# Brooks-type theorem for $r$-hued coloring of graphs

Stanislav Jendrol<EMAIL_ADDRESS>Alfréd Onderko
<EMAIL_ADDRESS>Institute of Mathematics, P. J. Šafárik
University, Jesenná 5, 040 01 Košice, Slovakia

###### Abstract

An $r$-hued coloring of a simple graph $G$ is a proper coloring of its
vertices such that every vertex $v$ is adjacent to at least
$\min\\{r,\deg(v)\\}$ differently colored vertices. The minimum number of
colors needed for an $r$-hued coloring of a graph $G$, the $r$-hued chromatic
number, is denoted by $\chi_{r}(G)$. In this note we show that

$\chi_{r}(G)\leq(r-1)(\Delta(G)+1)+2,$

for every simple graph $G$ and every $r\geq 2$, which in the case when
$r<\Delta(G)$ improves the presently known $\Delta(G)$-based upper bound on
$\chi_{r}(G)$, namely $r\Delta(G)+1$.

We also discuss the existence of graphs whose $r$-hued chromatic number is
close to $(r-1)(\Delta+1)+2$ and we prove that there is a bipartite graph of
maximum degree $\Delta$ whose $r$-hued chromatic number is $(r-1)\Delta+1$ for
every $r\in\\{2,\dots,9\\}$ and infinitely many values of $\Delta\geq r+2$; we
believe that $(r-1)\Delta(G)+1$ is the best upper bound on the $r$-hued
chromatic number of any bipartite graph $G$.

###### keywords:

vertex-coloring, dynamic coloring, $r$-hued coloring, $r$-hued chromatic
number, bipartite graph

###### MSC:

[2010] 05C15 , 05C10

##  1 Introduction

Throughout this note we use graph theory terminology according to the book
[3]. However, we recall the most frequent notions. In this paper, $G$ is a
connected simple graph with vertex set $V(G)$ and edge set $E(G)$. The degree
of a vertex $v$, denoted by $\deg(v)$, is the number of edges incident to $v$.
We use $N(v)$ for the set of neighbors of $v$ in G, and $N[v]$ for the set of
vertices $N(v)\cup\\{v\\}$. We use $\Delta(G)$ to denote the maximum vertex
degree of $G$.

A $k$-coloring of a given graph $G$ is an assignment
$\phi:V(G)\rightarrow\\{1,\dots,k\\}$. For a $k$-coloring $\phi$ and a subset
of vertices $U$ we use $\phi(U)$ to denote the set of colors used on vertices
of $U$, i.e., $\phi(U)=\\{\phi(u)\colon u\in U\\}$.

A proper $k$-coloring of a graph is a vertex $k$-coloring such that any two
adjacent vertices receive distinct colors. The minimum number of colors needed
for a proper coloring of a graph $G$, the chromatic number, is denoted by
$\chi(G)$. This concept is generally well known.

Wegner [24] in 1977 introduced $2$-distance coloring (also known as square
coloring) as a coloring in which vertices at distance at most two receive
distinct colors. The problem is to find the minimum number of colors in a
$2$-distance coloring of a graph $G$, which is equivalent to find the
chromatic number of the square of $G$ ($G$ with added edges between pairs of
vertices at distance 2). Because of this fact, and to prevent confusion with
the notation used for $r$-hued colorings, we use $\chi(G^{2})$ to denote the
minimum number of colors in a $2$-distance coloring of a graph $G$. The
concept of $2$-distance colorings is also well known and broadly studied, see,
for example, [5], [8], and [21].

In this paper we study $r$-hued colorings of graphs. An $r$-hued coloring of a
simple graph $G$ is a proper coloring $\phi$ of its vertices such that
$|\phi(N(v))|\geq\min\\{r,deg_{G}(v)\\}$ for every vertex $v\in V(G)$, i.e.,
at least $\min\\{r,\deg(v)\\}$ vertices in $N(v)$ are differently colored. The
minimum number of colors needed for an $r$-hued coloring of a graph $G$, the
$r$-hued chromatic number, is denoted by $\chi_{r}(G)$.

The concept of $r$-hued colorings (also known as $r$-dynamic colorings) was
introduced by Montgomery [20] in 2001. There is an extensive research
motivated by this concept. For a recent results and surveys see [7], [14], and
[18].

In this paper, in Section 2, we show how to color vertices of a graph $G$, one
by one, in order to obtain an $r$-hued coloring of $G$ with reasonably few
colors, namely $(r-1)(\Delta(G)+1)+2$ colors. To refer the state of a
partially colored graph in each step, we define partial coloring and partial
$r$-hued coloring of $G$.

A partial $k$-coloring of $G$ is any mapping $\phi$ from a subset of vertices
of $G$ to $\\{1,\dots k\\}$; similar to classical vertex colorings of $G$
(where all vertices are colored), we use $\phi(U)$ to denote the set of colors
used on vertices from $U$, i.e., $\phi(U)=\\{\phi(u)\colon u\in
U\cap\mathrm{dom}(\phi)\\}$, where $\mathrm{dom}(\phi)$ stands for the domain
of $\phi$.

A partial $r$-hued $k$-coloring of $G$ is a partial $k$-coloring $\phi$ of $G$
such that

i)

if $u$ and $v$ are adjacent vertices and $u,v\in\mathrm{dom(\phi)}$, then
$\phi(u)\neq\phi(v)$,

ii)

for each vertex $u$ of $G$,
$|\phi(N_{G}(u))|\geq\min\\{r,|N_{G}(u)\cap\mathrm{dom}(\phi)|\\}$.

In the Section 3 we use Steiner systems to show that for particular values of
$r$ but infinitely many $\Delta\geq r+2$ there are graphs with maximum degree
$\Delta$ and $r$-hued chromatic number $(r-1)\Delta+1$.

##  2 General upper bound

One of the most famous theorems in the chromatic graph theory, Brooks’ theorem
[6] (see also [3]), states the following:

######  Theorem 1.

[6] If $G$ is a simple graph of maximum degree $\Delta(G)$, then
$\chi(G)\leq\Delta(G)+1$, with equality if and only if $G$ is a complete graph
or an odd cycle.

This theorem, as well as its short proof provided by Lovász in [19] are up to
present inspiration to other authors concerning wide variety of vertex
coloring problems, namely those which are based on adding another condition to
a proper vertex coloring of a graph, and where the objective is to minimize
the total number of colors used.

For example, in the case of $2$-distance colorings, simple greedy algorithm
shows that $\chi(G^{2})\leq\Delta(G)^{2}+1$ for every simple graph $G$ (to
color vertex $v$ at most $\Delta(G)$ colors used on neighbors of $v$ are
forbidden to use, as well as at most $\Delta(G)(\Delta(G)-1)$ colors used on
neighbors of neighbors of $v$). Diameter two cages such as the $5$-cycles, the
Petersen graph and the Hoffman-Singleton graph (see [3], page 84) show that
there exist graphs that in fact require $\Delta^{2}+1$ colors for $\Delta=2,3$
and $7$, and possibly one for $\Delta=57$.

In the case of $r$-hued colorings, an easy observation shows that
$\chi_{1}(G)\leq\chi_{2}(G)\leq\dots\chi_{\Delta(G)}(G)=\chi_{\Delta(G)+1}(G)=\dots$.
And since, evidently, $\chi_{1}(G)=\chi(G)$ and
$\chi_{\Delta(G)}(G)=\chi(G^{2})$, for every positive integer $r$ we have
$\chi(G)\leq\chi_{r}(G)\leq\Delta(G)^{2}+1$.

There is a broad research concerning $r$-hued colorings of graphs of
particular families, improving the bounds on $r$-hued chromatic number and, in
some cases, providing the exact value of it; see a recent survey [7].

In this note we build upon the following result, which gives the upper bound
on the $r$-hued chromatic number of a simple graph $G$ with the maximum degree
$\Delta(G)$, $\chi_{r}(G)$, for a simple graph $G$ in general. The bound in
this theorem is sharp, however, only Moore graphs achieve it, and the authors
of [14] themselves ask for the best upper bound that holds for all but
finitely many graphs.

######  Theorem 2.

[14] Let $G$ be a graph. Then $\chi_{r}(G)\leq r\Delta(G)+1$, with equality
for $r\geq 2$ if and only if $G$ is regular with diameter $2$ and girth $5$.

######  Theorem 3.

[17] Let $G$ be a connected graph. If $\Delta(G)\geq 3$, then
$\chi_{2}(G)\leq\Delta(G)+1$.

For some other Brooks’ type bounds of $r$-hued chromatic number for specific
$r$ and (or) specific families of graphs see [16], [17], and [7].

Our main result improves the result of Theorem 2 in the case when
$\Delta(G)>r$; as $\chi_{r}(G)=\chi(G^{2})$ if $r\geq\Delta(G)$, the case when
$\Delta(G)>r$ is the interesting one. We have:

######  Theorem 4.

If $G$ is a simple graph and $r\geq 2$, then
$\chi_{r}(G)\leq(r-1)(\Delta(G)+1)+2$.

###### Proof.

For the sake of simpler formulae, let $\Delta=\Delta(G)$ and let
$A=\\{1,\dots,(r-1)(\Delta+1)+2\\}$.

First note that if $r>\Delta$ then $(r-1)(\Delta+1)+2>r\Delta+1$. Since
$\chi_{r}(G)\leq r\Delta+1$ by Theorem 2, $\chi_{r}(G)\leq(r-1)(\Delta+1)+2$
as well. Hence, throughout the following, we only consider the case when
$2\leq r\leq\Delta$.

We proceed in a greedy way. Let $U$ be the set of uncolored vertices; $U=V(G)$
at the beginning. In each step, until $U$ is empty, we pick a vertex from $U$
at random, and proceed to the coloring phase. We distinguish two major cases,
for which the following notation is used (note that $\phi$ is the partial
$r$-hued coloring of $G$ at the given step):

$\displaystyle S(v)$ $\displaystyle=\\{x\in N(v)\colon|\phi(N(x))|\geq r\\},$
$\displaystyle W(v)$ $\displaystyle=\\{x\in N(v)\colon|\phi(N(x))|\leq
r-1\\}.$

We say in the following that $u$ is a strong neighbor if $u\in S(v)$, and we
say that $u$ is a weak neighbor of $v$ if $u\in W(v)$.

Case 1. If $|W(v)|\leq r-1$, we color $v$ with a color which is different from
all colors of $\big{(}\phi(S(v))\cup\phi(W(v))\cup\phi(N(W(v)))\big{)}$. Since
at most $r-1$ colors are used on neighbors of each weak neighbor of $v$, we
have $|\phi(N(W(v)))|\leq|W(v)|(r-1)\leq(r-1)^{2}$. Moreover,
$|\phi(S(v))\cup\phi(W(v))|\leq|S(v)\cup W(v)|=\deg(v)\leq\Delta$, and thus at
most $\Delta+(r-1)^{2}$ colors cannot be used to color $v$. We briefly show
that $|A|\geq\Delta+(r-1)^{2}+1$, hence, there is a color to be used on $v$,
which use extends $\phi$ into a partial $r$-hued coloring of $G$ with more
vertices colored.

We have
$|A|-\big{(}\Delta+(r-1)^{2}+1\big{)}=\big{(}(r-1)(\Delta(G)+1)+2\big{)}-\big{(}\Delta+(r-1)^{2}+1\big{)}=-r^{2}+(3+\Delta)r-2-2\Delta=(\Delta+1-r)(r-2)\geq
0$, since $2\leq r\leq\Delta$.

We remove $v$ from $U$.

Case 2. If $|W(v)|\geq r$, we first remove colors from colored weak vertices
of $v$. We then color the vertex $v$ itself, and afterwards, we color each of
the weak neighbors of $v$, regardless whether the neighbor was colored or not
before this step. To color $v$, we omit colors from
$\big{(}\phi(S(v))\cup\phi(N(W(v)))\big{)}$; by doing so we add a new color to
the neighborhood of each weak neighbor, while the resulting partial coloring
remains proper. Evidently, $|\phi(S(v))\cup\phi(N(W(v)))|\leq\Delta(r-1)$, the
worst case being the case when $\deg(v)=\Delta$ and there is exactly $r-1$
colors on neighbors of each neighbor of $v$, i.e., $N(v)=W(v)$. Since $|A|\geq
1+\Delta(r-1)$, there is a color to be used on $v$.

