# Bounds on eigenfunctions of quantum cat maps

Elena Kim  Massachusetts Institute of Technology, Department of Mathematics,
Cambridge, MA 02142, USA<EMAIL_ADDRESS>and  Robert Koirala  University of
California San Diego, Department of Mathematics, La Jolla, CA 92093, USA
<EMAIL_ADDRESS>
###### Abstract.

We study $\ell^{\infty}$ norms of $\ell^{2}$-normalized eigenfunctions of
quantum cat maps. For maps with short quantum periods (constructed by Bonechi
and de Bièvre in [Bonechi-
DeBievre2000_Article_ExponentialMixingAndTimeScales]) we show that there
exists a sequence of eigenfunctions $u$ with $\|u\|_{\infty}\gtrsim(\log
N)^{-1/2}$. For general eigenfunctions we show the upper bound
$\|u\|_{\infty}\lesssim(\log N)^{-1/2}$. Here the semiclassical parameter is
$h=(2\pi N)^{-1}$. Our upper bound is analogous to the one proved by Bérard in
[berard] for compact Riemannian manifolds without conjugate points.

The authors gratefully acknowledge Semyon Dyatlov, who is partially supported
by NSF CAREER grant DMS-1749858, for suggesting and mentoring this project.
The first named author is supported by NSF GRFP under Grant No. 1745302. The
second named author was supported by MIT UROP funding.

##  1\. Introduction

In this paper, we build on an existing body of work that examines the extreme
values of eigenfunctions of classically chaotic quantum systems. Specifically,
we examine the quantum cat map, one of the best-studied models. First
introduced by Berry and Hannay in [HB1980], cat maps are the quanitzation of
hyperbolic linear maps in $\text{SL}(2,\mathbb{Z})$ on the 2-dimensional
torus.

The quantum cat map is a toy model in quantum chaos. Another standard class of
quantum chaos models is given by Laplacian eigenfunctions on negatively curved
compact manifolds $(M,g)$, satisfying $-\Delta_{g}u=\lambda^{2}u$ and
normalized so that $\|u\|_{L^{2}}=1$. Estimating the suprema of these
Laplacian eigenfunctions has been an area of great interest. For example,
Levitan, Avakumović, and Hörmander in [levitan], [avakumovic], [hormander],
respectively, proved the well-known result that on a compact Riemannian
manifold, $\|u\|_{L^{\infty}}\lesssim\lambda^{\frac{n-1}{2}}$ where $n=\dim
M$. However, it is conjectured that much stronger results hold. Specifically,
in [iwaniecsarnak], Iwaniec and Sarnak conjectured that for surfaces of
constant negative curvature,
$\|u\|_{L^{\infty}}\lesssim_{\varepsilon}\lambda^{\varepsilon}$ for all
$\varepsilon>0$; in the special case of Hecke eigenfunctions on arithmetic
surfaces they obtained the bound
$\|u\|_{L^{\infty}}\lesssim_{\varepsilon}\lambda^{\frac{5}{12}+\varepsilon}$.
The best known bound outside of the arithmetic cases is the result of Bérard
[berard]: when $(M,g)$ has no conjugate points,
$\|u\|_{L^{\infty}}=O(\lambda^{\frac{n-1}{2}}/\sqrt{\log\lambda})$.

In this paper, we are concerned with metaplectic transformations, the
quantizations of hyperbolic maps

(1) $A=\begin{bmatrix}a&b\\\
c&d\end{bmatrix}\in\text{SL}(2,\mathbb{Z}),\quad|a+d|>2,\quad ab,cd\in
2\mathbb{Z}.$

We decompose $L^{2}(\mathbb{R}^{n})$ into a direct integral of finite
dimensional spaces $\mathcal{H}_{N}(\theta)$ where $\theta\in\mathbb{T}^{2}$
and $N\in\mathbb{N}$. As shown in Section 2, the condition that $ab,cd$ be
even is needed to make sure that the metaplectic map associated to $A$
descends to a map from $\mathcal{H}_{N}(0)$ to itself. We denote the resulting
quantum cat map by $M_{N,0}$. Then, we prove the following theorems bounding
the supremum norm of eigenfunctions on $\mathcal{H}_{N}(0)$. See Figure 1 for
a numerical illustration. Note that due to the normalization in the spaces
$\ell^{2}$ and $\ell^{\infty}$, the standard bound on Laplacian eigenfunctions
$\|u\|_{L^{2}}\lesssim\|u\|_{L^{\infty}}\lesssim\lambda^{\frac{n-1}{2}}\|u\|_{L^{2}}$
becomes the bound
$\frac{1}{\sqrt{N}}\|u\|_{\ell^{2}}\leq\|u\|_{\ell^{\infty}}\leq\|u\|_{\ell^{2}}$.

For a sequence of $N$, we have the following lower bound.

######  Theorem 1.

Suppose $A$ is a matrix satisfying (1) with positive eigenvalues, even trace,
and coprime off-diagonal terms. Then we can find a sequence of odd integers
$N_{k}\to\infty$ such that for all $\varepsilon>0$, there exists $k_{0}$ such
that for all $k\geq k_{0}$, there exists an eigenfunction $u$ of $M_{N,0}$
with $\|u\|_{\ell^{2}}=1$ and

(2)
$\displaystyle\|u\|_{\ell^{\infty}}\geq\frac{1-\varepsilon}{\sqrt{2\log_{\lambda}N_{k}}}.$

Under more general assumptions, for all odd $N$, we have the following upper
bound.

######  Theorem 2.

Suppose $A$ is a matrix satisfying (1). Then for $\varepsilon>0$, there exists
$N_{0}$ such that for all odd $N\geq N_{0}$, if $u$ is an eigenfunction of
$M_{N,0}$ with $\|u\|_{\ell^{2}}=1$ then

(3)
$\displaystyle\|u\|_{\ell^{\infty}}\leq\frac{1}{\sqrt{(1-\varepsilon)\log_{\lambda}N}}.$

Figure 1. The plot of the maximal $\ell^{\infty}$-norm of an
$\ell^{2}$-normalized eigenfunction of $M_{N,0}$ where $M_{N,0}$ is associated
with $A=\begin{bmatrix}2&3\\\ 1&2\end{bmatrix}$. The lower and upper bounds
from Theorems 1 and 2, $(2\log_{\lambda}N)^{-1/2}$ and
$(\log_{\lambda}N)^{-1/2}$, respectively, give the solid red lines. The dotted
blue line is the trivial lower bound $N^{-1/2}$. Finally, the large red dots
correspond to the sequence $N_{k}$ from Theorem 1.

Analogous statements can be proven for a sequence of even $N$ and any
$\theta$, using similar proofs to those of Theorems 1 and 2. However, we
exclude these arguments as they are overly technical and do not introduce any
novel ideas.

The $\ell^{\infty}$ bounds on eigenfunctions of quantum cat maps have been
extensively studied in arithmetic quantum chaos, see [Kurlberg-Rudnick-1,
Kurlberg-1, Olofsson-1, Olofsson-2, Olofsson-3]. These works have focused on
_Hecke eigenfunctions_ , which are joint eigenfunctions of the quantum cat map
$M_{N,0}$ and the Hecke operators, constructed in this setting by
Kurlberg–Rudnick [Kurlberg-Rudnick-0]. There always exists an orthonormal
basis of $\mathcal{H}_{N}(0)$ consisting of Hecke eigenfunctions; however, due
to the possibility of large multiplicities of the eigenspaces of $M_{N,0}$
(see [Kurlberg-Rudnick-0, footnote 3]) an upper bound on the $\ell^{\infty}$
norm of Hecke eigenfunctions does not imply the same bound for general
eigenfunctions. We list below the known bounds on $\ell^{2}$-normalized Hecke
eigenfunctions $u$:

  * •

Kurlberg–Rudnick [Kurlberg-Rudnick-1] proved the upper bound
$\|u\|_{\ell^{\infty}}\lesssim_{\varepsilon}N^{-\frac{1}{8}+\varepsilon}$.

  * •

Building on [Kurlberg-Rudnick-1] (which handled roughly half of the prime
values of $N$), Kurlberg [Kurlberg-1] showed that for all but finitely many
_prime_ values of $N$ one has the upper bound $\|u\|_{\ell^{\infty}}\leq
2N^{-\frac{1}{2}}$, and deduced the upper bound
$\|u\|_{\ell^{\infty}}\lesssim_{\varepsilon}N^{-\frac{1}{2}+\varepsilon}$ for
_square-free_ values of $N$.

