# Arithmetic progressions among powerful numbers

Tsz Ho Chan

###### Abstract

In this paper, we study $k$-term arithmetic progressions $N,N+d,...,N+(k-1)d$
of powerful numbers. Under the $abc$-conjecture, we obtain
$d\gg_{\epsilon}N^{1/2-\epsilon}$. On the other hand, there exist infinitely
many $3$-term arithmetic progressions of powerful numbers with $d\ll N^{1/2}$
unconditionally. We also prove some partial results when $k\geq 4$ and pose
some open questions.

For any integer $k\geq 1$, a non-trivial $k$-term arithmetic progression
(abbreviated as $k$-AP) is a sequence of the form

$N,\;\;N+d,\;\;N+2d,\;\;...,\;\;N+(k-1)d$

with initial term $N$ and common difference $d>0$. Clearly, any one or two
members in a general sequence form an arithmetic progression. So, we will
assume $k\geq 3$ from now on. It is well-known that there are infinitely many
$3$-term arithmetic progressions among perfect squares (e.g. $1,25,49$) but
there is no $4$-term arithmetic progressions of perfect squares (discovered by
Fermat). One may ask about the existence of $k$-AP among other interesting
arithmetic and polynomial sequences. A recent breakthrough result of this sort
is that there are arbitrarily long arithmetic progressions among the prime
numbers by Green and Tao [4]. In this paper, we are interested in studying
arithmetic progressions of powerful numbers which are square-like.

###### Definition 1

A number $n$ is powerful if $p^{2}|n$ whenever $p|n$ (i.e. its prime
factorization $n=p_{1}^{a_{1}}p_{2}^{a_{2}}\cdots p_{r}^{a_{r}}$ satisfies
$a_{i}\geq 2$ for all $1\leq i\leq r$.)

For example, $72=2^{3}\cdot 3^{2}$ is powerful but $24=2^{3}\cdot 3$ is not.
Another common name for powerful number is squarefull number. A closely
related concept is squarefree number.

###### Definition 2

A number $n$ is squarefree if $p^{2}\nmid n$ for all prime $p|n$ (i.e. its
prime factorization $n=p_{1}^{a_{1}}p_{2}^{a_{2}}\cdots p_{r}^{a_{r}}$
satisfies $a_{i}=1$ for all $1\leq i\leq r$.).

For example, $30=2\cdot 3\cdot 5$ is squarefree but $24=2^{3}\cdot 3$ is not.
By unique prime factorization, one can show that any positive integer can be
factored uniquely as $n=a^{2}b$ and any powerful number can be written
uniquely as $n=a^{2}b^{3}$ for some integer $a\geq 1$ and squarefree number
$b\geq 1$. Unlike perfect squares, there are arbitrarily long arithmetic
progressions among powerful numbers.

###### Theorem 1

For any integer $k\geq 3$, there is a $k$-term arithmetic progression of
powerful numbers.

For $k=3$, there is a folklore conjecture concerning $3$-AP of powerful
numbers which seems to be first posed by Erdős [2].

###### Conjecture 1

There is no three consecutive powerful numbers. i.e. $N,N+1,N+2$ cannot all be
powerful.

Later, Mollin and Walsh [5] and Granville [3] reiterated the same conjecture
and provided evidence and some interesting consequences. Currently, we are far
from being able to prove it. However, the above conjecture follows from the
famous $abc$-conjecture.

######  Conjecture 2 ($abc$-conjecture)

For any $\epsilon>0$, there exists a constant $C_{\epsilon}>0$ such that, for
any integers $a,b,c$ with $a+b=c$ and $\text{gcd}(a,b)=1$, the bound

$\max\\{|a|,|b|,|c|\\}\leq C_{\epsilon}\kappa(abc)^{1+\epsilon}$

holds where

$\kappa(m):=\prod_{p|m}p$

stands for the squarefree kernel or radical of $m$.

In other words, there is no $3$-AP of powerful numbers with common difference
$d=1$ under the $abc$-conjecture. Recently, the author studied powerful
numbers in short intervals, and it can be deduced from [1] that, for any
$\epsilon>0$, there is no $3$-AP of powerful numbers with $d\leq
N^{1/4-\epsilon}$ for sufficiently large $N$ under the $abc$-conjecture. On
the other hand, one can easily check that, for integers $m\geq 1$,

$(2m^{2}-1)^{2},\;\;(2m^{2}+2m+1)^{2},\;\;(2m^{2}+4m+1)^{2}$ (1)

form a $3$-AP of perfect squares with common difference $d=8m^{3}+12m^{2}+4m$.
Hence, there are infinitely many $3$-AP of powerful numbers with $d\leq
6N^{3/4}$. Thus, we are led to the following natural question.

###### Question 1

We say that $0<\theta<1$ is an admissible exponent if there exists
$C_{\theta}>0$ such that there are infinitely many $3$-AP of powerful numbers
$N,N+d,N+2d$ with common difference $d\leq C_{\theta}N^{\theta}$. Find the
infimum of all such admissible exponents and call it $\theta_{3}$.

The above discussion yields $\frac{1}{4}\leq\theta_{3}\leq\frac{3}{4}$. We
shall prove the following optimal result.

###### Theorem 2

Assuming the $abc$-conjecture, we have $\theta_{3}=\frac{1}{2}$.

Analogously, one can define $\theta_{k}$ for $k$-AP of powerful numbers when
$k\geq 4$. We have the following partial results.

###### Theorem 3

Assuming the $abc$-conjecture, we have

$\frac{1}{2}\leq\theta_{4}\leq\frac{4}{5},\;\;\frac{1}{2}\leq\theta_{5}\leq\frac{9}{10},\;\;\text{
and }\;\;\frac{1}{2}\leq\theta_{k}\leq 1-\frac{1}{10\cdot 3^{k-5}}$

for $k\geq 5$.

Note that the upper bounds in Theorems 2 and 3 hold unconditionally and it is
their lower bounds that are conditional on the $abc$-conjecture.

It would be interesting to see if one can prove $\theta_{4}>1/2$ under the
$abc$-conjecture. Another future direction would be narrowing the above ranges
for $\theta_{k}$ when $k\geq 4$. One can also ask if it is possible to
construct infinitely many $3$-AP of powerful numbers with common difference
$d=o(\sqrt{N})$.

Notation. Throughout the paper, $N$, $k$, $m$, $n$ and $d$ stand for positive
integers while $p$, $p_{ij}$ and $q_{ij^{\prime}}$ stand for prime numbers.
The symbol $a|b$ means that $a$ divides $b$, the symbol $a\nmid b$ means that
$a$ does not divide $b$, and the symbol $p^{n}||a$ means that $p^{n}|a$ but
$p^{n+1}\nmid a$. The function $\nu_{p}(a)$ stands for the $p$-adic valuation
of $a$ (i.e. $\nu_{p}(a)=n$ where $p^{n}||a$). The symbols $f(x)\ll g(x)$ and
$g(x)\gg f(x)$ are equivalent to $|f(x)|\leq Cg(x)$ for some constant $C>0$.
$f(x)\ll_{\lambda}g(x)$ and $g(x)\gg_{\lambda}f(x)$ mean that the implicit
constant may depend on $\lambda$. Finally, $f(x)=o(g(x))$ means that
$\lim_{x\rightarrow\infty}f(x)/g(x)=0$.

##  1 Proof of Theorem 1: Long AP among powerful numbers

We apply induction on $k$. The base case $k=3$ follows from (1) on $3$-AP
among perfect squares. Suppose, for some $k\geq 3$, there is a $k$-AP among
powerful numbers, say

$a_{1}^{2}b_{1}^{3}<a_{2}^{2}b_{2}^{3}<\cdots<a_{k}^{2}b_{k}^{3}\;\;\text{
with common difference }\;d\geq 1.$

Consider the number $a_{k}^{2}b_{k}^{3}+d=a^{2}b$ for some integer $a$ and
squarefree number $b$. Then

$a_{1}^{2}b_{1}^{3}b^{2}<a_{2}^{2}b_{2}^{3}b^{2}<\cdots<a_{k}^{2}b_{k}^{3}b^{2}<a^{2}b^{3}$

is a $(k+1)$-AP of powerful numbers with common difference $db^{2}$. This
finishes the induction proof.

##  2 Proof of Theorem 2: $3$-AP upper bound

For the upper bound $\theta_{3}\leq 1/2$, we first consider the following
$3$-AP:

$x^{2}-2x-1,\;\;x^{2},\;\;x^{2}+2x+1.$

The last two terms are perfect squares. We want the first term to contain a
large square factor. Consider the Pell equation

$X^{2}-2Y^{2}=-1$

which has infinitely many integer solutions given by

$X_{m}+\sqrt{2}Y_{m}=(1+\sqrt{2})^{2m+1}\;\;\text{ with integer }m\geq 1.$

By setting $n=X$ and $\frac{x-1}{2}=Y$, the generalized Pell equation

$(x-1)^{2}-2n^{2}=2$ (2)

has infinitely many integer solutions given by

$x-1=2Y_{m}\;\;\text{ and }n=X_{m}.$

Then, (2) gives us infinitely many integers $x$ such that $x^{2}-2x-1=2n^{2}$
for some integer $n$. Therefore, we have infinitely many $3$-AP of powerful
numbers, namely

$N=2^{2}(x^{2}-2x-1)=2^{3}n^{2},\;\;N+d=2^{2}x^{2}=(2x)^{2},\;\;N+2d=2^{2}(x^{2}+2x+1)=(2(x+1))^{2}$

with common difference

$d=2^{2}(2x+1)=8x+4\leq 3\sqrt{N}.$

Hence, $\theta_{3}\leq 1/2$.

##  3 Proof of Theorem 2: $3$-AP lower bound

First, we need a simple observation.

###### Lemma 1

Suppose $a$ and $b$ are positive integers and $p^{\delta}|a^{2}b^{3}$ for some
prime $p$ and integer $\delta\geq 1$. Then $\nu_{p}(ab)\geq\delta/3$.

Proof: From the definitions of divisibility and valuation, we have $\delta\leq
2\nu_{p}(a)+3\nu_{p}(b)$. Dividing everything by $3$, we have $\delta/3\leq
2\nu_{p}(a)/3+\nu_{p}(b)\leq\nu_{p}(a)+\nu_{p}(b)=\nu_{p}(ab)$.

Consider $3$-AP of powerful numbers $N,N+d,N+2d$ with

$N=a_{1}^{2}b_{1}^{3},\;\;N+d=a_{2}^{2}b_{2}^{3},\;\text{ and
}\;N+2d=a_{3}^{2}b_{3}^{3}$

for some integers $a_{1},a_{2},a_{3}$ and squarefree numbers
$b_{1},b_{2},b_{3}$. If there is some prime $p$ dividing $b_{1}$, $b_{2}$ and
$b_{3}$, then we can consider the reduced $3$-AP of powerful numbers

$\frac{N}{p^{3}},\;\;\frac{N}{p^{3}}+\frac{d}{p^{3}},\;\;\frac{N}{p^{3}}+\frac{2d}{p^{3}}.$

If one could prove a lower bound $d/p^{3}\geq C_{\theta}(N/p^{3})^{\theta}$
with some $0<\theta<1$ and $C_{\theta}>0$ for the reduced $3$-AP, one would
also have $d\geq C_{\theta}N^{\theta}$ for the original $3$-AP. Hence, we may
reduce the situation to $\text{gcd}(b_{1},b_{2},b_{3})=1$.