We now proceed into coloring vertices from $W(v)$; let there be a linear
ordering of these vertices, say $w_{1},\dots,w_{k}$. Let $j$ be an integer,
$1\leq j\leq r$. To color the vertex $w_{j}$, assuming all vertices $w_{\ell}$
for $1\leq\ell\leq j-1$ are colored, we use the color from

$\displaystyle
A\setminus\big{(}\\{\phi(v)\\}\cup\phi(N(w_{j})\setminus\\{v\\})\cup\phi(W(N(w_{j})\setminus\\{v\\}))\cup\bigcup\limits_{\ell=1}^{j-1}\phi(w_{\ell})\big{)}.$
(1)

For the clarity of the proof, we count the sizes of the sets in (1). Evidently
$|\\{\phi(v)\\}|=1$, and since $w_{j}$ was (at the begining of this step) a
weak neighbor of $v$, $|\phi(N(w_{j})\setminus\\{v\\})|\leq r-1$. Then,
$|\phi(N(W(w_{j})\setminus\\{v\\}))|\leq(\Delta-1)(r-1)$, as the number of
weak neighbors of $w_{j}$ different from $v$ is at most $\Delta-1$, and each
weak neighbor sees at most $r-1$ colors on its respective neighbors. Finally,
$|\bigcup\limits_{\ell=1}^{j-1}\phi(w_{j})|\leq r-1$. Considering all of these
sets and their sizes, at most
$1+(r-1)+(\Delta-1)(r-1)+(r-1)=1+(\Delta+1)(r-1)$ colors are forbidden to use
on $w_{j}$ for each $j\in\\{1,\dots r\\}$. Since, however, $|A|\geq
2+(\Delta+1)(r-1)$, there is a color to be used on each such vertex $w_{j}$.

For each $j\in\\{r+1,\dots,k\\}$ we color the vertex $w_{j}$ with a color from

$A\setminus\big{(}\phi(v)\cup\phi(N(w_{j})\setminus\\{v\\})\cup\phi(W(N(w_{j})\setminus\\{v\\})\big{)}.$

Observe that in this case we do not have to explicitly forbid colors used on
other vertices from $W(v)$ as there are at least $r$ mutually distinct colors
used on neighbors of $v$, namely on $w_{1},\dots,w_{r}$. Since
$\big{|}\phi(v)\cup\phi(N(w_{j})\setminus\\{v\\})\cup\phi(W(N(w_{j})\setminus\\{v\\})\big{|}\leq
1+(r-1)+(\Delta-1)(r-1)$, there is a color to be used on $w_{j}$. Remove $v$
and all vertices from $W(v)$ from $U$.

Note that, in each step, if at most $r$ neighbors of $x$ are already colored,
then they are colored in a rainbow way, i.e., each of them has different
color. Moreover, in Case 2, weak neighbors of $v$ are colored in a way that at
least $r$ of them receive mutually different colors, hence, $r$-hued property
holds for the vertex $v$. It follows from these observations that the produced
coloring is in each step partial $r$-hued coloring of $G$; thus, the resulting
coloring is an $r$-hued coloring of $G$. ∎

Evidently, in the case when $\Delta(G)=r$ we have
$(r-1)(\Delta(G)+1)+2=r\Delta(G)+1$. Hence, the bound in Theorem 4 is sharp,
for the same reason as the bound in Theorem 2, i.e., 2-distance chromatic
number of a Moore graph $G$ is equal to $\Delta(G)^{2}+1$. However, as authors
of [14] mentioned, there is only a finite number of Moore graphs, and
therefore, there is a space for improvement of the currently known upper
bounds after excluding some finite family of graphs such as Moore graphs.

##  3 Bipartite graphs needing $(r-1)\Delta+1$ colors

As we mentioned before, only graphs needing $r\Delta+1$ colors are Moore
graphs. Moreover, only examples of graphs which are known to need $r\Delta$
colors in an $r$-hued coloring are cycles whose length is not divisible by 3,
in the case when $r=2$. In [14] a way to obtain a graph $G$ which needs
$r\Delta(G)-1$ colors is described. The construction involves removing some
edges from several copies of an $r$-regular Moore graph and then adding new
ones, resulting in an $r$-regular graph. Even though this construction yields
infinitely many such graphs, they all have their maximum degree equal to $r$,
which is 2, 3 or 7 (or possibly 57 if a 57-regular Moore graph exists). Hence,
for $r\notin\\{2,3,7,57\\}$, the problem of finding an infinite family of
graphs which would have their $r$-hued chromatic number close (up to a
constant) to $r\Delta+1$ or $(r-1)(\Delta+1)+2$ remains open.

In design theory, Steiner system $S(t,r,n)$ is a pair $(V,\mathcal{B})$ where
$V$ is an $n$-element set and $\mathcal{B}$ is a family of $r$-element subsets
of $V$, called blocks, such that each $t$-element subset of $V$ is contained
in exactly one block. The necessary conditions for the existence of a Steiner
system $S(t,r,n)$ with $b$ $k$-element subsets are: $n-1=k(r-1)$ and $nk=br$.

Consider now a Levi graph [10] (or incidence graph) of a Steiner system
$(V,\mathcal{B})=S(2,r,n)$, i.e., a bipartite graph $G$ with vertices
$V\cup\mathcal{B}$ and edges between every $v\in V$ and $B\in\mathcal{B}$ if
and only if $v\in B$. As there is some block $B\in\mathcal{B}$ which contains
both $v_{1}$ and $v_{2}$, for every pair of vertices from $V$, and
$\deg_{G}(B)=r$ for every $B\in\mathcal{B}$, vertices from $V$ are colored
with mutually distinct colors in any $r$-hued coloring of $G$. Thus,
$\chi_{r}(G)\geq|V|=n$, and, since evidently there is not any 4-cycle in $G$
(two points of $V$ are contained in exactly one block in $S(2,r,n)$), we have
the following:

######  Theorem 5.

Let $r\geq 2$ be a positive integer. If Steiner system $S(2,r,n)$ exists,
there is a biregular bipartite graph $G$ of girth at least 6 with
$\chi_{r}(G)\geq n$.

######  Theorem 6.

For every $r\in\\{2,\dots,9\\}$ and infinitely many values of $\Delta\geq
r+2$, there is a graph $G$ with $\Delta(G)=\Delta$ and

$\chi_{r}(G)=(r-1)\Delta(G)+1.$

###### Proof.

From Theorem 5 it follows that there are graphs, namely Levi graphs of Steiner
systems $S(2,r,n)$, whose $r$-hued chromatic number is at least $n$. Note that
if we were to ask the Levi graph of $(V,\mathcal{B})=S(2,r,n)$ to be of
maximum degree $\Delta$, then $n\leq\Delta(r-1)+1$, since the number of
distinct neighbors of neighbors of any vertex from $V$ is at most
$\Delta(r-1)$. It is easy to see that a Steiner system $S(2,2,n)$ exists for
every $n\geq 3$. In the case of $r\in\\{3,4,5\\}$ a complete characterization
of Steiner systems $S(2,r,n)$ was provided by Bose [4] and Skolem [22] for
$r=3$, by Hanani [12] for $r=4$, and by Hanani [13] for $r=5$. Namely:

$S(2,3,n)$ exists whenever $n\equiv 1\text{ or }3\mod 6$ [9],

$S(2,4,n)$ exists whenever $n\equiv 1\text{ or }4\mod 12$ [2],

$S(2,5,n)$ exists whenever $n\equiv 1\text{ or }5\mod 20$ [2].

For $r\in\\{6,7,8,9\\}$ sufficient conditions on existence of Steiner systems
$S(2,r,n)$ are known due to Abel and Greig [1] in the case when $r=6$, Janko
and Tonchev [15] in the case when $r=7$, and due to Greig [11] in the case
when $r\in\\{7,8,9\\}$:

$S(2,6,n)$ exists if $n\geq 802$, $n\equiv 1\text{ or }6\mod 15$,

$S(2,7,n)$ exists if $n\geq 2606$, $n\equiv 1\text{ or }7\mod 42$,

$S(2,8,n)$ exists if $n\geq 3754$, $n\equiv 1\text{ or }8\mod 56$,

$S(2,9,n)$ exists if $n\geq 16498$, $n\equiv 1\text{ or }9\mod 72$.

Hence, a biregular bipartite graph $G$ of girth at least 6 with
$\Delta(G)=\Delta$ and $\chi_{r}(G)\geq(r-1)\Delta+1$ exists for infinitely
many values of $\Delta$ and any $r\in\\{3,\dots,9\\}$. For example there is a
biregular bipartite graph $G$ with $\chi_{3}(G)\geq 2\Delta+1$ for every
$\Delta$ congruent to $0,1,3$ or $4\mod 6$; and there is a biregular bipartite
graph $G$ with $\chi_{4}(G)\geq 3\Delta+1$ for every $\Delta$ congruent to
$0,1,4,5,8$, or $9\mod 12$.

We now show that Levi graph of any Steiner system $S(2,r,(r-1)\Delta+1)$ for
$\Delta\geq r+2$ has its $r$-hued chromatic number equal to $(r-1)\Delta+1$.
Suppose to the contrary that there is a Levi graph $G$ of Steiner system
$(V,\mathcal{B})=S(2,r,(r-1)\Delta+1)$ with $\chi_{r}(G)=k\geq(r-1)\Delta+2$.
As $|V|=(r-1)\Delta(G)+1$ and, by the above, $|\phi(V)|=(r-1)\Delta(G)+1$,
there is an $r$-vertex $y\in\mathcal{B}$ with $\phi(y)\notin\phi(V)$.

We show that there is a color in $\phi(V)$ that can be used to recolor $y$ in
order to get an $r$-hued coloring of $G$ with less $r$-vertices having color
not in $\phi(V)$. We distinguish two cases.

Case 1. Let there be $\ell\geq 1$ and some vertices $z_{1},\dots,z_{\ell}$ in
$N(y)$ such that $|\phi(N(z_{i})\setminus\\{y\\})|\geq r$ for every
$i\in\\{1,\dots,\ell\\}$. Denote by $A$ the set of colors which cannot be used
on $y$, i.e.,

$A=\\{\phi(z_{1}),\dots,\phi(z_{\ell})\\}\cup\bigcup_{w\in
N(y)\setminus\\{z_{1},\dots,z_{\ell}\\}}\phi(N[w]\setminus\\{y\\}).$

To show that there is a color from $\phi(V)$ to be used on $y$ without
violating the $r$-hued condition, it is enough that the size of $A$ is smaller
than that of $\phi(V)$. But it certainly is, as $|A|\leq\ell+(r-\ell)r\leq
1+(r-1)r$, hence,

$|\phi(V)\setminus
A|\geq(r-1)\Delta(G)+1-\big{(}1+(r-1)r\big{)}=(r-1)(\Delta(G)-r)\geq 1.$

Case 2. Suppose that $|\phi(N(z)\setminus\\{y\\})|=r-1$ for every vertex $z\in
N(y)$. In this case, colors from $A^{*}=\bigcup_{w\in
N(y)\setminus\\{z\\}}\phi(N(w)\setminus\\{y\\})$ cannot be used on $y$. We
have $|A^{*}|\leq r^{2}$ and consequently

$|\phi(V)\setminus
A^{*}|\geq(r-1)\Delta(G)+1-\big{(}r+r(r-1)\big{)}=(r-1)(\Delta(G)-r-1)\geq 1,$

since $\Delta(G)=\Delta\geq r+2$. Thus, in this case, there is also a color
from $\phi(V)$ to be used on $y$ in order to obtain an $r$-hued coloring of
$G$ with less vertices colored with colors not in $\phi(V)$

If there is no other $r$-vertex $w$ with $\phi(w)\notin\phi(V)$, the obtained
$r$-hued coloring is a required $\big{(}(r-1)\Delta(G)+1\big{)}$-coloring of
$G$. If there is another $r$-vertex $w$ with $\phi(w)\notin\phi(V)$ we repeat
the above procedure until we obtain a required $r$-hued
$\big{(}(r-1)\Delta(G)+1\big{)}$-coloring of $G$.

∎

There are other two special Steiner systems $S(2,n+1,n^{2}+n+1)$ and
$S(2,n,n^{2})$ , known as projective and affine plane of order $n$,
respectively. Using Theorem 5, the existence of a projective plane of order
$r$ implies the existence of a bipartite biregular graph $G$ of girth at least
6 with $\Delta(G)=r$ and $\chi_{r}(G)\geq(r-1)\Delta+1$, and the existence of
an affine plane of order $r$ implies the existence of a bipartite biregular
graph $G$ of girth at least 6 with $\Delta(G)=r+1$ and
$\chi_{r}(G)\geq(r-1)\Delta+1$. It is known that if $n$ is a power of a prime,
there exists a projective plane of order $n$, and therefore an affine plane of
order $n$ as well (since it is a residual design to a projective plane), see
[23].

As mentioned before, $S(2,r,(r-1)\Delta+1)$ does not exist if necessary
conditions are not satisfied, i.e., $(r-1)\Delta\neq r(r-1)$. However if
$(r-1)\Delta\neq r(r-1)$, one could possibly construct a biregular bipartite
graph with bipartition $X$ and $Y$ such that $\deg(x)=r$ for every $x\in X$
and $\deg(y)=\Delta$ for every $y\in Y$, which would have the property that
vertices in $Y$ are colored with mutually different colors in any $r$-hued
coloring, and size of $Y$ is as close to $(r-1)\Delta+1$, as reasonably
possible. Namely, we ask

Question 1. Is it true that for every $r$ and $\Delta$, $r\leq\Delta$, there
is a biregular bipartite graph with $\chi_{r}(G)\geq(r-1)\Delta+1-z$, where
$z\in\\{0,\dots,r-1\\}$?

