# Lucas numbers that are palindromic concatenations of two distinct repdigits

Herbert Batte1,∗

###### Abstract

Let $\\{L_{n}\\}_{n\geq 0}$ be the sequence of Lucas numbers. In this paper,
we determine all Lucas numbers that are palindromic concatenations of two
distinct repdigits.

Keywords and phrases: Lucas numbers; linear forms in logarithms; Repdigits;
Baker–Davenport reduction method.

2020 Mathematics Subject Classification: 11B39, 11D61, 11J86

∗ Corresponding author

##  1 Introduction

###  1.1 Background

Consider the Lucas number sequence $\\{L_{n}\\}_{n\geq 0}$, which starts with
$L_{0}=2$, $L_{1}=1$, and follows the pattern $L_{n+2}=L_{n+1}+L_{n}$ for all
$n\geq 0$. The initial numbers in this sequence are

$2,\;1,\;3,\;4,\;7,\;11,\;18,\;29,\;47,\;76,\;123,\;199,\ldots.$

A repdigit in base 10 is a positive number $N$ made up of a single repeating
digit. Specifically, $N$ is written as

$N=\underbrace{\overline{d\cdots d}}_{\ell\text{
times}}=d\left(\frac{10^{\ell}-1}{9}\right),$

with positive integers $d$ and $\ell$, where $0\leq d\leq 9$ and $\ell\geq 1$.
Our study adds to the extensive research on the Diophantine characteristics of
certain sequences defined by recurrence relations. Particularly, we explore
how the terms of these sequences can be expressed as sequences within
themselves or as combinations thereof. The work of Luca and Banks [3], despite
its broad scope, yielded some limited results regarding the count of such
sequence terms. The case of Fibonacci numbers composed of two repdigits was
addressed in [1], with the largest identified as $F_{14}=377$.

Recent work has also looked into the relationship between linear recurrence
numbers and repdigits. For example, all repdigits formed by the addition of
two Padovan numbers were identified in [15]. This was expanded upon by the
Ddamulira in [10], who also looked at Padovan numbers that are the
concatenation of two different repdigits, finding the largest to be
$P_{21}=200$ in [11].

Further contributions to this field of research have been made by Bednářik and
Trojovská [4], Boussayoud et al. [5], Bravo and Luca [7], and others [9, 14,
19, 20, 21, 18]. The specific findings in [18] were revisited in [13], which
confirmed that the only Lucas numbers that can be formed by combining two
repdigits are

$11,~{}18,~{}29,~{}47,~{}76,~{}199,~{}322.$

This finding, also reported in [18], was derived through different methods. An
interesting follow-up to this work [13] would be to identify Lucas numbers
that are palindromic. In this context, a number is a palindrome if it reads
the same backwards as forwards. To begin exploring this, we currently examine
a more constrained Diophantine equation:

$\displaystyle L_{n}=\overline{\underbrace{d_{1}\ldots d_{1}}_{\ell\text{
times}}\underbrace{d_{2}\ldots d_{2}}_{m\text{ times}}\underbrace{d_{1}\ldots
d_{1}}_{\ell\text{ times}}},$ (1.1)

where $d_{1},d_{2}\in\\{0,1,2,\ldots,9\\},$ with $d_{1}>0$ and $d_{1}\neq
d_{2}$. Similar work has been done in [8] proving that $151$ and $616$ are the
only Padovan numbers that are palindromic concatenations of two distinct
repdigits.

Here, we present the following result.

###  1.2 Main Results

######  Theorem 1.1.

There is no Lucas number which is a palindromic concatenation of two distinct
repdigits.

##  2 Methods

###  2.1 Preliminaries

Here, we start with the well-known Binet formula for the sequence of Lucas
numbers. It is given by

$\displaystyle
L_{n}=\alpha^{n}+\beta^{n},~{}~{}~{}\text{where}~{}~{}\alpha=\dfrac{1+\sqrt{5}}{2},~{}~{}\beta=\dfrac{1-\sqrt{5}}{2}.$
(2.1)

Note that $\beta=-\alpha^{-1}$ and $|\beta|<1$. It was shown in [6] that

$\displaystyle\alpha^{n-1}\leq L_{n}\leq 2\alpha^{n},\quad\text{holds for
all}\quad n\geq 0.$ (2.2)

We go back and rewrite relation (1.1) as

$\displaystyle L_{n}$ $\displaystyle=\overline{\underbrace{d_{1}\ldots
d_{1}}_{\ell\text{ times}}\underbrace{d_{2}\ldots d_{2}}_{m\text{
times}}\underbrace{d_{1}\ldots d_{1}}_{\ell\text{ times}}}$
$\displaystyle=\overline{\underbrace{d_{1}\ldots d_{1}}_{\ell\text{
times}}\cdot 10^{\ell+m}+\underbrace{d_{2}\ldots d_{2}}_{m\text{ times}}\cdot
10^{\ell}+\underbrace{d_{1}\ldots d_{1}}_{\ell\text{ times}}}$
$\displaystyle=\dfrac{1}{9}\left(d_{1}\cdot 10^{2\ell+m}-(d_{1}-d_{2})\cdot
10^{\ell+m}+(d_{1}-d_{2})\cdot 10^{\ell}-d_{1}\right),$ (2.3)

where $d_{1},d_{2}\in\\{0,1,2,\ldots,9\\}$, $d_{1}>0$ and $d_{1}\neq d_{2}$.

By equation (2.2), assume for a moment that $n>1000$, then equation (2.2)
together with (2.1) imply that

$\displaystyle 2\alpha^{n}\geq L_{n}>10^{2\ell+m-1},$

and taking logarithms both sides yields $(2\ell+m-1)\log 10<n\log\alpha+\log
2$, which simplifies as

$\displaystyle(2\ell+m)\log 10<n\log\alpha+3<n,$

so that

$\displaystyle 2\ell+m<n,$ (2.4)

holds for all $n>1000$.

###  2.2 Linear forms in logarithms

We use three times Baker–type lower bounds for nonzero linear forms in three
logarithms of algebraic numbers. There are many such bounds mentioned in the
literature like that of Baker and Wüstholz from [2] or Matveev from [17].
Before we can formulate such inequalities we need the notion of height of an
algebraic number recalled below.

######  Definition 2.1.

Let $\gamma$ be an algebraic number of degree $d$ with minimal primitive
polynomial over the integers

$a_{0}x^{d}+a_{1}x^{d-1}+\cdots+a_{d}=a_{0}\prod_{i=1}^{d}(x-\gamma^{(i)}),$

where the leading coefficient $a_{0}$ is positive. Then, the logarithmic
height of $\gamma$ is given by

$h(\gamma):=\dfrac{1}{d}\Big{(}\log
a_{0}+\sum_{i=1}^{d}\log\max\\{|\gamma^{(i)}|,1\\}\Big{)}.$

In particular, if $\gamma$ is a rational number represented as $\gamma:=p/q$
with coprime integers $p$ and $q\geq 1$, then $h(\gamma)=\log\max\\{|p|,q\\}$.
The following properties of the logarithmic height function $h(\cdot)$ will be
used in the rest of the paper without further reference:

$\displaystyle h(\gamma_{1}\pm\gamma_{2})$ $\displaystyle\leq
h(\gamma_{1})+h(\gamma_{2})+\log 2;$ $\displaystyle
h(\gamma_{1}\gamma_{2}^{\pm 1})$ $\displaystyle\leq
h(\gamma_{1})+h(\gamma_{2});$ $\displaystyle h(\gamma^{s})$
$\displaystyle=|s|h(\gamma)\quad{\text{\rm valid for}}\quad s\in\mathbb{Z}.$

A linear form in logarithms is an expression

$\Lambda:=b_{1}\log\gamma_{1}+\cdots+b_{t}\log\gamma_{t},$ (2.5)

where for us $\gamma_{1},\ldots,\gamma_{t}$ are positive real algebraic
numbers and $b_{1},\ldots,b_{t}$ are nonzero integers. We assume, $\Lambda\neq
0$. We need lower bounds for $|\Lambda|$. We write
${\mathbb{K}}:={\mathbb{Q}}(\gamma_{1},\ldots,\gamma_{t})$ and $D$ for the
degree of ${\mathbb{K}}$. We start with the general form due to Matveev [17].

######  Theorem 2.1 (Matveev, [17]).

Put $\Gamma:=\gamma_{1}^{b_{1}}\cdots\gamma_{t}^{b_{t}}-1=e^{\Lambda}-1$.
Assume $\Gamma\neq 0$. Then

$\log|\Gamma|>-1.4\cdot 30^{t+3}\cdot t^{4.5}\cdot D^{2}(1+\log D)(1+\log
B)A_{1}\cdots A_{t},$

where $B\geq\max\\{|b_{1}|,\ldots,|b_{t}|\\}$ and
$A_{i}\geq\max\\{Dh(\gamma_{i}),|\log\gamma_{i}|,0.16\\}$ for $i=1,\ldots,t$.

###  2.3 Reduction methods

Typically, the estimates from Matveev’s theorem are excessively large to be
practical in computations. To refine these estimates, we employ a modified
approach based on the Baker–Davenport reduction method. Our adaptation follows
the method introduced by Dujella and Pethö ([12], Lemma 5a). When considering
a real number $r$, we use $\|r\|$ to represent the smallest distance between
$r$ and any integer, which is formally written as
$\min\\{|r-n|:n\in\mathbb{Z}\\}$.

######  Lemma 2.1 (Dujella & Pethö, [12]).

Let $\tau\neq 0$, and $A,B,\mu$ be real numbers with $A>0$ and $B>1$. Let
$M>1$ be a positive integer and suppose that $p/q$ is a convergent of the
continued fraction expansion of $\tau$ with $q>6M$. Let

$\varepsilon:=\|\mu q\|-M\|\tau q\|.$

If $\varepsilon>0$, then there is no solution of the inequality

$0<|m\tau-n+\mu|<AB^{-k}$

in positive integers $m,n,k$ with

$\frac{\log(Aq/\varepsilon)}{\log B}\leq k\quad\text{and}\quad m\leq M.$

Finally, we present an analytic argument which is Lemma 7 in [16].