Since $(N+d)^{2}=N(N+2d)+d^{2}$, we have
$a_{2}^{4}b_{2}^{6}=a_{1}^{2}b_{1}^{3}a_{3}^{2}b_{3}^{3}+d^{2}$. Let
$D^{2}=\text{gcd}(a_{2}^{4}b_{2}^{6},d^{2})$ which also equals
$\text{gcd}(a_{2}^{4}b_{2}^{6},a_{1}^{2}b_{1}^{3}a_{3}^{2}b_{3}^{3})$ and
$\text{gcd}(a_{1}^{2}b_{1}^{3}a_{3}^{2}b_{3}^{3},d^{2})$. Note that since
$D|a_{2}^{2}b_{2}^{3}$ and $D|d$, we also have $D|a_{1}^{2}b_{1}^{3}$ and
$D|a_{3}^{2}b_{3}^{3}$. Dividing everything by $D^{2}$, we have the equation

$\Bigl{(}\frac{a_{2}^{2}b_{2}^{3}}{D}\Bigr{)}^{2}=\Bigl{(}\frac{a_{1}^{2}b_{1}^{3}}{D}\frac{a_{3}^{2}b_{3}^{3}}{D}\Bigr{)}+\Bigl{(}\frac{d}{D}\Bigr{)}^{2}$

where the three terms are pairwise relatively prime. By the $abc$-conjecture,

$\frac{N^{2}}{D^{2}}\leq\Bigl{(}\frac{a_{2}^{2}b_{2}^{3}}{D}\Bigr{)}^{2}\leq
C_{\epsilon}\Bigl{(}\kappa\Bigl{(}\frac{a_{1}^{2}b_{1}^{3}}{D}\frac{a_{2}^{2}b_{2}^{3}}{D}\frac{a_{3}^{2}b_{3}^{3}}{D}\Big{)}\kappa\Bigl{(}\frac{d}{D}\Bigr{)}\Bigr{)}^{1+\epsilon}$
(3)

as $\kappa(mn)\leq\kappa(m)\kappa(n)$. If one simply bounds the right-hand
side of (3) by $\leq
C_{\epsilon}(a_{1}b_{1}a_{2}b_{2}a_{3}b_{3}d/D)^{1+\epsilon}\ll
C_{\epsilon}(N^{3/2}d/D)^{1+\epsilon}$, solves for $d$ and applies $D\leq d$
as in [1, Theorem1.6], one would get $d\gg_{\epsilon}N^{1/4-\epsilon}$ and
hence the lower bound $\theta_{3}\geq 1/4$ only. So, in order to prove Theorem
2, we need a finer analysis. We claim that

$\kappa\Bigl{(}\frac{a_{1}^{2}b_{1}^{3}}{D}\frac{a_{2}^{2}b_{2}^{3}}{D}\frac{a_{3}^{2}b_{3}^{3}}{D}\Big{)}\leq\frac{a_{1}b_{1}a_{2}b_{2}a_{3}b_{3}}{D}$
(4)

which follows from

$\nu_{p}\Bigl{(}\frac{a_{1}^{2}b_{1}^{3}}{D}\frac{a_{2}^{2}b_{2}^{3}}{D}\frac{a_{3}^{2}b_{3}^{3}}{D}\Big{)}\leq\nu_{p}(a_{1}b_{1}a_{2}b_{2}a_{3}b_{3})-\nu_{p}(D)$
(5)

for any prime $p$. Firstly, if a prime $p$ does not divide
$a_{1}b_{1}a_{2}b_{2}a_{3}b_{3}$, then (5) is true as both sides are $0$.
Secondly, if a prime $p|a_{1}b_{1}a_{2}b_{2}a_{3}b_{3}$ but $p\nmid D$, then
left hand side of (5) is exactly $1$ while the right-hand side of (4) is $\geq
1-0$. So, (5) is true for such primes. Thus, it remains to consider those
primes $p$ which divide both $a_{1}b_{1}a_{2}b_{2}a_{3}b_{3}$ and $D$. Notice
that the left-hand side of (5) is at most $1$ for such primes. Suppose we have
the following prime factorizations:

$\displaystyle b_{1}=p_{11}\cdots
p_{1r_{1}},\;\;a_{1}=p_{11}^{\alpha_{11}}\cdots
p_{1r_{1}}^{\alpha_{1r_{1}}}\cdot q_{11}^{\beta_{11}}\cdots
q_{1s_{1}}^{\beta_{1s_{1}}}$ $\displaystyle b_{2}=p_{21}\cdots
p_{2r_{2}},\;\;a_{2}=p_{21}^{\alpha_{21}}\cdots
p_{2r_{2}}^{\alpha_{2r_{2}}}\cdot q_{21}^{\beta_{21}}\cdots
q_{2s_{2}}^{\beta_{2s_{2}}}$ $\displaystyle b_{3}=p_{31}\cdots
p_{3r_{3}},\;\;a_{3}=p_{31}^{\alpha_{31}}\cdots
p_{3r_{3}}^{\alpha_{3r_{3}}}\cdot q_{31}^{\beta_{31}}\cdots
q_{3s_{3}}^{\beta_{3s_{3}}}$

for some integers $r_{1},r_{2},r_{3},s_{1},s_{2},s_{3}\geq 0$,
$\alpha_{ij}\geq 0$, $\beta_{ij^{\prime}}\geq 1$ and primes
$p_{ij},q_{ij^{\prime}}$ with $q_{ij^{\prime}}\neq p_{ij}$. Now consider a
fixed prime $p|D$ with $\delta:=\nu_{p}(D)$. Note that $p$ does not divide all
of the $b_{1},b_{2},b_{3}$ as $\text{gcd}(b_{1},b_{2},b_{3})=1$.

Case 1: $p$ does not divide any of the $b_{1},b_{2},b_{3}$. Since
$D|a_{i}^{2}b_{i}^{3}$ for $i=1,2,3$, we must have $p|a_{1},a_{2},a_{3}$ and
$p=q_{1j_{1}}=q_{2j_{2}}=q_{3j_{3}}$ for some $1\leq j_{m}\leq s_{m}$ for
$m=1,2,3$. As
$p^{2\delta}||D^{2}=\text{gcd}(a_{2}^{4}b_{2}^{6},a_{1}^{2}b_{1}^{3}a_{3}^{2}b_{3}^{3})$
and $\text{gcd}(p,b_{1}b_{2}b_{3})=1$, we must have
$2\delta=\min(4\beta_{2,j_{2}},2(\beta_{1,j_{1}}+\beta_{3j_{3}}))\geq 4$.
Thus, $2\leq\delta=\min(2\beta_{2j_{2}},\beta_{1j_{1}}+\beta_{3j_{3}})$.
Hence,

$\nu_{p}(a_{1}b_{1}a_{2}b_{2}a_{3}b_{3})-\nu_{p}(D)\geq\beta_{1j_{1}}+\beta_{2j_{2}}+\beta_{3j_{3}}-(\beta_{1j_{1}}+\beta_{3j_{3}})=\beta_{2j_{2}}\geq
1.$

Case 2: $p$ divides exactly one of the $b_{1},b_{2},b_{3}$.

Subcase 1: $\delta$ is even. Without loss of generality, suppose $p|b_{1}$,
$p\nmid b_{2}$, $p\nmid b_{3}$ as the other cases are similar. Then
$p|a_{2},a_{3}$ and $p=q_{2j_{2}}=q_{3j_{3}}$ for some $1\leq j_{2}\leq s_{3}$
and $1\leq j_{3}\leq s_{3}$. As
$p^{\delta}|a_{2}^{2}b_{2}^{3},a_{3}^{2}b_{3}^{3}$, we have
$\delta/2\leq\beta_{2j_{2}}$ and $\delta/2\leq\beta_{3j_{3}}$. Hence, by Lemma
1,

$\nu_{p}(a_{1}b_{1}a_{2}b_{2}a_{3}b_{3})-\nu_{p}(D)\geq\max\Bigl{(}\frac{\delta}{3},1\Bigr{)}+\beta_{2j_{2}}+\beta_{3j_{3}}-\delta\geq\left\\{\begin{array}[]{ll}1+\delta/2+\delta/2-\delta\geq
1,&\text{ when }\delta=2;\\\ \delta/3+\delta/2+\delta/2-\delta>1,&\text{ when
}\delta\geq 4.\end{array}\right.$

Subcase 2: $\delta$ is odd. If $p|b_{1}$, $p\nmid b_{2}$, $p\nmid b_{3}$, then
$p|b_{1},a_{2},a_{3}$. Hence, $\nu_{p}(a_{2}^{4}b_{2}^{6})\geq 4$ and
$\nu_{p}(a_{1}^{2}b_{1}^{3}a_{3}^{2}b_{3}^{3})\geq 5$ which gives $\delta\geq
3$ as
$p^{2\delta}||D^{2}=\text{gcd}(a_{2}^{4}b_{2}^{6},a_{1}^{2}b_{1}^{3}a_{3}^{2}b_{3}^{3})$
and $\delta$ is odd. If $p|b_{2}$, $p\nmid p_{1}$, $p\nmid p_{3}$, then
$p|b_{2},a_{1},a_{3}$. Hence, $\nu_{p}(a_{2}^{4}b_{2}^{6})\geq 6$ and
$\nu_{p}(a_{1}^{2}b_{1}^{3}a_{3}^{2}b_{3}^{3})\geq 4$ which also gives
$\delta\geq 3$ by similar reasoning. If $p|b_{3}$, $p\nmid b_{1}$, $p\nmid
b_{2}$, we also have $\delta\geq 3$ as it is similar to $p|b_{1}$, $p\nmid
b_{2}$, $p\nmid b_{3}$. Therefore, $\delta\geq 3$ in all circumstances.

Suppose $p\nmid b_{i},b_{i^{\prime}}$ for some $1\leq i<i^{\prime}\leq 3$.
Then $p|a_{i},a_{i^{\prime}}$ and $p=q_{ij_{i}}=q_{i^{\prime}j_{i^{\prime}}}$
for some $1\leq j_{i}\leq s_{i}$ and $1\leq j_{i^{\prime}}\leq
s_{i^{\prime}}$. Thus, $3\leq\delta\leq 2\beta_{ij_{i}}-1$ and
$3\leq\delta\leq 2\beta_{i^{\prime}j_{i^{\prime}}}-1$ as $\delta$ is odd. This
implies $\beta_{ij_{i}},\beta_{i^{\prime}j_{i^{\prime}}}\geq\delta/2+1/2$.
Hence,

$\nu_{p}(a_{1}b_{1}a_{2}b_{2}a_{3}b_{3})-\nu_{p}(D)\geq
1+\beta_{ij_{i}}+\beta_{i^{\prime}j_{i^{\prime}}}-\delta>1.$

Case 3: $p$ divides exactly two of the $b_{1},b_{2},b_{3}$.

