# Semi-Supervised Graph Representation Learning with Human-centric Explanation
for Predicting Fatty Liver Disease

So Yeon Kim1,2, Sehee Wang1, Eun Kyung Choe3

###### Abstract

Addressing the challenge of limited labeled data in clinical settings,
particularly in the prediction of fatty liver disease, this study explores the
potential of graph representation learning within a semi-supervised learning
framework. Leveraging graph neural networks (GNNs), our approach constructs a
subject similarity graph to identify risk patterns from health checkup data.
The effectiveness of various GNN approaches in this context is demonstrated,
even with minimal labeled samples. Central to our methodology is the inclusion
of human-centric explanations through explainable GNNs, providing personalized
feature importance scores for enhanced interpretability and clinical
relevance, thereby underscoring the potential of our approach in advancing
healthcare practices with a keen focus on graph representation learning and
human-centric explanation.

## Introduction

The integration of representation learning in machine learning and deep
learning within clinical settings, while promising, encounters significant
challenges. A primary obstacle is the scarcity of labeled training data. This
limitation is particularly crucial in representation learning, where the
efficacy of simpler models may surpass complex ones given sufficient training
samples. In clinical contexts, this issue is exacerbated by the challenges in
collecting sufficient number of patient samples.

To address these challenges, advancements in graph representation learning and
semi-supervised learning have become increasingly significant in healthcare
(Yi et al. 2022; Zhang et al. 2021). These techniques employ structural
information in graph-based models for supervision, making them highly suitable
for semi-supervised learning scenarios, a common occurrence in clinical
settings where data scarcity and the challenge of defining ground-truths are
prevalent. Utilizing graph representation learning allows for navigating these
limitations, thereby enhancing the accuracy and reliability of clinical
predictions.

A notable application of these advanced methodologies is in predicting fatty
liver disease using health checkup data. This disease, marked by excessive fat
in the liver, is a significant global health issue, linked to obesity,
diabetes, and metabolic disorders. Accurate prediction is vital as it can
progress to severe complications like cirrhosis and liver cancer, making it a
major cause of liver-related deaths. Predicting fatty liver disease is
complex, requiring analysis of diverse factors like demographics, environment,
and health indicators. Graph-based models, using similarity graphs from this
data, can identify risk patterns, and categorize individuals into risk groups,
providing human-centric explanations. This approach mirrors the decision-
making processes of clinicians.

Our research focuses on the use of graph neural networks (GNNs) in the context
of semi-supervised learning for the prediction of fatty liver disease,
particularly in scenarios with limited labeled data. We explore the
construction of a subject similarity graph and conduct a comparative study of
various state-of-the-art GNN approaches. Our findings demonstrate the
effectiveness of GNNs in accurately predicting fatty liver disease, even with
minimal labeled data.

Furthermore, we emphasize the importance of human-centric explanations in our
approach. By incorporating explainable AI techniques into our GNN model, we
ensure that our framework is not only efficient in prediction but also
provides interpretable and human-centric explanations. This is achieved by
offering personalized feature importance scores, allowing the model to be
clinically relevant and understandable. We observe that significant biomarkers
vary among different groups, correlating with the differentiation between
fatty liver disease patients and healthy individuals. This insight underscores
the potential of our approach in contributing meaningfully to healthcare
practices, especially in the realm of semi-supervised representation learning
with a focus on human-centric explanations and graph representation learning
techniques.

## Methods

### Dataset

Our study utilized data from the Gene-Environment Interaction and Phenotype
(GENIE) cohort at Seoul National University Hospital Healthcare System Gangnam
Center, which focuses on comprehensive health checkups within the Korean
population (Choe et al. 2022). Initially, the GENIE cohort comprised 10,349
subjects. To enhance the accuracy and relevance of our analysis, we first
excluded 2,244 individuals diagnosed with any cancer or metabolic syndrome, to
minimize confounding factors. Further refining our dataset, we removed ten
features closely associated with fatty liver disease. This included six
features integral to the fatty liver index calculation (BMI, triglyceride
levels, waist circumference, height, and weight) and four from liver function
tests (AST, ALT, ALP, and ALBUMIN). Despite the relevance of the fatty liver
index, it was not directly utilized in our research, but its associated
features were considered impactful and thus excluded.

Additionally, we tackled the issue of incomplete data. We eliminated 22
features with over 50% missing data and applied imputation using mean values
from the 10 nearest neighbors for other instances of missing data. As a
result, our final dataset included 8,104 samples, encompassing 119 features.
It comprised 5,545 normal samples (individuals with no diagnosed liver
conditions) and 2,559 samples from individuals with confirmed fatty liver
disease through abdominal sonography. We note that our application is designed
for binary classification, aiming to differentiate between individuals with
fatty liver disease and those who are normal. The descriptions of all 119
features are provided in Supplementary Table S1.

### Graph Construction

In our study, we constructed a graph, denoted as
$\mathcal{G}=(\mathcal{V},\mathcal{E})$, to represent the relationships and
similarities among individuals in the GENIE cohort. The vertex set
$\mathcal{V}=(v_{1},…,v_{n})$ corresponds to individuals who underwent health
checkups, with each individual, or subject $v_{i}$, characterized by a feature
vector $x_{i}$ that includes 119 normalized features from health examinations.
Normalization was performed to ensure unit variance, thus maintaining
consistency in our analysis.

