# A finite-bound partition equinumerosity result generalizing
a solution of a problem posed by Andrews and Deutsch

Michael J. Schlosser  Fakultät für Mathematik, Universität Wien
Oskar-Morgenstern-Platz 1, A-1090, Vienna, Austria
<EMAIL_ADDRESS>and  Nicolas Allen Smoot  Fakultät für
Mathematik, Universität Wien
Oskar-Morgenstern-Platz 1, A-1090, Vienna, Austria
<EMAIL_ADDRESS>
###### Abstract.

We introduce a finite-bound extension of a partition equinumerosity result
which was orignally proposed as a problem by Andrews and Deutsch in 2016, and
given a generalized form in 2018 by Smoot and Yang. We also give a simple
bijective proof, itself an extension of the proof given of the 2018
generalization.

###### Key words and phrases:

integer partitions; partition bijection

###### 2020 Mathematics Subject Classification:

Primary 05A19, Secondary 11P81

The first author was partially supported by Austrian Science Fund FWF
10.55776/P32305.

The second author was fully supported by Austrian Science Fund FWF
10.55776/PAT6428623.

##  1\. Background

In 2016 Andrews and Deutsch proposed the following problem:

[American Mathematical Monthly Problem 11908] Let $n,k$ be nonnegative
integers. Show that the number of partitions of $n$ having $k$ even parts is
the same as the number of partitions of $n$ in which the largest repeated part
is $k$ (defined to be 0 if the parts are all distinct).

As an example, they list the three partitions of 7 with two even parts:

$\displaystyle 4+2+1,\ \ \ \ \ 3+2+2,\ \ \ \ \ 2+2+1+1+1,$

and the three partitions of 7 in which the largest repeated part is 2:

$\displaystyle 3+2+2,\ \ \ \ \ 2+2+2+1,\ \ \ \ \ 2+2+1+1+1.$

The counting functions for each class of partitions has a natural generating
function, and Madhyastha gave a proof of the problem through manipulation of
their generating functions [6, Solution I], showing that

$\displaystyle\frac{q^{2k}}{(q^{2};q^{2})_{k}}\frac{(q^{2};q^{2})_{\infty}}{(q;q)_{\infty}}=\frac{q^{2k}}{(q;q)_{k}}\frac{(q^{2(k+1)};q^{2})_{\infty}}{(q^{k+1};q)_{\infty}}.$
(1.1)

Here, and in the first proof of Theorem 2.3, $q$ is a formal variable; for any
indeterminate $a$ and non-negative integer $k$ (we also allow $k=\infty$) the
$q$-shifted factorial is defined as $(a;q)_{k}=\prod_{0\leq j<k}(1-aq^{j})$
(empty products being $1$).

However, two additional discoveries have been made regarding this problem.
First, it may be generalized, as was discovered by Nicolas Allen Smoot and
Mingjia Yang [6, Solution II, Editorial comment], [4, Theorem 23]:

######  Theorem 1.1 (N. Smoot, M. Yang).

Let $n,k,d$ be nonnegative integers. The number of partitions of $n$ with $k$
parts divisible by $d$ is the same as the number of partitions of $n$ in which
$k$ is the largest part to occur at least $d$ times.

Moreover, a delightful proof by direct construction may be given for this
generalization [6, Solution II].

###### Proof.

Let $\lambda$ be a partition of $n$ with $k$ parts divisible by $d$. Write

$\displaystyle\lambda=\mu+o,$

in which $\mu$ is the subpartition containing the $k$ parts divisible by $d$,
and $o$ contains the remaining parts.

We map $\mu$ to its conjugate, which we denote $\mu^{\ast}$. Notice that
$\mu^{\ast}$ contains $k$ as its largest part, and that indeed $k$ must repeat
$dq_{k}$ times for some integer $q_{k}>0$.

Next we map $o$, which contains no parts divisible by $d$, to its
corresponding partition $\delta$ in which every part may only repeat up to
$d-1$ times—say, using Glaisher’s map [3]. Both of these actions constitute
bijections. Our new partition, then, is

$\displaystyle\kappa=\mu^{\ast}+\delta.$

Notice that any part $i$ of $\kappa$ which is less than or equal to $k$ will
occur $dq_{i}+r_{i}$ times, in which $dq_{i}$ comes from $\mu^{\ast}$ and
$q_{i}$ is a nonnegative integer (which is possibly 0 except when $i=k$), and
$r_{i}$, with $0\leq r_{i}\leq d-1$ comes from $\delta$. Thus parts less than
$k$ may repeat any number of times. On the other hand, any part greater than
$k$ must only have a contribution from $\delta$, and must therefore occur
strictly less than $d$ times.