  * •

Olofsson [Olofsson-1, Olofsson-2] showed an upper bound
$\|u\|_{\ell^{\infty}}\lesssim N^{-\frac{1}{4}}$ for _most_ values of $N$ (in
the sense of density as $N\to\infty$). On the other hand, these papers also
construct eigenfunctions satisfying a lower bound
$\|u\|_{\ell^{\infty}}\gtrsim N^{-\frac{1}{4}}$ for most values of $N$ which
are _not square-free_.

##  2\. Preliminaries

We begin with a review of the necessary definitions for this paper. First,
recall the semiclassical Weyl quantization. For
$a\in\mathscr{S}(\mathbb{R}^{2})$ and a semiclassical parameter $h\in(0,1]$,

$\text{Op}_{h}(a)f(x):=\frac{1}{2\pi
h}\int_{\mathbb{R}^{2}}e^{\frac{i}{h}(x-x^{\prime})\xi}a\left(\frac{x+x^{\prime}}{2},\xi\right)f(x^{\prime})dx^{\prime}d\xi,\quad
f\in\mathscr{S}(\mathbb{R}).$

Define the symbol class

$S(1)=\left\\{a\in
C^{\infty}\left(\mathbb{R}^{2}\right):\sup_{(x,\xi)\in\mathbb{R}^{2}}\left|\partial^{\alpha}_{(x,\xi)}a\right|<\infty\text{
for all }\alpha\in\mathbb{N}^{2}\right\\},$

which naturally induces the seminorms $\|a\|_{C^{m}}:=\max_{|\alpha|\leq
m}\sup_{\mathbb{R}^{2}}|\partial_{(x,\xi)}^{\alpha}a|$ for
$m\in\mathbb{N}_{0}$. From [z12semiclassical]*Theorem 4.16, we know for $a\in
S(1)$, $\text{Op}_{h}(a)$ acts on both $\mathscr{S}(\mathbb{R})$ and
$\mathscr{S}^{\prime}(\mathbb{R})$.

Now, let $\omega=(y,\eta),z=(x,\xi)\in\mathbb{R}^{2}$. Define the standard
symplectic form $\sigma$ on $\mathbb{R}^{2}$ by $\sigma(z,\omega):=\xi
y-x\eta$ and define the quantum translation by
$U_{\omega}:=\text{Op}_{h}(a_{\omega})$, where
$a_{\omega}(z):=\exp(\frac{i}{h}\sigma(\omega,z))$. Noting that
$a_{\omega}(z)\in S(1)$, we see $U_{\omega}$ is well-defined and acts on
$\mathscr{S}(\mathbb{R})$. In [z12semiclassical]*Theorem 4.7, it is shown that

$U_{\omega}f(x)=e^{\frac{i}{h}\eta x-\frac{i}{2h}y\eta}f(x-y).$

Thus, $U_{\omega}$ is a unitary operator on $L^{2}(\mathbb{R})$ that satisfies
the following exact Egorov’s theorem,

(4)
$U^{-1}_{\omega}\text{Op}_{h}(a)U_{\omega}=\text{Op}_{h}(\tilde{a})\quad\text{for
all }a\in S(1),\quad\tilde{a}(z):=a(z+\omega).$

From the fact that
$U_{\omega}U_{\omega^{\prime}}=e^{\frac{i}{2h}\sigma(\omega,\omega^{\prime})}U_{\omega+\omega^{\prime}}$,
we deduce the following commutator formula,

(5)
$U_{\omega}U_{\omega^{\prime}}=e^{\frac{i}{h}\sigma(\omega,\omega^{\prime})}U_{\omega^{\prime}}U_{\omega}.$

Now let $\text{Sp}(2,\mathbb{R})$ be the group of real symplectic $2\times 2$
matrices. In other words, $A\in\text{Sp}(2,\mathbb{R})$ if and only if
$\sigma(Az,A\omega)=\sigma(z,\omega)$. Note that in this specific
2-dimensional case, $\text{Sp}(2,\mathbb{R})=\text{SL}(2,\mathbb{R})$. For
each $A\in\text{SL}(2,\mathbb{R})$, denote by $\mathcal{M}_{A}$ the set of all
unitary transformations $M:L^{2}(\mathbb{R})\rightarrow L^{2}(\mathbb{R})$
satisfying the following exact Egorov’s theorem,

(6) $M^{-1}\text{Op}_{h}(a)M=\text{Op}_{h}(a\circ A)\quad\text{for all }a\in
S(1).$

From [z12semiclassical]*Theorem 11.9, we have both existence of these
transformations and uniqueness up to a unit factor.

Then, $\mathcal{M}:=\cup_{A\in\text{SL}(2,\mathbb{R})}\mathcal{M}_{A}$ is a
subgroup of unitary transformations of $L^{2}(\mathbb{R})$ called the
metaplectic group and the map $M\mapsto A$ is a group homomorphism
$\mathcal{M}\rightarrow\text{SL}(2,\mathbb{R})$. As a corollary of equation
(6), we obtain the following intertwining of the metaplectic and quantum
transformations: $M^{-1}U_{\omega}M=U_{A^{-1}\omega}$ for all
$M\in\mathcal{M}_{A}$, $\omega\in\mathbb{R}^{2}.$

We turn our attention to quantizations of functions on the torus
$\mathbb{T}^{2}:=\mathbb{R}^{2}/\mathbb{Z}^{2}$. Each $a\in
C^{\infty}(\mathbb{T}^{2})$ can be identified with a $\mathbb{Z}^{2}$-periodic
function on $\mathbb{R}^{2}$. Note that any $a\in C^{\infty}(\mathbb{T}^{2})$
is also an element of $S(1)$, therefore its Weyl quantization
$\text{Op}_{h}(a)$ is an operator on $L^{2}(\mathbb{R})$.

By equation (5), we have the following commutation relations,

(7) $\text{Op}_{h}(a)U_{\omega}=U_{\omega}\text{Op}_{h}(a)\quad\text{for all
}a\in C^{\infty}(\mathbb{T}^{2}),\quad\omega\in\mathbb{Z}^{2}.$

These commutation relations motivate a decomposition of $L^{2}(\mathbb{R})$
into a direct integral of finite dimensional spaces $\mathcal{H}_{N}(\theta)$,
where $\theta\in\mathbb{T}^{2}$, such that $\text{Op}_{h}(a)$ descends onto
these spaces. From [Bouzouina-deBievre]*Proposition 2.1, to ensure the these
spaces are nontrivial, for the rest of the paper, we assume

$h=(2\pi N)^{-1}\quad\text{where }N\in\mathbb{N}.$

We call $\mathcal{H}_{N}(\theta)$ the space of quantum states. Specifically,
for each $\theta\in\mathbb{T}^{2}$, set

$\mathcal{H}_{N}(\theta):=\left\\{f\in\mathscr{S}^{\prime}(\mathbb{R}):U_{\omega}f=e^{2\pi
i\sigma(\theta,\omega)+N\pi iQ(\omega)}f\text{ for all
}\omega\in\mathbb{Z}^{2}\right\\},$

where the quadratic form $Q$ on $\mathbb{R}^{2}$ is defined by
$Q(\omega)=y\eta$ for $\omega=(y,\eta)\in\mathbb{R}^{2}$. The following lemma
gives an explicit basis for $\mathcal{H}_{N}(\theta)$.

######  Lemma 3 ([dyatlov2021semiclassical], Lemma 2.5).

The space $\mathcal{H}_{N}(\theta)$ is $N$-dimensional with a basis
$\\{e_{j}^{\theta}\\}$ defined for $j\in\\{0,\ldots,N-1\\}$ and
$\theta=(\theta_{x},\theta_{\xi})\in\mathbb{R}^{2}$. In particular,

$e_{j}^{\theta}(x):=\frac{1}{\sqrt{N}}\sum_{k\in\mathbb{Z}}e^{-2\pi
i\theta_{\xi}k}\delta\left(x-\frac{Nk+j-\theta_{x}}{N}\right).$

We fix an inner product $\left\langle\cdot,\cdot\right\rangle_{\mathcal{H}}$
on each quantum state $\mathcal{H}_{N}(\theta)$ by requiring
$\\{e_{j}^{\theta}\\}$ to be an orthonormal basis. It can be shown using
translation idenitities for $e_{j}^{\theta}$ (see
[dyatlov2021semiclassical]*(2.35)) that although each $\\{e_{j}^{\theta}\\}$
depends on the choice of the representative $\theta_{x}\in\mathbb{R}$, the
inner product depends only on $\theta\in\mathbb{T}^{2}$. We often denote the
norm defined by this inner product by $\|\cdot\|_{\ell^{2}}$. Using the bases
$\\{e_{j}^{\theta}\\}$, we can consider the spaces $\mathcal{H}_{N}(\theta)$
as fibers of a smooth $N$ dimensional vector bundle over $\mathbb{T}^{2}$,
which we call $\mathcal{H}_{N}$.

