# A note on almost everywhere convergence along tangential curves to the
Schrödinger equation initial datum

Javier Minguillón

###### Abstract

In this short note, we give an easy proof of the following result: for $n\geq
2,$ $\underset{t\to 0}{\lim}\,e^{it\Delta}f\left(x+\gamma(t)\right)=f(x)$
almost everywhere whenever $\gamma$ is an $\alpha-$Hölder curve with
$\frac{1}{2}\leq\alpha\leq 1$ and $f\in H^{s}(\mathbb{R}^{n})$, with
$s>\frac{n}{2(n+1)}$. This is the optimal range of regularity up to the
endpoint.

††2020 Mathematics Subject Classification. 35Q41, 42B25, 42B37. ††Keywords.
Schrödinger equation, Schrödinger maximal function, almost everywhere
convergence, tangential convergence. †† Javier Minguillon was supported by
Grant PID2022-142202NB-I00 / AEI / 10.13039/501100011033.

##  1 Introduction

Consider the linear Schrödinger equation on $\mathbb{R}^{n}\times\mathbb{R}$,
$n\geq 1,$ given by

$\begin{cases}i\partial_{t}u(x,t)-\Delta_{x}u(x,t)=0,\\\
u(x,0)=f(x).\end{cases}$ (1)

Its solution can be formally expressed as

$e^{it\Delta}f\left(x\right)=\int_{\mathbb{R}^{n}}e^{2\pi ix\cdot\xi}e^{2\pi
it|\xi|^{2}}\widehat{f}(\xi)d\xi.$ (2)

It was first proposed by Carleson in 1980 [3] to find the values of $s>0$ for
which

$\lim_{t\to 0}e^{it\Delta}f\left(x\right)=f(x),\quad\text{a.e.}\quad
x\in\mathbb{R}^{n},$ (3)

holds true for all functions $f\in H^{s}(\mathbb{R}^{n})$. Carleson [3] proved
this convergence when $n=1$ and $s\geq\frac{1}{4}$. Later, in 2006, Dahlberg
and Kenig [6] showed that (3) was false whenever $s<\frac{1}{4}.$

Many researchers have worked in this problem throughout the years. Authors
such as Carbery, Cowling, Vega, Sjölin, Moyua, Vargas, Tao, S.Lee and Bourgain
to name a few. More recently, the problem has been solved in higher
dimensions, except for the endpoint. In 2016, Bourgain [1] proved the
necessity of $s\geq\frac{n}{2(n+1)}$ in order to have (3). In 2017, Du, Guth
and Li [7] proved the sufficiency of the condition $s>\frac{1}{3}$ when $n=2.$
Later, in 2019, Du and Zhang [8] proved the sufficiency of
$s>\frac{n}{2(n+1)}$ for general $n\geq 3.$ A more detailed history of the
problem can be found in [8] and the references therein.

Take a solution of (1). Consider a set of curves $\rho(x,t)=x+\gamma(t)$ that
are bi-Lipschitz in $x\in\mathbb{R}^{n}$ and $\alpha-$Hölder in
$t\in\mathbb{R}$. Cho, Lee and Vargas [4] proved in 2012 that
$u\left(\rho(x,t),t\right)$ converges to $f(x)$ almost everywhere as $t\to 0$
in $n=1$ when $s>\max\left\\{\frac{1}{2}-\alpha,\frac{1}{4}\right\\}$. They
also found this to be sharp up to the endpoint. Later, in 2021, Li and Wang
[11] proved that convergence in dimension $n=2$, for index
$\frac{1}{2}\leq\alpha\leq 1$ and the range $s>\frac{3}{8}$. In 2023, Cao and
Miao [2] gave a proof for general dimension $n$, index
$\frac{1}{2}\leq\alpha\leq 1$, and $s>\frac{n}{2(n+1)}$. Their proof followed
the argument presented in [8] and relied on techniques such as dyadic
pigeonholing, broad-narrow analysis and induction on scales.

Our objective is to give an easy proof of the result in [2] without using the
aforementioned techniques.

Fix $0<\alpha\leq 1$ and $\tau\geq 1.$ We consider the family of curves,

$\Gamma_{\tau}^{\alpha}:=\left\\{\gamma:[0,1]\to\mathbb{R}^{n}:\text{for all
}t,t^{\prime}\in[0,1],\;|\gamma(t)-\gamma(t^{\prime})|\leq\tau|t-t^{\prime}|^{\alpha}\right\\}.$
(4)

The convergence result follows from the maximal bound below. Let
$B_{r}^{n}(x_{0})$ denote the ball of radius $r>0$ centered at
$x_{0}\in\mathbb{R}^{n}$.

######  Theorem 1.1.

Let $n\geq 1$. Fix $\frac{1}{2}\leq\alpha\leq 1$ and $\tau\geq 1$. For any
$\varepsilon>0$, there exists a positive constant $C_{\varepsilon,\tau}$ such
that, for every $\gamma\in\Gamma_{\tau}^{\alpha}$,

$\left\|\sup_{0<t<1}\left|e^{it\Delta}f\left(x+\gamma(t)\right)\right|\right\|_{L^{2}\left(B^{n}_{1}(0)\right)}\leq
C_{\varepsilon,\tau}\|f\|_{H^{\frac{n}{2(n+1)}{\color[rgb]{0.12,0.56,1.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.12,0.56,1.0}}+\epsilon}\left(\mathbb{R}^{n}\right)},$
(5)

holds for all $f$ $\in H^{\frac{1}{2(n+1)}+\epsilon}(\mathbb{R}^{n}).$

######  Remark 1.2.

A change of variables shows that it is enough to consider the case $\tau=1$.
From now on we assume $\tau=1.$

Then, we can reduce Theorem 1.1 as in [8]. We begin with a definition.