Subcase 1: $\delta$ is even. Without loss of generality, suppose $p|b_{1}$,
$p|b_{2}$, $p\nmid b_{3}$ as the other cases are similar. Then $p|a_{3}$ and
$p=p_{1j_{1}}=p_{2j_{2}}=q_{3j_{3}}$ for some $1\leq j_{1}\leq r_{1}$, $1\leq
j_{2}\leq r_{2}$ and $1\leq j_{3}\leq s_{3}$. As
$p^{\delta}|a_{3}^{2}b_{3}^{3}$, we have $\delta/2\leq\beta_{3j_{3}}$. Also,
as $p^{\delta}|a_{1}^{2}b_{1}^{3},a_{2}^{2}b_{2}^{3}$, we have $\delta\leq
2\alpha_{1j_{1}}+2$ and $\delta\leq 2\alpha_{2j_{2}}+2$ since $\delta$ is
even. Hence,

$\nu_{p}(a_{1}b_{1}a_{2}b_{2}a_{3}b_{3})-\nu_{p}(D)\geq(\alpha_{1j_{1}}+1)+(\alpha_{2j_{2}}+1)+\beta_{3j_{3}}-\delta\geq\beta_{3j_{3}}\geq
1.$

Subcase 2: $\delta$ is odd. If $p|b_{1}$, $p|b_{2}$, $p\nmid b_{3}$, then
$p|b_{1},b_{2},a_{3}$. Hence, $\nu_{p}(a_{2}^{4}b_{2}^{6})\geq 6$ and
$\nu_{p}(a_{1}^{2}b_{1}^{3}a_{3}^{2}b_{3}^{3})\geq 5$ which gives $\delta\geq
3$ as
$p^{2\delta}||D^{2}=\text{gcd}(a_{2}^{4}b_{2}^{6},a_{1}^{2}b_{1}^{3}a_{3}^{2}b_{3}^{3})$
and $\delta$ is odd. If $p|b_{1}$, $p\nmid b_{2}$, $p|b_{3}$, then
$p|b_{1},a_{2},b_{3}$. Hence, $\nu_{p}(a_{2}^{4}b_{2}^{6})\geq 4$ and
$\nu_{p}(a_{1}^{2}b_{1}^{3}a_{3}^{2}b_{3}^{3})\geq 6$ which also gives
$\delta\geq 3$ by similar reasoning. If $p\nmid b_{1}$, $p|b_{2}$, $p|b_{3}$,
we also have $\delta\geq 3$ as it is similar to $p|b_{1}$, $p|b_{2}$, $p\nmid
b_{3}$. Therefore, $\delta\geq 3$ in all circumstances.

Suppose $p\nmid b_{i}$ for some $1\leq i\leq 3$. Then $p|a_{i}$ and
$p=q_{ij_{i}}$ for some $1\leq j_{i}\leq s_{i}$. Thus, $3\leq\delta\leq
2\beta_{ij_{i}}-1$ as $\delta$ is odd. This implies
$\beta_{ij_{i}}\geq\delta/2+1/2$. Hence, by Lemma 1,

$\nu_{p}(a_{1}b_{1}a_{2}b_{2}a_{3}b_{3})-\nu_{p}(D)\geq\frac{\delta}{3}+\frac{\delta}{3}+\beta_{ij_{i}}-\delta\geq\frac{1}{2}+\frac{\delta}{6}\geq\frac{1}{2}+\frac{3}{6}=1.$

Consequently, the left-hand side of (5) is at least $1$ in all of the above
cases. Therefore, (5) and, hence, (4) are true. Putting (4) into (3), we have

$\frac{N^{2}}{D^{2}}\leq
C_{\epsilon}\Bigl{(}\frac{a_{1}b_{1}a_{2}b_{2}a_{3}b_{3}d}{D^{2}}\Bigr{)}^{1+\epsilon}\ll
C_{\epsilon}\Bigl{(}\frac{N^{3/2}d}{D^{2}}\Bigr{)}^{1+\epsilon}$

as $a_{1}^{2}b_{1}^{3},a_{2}^{2}b_{2}^{3},a_{3}^{2}b_{3}^{2}\ll N$. This and
$D\geq 1$ imply $d\gg_{\epsilon}N^{1/2-2\epsilon}$ which gives $\theta_{3}\geq
1/2$ as $\epsilon$ can be arbitrarily small.

##  4 Proof of Theorem 3

For $k\geq 4$, the lower bound $\theta_{k}\geq 1/2$ follows from the
observation that $\theta_{k}\geq\theta_{3}$ and $\theta_{3}\geq 1/2$ from
Theorem 2 under the $abc$-conjecture.

For the upper bound $\theta_{4}\leq 4/5$, we construct $4$-AP of powerful
numbers as follows. With positive integer $a$, consider

$(x-a)^{3}(x+a)^{2},\;\;(x-a)^{2}x(x+a)^{2},\;\;(x-a)^{2}(x+a)^{3},\;\;(x-a)^{2}(x+a)^{2}(x+2a)$
(6)

which form an arithmetic progression with common difference
$d=a(x-a)^{2}(x+a)^{2}$. Note that the first and third terms give powerful
numbers for any integer $x$. If $x$ and $x+2a$ are powerful, then all four
polynomials would result in powerful numbers. We can pick $a=2$. Note that the
Pell equation

$X^{2}-2Y^{2}=1\;\;\text{ or }\;\;2X^{2}-4Y^{2}=2\;\;\text{ or
}\;\;4X^{2}=8Y^{2}+4$

has solutions

$X_{m}+\sqrt{2}Y_{m}=(3+2\sqrt{2})^{m}\;\;\text{ for positive integer }m.$

So, we can pick $x=8Y_{m}^{2}$ and $x+2a=x+4=4X_{m}^{2}$. These will make (6)
the desired $4$-AP of powerful numbers. Observe that the common difference

$d=2(x-2)^{2}(x+2)^{2}\leq 3((x-2)^{3}(x+2)^{2})^{4/5}=3N^{4/5}$

for large enough $m$ (and hence $N$) which gives $\theta_{4}\leq 4/5$.

For the upper bound $\theta_{5}\leq 9/10$, one can build upon our $3$-AP and
$4$-AP constructions. With positive integer $a$, consider

$(y-2a)(y-a)^{2}(y+a)^{2},\;\;(y-a)^{3}(y+a)^{2},\;\;(y-a)^{2}y(y+a)^{2},\;\;(y-a)^{2}(y+a)^{3},\;\;(y-a)^{2}(y+a)^{2}(y+2a)$
(7)

which form an arithmetic progression with common difference
$d=a(y-a)^{2}(y+a)^{2}$. Note that the second and fourth terms give powerful
numbers for any integer $y$. If $y-2a$, $y$ and $y+2a$ are powerful, then all
five terms would be powerful. From our $3$-AP construction, we can find
infinitely $3$-AP of powerful numbers

$y-2a=2^{2}(x^{2}-2x-1)=2^{3}n^{2},\;\;y=2^{2}x^{2}=(2x)^{2},\;\;y+2a=2^{2}(x^{2}+2x+1)=(2(x+1))^{2}$

with

$2a=2^{2}(2x+1)=8x+4.$

With these, we get the desired $5$-AP of powerful numbers with common
difference

$d=(4x+2)(4x^{2}-4x-2)^{2}(4x^{2}+4x+2)^{2}\leq
3\Bigl{(}(4x^{2}-8x-4)(4x^{2}-4x-2)^{2}(4x^{2}+4x+2)^{2}\Bigr{)}^{9/10}=3N^{9/10}$

for large enough $x$ (and hence $N$). Thus, $\theta_{5}\leq 9/10$.

For the general upper bound $\theta_{k}\leq 1-\frac{1}{10\cdot 3^{k-5}}$, we
use induction on $k\geq 5$ similar to Theorem 1. The base case $\theta_{5}\leq
1-\frac{1}{10\cdot 3^{5-5}}$ is true from above. Suppose, for some $k\geq 5$,
there are infinitely many $k$-APs among powerful numbers with $d\leq
C_{k}N^{1-\frac{1}{10\cdot 3^{k-5}}}$. Say one such AP is

$N=a_{1}^{2}b_{1}^{3}<a_{2}^{2}b_{2}^{3}<\cdots<a_{k}^{2}b_{k}^{3}\;\;\text{
with common difference }\;1\leq d\leq C_{k}N^{1-\frac{1}{10\cdot 3^{k-5}}}.$

Consider the number $a_{k}^{2}b_{k}^{3}+d=a^{2}b$ for some integer $a$ and
squarefree number $b$. Multiply everything by $b^{2}$,

$N_{1}:=Nb^{2}=a_{1}^{2}b_{1}^{3}b^{2}<a_{2}^{2}b_{2}^{3}b^{2}<\cdots<a_{k}^{2}b_{k}^{3}b^{2}<a^{2}b^{3}$

form a $(k+1)$-AP of powerful numbers with common difference $db^{2}$. Note
that

$b\leq a^{2}b=N+kd\leq(1+kC_{k})N.$

Hence,

$\displaystyle db^{\frac{2}{10\cdot 3^{k-4}}}$ $\displaystyle\leq
d(1+kC_{k})^{\frac{2}{10\cdot 3^{k-4}}}N^{\frac{2}{10\cdot 3^{k-4}}}$
$\displaystyle\leq C_{k}(1+kC_{k})^{\frac{2}{10\cdot
3^{k-4}}}N^{1-\frac{1}{10\cdot 3^{k-5}}+\frac{2}{10\cdot
3^{k-4}}}=C_{k}(1+kC_{k})^{\frac{2}{10\cdot 3^{k-4}}}N^{1-\frac{1}{10\cdot
3^{k-4}}}.$

This implies

$db^{2}\leq C_{k}(1+kC_{k})^{\frac{2}{10\cdot
3^{k-4}}}(Nb^{2})^{1-\frac{1}{10\cdot
3^{k-4}}}=:C_{k+1}N_{1}^{1-\frac{1}{10\cdot 3^{(k+1)-5}}}$

which completes the induction.

## References

  * [1] T.H. Chan, A note on powerful numbers in short intervals, Bull. Australian Math. Soc. https://doi.org/10.1017/S0004972722000995, 8 pp.
  * [2] P. Erdős, Consecutive integers, Eureka 38 (1975-76), 3–8.
  * [3] A. Granville, Powerful numbers and Fermat’s last theorem, C. R. Math. Rep. Acad. Sci. Canada 8 (1986), 215–218.
  * [4] B.J. Green and T.C. Tao, The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2) 167 (2008), no. 2, 481–547.
  * [5] R.A. Mollin and P.G. Walsh, A note of powerful numbers, quadratic fields and the Pellian, C. R. Math. Rep. Acad. Sci. Canada 8 (1986), 109–114.