For fixed $N\in\mathbb{N}$ and $a\in C^{\infty}(\mathbb{T}^{2})$, define the
quantization

$\text{Op}_{N,\theta}(a):=\text{Op}_{h}(a)|_{\mathcal{H}_{N}(\theta)}:\mathcal{H}_{N}(\theta)\rightarrow\mathcal{H}_{N}(\theta),\quad\theta\in\mathbb{T}^{2},$

which depends smoothly on $\theta$. This restriction holds by definition of
$\mathcal{H}_{N}(\theta)$ and the commutation relations given in equation (7).

We set

(8) $A=\begin{bmatrix}a&b\\\ c&d\end{bmatrix}\in\text{SL}(2,\mathbb{Z})$

and choose a metaplectic operator $M\in\mathcal{M}_{A}$. Recall that for
$\omega=(y,\eta),z=(x,\xi)\in\mathbb{Z}^{2}$, $\sigma(z,\omega)=\xi y-x\eta$
and $Q(\omega)=y\eta$. By [dyatlov2021semiclassical]*Lemma 2.9, there exists a
unique $\varphi_{A}\in(\mathbb{Z}/2)^{2}$ such that for all
$\omega\in\mathbb{Z}^{2}$,
$Q(A^{-1}\omega)-Q(\omega)=\sigma(\varphi_{A},\omega)\mod 2\mathbb{Z}$. Using
the definition of $\mathcal{H}_{N}(\theta)$ and that fact that
$M^{-1}U_{\omega}M=U_{A^{-1}\omega}$, we can verify that
$M(\mathcal{H}_{N}(\theta))\subset\mathcal{H}(A\theta+\frac{N\varphi_{A}}{2})$
for all $\theta\in\mathbb{T}^{2}$.

Denote
$M_{N,\theta}:=M|_{\mathcal{H}_{N}(\theta)}:\mathcal{H}_{N}(\theta)\rightarrow\mathcal{H}_{N}(A\theta+\frac{N\varphi_{A}}{2})$,
which depends smoothly on $\theta\in\mathbb{T}^{2}$. We require the domain and
range of $M_{N,\theta}$ to be the same, in other words, we must have

(9) $(I-A)\theta=\frac{N\varphi_{A}}{2}\mod\mathbb{Z}^{2}.$

Thus, when $\theta=0$, condition (9) is satisfied when $N$ is even or
$\varphi_{A}=0$.

We henceforth assume that $\varphi_{A}=0$, which gives
$\sigma(\varphi_{A},\omega)=0\mod\mathbb{Z}/2$. Noting that for all
$\omega=(y,\eta)$,

$Q(A^{-1}\omega)-Q(\omega)=(dy-b\eta)(-cy+a\eta)-y\eta=-dcy^{2}+2bcy\eta-
ba\eta^{2},$

we must have $dcy^{2}+ba\eta^{2}=0\mod 2\mathbb{Z}$. We conclude that $dc$ and
$ab$ must be even.

Assuming condition (9), we have the following exact Egorov’s theorem for all
$a\in C^{\infty}(\mathbb{T}^{2})$,

$M_{N,\theta}^{-1}\text{Op}_{N,\theta}(a)M_{N,\theta}=\text{Op}_{N,\theta}(a\circ
A).$

Essential to our proof of Theorem 2 is the following explicit formula for $M$.

######  Lemma 4.

When $A$ is given by (8) and $b\neq 0$, an element of $\mathcal{M}_{A}$ is
given by

(10) $Mu(x)=\frac{\sqrt{N}}{\sqrt{|b|}}\int_{\mathbb{R}}e^{2\pi
Ni\Phi(x,y)}u(y)dy,$

where

(11) $\Phi(x,y)=\frac{d}{2b}x^{2}-\frac{xy}{b}+\frac{a}{2b}y^{2}.$

We provide a short outline of this formula’s derivation; for further details
see Theorem 11.10 and its following remark in [z12semiclassical]. Set
$\Lambda=\\{(x,y,\xi,-\eta):(x,\xi)=A(y,\eta)\\}$ and note that $\Lambda$ is a
Lagrangian submanifold of $\mathbb{R}^{4}$. Additionally, as $b\neq 0$,
$\Lambda\ni(x,y,\xi,\eta)\mapsto(x,\eta)$ is surjective. Therefore, there
exists a generating function $\Phi(x,y)$ such that
$\Lambda=\\{(x,y,\partial_{x}\Phi,\partial_{y}\Phi)\\}$. As
$\eta=-\partial_{y}\Phi$ and $\xi=\partial_{x}\Phi$, we know $\Phi(x,y)$ is
given by (11). We then have (10), where the coefficient ensures $M$ is
unitary, up to a unit factor.

##  3\. Proof of Theorem 1

First, following the presentation in [Bonechi-
DeBievre2000_Article_ExponentialMixingAndTimeScales], in Section 3.1, we show
that for a sequence $N_{k}$, $M_{N_{k},0}$ has a large degenerate eigenspace.
In Section 3.2, we then reduce the proof of Theorem 1 to a linear algebra
argument.

###  3.1. Eigenspace of $M_{N_{k},0}$

Consider $A$ given by (8) such that $b$ and $c$ are coprime and
$\operatorname{Tr}A$ is even and greater than 2. Let $\lambda$ be the largest
eigenvalue of $A$. Then for each $t\in\mathbb{N}$,

(12) $A^{t}=p_{t}A-p_{t-1}I,\quad
p_{t+1}=\operatorname{Tr}(A)p_{t}-p_{t-1},\text{ where}\quad
p_{t}=\frac{\lambda^{t}-\lambda^{-t}}{\lambda-\lambda^{-1}}\in\mathbb{Z}.$

Set $T_{N}=\min\\{t:A^{t}=I\mod N\\}$ and, for $k\in\mathbb{N}$, define

$N^{\prime}_{k}:=\max\\{N:A^{k}=I\mod N\\}.$

Essential to our proof of Theorem 1 is the following statement from [Bonechi-
DeBievre2000_Article_ExponentialMixingAndTimeScales]. For the reader’s
convenience, we replicate their proof here.

######  Theorem 5 ([Bonechi-
DeBievre2000_Article_ExponentialMixingAndTimeScales], Prop. 11).

For each $k\in\mathbb{N}$, we have $N^{\prime}_{2k}=2p_{k}$,
$N^{\prime}_{2k+1}=p_{k}+p_{k+1}$, and $T_{N^{\prime}_{k}}=k$.

###### Proof.

Using equation (12), we see that $N^{\prime}_{k}$ is the greatest integer such
that

$\begin{bmatrix}p_{k}a-p_{k-1}-1&p_{k}b\\\
p_{k}c&p_{k}d-p_{k-1}-1\end{bmatrix}=\begin{bmatrix}0&0\\\
0&0\end{bmatrix}\mod N^{\prime}_{k}.$

Recall that we assumed $b$ and $c$ are coprime. Therefore, $p_{k}=0\mod
N^{\prime}_{k}$ and $p_{k-1}=-1\mod N^{\prime}_{k}$, which gives
$N^{\prime}_{k}=\gcd(p_{k},p_{k-1}+1)$. We claim that for $s=0,\ldots,k-1$,

(13) $N^{\prime}_{k}=\gcd(p_{k-s}+p_{s},p_{k-(s+1)}+p_{s+1}).$

We proceed by induction. Note that $p_{0}=0$ and $p_{1}=1$, therefore, (13)
clearly holds for $s=0$. Now suppose we know (13) for some $s\geq 0$. Using
(12) and the identity $\gcd(a,ca-b)=\gcd(a,b)$, we have

$\displaystyle N^{\prime}_{k}$
$\displaystyle=\gcd(\operatorname{Tr}Ap_{k-s-1}-p_{k-s-2}+p_{s},\text{
}p_{k-s-1}+p_{s+1})$
$\displaystyle=\gcd(\operatorname{Tr}A(p_{k-s-1}+p_{s+1})-\operatorname{Tr}Ap_{s+1}-p_{k-s-2}+p_{s},\text{
}p_{k-s-1}+p_{s+1})$ $\displaystyle=\gcd(p_{s+2}+p_{k-(s+2)},\text{
}p_{k-(s+1)}+p_{s+1}),$

which completes the induction. We set $k=2\ell$, $s=\ell$ in (13) to conclude

$N^{\prime}_{2\ell}=\gcd(2p_{\ell},p_{\ell-1}+p_{\ell+1})=\gcd(2p_{\ell},\operatorname{Tr}Ap_{\ell})=2p_{\ell},$

where the last equality follows from our assumption that $\operatorname{Tr}A$
is even. Similarly, setting $k=2\ell+1$ and $s=\ell$ in (13) gives
$N^{\prime}_{2\ell+1}=p_{\ell}+p_{\ell+1}$.

