# Jointprop: Joint Semi-supervised Learning for Entity and Relation Extraction
with Heterogeneous Graph-based Propagation

Zheng Yandan1,2, Hao Anran1 and Luu Anh Tuan1
1School of Computer Science and Engineering
2 Interdisciplinary Graduate Program-HealthTech
Nanyang Technological University, Singapore
{yandan002<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Semi-supervised learning has been an important approach to address challenges
in extracting entities and relations from limited data. However, current semi-
supervised works handle the two tasks (i.e., Named Entity Recognition and
Relation Extraction) separately and ignore the cross-correlation of entity and
relation instances as well as the existence of similar instances across
unlabeled data. To alleviate the issues, we propose Jointprop, a Heterogeneous
Graph-based Propagation framework for joint semi-supervised entity and
relation extraction, which captures the global structure information between
individual tasks and exploits interactions within unlabeled data.
Specifically, we construct a unified span-based heterogeneous graph from
entity and relation candidates and propagate class labels based on confidence
scores. We then employ a propagation learning scheme to leverage the
affinities between labelled and unlabeled samples. Experiments on benchmark
datasets show that our framework outperforms the state-of-the-art semi-
supervised approaches on NER and RE tasks. We show that the joint semi-
supervised learning of the two tasks benefits from their codependency and
validates the importance of utilizing the shared information between unlabeled
data.

##  1 Introduction

Named Entity Recognition (NER) and Relation Extraction (RE) are two crucial
tasks in Information Extraction. Supervised learning schemes have made
significant progress in NER and RE research by leveraging rich annotated data
(e.g., Lin et al. (2020); Yamada et al. (2020); Baldini Soares et al. (2019).
However, high-quality data annotation still involves extensive and expensive
labor. Moreover, training NER and RE models in various domains and
applications require diverse annotated data. Semi-supervised learning
approaches (SSL) employ a small amount of annotated data as a source of
supervision for learning powerful models at a lower cost.

(a) Before label propagation

(b) After label propagation

Figure 1: An example of label propagation. We represent the sentence as a
triplet ($entity_{1}$, relation, $entity_{2}$) which consists of an entity
pair (circle) and a relation (triangle) in a graph structure. The colored
nodes indicate labeled semantic units (entity or relation candidates), while
the uncolored nodes represent the unlabeled semantic units. Purple denotes
relation label Used-for, blue denotes for entity label Method, and orange
denotes another entity label label Task.

SSL in NER and RE have performed very well in recent years by employing
bootstrapping, distant supervision or graph-based approach (Batista et al.,
2015; Zeng et al., 2015; Delalleau et al., 2005). However, they either train a
NER (Yang et al., 2018; Chen et al., 2018; Lakshmi Narayan et al., 2019) or a
RE model (Lin et al., 2019; Hu et al., 2021a; Li et al., 2021). Therefore,
they neglect the underlying connections between entity recognition and
relation extraction under a semi-supervised learning scheme, making it harder
for the model to assign the accurate annotation to unlabeled data. For
instance, in Figure 1, the annotated entity entity “generative model” in
sentence S1 and the unannotated “probabilistic model” in sentence S2 are
syntactically similar. Likewise, the context phrases “uses… to” and “used in”
are also similar. If such similarities are ignored, the model may fail to draw
a syntactic analogy between “dependency parsing” and “alignment”, and thereby
miss labeling the latter as an entity that shares the same type with the
former. To the best of our knowledge, there is no universal framework to
integrate the semi-supervised learning for different tasks in IE, despite
evidence of the effectiveness of a joint or multi-task learning approach (Luan
et al., 2018a, 2019; Ye et al., 2021; Luan et al., 2018a, 2019; Lin et al.,
2020).

In addition, existing semi-supervised approaches devote considerable effort to
aligning labeled and unlabeled data but do not exploit similarities between
instance pairs that are structurally parallel, which exist across unlabeled
data. Consequently, they do not perform classification from the perspective of
global consistency. For example, given the sentences S1 to S3 in Figure 1, we
expect a system to recognize the entities and relations as (Method, Used-for,
Task) in triplet form. However, it is hard to infer the correct pseudo label
to the unlabeled entities “alignment” or “NLI alignment” from the annotated
entity “dependency parsing”. Because they are not semantically or lexically
similar. Likewise, the affinity between “uses to” and “apply” is not obvious;
and hence it would be difficult to extract the relation Used-for between
entities. Nonetheless, the “alignment” and “NLI alignment” pair are alike, and
so are the pair “probabilistic model” and “probabilistic model”. Exploiting
the relationships between unlabeled data would integrate the information
hidden in the text and make use of the large quantity of unlabeled data for
semi-supervised learning.

To address the above limitations, we propose a semi-supervised method based on
label propagation over a heterogeneous candidate graph to populate labels for
the two tasks (see Figure 3). More specifically, we introduce a joint semi-
supervised algorithm for the two tasks, where unannotated and annotated
candidates (entities and relations) are treated as nodes in a heterogeneous
graph, and labels are propagated across the graph through similarity-scored
edges. Our framework Jointprop considers the interactions among the unlabeled
data by constructing the graph using the union of labeled and unlabeled data
into one learning diagram. We evaluate Jointprop on multiple benchmark
datasets and our proposed framework achieve state-of-the-art results on both
semi-supervised NER and RE tasks. To the best of our knowledge, this is the
first work that performs semi-supervised learning for entity and relation
extraction in a unified framework to leverage unannotated data for both tasks.

Our contributions are summarized as following:

  * •

We propose a joint learning scheme using heterogeneous graph-based label
propagation for semi-supervised NER and RE. The model exploits the
interrelations between labeled and unlabeled data and the similarity between
unlabeled examples from both tasks by propagating the information across a
joint heterogeneous graph. To the best of our knowledge, this is the first
work that combines semi-supervised NER and RE.

  * •

We propose a unified semi-supervised framework for both entity and relation
extraction. The framework generates candidate spans from the unlabeled data,
automatically constructs a semantic similarity-based graph for all the
candidates, and performs label propagation across the graph.

  * •

We show that our proposed method can reliably generate labels for unlabeled
data and achieve good performance under a limited data scenario. Our model
outperforms strong baselines in two- and single-task settings and establishes
new state-of-the-art F1 on benchmark datasets.

##  2 Related Work

##### Joint Entity and Relation Extraction

Name Entity Recognition, and Relation Extractions are two essential problems
in information extraction (Grishman, 1997). Exploiting their
interrelationships, models that combine the identification of entities and
relations have attracted attention. Conventional joint extraction systems
combine the tasks in a pipelined approach (e.g., Ratinov and Roth (2009); Chan
and Roth (2011); Luu et al. (2014, 2015); Tuan et al. (2016)): first
identifying entities and employing the detected entity for relation
extraction. However, they overlook their inherent correlation. Recent works
have proposed coupling various IE tasks to avoid error propagation issues. For
example, joint extract entities and relations (Miwa and Sasaki, 2014; Li and
Ji, 2014; Luu et al., 2016) or end-to-end multi-task learning (Luan et al.,
2018a, 2019; Wadden et al., 2019; Lin et al., 2020; Zhang et al., 2017).
Despite evidence of the efficiency of joint or multi-task learning, there is
currently no framework that integrates semi-supervised learning for both tasks
in a joint entity and relation extraction system.

##### Semi-supervised learning

The Semi-Supervised learning seeks to enhance limited labeled data by
leveraging vast volumes of unlabeled data (Søgaard, 2013) which mitigate data-
hungry bottleneck and supervision cost. SSL has a rich history (Scudder,
1965). There have been substantial works in semi-supervised settings in NLP,
such as bootstrapping (Gupta and Manning, 2014, 2015; Batista et al., 2015),
co-training (Blum and Mitchell, 1998), distant supervision (Zeng et al., 2015;
Yang et al., 2018), and graph-based methods (Delalleau et al., 2005;
Subramanya and Bilmes, 2011; Subramanya et al., 2010; Luan et al., 2017).

In particular, graph-based SSL algorithms have gained considerable attention
(Zhu and Ghahramani, 2002; Seeger, 2001; Delalleau et al., 2005). There are
two underlying assumptions for the label propagation (Zhou et al., 2004).
First, similar training samples are more likely to belong to the same class.
Second, nodes in similar structures are likely to have the same label. Label
propagation exploits these assumptions by propagating label information to
surrounding nodes based on their proximity. The metric-based method had been
applied in a graph-based SSL setting for its ability to infer labels for
unseen classes directly during inference. For example, Luan et al. (2017)
propagates the label based on estimating the posterior probabilities of
unlabeled data. Meanwhile, Liu et al. (2019) sought to exploit the manifold
structure of novel class space in a transitive setting.

##  3 Methodology

##### Problem Definition

The input of the problem is a sentence $X=\\{x_{1},...,x_{n}\\}$ consisting of
n tokens, from which we derive $S=\\{s_{1},...,s_{d}\\}$, the set of all
possible within-sentence word sequence spans (up to length $L$) in the
sentence. Let START$(i)$ and END$(i)$ denote the start and end indices of span
$s_{i}$, $\mathcal{E}$ denote a set of predefined entity types, and
$\mathcal{R}$ denote the set of relational types. The full data is defined as
$D=(X,Y)$. In Jointprop, the goal is to learn from the small portion of
labelled data $D_{l}$ and generalize to the unlabelled portion of data
$D_{u}$. The labelled data $D_{l}$ and unlabelled data $D_{u}$ are originally
split from the training set $D_{train}$, where $D_{l}\cap D_{u}=\emptyset$.

The purpose of this work is to predict a possible entity type
$y_{e}(s_{i})\in\mathcal{E}$ for each span $s_{i}\in S$ while predicting a
possible relation types $y_{r}(s_{i},s_{j})\in R$ for every pair of spans
$s_{i}\in S,s_{j}\in S$ under SSL settings. The label can also be a ‘null’
label for a span (i.e. $y_{e}(s_{i})=\epsilon$) or a span pair (i.e.
$y_{r}(s_{i},s_{j})=\epsilon$). The output of the task are
$Y_{e}=\\{(s_{i},e):si\in S,e\in\mathcal{E}\\}$ and
$Y_{r}=\\{(s_{i},s_{j},r):s_{i},s_{j}\in S,r\in\mathcal{R}\\}$.