Mathematics Department
Kennesaw State University
Marietta, GA 30060
<EMAIL_ADDRESS>


# Ge-based Clinopyroxene series: first principles and experimental local probe
study

Ricardo P. Moreira  IFIMUP, Institute of Physics for Advanced Materials,
Nanotechnology and Photonics, Departamento de Física e Astronomia da Faculdade
de Ciências da Universidade do Porto, Rua do Campo Alegre s/n, 4169-007 Porto,
Portugal.  E. Lora da Silva  IFIMUP, Institute of Physics for Advanced
Materials, Nanotechnology and Photonics, Departamento de Física e Astronomia
da Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre s/n,
4169-007 Porto, Portugal.  High Performance Computing Chair, University of
Évora, Largo dos Colegiais 2, 7004-516 Évora, Portugal  Gonçalo N. P. Oliveira
IFIMUP, Institute of Physics for Advanced Materials, Nanotechnology and
Photonics, Departamento de Física e Astronomia da Faculdade de Ciências da
Universidade do Porto, Rua do Campo Alegre s/n, 4169-007 Porto, Portugal.
Pedro Rocha-Rodrigues  IFIMUP, Institute of Physics for Advanced Materials,
Nanotechnology and Photonics, Departamento de Física e Astronomia da Faculdade
de Ciências da Universidade do Porto, Rua do Campo Alegre s/n, 4169-007 Porto,
Portugal.  Alessandro Stroppa  CNR-SPIN c/o Università degli Studi
dell’Aquila, Via Vetoio 10, 67010 Coppito, L’Aquila, Italy.  Claire V. Colin
Université Grenoble Alpes, CNRS, Institut Néel, 38000, Grenoble, France
Céline Darie  Université Grenoble Alpes, CNRS, Institut Néel, 38000, Grenoble,
France  João G. Correia  C2TN, DECN, Instituto Superior Técnico, Universidade
de Lisboa, Bobadela, Portugal.  Lucy V. C. Assali  Instituto de Física,
Universidade de São Paulo, CP 66318, 05315-970, São Paulo-SP, Brazil  Helena
M. Petrilli  Instituto de Física, Universidade de São Paulo, CP 66318,
05315-970, São Paulo-SP, Brazil  Armandina M. L. Lopes  IFIMUP, Institute of
Physics for Advanced Materials, Nanotechnology and Photonics, Departamento de
Física e Astronomia da Faculdade de Ciências da Universidade do Porto, Rua do
Campo Alegre s/n, 4169-007 Porto, Portugal.  João P. Araújo<EMAIL_ADDRESS>IFIMUP, Institute of Physics for Advanced Materials, Nanotechnology and
Photonics, Departamento de Física e Astronomia da Faculdade de Ciências da
Universidade do Porto, Rua do Campo Alegre s/n, 4169-007 Porto, Portugal.

(August 28, 2024)

###### Abstract

The structural and electronic properties of the CaMnGe2O6 and SrMnGe2O6
clinopyroxene systems have been investigated by means of perturbed angular
correlation (PAC) measurements, performed at ISOLDE, combined with ab initio
electronic structure calculations within the density functional theory (DFT)
framework. The partial density of states (PDOS) of the CaMnGe2O6 and SrMnGe2O6
stable compounds has been determined, and it has been observed that the
requirement of including an on-site Hubbard-U potential was necessary in order
to describe the highly correlated Mn 3d-states. By considering
U${}_{\textrm{eff}}$=4 eV, we obtained a band gap width of 1.82 eV and 1.70
eV, for the CaMnGe2O6 and SrMnGe2O6, respectively. Combining electric field
gradient (EFG) first principles calculations, using a supercell scheme, with
experimental PAC results, we were able to infer that the Cd probe can replace
either the A (Ca, Sr) or the Mn sites in the crystalline structures. We also
showed that Cd substitution is expected to lead to a reduction in the width of
the band gap in these systems, evidencing opportunities for potential band-gap
engineering.

Density Functional Theory, Perturbed Angular Correlation, Clinopyroxenes

##  I Introduction

Pyroxenes are a class of materials with general formula AMX2O6, where A is a
monovalent or a divalent cation, M is respectively a trivalent or a divalent
cation, and X is typically either Si or Ge. They are well known in mineralogy
and geology as they are one of the main rock-forming minerals of the Earth’s
crust [1]. Within this rich family of compounds, monoclinic pyroxenes
(clinopyroxenes), with a $3d$ transition metal at the M-site, have recently
been the subject of interest in the field of condensed matter physics due to
the diversity of their magnetic properties, namely: multiferroicity in
SrMnGe2O6 [2, 3] and NaFe(Si/Ge)2O6 [4, 5]; magnetoelectric effect in
CaMnGe2O6 [6] and Li(Cr/Fe)Si2O6 [4]; and also ferrotoroidal ordering in
LiFeSi2O6.

In spite of their natural prevalence, pyroxenes have not been the subject of
much computational or theoretical investigations. It is noteworthy of
mentioning, that recent density functional theory (DFT) studies have shown
that substitution of Al3+ by Tl^3+ in NaAlSi2O6 results in a reduction in the
band gap width from 5.32 eV to 2.05 eV. A further reduction is observed when
replacing Na by the organic cation CH3SH2, showing the high tunability of the
band gap in these materials[7]. Other works have shown their potential
usefulness in batteries, demonstrating that pyroxenes CaFe/Mn/Si2O6 could
display high theoretical energy densities in Ca-based batteries[8], whereas
LiFeSi2O6 has been shown to undergo a reversible electrochemical reaction
against Li, thus exhibiting potential for use as an electrode material in Li-
ion batteries[9]. Lastly, the magnetic properties of CaMnGe2O6 were studied by
Temnikov et al. [10] who computed the exchange parameters and showed that the
magnetic frustration in the system is weak, which could explain the frequent
occurrence of commensurate collinear antiferromagnetic structures in
Ca2+-bearing pyroxenes.

The ground state structure for the clinopyroxenes considered in this study is
a monoclinic Bravais lattice with space group C2/c (#15), where the
$M$-cations are in octahedral sites (Wyckoff 4e atomic positions), the
$A$-cations are in 8-coordinated sites (Wyckoff 4e atomic positions) and the
$X$-cations are in tetrahedral sites (Wyckoff 8f atomic positions). Their
structure is characterised by quasi-one-dimensional chains with edge-sharing
MO6 octahedra running along the c-axis, and connected by corner-sharing XO4
tetrahedra, along the c-axis, as schematically represented in Fig.1.

Figure 1: Representation of the AMX2O6 clinopyroxene structure. Red spheres
represent the oxygen site, green spheres the A-site, grey tetrahedra the
X-site coordination polyhedra, and blue octahedra the M-site coordination
polyhedra.

When the M-site is occupied by a magnetic cation, such an arrangement leads to
low-dimensional magnetic properties and to magnetic frustration, due to the
competition between interchain and intrachain interactions [4]. It is the
existence and the possible interplay between the low dimensionality and
frustration that is thought to give rise to the aforementioned diversity of
magnetic properties in these materials [6]. Moreover, it has been suggested
that the magnetic frustration can lead to spiral spin structures, which may be
favourable towards magnetically driven ferroelectricity, making these
compounds good candidates for multiferroic behaviour [4]. Since these
properties are thought to arise from local features that are often not
accurately described by long-range order based techniques, such as X-ray
diffraction and Raman spectroscopy, it is important to utilise techniques that
are more sensitive to local order to aid in the characterisation of these
intriguing systems. Several works have, in the past, studied pyroxene systems
by means of nuclear techniques, such as nuclear magnetic resonance
spectroscopy [11, 12, 13] and Mössbauer spectroscopy [14, 15, 16]; however,
and to the best of the authors’ knowledge, this is the first work being
reported where the pyroxene systems are studied via perturbed angular
correlation (PAC) spectroscopy. In particular, none of the nuclear techniques
above mentioned have been used to characterise the systems here under study.

In the present work, the AMnGe2O6 germanate clinopyroxene series were
explored, where the A-site is occupied by periodic table group-II elements,
i.e., Be, Mg, Ca, and Sr. The former two compounds have so far not been
experimentally realised, and they were thus studied solely through ab initio
calculations. (Ca, Sr)1-xCdxMnGe2O6 and (Ca, Sr)Mn1-xCdxGe2O6 were explored by
means of ab initio computational simulations, from which we were able to
estimate the band gaps, as well as the effective masses of the electrons and
holes of the systems; and by local Cd probe experiments using PAC
spectroscopy, allowing one to closely follow the temperature evolution of the
electric field gradient (EFG) tensor.

##  II Methods

###  II.1 Experimental Methods

For the purpose of performing the PAC measurements, pelletised samples of
CaMnGe2O6 and SrMnGe2O6 materials were implanted at ISOLDE-CERN by a 30 keV
beam of 111mCd probes [111mCd$\,\rightarrow$111Cd, $t_{1/2}$ = 48.6 minutes],
which have a nuclear spin $I=5/2$ and an electric quadrupole moment of $Q$ =
0.83(13) b in the intermediate state of decay [17]. Small doses of the order
of 1011 atoms/cm2 were implanted and subsequently an annealing was performed
at a temperature of $\sim$1123 K for 20 minutes in air to recover from
implantation damage. Confirmation for the recovery of point defects and for
the incorporation of Cd into the lattice was given by the PAC measurements
themselves, which yielded well-defined experimental R(t) anisotropy functions.
The measurements were performed on 6-BaF2 detectors PAC spectrometers[18],
equipped with either a closed-cycle refrigerator or a special high-temperature
furnace for temperature control. Due to the relatively short lifetime of the
parent probe, each temperature point required the implantation of a new sample
for satisfactory activity to be maintained during measurement. The
experimental R(t) function was fitted with exact numerical methods that build
the expected observable by solving the hyperfine interaction Hamiltonian’s
characteristic equations[19, 20, 21, 22, 23]. The EFG, that characterizes the
surrounding charge density that interacts with a probe nucleus, can be
represented as a rank-2 symmetric and traceless tensor. As such it is always
possible to diagonalise it by rotating the axis system into the EFG
eigenbasis, that is referred to as the principal reference frame. The
experimentally observable EFG is conventionally described by two parameters,
the principal component $V_{zz}$ and the axial asymmetry parameter:

$\eta=\frac{V_{xx}-V_{yy}}{V_{zz}},$ (1)

with $V_{zz}$, $V_{yy}$ and $V_{xx}$ being the eigenvalues of the EFG tensor,
defined such that $|V_{zz}|\geq|V_{yy}|\geq|V_{xx}|$[19].

The signature for the interaction between the probe atoms and their local
environment is the perturbation function $G_{kk}(t)$. In the case of static
electric quadrupole interactions, the interaction can be written as[24]:

$\displaystyle
G_{kk}(t)=\sum_{n}s_{k_{n}}\cos(\omega_{n}t)e^{-\delta\omega_{n}t/2},$ (2)

where $s_{k_{n}}$ and $\omega_{n}$ are functions of $\eta$ and the fundamental
frequency $\omega_{Q}$. The exponential term arises from randomly distributed
vacancies, defects, and lattice strains that result in an attenuation of the
experimental R(t) function. In the case of the 111mCd probes, the intermediate
level in the decay has a nuclear spin of $I=5/2$, which leads to a split of
the intermediate level into three sub-levels due to the quadrupole interaction
and three transition frequencies, $\omega_{1}$, $\omega_{2}$, and
$\omega_{3}=\omega_{1}+\omega_{2}$. The fundamental frequency is usually
defined as [19]:

$\displaystyle\omega_{Q}=\frac{eQV_{zz}}{4I(2I-1)\hbar},$ (3)

where $Q$ is the nuclear quadrupole moment, $e$ is the electron charge, and
$\hbar$ is the reduced Planck constant.