######  Lemma 2.2 (Lemma 7 in [16]).

If $s\geq 1$, $T>(4s^{2})^{s}$ and $T>\displaystyle\frac{z}{(\log z)^{s}}$,
then

$z<2^{s}T(\log T)^{s}.$

SageMath 9.5 is used to perform all the computations in this work.

##  3 Proof of Theorem 1.1

###  3.1 The low range $n\leq 1000$

Using a basic SageMath script, we investigated all possible solutions to the
Diophantine equation (1.1). The parameters $d_{1},d_{2}$ were taken from the
set $\\{0,1,2,\ldots,9\\}$ with the conditions that $d_{1}>0$ and $d_{1}\neq
d_{2}$, and we restricted our search to $1\leq\ell,m\leq n\leq 1000$. This
search did not result in any solutions, see Appendix 1. From this point, we
will only consider cases where $n>1000$.

###  3.2 The case $n>1000$

In this subsection, we proceed to examine (2.1) in three different ways. We
first prove the following result.

######  Lemma 3.1.

Let $\ell$, $m$ and $n>1000$ be solutions to the Diophantine equation (1.1),
then

$\ell<5\cdot 10^{12}\log n.$

###### Proof.

We go back to (2.1) and rewrite it using (2.1) as

$\displaystyle L_{n}$ $\displaystyle=\dfrac{1}{9}\left(d_{1}\cdot
10^{2\ell+m}-(d_{1}-d_{2})\cdot 10^{\ell+m}+(d_{1}-d_{2})\cdot
10^{\ell}-d_{1}\right),$ $\displaystyle 9(\alpha^{n}+\beta^{n})$
$\displaystyle=d_{1}\cdot 10^{2\ell+m}-(d_{1}-d_{2})\cdot
10^{\ell+m}+(d_{1}-d_{2})\cdot 10^{\ell}-d_{1},$ $\displaystyle
9\alpha^{n}-d_{1}\cdot 10^{2\ell+m}$
$\displaystyle=-9\beta^{n}-(d_{1}-d_{2})\cdot 10^{\ell+m}+(d_{1}-d_{2})\cdot
10^{\ell}-d_{1}.$

Therefore, we have that

$\displaystyle\left|9\alpha^{n}-d_{1}\cdot 10^{2\ell+m}\right|$
$\displaystyle=\left|-9\beta^{n}-(d_{1}-d_{2})\cdot
10^{\ell+m}+(d_{1}-d_{2})\cdot 10^{\ell}-d_{1}\right|$ $\displaystyle\leq
9\alpha^{-n}+27\cdot 10^{\ell+m},\quad\text{since}\quad\beta=-\alpha^{-1},$
$\displaystyle<28\cdot 10^{\ell+m},$

where in the last inequality we have used the fact that $n>1000$. Now,
dividing both sides by $d_{1}\cdot 10^{2\ell+m}$, we get

$\displaystyle\left|\dfrac{9}{d_{1}}\cdot\alpha^{n}\cdot
10^{-2\ell-m}-1\right|$ $\displaystyle<28\cdot 10^{-\ell}.$ (3.1)

Let

$\Gamma=\dfrac{9}{d_{1}}\cdot\alpha^{n}\cdot 10^{-2\ell-m}-1=e^{\Lambda}-1.$

Notice that $\Lambda\neq 0$, otherwise we would have

$\displaystyle\alpha^{n}=\dfrac{d_{1}\cdot 10^{2\ell+m}}{9},$

which is impossible since the left–hand side is irrational and the right–hand
side is rational. The algebraic number field containing the following
$\gamma_{i}$’s is $\mathbb{K}:=\mathbb{Q}(\sqrt{5})$. We have $D=2$, $t:=3$,

$\displaystyle\gamma_{1}$
$\displaystyle:=9/d_{1},\quad\gamma_{2}:=\alpha,\quad\gamma_{3}:=10,$
$\displaystyle b_{1}$ $\displaystyle:=1,\quad\quad~{}~{}b_{2}:=n,\quad
b_{3}:=-2\ell-m.$

Since $h(\gamma_{1})=h(9/d_{1})\leq\log 9<2.2$,
$h(\gamma_{2})=h(\alpha)=0.5\log\alpha<0.25$ and $h(\gamma_{3})=h(10)=\log
10<2.31$, we take $A_{1}:=4.4$, $A_{2}:=0.5$ and $A_{3}:=4.62$. Next,
$B\geq\max\\{|b_{i}|:i=1,2,3\\}$. By equation (2.4), $2\ell+m<n$, so we take
$B:=n$. Now, by Theorem 2.1,

$\displaystyle\log|\Gamma|$ $\displaystyle>-1.4\cdot 30^{6}\cdot 3^{4.5}\cdot
2^{2}(1+\log 2)(1+\log n)\cdot 4.4\cdot 0.5\cdot 4.62$
$\displaystyle>-9.9\cdot 10^{12}(1+\log n).$ (3.2)

Comparing (3.1) and (3.2), we get

$\displaystyle\ell\log 10-\log 28$ $\displaystyle<9.9\cdot 10^{12}(1+\log n),$
$\displaystyle\ell$ $\displaystyle<4.3\cdot 10^{12}(1+\log n)+1.5$
$\displaystyle=4.3\cdot 10^{12}\log n\left(\dfrac{1}{\log
n}+1+\dfrac{1.5}{4.3\cdot 10^{12}\log n}\right),$

which leads to $\ell<5\cdot 10^{12}\log n$, for all $n>1000$. ∎

Next, we prove the following.

######  Lemma 3.2.

Let $\ell$, $m$ and $n>1000$ be solutions to the Diophantine equation (1.1),
then

$m<3\cdot 10^{25}(\log n)^{2}.$

###### Proof.

Again, we go back to (2.1) and rewrite it using (2.1) as

$\displaystyle L_{n}$ $\displaystyle=\dfrac{1}{9}\left(d_{1}\cdot
10^{2\ell+m}-(d_{1}-d_{2})\cdot 10^{\ell+m}+(d_{1}-d_{2})\cdot
10^{\ell}-d_{1}\right),$ $\displaystyle 9(\alpha^{n}+\beta^{n})$
$\displaystyle=d_{1}\cdot 10^{2\ell+m}-(d_{1}-d_{2})\cdot
10^{\ell+m}+(d_{1}-d_{2})\cdot 10^{\ell}-d_{1},$ $\displaystyle
9\alpha^{n}-d_{1}\cdot 10^{2\ell+m}+(d_{1}-d_{2})\cdot 10^{\ell+m}$
$\displaystyle=-9\beta^{n}+(d_{1}-d_{2})\cdot 10^{\ell}-d_{1}.$

Therefore, we have that

$\displaystyle\left|9\alpha^{n}-(d_{1}\cdot 10^{\ell}-(d_{1}-d_{2}))\cdot
10^{\ell+m}\right|$ $\displaystyle=\left|-9\beta^{n}+(d_{1}-d_{2})\cdot
10^{\ell}-d_{1}\right|$ $\displaystyle\leq 9\alpha^{-n}+18\cdot
10^{\ell},\quad\text{since}\quad\beta=-\alpha^{-1},$ $\displaystyle<19\cdot
10^{\ell},$

where in the last inequality we have used the fact that $n>1000$.

Now, dividing both sides by $(d_{1}\cdot 10^{\ell}-(d_{1}-d_{2}))\cdot
10^{\ell+m}$, we get

$\displaystyle\left|\dfrac{9}{(d_{1}\cdot
10^{\ell}-(d_{1}-d_{2}))}\cdot\alpha^{n}\cdot 10^{-\ell-m}-1\right|$
$\displaystyle<\dfrac{19}{(d_{1}\cdot 10^{\ell}-(d_{1}-d_{2}))}\cdot
10^{-m}<19\cdot 10^{-m}.$ (3.3)

Let

$\Gamma_{1}=\dfrac{9}{(d_{1}\cdot
10^{\ell}-(d_{1}-d_{2}))}\cdot\alpha^{n}\cdot
10^{-\ell-m}-1=e^{\Lambda_{1}}-1.$

Notice that $\Lambda_{1}\neq 0$, otherwise we would have

$\displaystyle\alpha^{n}=\dfrac{(d_{1}\cdot 10^{\ell}-(d_{1}-d_{2}))\cdot
10^{\ell+m}}{9},$

which is impossible since $\alpha^{n}$ is irrational and the right–hand side
is rational. The algebraic number field containing the following
$\gamma_{i}$’s is $\mathbb{K}:=\mathbb{Q}(\sqrt{5})$. Again, we have $D=2$,
$t:=3$,

$\displaystyle\gamma_{1}$ $\displaystyle:=\dfrac{9}{(d_{1}\cdot
10^{\ell}-(d_{1}-d_{2}))},\quad\gamma_{2}:=\alpha,\quad\gamma_{3}:=10,$
$\displaystyle b_{1}$
$\displaystyle:=1,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
b_{2}:=n,~{}~{}~{}b_{3}:=-\ell-m.$

In order to determine what $A_{1}$ will be, we need to find the find the
maximum of the quantities $h(\gamma_{1})$ and $|\log\gamma_{1}|$. We note that

