier learned from an
assumed fully labeled data set $\\{X_{l},X_{u}\\}$ with its label set
$\\{L_{l},L_{u}\\}$. Dual Teaching aims to train an effective classifier $f$
from a few labeled data $X_{l}$ with $L_{l}$ and amounts of unlabeled data
$X_{u}$, where $l<<u$. The classifier $f$ named the ”base learner” is a
supervised classification model parameterized by $\Theta$. Dual Teaching try
to train the base learner as effectively as the fully-labeled-data-trained
classifier.

Dual Teaching contains four base steps: zero-initialization, classification,
teacher testing & tuning and error estimation & re-training. Dual Teaching
adopts two external classifiers named ”FP Teacher” and ”FN Teacher” to
estimate the false positives and false negatives respectively from the outputs
of base learner. The assumptions of Dual Teaching are that the recall of each
teacher is greater than zero and the sum of precision is greater than one.
When the assumptions are valid, Dual Teaching can train a effective base
learner from partially labeled data.

Figure 1: The framework of Dual Teaching.

Step1: Zero-Initialization For convenience, set the parameters of the base
learner to zero, $\Theta^{0}=\textbf{0}$. In terms of Dual Teaching, how to
initialize the base learner does not influence the performance of the base
learner, which is different from Self-training and Co-training. Supposing $R$
represents the re-training data set for the base learner, it is an empty set
initially, $R^{0}=\emptyset$. This step is shown as Fig 1 $①$.

(a) precision/labeled

(b) precision/unlabeled

(c) recall/labeled

(d) recall/unlabeled data

(e) accuracy

Figure 2: Performance of different strategies on constructing teachers. L/U in
the legend represents the ratio of labeled data to unlabeled data in training
set. (a-e) use a same legend in (a). The experiment is done on a dataset named
heart scale.

Step2: Classification In every generation, all unlabeled examples are
classified by the base learner re-trained in last generation.
$y^{k}=f(x,\Theta^{k-1})$ for $x\in X_{u}$ where $k$ represents the $k$th
generation . This step is shown as Fig 1 $②$.

Step3: Teacher Testing & Tuning First, the auxiliary classifiers are trained
from half of the labeled data which is referred to as $TX_{l}$. The rest
labeled data are reserved as a validation set named $VX_{l}$. Then before
estimating the error of the base learner, the teachers are tested on the
validation set to make sure they match the assumptions. We estimate the true
precision and recall of auxiliary classifiers by their recall and precision on
validation set. If teachers do not match the assumptions, we will increase the
precision of teachers by reducing their recall. Take logistic regression for
instance, the threshold of discrimination, referred to as $T$ , is usually set
to 0.5. But if we want to improve the precision, we will set $T$ greater than
0.5 and the recall will decrease accordingly.

This step, Teacher Testing & Tuning, is shown as Fig 1 $③$. The specific
strategy is detailed as algorithm 1. In the algorithm 1, $FP$ and $FN$
represent the FP teacher and FN teacher respectively. $p1$ and $r1$ represent
the precision and recall of FP teacher. $p2$ and $r2$ represent the precision
and recall of FN teacher. Allowing for a certain error, we empirically set the
sum of precision should be greater than 1.1 rather than 1.

Algorithm 1 Teaching Tuning

Input: the validation set $VX_{l}$, the rest labeled data $TX_{l}$, the base
learner $f$, the parameters $\Theta^{k}$

Initialize: $T$=0.1

Data Division:

$V_{l+}=\\{x\in VX_{l}|f(x,\Theta^{k})==1\\}$

$V_{l-}=\\{x\in VX_{l}|f(x,\Theta^{k})==-1\\}$

$Y_{l+}=\\{x\in TX_{l}|f(x,\Theta^{k})==1\\}$

$Y_{l-}=\\{x\in TX_{l}|f(x,\Theta^{k})==-1\\}$

Teacher Training:

train $FP$ from $Y_{l+}$ with its true label

train $FN$ from $Y_{l-}$ with its true label

repeat

test $FP$ with $T$ on $V_{l+}$ and compute $p_{1}$,$r_{1}$

test $FN$ with $T$ on $V_{l-}$ and compute $p_{2}$,$r_{2}$

$T$ = $T$+0.05

until $(p1+p2>1.1\ and\ r1*r2>0)\ or\ (T>1)$

As the base learner improves, it would be harder for teachers to correctly
point out its mistakes because the proportion of mis-classified examples
reduces. The two teachers gradually face a class imbalance problem as the base
learner grows better. In practice, we construct each teacher by a balanced-
weight logistic regression model (?). The threshold of discrimination is set
to a small value initially for high recall. High recall means that teachers
can pick out as many mis-classified examples as they could in step4: error
estimation & re-training.

The proposed strategy on constructing teachers is compared with other two
strategies. One comparative strategy abbreviated to LR(equal weight) is to
construct the teachers using equal-weight logistic regression. Another one
abbreviated to LR(balanced weight)is to construct the teachers using balanced-
weight logistic regression with a constant threshold. LR(balanced weight+L/R
tuned) is short for the proposed strategy. The precision and recall of the
base learner on both labeled set and unlabeled set are recorded in every
generation. The performance of the base learner on unlabeled data is also
recorded.

As Fig 2 shows, the recall of the base learner when using LR(equal weight)
drops fast because the FP teacher and FN teacher can not find any mis-
classified example in early generation. The accuracy of the base learner when
using LR(balanced weight) shakes because the two teachers cannot match the
assumptions in every generation. The base learner when using LR(balanced
weight+L/R tuned) performs well because the teachers not only point out as
many mis-classified examples as they can but also do not make too many
mistakes (precision is always constant) as the base learner improves. Finally
all of the outputs of two teachers are -1 which means the teachers think that
the base learner makes no mistake. The performance of the base learner would
be stable because the re-training set for the base learner does not change in
next generation.

Step4: Error Estimation & Re-training Dual Teaching adopts two external
classifiers, FP Teacher and FN Teacher, to estimate the mis-classified
examples in every generation. FP Teacher tries to pick out the false positives
from the positive outputs of the base learner. FN Teacher tries to pick out
the false negatives from the negative outputs. Then ”add” these mis-classified
examples with their estimated labels to the re-training set $R$. The ”add” is
defined as follows: if the mis-classified example already exists in re-
training set $R$, just change its label; if the mis-classified example does
not exit in the re-training set $R$, add the example with its corrected label
to the re-training set. Finally, re-train the base learner from the updated
re-training data set $R^{k}$. This step is shown as Fig 1 $④$. Repeat steps
2-4 until the base learner is stable, then a strong classifier will be
trained.

## Effectiveness Analysis

To begin with, this section theoretically analyzes why Dual Teaching could
train an effective classifier from partially labeled data. Then, we detail how
to drive the assumptions Dual Teaching relies on. The assumptions are
summarized as that the recall of each external classifier is greater than zero
and the sum of precision is greater than one.

### Effectiveness

This part theoretically derives that Dual Teaching could train an effective
base learner from a few labeled data and amounts of unlabeled data just by two
external classifiers.

Define $\overrightarrow{e}(k)$ as the error of the base classifier in kth
generation,

$\overrightarrow{e}(k)=[\alpha(k),\beta(k)]^{T}$ (1)

where $\alpha(k)$ represents false positives and $\beta(k)$ represents false
negatives in the outputs of the base learner.

Define $P^{+}$,$R^{+}$ as the precision and the recall of FP Teacher
respectively. Similarly, define $P^{-}$,$R^{-}$ as the precision and the
recall of FN Teacher. Assume that if the mis-classified examples are pointed
out and put into re-training set, the base learner will not make same mistakes
in next generation. Assume that the base learner will classify the examples
incorrectly if the examples along with wrong labels are contained in the re-
training set. The recursive equations are shown as equation (2).

$\left\\{\begin{aligned}
\alpha(k+1)=\alpha(k)-R^{+}\cdot\alpha(k)+\frac{1-P^{-}}{P^{-}}R^{-}\cdot\beta(k)\\\
\beta(k+1)=\beta(k)-R^{-}\cdot\beta(k)+\frac{1-P^{+}}{P^{+}}R^{+}\cdot\alpha(k)\end{aligned}\right.$
(2)

where

  * •

$R^{+}\cdot\alpha(k)$: the false positives that the FP Teacher pick out
correctly.

