#  One City, Two Tales: Using Mobility Networks to Understand Neighborhood
Resilience and Fragility during the COVID-19 Pandemic

Hasan Alp Boz  Sabanci University, Faculty of Engineering and Natural
Sciences, Istanbul, 34956, Turkey  These authors contributed equally to this
work. Authors are listed alphabetically.  Mohsen Bahrami  Massachusetts
Institute of Technology, Institute for Data, Systems, and Society, Cambridge,
MA 02139, USA<EMAIL_ADDRESS>These authors contributed equally to this
work. Authors are listed alphabetically.  Selim Balcisoy  Sabanci University,
Faculty of Engineering and Natural Sciences, Istanbul, 34956, Turkey  Burcin
Bozkaya  New College of Florida, The Graduate Program in Applied Data Science,
5800 Bay Shore Rd, Sarasota, FL 34243, USA  Nina Mazar  Boston University,
Questrom School of Business, Boston, MA 02215, USA  Aaron Nichols  Boston
University, Questrom School of Business, Boston, MA 02215, USA  Alex Pentland
Massachusetts Institute of Technology, Institute for Data, Systems, and
Society, Cambridge, MA 02139, USA

###### Abstract

What predicts a neighborhood’s resilience and adaptability to essential public
health policies and shelter-in-place regulations that prevent the harmful
spread of COVID-19? To answer this question, in this paper we present a novel
application of human mobility patterns and human behavior in a network
setting. We analyze mobility data in New York City over two years, from
January 2019 to December 2020, and create weekly mobility networks between
Census Block Groups by aggregating Point of Interest level visit patterns. Our
results suggest that both the socioeconomic and geographic attributes of
neighborhoods significantly predict neighborhood adaptability to the shelter-
in-place policies active at that time. That is, our findings and simulation
results reveal that in addition to factors such as race, education, and
income, geographical attributes such as access to amenities in a neighborhood
that satisfy community needs were equally important factors for predicting
neighborhood adaptability and the spread of COVID-19. The results of our study
provide insights that can enhance urban planning strategies that contribute to
pandemic alleviation efforts, which in turn may help urban areas become more
resilient to exogenous shocks such as the COVID-19 pandemic.

## Introduction

Mobility in urban and metropolitan settings is the result of the dynamic
interaction, over space and time, of a large number of agents with diverse
goals and characteristics. Understanding mobility patterns is crucial for
predicting future social and economic well-being as well as growth
[chong2020economic, singh2015money], as it has been shown to reflect social
interactions [bettencourt2013origins, buera2020global], productivity
[sveikauskas1975productivity, bettencourt2007growth], and economic prosperity
and resilience [alvarez2013idea, steele2017mapping, pan2013urban,
schlapfer2021universal].

The dynamic interactions in mobility data are complex and are typically
represented by network structures. However, despite the ability for network
analyses to yield unique and critically important insights [lazer2009life],
researchers have yet to use network analyses to explore how the COVID-19
pandemic and the shelter-in-place and social distancing policies that it
triggered – two of the most effective non-pharmaceutical interventions (NPI)
aimed at reducing the spread of the Corona virus [flaxman2020estimating,
kraemer2020effect] (for a differing conclusion, see a recently published
article by Berry et al.[berry2021evaluating]) – influenced human mobility and
interaction patterns. Such interventions directly impact the human mobility
network by reducing levels of mobility and interactions among individuals and
points of interest (POIs) such as restaurants or supermarkets
[aleta2020modelling, schlosser2020covid]. The decline in movement activities
during the current COVID-19 pandemic has negatively impacted economies all
around the world, and these economic repercussions are predicted to continue
for years [chetty2020economic].

Although network analysis approaches are well suited to investigate the
multifaceted impact that policy changes have on mobility outcomes, early
studies have examined the short-term impacts of the COVID-19 pandemic on
various unidimensional mobility metrics such as travel distance, visit
patterns, and dwelling times in different countries and cities
[gao2020mapping, galeazzi2020human], or used mobility networks along with
epidemiological models to understand the effects of changes in mobility
patterns on the spread of the Corona virus [chang2021mobility,
aleta2020modelling]. Indeed, there is a lack of research investigating how the
COVID-19 pandemic and corresponding interventions impacted the structure of
mobility networks (e.g. the centrality metrics at a node and network level).
For example, investigating the _betweenness_ centrality metric may reveal the
role certain neighborhoods played during the pandemic, such as acting as a
bridge among the nodes in a mobility network, thereby producing a spreader
effect.

Given the close link between changes in mobility patterns and economic
outcomes, it is critical that research investigates how various environmental
and demographic factors have influenced the adaptability of mobility networks
during the COVID-19 pandemic. Using network science methodologies, the current
research aims to help scientists and policy-makers understand the factors that
contribute to economic resilience and adaptability in the wake of economic
shocks. In particular, we extend previous research about the impact of the
COVID-19 pandemic on human behavior by examining changes in mobility patterns
using a dynamic network analysis on one of the most important economic hubs in
the world: New York City (NYC). We create weekly mobility networks, from
January 2019 to December 2020, in which nodes represent neighborhoods (i.e.
Census Block Groups or CBGs), and the edges between them correspond to the
visitors from the source neighborhood to POIs such as restaurants or
supermarkets in the target neighborhood. For each neighborhood in the weekly
networks, we compute node and ego-network based features [netsimile] so the
resulting feature vectors not only capture the dynamics of local mobility but
also the relationship with the neighboring CBGs, allowing us to incorporate
the complexity of mobility into our analyses. We investigate the dissimilarity
of the resulting feature vectors for each neighborhood, between the same weeks
of 2019 and 2020, and break the results down by different socioeconomic
groups. Combining the mobility network metrics with data from various sources,
including census data and COVID-19 test results, we are able to reveal how
differences in NYC neighborhood characteristics predict dynamic structural
changes of mobility networks and behavior.

The results of our study indicate that the centrality metrics and geographic
attributes significantly predict neighborhood adaptability to shelter-in-place
orders. In addition to confirming the results of previous research
[chang2021mobility, chetty2020economic, hunter2021effect, heroy2021covid], our
findings reveal that not only are race, education, and income important
factors in predicting neighborhood adaptability to shelter-in-place orders,
but so are geographical attributes such as access to diverse amenities that
satisfy community needs. This indicates that in the same city, communities
with similar socioeconomic and demographic features may have different
mobility responses based on their neighborhoods’ urban structure.

Using the information extracted from the mobility network structure, we study
the case of COVID-19 hotspots to investigate which neighborhoods act as the
COVID-19 bridges among the hotspots and other neighborhoods, and uncover the
associated factors. We then utilize the well-known Huff gravity model to
perform a hypothetical scenario analysis to show how the higher levels of
access to essential businesses (e.g. grocery stores) could reduce the
interaction among the COVID-19 hotspots and other neighborhoods that could
potentially lead to a reduction in the infection rates and save more lives.
These novel results provide significant insights and recommend policies that
can enhance urban planning strategies that contribute to pandemic alleviation
efforts, which in turn may help urban areas become more resilient to exogenous
shocks such as the COVID-19 pandemic that restrict movements and interactions.

## Results

To demonstrate the impact of the COVID-19 pandemic on different socioeconomic
groups, we first analyze the change in network topologies in a weekly
resolution at the neighborhood level. To this end, we extract the node-level
feature vectors summarizing the statistical properties of their respective
ego-networks [netsimile]. The resulting node feature vectors are used to
compute the dissimilarities between paired weekly networks from 2019 and 2020.
Then, we illustrate the course of centrality metrics with respect to different
demographic groups and highlight their variability. Next, we analyze the
possible COVID-19 bridges, neighborhoods that frequently interact with
COVID-19 hotspots (CBGs with higher numbers of infected residents), by
focusing on the in- and out-going edges between CBGs over two-week periods,
which is considered as the virus incubation time [chang2021mobility]. Finally,
utilizing the Huff Gravity Model, we analyze the mobility in Staten Island
using hypothetical Grocery Store densities in order to observe the change in
visits to hotspot CBGs that frequently appear in the top new COVID-19 cases
quartile.