This mapping is well-defined, as well as easily reversable: for any partition
$\kappa$ in which $k$ is the largest part to occur at least $d$ times, take
all parts $i$ which occur $dq_{i}+r_{i}$ times with $q_{i}\geq 0$ (which is
positive for $i=k$), and define the subpartition $\mu^{\ast}$ by collecting
$i$ to occur $dq_{i}$ times. The largest part will be $k$. What is left will
be parts $i$ which repeat $r_{i}$ times for $0\leq r_{i}<d$. Collect these
terms into $\delta$.

We map $\mu^{\ast}$ to its conjugate $\mu$, which will contain $k$ parts, all
divisible by $d$. Also we map $\delta$ to $o$, which contains no multiples of
$d$. Our new partition is $\lambda=\mu+o$. ∎

##  2\. A finite-bound generalization of Theorem 1.1

The proof of Theorem 1.1 is especially satisfying, but it is not obvious that
anything more can be done with it. However, we have found that Theorem 1.1 may
be extended to a finite-bound result.

######  Definition 2.1.

Let $A(n,k,d,m)$ be the set of integer partitions of $n$ in which that exactly
$k$ parts are divisible by $d$, and all other parts are strictly less than
$md$.

######  Definition 2.2.

Let $B(n,k,d,m)$ be the set of integer partitions of $n$ such that:

  * •

For $m<k$, the largest part is $kd$, and all parts greater than $md$ are
divisible by $d$;

  * •

For $m\geq k$, the part $k$ occurs at least $d$ times, none of the parts
exceed $md$, and any part $i$ such that $k<i\leq m$ occurs less than $d$
times.

With these definitions, we have the following:

######  Theorem 2.3.

For $n,k,d,m\in\mathbb{Z}_{\geq 1}$,
$\left|A(n,k,d,m)\right|=\left|B(n,k,d,m)\right|$.

Of course, for $m\rightarrow\infty$ (indeed, for $m>n$), this result reduces
to Theorem 1.1.

We give two different proofs of Theorem 2.3. First, we demonstrate the
following identity, a $q$-series extension of (1.1) used in Madhyastha’s
proof:

$\displaystyle\frac{q^{dk}}{(q^{d};q^{d})_{k}}\frac{(q^{d};q^{d})_{m}}{(q;q)_{dm}}=\begin{cases}\frac{q^{kd}}{(q^{d(m+1)};q^{d})_{k-m}}\frac{1}{(q;q)_{md}},&\text{
if }m<k,\\\ &\\\
\frac{q^{kd}}{(q;q)_{k}}\frac{(q^{d(k+1)};q^{d})_{m-k}}{(q^{k+1};q)_{m-k}}\frac{1}{(q^{m+1};q)_{md-m}},&\text{
if }m\geq k.\end{cases}$ (2.1)

We first interpret (2.1). Examining the left-hand side,

$\displaystyle\frac{q^{dk}}{(q^{d};q^{d})_{k}}\frac{(q^{d};q^{d})_{m}}{(q;q)_{dm}},$
(2.2)

we notice that the first factor of (2.2) enumerates partitions into exactly
$k$ parts divisible by $d$. The second factor of (2.2) enumerates partitions
in which no parts are divisible by $d$, but moreover one in which all parts
are bounded above by $md$; indeed, because the part $md$ is specifically
excluded, the parts must be strictly less than $md$. Thus the left-hand side
of (2.1) enumerates partitions of $A(n,k,d,m)$.

Examining the right-hand side of (2.1) in the case that $m<k$, we note that
the very last factor, $1/(q;q)_{md}$, enumerates partitions into parts of size
up to $md$, while $1/(q^{d(m+1)};q^{d})_{k-m}$ enumerates partitions of size
larger than $md$ but smaller than $kd$, in which the parts are divisible by
$d$. Finally, the factor of $q^{kd}$ confirms that $kd$ does in fact occur.
Thus, we have partitions of type $B(n,k,d,m)$ for $m<k$ enumerated by the
right-hand side of (2.1).