  * •

$R^{-}\cdot\beta(k)$: the false negatives that the FN Teacher pick out
correctly.

  * •

$\frac{1-P^{+}}{P^{+}}R^{+}\cdot\alpha(k)$: examples incorrectly classified to
be false positives by FP Teacher.

  * •

$\frac{1-P^{-}}{P^{-}}R^{-}\cdot\beta(k)$: examples incorrectly classified to
be false negatives by FN Teacher.

Reconstruct equation (1) and (2),

$\overrightarrow{e}(k+1)=M\overrightarrow{e}(k)$ (3)

where $M=[\begin{array}[]{cc}1-R^{+}&\frac{1-P^{-}}{P^{-}}R^{-}\\\
\frac{1-P^{+}}{P^{+}}R^{+}&1-R^{-}\\\ \end{array}]$.

Based on the well founded theory of dynamic system (?),
$\overrightarrow{e}(k)$ would converge to zero if all magnitudes of
eigenvalues of the transformation matrix are smaller than one. The above
analysis refers to the error analysis in Kalal’s work (?; ?).

Originally we assume that the base learner would not make the same mistakes as
those picked out by two teachers in last generation, which is not always valid
in practice. For instance, a classifier trained from fully-labeled data with
their true labels would make mistakes on training set. Thus, the error of the
base learner $f$ would converge to a certain level, a level as same as that of
the fully-labeled-data-trained-classifier $F$.

Figure 3: Simulations on $|\lambda|_{max}$. (a) shows the relationship between
$|\lambda|_{max}$ and ($P^{+}$, $P^{-}$). The red line is $P^{+}+P^{-}=1$ and
$|\lambda|_{max}$ in the red line is definitely equal to one. When
$P^{+}+P^{-}>1$, $|\lambda|_{max}$ is definitely smaller than one. (b) shows
that no matter what the recalls of the two classifiers are, the maximum
magnitude of eigenvalue is definitely equal to one when $P^{+}+P^{-}=1$.

.

### Assumption Derivation

This part details the assumptions Dual Teaching makes. Shown as equation (3),
the error continues reducing when the magnitudes of eigenvalues of the
transformation matrix are smaller than 1. The eigenvalues
$\lambda_{1},\lambda_{2}$ are shown as equation (4).

$\lambda_{1},\lambda_{2}=\frac{(2-R^{-}-R^{+})\pm\sqrt{\bigtriangleup}}{2}$
(4)

$\bigtriangleup=(R^{+}-R^{-})^{2}+4\frac{(1-P^{+})(1-P^{-})}{P^{+}P^{-}}R^{+}R^{-}$
(5)

where $P^{+},R^{+},P^{-},R^{-}\in[0,1]$ and $\bigtriangleup$ is always greater
than or equal to 0.

So, the maximum absolute value of the two eigenvalues is shown as equation
(6).

$|\lambda|_{max}=\frac{(2-R^{-}-R^{+})+\sqrt{\bigtriangleup}}{2}$ (6)

Assuming that $|\lambda|_{max}$ is smaller than 1, we can derive a conclusion
as equation (7).

$\sqrt{\bigtriangleup}<R^{-}+R^{+}$ (7)

Simplify equation (7) and we can get equation (8).

$\frac{1-P^{+}-P^{-}}{P^{+}P^{-}}R^{+}R^{-}<0$ (8)

Furthermore, the assumptions Dual Teaching makes are derived as equation 9.

$\left\\{\begin{aligned} R^{+}R^{-}\neq 0\\\
P^{+}+P^{-}>1\end{aligned}\right.$ (9)

(a) bci

(b) g241c

(c) digit1

(d) usps

(e) heart scale

(f) car evaluation

(g) wine quality

(h) adult-a

(i) housing

(j) vehicle

(k) mnist1v7

(l) mnist4v9

Figure 4: Performance of different semi-supervised wrapper methods with
different base learners.

Equation (9) proposes the requirement on the two external classifiers. When
the recall of each classifier is greater than 0 and the sum of the precisions
is greater than 1, the base learner will be trained effectively by Dual
Teaching from partially labeled data.

## Experiments

This section proposes a series of evaluations of the proposed method and
previous works on a broad range of tasks including 4 semi-supervised benchmark
data sets111http://www.kyb.tuebingen.mpg.de/ssl-book/., _bci_ , _g241c_ ,
_digit1_ , _usps_ , and 6 UCI
tasks222http://archive.ics.uci.edu/ml/datasets.html., _heart scale_ , _car
evaluation_ , _wine quality_ , _adult_ , _housing_ , _vehicle_ , and 2 image
classification problems (?), _mnist1v7_ , _mnist4v9_. The size of data ranges
from 170 to more than 15000 and the dimensionality ranges from 6 to more than
600. The ratio of the number positive examples to that of negative examples
ranges from 0.25 to 1.149.

Every data set is separated into two parts. 75% data in each data set are
reserved as training data. The testing data set is composed by the rest 25%
data. Not all data in the training data set are labeled. Features in each data
set are normalized beforehand.

Table 1: Accuracy of svm and accuracy improvements of S3VM, S4VM and Dual Teaching on different numbers of labeled data Dataset | 5% labeled data | 10% labeled data | 20% labeled data
---|---|---|---
svm | s3vm | s4vm | svm/DT | svm | s3vm | s4vm | svm/DT | svm | s3vm | s4vm | svm/DT
bci | 56.2 | -1.0 | -1.1 | 1.8 | 62.7 | 1.2 | -0.6 | 0 | 64.3 | 0.9 | 0 | -0.9
g241c | 73.7 | 6.6 | 0.4 | 4.2 | 80.7 | 0.9 | 0.2 | 0.1 | 84.2 | 0.9 | 0.8 | 2.1
digit1 | 90.9 | 0.6 | 2.0 | 1.0 | 93.7 | 0.2 | 0 | 0.4 | 93.9 | 0 | 0.1 | 0.2
usps | 86.9 | -0.2 | 8.3 | 1.7 | 86.9 | 0 | 8.4 | 1.8 | 90.4 | 0 | 0.1 | 0.4
heart scale | 71.2 | 0.9 | 1.3 | 1.0 | 78.9 | -0.6 | 0.7 | 1.9 | 80.8 | 0.3 | 0.2 | 5.7
vehicle | 65.3 | 0.2 | 2.3 | 0.8 | 74.2 | 2.3 | 3.4 | -0.1 | 77.0 | 3.0 | -0.9 | -1.2
| s3vm | s4vm | svm/DT
Win/Tie/Loss against svm: | 12/3/3 | 13/2/3 | 14/1/3
Best/Worst performance over all: | 5/8 | 6/5 | 7/4

### Comparison with Classical Semi-supervised Wrapper Methods

This part compares Dual Teaching (DT) with previous semi-supervised wrapper
methods, Self-Training (ST) and Co-Training (CT). To show the generalization
of the proposed method, Logistical Regression (lr) (?), Support Vector Machine
with linear kernel (svm) (?) and Adaboost (ada) (?) serve as the base learner
separately for every wrapper method. Every specific method is abbreviated to
”the name of the base learner/the name of the wrapper method”. For instance,
”lr/ST” means the base learner is a logistic regression model and the wrapper
method is Self-Training. To show whether the wrapper methods can improve the
performance of the base learner from unlabeled data, supervised models with
the use of labeled data only, which are abbreviated to ”name of the supervised
model”, serve as the baseline approaches. For instance, ”lr” represents the
logistic regression model trained only from labeled data. This experiment
contains all data sets referred above. For every data set, each method works
on various numbers of labeled data and the performance on testing set is
recorded. The ratio of the number of labeled data to that of all data in
training set ranges from 0.1 to 1. When the ratio is equal to 1, the base
learner actually is a supervised classifier trained from fully-labeled data.