### Demographic Disparities: Temporal Changes in Mobility Networks

#### CBG-level dissimilarity analysis

We compute the node-level dissimilarity scores between paired weekly networks
of 2019 (pre-pandemic) and 2020 (pandemic) using the extracted ego-network
feature vectors (the components of these feature vectors and dissimilarity
score formulation are explained in the Methods section). The CBGs are then
ranked with respect to their dissimilarity scores at each time step (week). To
demonstrate the differences between the CBGs with distinct behaviors, we focus
on the CBGs in the top and bottom dissimilarity quartiles, and create two
cohorts of CBGs that frequently appear in those quartiles during the first
wave of the pandemic between March and June 2020. Figure 1 shows the spatial
and socioeconomic distribution of the resulting cohorts of CBGs that appear in
at least 60% of the time steps in the top and bottom quartiles. This threshold
(60%) is the highest frequency that yields a similar number of CBGs in each
group, enabling a better comparison between them. CBGs that ended up changing
their mobility pattern the most (i.e. the top dissimilarity quartile) are
primarily located in the Manhattan borough, the financial center of NYC. From
all CBGs in this cohort, 63% of them rank in the top quartile for income, 79%
in the top education quartile, 62% in the top white population percentage
quartile, and 52% in the bottom quartile for commute time, meaning they either
do not travel relatively long distances to get to their workplace or are
located in areas that have greater access to fast and frequent transportation.
There is no evident socioeconomic profile for the bottom dissimilarity
quartile (the CBGs that did not change their mobility patterns much), although
the distributions in quartiles delineate the residents to some extent.
However, there exists a decreasing trend from bottom to top quartiles in
income, education, and white population percentage.

Figure 1: The socioeconomic distribution of the census block groups (CBGs)
that changed their mobility patterns the most in comparison to the previous
year in at least 60% of the time steps, versus the least. CBGs that ended up
changing their mobility patterns the most are primarily located in the
financial center of NYC. Note that there are only significant socioeconomic
characteristics for the top quartiles but not for the bottom.

#### Node Degree and Centrality Metrics

Centrality metrics help us examine a node’s role in the network, such as
influence and information diffusion [deville2016scaling,
miritello2011dynamical]. In the proposed mobility network, since each node is
a CBG, the centrality metrics highlight the CBGs that are noteworthy in terms
of the flow of masses.

In this context, the temporal changes in centrality metrics reveal the
interaction patterns between different socioeconomic communities, which
consequently indicate complex mobility behaviors from a network perspective.
To this end, we focus on basic node degree centrality metrics and analyze
their course of change in distinct demographic groups (i.e. CBGs in the top
and bottom quartiles). We use the node _betweenness_ to illustrate a CBG’s
importance based on its connections and position in the network. Furthermore,
we use _degree_ centrality metrics to reveal the weekly incoming and outgoing
visitors among CBGs. Lastly, we use a custom metric named _self-visit ratio_
to represent the fraction of visits to the POIs inside the home CBG.

_Betweenness_ : This centrality score measures how frequently a CBG appears
along the shortest paths in a network, and is the only node centrality metric
that demonstrates a significant difference between the selected demographic
groups. As displayed in Figure 2-A, CBGs in the top income quartile held a
higher betweenness value until the beginning of March 2020 (start of the
pandemic), meaning that they played a critical role in terms of bridging the
flow of masses. However, an abrupt decrease of betweenness in the top income
CBGs took place after the start of the pandemic, while less affluent CBGs
gained higher betweenness scores. That is, less affluent CBGs increasingly
acted as connectors among the nodes in the mobility network but only until
September 2020, when the economic activity revived. The same relationship can
also be observed when focusing on education levels. CBGs with lower education
levels had a higher betweenness score in the same time interval (displayed in
the Supplementary Information (SI) Figure 4).

Figure 2: The temporal change in (A) betweenness, (B) total-degree, and (C)
self-visit ratio metrics in the top and bottom income quartiles. The vertical
line segments show a 95% confidence interval.

_Degree_ : Node degree analysis indicates that income and education play a
significant role in distribution of degree centrality values as well. Affluent
CBGs were more successful at lowering their mobility compared to less affluent
neighborhoods as shown in Figure 2-B.

_Self-Visit Ratio_ : Another metric we defined to investigate the change in
visit patterns is called self-visit ratio. Self-visit ratio is the fraction of
the visits made by the residents of a CBG to their home CBG over all visits
paid by them. Figure 2-C displays the course of the self-visit ratio with
respect to the top and bottom income quartiles. From March to June 2020,
during the first wave and the most striking decline in mobility, CBGs in the
top income quartile had a higher rate of visits to the POIs inside their home
CBGs, while on the contrary, the residents of lower income CBGs displayed a
lower self-visit ratio. However, starting in June 2020, as the re-opening of
economic activity commenced, the disparity of self-visit ratio between income
groups is narrowed.

### Analysis of COVID-19 Hotspots, Bridge CBGs & the Case of Staten Island

As explained in the Methods section, we define the COVID-19 hotspots as those
CBGs that frequently appear among the CBGs with highest weekly new cases.
Additionally, we refer to the CBGs that have a high level of interactions with
hotspots in the beginning of the virus incubation time as COVID-19 bridge
CBGs. Examining the COVID-19 bridge CBGs may reveal invaluable insights for
policy makers and urban planners attempting to prevent the spread of new
infections and build cities that are resilient to future pandemics.

To this end, we first obtain the CBGs in the top weekly new cases quartile for
each time step $t$. Then, we create a list of CBGs that had edges to the
previously obtained CBGs with the highest cases in time step $t-2$ considering
a period of two weeks as the incubation time for new cases to surface
[chang2021mobility]. Subsequently, we apply a frequency analysis on the
possible bridges to check how often they were connected to CBGs with the
highest weekly cases, and finally, consider the CBGs in the $75^{th}$
frequency percentiles as bridges, for further analyses.

As demonstrated in Figure 3, the majority of the resulting CBGs in the
$75^{th}$ frequency percentile are comprised of those in the lower quartiles
for income and education and higher quartiles for commuting time. Yet the
spatial distribution of the potential bridges in the $75^{th}$ frequency
percentile in Figure 3 unveils the special case of Staten Island, where 85% of
the CBGs located in Staten Island appeared in the bridges. Moreover, as the
threshold value is increased to the $95^{th}$ frequency percentile, the
demographic features begin to display Staten Island’s presence. Figure 4 shows
the box plot of the COVID-19 bridge CBGs at a borough level. Additionally, the
results of our OLS regression analysis considering occurrence in the bridge
CBG set as a dependent variable and borough code as an independent variable,
showed the boroughs are significant in predicting occurrence of CBGs in the
bridge set (the regression analysis results are provided in SI). This is
counter-intuitive and does not align with the previous observations, because
48% of the CBGs in Staten Island are from high income quartiles and their
residents are mainly white. That is, the CBGs in Staten Island display a
distinctive behavior compared to CBGs in other boroughs that are similarly in
high income and high white population quartiles. This observation is
particularly noteworthy since Staten Island is geographically relatively
isolated with a low level of connectivity to other boroughs (connected through
bridges and ferry; and not connected with the NYC subway system). Thus, if
anything, one may conclude the opposite, that is, Staten Island would have
been more shielded from the pandemic. Therefore, we followed up with
additional analysis at the borough level to try to shed light on this
observation.

Figure 3: Spatial and demographic distributions of the CBGs in the 75th (top)
and 95th (bottom) frequency percentiles as COVID-19 bridges. Staten Island
clearly stands out. Figure 4: Box plot of the COVID-19 bridge CBGs by boroughs

#### Borough Level Analyses Results

As depicted in Figure 5 from January 2019 to January 2021, residents in Staten
Island always had the highest mobility (average visit count per smartphone
user) among all NYC boroughs, regardless of the COVID-19 pandemic and the rise
of different containment policies (e.g. shelter-in-place or physical/social
distancing).

Figure 5: Average weekly number of visits per user at a borough level

Since the Safegraph mobility data has limited coverage on workplaces and
offices, we use Google’s COVID-19 Community Mobility Reports [googlemobility]
to investigate the mobility trends for places of work. As shown in Figure 6
Staten Island has the minimum relative change in mobility trends for workplace
among all NYC boroughs, indicating that the residents of Staten Island reduced
their mobility and visits to POIs noticeably less than the residents of other
boroughs.

