11institutetext: Research Center for Medical Image Analysis and Artificial
Intelligence, Department of Medicine, Danube Private University, Krems an der
Donau, Austria 22institutetext: Institute of Artificial Intelligence, Medical
University of Vienna, Vienna, Austria 33institutetext: Institute for
Pathophysiology and Allergy Research, Medical University of Vienna, Vienna,
Austria 44institutetext: Austrian Center for Medical Innovation and
Technology, Wiener Neustadt, Austria

# Improving Generalization Capability of Deep Learning-Based Nuclei Instance
Segmentation by Non-deterministic Train Time and Deterministic Test Time Stain
Normalization ††thanks: Partially supported by the Austrian Research Promotion
Agency (FFG), No. 872636.

Amirreza Mahbod 11 Georg Dorffner 22 Isabella Ellinger 33 Ramona Woitek 11
Sepideh Hatamikia 1144

###### Abstract

With the advent of digital pathology and microscopic systems that can scan and
save whole slide histological images automatically, there is a growing trend
to use computerized methods to analyze acquired images. Among different
histopathological image analysis tasks, nuclei instance segmentation plays a
fundamental role in a wide range of clinical and research applications. While
many semi- and fully-automatic computerized methods have been proposed for
nuclei instance segmentation, deep learning (DL)-based approaches have been
shown to deliver the best performances. However, the performance of such
approaches usually degrades when tested on unseen datasets.

In this work, we propose a novel approach to improve the generalization
capability of a DL-based automatic segmentation approach. Besides utilizing
one of the state-of-the-art DL-based models as a baseline, our method
incorporates non-deterministic train time and deterministic test time stain
normalization. We trained the model with one single training set and evaluated
its segmentation performance on seven test datasets. Our results show that the
proposed method provides up to 5.77%, 5.36%, and 5.27% better performance in
segmenting nuclei based on Dice score, aggregated Jaccard index, and panoptic
quality score, respectively, compared to the baseline segmentation model.

###### Keywords:

Digital Pathology Normalization Nuclei Segmentation Machine Learning Deep
Learning Medical Image Analysis.

##  1 Introduction

The evaluation of histopathological images by experts remains an integral part
of the diagnostic routine of many human diseases [12]. An essential element of
this process is the inspection of the appearance, morphology, and density of
cells, which is subsequently used, for example, to diagnose different types of
cancer or to assess the progression of certain diseases [25, 6, 3]. Another
important aspect in this context is the shape and structure of the cell nuclei
[34]. Nuclear segmentation- the task of finding all individual nuclei in
digitized histological images- is therefore a key feature of many automated
pathology frameworks, as it enables the subsequent extraction of important
information, including cell number or numbers related to the shape and
structure of the nuclei [27]. Since automated pathology frameworks can improve
diagnostic accuracy, reduce evaluation time, and create more efficient
workflows, there has been an increased effort to automate nuclei segmentation
with the goal of achieving more robust and objective segmentation results [41,
17, 21]. Many computerized methods have been proposed for unsupervised, semi-
supervised, and fully-supervised nuclei segmentation ranging from standard
image processing techniques to advanced machine learning (ML) and deep
learning (DL) algorithms [11, 26]. However, among these methods, supervised DL
algorithms, such as convolutional neural networks (CNN), have achieved the
best performances [11, 1]. Although there are many different nuclei instance
segmentation algorithms based on CNNs, the state-of-the-art strategies can be
broadly categorized into three different categories. There are detection-based
methods such as adapted or improved Mask R-CNN [2, 13], ternary segmentation
models such as deep contour-aware networks (DCAN) or Kumar et al. method [16,
4] and distance-based methods such as Hover-Net [9], two-stage U-Net
algorithms [24] or dual decoder U-Net-based (DDU-Net) model [22].
Additionally, these approaches can be jointly used in order to boost the
segmentation performance [2, 39]. Furthermore, considering the recent advances
of large language models, vision transformer-based architectures have also
been utilized in the encoder-decoder-based models for various histological
image analysis tasks, including nuclei instance segmentation [42, 33].
Comprehensive overviews of state-of-the-art methodologies for nuclei
segmentation can be found in the respective studies [3, 11, 26, 42].

Despite the fact that supervised CNNs achieve excellent nuclei segmentation
performances for individual datasets, the CNN performance usually diminishes
on external or unseen test data, because they have a different image space
than the training data [35, 32]. There are many sources of varying cell
appearance among histological images from different labs, including color
variations caused by minor stain deviations, different organs, or different
image acquisition devices. Thus, creating an algorithm, which performs well on
all datasets, is a challenging task. Although many efforts were made to
overcome these obstacles and improve the generalization capability of DL-based
methods, there is still a lack of robust algorithms with acceptable nuclei
segmentation performance for unseen test datasets [9, 19].

Normalization-based and augmentation-based approaches are the most well-known
applied methods to improve generalization [35, 29].

Normalization algorithms focus on reducing the variability of input images
from different sources. They often match certain properties of the target
image to those of a reference image [35, 29]. Various methods have been
proposed to perform normalization in histological images. These include
classical image processing techniques such as histogram matching [30], stain-
based color deconvolution such as Macenko et al. method [18], or neural
network-based approaches such as cycle generative adversarial networks [44].

Augmentation, on the other hand, exploits various geometrical or color
transformations in order to introduce more variety during the training. Common
techniques are rotations, mirroring, scaling, elastic deformations, or adding
small perturbations to the channels in different color spaces [35].