Fig 4 shows the comparison between Dual Teaching and other semi-supervised
wrapper methods. Dual Teaching performs much better than Self-Training and Co-
Training. Except that the training data are all labeled, there are 324
different cases (3 base learners, 12 data sets and 9 ratios of labeled data).
In 235(72.3%) cases, Dual Teaching makes more accurate predictions than the
other methods and only in 9(2.8%) cases, Dual Teaching is worse than others.
In addition, only in 11 cases, Dual Teaching is outperformed by the baseline
model using labeled data only. Furthermore, for almost all data sets and
different types of supervised models, Dual Teaching can train a base learner
from 20% labeled data and 80% unlabeled as well as a fully-labeled-data-
trained classifier. In summary, Dual Teaching can effectively improve the
performance of the base learner from unlabeled data.

### Comparison with Productive Semi-supervised Methods

The proposed method is also compared with some of productive semi-supervised
methods (no wrapper method): S3VM (?) and S4VM (?) with linear kernel. The
base learner of Dual Teaching is set to a SVM with linear kernel too. the
ratio of the number of labeled data to that of training data ranges in
$\\{5\%,\ 10\%,\ 20\%$}. To validate the safeness of the above semi-supervised
methods, a SVM classifier with linear kernel trained from labeled data only
serves as the baseline approach.

Table 1 presents the accuracy of SVM and accuracy improvements of S3VM, S4VM
and Dual Teaching on different numbers of labeled data. In the perspective of
no matter effectiveness or safeness, Dual Teaching is competitive in this
experiment. Different from S3VM and S4VM, Dual Teaching is a wrapper method
that can make full use of the supervised learning according to different data
sets. For example, as shown in Fig 4, Dual Teaching with adaboost as the base
learner performs better than Dual Teaching with svm as the base learner on
_bci_ and _vehicle_.

## Conclusion and Future Work

Dual Teaching’s good performance relies on its easy assumptions and novel
framework. The assumptions are easy to set up in practice. First, the
assumptions themselves are not critical. For instance, with the base learner
improves, the performance of teachers are usually worse than that in early
generations. But the teachers can still match the assumptions just by reducing
the recall to keep their precision in a satisfactory level. Then, the
assumptions are well quantified so that we can monitor and tune the external
teachers in the process of Dual Teaching to match the necessary conditions.
Dual Teaching can effectively improve the performance of supervised learner
from unlabeled data.

The above characters also improve the safety of Dual Teaching. By tuning the
teachers, the base learner in Dual Teaching is able to be not worse than a
same supervised model trained from labeled data only. When the recall of each
teacher is equal to 0, the re-training set does not change anymore and the
base learner is constant. To estimate the performance of teachers precisely,
Dual Teaching needs enough labeled data reserved (at least 10% empirically).
Unlike methods which focus on few labeled data training (?), Dual Teaching
does not perform well in the cases where only few labeled data exist.

The easy assumptions originate from the carefully designed framework. Dual
Teaching converts training an high performance base learner into training two
qualified external classifiers in every generation. The re-training set of the
base learner synthesizes the outputs of external classifiers in every
generation and boosts the performance of the base learner. Even though the
teachers in late generations are poor, only if they do not pollute the re-
training set too much, the base learner will not grow worse. For high
effectiveness, Dual Teaching sacrifices its efficiency. Dual Teaching re-
trains the base learner in every generation and the scale of re-training set
increases in the recursive process of Dual Teaching.

In the future, it is worthy to extend Dual Teaching to solve multiple
classification problems. Designing a more effective and efficient way to
estimate the precision and the recall of auxiliary classifiers is also worthy.

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# Comprehensive quantum transport analysis of M-Superlattice structures for
barrier infrared detectors

Anuja Singh  Swarnadip Mukherjee  Bhaskaran Muralidharan  Department of
Electrical Engineering, Indian Institute of Technology Bombay, Powai,
Mumbai-400076, India

(August 27, 2024)

###### Abstract

In pursuit of designing superior type-II superlattice barrier infrared
detectors, this study encompasses an exhaustive analysis of utilizing
M-structured superlattices for both the absorber and barrier layers through
proper band engineering and discusses its potential benefits over other
candidates. The electronic band properties of ideally infinite M-structures
are calculated using the eight band $k.p$ method which takes into account the
effects of both strain and microscopic interface asymmetry to primarily
estimate the bandgap and density-of-states effective mass and their variation
with respect to the thicknesses of the constituent material layers. In
contrast, for practical finite-period structures, the local density-of-states
and spectral tunneling transmission and current calculated using the Keldysh
non-equilibrium Green’s function approach with the inclusion of non-coherent
scattering processes offer deep insights into the qualitative aspects of
miniband and localization engineering via structural variation. Our key
results demonstrate how to achieve a wide infrared spectral range, reduce
tunneling dark currents, induce strong interband wavefunction overlaps at the
interfaces for adequate absorption, and excellent band-tunability to
facilitate unipolar or bipolar current blocking barriers. This study,
therefore, perfectly exemplifies the utilization of 6.1Å material library to
its full potential through the demonstration of band engineering in
M-structured superlattices and sets up the right platform to possibly replace
other complex superlattice systems for targeted applications.

##  I INTRODUCTION

The evolution of materials technology in the field of infrared (IR) detection
has experienced a rapid turnaround in the late nineties when traditional bulk
materials such as HgCdTe, InSb were replaced by quantum-engineered structures
as the former exhibited limited applicability due to the non-uniform growth
defects and their ineffectiveness to cover the full IR range Plis (2014);
Rogalski  _et al._ (2018); Martyniuk  _et al._ (2014a, b). Since then, type-II
superlattices (T2SL) composed of InAs/(In,Ga)Sb heterostructures have gained a
lot of attention for their thickness dependent bandgap tunabilities,
suppressed Auger recombination and high quantum efficiencies. This has enabled
them to become the latest industry-standard technology for next-generation IR
detectors and focal plane arrays (FPA) Plis (2014); Mukherjee  _et al._
(2021); Rogalski  _et al._ (2018); Ting  _et al._ (2020); Zavala-Moran  _et
al._ (2020); Martyniuk  _et al._ (2014a). Subsequently, one of the major
breakthroughs en-route achieving a higher operating temperature and background
limited IR detection has come in effect in the mid-2000s, when the concept of
barrier-based detectors was first introduced. This later laid out an ideal
platform to explore several novel device architectures of the form nBn, pBp,
pBiBn etc., through “band-diagram engineering” Rogalski  _et al._ (2017);
Nguyen  _et al._ (2007a, 2008); Plis (2014); Zavala-Moran  _et al._ (2020).
Despite numerous advantages, T2SL systems suffer from an inadequacy to produce
perfect hole barriers due to the strong adherence of hole minibands to the
bottom of GaSb valence bands and the lack of tunability with varying GaSb
layer thicknesses Razeghi and Nguyen (2010); Nguyen  _et al._ (2007b, 2009);
Zavala-Moran  _et al._ (2020); Lang and Xia (2013). Moreover, the spatial
separation of carrier localizations in these structures leads to smaller
absorption, affecting the photo response Ting  _et al._ (2020). To overcome
this, complex structures like M, W, N were introduced later by inserting an
additional high-bandgap AlSb layer in the original T2SL layout, which offered
higher degrees of freedom to achieve a full tailorability of miniband
alignment and better wavefunction engineering Ting  _et al._ (2020); Lang and
Xia (2013); Nguyen  _et al._ (2008); Maimon and Wicks (2006).
The placement of the AlSb layer determines the nomenclature of these
structures based on the shape formed on joining the conduction band edges Ting
_et al._ (2020); Nguyen  _et al._ (2008). In M-superlattice (MSL) structures,
the AlSb layer is inserted at the center of the GaSb layer which splits the
single hole quantum well into two and makes the valence band more sensitive to
the GaSb width Lang and Xia (2013); Ting  _et al._ (2020); Nguyen  _et al._
(2007a, b). Furthermore, due to the AlSb layer, the center of the hole
wavefucntion in the GaSb layer shifts closer to the center of electronic
wavefunction in the InAs layer which effectively increases the overlap between
them, thereby facilitating higher absorption Ting  _et al._ (2020); Lang and
Xia (2013). In addition, MSLs possess fine tuning of band alignments between
the absorber and the barrier and strongly provide an impediment to tunneling
transport of carriers, thereby knocking off their contribution to the SRH dark
current Nguyen  _et al._ (2007b, a, 2008, 2009). However, a unified study on
the key roles played by the AlSb layer width in modulating the bandgap,
density of states (DOS) effective masses, valence miniband offsets, interband
overlaps, and spectral transport are still scarce in the literature and thus
form the basis of this study.
This motivates us to perform a scrupulous theoretical investigation of MSL
systems and present a thorough comparison with the conventional T2SL systems
Li  _et al._ (2010); Livneh  _et al._ (2012). We show that the bandgap of such
systems can be accurately predicted by incorporating an appropriate interface
model that takes into account the effects of strain due to lattice mismatch
and microscopic interface asymmetry (MIA) corresponding to the alternate
arrangement of interface materials Li  _et al._ (2010); Szmulowicz (1996);
Szmulowicz  _et al._ (2004); Hong  _et al._ (2009); Krebs and Voisin (1996);
Mlinar  _et al._ (2005); Szmulowicz  _et al._ (2004). In addition, we
emphasize the non-trivial roles played by the inserted AlSb layer in tandem
with the original T2SL material layers in regulating the miniband edges and
carrier effective masses towards an efficient engineering of carrier transport
to optimize the dark current. In subsequent explorations, we compare finite-
period MSL and T2SL systems in terms of the tunneling transmission and local
density of states (LDOS) features evaluated from the Keldysh non-equilibrium
Green’s function (NEGF) approach and highlight key differences based on the
nature of carrier localization in the available subband states Priyadarshi
_et al._ (2018); Mukherjee and Muralidharan (2018); Mukherjee  _et al._
(2021); Tibaldi  _et al._ (2021); Bertazzi  _et al._ (2020). The strong
interband overlap around the interface region and a signature of spatially
continuous broad current spectra in MSL structures are respectively indicative
of enhanced absorption and phonon-assisted non-coherent miniband transport of
carriers. Finally, MSL structures are shown to demonstrate excellent and
precise band-tunability features through carefully introduced design
guidelines which makes them suitable to be used as carrier blocking unipolar
or bipolar barriers.
The rest of the paper is organized as follows. In Sec. II, we describe the
$\bf{k.p}$ and NEGF theory applied for modeling electronic band properties and
miniband characteristics of MSLs. We discuss the results in Sec. III which is
divided into three subsections. Section III.1 discusses the band structure
properties evaluated via the $\bf{k.p}$ method and explains the effect of
varying layer widths on the key band parameters. In Sec. III.2, a comparative
study between a finite MSL and a finite T2SL absorber is presented in terms of
the local density of states (LDOS) and spectral tunneling transmission
evaluated via the NEGF approach. Finally, Sec. III.3 depicts the potential of
MSL structures to be used as both electron and hole barriers with respect to
the T2SL based absorbers. The paper is concluded in Sec. IV.