Additionally, the POIs analysis results show that Staten Island has the lowest
number and diversity of POIs among all boroughs of NYC, where the majority of
visits to POIs inside the city originating from Staten Island are made to
Brooklyn and Manhattan, which are the neighboring boroughs of Staten Island.
This observation is in line with a report by the NYC government’s planning
department [nycinsouts] documenting that 24% of workers residing in Staten
Island have their workplaces located in Manhattan.

Figure 6: Relative change in mobility trends for workplace by borough.

In summary, in Staten Island due to the shortage of POIs and its relatively
isolated location (e.g., just one automobile bridge to Brooklyn, three
automobile bridges to New Jersey, one free ferry to Manhattan, no subway),
which promotes using personal cars[nycmobilityreport, nycinsouts], the
residents had to visit POIs in other boroughs to accommodate their needs and
thus traveled longer distances to workplaces and the majority of POI
categories. We conclude that this is likely why we observe the CBGs in Staten
Island displayed a distinct response behavior compared to their demographic
counterparts (i.e. relatively high income and percent white population CBGs),
and that this is what led to higher infection rates in its neighborhoods.

### Hypothetical Scenario Analysis

In order to understand the mobility patterns of CBGs in Staten Island under
hypothetical POI distributions, we employ the Huff Gravity Model
[huff1964defining] focusing on the distance between CBGs and POIs, and POI
areas in square feet (more details are provided in SI). In addition, instead
of including all POI categories, we only incorporated grocery stores into the
model to narrow down the scope to the stores that provide the most essential
needs of human daily life. Table 1 provides information about the number of
grocery stores per 1K residents and the median distance travelled by residents
to visit grocery stores in kilometers at a borough level.

Borough Name | Grocery Stores per 1K Residents | Median Distance Travelled (in km)
---|---|---
Manhattan | 0.582 | 0.90
Brooklyn | 0.470 | 1.35
Bronx | 0.435 | 1.26
Queens | 0.414 | 1.71
Staten Island | 0.332 | 2.66
Table 1: Number of grocery stores per 1K residents and the median distance
travelled from home to grocery stores (from March 22nd and June 8th 2020) by
residents of NYC boroughs.

As shown in the table, Staten Island has the lowest number of grocery stores
per resident and highest distance travelled by residents to visit a grocery
store among all NYC boroughs. Our aim is to simulate the mobility patterns of
Staten Island residents with a higher POI density equal to that of other
boroughs with similar demographic traits. We choose the analysis time frame to
be during the first wave of the pandemic between March 22nd (pause program
begins) and June 8th (phase I of the reopening) of 2020. Since NYC enforced a
strict citywide lockdown policy during this time frame [birge2022controlling],
most of the workplaces were closed and only essential businesses like grocery
stores were allowed to operate. Therefore, we contend that the most trips to
grocery stores during that time frame could be considered single-purpose trips
rather than multi-purpose ones [lucchini2021living], and thus we are able to
use distance between the store and customer home location in the model.
Moreover, due to lowered mobility within the lockdown period, we employ
_census tract_ level visit patterns to capture broader mobility observations.
We treat each census tract as an individual mobility center and generate a
model on the aggregated mobility patterns within the specified time range so
that the probabilities of residents within a census tract visiting given POIs
are obtained.

We add randomly generated hypothetical grocery stores with area equal to the
mean store area within Staten Island and continue adding until the new grocery
per 1K residents match the grocery store densities of Manhattan and Queens,
boroughs with more similar demographics. We use the existing mobility data to
simulate visit numbers with respect to the previously obtained probabilities
for the hypothetical setting and obtain the synthetic visit numbers for all
POIs. We aggregate the visits in each census tract both for the ground truth
and simulated cases, and then analyze the change in visits by residents of
Staten Island to hotspot CBGs, that frequently appear in the top new cases
quartile. The simulation results show that the visits to hotspot CBGs
decreased by 47% and 23% by Manhattan and Queens’ grocery store densities,
respectively, which in turn means less exposure to possible COVID-19 spreaders
and lower risk of contamination and reduced mortality (for more details see SI
Notes 2).

## Discussion

In this research we build on the tradition of using network structures to
explain human behavior in socioeconomic settings by examining the complex
relationship between human mobility, socioeconomic outcomes, and demographic
attributes within the context of the COVID-19 pandemic. In particular, we take
a network analysis approach to understanding the impact of the COVID-19
pandemic on the mobility patterns of the residents of NYC within its five
boroughs and 6,493 census block groups (CBGs) during 2020. The network nodes
represent CBGs, and the links correspond to visits between pairs of CBGs using
visits to POIs by residents. We investigate node-specific and ego-network
based structural features to compute dissimilarity scores between weekly
networks year-on-year for 2019 and 2020 to understand the magnitude of change
in the network structure.

In addition, we analyze the temporal changes in node and degree centrality
measurements across different socioeconomic groups. Our approach and findings
reveal that while the COVID-19 response measures resulted in structural
changes in mobility network, the CBGs that changed their ego-network structure
the least had higher COVID-19 infection rates. The majority of such nodes are
from low income and low education level neighborhoods with higher rates of
front-line workers [nycfrontline] (e.g., workers in healthcare, grocery,
convenience and drug stores, child care, food and family services, public
transport, trucking, warehouse and postal services) who could not reduce their
mobility by as much. Arguably this is because of their job type that requires
working outside their home and commuting frequently.

Our CBG-level dissimilarity analysis suggests a clear demographic distinction
of residents who live in top-dissimilarity CBGs that appeared in more than 60%
of the weekly patterns we analyzed: high-income, higher education level, and
mostly white population. We argue that these are residents whose job types can
most suitably respond to shelter-in-place and physical/social distancing
orders through remote work, unlike the residents in neighborhoods with other
job types. Hence, in line with other studies [chetty2020economic,
chang2021mobility, hunter2021effect] we find that these top-dissimilarity CBGs
and their residents are more resilient under the conditions of a pandemic such
as COVID-19.

#### Two-sides of the same city

Using a network perspective, we extend the previous work by finding that the
adaptability of mobility patterns varies significantly not only by
socioeconomic and demographic features but also by geographical attributes of
neighborhoods. Quantifying the ego-network dissimilarity, we show that the
less affluent and less educated neighborhoods had less adaptability to policy
interventions aimed at reducing their mobility level. In addition, we find
that the neighborhoods with higher income and education can show a similar
behavior to less affluent and less educated neighborhoods if they have
relatively limited access to public transport, workplaces, shops, and a set of
other diverse amenities, as is the case with Staten Island. Even though there
exist very few physical connections between Staten Island and other boroughs
of NYC and New Jersey, which could conceivably mean that the existing mobility
network structure would be expected to break down when there is an extreme
exogenous shock, in contrast we find that the network changes were the
opposite: They were very minimal. Thus, Staten Island was relatively fragile
to the COVID-19 pandemic in terms of infections. Exploring similar
neighborhoods or isolated geographical blocks in other urban areas could be
the subject of future research to help policy makers develop effective
policies to alleviate the burden of a pandemic in such areas and increase
their resilience.

#### Opportunities for urban planning

Our study provides valuable insights for urban planning and policy: Both
socioeconomic as well as geographic characteristics of neighborhoods (the
physical structures of neighborhoods and communities in an urban area) are
important considerations when trying to increase neighborhoods’ resilience to
future exogenous shocks. With regards to the latter and based on our
simulation results, convenient access to POIs that provide daily essential
needs (e.g. grocery stores), workplaces, and centers of attraction that offer
a variety of amenities through a diverse set of POIs would be expected to
reduce the need for traveling long distances. Using a Huff Gravity Model we
performed a hypothetical scenario analysis to estimate the number of visits to
COVID-19 hotspot areas with hypothetically added grocery stores for each CBG
in Staten Island. Although we only consider customer-store distance and POI
floor area in our model, the results yield valuable insights on the simulated
mobility. Our findings indicate that as a result of increase in access to the
essential POIs, residents are less exposed to hotspot districts and are able
to satisfy their needs without traveling further distances. In cases like the
COVID-19 pandemic, increased level of access to POIs could potentially
decrease the level of infections and save more lives. As the continued
pandemic conditions reveal, these factors might be a major influence on future
work habits and trends, and resilient communities organized and structured
along these lines might be the preferred choice for many members of the
population. As a future research direction, we aim to extend our simulation
analysis by incorporating more POI categories with additional POI attraction
features such as a neighborhood’s physical characteristic into the model to
evaluate the results within different settings.

