# Smooth _l_ -Fano weighted complete intersections

Anastasia V. Vikulova  Steklov Mathematical Institute of Russian Academy of
Sciences, 8 Gubkin str., Moscow, 119991, Russia Laboratory of Algebraic
Geometry, National Research University Higher School of Economics, 6 Usacheva
str., Moscow, 119048, Russia.
<EMAIL_ADDRESS>
###### Abstract.

In this paper we prove that for $n$-dimensional smooth $l$-Fano well formed
weighted complete intersections, which is not isomorphic to a usual projective
space, the upper bound for $l$ is equal to $\lceil\log_{2}(n+2)\rceil-1.$ We
also prove that the only $l$-Fano of dimension $n$ among such manifolds with
inequalities $\lceil\log_{3}(n+2)\rceil\leqslant
l\leqslant\lceil\log_{2}(n+2)\rceil-1$ is a complete intersection of quadrics
in a usual projective space.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Chern characters
  4. 4 Proof of Theorem 1.7
  5. 5 Proof of Theorem 1.8
  6. A Monotonicity and boundedness
  7. B Inequalities on sums of $l$-th degrees

##  1\. Introduction

A smooth projective variety $X$ is called a Fano variety, if the first Chern
class of the tangent bundle $\text{c}_{1}(X)$ is ample. Varieties which are
called $l$-Fano varieties and which generalize Fano varieties were introduced
in the papers [1] and [18]. In the sequel, we always work over the field of
complex numbers.

######  Definition 1.1.

We say that the Chern character $\text{ch}_{i}(X)$ of a smooth projective
variety $X$ is positive if $\text{ch}_{i}(X)\cdot Z>0$ for every effective
$i$-cycle $Z.$

######  Definition 1.2.

A smooth Fano variety $X$ is called an _l-Fano_ variety if the Chern
characters $\text{ch}_{i}(X)$ are positive for all $2\leqslant i\leqslant l.$

######  Example 1.3.

It is obvious that a usual projective space $\mathbb{P}^{n}$ is an $l$-Fano
variety for $l=1,\ldots,n$. Indeed, we have
$\text{ch}_{k}(\mathbb{P}^{n})\,=\,\frac{n+1}{k!}H^{k},$ where $H$ is a
hyperplane class in $\mathbb{P}^{n}.$

######  Example 1.4.

If $X\subset\mathbb{P}^{n}$ is a smooth complete intersection of hypersurfaces
of degrees $d_{1},\ldots,d_{k},$ then $X$ is an $l$-Fano variety if and only
if $\sum_{i=1}^{k}d_{i}^{l}<N+1$ (see, for example, [17, Section 2, Example
3]). In particular, we immediately get that a smooth complete intersection of
$k$ quadrics $X$ in a projective space is $l$-Fano if and only if
$l\leqslant\lceil\log_{2}(\frac{N+1}{k})\rceil-1.$ In other words, a complete
intersection of $k$ quadrics in $\mathbb{P}^{N}$ is $l$-Fano variety if and
only if $k\leqslant\lceil\frac{N+1}{2^{l}}\rceil-1.$ In the same way we get
that a smooth cubic hypersurface in a projective space is $l$-Fano if and only
if $l\leqslant\lceil\log_{3}(N+1)\rceil-1.$

Definition 1.2 is motivated by the following observations. While a general
point of a Fano variety is contained in a rational curve (see [8, Chapter 5,
Theorem 1.6.1]), a general point of a 2-Fano variety is contained in a
rational surface under some mild assumptions (see [18]). Another property of
Fano varieties is Tsen theorem which states that a smooth Fano hypersurface
$X\subset\mathbb{P}^{N}_{K}$ over a field $K/k$ of transcendence degree $1$
over an algebraically closed field $k$ always has a $K$-point. At the same
time, Tsen–Lang theorem states that a smooth $l$-Fano hypersurface
$X\subset\mathbb{P}^{N}_{K}$ over a field $K/k$ of transcendence degree $l$
over an algebraically closed field $k$ always has a $K$-point (see [7, Theorem
6]).

The following conjecture was suggested in [1]:

######  Conjecture 1.5 ([1, Conjecture 1.7]).

If $X$ is an n-dimensional smooth $l$-Fano variety with
$l\geqslant\lceil\log_{2}(n+2)\rceil,$ then $X\simeq\mathbb{P}^{n}.$

There is some progress in proving this conjecture. In [1, Theorems 1.3 and
1.5] there is a classification of $2$-Fano varieties among rational
homogeneous spaces of Picard rank 1 and a classification of $3$-Fano varieties
of dimension $n$ and index at least $n-2$ (here the index is the maximal
positive integer which divides the canonical class), and it was shown that for
these two classes of varieties Conjecture 1.5 is true. In the paper [14] by
computer calculations it was shown that the only smooth toric varieties of
dimension at most $8$ with positive second Chern character is a projective
space. Thus, for toric varieties of dimension at most $8$ Conjecture 1.5 is
true.

The purpose of this paper is to study Conjecture 1.5 in the case of smooth
weighted complete intersections (see [6] or Section 2 below for definitions
and basic properties of these varieties).

######  Remark 1.6.

Note that we assume that codimension of a smooth weighted complete
intersection is positive, i.e. a weighted projective space is not considered
as a weighted complete intersection in itself. Note also that by [13, Theorem
2.7] a smooth well formed weighted complete intersection which is not an
intersection with a linear cone is not isomorphic to a projective space.

We are going to prove the following theorem:

######  Theorem 1.7.

Let $X\subset\mathbb{P}(a_{0},\ldots,a_{N})$ be a smooth $n$-dimensional well
formed weighted complete intersection which is not an intersection with a
linear cone. Suppose that $X$ is $l$-Fano. Then $l<\lceil\log_{2}(n+2)\rceil.$

In other words, Theorem 1.7 states that Conjecture 1.5 holds for smooth well
formed weighted complete intersections which are not intersections with a
linear cone. Moreover, in this case there is the following strengthening of
Theorem 1.7:

######  Theorem 1.8.

Let $X\subset\mathbb{P}=\mathbb{P}(a_{0},\ldots,a_{N})$ be a smooth
n-dimensional well formed weighted complete intersection which is not an
intersection with a linear cone. Suppose that $X$ is $l$-Fano, and

$\lceil\log_{3}(n+2)\rceil\leqslant l<\lceil\log_{2}(n+2)\rceil.$

Then

$\mathbb{P}=\mathbb{P}(1,\dots,1)=\mathbb{P}^{N}$

and $X\subset\mathbb{P}$ is a complete intersection of at most
$\lceil\frac{N+1}{2^{l}}\rceil-1$ quadrics.

######  Remark 1.9.

By [17, Section 2, Example 3] a smooth weighted complete intersection
$X\subset\mathbb{P}(a_{0},\ldots,a_{N})$ of multidegree $(d_{1},\ldots,d_{k})$
is $l$-Fano if and only if

(1.1) $\sum_{j=0}^{N}a_{j}^{m}>\sum_{i=1}^{k}d_{i}^{m}\;\;\text{for
all}\;\;m=1,\ldots,l.$

In Section 3 we show that in order to check $l$-Fano condition it is
sufficient to show that the inequality (1.1) holds for $m=l.$

The plan of the paper is as follows. In Section 2 we collect some basic facts
about weighted complete intersections. In Section 3 we study the Chern
characters of Fano varieties and prove Theorem 3.1 about monotonocity property
of smooth $l$-Fano varieties; namely, we show that if the $(l+1)$-th Chern
character of smooth well formed weighted complete intersection which is not an
intersection with a linear cone is positive, then the $l$-th one is positive
as well. In Section 4 we prove Theorem 1.7. In Section 5 we prove Theorem 1.8.
In Appendices A and B we perform some auxiliary elementary computations.

Acknowledgment. This work was performed at the Steklov International
Mathematical Center and supported by the Ministry of Science and Higher
Education of the Russian Federation (agreement no. 075-15-2022-265) and it was
partially supported by the HSE University Basic Research Program.

The author wish to warmly thank C.A.Shramov for suggesting this problem, for
constant attention to this paper and for important remarks. Also the author is
grateful to A.S.Trepalin for useful suggestions and A.G.Kuznetsov for
interesting discussions.

##  2\. Preliminaries

First of all, we recall some definitions.

######  Definition 2.1.

The variety
$\mathbb{P}(a_{0},\ldots,a_{N})=\text{Proj}\,\mathbb{C}[x_{0},\ldots,x_{N}],$
where the weight of the variable $x_{i}$ is equal to $a_{i},$ is called a
weighted projective space.