The edge set $\mathcal{E}=\\{e_{ij}\\}$ captures the relationships between
pairs of subjects. The weights of these edges reflect the similarity between
individuals. We measured the similarity between any two subjects $v_{i}$ and
$v_{j}$ as $\mathcal{W}(i,j)$, using a scaled exponential similarity kernel.
This kernel is based on the probability density function of the normal
distribution and is expressed as follows:

$W(i,j)=\frac{1}{\sqrt{2\pi{\sigma}^{2}}}\exp{-\frac{{\rho}^{2}(x_{i},x_{j})}{2{\sigma}^{2}}}$

$\sigma=\mu\frac{\overline{\rho}(x_{i},N_{i})+\overline{\rho}(x_{j},N_{j})+\rho(x_{i},x_{j})}{3}$

Here, ${\rho}(x_{i},x_{j})$ denotes the squared Euclidean distance between the
feature vectors of subjects $x_{i}$ and $x_{j}$. The term
$\overline{\rho}(x_{i},N_{i})$ represents the average distance between $x_{i}$
and its nearest neighbors $N_{1...k}$, with $k$ denoting the number of
neighbors considered. The hyperparameter $\mu$ modulates the scale of the
similarity kernel $W$, and its value ranges between 0 and 1. Based on previous
research (Wang et al. 2014), we used a default value of 20 for $k$ and,
recognizing the sensitivity of $\mu$, empirically set it within the range
[0.1, 1.0].

An edge $e_{ij}$ is assigned a weight $W(i,j)$ if it exceeds the threshold of
$10^{-9}$. Employing this methodology, we constructed a subject similarity
graph for the GENIE cohort, featuring 8,104 vertices, 8,292 edges, and 119
node features. This graph serves as a fundamental structure for analyzing the
patterns and relationships among individuals in the health checkup data.

### Diffusion-based Graph Transformers

To effectively learn similarity structures within our graph, we adapted the
Diffusion-based Transformers (DIFFormer) (Wu et al. 2023), known for its
superior performance in both semi-supervised and supervised contexts.
DIFFormer is particularly adept at discerning latent graph structures, even in
the absence of explicitly observed graph structures.

The DIFFormer model initiates with initial embeddings $Z^{(0)}$, which evolve
through $L$ layers into $Z^{(1)},...,Z^{(L)}$. These evolving embeddings
enable all-pair propagation in the $h$-th head of the $k$-th layer, denoted as
$P^{(k,h)}$. DIFFormer is available in two versions: DIFFormer-s, featuring a
simple diffusivity model, and DIFFormer-a, which employs an advanced
diffusivity model. While DIFFormer-s maintains a linear complexity of
$O(Nd^{2})$, DIFFormer-a exhibits a higher complexity, $O(Nd^{2}+N^{2}d)$.

For our specific requirements, we tailored DIFFormer-s to integrate input
graphs while preserving its linear complexity. This adaptation facilitated an
enhanced comprehension of similarity structures within our graph. In
DIFFormer’s propagation layer, we introduced attention coefficients to the
input graph. These attention coefficients, $e_{ij}$, for the embeddings
$Z^{(k)}$ at layer $k$ are computed as follows:

$e_{ij}^{(k,h)}=\sigma\left(\mathbf{a}^{T}\left[\mathbf{W}^{(k,h)}_{Z}Z_{i}^{(k)}\|\mathbf{W}^{(k,h)}_{Z}Z_{j}^{(k)}\right]\right)$

Here, $\mathbf{a}$ is a learnable parameter vector, $\mathbf{W}^{(k,h)}_{Z}$
is a learnable weight matrix for the embeddings of the $h$-th head at the
$k$-th layer, $\|$ denotes concatenation, and $\sigma$ represents the
LeakyReLU activation function. These coefficients are then normalized using
the softmax function:

$\alpha_{ij}^{(k,h)}=softmax\left(e_{ij}^{(k,h)}\right)$

These coefficients are subsequently integrated into the propagation equation
by combining them with the adjacency matrix $A$. The all-pair propagation
matrix $P^{(k,h)}$ is updated accordingly:

$\bar{P}^{(k,h)}=P^{(k,h)}+D^{-\frac{1}{2}}\left(\alpha_{ij}^{(k,h)}\odot
A\right)D^{-\frac{1}{2}}V^{(k,h)}$

In this equation, $\odot$ signifies element-wise multiplication and $D$ is the
diagonal degree matrix. The transformed embeddings $V^{(k,h)}$ derived from
the DIFFormer model are used in conjunction with the attention coefficients
and the adjacency matrix to update the propagation matrix. This maintains the
linear complexity of the original model while enhancing its ability to capture
complex similarity structures within the graph through an attention mechanism.

This adaptation maintains the linear complexity of the DIFFormer model while
enhancing its ability to capture complex similarity structures within our
similarity graph through an attention mechanism. We have named this attention-
enhanced version of DIFFormer as DIFFormer-attn for our study. Further details
on the algorithms before and after these adjustments are described in (Wu et
al. 2023).