##  II THEORETICAL APPROACH

###  II.1  $\bf{k.p}$ method

The electronic band structure of the non-common atom (NCA) interface InAs/GaSb
based T2SL is commonly calculated by the k.p based envelope function
approximation (EFA) approach Szmulowicz (1996, 1998). To account for the
strain and interface effects, an additional layer of InSb is inserted at the
interface of InAs and GaSb Delmas  _et al._ (2019); Mukherjee  _et al._
(2021); Livneh  _et al._ (2012).

Figure 1: Energy band alignment of the 6.1Å family of compound semiconductors
(InAs, GaSb and AlSb): The given arrangement of materials as shown here, forms
a M-shaped structure within a typical unit cell of the periodic SL. In the
absence of the AlSb layers, the structure becomes a conventional InAs/GaSb
T2SL.

However, the EFA model is incapable of distinguishing the CA and NCA type
interfaces leading to an overestimation of the bandgap Li  _et al._ (2010).
Therefore, to investigate these NCA superlattices, we include the model
proposed by Krebs and Voisin with the EFA model, which distinguishes the
interface chemical bonds stacked in forward and backward directions Krebs and
Voisin (1996); Li  _et al._ (2010). In order to take the MIA effect into
consideration within the $\bf{k.p}$ framework, the two distinct interfaces,
GaSb-on-InAs and InAs-on-GaSb are considered Li  _et al._ (2010); Hong  _et
al._ (2009); Krebs and Voisin (1996); Mlinar  _et al._ (2005); Szmulowicz  _et
al._ (2004). The explicit Hamiltonians for both interface and strain are added
with the original $\bf{k.p}$ Hamiltonian. The total Hamiltonian of the system
can thus be written as $H(k)=H_{k}+H_{S}+H_{IF}+H_{strain}$, where, $k$ is the
in-plane wave vector, $H_{k}$ are $H_{S}$ are respectively the $\bf{k}$ and
spin dependent terms, $H_{strain}$ stands for the substrate induced strain
effects due to the lattice mismatch Li  _et al._ (2010); Livneh  _et al._
(2012); Delmas  _et al._ (2019); Becer  _et al._ (2019), and $H_{IF}$ is the
interface term which accounts for the NCA interface MIA effects, given by

$H_{4}^{I}=H_{XY}^{AB/BA}\begin{bmatrix}0&0&0&0\\\ 0&1&1&0\\\ 0&1&1&0\\\
0&0&0&1\end{bmatrix},$ (1) $H_{IF}=\begin{bmatrix}H_{4}^{I}&0\\\
0&H_{4}^{I}\end{bmatrix}.$ (2)

Here, $H_{IF}$ is added at each interface by the delta interface strength
potentials i.e $H_{XY}^{BA}=490meV$ and $H_{XY}^{AB}=870meV$, respectively
denoting the InSb (GaSb-on-InAs) and GaAs (InAs-on-GaSb) like strength
potentials at the interface Li  _et al._ (2010); Livneh  _et al._ (2012); Li
_et al._ (2010); Lang and Xia (2011); Le Thi  _et al._ (2019). For MSL, the
GaSb layer is treated as unstrained and the other layers i.e InAs and AlSb as
strained to attain the GaSb lattice constant Lang and Xia (2011). The strain
Hamiltonian contains the Pikus-Bir deformation potential (b, ac, av) Livneh
_et al._ (2012); Lang and Xia (2011) and the strain terms, given by

$\centering\epsilon_{xx}=\epsilon_{yy}=\frac{a_{GaSb}-a}{a},\@add@centering$
(3)
$\centering\epsilon_{zz}=-2\frac{C_{11}}{C_{12}}\epsilon_{xx},\@add@centering$
(4)

where, $a_{GaSb}$ is the lattice constant of GaSb and $a$ is the lattice
constant of the layer for which strain parameters are to be calculated,
$\epsilon_{xx}$, $\epsilon_{yy}$, and $\epsilon_{zz}$ are strains along x, y
and z direction, and $C_{11}$, $C_{12}$ are the elastic stiffness coefficients
Livneh  _et al._ (2012). The interband momentum matrix Kane’s parameter, being
ideally taken same throughout the lattice, is assumed to be an weighted
average of Kane’s energy of InAs, GaSb and AlSb with respect to their
thicknesses Lang and Xia (2011). Also, a finite difference discretization
technique is employed with the periodic boundary conditions to solve the slow
varying envelope functions Mukherjee  _et al._ (2021); Galeriu (2005). The
schematic of the MSL structure as shown in Fig. 1, contains the additional
AlSb layers at the center of each GaSb layer throughout the lattice. The
interface matrix ($H_{IF}$) is only considered at the NCA interfaces, and the
other material parameters for InAs, GaSb, and AlSb taken for the simulation of
MSL electronic band structure are provided in Table 1.