######  Definition 2.2.

A weighted complete intersection $X\subset\mathbb{P}(a_{0},\ldots,a_{N})$ of
multidegree $(d_{1},\ldots,d_{k}),$ where $k\geqslant 1$ is an intersection of
hypersurfaces of degrees $d_{1},\ldots,d_{k}$ such that the codimension of
every irreducible component of $X$ is equal to $k.$

######  Definition 2.3.

We say that a weighted projective space $\mathbb{P}(a_{0},\ldots,a_{N})$ is
well formed if $\gcd(a_{0},\ldots,\widehat{a_{i}},\ldots,a_{N})\,=\,1$ for all
$i.$ We say that a weighted complete intersection $X$ in a well formed
projective space $\mathbb{P}(a_{0},\ldots,a_{N})$ is well formed if

$\text{codim}_{X}(X\cap\text{Sing}\,\mathbb{P}(a_{0},\ldots,a_{N}))\geqslant
2.$

######  Definition 2.4.

Let $X\subset\mathbb{P}(a_{0},\ldots,a_{N})$ be a weighted complete
intersection of multidegree ($d_{1},\ldots,d_{k})$ in a well formed weighted
projective space. We say that $X$ is not an intersection with a linear cone,
if $d_{i}\neq a_{j}$ for all $i$ and $j.$

######  Remark 2.5.

It is reasonable to consider $X$ such that it is not an intersection with a
linear cone because of the following observation. Assume that
$X\subset\mathbb{P}(a_{0},\ldots,a_{N})$ is a degree $d$ hypersurface which is
an intersection with a linear cone, i.e. $d=a_{i}$ for some $i.$ Then the
general hypersurface is the zero set of a polynomial $f=x_{i}+g.$ Such
hypersurfaces are parametrized by a Zariski open subset in the space of degree
$d$ hypersurfaces (see [6, Chapter 6]). Thus, if
$X\subset\mathbb{P}(a_{0},\ldots,a_{N})$ is a general smooth complete
intersection of multidegree $(d_{1},\ldots,d_{k})$ and $d_{r}=a_{s}$ for some
$r$ and $s,$ then $X\simeq
X^{\prime}\subset\mathbb{P}(a_{0},\ldots,\widehat{a_{s}},\ldots,a_{N}),$ where
$X^{\prime}$ is a smooth weighted complete intersection of multidegree
$(d_{1},\ldots,\widehat{d_{r}},\ldots,d_{k}).$

The main auxiliary result used in the proofs of Theorems 1.7, 1.8 and 3.1 is
the following:

######  Theorem 2.6.

Let $X\subset\mathbb{P}(a_{0},\ldots,a_{N})$ be a smooth well formed Fano
weighted complete intersection of multidegree $(d_{1},\ldots,d_{k})$ which is
not an intersection with a linear cone. Assume that

$d_{1}\leqslant d_{2}\leqslant\ldots\leqslant d_{k}\qquad\text{and}\qquad
a_{0}\leqslant a_{1}\leqslant\ldots\leqslant a_{N}.$

Then

  1. (1)

$d_{i}\,>\,a_{N-k+i}\quad\text{for}\quad 1\leqslant i\leqslant k-1;$

  2. (2)

$\\#\\{j\mid a_{j}\,=\,1\\}\geqslant k+1;$

  3. (3)

$\\#\\{j\mid a_{j}\,=\,1\\}\geqslant\sum_{j=0}^{N}a_{j}-\sum_{i=1}^{k}d_{i};$

  4. (4)

$d_{i}\neq a_{j}\quad\text{for all}\quad i,j;$

  5. (5)

$d_{i}\neq 1\quad\text{for all}\quad i;$

  6. (6)

$N\geqslant 2k;$

  7. (7)

$\\#\\{j\mid b\;\;\text{divides}\;\;a_{j}\\}\leqslant\\#\\{i\mid
b\;\;\text{divides}\;\;d_{i}\\}\quad\text{for all}\quad b\in\mathbb{N}_{>1};$

  8. (8)

$d_{1}\cdot\ldots\cdot d_{k}\;\;\text{is divisible
by}\;\;a_{0}\cdot\ldots\cdot a_{N};$

  9. (9)

$d_{k}\geqslant 2a_{N}.$

###### Proof.

We refer to [6, Lemma 18.14] for (1); we refer to [10, Corollary 5.3 and 5.11]
for (2) and (3); assertions (4) and (5) follow from assertion (2) and from the
assumption that $X$ is not an intersection with a linear cone; for (6) we
refer to [3, Theorem 1.3]; for (7) we refer to [11, Lemma 2.15] and (8)
follows from (7).

Assertion (9) can be proved using (5) and (7). Indeed, if all $a_{j}\,=\,1,$
then by (5) we have

$d_{k}\geqslant 2=2a_{N}.$

If $a_{N}\neq 1,$ then according to (7) there is $d_{t}$ such that
$d_{t}=ma_{N}$ for some integer $m>1.$ Thus we obtain

$d_{k}\geqslant d_{t}=ma_{N}\geqslant 2a_{N}.$

∎

##  3\. Chern characters

Let $X\subset\mathbb{P}=\mathbb{P}(a_{0},\ldots,a_{N})$ be a smooth well
formed weighted complete intersection of multidegree $(d_{1},\ldots,d_{k}).$
It is well known (for example, see [1, Section 6]) that the Chern characters
can be represented in a simple way as

(3.1)
$\text{ch}_{l}(X)\,=\,\frac{1}{l!}(\sum_{j=0}^{N}a_{j}^{l}-\sum_{i=1}^{k}d_{i}^{l})H^{l}|_{X},$

where $H$ is a generator of class group $\text{Cl}(\mathbb{P}).$ This follows
immediately from the exact sequence

(3.2) $0\to T_{X}\to
T_{\mathbb{P}}|_{X}\to\bigoplus_{j=1}^{k}\mathcal{O}_{\mathbb{P}}(d_{j})|_{X}\to
0,$

where $T_{\mathbb{P}}=\Omega_{\mathbb{P}}^{\vee}$ is the dual sheaf of the
sheaf of Kähler differentials. Let us remark that (3.2) is well defined,
because we assume that $X$ is well formed and thus by [11, Proposition 2.11]
it does not pass through the singular locus of $\mathbb{P}.$ So we have the
following exact sequence (for example, see [5, Theorem 8.1.6])

$0\to\mathcal{O}_{\mathbb{P}}\to\bigoplus_{j=0}^{N}\mathcal{O}_{\mathbb{P}}(a_{j})\to
T_{\mathbb{P}}\to 0,$

which gives us the Chern characters of $T_{\mathbb{P}}.$

We will also prove the following theorem for the Chern characters:

######  Theorem 3.1.

Let $X\subset\mathbb{P}(a_{0},\ldots,a_{N})$ be a smooth well formed Fano
weighted complete intersection which is not an intersection with a linear
cone. Then $X$ is $l$-Fano if and only if $\emph{\text{ch}}_{l}(X)$ is
positive.

###### Proof.

We should prove that if $\text{ch}_{l}(X)$ is positive, then all
$\text{ch}_{m}(X)$ for $2\leqslant m\leqslant l-1$ are positive as well. So we
have to prove that $\text{ch}_{m}(X)\cdot Z>0$ for every effective $m$-cycle
$Z$ and all $2\leqslant m\leqslant l-1.$ But by (3.1) we know that
$\text{ch}_{m}(X)$ is a multiple of $H^{m}|_{X}$ which is a power of an ample
divisor. Thus, we have that $H^{m}|_{X}\cdot Z>0$ for every effective
$m$-cycle $Z.$ So we have to prove that the coefficients at $H^{m}|_{X}$ in
the expression for $\text{ch}_{m}(X)$ (see (3.1)) are positive for $m\leqslant
l.$ For this we will prove that

(3.3)
$\sum_{j=0}^{N}a_{j}^{m}-\sum_{i=1}^{k}d_{i}^{m}>\sum_{j=0}^{N}a_{j}^{m+1}-\sum_{i=1}^{k}d_{i}^{m+1}$

for all integers $m\geqslant 1.$

From Theorem 2.6(3) and Fano condition we get that if

$s\,=\,\\#\\{j\mid a_{j}\,=\,1\\},$

then

$\sum_{j=s}^{N}a_{j}\,\leqslant\,\sum_{i=1}^{k}d_{i}\,<\,\sum_{j=s}^{N}a_{j}+s.$

As all assumptions of Lemma A.4 hold in the theorem, thus we get that
inequality (3.3) is true.

∎

It is not true in general that if the $l$-th Chern character of smooth Fano
variety is positive, then the ($l-1$)-th one is positive as well. Let us
consider the following example.

######  Example 3.2.

Let $X$ be a blow-up of a point on the projective space $\mathbb{P}^{n}.$ Its
Picard group is $\text{Pic}(X)=\mathbb{Z}H\oplus\mathbb{Z}E,$ where $H$ is the
pullback of the hyperplane class in $\mathbb{P}^{n},$ and $E$ is an
exceptional divisor. We have
$X\simeq\mathbb{P}(\mathcal{O}_{\mathbb{P}^{n-1}}\oplus\mathcal{O}_{\mathbb{P}^{n-1}}(-1))$
and there is a natural $\mathbb{P}^{1}$-fibration $\pi:X\to\mathbb{P}^{n-1}.$
If $H^{\prime}$ is a hyperplane class in $\mathbb{P}^{n-1},$ then
$\pi^{*}(H^{\prime})=H-E.$

According to [4, Lemma 2.3] the cone of effective $k$-cycles of $X$ for
$1\leqslant k<n$ is generated by

  * •

$X_{k}=H^{n-k}-(-1)^{n-k+1}E^{n-k},$

  * •

$Y_{k}=(-1)^{n-k+1}E^{n-k}.$

Also note that the following relations on $H$ and $E$ hold:

(3.4) $H\cdot E=0,\;\;H^{n}=1,\;\;E^{n}=(-1)^{n+1}.$

We have the exact sequence

$0\to L\to T_{X}\stackrel{{\scriptstyle
d\pi}}{{\longrightarrow}}\pi^{*}T_{\mathbb{P}^{n-1}}\to 0,$

where $L\simeq K_{X}^{-1}\otimes\pi^{*}K_{\mathbb{P}^{n-1}}$ is an invertible
sheaf on $X.$ So we get that the Chern character of $X$ is

$\text{ch}(X)=\text{ch}(K_{X}^{-1})\text{ch}(\pi^{*}K_{\mathbb{P}^{n-1}})+\text{ch}(\pi^{*}T_{\mathbb{P}^{n-1}}).$

One we can rewrite it as

$\text{ch}(X)=\exp(H+E)+n\exp(H-E)-1.$

By (3.4) we get

$\text{ch}_{k}(X)=\frac{(H+E)^{k}}{k!}+n\frac{(H-E)^{k}}{k!}=\frac{1}{k!}((n+1)H^{k}+(-1)^{k}(n+(-1)^{k})E^{k}).$

So we can calculate the intersection of $\text{ch}_{k}$ with effective
$k$-cycles for $k<n$:

$\text{ch}_{k}(X)\cdot X_{k}=\frac{1}{k!}\left((n+1)-(n+(-1)^{k})\right)$

and

$\text{ch}_{k}(X)\cdot Y_{k}=\frac{1}{k!}\left(n+(-1)^{k}\right).$

For $k=n$ we obtain

$\text{ch}_{n}(X)=\frac{1}{n!}\left(1-(-1)^{n}\right).$

Thus we get that $\text{ch}_{k}(X)$ is positive for odd $k$ and nef but not
positive for even $k.$ In particular, $X$ is not a $2$-Fano variety.