Now note that for each $k$, we have

(14) $A^{k}=1\mod N^{\prime}_{k},$ (15) $A^{T_{N^{\prime}_{k}}}=1\mod
N^{\prime}_{k},\quad\text{and}\quad A^{T_{N^{\prime}_{k}}}=1\mod
N^{\prime}_{T_{N^{\prime}_{k}}}.$

From the definition of $T_{N^{\prime}_{k}}$ and (14), we see that
$T_{N^{\prime}_{k}}\leq k$. From the definition of $N^{\prime}_{k}$ and (15),
we see that $N^{\prime}_{k}\leq N^{\prime}_{T_{N^{\prime}_{k}}}$. As
$\\{N^{\prime}_{k}\\}$ is increasing, we conclude that $T_{N^{\prime}_{k}}\geq
k$. Therefore, $T_{N^{\prime}_{k}}=k$. ∎

Now, let $n(N)$ denote the period of $M_{N,0}$; specifically

$n(N):=\min\\{t:M_{N,0}^{t}=e^{i\varphi}\text{ for some
}\varphi\in\mathbb{R}\\}.$

Suppose $A_{N}$ is the matrix with integer entries that satisfies
$A^{T_{N}}=1+NA_{N}$. From [HB1980]*(36)-(46), we know that $n(N)=T_{N}$ if
$N$ is odd or if $N$ is even and $(A_{N})_{12}$ and $(A_{N})_{21}$ are even.
Otherwise, $n(N)=2T_{N}$.

Using this formula for $n(N)$ and Theorem 5, following [Bonechi-
DeBievre2000_Article_ExponentialMixingAndTimeScales], we show an upper bound
for $n(N)$ that depends only on $\lambda$ and $N$.

(a) $N=991$ (b) $N=989$

Figure 2. The plots of a maximal $\ell^{\infty}$-norm, $\ell^{2}$-normalized
eigenfunction of $M_{N,0}$, where $M_{N,0}$ corresponds to
$A=\begin{bmatrix}2&3\\\ 1&2\end{bmatrix}$. Specifically, each plot point
corresponds to the absolute value of the $i$th coordinate of the eigenfunction
for $0\leq i\leq N-1$. Note that $N=989$ is an element of the sequence $N_{k}$
in Corollary 6, while $N=991$ is not.

######  Corollary 6.

Let $\lambda$ be the largest eigenvalue of $A$. Then, there exists a sequence
of odd $N_{k}$ such that $2\log_{\lambda}N_{k}+1\geq n(N_{k})$.

###### Proof.

Using our assumption that $\operatorname{Tr}A$ is even, we note that $p_{2k}$
is even and $p_{2k+1}$ is odd. Thus,
$n(N^{\prime}_{2k+1})=T_{N^{\prime}_{2k+1}}=2k+1$.

We now formulate a bound for $n(N^{\prime}_{2k+1})$ in terms of
$N^{\prime}_{2k+1}$. As $\operatorname{Tr}A>2$, we know $\lambda>1$.
Additionally using Theorem 5,
$N^{\prime}_{2k+1}=\frac{\lambda^{k+1}+\lambda^{k}-\lambda^{-k-1}-\lambda^{-k}}{\lambda-\lambda^{-1}}\geq\lambda^{k}$.
Therefore, $\log_{\lambda}N^{\prime}_{2k+1}\geq k$, which gives
$2\log_{\lambda}(N^{\prime}_{2k+1})+1\geq 2k+1=n(N^{\prime}_{2k+1})$.
Labeling, $N^{\prime}_{2k+1}$ as $N_{k}$, we are done. ∎

Note that the same proof ideas can be adapted for a sequence of even $N$,
using the fact that $N^{\prime}_{2k}$ is even. However, as the odd case is the
simpler of the two, our main theorem is proven for a sequence of odd $N$.

For ease of notation, again using $N_{k}:=N^{\prime}_{2k+1}$, set

$t_{k}:=n(N_{k}).$

Therefore, if $u$ is an eigenfunction of $M_{N_{k},0}$ with eigenvalue
$\lambda$, $M_{N_{k},0}^{t_{k}}u=cu=\lambda^{t_{k}}cu$ for some $|c|=1$. Thus,
each eigenvalue of $M_{N_{k},0}$ satisfies $\lambda^{t_{k}}=1$, giving an
eigenvalue whose multiplicity is at least $\frac{N_{k}}{t_{k}}$. In other
words, $M_{N_{k},0}$ has a degenerate eigenspace with dimension at least
$\frac{N_{k}}{t_{k}}.$

###  3.2. Lower Bound

Now we are ready to state the lower bound in linear algebraic terms.

######  Proposition 7.

Suppose $V$ is subspace of $\mathbb{C}^{N}$ with dimension at least
$\frac{N}{t}$. Then,

(16) $\displaystyle\frac{1}{\sqrt{t}}\leq\max_{u\in V,\
\|u\|_{\ell^{2}}=1}\|u\|_{\ell^{\infty}}.$

###### Proof.

Define $\Pi:\mathbb{C}^{N}\to V$ to be the orthogonal projection onto $V$ and
let $e_{j}$ denote the $j$th coordinate vector. Note that

$\sum_{j=1}^{N}\|\Pi
e_{j}\|_{\ell^{2}}^{2}=\operatorname{Tr}(\Pi^{*}\Pi)=\operatorname{Tr}\Pi=\dim
V\geq\frac{N}{t}.$

Therefore, there exists a $j$ such that $\|\Pi
e_{j}\|_{\ell^{2}}\geq\frac{1}{\sqrt{t}}$. We then have

$\sup_{\begin{subarray}{c}u\in V\\\
\|u\|_{\ell^{2}}=1\end{subarray}}\|u\|_{\ell^{\infty}}=\max_{j}\sup_{\begin{subarray}{c}u\in\mathbb{C}^{N}\\\
\|u\|_{\ell^{2}}=1\end{subarray}}|\left\langle\Pi
u,e_{j}\right\rangle|=\max_{j}\sup_{\begin{subarray}{c}u\in\mathbb{C}^{N}\\\
\|u\|_{\ell^{2}}=1\end{subarray}}|\left\langle u,\Pi
e_{j}\right\rangle|=\max_{j}\|\Pi e_{j}\|_{\ell^{2}}\geq\frac{1}{\sqrt{t}},$

which completes the proof. ∎

We claim that Proposition 7 implies Theorem 1. In particular, fixing $V$ to be
the degenerate eigenspace of $M_{N_{k},0}$ with dimension at least
$\frac{N_{k}}{t_{k}}$ and using the fact that
$\mathbb{C}^{N_{k}}\simeq\mathcal{H}_{N_{k}}(0)$, Proposition 7 implies

(17)
$\displaystyle\frac{1}{\sqrt{2\log_{\lambda}N_{k}+1}}\leq\frac{1}{\sqrt{t_{k}}}\leq\max_{u\in
V,\ \|u\|_{\ell^{2}}=1}\|u\|_{\ell^{\infty}}.$

##  4\. Proof of Theorem 2

We examine $A\in\text{SL}(2,\mathbb{Z})$ of the form (1). Note that the
conditions on $A$ are less restrictive than those in Section 3. Let $N$ be
odd. We begin with the following dispersive estimate.