Table 1: Semi-supervised fatty liver prediction performance of GNN models trained with different numbers of labeled samples per class ($\ell$). The highest AUC is highlighted in bold. | Basic models | GNNs | Diffusion-based GNNs | Graph Transformers
---|---|---|---|---
$\ell$ | LR | MLP | GCN | GAT | DGCEuler | GRAND++ | DIFFormer | DIFFormer-attn
1 | 59.21 $\pm$ 1.26 | 56.53 $\pm$ 0.56 | 54.28 $\pm$ 0.53 | 58.76 $\pm$ 0.90 | 58.17 $\pm$ 1.28 | 66.98 $\pm$ 0.44 | 64.13 $\pm$ 1.69 | 63.24 $\pm$ 1.96
2 | 62.39 $\pm$ 0.88 | 56.76 $\pm$ 0.49 | 54.12 $\pm$ 0.57 | 60.95 $\pm$ 0.76 | 60.87 $\pm$ 1.22 | 67.12 $\pm$ 0.44 | 67.82 $\pm$ 1.07 | 66.76 $\pm$ 1.44
5 | 64.80 $\pm$ 0.68 | 57.72 $\pm$ 0.61 | 55.49 $\pm$ 0.50 | 63.03 $\pm$ 0.73 | 64.52 $\pm$ 0.66 | 67.69 $\pm$ 0.39 | 69.43 $\pm$ 1.02 | 67.85 $\pm$ 1.74
10 | 65.86 $\pm$ 0.75 | 59.44 $\pm$ 0.61 | 56.94 $\pm$ 0.51 | 63.03 $\pm$ 0.73 | 65.75 $\pm$ 0.76 | 68.86 $\pm$ 0.46 | 70.81 $\pm$ 1.43 | 71.65 $\pm$ 1.34
20 | 67.47 $\pm$ 0.62 | 62.01 $\pm$ 0.66 | 61.33 $\pm$ 0.56 | 63.44 $\pm$ 0.66 | 67.71 $\pm$ 0.55 | 71.19 $\pm$ 0.43 | 71.69 $\pm$ 1.13 | 73.28 $\pm$ 0.81
50 | 69.55 $\pm$ 0.39 | 66.62 $\pm$ 0.48 | 65.59 $\pm$ 0.37 | 65.58 $\pm$ 0.69 | 68.16 $\pm$ 0.41 | 72.89 $\pm$ 0.36 | 74.39 $\pm$ 0.80 | 74.62 $\pm$ 0.58
100 | 71.40 $\pm$ 0.32 | 69.94 $\pm$ 0.42 | 67.95 $\pm$ 0.41 | 70.18 $\pm$ 0.67 | 70.43 $\pm$ 0.43 | 74.65 $\pm$ 0.32 | 76.38 $\pm$ 0.68 | 76.18 $\pm$ 0.65
Figure 1: Performance analysis on the varying sizes of model depth in
DIFFormer and its variations, and baseline models

## Results

### Experimental Setup

Our experimental study conducted an extensive evaluation of two DIFFormer
model variants, alongside a range of other models, to predict fatty liver
disease. We compared these advanced Graph Transformer models — DIFFormer (Wu
et al. 2023) and DIFFormer-attn (ours) — against basic models like Logistic
Regression (LR) and Multi-Layer Perceptron (MLP), which rely on feature
matrices, and standard Graph Neural Networks (GNNs) such as Graph
Convolutional Network (GCN) (Kipf and Welling 2016) and Graph Attention
Network (GAT) (Veličković et al. 2017). Additionally, diffusion-based GNNs
like DGCEuler (Wang et al. 2021), GRAND++ (Thorpe et al. 2022) and NodeFormer
(Wu et al. 2022) were included.

We employed a dataset divided randomly into a fixed number ($\ell$) of labeled
training samples per class, supplemented by a validation set of 500 samples
and a test set of 1000 samples. This distribution was repeated 50 times to
validate robustness, with the models’ performance evaluated using the mean and
standard error of the Area Under the ROC Curve (AUC) metric.

To ensure consistency, we standardized common hyperparameters across most
models. These included 1000 epochs, a learning rate of 0.001, weight decay of
0.01, dropout rate of 0.5, and a hidden layer size of 64. Batch normalization
was universally applied, an alpha value of 0.8 was set for residual
connections, and the number of attention heads was fixed at 4. Additional
specific configurations were tailored to optimize the performance of certain
models. The NodeFormer model incorporated 30 random features, utilized 10
samples for gumbel softmax for message passing, and had specific settings of
0.25 for the softmax temperature, a weight of 1 for edge regularization loss,
and an order of 2 for relational bias. DGCEuler had distinct settings for its
parameters, 5.15 for T and 4 for K.

### Semi-Supervised Fatty Liver Prediction

In our study, we evaluated the efficacy of various machine learning models for
semi-supervised prediction of fatty liver disease, focusing on how they
performed across different quantities of labeled data, denoted as $\ell$. We
varied $\ell$ through a series, specifically at 1, 2, 5, 10, 20, 50, and 100,
to assess model performance under varying data availability.

As shown in Table 1, our findings highlighted the exceptional performance of
advanced GNN models, notably GRAND++, DIFFormer, and our enhanced version,
DIFFormer-attn. These models demonstrated robust predictive capabilities
across all ranges of labeled data. The DIFFormer-attn, in particular,
consistently achieved high AUC scores, showcasing the effectiveness of its
integrated attention mechanism. This mechanism, which allows the model to
focus on the most relevant features and relationships within the graph,
significantly improves prediction accuracy. The superiority of the attention
mechanism becomes apparent when comparing the performance of GCN and GAT, GAT
consistently demonstrates enhanced performance over GCN.

A key aspect of DIFFormer-attn is its ability to learn attention coefficients
directly from the input graph, unlike the original DIFFormer model. This
feature is instrumental in refining the model’s understanding and
representation of complex clinical data. When compared with GRAND++, a leading
diffusion-based GNN model, DIFFormer-attn exhibited superior performance,
particularly in scenarios with more than 10 labeled samples. While GRAND++
excelled with very limited labeled data like 1 to 5, DIFFormer and DIFFormer-
attn were more effective in scenarios with 10 to 100 labeled samples,
demonstrating consistent and reliable performances.

Figure 2: Heatmap displaying the distribution of importance scores from
features (rows) across individuals (columns), generated from the DIFFormer-
attn model.

### Model Depth Analysis in DIFFormer Models

We investigated the impact of model depth on the performance of the DIFFormer
and its variants. This exploration included NodeFormer and a DIFFormer model
without a similarity graph (DIFFormer w/o graph). We maintained a consistent
comparison approach by using only 10 labeled samples and varying the model
depth across 2, 4, 6, and 8 layers. The findings, presented in Figure 1,
revealed that the DIFFormer models, both with and without the similarity
graph, demonstrated robust performance across varying depths. Specifically,
the DIFFormer-attn model, which integrates an attention mechanism,
consistently achieved high AUC scores across all tested model depths.