######  Remark 3.3.

There are other examples of varieties with the same behavior of the Chern
characters (see, for example, [16, Example 4.1]). The is a conjecture which
states that all smooth toric varieties except $\mathbb{P}^{n}$ are not
$2$-Fano. See, for example, the paper [16]. However, if we consider
$\mathbb{Q}$-factorial terminal varieties, then there is an example of a toric
$2$-Fano variety which is not isomorphic to $\mathbb{P}^{n}$ (see [15]).

##  4\. Proof of Theorem 1.7

Without loss of generality we can assume that

$d_{1}\leqslant d_{1}\leqslant\ldots\leqslant d_{k}\qquad\text{and}\qquad
a_{0}\leqslant a_{1}\leqslant\ldots\leqslant a_{N}.$

Also we can assume that $X$ is a Fano variety, because otherwise there is
nothing to prove. It is enough to prove that for
$l=\lceil\log_{2}(N-k+2)\rceil$ the $l$-th Chern character is not positive,
i.e.

$\sum_{i=1}^{k}d_{i}^{l}\geqslant\sum_{j=0}^{N}a_{j}^{l}.$

From Lemma A.4 it is enough to prove this for $l=\log_{2}(N-k+2)$. Assuming
that $\sum_{i=1}^{k}d_{i}^{l}<\sum_{j=0}^{N}a_{j}^{l}.$ Taking into account
assertions (2), (5) and (9) of Theorem 2.6 and the fact that $a_{N}\geqslant
a_{j}$ for $k+1\leqslant j\leqslant N,$ we get

$k-1+2^{l}a_{N}^{l}\,<\,\sum_{i=1}^{k}d_{i}^{l}\,<\,\sum_{j=0}^{N}a_{j}^{l}\leqslant
k+1+a_{N}^{l}(N-k).$

We obtain

$k-1+(N-k+2)a_{N}^{l}\,<\,k+1+a_{N}^{l}(N-k),$

so we get $a_{N}^{l}<1,$ which is impossible.

##  5\. Proof of Theorem 1.8

Let us prove some preliminary lemmas.

######  Lemma 5.1.

Let $X$ be a smooth well formed Fano weighted complete intersection of
codimension $k$ in $\mathbb{P}(a_{0},\ldots,a_{N}),$ which is not an
intersection with a linear cone. Assume that $\lceil\log_{3}(N-k+2)\rceil=1$.
Then $N=2,$ $k=1$ and $X$ is a conic in $\mathbb{P}(1,1,1)=\mathbb{P}^{2}.$

###### Proof.

As $\lceil\log_{3}(N-k+2)\rceil=1,$ we have that $N-k+2\leqslant 3,$ or,
equivalently, $N-k\leqslant 1.$ The case when $N-k=0$ is impossible by Theorem
2.6(6). Thus, we have $N-k=1.$ According to assertions (2) and (6) of Theorem
2.6 this holds if and only if $N=2,$ $k=1$ and $s\geqslant 2.$ But the only
smooth Fano variety in this case should be a curve
$X\subset\mathbb{P}(1,1,a_{2})$ of degree $d$ such that $2+a_{2}>d.$ However,
by Theorem 2.6(9) we have $2+a_{2}>d\geqslant 2a_{2}.$ Thus, we get $a_{2}=1$
and $d=2,$ i.e. $X$ is a conic in $\mathbb{P}^{2}.$

∎

######  Lemma 5.2.

Let $X\subset\mathbb{P}^{N}$ be a smooth Fano complete intersection of
codimension $k$ which does not lie in a hyperplane. Assume that $X$ is not an
intersection of quadrics. Then for $l=\log_{3}(N-k+2)$ the inequality

(5.1) $\sum_{i=1}^{k}d_{i}^{l}\geqslant N+1$

holds true.

###### Proof.

From our assumptions there is $d_{i}$ such that $d_{i}>2.$ Suppose that the
inequality (5.1) is false, i.e.

$\sum_{i=1}^{k}d_{i}^{l}\,<\,N+1.$

We have

$(k-1)2^{l}+3^{l}\leqslant\sum_{i=1}^{k}d_{i}^{l}\,<\,N+1.$

Thus we get

$(k-1)2^{l}+N-k+2\,<\,N+1,$

which is equivalent to $(k-1)\cdot 2^{l}\,<\,k-1,$ but it is impossible,
because $l>1$ by Lemma 5.1.

∎

Now we are ready to prove Theorem 1.8.

###### Proof of Theorem 1.8.

Let $X\subset\mathbb{P}(a_{0},a_{1},\ldots,a_{N})$ be a smooth well formed
weighted complete intersection of multidegree $(d_{1},\ldots,d_{k})$ which is
not an intersection with a linear cone. Without loss of generality we can
assume that

$d_{1}\leqslant d_{2}\leqslant\ldots\leqslant d_{k}\qquad\text{and}\qquad
a_{0}\leqslant a_{1}\leqslant\ldots\leqslant a_{N}.$

We can also assume that $X$ is a Fano variety, because otherwise there is
nothing to prove. Also we assume that $X$ is not a complete intersection of
quadrics in $\mathbb{P}^{N}$ by Example 1.4. It is enough to prove that for
$l=\lceil\log_{3}(N-k+2)\rceil$, which is at least 2 by Lemma 5.1, the $l$-th
Chern character is not positive, i.e.

(5.2) $\sum_{i=1}^{k}d_{i}^{l}\geqslant\sum_{j=0}^{N}a_{j}^{l}.$

Let $s\,=\,\\#\\{a_{j}\mid a_{j}\,=\,1\\}.$ In the case $s=N+1$ Theorem 1.8
holds true by Lemma 5.2. So assume that $s\leqslant N.$

Suppose that $a_{N}=2.$ Then we have $a_{j}=2$ for all $s\leqslant j\leqslant
N.$ Thus, by Theorem 2.6(7) we get $k\geqslant N-s+1.$ So by Lemma B.1 the
inequality (5.2) holds.

Otherwise, if $a_{N}\geqslant 3,$ the assumptions of Lemma B.3 hold by Theorem
2.6. Thus, the inequality (5.2) is implied by this lemma.

∎

##  Appendix A Monotonicity and boundedness

In this appendix we prove some elementary lemmas about monotonicity and
boundedness of functions which are needed in Theorems 1.8 and 3.1.

######  Lemma A.1.

If $a\geqslant b\geqslant c>0$, then the function $f(l)=a^{l}+b^{l}$ grows
slower than the function $g(l)=(a+c)^{l}+(b-c)^{l}$ for $l\geqslant 1$ in
$\mathbb{R};$ in other words, the function $g(l)-f(l)$ in non-increasing for
$l\geqslant 1.$

###### Proof.

We have

(A.1) $h(l)=g(l)-f(l)=(a+c)^{l}+(b-c)^{l}-a^{l}-b^{l}.$

If $b=c,$ then we are done, because

$h(l)=(a+b)^{l}-a^{l}-b^{l}=b^{l}\left(\left(\frac{a+b}{b}\right)^{l}-\left(\frac{a}{b}\right)^{l}-1\right)$

which is monotonically increasing, as a product of monotonically increasing
non-negative functions. So assume that $b>c.$ We have

(A.2)
$h(l)=(b-c)^{l}\left(\left(\frac{a+c}{b-c}\right)^{l}-\left(\frac{a}{b-c}\right)^{l}-\left(\frac{b}{b-c}\right)^{l}+1\right).$

Put $y=\frac{a}{b-c}$ and $z=\frac{b}{b-c}.$ So we can rewrite $h(l)$ as

$h(l)=(b-c)^{l}\left((y+z-1)^{l}-y^{l}-z^{l}+1\right)=(b-c)^{l}t(l,y,z).$

The function $(b-c)^{l}$ is monotonically increasing. So it is enough to prove
that $t(l,y,z)$ is monotonically increasing with respect to $l.$ Let us note
that

$y+z-1>y\geqslant z>1$

by our conditions. We have

(A.3) $s(l,y,z)=\frac{\partial t(l,y,z)}{\partial
l}=\ln(y+z-1)\cdot(y+z-1)^{l}-\ln(y)\cdot y^{l}-\ln(z)\cdot z^{l}.$

For the sake of proving that $t(l,y,z)$ is monotonically increasing, we need
to show that $s(l,y,z)$ is monotonically increasing with respect to $z$ and
get that

$s(l,y,z)>s(l,y,1)=0.$

Indeed,

$\frac{\partial s(l,y,z)}{\partial
z}=(y+z-1)^{l-1}-z^{l-1}+l\ln(y+z-1)\cdot(y+z-1)^{l-1}-l\ln(z)\cdot
z^{l-1}>0.$

Thus, $h(l)$ is monotonically increasing.

∎

######  Lemma A.2.

Let $1\leqslant d\leqslant a\leqslant b$ be real numbers. Then we have

$a+b\leqslant\frac{a}{d}+bd.$

###### Proof.

Indeed, we have

$bd+\frac{a}{d}-a-b=\frac{a}{d}\left(\frac{bd^{2}}{a}+1-d-\frac{bd}{a}\right)=\frac{a}{d}\left(d-1\right)\left(\frac{bd}{a}-1\right)\geqslant
0.$

∎

######  Lemma A.3.

Let $f(l)\,=\,4^{l}-2^{l}-3^{l},$ where $l\in\mathbb{R}.$ Then $f(l)\,>\,0$
for $l\geqslant 2.$

###### Proof.

Indeed, the function $\frac{f(l)}{2^{l}}$ is monotonically increasing, because

$\frac{\partial}{\partial l}\left(\frac{f(l)}{2^{l}}\right)=2^{l}\ln
2-\left(\frac{3}{2}\right)^{l}\ln\left(\frac{3}{2}\right)>\ln
2\left(2^{l}-\left(\frac{3}{2}\right)^{l}\right)>0.$

Moreover, we have $f(2)=3>0.$ Thus, $f(l)$ is positive for all $l\geqslant 2.$

∎

Now we prove the lemma that we use in Theorem 3.1.