$\displaystyle h(\gamma_{1})$ $\displaystyle=h\left(\dfrac{9}{d_{1}\cdot
10^{\ell}-(d_{1}-d_{2})}\right)\leq h(9)+h(d_{1})+\ell
h(10)+h(d_{1}-d_{2})+\log 2\leq 4\log 9+\ell\log 10$ $\displaystyle<4\log
9+(5\cdot 10^{12}\log n)\log 10<1.2\cdot 10^{13}\log n,$

where we used Lemma 3.1 and the fact that $n>1000$. On the other note,

$\displaystyle|\log\gamma_{1}|$
$\displaystyle=\left|\log\left(\dfrac{9}{d_{1}\cdot
10^{\ell}-(d_{1}-d_{2})}\right)\right|\leq\log 9+\left|\log(d_{1}\cdot
10^{\ell}-(d_{1}-d_{2}))\right|$ $\displaystyle\leq\log 9+\log\left(d_{1}\cdot
10^{\ell}\right)+\left|\log\left(1-\dfrac{(d_{1}-d_{2})}{d_{1}\cdot
10^{\ell}}\right)\right|$ $\displaystyle\leq\log 9+\log d_{1}+\ell\log
10+\left|\frac{|d_{1}-d_{2}|}{d_{1}\cdot
10^{\ell}}+\frac{1}{2}\left(\frac{|d_{1}-d_{2}|}{d_{1}\cdot
10^{\ell}}\right)^{2}+\cdots\right|$ $\displaystyle\leq 2\log 9+\ell\log
10+\frac{1}{10^{\ell}}+\frac{1}{2\cdot 10^{2\ell}}+\cdots$
$\displaystyle<2\log 9+(5\cdot 10^{12}\log n)\log
10+\frac{1}{10^{\ell}-1}<1.16\cdot 10^{13}\log n,$

where still we used Lemma 3.1 and the fact that $n>1000$. Since
$Dh(\gamma_{1})>|\log\gamma_{1}|$, we can take $A_{1}:=2.4\cdot 10^{13}\log
n$. As before in the proof of Lemma 3.1, we can still take $A_{2}:=0.5$,
$A_{3}:=4.62$ and $B:=n$. Now, by Theorem 2.1,

$\displaystyle\log|\Gamma|$ $\displaystyle>-1.4\cdot 30^{6}\cdot 3^{4.5}\cdot
2^{2}(1+\log 2)(1+\log n)\cdot 0.5\cdot 4.62\cdot 2.4\cdot 10^{13}\log n$
$\displaystyle>-6.2\cdot 10^{25}(\log n)^{2}.$ (3.4)

Comparing (3.3) and (3.2), we get

$\displaystyle m\log 10-\log 19$ $\displaystyle<6.2\cdot 10^{25}(\log n)^{2},$
$\displaystyle m$ $\displaystyle<2.7\cdot 10^{25}(\log n)^{2}+1.3=2.7\cdot
10^{25}(\log n)^{2}\left(1+\dfrac{1.3}{2.7\cdot 10^{25}(\log n)^{2}}\right),$

which leads to $m<3\cdot 10^{25}(\log n)^{2}$, for all $n>1000$. ∎

Lastly in this subsection, we prove the following.

######  Lemma 3.3.

Let $\ell$, $m$ and $n>1000$ be solutions to the Diophantine equation (1.1),
then

$\ell<5.3\cdot 10^{14},\quad m<3.4\cdot 10^{29}\quad\text{and}\quad n<9\cdot
10^{45}.$

###### Proof.

Once more, we revisit equation (2.1) and rewrite it using (2.1) as

$\displaystyle 9\alpha^{n}-d_{1}\cdot 10^{2\ell+m}+(d_{1}-d_{2})\cdot
10^{\ell+m}-(d_{1}-d_{2})\cdot 10^{\ell}$ $\displaystyle=-9\beta^{n}-d_{1}.$

Therefore, we have that

$\displaystyle\left|9\alpha^{n}-(d_{1}\cdot 10^{\ell+m}-(d_{1}-d_{2})\cdot
10^{m}+(d_{1}-d_{2}))\cdot 10^{\ell}\right|$
$\displaystyle=\left|-9\beta^{n}-d_{1}\right|\leq 9\alpha^{-n}+9<10,$

where we have used the fact that $\beta=-\alpha^{-1}$ and $n>1000$. Now,
dividing sides both by $9\alpha^{n}$, we get

$\displaystyle\left|\dfrac{(d_{1}\cdot 10^{\ell+m}-(d_{1}-d_{2})\cdot
10^{m}+(d_{1}-d_{2}))}{9}\cdot\alpha^{-n}\cdot 10^{\ell}-1\right|$
$\displaystyle<\dfrac{10}{9\alpha^{n}}.$ (3.5)

Let

$\Gamma_{2}=\dfrac{(d_{1}\cdot 10^{\ell+m}-(d_{1}-d_{2})\cdot
10^{m}+(d_{1}-d_{2}))}{9}\cdot\alpha^{-n}\cdot 10^{\ell}-1=e^{\Lambda_{2}}-1.$

Notice that $\Lambda_{2}\neq 0$, otherwise we would have

$\displaystyle\alpha^{n}=\dfrac{(d_{1}\cdot 10^{\ell+m}-(d_{1}-d_{2})\cdot
10^{m}+(d_{1}-d_{2}))}{9}\cdot 10^{\ell},$

which is impossible since $\alpha^{n}$ is irrational and the right–hand side
is rational. The algebraic number field containing the following
$\gamma_{i}$’s is $\mathbb{K}:=\mathbb{Q}(\sqrt{5})$. Again, we have $D=2$,
$t:=3$,

$\displaystyle\gamma_{1}$ $\displaystyle:=\dfrac{(d_{1}\cdot
10^{\ell+m}-(d_{1}-d_{2})\cdot
10^{m}+(d_{1}-d_{2}))}{9},\quad\gamma_{2}:=\alpha,\quad\gamma_{3}:=10,$
$\displaystyle b_{1}$
$\displaystyle:=1,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad~{}~{}b_{2}:=-n,~{}~{}b_{3}:=-\ell.$

Again, in order to determine what $A_{1}$ will be here, we need to find the
find the maximum of the quantities $h(\gamma_{1})$ and $|\log\gamma_{1}|$. We
note that

$\displaystyle h(\gamma_{1})$ $\displaystyle=h\left(\dfrac{d_{1}\cdot
10^{\ell+m}-(d_{1}-d_{2})\cdot 10^{m}+(d_{1}-d_{2})}{9}\right)$
$\displaystyle\leq
h(9)+h(d_{1})+(\ell+m)h(10)+h(d_{1}-d_{2})+mh(10)+h(d_{1}-d_{2})+3\log 2$
$\displaystyle\leq 7\log 9+(\ell+m)\log 10+m\log 10$ $\displaystyle<7\log
9+\left(3.1\cdot 10^{25}(\log n)^{2}\right)\log 10+\left(3\cdot 10^{25}(\log
n)^{2}\right)\log 10<1.41\cdot 10^{26}(\log n)^{2},$

where we used Lemmas 3.1, 3.2 and the fact that $n>1000$.

On the other note,

$\displaystyle|\log\gamma_{1}|$
$\displaystyle=\left|\log\left(\dfrac{d_{1}\cdot
10^{\ell+m}-(d_{1}-d_{2})\cdot 10^{m}+(d_{1}-d_{2})}{9}\right)\right|$
$\displaystyle\leq\log 9+\left|\log(d_{1}\cdot 10^{\ell+m}-(d_{1}-d_{2})\cdot
10^{m}+(d_{1}-d_{2}))\right|$ $\displaystyle\leq\log 9+\log\left(d_{1}\cdot
10^{\ell+m}\right)+\left|\log\left(1-\dfrac{(d_{1}-d_{2})(10^{m}-1)}{d_{1}\cdot
10^{\ell+m}}\right)\right|$ $\displaystyle\leq\log 9+\log d_{1}+(\ell+m)\log
10+\left|\dfrac{(d_{1}-d_{2})(10^{m}-1)}{d_{1}\cdot
10^{\ell+m}}+\frac{1}{2}\left(\dfrac{(d_{1}-d_{2})(10^{m}-1)}{d_{1}\cdot
10^{\ell+m}}\right)^{2}+\cdots\right|$ $\displaystyle\leq 2\log 9+(\ell+m)\log
10+\frac{1}{10^{\ell}}+\frac{1}{2\cdot 10^{2\ell}}+\cdots$
$\displaystyle<2\log 9+\left(3.1\cdot 10^{25}(\log n)^{2}\right)\log
10+\frac{1}{10^{\ell}-1}<7.2\cdot 10^{25}(\log n)^{2},$

where still we used Lemmas 3.1, 3.2 and the fact that $n>1000$. Since
$Dh(\gamma_{1})>|\log\gamma_{1}|$, we can take $A_{1}:=2.82\cdot 10^{26}(\log
n)^{2}$. As before in the proof of Lemma 3.1, we can still take $A_{2}:=0.5$,
$A_{3}:=4.62$ and $B:=n$. Now, by Theorem 2.1,

$\displaystyle\log|\Gamma|$ $\displaystyle>-1.4\cdot 30^{6}\cdot 3^{4.5}\cdot
2^{2}(1+\log 2)(1+\log n)\cdot 0.5\cdot 4.62\cdot 2.82\cdot 10^{26}(\log
n)^{2}$ $\displaystyle>-7.24\cdot 10^{38}(\log n)^{3}.$ (3.6)

Comparing (3.5) and (3.2), we get

$\displaystyle n\log\alpha-\log(10/9)$ $\displaystyle<7.24\cdot 10^{38}(\log
n)^{3},$ $\displaystyle n$ $\displaystyle<1.51\cdot 10^{39}(\log
n)^{3}+0.22=1.51\cdot 10^{39}(\log n)^{3}\left(1+\dfrac{0.22}{1.51\cdot
10^{39}(\log n)^{3}}\right),$

which leads to $n<1.52\cdot 10^{39}(\log n)^{3}$, for all $n>1000$. To proceed
from here, let $s=3\geq 1$, $T=1.52\cdot 10^{39}>(4s^{2})^{s}=46656$ and
$z=n$, then Lemma 2.2 implies that $n<2^{3}\cdot 1.52\cdot 10^{39}(\log
1.52\cdot 10^{39})^{3}$, or $n<9\cdot 10^{45}$.

Moreover, Lemma 3.1 gives $\ell<5\cdot 10^{12}\log n<5\cdot 10^{12}\log(9\cdot
10^{45})<5.3\cdot 10^{14}$ and Lemma 3.2 gives $m<3\cdot 10^{25}(\log
n)^{2}<3\cdot 10^{25}(\log(9\cdot 10^{45}))^{2}<3.4\cdot 10^{29}$. ∎

The bounds established in Lemma 3.3 exceed practical computational utility and
require reduction. This process is detailed in Subsection 3.3.