##### Model Overview

Figure 2: Overview of our proposed framework for semi-supervised joint
learning. Span feature generation, Heterogeneous Graph Construction, Joint
Label Propagation are represented in  red arrows, and Model optimization is
illustrated in black arrows

.

Figure 2 illustrates an overview architecture of the proposed Jointprop
framework. Our framework consists of 1) Span feature generation that learns
the discriminative contextualized features for labelled data $D_{l}$ and
unlabeled span $D_{u}$; 2) Heterogeneous Graph Construction which maps both
labelled-unlabeled, labelled-labelled and unlabeled-unlabeled relationships
for both entities and relations; 3) Joint Label Propagation which disseminates
labels over the whole heterogeneous graph is produced by unlabeled nodes, and
4) Model decode and fine-tune module that decodes and select the refined
propagated pseudo labels to perform fine-tuning.

###  3.1 Span feature generation

Our feature extractor is a standard span-based model following prior work
(Wadden et al., 2019; Luan et al., 2018a, b). For each input token $x_{k}$, we
obtain contextualized representations $\mathbf{x}_{k}$ using a pre-trained
language model (e.g., BERT (Devlin et al., 2019)). For the i-th span $s_{k}\in
S$, the span representation $\boldsymbol{h}_{e}(s_{i})$ is as follows:

$\boldsymbol{h}_{e}(s_{i})=[\mathbf{x}_{START(i)};\mathbf{x}_{END(i)};\phi(s_{i})]$
(1)

where $\phi(s_{i})\in\mathbb{R}^{1\times d_{F}}$ denotes the learned
embeddings of span width features.

For each pair of spans input $s_{i},s_{j}\in S$, the span pair representation
is defined as:

$\boldsymbol{h}_{r}(s_{i},s_{j})=[\boldsymbol{h}_{e}(s_{i});\boldsymbol{h}_{e}(s_{j});\mathcal{F}_{af}]$
(2)

where
$\mathcal{F}_{af}=\boldsymbol{h}_{e}(s_{i})\cdot\boldsymbol{h}_{e}(s_{j})$
refers to the entity affinity function of $e(s_{i})$ and $e(s_{j})$.

Both pairwise span feature $\boldsymbol{h}_{r}(s_{i},s_{j})$ and span feature
$\boldsymbol{h}_{e}(s_{i})$ will be fed into feedforward neural networks
(FFNNs) respectively. The probability distribution of entity is denoted as
$P_{e}(e|s_{i}),(e\in\mathcal{E}\cup\epsilon)$ and entity pairs is denoted as
$P_{r}(r|s_{i},s_{j}),(r\in\mathcal{R}\cup\epsilon)$.

The classification loss will be defined as:

$\mathcal{L}=\sum_{t\in T}w_{t}{\mathcal{L}_{t}}$ (3)

where $w_{t}$ is the predefined weight of a task $t$ and T is the total number
of tasks.

We then use labelled data $D_{l}$ to train the classifier $C_{l}$. The $C_{l}$
can generate contextualized span or span pair feature from Equation 1 and
Equation 2 which converts unlabeled data $D_{u}$ into query entity
presentation $\boldsymbol{h}_{u,e}$ or query entity pair representation
$\boldsymbol{h}_{u,r}$. For labelled data $D_{l}$, we denote the $C_{l}$
generated support entity presentation as $\boldsymbol{h}_{l,e}$ and support
entity pair representation as $\boldsymbol{h}_{l,r}$.

###  3.2 Joint Semi-supervised Learning

##### Heterogeneous Graph Construction

We construct the heterogeneous graph to exploit the manifold structure of the
class space and exploit the combination of labelled data $D_{l}$ and unlabeled
data $D_{u}$. Specifically, we first examine the similarity relations among
pairs of unlabeled query data as well as the similarity relationships between
the labelled support data in order to take advantage of the smoothest
constraints among neighbouring query unlabelled data in our semi-supervised
joint entity and relation extraction task. Based on the embedding space, we
propose the use of transductive label propagation to construct a graph from
the labelled support set and unlabeled query set, and then propogate the
labels based on random walks to reason about relationships in labelled support
and unlabeled query sets. Figure 3 illustrates the whole process of
heterogeneous graph-based propagation $\mathcal{G}$. The circle node is the
entity span representation and the triangle node is the relation
representation.

For computational efficiency, we construct a k Nearest Neighbor (kNN) graph
instead of a fully-connected graph. Let $N$ be the number of labelled entity
representations and let $M$ be the number of unlabelled entity
representations. Specifically, we take $N$ entity representations and $M$
unlabelled entity representations as nodes of an entity graph with size
$T_{e}=N+M$. For the relation graph, we take span pair representation as nodes
with size $T_{r}=((N+M)\times(N+M))$. We construct a sparse affinity matrix,
denoted as $\mathbf{A}\in\mathbb{R}^{\mathbf{T}\times\mathbf{T}}$, where
$T_{e},T_{r}\in\mathbf{T}$ by computing the Gaussian similarity function
between each node:

$\mathbf{A}_{ab}=exp(-\frac{||(\textbf{h}_{a},\textbf{h}_{b})||_{2}^{2}}{2\sigma^{2}})$
(4)

where $\textbf{h}_{a}$ denotes the a-th entity representation or pairwise
entity representation (i.e.
$\\{h_{r}(s_{i},s_{j}),h_{e}(s_{i}),h_{e}(s_{j})\\}\in\textbf{h}_{a}$). The
$\sigma$ is the length scale parameter.

Subsequently, we symmetrically normalize the non-negative and symmetric matrix
$\mathbf{O}=\mathbf{A}+\mathbf{A}^{T}$ by applying Normalized Graph Laplacian
on $O$:

$S=H^{(-1/2)}OH^{(-1/2)}$ (5)

where $H$ is a diagonal matrix with its ($i,i$)-value to be the sum of the
i-th row of $O$.

For pairwise span representation $\boldsymbol{h}_{r}(s_{i},s_{j})$ is
essentially a function of $\boldsymbol{h}_{e}(s_{i})$ and
$\boldsymbol{h}_{e}(s_{i}j)$. The entity nodes and the relation nodes are
automatically associated via their representation.

Figure 3: A conceptual demonstration of the label propagation process. Through
the heterogeneous graph, our proposed joint semi-supervised learning method
propagates labels to entity or relation candidates in the unlabeled data
alternatively. As shown in the figure, the pseudo label for entities or
relations will be refined every time t until stabilized.

##### Label propagation

We define a label matrix $Z\in\mathbb{R}^{V\times U}$ where U is either the
size of entity types or relation types $U=\\{\mathcal{E};\mathcal{R}\\}$. For
label matrix, $Z$, the corresponding labelled data are one-hot ground truth
labels and the rest are 0. Additionally, we denote $Y_{t}$ as a representation
of the predicted label distributions at iteration t. Initially, we set the
rows in $Y_{0}=Z$. Starting from $Y_{0}$, message passing via label
propagation in an iterative manner selects the type of the span or span pairs
in the unlabeled set $D_{u}$ according to the graph structure according to the
following operation:

$Y_{t+1}=cSY_{t}+(1-c)Z$ (6)

where $c\in(0,1)$ controls the probability of information being obtained from
a node’s adjacency nodes or its initial label. $Y_{t}$ refers to the predicted
labels at time $t$.

Given $Y_{0}=Z$, and equation (6) , we have:

$Y_{t}=(cS)^{(}t-1)Z+(1-c)\sum^{t-1}_{i=0}(cS)^{i}Y$ (7)

As the parameter $c\in(0,1)$, taking the limit of equation (7) $(t->\infty)$
we have:

$\lim_{t\to\infty}Y_{t}=(1-c)(1-cS)^{-1}=Y_{converge}$ (8)

The label propagation will converge to $Y_{converge}$.

###  3.3 Model optimization

After we obtain the $Y_{converge}$, we use the $softmax$ function followed by
a standard `argmax` operation to determine the pseudo labels $\\{\hat{y}\\}$
for all the instances in the unlabeled set based on the final label
probability matrix $Y_{converge}$. After generating the pseudo labels
$\\{\hat{y}\\}$ for all the labelled data $D_{l}$, we filter those of lower
quality with a confidence threshold of $g$ and combine the rest (of confidence
above the threshold) with the labelled data $D_{l}$ to retrain the
classification model:

$\displaystyle\\{\hat{y}\\}=\\{\hat{y}\text{ | confidence}(y)\geq g\\}$ $\displaystyle(X,Y)=(X,Y)_{D_{l}}+\\{(x,\hat{y})|x\in D_{u}\\}$

As shown in the Figure 2, the final step in our proposed joint semi-supervised
learning framework is re-training. The retraining model is exactly the same as
the baseline model and the joint NER-RE objectives also remain the same

##  4 Experiments

We evaluate the effectiveness of Jointprop against models from two lines of
work: semi-supervsied NER and semi-supervsied RE. We also provided a detailed
analysis to demonstrate the benefits of our framework. For implementation
details and dataset descriptions please refer to Appendix A and Appendix B.

##### Datasets

We perform experiments to assess the efficacy of our framework on four public
datasets: SciERC (Luan et al., 2018b), ACE05 (Walker et al., 2006), SemEval
(Hendrickx et al., 2010) and ConLL (Tjong Kim Sang and De Meulder, 2003).

Settings% labeled Data | Task | 5% | 10% | 20% | 30%
---|---|---|---|---|---
P | R | F1 | P | R | F1 | P | R | F1 | P | R | F1
Beforeprop(baseline) | NER | 46.78 | 47.25 | 47.01 | 52.44 | 59.80 | 55.94 | 55.80 | 62.37 | 58.90 | 60.42 | 67.56 | 63.79
RE | 20.89 | 15.40 | 17.73 | 35.75 | 16.74 | 22.80 | 38.68 | 23.51 | 29.25 | 43.41 | 29.77 | 35.32
Jointprop | NER | 52.67 | 48.46 | 51.02 | 60.15 | 61.95 | 61.04 | 62.03 | 64.52 | 63.25 | 66.55 | 65.73 | 66.19
RE | 40.82 | 33.78 | 36.97 | 44.42 | 26.34 | 39.98 | 44.55 | 45.28 | 44.91 | 57.94 | 39.32 | 46.85

Table 1: Performance on SciERC with various amount of labeled data.