The main contribution of this work is proposing and developing a hybrid
approach by combining these two techniques in the instance segmentation
workflow. We use DDU-Net [22], one of the state-of-the-art DL-based nuclei
instance segmentation models, as the baseline. For training, we incorporate
non-deterministic normalization based on Macenko et al. method as an online
augmentation. Instead of using a single reference image and performing
normalization as an offline pre-processing, we select multiple reference image
candidates from different organs. We normalize input images randomly as an
online augmentation during the training phase. In the inference phase, we use
deterministic test time normalization and results ensemble to have the final
instance segmentation output. Our results on several unseen test datasets show
that the proposed method has improved nuclei instance segmentation performance
compared to the baseline model.

##  2 Materials and Methods

###  2.1 Datasets

In our study, we used one single dataset for training and multiple datasets
for testing to evaluate the generalization capability of our proposed approach
(i.e., one single trained model was tested on multiple unseen images from
various test datasets). The training was done on the training set from the
MoNuSeg challenge [17], as it encompasses a large variety of nuclei from
different organs and has been widely used as a benchmark dataset in previous
studies [9, 43, 22]. It includes a total of 21,623 nuclei, found in 30
($1000\times 1000$ pixels) Hematoxylin & Eosin(H&E)-stained image patches
extracted from whole slide images from the cancer genome atlas (TCGA)
repository111https://portal.gdc.cancer.gov/repository. The dataset includes
images from seven organs, namely liver, breast, kidney, bladder, prostate,
stomach, and colon. For testing, we utilized seven test datasets. We used
MoNuSeg test data [17], TNBC [28], CryoNuSeg [21], CPM-15 [38], CPM-17 [38],
CoNSeP [9] and NuInsSeg dataset [20], all include image patches with H&E
staining. Further depiction of the attributes of each dataset can be found in
Table 1.

Table 1: Summary of the used datasets for training (first row) and testing (other rows). Datasets | # Nuclei | # Images | Image Size | # Organs
---|---|---|---|---
MoNuSeg Training Set | 21,623 | 30 | $1000\times 1000$ | 7
MoNuSeg Test Set | 7,223 | 14 | $1000\times 1000$ | 9
TNBC | 4,022 | 50 | $512\times 512$ | 1 (breast)
CryoNuSeg | 7,596 | 30 | $512\times 512$ | 10
CPM-15 | 2,905 | 15 | various sizes | 2
CPM-17 | 7,570 | 32 |  $500\times 500$, $600\times 600$ | 4
CoNSeP | 24,319 | 41 | $1000\times 1000$ | 1 (colon)
NuInsSeg | 30,698 | 665 | $512\times 512$ | 31

###  2.2 Segmentation model

This work is based on the DDU-Net [22] as the baseline nuclei instance
segmentation model. This model has shown excellent performance on the nuclear
segmentation task on various datasets (e.g., it achieved the first rank on the
MoNuSAC post-challenge leaderboard for multi-organ nuclear segmentation and
classification challenge [37, 7]). It uses a U-Net-alike encoder-decoder
structure [31], such that the images are fed into a shared encoder path, whose
intermediate results are consequently passed to two different decoder
branches. These decoders are designed to predict nuclear pixels (first
decoder) and distance maps (second decoder) of all instances in a given image.
The shared encoder consists of convolutional, drop-out, and max pooling
layers, while the decoders consist of convolutional, drop-out, and transpose
convolutional layers. The only difference between the two decoders is the last
layer, where a sigmoid activation function and a linear activation function
are used for the first and second decoders, respectively. For the first
decoder, a combined Dice loss and a binary cross-entropy loss were used, and
for the second decoder, a mean square error loss function was utilized. The
general workflow of the utilized DDU-Net is shown in Fig. 1. The results from
decoders are post-processed using a Gaussian smoothing filter, a watershed
algorithm, and morphological operations to form the final instance
segmentation results. Further details about model architecture and workflow
can be found in the respective study [22].

Figure 1: The generic workflow of DDU-Net [22] for nuclei instance
segmentation.

###  2.3 Non-deterministic stain normalization in training

In this work, we made use of the normalization algorithm introduced by Macenko
et al. [18], which has been widely used in the literature to normalize
H&E-stained histological images [8, 10]. This method is based on deconvoluting
images to retrieve the pure hematoxylin (i.e., nuclei) and eosin (i.e. other
cell parts) components. After applying the negative logarithm to convert the
images into the optical density (OD) space, it is possible to model the stains
with the following equation:

$OD=S*V$ (1)

Where $S$ and $V$ are matrices containing the stain saturations (S) and the
stain vectors (V). The stain vector, however, is not known and is hence
approximated. Macenko et al. method achieves this by utilizing singular value
decomposition. Afterward, the saturations of the single stains can be
calculated using the formula above. While this is shown to preserve
histological information [29], the results rely on the sophisticated selection
of the reference image, a step that is usually performed manually.

We introduced a new strategy for randomly applying the Macenko et al. method
during training to improve the generalization capability, paired with an
automated reference image selection algorithm, reducing the impact of
reference image selection and creating more variety in the training data.

To select the reference images automatically, we used the MoNuSeg training
data histograms. We calculated the mean intensity value for nuclei regions and
background regions using the provided binary segmentation masks for the
MoNuSeg training data. We sorted all images by the absolute difference between
the mean tissue intensity and the mean nuclear intensity and selected the
images with the largest differences between the means (one per organ). See
Fig. 2 for examples of selected and non-selected reference images. This
allowed us to automatically select images, such that the colors in the tissue
are different from the ones found in the nuclei, ensuring a good contrast of
the nuclear boundaries and high visibility of the nuclei. We selected seven
reference images (one per organ) using the described technique. The chosen
reference images from the MoNuSeg training data are shown in Fig. 3.