Conversely, other models like GAT, GCN and MLP displayed a significant
decrease in performance with an increase in layers, with GCN and GAT
particularly showing the lowest performance among all. These results highlight
the superiority of the DIFFormer and DIFFormer-attn models in accurately
predicting fatty liver disease across shallow to deep models, even with a
limited amount of labeled data. The effectiveness of these models, especially
the DIFFormer-attn, is attributed to their ability to utilize a similarity
graph effectively. This graph aids in capturing meaningful relationships among
subjects, thereby enhancing the distinction between normal individuals and
those with fatty liver disease. The consistent performance of these models
across various depths, coupled with the insights gleaned from the similarity
graph, emphasizes the potential of advanced GNN models in improving disease
prediction accuracy in clinical settings.

### Population-centric Explanation

To elucidate the DIFFormer-attn model’s efficacy in distinguishing between
normal individuals and fatty liver disease patients, we applied GNNExplainer
(Ying et al. 2019) for interpretative analysis, focusing on the model’s best-
performing scenario with 100 labeled samples per class. As shown in Figure 2,
heatmap analysis of feature importances provided insights into how the model
discerns between groups based on specific feature patterns, particularly in
cases of correct predictions. This analysis revealed two prominent clusters of
features: Group A, representing characteristics closely related and possibly
linked to a common biological or clinical pathway associated with the disease,
and Group B, suggesting a distinct pathway or mechanism, highlighting the
multifaceted nature of fatty liver disease.

Further, the heatmap identified three unique groups of patient samples, each
characterized by distinct patterns of feature importance. One group showed
minimal impact across most features, suggesting a disease driven by a few key
factors. Another group appeared heavily influenced by Group B features,
potentially representing a specific disease subtype or a patient demographic
impacted by these factors. The third group exhibited high importance across a
broad range of features, indicating a systemic manifestation of the disease.
These insights emphasize the complexity of fatty liver disease and the need
for personalized diagnostic and treatment approaches. The variability in
feature importance across patient samples underscores the importance of
considering individual disease profiles, highlighting the value of advanced
machine learning in enabling more precise and effective healthcare
interventions.

## Conclusion

This study highlights the effectiveness of graph representation learning and
semi-supervised learning in healthcare, as shown by the DIFFormer-attn model’s
success in predicting fatty liver disease. The incorporation of attention
mechanisms enhances the model’s ability to process complex data, leading to
improved outcomes. The research’s use of human-centric explanations,
particularly heatmap analysis, sheds light on fatty liver disease and the
model’s discriminative power between affected individuals and healthy
subjects. These findings demonstrate the importance of explainable AI in
building trust in AI diagnostics and aiding clinicians. Additionally, while
centered on fatty liver disease, the methods and approaches used here show
promise for broader applications in other diseases, illustrating the potential
of these advanced techniques in providing more precise and personalized
medical treatments.

## References

  * Choe et al. (2022) Choe, E. K.; Shivakumar, M.; Verma, A.; Verma, S. S.; Choi, S. H.; Kim, J. S.; and Kim, D. 2022.  Leveraging deep phenotyping from health check-up cohort with 10,000 Korean individuals for phenome-wide association study of 136 traits.  _Scientific Reports_ , 12(1): 1930.
  * Kipf and Welling (2016) Kipf, T. N.; and Welling, M. 2016.  Semi-supervised classification with graph convolutional networks.  _arXiv preprint arXiv:1609.02907_.
  * Thorpe et al. (2022) Thorpe, M.; Nguyen, T. M.; Xia, H.; Strohmer, T.; Bertozzi, A.; Osher, S.; and Wang, B. 2022.  GRAND++: Graph neural diffusion with a source term.  In _International Conference on Learning Representation (ICLR)_.
  * Veličković et al. (2017) Veličković, P.; Cucurull, G.; Casanova, A.; Romero, A.; Lio, P.; and Bengio, Y. 2017.  Graph attention networks.  _arXiv preprint arXiv:1710.10903_.
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# Sub 20 meV Schottky barriers in metal/MoTe2 junctions

Nicola J Townsend1,+, Iddo Amit1,+, Monica F Craciun2 and Saverio Russo1 1
University of Exeter, School of Physics, College of Engineering, Mathematics
and Physical Sciences, Exeter, EX4 4QL, United Kingdom  2 University of
Exeter, School of Engineering, College of Engineering, Mathematics and
Physical Sciences, Exeter, EX4 4QF, United Kingdom  + these authors
contributed equally to this work<EMAIL_ADDRESS>
###### Abstract

The newly emerging class of atomically-thin materials has shown a high
potential for the realisation of novel electronic and optoelectronic
components. Amongst this family, semiconducting transition metal
dichalcogenides (TMDCs) are of particular interest. While their band gaps are
compatible with those of conventional solid state devices, they present a wide
range of exciting new properties that is bound to become a crucial ingredient
in the future of electronics. To utilise these properties for the prospect of
electronics in general, and long-wavelength-based photodetectors in
particular, the Schottky barriers formed upon contact with a metal and the
contact resistance that arises at these interfaces have to be measured and
controlled. We present experimental evidence for the formation of Schottky
barriers as low as 10 meV between MoTe2 and metal electrodes. By varying the
electrode work functions, we demonstrate that Fermi level pinning due to metal
induced gap states at the interfaces occurs at 0.14 eV above the valence band
maximum. In this configuration, thermionic emission is observed for the first
time at temperatures between 40 K and 75 K. Finally, we discuss the ability to
tune the barrier height using a gate electrode.