###  3.3 The reduction process

Here, we apply Lemma 2.1 as follows. First, we return to the inequality (3.1)
and put

$\Lambda_{1}:=(2\ell+m)\log 10-n\log\alpha+\log\left(\frac{d_{1}}{9}\right).$

Inequality (3.1) can be rewritten as $|\Gamma_{1}|=|e^{\Lambda_{1}}-1|<28\cdot
10^{-\ell}$. If we assume that $\ell\geq 2$, then the right–hand side of this
inequality is at most $0.28<1/2$. The inequality $|e^{\Lambda_{1}}-1|<x$ for
real values of $x$ and ${\Lambda_{1}}$ implies that $|{\Lambda_{1}}|<2x$.
Thus, $|\Lambda_{1}|<56\cdot 10^{-\ell}$. This implies that

$\left|(2\ell+m)\log
10-n\log\alpha+\log\left(\frac{d_{1}}{9}\right)\right|<56\cdot 10^{-\ell}.$

Dividing through the above inequality by $\log\alpha$ gives

$\left|(2\ell+m)\frac{\log
10}{\log\alpha}-n+\left(\frac{\log(d_{1}/9)}{\log\alpha}\right)\right|<\frac{56}{\log\alpha}\cdot
10^{-\ell}.$

So, we apply Lemma 2.1 with the quantities:

$\tau:=\frac{\log
10}{\log\alpha},\quad\mu(d_{1}):=\frac{\log(d_{1}/9)}{\log\alpha},\quad 1\leq
d_{1}\leq 9,\quad A:=\frac{56}{\log\alpha},\quad B:=10.$

Let
$\tau=[a_{0};a_{1},a_{2},\ldots]=[4;1,3,1,1,1,6,4,2,1,10,1,4,46,3,1,2,1,2,\ldots]$
be the continued fraction expansion of $\tau$. We set $M:=10^{46}$ which is an
upper bound of $2\ell+m$. With the help of SageMath in Appendix 2, we find
that the convergent

$\dfrac{p}{q}=\dfrac{p_{98}}{q_{98}}=\dfrac{1645685064668785741047746957258993430046006088389}{343927838259763182336125476035118084206130771252},$

is such that $q=q_{98}>6M$. Furthermore, it gives $\varepsilon>0.4614141430$,
and thus,

$\ell\leq\frac{\log((56/\log\alpha)q/\varepsilon)}{\log 10}<52.$

Therefore, we have that $\ell<52$. The case $\ell<2$ also holds because
$\ell<2<52$.

Next, for fixed $d_{1},d_{2}\in\\{0,1,2,\ldots,9\\}$, $d_{1}>0$, $d_{1}\neq
d_{2}$ and $1\leq\ell<52$, we return to inequality (3.3) and put

$\Lambda_{2}:=(\ell+m)\log 10-n\log\alpha+\log\left(\frac{d_{1}\cdot
10^{\ell}-(d_{1}-d_{2})}{9}\right).$

From inequality (3.3), we have that $|\Gamma_{2}|=|e^{\Lambda_{2}}-1|<19\cdot
10^{-m}.$ Assume that $m\geq 2$, then the right–hand side of this inequality
is at most $0.19<1/2$. Thus, $|\Lambda_{2}|<38\cdot 10^{-m}$, which implies
that

$\left|(\ell+m)\log 10-n\log\alpha+\log\left(\frac{d_{1}\cdot
10^{\ell}-(d_{1}-d_{2})}{9}\right)\right|<38\cdot 10^{-m}.$

Dividing through the above inequality by $\log\alpha$ gives

$\left|(\ell+m)\frac{\log 10}{\log\alpha}-n+\frac{\log((d_{1}\cdot
10^{\ell}-(d_{1}-d_{2}))/9)}{\log\alpha}\right|<\frac{38}{\log\alpha}\cdot
10^{-m}.$

Thus, we apply Lemma 2.1 with the quantities:

$\mu(d_{1},d_{2}):=\frac{\log((d_{1}\cdot
10^{\ell}-(d_{1}-d_{2}))/9)}{\log\alpha},\quad A:=\frac{38}{\log\alpha},\quad
B:=10.$

We take the same $\tau$ as before and its convergent $p/q=p_{98}/q_{98}$.
Since $\ell+m<2\ell+m$, we set $M:=10^{46}$ as an upper bound on $\ell+m$.
With the help of a simple computer program in SageMath (Appendix 3), we get
that $\varepsilon>0.4906425804$, and therefore,

$m\leq\frac{\log((38/\log\alpha)/\varepsilon)}{\log 10}<54.$

Hence, we have that $m<54$. The case $m<2$ holds as well since $m<2<54$.

Lastly, for fixed $d_{1},d_{2}\in\\{0,1,2,\ldots,9\\}$, $d_{1}>0$, $d_{1}\neq
d_{2}$, $1\leq\ell<52$ and $1\leq m<54$, we return to inequality (3.5) and put

$\Lambda_{3}:=\ell\log 10-n\log\alpha+\log\left(\frac{d_{1}\cdot
10^{\ell+m}-(d_{1}-d_{2})\cdot 10^{m}+(d_{1}-d_{2})}{9}\right).$

From inequality (3.5), we have that
$|\Gamma_{3}|=|e^{\Lambda_{3}}-1|<10/9\alpha^{n}$. Since $n>1000$, the
right–hand side of this inequality is less than $1/2$. Thus, the above
inequality implies that $|\Lambda_{3}|<20/9\alpha^{n},$ which leads to

$\left|\ell\log 10-n\log\alpha+\log\left(\frac{d_{1}\cdot
10^{\ell+m}-(d_{1}-d_{2})\cdot
10^{m}+(d_{1}-d_{2})}{9}\right)\right|<\frac{20}{9\alpha^{n}}.$

Dividing through the above inequality by $\log\alpha$ gives,

$\left|\ell\frac{\log 10}{\log\alpha}-n+\frac{\log\left((d_{1}\cdot
10^{\ell+m}-(d_{1}-d_{2})\cdot
10^{m}+(d_{1}-d_{2}))/9\right)}{\log\alpha}\right|<\frac{20}{9\log\alpha}\cdot\alpha^{-n}.$

Again, we apply Lemma 2.1 with the quantities:

$\mu(d_{1},d_{2}):=\frac{\log\left((d_{1}\cdot 10^{\ell+m}-(d_{1}-d_{2})\cdot
10^{m}+(d_{1}-d_{2}))/9\right)}{\log\alpha},\quad
A:=\frac{20}{9\log\alpha},\quad B:=\alpha.$

We can still take the same $\tau$ and its convergent $p/q=p_{98}/q_{98}$ as
before. Since $\ell<2\ell+m$, we choose $M:=10^{46}$ as an upper bound for
$\ell$. With the help of a simple computer program in SageMath (Appendix 4),
we get that $\varepsilon>0.4929934686$, and thus,

$n\leq\frac{\log((20/9\log\alpha)/\varepsilon)}{\log\alpha}<269,$

contradicting our working assumption that $n>1000$, hence Theorem 1.1 holds.∎

## Conclusion

In this work, we showed that there is no Lucas numbers that is a palindromic
concatenation of two distinct repdigits. It remains an open problem to
determine such palindromes in generalized $k-$Lucas numbers.

## Acknowledgments

The author thanks the Eastern Africa Universities Mathematics Programme
(EAUMP) for funding his doctoral studies.

## References

  * [1] Alahmadi, A., Altassan, A., Luca, F., Shoaib, H.: Fibonacci numbers which are concatenations of two repdigits. Quaest. Math. 44(2), 281–290 (2021).
  * [2] Baker, A., & Wüstholz, G.: Logarithmic forms and group varieties, 19–62 (1993).
  * [3] Banks, W.D., Luca, F.: Concatenations with binary recurrent sequences. J. Integer Seq. 8(5), Art. 05.1.3 (2005).
  * [4] Bednářik, D., Trojovská, E.: Repdigits as product of Fibonacci and Tribonacci numbers. Mathematics 8(10), 1720 (2020).
  * [5] Boussayoud, A., Boughaba, S., Kerada, M., Araci, S., Acikgoz, M.: Generating functions of binary products of $k$-Fibonacci and orthogonal polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113(3), 2575–2586 (2019).
  * [6] Bravo, J. J., & Luca, F. (2014). Repdigits in $k$–Lucas sequences. Proceedings-Mathematical Sciences 124, 141–154.
  * [7] Bravo, J.J., Luca, F.: On a conjecture about repdigits in $k$-generalized Fibonacci sequences. Publ. Math. Debrecen 82(3–4), 623–639 (2013).
  * [8] Chalebgwa, Taboka P., Mahadi Ddamulira.: Padovan numbers which are palindromic concatenations of two distinct repdigits. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 115(3) (2021): 108.
  * [9] Ddamulira, M.: Tribonacci numbers that are concatenations of two repdigits. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114(4), Paper No. 203, 10 pp (2020).
  * [10] Ddamulira, M.: Repdigits as sums of three Padovan numbers. Bol. Soc. Mat. Mex. 26(2), 247–261 (2020).
  * [11] Ddamulira, M.: Padovan numbers that are concatenations of two distinct repdigits. Math. Slovaca 71(2), 275–284 (2021).
  * [12] Dujella, A., Pethö, A.: A generalization of a theorem of Baker and Davenport. Q. J. Math. Oxford Ser. 2 49(195), 291–306 (1998).
  * [13] Erduvan, F., Keskin, R.: Lucas numbers which are concatenations of two repdigits. Boletín de la Sociedad Matemática Mexicana, 27, pp.1-11 (2021).
  * [14] Erduvan, F., Keskin, R.: Lucas numbers which are concatenations of three repdigits. Results Math. 76(1), 13 (2021).
  * [15] García Lomelí, A.C., Hernández Hernández, S.: Repdigits as sums of two Padovan numbers. J. Integer Seq. 22(2), Art. 19.2.3 (2019)
  * [16] Gúzman Sánchez, S., Luca, F.: Linear combinations of factorials and $s$–units in a binary recurrence sequence. Ann. Math. Qué. 38(2), 169–188 (2014).
  * [17] Matveev, E. M.: An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers. II. Izvestiya: Mathematics, 64(6), 1217 (2000).
  * [18] Qu, Y., Zeng, J.: Lucas numbers which are concatenations of two repdigits. Mathematics 8(8), 1360 (2020).
  * [19] Rayaguru, S.G., Panda, G.K.: Balancing numbers which are concatenations of two repdigits. Bol. Soc. Mat. Mex. 26(3), 911–919 (2020).
  * [20] Trojovský, P.: Fibonacci numbers with a prescribed block of digits. Mathematics 8(4), 639 (2020).
  * [21] Trojovský, P.: On repdigits as sums of Fibonacci and Tribonacci numbers. Symmetry 12(11), 1774 (2020).