Original image | Nuclei image | Tissue image | Histogram
---|---|---|---
|  |  |
|  |  |
Figure 2: Examples of a non-selected (first row) and a selected (second row) reference images.  Liver | Breast | Kidney | Bladder | Prostate | Stomach | Colon
---|---|---|---|---|---|---
|  |  |  |  |  |
Figure 3: Selected reference images from the MoNuSeg training data based on
histogram analysis.

We applied non-deterministic stain normalization as an online augmentation
based on the selected reference images. For a given input image in the
training phase, we either sent the unmodified image to the model (probability
of 50%) or sent it to the normalization pipeline (probability of 50%). In the
latter case, the image was subsequently normalized against one of the
reference images based on the Macenko et al. method, introducing another non-
deterministic component to the workflow. The probability for each path was
chosen equally (i.e., 7.14% for each path). This step for a sample input image
is depicted in Fig. 4.

Figure 4: During non-deterministic stain normalization, the input training
images were randomly passed either directly to the segmentation model or into
the normalization pipeline, where they were normalized to one of seven
reference images. The probability for each path in the normalization pipeline
was chosen equally.

###  2.4 Test time normalization

Test time augmentation (TTA) has been shown to boost the segmentation and
classification performance for various medical image analysis tasks in former
studies [40, 23, 36]. In this work, besides morphological TTA (90-degree
rotation and horizontal flipping), we propose to use test time stain
normalization (TTSN) as shown in Fig. 5. In the inference phase, the original
test image and seven normalized test images (identical normalization as
performed in training) were sent to the trained model. The outputs were then
merged using a weighted average scheme. The weights were chosen based on the
exploited probabilities in the training phase (i.e., 50 for the original input
test image and 7.14 for each of the seven normalized images).

Figure 5: Proposed test time normalization.

###  2.5 Evaluation

To evaluate the performance of the nuclei instance segmentation, we used the
Dice, aggregate jaccard index (AJI), and panoptic quality (PQ) scores [15, 9].
These metrics have been widely used to evaluate and compare the performance of
different nuclei segmentation methods [37, 17]. While the Dice score shows the
general performance of semantic segmentation, AJI and PQ score are sensitive
to the capability of the model to separate touching objects (show the instance
segmentation performance) and hence are more critical metrics in this study.

###  2.6 Experimental setup

To show the effectiveness and generalization power of the proposed method, we
designed five experiments. The schematic workflow of the experiments is shown
in Fig. 6. The details of the experiments are as follows:

Figure 6: Schematic design of the experiments. The selected shown image and
normalized images are derived from the MoNuSeg training set for visualization.

  * •

No normalization (baseline): In this experiment, we did not apply any
normalization and just used the raw MoNuSeg training dataset to train the DDU-
Net model. To evaluate the model on the test datasets, raw test images were
used.

  * •

Offline normalization: In this setup, one single image was chosen as the
reference image, and all other images (from the MoNuSeg training set and all
test images from all test datasets) were normalized using the Macenko et al.
approach. To choose the reference image, we used the histogram analysis
described in Section 2.3.

  * •

Extended offline normalization: In this setup, we merged the raw and
normalized MoNuseg training data to train the model (hence the size of the
training set was doubled). In the inference phase, we sent the raw and
normalized test images to the trained model and got the average over the
results to have the final segmentation mask. Similar to the previous
experiment, histogram analysis was used to select the reference image from the
MoNuSeg training dataset.

  * •

Non-deterministic stain normalization: In this experiment, we applied the
proposed non-deterministic stain normalization as described in Section 2.3. In
the inference phase, the raw test images were sent to the trained model.

  * •

Non-deterministic normalization + TTSN: In this experiment, we used both non-
deterministic stain normalization and TTSN as described in Section 2.3 and
Section 2.4.

Besides the differences mentioned above, all other parameters were kept
identical in all experiments. We used 5-fold cross-validation to train the
DDU-Net in all experiments and generated five models from cross-validation
ensembled in the inference phase. Each model was trained for 200 epochs with
an initial learning rate (LR) of 0.001. We used a LR scheduler (dropped by
half after every 30 epochs). Random cropped images with the size of $992\times
992$ pixels were used to train the models in the experiments. In the inference
phases, all test datasets except CPM-15, test images were white-padded to form
$1024\times 1024$ images and then sent to the train models. The original part
was then cropped from the results. For the CPM-15 dataset, all images were
white-padded to $1056\times 1056$ pixels, as one of the images in the CPM-15
dataset has a larger size than $1024$ in one dimension ($1032\times 808$
pixels). Moreover, Adam optimizer [14], batch size of four, and a threshold of
0.5 (to convert probabilities to binary values) were used in training and
testing. All models were trained using a single workstation with an Intel Core
i7-8700 3.20 GHz CPU, 32 GB of RAM, and a TITIAN V Nvidia GPU card with 12 GB
of installed memory. The DDU-Net was trained and evaluated using Tensorflow
(version 2.11) DL framework.

##  3 Results and Discussion

In this section, we report the results for each test dataset in a separate
table. Each table contains the results based on different experimental setups
explained in Section 2.6 including baseline segmentation results (first row),
results from offline normalization technique (second row), results from
extended offline normalization (third row), and the results from the proposed
method with or without TTSN (fourth and fifth rows).