††:

Keywords: MoTe2, TMDCs, Schottky barriers, 2D materials, low temperature,
thermionic emission

##  1 Introduction

The emerging class of atomically thin materials has captured the interest of
the research community due to the versatile physical phenomena that they
exhibit[1, 2]. To name a few, the indirect to direct band gap transition as a
function of layer number in atomically thin semiconductors[3], gate-controlled
modulation of the band structure in few layer graphene[4] and phase structure
transitions[5] hold the key to future innovations in electronic devices.
Within this broad spectrum of materials, semiconducting transition metal
dichalcogenides (TMDCs) are a major focal point for the wide scientific
community working on fundamental and applied aspects of device physics due to
their energy gaps of 1-2 eV.[6, 7] These gap values are ideally suited for
electronics and optoelectronics, making TMDCs the prime candidates to replace
bulk semiconductors in applications where added functionality, such as
mechanical flexibility, is required. Atomically thin semiconductors also have
the potential to readily form previously hard to access energy barriers,
mostly in the few meV regime.[8] This feature stems from the wide variation in
properties (e.g. work function and band gap values[6, 7]) found within the
layered materials family. As such, by tailoring the energy barrier formed at
the contacts to specific applications, TMDCs can be extremely valuable for
light detection in the far infrared regime as internal photoemission
diodes,[9] replacing other structures that are costly, or difficult to
fabricate by any other means.

One approach to realise low energy barriers is by forming a Schottky junction
between a TMDC and a metal. Several recent studies on few-layer TMDCs, such as
MoS2, WSe2 and MoTe2 have identified Schottky barriers as the primary
contributors to the observed contact resistance.[10, 5, 11, 12, 13, 14, 15]
Indeed, Guo et al. showed that metal induced gap states (MIGS) are the primary
cause for barrier formation and that the Schottky barrier height (SBH) cannot
be significantly altered by changing the work function of the contact
metal.[10] In that work the formation of the Schottky barrier has been
attributed mainly to the lack of dangling bonds normal to the crystallographic
plane of TMDCs. The self terminating plane means that strong bonds between
metal electrodes and the semiconductor cannot be easily formed, a fact that
contributes to the increased contact resistance.[11] Experimentally, SBH
values of 80 meV were reported with the thermionic emission process being
dominant down to 100 K for MoTe2[13, 14] and 60 K for MoS2.[12] However, lower
SBH values have, so far, not been achieved at cryogenic temperatures. At
temperatures between 200 and 400 ${}^{\circ}C$, a barrier height of 23 meV has
been reported where the MoTe2 flake is transferred onto pre-defined gold
contacts, which may be due to a strain induced structural phase change from
semiconducting 2H phase to a metallic 1T’ phase at the contacts.[16]

Here we report the formation of ultra-low effective SBHs down to 10 meV on
MoTe2/metal structures, that is aided by Fermi level pinning at the
interfaces. MoTe2 is a TMDC with three different structural phases, the
semiconducting 2H, and the semimetalic T${}_{\scriptsize\textrm{d}}$ and 1T’
phases. The 2H phase consists of three hexagonal planes in a “sandwich”
formation, with Te at the outer planes and Mo at the centre. These planes are
covalently bonded in a trigonal prismatic configuration that constitutes a
single layer, and the layers are held together by van der Waals forces.[17,
18] The observed ultra-low effective SBHs in MoTe2 play a minor role in the
conduction at room temperature, but become significant at cryogenic
temperatures and manifest in a strongly rectifying behaviour. By analysing the
current-voltage characteristics of the conducting channel at different
temperatures and various gate biases, we are able to extract its SBH and
conductivity values. Within these systems, we find that thermionic emission
persists as the dominant mechanism of transport at temperatures that range
between 40 K and 75 K, whereas at higher temperatures (T $\geq$ 80 K) other
transport mechanisms become more prominent. By varying the electrode material,
we show that MIGS pin the Fermi level at the interface at a level of 0.14 eV
over the valence band maximum. Finally, we briefly discuss the modulation of
the effective SBH with applied gate bias. Our results pave the way for the
realisation of far infrared devices in TMDCs and provide insight on the
mechanisms of transport in MoTe2 at cryogenic temperatures.

##  2 Methods

Two terminal field effect devices, schematically shown in figure 1a were
fabricated by mechanically exfoliating MoTe2 flakes from a synthetic crystal
(HQ Graphene) and transferring them onto highly doped silicon substrate with a
high quality thermally grown oxide layer. The Si/SiO2 serves as a global gate
electrode and gate dielectric, respectively. Metal contacts were patterned
using a regular electron beam lithography procedure immediately prior to
metallisation. The devices were then thermally annealed for 2 hours in a H2/Ar
environment at 200 ∘C.

MoTe2 flakes were characterised using Raman spectroscopy on a Renishaw Raman
microspectrometer using a 532 nm laser. Atomic force microscopy (AFM)
micrographs were obtained on an Innova AFM system (Bruker Inc.) working in the
“tapping mode”, using a Nanosensors high-reflectivity probes operating at a
resonance frequency of $\sim 320$ kHz, with a nominal radius of curvature of
10 nm or smaller. Once the quality of the flakes were established, the sample
was loaded into a helium-3 cryostat, where the source-drain current vs source-
drain voltage
(I${}_{{\scriptsize\textrm{ds}}}$-V${}_{{\scriptsize\textrm{ds}}}$) response
curves and the source-drain current vs gate-source voltage
(I${}_{{\scriptsize\textrm{ds}}}$-V${}_{{\scriptsize\textrm{gs}}}$) were
recorded at decreasing temperatures between 80 K and 40 K using a home-built
electrical characterisation setup.

##  3 Results and Discussion

Figure 1: Material characterisation
a Schematic of an FET device using an MoTe2 flake as the channel. The drain is
the biased electrode and the current is measured at the source contact. The
silicon layer acts as a global gate electrode with the silicon dioxide acting
as the gate dielectric. b Crystal structure of 2H-MoTe2 showing tellerium
atoms (in yellow) covalently bonded to the molybdenum atoms (blue) in a
“sandwich” formation. The covalently bound layers are held together by van der
Waals forces. c Raman spectrum of typical flake showing
$A_{\scriptsize\textrm{1g}}$, $E_{\scriptsize\textrm{2g}}$ and
$B_{\scriptsize\textrm{2g}}$ peaks associated with few layer, 2H-MoTe2. d AFM
micrograph of a typical device with the height profile in the inset showing
the flake has a thickness of 8 nm, equivalent to around 10 layers.