######  Lemma A.4.

Let $N,k,d_{1},\ldots,d_{k},a_{0},a_{1},\ldots,a_{N}$ be positive integers
such that

$\displaystyle d_{1}\leqslant d_{2}\leqslant\ldots\leqslant d_{k-1}\leqslant
d_{k};$ $\displaystyle 1=a_{0}=a_{1}=\ldots
a_{s-1}<a_{s}\leqslant\ldots\leqslant a_{N-1}\leqslant a_{N}.$

Suppose the following conditions are satisfied:

(A.4) $\displaystyle k\leqslant N+1;$ (A.5) $\displaystyle d_{i}\geqslant
2\quad\text{for all}\;\;1\leqslant i\leqslant k;$ (A.6) $\displaystyle
d_{i}>a_{N-k+i};$ (A.7) $\displaystyle s=\\#\\{a_{j}\mid a_{j}=1\\}\geqslant
1;$ (A.8)
$\displaystyle\sum_{j=s}^{N}a_{j}\,\leqslant\,\sum_{i=1}^{k}d_{i}\,<\,\sum_{j=s}^{N}a_{j}+s.$

Let

$f(l)\,=\,\sum_{i=1}^{k}d_{i}^{l}-\sum_{j=0}^{N}a_{j}^{l},$

where $l\in\mathbb{R}.$ Then $f(l)$ is monotonically increasing for
$l\geqslant 1.$

###### Proof.

To prove the lemma we launch an algorithm such that on every step we change
$a_{j}$, $d_{i}$, $N$, $k$, $s$ by $a^{\prime}_{j}$, $d^{\prime}_{i}$,
$N^{\prime}$, $k^{\prime}$, $s^{\prime}$, respectively, such that the new
function

$f^{\prime}(l)\,=\,\sum_{i=1}^{k^{\prime}}(d_{i}^{\prime})^{l}-\sum_{j=0}^{N^{\prime}}(a_{j}^{\prime})^{l},$

grows slower than $f(l).$ Dropping the primes from notation for simplicity, we
apply the algorithm again, until we obtain $s=N$ or $s=N+1$. Then we prove
that after the algorithm stops we obtain the monotonically increasing
function. Thus, the function $f(l)$ we have started with is also monotonically
increasing. Let us describe the step of the algorithm.

We take $a_{N}$, $a_{s}$ (i.e. the minimal $a_{j}$ which is not equal to 1)
and we increase and decrease them, respectively, by the same integer $c$ with
the properties

(A.9) $\displaystyle a_{s}-c\geqslant 1;$ (A.10) $\displaystyle
a_{N}+c\leqslant d_{k}.$

Let us take the maximal integer number $c$ such that at least one of the
inequalities (A.9)–(A.10) is an equality and the other inequality is true,
i.e.

$c=\min\\{a_{s}-1,d_{k}-a_{N}\\}.$

If (A.9) is an equality and (A.10) is a strict inequality, we increase $s$ by
1, i.e. put $s^{\prime}=s+1$, $N^{\prime}=N$, $k^{\prime}=k$, change $a_{j}$,
$d_{i}$ by $a^{\prime}_{j}$, $d^{\prime}_{i},$ respectively, which are defined
as follows:

$\displaystyle
a^{\prime}_{s^{\prime}-1}=1;\;\;a^{\prime}_{s^{\prime}}=a_{s+1};\;\;a^{\prime}_{j}=a_{j}\;\;\text{for}\;\;j\neq
s^{\prime}-1,s^{\prime},N^{\prime};\;\;a^{\prime}_{N^{\prime}}=a_{N}+c;$
$\displaystyle d^{\prime}_{i}=d_{i}\;\;\text{for}\;\;1\leqslant i\leqslant
k^{\prime}.$

If (A.9) is a strict inequality and (A.10) is an equality, we decrease $k$ and
$N$ by 1, i.e. put $k^{\prime}=k-1$, $N^{\prime}=N-1$, $s^{\prime}=s$, change
$a_{j}$, $d_{i}$ by $a^{\prime}_{j}$, $d^{\prime}_{i}$, respectively, which
are defined as follows:

$a^{\prime}_{s^{\prime}}=a_{s}-c;\;\;a^{\prime}_{j}=a_{j}\;\;\text{for}\;\;j\neq
s^{\prime};\;\ d^{\prime}_{i}=d_{i}\;\;\text{for}\;\;1\leqslant i\leqslant
k^{\prime}.$

If both (A.9) and (A.10) are equalities, we increase $s$ by 1, decrease $k$
and $N$ by 1, i.e. put $k^{\prime}=k-1$, $N^{\prime}=N-1$, $s^{\prime}=s+1$,
change $a_{j}$, $d_{i}$ by $a^{\prime}_{j}$, $d^{\prime}_{i}$, respectively,
which are defined as follows:

$a^{\prime}_{s^{\prime}-1}=1;\;\;a^{\prime}_{s^{\prime}}=a_{s+1};\;\;a^{\prime}_{j}=a_{j}\;\;\text{for}\;\;j\neq
s^{\prime}-1,s^{\prime};\;\;d^{\prime}_{i}=d_{i}\;\;\text{for}\;\;1\leqslant
i\leqslant k^{\prime}.$

By Lemma A.1 the function $-(a_{N}+c)^{l}-(a_{s}-c)^{l}$ grows slower than
$-a_{N}^{l}-a^{l}_{s}$. Thus, in each of the above three cases the new
function

$f^{\prime}(l)\,=\,\sum_{i=1}^{k^{\prime}}(d_{i}^{\prime})^{l}-\sum_{j=0}^{N^{\prime}}(a_{j}^{\prime})^{l},$

grows slower than $f(l).$ Also note that all the conditions of the lemma are
satisfied for
$a^{\prime}_{j},d^{\prime}_{i},N^{\prime},k^{\prime},s^{\prime}.$

The algorithm stops when we have either $s=N$, or $s=N+1$. Note that the case
$s=N+1$ is possible if and only if at the last step both (A.9) and (A.10) are
equalities. From (A.8) we get that after the algorithm stops we have
$k\geqslant 1.$ In the first case we need to prove that
$f(l)=\sum^{k}_{i=1}d^{l}_{i}-a^{l}_{N}-s$ is monotonically increasing. We can
rewrite this function as

$f(l)=\left(d_{k}^{l}-a^{l}_{N}\right)+\sum^{k-1}_{i=1}d_{i}^{l}-s$

and get that $f(l)$ is monotonically increasing function as a sum of
monotonically increasing functions. In the second case we need to rpove that
$f(l)=\sum^{k}_{i=1}d^{l}_{i}-N-1$ is monotonically increasing, which is
obviously true.

∎

##  Appendix B Inequalities on sums of $l$-th degrees

In this appendix we prove some computational lemmas which are needed in
Section 5.

######  Lemma B.1.

Let $N$ and $k$ be positive integer numbers, $a_{0},a_{1},\ldots,a_{N}$ be
positive rational numbers and $d_{1},d_{2},\ldots,d_{k}$ be positive integer
numbers such that the following inequalities hold

(B.1) $\displaystyle N-k\geqslant 2;$ (B.2) $\displaystyle
1=a_{0}=a_{1}=\ldots=a_{s-1}<a_{s}\leqslant\ldots\leqslant a_{N-1}\leqslant
a_{N};$ (B.3) $\displaystyle a_{N}\geqslant 2;$ (B.4) $\displaystyle
d_{k}\geqslant 2a_{N};$ (B.5) $\displaystyle
d_{i}>a_{N-k+i}\quad\text{for}\quad 1\leqslant i\leqslant k-1.$

Assume that $1\leqslant\\#\\{a_{j}\mid a_{j}\neq 1\\}\leqslant k.$ Then for
$l=\lceil\log_{3}(N-k+2)\rceil$ the inequality

(B.6) $\sum_{i=1}^{k}d_{i}^{l}\geqslant\sum_{j=0}^{N}a_{j}^{l}$

holds true.

###### Proof.

We have $N-s+1\leqslant k.$ We need to prove that

(B.7) $\sum_{i=1}^{k}d_{i}^{l}-\sum_{j=s}^{N}a_{j}^{l}\geqslant s.$

By (B.4) we have

$\sum_{i=1}^{k}d_{i}^{l}-\sum_{j=s}^{N}a_{j}^{l}\geqslant
a_{N}^{l}(2^{l}-1)+\sum_{i=1}^{k-1}d_{i}^{l}-\sum_{j=s}^{N-1}a_{j}^{l}$

Taking into account (B.5) and $N-s+1\leqslant k$ we obtain

(B.8) $\sum_{i=1}^{k-1}d_{i}^{l}-\sum_{j=s}^{N-1}a_{j}^{l}>k-1.$

Thus, using (B.3) and (B.8), we get

$a_{N}^{l}(2^{l}-1)+\sum_{i=1}^{k-1}d_{i}^{l}-\sum_{j=s}^{N-1}a_{j}^{l}\geqslant
2^{l}(2^{l}-1)+k-1.$

By (B.1) we obtain $l\geqslant 2.$ So by Lemma A.3 and the definition of $l$
we have

(B.9) $2^{l}(2^{l}-1)\geqslant 3^{l}\geqslant N-k+2.$

So by (B.9) we have

(B.10) $2^{l}(2^{l}-1)+k-1\geqslant N-k+2+k-1=N+1\geqslant s,$

which proves the inequality (B.7).