######  Definition 1.3.

Fixed $0<\alpha\leq 1$ and $R>1$, we define

$\Gamma^{\alpha}\left(R^{-1}\right):=\left\\{\gamma:[0,R^{-1}]\to\mathbb{R}^{n}:\text{for
all
}t,t^{\prime}\in[0,R^{-1}],\;|\gamma(t)-\gamma(t^{\prime})|\leq|t-t^{\prime}|^{\alpha}\right\\}.$
(6)

By Littlewood-Paley decomposition, the time localization lemma (e.g. Lemma 3.1
in S.Lee [9]) and parabolic rescaling, Theorem 1.1 can be reduced to the
following Theorem 1.4.

######  Theorem 1.4.

Let $n\geq 1$ and $\frac{1}{2}\leq\alpha<1$. For any $\varepsilon>0$, there
exists a constant $C_{\varepsilon}$ such that, for all
$\gamma\in\Gamma^{\alpha}\left(R^{-1}\right)$,

$\left\|\sup_{0<t\leq
R}\left|e^{it\Delta}f\left(x+R\gamma\left(\frac{t}{R^{2}}\right)\right)\right|\right\|_{L^{2}\left(B^{n}_{R}(0)\right)}\leq
C_{\varepsilon}R^{\frac{n}{2(n+1)}+\varepsilon}{\color[rgb]{0.12,0.56,1.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.12,0.56,1.0}}\|f\|_{2}.$
(7)

holds for all $R\geq 1$ and all $f$ with
$\operatorname{supp}\widehat{f}\subset
A(1)=\left\\{\xi\in\mathbb{R}^{n}:|\xi|\sim 1\right\\}$.

##  2 Intermediate results

We consider the following result from [8].

######  Theorem 2.1 (Corollary 1.7 in [8]).

Let $n\geq 1$. For any $\varepsilon>0$, there exists a constant
$C_{\varepsilon}$ such that the following holds for all $R\geq 1$ and all $f$
with $\operatorname{supp}\widehat{f}\subset B_{1}^{n}(0)$. Suppose that
$X=\cup_{k}B_{k}$ is a union of lattice unit cubes in $B_{R}^{n+1}(0)$. Let
$1\leq\beta\leq n+1$ and

$\phi:=\phi_{X,\beta}:=\max_{\begin{subarray}{c}B_{r}^{n+1}(x^{\prime})\subset
B_{R}^{n+1}(0)\\\ x^{\prime}\in\mathbb{R}^{n+1},r\geq
1\end{subarray}}\frac{\\#\left\\{B_{k}:B_{k}\subset
B^{n+1}_{r}\left(x^{\prime}\right)\right\\}}{r^{\beta}}.$ (8)

Then

$\left\|e^{it\Delta}f\right\|_{L^{2}(X,dxdt)}\leq
C_{\varepsilon}\phi^{\frac{1}{n+1}}R^{\frac{\beta}{2(n+1)}+\varepsilon}\|f\|_{2}.$
(9)

We generalize the above result to include $\alpha-$Hölder curves.

######  Theorem 2.2.

Let $n\geq 1$ and $\frac{1}{2}\leq\alpha\leq 1$. For any $\varepsilon>0$,
there exists a constant $C_{\varepsilon}$ such that the following holds for
any $R\geq 1$, every $\gamma\in\Gamma^{\alpha}\left(R^{-1}\right)$ and all $f$
with $\operatorname{supp}\widehat{f}\subset B_{1}^{n}(0)$. Suppose that
$X=\cup_{k}B_{k}$ is a union of lattice unit cubes in $B_{R}^{n+1}(0)$. Let
$1\leq\beta\leq n+1$ and $\phi$ be given by (8). Then

$\left\|e^{it\Delta}f\left(x+R\gamma\left(\frac{t}{R^{2}}\right)\right)\right\|_{L^{2}(X,dxdt)}\leq{\color[rgb]{0.12,0.56,1.0}\definecolor[named]{pgfstrokecolor}{rgb}{0.12,0.56,1.0}}C_{\varepsilon}\phi^{\frac{1}{n+1}}R^{\frac{\beta}{2(n+1)}+\varepsilon}\|f\|_{2}.$
(10)

###### Proof of Theorem 2.2.

Denote

$\theta(t):=\theta_{R}(t):=R\gamma\left(\frac{t}{R^{2}}\right).$ (11)

We begin with

$\displaystyle\left\|e^{it\Delta}f\left(x+R\gamma\left(\frac{t}{R^{2}}\right)\right)\right\|_{L^{2}(X,dxdt)}^{2}$
$\displaystyle=\sum_{k}\int_{B_{k}}\left|e^{it\Delta}f\left(x+\theta(t)\right)\right|^{2}dxdt.$
(12) Denote $(x_{k},t_{k})$ to be the center of $B_{k}.$ Then,
$\displaystyle\leq\sum_{k}\int_{t_{k}-1{\color[rgb]{0.0,0.85,0.85}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.85,0.85}}}^{t_{k}+1{\color[rgb]{0.0,0.85,0.85}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.85,0.85}}}\int_{B^{n}_{1}(x_{k})}\left|e^{it\Delta}f\left(x+\theta(t)\right)\right|^{2}dxdt$
(13)
$\displaystyle=\sum_{k}\int_{t_{k}-1}^{t_{k}+1}\int_{B^{n}_{1}\left(x_{k}+\theta(t)\right)}\left|e^{it\Delta}f\left(y\right)\right|^{2}dydt.$
(14) Recall that $\gamma\in\Gamma^{\alpha}\left(R^{-1}\right)$, and
$\alpha\geq\frac{1}{2}$. Thus, if $t\in(t_{k}-1,t_{k}+1),$ then
$\left|\theta(t)-\theta(t_{k})\right|\leq 1$. Hence,
$\displaystyle\leq\sum_{k}\int_{t_{k}-1}^{t_{k}+1}\int_{B^{n}_{3}\left(x_{k}+\theta(t_{k})\right)}\left|e^{it\Delta}f\left(y\right)\right|^{2}dydt$
(15)
$\displaystyle\leq\int_{\mathbb{R}^{n}}\sum_{k}\chi_{B^{n+1}_{4}\left(x_{k}+\theta(t_{k}),t_{k}\right)}(y)\left|e^{it\Delta}f\left(y\right)\right|^{2}dydt$
(16) $\displaystyle\leq C\int_{\bigcup
B^{n+1}_{4}(x_{k}+\theta(t_{k}),t_{k})}\left|e^{it\Delta}f\left(y\right)\right|^{2}dydt,$
(17)