###  II.2 Computational Methodology

DFT calculations were performed by using the Quantum Espresso (QE) [25, 26]
suite with Projector-Augmented Wave (PAW) datasets[27, 28]. The atomic valence
configurations that were considered for the calculations were: Be[2s2],
Ca[3s23p64s2], Cd[4d9.55s25p0.5], Ge[3d104s24p2], Mg[2s22p63s2],
Mn[3s23p63d54s2], O[2s22p4], and Sr[4s24p65s2]. The kinetic energy cutoffs
were converged to the values of 70 and 540 Ry for the wavefunctions and the
charge density, respectively. However, in order to converge the EFGs to the
experimental data, and for the calculations that included the Cd
substitutional atom, these values were raised to 113 and 900 Ry, for the
wavefunctions and the charge density, respectively. Regarding the exchange-
correlation (XC) functional, the generalised gradient approximation (GGA) with
Perdew-Burke-Ernzerhof[29] (PBE) parametrization was employed. In order to
treat the strongly correlated 3d-states of Mn, an effective Hubbard-U
potential was considered by defining $U_{eff}=4$ eV [30, 31], by employing the
simplified version of Cococcioni and de Gironcoli [32], with PBE (PBE+U).

Figure 2: Representation of the AMX2O6 clinopyroxene structure. Red spheres
represent the oxygen site, green spheres the A-site, grey tetrahedra the
X-site coordination polyhedra, and blue octahedra the M-site coordination
polyhedra.

In order to obtain the ground-state properties, total energy minimization with
respect to the changes in the atomic positions and cell parameters were taken
into account, until the total force did not exceed 0.05 eV/Å, and constraining
the system to zero internal pressure, with the variable cell shape relaxation,
considering the damped Beeman ionic dynamics and the Wentzcovitch extended
Lagrangian for the cell dynamics [33, 34, 35]. For the four compounds, the
enthalpies of formation were determined by comparing respective total energies
to that of the respective DFT total energy of the AGeO3 and MnGeO3 stable
constituent oxides (Eq. 6 in Appendix A). With respect to the CaMnGe2O6 and
SrMnGe2O6 compounds, the EFGs were evaluated for the pristine unit-cell, as
well as for the unit-cell and the $1\times 1\times 2$ supercell by considering
Cd substitution of either A or Mn cation sites (Fig. 2), in order to compare
with previously obtained PAC measurements. The computation of the EFGs was
performed by using the Gauge Including Projector Augmented Waves (GIPAW)[36]
package routine implemented in QE.

For the supercell calculations, only the atomic positions were relaxed, by
using the lattice parameters obtained from the pristine unit-cell structural
minimization(by employing the variable cell shape relaxations). The Brillouin-
zones (BZ) were sampled with a $(9\times 9\times 18)$ Monkhorst-Pack k-point
grids for unit-cell calculations, and a $(9\times 9\times 9)$ for calculations
on the $1\times 1\times 2$ supercells; whereas for the calculation of EFGs and
PDOS, k-point grids of $(11\times 11\times 22)$ and $(11\times 11\times 11)$
were used, for the unit-cells and supercells, respectively.

##  III Results and Discussion

###  III.1 Perturbed Angular Correlation

The PAC measurements on the SrMnGe2O6 (SMGO) system were performed at selected
temperatures in the 13-902 K range. Figure 3 shows a few representative
experimental $\gamma-\gamma$ R(t) perturbation functions obtained for the
SrMnGe2O6 compound along with their corresponding Fourier transforms. Several
models were tested in order to numerically obtain the best fit to the
experimental R(t) spectra before arriving at the presented model, wherein two
distinct fractions (two distinct local environments) are evident across the
whole temperature range, indicating the probes interacted with two distinct
EFG distributions, henceforth referred to as EFGSMGO1 and EFGSMGO2, and
represented by the green and blue lines in figure 3, respectively.

Figure 3: Representative R(t) spectra of SrMnGe2O6 and corresponding fits
(left), as well as Fourier transforms (right). Figure 4: a) Experimental
$V_{zz}$ for SrMnGe2O6; b) Experimental axial asymmetry for SrMnGe2O6. The
green circles correspond to EFGSMGO1 fraction and blue triangles to EFGSMGO2
fraction. The full red line corresponds to a fit of the temperature dependence
of $V_{zz}^{\text{SMGO}_{1}}$ by the Bayer-Kushida model [37, 38, 39], while
the dashed lines are merely a guide to the eye.

The temperature behaviour of the experimental R(t) fit parameters of SMGO is
displayed in figure 4. One can observe, in figure 4 a), green circles, that
the $V_{zz}$ value for the EFGSMGO1 fraction has a noticeable increase at low
temperatures, with increasing temperature, reaching a maximum; whereupon it
starts to decrease. On the other hand, the axial asymmetry $\eta(1)$ parameter
increases as the temperature increases, as displayed in figure 4 b) by the
green circles. The EFG aforementioned behaviour is typical for a phonon
contribution to $V_{zz}$ and, accordingly, a temperature dependence least-
squares fit by the Bayer-Kushida[37, 38, 39] model, with a single phonon
frequency, was done, according to

$\displaystyle\small
V_{zz}(T)=V_{zz}(0)\left\\{1-\frac{3}{8\pi^{2}}\frac{hA}{\nu}\left(\frac{1}{2}+\frac{1}{e^{\frac{h\nu}{kT}}-1}\right)\right\\}\;,$
(4)

where $h$ is the Planck constant, $\nu$ is the phonon frequency, $k$ is the
Boltzmann constant and $A$ is a quantity that in general can only be said to
have the dimensions and order of magnitude of the inverse of the moment of
inertia associated to the molecular vibration, except in specific cases where
it is indeed the inverse of the moment of inertia associated with a normal
mode of molecular vibration[39]. From this fitting, we have obtained
$\nu=8.01(7)\times 10^{12}$ Hz, which is within the range of frequencies that
are typical for intra-molecular vibrations and is comparable to the value of
$6.60(85)\times 10^{12}$ Hz reported for Ca2MnO4[40], suggesting that the
$V_{zz}$ temperature behaviour for the EFGSMGO1 local environment is indeed
due to a phonon contribution. Regarding the axial asymmetry parameter
$\eta(1)$ at the EFGSMGO1 site, it displays an intermediate asymmetry, which
increases as temperature raises.

For the EFGSMGO2 fraction, depicted by the blue triagles in figure 4 a), the
$V_{zz}$ shows an atypical behaviour of increasing as temperature also
increases, though such behaviour has nevertheless been reported in other
compounds. Specifically, analogous behaviour was experimentally observed in
the A21am phase of Ca3Mn2O7 for which DFT simulations indicated that such
behaviour is actually due to structural effects, that prevail over the phonon
contributions [41]. The asymmetry parameter $\eta(2)$ for the EFGSMGO2
fraction, as displayed in figure 4 b) by the blue triangles, on the other
hand, remains approximately constant, up to 474 K, whereupon an anomalous
increase occurs. It is unclear if such behaviour could be related to
structural phenomena, as there are no measurements reported for temperatures
above 350 K. It is worth noting that the probes distribution between both
fraction remained fairly constant across the whole temperature range,
suggesting that they are related to the probes occupying two distinct lattice
sites.

With respect to the CaMnGe2O6 (CMGO) system, the PAC measurements were
likewise performed at selected temperatures, in the 11-898 K range.
Representative fits to the experimental $R(t)$ perturbation functions for CMGO
are presented in figure 5, along with the corresponding Fourier transforms.
Similar to how the data from the previous compound was treated, several models
were also tested in order to numerically obtain the best fit to the
experimental R(t) spectra. Distinctly from SrMnGe2O6, wherein two noticeable
fractions are present across the whole temperature range, for the CaMnGe2O6
compound there is only one fraction present from low temperatures, up to 351
K, hereby referred to as EFGCMGO1 and represented by the green lines in figure
5. It can be noticed in this figure that a second fraction, labeled EFGCMGO2,
appears for temperatures above 351 K, represented by the blue lines. However,
it is likely that this second fraction is likewise present at lower
temperatures, but owing to the very small observable frequency associated with
it, that is not well defined in the time window available in the PAC
experiments, it is not possible to fit this fraction at lower temperatures.
Even at high temperatures the EFGCMGO2 fitting parameters have nonetheless an
associated high uncertainty.

Figure 5: Representative R(t) spectra of CaMnGe2O6 and corresponding fits
(left), as well as Fourier transforms (right).

The temperature behaviour of the fit parameters relative to the CaMnGe2O6
system are shown in figure 6. One can observe, in figure 6 a), green circles,
that the EFGCMGO1 distribution has a $V_{zz}$ value of similar magnitude to
EFGSMGO1 (figure 4 a)), with an analogous decrease as temperature raises.

Figure 6: a): Experimental $V_{zz}$ for CaMnGe2O6. b): Experimental axial
asymmetry $\eta$ for CaMnGe2O6. The green circles correspond to EFGCMGO1
fraction and blue triangles to EFGCMGO2 fraction. The continuous red line
corresponds to a fit of the temperature dependence of
$V_{zz}^{\text{CMGO}_{1}}$ by the Bayer-Kushida model[37, 38, 39], while the
dashed lines are merely meant to guide the eye.

Also $\eta(1)$, displayed in figure 6 b), green circles, similarly increases
with increasing temperature, when compared to SMGO $\eta(1)$ variation (figure
4 b)). As was done in the case of the SrMnGe2O6 compound, a least-squares fits
was likewise performed to $V_{zz}$, according to equation 4. From the fit, a
frequency of $\nu=10.0(2)\times 10^{12}$ Hz was obtained, once more indicating
that this behaviour is likely attributable to a phonon contribution to
$V_{zz}$. Similarly to what was observed for SrMnGe2O6, we note that the
magnetic hyperfine interactions were not apparent in the fits to the
experimental R(t) functions.

Table 1: Experimental [6, 2] and theoretical lattice parameters for SrMnGe2O6 and CaMnGe2O6 compounds, obtained by using the PBE XC functional, without and with an effective Hubbard-$U$ potential correction to describe the strongly correlated Mn $3d$-states, labeled as PBE and PBE$+U$, respectively.  System | Research type |  $a\,$(Å) |  $b\,$(Å) |  $c\,$(Å) | $\beta\,(^{\circ})$
---|---|---|---|---|---
SrMnGe2O6 | Expt. |  10.3511(6) |  9.4204(5) |  5.5093(3) |  104.700(2)
Theo. PBE | 10.5295 | 9.4912 | 5.5992 | 105.029
| Theo. PBE+$U$ | 10.5638 | 9.5795 | 5.6230 | 104.906
CaMnGe2O6 | Expt. | 10.2794(3) | 9.1756(3) | 5.4714(2) | 104.244(2)
Theo. PBE | 10.4006 | 9.2248 | 5.5194 | 103.921
| Theo. PBE$+U$ | 10.4495 | 9.3204 | 5.5472 | 104.109

###  III.2 Density Functional Theory Calculations

####  III.2.1 Structural Parameters

We begin this section by summarizing in Table 1 the calculated lattice
parameters for the SrMnGe2O6 and CaMnGe2O6 systems, computed by applying both
the PBE and PBE$+U$ methodologies, as well as the reported experimental
values,[6, 2] for comparison purposes.