Settings% labeled Data | Task | 5% | 10% | 20% | 30%
---|---|---|---|---|---
P | R | F1 | P | R | F1 | P | R | F1 | P | R | F1
Beforeprop (baseline) | NER | 78.32 | 76.88 | 77.59 | 80.81 | 81.68 | 81.24 | 81.01 | 85.17 | 83.04 | 84.51 | 86.98 | 85.72
RE | 46.33 | 20.85 | 28.76 | 49.10 | 30.76 | 37.82 | 46.71 | 46.31 | 46.51 | 57.59 | 48.78 | 52.83
Jointprop | NER | 81.91 | 78.39 | 80.11 | 83.38 | 82.76 | 83.07 | 86.82 | 83.76 | 85.27 | 87.69 | 86.40 | 87.04
RE | 48.10 | 29.63 | 36.67 | 48.89 | 36.23 | 42.00 | 60.26 | 44.67 | 51.30 | 61.54 | 48.65 | 54.34

Table 2: Performance on ACE05 with various amounts of labelled data.

Methods / % labeled Data | 5% | 10% | 30%
---|---|---|---
P | R | F1 | P | R | F1 | P | R | F1
Mean-Teacher | 70.33 | 68.55 | 69.05 | 74.01 | 72.08 | 73.37 | 79.09 | 82.23 | 80.61
Self-Training | 73.10 | 70.01 | 71.34 | 75.54 | 73.00 | 74.25 | 80.92 | 82.39 | 81.71
DualRE | 73.32 | 77.01 | 74.35 | 75.51 | 78.81 | 77.13 | 81.30 | 84.55 | 82.88
MRefG | 73.04 | 78.29 | 75.48 | 76.32 | 79.76 | 77.96 | 81.75 | 84.91 | 83.24
MetaSRE | 75.59 | 81.40 | 78.33 | 78.05 | 82.29 | 80.09 | 82.01 | 87.95 | 84.81
GradLRE | 75.96 | 83.72 | 79.65 | 78.90 | 82.94 | 81.69 | 82.74 | 88.49 | 85.52
Jointprop$\dagger$ | 76.09 | 86.35 | 80.89 | 79.10 | 88.64 | 83.60 | 83.62 | 89.35 | 86.39
Gold labels | - | - | 84.64 | - | - | 84.40 | - | - | 87.08

Table 3: Performance on SemEval with various labelled data and 50% unlabeled
data. We provide the Gold labels serves as the upper bound of the model.
($\dagger$ indicates our framework.)

Methods / % labeled Data | 5% | 10% | 30%
---|---|---|---
P | R | F1 | P | R | F1 | P | R | F1
VSL-GG-Hier | 84.13 | 82.64 | 83.38 | 84.90 | 84.52 | 84.71 | 85.37 | 85.67 | 85.52
MT + Noise | 83.74 | 81.49 | 82.60 | 84.32 | 82.64 | 83.47 | 84.98 | 84.78 | 84.88
Semi-LADA | 86.93 | 85.74 | 86.33 | 88.61 | 88.95 | 88.78 | 89.98 | 90.52 | 90.25
Jointprop$\dagger$ | 89.88 | 85.98 | 87.68 | 88.76 | 90.25 | 88.89 | 91.16 | 90.58 | 90.87

Table 4: Performance on CoNLL 2003 with various labelled data. ($\dagger$
indicates our framework.)

###  4.1 Main Results

Tables 1 to 4 provide the framework performance on the joint entity and
relation extraction task, the NER task, and the RE task, respectively. Note
that Beforeprop only trains using the labelled corpus. (i.e., The Beforeprop
only trains with $5\%$, $10\%$ and $30\%$ training data.) As no unlabeled data
are used in the training, this indicates the lower bound performance and
establishes a new baseline.

##### Results on SciERC

Table 1 illustrate our main results on semi-supervised joint learning on the
SciERC dataset. We observed Jointprop improve significantly on both entity
recognition and relation extraction. Jointprop achieves 3.97% and 15.89% F1
improvements, respectively, comparing to Beforeprop. This improvement
validates the robustness of Jointprop by performing joint learning on NER and
RE.

##### Results on ACE05

Table 2 we summarize the results of comparing to the baseline performance. As
can be seen from the table, Jointprop improves by around 2% and 5% on F1 for
entity recognition and relation extraction task respectively. The results of
this study provide further evidence of the consistency of the framework for
multitask datasets.

##### Results on SemEval

Table 3 summarizes the experimental results on the SemEval dataset using
various labelled data and 50% unlabeled data. Jointprop improves on the
Beforeprop by 5.47% on average. We can observe that Jointprop attains 1.24%,
1.91% and 0.81% F1 improvements over the state-of-the-art model GradLRE (Hu et
al., 2021b) with 5%, 10% and 30% training data. Moreover, the model’s
performance consistently improves while narrowing down the gap towards the
upper bound as the proportion of labelled data increases. Jointprop
establishes a new state-of-the-art result, indicating that our framework is
relatively robust even when performing a single task: semi-supervised RE.

##### Results on CoNLL

Experimental results on CoNLL dataset are shown in Table 4. Semi-LADA (Chen et
al., 2020) is the current state-of-the-art semi-supervised NER model. In
multiple training data settings, Jointprop achieves an average improvement of
0.9% over Semi-LADA. Semi-LADA reports a 91.83% F1 score in a fully supervised
setting, as the upper bound of the semi-supervised model. Jointprop achieves
90.87% in F1 score with 30% of training data. The difference between the upper
bound and the model performance narrows to less than 1%. Moreover, Jointprop
surpasses the current state-of-the-art semi-supervised NER model, showing our
model’s effectiveness on another single task: semi-supervised NER.

###  4.2 Analysis

| Name Entity Recognition
---|---
Model / Task | 5% | 10% | 20% | 30%
| P | R | F1 | P | R | F1 | P | R | F1 | P | R | F1
Beforeprop | 46.78 | 47.25 | 47.01 | 52.44 | 59.80 | 55.94 | 55.80 | 62.37 | 58.90 | 60.42 | 67.56 | 63.79
w/o REprop | 51.82 | 45.10 | 48.23 | 58.92 | 53.46 | 56.06 | 61.55 | 57.77 | 59.60 | 64.71 | 63.32 | 64.01
Jointprop | 52.67 | 48.46 | 51.02 | 60.15 | 61.95 | 61.04 | 62.03 | 64.52 | 63.25 | 66.55 | 65.73 | 66.19

Table 5: Ablation study on pure NER task on SciERC dataset.

| Relation Extraction
---|---
Model / Task | 5% | 10% | 20% | 30%
| P | R | F1 | P | R | F1 | P | R | F1 | P | R | F1
Beforeprop | 20.89 | 15.40 | 17.73 | 35.75 | 16.74 | 22.80 | 38.68 | 23.51 | 29.25 | 43.41 | 29.77 | 35.32
w/o NERprop | 38.92 | 13.35 | 19.88 | 19.20 | 44.97 | 26.91 | 22.27 | 62.53 | 32.74 | 32.12 | 44.56 | 37.33
Jointprop | 40.82 | 33.78 | 36.97 | 44.42 | 26.34 | 39.98 | 44.55 | 45.28 | 44.91 | 57.94 | 39.32 | 46.85

Table 6: Ablation study on pure RE task on SciERC dataset.

Sentence |  Semi-LADA |  GradLRE |  Jointprop
---|---|---|---
S1: We propose a Cooperative Model for natural language understanding in a dialogue system. |  e1: Method
e2: Task
R: - |  e1: -
e2: -
R: Used-for |  e1: Method
e2: Task
R: Used-for
S2: We address appropriate user modelling in order to generate cooperative responses to each user in spoken dialogue systems. |  e1: Method
e2: Task
R: - |  e1: -
e2: -
R: Used-for (x) |  e1: Method
e2: Task
R: Part-of
S3: We explore correlation of dependency relation paths to rank candidate answers in answer extraction. |  e1: - (x)
e2: Task
R: - |  e1: -
e2: -
R: no_relation (x) |  e1: OST
e2: Task
R: Used-for
S4: We present a syntax-based constraint for word alignment, known as the cohesion constrain. |  e1: Generic (x)
e2: Generic (x)
R: - |  e1: -
e2: -
R: Hyponym-of |  e1: OST
e2: OST
R: Hyponym-of

Table 7: Case study of Jointprop. The red marked span denotes the head ($e1$)
entity while the blue marked span represents the tail ($e2$) entity. Semi-LADA
performs OtherScientificTerm abbreviated as OST. (x) indicates the wrong
prediction and - means the model does not have certain predictions (i.e., The
model does not predict entity type or relation type).

####  4.2.1 Ablation Studies

This section provides comprehensive ablation studies to show the efficacy of
Jointprop frameworks. Tables 5 and 7 show the effect of joint label
propagation on single-task (NER or RE) prediction accuracy. w/o REprop denotes
ablating the relation propagation while w/o NERprop denotes ablating the
entity propagation. As a lower bound to the framework, we provide the
Beforeprop result, which is the base model without any propagation. As shown
in Table 5, although w/o REprop achieved an average 0.85% improvement on F1
compared to Beforeprop. The Jointprop further improve the performance
significantly by 4.01%, 4.98%, 3.65% and 2.19% across 5%, 10%, 20% and 30%
training data, respectively. From Table 7, we observed that w/o REprop attain
an average of 2.94% performance gain in F1 compared to Jointprop. Though w/o
REprop shows its effectiveness, w/o NERprop has 7.03% further overall across
different proportions of training data. In general, we observe that joint
label propagation is very helpful to Jointprop performance, especially for
relation extraction tasks.