for some $C>0.$ Note that, if $B_{4}(x_{k}+\theta(t_{k}),t_{k})\cap
B_{4}(x_{i}+\theta(t_{i}),t_{i})\neq\emptyset,$ then $|t_{k}-t_{i}|\leq 8.$
Since $\alpha\geq\frac{1}{2},$ we have $|\theta(t_{k})-\theta(t_{i})|\leq 8$.
Hence, $|x_{k}-x_{i}|\leq 16.$ Therefore, the balls
$\\{B_{4}(x_{k}+\theta(t_{k}),t_{k})\\}_{k}$ have finite overlap $C=C_{n}$.

Define $Y=\bigcup_{l}Q_{l}$ to be the minimal union of lattice unit cubes
satisfying that $\bigcup_{k}B_{4}^{n+1}(x_{k}+\theta(t_{k}),t_{k})\subset Y.$
We have proven that

$\displaystyle\left\|e^{it\Delta}f\left(x+\theta(t)\right)\right\|_{L^{2}(X,dxdt)}$
$\displaystyle\leq
C_{n}\left\|e^{it\Delta}f\left(x\right)\right\|_{L^{2}(Y,dxdt)}.$ (18) Hence
by Theorem 2.1, $\displaystyle\leq
C_{\epsilon,n}\phi_{Y,\beta}^{\frac{1}{n+1}}R^{\frac{\beta}{2(n+1)}+\epsilon}\|f\|_{2}.$
(19)

We claim that

$\phi_{Y,\beta}\leq c_{n}\phi_{X,\beta},$ (20)

for some $C>0.$ This would conclude the proof.

To prove (20), note that, if $Q_{l}\subset B_{r}^{n+1}(y_{0},s_{0})$, $r\geq
1,$ and $Q_{l}\cap B_{4}^{n+1}(x_{k}+\theta(t_{k}),t_{k})\neq\emptyset$, then
$B_{k}=B_{1}^{n+1}(x_{k},t_{k})\subset
B_{r+5}^{n+1}(y_{0}-\theta(s_{0}),s_{0}).$ Therefore,

$\dfrac{\\#\\{Q_{l}:Q_{l}\subset B_{r}^{n+1}(y_{0},s_{0})\\}}{r^{\beta}}\leq
c_{n}\dfrac{\\#\left\\{B_{k}:B_{k}\subset
B_{r+5}^{n+1}(y_{0}-\theta(s_{0}),s_{0})\right\\}}{(r+5)^{\beta}}\cdot\dfrac{(r+5)^{\beta}}{r^{\beta}}\leq
c_{n}\phi_{X,\beta}.$ (21)

∎

##  3 Proof of Theorem 1.4

Before the proof, let us introduce a stability property of the Schrödinger
operator. More general versions of the following appeared in an article of Tao
[12] from 1999 and an article of Christ [5] from 1988.

Suppose that $\widehat{f}$ is supported inside a ball of radius 1. If
$|x^{\prime}-y^{\prime}|\leq 4$ and $|t^{\prime}-s^{\prime}|\leq 4,$ then,

$\left|e^{it^{\prime}\Delta}f(x^{\prime})\right|\leq\sum_{\mathfrak{l}\in\mathbb{Z}^{n}}\frac{1}{(1+|\mathfrak{l}|)^{n+1}}\left|e^{is^{\prime}\Delta}f_{\mathfrak{l}}(y^{\prime})\right|,$
(22)

where $\widehat{f_{\mathfrak{l}}}(\xi)=e^{2\pi
i\mathfrak{l}\xi}\widehat{f}(\xi)$.

Now, fix $\alpha\geq 1/2$ and $\gamma\in\Gamma^{\alpha}\left(R^{-1}\right)$.
Define $\theta$ as in (11). Whenever $|x-y|\leq 2$ and $|t-s|\leq 2$, we have
that $\left|x+\theta(t)-(y+\theta(s))\right|\leq 4.$ Thus,

$\left|e^{it\Delta}f(x+\theta(t))\right|\leq\sum_{\mathfrak{l}\in\mathbb{Z}^{n}}\frac{1}{(1+|\mathfrak{l}|)^{n+1}}\left|e^{is\Delta}f_{\mathfrak{l}}(y+\theta(s))\right|.$
(23)

Therefore, if $|x-x_{0}|\leq 1$ and $|t-t_{0}|\leq 1$, then,

$\left|e^{it\Delta}f(x+\theta(t))\right|\leq\sum_{\mathfrak{l}\in\mathbb{Z}^{n}}\frac{1}{(1+|\mathfrak{l}|)^{n+1}}\int_{t_{0}}^{t_{0}+1}\int_{B_{1}(x_{0})}\left|e^{is\Delta}f_{\mathfrak{l}}(y+\theta(t))\right|dyds.$
(24)

###### Proof of Theorem 1.4..

For the sake of briefness, given $(x,t)\in\mathbb{R}^{n+1}$ let us denote

$E^{\prime}f(x,t):=E^{\prime}_{\gamma,R}f(x,t):=e^{it\Delta}f\left(x+R\gamma\left(\frac{t}{R^{2}}\right)\right).$
(25)