Regarding the SrMnGe2O6 system, we can observe that the lattice parameters
computed with the PBE functional are overestimated, with respect to the
experimental values, corresponding to a relative increase of $\sim$1.7%,
$\sim$0.8%, and $\sim$1.6%, respectively, for a, b, and c parameters, which is
in line with the well-known tendency of the PBE functional.[42] The inclusion
of the on-site effective Hubbard-U potential results in a further relative
increase of these parameters with $\sim$2.1%, $\sim$1.7%, and $\sim$2.1%, when
compared to the experimental values. Though the $\beta$ monoclinic angle is
overestimated for both PBE and PBE$+U$ approximations, when comparing to
experimental data, the difference is in fact smaller when considering the
PBE$+U$ calculations, resulting in a value closer to the experimental one.

With respect to CaMnGe2O6, the a and b theoretical lattice parameters obtained
from PBE calculations likewise are overestimated, relative to experimental
data, by $\sim$1.2% and $\sim$0.5%, respectively. On the other hand we
observed that the c parameter is underestimated by $\sim$0.9%. Analogously to
what was observed for SrMnGe2O6, the $a$ and $b$ lattice parameters that
resulted from the PBE$+U$ approach display an increase with respect to the PBE
values, presenting an overestimation of $\sim$1.7% and $\sim$1.6%,
respectively, with respect to experimental results; whereas c parameter is
still underestimated, though smaller than the one obtained by the PBE
functional, of $\sim$0.4%. Also, and distinctly from what was observed for
SrMnGe2O6, the theoretical monoclinic angle is underestimated relative to the
experimental angle, though again, results from the PBE$+U$ approximation is
closer to the experimental value.

####  III.2.2 Enthalpies of Formation

The enthalpy of formation of the constituent oxides of the AMnGe2O6 series has
also been examined through DFT calculations and we have determined that only
the CaMnGe2O6 and SrMnGe2O6 clinopyroxene compounds are thermodynamically
stable. This is evidenced by their negative enthalpies of formation of -0.19
eV/f.u. and -0.11 eV/f.u., respectively (calculated by considering equation 6
of Appendix A). On the other hand, the BeMnGe2O6 and MgMnGe2O6 systems both
evidenced positive enthalpies of formation of 0.12 eV/f.u., supporting that
these are not stable against dissociation into the respective stable
constituent oxides BeGeO3 or MgGeO3 and MnGeO3. This observation is consistent
with the fact that only the former two compounds have been experimentally
synthesised, whereas the synthesis of the latter two has not yet been
reported. Such observation implies that different reaction pathways towards
synthesis will have to be explored if these systems were to be experimentally
realised. Details on how the enthalpies of formation were obtained are
presented in the Appendix A.

####  III.2.3 Partial Density of States

The PDOS for the CaMnGe2O6 and SrMnGe2O6 compounds, obtained considering the
PBE calculation, without the Hubbard-$U$ correction (Ueff = 0 eV), were
computed and are displayed in figure 7, in which the valence band maximum
(VBM) is aligned at the Fermi energy and is set to zero. Two narrow bands form
the VBM, and these are mainly composed by the Mn occupied $3d$-states,
hybridized with a smaller contribution from the O p-states. Energetically
below these narrow bands ($<$ -2.5 eV), a broader band is observed and it is
mostly dominated by the O $p$-states, with a small mixture of $3d$-states from
Mn. The conduction band minimum (CBM), on the other hand, presents low density
of states and hybridization between the $p$-states of O and Ge. For increasing
energies, the DOS significantly increases forming a narrow resonant band,
between 2.5 and 3.0 eV, that is mainly composed by the Mn $3d$-states
unoccupied states.

For both studied systems the PDOS are qualitatively very similar, although at
the conduction band the observed resonant peaks of Mn for SrMnGe2O6 are much
broader when compared to those of CaMnGe2O6.

Figure 7: Partial density of states of (a) CaMnGe2O6 and (b) SrMnGe2O6,
computed with Ueff = 0 eV, and considering antiferromagnetic ordering. The
Fermi level is set to zero and corresponds to the VBM.

The electronic band gaps are also noted to be fairly narrow, being 0.46 eV and
0.51 eV, respectively, for CaMnGe2O6 and SrMnGe2O6, which is narrower than
what has been reported for similar Si-based compounds[43, 7].

Since the band gap widths are likely underestimated in the absence of the
Hubbard-$U$ correction, we have also considered the PDOS by including the Ueff
= 4 eV, applied to the Mn $3d$-states. These are shown in figure 9 along with
the respective electronic band structures in the high-symmetry directions of
the primitive cell first Brillouin zone (figure 8).

Figure 8: First Brillouin zone of the base-centered monoclinic lattice and
respective high-symmetry points and directions. Adapted from reference 44.

When including the Ueff = 4 eV, and comparing to the PDOS results when Ueff =
0 eV, differences are mainly evidenced at the VBM, with the Mn $3d$-states.
The two narrow bands observed in figure 7, now overlap and form one narrow
band for CaMnGe2O6 (figure 9 a)). As for the SrMnGe2O6 system, the narrow
bands are energetically very close, almost forming also one narrow band
(figure 9 b)). Moreover, these narrow bands are now dominated by a stronger
hybridization between Mn d-states with the O p-states.

Figure 9: Bandstructures in the high-symmetry directions of the primitive cell
first Brillouin zone (figure 8) (left) and partial density of states (right)
of a) CaMnGe2O6 and b) SrMnGe2O6, both computed with Ueff = 4 eV to treat the
strongly correlated Mn $3d$-states, and considering antiferromagnetic
ordering. The Fermi level is set to zero and matches the VBM.

Since the inclusion of the $U$ parameter localises and positions the
unoccupied narrow $3d$-states of Mn higher in energy (between 5.0 and 5.5 eV),
the CBM shows slightly increased $p$ hybridized states of O and Ge, than when
$U$ is not considered. The PBE$+U$ theoretical band gaps are now wider, when
compared to the previous case, with 1.82 eV and 1.70 eV for the CaMnGe2O6 and
SrMnGe2O6 compounds, respectively. Despite this, the gap widths are still
below the range of other similar Si-based clinopyroxenes[43, 7].

####  III.2.4 Electronic Bandstructures

The electronic bandstructures for the pristine CaMnGe2O6 and SrMnGe2O6
systems, as well as the respective Cd-containing supercells were also computed
through DFT calculations, in a path along the high-symmetry points depicted in
figure 8, generated with the SeeK-path tool.[44, 45]

The bandstructures for the pristine systems are presented in figure 9, whilst
those for the supercells can be found in Appendix C.

Differences between both pristine systems is evidenced by the character of the
band gap: while SrMnGe2O6 exhibits a direct band gap, CaMnGe2O6 has an
indirect gap. We must stress that subtleties persist with respect to the
analysis of the VBM, since the valence bands are quite flat, and thus the
electronic bands along the different high symmetry segments/points are
energetically very close.

From figure 9, one can observe that the top of the valence band is much
flatter than the bottom of the conduction band, which shows narrow parabolas
around the zone-centre, indicating that the effective mass of the holes,
$m^{*}_{h}$, is much larger than the effective mass of electrons, $m^{*}_{e}$.
Such an observation is an indication of a higher mobility of the electrons
comparatively to the holes of the system. Since the effective mass of the
charge carriers is an essential parameter for the performance of photoelectric
and photo-catalytic activities[46], and also for semiconductor radiation
detectors[47]; we have considered further analysing respective properties
(table 2). Therefore and to quantify the effective masses of holes and
electrons, we employ the well-known definition of the effective mass as

$m^{*}=\frac{\hbar^{2}}{\frac{\partial^{2}E}{\partial k^{2}}},$ (5)

with the second derivative of the electronic dispersion relation being
obtained from a quadratic fit around the energy range where the band edges are
located.

In the case of CaMnGe2O6, we obtain a $m^{*}_{e}$, around the $\Gamma$-point,
very close to that of the free electrons, with $m^{*}_{e}=1.028m_{0}$ (where
$m_{0}$ is the mass of the free electron with 9.11$\times$10-31 kg). While for
the hole carriers, and as expected by considering the flatness-shape of the
VBM, the mass of holes proved to be substantially larger (in absolute terms)
with $m^{*}_{h}=26.125m_{0}$, and calculated along the high-symmetry segment
of $\overline{A\Gamma}\rightarrow A$ (table 2). Regarding the Cd-containing
supercells of CaMnGe2O6, when Cd substitution was considered at the Ca site,
we found that the effective masses of the electrons only yielded mild
variation when compared to the pristine system. For the electrons, the
$m^{*}_{e}$ showed very little to no increase with $m^{*}_{e}=0.996m_{0}$ and
$m^{*}_{e}=1.028m_{0}$, for the Ca3/4Cd1/4MnGe2O6 and Ca7/8Cd1/8MnGe2O6
systems, respectively. In the case of the holes, we observe that for the
Ca3/4Cd1/4MnGe2O6 cell, and by comparing to the pristine system, $m^{*}_{h}$
slightly increases along the $\overline{A\Gamma}\rightarrow\Gamma$ with
$m^{*}_{h}=45.259m_{0}$, although showing similar values along
$\overline{A\Gamma}\rightarrow A$ with $m^{*}_{h}=27.875m_{0}$.

With respect to the larger supercell (CaMn7/8Cd1/8Ge2O6), we observe a very
high increase with respect to the pristine system, with
$m^{*}_{h}=67.562m_{0}$, along the same high-symmetry segment.

On the other hand, when the Cd atom was placed at the Mn site, we found that
the dopant affected the valence band edge even further, leading to a further
flattening of the VBM; however, and once again, the $m^{*}_{e}$ remained very
similar to the pure unit-cell. A significant increase of the hole effective
mass was found for the CaMn3/4Cd1/4Ge2O6 structure ($V_{2}\rightarrow\Gamma$
direction), with $m^{*}_{h}=118.325m_{0}$. Moreover, for the more diluted
system we observed that along $\overline{A\Gamma}\rightarrow A$ the effective
mass decreases slightly ($m^{*}_{h}=28.65m_{0}$), showing an increase of
$m^{*}_{h}$ in the direction of $\overline{A\Gamma}\rightarrow A$
($m^{*}_{h}=68.128m_{0}$).

Table 2: Electronic properties of the studied systems without and with Cd,
considered as dopant, and applying Ueff = 4 eV to treat the Mn $3d$-states.
The values of the effective masses are given in units of the electron mass:
$m_{0}=9.11\times 10^{-31}$ kg.)