The results in Table.2 are derived from the MoNuSeg test data. Although the
test data and train data came from the same distribution (MoNuSeg dataset) in
this experiment, the results indicate a superior segmentation performance of
the proposed method (with or without TTSN) in comparison to the baseline
results. The proposed method with TTSN also delivered better semantic and
instance segmentation performance compared to offline normalization of
extended offline normalization approaches. Comparing the fourth and fifth rows
shows that using TTSN slightly improves the segmentation performance for all
evaluation indices.

The results of Table 3 to Table 8 show the generalization capability of the
models in different experimental setups for unseen test datasets. A number of
observations can be inferred from the results of these tables.

First of all, for all test datasets, the proposed method (with or without
TTSN) consistently delivers better segmentation performance in comparison to
the baseline model for all evaluation metrics.

Secondly, for most cases, adding TTSN to the workflow (i.e., comparison
between the fourth and fifth rows in the tables) improves the segmentation
performance in most test datasets (5 out 7 based on Dice score, 6 out of 7 for
AJI, 4 out of 7 for PQ score) and delivers very competitive performance in
other cases. However, it is worth mentioning that adding TTSN to the workflow
would increase the test time by eight folds (instead of sending one single
test image to the models, eight images have to be sent to the model as shown
in Fig. 5). This could be a barrier of using TTSN when limited computational
resources are available to analyze whole slide histological images. However,
with proper computational resources and parallel processing, this should not
be an issue.

Thirdly, for most cases (all except the CryoNuSeg dataset), the proposed
method with TTSN delivers superior semantic and instance segmentation
performance compared to other normalization methods discussed in this study
(offline normalization and extended offline normalization).

Table 2: Segmentation results for the MoNuSeg test data. Method | Dice (%) |  AJI (%) | PQ (%)
---|---|---|---
No normalization (baseline) | 82.84 | 65.76 | 63.40
Offline normalization | 83.24 | 66.10 | 63.29
Extended offline normalization | 83.10 | 66.65 | 64.21
Non-det. stain normalization (proposed) | 82.96 | 66.47 | 64.25
Non-det. stain normalization + TTSN (proposed) | 83.32 | 66.67 | 64.56
Table 3: Segmentation results for the TNBC test data. Method | Dice (%) |  AJI (%) | PQ (%)
---|---|---|---
No normalization (baseline) | 80.57 | 62.72 | 60.53
Offline normalization | 79.27 | 61.56 | 58.47
Extended offline normalization | 78.43 | 60.92 | 60.30
Non-det. stain normalization (proposed) | 81.02 | 64.51 | 62.51
Non-det. stain normalization + TTSN (proposed) | 81.22 | 64.59 | 61.81
Table 4: Segmentation results for the CryoNuSeg test data. Method | Dice (%) |  AJI (%) | PQ (%)
---|---|---|---
No normalization (baseline) | 78.71 | 50.76 | 47.20
Offline normalization | 78.45 | 50.61 | 47.24
Extended offline normalization | 79.17 | 52.20 | 48.95
Non-det. stain normalization (proposed) | 78.93 | 52.01 | 49.03
Non-det. stain normalization + TTSN (proposed) | 78.87 | 51.97 | 48.89
Table 5: Segmentation results for the CPM-15 test data. Method | Dice (%) |  AJI (%) | PQ (%)
---|---|---|---
No normalization (baseline) | 73.80 | 53.23 | 49.28
Offline normalization | 73.80 | 52.84 | 48.50
Extended offline normalization | 73.03 | 52.79 | 48.72
Non-det. stain normalization (proposed) | 74.68 | 53.68 | 49.39
Non-det. stain normalization + TTSN (proposed) | 75.10 | 54.07 | 50.02
Table 6: Segmentation results for the CPM-17 test data. Method | Dice (%) |  AJI (%) | PQ (%)
---|---|---|---
No normalization (baseline) | 81.31 | 62.52 | 58.08
Offline normalization | 81.83 | 63.31 | 58.33
Extended offline normalization | 80.31 | 61.82 | 57.64
Non-det. stain normalization (proposed) | 81.66 | 63.46 | 58.80
Non-det. stain normalization + TTSN (proposed) | 81.98 | 63.86 | 58.93
Table 7: Segmentation results for the ConSep test data. Method | Dice (%) |  AJI (%) | PQ (%)
---|---|---|---
No normalization (baseline) | 69.14 | 40.36 | 39.46
Offline normalization | 66.51 | 36.99 | 35.31
Extended offline normalization | 67.68 | 40.75 | 40.34
Non-det. stain normalization (proposed) | 72.78 | 44.12 | 43.80
Non-det. stain normalization + TTSN (proposed) | 71.54 | 42.97 | 42.00

The results in Table 8 show the segmentation performance for the NuInsSeg
dataset. NuInSeg is the largest test dataset used in this study, with 665
images that are derived from 31 different human and mouse organs. Therefore,
this dataset can be considered the most important dataset to show the
generalization capability of the proposed method, and hence we describe its
results separately. As the results show, we observe the same trend of superior
performance of the proposed method (both fourth and fifth rows) in comparison
to the baseline model. However, the difference between baseline and prospered
segmentation performance is more evident for the NuInsSeg dataset (5.77%,
5.36%, and 5.27% for Dice, AJI, and PQ score, respectively). The proposed
method with TTSN also outperforms the other applied normalization techniques
(1.95% for Dice, 1.75% for AJI, and 0.93% for PQ score, respectively for the
offline normalization method, and 5.92% for Dice, 6.01% for AIJ and 6.33% for
PQ score, respectively for the extended offline normalization strategy). For
qualitative comparison, we show sample segmentation results from the baseline
model and the proposed method in Fig. 7.