And, concerning the exact number of colors in such graphs:

Question 2. Is it true, that $\chi_{r}(G)\leq(r-1)\Delta(G)+1$ for every
bipartite graph $G$?

We believe that answers to both of these questions are positive. Note that
Question 1 is equivalent to the problem of existence of some not necessarily
balanced designs. The exact number of colors in a Levi graph of any Steiner
system $S(2,r,(r-1)\Delta+1)$ is $(r-1)\Delta+1$, which supports the possible
positive answer in Question 2.

Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work on this
paper.

Acknowledgments

The authors are thankful to professor Alex Rosa for sharing his knowledge on
problems concerning the theory of Steiner systems.

This work was supported by the Slovak Research and Development Agency under
the Contract No. APVV-19-0153, by the Science Grant Agency - project VEGA
1/0574/21, and by UPJŠ internal grant vvgs-pf-2022-2137.

## References

  * [1] R. J. R. Abel, M. Greig, Some new RBIBDs with block size 5 and BIBDs with block sizes $\equiv$ 1 mod 5, Australas. J. Comb., 15 (1997) 177–202.
  * [2] R. J. R. Abel, M. Greig, BIBDs with small sizes, Chapter II. 3. in J. Colbourn, J. H. Dinitz, Handbook of Combinatorial Design, Second Edition, Chaptman and Hall/CRC, Boca Raton, London, New York, 2006.
  * [3] J. A. Bondy, U. S. R. Murty, Graph Theory, Graduate text in Mathematics 244 (Third Edition), Springer, New York, 2008.
  * [4] R. C. Bose, On the construction of balanced incomplete block designs, Annals of Eugenics, 9(4) (1939) 353–399.
  * [5] N. Bousquet, Q. Deschamps, L. de Meyer, T. Pierron, Improved square coloring of planar graphs, arXiv:2112.12512v1 [math.CO] 23 Dec 2021.
  * [6] R. L. Brooks, On coloring the nodes of a network, Proc. Cambridge Phil. Soc. 37 (1941) 194–197.
  * [7] Y. Chen, S. Fan, H.-J. Lai, M. Xu, Graph $r$-hued coloring - A survey, Discrete Appl. Math. 321 (2022) 24–48.
  * [8] I. Choi, D. W. Cranston, T. Pierron, Degeneracy and Colorings of Squares of planar graphs without $4$-cycles, arXiv:1806.07204v2 [math.CO] 24 Sep 2019.
  * [9] C. J. Colbourn, Triple systems, Chapter II. 2. in J. Colbourn, J. H. Dinitz, Handbook of Combinatorial Design, Second Edition, Chaptman and Hall/CRC, Boca Raton, London, New York, 2006.
  * [10] H. S. Coxeter, Self-dual configurations and regular graphs, Bull. Am. Math. Soc, 56(5) (1950) 413–455.
  * [11] M. Greig, Recursive constructions of balanced incomplete block designs with block size of 7, 8, or 9, Ars Combin, 60 (2001) 3–54.
  * [12] H. Hanani, The existence and construction of balanced incomplete block designs, Ann. Math. Stat., 32(2) (1961) 361–386.
  * [13] H. Hanani, Balanced incomplete block designs and related designs, Discrete Math., 11(3) (1975) 255–369.
  * [14] S. Jahanbekan, J. Kim, O. Suil, D. B. West, On the $r$-dynamic coloring of graphs, Discrete appl. Math. 206 (2016) 65–72.
  * [15] Z. Janko, V. D. Tonchev, New designs with block size 7. J. Comb. Theory Ser. A., 83(1) (1998) 152–157.
  * [16] D. V. Karpov, Dynamic proper coloring of graphs, J. Math. Sci. (N.Y.) 179 (5) (2011) 601–615.
  * [17] H.-J. Lai, B. Montgomery, H. Poon, Upper bounds of dynamic chromatic number, Ars Combin. 68 (2003) 193–2001.
  * [18] S. Loeb, T. Mahoney, B. Reiniger, J. Wise, Dynamic coloring parameters for graphs with given genus, Discrete Appl. Math. 235 (2018) 129–141.
  * [19] L. Lovász, Three short proofs in graph theory, J. Comb. Theory. Ser. B, 19 (3) (1975) 269-271.
  * [20] B. Montgomery: Dynamic coloring of graphs, ProQuest LLC, Ann Arbor, MI 2001, p.52, West Virginia University.
  * [21] M. Molloy, M. R. Salavatipour, A bound on the chromatic number of the square of a planar graph, J. Combin. Theory Ser. B, 94 (2005) 189–2013.
  * [22] T. Skolem, Some remarks on the triple systems of Steiner, Math. Scand., (1958) 273-280.
  * [23] O. Veblen, W. H. Bussey, Finite projective geometries. Transactions of the American mathematical society, 7(2) (1906), 241–259.
  * [24] G. Wegner, Graphs with given diameter and a coloring problem, Technical Report, University of Dortmund, 1977.


# Sensitivity of superconducting states to the impurity location in layered
materials

Bastian Zinkl  Institute for Theoretical Physics, ETH Zurich, CH-8093 Zurich,
Switzerland  Aline Ramires  Condensed Matter Theory Group, Paul Scherrer
Institute, CH-5232 Villigen PSI, Switzerland

###### Abstract

The family of multi-layered superconductors derived from the doped topological
insulator Bi2Se3 has been found to be unusually robust against non-magnetic
disorder. Recent experimental studies have highlighted the fact that the
location of impurities could play a critical role for this puzzling
robustness. Here we investigate the effects of four different types of
impurities, on-site, interstitial, intercalated and polar, on the
superconducting critical temperature. We find that different components of the
scattering potential are active depending on the impurity configuration and
choice of orbitals for the effective low-energy description of the normal
state. For the specific case of Bi2Se3-based superconductors, we find that
only the symmetric share of impurity configurations contribute to scattering,
such that polar impurities are completely inactive. We also find that a more
dominant mass-imbalance term in the normal-state Hamiltonian can make the
superconducting state more robust to intercalated impurities, in contrast to
the case of on-site or interstitial impurities.

##  I Introduction

The effects of impurities in simple superconductors are well understood in
terms of elementary symmetry arguments, elegantly summarized in what is known
as Anderson’s theorem [1, 2]. In fully-gapped conventional superconductors,
pairs of electrons are formed related by time-reversal symmetry. As a
consequence, impurities have a detrimental effect to superconductivity only if
they break this key symmetry, namely, if the impurities are magnetic. For
unconventional nodal superconductors both magnetic and non-magnetic disorder
have a negative effect on the critical temperature [3, 4]. The sensitivity of
superconductors to non-magnetic disorder has therefore been taken as a strong
indication of the unconventional nature of the order parameter, as observed in
UPt3 [5], Sr2RuO4 [6], and in the cuprates [7], to name a few.

In contrast, the effects of impurities in complex superconductors have only
recently started to attract attention, mostly motivated by the phenomenology
of iron-based superconductors [8, 9, 10]. Complex superconductors are
characterized by multiple Fermi surfaces, or by single Fermi surfaces emerging
from a combination of multiple internal degrees of freedom. In this context,
one result that goes beyond the prediction of Anderson’s theorem concerns
fully-gapped superconductors in multi-band systems. A two-band system with
full gaps of opposite signs is known to be sensitive to non-magnetic disorder
[8, 9, 10]. Naturally, extra sensitivity to disorder of such fully-gaped
systems is in principle not a desirable feature to explore potential
applications of these materials.

Intriguingly, the traditional picture has recently been challenged by the
observation of unconventional superconductors that are unusually robust
against non-magnetic doping. For instance, the superconductor
Cux(PbSe)5(Bi2Se3)6 (CPSBS) [11], showing nematic properties and a nodal gap
structure [12], survives scattering rates much larger than anticipated by
Anderson’s theorem [13]. Other materials in the same family, CuxBi2Se3 [14,
15, 16] and NbxBi2Se3 [17, 18, 19], also display unusual robustness against
non-magnetic disorder. In-doped SnTe remarkably shows a larger critical
temperature for samples with high residual resistivity [20]. Irradiated PdTe2
also shows robustness against non-magnetic disorder, with its critical
temperature suppressed at a rate that is about sixteen times slower than
predicted by standard estimations [21].

The first theories developed to address this unusual robustness were based on
the presence of strong spin-orbit coupling (SOC) [22, 23]. Later, it became
clear that for complex superconductors (with extra internal degrees of freedom
such as orbitals or sublattices), the concept of superconducting fitness
allows for a generalization of Anderson’s theorem, providing an universal
framework and explanation for the unusual robustness of unconventional
superconducting states [13, 21]. Specific results for Cu-doped Bi2Se3 report
that the robustness of the superconducting state depends not only on the
superconducting order parameter, but on details of the electronic structure in
the normal state [24]. Recently, this understanding was corroborated by a more
complete analysis of the sensitivity of pairing states to various scattering
potentials in two-orbital systems [25, 26].

Here we focus on superconductors derived from Bi2Se3. Common among these
materials is the basic crystallographic unit consisting of quintuple layers
(Se - Bi - Se′′ \- Bi′ \- Se′) [27], as schematically depicted in Fig. 1.
Superconductivity emerges in these systems only after doping, which
intrinsically also introduces disorder. The location of impurities within
these layers has attracted some interest, since evidence has accumulated that
their distribution, and not just the electron donation, plays a decisive role
for the formation of unconventional superconductivity [28]. Density-functional
theory calculations suggest that the most energetically favorable location for
Cu [29] and Sr [30] dopants should be between the quintuple layers, i.e. in
the van der Waals (vdW) gap. X-ray diffraction experiments seem to support
this suggestion, since an expansion of the $c$-axis has been observed both in
CuxBi2Se3 [31, 32, 33] and SrxBi2Se3 [34, 35]. However, dopants in the vdW gap
have never been directly detected so far, neither with neutron-scattering
experiments in the case of CuxBi2Se3 [33] nor with transmission electron
microscopy in the case of SrxBi2Se3 [30]. In the latter compound the vertical
position of the impurities has only recently been determined using normal
incidence x-ray standing wave measurements [28]. It was found that the dopants
lie close to the Se and Se′ sites with a small vertical displacement towards
the center of the quintuple layer, hence not in the vdW gap.

Motivated by these observations, we investigate in detail how the location of
impurities influence the renormalization of the critical temperature in
layered superconductors. Using Bi2Se3 as our example, we derive the scattering
matrices for four different scenarios: substitutional (on-site), interstitial
(in-between two sites), intercalated (in the vdW gap), and polar defects, as
illustrated in Fig. 2. We start with an effective microscopic model for the
electronic states in the quintuple layers and analyze the effects of
impurities in case different pairs of orbitals dominate the effective low-
energy electronic structure. Here we are guided by the concept of
superconducting fitness and the generalized Anderson’s theorem to discuss the
robustness of superconducting states in multiple scenarios.

This paper is organized as follows. In Section II we review the microscopic
description of the electronic structure of materials in the family of doped
Bi2Se3 and the possible s-wave superconducting states. In Section III after
modelling different impurity configurations in the layer basis we discuss how
these manifest in effective two-orbital models. In Section IV we then evaluate
the effective scattering rates for different impurity distributions and
effective model scenarios. Finally, in Section V we summarize our findings and
how these can help us understand the unusual robustness of superconducting
states in layered materials.

##  II Effective microscopic description

In this section, we introduce and discuss the microscopic structure of the
normal-state Hamiltonian and the superconducting order parameters.

###  II.1 Normal state

Motivated by the phenomenology of Bi2Se3-based superconductors, we start
modelling the electronic structure in the quintuple-layer of these materials.
First principles calculations suggest that the main orbitals contributing to
the low-energy electronic structure of Bi2Se3 stem from $p_{z}$ orbitals
located at the four outermost layers (Se - Bi - Bi′ \- Se′), here labelled as
$\ket{S_{z}}$, $\ket{B_{z}}$, $\ket{B^{\prime}_{z}}$, $\ket{S^{\prime}_{z}}$,
respectively [27]. It is convenient to combine these in bonding and anti-
bonding configurations,

$\displaystyle\ket{P1_{z}^{\pm}}$
$\displaystyle=\frac{1}{\sqrt{2}}(\ket{B_{z}}\mp\ket{B_{z}^{\prime}}),$ (1)
$\displaystyle\ket{P2_{z}^{\pm}}$
$\displaystyle=\frac{1}{\sqrt{2}}(\ket{S_{z}}\mp\ket{S_{z}^{\prime}}),$ (2)

which are schematically indicated in Fig. 1. Note that the upper index
corresponds to the parity of the state.

Figure 1: Side view of the quintuple layer unit cell of Bi2Se3. We illustrate
the $PI_{z}^{\alpha}$ orbitals as symmetric ($\alpha=+$) or anti-symmetric
($\alpha=-$) combinations of the $p_{z}$ orbitals in each layer. We also
highlight which scattering matrices remain relevant, if two specific states
are chosen as the low-energy basis. As examples, we consider the combinations
$\\{P1_{z}^{+},P2_{z}^{-}\\}$ (top) and $\\{P1_{z}^{+},P1_{z}^{-}\\}$
(bottom). More details on the matrix structures are given in Table 4.