######  Proposition 8.

For $N$ odd and $M_{N,0}:\mathcal{H}_{N}(0)\rightarrow\mathcal{H}_{N}(0)$, we
have

$\displaystyle\left|\left\langle
M_{N,0}e_{j}^{0},e_{k}^{0}\right\rangle_{\mathcal{H}}\right|\leq\frac{\sqrt{|b|}}{\sqrt{N}}.$

###### Proof.

We begin by calculating $M_{N,0}e_{j}^{0}$. By Lemma 4, for
$\Phi(x,y)=\frac{d}{2b}x^{2}-\frac{xy}{b}+\frac{a}{2b}y^{2}$,

$M_{N,0}e^{0}_{j}(x)=\frac{1}{\sqrt{|b|}}\int_{\mathbb{R}}e^{2\pi
iN\Phi(x,y)}\sum_{k\in\mathbb{Z}}\delta\left(y-\frac{Nk+j}{N}\right)dy=\frac{1}{\sqrt{|b|}}\sum_{k\in\mathbb{Z}}e^{2\pi
iN\Phi\left(x,\frac{Nk+j}{N}\right)}.$

Thus, we focus on calculating $\Phi\left(x,\frac{Nk+j}{N}\right)$ mod
$N\mathbb{Z}$. In the following, we set $k=mb+r$ for $0\leq r\leq|b|-1$. We
have

$\Phi\left(x,\frac{Nk+j}{N}\right)=\frac{d}{2b}x^{2}-xm-\frac{xr}{b}-\frac{xj}{Nb}+\frac{ar^{2}}{2b}+\frac{arj}{bN}+\frac{aj}{2bN^{2}}\mod
N\mathbb{Z}.$

Thus, using the Poisson summation formula and the fact that
$\delta(x-x_{0})f(x)=\delta(x-x_{0})f(x_{0})$, we know

$\displaystyle M_{N,0}e_{j}^{0}(x)$
$\displaystyle=\frac{1}{\sqrt{|b|}}\sum_{r=0}^{|b|-1}e^{\frac{2\pi
i}{b}\left(\frac{aNr^{2}}{2}+arj+\frac{aj}{2N}\right)}\sum_{m\in\mathbb{Z}}e^{\frac{2\pi
iN}{b}\left(\frac{d}{2}x^{2}-xr-\frac{xj}{N}\right)}e^{-2\pi iNxm}$
$\displaystyle=\frac{1}{N\sqrt{|b|}}\sum_{l\in\mathbb{Z}}\sum_{r=0}^{|b|-1}e^{\frac{2\pi
i}{b}\left(\frac{dl^{2}}{2N}-lr-\frac{lj}{N}+\frac{ar^{2}N}{2}+arj+\frac{aj}{2N}\right)}\delta\left(x-\frac{l}{N}\right).$

Now setting

$c_{l}=\sum_{r=0}^{|b|-1}e^{\frac{2\pi
i}{b}\left(\frac{dl^{2}}{2N}-lr-\frac{lj}{N}+\frac{ar^{2}N}{2}+arj+\frac{aj}{2N}\right)},$

we want to show that $c_{l}$ has period $N$, in other words $c_{l}=c_{l+N}$.
We know that

$c_{l+N}=e^{\frac{2\pi
i}{b}\left(\frac{dl^{2}}{2N}+dl+\frac{dN}{2}-\frac{lj}{N}-j+\frac{aj}{2N}\right)}\sum_{r=0}^{|b|-1}e^{\frac{2\pi
i}{b}\left(-lr-
Nr+\frac{ar^{2}N}{2}+arj\right)}=\alpha_{l+N}\sum_{r=0}^{|b|-1}\beta_{l+N,r}.$

Recall in Section 2, we showed in order for $M_{N,0}$ to descend
$\mathcal{H}_{N}(0)$ to itself, $ab$ must even. Therefore,

$\beta_{l,r+b}=e^{\frac{2\pi i}{b}\left(-lr-
lb+\frac{ar^{2}N}{2}+abrN+\frac{ab^{2}N}{2}+arj+abj\right)}=e^{\frac{2\pi
i}{b}\left(-lr+\frac{ar^{2}N}{2}+arj\right)}=\beta_{l,r}.$

Therefore, $c_{l+N}=\alpha_{l+N}\sum_{r=d}^{|b|-1+d}\beta_{l+N,r}$. As $ad-
bc=1$, we know $ad=1$ mod $b$. Then,

$c_{l+N}=\alpha_{l+N}\sum_{r=d}^{|b|-1+d}e^{\frac{2\pi i}{b}\left(-lr-
Nr+\frac{ar^{2}N}{2}+arj\right)}=\alpha_{l+N}\sum_{r=0}^{|b|-1}e^{\frac{2\pi
i}{b}\left(-lr-ld-Nd+\frac{ar^{2}N}{2}+\frac{dN}{2}+arj+j\right)}=c_{l}.$

Using the fact that $c_{k}=c_{k+lN}$ for all $l\in\mathbb{Z}$ and that
$\\{e_{j}^{0}\\}$ is an orthonormal basis, we have

$\displaystyle\left\langle
M_{N,0}e_{j}^{0}(x),e_{k}^{0}(x)\right\rangle_{\mathcal{H}}$
$\displaystyle=\frac{1}{\sqrt{N|b|}}\left\langle\frac{1}{\sqrt{N}}\sum_{l\in\mathbb{Z}}c_{k+lN}\delta\left(x-\frac{Nl+k}{N}\right),\frac{1}{\sqrt{N}}\sum_{l\in\mathbb{Z}}\delta\left(x-\frac{Nl+k}{N}\right)\right\rangle_{\mathcal{H}}$
$\displaystyle=\frac{1}{\sqrt{N|b|}}\sum_{r=0}^{|b|-1}e^{\frac{2\pi
i}{b}\left(\frac{ar^{2}N}{2}+arj+\frac{aj}{2N}+\frac{dk^{2}}{2N}-kr-\frac{kj}{N}\right)}.$

Using the triangle inequality, we conclude $|\left\langle
M_{N,0}e_{j}^{0}(x),e_{k}^{0}(x)\right\rangle_{\mathcal{H}}|\leq\frac{\sqrt{|b|}}{\sqrt{N}}$.
∎

Figure 3. The plot of $\|M_{N,0}^{j}\|_{\ell^{1}\rightarrow\ell^{\infty}}$ for
$0\leq j\leq 50$ and several values of $N$. $M_{N,0}$ corresponds to
$A=\begin{bmatrix}2&3\\\ 1&2\end{bmatrix}$. We also plot the upper bound
$\sqrt{|b|/N}$ for $N=855$.

For $j>0$, we use the notation $A^{j}=\begin{bmatrix}a_{j}&b_{j}\\\
c_{j}&d_{j}\end{bmatrix}$. As a direct consequence of Proposition 8, for
$j>0$,

$\left\|M_{N,0}^{j}\right\|_{\ell^{1}\rightarrow\ell^{\infty}}\leq\frac{\sqrt{|b_{j}|}}{\sqrt{N}}.$

Figure 3 compares this upper bound to actual values of
$\|M_{N,0}^{j}\|_{\ell^{1}\rightarrow\ell^{\infty}}$. Now as
$\operatorname{Tr}A>2$, $A$ has eigenvalues $\lambda$, $\lambda^{-1}$ with
$\lambda>1$. Then we have $|b_{j}|\sim\lambda^{j}$, giving

$\left\|M_{N,0}^{j}\right\|_{\ell^{1}\rightarrow\ell^{\infty}}\leq\frac{C\lambda^{j/2}}{\sqrt{N}}.$

Suppose $u$ is an eigenfunction of $M_{N,0}$ with eigenvalue $\mu$. As
$M_{N,0}$ is unitary, $|\mu|=1$. We have $u=\mu^{-n}M_{N,0}^{n}u$, which gives
$u=\frac{1}{T}\left(\sum_{n=0}^{T-1}\mu^{-n}M^{n}_{N,0}\right)u$.