## Address

1 Department of Mathematics, School of Physical Sciences, College of Natural
Sciences, Makerere University, Kampala, Uganda

Email<EMAIL_ADDRESS>
## Appendices

### Appendix 1

    def generate_lucas_numbers(limit):
        lucas = [2, 1]  # Initial Lucas numbers
        while len(lucas) < limit:
            lucas.append(lucas[-1] + lucas[-2])
        return set(lucas)  # Use a set for efficient membership checking

    def construct_palindromic_numbers():
        palindromic_numbers = set()
        for d1 in range(1, 10):  # d1 is a nonzero digit
           for d2 in range(10):  # d2 can be any digit, including zero
              if d1 != d2:
                 for length in range(1, 100):  # Reasonable lengths for repdigits
                      first_part = str(d1) * length
                      second_part = str(d2) * length
                      palindromic_number = int(first_part + second_part + first_part)
                      palindromic_numbers.add(palindromic_number)
        return palindromic_numbers

    # Generate Lucas numbers and palindromic numbers
    lucas_numbers = generate_lucas_numbers(1001)
    palindromic_numbers = construct_palindromic_numbers()

    # Find the intersection of the two sets
    palindromic_lucas = lucas_numbers.intersection(palindromic_numbers)

    print("Lucas numbers which are palindromic concatenations of two distinct repdigits:")
    print(sorted(palindromic_lucas))

### Appendix 2

    from sage.all import *

    # Constants
    a = golden_ratio.n(digits=1000)  # Using the golden ratio as an approximation for the
                                     # root of x^2 - x - 1
    tau = (log(10) / log(a)).n(digits=1000)
    A = (56 / log(a))
    B = 10
    M = 1 * 10^46

    # Continued Fraction and Convergents
    cf_tau = continued_fraction(tau)
    convergents = cf_tau.convergents()

    for d1 in range(1, 10):  # Iterate through d1 from 1 to 9
        mu = (log(d1/9) / log(a)).n(digits=1000)

        DD = []  # Initialize empty list for results for each d1

        # Dujella and Peth Reduction Method
        for i, convergent in enumerate(convergents):
            p, q = convergent.numerator(), convergent.denominator()
            ep = abs(mu * q - round(mu * q)) - M * abs(tau * q - round(tau * q))

            if q > 6 * M and ep > 0:
                log_expr_a = (log(A * q / ep) / log(B)).n(digits=10)
                DD.append((i, ep.n(digits=10), log_expr_a))
                print(f"d1 = {d1}, p_{i}/q_{i} = {p}/{q}")
                break  # Stop after finding the first suitable convergent for this d1

        # Results for each d1
        if DD:
            print(f"Results for d1 = {d1}:")
            print("First few elements of DD:", DD[:1])
        else:
            print(f"No suitable convergent found for d1 = {d1}.")
    print("Continued fraction expansion of tau:", cf_tau[:20])

### Appendix 3

    from sage.all import *

    # Constants
    a = golden_ratio.n(digits=1000)  # Using the golden ratio
    tau = (log(10) / log(a)).n(digits=1000)
    A = (38 / log(a))
    B = 10
    M = 1 * 10^46

    # Continued Fraction and Convergents
    cf_tau = continued_fraction(tau)
    convergents = cf_tau.convergents()

    # Variables to store maximum values
    max_ep = -Infinity
    max_log_expr_a = -Infinity

    for d1 in range(1, 10):
        for d2 in range(0, 10):
            if d1 == d2:
               continue  # d1 should not be equal to d2
            for l in range(1, 52):
               mu = (log((d1 * 10^l - (d1 - d2)) / 9) / log(a)).n(digits=1000)

               # Dujella and Peth Reduction Method
               for i, convergent in enumerate(convergents):
                   p, q = convergent.numerator(), convergent.denominator()
                   ep = abs(mu * q - round(mu * q)) - M * abs(tau * q - round(tau * q))

                   if q > 6 * M and ep > 0:
                      log_expr_a = (log(A * q / ep) / log(B)).n(digits=10)
                      if ep > max_ep:
                         max_ep = ep.n(digits=10)
                      if log_expr_a > max_log_expr_a:
                         max_log_expr_a = log_expr_a
                      break  # Stop after finding the first suitable convergent

    # Print maximum values
    print("Maximum ep across all combinations:", max_ep)
    print("Maximum log_expr_a across all combinations:", max_log_expr_a)

### Appendix 4

    from sage.all import *

    # Constants
    a = golden_ratio.n(digits=1000)  # Using the golden ratio
    tau = (log(10) / log(a)).n(digits=1000)
    A = (20 / (9 * log(a)))
    B = a
    M = 1 * 10^46

    # Continued Fraction and Convergents
    cf_tau = continued_fraction(tau)
    convergents = cf_tau.convergents()

    # Variables to store maximum values
    max_ep = -Infinity
    max_log_expr_a = -Infinity

    # Iterate through combinations of d1, d2, l, and m
    for d1 in range(1, 10):
        for d2 in range(0, 10):
          if d1 == d2:
            continue  # d1 should not be equal to d2
          for l in range(1, 52):
             for m in range(1, 54):
                mu = (log((d1 * 10^(l+m) - (d1 - d2) * 10^m +
                (d1 - d2)) / 9) / log(a)).n(digits=1000)

                # Dujella and Peth Reduction Method
                for convergent in convergents:
                    p, q = convergent.numerator(), convergent.denominator()
                    ep = abs(mu * q - round(mu * q)) - M * abs(tau * q - round(tau * q))

                    if q > 6 * M and ep > 0:
                       log_expr_a = (log(A * q / ep) / log(B)).n(digits=10)
                       if ep > max_ep:
                          max_ep = ep.n(digits=10)
                       if log_expr_a > max_log_expr_a:
                          max_log_expr_a = log_expr_a
                       break  # Stop after finding the first suitable convergent

    # Print maximum values
    print("Maximum ep across all combinations:", max_ep)
    print("Maximum log_expr_a across all combinations:", max_log_expr_a)Ψ


# Bulk conducting states of intrinsically doped Bi2Se3

R. T. Paulino  Present Address: Department of Physics, McGill University,
Montreal, Quebec, H3A 2T8 Canada.  Centro de Ciências Naturais e Humanas,
Universidade Federal do ABC, Santo André, SP, 09210-580 Brazil  M. A. Avila
Centro de Ciências Naturais e Humanas, Universidade Federal do ABC, Santo
André, SP, 09210-580 Brazil

(August 28, 2024)

###### Abstract

With a large band gap and a single Dirac cone responsible for the topological
surface states, Bi2Se3 is widely regarded as a prototypical 3D topological
insulator. Further applications of the bulk material has, however, been
hindered by inherent structural defects that donate electrons and make the
bulk conductive. Consequently, controlling these defects is of great interest
for future technological applications, and while past literature has focused
on adding external doping elements to the mixture, a complete study on undoped
Bi2Se3 was still lacking. In this work, we use the self-flux method to obtain
high-quality Bi2Se3 single-crystals in the entire concentration range
available on the phase-diagram for the technique. By combining basic
structural characterization with measurements of the resistivity, Hall effect
and Shubnikov-de Haas (SdH) quantum oscillations, the effects of these
impurities on the bulk transport are investigated in samples with electron
densities ranging from $10^{17}$ cm-3 to $10^{19}$ cm-3, from Se-rich to Bi-
rich mixtures, respectively, evidencing the transition into a degenerate
semiconductor regime. We find that electron-donor impurities, likely Se
vacancies, unavoidably shift the Fermi level up to $200$ meV inside the
conduction band. Other impurities, like interstitial Bi and Se, are shown to
play a significant role as scattering centres, specially at low temperatures
and in the decoherence of the SdH oscillations. Previous open questions on
Bi2Se3, such as the upturn in resistivity below $30$ K, the different
scattering times in transport and quantum oscillations, and the presence of
additional low mobility bands, are addressed. The results outlined here
provide a concise picture on the bulk conducting states in flux-grown Bi2Se3
single crystals, enabling better control of the structural defects and
electronic properties.

Topological Insulators, Electrical Transport, Quantum Oscillations, Single-
Crystal Growth

##  I Introduction

The application of topology in the well established quantum band theory
enabled material scientists to explore a wide array of novel phases of matter.
The first and most widely studied are the topological insulators (TI),
initially discovered in graphene sheets [1, 2, 3] but soon predicted in other
two- and three-dimensional materials [4, 5, 6, 7, 8]. TIs have the distinct
property of hosting conducting edge states while the bulk remains electrically
insulating, promising direct applications ranging from quantum information to
spintronics [6, 9, 10, 11].

Among the first 3D TIs found, the Bi2Se3 and related family of materials
quickly became the prototypical platform for exploring the properties of Dirac
fermions, as only a single Dirac cone is responsible for the topological
surface states [12]. Bi2Se3 gained special attention due to a large direct
bulk band gap of over $200$ meV [13], allowing relativistic quasiparticles to
persist even at room temperature [14]. However, further applications of
topological phenomena have been limited, as bulk electrons originating from
structural defects shift the Fermi level of Bi2Se3 into the conducting band
and completely dominate the surface states in most transport experiments.