Table 1:  Material parameters of InAs, GaSb and AlSb used for the $\bf{k.p}$ electronic band structure calculation at a temperature of $77K$ Becer  _et al._ (2019); Livneh  _et al._ (2012); Delmas  _et al._ (2019); Qiao  _et al._ (2012); Mukherjee  _et al._ (2021); Vurgaftman  _et al._ (2001) Parameters | InAs | GaSb | AlSb
---|---|---|---
Lattice constant(Å) | 6.0584 | 6.0959 | 6.1297
Energy band gap at 0K ($eV$) | 0.418 | 0.814 | 2.386
Elastic stiffness constant ($C_{11}$) | 8.329 | 8.842 | 8.769
($10^{11}dyne/c{m^{2}}$) |  |  |
Elastic stiffness constant ($C_{12}$) | 4.526 | 4.026 | 4.341
($10^{11}dyne/c{m^{2}}$) |  |  |
Deformation potential $ac$ ($eV$) | -5.08 | -7.5 | -4.5
Deformation potential $av$ ($eV$) | 1 | 0.8 | 1.4
Deformation potential $b$ ($eV$) | -1.8 | -2 | -1.35
Varshini Parameter $\alpha$ $[meV/K$] | 0.276 | 0.417 | 0.42
Varshini Parameter $\beta$ [$K$] | 93 | 140 | 140
Effective mass electron ($m_{e}^{*}$) | 0.022 | 0.0412 | 0.14
Luttinger parameter $\gamma 1$ | 19.4 | 11.84 | 4.15
Luttinger parameter $\gamma 2$ | 8.545 | 4.25 | 1.28
Luttinger parameter $\gamma 3$ | 9.17 | 5.01 | 1.75
Interband mixing parameter Ep [$eV$] | 21.5 | 22.4 | 18.7
Spin orbit splitting (SO) [$eV$] | 0.38 | 0.76 | 0.65
Valence band offset (VBO) [$eV$] | -0.56 | 0 | -0.38

###  II.2 Keldysh NEGF method

In quantum-confined finite SL structures, the LDOS and electronic transmission
function offer a concrete and qualitative understanding on the quantum
mechanical nature of the carrier localization profile Aeberhard  _et al._
(2018); Galvani  _et al._ (2017) and spectral current flow under different
biasing conditions. In this work, we employ the quantum transport based NEGF
formalism Datta (2012, 2005) as a mathematical tool to numerically compute
these parameters, incorporating the concerned device physics models Henrickson
(2002); Aeberhard (2011); Haldaś  _et al._ (2011); Aeberhard and Morf (2008);
Stewart and Léonard (2004); Mukherjee  _et al._ (2018); Mukherjee and
Muralidharan (2019, 2018); Priyadarshi  _et al._ (2018); Mukherjee  _et al._
(2021). The retarded Green’s function ($G$) with the proper information of
self energies corresponding to the macroscopic contacts and scattering
phenomena along the longitudinal energy (E) is defined as Cavassilas  _et al._
(2014); Dehdashti Akhavan  _et al._ (2016, 2018); Kolek (2016); Foreman
(1993),

$G(z,z^{\prime},E)=[E^{+}I-H-U-\sum_{j}\Sigma^{C}_{j}(E)-\Sigma^{S}(E)]^{-1},$
(5)

where, $E^{+}=E+i\eta^{+}$ where $\eta^{+}$ is a small positive number,
$z,z^{\prime}$ are the position indices, $I$ denotes the Identity matrix, $H$
represents the 1-D tight-binding Hamiltonian matrix of the SL in real space
and $U$ is the potential profile calculated from the self-consistent NEGF-
Poisson solver, $\Sigma^{C}_{j}$ is the self-energy of the $j^{th}$ contact
(where $j\in R$ (Right), $L$ (Left)), which are calculated from the
corresponding broadening functions ($\Gamma_{1,2}$), and $\Sigma^{S}$ is the
scattering self-energy evaluated using the self-consistent Born approach
Henrickson (2002); Aeberhard (2011); Haldaś  _et al._ (2011); Aeberhard and
Morf (2008); Stewart and Léonard (2004); Mukherjee  _et al._ (2018); Mukherjee
and Muralidharan (2019, 2018); Priyadarshi  _et al._ (2018); Mukherjee  _et
al._ (2021). We calculate the LDOS as the diagonal elements of the spectral
function, given by Mukherjee  _et al._ (2021); Datta (2005)

$\mathcal{A}(z,z^{\prime},E)=i\left[G(z,z^{\prime},E)-G^{\dagger}(z,z^{\prime},E)\right],$
(6)

for the entire energy ($E$) range of interest. In the ballistic transport
limit, the electronic transmission probability is calculated as Datta (2012,
2005)

$T(E)=Re[Tr(\Gamma_{1}G\Gamma_{2}G^{\dagger})].$ (7)

However, in practical devices, scattering effects caused due to the
fluctuations at the atomic level, have a deep impact on the carrier transport.
These effects destroy the system coherence and are included in the simulation
setup through an additional self-energy coupled to the electron and hole
correlation functions ($G^{n,p}$) which are given by

$\begin{split}&G^{n}(E)=G(E)\Sigma^{in}(E)G^{\dagger}(E),\\\
&G^{p}(E)=G(E)\Sigma^{out}(E)G^{\dagger}(E),\end{split}$ (8)

where
$\Sigma^{in}(E)=\sum_{j=L,R}\Gamma^{C}_{j}(E)f_{j}(E-\mu_{j})+\Sigma^{in}_{S}(E)$
and
$\Sigma^{out}(E)=\sum_{j=L,R}\Gamma^{C}_{j}(E)\left(1-f_{j}(E-\mu_{j})\right)+\Sigma^{out}_{S}(E)$.
In particular, $f_{j}(E-\mu_{j})$ is the equilibrium Fermi function
corresponding to the $j^{th}$ contact with the electrochemical potential
$\mu_{j}$ and $\Sigma_{S}^{in(out)}$ signifies the in (out) scattering
functions related to the scattering event $S$.
The scattering matrices corresponding to the momentum and phase relaxed
elastic scattering events are considered as $\Sigma_{S}^{in(out)}=DG^{n(p)}$,
where the scaling factor D signifies the strength of scattering which in our
simulation is taken as $D=10^{-5}eV^{2}$. In contrast, the high temperature
energy relaxing inelastic scattering processes are associated with the
absorption and emission of phonons. The scattering rates in such cases are
determined from the first order self-consistent Born’s approximation (SCBA),
given by

$\Sigma_{S}^{in}(E)=D_{0}(N_{w}+1)G^{n}(E+\hbar*w)+D_{0}N_{w}G^{n}(E-\hbar*w),$
(9)
$\Sigma_{S}^{out}(E)=D_{0}(N_{w}+1)G^{p}(E-\hbar*w)+D_{0}(N_{w})G^{p}(E+\hbar*w)$
(10)

where, $N_{w}=(\exp(\frac{\hbar*w}{K_{B}T})-1)^{-1}$ represents equilibrium
phonon occupation number at temperature T with frequency $w$. Here, the
inelastic scattering coefficient $D_{0}$ is taken as $10^{-2}$ $eV^{2}$. In
such cases, (7) no longer remains valid and the spectral current flowing from
point $z_{i}$ to $z_{i+1}$ is then given by

$I^{sp}_{e(h)}(E)=\frac{iq}{\pi\hbar}\left[H_{i,i+1}G^{n(p)}_{i+1,i}(E)\\\
-H_{i+1,i}G^{n(p)}_{i,i+1}(E)\right].$ (11)

A detailed description of the carrier scattering and their modeling in the
NEGF framework can be found in Refs. Nikonov  _et al._ (2009a, b); Datta
(2005). In our study, we consider both the elastic and inelastic models of
scattering based on their importance and validity.