#### Limitations

Our study has limitations that arise mainly from the datasets used. The
mobility patterns and visits to POIs are aggregated by time (e.g., weekly) and
neighborhood (e.g., CBGs). While this method preserves the privacy of
smartphone users, we were not able, as a result, to capture the primary aim of
users’ trips to different destination CBGs (e.g., for work, shopping,
entertainment or other purposes). This additional level of detail would allow
us to distinguish between essential and non-essential trips made by residents
and match it with socioeconomic characteristics of neighborhoods. Originating
from the same limitation, we are only able to see which neighborhoods are
connected to one another in the form of visits and with what frequency, but
not how (i.e. mode of transportation) they are connected. While it might be
possible to infer mostly-used transportation modes and routes between CBGs
using other datasets, the resolution of our data does not allow us to link it
with individual trips, preventing us from conducting additional analysis on
what portion of trips are made in what modes and its relations with the
spreading of the virus and the resilience of communities.

Furthermore, the POI dataset we use in our study mostly includes places where
financial transactions take place (e.g. supermarkets, dentist offices,
restaurants, etc.) and has a low coverage of workplaces and offices (e.g.
corporate buildings, co-work spaces, etc.). The latter would, again, help us
identify additional types of essential travel (e.g., workplaces) and better
fit the gravity model for simulation purposes. We tried to address this
limitation partially by using complementary datasets such as Google’s mobility
report and the mobility survey results published by NYC officials.

Finally, despite all the limitations, our study makes at least two
contributions. First, our study results contribute substantive insights into
the heterogeneity of mobility patterns among different neighborhoods during
the COVID-19 pandemic through leveraging network science approaches and
quantitative scenario analysis. Second, our study illuminates the factors
associated with a neighborhood’s resilience and fragility. Understanding the
factors associated with a neighborhood’s resilience and fragility can help
urban planners and authorities recommend sustainable policies, make better
intervention decisions and be prepared to react more effectively to future
exogenous shocks like the COVID-19 pandemic and save more lives.

## Methods

### Datasets

#### SafeGraph Mobility and Places Data

The dataset released by _SafeGraph_ (www.safegraph.com) provides fine-grained
user geo-location information collected through the smartphone applications of
millions of users who consented to share their live location. The data is
anonymized and then aggregated by the total visits from CBGs to POIs of
various categories (e.g. restaurants, grocery stores, and supermarkets) within
a defined time period (i.e. weekly and monthly). More specifically, we use the
_Weekly Patterns_ [safegraph] dataset that provides the weekly aggregated
visits from January 2019 to December 2020. The patterns dataset is
geographically filtered with respect to the administrative boundaries of the
New York Metropolitan Area, which resulted in 6,463 CBGs (in New York City)
and 333,241 POIs in total.

#### Google COVID-19 Community Mobility Reports

This dataset is prepared by Google to provide insights into mobility trends in
order to help understand communities’ response to interventions against
COVID-19 pandemic. The data is collected from various smartphone applications
including Google Maps. The data shows dynamics of mobility trends by region,
across different POI categories compared to a baseline day that represents a
normal day before the pandemic. The baseline is the median value from the
five-week period from January 3rd to February 6th, 2020. The POI categories
include retail and recreation, groceries and pharmacies, parks, transit
stations, workplaces, and residential buildings.

#### COVID-19 Cases

The Center for Systems Science and Engineering (CSSE) at Johns Hopkins
University [Badr2021.05.05.21256712] provides a COVID-19 data catalog
presenting the latest updates all around the world with differing granularity
levels. In NYC, the statistics for variables such as new cases, test counts,
and deaths, are provided by ZIP Code Tabulation Areas (ZCTAs). A CBG may be
located inside the borders of multiple ZCTAs. In order to work with the
estimated weekly cases per CBG, the numbers were inferred by taking a weighted
average based on the ratio of population of a CBG in each ZCTA and its
corresponding COVID-19 case rate. Assume $S$ is the set of ZCTAs that
geographically intersect with $CBG_{i}$. Then the number of COVID-19 cases for
$CBG_{i}$ at time $t$ are estimated using the following formulations.

$w_{ij}=\frac{population\;of\;\;CBG_{i}\;\;in\;\;ZCTA_{j}}{population\;\;of\;\;ZCTA_{j}}$
(1)

$CCBG^{t}_{i}=\sum_{j\in S}(w_{ij}\times case^{t}_{j})$ (2)

$CCBG^{t}_{i}$ denotes the estimated COVID-19 cases in $CBG_{i}$ at time $t$,
and $case^{t}_{j}$ is the number of COVID cases in $ZCTA_{j}$ at time $t$.

#### The United States Census Data

The American Community Survey (ACS) of the U.S. Census Bureau reports the
estimates of demographic features at a CBG level. The most recently published
5-year ACS data collected in 2019, is utilized to extract the demographic
features, namely: total population, median household income, education level,
commuting time, and population distribution by race for each CBG. The
resulting features are also represented with corresponding percentile levels.

#### New York Metropolitan Area

In 2019, New York Metropolitan Area hosted the highest population at 19.22
million as one of the leading economies in the US with 1.522 Billion
Dollars[ny-statista]. The metropolitan area consists of counties from four
different states, New York, New Jersey, Connecticut, and Pennsylvania. In
total, 7,809 census tracts and 23 counties are represented. POIs provided by
SafeGraph Patterns are filtered by their location with respect to the New York
Metropolitan area. New York City (NYC) with about 9 million residents is the
largest city in the New York Metropolitan area containing 6,493 CBGs. NYC is
also divided into five administrative boundaries called boroughs. Those five
boroughs are: Manhattan, Brooklyn, Bronx, Queens, and Staten Island.

### Mobility Networks

We model the mobility patterns among CBGs as weighted directed networks,
$G^{(t)}\,=\,(V^{(t)},E^{(t)})$, for each time step $t$, in which the nodes
$V^{(t)}$ correspond to CBGs and edges $E^{(t)}$ capture the weekly number of
visits initiating from a CBG to another with weight $w_{ij}^{(t)}$
representing the number of visits from $CBG_{i}$ to $CBG_{j}$ in time step
$t$. Time steps are weeks from January 2019 and December 2020. In the _Weekly
Patterns_ dataset, the aggregated number of visits from CBGs to a particular
POI is stored. Here, the CBG where the destination POI is located, is defined
as the target CBG. This means the POIs $P\,=\,\\{p_{1},...,p_{n}\\}$ inside
$CBG_{i}$ are used as a proxy to obtain the in-flow mobility, and their
aggregated number of visitors constitutes the weights of the incoming edges
for $CBG_{i}$. Furthermore, the road network distances between CBGs are also
added as edge attributes into the network metrics.

### Topological changes over time

In order to display the temporal changes in the network topology, we first
focus on the dissimilarity analysis between paired weekly networks of 2019 and
2020 and further break the nodes down based on demographic groups. Next, we
analyze the relationship between centrality metrics and demographic groups.

#### Dissimilarity analysis

To form the feature vectors of the aligned weekly networks of 2019 and 2020,
we consider the following ego-network based node features as proposed by
Berlingerio et al. [netsimile].

  * •

Node In & Out Degree

  * •

Node Strength

  * •

Clustering Coefficient

  * •

Average Degree & Average Clustering Coefficient of Node’s Neighborhood

  * •

Number of Edges & Alters in Node’s Ego-Network

  * •

Number of Out-going Edges from Node’s Ego-Network

To compute the overall dissimilarity score between the paired weekly networks,
the resulting node feature vectors are merged into a single vector, in which
the network features are obtained by applying statistical aggregations, such
as standard deviation, median, kurtosis, and skewness on the node features.
Next, in the node level dissimilarity analysis, paired node feature vectors
are utilized without any aggregation and a score is generated for each node.
For both analyses, Canberra distance [lance1967mixed] is applied on the paired
feature vectors $P$ and $Q$ using Equation 3.