We investigate a real and illustrative example in Figure 1. Given sentences S1
to S3. w/o REprop is unable to identify the label of "alignment" in S2 and
"NLI alignment" in S3. Moreover, w/o NERprop tends to miss predict the pseudo
label as no_relation. More specifically, in annotated S1, the entity
"dependency parsing" has no direct link to the entity "alignment" in S2 and
entity "NLI alignment" in S3. Consequently, w/o REprop makes the wrong
prediction. Similar to w/o REprop, the relation indicator "uses..to" in
annotated S1 is semantically similar to "used in" in S2 but not akin to
"apply..for.." in S3, hence w/o NERprop miss identify the label of
$r^{\prime\prime\prime}$. Whereas Jointprop can assign the correct pseudo
label to entities and relations in all three sentences for it benefits from
the shared information from NER and RE. The results indicate that our
framework Jointprop could leverage the interactions across the two tasks and
derive useful information from a broader context. Therefore achieve
significant improvement across NER and RE tasks.

####  4.2.2 Case study

We perform a case study examining our framework’s performance on four
sentences (i.e., S1, S2, S3, and S4) in comparison to the benchmark models
Semi-LADA and GradLRE. Semi-LADA performs semi-supervised NER task while
GradLRE performs semi-supervised RE task. Meanwhile Jointprop performs the
semi-supervised style joint for NER and RE.

S1 has a simple structure, and all three models correctly classify the label
for relation and entity. For S2, the GradLRE misclassifies the "Statistical
machine translation" entity as Task. Most of the labelled samples with given
entity pair are likely as in (e1: Method, e2: Task), plus there is a relation
indicator "in order to," which misguides the GradLRE into the wrong
prediction. Similarly, in S4, Semi-LADA predicts the entity as Generic, the
dominant class in the training set. Jointprop can assign the correct label
without being sensitive to the label distribution in the training data.

Moreover, Semi-LADA fails to recognize the entity "correlation of dependency
relation paths" in S3, while GradLRE cannot identify the relation Used-for.
One possible reason is that there were not many similar long sequences in the
training data. Consequently, Semi-LADA is insufficient in entity learning,
especially for long lines, while the GradLRE fails to establish edges with
samples in the training set. Jointprop not only builds a connection between
labelled and unlabeled data but also within labelled/unlabeled data. The extra
connections hence help our model to make the correct prediction.

####  4.2.3 Qualitative Analysis

Table 8 shows the qualitative results of our proposed method Joint Semi-
supervised Learning for Entity and Relation Extraction with Heterogeneous
Graph-based Propagation. We show the performance of the propagated pseudo
labels with the ground truths under 10% split training set on ACE05 dataset.
As we can see from the performance Table 8, in both NER and RE, the recall of
the predictions indicates that most of the positive candidates have been
propagated a positive label.Meanwhile, the precision of the predictions for
the NER task is also high. However, the precision for the RE task is low,
showing that almost half of the null candidates have been assigned a positive
label. The propagation of RE tasks is still quite challenging.

% | P | R | F1
---|---|---|---
NER | 86.23 | 92.78 | 89.34
RE | 52.17 | 98.82 | 68.57
Table 8: Qualitative results of our method in 10% split on ACE05 dataset.
(Average F1)

In spite of this, our method still generally produces more accurate
predictions. Given a sentence in ACE05: ‘Although the Russian government…’.
Our model prediction for the phrase "Russian government" is "Organization",
which is more accurate than the ground truth GPE-Geographic Entities.

##  5 Conclusion

In this paper, we propose a novel heterogeneous graph-based propagation
mechanism for joint semi-supervised learning of entity and relation
extraction. For the first time, we explore the interrelation between different
tasks in a semi-supervised learning setting. We show that the joint semi-
supervised learning of two tasks benefits from their codependency and
validates the importance of utilizing the shared information between unlabeled
data. Our experiments show that combining the two tasks boosts system
performance. Moreover, by leveraging the inherent We also evaluate three
public datasets over competitive baselines and achieve state-of-the-art
performance. We also conduct ablation studies of our proposed framework, which
demonstrate the effectiveness of our model. We further present case studies of
our system output.

##  6 Limitations

##### May extend to other domains

In this paper, we present a generic framework and evaluate the effectiveness
of our proposed model Jointprop on three public datasets. We may further
extend the framework to various datasets in different domains. For example,
ACE05 (Walker et al., 2006) in social networks, journalism, and broadcasting,
as well as GENIA corpus (Ohta et al., 2002) in biomedical research.

##### May extend to other NLP tasks

Our proposed model focus on two tasks, namely NER and RE. We may extend our
framework to include more information extraction tasks, such as coreference
resolution and event extraction. Moreover, we may contract knowledge graphs
from extracted structural information.

## Acknowledgment

This research is supported by Nanyang Technological University, under SUG
Grant (020724-00001)

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##  Appendix A Experimental Settings

##### Framework

We show our overall framework Jointprop in Figure 2. Following (Liu et al.,
2019), the hyper-parameter $c$ in Equation 6 is set to 0.99. According to our
empirical findings, the best values for the settings of k and $\sigma$ in
graph construction in Section 3.2 are varied in datasets. We select $\sigma$
as two and the k as 50. Meanwhile, We adopt the affinity function
$\mathcal{F}_{af}$ with all the generated spans between relation spans.
Moreover, we perform average pooling for them. The optimal hyperparameters and
settings are selected based on the model’s performance.

##### Training

We employ the BERT-cased as an encoder for SemEval and ConLL datasets and
adopt the SciBERT-SCIVOCAB-cased (Beltagy et al., 2019) encoder for the SciERC
dataset as suggested in (Luan et al., 2019). The rest will be treated as an
unlabeled set. To maximize the loss, we use BERTAdam with a 1e-3 learning
rate. The maximum span width is set at 8.

##  Appendix B Datasets and baselines

###  B.1 Dataset implementation

For semi-supervised joint task, we consider SciERC (Luan et al., 2018b) and
ACE05(Walker et al., 2006) datasets and follow the pre-processing steps in
(Wadden et al., 2019). For a single task, we conduct experiments against
models from two types of work: semi-supervised NER, and semi-supervised RE.
For the semi-supervised NER task, we consider ConLL 2003 (ConLL) (Tjong Kim
Sang and De Meulder, 2003) and adopt the pre-processing in (Chen et al.,
2020). For semi-supervised RE we evaluate our approach on SemEval 2010 Task 8
(SemEval) (Hendrickx et al., 2010) dataset and adopt the pre-processing in (Hu
et al., 2021b). Note that the entity mentioned in the sentences in the SemEval
has been identified and marked in advance.

Table 9 shows the statistics of each dataset.

Dataset | Sentences | Types
---|---|---
Train | Dev | Test | # E | # R
SciERC | 1861 | 275 | 551 | 6 | 7
ACE05 | 10051 | 2424 | 2050 | 7 | 6
ConLL | 14,987 | 3466 | 3684 | 4 | -
SemEval | 7199 | 800 | 1864 | - | 19
Table 9: Statistics for the SciERC, ACE05, ConLL and SemEval datasets. #E:
Number of entity classes. #R: Number of relation classes.

##### Data split for semi-supervised settings

We follow split settings in (Chen et al., 2020), (Hendrickx et al., 2010)
respectively for ConLL and SemEval and generate different proportions (5%, 10%
and 30%) of training data to investigate how training set size impacts
performance and to retain the original development set and test set for
evaluation purposes. Noted that we sample 50% of the training set as the
unlabeled set as (Hendrickx et al., 2010) for fair comparisons. For ACE05 and
SciERC datasets, we split the training data based on documents and generate
5%, 10%l 20% and 30% of training data. In particular, we endeavour to ensure
that each proportion of data contains as many types of entity types and
relation types possible.

###  B.2 Evaluation Metrics

We consider the same criteria to apply as previous works (Hu et al., 2021b, a;
Li et al., 2021; Lin et al., 2019) where precision and recall serve as
supplementary metrics, while F1 score serves as the primary evaluation
metrics. Note that the evaluation excludes the accurate prediction for
no_relation.

###  B.3 Compared baselines

##### Semi-supervised joint learning

For joint learning, because there is no prior study on semi-supervise joint
learning, we use DYGIE++ (Wadden et al., 2019) (i.e. Beforeprop) as our
baseline model to train.

##### Semi-supervised NER

In order to show that our Jointprop framework works with unlabeled data, we
compared it to three recent state-of-the-art semi-supervised NER models that
were already in use:

  * •

VSL-GG-Hier (Chen et al., 2018) introduced a hierarchical latent variables
models into semi-supervised NER learning.

  * •

MT + Noise (Lakshmi Narayan et al., 2019) explored different noise strategies
including word-dropout, synonym-replace, Gaussian noise and network-dropout in
a mean-teacher framework.

  * •

Semi-LADA (Chen et al., 2020) proposes a local additivity based data
augmentation method which uses the back-translation technique.

##### Semi-supervised RE

We compared our Jointprop framework with the following 6 representative semi-
supervised relation models:

  * •

Mean-Teacher promotes the model’s variants to generate consistent predictions
for comparable inputs.

  * •

DualRE (Lin et al., 2019) trains a prediction and retrieval module in
conjunction to choose samples from unlabeled data.

  * •

MRefG (Li et al., 2021) constructs reference graphs to semantically relate
unlabeled data to labelled data.

  * •

MetaSRE (Hu et al., 2021a) constructs pseudo labels on unlabeled data using
meta-learning from the successfulness of the classifier module as an extra
meta-objective.

  * •

GradLRE (Hu et al., 2021b) is the state-of-the-art approach that encourages
pseudo-labeled data to mimic the gradient descent direction on labelled data
and boost its optimization capabilities via trial and error.

  * •

Gold labels train annotated (i.e., sampled 5%, 10% or 30% training data) and
unlabeled data with their gold labels indicating the model upper bound.


# Influence of chirality on the electron transmission through step-like
potential in zigzag, armchair, and (2m,m) carbon nanotubes

Lyuba Malysheva<EMAIL_ADDRESS>Bogolyubov Institute for Theoretical
Physics, 03680 Kiev, Ukraine

###### Abstract

We report the one-electron spectrum and eigenstates of infinite achiral and
chiral $(2m,m)$ carbon nanotubes found by using the analytic solution to the
Schrödinger equation for the tight-binding Hückel-type Hamiltonian. With the
help of matching the wave functions on the interfaces between the regions,
where electrons have different site energies, we find and compare the
transmission coefficients for zigzag, armchair and chiral nanotubes subjected
to the action of an applied step-like potential. The correspondence between
the nanotube band structure and the energy dependence of the transmission
coefficient is demonstrated. It is shown that the $(2m,m)$ nanotubes with a
medium chiral angle reveal intermediate transport properties as compared with
the achiral armchair, and zigzag nanotubes.