The states written down in Eqs. (1, 2) constitute what we call the “orbital”
basis, denoted by
$\\{P1_{z}^{+},~{}P1_{z}^{-},~{}P2_{z}^{+},~{}P2_{z}^{-}\\}$. In the “layer”
basis {Se, Bi, Bi′, Se′} they take the form

$\displaystyle\ket{P1_{z}^{\pm}}=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ 1\\\
\mp 1\\\
0\end{pmatrix},~{}\ket{P2_{z}^{\pm}}=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
0\\\ 0\\\ \mp 1\end{pmatrix}.$ (3)

The unitary transformation matrix corresponding to this basis change is

$\displaystyle\hat{U}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&1&1\\\ 1&1&0&0\\\
-1&1&0&0\\\ 0&0&-1&1\end{pmatrix},$ (4)

which will be important in Sect. III, where we analyze the structure of
different impurity configurations in different reduced orbital bases.

For Bi2Se3-based materials, it is clear from first principles that only two of
these orbitals contribute to the Fermi surface. In particular, it is suggested
that these orbitals are $\\{P1_{z}^{+},P2_{z}^{-}\\}$ [27]. More generally, we
can think of any pair of orbitals effectively contributing to the Fermi
surface. There are three fundamentally distinct scenarios: i) two orbitals
with opposite parity (OP) coming from different types of atoms (this would be
the case of Bi2Se3-based materials just mentioned, but also
$\\{P2_{z}^{+},P1_{z}^{-}\\}$); ii) two orbitals with OP coming from the same
type of atoms ($\\{P1_{z}^{+},P1_{z}^{-}\\}$ or
$\\{P2_{z}^{+},P2_{z}^{-}\\}$); iii) two orbitals with equal parity (EP),
necessarily associated with different atoms ($\\{P1_{z}^{+},P2_{z}^{+}\\}$ or
$\\{P1_{z}^{-},P2_{z}^{-}\\}$).

For all these cases the low-energy Hamiltonian of the system can be
parametrized as

$\displaystyle\hat{H}(\boldsymbol{k})=\sum_{a,b}h_{ab}(\boldsymbol{k})(\hat{\tau}_{a}\otimes\hat{\sigma}_{b}),$
(5)

where $\hat{\tau}_{a}$ and $\hat{\sigma}_{b}$, with
$\\{a,b\\}\in\\{0,1,2,3\\}$, are Pauli matrices encoding the orbital and spin
degrees of freedom, respectively. For the OP scenario, the symmetry-allowed
terms in the Hamiltonian are the ones summarized in Table 1. Analogously, for
the EP scenario, we obtain the terms listed in Table 2. Naturally, the
explicit momentum dependence of the $h_{ab}(\boldsymbol{k})$ terms in the
normal-state Hamiltonian depends on the point group symmetry of the system
(here $D_{3d}$) and the relative parity of the low-energy states. The explicit
form of their momentum dependence around the $\Gamma$ point is given in Tables
1 and 2, for the OP and EP scenarios, repectively.

$(a,b)$ | Irrep | Process | $h_{ab}({\boldsymbol{k}})$
---|---|---|---
$(0,0)$ | $A_{1g}$ | intra-orbital hopping | $C_{0}+C_{1}k_{z}^{2}+C_{2}(k_{x}^{2}+k_{y}^{2})$
$(3,0)$ | $M_{0}+M_{1}k_{z}^{2}+M_{2}(k_{x}^{2}+k_{y}^{2})$
$(1,3)$ | $A_{1u}$ | SOC | $R_{1}k_{x}(k_{x}^{2}-3k_{y}^{2})$
$(2,0)$ | $A_{2u}$ | inter-orbital hopping | $B_{0}k_{z}$
$(1,1)$ | $E_{u}$ | SOC | $-A_{0}k_{y}$
$(1,2)$ | $A_{0}k_{x}$
Table 1:  Parametrization of the normal-state Hamiltonian [Eq. (5)] for materials in the family of Bi2Se3 assuming orbitals with OP. For each pair of indexes $(a,b)$ corresponding to the basis matrices $\hat{\tau}_{a}\otimes\hat{\sigma}_{b}$, the table highlights the irreducible representations (Irrep) and the physical process that originates them. The last column gives the expansion of the form factors $h_{ab}({\boldsymbol{k}})$ for small momentum.  $(a,b)$ | Irrep | Process | $h_{ab}({\boldsymbol{k}})$
---|---|---|---
$(0,0)$ | $A_{1g}$ | intra-orbital hopping | $C^{\prime}_{0}+C^{\prime}_{1}k_{z}^{2}+C^{\prime}_{2}(k_{x}^{2}+k_{y}^{2})$
$(3,0)$ | $M^{\prime}_{0}+M^{\prime}_{1}k_{z}^{2}+M^{\prime}_{2}(k_{x}^{2}+k_{y}^{2})$
$(1,0)$ | inter-orbital hopping | $N^{\prime}_{0}+N^{\prime}_{1}k_{z}^{2}+N^{\prime}_{2}(k_{x}^{2}+k_{y}^{2})$
$(2,3)$ | $A_{2g}$ | SOC | $R^{\prime}_{1}k_{x}k_{z}(k_{x}^{2}-3k_{y}^{2})$
$(2,1)$ | $E_{g}$ | SOC | $-A^{\prime}_{0}k_{y}k_{z}$
$(2,2)$ | $A^{\prime}_{0}k_{x}k_{z}$
Table 2:  Parametrization of the normal-state Hamiltonian [Eq. (5)] for the EP
scenario. Same description as in Table 1.

###  II.2 Superconducting state

The order parameter of the superconducting state can be written as

$\displaystyle\hat{\Delta}(\boldsymbol{k})=\sum_{a,b}d_{ab}(\boldsymbol{k})\left[\hat{\tau}_{a}\otimes\hat{\sigma}_{b}\left(i\hat{\sigma}_{2}\right)\right].$
(6)

From now on we implicitly assume that the order parameter is normalized, i.e.
$\braket{}{\hat{\Delta}(\boldsymbol{k})}{{}^{2}}_{\rm FS}=1$, where
$\braket{\dots}_{\rm FS}$ denotes the average over the FS, and
$||\hat{M}||^{2}=\rm{Tr}[\hat{M}\hat{M}^{\dagger}]/4$ is the Frobenius norm of
the matrix $\hat{M}$. Fermionic anti-symmetry requires
$\hat{\Delta}({\boldsymbol{k}})=-\hat{\Delta}^{T}(-{\boldsymbol{k}})$, what
means that even-${\boldsymbol{k}}$ (odd-${\boldsymbol{k}}$) order parameters
are necessarily accompanied by anti-symmetric (symmetric) matrices. For
simplicity, in the following we restrict ourselves to
$\boldsymbol{k}$-independent $d_{ab}(\boldsymbol{k})$, also referred to as
s-wave superconducting states. This reduces the space of order parameters to
the six anti-symmetric basis matrices listed in Table 3. Note that they are
labelled as $[a,b]$ (rectangular brackets are used to label the
superconducting states, while round brackets label the terms in the normal-
state Hamiltonian) and that they are independent of the choice of EP or OP
orbitals contributing to the Fermi surface. However, as summarized in Table 3,
the parity of the orbitals influences the symmetry of the order parameter and
its associated irreducible representation.

$[a,b]$ | Spin | Orbital | Irrep (OP) | Irrep (EP)
---|---|---|---|---
$[0,0]$ | Singlet | Trivial | $A_{1g}$ | $A_{1g}$
$[3,0]$
$[2,3]$ | Triplet | Singlet | $A_{1u}$ | $A_{2g}$
$[1,0]$ | Singlet | Triplet | $A_{2u}$ | $A_{1g}$
$[2,1]$ | Triplet | Singlet | $E_{u}$ | $E_{g}$
$[2,2]$
Table 3:  Momentum-independent superconducting order parameters for two-
orbital models. We highlight here the spin and orbital characters as well as
the irreducible representation of the respective gap matrix for the odd parity
(OP) and even parity (EP) cases.

Having clarified our microscopic model, we consider in the following different
scattering potentials out of non-magnetic impurities.

##  III Impurity configurations in the quintuple layer and orbital bases

To unveil the effects of the impurity location in materials belonging to the
family of Bi2Se3, we distinguish between on-site (on), interstitial (is),
intercalated (ic) and polar (po) configurations, which are illustrated in Fig.
2.

The scattering matrices written in the layer basis {Se, Bi, Bi′, Se′} have an
entry on the diagonal, if the corresponding layer is affected by the impurity.
For instance, the on-site configurations shown in Fig. 2 (a) are given by

$\displaystyle\hat{V}_{\rm on}^{\rm(i)}=\begin{pmatrix}0&0&0&0\\\ 0&V&0&0\\\
0&0&0&0\\\ 0&0&0&0\end{pmatrix},~{}\hat{V}_{\rm
on}^{\rm(ii)}=\begin{pmatrix}0&0&0&0\\\ 0&V&0&0\\\ 0&0&V&0\\\
0&0&0&0\end{pmatrix},$ (7)

where $V>0$ denotes the impurity scattering strength, which we take to be the
same for all impurities. If the impurities are located between the Bi and Se
sites, as in Fig. 2 (b), we assume that their presence affects both
neighboring layers, which leads to

$\displaystyle\hat{V}_{\rm is}^{\rm(i)}=\begin{pmatrix}V&0&0&0\\\ 0&V&0&0\\\
0&0&0&0\\\ 0&0&0&0\end{pmatrix},~{}\hat{V}_{\rm
is}^{\rm(ii)}=\begin{pmatrix}0&0&0&0\\\ 0&V&0&0\\\ 0&0&V&0\\\
0&0&0&0\end{pmatrix}.$ (8)

Figure 2: Schematic illustration of (a) on-site, (b) interstitial, (c)
intercalated and (d) polar impurity configurations (marked as dark grey
circles). In Eqs. (15) to (21) we show that the scattering potential can be
always decomposed into a superposition of a symmetric and an anti-symmetric
impurity potentials. The polar configuration consists of oppositely charged
impurities in opposite layers, which leads to a fully anti-symmetric potential
[Eq.(23)].

Correspondingly, we find for the intercalated scenario, depicted in Fig. 2
(c),

$\displaystyle\hat{V}_{\rm ic}^{\rm(i)}=\begin{pmatrix}V&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\end{pmatrix},~{}\hat{V}_{\rm
ic}^{\rm(ii)}=\begin{pmatrix}V&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&0&0&V\end{pmatrix},$ (9)

and for the polar case, exemplified in Fig. 2 (d),

$\displaystyle\hat{V}_{\rm po}^{\rm(i)}=\begin{pmatrix}V&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\\\ 0&0&0&-V\end{pmatrix},~{}\hat{V}_{\rm
po}^{\rm(ii)}=\begin{pmatrix}0&0&0&0\\\ 0&V&0&0\\\ 0&0&-V&0\\\
0&0&0&0\end{pmatrix}.$ (10)

There are two linearly independent symmetric (S) configurations, which are
even under inversion. They take the form

$\displaystyle\hat{V}_{\rm S,1}=\begin{pmatrix}V&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\\\ 0&0&0&V\end{pmatrix},~{}\hat{V}_{\rm S,2}=\begin{pmatrix}0&0&0&0\\\
0&V&0&0\\\ 0&0&V&0\\\ 0&0&0&0\end{pmatrix}.$ (11)

The anti-symmetric (A) counterparts are given by

$\displaystyle\hat{V}_{\rm A,1}=\begin{pmatrix}V&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\\\ 0&0&0&-V\end{pmatrix},~{}\hat{V}_{\rm
A,2}=\begin{pmatrix}0&0&0&0\\\ 0&V&0&0\\\ 0&0&-V&0\\\ 0&0&0&0\end{pmatrix}.$
(12)

Every impurity configuration can be split up into symmetric and anti-symmetric
parts using Eqs. (11, 12). For example, the configurations introduced above
can be rewritten as

$\displaystyle\begin{gathered}\hat{V}_{\rm
on}^{\rm(i)}=\frac{1}{2}\left(\hat{V}_{\rm S,2}+\hat{V}_{\rm A,2}\right),\\\
\hat{V}_{\rm on}^{\rm(ii)}=\hat{V}_{\rm S,2},\end{gathered}$ (15)
$\displaystyle\begin{gathered}\hat{V}_{\rm
is}^{\rm(i)}=\frac{1}{2}\left(\hat{V}_{\rm S,1}+\hat{V}_{\rm S,2}+\hat{V}_{\rm
A,1}+\hat{V}_{\rm A,2}\right),\\\ \hat{V}_{\rm is}^{\rm(ii)}=\hat{V}_{\rm
S,2},\end{gathered}$ (18) $\displaystyle\begin{gathered}\hat{V}_{\rm
ic}^{\rm(i)}=\frac{1}{2}\left(\hat{V}_{\rm S,1}+\hat{V}_{\rm A,1}\right),\\\
\hat{V}_{\rm ic}^{\rm(ii)}=\hat{V}_{\rm S,1},\end{gathered}$ (21)

and

$\displaystyle\begin{gathered}\hat{V}_{\rm po}^{\rm(i)}=\hat{V}_{\rm
A,1},~{}\hat{V}_{\rm po}^{\rm(ii)}=\hat{V}_{\rm A,2}.\end{gathered}$ (23)

Using this framework, we can decompose every possible impurity distribution
for a single unit cell in terms of symmetric and anti-symmetric scattering
matrices. Assuming a system with $N_{uc}$ unit cells, each with four layers,
the maximum number of impurities is equal to $N_{\rm max}=4N_{uc}$, $3N_{uc}$
or $2N_{uc}$ for on-site, interstitial or intercalated and polar
configurations, respectively. For a fixed number of impurities it is a
straightforward combinatorial problem to find all possible impurity
configurations. To give a concrete example, we show all possible, on-site
configurations for $N_{uc}=1$ in Fig. 3.