∎

######  Lemma B.2.

Let $d$ and $N\geqslant 3$ be positive integer numbers,
$a_{0},a_{1},\ldots,a_{N}$ be rational numbers. Assume that the following
holds

(B.11) $\displaystyle
1=a_{0}=a_{1}=\ldots=a_{s-1}<a_{s}\leqslant\ldots\leqslant a_{N-1}\leqslant
a_{N};$ (B.12) $\displaystyle d=m\cdot a_{0}\cdot\ldots\cdot
a_{N}\;\;\text{for some positive integer}\;\;m;$ (B.13) $\displaystyle
a_{N}\geqslant 3;$ (B.14) $\displaystyle d\geqslant 2a_{N}.$

Assume also that all $a_{j}$ with a possible exception of $a_{s}$ are integers
and $s\leqslant N-1.$ Then if $l=\lceil\log_{3}(N+1)\rceil,$ we have

(B.15) $d^{l}\geqslant\sum_{j=0}^{N}a_{j}^{l}.$

###### Proof.

Since $N\geqslant 3,$ we have $l\geqslant 2.$ So first of all, assume that
$d\geqslant 3a_{N}.$ We have to prove that

(B.16) $3^{l}a_{N}^{l}-\sum_{j=s}^{N}a_{j}^{l}\geqslant s.$

By (B.11) we have

$\sum_{j=s}^{N}a_{j}^{l}\leqslant a_{N}^{l}(N-s+1).$

So we have

$3^{l}a_{N}^{l}-\sum_{j=s}^{N}a_{j}^{l}\geqslant
3^{l}a_{N}^{l}-a_{N}^{l}(N-s+1).$

As $3^{l}\geqslant N+1,$ we get

$3^{l}a_{N}^{l}-a_{N}^{l}(N-s+1)\geqslant sa_{N}^{l}\geqslant s,$

which proves the inequality (B.16).

Now we assume that $3a_{N}>d\geqslant 2a_{N}.$ By (B.12) we have

$d=a_{N}(ma_{N-1}\ldots a_{s})=a_{N}\cdot T,$

where $m$ is a positive integer and $T=ma_{N-1}\ldots a_{s}.$ The number $T$
is at least 3, unless either $s=N-2$ with $a_{N-1}=2$ and $a_{N-2}<2,$ or
$s=N-1$ and $a_{N-1}<3.$

First of all, consider the case when $s=N-2$ with $a_{N-1}=2$ and $a_{N-2}<2.$
We need to prove that

$2^{l}a_{N}^{l}\geqslant a_{N}^{l}+2^{l}+2^{l}+N-2,$

or, equivalently,

$(2^{l}-1)a_{N}^{l}-2^{l}-2^{l}-N+2\geqslant 0.$

By (B.13) we get that

$(2^{l}-1)a_{N}^{l}-2^{l}-2^{l}-N+2\geqslant 3^{l}(2^{l}-1)-2\cdot 2^{l}-N+2.$

As $3^{l}\geqslant N+1,$ we have

$\displaystyle 3^{l}(2^{l}-1)-2\cdot 2^{l}-N+2\geqslant(N+1)(2^{l}-1)-2\cdot
2^{l}-N+2=$ $\displaystyle=2^{l}(N-1)-2(N-1)-1\geqslant 0,$

because $l\geqslant 2.$

Now consider the case when $s=N-1$ and $a_{N-1}<3.$ We need to prove that

$2^{l}a_{N}^{l}\geqslant a^{l}_{N}+3^{l}+N-1.$

In other words, as $a_{N}\geqslant 3,$ we need to prove that

$3^{l}(2^{l}-2)\geqslant N-1,$

which is true, because $l\geqslant 2$ and $3^{l}\geqslant N+1.$ ∎

######  Lemma B.3.

Let $N,k,d_{1},d_{2},\ldots,d_{k}$ be positive integer numbers,
$a_{0},a_{1},\ldots,a_{N}$ be rational numbers such that the following
inequalities hold

(B.17) $\displaystyle N-k\geqslant 2;$ (B.18) $\displaystyle d_{1}\leqslant
d_{2}\leqslant\ldots\leqslant d_{k-1}\leqslant d_{k};$ (B.19) $\displaystyle
1=a_{0}=a_{1}=\ldots=a_{s-1}<a_{s}\leqslant\ldots\leqslant
a_{N-1}\;\;\text{and}\;\;a_{N-2}\leqslant a_{N};$ (B.20) $\displaystyle
d_{i}\,>\,a_{N-k+i}\quad\text{for}\quad 1\leqslant i\leqslant k-1;$ (B.21)
$\displaystyle d_{i}\neq 1\quad\text{for all}\quad i;$ (B.22) $\displaystyle
a_{N}\geqslant 3;$ (B.23) $\displaystyle d_{1}\cdot\ldots\cdot d_{k}=m\cdot
a_{0}\cdot\ldots\cdot a_{N}\;\;\text{for some positive integer}\;\;m;$ (B.24)
$\displaystyle d_{k}\geqslant 2a_{N}.$

Assume that all $a_{j}$ with a possible exception of $a_{s}$ and $a_{N-1}$ are
integer numbers. If $l=\lceil\log_{3}(N-k+2)\rceil,$ then we have

(B.25) $\sum_{i=1}^{k}d_{i}^{l}\geqslant\sum_{j=0}^{N}a_{j}^{l}.$

###### Proof.

To prove the lemma we launch an algorithm such that on every step we change
the numbers $a_{j}$, $d_{i}$, $N$, $k$, $s$ by $a^{\prime}_{j}$,
$d^{\prime}_{i}$, $N^{\prime}$, $k^{\prime}$, $s^{\prime}$, respectively, such
that

$\sum_{i=1}^{k}d_{i}^{l}-\sum_{j=0}^{N}a_{j}^{l}\geqslant\sum_{i=1}^{k^{\prime}}(d^{\prime}_{i})^{l}-\sum_{j=0}^{N^{\prime}}(a^{\prime}_{j})^{l}.$

Then dropping the primes from notation for simplicity we apply the algorithm
again, until we obtain $a_{j},$ $d_{i},$ $N,$ $k,$ $s$ satisfying certain
strong restrictions. We prove that after the algorithm stops we obtain non-
negative value

$\sum_{i=1}^{k}d_{i}^{l}-\sum_{j=0}^{N}a_{j}^{l}\geqslant 0.$

Thus, the value we have started with is also non-negative. Let us describe the
step of the algorithm. We take $a_{N-1}$ and $a_{s}$, then we multiply and
divide them, respectively, by the same rational number $p$ with the properties

(B.26) $\displaystyle\frac{a_{s}}{p}\geqslant 1;$ (B.27) $\displaystyle
a_{N-1}\cdot p\leqslant d_{k-1}.$

Let us take the maximal $p$ such that at least one of the inequalities (B.26)
and (B.27) is an equality and the other inequality is true, i.e.

$p=\min\\{a_{s},\frac{d_{k-1}}{a_{N-1}}\\}.$

If (B.26) is an equality and (B.27) is a strict inequality, we increase $s$ by
1, i.e. put $s^{\prime}=s+1$, $N^{\prime}=N$, $k^{\prime}=k$, change $a_{j}$,
$d_{i}$ by $a^{\prime}_{j}$, $d^{\prime}_{i},$ respectively, which are defined
as follows:

$\displaystyle
a^{\prime}_{s^{\prime}-1}=1;\;\;a^{\prime}_{s^{\prime}}=a_{s+1};\;\;a^{\prime}_{j}=a_{j}\;\;\text{for}\;\;j\neq
s^{\prime}-1,s^{\prime},N^{\prime}-1;\;\;a^{\prime}_{N^{\prime}-1}=p\cdot
a_{N-1};$ $\displaystyle d^{\prime}_{i}=d_{i}\;\;\text{for}\;\;1\leqslant
i\leqslant k^{\prime}.$

After the step of the algorithm we have $N-k+2=N^{\prime}-k^{\prime}+2$ and
all conditions of the lemma still hold true.

If (B.26) is a strict inequality and (B.27) is an equality, we decrease $k$
and $N$ by 1, i.e.

$k^{\prime}=k-1;\;\;N^{\prime}=N-1;\;\;s^{\prime}=s;$

change $a_{j}$, $d_{i}$ by $a^{\prime}_{j}$, $d^{\prime}_{i},$ respectively,
which are defined as follows:

$a^{\prime}_{s^{\prime}}=\frac{a_{s}}{p};\;\;a^{\prime}_{j}=a_{j}\;\;\text{for}\;\;j\neq
s^{\prime},N^{\prime};\;\;a^{\prime}_{N^{\prime}}=a_{N};\;\;d^{\prime}_{i}=d_{i}\;\;\text{for}\;\;1\leqslant
i\leqslant k^{\prime}.$

After the step of the algorithm we have $N-k+2=N^{\prime}-k^{\prime}+2$ and
the conditions of the lemma still hold true, and, additionally, we have the
strengthening of the condition (B.19):

(B.28) $a^{\prime}_{0}\leqslant a^{\prime}_{1}\leqslant\ldots\leqslant
a^{\prime}_{N-1}\leqslant a^{\prime}_{N}.$

and $a_{N^{\prime}-1}$ an integer number, unless $s^{\prime}=N^{\prime}-1.$

If both (B.26) and (B.27) are equalities, we increase $s$ by 1, decrease $k$
and $N$ by 1, i.e. $k^{\prime}=k-1$, $N^{\prime}=N-1$, $s^{\prime}=s+1$,
change $a_{j}$, $d_{i}$ by $a^{\prime}_{j}$, $d^{\prime}_{i},$ respectively,
which are defined as follows:

$\displaystyle
a^{\prime}_{s^{\prime}-1}=1;\;\;a^{\prime}_{s^{\prime}}=a_{s+1};\;\;a^{\prime}_{j}=a_{j}\;\;\text{for}\;\;j\neq
s^{\prime}-1,s^{\prime},N^{\prime};\;\;a^{\prime}_{N^{\prime}}=a_{N};$
$\displaystyle d^{\prime}_{i}=d_{i}\;\;\text{for}\;\;1\leqslant i\leqslant
k^{\prime}.$

After the step of the algorithm we have $N-k+2=N^{\prime}-k^{\prime}+2$ and
the conditions of the lemma still hold true, and, additionally, we have the
strengthening of the condition (B.19):

(B.29) $a^{\prime}_{0}\leqslant a^{\prime}_{1}\leqslant\ldots\leqslant
a^{\prime}_{N-1}\leqslant a^{\prime}_{N}.$

and $a_{N^{\prime}-1}$ an integer number, unless $s^{\prime}=N^{\prime}-1.$

In any case, by Lemma A.2 we have

$\left(a_{N-1}\cdot p\right)^{l}+\left(\frac{a_{s}}{p}\right)^{l}\geqslant
a_{N-1}^{l}+a_{s}^{l}.$

This implies

$\sum_{i=1}^{k^{\prime}}(d^{\prime}_{i})^{l}-\sum_{j=0}^{N^{\prime}}(a^{\prime}_{j})^{l}\leqslant\sum_{i=1}^{k}d_{i}^{l}-\sum_{j=0}^{N}a_{j}^{l}.$

Note also that the algorithm does not change $a_{N}$ and $d_{k}.$

The algorithm stops if one of the following cases is satisfied:

  * 1)

$s=N-1$ and $k>1$;

  * 2)

$s=N$ for arbitrary $k$;

  * 3)

$k=1$ and $s\leqslant N-1$.

Let us prove that in all these cases the inequality (B.25) holds true. In the
first two cases we are done by Lemma B.1. In the third case the equality
$a_{N-1}\cdot p=d_{k-1}$ holds in the last step of the algorithm. Thus, the
inequality (B.29) holds and $a_{N-1}$ is an integer, unless $s=N-1.$ So in
this case we are done by Lemma B.2.

∎

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# Enhanced thermoelectric properties in phosphorene nanorings

Fatemeh Moghadasi Borojeni1 Esmaeil Taghizadeh Sisakht1 Farhad Fazileh1,
<EMAIL_ADDRESS>F. M. Peeters2,3 1Department of Physics, Isfahan University
of Technology, Isfahan 84156-83111, Iran  2HSE University, Moscow, 10100
Russia  3Departamento de Fisica, Universidade Federal do Ceara, 60455-760
Fortaleza, Ceara, Brazil

###### Abstract

Using the tight-binding approach, we investigate the thermoelectric properties
of rectangular phosphorene nanorings for both symmetrically and asymmetrically
attaching to phosphorene nanoribbon leads. We design our phosphorene-based
nanostructures to enhance the thermoelectric performance in the absence and
the presence of perpendicular magnetic fields. Our results show that when
zigzag phosphorene nanoribbons (ZPNRs) are coupled symmetrically to
rectangular rings, a comparatively large band gap is induced in the electronic
conductance due to the suppression of the contribution of edge states. This
gives rise to a remarkable increase in the thermopower response compared to
the case of pristine ZPNRs. More intriguingly, we found that though the
maximum power factor in this system is about the same as the one for its ZPNR
counterpart, the much smaller electronic thermal conductance of this
phosphorene-based nanostructure can remarkably contribute to the improvement
of the figure of merit. Also, we found that the symmetry/asymmetry of our
designed nanostructures, the geometrical characteristics of the ring, and the
magnetic flux are three important factors that control the thermoelectric
properties of phosphorene quantum rings. Our numerical calculations show that
by changing the magnetic flux through the nanoring, a drastic increase in the
thermopower is observed near an antiresonance point. We demonstrate the
tunability of the thermopower and the possibility to switch on and off the
thermoelectric response of phosphorene nanorings with the magnetic flux.
Moreover, for asymmetric connection configurations with armchair-edged leads,
we found that though the thermopower is almost intact, a remarkable reduction
of the electronic thermal conductance can lead to a notable improvement in the
figure of merit. Our results suggest phosphorene nanorings as promising
candidate nanostructures for thermoelectric applications.

###### pacs:

73.22.-f,71.70.Ej,73.63.-b

##  I Introduction

Designing devices with optimal thermoelectric (TE) properties is highly
desirable for the future of energy harvesting and environmental issues
Petsagkourakis  _et al._ (2018). TE materials occupy a special place in clean
energy research field, since they directly convert heat into electrical energy
and vice versa. They are promising candidate materials for the future of clean
and efficient electrical power generators and cooling (heating) devices. The
TE efficiency of a material is often assessed by dimensionless figure of merit
$ZT=\frac{S^{2}\sigma T}{\kappa}$; where $S$ is the Seebeck coefficient (SC)
or thermopower, $\sigma$ and $\kappa$ are the electrical and thermal
conductivities, respectively and $T$ is the temperature. This relation
suggests that a reduction in thermal conductivity (while the electrical
conductivity is kept almost the same) leads to a strong improvement of the TE
efficiency. One way to accomplish this task is to make nanostructures out of
the bulk TE materials Szczech  _et al._ (2011). Here, the enhanced TE factor
of merit $ZT$ is due to phonon scattering with boundaries and quantum
confinement effects. The TE efficiency can be further enhanced by making
nanorings out of the materials Hozana and Latgé (2019); Mazzamuto  _et al._
(2011). Using nanorings made of nanoribbons of the pristine materials we can
take advantage of the Aharonov-Bohm (AB) effect when we apply an external
magnetic field to tune the magnetoresistance of the system Zhang  _et al._
(2017) and further enhance the TE efficiency Liu and Yang (2010); Haack and
Giazotto (2019).

Among nanostructured semiconducting materials, two dimensional (2D)
phosphorene has recently received remarkable attention due to its great
transport properties and potential applications Samuel Reich (2014); Li  _et
al._ (2016, 2014); Liu  _et al._ (2014); Xia  _et al._ (2014); Castellanos-
Gomez  _et al._ (2014); Koenig  _et al._ (2014); Qiao  _et al._ (2014). This
structure consists of a single- or few-layer of black phosphorus and has been
successfully fabricated by researchers Li  _et al._ (2014); Liu  _et al._
(2014); Koenig  _et al._ (2014). From these studies, it is suggested that
because of its remarkable electronic, mechanical, and optical properties, it
offers great promise for applications in electronic and optoelectronic devices
Buscema  _et al._ (2014); Das  _et al._ (2014); Lu  _et al._ ; Rodin  _et al._
(2014); Chen  _et al._ (2015); Tayari  _et al._ (2015); Zou  _et al._ (2016);
Wei and Peng (2014); Island  _et al._ ; Ezawa (2014). In addition, due to
superior thermoelectric characteristics, it is also proposed that phosphorene
nanostructures are highly promised for thermoelectric devices and it could be
their key application in the future Castellanos-Gomez (2015). Various factors
may significantly affect the three parameters that are used to quantify the
figure of merit in this structure. For single-crystals of bulk black
phosphorus, the experimental measurements revealed that the value of SC is
about 340 $\mu$V/K at room temperature Flores  _et al._ (2015). It was shown
that gate-tuning is a successful way to control the TE power coefficient in a
thin flake of black phosphorus Saito  _et al._ (2016). These experimental
measurements verified that the SC of the ion-gated bulk black phosphorus can
reach 510 $\mu$V/K at 210 K which may result in a great increase of $ZT$
compared to the bulk single crystal at room temperature. The puckered
structure of monolayer phosphorene has led to anisotropy in its electrical and
thermal properties Carvalho  _et al._ (2016). Theoretical studies showed that
this distinct feature gives rise to orthogonal electrical and thermal
conductances Fei  _et al._ (2014). This results in a higher $\sigma/\kappa$
ratio and thus the rather large figure of merit $ZT\sim$1 (at room
temperature) along the armchair direction of monolayer phosphorene Fei  _et
al._ (2014). Moreover, it is traditionally believed that the utilization of
the nanoribbons of such 2D materials is a way to enhance the TE efficiency.
Theoretical calculations predicted that using phosphorene nanoribbons (PNRs)
as thermoelectric material can lead to improved TE efficiencies Zhang  _et
al._ (2014). In addition, the effect of the edge states on the thermopower in
zigzag phosphorene nanoribbons (ZPNRs) was studied Ma  _et al._ (2016). It was
shown that by applying a transverse electric field, one can completely push
the edge modes into the bulk bands and maximize the bulk energy gap which
results in enhanced thermoelectric power in PNRs Ma  _et al._ (2016). Also, it
was found that the passivation of edge phosphorous atoms with hydrogen in both
types of nanoribbon edges Zhang  _et al._ (2014), and even oxidation in
phosphorene oxide Lee  _et al._ (2021) enhances the thermopower.
These studies confirm that nanoribbon-based structures of monolayer
phosphorene are likely candidates to improve the TE properties and the advent
of efficient TE nanodevices. Recent studies reported the production of high-
quality PNRs with relatively large and uniform lengths that may renew the
interest in the study of thermoelectric properties in nanoribbons and
nanostructured systems made of phosphorene Watts  _et al._ (2019). Nanorings
made of two dimensional materials are possible nanostructures that may be used
to improve TE efficiency. For example, TE properties of rectangular graphene
nanorings have been investigated for different conditions and are proposed to
build tunable TE generators Saiz-Bretín  _et al._ (2015); Hozana and Latgé
(2019). In such systems, the improvement in the TE performance is attributed
to the emergence of Fano line shapes or Breit-Wigner line shapes in the
transmission coefficient that their behavior depends on the geometrical
characteristics of the ring and the applied side-gate voltage Hozana and Latgé
(2019). However, despite the numerous works Saito  _et al._ (2016); Carvalho
_et al._ (2016); Fei  _et al._ (2014); Zhang  _et al._ (2014); Ma  _et al._
(2016); Lee  _et al._ (2021) on the TE properties of phosphorene and
phosphorene nanoribbons, the TE performance of rectangular phosphorene
nanorings has remained elusive so far.
In the present work, inspired by the mentioned methods to improve the TE
performance in graphene nanorings, we investigate the thermoelectric
characteristics of rectangular phosphorene nanorings for both symmetrically
and asymmetrically attachment to the leads. We investigate the effect of an
applied external magnetic field on the TE properties of these rectangular
rings and show how the thermoelectric coefficients are controllable in our
designed systems by the magnetic field. Also, we examine the role of ring size
and the symmetry of the connection configuration of leads on the TE response
of these systems. Our results will reveal the desirability of these
phosphorene-based nanostructures in designing efficient TE devices.