System | Band Character | Band gap (eV) |  $m^{*}_{e}$ ($m_{0}$) |  $m^{*}_{h}$ ($m_{0}$)
---|---|---|---|---
SrMnGe2O6 | Direct | 1.70 | 1.023 | 9.380 ($\Gamma\rightarrow D$)
Sr7/8Cd1/8MnGe2O6 | Indirect | 1.55 | 1.028 | 58.223 ($\overline{C_{2}\Gamma}\rightarrow C_{2}$)
Sr3/4Cd1/4MnGe2O6 | Indirect | 1.43 | 1.106 | 58.287 ($\overline{C_{2}\Gamma}\rightarrow C_{2}$)
SrMn7/8Cd1/8Ge2O6 | Indirect | 1.56 | 1.057 | 40.062 ($\overline{C_{2}\Gamma}\rightarrow C_{2}$)
SrMn3/4Cd1/4Ge2O6 | Indirect | 1.50 | 1.068 | 46.882 ($V_{2}\rightarrow\Gamma)$
CaMnGe2O6 | Indirect | 1.82 | 1.028 | 26.125 ($\overline{A\Gamma}\rightarrow A$)
Ca7/8Cd1/8MnGe2O6 | Indirect | 1.59 | 1.028 | 67.562 ($\overline{A\Gamma}\rightarrow A$)
Ca3/4Cd1/4MnGe2O6 | Indirect | 1.42 | 0.996 | 27.875 ($\overline{A\Gamma}\rightarrow A$)
45.259 ($\overline{A\Gamma}\rightarrow\Gamma$)
CaMn7/8Cd1/8Ge2O6 | Indirect | 1.58 | 1.033 | 18.704 ($\overline{A\Gamma}\rightarrow A$)
68.128 ($\overline{A\Gamma}\rightarrow\Gamma$)
CaMn3/4Cd1/4Ge2O6 | Indirect | 1.61 | 1.058 | 118.325 ($V_{2}\rightarrow\Gamma$)

Regarding the pristine SrMnGe2O6 structure, we found a similar value for the
electron effective mass with $m^{*}_{e}=1.023m_{0}$. However, a significantly
more parabolic dispersion relation at the VBM when compared to CaMnGe2O6,
leading to a hole effective mass of $m^{*}_{h}=9.380m_{0}$, when considering
the $\Gamma\rightarrow D$ segment. We must however state that since the
valence band dispersion is quite asymmetric, the interpolation with a
quadratic function did not yield a very accurate fit to the dispersion
relation; therefore the value of the $m^{*}_{h}$ should be understood more as
an order of magnitude rather than a quantitative value. Once again, we observe
that with the Cd substitution, there is only a mild variation of $m^{*}_{e}$,
leading to a slightly increased value of $m^{*}_{e}=1.028m_{0}$ for
Sr7/8Cd1/8MnGe2O6, and a more pronounced increase for the Sr3/4Cd1/4MnGe2O6
compound with $m^{*}_{e}=1.106m_{0}$. As for the SrMn3/4Cd1/4Ge2O6 and
SrMn7/8Cd1/8Ge2O6 systems, a slight increase of the effective masses is
observed to $m^{*}_{e}=1.068m_{0}$ and $m^{*}_{e}=1.057m_{0}$, respectively.
With respect to the $m^{*}_{h}$, we again note that due to the flat-shape of
the bands, a parabolic interpolation does not provide a dependable
quantitative value, so we can only state with confidence that this value
should remain of the same order as that of the pristine SrMnGe2O6 compound. We
must state that such observed differences of the masses between the charge
carriers is quite interesting because of the increased possibility to enhance
the separation of the electron–hole pairs, and even of evidencing large
electron and hole drift lengths[47], thus affecting the opto-electronic
properties of the studied clinopyroxenes [48]. The band gaps were also
determined for all systems from the bandstructure calculations, and are
summarised in Table 2. As already observed for the PDOS, the band gaps for all
systems lie between 1.70 eV and 1.82 eV for the pristine systems, SrMnGe2O6
and CaMnGe2O6, respectively; whereas the doped structures show narrower band
gaps. All the doped structures show indirect band gap character, even for the
Sr-based strucutures, where the pristine counterpart evidences a direct one
(figure 9 and figures C1, C2, C3, C4, C5, C6, C7, C8 and C9 in Appendix C).

####  III.2.5 Ab Initio Electric Field Gradients

The DFT EFGs calculations were performed for the pristine unit-cells, along
with the EFGs for the unit-cells and $1\times 1\times 2$ supercells with Cd
substitution, either at the A\- or the Mn-sites. Results were compared with
those obtained for the experimental EFG fractions and are summarised in Table
3, whilst a more comprehensive list of computed values can be found in
appendix B, Table B1. It should be noted that in the cases where Cd was placed
at a Mn site, the systems were treated as ferromagnetic, for the sake of
computational simplicity. An antiferromagnetic calculation for the
CaMn7/8Cd1/8Ge2O6 system was nevertheless performed to check how much of an
effect the magnetic ordering might have on the computed EFG. Very similar
results were obtained in both the ferromagnetic and antiferromagnetic
calculations, with $V_{zz}$=12 V/Å2 and $\eta$=0.21, in the former case, and
$V_{zz}$=13 V/Å2 and $\eta$=0.37, in the latter, thus showing that this
approximation is reliable.

Table 3: Experimental (highlighted in gray shaded cells, nomenclature as defined above in figures 5 and 3) and computational (absolute values) EFG parameters for SrMnGe2O6 and CaMnGe2O6 related compounds. System |  $|V_{zz}|$ (V/Å2) | $\eta$
---|---|---
gray!30 |  gray!30 |  gray!30
gray!30 EFGSMGO1 (13 K) |  gray!30120(1) |  gray!300.28(1)
gray!30 |  gray!30 |  gray!30
gray!30 EFGSMGO2 (13 K) |  gray!3020(1) |  gray!300.81(6)
SrMnGe2O6 Sr-site | 64 | 0.54
Sr7/8Cd1/8MnGe2O6 | 108 | 0.14
Sr3/4Cd1/4MnGe2O6 | 103 | 0.14
SrMnGe2O6 Mn-site | 14 | 0.32
SrMn7/8Cd1/8Ge2O6 | 21 | 0.74
SrMn3/4Cd1/4Ge2O6 | 22 | 0.67
gray!30 |  gray!30 |  gray!30
gray!30 EFGCMGO1 (11 K) |  gray!30130(1) |  gray!300.15(1)
gray!30 |  gray!30 |  gray!30
gray!30 EFGCMGO2 (351 K) |  gray!3010(2) |  gray!30—
CaMnGe2O6 Ca-site | 34 | 0.52
Ca7/8Cd1/8MnGe2O6 | 109 | 0.01
Ca3/4Cd1/4MnGe2O6 | 108 | 0.02
CaMnGe2O6 Mn-site | 8 | 0.49
CaMn7/8Cd1/8Ge2O6 | 12 | 0.21
CaMn3/4Cd1/4Ge2O6 | 12 | 0.31

Comparing the EFG parameters for the experimental EFGSMGO1 fraction of
SrMnGe2O6 with those computed for the supercells with Cd substitution at the
Sr-site, where the 1/4 and 1/8 Cd dilution was considered (unit-cell and
$1\times 1\times 2$ supercell, respectively), we may observe a very good
agreement between $V_{zz}$ and $\eta$. From the PAC measurements we obtain for
EFGSMGO1the values of $V_{zz}$=120 V/Å2 and $\eta$=0.28; for the Cd probe
nucleus for Sr3/4Cd1/4MnGe2O6 they are $V_{zz}$=103 V/Å2 and $\eta$=0.14 and
for Sr7/8Cd1/8MnGe2O6 these result as $V_{zz}$=108 V/Å2 and $\eta$=0.14.
Likewise, good agreement is also observed for the EFGSMGO2 fraction of
SrMnGe2O6 for the case of the Cd substitution at the Mn-site. We obtain for
the EFGSMGO2 fraction the values of $V_{zz}$=20 V/Å2 and $\eta$=0.81. From the
DFT results we get $V_{zz}$=22 V/Å2 and $\eta$=0.67, and $V_{zz}$=21 V/Å2 and
$\eta$=0.72, for the SrMn3/4Cd1/4Ge2O6 and SrMn7/8Cd1/8Ge2O6 cells,
respectively. Based on the obtained theoretical results, we may conclude that
the supercells are the systems that simulate quite well the experimental data
when the Cd probe is considered, with the EFGs being in good agreement to the
EFGSMGO1 and EFGSMGO2 experimental fractions, which correspond to the Cd probe
replacing the Sr-sites and the Mn-sites, respectively, in the crystalline
structure. Similarly, this same correspondence seems to hold true for the
CaMnGe2O6 compound. For the EFGCMGO1 experimental fraction, we observe the
values of $V_{zz}$=130 V/Å2 and $\eta$=0.15, and for the DFT data we obtain
values of $V_{zz}$=108 (109) V/Å2 and $\eta$=0.01 (0.02) for Cd concentration
of 1/4 (1/8) in the Ca atom position. The obtained theoretical EFGs are
slightly lower than those of the experimental fraction; such is mostly evident
for the asymmetry parameter, $\eta$. As for the EFGCMGO2 fraction from the PAC
experiments, the EFG result in $V_{zz}$=10 V/Å2 and an indeterminate $\eta$,
whereas for the DFT diluted supercell of CaMn7/8Cd1/8Ge2O6 we have $V_{zz}$=12
V/Å2 and $\eta$=0.21, similar to those obtained for CaMn3/4Cd1/4Ge2O6. These
results indicate that first-principles calculations combined with PAC
spectroscopy provide a valuable tool to explain the observed two distinct
local environments, indicating that Cd probes are distributed between the Sr-
or Ca-sites and the Mn-sites in the SrMnGe2O6 and CaMnGe2O6 basic compounds,
respectively.

##  IV Conclusions

We have undertaken DFT calculations to obtain the structural and electronic
properties of the AMnGe2O6 clinopyroxene systems, with
$\textit{A}=\rm{Be,Mg,Ca,Sr}$. We found, through enthalpy of formation
calculations, that the Be- and Mg-based systems are not thermodynamically
stable against dissociation into the stable constituent oxides BeGeO3 or
MgGeO3 and MnGeO3. On the other hand, the Ca- and Sr-based systems are
thermodynamically stable against dissociation and, accordingly, have been
realized experimentally. From the PDOS, we found that the strongly correlated
Mn $3d$-states form the narrow band close to the VBM together with the O
$p$-states. The CBM is composed mainly of low density hybridized $p$-states of
O and Ge and, for higher energies, the density of states significantly
increases forming a narrow resonant band that is mainly composed of the anti-
bound unnocupied Mn $3d$-states. The theoretical band gaps of CaMnGe2O6 and
SrMnGe2O6 compounds, computed using the PBE+$U$ approximation, are 1.82 eV and
1.70 eV, respectively. The calculated effective masses of the charge carriers
of the two studied pristine compounds show considerable differences, which
implies that the separation of the electron–hole pairs can be enhanced, thus
improving the photoelectric efficiency. By considering the Cd dopants, we do
not observe much variation on the value of the charge carrier masses, whereas
a decrease of the band-gap widths is observed.

The temperature behavior of the experimental EFG $V_{zz}$ fractions, labeled
EFGSMGO1 and EFGCMGO1, respectively related to the Cd probe on the SrMnGe2O6
and CaMnGe2O6 compounds, fits precisely, according to the Bayer-Kushida model
[37, 38, 39], with a single phonon frequency, producing frequencies of
$8.01(7)\times 10^{12}$ and $10.0(2)\times 10^{12}$ respectively, suggesting
that their behavior can be attributed to a phonon contribution to $V_{zz}$.

The hyperfine parameters were computed to obtain the EFGs and these were
compared to PAC measurement results. Data between both sets are consistent
with the two obtained experimental fractions, EFGSMGO1 and EFGSMGO2, when
considering a supercell approach with a Cd ion replacing Sr and Mn sites,
respectively. We also find that this same correspondence seems to be true for
the CaMnGe2O6 compound, where the EFGCMGO1 and EFGCMGO2 experimental fractions
can be atributed, respectively, to Cd replacing Ca and Mn ions. Furthermore,
through the combination of theoretical ab-initio calculations and PAC results,
we could infer that the Cd probe in the AMnGe2O6 clinopyroxene compounds, with
$\textit{A}=\rm{Ca,Sr}$, can replace the ion located at the $A$-site as well
as the Mn ion, offering new perspectives for new systems synthesis with
possibly improved funcionalities.