Figure 3: Overview of all possible on-site impurity configurations for a
single unit cell with four layers. The maximum number of impurities (indicated
by the grey circles) is $N_{\rm max}=4$. For each number of impurities $N$ we
included the corresponding symmetric share $S(n=N/N_{\rm max})$.

For a single impurity, $n=N/N_{\rm max}=1/4$, there are only asymmetric
configurations. These can be decomposed into a sum of symmetric and anti-
symmetric potentials, what leads to a symmetric share $S(n)=1/2$. The
symmetric share increases with growing $N$ until it reaches one for the
completely filled unit cell ($n=1$). By calculating the symmetric share
numerically for all different configurations and for an arbitrary large number
of sites we find that $S(n)$ quickly converges to a universal result, which is
depicted by the solid lines in Fig. 4. The dashed lines represent the anti-
symmetric shares $A(n)=1-S(n)$. Note that for polar impurities, whose
scattering potential is completely anti-symmetric, we simply obtain $A_{\rm
po}(n)=1$. Furthermore, as indicated in Fig. 4, the shares for on-site and
intercalated impurities evolve in the same way as a function of the filling
$n$.

Figure 4: Evolution of the symmetric share of impurity configurations, $S(n)$
(solid lines), as a function of the filling $n=N/N_{\rm max}$ for $N_{uc}\gg
1$. The difference between on-site (on), interstitial (is), and intercalated
(ic) configurations is illustrated in Fig. 2. The dashed lines represent the
anti-symmetric shares given by $A(n)=1-S(n)$.

We now examine which orbital wave functions are affected by the different
impurity scattering matrices. To this extent we transform the symmetric and
anti-symmetric matrices [Eqs. (11, 12)] to the
$\\{P1_{z}^{+},~{}P1_{z}^{-},~{}P2_{z}^{+},~{}P2_{z}^{-}\\}$ basis by
evaluating $\hat{U}^{\dagger}\hat{V}\hat{U}$, with $\hat{U}$ given by Eq. (4).
We obtain for the symmetric scattering matrices,

$\displaystyle\hat{V}^{\prime}_{\rm S,1}=\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\
0&0&V&0\\\ 0&0&0&V\end{pmatrix},~{}\hat{V}^{\prime}_{\rm
S,2}=\begin{pmatrix}V&0&0&0\\\ 0&V&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix},$
(24)

while the purely anti-symmetric matrices are

$\displaystyle\hat{V}^{\prime}_{\rm A,1}=\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\
0&0&0&V\\\ 0&0&V&0\end{pmatrix},~{}\hat{V}^{\prime}_{\rm
A,2}=\begin{pmatrix}0&V&0&0\\\ V&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix}.$
(25)

As discussed in the previous section, the ultimate low-energy description of
the electronic states can be cast in terms of two orbitals. From Eqs. (24) and
(25), we see that the effective scattering matrices are going to depend on the
choice of orbitals. In particular, some of the scattering matrices vanish in
the low-energy basis. A summary of the reduced scattering matrices for all
possible choices of two low-energy states is given in Table 4.

Two-orbital basis | $\hat{V}^{\prime}_{\rm S,1}$ | $\hat{V}^{\prime}_{\rm S,2}$ | $\hat{V}^{\prime}_{\rm A,1}$ | $\hat{V}^{\prime}_{\rm A,2}$
---|---|---|---|---
$\\{P1_{z}^{+},P1_{z}^{-}\\}$ | $0$ | $\hat{\tau}_{0}$ | 0 | $\hat{\tau}_{1}$
$\\{P2_{z}^{+},P2_{z}^{-}\\}$ | $\hat{\tau}_{0}$ | 0 | $\hat{\tau}_{1}$ | 0
$\\{P1_{z}^{\alpha},P2_{z}^{\alpha^{\prime}}\\}$ | $\hat{\tau}_{0}-\hat{\tau}_{3}$ | $\hat{\tau}_{0}+\hat{\tau}_{3}$ | 0 | 0
Table 4: Explicit form of the symmetric and anti-symmetric scattering matrices
for a particular two-orbital basis chosen from
$\\{P1_{z}^{+},~{}P1_{z}^{-},~{}P2_{z}^{+},~{}P2_{z}^{-}\\}$, with
$\alpha,\alpha^{\prime}\in\\{+,-\\}$. Pauli matrices acting in orbital space
are denoted by $\hat{\tau}_{i}$ with $i\in\\{0,1,2,3\\}$, where
$\hat{\tau}_{0}$ corresponds to the identity matrix.

Note that for orbitals involving different types of atoms,
$\\{P1_{z}^{\alpha},P2_{z}^{\alpha^{\prime}}\\}$, all anti-symmetric
scattering matrices are zero in the low-energy basis. On the contrary, if the
combinations involve the same type of atom, meaning
$\\{P1_{z}^{+},P1_{z}^{-}\\}$ and $\\{P2_{z}^{+},P2_{z}^{-}\\}$, both the
symmetric and anti-symmetric matrices are finite, as illustrated in Fig. 1. In
the next section, we will analyze the implications of these findings regarding
the renormalization of the superconducting temperature.

##  IV Renormalization of $\boldsymbol{T_{c}}$

Within the standard self-consistent Born approximation, the reduction of the
critical temperature in the presence of impurities for simple superconductors
can be cast as [36]:

$\displaystyle\log\left(\frac{T_{c}}{T_{c0}}\right)=\Psi\left(\frac{1}{2}\right)-\Psi\left(\frac{1}{2}+\frac{1}{4\pi\tau_{\rm
eff}T_{c}}\right),$ (26)

with $\Psi(x)$ denoting the digamma function and $\tau_{\rm eff}$ the
effective scattering rate defined as

$\displaystyle\tau_{\rm eff}^{-1}=\tau_{\rm n}^{-1}-\tau_{\rm sc}^{-1},$ (27)

in terms of the normal ($\tau_{\rm n}$) and superconducting ($\tau_{\rm sc}$)
scattering rates. Below we begin by discussing how $\tau_{\rm n}$ and
$\tau_{\rm sc}$ are obtained for complex superconductors with two orbital
degrees of freedom contributing to a single Fermi surface. In Subsect. IV.3,
we then discuss different scenarios with specific choices of orbitals and
impurity location.

###  IV.1 The normal scattering rate

The influence of impurity scattering on the electronic system is described by
the renormalization of the Green’s function of the system, which we calculate
by solving Dyson’s equation [37],

$\displaystyle\hat{G}^{-1}(\boldsymbol{k},i\omega_{n})=\hat{G}_{0}^{-1}(\boldsymbol{k},i\omega_{n})-\hat{\Sigma}_{1}(\boldsymbol{k},i\omega_{n}).$
(28)

Here we introduced the bare Green’s function
$\hat{G}_{0}(\boldsymbol{k},i\omega_{n})$ and self-energy
$\hat{\Sigma}_{1}(\boldsymbol{k},i\omega_{n})$ of the normal state, which
depend in general on the wave vector $\boldsymbol{k}$ and the fermionic
Matsubara frequencies $\omega_{n}=(2n+1)\pi k_{B}T$. The bare Green’s function
is defined as

$\displaystyle\hat{G}^{-1}_{0}(\boldsymbol{k},i\omega_{n})=i\omega_{n}(\hat{\tau}_{0}\otimes\hat{\sigma}_{0})-\hat{H}(\boldsymbol{k}),$
(29)

with $\hat{H}(\boldsymbol{k})$ given by Eq. (5). Using the Born approximation
and assuming isotropic scattering, the self-energy has no
$\boldsymbol{k}$-dependence and takes the form

$\displaystyle\hat{\Sigma}_{1}(i\omega_{n})=\sum_{i}n_{i}\hat{V}_{i}\int\frac{d^{3}k}{(2\pi)^{3}}\hat{G}(\boldsymbol{k},i\omega_{n})\hat{V}_{i},$
(30)

where $\hat{V}_{i}$ are the different scattering matrices in the two-orbital
basis, as summarized in Table 4. After inverting Eq. (29), inserting the
result in the equation for the self-energy and solving the Dyson’s equation
self-consistently, we end up with a renormalization of the Matsubara
frequencies. This allows us to identify the normal-state scattering rate [26],

$\displaystyle\tau^{-1}_{\rm n}$
$\displaystyle=\sum_{i}\frac{\pi}{2}V_{i}^{2}N(0)n_{i}X_{i}(n)$
$\displaystyle\hskip
28.45274pt\times\Bigg{[}1+\sum_{a,b}C_{\text{n},i}^{ab}\langle\hat{h}_{ab}(\boldsymbol{k})\rangle^{2}\Bigg{]},$
(31)

with $N(0)$ denoting the density of states at the Fermi level, $V_{i}$ the
magnitude of the scattering potential, and $X_{i}(n)=\\{S_{i}(n),A_{i}(n)\\}$
depending on the specific impurity potential $\hat{V}_{i}$, as indicated in
Table 4. The factors $C_{\text{n},i}^{ab}$ are equal to $+1$ ($-1$), if the
corresponding term $(a,b)$ in the normal-state Hamiltonian [Eq. (5)] commutes
(anti-commutes) with the scattering potential $\hat{V}_{i}$. Here $\langle
f(\boldsymbol{k})\rangle$ denotes the average over the Fermi surface of the
function $f(\boldsymbol{k})$. Note that the sum of all symmetry-allowed pairs
$(a,b)$ excludes $(0,0)$ and that the coefficients are normalized,

$\displaystyle\sum_{a,b}\hat{h}^{2}_{ab}(\boldsymbol{k})=1.$ (32)

Moreover, we implicitly assume a weak momentum-dependence of the scattering
rate such that we can simply consider its FS average. In Table 5 we list the
values of $C_{\text{n},i}^{ab}$ for the all scattering potentials
$\hat{V}_{i}=\hat{\tau}_{0},\hat{\tau}_{1},\hat{\tau}_{3}$. Note that we have
to distinguish between OP and EP basis states, since the structure of the
normal-state Hamiltonian is different for the EP or OP scenarios, as can be
seen from Tables 1 and 2.

###  IV.2 The superconducting scattering rate

In the superconducting state, we additionally have to take into account the
renormalization of the anomalous Green’s function
$\hat{F}(\boldsymbol{k},i\omega_{n})$, which incorporates the pairing
potential. Since we are only interested in the change of $T_{c}$, we can
restrict ourselves to linear order in the gap function. The anomalous Green’s
function is then simply given by

$\displaystyle\hat{F}(\boldsymbol{k},i\omega_{n})\approx-\hat{G}(\boldsymbol{k},i\omega_{n})$
$\displaystyle\left[\hat{\Delta}(\boldsymbol{k})+\hat{\Sigma}_{2}(i\omega_{n})\right]$
$\displaystyle\hskip
28.45274pt\times\hat{G}^{T}(-\boldsymbol{k},-i\omega_{n}),$ (33)

where $\hat{\Delta}(\boldsymbol{k})$ is given by Eq. (6). We introduced the
self-energy of the superconducting state $\hat{\Sigma}_{2}(i\omega_{n})$,
which in the Born approximation is defined as

$\displaystyle\hat{\Sigma}_{2}(i\omega_{n})=-\sum_{i}n_{i}\hat{V}_{i}\int\frac{d^{3}k}{(2\pi)^{3}}\hat{F}(\boldsymbol{k},i\omega_{n})\hat{V}_{i}^{\dagger}.$
(34)

Analogously to before, we calculate first the anomalous Green’s function and
the corresponding self-energy. Solving then everything self-consistently
enables us to find the renormalization of the pairing potential. From here on,
we restrict ourselves to purely unconventional states, i.e. the non-$A_{1g}$
gap functions. This allows us to directly obtain a closed-form solutions for
the scattering rates. Of course, the framework can also be extended to the
$A_{1g}$ channel. However, one then has to account for a superposition of
multiple superconducting components, which we want to avoid here. Neglecting
interband pairing contributions we finally obtain for the superconducting
scattering rate [26],

$\displaystyle\tau^{-1}_{\rm
sc}=\sum_{i}\frac{\pi}{2}V_{i}^{2}N(0)n_{i}X_{i}(n)C_{\text{sc},i}^{ab}\left[1-\langle
F_{C}\rangle\right],$ (35)

where $C_{\text{sc},i}^{ab}$ is equal to $+1$ ($-1$), if the scattering
potential $\hat{V}_{i}$ commutes (anti-commutes) with the particular gap
function labelled as $[a,b]$ [Eq. (6)]. The values for all different $s$-wave
superconducting states and scattering potentials are summarized in Table 5.
Note that the EP scenario has less active scattering matrices if compared to
the OP scenario. In Eq. (35) we introduced the normalized average of the
Fitness function,

$\displaystyle\braket{{F}_{C}}=\left\langle\frac{||\hat{F}_{C}(\boldsymbol{k})||^{2}}{\sum_{(a,b)\neq(0,0)}h_{ab}^{2}(\boldsymbol{k})}\right\rangle,$
(36)

with the Fitness matrix [38, 39] given by

$\displaystyle\hat{F}_{C}(\boldsymbol{k})$
$\displaystyle=\hat{H}(\boldsymbol{k})\hat{\Delta}(\boldsymbol{k})-\hat{\Delta}(\boldsymbol{k})\hat{H}^{T}(-\boldsymbol{k}).$
(37)

In Table 6 we provide an overview of the terms $(a,b)$ in the normal-state
Hamiltonian, which contribute to a finite Fitness function for the different
pairing channels of interest labelled as $[a,b]$. As before, we also include
the possibility of OP and EP basis states.