In fact, this family has long been known as degenerate semiconductors, and
Bi2Te3 specifically can be readily made $n-$ or $p-$type depending on the
defect favoured through the growth procedure [15, 16, 17]. This, in fact, has
made Bi2Te3 the parent compound on the majority of currently used Peltier
devices for near room-temperature cooling [18]. Bi2Se3, on the other hand, has
Se vacancies (V${}_{\text{Se}}$) as the main defect present in stoichiometric
samples, unavoidably donating electrons into the system [19, 20, 21] and
making a $p-$type Bi2Se3 hard to obtain.

As a consequence, Bi2Se3 received much less attention until recent years, when
the prediction of a 3D TI phase brought interest into suppressing these bulk
charge carriers and tuning the Fermi level into the bulk band gap and
topological Dirac band crossing. It is known that the charge carrier density
can be reduced to $10^{17}$ cm-3 by increasing the Se concentration during
growth [22, 19, 23, 24, 25, 26], and it can be further reduced to $10^{16}$
cm-3 by Sb-doping the Se-rich samples [27]. Recently, it has been shown that
the Fermi level can be lowered further to yield a $p$-type or nonconducting
Bi2Se3 by doping it with Ca [21, 28]. In such samples, however, the mean free
path seems to be reduced by an order of magnitude [21], and additional bulk
in-gap states of unknown origin become apparent [28]. This suggests that a
chemically simpler $p$-type Bi2Se3 is desirable to resolve this effect, either
as an intrinsic property of the material or originated solely due to extrinsic
doping defects.

The coexistence of a wide array of defects in Bi2Se3 adds an extra layer of
complexity to the origin of the bulk conductivity. In stoichiometric and Se-
rich samples, STM measurements have shown that the density of electrons can be
largely related to the density of V${}_{\text{Se}}$ [29]. Theoretical
calculations have supported V${}_{\text{Se}}$ as the structural defect with
lowest formation energy in a wide range of Bi:Se proportions [15, 30, 31, 32],
but rather surprisingly, Zhang et al. predicted that a $p$-type Bi2Se3 might
be obtained in extremely Bi-rich growth conditions, where the acceptor-type
substitution of Bi in the Se site (Bi${}_{\text{Se}}$) becomes the most stable
defect [15]. Bi${}_{\text{Se}}$ substitution has in fact been reported in
samples grown with slight Bi excess [33, 26], but due to band-bending effects
near the surface of Bi2Se3, the bulk energy states associated to each defect
cannot be determined [33, 25]. More so, with the maximum reported Bi
proportion of 45 % [33] in the initial growth mixture, the prediction of a
$p$-type Bi2Se3 in the Bi-rich portion of the phase diagram remains
unrealized.

In this work, we performed a systematic study of Bi2Se3 on single-crystalline
samples grown in the entire Bi:Se phase diagram range available for the self-
flux method. Following structural characterization through powder X-ray
diffraction (PXRD) and energy dispersive spectroscopy (EDS), the electronic
transport properties were determined through measurements of the resistivity,
Hall effect and Shubnikov-de Haas oscillations, allowing insights on the
defect formations. With recent works taken into consideration, our results
provide a cohesive picture of the bulk properties of Bi2Se3, addressing
previously overlooked issues in literature like the possibility of a Bi-rich
$p$-type Bi2Se3 [15] and a topological phase transition proposed to explain a
nonzero Berry phase in the bulk band [34].

##  II Experimental Methods

High-quality single crystals of Bi2Se3 were obtained through the self-flux
method. High-purity elemental Bi (Alfa-Aesar, 99.999%) and Se (Sigma-Aldrich,
99.999%) were loaded in evacuated quartz ampoules, kept at
$750\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for $12$ h and then slowly cooled
from $710\text{\,}\mathrm{\SIUnitSymbolCelsius}$ down to
$630\text{\,}\mathrm{\SIUnitSymbolCelsius}$ at a
$1\text{\,}\mathrm{\SIUnitSymbolCelsius}\mathrm{/}\mathrm{h}$ rate, at which
point the ampoules were centrifuged to separate excess flux from the grown
crystals. Being a congruently melting phase, the compound can be obtained
through a Bi-Se initial mixture with concentrations ranging from 49 % up to 72
% of Se [35], so $10$ different batches were prepared with concentrations
evenly spaced between 52% and 70% of Se. Plate-like single crystals with
dimensions as large as $10\times 8\times 2$ mm3 were systematically obtained,
particularly in Bi-rich growths (lower Se %). Hereafter, samples will be
referred by the nominal Se proportion used in the initial mixture, e.g., a
sample grown from the stoichiometric Bi:Se proportion of 40:60 is denoted as
Se60.

The structural details of the obtained phase in each growth were analyzed
through powder X-ray diffraction (PXRD), using a Stoe STADI-P diffractometer
with Cu-Kα1 radiation. Elemental analysis was performed through energy-
dispersive X-ray spectroscopy (EDS) on freshly cleaved surfaces of the single
crystals, using the EDS module of a Compact JEOL JSM-6010LA Scanning Electron
Microscope (SEM).

Bi2Se3 is reported to show slightly different carrier densities in samples
from the same batch [36], so the $5$-probe method was used to simultaneously
measure longitudinal resistivity ($\rho_{xx}$), magnetoresistance (MR) and
transverse resistivity ($\rho_{xy}$) on the same sample. The measurements were
performed in a $9$ T EverCool-II Physical Property Measurement System (PPMS)
from Quantum Design, on $2$ single-crystals from each batch with the magnetic
field parallel to the $c$-axis. All data obtained under a magnetic field was
symmetrized by removing odd (even) components from the raw MR ($\rho_{xy}$)
signal, and the results at $2.5$ K were checked to match the standard approach
of relating values at positive and negative magnetic fields [37]. In the
analysis of the Shubnikov-de Haas effect, _ad-hoc_ weights were attributed to
the data to ensure proper background subtraction [27].

##  III Experimental Results

###  III.1 Crystal Structure

Bi2Se3 and the related family of materials (Bi2Te3, Sb2Se3, etc) crystallizes
in a rhombohedral primitive cell with centrosymmetric $R\bar{3}m$ space group
[15]. Planes formed by the individual atoms stack on top of each other in the
$z$-direction, forming a so-called quintuple layer (QL) of Se-Bi-Se-Bi-Se
atoms. Two adjacent QLs are bounded by weak van der Waals forces between the
adjacent Se planes, resulting in plane-like single crystals where the
$xy$-planes are easily exposed through exfoliation of the weakly bonded QLs.

Figure 1: (a) Powder X-ray diffraction of the Se64 Bi2Se3 sample, with the
measured data points, the calculated pattern and their difference. The inset
shows the normalized diffraction in a shorter angle range for Se68, Se64,
Se60, Se56 and Se52, where additional peaks related to the (Bi2Se3)x(Bi2)y
family are marked with *. (b) In blue, the crystallographic $c$-axis as a
function of initial Se concentration at growth, relating the decreasing QL
distance with Se %. In red, the effective Se compositions in the grown
crystals, measured with EDS. (c) Density of electrons measured through the
Hall effect at $200$ K as a function of Se concentration at growth. Dotted
line is an exponential trend line. The colour code used in this article is
shown here, with orange for Se-rich samples, blue for Bi-rich and red for
close to stoichiometric.

Rietveld refinement of the powder X-Ray diffraction patterns (Fig. 1a)
confirmed the expected crystalline structure of the obtained phase. For the
stoichiometric and Se rich mixtures, Bi2Se3 is the only detected phase with a
refinement $\chi^{2}$ reaching as low as 2.00. The inset of Fig. 1a shows,
however, extra peaks in Bi-rich samples, confirming the presence of secondary
phases of the (Bi2Se3)x(Bi2)y family of materials, formed by periodically
inserted Bi2 layers between the QLs [38, 39].

Figure 1b shows the $c$ unit cell parameter decreasing with an increasing Se
proportion in the mixture. Since the $c$ parameter is directly dependent on
the interplanar distance between QLs, this monotonic increase of the QL
distance in Bi rich mixtures is a direct consequence of the different defects
intrinsically present in Bi2Se3, as will be discussed in Section IV.

Conversely, the effective Se concentrations as measured by EDS (Fig. 1b),
grows with the nominal Se proportion, as expected. Since the majority of the
pertinent defects (Se vacancies V${}_{\text{Se}}$, Bi substitution on the Se
site Bi${}_{\text{Se}}$ and interstitial Bi [26, 15, 29, 33]) would reduce the
concentration of Se from stoichiometry, the effective values above 60 % are
attributed to the limited accuracy of the EDS experiments in determining
absolute concentration values, only the trends. An alternative explanation,
like the presence of interstitial Se atoms between the QLs [26], seems
unlikely considering the clear signatures of Bi interstitial atoms in our XRD
results.

###  III.2 Charge Carrier Properties

The negative and constant slope observed in the transverse resistivity
$\rho_{xy}$ under an applied magnetic field confirms that all our single-
crystals have $n$-type conductivity dominated by a single band of electrons,
implying that an undoped $p$-type Bi2Se3 grown through the flux method remains
elusive.

The Hall coefficient $R_{H}$ was determined from the linear fit of the low-
field $\rho_{xy}$ and, by considering the relation $R_{H}=r/(-en_{e})$, where
$-e$ is the electron charge and $r$ is the Hall scattering factor (taken as
unity), the electron density $n_{e}$ was obtained. As expected for an
extrinsic semiconductor whose conductance comes mainly from doping defects,
the charge carrier density does not depend significantly on temperature, but
it can be tuned by the elemental proportion of the initial growth mixture.
Figure 1c shows that the electron density ranges from $10^{17}$ cm-3 up to
$10^{19}$ cm-3 in Se and Bi-rich mixtures, respectively, consistent with
previous reports [22, 19, 24, 25].

Figure 2:  (a) Resistivity as a function of temperature for $4$ representative
Bi2Se3 crystals (Se56, Se60, Se64 and Se68). Black lines are fits to Eq. 1.
The inset displays the normalized increase in resistivity below $30$ K for
samples grown with high Se concentration. (b) Mobility at $2.5$ K as a
function of initial Se % used at growth. (c) Mobility as a function of
temperature for Bi2Se3. The carrier density $n_{e}$ of each sample is
displayed, showing the transition from typical semiconductor behaviour in LCC
to a temperature independent metallic behaviour at HCC.