##  III RESULTS AND DISCUSSIONS

###  III.1 Electronic band properties

The peak absorption properties and the spectral range of operation are the
most crucial features of any IR detector setup which can be determined from
the knowledge of the band structure of the absorber material. In particular,
the bandgap, bandwidth and the other key parameters derived from the band
structure hold much significance owing to their direct connection with the
aforementioned detector parameters.
We, therefore, start the discussion of this section by first evaluating the
band structure properties of a MSL structure and compare the same with that of
a T2SL having almost similar bandgap and electron effective mass for the sake
of a comprehensive understanding about their key differences Lang and Xia
(2013). In order to achieve the correct bandgap values in these broken band
alignment structures having interfaces formed by NCA, the appropriate
interface treatment is required to account for the MIA effects Li  _et al._
(2010). For this, we have considered the interface matrix at the junction of
InAs and GaSb layers, as discussed in section II.1. In order to validate the
k.p model with interface consideration for MSL, we calculate the band gap for
some MSL configurations, and found that the obtained bandgaps with the MIA
effects are in close agreement with the experimental values as shown in Fig.
2. Next, employing the same model, we calculate the electronic band structures
of 6ML/20ML T2SL and 7ML/6ML/3ML/6ML MSL structures having almost similar
bandgaps and plot the obtained $E-k$ dispersion results at T=77K in Figs. 3
and 3. We consider a supercell of three periods to capture the impact of the
alternative interface potentials Li  _et al._ (2010). Here, due to the double
degeneracy Becer  _et al._ (2019), each state splits into two states. The
obtained effective masses from the curvature of dispersion plots for T2SL are
$m^{*}_{e}=0.0466m_{0}$ and $m_{h}^{*}=0.258m_{0}$, similarly for MSL,
$m_{e}^{*}=0.0588m_{0}$ and $m_{h}^{*}=0.26m_{0}$.

Figure 2: Comparison between measured and theoretically predicted bandgaps of
eight MSL samples. Without MIA effects the bandgaps were largely
overestimated, whereas with the inclusion of MIA effects, they are obtained in
close proximity to the experimental values.

Figure 3: Electronic band structure of T2SL and MSL at T = 77 K using 8 band
$k.p$ method: In-plane and out-of-plane E-k dispersion within the 1st
Brillouin zone are calculated for (a) 6ML/20ML T2SL and (b) 7ML/6ML/3ML/6ML
MSL using the periodic boundary condition with $H_{IF}$ matrix added at the
interface. The obtained plots depict strong anisotropy between the in-plane
and out-of-plane directions of the both the structures having almost similar
bandgaps.

In T2SL, the electrons (holes) are spatially confined in InAs (GaSb) layers
and their eigen energies vary with the width of that layer Lang and Xia
(2013); Ting  _et al._ (2020); Delmas  _et al._ (2019). It is evident that the
change in conduction band energy level is eminent and sensitive with respect
to the thickness of the InAs layer, leading to a robust control on the
conduction band offset Lang and Xia (2013); Ting  _et al._ (2020). However,
the variation of GaSb layer width does not have adequate control on the tuning
of valence band due to the large heavy hole effective mass Lang and Xia
(2013); Nguyen  _et al._ (2008); Haugan  _et al._ (2004); Mukherjee  _et al._
(2021). Therefore, the T2SL system strongly suffers from valence band
tunability which is overcome in MSL structures by inserting a thin AlSb layer
within the GaSb layer Lang and Xia (2013); Ting  _et al._ (2020).
The insertion of AlSb divides the GaSb hole quantum well into two quantum
wells, which leads to a reduction of the individual well widths. This suggests
that both the GaSb and AlSb layers have a major role to play in tuning the
valence band maxima (VBmax) and thereby tailoring the bandgap. Figure. 4
depicts the variation of bandgap ($E_{g}$) with respect to $d_{GaSb}$ and
$d_{AlSb}$ in a 2D color plot with a constant $d_{InAs}=12$ML . Similarly,
Fig. 4 shows the $E_{g}$ variation with $d_{InAs}$ and $d_{AlSb}$ with a
constant $d_{GaSb}=5$ML. The variation of conduction band minima (CBmin) and
VBmax are also shown with respect to the same parameters in the insets of Fig.
4 and 4. The presence of the additional AlSb barrier reduces the interaction
between the electrons confined in adjacent InAs layers and as a consequence
CBmin moves upwards due to the lowering of conduction band split Haugan  _et
al._ (2004) as evidenced from Fig. 4. Moreover, the electron wavefunction
becomes more localized in InAs wells which effectively gives rise to a higher
electron effective mass in MSL Lang and Xia (2013); Nguyen  _et al._ (2008);
Haugan  _et al._ (2004) as shown in Fig. 5. In Fig. 4, VBmax shifts up (down)
with the increase in $d_{GaSb}$ ($d_{AlSb}$) when $d_{AlSb}$ ($d_{GaSb}$) is
kept constant. Furthermore, it is observed that as the AlSb width goes up, the
lowest attainable limit of VBmax increases even when GaSb thickness remains
constant. For instance, at $d_{AlSb}=5ML$ and GaSb varying from 3ML to 10ML,
the lowest value attained by VBmax is found to be around $-0.2eV$, whereas at
$d_{AlSb}=1ML$ this value reaches only $-0.125eV$. $d_{InAs}$ has more
pronounced effect on CBmin than $d_{AlSb}$, therefore, CBmin shown in the
inset of Fig. 4, shifts downwards with the increase in $d_{InAs}$, and there
is almost no change with $d_{AlSb}$.

Figure 4: Bandgap and band offsets of MSL with reference to zero energy level
at $T=77K$ in 2D color plots: variation of $E_{g}$ with respect to widths of
(a) $d_{AlSb}$ and $d_{GaSb}$ when $d_{InAs}$ is kept constant, and (b)
$d_{AlSb}$ and $d_{InAs}$ at constant $d_{GaSb}$. In the former case, VBmax
and CBmin shift in opposite directions with the increase in AlSb thickness,
and CBmin is less sensitive with the change in AlSb thickness unlike VBmax.
The latter case portrays the inverse effect of InAs width on CBmin, while AlSb
layer width has direct impact on VBmax, offering a superior tuning of band
edge alignments.

Figure 5: MSL electron and hole effective masses with 2D color plots at
$T=77K$. (a) and (b) Varying $d_{GaSb}$ and $d_{AlSb}$ while keeping
$d_{InAs}$ as constant. (c) and (d) Varying $d_{InAs}$ and $d_{AlSb}$ while
keeping GaSb as constant. AlSb barrier makes electron wavefunction more
localised in InAs wells, therefore $m^{*}_{e}$ increases. $m^{*}_{h}$
increases marginally with AlSb at lower widths of InAs and GaSb, but rises
sharply at higher widths of InAs and GaSb. $m^{*}_{e}$ and $m^{*}_{h}$
increase with AlSb thickness, therefore the tunnelling probability of carriers
will be less in these MSL configurations.

It appears in Fig. 4 that at a constant $d_{AlSb}$ and increasing $d_{InAs}$
from 6ML to 18ML, VBmax remains unchanged, which is similar to the trend
noticed in T2SL Delmas  _et al._ (2019); Mukherjee  _et al._ (2021). Here,
VBmax moves from -0.08 $eV$ to -0.115$eV$ as $d_{AlSb}$ is increased from 1ML
to 5ML for all values of $d_{InAs}$ under consideration. This suggests that
VBmax changes sharply with both $d_{AlSb}$ and $d_{GaSb}$, however, remains
nearly invariant with $d_{InAs}$. Furthermore, at a constant $d_{AlSb}$ and
with increasing $d_{InAs}$, as CBmin is pulled down with VBmax remains
unchanged, the effective bandgap decreases as observed in Fig. 4. Such a wide
tunable range in valence band edge is key to design hole blocking barriers in
hetero-structure based photodetectors to suppress the SRH processes in the
depletion region which in turn alleviates dark current Nguyen  _et al._
(2007b); Zavala-Moran  _et al._ (2020); Rodriguez  _et al._ (2007). The
obtained bandgap values in Fig. 4 and Fig. 4 corresponds to the wavelength
range between 3$\mu m$ to 12$\mu m$.
Next, we turn our attention towards the DOS effective masses of electrons and
holes which are calculated from the band structure data using the relation
$m_{e(h)}^{*}=\left(m_{e(h)\parallel}^{*}\right)^{2/3}\left(m_{e(h)\perp}^{*}\right)^{1/3}$
and are plotted in 2D color plots with respect to the variation in the layer
thicknesses. It is seen from Fig. 5 and Fig. 5 that the obtained electron
effective masses of MSL are higher than that of T2SL for a similar wavelength
range Mukherjee  _et al._ (2021); Delmas  _et al._ (2019) due to the strong
electron localization in InAs layer is fuelled by the additional AlSb layer
Lang and Xia (2013); Nguyen  _et al._ (2008); Haugan  _et al._ (2004). In
particular, when we consider the bandgap range from $0.1eV$ to $0.3eV$, the
maximum electron mass obtained in case of T2SL is approximately $0.034m{0}$
Mukherjee  _et al._ (2021); Delmas  _et al._ (2019), while in MSL, for the
same range, the electron masses are varying from $0.06m_{0}$ to $0.08m_{0}$.
It is also evident that higher electron effective masses can be achieved even
with thin GaSb layer, as they are more sensitive to $d_{AlSb}$ Lang and Xia
(2013); Haugan  _et al._ (2004). From Fig. 5 and Fig. 5, it is observed that
the rise in hole effective masses with the increase in AlSb width is prominent
only at higher GaSb and InAs thicknesses. The acquired higher electron masses
are important for the p-$\pi$-M-n type structures, in which, an additional MSL
is inserted between the $\pi$ and n region of traditional p-$\pi$-n structure
to restrict the carrier transport due to diffusion and tunneling at depletion
region to lessen the dark current Nguyen  _et al._ (2007b, a); Delmas  _et
al._ (2019).