The devices, shown schematically in figure 1a, were characterised using Raman
spectroscopy in ambient conditions to ensure that the flakes crystalline
structure is the 2H phase (see figure 1b). The three peaks in the Raman
spectrum (figure 1c), at $\sim 170$ cm-1, $\sim 230$ cm-1 and $\sim 290$ cm-1
are the finger print modes of the 2H phase, namely the
$A_{{\scriptsize\textrm{1g}}}$, $E_{{\scriptsize\textrm{2g}}}$ and
$B_{{\scriptsize\textrm{2g}}}$ modes respectively.[20, 19, 21] The
$B_{\scriptsize{\textrm{g}}}$ ($\sim 163$ cm-1) and
$A_{\scriptsize\textrm{g}}$ ($\sim 260$ cm-1) peaks, which are associated with
the 1T’ phase[19] are absent indicating that the crystalline phase is indeed
the semiconducting trigonal structure. As the T${}_{\scriptsize\textrm{d}}$
phase does not exist in room temperature, its Raman peaks are not considered
here.[22]

The number of layers was determined from the thickness of the flake as
measured using atomic force microscopy (AFM). The device topography and
structure are shown in the AFM micrograph (figure 1d). The cross section
profile in the inset of the figure shows that the thickness of the flake is 8
nm, corresponding to 10 layers.[21] All the flakes used in this work were in
the range of 5-10 layers.

Figure 2: Response curves at different temperatures
a The response
(I${}_{{\scriptsize\textrm{ds}}}$-V${}_{{\scriptsize\textrm{ds}}}$) curve for
a device with Au contacts taken at room temperature while the device is in the
“open” state at V${}_{{\scriptsize\textrm{gs}}}=-40$ V. b The response curve
for the same device taken at 40 K while the rest of the parameters are
constant.

The few-layer MoTe2 FET bears two identical metal contacts and should present
current response characteristics that are symmetric about the zero source-
drain bias (V${}_{\scriptsize\textrm{ds}}=0$ V). However, the response curve
(I${}_{\scriptsize\textrm{ds}}$-V${}_{\scriptsize\textrm{ds}}$) shown in
figure 2a is sub-linear and slightly rectifying, a trend that becomes more
pronounced at lower temperatures (figure 2b). The increasingly rectifying
response to V${}_{\scriptsize\textrm{ds}}$ with decreasing temperature is
indicative of the formation of asymmetric Schottky barriers at the interfaces
between the metal contacts and the semiconducting channel (see supplementary
information section II online). While the semiconductor interface potential at
the grounded (source) electrode is pinned by the bottom gate, and is
effectively a fixed energy barrier, the electrostatic potential of the biased
(drain) interface decreases the barrier height with forward bias.[23]

To gain further insights into the response characteristics of the devices,
additional electrical characterisation was carried out in the temperature
range from 80 K to 40 K. These measurements were performed at decreasing
temperatures to avoid thermal emission of captured charge carriers which would
significantly alter the barrier height. The source-drain current vs source-
drain bias (I${}_{\scriptsize\textrm{ds}}$-V${}_{\scriptsize\textrm{ds}}$)
response curves presented in figure 3a were measured on a device bearing Au
contacts, using an applied gate bias (V${}_{\scriptsize\textrm{gs}}$) of -40
V, which is in the devices “open” state. It is apparent that the device
exhibits a highly rectifying diode-like behaviour, which persists at lower
temperatures. A graph on a semi-logarithmic scale, plotted in figure 3b,
further emphasises the saturation of the source-drain current when the device
is in reverse bias with respect to the source barrier. Other devices displayed
similar trends and are shown in supplementary figure S1 online.

Figure 3: Electrical characterisation
a Response curves for a device with Au contacts at temperatures between 80 K
and 40 K, measured with the devices in the “open” state at
V${}_{\scriptsize\textrm{gs}}=-40$ V. The curves exhibit a diode-like
rectifying characteristics, as is further emphasised in the semi-logarithmic
scale in b. c Transfer curves of the Au contacted device showing the
characteristics of an enhancement type, p-channel semiconductor with a forward
bias of V${}_{\scriptsize\textrm{ds}}$ = +20 V. The semi-logarithmic scale in
d shows a constant current in the subthreshold region.

Fixing the source-drain bias to a “forward bias” configuration
(V${}_{\scriptsize\textrm{ds}}=+20$ V), the source-drain current vs gate bias
(I${}_{\scriptsize\textrm{ds}}$-V${}_{\scriptsize\textrm{gs}}$) transfer
curves were measured, and are plotted in figure 3c. When the device is forward
biased, it exhibits the characteristics of an enhancement type, p-channel
transistor with its threshold voltage (V${}_{\scriptsize\textrm{th}}$) growing
increasingly negative with decreasing temperature. The semi-logarithmic scale
plot, shown in figure 3d, shows the subthreshold region where the FET channel
switches from its “closed” state to its “open” state and enters the linear
regime. At 80 K, the two-terminal hole mobility extracted from the linear
region of the transfer graph is around 0.1 cm2 V-1 s-1, which is in agreement
with our previously published devices,[24] but lower than the values reported
by other groups.[25] We attribute the lower mobility to a high density of
interface states, present either at the contacts or across the semiconducting
flake. As we will show, this high density of traps contributes to the pinning
of the Fermi level at the contacts and is instrumental in the realisation of
ultra-low Schottky barriers. Other devices discussed in this report showed
similar trends, see supplementary figure S2 online. Contrary to MoS2,[26] in
MoTe2 p-type behaviour dominates the transistor characteristics among all the
metals used, with the exception of Ti which induces an asymmetric ambipolar
behaviour, with p-doping at V${}_{\scriptsize\textrm{gs}}$ = 0 V.