For $m\geq k$, $1/(q;q)_{k}$ enumerates partitions into parts of size up to
and including $k$ with no additional restrictions. To insure that $k$ occurs
at least $d$ times, we have the factor $q^{kd}=q^{k+k+\cdots+k}.$ On the other
hand, the factor $1/(q^{m+1};q)_{md-m}$ enumerates partitions into parts of
size strictly greater than $k$ and less than or equal to $md$, but with no
other restrictions. Finally, the factor

$\displaystyle\frac{(q^{d(k+1)};q^{d})_{m-k}}{(q^{k+1};q)_{m-k}}$

consists of factors of the form

$\displaystyle\frac{1-q^{dj}}{1-q^{j}}=1+q^{j}+q^{2j}+\cdots+q^{(d-1)j},$

for $k<j\leq m$. Thus, the parts strictly larger than $k$ and less than or
equal to $m$ must occur fewer than $d$ times. Thus the right-hand side of
(2.1) enumerates $B(n,k,d,m)$ in both cases.

###### Proof.

We have two cases to prove; but in both cases, we begin with (2.2). First we
suppose that $m<k$. Then we have

$\displaystyle(q^{d};q^{d})_{k}=(q^{d};q^{d})_{m}(q^{d(m+1)};q^{d})_{k-m}.$

Thus substituting this for $(q^{d};q^{d})_{k}$ in (2.2), we have

$\displaystyle\frac{q^{dk}}{(q^{d};q^{d})_{k}}\frac{(q^{d};q^{d})_{m}}{(q;q)_{dm}}$
$\displaystyle=\frac{q^{dk}}{(q^{d};q^{d})_{m}(q^{d(m+1)};q^{d})_{k-m}}\frac{(q^{d};q^{d})_{m}}{(q;q)_{dm}}$
$\displaystyle=\frac{q^{dk}}{(q^{d(m+1)};q^{d})_{k-m}}\frac{1}{(q;q)_{dm}}.$

Similarly, for $m\geq k$,

$\displaystyle(q^{d};q^{d})_{m}=(q^{d};q^{d})_{k}(q^{d(k+1)};q^{d})_{m-k}.$

Substituting into (2.2),

$\displaystyle\frac{q^{dk}}{(q^{d};q^{d})_{k}}\frac{(q^{d};q^{d})_{m}}{(q;q)_{dm}}$
$\displaystyle=\frac{q^{dk}}{(q^{d};q^{d})_{k}}\frac{(q^{d};q^{d})_{k}(q^{d(k+1)};q^{d})_{m-k}}{(q;q)_{dm}}$
(2.3) $\displaystyle=q^{dk}\frac{(q^{d(k+1)};q^{d})_{m-k}}{(q;q)_{dm}}.$ (2.4)

Next, we note that

$\displaystyle(q;q)_{dm}=(q;q)_{k}(q^{(k+1)};q)_{m-k}(q^{(m+1)};q)_{md-m}.$

With this substitution into (2.3)–(2.4) we have

$\displaystyle q^{dk}\frac{(q^{d(k+1)};q^{d})_{m-k}}{(q;q)_{dm}}$
$\displaystyle=q^{dk}\frac{(q^{d(k+1)};q^{d})_{m-k}}{(q;q)_{k}(q^{(k+1)};q)_{m-k}(q^{(m+1)};q)_{md-m}}$
$\displaystyle=\frac{q^{dk}}{(q;q)_{k}}\frac{(q^{d(k+1)};q^{d})_{m-k}}{(q^{(k+1)};q)_{m-k}}\frac{1}{(q^{(m+1)};q)_{md-m}}.\hfill\qed$

However, we have also discovered a corresponding extension of the direct
constructive proof that we already gave. We note that our new proof depends on
the proof of a finite version of Glaisher’s theorem which can be found in [5];
the latter is itself an extension of the finite version of Euler’s partition
identity proved by Andrews in [1]. Both of these finite bound theorems play a
substantial role in achieving finite-bound equinumerosity results, e.g., [2].

###### Proof.

Let $\lambda\in A(n,k,d,m)$. We write

$\displaystyle\lambda=\mu+o,$

in which $\mu$ contains the $k$ parts divisible by $d$, and $o$ contains the
remaining parts.

We begin by considering the subpartition $\mu$. Here we map $\mu$ to its
conjugate, $\mu^{\ast}$, in which $k$ occurs at least $d$ times (indeed, the
number of occurrences of $k$ is a positive multiple of $d$). We can therefore
write $\mu^{\ast}$ as

$\displaystyle\mu^{\ast}:\sum_{i=1}^{k}dq_{i}\cdot(i),$

in which we interpret the parts as $i$, and the number of occurrences as
$dq_{i}$ for some $q_{i}\in\mathbb{Z}_{\geq 0}$ (with $q_{k}>0$).