Now, we can write

$\displaystyle\left\|\sup_{0<t\leq
R}\left|e^{it\Delta}f\left(x+R\gamma\left(\frac{t}{R^{2}}\right)\right)\right|\right\|_{L^{2}\left(B^{n}_{R}(0)\right)}^{2}=\left\|\sup_{0<t\leq
R}|E^{\prime}f(x,t)|\right\|_{L^{2}\left(B^{n}_{R}(0)\right)}^{2}$ (26)
$\displaystyle=\int_{B^{n}_{R}(0)}\left(\sup_{\begin{subarray}{c}0<t\leq
R\end{subarray}}|E^{\prime}f(x,t)|^{2}\right)dx$ (27)
$\displaystyle=\sum_{\begin{subarray}{c}x_{0}\in\mathbb{Z}^{n}\\\
|x_{0}|<R\end{subarray}}\int_{B^{n}_{1}(x_{0})}\left(\sup_{0<t\leq
R}|E^{\prime}f(x,t)|^{2}\right)dx$ (28) $\displaystyle\leq
C_{n}\sum_{\begin{subarray}{c}x_{0}\in\mathbb{Z}^{n}\\\
|x_{0}|<R\end{subarray}}\left[\sup_{|x-x_{0}|\leq
1}\left(\sup_{\begin{subarray}{c}0<t\leq
R\end{subarray}}|E^{\prime}f(x,t)|^{2}\right)\right].$ (29)

For each $x_{0}$, there exists
$\widetilde{t_{0}}=\widetilde{t_{0}}(x_{0})\in\mathbb{Z}\cap[0,R]$ such that
the supremum on each term of the above sum is almost attained inside
$B^{n+1}_{1}(x_{0},\widetilde{t_{0}})$. Therefore,

$\displaystyle\left\|\sup_{0<t\leq
R}|E^{\prime}f(x,t)|\right\|_{L^{2}(B^{n}_{R}(0))}^{2}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}x_{0}\in\mathbb{Z}^{n}\\\
|x_{0}|<R\end{subarray}}\sup_{(x,t)\in
B_{1}^{n+1}(x_{0},\widetilde{t_{0}})}\left|E^{\prime}f(x,t)\right|^{2},$ (30)
which is, by (24),
$\displaystyle\lesssim\sum_{\begin{subarray}{c}x_{0}\in\mathbb{Z}^{n}\\\
|x_{0}|<R\end{subarray}}\left|\sum_{\mathfrak{l}\in\mathbb{Z}^{n}}\frac{1{\color[rgb]{0.0,0.85,0.85}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.85,0.85}}}{(1+|\mathfrak{l}|)^{N}}\int_{B^{n}_{1}\left(x_{0}\right)}\int_{\widetilde{t_{0}}}^{\widetilde{t_{0}}+1}\left|E^{\prime}f_{\mathfrak{l}}(y,s)\right|dsdy\right|^{2}.$
(31) Therefore, denoting $C_{\mathfrak{l}}=\frac{1}{(1+|\mathfrak{l}|)^{N}}$
and using Cauchy-Schwarz,
$\displaystyle\lesssim\sum_{\begin{subarray}{c}x_{0}\in\mathbb{Z}^{n}\\\
|x_{0}|<R\end{subarray}}\sum_{\mathfrak{l}\in\mathbb{Z}^{n}}C_{\mathfrak{l}}\left\|E^{\prime}f_{\mathfrak{l}}(y,t)\right\|^{2}_{L^{2}(B^{n+1}_{2}(x_{0},\widetilde{t_{0}}))}.$
(32)

Let us choose $X=\bigcup_{\begin{subarray}{c}x_{0}\in\mathbb{Z}^{n}\\\
|x_{0}|<R\end{subarray}}B^{n+1}_{2}\left(x_{0},\widetilde{t_{0}}\right)$. The
above lets us deduce that

$\displaystyle\left\|\sup_{0<t\leq
R}|E^{\prime}f(x,t)|\right\|_{L^{2}(B^{n}_{R}(0),dx)}^{2}$
$\displaystyle\lesssim\sum_{\mathfrak{l}\in\mathbb{Z}^{n}}C_{\mathfrak{l}}\left\|E^{\prime}f_{\mathfrak{l}}(y,t)\right\|_{L^{2}(X)}^{2}.$
(33) By Theorem 2.2, this is, $\displaystyle\lesssim
C_{\epsilon}{\color[rgb]{0.0,0.85,0.85}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.85,0.85}}{\color[rgb]{0.0,0.85,0.85}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.85,0.85}}\sum_{\mathfrak{l}\in\mathbb{Z}^{n}}C_{\mathfrak{l}}(\phi_{X,n})^{\frac{2}{n+1}}\|f_{\mathfrak{l}}\|_{L^{2}(\mathbb{R}^{n})}^{2}R^{\frac{2n}{2(n+1)}+\epsilon}.$
(34)

Recall that, given $x_{0}\in\mathbb{Z}^{n}\cap B_{R}^{n}(0),$ we have chosen
exactly one $\widetilde{t_{0}}\in\mathbb{Z}\cap[0,R]$. Consequently,
$\phi_{X,n}\leq 1$. Therefore, the above inequalities yield

$\left\|\sup_{0<t\leq
R}|E^{\prime}f(x,t)|\right\|_{L^{2}(B^{n}_{R}(0),dx)}^{2}\lesssim
C_{\epsilon}R^{\frac{2n}{2(n+1)}+\epsilon}\|f\|_{2}^{2}.$ (35)

∎

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JAVIER MINGUILLÓN, DEPARTMENT OF MATHEMATICS, UNIVERSIDAD AUTÓNOMA DE MADRID,
28049 MADRID, SPAIN

E-mail address<EMAIL_ADDRESS>


# Thermoelectricity and electronic correlations enhancement in FeS by slight
Se substitution

Yu Liu,1,†,‡,∗ Aifeng Wang,1,∥,§,∗ V. N. Ivanovski,2 Qianheng Du,1,3,♯ V.
Koteski,2 and C. Petrovic1,3,¶ 1Condensed Matter Physics and Materials Science
Department, Brookhaven National Laboratory, Upton, New York 11973, USA
2Department of Nuclear and Plasma physics, Vinca Institute of Nuclear Sciences
- National Institute of the Republic of Serbia, University of Belgrade,
Belgrade, Serbia
3Department of Materials Science and Chemical Engineering, Stony Brook
University, Stony Brook, New York 11790, USA

###### Abstract

We report thermoelectric studies of FeS1-xSex ($x$ = 0, 0.06) superconducting
single crystals that feature high irreversibility fields and critical current
density $J_{c}$ comparable to materials with much higher superconducting
critical temperatures ($T_{c}$’s). The ratio of $T_{c}$ to the Fermi
temperature $T_{F}$ is very small indicating weak electronic correlations.
With a slight selenium substitution on sulfur site in FeS both $T_{c}$/$T_{F}$
and the effective mass $m^{*}$ rise considerably, implying increase in
electronic correlation of the bulk conducting states. The first-principle
calculations show rise of the density of states at the Fermi level in
FeS0.94Se0.06 when compared to FeS which is related not only to Fe but also to
chalcogen-derived electronic states.