Setting $B=\frac{1}{T}\sum_{n=0}^{T-1}\mu^{-n}M_{N,0}^{n}$, we see $u=Bu$.
Again, as $M_{N,0}$ is unitary,

$B^{*}B=\frac{1}{T^{2}}\sum_{m,n=0}^{T-1}\mu^{m-n}M^{n-m}_{N,0}.$

Then

$\displaystyle\|B^{*}B\|_{\ell^{1}\rightarrow\ell^{\infty}}$
$\displaystyle\leq\frac{1}{T}+\frac{1}{T^{2}}\sum_{\begin{subarray}{c}m\neq
n\\\ 0\leq m,n\leq
T-1\end{subarray}}\|M^{n-m}_{N,0}\|_{\ell^{1}\rightarrow\ell^{\infty}}$
$\displaystyle\leq\frac{1}{T}+\frac{C}{T^{2}}\sum_{\begin{subarray}{c}m\neq
n\\\ 0\leq m,n\leq T-1\end{subarray}}\frac{\lambda^{\frac{n-m}{2}}}{\sqrt{N}}$
$\displaystyle\leq\frac{1}{T}+\frac{C}{\sqrt{N}}\lambda^{\frac{T}{2}}.$

We set $T=(1-\varepsilon/2)\log_{\lambda}N$ to get

$\|B^{*}B\|_{\ell^{1}\rightarrow\ell^{\infty}}\leq\frac{1}{(1-\varepsilon/2)\log_{\lambda}N}+\frac{C}{\sqrt{\lambda
N^{\varepsilon}}}.$

Finally, we know that
$\|B\|^{2}_{\ell^{2}\rightarrow\ell^{\infty}}=\|B^{*}B\|_{\ell^{1}\rightarrow\ell^{\infty}}$.
Therefore, for $0<\varepsilon<1$, there exists an $N_{0}$ such that for odd
$N\geq N_{0}$,

$\|u\|_{\ell^{\infty}}\leq\|B\|_{\ell^{2}\rightarrow\ell^{\infty}}\|u\|_{\ell^{2}}\leq\frac{1}{\sqrt{(1-\varepsilon)\log_{\lambda}N}}.$

## References


# Two-dimensional pentagonal material Penta-PdPSe: A first-principle study

A. Bafekry<EMAIL_ADDRESS>Department of Radiation Application, Shahid
Beheshti University, Tehran 1983969411, Iran  M. M. Fadlallah  Department of
Physics, Faculty of Science, Benha University, 13518 Benha, Egypt  M. Faraji
Micro and Nanotechnology Graduate Program, TOBB University of Economics and
Technology, Sogutozu Caddesi No 43 Sogutozu, 06560, Ankara, Turkey  A.
Shafique  Department of Physics, Lahore University of Management
Sciences,Lahore, Pakistan  H. R. Jappor  Department of Physics, College of
Education for Pure Sciences, University of Babylon, Hilla, Iraq  I.
Abdolhoseini Sarsari  Department of Physics, Isfahan University of Technology,
Isfahan, 84156-83111, Iran  Yee Sin Ang  Science, Mathematics and Technology
(SMT) Cluster, Singapore University of Technology and Design, Singapore 487372
M. Ghergherehchi  Department of Electrical and Computer Engineering,
Sungkyunkwan University, 16419 Suwon, Korea

###### Abstract

Low-symmetry Penta-PdPSe with intrinsic in-plane anisotropy synthesized
successfully [P. Li et al., Adv. Mater., 2102541 (2021)]. Motivated by this
experimental discovery, we investigate the structural, mechanical, electronic,
optical and thermoelectric properties of PdPSe monolayer via density
functional theory calculations. The phonon dispersion, molecular dynamics
simulation, the cohesive energy mechanical properties of the _penta_ -PdPSe
monolayer is verified to confirm its stability. The phonon spectrum represents
a striking gap between the high-frequency and the low-frequency optical
branches and an out-of-plane flexure mode with a quadratic dispersion in the
long-wavelength limit. The Poisson’s ratio indicates that _penta_ -PdPSe is a
brittle monolayer. The _penda_ -PdPSe monolayer is an indirect semiconductor
with bandgap of 1.40 (2.07) eV using PBE (HSE06) functional. Optical
properties simulation suggests that PdPSe is capable of absorbing a
substantial range of visible to ultraviolet light. Band alignment analysis
also reveals the compatibility of PdPSe for water splitting photocatalysis
application. By combining the electrical and thermal transport properties of
PdPSe, we show that a high $PF$ is achievable at room temperature, thus making
PdPSe a candidate material for thermoelectric application. Our findings reveal
the strong potential of _penta_ -PdPSe monolayer for a wide array of
applications, including optoelectronic, water splitting and thermoelectric
device applications.

##  I Introduction

Two-dimensional (2D) materials have drawn much interests in recent years due
to their superb optoelectronic, mechanical, and magnetic properties, extremely
high surface areas, and narrow thickness at the nanoscale [1, 2, 3, 45]. It is
well known that graphene is one of the most promising types of 2D materials
thanks to its excellent Young’s modulus and carrier mobility [6, 7, 8].
However, the absence of a bandgap in graphene greatly limits its application
in electronics device applications. As a material with configurable and
suitable magnitude of band gap is much desirable for nanodevices applications,
myriads of 2D materials have been fabricated and investigated one after the
other in the past decade. Examples include graphene [9], phosphorene [10],
silicene [11], antimonene [12], transition metal dichalcogenides (MoS2, WS2,
WSe2, PtS2, and PtSe2) [13, 14], metal dichalcogenides (SnSe2 and SnS2)[15,
16, 17], carbon nitrides (C3N, C3N4, C6N, C6N6 and C6N7) [18, 19, 20], Janus
monolayers (Ga2SSe, Ga2STe, and Ga2SeTe) [21], monochalcogenides (GaSe, GaS,
SnTe, InSe, and InS) [22, 23, 24] and MA2Z4 (MoSi2N4 and WSi2N4) [25, 26, 27,
28]. These 2D materials have been widely explored for potential used in a
colossal range of applications ranging from the storage and conversion of
energy to electronics devices.

Although the palladium phosphochalcogenides PdPX (X=Se, S) belonging to the
category of layered inorganic materials and ternary pnictide chalcogenides
have been characterized and synthesised their bulk materials since four
decades ago [29, 30], the 2D monolayer form of PdPX (X=Se, S) have received
less attention in comparison with other 2D materials. Prior studies have
revealed that the crystals structure of compounds PdPSe and PdPS are closer to
that of PdSe2 and PdS2 [31] and crystallized with orthorhombic structures and
Pbcn space group [32]. Besides, it was found that the PdPSe and PdPS compounds
are semiconducting material with moderate band gaps of 1.28 eV [33] and 1.45
eV [34], respectively. Recently, Jing et al. [35] studied thin layers of PdPS
and PdPSe with orthogonal lattices and demonstrated that the exfoliation of
single-layer PdPX can be done from their bulk material with a cleavage energy
of approximately 0.6 Jm-2. They also confirmed that the thin layers of PdPS
and PdPSe are kinetically, thermodynamically and mechanically stable. However,
most of the described 2D materials have hexagonal or orthorhombic structures.
Notwithstanding the tremendous potential of 2D materials in technological
applications, obtaining stable 2D hexagonal structures remains a difficult
task in practical devices. Consequently, there is an enormous growth of
interests in producing 2D materials of different structures in order to cover
a wider range of physical properties for various device applications.
Particularly, 2D materials in pentagonal lattice structures is relatively less
explored compared to other high-symmetry lattices. Unlike the hexagonal
systems, most penta-2D materials such as penta-SnS2 [36], penta-graphene [37],
and penta-As2C [38] are puckered and buckled in a regularly curved way to
retain the symmetry. Amazingly, although most penta-2D materials have been
predicted theoretically [39, 40], some of them have been recently tested
experimentally. For example, 2D monolayer of PdSe2 with puckered pentagonal
structure and isotropic demeanour have been characterized and fabricated by
Oyedele et al [41].

Very recently, 2D puckered penta-PdPSe with a new low-symmetry pentagonal
arrangement is experimentally exfoliated by Li et al [42]. Surprisingly, Se
and P atoms form an unusual [Se-P-P-Se]4- polyanions, which is currently the
largest polyanion in the 2D systems. The robust intrinsic anisotropy of the
penta-PdPSe leads to unusual electronic and optical properties. The penta-
PdPSe layers were observed to have a high photoresponse of 5070 A/W at 635 nm,
strong anisotropic electrical conductivity and an excellent electron mobility
of 21.37 cm2V-1s-1. These results concretely establish penta-PdPSe as a
promising candidate material for building the next-generation of anisotropic
electronics and optoelectronics device with novel functionalities. It should
be noted that the wrinkled pentagons that make up the PdPSe-sublayers are
uncommon in nature. Within the same sublayer, every Pd atom is linked to two
Se atoms and two P atoms, producing a puckered tetracoordinate pattern. Hence,
the atom coordinations of Pd, P and Se in penta-PdPSe differ significantly
from that of noble metal dichalcogenides and it is thus expected that penta-
PdPSe shall shed new lights on the physical properties of 2D materials not
commonly found in 2D nobel metal dichalcogenides.