We will in turn break $\mu^{\ast}=\mu^{\ast}_{0}+\epsilon$ into two smaller
subpartitions:

$\displaystyle\mu^{\ast}_{0}$
$\displaystyle:\sum_{i=1}^{\mathrm{min}(m,k)}dq_{i}\cdot(i),\ \ \ \ \
\epsilon:\sum_{m<i\leq k}q_{i}\cdot(di).$

For $\epsilon$ we reinterpret the terms: the parts are now $di$ as $i$ ranges
from 1 to $m$. Notice that for $m<k$, the largest part from either of these
subpartitions will be $dk$, and it will come from $\epsilon$. Note also that
$\epsilon$ is empty for $m\geq k$.

Next, we consider the subpartition $o$. Notice that these parts are bounded
strictly above by $md$, and that none of them are divisible by $d$. Write

$\displaystyle o:\sum_{\begin{subarray}{c}1\leq j<md,\\\ d\nmid
j\end{subarray}}N_{j}\cdot(j),$

in which $j$ is a given part, and $N_{j}\geq 0$ the number of occurrences of
that part.

For each $j\leq md$ there exists a unique integer $L_{j}\geq 0$ such that

$\displaystyle m<jd^{L_{j}}\leq md.$

We write $N_{j}$ in base $d$, but we collect together all terms corresponding
to powers of $d$ which match or exceed $L_{j}$:

$\displaystyle N_{j}$ $\displaystyle=\sum_{l\geq 0}a_{j,l}\cdot
d^{l}=\sum_{0\leq l\leq L_{j}-1}a_{j,l}\cdot d^{l}+M_{j}\cdot d^{L_{j}}.$
(2.5)

Here for $l\geq 0$, $0\leq l\leq L_{j}-1$ we have $0\leq a_{j,l}\leq d-1$. Now
we re-examine our original subpartition $o$:

$\displaystyle\sum_{\begin{subarray}{c}1\leq j<md,\\\ d\nmid
j\end{subarray}}N_{j}\cdot(j)=\sum_{\begin{subarray}{c}1\leq j<md,\\\ d\nmid
j\end{subarray}}\left(\sum_{0\leq l\leq L_{j}-1}a_{j,l}\cdot d^{l}+M_{j}\cdot
d^{L_{j}}\right)\cdot(j).$

We thus have our new subpartition

$\displaystyle\delta:\sum_{\begin{subarray}{c}1\leq j<md,\\\ d\nmid j,\\\
0\leq l\leq L_{j}-1\end{subarray}}a_{j,l}\cdot\left(j\cdot
d^{l}\right)+\sum_{\begin{subarray}{c}1\leq j<md,\\\ d\nmid
j\end{subarray}}M_{j}\cdot\left(j\cdot d^{L_{j}}\right).$ (2.6)

The first sum contributes parts of the form $i=j\cdot d^{l}$ for $0\leq l\leq
L_{j}-1$, and therefore $1\leq i\leq m$; indeed, by unique factorization, any
possible part between 1 and $m$ must have this form. Moreover, each $i$ occurs
less than $d$ times.

The parts which are permitted to repeat arbitrarily many times are $i=j\cdot
d^{L_{j}}$, with $m<i\leq md$. Again, by the possible range of $j$ and through
unique factorization, every possible part between $m$ and $md$ must have this
form. These parts repeat $M_{j}$ times, for any integer $M_{j}\geq 0$.

We will send $\lambda$ to the following partition:

$\displaystyle\kappa=\mu^{\ast}_{0}+\epsilon+\delta.$

We need to show that $\kappa\in B(n,k,d,m)$.

Case: $m<k$
In this case, $\epsilon$ is not empty, and has largest part $dk$. The parts of
$\mu^{\ast}_{0}$ are all $\leq m<k$, and the parts of $\delta$ are all $\leq
md<kd$. Thus our largest part will be $kd$.

A part $i$ with $1\leq i\leq m$ will have the form $i=jd^{l}$ for some $j$
between 1 and $m$, and some nonnegative power $l$ of $d$. It occurs
$dq_{i}+a_{j,l}$ times; here $dq_{i}$ comes from $\mu^{\ast}_{0}$, and
$a_{j,l}$ comes from $\delta$. Per the conditions of the division algorithm,
this is an unrestricted nonnegative integer.