##  I INTRODUCTION

Since the discovery of La(O1-xFx)FeAs ($x$ = 0.05-0.12) with a superconducting
transition temperature $T_{c}$ = 26 K Hosono , the layered iron-based
superconductors have been extensively studied. Stoichiometric tetragonal FeS
showing a $T_{c}$ below 5 K was synthesized in 2015 by using a hydrothermal
method Lai . It shares the same simplest PbO-type structure with FeSe FeSe ,
which motivated many subsequent studies devoted to investigation of its
superconducting state and relation to crystal structure details.

In contrast to FeSe, the X-ray diffraction experiment indicated no structural
transition down to 10 K in FeS Pachmayr ; Kuhn . The $T_{c}$ of FeS
continuously decreases for hydrostatic pressures up to 2.2 GPa Holenstein ,
while a second superconducting dome for higher pressures from 5 to 22.3 GPa
has been observed with an enhanced $T_{c}$ Zhang . The neutron scattering
experiment and angle-resolved photoemission spectroscopy (ARPES) measurement
indicated that FeS is less correlated when compared to FeSe Man ; Reiss ,
however the scanning tunneling microscopy (STM) measurement suggested a
strong-coupling superconductivity with a large gap ratio of $\sim$ 4.65 in FeS
Yang1 . The thermal conductivity and heat capacity measurements suggested a
nodal or highly anisotropic superconducting gap structure in FeS Ying ; Xing ,
while a $\mu$SR study proposed that a full-gap state coexists with low-moment
disordered magnetism Holenstein . A large upper critical field anisotropy
$H_{c2}^{\parallel ab}$(0)/$H_{c2}^{\parallel c}$(0) $=$ 10 has been observed
in angle-dependent magnetoresistance and quantum oscillations, in line with
the band structure calculation Borg ; Terashima ; Lin ; Yang2 . The nonlinear
behavior of Hall resistivity indicated multiband conductivity Lin .
Interestingly, the critical current density $J_{c}$ in FeS is comparable to
other iron-based superconductors with much higher $T_{c}$, and it can be
enhanced three times by 6% Se doping AF1 ; AF2 , pointing that FeS-based
materials with higher $T_{c}$ are promising for high-magnetic-field
applications.

Thermopower is an effective parameter to characterize the nature and sign of
carries as well as the correlation strength in superconductors Sefat ; Kang ;
Mun ; Pourret ; Matusiak ; Butch ; Kefeng1 ; Kefeng2 ; Collignon .
Fundamentally, the thermopower is entropy per carrier. To evaluate the
correlation strength of electronic states in the bulk that may influence
anisotropy of both normal state and the $J_{c}$ as well as the current-
carrying capability in grain boundaries AndersenBM ; WolfFA ; YoYJ ; HecherJ ;
YuSL ; SkornyakovSL , in this work we perform thermal transport studies of
superconducting FeS1-xSex ($x$ = 0 and 0.06) single crystals. The Fermi
temperature $T_{F}$ in both crystals is about an order of magnitude smaller
when compared to common values in metals. Rather small $T_{c}$/$T_{F}$
suggests weak electronic correlations. Yet, our combined experimental and
theoretical analysis indicates an increase of electronic correlations for a
change in $x$ as small as 0.06, i.e. in FeS0.94Se0.06 when compared to FeS.

##  II EXPERIMENTAL DETAILS

Single crystals of FeS1-xSex ($x$ = 0, 0.06) were synthesized by a
hydrothermal method and characterized as described previously AF1 ; AF2 .
X-ray diffraction (XRD) data were acquired on a Rigaku Miniflex powder
diffractometer with Cu $K_{\alpha}$ ($\lambda=0.15418$ nm). The in-plane
resistivity $\rho$ and thermopower $S$ were measured in a Quantum Design
PPMS-9 system with standard four-probe technique. Thermopower was measured by
using one-heater-two-thermometer setup with hooked copper leads using silver
paint contact directly on crystals with typical dimensions $\sim$
4$\times$3$\times$0.3 mm3. Continuous measuring mode was adopted for
thermopower measurement with the maximum heater power and period set as 50 mW
and 1430 s along with the maximum temperature rise of 3$\%$. The relative
error in our measurement for thermopower was below 5% based on Ni standard
measured under identical conditions. The Hall resistivity $\rho_{xy}$ was
measured using standard four-probe method with the current flowing in the $ab$
plane and the magnetic field along the $c$ axis. In order to effectively
eliminate the longitudinal resistivity contribution due to voltage probe
misalignment, the Hall resistivity was obtained by the difference of
transverse resistance measured at positive and negative fields, i.e.
$\rho_{xy}=[\rho(+\mu_{0}H)-\rho(-\mu_{0}H)]/2$. The sample dimensions were
measured by an optical microscope Nikon SMZ-800 with 10 $\mu$m resolution. The
heat capacity was measured on warming procedure between 0.35 and 7 K by the
heat pulse relaxation method in a Quantum Design PPMS-9.

The density of states of both FeS and Fe16S15Se1 were calculated by the VASP
code in the gradient corrected (GGA) implementation PhysRevB.54.11169  using
the experimental value of FeS lattice parameter $c$. Lattice parameter $a$ and
internal parameter $u$ were DFT-relaxed. The non-magnetic ground states were
obtained via the Methfessel-Paxton order 1 integration method of the Brillouin
zone using a grid of 13 $\times$ 13 $\times$ 9 $k$-points. The maximum forces
are less than 0.02 eV Å-1 and the energy criterion was set to 10-5 eV.