$d(P,Q)=\sum_{i=1}\,\frac{|P\,-\,Q|}{|P|+|Q|}$ (3)

#### Centrality Evolution

Neighborhoods with different demographic backgrounds manifest varying
repercussions to the enforced NPIs over time, due to their distinctive needs
and socioeconomic dynamics. In parallel, CBGs with ample POIs became frequent
destinations. To this end, centrality metrics are employed to demonstrate the
temporal change in the topological importance of the CBGs. In particular, we
focus on _betweenness_ , _in-degree_ , _out-degree_ and _self-visit ratio_. In
order to estimate the number of visits to the POIs in the home CBG, the self-
visit ratio is defined as:

$S_{c}^{t}\,=\,\frac{W_{l}^{t}}{W_{l}^{t}\,+\,W_{o}^{t}}$ (4)

In Equation 4 $W_{l}^{t}$ is the sum of weights on self-loops and $W_{o}^{t}$
is the sum of weights on outgoing edges in time step $t$ for CBG $c$. The
self-visit ratio is considered as an indicator of locality of residents
visits.

### COVID-19 Hotspots and Bridge CBGs

The mobility per CBG is a function of multiple parameters which are heavily
affected by intricate social dynamics. Because of the evident correlation
between mobility and spreading of COVID-19 infections [chang2021mobility,
iacus2020human, xiong2020mobile], CBGs with higher mobility rates may cause a
potential threat to restraining the disease prevalence. We refer to the CBGs
that rank in the top weekly new COVID-19 cases as COVID-19 hotspots, and the
CBGs that interact frequently with the hotspots as COVID-19 bridge CBGs. To
detect and analyze the potential spreaders or bridges, we utilize the visit
frequencies between CBGs. First, we consider two weeks long time span as the
incubation period for new cases to emerge [chang2021mobility, iacus2020human].
Starting from March 2nd, 2020, when the first case in NYC was reported, we
isolate the CBGs in the top weekly new cases quartile in week $t$ (hotspots).
Next, the CBGs that have an outgoing edge to the isolated group in week $t-2$
are recorded. To determine the COVID-19 bridge CBGs, a percentile based
frequency filtering is applied. To this end, we investigate CBGs with
occurrence frequencies in the $75^{th}$ percentile as possible bridges. The
aim of this analysis is to find out if the bridge CBGs have special
socioeconomic and demographic features and to understand why they show such
mobility patterns.

### Huff Gravity Model

The Huff gravity model [huff1964defining] is a well-known and widely used
market share estimation model that focuses on the relationship between retail
stores and customers by modelling it as a function of distance and store
attractiveness traits such as store area. Equation 5 formalizes the basic
version of the gravity model that we use in our analyses.

$P_{ij}=\frac{\frac{A_{j}^{\alpha}}{D_{ij}^{\beta}}}{\sum_{k=1}^{n}\frac{A_{k}^{\alpha}}{D_{ik}^{\beta}}}$
(5)

$P_{ij}$ corresponds to the probability of customer $i$ choosing POI $j$ for
shopping among all available stores in her choice set $k$. The exponents
$\alpha$ and $\beta$ control the weight of distance and store area. Since
there was a significant decline in movement activities during the COVID-19
pandemic, in order to have reasonable amount of observations that can provide
a better fit for the model, we aggregate the visits from CBGs into a census
tract level. We consider each census tract as an individual mobility center
and estimate a pair of parameters $\alpha$ and $\beta$ for each census tract
using Particle Swarm Optimization technique[suhara2021validating,
liang2020calibrating, bahrami2022using].

### Availability of Data and Code

All the datasets and scripts used for this study are freely available for
further research and replication purposes.
The Safegraph mobility dataset is freely available for academic research
purposes through request at:
https://www.safegraph.com/academics
The Google COVID-19 Community Mobility Reports data is available at:
https://www.google.com/covid19/mobility/
All scripts are available at the project github repository:
https://github.com/alppboz/safegraph-covid19-mobility

### Abbreviations

CBG: Census Block Group
NPI: Non-pharmaceutical Intervention
NYC: New York City
POI: Point of Interest

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## Acknowledgements

The authors would like to thank SafeGraph for making the mobility data
available to this research project.

## Author contributions

Conceptualized the initial idea for the research: HAB, MB, NM, SB, BB, AP
Developed/designed research: HAB, MB, NM, SB, AN, BB
Prepared and analyzed the data: HAB, MB
Created figures/tables, and Github materials: HAB
Supervised the work: AP, SB
Wrote first draft: HAB, MB
Edited the draft: HAB, MB, NM, SB, AN, BB
All authors approved the final version.

## Competing Interests

The authors declare no competing interests.


Superconductivity in ThMo2Si2C with Mo2C Square Net

# Superconductivity in ThMo2Si2C with Mo2C Square Net

Zichen Liu,1 Baizhuo Li,2 Yusen Xiao,1 Qingchen Duan,1 Yanwei Cui,3 YuXue Mei1
Qian Tao,2 Shuli Wei,1 Shugang Tan,1 Qiang Jing,1 Qing Lu,1 Yuping Sun,1
Yunyan Liu,1 Shenggui Fu,1 Hao Jiang,4 Zhi Ren,3 Zhu’an Xu,2 Cao
Wang,1111E-mail<EMAIL_ADDRESS>Guanghan Cao.2 1School of Physics and
Optoelectronic Engineering, Shandong University of Technology, Zibo 255000, P.
R. China
2Department of Physics, Zhejiang Province Key Laboratory of Quantum Technology
and Devices, Interdisciplinary Center for Quantum Information, and State Key
Lab of Silicon Materials, Zhejiang University, Hangzhou 310027, P. R. China
3School of Science, Westlake University, 18 Shilongshan Road, Hangzhou 310064,
P. R. China
4School of Physics and Optoelectronics, Xiangtan University, Xiangtan 411105,
P. R. China

###### Abstract

We report the superconductivity of a new quaternary compound ThMo2Si2C,
synthesized with the arc-melting technique. The compound crystallizes in a
tetragonal CeCr2Si2C-type structure with cell parameters of $a$ = 4.2296 Å and
$c$ = 5.3571 Å. An interlayer Si–Si covalent bonding is suggested by the
atomic distance. The electrical resistivity and magnetic susceptibility
measurements indicate a Pauli-paramagnetic metal with dominant
electron–electron scattering in the normal-state. Bulk superconductivity at
2.2 K is demonstrated with a dimensionless specific-heat jump of
${\Delta}C/\gamma_{\rm n}T$ = 0.98. The superconducting parameters of the
critical magnetic fields, coherence length, penetration depth, and
superconducting energy gap are given.

##  I Introduction

Since the discovery of high-temperature cuprate superconductors, the
exploration of new superconductors has aroused enormous interest in layered
compounds, including $R$Ni2B2C($R$ for rare earth),RNi2B2C  MgB2,MgB2
Sr2RuO4,Sr2RuO4  iron pnictides/chalcogenides,LaFeAsO ; FeSe  and BiS2-based
compounds.BiS2  Recently, a new quaternary superconductor BaTi2Sb2O has been
reported, consisting of alternately stacked Ba planes and Ti2Sb2O layers. Upon
cooling, the compound undergoes a density-wave-like (DW) transition at 54 K
and superconducting transition at 1.2 K.BaTi2Sb2O ; BaTi2Sb2O-NMR ;
BaTi2Sb2O-muSR ; BaTi2Sb2O-FDL  The DW ordering can be suppressed by both hole
doping in Ba1-xNaxTi2Sb2O and isovalent substitution in BaTi2Sb2-xBixO, where
the optimal superconducting $T_{\rm c}$s reaches 5.5 K and 3.7 K,
respectively.BaTi2Sb2O-Na ; BaTi2Sb2O-Bi  Superconductivity of 4.6 K has also
been observed in the undoped end member BaTi2Bi2O.BaTi2Bi2O  All these
discoveries mark the birth of a new superconducting family based on the
Ti${}_{2}Pn_{2}$O ($Pn$ = Sb, Bi) block layers.Ti2Pn2O-review  If the single-
layered Ba atom in the lattice is replaced by fluorite-like Sr2F2 or Sm2O2
layers, no superconductivity is observed while the DW state prevails.BaTi2Bi2O
; Sm2O2Ti2Sb2O  In addition, the coexistence of superconductivity and DW
transition was observed in the intergrowth compound Ba2Ti2Fe2As4O, where the
superconductivity and the DW instability were ascribed to the Fe2As2 and
Ti2As2O layers, respectively.SunYL