$C_{\text{n},i}^{ab}$
---
$(a,b)$ |  $\hat{V}^{\prime}_{i}$
$\hat{\tau}_{0}$ | $\hat{\tau}_{1}$ | $\hat{\tau}_{3}$
$(3,0)$ | $+1$ | $-1$ | $+1$
$(1,0)$ | $+1$ | $+1$ | $-1$

$C_{\text{sc},i}^{ab}$
---
$[a,b]$ |  $\hat{V}^{\prime}_{i}$
$\hat{\tau}_{0}$ | $\hat{\tau}_{1}$ | $\hat{\tau}_{3}$
$[0,0]$ | $+1$ | $+1$ | $+1$
$[3,0]$ | $+1$ | $-1$ | $+1$
$[2,3]$ | $+1$ | $-1$ | $-1$
$[1,0]$ | $+1$ | $+1$ | $-1$
$[2,1]$ | $+1$ | $-1$ | $-1$
$[2,2]$ | $+1$ | $-1$ | $-1$

Table 5: Explicit values of the factors $C_{\text{n},i}^{ab}=\pm 1$ (left) and
$C_{\text{sc},i}^{ab}=\pm 1$ (right), which encode the commutation relations
between the scattering potentials $\hat{V}^{\prime}_{i}$ and
$h_{ab}(\boldsymbol{k})$ or $d_{ab}$, respectively. These factors appear in
the equations of the normal-state [Eq. (31)] and superconducting [Eq. (35)]
scattering rates. We assume an effective two-orbital model with OP or EP basis
states. The scattering matrices in the low-energy model for each of these
cases are contained in Table 4. Note that in the left table we have only
included the $(a,b)$ terms, which can belong to the $A_{1g}$ representation,
since they do not vanish after a FS average and ultimately contribute to the
normal-state scattering rate.  $\hat{F}_{C}^{\rm OP}(\boldsymbol{k})$
---
$[a,b]$ | Irrep |  $(a,b)$
$[0,0]$ | $A_{1g}$ | $-$
$[3,0]$ | $(2,0),(1,1),(1,2),(1,3)$
$[2,3]$ | $A_{1u}$ | $(3,0),(1,3)$
$[1,0]$ | $A_{2u}$ | $(3,0),(2,0)$
$[2,1]$ | $E_{u}$ | $(3,0),(1,2)$
$[2,2]$ | $(3,0),(1,1)$
$\hat{F}_{C}^{\rm EP}(\boldsymbol{k})$
$[0,0]$ | $A_{1g}$ | $-$
$[3,0]$ | $(1,0),(2,1),(2,2),(2,3)$
$[1,0]$ | $(3,0),(2,1),(2,2),(2,3)$
$[2,3]$ | $A_{2g}$ | $(1,0),(3,0),(2,1),(2,2)$
$[2,1]$ | $E_{g}$ | $(1,0),(3,0),(2,2),(2,3)$
$[2,2]$ | $(1,0),(3,0),(2,1),(2,3)$
Table 6: The terms $(a,b)$ of the normal-state Hamiltonian [Eq. (5)],
contributing to the finite Fitness function as
$F_{C}(\boldsymbol{k})=\sum_{a,b}\hat{h}_{ab}^{2}(\boldsymbol{k})$, for the
corresponding s-wave superconducting states labelled as $[a,b]$ [Eq. (6)] in
both OP and EP scenarios.

###  IV.3 Analysis of three scenarios

In the following, we will use these results to analyze qualitatively the
impurity-induced renormalization of the superconducting critical temperature
for three generic scenarios of multi-layered superconductors. The first
example, which is about OP basis states originating from distinct types of
atoms, is most closely related to the Bi2Se3-related superconductors. In the
second and third examples, we then discuss how the renormalization changes if
the states have EP or have OP but originate from the same type of atoms. For
all of these instances we consider on-site, interstitial, intercalated and
polar impurity configurations.

####  IV.3.1 OP orbitals from different atoms

For concreteness, here we assume the low-energy orbitals are
$\\{P1_{z}^{+},P2_{z}^{-}\\}$, the first associated with an even parity
orbital originating from Bi atoms, the second associated with an odd parity
orbital stemming from the Se atoms. From Table 4, we conclude that only the
symmetric impurity configurations contribute to scattering. Thus, our first
conclusion is that in the presence of purely polar impurities, which are
entirely anti-symmetric, the superconducting state is left untouched.

If the scattering is due to on-site or interstitial defects, the situation
changes and we have two finite scattering processes in the effective low-
energy model:

$\displaystyle\hat{V}^{\prime}_{S,1}$
$\displaystyle=\hat{\tau}_{0}-\hat{\tau}_{3}$ (38)
$\displaystyle\hat{V}^{\prime}_{S,2}$
$\displaystyle=\hat{\tau}_{0}+\hat{\tau}_{3}.$ (39)

The overall impurity potential, which effectively renormalizes the order
parameter, corresponds to a superposition of these. For random impurity
distributions, the two scattering matrices are equally likely, and their
average is ultimately given by the identity matrix $\hat{\tau}_{0}$. This
leads to scattering rates given by

$\displaystyle\tau^{-1}_{\text{n},i}$
$\displaystyle=\frac{\pi}{2}V_{i}^{2}N(0)nS_{i}(n)\left(1+\braket{\hat{h}_{30}}^{2}\right),$
(40)

and

$\displaystyle\tau^{-1}_{\text{sc},i}$
$\displaystyle=\frac{\pi}{2}V_{i}^{2}N(0)nS_{i}(n)\left(1-\braket{F_{C}^{\rm
OP}}\right),$ (41)

where $i=\\{{\rm on},{\rm is}\\}$. Note that above the
$\boldsymbol{k}$-dependence is implicit inside the brakets denoting the
averages over the Fermi surface. Combining both equations leads to an
effective scattering rate equal to

$\displaystyle\tau_{\text{eff},i}^{-1}=\frac{\pi}{2}V_{i}^{2}N(0)nS_{i}(n)\left(\braket{\hat{h}_{30}}^{2}+\braket{F_{C}^{\rm
OP}}\right).$ (42)

The first term inside of the brackets is the same for all pairing symmetries.
Thus, the difference between the effective scattering rate for different
superconducting states is entirely governed by the average of the respective
fitness function. As we can infer from Table 6, all gap functions have a
finite fitness measure, whose size depends on the details of the microscopic
Hamiltonian. This aspect was already emphasized in previous works [25, 26,
24]. However, we also observe that the evolution of the share of symmetric
configurations, $S_{i}(n)$, affects the scattering rate, which is a feature
that does not depend on the structure of the Hamiltonian, but on the
distribution of impurities. The influence of $S_{i}(n)$ on the renormalization
of $T_{c}$ is illustrated in Fig. 5.

Figure 5: Suppression of the normalized superconducting transition temperature
$T_{c}/T_{c0}$ versus the impurity concentration $n$, if only the symmetric
share $S_{i}(n)$ of impurity distributions contributes to scattering,
calculated from Eqs. (26, 42). The standard case corresponds to the situation
where the share of symmetric configurations stays constant as a function of
the filling, i.e. $S_{\rm stand}=1$. For the top (bottom) figure we choose a
small (large) prefactor to the scattering rate, corresponding to a weak
(strong) scattering potential.

As expected from the shape of $S_{i}(n)$ [Fig. 4], we obtain that all
superconducting states are more sensitive to interstitial than on-site defects
(assuming same magnitude of the scattering potential on the sites).
Furthermore, we note that in case of strong impurity potentials, for which the
superconducting state is suppressed by small impurity concentrations, there is
an enhancement of the critical concentration by a maximum factor of
approximately two (illustrated by Fig. 5 b).

For intercalated configurations, the impurities can only occupy the first and
fourth layer. Hence, the only finite scattering matrix in the low-energy model
is given by $\hat{V}_{S,1}$ [Eq. (38)], which does not square to the identity
matrix. This has important consequences for the scattering rates. First of
all, since the gap functions not belonging to $A_{1g}$ commute (anti-commute)
with $\hat{\tau}_{0}$ ($\hat{\tau}_{3}$), the superconducting scattering rate
vanishes, i.e. $\tau^{-1}_{\rm sc,ic}=0$. In contrast, the normal-state
scattering rate is finite because all terms of the averaged Green’s function
fully commute with the scattering potential. The effective scattering rate is
therefore equal to

$\displaystyle\tau_{\text{eff},\text{ic}}^{-1}=\tau_{\text{n},\text{ic}}^{-1}$
$\displaystyle=\pi V_{\rm ic}^{2}N(0)nS_{\rm
ic}(n)\left(1-\braket{\hat{h}_{30}}\right)^{2}.$ (43)

We find that the normalized mass imbalance-term in the Hamiltonian,
$\hat{h}_{30}$, is again an important factor that controls the renormalization
of the superconducting state in presence of impurities. However, in contrast
to the cases of interstitial or on-site impurities, we obtain that a larger
contribution of $\hat{h}_{30}$ to the normal-state Hamiltonian actually leads
to a smaller effective scattering rate. Depending on the details of the
normal-state Hamiltonian this could have profound consequences on the
suppression of $T_{c}$.

In order to make better connections with experiments, we would like to
emphasize here that the critical temperature is usually investigated as a
function of the residual resistivity in the normal state [16, 19]. The
residual resistivity is proportional to the normal-state scattering rate, so
it is sensible to study the evolution of the critical temperature not only
with respect to the impurity concentration, but also with respect to the
normal state scattering rate. We can write

$\displaystyle\log\left(\frac{T_{c}}{T_{c0}}\right)=\Psi\left(\frac{1}{2}\right)-\Psi\left(\frac{1}{2}+\frac{1}{4\pi\tau_{\rm
n}T_{c}}\frac{\tau_{\rm n}}{\tau_{\rm eff}}\right),$ (44)

such that the ratio $\tau_{\rm n}/\tau_{\rm eff}$ can be thought of as the
_effectiveness_ of the impurities in the superconducting state. If this ratio
is large (small) the superconducting critical temperature is suppressed faster
(slower) than naively expected for a single-band superconductor. In the case
of on-site and interstitial scenarios, we find for superconducting states in
non-trivial irreps

$\displaystyle\frac{\tau_{\rm n}}{\tau_{\rm eff}}=\frac{\langle
F_{C}^{OP}\rangle+\langle\hat{h}_{30}\rangle^{2}}{1+\langle\hat{h}_{30}\rangle^{2}}.$
(45)

From this ration, let us discuss two extreme limits. The first corresponds to
$\braket{\hat{h}_{30}}\rightarrow 0$ such that $\tau_{\rm n}/\tau_{\rm
eff}\rightarrow\langle F_{C}^{OP}\rangle$. In this limit, assuming OP orbitals
we obtain $\braket{F_{C}^{OP}}=\braket{\hat{h}_{ab}^{2}}$ for a single
$(a,b)$-term in the normal-state Hamiltonian (according to Table 6). If this
term is dominant, $\tau_{\rm n}/\tau_{\rm eff}\rightarrow 1$ and the
superconductor behaves as expected for simple single-band scenarios. On the
other hand, if the specific $(a,b)$ term is negligible, $\tau_{\rm
n}/\tau_{\rm eff}\rightarrow 0$ and the superconducting state is suppressed at
a much slower rate than expected from a naive estimation. The second limit
corresponds to $\braket{\hat{h}_{30}}\rightarrow 1$, with all other
$\braket{\hat{h}_{ab}^{2}}\rightarrow 0$. In this case, the ratio $\tau_{\rm
n}/\tau_{\rm eff}\rightarrow 1$ and the superconducting state is suppressed at
the usual rate in terms of the normal state scattering rate. For intercalated
impurities the ratio $\tau_{\rm n}/\tau_{\rm eff}=1$, which is also in
accordance with the expectation for simple superconductors.