##  III RESULTS AND DISCUSSIONS

Figure 1(a) shows the single crystal XRD patterns of tetragonal FeS1-xSex ($x$
= 0 and 0.06). Only (00l) reflections were observed for both crystals,
yielding the $c$-axis lattice parameter $c$ = 5.035(5) and 5.061(22) Å for FeS
and FeS0.94Se0.06, respectively. The crystal structure as presented in the
inset in Fig. 1(a); it is composed of a stack of edge-sharing FeS4 tetrahedra
layer by layer, without a spacer layer Lai . The FeS4 tetrahedron is nearly
regular with two S-Fe-S angles very close to the ideal value of 109.5∘ Lee .

The in-plane resistivity $\rho$(T) for FeS1-xSex ($x$ = 0 and 0.06) [Fig.
1(b)] monotonically decreases with decreasing temperature showing good
metallic conductivity. The residual resistivity ratio $RRR$ = $\rho$(300
K)/$\rho$(5 K) is $\sim$ 68 for FeS, indicating low defect scattering. The
value of $RRR$ decreases to $\sim$ 27 for FeS0.94Se0.06 with S/Se substitution
disorder. The $\rho$(T) of FeS1-xSex ($x$ = 0 and 0.06) shows a positive
curvature all the way up to room temperature, in contrast to FeSe where a
negative curvature is generally observed at high temperatures FeSe . An abrupt
resistivity drop can be clearly seen [Fig. 1(c)], signaling the onset of
superconductivity. Zero resistivity is observed at $T_{c}$ = 4.5(1) K for FeS,
in agreement with the previous reports Lin . The $T_{c}$ is slightly lower
with a value of 4.1(1) K for FeS0.94Se0.06.

Figure 1: (Color online) (a) Single crystal X-ray diffraction (XRD) patterns
of FeS1-xSex ($x$ = 0, 0.06). Inset shows the crystal structure with the space
group $P4/nnm$. (b) Temperature dependence of in-plane resistivity $\rho$(T)
for FeS1-xSex ($x$ = 0, 0.06) single crystals. (c) The enlargement of
superconducting transitions at low temperature.

Figure 2: (Color online) Temperature dependence of in-plane (a) thermopower
$S$(T) and (b) $S$/$T$ for FeS1-xSex ($x$ = 0 and 0.06) single crystals. (c)
The Moriya-Ueda plot: $T_{c}$ as a function of $T_{F}$ for indicated
superconductors Pourret ; Sarrao ; Moriya ; Four blue solid squares represent
K0.64Fe1.44Se2, K0.70Fe1.55Se1.01S0.99 Kefeng2 , and FeS1-xSex ($x$ = 0,
0.06).

Figure 2(a) shows the temperature dependence of in-plane thermopower $S$(T)
for FeS1-xSex ($x$ = 0 and 0.06). At room temperature, the $S$(T) for both
samples shows a negative value, consistent with dominant negative charge
carriers Lin . It is interesting that the value of $S$(T) decreases with
decreasing temperature and becomes positive at $\sim$ 122(4) and 228(4) K for
FeS and FeS0.94Se0.06, respectively. For pure FeS, the positive $S$(T)
gradually increases with a much smaller slope and features a broad peak
[$\sim$ 1.3(2) $\mu$V K-1] around 28(9) K, then decreases again as temperature
is lowered. The sign change of $S$(T) implies multiband transport in FeS, in
agreement with the nonlinear Hall resistivity analysis Lin . Even though the
Hall coefficient $R_{H}$ is unchanged in this temperature range Lin , the sign
of $S$(T) might change due to a different dependence on the carrier density
$n_{e}$ ($n_{h}$), mobility $\mu_{e}$ ($\mu_{h}$), and $S_{e}$ ($S_{h}$) in
the two-band model Smith :

$R_{H}=\frac{1}{e}\frac{n_{h}\mu_{h}^{2}-n_{e}\mu_{e}^{2}}{(n_{h}\mu_{h}+n_{e}\mu_{e})^{2}},$
(1) $S=\frac{S_{e}n_{e}\mu_{e}+S_{h}n_{h}\mu_{h}}{n_{e}\mu_{e}+n_{h}\mu_{h}}.$
(2)

The positive $S$(T) of FeS0.94Se0.06 changes its slope from negative to
positive temperature dependence at 107(3) K with a larger maximum value of
7.22(1) $\mu$V K-1. With further decrease in temperature, the S(T) for both
samples becomes negative again and vanishes at $T_{c}$ since the Cooper pairs
carry no entropy. The $S$(T) of FeSe is positive at room temperature of $\sim$
10 $\mu$V K-1, then decreases with decreasing temperature and changes its sign
to be negative below 200 K and back to be positive again near 30 K above
$T_{c}$, featuring a minimum value of -10 $\sim$ -15 $\mu$V K-1 around 115 K
YJSong ; Caglieris ; Pallecchi . The similar multiple sign changes in $S$(T)
suggests that almost compensated electrons and holes complete in FeS1-xSex.
From the band structure calculations Subedi , indeed, both the hole and
electron pockets appear in the identical Fermi-surface area. However, the
complex details of $S$(T) make it difficult to be reproduced by the theory
calculation using constant relaxation time approximation Caglieris , calling
for further improved method taken into consideration the correlation effects
in this system.

Figure 3: (a) Field dependence of the Hall resistivity $\rho_{xy}$ for
FeS0.94Se0.06 at various temperatures. (b) Plots of $\rho_{xy}$ vs $\mu_{0}H$
below 2 T with solid linear fitting curves. (c) Temperature dependence of Hall
coefficient $R_{H}$ derived from linear fit in (b) below 2 T.