The newly discovered superconductors containing Ti${}_{2}Pn_{2}$O layers
constitute a small fraction of a much larger compound family. Among them, the
key structural motif is the anti-CuO2-layer-like $M_{2}$O square planes, such
as Ti2O,Ti2Pn2O-review  V2O,CsV2S2O ; RbV2Te2O  Mn2O,La2O3Mn2Se2
Fe2O,La2Fe2Se2O3 ; Sr2F2Fe2Se2O ; Na2Fe2Se2O ; BaFe2Se2O ; Ca2O3Fe3S2  and
Co2O.La2Co2Se2O3  The $M_{2}$O planes are sandwiched by the $X$ (chalcogen,
pnictogen) monolayers, forming unique $M_{2}X_{2}$O structural units. We note
that a similar $M_{2}X_{2}$O-type structure can also be found in oxygen-free
compounds, such as LaRu2Al2B and $RM_{2}$Si2C ($M$ = Cr, Mo).LaRu2Al2B ;
CeCr2Si2C ; RCr2Si2C ; CeMo2Si2C ; PrMo2Si2C  The most distinctive feature of
these oxygen-free compounds, compared to oxygen-containing ones, is the
$X$–$X$ interlayer covalent bond, which results in lattice collapse along the
$c-$axis. All these compounds with a collapsed structure show metallic
behavior, and the electronic states observed in the vicinity of the Fermi
level are mainly dominated by $d-$orbital electrons.LaRu2Al2B ; CeMo2Si2C ;
RCr2Si2C-PRB  To date, no superconductivity has been reported in these
compounds.

Here, we report the synthesis, structure, and physical properties of a new
layered carbide ThMo2Si2C. Powder X-ray diffraction (XRD) indicates that the
compound crystallizes in the CeCr2Si2C-type structure with interlayer Si–Si
covalent bonds. The compound exhibits metallic behavior with Pauli
paramagnetism. Both magnetic susceptibility and electronic resistivity reveal
superconductivity of $T_{\rm c}$ = 2.2 K. No charge/spin-density-wave anomaly
is observed on the $\chi-T$ and $\rho-T$ curves, which marks the difference to
BaTi2Sb2O. The compound is characterized as a phonon-mediated superconductor
with considerable electron–electron interaction.

##  II Experimental

Figure 1: (Color online) Rietveld refinement profile of ThMo2Si2C
polycrystalline sample.

A polycrystalline sample of ThMo2Si2C was synthesized using powders of Th,
Mo(Alfa 99.9%), Si (Alfa 99.99%), and graphite (99.95%) as starting materials.
The preparation of thorium metal is described elsewhere.ThFeAsN  The
stoichiometric mixture of the materials were sufficiently mixed in an agate
mortar and cold-pressed into a pellet. All these procedures are carried out in
a glove box protected by high purity argon. The pellet was then melted by the
standard arc-melting technique on a water-cooled copper hearth. The sample was
melted five times and was flipped after each melting process to ensure
homogeneity. The arc-melted button was placed in an alumina crucible, sealed
in an evacuated silica ampoule, and annealed at 1100 ∘C for two weeks.

Powder XRD was carried out at room temperature on a PANalytical X-ray
diffractometer (model EMPYREAN) with monochromatic Cu K$\alpha 1$ radiation.
The crystal structure data were obtained by Rietveld refinement using the
step-scan XRD data with $10^{\circ}\leq 2\theta\leq 120^{\circ}$, and the
diffraction data were refined via the Rietveld method using the program
RIETAN-2000.RIETAN-2000  The structural parameters were obtained assuming that
the occupancy was the same as the chemical composition. The electrical
resistivity was measured in a Quantum Design Physical Property Measurement
System (PPMS-9 Dynacool), using a standard four-probe method down to 1.8 K,
and then, if necessary, from 1.8 to 0.4 K with an adiabatic dilution
refrigerator option. The measurement of specific heat down to 0.4 K with 3He
subassembly was carried out on a Quantum Design PPMS-9 Evercool II using a
relaxation method. The dc magnetization measurements down to 0.4 K were
performed on a Quantum Design magnetic property measurement system (MPMS3)
with a 3He cryostat.

##  III Results and discussion

Table 1: Summary of crystallographic data and refinement parameters for ThMo2Si2C Data set |  |  | Parameters |
---|---|---|---|---
space group |  |  | $P4/mmm$ |
$a$ (Å) |  |  | 4.2296(3) |
$c$ (Å) |  |  | 5.3571(4) |
$V$ (Å3) |  |  | 95.84(1) |
$R_{\rm{wp}}$ (%) |  |  | 8.346 |
$S$ |  |  | 1.21 |
$Z$ |  |  | 1 |

Figure 1 shows the XRD pattern of ThMo2Si2C together with the Rietveld
refinement profile for the CeCr2Si2C-type structure (space group $P4/mmm$).
The obtained lattice parameters for ThMo2Si2C are $a$ = 4.2296(3) Å and $c$ =
5.3571(4) Å. A small amount of ThO2 (3.7 wt.%, according to refinement) is
present, which can be ascribed to oxygen contamination during the sample
transfer from the glove box to the arc-melting equipment. Table 1 lists the
refined crystal structure parameters. Table 2 shows the coordinates and
temperature factors for all the atoms in the lattice. Figure 2 shows the
crystal lattice of ThMo2Si2C and the details of the Mo2Si2C block layer. Table
3 compares the lattice parameters of ThMo2Si2C with those of the isostructural
compounds CeMo2Si2C and PrMo2Si2C. According to the refinement, the
interlayer’s nearest Si–Si distance ($d_{\rm{Si-Si}}$) is 2.429 Å. This Si–Si
distance is similar to that of silicon with a diamond-type structure(2.352 Å),
and thus should be considered a single Si–Si bond. Therefore, the structure of
ThMo2Si2C is of the CeCr2Si2C-type collapsed phase. As is well known, the
ionic radius of Th4+ (0.94 Å, $CN$ = 6) is smaller than that of Pr3+ (0.99 Å)
or Ce3+ (1.01 Å).radii  However, we note that the cell volume does not follow
the radius trend of the rare-earth elements. Moreover, the $a-$axis of
ThMo2Si2C is the longest among the three compounds. This can be understood as
an electron-doping effect, as Th4+ contributes more electrons than Pr3+ or
Ce3+. On the one hand, the electron transferred to the Mo2C plane expands the
$a-$axis by elongating the Mo–C bond. On the other hand, the charge transfer
strengthens the interlayer Coulomb attraction and results in the shrinkage of
the $c-$axis. We note that similar phenomena are quite common in iron arsenide
superconductors, where electron doping can cause a decrease in axial ratio
$c/a$.Gd1111-Th ; La1111-Co

Table 2: Atomic coordinates and temperature factors for all the elements in ThMo2Si2C atom | site | $x$ | $y$ | $z$ | $B$
---|---|---|---|---|---
Th | 1a | 0 | 0 | 0 | 0.1(fixed)
Mo | 2e | 0 | 1/2 | 1/2 | 0.15(3)
Si | 2h | 1/2 | 1/2 | 0.2267(6) | 0.17(7)
C | 1b | 0 | 0 | 1/2 | 1.9(5)
Table 3: Lattice parameters of $R$Mo2Si2C ($R$ = Ce, Pr, Th), where $h_{\rm Si}$ and $d_{\rm{Si-Si}}$ represent the vertical distance from Si to the Mo2C plane and the interlayer Si–Si distance. Compound |  $a-$axis (Å) |  $c-$axis (Å) | $c/a$ |  $V_{\rm{cell}}$ (Å3) |  $h_{\rm Si}$ (Å) |  $d_{\rm{Si-Si}}$ (Å)
---|---|---|---|---|---|---
CeMo2Si2CCeMo2Si2C  | 4.213 | 5.376 | 1.276 | 95.42 | 1.512 | 2.353
PrMo2Si2CPrMo2Si2C  | 4.2139 | 5.4093 | 1.284 | 96.05 | 1.508 | 2.393
ThMo2Si2C | 4.2296 | 5.3571 | 1.267 | 95.84 | 1.464 | 2.429
Figure 2: (Color online) (a) Crystal structure of ThMo2Si2C. (b) Schematic of
the distorted MoSi4C2 octahedron. (c) Bird’s-eye view of the Mo2C square
lattice.