The conductivity dependence on the temperature, determined by the response and
transfer curves, which were acquired at different temperatures, reveals that
the charge transport is dominated by a thermionic emission over an energy
barrier. There are three main mechanisms of transport across an energy
barrier: (1) thermionic emission from the high-end tail of Boltzmann
distributed particles, (2) diffusion and (3) tunnelling (or field emission)
through the barrier.[27, 11] For thermionic emission of charge carriers into a
three-dimensional semiconductor, the current dependence with temperature ($T$)
is given by the Schottky diode equation
$I=AA^{*}T^{2}\exp(-E_{A}/kT)\left[\exp(qV/nk_{B}T)-1\right]$, where $A$ is
the contact area, $A^{*}=4\pi m^{*}qk_{B}^{2}/h^{3}$ is the modified
Richardson constant, $E_{A}$ is the activation energy required to overcome the
barrier, $n$ is the ideality factor, $q$ is the basic charge and $k_{B}$ is
the Boltzmann constant. Other mechanisms of charge transport differ mainly in
the exponent of $T$, which is 0 for diffusion and 1 for tunnelling. To
determine the transport mechanism, the MoTe2 FET current vs. temperature plot
shown in figure 4a is fitted with I${}_{\scriptsize\textrm{ds}}=B\cdot$
T${}^{\alpha}e^{-C/T}$, where $\alpha$ is the exponent of T used as a fitting
parameter and $B$ and $C$ are arbitrary constants. The use of the pre-
exponential constant is justified by plotting the current at different
temperatures with fixed V${}_{\scriptsize\textrm{ds}}$ and fixed
V${}_{\scriptsize\textrm{gs}}$. The extracted $\alpha=2.082\pm 0.211$ is in
good agreement with the temperature exponent for thermionic emission,
suggesting that this is the dominant transport mechanism governing the
transport in the device, while diffusion and tunnelling of charges play a
negligible role in the temperature range between 40 K and 80 K. This is
further supported by previous work that has shown the channel is depleted of
its majority carriers,[24] leading to the formation of barriers as wide as 10
$\mu$m in MoTe2,[28] a fact which renders the efficiency of the tunnelling
process insignificant. Further discussion on the additional transport
mechanisms are included in supplementary information section III online.

Figure 4: Barrier height determination
a Current-temperature dependence for an Au contacted device in the “open”
state, showing a good agreement with the thermionic emission model for
temperatures below 80 K. b Schematic energy band diagram of the metal/p-type
semiconductor interface showing band bending of the conduction band edge
$E_{C}$, the valence band edge $E_{V}$ and the local vacuum level $LVL$ when
the Fermi levels $E_{F}$ are aligned and the system is in thermal equilibrium.
The metal work function $\phi_{M}$, MoTe2 work function $\phi_{S}$, electron
affinity $\chi$, band gap $E_{g}$ and barrier height $\phi_{Bp}$ are noted in
the diagram. c A Richardson plot for the different devices at
V${}_{\scriptsize\textrm{gs}}-$V${}_{\scriptsize\textrm{th}}=-11$ V. d The
relation between barrier height and metal work function from which the extent
of $E_{F}$ pinning can be determined.

For thermionic emission to occur at low temperatures the thermal width of the
Fermi-Dirac distribution ($\Delta E$) has to be comparable to or lower than
the barrier height. $\Delta E$ for the majority (80%) of charge carriers can
be found using the Fermi-Dirac distribution
($f(E)=1/(1+\exp((E-E_{F})/k_{B}T))$)[27] by considering the region between
$f=0.1$ and $f=0.9$. Within this range, the thermal distribution width is
given by $\Delta E=4.4k_{B}T$. Supplementary figure S6 online shows the
thermal distribution width as a function of temperature. Between the
temperatures of 40 K and 80 K, the thermal distribution width ranges from 15
meV to 30 meV. When the thermal energy distribution grows larger than the
barrier height, additional mechanisms of charge transfer over an energy
barrier, such as tunnelling currents and diffusion, become more prominent,
rendering the thermionic emission mechanism irrelevant at higher temperatures.

As previously discussed, the thermionic emission occurs over the
source/channel energy barrier that forms when the Fermi energies of the
semiconductor and metal reach thermal equilibrium. The surface potential of
the metal depletes the adjacent segment of the channel, effectively bending
the energy bands of the semiconductor, as shown in figure 4b, forming the
Schottky barrier. In the ideal case the SBH ($\phi_{Bp}$) is a function of the
metal work function ($\phi_{M}$), the semiconductor electron affinity ($\chi$)
and (for p-type materials) the band gap ($E_{g}$), given by the Schottky-Mott
rule, $\phi_{Bp}=E_{g}+\chi-\phi_{M}$.[10] To tune the barrier height, four
different metals were used for the contacts, Ti, Cr, Au and Pd with work
functions of 4.3[13], 4.5,[29] 5.1[29] and 5.6 eV[30] respectively. These work
functions should theoretically form barrier heights of 480, 280, -320 and -820
meV respectively, based on an electron affinity of 3.78 eV[31] and a band gap
of 1.0 eV[32] for few-layer MoTe2. Within this model a negative SBH is
indicative of Ohmic contacts. In a realistic interface, mid-gap states, that
may be formed by vacancies,[10] oxidation,[25] and MIGS,[33] increase the
quantum capacitance of the band gap, an effect known as “Fermi level pinning”,
effectively altering the barrier height. In the extreme case, known as the
Bardeen limit, a large density of mid-gap states can pin the barrier height
completely.[10]