Generally, the phonon drag $S_{\textrm{drag}}$ related to electron-phonon
coupling gives $\propto T^{3}$ dependence for $T\ll\Theta_{\textrm{D}}$,
$\propto T^{-1}$ for $T\gg\Theta_{\textrm{D}}$, and a peak structure at
$\sim\Theta_{\textrm{D}}/5$, where $\Theta_{\textrm{D}}$ is the Debye
temperature Barnard ; Cohn . The derived $\Theta_{\textrm{D}}$ is about 230 K
for FeS Xing . Since there is no peak structure at $\Theta_{\textrm{D}}/5$ =
50.7(1) K for FeS and thermopower is linear for FeS0.94Se0.06 around 49.1(1) K
(see below for the values of $\Theta_{\textrm{D}}$), it is unlikely that the
sign and slope change in $S$(T) is related to the phonon drag effect.

Figure 2(b) shows the temperature dependent of $S$(T) divided by temperature
for FeS1-xSex ($x$ = 0 and 0.06). The zero-temperature extrapolated value of
$S/T$ increases with Se doping; it is $\sim$ -0.15(1) $\mu$V K-2 for FeS and
reaches -0.34(3) $\mu$V K-2 for FeS0.94Se0.06, with a magnitude proportional
to the strength of electron correlation Behnia . The $S$ is usually given by
Barnard ; Miyake ; Behnia ,

$\frac{S}{T}=\pm\frac{\pi^{2}}{2}\frac{k_{\textrm{B}}}{e}\frac{1}{T_{\textrm{F}}}=\pm\frac{\pi^{2}}{3}\frac{k_{\textrm{B}}^{2}}{e}\frac{N(\varepsilon_{\textrm{F}})}{n},$
(3)

where $e$ is the electron charge, $k_{\textrm{B}}$ is the Boltzmann constant,
$T_{\textrm{F}}$ is the Fermi temperature, which is related to the Fermi
energy $\varepsilon_{\textrm{F}}$ and the density of states
$N(\varepsilon_{\textrm{F}})$ as $N(\varepsilon_{\textrm{F}})$ =
$3n/2\varepsilon_{\textrm{F}}$ = $3n/k_{\textrm{B}}T_{\textrm{F}}$, and $n$ is
the carrier concentration (the positive sign is for hole and the negative sign
is for electron). In a multiband system, it gives the upper limit of
$T_{\textrm{F}}$ of the dominant band. Therefore we can extract
$T_{\textrm{F}}$ = $2.8(2)\times 10^{3}$ K for FeS and $1.25(11)\times 10^{3}$
K for FeS0.94Se0.06, respectively. The ratio of
$T_{\textrm{c}}/T_{\textrm{F}}$ characterizes the correlation strength in
superconductors. For example, the $T_{\textrm{c}}/T_{\textrm{F}}$ is close to
0.1 in Fe1+yTe1-xSex Pourret , pointing to the importance of electronic
correlation, while it is $\sim$ 0.02 in BCS superconductor, such as LuNi2B2C
Pourret . The value of $T_{\textrm{c}}/T_{\textrm{F}}$ is $\sim$ 0.0016(1) for
FeS and $\sim$ 0.0033(2) for FeS0.94Se0.06, indicating weak electronic
correlations but also a substantial enhancement for rather small amount of Se
substitution on S atomic site. The value of $T_{\textrm{c}}/T_{\textrm{F}}$ is
also smaller than that of KxFe2-ySe2 $\sim$ 0.04 Kefeng1 , but comparable with
that of KxFe2-ySe1-zSz $\sim$ 0.005 Kefeng2 , which are also added into the
Moriya-Ueda plot for comparison [Fig. 2(c)] Pourret ; Sarrao ; Moriya .

Figure 3 shows the Hall resistivity $\rho_{xy}$ of FeS0.94Se0.06. At low
temperatures, the $\rho_{xy}$ shows a nonlinear correlation with the magnetic
field, similar to FeS Lin , confirming a multiband effect. With increasing
temperature, the nonlinear $\rho_{xy}$ gradually evolves to linear at high
temperatures (above 110 K). Then we estimated the Hall coefficient $R_{H}$ by
fitting the low field data in Fig. 3(b), and plotted the temperature-dependent
$R_{H}$ in Fig. 3(c). The $R_{H}$ is negative in the whole temperature range,
indicating dominant electron carriers, as well as a nonmonotonic temperature
dependence, similar with that in FeS Lin .

Figure 4: (a) Temperature dependence of heat capacity $C_{\textrm{p}}/T$ vs
$T^{2}$ for for FeS1-xSex ($x$ = 0 and 0.06) single crystals. The solid curve
represents the fittings using $C_{\textrm{p}}/T=\gamma+\beta T^{2}$. (b) The
low temperature electronic part $C_{e}/T$ vs $T^{2}$ in zero field.

The temperature-dependent heat capacity $C_{\textrm{p}}/T$ in the low-
temperature range as a function of $T^{2}$ in zero magnetic field for
FeS1-xSex ($x$ = 0, 0.06) is depicted in Fig. 4(a). A clear specific heat jump
is observed at $T_{c}$ for both samples, in line with the resistivity and
thermopower data, indicating bulk superconductivity. The data from 4.5 to 7 K
can be well fitted by $C_{\textrm{p}}/T=\gamma+\beta T^{2}$ [Fig. 4(a)], where
the first term is the Sommerfeld electronic specific heat coefficient, the
second term is low-temperature limit of lattice heat capacity. The derived
$\gamma$ is 5.56(8) and 12.60(9) mJ mol-1 K-2 for FeS and FeS0.94Se0.06,
respectively, comparable with that of FeSe. The Debye temperature
$\Theta_{\textrm{D}}=(12\pi^{4}\textrm{NR}/5\beta)^{1/3}$, where $N$ = 2 is
the number of atoms per formula unit and $R$ = 8.314 J mol-1 K-1 is the molar
gas constant. From the value of $\beta$ $\sim$ 0.238(2) and 0.262(2) mJ mol-1
K-3, we can calculate the value of $\Theta_{\textrm{D}}$ is 253.7(3) and
245.7(3) K for FeS and FeS0.94Se0.06, respectively, larger than that of FeSe
($\sim$ 210 K) XLin . According to the McMillan formula for electron-phonon
mediated superconductivity, the electron-phonon coupling constant $\lambda$
can be deduced by