From this discussion, we can conclude that an arbitrarily robust
unconventional superconducting state is possible in layered materials with OP
orbitals stemming from distinct types of atoms under the condition that the
$(a,b)$-terms enumerated in Table 6 (top) for each superconducting state are
the least dominant terms in the normal-state Hamiltonian. Let us analyze next
how these qualitative results change for EP basis states.

####  IV.3.2 EP orbitals from different atoms

According to Table 4, for basis states with EP, such as
$\\{P1_{z}^{+},P2_{z}^{+}\\}$, we obtain the same scattering matrices as in
the previous subsection. This means that polar impurities play again no role
in the renormalization of the superconducting state. The only differences
compared to the OP case arise due to the changes in the structure of the
Hamiltonian. Naturally, these differences are more of quantitative than of
qualitative nature.

For example, for on-site or interstitial distributions we find

$\displaystyle\tau^{\prime-1}_{\text{n},i}$
$\displaystyle=\frac{\pi}{2}V_{i}^{2}N(0)nS_{i}(n)\left(1+\braket{\hat{h}_{30}}^{2}+\braket{\hat{h}_{10}}^{2}\right),$
(46)

and

$\displaystyle\tau^{\prime-1}_{\text{sc},i}$
$\displaystyle=\frac{\pi}{2}V_{i}^{2}N(0)nS_{i}(n)\left(1-\braket{F_{C}^{\rm
EP}}\right),$ (47)

with $i=\\{{\rm on},{\rm is}\\}$. Consequently, the effective scattering rate
becomes

$\displaystyle\tau_{\text{eff},i}^{\prime-1}$
$\displaystyle=\frac{\pi}{2}V_{i}^{2}N(0)nS_{i}(n)$ $\displaystyle\hskip
28.45274pt\times\left(\braket{\hat{h}_{30}}^{2}+\braket{\hat{h}_{10}}^{2}+\braket{F_{C}^{\rm
EP}}\right).$ (48)

The only difference to Eq. (42) is therefore a different prefactor determined
by the microscopic structure of the Hamiltonian. Surprisingly, for
intercalated impurities we obtain exactly the same results as before, since
the $\hat{h}_{10}$-term of the Hamiltonian anti-commutes with
$\hat{\tau}_{3}$. Consequently, its contribution vanishes in the normal-state
scattering rate and we again end up with Eq. (43). From this we can deduce
that disorder located in the vdW gap has the same effect in OP and EP two-
orbital models stemming from distinct types of atoms.

For the EP orbitals scenario, the discussion of the evolution of the critical
temperature as a function of the normal-state scattering rate is similar to
the one given above for the OP scenario. We can again conclude that an
arbitrarily robust unconventional superconducting state is theoretically
possible in layered materials with EP orbitals stemming from distinct types of
atoms under the condition that the $(a,b)$-terms enumerated in Table 6
(bottom) are not the dominant terms in the normal-state Hamiltonian. Note that
the number of terms in Table 6 is four, in contrast to two for the OP
scenario. This means that superconductors stemming from Fermi surfaces formed
by EP orbitals are generally much less likely to be robust than the ones
stemming from OP orbitals. The condition for robustness would require four out
of the six symmetry allowed terms in the normal-state Hamiltonian to be
negligible.

####  IV.3.3 OP orbitals from same atoms

If the states occupy the same type of atoms, as it is the case for
$\\{P2_{z}^{+},P2_{z}^{-}\\}$, the renormalization changes substantially. From
Table 4, we see that polar impurities, with scattering potential in the low-
energy basis is given by $\hat{\tau}_{1}$, now suppress the superconducting
state. The corresponding scattering rates read

$\displaystyle\tau^{\prime\prime-1}_{\text{n},\text{po}}$
$\displaystyle=\frac{\pi}{2}V_{\text{po}}^{2}N(0)nA_{\text{po}}(n)\left(1-\braket{\hat{h}_{30}}^{2}\right),$
(49)

and

$\displaystyle\tau^{\prime\prime-1}_{\text{sc},\text{po}}$
$\displaystyle=C_{\rm
sc,po}^{ab}\frac{\pi}{2}V_{\text{po}}^{2}N(0)nA_{\text{po}}(n)\left(1-\braket{F_{C}^{\rm
OP}}\right),$ (50)

where we obtain a different overall sign in the last equation due to the
$C_{\rm sc,po}^{ab}$ factors [Table 5]. Subtracting the scattering rates leads
now to

$\displaystyle\tau_{\text{eff},\text{po}}^{\prime\prime-1}$
$\displaystyle=\frac{\pi}{2}V_{\rm po}^{2}N(0)nA_{\rm po}(n)$
$\displaystyle\hskip 14.22636pt\times\left[1-\braket{\hat{h}_{30}}^{2}-C_{\rm
sc,po}^{ab}\left(1-\braket{F_{C}^{\rm OP}}\right)\right].$ (51)

Note that again the scattering rates for different gap functions merely differ
by the overall factor in the square brackets. Assuming the system is purely
populated by polar impurities such that the share of anti-symmetric
configurations stays constant, i.e. $A_{\rm po}(n)=1$, Eq. (51) yields a
renormalization which only depends on the details of the microscopic
Hamiltonian. In Fig. 6, we show how this feature can influence the robustness
of the superconducting state irrespective of the expansion coefficients of the
normal-state Hamiltonian.

Figure 6: Supression of the normalized critical temperature $T_{c}/T_{c0}$ as
a function of the impurity filling fraction $n$, if only the anti-symmetric
share $A(n)$ of impurity distributions contributes, calculated from Eqs. (26,
55). We used the anti-symmetric shares $A_{\rm on/is}$ calculated in Sect.
III. We again choose a small and large prefactor to illustrate the effects of
a weak and strong scattering potential, respectively.

For on-site, interstitial and intercalated impurities the scattering rates
contain both symmetric and anti-symmetric shares. We obtain

$\displaystyle\tau^{\prime\prime-1}_{\text{n},i}$
$\displaystyle=\frac{\pi}{2}V_{i}^{2}N(0)n\Big{[}S_{i}(n)\left(1+\braket{\hat{h}_{30}}^{2}\right)$
$\displaystyle\hskip
71.13188pt+A_{i}(n)\left(1-\braket{\hat{h}_{30}}^{2}\right)\Big{]},$ (52)

and

$\displaystyle\tau^{\prime\prime-1}_{\text{sc},i}$
$\displaystyle=\frac{\pi}{2}V_{i}^{2}N(0)n\Big{[}S_{i}(n)\left(1-\braket{F_{C}^{\rm
OP}}\right)$ $\displaystyle\hskip
71.13188pt+C_{\text{sc},i}^{ab}A_{i}(n)\left(1-\braket{F_{C}^{\rm
OP}}\right)\Big{]},$ (53)

with $i=\\{\text{on},\text{is},\text{ic}\\}$. Note that not all non-A1g gap
functions commute with $\hat{\tau}_{1}$, hence we have to include the
different $C_{\text{sc},i}^{ab}$ in the second line of the last equation. The
effective scattering rate is

$\displaystyle\tau_{\text{eff},i}^{\prime\prime-1}$
$\displaystyle=\frac{\pi}{2}V_{i}^{2}N(0)n\Big{\\{}1+\left[S_{i}(n)-A_{i}(n)\right]\braket{\hat{h}_{30}}^{2}$
$\displaystyle\hskip 28.45274pt-\left(1-\braket{F_{C}^{\rm
OP}}\right)\left[S_{i}(n)+C_{\text{sc},i}^{ab}A_{i}(n)\right]\Big{\\}},$ (54)

where we used $S_{i}(n)+A_{i}(n)=1$. We observe that if the states originate
from the same layers, we cannot in general separate the symmetric and anti-
symmetric shares from the microscopic details of the Hamiltonian. Similar to
the previous cases, we find that the explicit structure of the shares
influences the scattering rate. Analyzing to what extent, however, requires a
more detailed calculation of the microscopic parameters, which is not the
scope of this paper. Nevertheless, we can deduce from Eq. (54) that there
might be a fine-tuned value of parameters, for which only the anti-symmetric
share contributes. Rewriting Eq. (54),

$\displaystyle\tau_{\text{eff},i}^{\prime\prime-1}$
$\displaystyle=\frac{\pi}{2}V_{i}^{2}N(0)n\Big{\\{}S_{i}(n)\left(\braket{\hat{h}_{30}}^{2}+\braket{F_{C}^{\rm
OP}}\right)$
$\displaystyle+A_{i}(n)\left[1-\braket{\hat{h}_{30}}^{2}-\left(1-\braket{F_{C}^{\rm
OP}}\right)C_{\text{sc},i}^{ab}\right]\Big{\\}},$ (55)

we see that this is the case if

$\displaystyle\braket{\hat{h}_{30}}^{2}+\braket{F_{C}^{\rm OP}}=0.$ (56)

Assuming $\braket{F_{C}^{\rm OP}}\rightarrow 0$ and $C_{\text{sc},i}^{ab}=-1$,
the renormalization of the critical temperature is governed by the anti-
symmetric share of scattering potentials, as shown in Fig. 6. Intriguingly, we
observe that the robustness of the superconducting state increases drastically
for small scattering potential, since the share of anti-symmetric
configurations approaches zero if the system is gradually filled with on-site
or interstitial impurities. As an aside, we also have to emphasize that for
intercalated impurities only our basis choice of $\\{P2_{z}^{+},P2_{z}^{-}\\}$
leads to the expression given in Eq. (54). Naturally, choosing a basis which
originates from the inner layers yields no renormalization for intercalated
distributions.

Continuing with the discussion of the evolution of the critical temperature as
a function of the normal-state scattering rate, we find now according to Eq.
(44),

$\displaystyle\frac{\tau_{\rm n}}{\tau_{\rm
eff}}=\frac{1+\left[S_{i}(n)-A_{i}(n)\right]\braket{\hat{h}_{30}}^{2}-\left(1-\braket{F_{C}^{\rm
OP}}\right)\left[S_{i}(n)+C_{\text{sc},i}^{ab}A_{i}(n)\right]}{1+\left[S_{i}(n)-A_{i}(n)\right]\braket{\hat{h}_{30}}^{2}}$
(57)

As before, we want to consider two extreme limits. The first concerns
$\braket{\hat{h}_{30}}\rightarrow 0$, in which case the ratio simplifies to

$\displaystyle\frac{\tau_{\rm n}}{\tau_{\rm
eff}}={1-\left(1-\braket{F_{C}^{\rm
OP}}\right)\left[S_{i}(n)+C_{\text{sc},i}^{ab}A_{i}(n)\right]},$ (58)

and becomes equal to $\braket{F_{C}^{\rm OP}}$ for $C_{\text{sc},i}^{ab}=+1$.
As the normalized fitness parameter satisfies $0<\braket{F_{C}^{\rm OP}}<1$,
the ratio is necessarily smaller than one, and the superconducting state is
generally less suppressed than in the naive single band scenario. For
$C_{\text{sc},i}^{ab}=-1$, the form of the ratio is not as simple, but the
conclusion, based on the fact that $0\leq\tau_{\rm n}/\tau_{\rm eff}\leq 1$,
is the same. Overall, we find that for the scenario of two OP orbitals
stemming from the same type of atoms, the conclusion that the superconducting
state can be arbitrarily robust as long as the normal-state Hamiltonian terms
$(a,b)$ contributing to $\braket{F_{C}^{\rm OP}}$ are negligible, is
essentially the same as for the case of two OP orbitals stemming from distinct
types of atoms.

##  V Conclusion

Inspired by the open questions concerning the robustness of the
superconducting state found in materials belonging to the family of doped
Bi2Se3, we have investigated the effects of the impurity location and orbital
content at the Fermi surface on the critical temperature of layered
superconductors. We started revisiting the microscopic description in the
layer basis to faithfully account for four distinct impurity configurations:
on-site, interstitial, intercalated and polar. After moving to the orbital
basis by projecting our model into two low-lying degrees of freedom, we found
three fundamentally distinct effective two-orbital bases. This allowed us to
discuss whether the symmetric and anti-symmetric shares of the scattering
potentials were active or inactive in each of these cases. In particular, we
find that choosing two orbitals stemming from distinct types of atoms, the
anti-symmetric part of the scattering potential is inactive, irrespective of
the relative parity of the orbitals. We then elaborated on the renormalization
of the superconducting critical temperature within the self-consistent Born
approximation by providing closed-form expressions for the effective
scattering rate. For simplicity, we restricted ourselves here to the
unconventional states not belonging to the $A_{1g}$ representation. We obtain
that the effective scattering rate depends on three important properties: i)
the commutation/anti-commutation relation between the normal-state Hamiltonian
and the scattering potential; ii) the commutation/anti-commutation relation
between the superconducting order parameter and the scattering potential; and
iii) the commutation/anti-commutation relation between the normal-state
Hamiltonian and the superconducting order parameter (known as the
superconducting fitness measure).