##  I Introduction

The unique spectroscopic and electron-transport properties of carbon nanotubes
(CNTs) explain the continuous efforts of experimentalists and theoreticians
aimed at the production and investigation of CNT- and graphene-based
electronic devices. Discovered in the early nineties [1], the carbon nanotubes
are now widely used in biomedical applications [2] and in energy storage
facilities [3], thin-film electronics [4], as well as in the production of
carbon nanotube computers, transistors, and sensors [5, 6, 7].

The comprehensive theoretic investigations of the electron and transport
properties of CNTs [8, 9] started shortly after their experimental discovery.
In recent years, the continued theoretical investigations of CNTs have been
mainly performed within the Dirac relativistic [10, 11, 12] and Schrödinger
nonrelativistic [13, 14, 15, 16, 17, 18, 19, 20] one-particle theories. In
this report, we use the nonrelativistic approach: the analytic solution to the
Schrödinger equation for the tight-binding Hückel-type Hamiltonian. Our
interest is mainly focused on the electronic structure and transport
properties both of the well-studied achiral armchair and zigzag CNTs, as well
as of the much less investigated chiral nanotubes. To be precise, we consider
medium-chiral-angle $(2m,m)$ CNTs with chirality angle equal to 19.1∘. This
choice is explained both by the relative simplicity of this structure and by
the existence of the well-known and experimentally realized procedure of
getting nanotubes with chiral indices $(2m,m)$ [21, 22].

Our goal is to find the transmission coefficients for the achiral and chiral
CNTs in a potential that has a step-like profile. We start with the solution
of the stationary Schrödinger equation ${\mathbf{H}}\Psi=E\Psi$ with the
nearest-neighbor tight-binding Hamiltonian for infinite achiral and chiral
nanotubes presented in Fig. 1. We use the one-electron approximation, and the
model represents an ideal $\pi$ electron system. In the nearest-neighbor
approximation, for the lattice site enumeration explained in Fig. 1, the
electronic properties of CNTs can be well described by the Hückel-type
Hamiltonian

${\mathbf{H}}=\sum_{\bf r^{\prime}-r=q}\sum_{\bf r}[\varepsilon_{\bf
r}(1-\delta_{\bf q,0})-\beta]c^{+}_{\bf r}c_{\bf r+q},$ (1)

where $c^{+}_{\bf r}$ ($c_{\bf r}$) is the electron creation (annihilation)
operator at ${\bf r}=n_{a},n_{z},\alpha$; $\varepsilon_{\bf r}$ and $-\beta$
($\beta>0$) are, respectively, the electron site energy and the C–C hopping
integral. The wave function of $\pi$ electron reads $\Psi=\sum_{\bf
r}\psi_{\bf r}c^{+}_{\bf r}|0\rangle$, where the expansion coefficients obey
the corresponding sets of equations depending on the type of CNTs. The Fermi
energy of $\pi$ electrons is equal to zero and serves as the reference. In
what follows, energy is always expressed in the units of $\beta$. To solve the
problem of particle transmission under a potential with step-like profile, we
consider the electron transmission through CNTs with site energies taking the
values 0 from the left and $U$ from the right of the interface, as shown in
Figs. 1 a–c.

It follows from Fig. 1 that a unit cell of a CNT contains the atoms with
labels $\alpha=l,r,\lambda,\rho$ for the zigzag and armchair CNTs, and the
atoms with $\alpha=l,r,\lambda,\rho,\omega,\epsilon$ for ($2m,m$) CNTs. The
zigzag nanotube (zCT) can be regarded as a periodic sequence of
polyparaphenylene molecules, the armchair nanotube (aCT) consist of the acene
chains, and the ($2m,m$) CNT is formed by the acene chains connected with a
simple carbon chain (the building blocks of all three CNTs are depicted as
blue frames in Fig. 1). As shown earlier [14, 15, 23], the states of $\pi$
electrons in the considered CNTs can be arranged in $\cal N$ (zCT), $N$ (aCT),
and $\mathrm{N}$ (($2m,m$) CNT) conduction bands and equal number of valence
bands, each of which is subdivided into two (zCT and aCT) and three (($2m,m$)
CNT) subbands. Due to the orthogonality of transverse wave functions (see Eqs.
(4) and (13) ), there is no interband transition, and, hence, the problem of
finding the transmission coefficient is reduced to the problem of obtaining
the scattering amplitudes for each $j$th (zCT) or $\nu$th (aCT and ($2m,m$)
CNT), see Fig. 1, conduction band. Labeling the wave function to the left and
right of the interface by $L$ and $R$, respectively, we can write the solution
to the Shrödinger equation for any incident mode $\mu$ as follows (the precise
form of these relations is specified for zCTs, aCTs, and ($2m,m$) CNTs in
Sections II and III):

$\displaystyle\psi^{L}_{n_{a},n_{z},\alpha}$
$\displaystyle=\psi_{n_{a},n_{z},\alpha}(\overset{\rightarrow}{k_{\mu}})+\sum_{\mu_{r}}{R}_{\mu_{r}}\psi_{n_{a},n_{z},\alpha}(\overset{\leftarrow}{k_{\mu_{r}}}),$
$\displaystyle\psi^{R}_{n_{a},n_{z},\alpha}$
$\displaystyle=\sum_{\mu_{t}}{T}_{\mu_{t}}\psi_{n_{a},n_{z},\alpha}(\overset{\rightarrow}{\bar{k}_{\mu_{t}}}).$
(2)

In Eqs. (I), the wave vectors are given in $a^{-1}$ units (lattice constant in
CNTs $a\approx 2.5\AA$), summation is carried out over reflected ($\mu_{r}$)
and transmitted ($\mu_{t}$) modes, and $\psi_{n_{a},n_{z},\alpha}$ (for any
mode) is an eigenstate. The eigenvalues $E(k)$ are found from the dispersion
relation. The wave vector marked by the overbar ($\bar{k}$), as well as the
corresponding eigenenergy $\overline{E}$, refer to the nanotubes with site
energies shifted by $U$: $\overline{E}=E-U$. In what follows, we consider only
positive energies $E\geq 0$, while the sign of $\overline{E}$ depends on $U$.

Figure 1: Fragments of infinite zigzag (a), armchair (b) and ($2m,m$) CNTs
subjected to the step-like potential. The grey and red circles indicate zero
and $U$-shifted site energies of carbon atoms. The labels
$\alpha=\lambda,\rho,l,r$ for (a), (b) and
$\alpha=\lambda,\rho,l,r,\omega,\epsilon$ for (c) are shown for the atoms with
$n_{z}=0$ for (a) and with $n_{a}=0$ for (b) and (c). The atoms in the frame
have one and the same value of the coordinate $n_{a}$ in (a) and $n_{z}$ in
(b) and (c).

##  II Achiral (zigzag and armchair) CNTs

The system of equations for the wave functions $\psi_{n_{a},n_{z},\alpha}$
(the meaning of labels is explained in Fig. 1) for both achiral CNTs has the
same form

$\begin{split}E\psi_{n_{a},n_{z},l}&=-\psi_{n_{a},n_{z},\lambda}-\psi_{n_{a}+1,n_{z},\lambda}-\psi_{n_{a},n_{z}-1,r},\\\
E\psi_{n_{a},n_{z},\lambda}&=-\psi_{n_{a},n_{z},l}-\psi_{n_{a}-1,n_{z},l}-\psi_{n_{a},n_{z},\rho},\\\
E\psi_{n_{a},n_{z},\rho}&=-\psi_{n_{a},n_{z},r}-\psi_{n_{a}-1,n_{z},r}-\psi_{n_{a},n_{z},\lambda},\\\
E\psi_{n_{a},n_{z},r}&=-\psi_{n_{a},n_{z},\rho}-\psi_{n_{a}+1,n_{z},\rho}-\psi_{n_{a},n_{z}+1,l},\end{split}$
(3)

while the boundary conditions are different for zigzag and armchair nanotubes.
We start with the simplest case, a zigzag CNT.

###  II.1 Zigzag CNT

A fragment of a zigzag CNT is drawn in Fig. 1a. This nanotube has chiral angle
$\theta=0^{\circ}$. Using the presented notation, the periodic boundary
conditions can be written as $\psi_{0,n_{z},l(r)}=\psi_{{\cal N},n_{z},l(r)}$,
$\psi_{{\cal N}+1,n_{z},\lambda(\rho)}=\psi_{1,n_{z},\lambda(\rho)}$, where
$n_{a}=1,\dots,{\cal N},$ and $n_{z}=-\infty,\dots,\infty$, which implies the
following form of solutions to the system (3) for $\alpha=\lambda,\rho,l,r$:

$\begin{split}\psi_{n_{a},n_{z},\alpha}={\alpha}e^{i\xi_{j}n_{a}}e^{i\kappa
n_{z}},\quad\xi_{j}\equiv\dfrac{2\pi j}{\cal N},\;j=1,\dots,{\cal
N},\end{split}$ (4)

and the wave vector $\kappa$, $0\leq k\leq\pi$, is in the units of $a^{-1}$.
The electron structure and quantum transport in the zCTs (and their
counterparts, armchair nanoribbons) were considered in a number of works [15,
14, 24, 25, 26, 27, 28, 29, 30], and the eigenvalues and eigenstates
$\psi_{n_{a},n_{z},\alpha}$ were found in different representations. The
eigenfunctions $\psi^{(j)}_{m,n,\alpha}(k)$ can be found by using different
approaches [15, 16, 24]. We use here the following form for the coefficients
$\alpha$ in (4):

$\begin{split}l&=\mp\sigma_{j}\dfrac{z^{\pm}_{j}}{E^{\pm}_{j}}e^{-i\kappa_{j}^{\pm}/2},\qquad\quad
r=1,\\\ \lambda&=\pm\sigma_{j}e^{-i{\pi j}/{{\cal
N}}}e^{-i\kappa_{j}^{\pm}/2},\quad\rho=-e^{-i{\pi j}/{{\cal
N}}}\dfrac{z^{\pm}_{j}}{E^{\pm}_{j}},\end{split}$ (5)

and the corresponding dispersion relation (for zero site energy):