###  III.2 Comparison of finite MSL and T2SL absorber

Our next step is to optimize the absorber layer configuration for better
optical properties through the enhancement of interband carrier overlap Ting
_et al._ (2020). Due to the spatial separation of carriers, T2SL has less
oscillator strength, which leads to weaker absorption Ting  _et al._ (2020);
Razeghi and Nguyen (2010). Whereas, the optical properties of MSL are expected
to be better than T2SL, and even comparable to short period T2SL Taalat  _et
al._ (2013); Lang and Xia (2013). To model finite superlattice structures in
the quantum mechanical framework, we implement single band NEGF approach using
the effective mass Hamiltonian and on the basis of obtained properties, we
compare the properties of few-period MSL and T2SL structures Mukherjee  _et
al._ (2021); Priyadarshi  _et al._ (2018); Dehdashti Akhavan  _et al._ (2016);
Akhavan  _et al._ (2016); Aeberhard  _et al._ (2018). The parameters used for
the simulation are provided in Table 2.

Table 2:  Material parameters used in the NEGF simulation Livneh  _et al._ (2012); Vurgaftman  _et al._ (2001) Parameters | InAs | GaSb | AlSb
---|---|---|---
Electron effective mass ($m_{e}^{*}$) | 0.023 | 0.041 | 0.14
Heavy hole effective mass ($m_{h}^{*}$) | 0.4 | 0.4 | 0.9
VBO [$eV$] @300K | -0.50 | 0 | -0.44
VBO [$eV$] @77K | -0.56 | 0 | -0.38

Figure 6: Transmission function and miniband formation at $T=77K$: (a)
Electron and hole transmission probabilities of ten period 6ML/20ML T2SL. (b)
LDOS of ten period 6ML/20ML T2SL. (c) Electron and hole transmission
probabilities of a 10 period 7ML/6ML/3ML/6ML MSL. (d) LDOS of ten period
7ML/6ML/3ML/6ML MSL. Transmission probabilities are plotted with respect to
energy, and LDOS plotted in a grey scale 2-D plot in the position and energy
space. The transmission probability in MSL is less than T2SL in valence band.
The conduction band bandwidth of MSL is less than T2SL, also there is a
formation of two quantum wells in GaSb due to the AlSb layer in MSL.

Figure 6 and 6 depict the electron and heavy hole transmission probabilities
in the ballistic limit for the same configuration of T2SL and MSL used in k.p
calculations, respectively Priyadarshi  _et al._ (2018); Mukherjee and
Muralidharan (2018). To obtain absolute transmission, the conduction and
valence band edges of the contacts are assumed at their respective lowest and
highest values. In Fig. 6 and Fig. 6, LDOS obtained via (6) within the elastic
limit of scattering are shown for both T2SL and MSL Priyadarshi  _et al._
(2018); Mukherjee  _et al._ (2021). The electron (hole) localization in InAs
(GaSb) layer is fairly evident from the LDOS plots. In particular, the hole
confinement in the two adjacent GaSb quantum wells formed due to the insertion
of AlSb, can be distinctly seen in MSL. In addition, the first conduction mini
bandwidth observed in MSL is lower than that of T2SL, which implies for higher
localization of electrons in MSL. The bandgaps obtained through the NEGF
simulation for the two structures are similar and in well agreement with the
$k.p$ results, discussed in Sec. III.1.
In Fig. 7 and Fig. 7, we plot the total number of states in the first
conduction ($A_{C1}$) and heavy hole ($A_{HH1}$) bands for T2SL and MSL,
respectively, as a function of position by integrating the LDOS over the
energy range of interest Mukherjee  _et al._ (2021); Priyadarshi  _et al._
(2018). In continuation, we further plot the spatial product
($A_{C1}$*$A_{HH1}$) of them in Fig. 7 and Fig. 7 to understand the nature of
interband overlap responsible for carrier transition Mukherjee  _et al._
(2021).

Figure 7:  Available states in conduction and valence band and their spatial
product at $T=77K$. (a) and (b) number of available states in conduction band
and valence band with respect to the position for T2SL and MSL. (c) and (d)
The spatial product of number of available states in C1 and HH1 for T2SL and
MSL. For MSL, insertion of a thin AlSb layer in the middle of each GaSb hole
quantum well pushes the HH1 wave function out for stronger overlap with the C1
wave function, therefore the interband overlap is higher in MSL, which
indicates more absorption. Figure 8:  1D-DOS for first conduction and first
heavy hole band for 6ML/20ML T2SL (red) and 7ML/6ML/3ML/6ML MSL (blue) at
$T=77K$. The higher hole densities in T2SL suggests higher probability of
auger recombination than MSL.

This spatial product in MSL is found to be quite higher than that in T2SL
especially at the interface region, which clearly indicates a stronger overlap
between C1 and HH1 wavefunctions. The AlSb layer in MSL pushes the HH1
wavefunction towards the newly created wells. As a consequence, the centre of
HH1 wavefunction is shifted more towards the center of C1 wavefunction which
leads to an enhancement in the interband overlap, and hence provides higher
absorption in MSL Ting  _et al._ (2020). In Fig. 8, we plot the 1D-DOS
calculated by integrating the LDOS over the entire T2SL and MSL lengths
Mukherjee  _et al._ (2021). It is noticed that the 1D-DOS for VB holes are
higher in T2SL than in MSL. Therefore, by comparing the maximum hole density
values in T2SL and MSL, we predict that Auger recombination will be less in
MSL, causing lesser dark current Flatté and Grein (1998). Also, higher
electron localization in MSL is an indicative for reduced tunneling dark
current in MSL absorber.
Having evaluated and compared the LDOS properties of MSL and T2SL structures,
it is now customary to examine the nature of carrier transport through these
structures in order to gain a qualitative insight on the dark current at any
given operating point and establish its connection to the LDOS. In doing so,
we plot the normalized spectral current, given by (11), of electrons and holes
with respect to the position at an applied voltage of $0.015V$ and $T=300K$ in
Fig. 9 and Fig. 9 for the T2SL and MSL structures, respectively. For a fair
comparison, we maintain a nearly similar electric field corresponding to the
applied bias of $0.015V$ across both these structures by considering ten
periods of 6ML/20ML T2SL and twelve periods of 7ML/6ML/3ML/6ML MSL. The
contacts are assumed to be carrier selective with one for the electron
injection and the other for the hole. We also bring in the non-coherent
transport features in our simulation model by including the self-consistent
inelastic electron-phonon scattering model Nikonov  _et al._ (2009a) with a
phonon energy of $30meV$ Deacon  _et al._ (2005); Li  _et al._ (2010) to look
for possible broadening of the current spectra, although they are less likely
in the near-equilibrium regime of transport considered here. The bright
stripes observed in both the figures are distinctly indicative of the miniband
transport of electrons and holes for both the T2SL and MSL. However, a careful
observation reveals that the T2SL current spectrum is broader than the MSL,
especially near the contacts, which fairly justifies the strong localization
in MSL as discussed earlier. Furthermore, the amount of broadening observed
around the contacts with inelastic scattering largely vanishes when only the
ballistic and elastic scattering effects are considered. This clearly points
towards the existence of phonon-actuated energy relaxation processes of
carriers occurring at higher temperatures. These processes become predominant
at higher built-in and applied field and play a key role in tailoring the
carrier transport. Therefore, this study holds much significance in the
context of understanding the miniband and phonon-mediated hopping transport of
carriers through heterostructures and can be extremely useful to predict the
dark and photo current of T2SL systems.