For the main case of interest, OP orbitals stemming from distinct types of
atoms, we observe that purely polar impurities do not suppress the
superconducting state. Moreover, on-site and interstitital impurities behave
qualitatively the same: the superconducting state is more robust the smaller
the $(3,0)$-term in the normal-state Hamiltonian. Nevertheless, on-site
impurities are generally less effective in suppressing the superconducting
state compared to interstitial impurities (assuming the magnitude of the
scattering potential and the normal-state Hamiltonian are the same). This is
due to the distinct evolution of the symmetric share as a function of doping.
The superconducting state is also possibly more robust than usual for
intercalated impurities, but now the robustness is enhanced if the
contribution of the $(3,0)$-term to the normal-state Hamiltonian increases.

We also conclude that superconductors emerging from Fermi surfaces formed by
two OP orbitals are generally more robust than superconductors stemming from
Fermi surfaces with EP orbitals. This effect is purely controlled by the
fitness measure $\braket{F_{C}}$. Furthermore, considering two OP orbitals
stemming from the same type of atoms as the basis states, we conclude that
these are affected by purely polar impurities, in contrast to the scenario
where the orbitals stem from different types of atoms.

In addition, within the scenario of OP orbitals, we discovered a new mechanism
to enhance the robustness of superconducting states. In case the anti-
symmetric share of the impurity potential is dominant, the superconducting
state can also be exceptionally robust to on-site, interstitial and
intercalated impurities since the anti-symmetric share decreases with doping.

Finally, we highlighted the difference in discussing the robustness of the
superconducting state as a function of the impurity concentration or as a
function of the normal-state scattering rate. Our findings show that an
enhanced robustness can be observed for both on-site and interstitial
impurities when analyzing the critical temperature either as a function of
impurity concentration or $\tau_{\rm n}^{-1}$. On the other hand, an enhanced
robustness for intercalated impurities only appears when analyzing the
evolution of the superconducting critical temperature with the impurity
concentration. These results can have important implications for the
interpretation of experiments and the identification of the impurity location
in layered materials.

## Acknowledgements

We would like to thank Manfred Sigrist, Philip Brydon, and David C. Cavanagh
for helpful discussions. A.R. acknowledges the financial support of the Swiss
National Science Foundation (SNSF) through an Ambizione Grant No. 186043 and
B.Z. the financial support of the SNSF through Division II (No. 184739).

## References

  * Anderson [1959] P. Anderson, Theory of dirty superconductors, J. Phys. Chem. Solids 11, 26 (1959).
  * Anderson [1984] P. W. Anderson, Structure of ”triplet” superconducting energy gaps, Phys. Rev. B 30, 4000 (1984).
  * Abrikosov and P. [1959] A. Abrikosov and G. L. P., On the theory of superconducting alloys; i. the electrodynamics of alloys at absolute zero, JETP 8, 1090 (1959).
  * Suhl and Matthias [1959] H. Suhl and B. T. Matthias, Impurity scattering in superconductors, Phys. Rev. 114, 977 (1959).
  * Dalichaouch  _et al._ [1995] Y. Dalichaouch, M. C. de Andrade, D. A. Gajewski, R. Chau, P. Visani, and M. B. Maple, Impurity scattering and triplet superconductivity in UPt3, Phys. Rev. Lett. 75, 3938 (1995).
  * Mackenzie  _et al._ [1998] A. P. Mackenzie, R. K. W. Haselwimmer, A. W. Tyler, G. G. Lonzarich, Y. Mori, S. Nishizaki, and Y. Maeno, Extremely strong dependence of superconductivity on disorder in Sr2RuO4, Phys. Rev. Lett. 80, 161 (1998).
  * Fujita  _et al._ [2005] K. Fujita, T. Noda, K. M. Kojima, H. Eisaki, and S. Uchida, Effect of disorder outside the CuO2 planes on ${T}_{c}$ of copper oxide superconductors, Phys. Rev. Lett. 95, 097006 (2005).
  * Senga and Kontani [2008] Y. Senga and H. Kontani, Impurity effects in sign-reversing fully gapped superconductors: Analysis of FeAs superconductors, Journal of the Physical Society of Japan 77, 113710 (2008), https://doi.org/10.1143/JPSJ.77.113710 .
  * Onari and Kontani [2009] S. Onari and H. Kontani, Violation of anderson’s theorem for the sign-reversing $s$-wave state of iron-pnictide superconductors, Phys. Rev. Lett. 103, 177001 (2009).
  * Wang  _et al._ [2013] Y. Wang, A. Kreisel, P. J. Hirschfeld, and V. Mishra, Using controlled disorder to distinguish ${s}_{\pm{}}$ and ${s}_{+}{}_{+}$ gap structure in Fe-based superconductors, Phys. Rev. B 87, 094504 (2013).
  * Sasaki  _et al._ [2014] S. Sasaki, K. Segawa, and Y. Ando, Superconductor derived from a topological insulator heterostructure, Phys. Rev. B 90, 220504 (2014).
  * Andersen  _et al._ [2018] L. Andersen, Z. Wang, T. Lorenz, and Y. Ando, Nematic superconductivity in ${\mathrm{Cu}}_{1.5}{(\mathrm{PbSe})}_{5}{({\mathrm{Bi}}_{2}{\mathrm{Se}}_{3})}_{6}$, Phys. Rev. B 98, 220512 (2018).
  * Andersen  _et al._ [2020] L. Andersen, A. Ramires, Z. Wang, T. Lorenz, and Y. Ando, Generalized Anderson’s theorem for superconductors derived from topological insulators, Sci. Adv. 6 (2020).
  * Hor  _et al._ [2010] Y. S. Hor, A. J. Williams, J. G. Checkelsky, P. Roushan, J. Seo, Q. Xu, H. W. Zandbergen, A. Yazdani, N. P. Ong, and R. J. Cava, Superconductivity in ${\mathrm{Cu}}_{x}{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ and its implications for pairing in the undoped topological insulator, Phys. Rev. Lett. 104, 057001 (2010).
  * Yonezawa  _et al._ [2017] S. Yonezawa, K. Tajiri, S. Nakata, Y. Nagai, Z. Wang, K. Segawa, Y. Ando, and Y. Maeno, Thermodynamic evidence for nematic superconductivity in ${\mathrm{Cu}}_{x}{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$, Nature Physics 13, 123 (2017).
  * Kriener  _et al._ [2012] M. Kriener, K. Segawa, S. Sasaki, and Y. Ando, Anomalous suppression of the superfluid density in the CuxBi2Se3 superconductor upon progressive Cu intercalation, Phys. Rev. B 86, 180505 (2012).
  * Qiu  _et al._ [2015] Y. Qiu, K. N. Sanders, J. Dai, J. E. Medvedeva, W. Wu, P. Ghaemi, T. Vojta, and Y. S. Hor, Time reversal symmetry breaking superconductivity in topological materials (2015), arXiv:1512.03519 .
  * Smylie  _et al._ [2016] M. P. Smylie, H. Claus, U. Welp, W.-K. Kwok, Y. Qiu, Y. S. Hor, and A. Snezhko, Evidence of nodes in the order parameter of the superconducting doped topological insulator ${\mathrm{Nb}}_{x}{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ via penetration depth measurements, Phys. Rev. B 94, 180510 (2016).
  * Smylie  _et al._ [2017] M. P. Smylie, K. Willa, H. Claus, A. Snezhko, I. Martin, W.-K. Kwok, Y. Qiu, Y. S. Hor, E. Bokari, P. Niraula, A. Kayani, V. Mishra, and U. Welp, Robust odd-parity superconductivity in the doped topological insulator ${\mathrm{Nb}}_{x}{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$, Phys. Rev. B 96, 115145 (2017).
  * Novak  _et al._ [2013] M. Novak, S. Sasaki, M. Kriener, K. Segawa, and Y. Ando, Unusual nature of fully gapped superconductivity in in-doped SnTe, Phys. Rev. B 88, 140502 (2013).
  * Timmons  _et al._ [2020] E. I. Timmons, S. Teknowijoyo, M. Kończykowski, O. Cavani, M. A. Tanatar, S. Ghimire, K. Cho, Y. Lee, L. Ke, N. H. Jo, S. L. Bud’ko, P. C. Canfield, P. P. Orth, M. S. Scheurer, and R. Prozorov, Electron irradiation effects on superconductivity in ${\mathrm{PdTe}}_{2}$: An application of a generalized Anderson theorem, Phys. Rev. Research 2, 023140 (2020).
  * Michaeli and Fu [2012] K. Michaeli and L. Fu, Spin-orbit locking as a protection mechanism of the odd-parity superconducting state against disorder, Phys. Rev. Lett. 109, 187003 (2012).
  * Nagai [2015] Y. Nagai, Robust superconductivity with nodes in the superconducting topological insulator CuxBi2Se3: Zeeman orbital field and nonmagnetic impurities, Phys. Rev. B 91, 060502 (2015).
  * Sato and Asano [2020] T. Sato and Y. Asano, Superconductivity in cu-doped ${\mathrm{bi}}_{2}{\mathrm{se}}_{3}$ with potential disorder, Phys. Rev. B 102, 024516 (2020).
  * Dentelski  _et al._ [2020] D. Dentelski, V. Kozii, and J. Ruhman, Effect of interorbital scattering on superconductivity in doped dirac semimetals, Phys. Rev. Research 2, 033302 (2020).
  * Cavanagh and Brydon [2021] D. C. Cavanagh and P. M. R. Brydon, General theory of robustness against disorder in multiband superconductors, Phys. Rev. B 104, 014503 (2021).
  * Liu  _et al._ [2010] C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.-C. Zhang, Model hamiltonian for topological insulators, Phys. Rev. B 82, 045122 (2010).
  * Lin  _et al._ [2021] Y.-R. Lin, M. Bagchi, S. Soubatch, T.-L. Lee, J. Brede, F. C. Bocquet, C. Kumpf, Y. Ando, and F. S. Tautz, Vertical position of sr dopants in the ${\mathrm{Sr}}_{x}{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ superconductor, Phys. Rev. B 104, 054506 (2021).
  * Wang  _et al._ [2011] Y.-L. Wang, Y. Xu, Y.-P. Jiang, J.-W. Liu, C.-Z. Chang, M. Chen, Z. Li, C.-L. Song, L.-L. Wang, K. He, X. Chen, W.-H. Duan, Q.-K. Xue, and X.-C. Ma, Structural defects and electronic properties of the Cu-doped topological insulator Bi${}_{2}{\mathrm{Se}}_{3}$, Phys. Rev. B 84, 075335 (2011).
  * Li  _et al._ [2018] Z. Li, M. Wang, D. Zhang, N. Feng, W. Jiang, C. Han, W. Chen, M. Ye, C. Gao, J. Jia, J. Li, S. Qiao, D. Qian, B. Xu, H. Tian, and B. Gao, Possible structural origin of superconductivity in Sr-doped $\mathrm{B}{\mathrm{i}}_{2}\mathrm{S}{\mathrm{e}}_{3}$, Phys. Rev. Materials 2, 014201 (2018).
  * Shirasawa  _et al._ [2014] T. Shirasawa, M. Sugiki, T. Hirahara, M. Aitani, T. Shirai, S. Hasegawa, and T. Takahashi, Structure and transport properties of Cu-doped ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ films, Phys. Rev. B 89, 195311 (2014).
  * Wang  _et al._ [2016] M. Wang, Y. Song, L. You, Z. Li, B. Gao, X. Xie, and M. Jiang, A combined method for synthesis of superconducting Cu doped Bi2Se3, Sci. Rep. 6, 22713 (2016).
  * Fröhlich  _et al._ [2020] T. Fröhlich, Z. Wang, M. Bagchi, A. Stunault, Y. Ando, and M. Braden, Crystal structure and distortion of superconducting ${\mathrm{Cu}}_{x}{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$, Phys. Rev. Materials 4, 054802 (2020).
  * Shruti  _et al._ [2015] Shruti, V. K. Maurya, P. Neha, P. Srivastava, and S. Patnaik, Superconductivity by Sr intercalation in the layered topological insulator ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$, Phys. Rev. B 92, 020506 (2015).
  * Liu  _et al._ [2015] Z. Liu, X. Yao, J. Shao, M. Zuo, L. Pi, S. Tan, C. Zhang, and Y. Zhang, Superconductivity with topological surface state in SrxBi2Se3, J. Am. Chem. Soc. 137, 10512 (2015).
  * Mineev and Samokhin [1999] V. P. Mineev and K. Samokhin,  _Introduction to Unconventional Superconductivity_ (Gordon and Breach, 1999).
  * Maki [2018] K. Maki, Gapless superconductivity, in  _Superconductivity_ (Routledge, 2018) pp. 1035–1105.
  * Ramires and Sigrist [2016] A. Ramires and M. Sigrist, Identifying detrimental effects for multiorbital superconductivity: Application to Sr2RuO4, Phys. Rev. B 94, 104501 (2016).
  * Ramires  _et al._ [2018] A. Ramires, D. F. Agterberg, and M. Sigrist, Tailoring ${T}_{c}$ by symmetry principles: The concept of superconducting fitness, Phys. Rev. B 98, 024501 (2018).