The longitudinal resistivity $\rho$ shows typical metallic behaviour, with a
linear decrease upon cooling from high temperatures until freezing below $30$
K at a residual resistivity of order $10^{-1}$ m$\Omega$cm (Fig. 2a). This
trend is typical of a dominating electron-phonon scattering, that can be
quantified with a Bloch-Grüneisen (BG) model [40] considering an additional
$T^{2}$ dependence that accounts for the Fermi liquid electron-electron
scattering [40, 41, 42]

$\begin{split}\rho=&\rho_{0}+B\quantity(\frac{T}{\theta_{BG}})^{2}+\\\
&+A\quantity(\frac{T}{\theta_{BG}})^{5}\int^{\theta_{BG}/T}_{0}\frac{x^{5}}{(e^{x}-1)(1-e^{-x})}dx,\end{split}$
(1)

where $\rho_{0}$ is the residual resistivity and $\theta_{BG}$ is the BG
temperature, related to the Debye temperature [40]. $A$ and $B$ are fitting
parameters scaling the electron-phonon and electron-electron scattering
mechanisms, respectively. Low carrier concentration samples (LCCs) present two
features that will be further discussed in Section IV but are worth
mentioning: (i) the resistivity below $\sim 30$ K shows an upturn that has
been previously attributed to impurity bands (inset Fig. 2a) [25, 23, 43], but
we will propose a simpler mechanism based on an increase in mobility; (ii) the
electron-phonon coupling (parameter $A$) is greatly enhanced (Fig. 4b) as a
consequence of the reduced defect density.

The Hall mobility $\mu$ of the charge carriers at a given temperature is
defined as $\mu=|R_{H}|/\rho$. Figure 2b shows the calculated values at
$T=2.5$ K, where low electron mobilities of order $10^{2}$ cm2/V s are found
in high carrier concentration samples (HCCs) and high mobilities of order
$10^{4}$ cm2/V s are seen on LCCs. As this is obtained at low temperatures
without active phonons, it is a clear indication of increased scattering from
structural defects in HCCs, supporting the idea that the defect density
increases at lower Se % and that these are responsible for donating charge
carriers as well as serving as scattering centres.

Figure 2c shows the temperature dependence of the mobility for five
representative samples. With the number of data points taken, the mobility
seems to be constant at low temperatures but starts decaying at $T>50$ K, with
the temperature dependence going from $\mu\propto T^{-1.2}$ for LCCs (high Se
%) down to $\mu\propto T^{-0.2}$ on HCCs (low Se %). This is similar to
previously reported results [20, 23, 44], but samples with charge carrier
densities on the order of $10^{16}$ cm-3 show an increase in the mobility at
low temperatures, attributed to ionized impurity scattering [23]. For this
reason, theoretical models for Bi2Se3 have been developed in the past that
consider scattering from ionized impurities dominating at low temperatures and
acoustic phonons at high temperatures [20]. In standard semiconductors, these
would result in $\mu\propto T^{+3/2}$ and $\mu\propto T^{-3/2}$ respectively
[40], but deviations from these values into a temperature independent mobility
are expected in highly degenerate semiconductors and metals [45, 46]. Although
our results are consistent with this picture at high temperatures, with the
exponent approaching the expected $T^{+3/2}$ dependence for LCC crystals, the
constant mobility at low temperatures across all samples may indicate that the
scattering is not solely from charged defects, but also from charge-neutral
interstitial impurities that contribute to temperature independent behaviour
[47, 48, 49, 50].

###  III.3 Shubnikov-de Haas oscillations

After subtracting a background from the raw magnetoresistance (MR),
oscillations due to the Shubnikov-de Haas effect (SdH) are clearly revealed in
the low temperature resistivity of LCCs. SdH oscillations could not be
resolved in HCCs since magnetic fields above $B=9$ T are required, due to the
reduced carrier mobility [51]. The amplitude of the oscillations decays
rapidly with increasing temperature and exponentially with $B^{-1}$, as shown
in the inset of Fig. 3a for a Se68 sample with carrier density of $10^{18}$
cm-3. The frequency is obtained by fitting a peak function to the Fast Fourier
Transform (FFT) of the signal (Fig. 3a), where both the frequency and
amplitude of the oscillations are significantly sample dependent. The
uncertainty is taken as the half width at half maximum. Theoretically, the
frequency of oscillation $F$ is determined by the Onsager relation
$F=(\hbar/2\pi e)A_{F}$, where $\hbar$ is the reduced Planck constant and
$A_{F}$ is the cross-sectional area of the Fermi surface perpendicular to $B$
(in this case, orthogonal to the $k_{z}$ direction of the Brillouin Zone). In
Section IV a model will be constructed for the Fermi surface, enabling the
relation between the frequency $F$ and the electron density $n_{e}$.

Figure 3:  (a) Fast Fourier Transform of the Shubnikov-de Haas oscillations
for Bi2Se3 crystals obtained with different Se% at growth at $2.5$ K. The
inset shows the periodic oscillations in resistance as a function of inverse
magnetic field at different temperatures, for a Se68 sample. (b) Effective
masses as a function of Fermi surface area perpendicular to $B$, calculated
through the temperature dependence of the oscillations. The inset shows the
fit of the peak to trough amplitude as a function of temperature for a Se68
sample. (c) Dingle temperature determined from the SdH effect as a function of
the inverse Hall mobility. The solid line represents the expected trend,
showing a clear offset. The inset shows the Dingle plot for Se64 and Se66
samples.

The overall shape of the SdH oscillation in the resistance $R$ is given by
[52]

$\frac{\Delta
R(B)}{R(0)}=\quantity(\frac{B}{F})^{1/2}R_{T}R_{D}R_{S}\text{cos}\quantity[2\pi\quantity(\frac{F}{B}-\phi)],$
(2)

where $R_{T}$, $R_{D}$ and $R_{S}$ are respectively the temperature, Dingle
and spin damping factors. The phase of oscillation $\phi$ is given by
$\phi=1/2-\beta/2\pi-\delta$, where $\delta=-1/8,+1/8,0$ for 3D electrons, 3D
holes or 2D carriers, respectively. The Berry phase $\beta$ is $\pi$ when the
Fermi surface encloses a Dirac crossing in a topological material, but is null
for trivial bands [53, 54].

The temperature dependence of the SdH oscillations is determined by the
temperature damping factor $R_{T}$, defined as
$R_{T}=\frac{\xi(B,T)}{\text{sinh}\quantity[\xi(B,T)]}$, where
$\xi(B,T)=2\pi^{2}k_{B}Tm^{*}/(\hbar eB)$, $k_{B}$ is the Boltzmann constant
and $m^{*}$ is the charge carrier’s effective mass. By fitting $R_{T}$ to the
amplitude peaks for different temperatures (inset Fig. 3b), the effective mass
$m^{*}$ of all samples average to $m^{*}=0.124(5)m_{0}$, with $m_{0}$ as the
free electron’s rest mass.

The Dingle factor $R_{D}$ describes the damping of the oscillations from the
scattering of the charge carriers. It is defined as
$R_{D}=\text{exp}\quantity[-\xi(B,T_{D})]$, where the Dingle temperature
$T_{D}$ is inversely proportional to the finite carrier lifetime $\tau$ as
$T_{D}=\hbar/(2\pi k_{B}\tau)$. Considering the lower mobilities in HCC Bi2Se3
samples, lower $\tau$ explains why these samples have SdH oscillations that
are harder to resolve. Considering the oscillations at a fixed temperature,
the maxima and minima of Eq. 2 can be linearized (inset Fig. 3c) to find the
Dingle temperatures, which result between $8$ K and $25$ K, consistent with
previous reports between $7$ K and $30$ K [55, 22, 56].

The scattering time $\tau$ is defined as $\tau=m^{*}\mu/e$, so $T_{D}$ is
predicted to be proportional to $\mu^{-1}$ (solid line in Fig. 3c). The
experimental data points do seem to follow this trend, but the Dingle
temperatures are much larger than the predicted values. For these $T_{D}$
values, the mobility of the charge carriers in the Landau Levels (LLs) is up
to six times smaller than those measured through the Hall effect. This reduced
mobility has previously been reported for Bi2Se3 [56] as well as for many
semiconductors [57] and can be attributed to the difference in scattering
times governing transport phenomena (e.g. Hall effect) and phase decoherence
in quantum oscillations such as SdH [56, 58]. When small angle scattering is
dominant, as is generally the case at low temperatures [56], the loss of
coherence happens faster. Furthermore, this difference has been theoretically
predicted to increase exponentially in 2D materials with an additional
scattering layer separated from the conducting layers [58]. In Bi2Se3 this
scenario points to the significance of charge scattering from the interstitial
impurities found between the conducting QLs and will be further discussed in
Sec. IV.

We now address the phase of the SdH oscillations by constructing the Landau
Fan Diagram (LFD), where LL indices $n$ are assigned to each maxima of the SdH
oscillations and plotted as a function of the inverse field values in which
they occur. The choice of using the maxima [56, 59, 60, 61, 62] instead of the
minima [34, 63, 64, 65] of $\rho$ to construct the LFD was made based on
recent theoretical developments, where both 2D [54] and 3D [66] structures
have been shown to present a maximum in $\rho$ (or a sharp peak in the Quantum
Hall Effect [6]) when LLs crosses the Fermi level.

Figure 4a shows the LFDs for the samples that presented clear SdH
oscillations, where the intercept of the solid lines with the ordinary axis is
the phase $-\phi$. It is important to consider the Zeeman splitting through
the spin damping factor $R_{S}$, defined as
$R_{S}=\text{cos}\quantity(\frac{\pi}{2}g\mu^{*})$ where $g$ is the electron
spin $g$-factor, since $R_{S}<0$ would change $\phi$ by $\frac{\pi}{2}$ and
render the determination of the topological Berry phase useless. Fortunately,
several works have shown that for the main bulk band of Bi2Se3, the Zeeman
splitting energy $g\mu_{B}B$, where $\mu_{B}=e\hbar/2m_{0}$ is the Bohr
magneton, is precisely double the LL splitting energy $\hbar eB/m^{*}$ [67,
68, 69, 55]. This fixes $g=4m_{0}/m^{*}$ and results in $R_{S}=1$.