$T_{c}=\frac{\Theta_{D}}{1.45}\exp[-\frac{1.04(1+\lambda)}{\lambda-\mu^{\ast}(1+0.62\lambda)}],$
(4)

where $\mu^{\ast}\approx$ 0.13 is the common value for Coulomb pseudo-
potential McMillan . By using the $T_{c}$ = 4.5(1) K and $\Theta_{D}$ =
253.7(3) K for FeS, we obtain $\lambda\approx$ 0.65(11), a typical value of an
intermediate-coupled BCS superconductor. However, the electronic specific heat
jump at $T_{c}$ [Fig. 4(b)], $\Delta$Ce/$\gamma T_{c}\approx$ 0.43, is much
smaller than the weak coupling value of 1.43 Xing ; McMillan .

Figure 5: (Color online) The density of states in the vicinity of the Fermi
level of the tetragonal Fe16S16 supercell (blue) and the tetragonal Fe16S15Se1
supercell (red) for: (a) Total; (b) Total Fe part; (c) Total chalcogen part;
and (d) S ion in the Fe16S16 supercell (cyan) and Se ion the Fe16S15Se1
supercell (magenta).

The electronic specific heat can be also expressed as:

$\gamma=\frac{\pi^{2}}{2}k_{\textrm{B}}\frac{n}{T_{\textrm{F}}}=\frac{\pi^{2}}{3}k_{\textrm{B}}^{2}N(\varepsilon_{\textrm{F}}).$
(5)

Combining equations (3) and (5) yields: $S/T=\pm\gamma/ne$, where the units
are V K-1 for $S$, J K-2 m-3 for $\gamma$, and m-3 for $n$, respectively. This
relation was shown to hold in the $T$ = 0 limit for a large variety of
materials, even in the presence of strong correlations, including heavy
fermion metals, organic conductors, and cuprates Behnia . Then we can define a
dimensionless quantity

$q=\frac{S}{T}\frac{N_{\textrm{A}}e}{\gamma},$ (6)

where $N_{\textrm{A}}$ is the Avogadro constant. The constant
$N_{\textrm{A}}e$ = 9.6$\times$104 C mol-1 is the Faraday constant. The $q$
gives the number of carriers per formula unit (proportional to $1/n$ where $n$
is the carrier concentration) Behnia . The derived $q$ is $\sim$ 2.6(1) for
FeS, which is almost unchanged for FeS0.94Se0.06 $\sim$ 2.6(2), in line with
equivalent doping S at Se sites. Given the volume of unit cell $\sim$ 0.068
nm3, we obtain the carrier density per volume $n\approx 5.7(3)\times 10^{21}$
cm-3 and the Fermi momentum $k_{\textrm{F}}=(3\pi^{2}n)^{1/3}\approx 5.5(1)$
nm-1 for FeS, which are almost unchanged with $n\approx 5.7(5)\times 10^{21}$
cm-3 and $k_{\textrm{F}}\approx 5.5(2)$ nm-1 for FeS0.94Se0.06, respectively.
The effective mass $m^{*}$, derived from
$k_{\textrm{B}}T_{\textrm{F}}=\hbar^{2}k_{\textrm{F}}^{2}/2m^{*}$, increases
from 4.8 $m_{e}$ for FeS to 9.8 $m_{e}$ for FeS0.94Se0.06, consistent with the
increase of correlation strength with Se substitution. This confirms
substantial increase in electron correlations in FeS0.94Se0.06. Furthermore,
we can derive the Fermi velocity $\nu_{\textrm{F}}$ using $\hbar
k_{\textrm{F}}=m^{*}\nu_{\textrm{F}}$; it decreases from 133 km s-1 for FeS to
65 km s-1 for FeS0.94Se0.06.

First-principle calculations (Fig. 5) show that the Fermi level is on a
pseudo-gap-like slope of the low energy side for both compounds. The density
of states (DOS) in the vicinity of Fermi level for both FeS and FeS0.94Se0.06
is presented in Fig. 5(a). Partial contribution of Fe and chalcogen atoms is
presented in Fig. 5(b,c); the result confirms that the Fe electronic states
are dominant at the Fermi level but also that there is a small increase in
partial contribution of chalcogen atoms to DOS for Se-doped crystal [Fig.
4(c)] Yang2 ; MiaoJ . Close inspection of the region near the Fermi level for
small Se substitution on S atomic site, i.e. for FeS0.94S0.06 [Fig. 5(d)]
indicates increase in DOS. This confirms experimental observation but also
points out that DOS changes are connected with orbital states that are
partially chalcogen-derived and whose contribution is not negligible Yang2 .

##  IV CONCLUSIONS

In summary, thermopower analysis of tetragonal FeS and FeS0.94Se0.06 indicates
weak electron correlation. Both $T_{c}$/$T_{F}$ and effective mass $m$∗
substantially increase with slight Se substitution as small as 0.06. Since
drastic changes in electronic structure and emergence of orbital-selective
correlations associated with Lifshitz transition have been proposed to emerge
with small lattice expansion in FeS SkornyakovSL , our results call for
investigation of electronic structure in FeS1-xSex crystals.

## Acknowledgements

Work at Brookhaven National Laboratory (BNL) is supported by the Office of
Basic Energy Sciences, Materials Sciences and Engineering Division, U.S.
Department of Energy (DOE) under Contract No. DE-SC0012704. First principle
calculations were supported by the Ministry of Education, Science and
Technological Development of the Republic of Serbia.
∗Y.L. and A.W. contributed equally to this work.
†Present address: Los Alamos National Laboratory, Los Alamos, New Mexico
87545, USA
∥Present address: College of Physics, Chongqing University, Chongqing 401331,
China
♯Present address: Material Science Division, Argonne National Laboratory,
Lemont, Illinois 60439, USA
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
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