Table 8: Segmentation results for the NuInsSeg test data. Method | Dice (%) |  AJI (%) | PQ (%)
---|---|---|---
No normalization (baseline) | 65.47 | 41.74 | 34.88
Offline normalization | 69.29 | 45.35 | 39.22
Extended offline normalization | 65.32 | 41.09 | 33.82
Non-det. stain normalization (proposed) | 69.60 | 45.43 | 38.76
Non-det. stain normalization + TTSN (proposed) | 71.24 | 47.10 | 40.15
Original image | Ground truth | Base prediction | Proposed prediction
---|---|---|---
|  |  |
|  |  |
|  |  |
Figure 7: Qualitative comparison between baseline segmentation model and
proposed approach. The first column shows some example test images from the
NuInsSeg dataset (from the human bladder, human placenta, and human cerebellum
in the first, second, and third rows, respectively). The second column shows
the ground truth segmentation masks. The third column shows the prediction by
the baseline DDU-Net model, and the fourth column shows the results of the
proposed approach. The red bounding boxes in columns two to four show some
example nuclei where the proposed method delivered a superior segmentation
performance (better semantic segmentation performance in the first row and
better instance segmentation performance in the second and third rows)
compared to the baseline segmentation model.

There are some limitations in this study that can be addressed in future
works. First, while we propose a generalizable framework for the nuclei
instance segmentation task, we only used one of the state-of-the-art DL-based
models (DDU-Net) in our study. Although DDU-Net has shown excellent nuclei
instance segmentation performance [7, 22, 36], DDU-Net can be replaced by
other state-of-the-art segmentation models in future research. However, we
would like to emphasize that most other state-of-the-art DL-based models for
nuclei instance segmentation (such as triple U-Net, HoverNet, or attention
augmented distance regression model [43, 9, 5]) have similar encoder-decoder-
based architectures to the utilized DDU-Net model. Second, in this study, we
focused on the nuclei instance segmentation task, but the proposed framework
can be evaluated for nuclei detection or nuclei instance segmentation and
classification or nuclei detection tasks in the future as well.

##  4 Conclusion

While many ML-based and DL-based approaches have been proposed for nuclei
instance segmentation in histological images, their performance usually
degrades when tested on unseen new images. We proposed a framework for
generalized nuclei instance segmentation with non-deterministic train time and
deterministic test time stain normalization. Applied on seven independent test
datasets, the results showed the superior performance of the proposed method
compared to the baseline segmentation model. Therefore, the proposed approach
can be considered a generalized framework for nuclei instance segmentation.

####  4.0.1 Acknowledgements

we thank NVIDIA corporation for the generous GPU donation. this project was
partially supported by the Austrian Research Promotion Agency (FFG), No.
872636.

####  4.0.2 Competing interests

The authors declare no competing interests.

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# Thermal transport in ultrathin Si nanowires: a first principles study

Konstanze R. Hahn<EMAIL_ADDRESS>Department of Physics,
University of Cagliari, Cittadella Universitaria, 09042 Monserrato (CA), Italy
Claudio Melis  Department of Physics, University of Cagliari, Cittadella
Universitaria, 09042 Monserrato (CA), Italy  Fabio Bernardini  Department of
Physics, University of Cagliari, Cittadella Universitaria, 09042 Monserrato
(CA), Italy  Lorenzo Paulatto  Sorbonne Unversité, CNRS, Institut de
Minéralogie, de Physiques des Matériaux et de Cosmochimie, IMPMC, UMR 7590, 4
place Jussieu, 75252 Paris Cedex 05, France  Luciano Colombo  Department of
Physics, University of Cagliari, Cittadella Universitaria, 09042 Monserrato
(CA), Italy

###### Abstract

Phonon properties of small Si nanowires in [110] direction have been analyzed
using density functional perturbation theory. Several samples with varying
diameters ranging from 0.38 to 1.5 nm have been calculated. It is found that
the frequency of optical phonons at the zone center tend to decrease with
increasing size of the nanowire. Investigation of the phonon scattering rates
has revealed very high values in the smallest sample which decrease with
increasing nanowire size. A remarkable change in scattering rates is shown for
increasing diameter from 0.53 and 0.78 nm to 0.86 nm. The higher phonon
scattering could be attributed to an alignment of phonon modes at a specific
frequency. Results of the thermal conductivity are lower with respect to bulk
Si and are found between 15 and 102 W/mK. A trend of increasing thermal
conductivity with increasing diameter can be observed. This effect is
attributed to several changes in the phonon dispersion which are not
necessarily correlated to the wire size. These explicit results have been
compared to the thermal conductivity when boundary effect is approximated with
Casimir scattering. The Casimir method substantially underestimates the
results for explicit nanowires.

Silicon nanowires have been extensively investigated [1] since they represent
an excellent test bed to study the fundamental physics of low-dimensional
systems where confinement plays a major role. In addition, their abundance and
physical properties make them promising candidates for numerous next-
generation technologies [2, 3]. In this perspective, their thermal transport
properties represent a key issue for many applications [4]. For instance,
reducing the material dimensions to the nanoscale can lower the thermal
conductivity and increase the thermoelectric efficiency. Control of thermal
transport is of great importance as well in photovoltaic and microelectronic
devices [5, 6, 7] where a high thermal conductivity is required. In addition,
Si nanowires have also been studied as potential materials for thermal diodes
and transistors [8].