Considering that the main band is composed of 3D electrons, we can take
$\delta=-1/8$ and obtain the Berry phase (inset Fig. 4a). The results agree
with the majority of reports in literature [70, 71], where $\beta$ approaches
$0$ and the main conductive band is topologically trivial in all samples. A
recent contradicting claim of a topological semimetal band, made after an
obtained Berry phase of $\beta=\pi$ [34], can be questioned since it was
determined by assigning the minima of the oscillations to the LL crossing in
the LFD, which changes the calculated Berry phase by $\pi$.

##  IV Discussion

Figure 4: (a) Landau fan diagram for the different Bi2Se3 samples. The inset
shows the calculated Berry phase as a function of the Fermi level, where it is
shown to be close to $0$ for all samples. (b) The $A$ parameter of Eq. 1 as a
function of electron density, showing the decreasing electron-phonon
interaction for higher electron densities. The inset displays the magnitude of
the resistivity upturn, increasing for lower density samples. (c) Electron
density measured through the Hall effect as a function of the Fermi level
obtained through the SdH. The solid line shows the prediction of Eq. 3,
evidencing that most electrons come from the main conduction band.

The increase in the carrier density from $10^{17}$ cm-3 to $10^{19}$ cm-3
(Fig. 1c) for lower Se (higher Bi) concentrations is consistent with the
picture of charged Se vacancies (V${}_{\text{Se}}$) being the main source of
conducting electrons in Bi2Se3 [19, 20, 21, 26, 29]. The reduced mobility
(Fig. 2b) can also be attributed to enhanced scattering from a larger density
of structural defects in Bi rich growths, but the nature of these defects
cannot be solely attributed to V${}_{\text{Se}}$. The extra peaks in the PXRD
(inset Fig. 1a) support the presence of interstitial Bi layers, and the
increase in the interlayer distance (left axis in Fig. 1b) is also predicted
from other defects such as Bi substitution on the Se site Bi${}_{\text{Se}}$
[15, 26, 33] and Se on the Bi site Se${}_{\text{Bi}}$ [15].

The presence of multiple types of defects is further supported by the
scattering processes of the charge carriers, manifested in the temperature
dependence of transport experiments. A temperature independent Hall mobility
(Fig. 2c) is expected at low temperatures when charge carriers are scattered
on electrically neutral defects [47, 48, 49, 50] and, since this is found even
in LCCs samples (rich in Se), it cannot not be solely attributed to the
degeneracy of the system. Furthermore, the significantly reduced lifetime of
the electrons generating the SdH oscillations, evidenced by the larger than
expected Dingle temperatures $T_{D}$ (Fig. 3c) is an indication that defects
outside the conducting layer are responsible for scattering [58]. These
results point to the significance of interstitial atoms in transport phenomena
for both Bi and Se-rich Bi2Se3 samples, signalling the presence of not only
the interstitial Bi layers found in the PXRD, but also of interstitial Se
atoms in Se-rich samples, as previously reported in flux-grown single crystals
[26].

The absence of a $p$-type single crystal even in very Bi-rich initial mixtures
shows that the Bi${}_{\text{Se}}$ substitution either appears in very low
concentration (unable to compensate for the high density of V${}_{\text{Se}}$
defects) or it is not an acceptor-like defect as was predicted by Zhang [15].
Note that Bi${}_{\text{Se}}$ as a donor-like substitution has been suggested
in other theoretical works [32, 30].

The structural defects also play an important role on the propagation of
phonons, as found in the temperature dependence of the resistivity $\rho$
through the parameter $A$ of the BG equation (Eq. 1). Being a measure of the
electron-phonon (e-ph) scattering, the decrease of $A$ in HCCs can be
attributed to the larger amount of structural defects effectively reducing the
distance that lattice distortions can propagate coherently. Considering that
the e-ph coupling increases with this distance [72], this shows that the e-ph
coupling can be controlled through the density of structural defects, or
experimentally through the growth procedure.

We now address the upturn in resistivity seen in Fig. 2a for LCC Bi2Se3. This
effect has previously been attributed to an impurity band with thermally
activated charge carriers, where its conductivity only becomes apparent when
electrons from the main bulk band "freeze" at low temperatures [25, 23, 43].
In-gap impurity states have in fact been reported in thin films [73, 74], and
in carefully doped samples with very low carrier concentration (1016 cm-3)
[28], but the linear Hall resistivity $\rho_{xy}$ found means that the
mobility of these impurity carriers has to be considerably lower than the ones
from the main band. Additionally, since the electron density seems temperature
independent and the exponentially decaying $\rho$, expected for thermally
activated carriers [75], is absent, we propose an alternate simpler
explanation based on the effects of ionized impurity scattering.

Following the discussion in Sec. III.2, the scattering at low temperatures is
likely a combination of neutral and ionized impurities, and these effects are
expected to become more apparent in LCCs. A non-degenerate semiconductor
dominated by ionized defect scattering presents a mobility that increases with
temperature when there are no active phonons, and with a constant charge
carrier density, the resistivity would decrease with temperature (Fig. 2a).
Combined with the fact that the absolute upturn in $\rho$ decreases somewhat
exponentially with charge density $n_{e}$ (inset Fig. 4b), Bi2Se3 can be
regarded as an extrinsic semiconductor where the effects of charged defect
scattering become apparent first through the resistivity upturn in low carrier
density samples.

The Fermi level $E_{F}$ of the samples can be calculated directly from the
cross sectional area of the Fermi surface $A_{F}$ obtained in the SdH
oscillations. Considering the constant effective mass and the topologically
trivial nature of the main conduction band found through the SdH oscillations,
a parabolic dispersion can be considered in the $k_{z}=0$ plane, resulting in
$E_{F}=\hbar^{2}A_{F}/2\pi m^{*}$. The Fermi level of our samples (Fig. 4c)
lies between $25$ meV and $200$ meV from the bottom of the conduction band.
From previous angle-dependent SdH oscillation studies [76, 22, 56], it is
known that the Fermi surface of Bi2Se3 has the shape of an ellipsoid elongated
in the $k_{z}$ direction of the Brillouin zone. Despite the parabolic
dispersion in the $k_{x}-k_{y}$ plane, there is a Fermi level-dependent
flattening of the FS in the $k_{z}$ direction at higher $E_{F}$ [22] that can
be accounted for by an anisotropy factor $\eta=A_{zF}/A_{F}$. Following
Mukhopadhyay et al. [55], $\eta$ can be fitted to $E_{F}$ through a power law
as $\eta=1+a_{\eta}E_{F}^{b_{\eta}}$, where $a_{\eta}=1.32(9)$ and
$b_{\eta}=0.27(3)$ is obtained using previous reports [55, 22, 51, 56]. From
the density of states of a spin-degenerate 3D Fermi gas [77], the effective
mass in the $k_{z}$ direction ($m_{z}^{*}$) must be evaluated before the
electron density can be obtained. As a first approximation, a phenomenological
ellipsoidal model can be considered as $m_{z}^{*}=m^{*}\eta^{2}$ [55, 22],
resulting in the electron density for Bi2Se3 as

$n_{e}=\frac{1}{3\pi^{2}}\quantity(\frac{2m^{*}E_{F}}{\hbar^{2}})^{3/2}\quantity(1+\frac{3a_{\eta}E_{F}^{b_{\eta}}}{2b_{\eta}+3}),$
(3)

represented by the solid curve in Fig. 4c, which shows fair agreement with the
experimental data obtained from Hall effect.

Note that by taking this phenomenological model into consideration,
discrepancies previously seen between the experimental carrier concentration
and the one predicted from the main bulk band have been largely reduced [22,
56]. Yet, the model still seems to predict slightly lower electron densities
than what is found through the Hall effect, and although this is a simplified
model, the presence of very low mobility carriers not originated from the main
conduction band cannot be discarded. Indeed, this could be attributed to
temperature-independent electrons from extrinsic impurities, inherent to all
Bi2Se3 samples grown here, or a secondary conduction band of low mobility
electrons, as has been shown to be populated above $40$ meV for Bridgman-grown
samples doped with Sb [51].

##  V Conclusion

The bulk conducting states of the topological insulator Bi2Se3 were studied by
means of transport experiments and quantum oscillations. High-quality single
crystals were obtained in the entire concentration range available in the
phase diagram for flux growths, allowing the construction of a concise picture
of all undoped Bi2Se3 that can be grown through the flux method.

Despite theoretical predictions [15], electron donor-type defects, likely
V${}_{\text{Se}}$, dominate all samples and shift the Fermi level into the
main conduction band. Other types of impurities, however, have a significant
role on the scattering of the charge carriers, both directly, as can be seen
in the low temperature mobility and scattering time, and indirectly through
the reduction of the electron-phonon coupling at higher temperatures.

Careful analysis of the Shubnikov-de Haas effect confirmed the bulk electrons
as topologically trivial, clarifying earlier discrepancies [34]. A
phenomenological model allowed the determination of the origin of the vast
majority of these electrons from the main conduction band, but the presence of
additional low mobility bands cannot be fully disregarded. Further theoretical
investigations considering the complex scattering mechanisms are required.

As a thorough overview of transport properties of bulk Bi2Se3, this work helps
to shed light on the complex interplay of the inherent impurities and provides
a reference for experimental fine-tuning the bulk conducting states of this
flagship topological insulator.

## Acknowledgements

We are grateful for J. Munevar, L. Mendonça-Ferreira, A.R.V. Benvenho, S. K.
Singh, G. Vasquez and C. A. Sonego for fruitful discussions. We would also
like to thank K. K. F. Barbosa and F. F. Ferreira for the PXRD measurements,
and the UFABC multiuser central facilities for the experimental support. This
work was funded by Brazilian agencies FAPESP (grant number 2017/10581-1),
CAPES and CNPq.

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