Figure 9: Spatially and energetically resolved normalized dark current for ten
periods of 6ML/20ML T2SL and twelve periods of 7ML/6ML/3ML/6ML MSL with the
inclusion of optical phonons with energy $30meV$ at $T=300K$ with the applied
voltage of 0.015 V (a) Dark current spectrum in conduction and valence bands
of T2SL (b) Dark current spectrum in conduction and valence bands of MSL. The
dark current spectrum at lead-device interfaces reflects the spectrum of
states from which the carriers injects from the contacts.

###  III.3 MSL as unipolar or bipolar barrier

Figure 10: CB and VB offset tuning using MSL in 2D color LDOS plot at $T=77K$:
unipolar (a) electron barrier ($B_{e}$) having CB offset of 0.27$eV$ using
5ML/6ML/3ML/6ML MSL and (b) hole barrier ($B_{h}$) with 0.13$eV$ VB offset
using 10ML/3ML/3ML/3ML MSL are shown with respect to a 9ML/9ML T2SL absorber
($A_{b}$). (c) Bipolar barrier using $B_{e}A_{b}B_{h}$ design having both
electron and hole blocking barriers. Specific band offsets can be attained by
the appropriate tuning of thicknesses of multiple layers.

Photodetectors suffer from noise-inducing currents such as generation currents
related to SRH processes in the depletion region, current due to thermal
generation of carriers in the absorber region and their diffusion to the
contact layers, and surface currents Maimon and Wicks (2006); Rakovska  _et
al._ (2000). SRH current, which dominates at lower temperature through the
activated midgap traps, is minimized by utilizing barrier-based structures
like nBn, XBp, XBn etc. The barrier blocks the majority carriers flow to
inhibit the SRH processes in the depletion region Zavala-Moran  _et al._
(2020); Rodriguez  _et al._ (2007). These barriers are usually made up of bulk
materials which have limited applications due to their inadequacy to provide
specific conduction and valence band tunability, and large dark currents at
elevated temperatures Zavala-Moran  _et al._ (2020); Ting  _et al._ (2020).
The design of barriers in the photodetector is noteworthy as the barrier
height and width mutually determine the ability to block the thermal
excitation of majority carriers from the contact layers, and the potential to
impede the electron tunneling through it Maimon and Wicks (2006); Zavala-Moran
_et al._ (2020); Nguyen  _et al._ (2007a, b, 2009). Here, we construct MSL
based barriers as the insertion of AlSb layer provides an additional degree of
freedom to design and control the band offsets as per requirement. Moreover,
MSL structures due to their high effective mass as discussed in Sec. III.1,
become more resistant to the diffusion and tunneling transport in the
depletion region Nguyen  _et al._ (2009, 2007b, 2007a); Lang and Xia (2013).
In this work, we present three configurations for $XB_{e}A_{b}$, $XB_{h}A_{b}$
and $XB_{e}A_{b}B_{h}X$ structures, where $B_{e}$ ($B_{h}$) is the electron
(hole) barrier layer made of MSL, $A_{b}$ is the T2SL absorber layer and $X$
is the contact layer which can be composed of T2SL or bulk material. The band
offsets pertaining to the electron and hole barriers with respect to a 9ML/9ML
T2SL absorber ($A_{b}$), as obtained from the outcome of LDOS calculated via
NEGF at 77$K$, are respectively plotted in Fig. 10 and Fig. 10. Here, $B_{e}$
and $B_{h}$ are respectively modeled using 5ML/6ML/3ML/6ML and
10ML/3ML/3ML/3ML MSL configurations. It is noted from Fig. 10 that the $B_{e}$
layer provides a conduction band offset of approximately 0.27 $eV$ above and a
nearly zero valence band discontinuity with respect to $A_{b}$, making it an
ideal unipolar electron barrier that blocks the majority carriers electrons
from the contacts and allows the minority holes to pass through it,
functioning similar to a pn junction space charge region Zavala-Moran  _et
al._ (2020); Maimon and Wicks (2006). These barriers are usually intrinsic or
have the similar doping as in the absorber region. Therefore, most of the
depletion region lies within them, resulting in a reduced SRH dark current due
to their high bandgap Zavala-Moran  _et al._ (2020); Maimon and Wicks (2006).
Similarly, $B_{h}$ in Fig. 10 offers a valence band offset of 0.13$eV$ with
respect to $A_{b}$, while having a negligible conduction band discontinuity.
Such a unipolar hole barrier opposes the flow of majority holes without
affecting the minority electrons flow. Combining these two, one can design a
$B_{e}A_{b}B_{h}$ bipolar barrier structure as depicted in Fig. 10 where
$B_{e}$ ($B_{h}$) is sandwiched between the p-type (n-type) contact and
absorber layer, which blocks the minority diffusion electron (hole) current
from the p-type (n-type) contact Gautam  _et al._ (2013); Zavala-Moran  _et
al._ (2020); Maimon and Wicks (2006).
The design strategy to achieve such MSL barriers although appears to be quite
challenging from an engineering perspective, follows from a definite physics-
based guideline demonstrated earlier while discussing the band structure in
Sec. III.1. With reference to the 9ML/9ML T2SL, the AlSb layer in MSL splits
the GaSb hole quantum well into two and reduces their effective width Lang and
Xia (2013). This pushes VBmax of MSL down with respect to the T2SL, giving
rise to a VB offset for the configuration 10ML/3ML/3ML/3ML shown in Fig.10. In
this case, for a zero CB offset, the thickness of the InAs layer in MSL should
be kept slightly larger (10ML) than in T2SL to compensate the rise of CBmin
caused due to the presence of the AlSb electron barrier layer. Similarly, the
lowering of InAs thickness in MSL pushes the CBmin up and gives rise to a CB
offset for the configuration 5ML/6ML/3ML/6ML as shown in Fig. 10. To ensure
the alignment of VB in this case to have zero offset, one should increase the
thickness of the two GaSb layers (6ML each) to pull the VBmax up to compensate
the downshift in VBmax due to AlSb. This discussion, as a whole, should serve
as a predictive and robust guideline to design efficient barriers using
complex T2SL structures.

##  IV Conclusion

This study provided comprehensive design guidelines of utilizing M-structured
superlattices for both the absorber and barrier layers through proper band
engineering and discussed its potential benefits over conventional T2SL
structures. Our detailed calculations carefully took into account the effects
of both strain and microscopic interface asymmetry to primarily estimate the
bandgap and density-of-states effective mass and their variation with respect
to the thicknesses of the constituent material layers. In contrast, for
practical finite-period structures, the local density-of-states and spectral
tunneling transmission and current calculated using the Keldysh non-
equilibrium Green’s function approach with the inclusion of non-coherent
scattering processes offered deep insights into the qualitative aspects of
miniband and localization engineering via structural variation. Our key
results demonstrated how to achieve a wide infrared spectral range, reduce
tunneling dark currents, induce strong interband wavefunction overlaps at the
interfaces for adequate absorption, and excellent band-tunability to
facilitate unipolar or bipolar current blocking barriers. This study,
therefore, exemplifies the utilization of 6.1Å material library to its full
potential through the demonstration of band engineering in M-structured
superlattices and sets up the right platform to possibly replace other complex
superlattice systems for targeted applications.

## Acknowledgments

The authors acknowledge the funding from the PMRF PhD scheme of Ministry of
Education, Government of India. This work is also supported by ISRO-IIT Bombay
Space Technology Cell. The research and development work undertaken in the
project under the Visvesvaraya Ph.D Scheme of the Ministry of Electronics and
Information Technology (MEITY), Government of India, is implemented by Digital
India Corporation (formerly Media Lab Asia).

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