##  II  MODEL AND FORMALISM

Figure 1:  (a) and (b) show the schematic of symmetric and asymmetric coupling
of a rectangular phosphorene ring to two leads. The geometrical parameters of
the system are shown (c) Lattice structure of monolayer phosphorene (top
view). There are four P atoms in the rectangular primitive unit cell of
monolayer phosphorene. Red and blue balls denote the puckered structure of the
system. $t_{1}$-t5 denote the used hopping parameters in our TB model.

The system under study is a rectangular phosphorene nanoring with a symmetric
or asymmetric coupling to two leads as shown in Figs. 1(a) and (b). In each
configuration, two semi-infinite zigzag- or armchair-edged nanoribbon of
phosphorene connect to a nanoring. The geometry of a rectangular quantum ring
is determined by the inner and outer sides Lin, Win, Lout, and Wout. Ma et al.
Ma  _et al._ (2016) showed that by adjusting the bias voltage of a zigzag
phosphorene nanoribbon, the midgap bands completely split and the
corresponding localized states give rise to an enhancement in the thermopower.
We connect two ZPNRs to the phosphorene nanoring to show that a wide band gap
can be induced. This is accompanied by a drastic increase in the magnitude of
thermopower compared with the case without the nanoring and even compared with
graphene nanorings. In monolayer phosphorene (MLP), each phosphorus atom (P)
is covalently bonded to three neighbor phosphorus atoms and form a puckered
structure as depicted in Fig. 1 (c). The lattice constants of MLP are
$a=4.38$Å, and $b=3.31$Å. There are four P atoms in the rectangular primitive
unit cell of MLP. The generally accepted low-energy tight-binding (TB)
Hamiltonian for MLP is given by Rudenko and Katsnelson (2014)

$H_{c}=\sum_{i\neq j}t_{ij}c_{i}^{\dagger}c_{j},$ (1)

which can also be used to construct the Hamiltonian of PNRs with bare edges.
By applying a perpendicular magnetic field $B$ to the plane of a MLP nanoring,
the hopping parameters $t_{ij}$ in Eq. (1) are substituted by

$t_{ij}\rightarrow t_{ij}\exp\left(i\frac{2\pi
e}{h}\int_{\bf{r_{i}}}^{\bf{r_{j}}}\bf{A}\cdot\bf{dl}\right),$ (2)

to include Peierl’s phase factor, where $\bf{A}$ is the vector potential. To
calculate the quantum conductance of a MLP nanoring, we utilize the Landauer-
Buttiker formalism as implemented in the KWANT Groth  _et al._ (2014) package.
In this approach, the ballistic transport of charge carriers through a
nanoring is given by

$G=\frac{2e^{2}}{h}\int\left(-\frac{\partial f_{0}}{\partial
E}\right)T_{LR}(E)dE.$ (3)

In this equation, $f_{0}$ is the Fermi distribution function
$1/[e^{(E-E_{f})/k_{B}T}+1]$ at temperature $T$, and $T_{LR}(E)$ denotes the
the transmission coefficient of our lead-nanoring-lead system Imry and
Landauer (1999); Datta (1997). The thermopower $S$ and the electronic
contribution to thermal conductance $\kappa_{e}$ are expressed in terms of the
moments of the transmission coefficient Lv and Li (2012)

$K_{n}=\frac{2}{h}\int_{-\infty}^{\infty}\ \left(-\frac{\partial
f_{0}}{\partial E}\right)(E-\mu)^{n}T_{LR}(E)dE,$ (4)

via the following equations

$S=-\frac{1}{eT}\frac{K_{1}}{K_{0}},$ (5)

$\kappa_{e}=\frac{1}{T}\left[K_{2}-\frac{K_{1}^{2}}{K_{0}}\right].$ (6)

To construct the TB models in our systems, we employed the Pybinding code
package Moldovan and Peeters (2017).

##  III  RESULTS

We design our phosphorene-based nanostructures to enhance the TE performances
compared to the TE responses of pristine PNRs. So, before proceeding with our
main systems, we first examine the electronic conductance $G$, the Seebeck
coefficient $S$, the electronic thermal conductance $\kappa_{e}$, and the
thermoelectric power factor PF=$GS^{2}$ for pristine PNRs. Figures 2(a1)-(a2)
and 2(b1)-(b2) show the calculated conductance and thermopower for a typical
armchair phosphorene nanoribbon (APNR) and ZPNR, respectively. We set the
width of ribbons to $W=8$ nm for both APNRs and ZPNRs. The intriguing
electronic properties of APNRs and ZPNRs have a direct consequence on their
transport characteristics. As shown in Figs. 2(a1) and (b1), the conductance
of both APNRs and ZPNRs exhibit quantized plateaus $G=ne^{2}/h$, where
$e^{2}/h$ is the conductance quantum and $n$ denotes the number of available
transport modes at energy $E$. As seen, the ZPNR shows a conductance plateau
of value $G=2e^{2}/h$ near the zero energy region because of the contribution
of edge-propagating states along the zigzag boundaries. On the other hand, due
to the absence of propagating edge modes in the APNR, there exists no
quantized plateau in the midgap region.

Figure 2:  (a1)-(a4) electronic conductance, thermopower, electronic thermal
conductance, and power factor a function of chemical potential for a APNR with
$W=8$ nm. The temperature is set to $k_{B}T=0.03|t_{1}|$ in our calculations.
(b1)-(b4) the same quantities for a ZPNR with $W=8$ nm and at
$k_{B}T=0.03|t_{1}|$.  Figure 3:  (a1)-(a4) electronic conductance,
thermopower, electronic thermal conductance, and power factor a function of
chemical potential for a symmetric PRZL. The parameters are set to
$k_{B}T=0.03|t_{1}|$, $B=0$, $W_{in}=L_{in}=W_{out}/2=L_{out}/2=W_{L}=8$ nm.
(b1)-(b4) the same quantities for an asymmetric PRZL with the same set of
parameters. Figure 4: Spatial LDOS for (a) a symmetric and (b) an asymmetric
PRZL at $E_{f}=0~{}eV$. The color bar denotes the electronic density
distributions across the device.

Furthermore, the lack of electron-hole symmetry in the electronic band
structures of PNRs leads to an asymmetry of their quantum conductance with
respect to $E=0$. Figures 2(a2) and (b2) depict the numerically calculated
thermopower $S$. A comparison between the thermopower of the APNR and ZPNR
shows that the presence of mid-gap edge modes drastically affects the SC, and
hence it is sensitive to the topology of the boundaries in a PNR. We have
shown in Figs. 2(a3) and (b3) the peaks of the power factor for the condition
in which chemical potential sets close to the transmission gap. Also, Figs.
2(a4) and (b4) display the electronic thermal conductance $\kappa_{e}$ as a
function of the chemical potential $\mu$. As seen, due to the lack of
electron-hole symmetry, $\kappa_{e}$ is also asymmetric around $\mu=0$. To
enhance the SC in phosphorene, one way is to induce a large band gap in
phosphorene nanostructures with zigzag edges that have metallic
characteristic. This can be achieved by connecting two narrow leads of PNR to
a rectangular phosphorene nanoring as schematically depicted in Fig. 1. We
couple two PNRs with zigzag edges to a rectangular ring to present the
mentioned quantities for both symmetric (Figs. 3(a1)-(a4)) and asymmetric
(Figs. 3(b1)-(b4)) configurations. It is found that for a symmetric connection
configuration, the conductance and thermopower are considerably affected
compared to the case of pristine PNRs. A comparison between the SCs in Figs.
3(a2) and 2(b2) shows that in this case, the absolute value of the maxima is
notably greater than that of the ZPNR counterpart. By comparing Figs. 2(b1)
and 3(a1), one can see that for the symmetric case, the first conductance
plateau in ZPNR disappears completely, which is understood as an induced large
band gap in the electronic conductance due to the suppression of the
contribution of edge states. We have shown in Fig. 4(a1) the electronic local
density of states (LDOS) for a symmetric PRZL at $E=0$ eV to clearly show the
suppression of the edge modes contribution to the electronic conductance near
connection regions. As a result, the peak value of the thermopower for this
PRZL around zero energy is enhanced to $23~{}k_{B}/e$ ($1978~{}\mu$V/K). On
the other hand, for an asymmetric PRZL, the situation is different. Figure.
4(a2) displays the LDOS of electrons for this configuration at $E=0$ eV. As
seen, in this ring-like geometry a new path between the electrodes is observed
and the original first conductance plateau still exists (see Figs. 3(b1)
2(b1)), though its height decreases due to blocking the other path of edge
modes propagation. To account for these observations, note that for the
asymmetric PRZL (symmetric PRZL), the high-energy modes in the terminal are
mixed with the high-energy (low-energy) modes in the middle zigzag PNRs
(middle armchair PNRs). Therefore, the transporting modes are more
significantly affected by scattering for the symmetric than for asymmetric
nanorings Wu  _et al._ (2010). Although the maximum power factor in Fig. 2(b3)
is about the same as the one in Fig. 3(a3), the crucial point here is that by
making such nanorings out, we observe a much less electronic thermal
conductance $\kappa_{e}$ (see Fig. 3(a4)) compared to ZPNRs. Therefore, our
designed phosphorene-based nanostrutre can remarkably contribute to the
improvement of the figure of merit $ZT$.