In all applications a detailed description of the heat transport is crucial,
in particular, for very small dimensions where bulk-like approximations are
questionable. However, despite the large number of studies, no consensus has
yet been reached on the thermal conductivity in ultrathin Si nanowires,
neither of absolute values nor of the overall size-dependent behavior.
Experimental studies have shown a decrease of thermal conductivity with
decreasing nanowire size reporting values from 10 to 40 W/mK for Si nanowires
of size from 20 to 100 nm [9, 10, 11]. A drastic reduction has been found down
to 5 and 1.2 W/mK, respectively, if such nanowires are corrugated [11] or have
rough surfaces [10].

From a theoretical point of view, several studies have been performed based on
molecular dynamics (MD) simulations. Using the Green-Kubo formalism and the
Stillinger-Weber potential, MD calculations have shown a steady decrease of
the thermal conductivity down to about 1 W/mK with decreasing wire size from 5
to 1.6 nm at 300 K [12]. Conflicting results, however, have been found: other
authors using the same formalism but the Tersoff potential have reported a
thermal conductivity larger than 45 W/mK and the opposite trend for a size
reduction from 2 to 1.1 nm [13]. Similar size effects have been obtained using
generalized tight-binding molecular dynamics simulations together with the
Stillinger-Weber potential [14]. It is unclear, however, where the discrepancy
between different studies using similar methods originates from. Volz et al.
[12] and Donadio et al. [13] used the Green-Kubo formalism but different
interatomic force potentials. Instead, Ponomareva et al. [14] applied the same
force field (Stillinger-Weber) as Volz et al. [12], but they calculated the
thermal conductivity based on tight-binding simulations.

Different results have been obtained by Wang et al. [15] who used accurate
phonon dispersions [16] to obtain the thermal conductivity. Their results show
a steady decrease of thermal conductivity with decreasing size down to 1 nm
and 0.33 W/mK similar to Volz et al. [12] and good agreement with experimental
results [17] for larger nanowires.

This rather confusing scenario is graphically captured in Fig. 1 and has been
addressed recently by Zhou et al. [18]. Their study is again based on Green-
Kubo MD simulations exploring different force fields. They find a similar
trend of increasing thermal conductivity with decreasing size for very small
nanowires in all of their calculations, but the absolute values vary
remarkably (up to one order of magnitude) with the use of different force
fields. With such huge differences the reliability of these methods is
questionable, in particular, since the adopted potentials have previously
shown to approximate other properties reasonably well [19], however, leaning
towards an overestimation of the thermal conductivity [20, 21, 22].

Figure 1: Thermal conductivity in small Si nanowires obtained by various
computational approaches. Grey symbols are from MD simulations using the
Green-Kubo approach and different force potentials [18], black crosses have
been obtained by Donadio et al. using a similar method and the Tersoff
potential [13]. The black broken line (without symbols) shows data from Wang
et al. where the thermal conductivity has been obtained from integration over
all phonon branches [15] and black downward triangles resulted as well from MD
calculations using the Stillinger-Weber potential [12]. The inset is an
amplification for better visualization of the results from references [15] and
[12].

This state of affairs questions the reliability and accuracy of the results
published so far. On the other hand, experimental data does not (yet) exist
for Si nanowires of such small dimensions. Accordingly, a detailed study and
understanding of the mechanisms directing the thermal conductivity in
ultrathin nanowires is of great potential and can possibly predict structures
with tailored thermal properties.

Motivated by this, we have developed a detailed study on ultrathin Si
nanowires, explored by first-principles calculations. Density functional
theory (DFT) calculations have been applied to obtain the ground state of Si
nanowires in [110] direction with diameters ranging from $d=$0.38 to 1.5 nm.
Phonon dispersions have been calculated using density functional perturbation
theory (DFPT). By further calculating the third-order interatomic force
constants, anharmonic scattering terms have been evaluated, eventually leading
to phonon scattering rates and the thermal conductivity for nanowires up to
$d=$0.86 nm.

Si nanowires with very small diameters have been generated using a lattice
spacing of $a$=5.3803 Å in the direction of the wire axis, corresponding to
the value optimized for bulk Si with the computational settings used here [23,
24]. The wire axis has been set to the [110] direction which has been shown to
be favored in various fabrication processes, in particular for small sizes
[25, 26, 27]. Si atoms at the surface of the nanowire have been saturated with
hydrogen atoms. The size of the generated samples range from 0.38 to 1.5 nm
corresponding to a total of 14 (6 Si, 8 H) to 62 (42 Si, 20 H) atoms in the
unit cell, respectively. Calculation of the thermal conductivity was
computationally feasible only for samples with a diameter up to 0.86 nm (28
atoms). The cross-section of theses samples are shown as insets in Fig. 4.