Figure 3 shows the resistivity data of the ThMo2Si2C sample. The zero-field
resistivity at 300 K is 1.19 m$\Omega\cdot$cm, which is similar to that of
CeMo2Si2C.CeMo2Si2C  Upon cooling, the resistivity exhibits metallic behavior
without anomaly. A superconducting transition is revealed by an abrupt
resistivity drop at 2.2 K. The residual resistivity is high, suggesting that
the sample may be in the dirty limit. The normal-state resistivity below 40 K
can be fitted using the equation

$\rho(T)=\rho_{0}+AT^{2}+\alpha\left(\frac{T}{\Theta_{R}}\right)^{n}\int^{\Theta_{R}/T}_{0}\frac{x^{n}}{(e^{x}-1)(1-e^{-x})}dx$
(1)

where $\Theta_{R}$ is the Debye temperature obtained from the resistivity
measurements and matches very closely with that obtained from the specific
heat. The best fit appears while the constant $n$ in the last term of the
equation (contribution of electron–phonon scattering) is 2. The fitting
parameters are $\rho_{0}$ = 0.35 m$\Omega\cdot$cm and $A$ = 2.75$\times
10^{-5}$ $m\Omega\cdot\rm{cm}\cdot K^{-2}$. Figure 3(b) shows the temperature
dependence of resistivity under static magnetic fields (0 T, 0.05 T, 0.1 T,
0.15 T, 0.2 T, 0.25 T). We define $T_{\rm c}(H)$ as the temperature where the
resistivity falls to 90% of the normal-state value. As shown in Figure 3(c),
the temperature dependence of the upper critical field ($H_{\rm c2}$)
increases approximately linearly upon cooling to the lowest temperature
measured. Thus, the upper critical field at 0 K is estimated from linear
extrapolation to be $H_{\rm c2}(0)$ = 2660 Oe, and the coherent length can be
calculated as $\xi$ = 352 Å according to the relation $\mu_{0}H_{\rm
c2}(0)=\Phi_{0}/2\pi\xi^{2}$, where $\Phi_{0}$ is the magnetic flux quantum.

Figure 3: (Color online) (a) Temperature dependence of resistivity at zero
magnetic field for the ThMo2Si2C polycrystalline sample. The inset shows the
normal-state resistivity below 40 K. The blue line fits the data using the
equation $\rho(T)=\rho_{0}+AT^{2}+\rho_{\rm e-ph}$, where $\rho_{\rm e-ph}$
represents the contribution of electron–phonon scattering. (b) Temperature
dependence of resistivity near the superconducting transition measured in
selected magnetic fields. (c) Temperature dependence of the upper critical
field.

Figure 4(a) plots the temperature dependence of magnetic susceptibility at $H$
= 10 Oe. Both zero-field-cooling (ZFC) and field-cooling (FC) protocols are
employed in the measurement. A strong diamagnetic signal below 2.2 K confirms
the superconducting transition observed in the resistivity measurement. The
magnetic shielding fraction at 0.4 K exceeds 100% due to the demagnetization
effect, indicating bulk superconductivity. The shoulder-like susceptibility
(ZFC) around 1.5 K is probably due to the intergrain weak-link effect. The
susceptibility measured at $H$ = 104 Oe is shown in Figure 4(b). The room
temperature susceptibility is 1.38$\times 10^{-4}$ emu$\cdot$mol-1. The
susceptibility increases gradually upon cooling. We fit the susceptibility
data using the extended Curie–Weiss law $\chi=\chi_{0}+C/(T-\theta)$ over a
temperature region of 10 K $\leq T\leq$ 300 K. The derived parameters are
$\chi_{0}$ = $1.3\times 10^{-4}$ emu$\cdot$mol-1, $C$ = $2.01\times 10^{-3}$
emu$\cdot$K$\cdot$mol-1, and $\theta$ = $-1.9$ K. The results suggest that
ThMo2Si2C is a metal with Pauli paramagnetism. The Curie constant corresponds
to a tiny effective magnetic moment of 0.12 $\mu_{\rm B}$/f.u., which may be
due to a small amount of paramagnetic impurities.

Figure 5 shows the field dependence of the zero-field cooled magnetization at
various temperatures for ThMo2Si2C. The lower critical field $H_{c1}$ is
plotted as a function of ($T/T_{\rm c}$)2. We fit the data using the
Ginzburg–Landau theory $H_{\rm c1}(T)$ = $H_{\rm c1}(0)[1-(T/T_{\rm c})^{2}]$
(where $T_{\rm c}$ = 2.2 K) and obtain the lower critical field $H_{\rm
c1}(0)$ = 16 Oe. Then the penetration depth and the Ginzburg–Landau parameter
are derived as $\lambda$ = 5280 Å and $\kappa$ = 15 from the relation
$\mu_{0}H_{\rm c1}(0)$ = $ln(\lambda/\xi)\Phi_{0}/(4\pi\lambda^{2})$ and
$\kappa$ = $\lambda/\xi$, respectively. Also, the thermodynamic critical field
at zero temperature $H_{\rm c}$(0) is estimated as 183 Oe.

Figure 4: (Color online) (a) Temperature dependence of magnetic susceptibility
measured at $H$ = 10 Oe for ThMo2Si2C. (b) Magnetic susceptibility measured at
$H$ = 104 Oe. The orange solid line fits the normal-state $\chi-T$ curve using
the equation $\chi(T)=\chi_{0}+C/(T-\theta)$.

Figure 6 shows the temperature dependence of the specific heat measured under
zero magnetic field. To make the specific heat of normal-state more intuitive,
we plot $C/T$ as a function of $T^{2}$ from 5 K down to 0.4 K. The data show
good linear $T^{2}$ dependence of $C/T$ above superconducting $T_{\rm c}$,
indicating that the specific heat comprises two contributions: the electronic
part, which is proportional to $T$, and the phonon part, which is proportional
to $T^{3}$ at low temperatures. We fit the specific heat of the normal-state
below 5 K using the relation $C=\gamma_{\rm n}T+{\beta}T^{3}$. The parameters
of the electron and phonon are determined as $\gamma_{\rm n}$ = 11.4
mJ${\cdot}$mol${}^{-1}{\cdot}$K-2 and $\beta$ = 2.7
mJ$\cdot$mol${}^{-1}{\cdot}$K-4 respectively. According to the formula
$\Theta_{\rm D}$ = $[12\pi^{4}NR/(5\beta)]^{1/3}$, the Debye temperature
$\Theta_{\rm D}$ is estimated to be 351 K.

Figure 5: (Color online) (a) Field dependence of magnetization for ThMo2Si2C
at various temperatures. (b) The lower critical field $H_{\rm c1}$ and the
thermodynamic critical field $H_{\rm{c}}$ for ThMo2Si2C as a function of
$(T/T_{\rm c})^{2}$.