$\begin{split}(E^{\pm}_{j})^{2}&=4\cos^{2}\dfrac{\pi j}{{\cal N}}+1\pm
4\sigma_{j}\cos\dfrac{\pi j}{{\cal
N}}\cos\dfrac{\kappa}{2}=\big{|}z_{j}^{\pm}\big{|}^{2},\end{split}$ (6)

where

$z_{j}^{\pm}\equiv 2\cos\dfrac{\pi j}{{\cal
N}}\pm\sigma_{j}e^{-i\frac{\kappa}{2}},\quad\sigma_{j}\equiv{\rm sing}({\cal
N}/2-j).$ (7)

The sign $\pm$ in Eq. (6) marks the ”plus” and ”minus” branches of the
dependence $E_{j}(\kappa)$. Note that the mode with the lowest absolute energy
belongs to the ”minus” branch. It is easy to see that for any $j$, there are
two positive $\kappa$ (that we denote by $\kappa_{j}^{\pm}$) satisfying
Eq.(6):

$\begin{split}\kappa_{j}^{\pm}&=2\arccos\Big{[}s\dfrac{(E_{j}^{\pm})^{2}-4\cos^{2}\frac{\xi_{j}}{2}-1}{4\sigma_{j}\cos\frac{\xi_{j}}{2}}\Big{]},\\\
s&\equiv\left\\{\begin{array}[]{cl }1,&\text{for}\quad E^{+}_{j}\\\\[7.0pt]
-1,&\text{for}\quad E^{-}_{j}.\end{array}\right.\end{split}$ (8)

Evidently, if some $\kappa$ satisfies Eq.(6), the same is true for $-\kappa$.
Differentiating $E^{\pm}_{j}$ by $\kappa$, we find the group velocities of the
corresponding eigenstates:

$\hbar
v_{\pm}=\frac{dE^{\pm}_{\nu}}{d\kappa}=\mp\dfrac{1}{E}\left|\cos\dfrac{\xi_{j}}{2}\right|\sin\frac{\kappa_{j}^{\pm}}{2}.$
(9)

Positive velocity corresponds to the wave going from left to right. Then, for
the ”plus” energy branch, $-\kappa_{j}^{\pm}$ is chosen for the incident mode,
while $\kappa_{j}^{\pm}$ corresponds to the reflected wave, and for the
”minus” branch the situation is opposite. Thus, for any incident mode $j$,
relations (I) simplify to

$\displaystyle\psi^{L}_{n_{a},n_{z},\alpha}$
$\displaystyle=\psi_{n_{a},n_{z},\alpha}(-s\kappa_{j}^{\pm})+{R}_{j}^{\pm}\psi_{n_{a},n_{z},\alpha}(s\kappa_{j}^{\pm}),$
$\displaystyle\psi^{R}_{n_{a},n_{z},\alpha}$
$\displaystyle={T}_{j}^{\pm}\psi_{n_{a},n_{z},\alpha}(-\bar{s}\bar{\kappa}_{j}^{\pm})$
(10)

(to recall, the wave vector marked by the overbar ($\bar{\kappa}_{j}^{\pm}$)
refer to the zCTs with site energies shifted by $U$:
$\overline{E}_{j}^{\pm}=E_{j}^{\pm}-U$.) Substituting these relations in the
first (for $n_{z}=1$) and last (for $n_{z}=0$) equations of system (3), we
find after some algebra (for details, see [24, 25])

$\begin{split}\mathcal{T}_{j}^{\pm}&=|{T}_{j}^{\pm}|^{2}\dfrac{E_{j}^{\pm}\sin\frac{\bar{\kappa}_{j}^{\pm}}{2}}{\overline{E}_{j}^{\pm}\sin\frac{\kappa_{j}^{\pm}}{2}}=\dfrac{16\cos^{2}\frac{\pi
j}{{\cal
N}}\sin\frac{\kappa_{j}^{\pm}}{2}\sin\frac{\bar{\kappa}_{j}^{\pm}}{2}}{\Big{|}-U^{2}+16\cos^{2}\frac{\pi
j}{{\cal N}}Q^{\pm}\Big{|}},\\\\[7.0pt]
Q^{\pm}&=\left\\{\begin{array}[]{ll}\sin^{2}\frac{\kappa_{j}^{\pm}+\bar{\kappa}_{j}^{\pm}}{4},&s=\bar{s},\\\\[7.0pt]
\cos^{2}\frac{\kappa_{j}^{\pm}+\bar{\kappa}_{j}^{\pm}}{4},&s=-\bar{s},\end{array}\right.\end{split}$
(11)

where $\bar{s}=1$ for $\overline{E}_{j}^{+}$ and $\bar{s}=-1$ for
$\overline{E}_{j}^{-}$. The main properties of the transmission coefficient,
which is equal to the sum of all transmission amplitudes of the propagating
transverse modes

$\mathcal{T}(E)=\sum_{j=1}^{\cal
N}\big{(}\mathcal{T}_{j}^{+}+\mathcal{T}_{j}^{-}\big{)}$ (12)

are discussed in Sec.III.

###  II.2 Armchair CNT

A fragment of infinite armchair CNT (aCT), a structure having chiral angle
$\theta=30^{\circ}$, is drawn in Fig. 1b. For this case, the periodic boundary
conditions in the transverse direction can be written as
$\psi_{n_{a},0,r}=\psi_{n_{a},N,r}$, $\psi_{n_{a},N+1,l}=\psi_{n_{a},1,l}$,
where $n_{a}=-\infty,\dots,\infty,$ and $n_{z}=1,\dots,N$, which implies the
following form of solutions to the system (3) for $\alpha=\lambda,\rho,l,r$:

$\begin{split}\psi_{n_{a},n_{z},\alpha}={\alpha}e^{ikn_{a}}e^{i\xi_{\nu}n_{z}},\quad\xi_{\nu}\equiv\dfrac{2\pi\nu}{N},\;\nu=1,\dots,N.\end{split}$
(13)

The energy eigenvalues are found from the condition of solvability of system
(3) in the form

$\begin{split}(E^{\pm}_{\nu})^{2}&=4\cos^{2}\dfrac{k_{\nu}^{\pm}}{2}+1\pm
4\sigma_{\nu}\cos\dfrac{k_{\nu}^{\pm}}{2}\cos\dfrac{\xi_{\nu}}{2}=\big{|}z_{\nu}^{\pm}\big{|}^{2},\end{split}$
(14)

where

$z_{\nu}^{\pm}\equiv
2\cos\dfrac{k_{\nu}^{\pm}}{2}\pm\sigma_{\nu}e^{-i\frac{\xi_{\nu}}{2}},\quad\sigma_{\nu}\equiv{\rm
sing}(N/2-\nu).$ (15)

The corresponding eigenstates evidently coincide with Eqs. (5) with the
interchange of transverse and longitudinal quantum numbers:
$\xi_{j}\leftrightarrow k_{\nu}^{\pm}$,
$\kappa_{j}^{\pm}\leftrightarrow\xi_{\nu}$. The same symmetry exists for the
eigenvalues, which is clearly seen from the comparison of Eqs. (6) and (14).
However, there is an essential difference between zCT and aCT: for the later,
for any energy $E$ and $\nu$, the values of $k_{\nu}$ should be found from the
solution to the quadratic equation (14):

$2\cos\dfrac{k_{\nu}^{1,2}}{2}=\left|\sqrt{E^{2}-\sin^{2}\dfrac{\xi_{\nu}}{2}}\pm\sigma_{\nu}\cos\dfrac{\xi_{\nu}}{2}\right|.$
(16)

Relation (16) gives the folllowing interval where $k_{1,2}$ are real:

$\sin\dfrac{\xi_{\nu}}{2}\leq E\leq\sqrt{5+4\sigma\cos\dfrac{\xi_{\nu}}{2}}.$
(17)

For this energy range, we have or four ($\pm k_{1,2}$), or two ($\pm k_{1}$)
real values of $k$. From these $k$, we choose only those for which the
velocity is positive (we suppose that the particles moves from left to right).
For two energy branches we get two electron velocities:

$\hbar
v_{\pm}=\frac{dE^{\pm}_{\nu}}{dk}=-\dfrac{1}{E}\,\sin\dfrac{k^{\pm}_{\nu}}{2}\left(2\cos\dfrac{k^{\pm}_{\nu}}{2}\pm\sigma_{\nu}\cos\dfrac{\xi_{\nu}}{2}\right).$
(18)

For the ”$+$” energy branch, the velocity is positive for negative $k$, and
vice versa. For the ”$-$” energy branch, the velocity is negative for $k$
starting from $-\pi$ up to $k$ satisfying

$2\cos\dfrac{k}{2}=\sigma_{\nu}\cos\dfrac{\xi_{\nu}}{2},$ (19)

for which the velocity becomes positive, up to $k=0$. Then for $k>0$, up to
$k$ satisfying equality (19), the velocity is negative, and for larger $k$ up
to $k=\pi$ it is positive.

Thus, to determine the transmission amplitudes, we first find four solutions
to Eq.(16), choose only real solutions (two or four), and choose those $k$ for
which the velocity (18) is positive. They are denoted by
$\overset{\rightarrow}{k_{1}},\overset{\rightarrow}{k_{2}}$ and/or
$\overset{\rightarrow}{\bar{k}_{1}},\overset{\rightarrow}{\bar{k}_{2}}$. If
some $k_{\nu}$ and/or $\bar{k}_{\nu}$ are complex, then for the wave going
from left to right, we choose those $k$ whose imagine parts are positive
(then, these evanescent waves vanish with increasing $n$ and do not
participate in the process of transmission). For reflection, we choose the
waves going in the opposite direction:
$\overset{\leftarrow}{k_{1}}=-\overset{\rightarrow}{k_{1}},\overset{\leftarrow}{k_{2}}=-\overset{\rightarrow}{k_{2}}$,
for which the velocities (18) are negative (real $k$) or whose imagine parts
are negative (complex $k$).