The MoTe2 FET exhibits a diode-like behaviour, due to the pinned
source/channel potential. As such, we may regard it as a single junction
device, while treating the second junction as a fixed resistor. This point is
further demonstrated in section II of the supplementary information online. In
a single diode device, the height of the barrier can be extracted from
temperature dependent transport measurements using a Richardson plot.[15] To
this end the response curves are fitted with the diode equation
$I_{\scriptsize\textrm{ds}}=I_{\scriptsize\textrm{S}}\exp(-q(V_{\scriptsize\textrm{ds}}-I_{\scriptsize\textrm{ds}}R_{\scriptsize\textrm{S}})/nk_{\scriptsize\textrm{B}}T)-V_{\scriptsize\textrm{ds}}/R_{\scriptsize\textrm{P}}$,
where $R_{\scriptsize\textrm{S}}$ is a series resistance that includes the
channel resistance and the contact resistance and $R_{\scriptsize\textrm{P}}$
is a parallel shunt resistance. $I_{\scriptsize\textrm{S}}$ is the saturation
current which is given by
$I_{\scriptsize\textrm{S}}=AA^{*}T^{2}\exp(-\phi_{{\scriptsize\textrm{Bp}}}/k_{\scriptsize\textrm{B}}T)$
for thermionic emission. The extracted $I_{\scriptsize\textrm{S}}$ values are
then plotted as $\ln(I_{\scriptsize\textrm{S}}/T^{2})$ vs $1/T$, shown in
figure 4c and supplementary figure S7, which yield effective barrier heights
of 41.1, 40.3, 30.3, and 10.2 meV for the Ti, Cr, Au and Pd contacts
respectively.

The extraction of Schottky barrier heights as low as a few meVs can cause
numerous difficulties due to competing mechanisms of transport. At low
temperatures, thermionic field emission can be the dominant mechanism for
charge injection into the channel.[34] However, our analysis of the source-
drain current vs temperature at fixed source-drain bias and gate bias in
figure 4a clearly show that thermionic emission is the dominant mechanism. In
the determination of the barrier height, we used non-linear implicit function
fitting tool to allow the physical parameters $I_{S}$, $R_{S}$, $R_{P}$ and
$n$ to reach a global minimum of residuals in the parameters space. For this,
we have used a step-wise fit[35] to obtain the initial values of the
parameters, which were then allowed to reach a higher degree of accuracy.
Indeed, the largest source of error in the fittings came from the noise level
of the instruments with the small current signal at the lower temperatures.

It is important to note two major consequences of the measured results. First,
while the resulting trend is in excellent qualitative agreement with the
theoretical case where the barrier height decreases with an increase in metal
work function, the extracted effective barrier heights differ profoundly from
the calculated values. This indicates significant pinning of the Fermi level
by mid-gap states. Second, the measured Schottky barrier heights are
comparable with the Fermi Dirac thermal distribution width, thus confirming
that thermionic emission over an energy barrier is the dominant current
mechanism.

The extent of Fermi level pinning can be quantified by comparing the barrier
height with the metal work function using
$\phi_{\scriptsize\textrm{Bp}}=\phi_{0}+S(\phi_{\scriptsize\textrm{M}}-\phi_{0})$,[10,
36] where $\phi_{0}$ is the reference energy of the mid-gap states. The
pinning factor $S$ is equivalent to -1 in the ideal case with no pinning
(Schottky limit) and 0 if $E_{\scriptsize\textrm{F}}$ is completely pinned
(Bardeen limit). Figure 4d shows that the linear decrease in effective barrier
height with increasing metal work function, has a gradient of -0.02 that
indicates very strong pinning of the Fermi level. From the intercept of the
linear fit, the reference energy of the mid-gap states is found to be 0.14 eV
above the valence band maximum. The strong pinning also means that gate
tunability of the effective Schottky barrier height is considerably suppressed
in these devices. Indeed, only the Au and Pd contacted devices show some
tunability, exhibiting SBH modulation in the order of up to 40 meV for gate
bias shifts of 15 V. This effect is most likely due to image force barrier
lowering, and is discussed in further detail in supplementary figure S8 and
the following discussion online in supplementary information Sec. VI, however
further work needs to be carried out to fully elucidate this mechanism.

Considering the low charge carrier mobility and the strong Fermi level
pinning, we attribute the measured effective barrier heights to the high
density of mid gap states present at the metal/semiconductor interface. We
have previously shown that slow charge carrier dynamics in MoTe2 strongly
affects the performance of FETs.[24] However, for photodetectors that are
based on internal photoemission of charge carriers between the contact and the
channel at that interface, the slow drift of charge carriers does not impede
the performance of the detector, and can in fact serve as a short-lived
reservoir for excited holes.

##  4 Conclusion

MoTe2 FETs that exhibit ultra-low effective SBH were presented for the first
time. The effective SBH can be controlled by changing the contact metal to
some extent, but is dominated mainly by pinning of the Fermi level at the
interface. The reference energy of the mid-gap states, which is a measure of
the highest energy of occupied mid-gap states, was found to be 0.14 eV above
the valence band maximum. The tunability of the barrier may assist in the
design of internal photoemission based photodetectors, particularly for near-
infrared applications.

NJT and SR acknowledge DSTL grant scheme Sensing and Navigation using quantum
2.0 technologies. IA acknowledges financial support from The European
Commission Marie Curie Individual Fellowships (Grant number 701704). SR and
MFC acknowledge financial support from EPSRC (Grant no. EP/J000396/1,
EP/K017160/1, EP/K010050/1, EP/G036101/1, EP/M001024/1, EP/M002438/1), from
Royal Society international Exchanges Scheme 2016/R1 and from The Leverhulme
trust (grant title ”Quantum Drums” and ”Room temperature quantum
electronics”).

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See pages - of Supplementary1