Despite the recent experimental breakthrough in the fabrication and the
characterization of penta-PdPSe, a comprehensive understanding on the
mechanical, optical and electronic characteristics of penta-PdPSe monolayer
remains incomplete thus far. In this work, we attempt to fill in this gap by
calculating the optical, electronic, transport and structural properties of
the penta-PdPSe monolayer using first-principle density functional theory
(DFT) calculations. Based on the phonon dispersion, molecular dynamics
simulation, and the cohesive energy mechanical properties, the stability of
_penta_ -PdPSe monolayer is confirmed, and the calculated Poisson’s ratio
indicates that _penta_ -PdPSe is a brittle monolayer. The _penda_ -PdPSe
monolayer is an indirect semiconductor with bandgap of 1.40 (2.07) eV using
PBE (HSE06) functional. PdPSe is predicted to absorb a substantial range of
visible to ultraviolet light. Band alignment analysis also reveals the
compatibility of PdPSe for water splitting photocatalysis application. We
further show that a high $PF$ is achievable at room temperature, thus making
PdPSe a candidate material for thermoelectric application. These findings
shall pave the way for designing devices based on 2D penta-PdPSe for a wide
range of device applications.

Figure 1:  (a) Atomic structure, (b) different charge density and (c)
potential average of the PdPSe monolayer. Rectangular primitive unit cell
indicated by the red square. blue (yellow) color refers to the accumulation
(depletion) electron density.

##  II Method

The first-principles electronic structure calculations is performed within the
density functional theory (DFT) defined in the Vienna ab-initio simulation
package (VASP) [45, 46]. To conduct the DFT calculations, the plane-wave basis
projector augmented wave (PAW) method accompanied with the Perdew-Burke-
Ernzerhof(PBE) [43, 44] is imposed. Furthermore, the Heyd-Scuseria-Ernzerhof
(HSE06)[47] functional is employed for more accurate calculation of the
bandgap value. We adjust the kinetic energy cut-off equal to 500 eV for the
plane-wave expansion, then the energy of the system is converged till
fluctuations in the energy is below 10-5 eV. Optimization of the structures is
performed with the total Hellmann-Feynman forces are decreased to 0.05 eV/Å.
21$\times$21$\times$1 $\Gamma$ centered k-point grid was employed for the
primary unit cells via taking the Monkhorst-Pack [48]. Charge migration
analysis is performed by utilizing the Bader method [49]. Here, the long-range
vdW interactions can be explained by the Van der Waals (vdW) corrections [50].
The phonon characteristics were derived from the small displacement technique
as employed in the PHONOPY code [51]. The thermal stability is also examined
by conducting the AIMD simulations at 600K.The training set is prepared by
conducting ab-initio molecular dynamics (AIMD) simulations over $3\times
3\times 1$ supercells with $2\times 2\times 1$ k-point grids. AIMD simulations
are carried out at 600K, each for 5000 time steps. The electronic transport
coefficients are calculated using the Boltzmann transport theory under the
rigid band and constant relaxation approximation.

The transport transport coefficients are calculated using the Boltztrap2 code
[60]. The Monkhorst–Pack k-mesh of $30\times 30\times 1$ is used to accurately
calculate the electronic transport coefficients.

##  III Results and Discussions

###  III.1 Structural and Mechanical properties

Figure 2:  (a) Phonon band spectrum of PdPSe monolayer. (b) Ab initio molecular dynamics (AIMD) for PdPSe monolayer at 600 K temperature. Top view of the structure after 5 ps of simulation demonstrated as insets. Figure 3:  (a) Electronic band structure and (b) PDOS and (c) electron localization function (ELF) of PdPSe monolayer. The zero of energy is set to Fermi-level. Table 1:  Structural, electronic and mechanical parameters of PdPSe monolayer, as shown in Fig. 1, including: lattice constants a,b; bond lengths between P-Se ($d_{1}$), Pd-Se ($d_{2}$), Pd-P ($d_{3}$), and Se-Se ($d_{4}$), bond angles between Pd-Se-Pd ($\theta_{1}$), P-Se-P ($\theta_{2}$), P-Se-Pd ($\theta_{2}$) and P-Se-Se ($\theta_{3}$), buckling ($t$); cohesive energy per atom, $(E_{coh})$; charge transfer $\Delta{Q}$; work function $\Phi$; band gap $E_{g}$ with PBE (HSE06); bulk modulus (B); shear modulus (S); Young’s modulus (Y); and Poisson’s ratio ($\nu)$) respectively.  Sys. | a,b | ${d_{1,2,3,4}}$ | ${\theta}{{}_{1,2,3}}$ | $t$ | $E_{coh}$ | $\Delta{Q}$ | $\Phi$ | $E_{g}$ | B | S | Y | $\nu$
---|---|---|---|---|---|---|---|---|---|---|---|---
| (Å) | (Å) | (∘) | Å | (eV/atom) | (e) | (eV) | (eV) | GPa | GPa | GPa |
PdPSe | 5.86,5.79 | 2.85,2.48,2.31,2.20 | 114.86,103.83,112.19 | 1.46 | -0.18 | 0.18 | 5.90 | 1.40 (2.07) | 21.43 | 15.36 | 37.19 | 0.21

The atomic structure of the PdPSe monolayer in the diverse views is
represented in Figs. 1(a). In the PdPSe, the lattice with a pentagonal
primitive unit cell is determined by eight atoms (2Pd, 2 P, and 2 Se), and the
primitive unit cell is shown as a red square. The pentagonal lattice of PdPSe
belongs to the $Pbca$ (60) space group with a unit cell lattice constants of
5.86 Å (a) and 5.79 Å (b). The bond lengths are determined to be: 2.85 Å
($d_{1}$), 2.48 Å ($d_{2}$), 2.31Å ($d_{3}$) and 2.20 Å ($d_{4}$). While the
bond angles of $\theta_{2}$), $\theta_{2}$ and $\theta_{3}$ are calculated
114.86, 103.83 and 112.19 ∘,respectively. The PdPSe monolayer is not a planar
structure. The buckling is determined as 1.46 Å. The structural parameters are
summarized in Table I.

The cohesive energy per atom, the energy obtained by adjusting the atoms in a
crystalline compared to the gas state, that is quantifying the stability of
materials, was computed applying the following equation:

$E_{coh}=\frac{E_{tot}-(n_{Pd}E_{Pd}+(n_{P}E_{P}+n_{Se}E_{Se})}{n_{Pd}+n_{P}+n_{Se}},$
(1)

where $E_{Pd}$, $E_{P}$ and $E_{Se}$ represent the energies of isolated single
Pd, P and Se atoms; $E_{tot}$ shows the total energy of the PdPSe.
Additionally, nPd, nP, and nSe determines the number of Pd, P, and Se atoms in
the primitive unit cell, respectively. Our results show that the cohesive
energies are -0.18 eV/atom. The negative cohesive energy obtained for this
layer suggests that the free-standing state of the nanostructure remains
likely to be stable.

The difference charge density showed in Fig. 1(b) is specified as:

$\Delta\rho=\rho_{tot}-\rho_{Pd}-\rho_{P}-\rho_{Se},$ (2)

where $\rho_{tot}$, $\rho_{Pd}$, $\rho_{P}$ and $\rho_{Se}$ depicts the charge
densities of the isolated atoms and PdPSe monolayer, respectively. The charge
depletion and accumulation are illustrated by coloring schemes with yellow and
blue regions, correspondingly. C and N atoms are negatively charged and are
enclosed by the positively charged B atoms. Based on the charge transfer
($\Delta\rho$) analysis, the Se atoms gain as 0.18 from the nearby Pd and P
atoms. The electrostatic potential of the PdPSe monolayer is flat in the
vacuum area (Fig. 1(c)). Applying the formula $\Phi=E_{vacuum}-E_{F}$, the
work function of the PdPSe was defined as 5.90 eV.

###  III.2 Dynamics and thermal stability

The phonon dispersion spectrum through the whole BZ is shown in Figs. 2(a).
The phonon spectrum represents an in-plane transverse (TA) and a longitudinal
(LA) optical mode, with a linear dispersion, and an out-of-plane flexure mode
(ZA) with quadratic dispersion in the long-wavelength limit. The quadratic
dispersion results in the large density of modes for ZA flexural phonons,
which carry the most heat comparison LA and TA phonon modes. We may expect the
other way around because the flexural acoustic ZA phonons have vanishing group
velocities for a wave vector near zero. The selection rule for three-phonon
scattering can also explain this unexpected result, which strongly restricts
the phase space for this scattering [52]. Flexural modes play a vital role in
contributing to the thermal conductivity of graphene and structures with
flexural modes, both as carriers [52] and as scatterers [53]. Flexural modes
are perceived in the eminent 2DMs such as graphene[52], hexagonal boron
nitride (h-BN)[54], and arsenene[55].