All calculations have been carried out with the Quantum Espresso program
package [28, 29], which, including its plug-ins, provides all elements
necessary for the calculation of the third-order interatomic force constants
(IFC) and related phonon scattering rates providing the lattice thermal
conductivity [30, 31, 32]

$\kappa_{\mathrm{L}}^{\alpha\beta}=\frac{\hbar^{2}}{N_{0}\Upomega
k_{B}T^{2}}\sum\limits_{\mathbf{q}j}c_{\mathbf{q}j}^{\alpha}c_{\mathbf{q}j}^{\beta}\omega_{\mathbf{q}j}^{2}n_{\mathbf{q}j}\left(n_{\mathbf{q}j}+1\right)\tau_{\mathbf{q}j}$
(1)

where $c_{\mathbf{q}j}^{\alpha,\beta}$ are the branch-depending components of
phonon group velocities in Cartesian coordinates $\alpha$ and $\beta$,
$\Upomega$ is the unit cell volume, $k_{B}$ is the Boltzmann constant and
$\omega_{\mathbf{q}j}$, $n_{\mathbf{q}j}$ and $\tau_{\mathbf{q}j}$ are the
vibrational frequencies, the occupation and the relaxation time, respectively,
of the corresponding phonon mode. The relaxation time $\tau_{\mathbf{q}j}$ is
obtained as the inverse of the sum of all considered scattering terms. For the
calculation of scattering terms in explicit nanowire samples, only anharmonic
scattering rates are considered (boundary scattering is implied in the
explicit geometry of the samples). A detailed description of the method has
been provided previously [31, 24]. The volume $\Upomega$ has been adjusted in
order to represent the volume of the nanowire without the vacuum present in
the unit cell.

The local density approximation and the PZ (Perdew-Zunger) [33] functional
have been applied to describe exchange and correlation interactions together
with norm-conserving Trouiller-Martins type pseudopotentials [34] for the
description of interactions between the frozen cores. Electronic ground state
calculations have been carried out using a 1x1x8 k-point grid. Phonon
dispersion relations have been obtained using the PH module [35] of Quantum
Espresso with a 1x1x8 $q$-point grid in the direction of the axis of the
nanowire which has been verified to be sufficient for converged results.
Calculation of the scattering mechanisms and the thermal conductivity has been
carried out using the D3Q plug-in [31] which provides the possibility to
evaluate third-order interatomic force constants from density functional
perturbation theory [36, 37, 38, 39, 40]. For the determination of scattering
terms, a 1x1x120 grid has been applied and the smearing parameter for energy
conservation has been set to 1 cm-1.

In the one-dimensional structure of a nanowire there are four acoustic modes
which can be assigned to one rotational mode, one longitudinal mode and two
bending modes. The rotational mode, which corresponds to a torsion of the
nanowire, and the longitudinal mode, which corresponds to a displacement along
the wire axis, have a linear dispersion in the vicinity of $\Upgamma$. The
other two modes, corresponding to displacements orthogonal to the wire axis
(flexural modes), are quadratic close to $\Upgamma$.

Fig. 2 shows the phonon band structure of a Si [110] nanowire with a diameter
of 0.86 nm in comparison to the band structure in bulk Si in the corresponding
direction. For better visualization we show phonon frequencies only up to 8
THz, keeping in mind that low frequency phonons are dominant in thermal
transport in such systems.

Figure 2: Phonon band structure of Si [110] nanowires (red lines) with
$d=$0.86 nm in comparison to the phonon band structure in bulk Si (blue lines)
in the corresponding direction.

Vibrational frequencies of acoustic modes in the nanowire are lower with
respect to bulk Si and show good agreement to previous computational studies
[41, 18]. Acoustic modes at the zone edge are two-fold degenerate at a
frequency of 2.5 and 3 THz, respectively.

If dimensions are above a certain threshold (around 1-2 nm), it is commonly
agreed that a reduction of characteristic dimensions of the material reduces
its thermal conductivity. This effect has been observed for nanoparticles [42,
43], porous structures [44], thin films or superlattices [45, 46, 47] and
nanowires [12, 9, 15, 48]. Recent theoretical studies, however, have shown the
opposite trend for very small wire dimensions ($<2$ nm) [18, 49] which has
mainly been attributed to an increase in frequency of the lowest optical modes
at the zone center with decreasing wire diameter which reduces the available
channels for Umklapp scattering at low frequencies.

This behavior in the phonon dispersion has been verified as shown in Fig. 3
for diameters from 0.38 to 1.5 nm. Red arrows indicate the position of the
lowest optical frequency at the zone center. In particular, it is evident for
an increase in diameter from 0.86 to 1.5 nm where the zone center optical
frequency is found to decrease from 2.2 to 1.4 THz, respectively. It should be
noted that negative frequencies in particular for the two smallest samples
occur since the structures at the bulk lattice spacing of $a$=5.3803 Å in wire
direction is not fully relaxed. Ill-defined modes disappear when the lattice
spacing in wire direction is increased. For a consistent comparison, however,
we chose the bulk lattice spacing for all samples. Negative phonon modes have
been ignored for the calculation of the thermal conductivity.

Figure 3: Phonon dispersion of Si nanowires in [110] direction with an average
diameter ranging from 0.38 to 1.5 nm. Red arrows indicate optical phonons with
the lowest energy at $\Upgamma$ and the blue horizontal line in b) and c)
highlight the alignment of various phonon modes at the same frequency.

Explicit calculation of the phonon dispersion in nanowires and subsequent
calculation of scattering terms using DFPT, as it has been done here, is
computationally highly demanding. The number and diameter of the calculated
samples had therefore been reduced to a feasible number with the largest
diameter of 0.86 nm.

It was expected to find less scattering channels with decreasing nanowire
diameter as a result of the higher optical bands as has been described above
based on recent studies [18, 49]. However, the contrary has been found for
decreasing diameters from 0.86 to 0.38 nm. An explanation can be derived from
the phonon dispersion. As evidenced by the blue horizontal line for $d$=0.53
and 0.78 nm (Fig. 3), the lowest optical band is almost dispersionless and
crosses with acoustic modes at about 0.2 $\pi/a$. The alignment of different
phonon modes at the same frequency and the crossing of acoustic and optical
modes gives rise to enhanced scattering channels. This effect is demonstrated
in Fig. 4 where the scattering rates are shown as a function of the phonon
frequency for Si nanowires of different size. Enhanced scattering events are
observed at this specific frequency (highlighted with a blue vertical line).
In nanowires with $d$=0.53 and 0.78 nm this frequency is found at $\sim$2.2
and 1.8 THz, respectively.