Since for any energy and $\nu$ there are at most two incident modes (numbered
by $\nu_{0}\leq 2$), solution to Eqs. (3) for each $\nu_{0}$ can be written in
a single-mode form

$\begin{split}\psi^{L}_{n_{a},n_{z},\alpha}&=\psi_{n_{a},n_{z},\alpha}(\overset{\rightarrow}{k_{\nu}})+\sum_{\mu=1}^{2}{R}^{\mu}_{\nu}\psi_{n_{a},n_{z},\alpha}(\overset{\leftarrow}{k_{\mu}}),\\\
\psi^{R}_{n_{a},n_{z},\alpha}&=\sum_{\mu=1}^{2}{T}^{\mu}_{\nu}\psi_{n_{a},n_{z},\alpha}(\overset{\rightarrow}{\bar{k}_{\mu}}).\end{split}$
(20)

Substituting relations (20) in the following system obtained from (3):

$\begin{split}E\psi^{L}_{0,n_{z},l}&=-\psi^{L}_{0,n_{z},\lambda}-\psi^{R}_{1,n_{z},\lambda}-\psi^{L}_{0,n_{z}-1,r},\\\
\overline{E}\psi^{R}_{1,n_{z},\lambda}&=-\psi^{R}_{1,n_{z},l}-\psi^{L}_{0,n_{z},l}-\psi^{R}_{1,n_{z},\rho},\\\
\overline{E}\psi^{R}_{1,n_{z},\rho}&=-\psi^{R}_{1,n_{z},r}-\psi^{L}_{0,n_{z},r}-\psi^{R}_{1,n_{z},\lambda},\\\
E\psi^{L}_{0,n_{z},r}&=-\psi^{L}_{0,n_{z},\rho}-\psi^{R}_{1,n_{z},\rho}-\psi^{L}_{0,n_{z}+1,l},\end{split}$
(21)

we get four linear equations for four unknowns ${R}_{\nu}^{\mu}$,
${T}_{\nu}^{\mu}$, $\mu=1,2$. They allow us to determine the transmission
coefficient

$\mathcal{T}(E)=\sum_{\nu=1}^{N}\big{(}\mathcal{T}_{\nu}^{1}+\mathcal{T}_{\nu}^{2}\big{)},\quad\mathcal{T}_{\nu}^{\mu}=\dfrac{v_{\nu}}{v_{\nu}^{\mu}}\left|{T}^{\mu}_{\nu}\right|^{2}.$
(22)

To elucidate the role of chirality in the electron transport through step-like
potential, in the next section we perform the derivation of $\mathcal{T}(E)$
for the chiral ($2m,m$) CNTs.

##  III Chiral ($2m,m$) CNT

The system of equations for the wave functions for infinite ($2m,m$) CNTs,
whose fragment is depicted in Fig. 1c, can be written as follows [23]:

$\begin{split}E\psi_{n_{a},n_{z},l}&=-\psi_{n_{a},n_{z},\lambda}-\psi_{n_{a}+1,n_{z},\lambda}-\psi_{n_{a},n_{z}-1,\epsilon},\\\
E\psi_{n_{a},n_{z},\lambda}&=-\psi_{n_{a},n_{z},l}-\psi_{n_{a}-1,n_{z},l}-\psi_{n_{a},n_{z},\rho},\\\
E\psi_{n_{a},n_{z},\rho}&=-\psi_{n_{a},n_{z},r}-\psi_{n_{a}-1,n_{z},r}-\psi_{n_{a},n_{z},\lambda},\\\
E\psi_{n_{a},n_{z},r}&=-\psi_{n_{a},n_{z},\rho}-\psi_{n_{a}+1,n_{z},\rho}-\psi_{n_{a},n_{z},\omega},\\\
E\psi_{n_{a},n_{z},\omega}&=-\psi_{n_{a},n_{z},\epsilon}-\psi_{n_{a}-1,n_{z},\epsilon}-\psi_{n_{a},n_{z},r},\\\
E\psi_{n_{a},n_{z},\epsilon}&=-\psi_{n_{a},n_{z},\omega}-\psi_{n_{a}+1,n_{z},\omega}-\psi_{n_{a},n_{z}+1,l},\end{split}$
(23)

where $n_{a}=-\infty,\dots,\infty,$ with the periodic boundary conditions

$\psi_{n_{a},0,\omega}=\psi_{n_{a},{\mathrm{N}},\omega},\quad\psi_{n_{a},{\mathrm{N}}+1,l}=\psi_{n_{a},1,l},\,n_{z}=1,\dots,{\mathrm{N}}.$
(24)

To satisfy these boundary conditions, the solution is sought in the form (13)
with $\alpha=\lambda,\rho,l,r,\omega,\epsilon$, $\nu=1,\dots,{\mathrm{N}}$,
$\pi\leq k\leq\pi$. Substituting this solution in (23), we obtain the
dispersion relation for ($2m,m$) CNTs [23] in the form of three energy
branches:

$\begin{split}(E_{\nu}^{\mu})^{2}&=1+4\cos^{2}\dfrac{k}{2}+4\cos\dfrac{k}{2}\cos\left(-\dfrac{k}{6}+\dfrac{\xi_{\nu}}{3}+y_{\mu}\right)\\\
&=\big{|}z_{\nu}^{\mu}\big{|}^{2},\quad\mu=1,2,3,\quad\xi_{\nu}=\dfrac{2\pi\nu}{{\mathrm{N}}},\\\
y_{1}&=0,\quad y_{2}=-\dfrac{2\pi}{3},\quad y_{3}=\dfrac{2\pi}{3},\\\
z_{\nu}^{\mu}&\equiv
2\cos\frac{k}{2}+e^{-i\eta_{\nu}^{\mu}},\quad\eta_{\nu}^{\mu}\equiv-\dfrac{k}{6}+\dfrac{\xi_{\nu}}{3}+y_{\mu}.\end{split}$
(25)

It is easy to see that the wave numbers $k$ satisfying Eq.(25) can be found
from the following equation of the 6th order:

$\begin{split}e^{6ik}&+c_{1}e^{5ik}+c_{2}e^{4ik}+c_{3}e^{3ik}+\overline{c_{2}}e^{2ik}+\overline{c_{1}}e^{ik}+1=0,\\\
c_{1}&=6-3E^{2}+e^{-i\xi_{\nu}},\\\
c_{2}&=3(5-5E^{2}+E^{4})+3e^{-i\xi_{\nu}}+e^{i\xi_{\nu}},\\\
c_{3}&=21-27E^{2}+9E^{4}-E^{6}+6\cos\xi_{\nu},\end{split}$ (26)

where the overbar denotes complex conjugate. The corresponding eigenstates can
be represented as follows:

$\begin{split}l&=-e^{-i\xi_{\nu}}e^{i\eta_{\nu}^{\mu}}\dfrac{z_{\nu}^{\mu}}{E_{\nu}^{\mu}},\quad\lambda=e^{-i\xi_{\nu}}e^{i\eta_{\nu}^{\mu}}e^{-ik/2},\\\
\rho&=-e^{-i\xi_{\nu}}e^{2i\eta_{\nu}^{\mu}}e^{-ik/2}\dfrac{z_{\nu}^{\mu}}{E_{\nu}^{\mu}},\\\\[5.0pt]
r&=e^{-i\eta_{\nu}^{\mu}}e^{-ik/2},\quad\omega=-e^{-ik/2}\dfrac{z_{\nu}^{\mu}}{E_{\nu}^{\mu}},\quad\epsilon=1.\end{split}$
(27)

The electron velocity for the $\mu$th energy branch is obtained by
differentiating Eq.(25):

$\hbar
v_{\mu}=\frac{dE^{\mu}_{\nu}}{dk}=\dfrac{1}{E^{\mu}_{\nu}}\Big{(}\dfrac{1}{3}\cos\dfrac{k}{2}\sin\eta_{\nu}^{\mu}-\sin
k-\sin\dfrac{k}{2}\cos\eta_{\nu}^{\mu}\Big{)}.$ (28)

Thus, the procedure of finding the transmission coefficient is similar to that
described in Sec. II: we find six roots of Eq.(26), choose only real $k$, find
to which branch they belong, and, to choose the waves going from left to
right, choose those $k$ for which the velocity $v_{\mu}$ (28) is positive. The
maximal number of such $k$ is 3. They are denoted by
$\overset{\rightarrow}{k_{1}},\overset{\rightarrow}{k_{2}},\overset{\rightarrow}{k_{3}}$
and/or
$\overset{\rightarrow}{\bar{k}_{1}},\overset{\rightarrow}{\bar{k}_{2}},\overset{\rightarrow}{\bar{k}_{3}}$.
For complex $k$, we choose those whose imagine parts are positive (then we get
the evanescent waves vanishing with increasing $n$ and not participating in
transmission). For reflection, we choose the waves going from right to left:
$\overset{\leftarrow}{k_{1}},\overset{\leftarrow}{k_{2}},\overset{\leftarrow}{k_{3}}$,
for which $v_{\mu}$ are negative (real $k$) or whose imagine parts are
negative (complex $k$).