Below the $T_{\rm c}$, a distinct peak in specific heat is observed. We
subtract the normal-state specific heat using the aforementioned fitted data
$C=\gamma_{\rm n}T+{\beta}T^{3}$. The inset of Figure 6 plots
${\Delta}C/\gamma_{\rm n}T$ versus $T/T_{\rm c}^{\rm th}$, where $T_{\rm
c}^{\rm th}$ = 2 K denotes the thermodynamic critical temperature considering
entropy conservation. Supposing that an isotropic single-gap model can be
adopted here, we fit the specific heat below 1.6 K with the exponential law
$C_{\rm e}\sim{\rm exp}(-\Delta_{0}/k_{\rm B}T)$. The derived energy gap is
$\Delta_{0}^{\rm exp}$ = 0.29 meV, which matches the Bardeen–Cooper–Schrieffer
(BCS) value of 0.30 meV based on the relation $\Delta_{0}^{\rm BCS}$ =
$1.76k_{\rm B}T_{\rm c}^{\rm th}$. Further, the electronic specific heat jump
at $T_{\rm c}^{\rm th}$ brings ${\Delta}C/\gamma_{\rm n}T$ = 0.98, which is
smaller than the BCS value of ${\Delta}C/\gamma_{\rm n}T$ = 1.43. A possible
explanation for this deviation is the presence of nonsuperconducting
impurities (e.g., ThO2) and/or amorphous grain-boundary that does not
contribute to the signal of superconducting transition. Assuming that the
electron–phonon coupling dominates the origin of the superconductivity in
ThMo2Si2C, the coupling strength $\lambda_{\rm ph}$ can be estimated from the
McMillan formula

$\lambda_{\rm ph}=\frac{1.04+\mu^{*}\ln(\frac{\Theta_{\rm D}}{1.45T_{\rm
c}})}{(1-0.62\mu^{*})\ln(\frac{\Theta_{\rm D}}{1.45T_{\rm c}})-1.04}$ (2)

where $\mu^{*}$ = 0.13 is the Coulomb pseudopotential for polyvalent
transition metals.McMillan  Using this formula, $\lambda_{\rm ph}$ was
evaluated to be 0.49, suggesting that the superconductivity in ThMo2Si2C is in
the weak or intermediate coupling regime.

The Wilson ratio ($R_{\rm W}$) that reflects the electron–electron
correlations in a material can be calculated using the formula $R_{\rm
W}=\chi_{\rm{P}}/(3\gamma_{\rm n})\times(\pi{k_{\rm B}}/\mu_{\rm B})^{2}$,
where $\chi_{\rm{P}}$ represents the Pauli paramagnetic susceptibility.wilson
During the Curie-Weiss fitting of the magnetic susceptibility, we get the
temperature independent susceptibility $\chi_{0}$ = $1.3\times 10^{-4}$
emu$\cdot$mol-1. Note that the fitted $\chi_{0}$ includes the orbital
diamagnetism contribution due to ion cores $\chi_{\rm{core}}$. If the nominal
valences can be represented by the formula
Th4+Mo${}^{3+}_{2}$Si${}^{3-}_{2}$C4-, then the diamagnetic contribution for
ThMo2Si2C can be estimated as $\chi_{\rm{core}}=-1.01\times 10^{-4}$
emu$\cdot$mol-1 using the data listed by G. A. Bain and J. F. Berry.Pascal
Thus the Pauli-paramagnetic contribution reaches $\chi_{\rm{P}}=2.3\times
10^{-4}$ emu$\cdot$mol-1. Then, the Wilson ratio is determined as $R_{\rm W}$
= 1.48, which should be close to 1 in a noninteracting Fermi-liquid system.
This indicates that the compound contains considerable electron–electron
interaction.

Figure 6: (Color online) The specific heat coefficient $C$/$T$ of ThMo2Si2C as
a function of $T^{2}$. The inset plots the normalized electronic specific heat
${\Delta}C/\gamma_{\rm n}T$ after subtracting the normal-state contribution.
The solid red line presents the fit with the conventional BCS model.

Another important parameter characterizing Fermi-liquid properties is the
Kadowaki–Woods ratio ($R_{\rm{KW}}=A/\gamma^{2}$), which presents the
relationship between the electron–electron scattering rate and the electron-
mass renormalization.KW-TM ; KW-HF ; KW-oxide ; KW-normalized  This ratio is
found to be approximately constant within a certain class of materials. For
transition metals and heavy-fermion compounds, $R_{\rm{KW}}$s are typically
0.4 and 10 $\mu\Omega\cdot$cm$\cdot$mol${}^{2}\cdot$K${}^{2}\cdot$J-2,
respectively. Interestingly, our results provide a much larger $R_{\rm{KW}}$ =
212 $\mu\Omega\cdot$cm$\cdot$mol${}^{2}\cdot$K${}^{2}\cdot$J-2 for ThMo2Si2C.
Note that $R_{\rm{KW}}$ is sometime material-dependent and sensitive to
dimensionality, electron density, multiband effect, and the interlayer hopping
integral.KW-oxide ; KW-normalized ; NaCoO  To our knowledge, large
$R_{\rm{KW}}$ values are usually observed in strongly correlated transition-
metal oxides or organic superconductors, such as La1.7Sr0.3CuO4 (52
$\mu\Omega\cdot$cm$\cdot$mol${}^{2}\cdot$K${}^{2}\cdot$J-2), Na0.7CoO2 (600
$\mu\Omega\cdot$cm$\cdot$mol${}^{2}\cdot$K${}^{2}\cdot$J-2), and $\beta$-ET2I3
(347 $\mu\Omega\cdot$cm$\cdot$mol${}^{2}\cdot$K${}^{2}\cdot$J-2).KW-oxide ;
LaSrCuO ; NaCoO ; organic-1 ; organic-2  Aside from the intrinsic mechanism
for large $R_{\rm{KW}}$s, we note that the magnetic impurities in a
polycrystalline sample may affect the fitting parameter $A$ in the $\rho(T)$
equation, which in turn can lead to overestimation of $R_{\rm{KW}}$. Thus,
high-quality samples, preferably single crystals, are needed to clarify this
issue.

Now let us discuss the electronic states in the CeCr2Si2C-type family. So far,
superconductivity has only been observed in members containing Ti2O or Mo2C
planes. For Ti2O-based compounds, superconductivity competes with the DW
ordering state, which is ascribed to Fermi-surface nesting or $p$–$d$ orbital
ordering.BaTi2Sb2O-FDL ; BaTi2As2O-CXH  For compounds containing Mo2C-plane,
the DW ordering is absent. We note that the DW state is also absent among the
members containing Cr2C-plane. This suggests the origin of superconductivity
in the carbides is different from that in the Ti2O-based compounds. Very
recently, neutron diffraction works on the nonsuperconductive compound
UCr2Si2C identified an antiferromagnetic ordering of Cr magnetic moment
($\sim$0.6 $\mu_{\rm B}$) at 300 K.UCr2Si2C  However, similar experiments on
the isostructural compounds $Ln$Cr2Si2C ($Ln$ for lanthanides) did not
recognize any Cr moment.RCr2Si2C-PRB  For the compounds based on Mo2Si2C
layers, superconductivity was only observed in ThMo2Si2C rather than
$Ln$Mo2Si2C.CeMo2Si2C ; PrMo2Si2C  Considering the different nominal valence
of (U/Th)4+ and $Ln^{3+}$, one may conclude that the electron configuration
plays a crucial role in determining the $d-$orbital states in Cr2C and Mo2C
planes. As the chemical valence of Cr/Mo is also affected by the interlayer
Si–Si bond order, which is sensitive to the Si–Si bond length, additional
theoretical and experimental works are needed to determine the configuration
of the $d-$orbital in the materials.

##  IV Concluding Remarks

We have synthesized a new quaternary compound ThMo2Si2C with superconducting
$T_{\rm c}$ = 2.2 K. This is the first superconductor among the CeCr2Si2C-type
carbides. The lower, upper, and thermodynamic critical fields at zero
temperature are determined to be $H_{\rm{c1}}$ = 16 Oe, $H_{\rm{c2}}$ = 2660
Oe and $H_{\rm c}$ = 183 Oe, respectively. Then, the penetration depth,
coherent length, and Ginzburg–Landau parameter are calculated accordingly as
$\lambda$ = 5280 Å, $\xi$ = 352 Å, and $\kappa$ = 15. Although the jump of
specific heat due to the superconducting transition deviates from that
predicted by the BCS theory, the superconducting energy gap derived by fitting
the specific heat below the $T_{\rm c}$ precisely matches the theoretical
value. The Wilson ratio and Kadowaki–Woods ratio are obtained as $R_{\rm W}$ =
1.48 and $R_{\rm KW}$ = 212. Our data indicate that ThMo2Si2C is a BCS
superconductor with considerable electron–electron interaction. More efforts
should be made to explore new superconductors among compounds with similar
crystal structures.

###### Acknowledgements.

This work was supported by the National Key Research and Development Program
of China (Grant No. 2017YFA0303002) and the Natural Science Foundation of
Shandong Province (Grants No. ZR2019MA036 and ZR2016AQ08).

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