Figure 4: Phonon scattering rates of acoustic (crosses) and optical (circles)
modes in Si [110] nanowires with an average diameter of 0.38 nm, 0.53 nm, 0.78
nm and 0.86 nm. Red arrows indicate optical phonons with the lowest energy at
$\Upgamma$ and the vertical blue lines highlight the frequency where phonon
modes shown notable alignment (see Fig. 3). Insets show the cross-section of
the corresponding sample.

Very high scattering rates (up to 11 1/ps) are present in the smallest sample
with $d$=0.38 nm. No particular phonon frequency alignment of phonon modes or
crossing of optical and acoustic modes is observed in contrast to the samples
with $d$=0.53 and 0.78 nm. However, a peak in scattering is present around 1.5
THz which coincides with an overlap of acoustic modes at the zone border
($>0.5\pi/a$).

Based on the phonon dispersions and the scattering rates described above, the
thermal conductivity has been calculated for diameters ranging from $d$= 0.38
to 0.86 nm. For these diameters, the thermal conductivity is found to increase
from 15 to 102 W/mK (Fig. 5).

This is in contrast to previous studies, that showed an increase in thermal
conductivity with decreasing size in ultrathin nanowires [18, 49]. These two
studies attributed the behavior to a decrease in Umklapp scattering as a
result of increasing optical bands with decreasing diameter as described
above. The decrease in Umklapp scattering leads to dominating Normal
scattering processes which promote hydrodynamic phonon flow, resulting in an
increase in thermal conductivity. However, it should be noted that diameters
of the nanowires calculated here are below the smallest size of the previous
studies. In fact, the thermal conductivity of the 0.86 nm sample, is in
reasonable agreement to Donadio et al. [13] (see inset of Fig. 5). As has been
discussed above, additional and increased scattering is found in the smallest
samples of our study which supposedly dominate over Normal scattering events
and thus suppressing hydrodynamic flow. Given our results of the thermal
conductivity, we suggest that the largest sample ($d$=0.86 nm) lies in the
regime of hydrodynamic flow while in all other samples additional scattering
events are dominant thus reducing again the thermal conductivity. Larger
systems, which could support this explanation, were not feasible for the
calculation of the thermal conductivity within the DFPT method.

In order to estimate the effect of boundary scattering, the calculated thermal
conductivity is compared to results using the Casimir model [50] (Fig. 5).
Within the Casimir approximation, anharmonic phonon scattering is obtained
from the bulk phonon structure, while size effects are described as additional
boundary scattering terms. As shown in Fig. 5, the Casimir method notably
underestimates the thermal conductivity calculated in explicit nanowires. This
confirms that the sole consideration of boundary scattering is not capable of
describing all effects that contribute to thermal transport in low-dimensional
systems. Instead, an accurate representation of the phonon dispersion is
necessary to be able to describe effects from anharmonic phonon interactions.

In fact, the decrease of thermal conductivity for increasing diameter for 0.53
to 0.78 nm can be explained by the lower frequency of the dispersionless
optical mode which gives rise to increased scattering channels (see Fig. 3).
On the other hand, the phonon dispersion in the $d$=0.86 nm sample does not
show an alignment of modes at a certain frequency and the frequency of the
lowest optical modes at the zone center is higher with respect to the 0.78 nm
sample, indicating the presence of hydrodynamic flow and resulting in a
remarkable increase in the thermal conductivity to 102 W/mK.

Figure 5: Thermal conductivity in ultrathin Si nanowires in [110] direction
calculated explicitely (red diamonds) in comparison to results using the
Casimir model (green broken line) where only boundary scattering is
considered. The inset shows literature values in comparison to the largest
sample studied here ($d$=0.86 nm). For legend and references of the inset see
Fig. 1.

In summary, ultrathin Si nanowires and their phonon characteristics have been
investigated using density functional perturbation theory. The phonon
dispersion is discussed for nanowires in [110] direction with a size ranging
from 0.38 to 1.5 nm. A decrease in frequency for optical modes at the zone
center is observed with increasing size.

In nanowires with $d=$0.53 and 0.78 nm, alignment of phonon modes at a certain
frequency is observed which results in increased scattering channels at this
frequency and subsequently to a decrease in thermal conductivity with respect
to the sample with $d$=0.86 nm where such alignment is not present.

As general trend it has been found that the thermal conductivity decreases
with decreasing nanowire size, which can partly be attributed to such
alignment of phonon modes. However, it is shown that small changes in the
geometry of ultrathin nanowires can lead to drastic changes in the phonon
dispersion which eventually alter the scattering mechanisms on various levels.
For an accurate description of thermal transport properties in ultra-thin
nanowires it is therefore crucial to rely on an explicit and correct
description of the phonon dispersion. This conclusion implies that thermal
properties in such ultrathin nanowires cannot be attributed solely to the wire
size but are highly affected by the geometry and the resulting phonon
dispersion.

###### Acknowledgements.

This work is financed by Ministero dell’Università e Ricerca (MIUR) under the
Piano Operativo Nationale 2014-2020 asse I, action I.2 ”Mobilità dei
Ricercatori” (PON 2014-2020, AIM), through project AIM1809115-1.

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