As a result, for any $\nu$, there are at most three incident modes, that are
numbered by $\mu_{0}\leq 3$. For each $\mu_{0},$ solution to Eqs. (1) in a
single-mode form is:

$\displaystyle\psi^{L}_{n_{a},n_{z},\alpha}$
$\displaystyle=\psi_{n_{a},n_{z},\alpha}(\overset{\rightarrow}{k_{\mu_{0}}})+\sum_{\mu=1}^{3}{R}^{\mu}_{\nu}\psi_{n_{a},n_{z},\alpha}(\overset{\leftarrow}{k_{\mu}})$
(29) $\displaystyle\psi^{R}_{n_{a},n_{z},\alpha}$
$\displaystyle=\sum_{\mu=1}^{3}{T}^{\mu}_{\nu}\psi_{n_{a},n_{z},\alpha}(\overset{\rightarrow}{\bar{k}_{\mu}}).$
(30)

Substituting these solutions in the system

$\begin{split}E\psi^{L}_{0,n_{z},l}&=-\psi^{L}_{0,n_{z},\lambda}-\psi^{R}_{1,n_{z},\lambda}-\psi^{L}_{0,n_{z}-1,\epsilon},\\\
\overline{E}\psi^{R}_{1,n_{z},\lambda}&=-\psi^{R}_{1,n_{z},l}-\psi^{L}_{0,n_{z},l}-\psi^{R}_{1,n_{z},\rho},\\\
\overline{E}\psi^{R}_{1,n_{z},\rho}&=-\psi^{R}_{1,n_{z},r}-\psi^{L}_{0,n_{z},r}-\psi^{R}_{1,n_{z},\lambda},\\\
E\psi^{L}_{0,n_{z},r}&=-\psi^{L}_{0,n_{z},\rho}-\psi^{R}_{1,n_{z},\rho}-\psi^{L}_{0,n_{z},\omega},\\\
\overline{E}\psi^{R}_{1,n_{z},\omega}&=-\psi^{R}_{1,n_{z},\epsilon}-\psi^{L}_{0,n_{z},\epsilon}-\psi^{R}_{1,n_{z},r},\\\
E\psi^{L}_{0,n_{z},\epsilon}&=-\psi^{L}_{0,n_{z},\omega}-\psi^{R}_{1,n_{z},\omega}-\psi^{L}_{0,n_{z}+1,l},\end{split}$
(31)

we get six linear equations for six unknowns ${R}_{\nu_{0},\nu}^{\mu}$,
${T}_{\nu_{0},\nu}^{\mu}$, $\mu=1,2,3$. They allow us to determine the
transmission coefficient

$\mathcal{T}(E)=\sum_{\nu}\big{(}\mathcal{T}_{\nu}^{1}+\mathcal{T}_{\nu}^{2}+\mathcal{T}_{\nu}^{3}\big{)},\quad\mathcal{T}_{\nu}^{\mu}=\dfrac{v_{\nu}}{v_{\nu}^{\mu}}\left|{T}^{\mu}_{\nu}\right|^{2}.$
(32)

Figure 2: Electron structure and transmission coefficient for, from left to
right, zCT (5,0), aCT (3,3) and chiral (4,2) CNT. Double degenerate modes are
drawn by thicker lines. Black curves in the right panel: $E_{\nu}^{1}$; red
curves: $E_{j}^{+}$ (a), $E_{\nu}^{+}$ (b), and $E_{\nu}^{2}$ (c); blue
curves: $E_{j}^{-}$ (a), $E_{\nu}^{-}$ (b), and $E_{\nu}^{3}$ (c).

Now we have at hand all the necessary expressions for the analysis of the
properties of transmission coefficients for all three types of CNTs.

##  IV Transmission coefficients

###  IV.1 Zero potential

It is instructive to start with the discussion of the dependence
$\mathcal{T}(E)$ calculated for $U=0$, when this function is simply equal to
the number of ”open” modes. In Fig. 2, we plot the dependences $E(k)$ and
$E(\kappa)$ and, on the same panel, the corresponding dependences
$\mathcal{T}(E)$.

zCTs. For these CNTs, as is well seen in the left panel, that there are no
incident modes (and $\mathcal{T}=0$) if $E<1-2\cos(2\pi/5)=0.38$, in
accordance with Eqs. (6), (7) for $\kappa=0$. For larger energies, two modes
open, since the mode $j=2$ is doubly degenerate. When $E>|1-2\cos(\pi/5)|$,
there are four open modes, and, thus, $\mathcal{T}=4$. The maximal value of
$\mathcal{T}$ is equal to 5, and it starts to decrease for
$E>1+2\cos(2\pi/5)=1.62$.

aCTs. Similar trends are observed for the aCT (middle panel), with an
essential difference: since the band gap for any aCT is zero, two modes open
immediately when $E$ becomes non-zero, and $\mathcal{T}=2$ up to
$E=\sin(\pi/3)=0.87$, the minimum of the double degenerate ”minus” energy
branch $E_{1}^{-}$ (see Eq.(17)). When $E$ becomes larger than $\sin(\pi/3)$,
four modes provided by this energy branch become open and $\mathcal{T}=6$. For
$E>1$, the number of open modes equals 5 (two from $E_{1}^{-}=E_{2}^{-}$, two
from $E_{1}^{+}=E_{2}^{+}$, and one from $E_{3}^{+}$), up to the maximum of
$E_{1}^{-}$. With further increase of energy the function $\mathcal{T}(E)$
decreases in a step-like manner, in accordance with the number of open modes.

($2m,m$) CTNs. All six modes of (4,2) CNT (right panel) are non-degenerate. As
it follows from Eq.(25), the maximal energy value is attained at $k=0$ for the
$\mu=2$ subbranch with $\nu=\mathrm{N}$: $E^{\mu=2}_{\nu=\mathrm{N}}(k=0)=3$.
The energy gap can be estimated as $2E^{\mu=3}_{\nu=1}(k=2\pi/3)\approx 0.7$.
Thus, as is seen in Fig. 2c, $\mathcal{T}(E)=0$ for $E\lesssim 0.35$. With
further increase of $E$, $\mathcal{T}(E)=2$ up to the energy equal to the
minimum of $E^{3}_{2}$ (or, which is the same, the minimum of $E^{1}_{2}$).
The transmission coefficient is maximal, $\mathcal{T}(E)=5$, for the energies
between $E=1$ and the maximum of $E^{3}_{2}$ (or $E^{1}_{2}$), when all modes
are open except $E^{1}_{3}$. For larger energies, the step-like decrease of
$\mathcal{T}$ for all three structure is rather similar.

Thus, as the common features of transmission in all three structure, we can
mention the step-like increase with increase of energy up to $E\approx 1$, or,
since we use the hopping integral energy units, $E\approx 2.7$eV. The maximum
of $\mathcal{T}$ equals (or is less by 1 then) the whole number of conducting
modes. The differences in the behavior of $\mathcal{T}$ for zCTs, aCTs, and
chiral CNTs can be expected in the energy region $E\lesssim 1$, while for
$E\gtrsim 1$, the three conductance ladders become more and more similar with
the energy increase. It can be expected that these regularities remain much
the same for small potentials, $U<<1$. In the next section, we will see what
happens with $\mathcal{T}$ for moderate values of $U$, taking $U=0.25$ and
$U=0.5$ as examples.

Figure 3: Transmission coefficient $\mathcal{T}(E)$ for, from left to right,
zCT (9,0), aCT (6,6) and chiral (6,3) CNT. Dashed curves: $U=0$, solid curves:
$U=0.25$. Insets: the corresponding dispersion curves (the colors have the
same meaning as in Fig/ 2) for site energies unshifted (left) and shifted by
$U=0.25$ (right).

###  IV.2 Non-zero potential

Due to the symmetry of the problem, we restrict ourselves to the case $U>0$
without loss of generality. In Fig. 3 we demonstrate the changes in
dependences $\mathcal{T}(E)$ under the step-like potential $U=0.25$ (i.e.,
$\sim 0.7$ eV). Note that the plots of functions $\mathcal{T}(E)$ for zero
potential coincide with the corresponding dependences calculated in [31] with
the help of the Green function matching technique, providing a useful cross-
check on our calculations. Figures 3 and 4 demonstrate that the transmission
coefficient $\mathcal{T}(E)$ is always smaller for larger values of $U$.

Figure 4: Transmission coefficient $\mathcal{T}(E)$ for, from bottom to top,
zCT (15,0), chiral (12,6) CNT, and aCT (9,9). All three CNTs have close
diameters $d\sim 12\AA$. Dashed curves: $U=0.25$, solid curves: $U=0.5$.

As is seen in Fig. 3, for all three types of nanotubes considered here,
dependences $\mathcal{T}(E)$ are the same for $E\lesssim 0.5$, namely,
$\mathcal{T}(E)=2$ with a good accuracy for both values $U=0$ and $U=0.25$.
This is due to the fact that, as shown in the insets, for moderate values of
the potential $U$, exactly two incident modes belonging to the zero-gap energy
band are transmitted to the same band via two conducting channels. The same
behavior of the transmission coefficient for small energies can be expected
for all other metallic CNTs.

For large energies, $E\gtrsim 1.5$, we also observe a close similarity of
$\mathcal{T}(E)$ for both values of the potential and all presented CNTs. This
can be explained by the simple structure of the electron spectrum for large
energy values: it is seen that the upper subbands of the energy bands have
similar bell-like shapes. At the same time, for energy lying in the interval
$0.5\lesssim E\lesssim 1.5$, the energy dependences of $\mathcal{T}$ are
essentially different for different types of nanotubes. Moreover, as
demonstrated in Fig. 4 for the CNTs of three considered types and close
diameters $d\sim 12\AA$ (to recall, $d=a\sqrt{n^{2}+m^{2}+nm}/\pi$ for $(n,m)$
CNTs [9]), the changes in behavior of $\mathcal{T}(E)$ can be organized in the
following sequence: zCT $\rightarrow$ ($2m,m$) CNT $\rightarrow$ aCT. This is
not at all surprising, taking into account that the corresponding sequence of
the chiral angles is $0^{\circ}\rightarrow 19^{\circ}\rightarrow 30^{\circ}$.
Thus, the differences in the behavior of electron conductance in the three
types of CNT should be observed in measurements of source-to-drain current
under varied gate voltage (playing the role of step potential) only in the
energy range $1.3\div 4$ eV. Our calculations show that the increase of the
diameter of CNT leads to the smoothness of the dependences of $\mathcal{T}(E)$
and, therefore, reduces the differences between the process of electron
transmission in different nanotubes. Thus, observable variations in
$\mathcal{T}(E)$ and, therefore, electron conductance with nanotube types can
be expected only for rather narrow nanotubes with diameters of up to $30\AA$.

##  V Conclusions

In conclusion, we note that we considered the transmission coefficients for
achiral an chiral CNTs of any diameter with the help of matching the wave
functions on the interface between the regions with zero and nonzero site
energies. We used the exact solutions of the model Hamiltonian for the
description of $\pi$ electron states in ideal carbon achiral and chiral
nanotubes. It is shown that the difference between the dependences of the
transmission coefficients for different types of CNTs is well pronounced
within the energy range $1.3\div 4$ eV for narrow nanotubes with diameters of
up to $30\AA$ and, most likely, can be observed in the course of the
experimental investigations.

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