Next, we examine the impact of the nanoring size on the electronic conductance
and thermopower of PRZLs. We focus on the symmetric configuration. Fig. 5
depicts the calculated conductance, SC, power factor, and electronic thermal
conductance of two PRZLs with different sets of structure parameters. Figs.
5(a1)-(a4) and (b1)-(b4) show the results for the first set of parameters
$W_{in}=2$ nm, $L_{in}=74$ nm, $W_{out}=8$ nm, $L_{out}=80$ nm, and $W_{L}=4$
nm, and the second set of structure parameters
$W_{in}=L_{in}=W_{out}/2=L_{out}/2=W_{L}=4$ nm, respectively. The results
clearly show that the thermopower profoundly depends on the size of considered
nanorings. In general, the smaller ring we consider, the better TE response we
get. So, in the second case, the maxima $S$ enhances and leads to a peak value
of $33~{}k_{B}/e$ ($2830~{}\mu$V/K) a the zero energy.

Figure 5: (a1)-(a4) electronic conductance, thermopower, electronic thermal
conductance, and power factor a function of chemical potential for a symmetric
PRZL with set of parameters $k_{B}T=0.03|t_{1}|$, $B=0$, $W_{in}=2$ nm,
$L_{in}=74$ nm, $W_{out}=8$ nm, $L_{out}=80$ nm, $W_{L}=4$ nm. (b1)-(b4) the
same quantities for another symmetric PRZL with the set of parameters
$k_{B}T=0.03|t_{1}|$, $B=0$, $W_{in}=L_{in}=W_{out}/2=L_{out}/2=W_{L}=4$ nm.
Figure 6:  (a1) and (a2) are zoomed-in views of Figs. 5(b1) and (b2) in the
energy range of 1.65 to 1.78 eV, respectively. (a3) LDOS of the Fano resonance
peak.

In addition, the SC spectrum of the PRZL exhibits some other small
oscillations at higher chemical potentials yielding changes in the sign.
Figures 6(a1) and (a2) show a zoomed-in view of Figs. 5(b1) and (b2) in the
energy range of 1.65 to 1.78 eV. As seen, the transmission coefficient shows
Fano line shapes that do not exist in ZPNRs. We have shown in Fig. 6(a3) the
LDOS of the Fano resonance peak. The reason behind the appearance of Fano
resonance peaks is the interaction between the electronic states of the ring
and connected leads. As seen, the mentioned LDOS shows localized states in
some regions of the ring implying that the Fano resonances are closely related
to the appearance of these bound states Zhang  _et al._ (2018). Interestingly,
when the chemical potential is tuned near a Fano resonance in the
transmission, an enhancement in the thermopower of the nanoring is observed.
This is justified as follows. We suppose that the chemical potential locates
near a Fano-like antiresonance. According to Eq. (5), the thermopower $S$ is
proportional to $K_{1}$ whose value is assessed by the integrand
$(-\frac{\partial f_{0}}{\partial E})(E-\mu)T_{LR}(E)$ where $(E-\mu)$ is an
odd function around $\mu$. As a result, a larger asymmetry of transmission
coefficient $T_{LR}(E)$ leads to an increase in $K_{1}$, and thus a higher TE
response.

Now, let us examine the effect of applying a perpendicular magnetic field on
the TE performance of a PRZL. It has been shown Zhang  _et al._ (2017) that by
applying a perpendicular magnetic field to PNRs, the conductance experiences
dramatic oscillations at special Fermi energies, leading to a giant
magnetoresistance (MR) in zigzag PNRs. Here, we show that the TE performance
of a PRZL highly depends on the magnetic flux.

Figure 7:  (a) Contour plot of the electronic conductance as a function of the
applied magnetic field and the Fermi energy at temperature T = 0 for a
symmetric PRZL. The set of parameters are $k_{B}T=0.03|t_{1}|$, $W_{in}=2$ nm,
$L_{in}=74$ nm, $W_{out}=8$ nm, $L_{out}=80$ nm, and $W_{L}=4$ nm. (b) The
electronic conductance of the symmetric PRZL at $E_{f}=0.732$ eV and $T=0$.

We only focus on the symmetric case because in this configuration the
thermopower changes more rapidly compared to the ZPNRs. Fig. 7(a) depicts the
contour plot of the conductance as a function of the applied magnetic field
and the Fermi energy for a symmetric PRZL with structure parameters $W_{in}=2$
nm, $L_{in}=74$ nm, $W_{out}=8$ nm, $L_{out}=80$ nm, and $W_{L}=4$ nm. Note
that here we have increased the size of the nanoring to reduce its oscillation
period in the presence of the magnetic field. In this situation, the effective
area of the nanoring is relatively large, and as a result, the period of AB
oscillations is small which makes it experimentally feasible. We approximate
the period of oscillation as $\Delta B=2\pi\phi_{0}/\bar{S}$, where
$\bar{S}=(S_{in}+S_{out})/2$ is the average area of the outer and inner
nanorings. In our case, the average area is $472.41~{}nm^{2}$, yielding the
oscillation period of $\Delta B\thickapprox 8.73~{}T$. As seen in Fig. 7(a),
the conductance exhibits an oscillative behavior as a function of the applied
magnetic field with a period of $\sim 8.7$ T which agrees very well with our
theoretical prediction. As seen from Figs. 7(b) and (c), the electronic
conductance $G$ and the thermopower $S$ also have a periodic character by
changing the magnetic field. This leads to a drastic increase in the
thermopower $S$ near an antiresonance point. Two salient features that we
realize from these figures are the high tunability of the thermopower and the
possibility to switch on and off the thermoelectric response of the
phosphorene nanoring with the magnetic flux. Remarkably, the differential SC
can be about eight times larger than the one in the absence of the magnetic
field. This confirms the excellent thermoelectric response of this phase-
coherent mesoscopic device, making it a promising quantum heat engine in a
closed-circuit configuration Li  _et al._ (2017).

Figure 8:  The same quantities as in Fig. 3 for both (a1)-(a4) symmetric and
(b1)-(b4) asymmetric coupling of two armchair leads two the ring.

Finally, we examine the thermoelectric performance of a phosphorene ring with
armchair leads (PRAL). Figure 8 depicts the same quantities as in Fig. 3 for
both symmetric ((a1)-(a4)) and asymmetric ((b1)-(b4)) connection
configurations. A comparison between the two nanostructures shows that for
PRALs, the conductance increases in the asymmetric case. The reason for this
observation is that for the asymmetric PRAL, the high-energy modes in the
terminal are mixed with the high-energy modes of the middle armchair PNRs.
This phenomenon occurs oppositely for a quantum dot ring geometry. To further
clarify our statement, we have shown in Fig. 9 the spatial LDOS of the
transport modes at $E=2$ eV. As seen, in the symmetric case (Fig. 9 (a2)), the
wave packets of electrons are mainly bounded in the attached left and right
leads, and one can find fewer bound states in the central device. This reduces
the propagation probability of electrons from the left lead to the right one
or vice versa. On the other hand, the LDOS of the asymmetric configuration
exhibits more bounding states in the central device, and thus increase the
probability of electron transmission which emerges as conductance peaks. The
SCs for both the symmetric and asymmetric PRALs are about the same as that of
APNRs. Here, the reason is that we have no suppressed edge modes in the
ribbons, and thus the band gap remains almost intact. Intriguingly, for PRALs
the electronic thermal conductance (see Figs. 8(a4) and (b4)) can be about ten
times smaller than that of APNRs (see Fig. 2(a4)). As a result, one expects to
observe a remarkable improvement in the figure of merit ZT for these
phosphorene-based nanostructures. The behavior of PRALs by changing the size
of the ring and by applying the perpendicular magnetic field is similar to
PRZLs and we do not present the results here.

Figure 9:  Spatial LDOS for (a) a symmetric and (b) an asymmetric PRAL at
$E_{f}=2$ eV.

##  IV  summary and conclusion

In summary, we investigated the thermoelectric properties of different
phosphorene nanorings with symmetrical or asymmetrical connection
configurations. We utilized the effective low-energy TB model of monolayer
phosphorene to construct the TB Hamiltonian of our designed devices and to
characterize their electronic conductance, thermopower, and thermal
conductance within the Landauer-Buttiker formalism. We first examined the TE
performance of phosphorene nanoribbons to validate our model by comparison
with other works and then studied the conductance and TE properties of our
systems. Our results showed that the quantum interference of localized
electronic states in the nanorings and the electronic wave packets of leads
profoundly affect the TE properties of the system. We found that for a
symmetrical connection of zigzag leads to rectangular rings, the original
first conductance plateau completely collapses due to the suppression of the
contribution of edge states. This induces a wide gap in the system, giving
rise to dramatically enhanced peak values in thermopower of such
configurations. We also have shown that the TE performance of these systems
depends on the size of the ring and one can reach the maximum amount of
$33k_{B}/e$ (2830 $\mu$ V/K) for the thermopower. The appeared Fano
antiresonances in the quantum conductance of phosphorene nanorings lead to
characteristic features in the Seeback coefficient. We showed the tunability
of the thermopower and the possibility to switch on and off the thermoelectric
response of phosphorene nanorings by controlling the applied magnetic field.
For nanorings with attached armchair phosphorene nanoribbons, our calculations
revealed that though the thermopower almost remains intact, a remarkable
decrease in the electronic conductivity occurs. This can lead to a notable
improvement in the figure of merit of these systems. Our study confirmed the
promising thermoelectric properties of these phase-coherent mesoscopic devices
and propose them as potential thermoelectric candidates.

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