###### Acknowledgements.

The authors acknowledge the support of the technical teams at ISOLDE for their
exceptional work in delivering high-quality beams for the presented PAC
measurements. The authors also acknowledge project NECL under
NORTE-01-0145-FEDER-022096, and FCT projects UIDP/04968/2020
(doi.org/10.54499/UIDP/04968/2020), UIDB/04968/2020 (doi.
org/10.54499/UIDB/04968/2020), LaP-MET LA/P/0095/2020,
POCI-01-0145-FEDER-029454, POCI-01-0145-FEDER-032527 and CERN/FIS-
TEC/0003/2021. Also, BMBF through grants 05K16PGA and 05K22PGA, EU Horizon
Europe Framework research and innovation programme under grant agreement no.
101057511 (EURO-LABS) for supporting IS679 ISOLDE-CERN experiment, and PRACE
project with reference 2021240118, with access to the Irene Skylake computer.
RPM acknowledges support from the Project HPC-EUROPA3 (INFRAIA-2016-1-730897),
with the support of the EC Research Innovation Action under the H2020
Programme, with reference HPC171030W; in particular, the support of Dr. A.
Stroppa and the hospitality provided by CNR-SPIN at the Department of Physical
and Chemical Science of the University of L’Aquila (Italy) during the visit
from 15/10/2021 to 20/12/2021 as well as computer resources and technical
support provided by CINECA. RPM also acknowledges computer resources and
technical support from Minho Advanced Computing Center through project
CPCA/A1/460622/2021, financed by FCT. LVCA (Project 314884/2021-1) and HMP
(Project 308438/2022-1) acknowledge funding from CNPq and support from FAPESP
(Project 2022/10095-8). AMLL acknowledge the FCT 2021.04084.CEECIND
(doi.org/10.54499/2021.04084.CEECIND/CP1655/CT0002) and ELdS the
2022.00082.CEECIND (doi.org/10.54499/2022.00082.CEECIND/CP1719/CT0001) grants.
ELdS further acknowledges the High Performance Computing Chair - a R&D
infrastructure (based at the University of Évora; PI: M. Avillez), endorsed by
Hewlett Packard Enterprise, and involving a consortium of higher education
institutions, research centers, enterprises, and public/private organizations.

##  Appendix A Calculation of Formation Energies

The formation energies for the studied oxides of the AMnGe2O6 (A = Be, Mg, Ca,
Sr) clinopyroxene series, were calculated according to the following equation:

$\displaystyle
E_{form}=E_{\ce{\textit{A}MnGe2O6}}-E_{\ce{\textit{A}GeO3}}-E_{\ce{MnGeO3}},$
(6)

where $E_{\ce{\textit{A}MnGe2O6}}$, $E_{\ce{\textit{A}GeO3}}$,
$E_{\ce{MnGeO3}}$ are the total electronic energies obtained from DFT+$U$
calculations, per formula unit (f.u.), which are summarised in table
LABEL:table:energies.

Table A1: Electronic total energies of the AMnGe2O6 clinopyroxene compounds (A = Be, Mg, Ca, Sr) and their respective constituent AGeO3 and MnGeO3 oxides. System |  Energy (eV/f.u.)
---|---
SrMnGe2O6 | -19433.62
CaMnGe2O6 | -17709.36
MgMnGe2O6 | -17815.41
BeMnGe2O6 | -15984.00
SrGeO3 | -9540.40
CaGeO3 | -7816.06
MgGeO3 | -7922.42
BeGeO3 | -6091.01
MnGeO3 | -9893.11

##  Appendix B Electric Field Gradient Parameters

Table B1: Computational EFG parameters at the Sr, Ca, Mn, Ge, and Cd nuclei
for $\rm{SrMnGe_{2}O_{6}}$, $\rm{SrMnGe_{2}O_{6}}$:$\rm{Cd_{Sr}}$, and
$\rm{SrMnGe_{2}O_{6}}$:$\rm{Cd_{Mn}}$ compounds, and for
$\rm{CaMnGe_{2}O_{6}}$, $\rm{CaMnGe_{2}O_{6}}$:$\rm{Cd_{Ca}}$, and
$\rm{SrMnGe_{2}O_{6}}$:$\rm{Cd_{Mn}}$ compounds.

System | EFG | Cd | Sr(1) | Sr(2) | Sr(3) | Sr(4) | Mn(1) | Mn(2) | Mn(3) | Mn(4) | Ge(1) | Ge(2) | Ge(3) | Ge(4)
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\rm SrMnGe_{2}O_{6}$ |  $|V_{zz}|$ (V/Å2) | - | 64 | - | - | - | 14 | - | - | - | 99 | - | - | -
$\eta$ | - | 0.54 | - | - | - | 0.32 | - | - | - | 0.70 | - | - | -
$\rm Sr_{7/8}Cd_{1/8}MnGe_{2}O_{6}$ | $|V_{zz}|$ | 108 | 61 | 62 | 63 | 66 | 10 | 19 | 17 | 12 | 101 | 97 | 102 | 101
$\eta$ | 0.14 | 0.46 | 0.71 | 0.53 | 0.83 | 0.64 | 0.98 | 0.44 | 0.84 | 0.60 | 0.64 | 0.64 | 0.69
$\rm Sr_{3/4}Cd_{1/4}MnGe_{2}O_{6}$ | $|V_{zz}|$ | 103 | 59 | 65 | 64 | - | 19 | 17 | 14 | 15 | 99 | 103 | 91 | 85
$\eta$ | 0.14 | 0.88 | 0.82 | 0.66 | - | 0.42 | 0.75 | 0.60 | 0.16 | 0.55 | 0.64 | 0.55 | 0.67
$\rm{SrMn_{7/8}Cd_{1/8}Ge_{2}O_{6}}$ | $|V_{zz}|$ | 21 | 57 | 67 | 64 | 68 | 14 | 14 | 11 | 14 | 99 | 99 | 101 | 100
$\eta$ | 0.74 | 0.47 | 0.63 | 0.57 | 0.59 | 0.28 | 0.46 | 0.63 | 0.35 | 0.71 | 0.69 | 0.69 | 0.69
$\rm{SrMn_{3/4}Cd_{1/4}Ge_{2}O_{6}}$ | $|V_{zz}|$ | 22 | 59 | 64 | 69 | 66 | 15 | 10 | 15 | - | 99 | 101 | 100 | 102
$\eta$ | 0.67 | 0.56 | 0.59 | 0.63 | 0.64 | 0.58 | 0.61 | 0.23 | - | 0.71 | 0.67 | 0.73 | 0.69
System | EFG | Cd | Ca(1) | Ca(2) | Ca(3) | Ca(4) | Mn(1) | Mn(2) | Mn(3) | Mn(4) | Ge(1) | Ge(2) | Ge(3) | Ge(4)
$\rm CaMnGe_{2}O_{6}$ | $|V_{zz}|$ | - | 34 | - | - | - | 8 | - | - | - | 96 | - | - | -
$\eta$ | - | 0.52 | - | - | - | 0.49 | - | - | - | 0.64 | - | - | -
$\rm Ca_{7/8}Cd_{1/8}MnGe_{2}O_{6}$ | $|V_{zz}|$ | 109 | 33 | 35 | 34 | 38 | 5 | 8 | 11 | 9 | 99 | 97 | 98 | 99
$\eta$ | 0.01 | 0.45 | 0.34 | 0.45 | 0.33 | 0.43 | 0.30 | 0.69 | 0.93 | 0.52 | 0.60 | 0.60 | 0.60
$\rm Ca_{3/4}Cd_{1/4}MnGe_{2}O_{6}$ | $|V_{zz}|$ | 108 | 35 | 38 | 35 | - | 2 | 12 | 11 | 8 | 99 | 100 | 96 | 86
$\eta$ | 0.02 | 0.13 | 0.27 | 0.29 | - | 0.09 | 0.11 | 0.15 | 0.95 | 0.50 | 0.58 | 0.44 | 0.67
$\rm{CaMn_{7/8}Cd_{1/8}Ge_{2}O_{6}}$ | $|V_{zz}|$ | 12 | 30 | 37 | 36 | 36 | 7 | 7 | 10 | 7 | 98 | 97 | 99 | 97
$\eta$ | 0.21 | 0.42 | 0.41 | 0.38 | 0.39 | 0.76 | 0.92 | 0.34 | 0.82 | 0.63 | 0.61 | 0.60 | 0.66
$\rm{CaMn_{3/4}Cd_{1/4}Ge_{2}O_{6}}$ | $|V_{zz}|$ | 12 | 31 | 37 | 38 | 37 | 7 | 10 | 8 | - | 97 | 99 | 98 | 100
$\eta$ | 0.31 | 0.42 | 0.39 | 0.40 | 0.39 | 0.37 | 0.38 | 0.96 | - | 0.63 | 0.58 | 0.64 | 0.59

The calculated EFG parameters are presented in table B1. From these results we
can observe that the EFGs at the Ge nuclei were only slightly affected by the
Cd substitution, as evidenced by the fact that the symmetry of the Ge sites
remained quite similar to SrMnGe2O6 and CaMnGe2O6 basic compounds after Cd
substitution. The same cannot be said for the Sr/Ca and Mn sites, which in
many cases deviated substantially.

##  Appendix C Electronic Band Dispersion of the Cd Doped Systems

The bandstructure and the density of states for the AMnGe2O6 (A = Ca, Sr)
clinopyroxene compounds with Cd substitution in $A$\- and Mn-sites are
presented in figures C1, C2, C3, C4, C5, C6, C7, C8 and C9.

Figure C1: Electronic bandstructure (left) and density of states (right) for
Sr7/8Cd1/8MnGe2O6 with antiferromagnetic ordering. Figure C2: Electronic
bandstructure (left) and density of states (right) for Sr3/4Cd1/4MnGe2O6 with
antiferromagnetic ordering. Figure C3: Electronic bandstructure (left) and
density of states (right) for SrMn7/8Cd1/8Ge2O6 with ferromagnetic ordering.
Figure C4: Electronic bandstructure (left) and density of states (right) for
SrMn3/4Cd1/4Ge2O6 with ferromagnetic ordering. Figure C5: Electronic
bandstructure (left) and density of states (right) for Ca7/8Cd1/8MnGe2O6 with
antiferromagnetic ordering. Figure C6: Electronic bandstructure (left) and
density of states (right) for Ca3/4Cd1/4MnGe2O6 with antiferromagnetic
ordering. Figure C7: Electronic bandstructure (left) and density of states
(right) for CaMn7/8Cd1/8Ge2O6 with ferromagnetic ordering Figure C8:
Electronic bandstructure and density of states for CaMn3/4Cd1/4Ge2O6 with
antiferromagnetic ordering. Figure C9: Electronic bandstructure (left) and
density of states (right) for CaMn3/4Cd1/4Ge2O6 with ferromagnetic ordering.

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