The phonon dispersion spectrum is free from any imaginary frequency,
discovering the dynamical stability of the PdPSe monolayer. In addition, a
remarkable gap between the high-frequency and low-frequency optical branches
is present. The splitting between the longitudinal and the transversal optical
phonon frequencies (LO-TO splitting) presented at the $\Gamma$ point predicts
the effects of the dipole-dipole interactions and the high-frequency
dielectric constants [56].

We analyze the thermal stability of the PdPSe monolayer by the ab initio
molecular dynamics (AIMD) trajectories at 600 K. The AIMD computational
results for the PdPSe monolayer at 600 K is illustrated in Fig. 2(b). The
inset in the panel of Fig. 2(b) demonstrates the optimized structure after
five ps of simulation. Analysis of the AIMD trajectories shows that the 2D
structure could stay intact at 600 K, with very stable temperature and energy
profiles, proving the thermal stability of the PdPSe monolayer which is
beneficial for the experimental synthesis of monolayer PdPSe.

Next, we investigate the mechanical stability and properties of the PdPSe. The
linear elastic constants are calculated using the harmonic approximation. We
find that the elastic parameters agree with the twelve Born criteria [61]. We
calculate the bulk (B), shear (S) and Young’s moduli (Y) of PdPSe monolayer as
21.43 GPa, 15.36 GPa, and 37.19, respectively. The Poisson’s ratio
($\nu)$=0.21) suggests that the PdPSe monolayer is a brittle structure as it
is less than 0.33 [62].

###  III.3 Electronic properties

Figure 4: (a) Real, (b) imaginary parts of the dielectric function, (c) the
optical absorption spectrum as a function ofas a function of photon energy for
PdPSe monolayer using the RPA+HSE06 approaches and (d) Band alignments of
PdPSe monolayer for photocatalytic water splitting and carbon dioxide
reduction. The band edges are given with respect to the NHE (normal hydrogen
electrode) potential (in Volts).

The electronic band structure show that the PdPSe monolayer is an indirect
bandgap semiconductor with a forbidden gap, Eg, of 1.26 eV and 2.07 eV within
the PBE and HSE06 functional, respectively (see Fig. 3). The HSE06 hybrid
functional does not alter the type of indirect bandgap in the PdPSe monolayer.
Notably, the band gap of PdPSe is similar to that of MoS2 and WS2 monolayers
[tmd_bg]. The valance band minimum (VBM) and conduction band maximum (CBM) are
located at the $\Gamma$ point and S point, respectively. The VBM and the CBM
are almost isotropic around the $\Gamma$ point. Based on the partial density
of state (PDOS) analysis, as depicted in Fig. 3(b), the VBM are composed
mainly of the Pd-$d_{z^{2}}$-$d_{xy}$-$d_{x^{2}-y^{2}}$, P-$s,p_{x}$ and
Se-$p_{x,z}$ orbitals states, while the CBM are originated from $d_{xz,yz}$,
$p_{x,z}$ and $p_{x,y,z}$ of Pd, P and Se atoms of PdPSe, respectively.

The electron localization function (ELF) along the (0 0 1) plane held the
monolayer as depicted in Fig. 3(d). The red (blue) color means high (low)
electron density in the ELF. ELF takes a value between 0 and 0.5, where ELF =
0.5 corresponds to the perfect localization. Electron localization occurs on
the Se atoms. The maximum ELF value between Pd and P/Se atoms is approximately
0.50, representing a large proportion of the covalent bond in the PdPSe
monolayer.

Figure 5:  The electronic transport coefficients of the PdPSe monolayer as
function carrier concentration at 300 K, 500 K, and 700 K, (a,b) Seebeck
coefficient ($S$), (c,d) relaxation time-scaled electrical conductivity
($\sigma/\tau$), (e,f) electronic thermal conductivity ($\kappa_{e}/\tau$),
and (g,h) power factor ($PF$). The solid and dashed lines represent the
$p$-type and $n$-type doping, respectively.

###  III.4 Optical properties and water splitting

We now investigate the optical properties of the PdPSe monolayer. The
dielectric function (the real and imaginary parts) is calculated using the RPA
method over the HSE06. Figs. 4(a,b) shows the real and imaginary parts of the
dielectric function which describe the interaction between the electromagnetic
field and monolayer PdPSe monolayer. The static value of the dielectric
constant is the value of the real part at the zero energy which is found to be
5.73 and 5.63 for the xx and yy components, respectively. The dielectric
function in the zz direction is very small as compared to the corresponding
function in xx and yy components due to the monolayered nature of PdPSe. The
dielectric function is anisotropic along the xx, yy, and zz directions. The
first plasma frequencies appear at $\sim$ 3.01 eV and 2.97 eV along xx and yy
direction, respectively. Other plasma frequencies appear at 8.95 eV and 9.10
eV along xx and yy direction, respectively. The imaginary part of the
dielectric function represents the sum of all transitions from the valence
bands to the conduction bands. It starts to increase above $\sim$ 2.1 eV which
is related to the band gap value. The imaginary part increases from 2.1 eV to
4 eV which are in the visible and the ultraviolet ranges of light. Figure 4(c)
shows the absorption spectrum for PdPSe monolayer. The onset of significant
absorption occurs at $\sim$ 2.0 eV which is related to the onset of the
imaginary part and the band gap of PdPSe. The absorption spectrum suggests
that the PdPSe has the ability to absorb the visible and ultraviolet light
range. Fig. 4(d) illustrates the band alignment of the oxidation and reduction
potentials for water splitting with respect to the band edges of PdPSe
monolayer suggesting that it is a potential candidate for photocatalytic water
splitting. Most importantly, PdPSe monolayer exhibits a suitable band gap
$\sim$ 2.07 eV and the band edges CBE and VBE must be higher (more negative)
and lower (more positive) than the hydrogen reduction potential of H+/H2 and
the water oxidation potential of H2O/O2, respectively, for efficient water
splitting to take place. The CBE is computed from the relation
$E_{CBE}=X-0.5E_{gap}-4.5$ eV [64, 65, 66], then the valence band edge is
calculated from $E_{VBE}=E_{CBE}+E_{gap}$, where $X$ is the geometric mean of
the electro-negativity of the ingredient atoms [67] and $4.5$ eV is the free
energy of the electron (with respect to the vacuum level).

###  III.5 Thermoelectric properties

The calculated $S$, $\sigma$, $\kappa_{e}$ and power factor
($PF=S^{2}\sigma/\begin{tabular}[]{c}$\tau$\end{tabular}$) for both $p$\- and
$n$-type for monolayer PdPSe are shown in Fig. 5. The $S$ linearly decreases
(increases) with increasing carrier concentration (temperature) and which is
agreement with previous work [69]. At 300 K, the $S$ for the hole (electron)
doping of 1011 are 702 (668) and 669 (692) $\mu$VK-1 along the $x$\- and
$y$-direction, respectively. The $S$ does not varry significantly with both
doping and transport directions. In general, $\sigma/\tau$ and
$\kappa_{e}/\tau$ increase with an increasing carrier concentration. The
$\sigma/\tau$ for the holes along the $x$-direction is lower than that of the
electrons due to the flat valence band while that of both electron and hole
are similar along the $y$-direction. The $PF$ is related to the interplay
between $S$ and $\sigma/\tau$. As shown in Figs. 5(g) and (f), the $PF$
increases with an increasing carrier concentration initially to reach a peak
before decreases with higher carrier concentration. The maximum $PF$ values of
7.9 and 5.2 are achieved for $n$-type along the $x$-direction and $p$-type
along $y$-direction, respectively. Such high values of $PF$ demonstrating that
PdPSe is a promising material for thermoelectricity.

##  IV Conclusion

In summary, first-principle investigations is carried out study the physical
properties of penta-PdPSe monolayer. The stability is confirmed based on the
dynamical, phonon dispersions and mechanical studies. Flexural modes and LO-TO
splitting are observed in the phonon dispersion of the PdPSe monolayer. The
monolayer has brittle behaviour. The electronic structure of the PdPSe
monolayer calculated using the HSE06 reveals that the monolayer is an indirect
semiconductor with a bandgap of 2.07 eV. The optical properties reveal the
PdPSe monolayer can absorb the visible and ultraviolet the light. High $PF$ is
achievable at room temperature. Furthermore, penta-PdPSe is predicted to be a
photocatalytic material for water splitting application. Our results highlight
the strong application potential of penta-PdPSe monolayer.

##  V Conflicts of interest

The authors declare that there are no conflicts of interest regarding the
publication of this paper.

##  VI ACKNOWLEDGMENTS

This work was supported by the National Research Foundation of Korea (NRF)
grant funded by the Korea government (MSIT) (NRF-2015M2B2A4033123).

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