Any part $i$ with $m<i\leq md$, has the form $jd^{L_{j}}$ and also occurs an
unrestricted number $M_{j}$ of times, coming from $\delta$. Neither
$\mu^{\ast}_{0}$ nor $\epsilon$ contribute to these parts.

Finally, for parts greater than $md$, we have only terms $dj$ from $\epsilon$.
The only restrictions is by size (the parts must be greater than $md$), and
divisibility by $d$.

We thus have a well-defined partition $\kappa\in B(n,k,d,m)$ with no set
restrictions beyond those of $B(n,k,d,m)$.

Case: $m\geq k$
In this case, $\epsilon$ is empty, and the largest part from $\mu^{\ast}_{0}$
is $k$. Indeed, any part $i$ with $1\leq i\leq k$ occurs $dq_{i}$ times from
$\mu^{\ast}_{0}$. It may also occur in $\delta$; however, because $i\leq k\leq
m$, it has the form $i=jd^{l}$ with $d\nmid j$ and $0\leq l\leq L_{j}-1$, and
can only occur $a_{j,l}$ times, i.e., no more than $d-1$ times. Thus the
number of occurrences of $i$ in $\kappa$ has the form $dq_{i}+a_{j,l}$; it is
therefore arbitrary, per the conditions of the division algorithm.
Specifically for $i=k$, we have the additional condition that $q_{k}>0$, so
that $k$ must occur at least $d$ times.

Similary, all parts $i>m$ can only come from $\delta$, and be $\leq md$.
Moreover, we have already shown that such an $i$ may repeat $M_{j}$ times,
with $i=jd^{L_{j}}$, $d\nmid j$.

Only the parts $i$ with $k<i\leq m$ have a restriction on their occurrences,
appearing less than $d$ times.

Finally, the largest parts in $\mu^{\ast}$ are $k\leq m\leq md$, and the
largest possible parts of $\delta$ are $md$. Thus $\kappa\in B(n,k,d,m)$. ∎

For the sake of illustration, we give two examples. First, we take $d=3$,
$k=7$, $m=4$, $n=123$. Notice that $m<k$. Moreover, we have $md=12$ and
$kd=21$. We consider the partition

$\displaystyle\lambda$ $\displaystyle:=(2)15+12+11+9+8+(4)7+(2)6+5+3+(2)2+1.$

The parts divisible by 3 are 15 (occuring twice), 12, 9, 6 (occuring twice),
and 3. That is, there are $k=7$ parts divisible by $d=3$. The largest part not
divisible by 3 is $11<12=md$. Thus, $\lambda\in A(123,7,3,4)$.

We build our bijection first by separating $\lambda$ into its key
subpartitions:

$\displaystyle\mu:\ $ $\displaystyle(2)15+12+9+(2)6+3,$ $\displaystyle o:\ $
$\displaystyle 11+8+(4)7+5+(2)2+1.$

First,we map $\mu$ to its conjugate $\mu^{\ast}$:

$\displaystyle(2)15+12+9+(2)6+3\longleftrightarrow(3)7+(3)6+(3)4+(3)3+(3)2.$

Notice that each part $i$ of $\mu^{\ast}$ lies between 1 and $k=7$, and occurs
a multiple of 3 times (possibly 0).

In turn we split $\mu^{\ast}$ into $\mu^{\ast}_{0}$ and $\epsilon$. Because
$\mathrm{min}(m,k)=m$, $\epsilon$ is not empty. We construct $\epsilon$ by
examining the parts of $\mu^{\ast}$ which are greater than $m=4$. The factor
of 3 in the number of occurences is now multiplied by our part to produce a
new part. What is left of the number of occurences after dividing out 3 is our
new occurence:

$\displaystyle(3)7+(3)6$ $\displaystyle=(3\cdot 1)7+(3\cdot
1)6\longleftrightarrow(1)21+(1)18=\epsilon.$

The remaining parts of $\mu^{\ast}$ give us $\mu^{\ast}_{0}$:

$\displaystyle\mu^{\ast}_{0}=(3)4+(3)3+(3)2.$

We now turn our attention to

$\displaystyle o:\ $ $\displaystyle 11+8+(4)7+5+(2)2+1.$

For $1\leq j<12,3\nmid j$, there exists a unique $L_{j}$ such that
$4<jd^{L_{j}}\leq 12$. We may compute $L_{j}$ for each part $j$ of $o$:

$\displaystyle\text{For }j=11,8,7,5,L_{j}=0.$ $\displaystyle\text{For
}j=2,L_{j}=1.$ $\displaystyle\text{For }j=1,L_{j}=2.$

Thus for $j=1,2$, we can expand the number of occurences by base $3$ up to the
power of 2 for $j=1$, and $1$ for $j=2$. But the occurences for these parts in
$o$ are 1 and 2, respectively, with resulting maximum power being 0 in the
base $3$ expansions. Therefore we can simply apply Glaisher’s bijection. In
this case, it is a trivial mapping:

$\displaystyle(2)2+(1)1\longrightarrow(2)2+(1)1.$

Since $L_{j}=0$ for the remaining parts of $o$, we will not apply Glaisher’s
bijection at all; we should multiply each $j$ by $3^{L_{j}}$. However, this
factor is simply 1, and we may therefore simply reproduce the partition $o$:

$\displaystyle\delta:\ $ $\displaystyle 11+8+(4)7+5+(2)2+1.$

Thus we have

$\displaystyle\kappa$
$\displaystyle=(1)21+(1)18+(3)4+(3)3+(3)2+11+8+(4)7+5+(2)2+1$
$\displaystyle=21+18+11+8+(4)7+5+(3)4+(3)3+(5)2+1.$

Our largest part is $kd=(7)(3)=21$, and the parts greater than $md=12$ are 21,
18, both divisible by $d=3$. Thus $\kappa\in B(123,7,3,4)$.

For our second example, we consider the $d=3$, $k=4$, $m=7$, $n=189$. Notice
that $m\geq k$. Moreover, we have $md=21$ and $kd=12$. Let us take, say,

$\displaystyle\lambda:=24+21+20+17+15+(4)14+9+(2)7+(5)2+(3)1.$

Here, the parts divisible by 3 are 24, 21, 15, 9. Thus there are $k=4$ parts
divisible by $d=3$. The largest part not divisibly by 3 is 20, which is less
than $md=21$. Thus, $\lambda\in A(189,4,3,7)$. Again, we split $\lambda$ into

$\displaystyle\mu$ $\displaystyle:\ 24+21+15+9,$ $\displaystyle o$
$\displaystyle:\ 20+17+(4)14+(2)7+(5)2+(3)1.$

We map $\mu$ to its conjugate $\mu^{\ast}$:

$\displaystyle 24+21+15+9\longleftrightarrow(9)4+(6)3+(6)2+(3)1.$

Because $m>k$, the subpartition $\epsilon$ is empty, and
$\mu^{\ast}_{0}=\mu^{\ast}$.

Next we take

$\displaystyle o:\ $ $\displaystyle 20+17+(4)14+(2)7+(5)2+(3)1.$

For $1\leq j<md,d\nmid j$, we compute $L_{j}$ as before, and we find that

$\displaystyle\text{For }j=20,17,14,L_{j}=0.$ $\displaystyle\text{For
}j=7,L_{j}=1.$ $\displaystyle\text{For }j=2,1,L_{j}=2.$

Now the mapping is nontrivial, and using (2.5)–(2.6) we get

$\displaystyle[(1)20+(1)17+(4)14]+[(2)7+(2\cdot 3^{0}+1\cdot 3^{1})2+(1\cdot
3^{1})1]$ $\displaystyle=$
$\displaystyle[(1)20+(1)17+(4)14]+[(2)7+(2)2+(1)6+(1)3].$

Putting our new subpartitions back together, we get

$\displaystyle[(9)4+(6)3+(6)2+(3)1]+[(2)7+(2)2+(1)6+(1)3]$
$\displaystyle+[(1)20+(1)17+(4)14],$ $\displaystyle\kappa$
$\displaystyle=20+17+(4)14+(2)7+6+(9)4+(7)3+(8)2+(3)1.$

Notice that $k=4$ occurs at least $d=3$ times (indeed, it occurs 9 times), and
all parts are bounded above by $md=21$. Of particular interest are the parts
larger than $k=4$ and less than or equal to $m=7$: $(2)7+6$. These parts occur
less than 3 times. Thus, $\kappa\in B(189,4,3,7)$.

## References

  * [1] G.E. Andrews, “Euler’s Partition Identity—Finite Version,” The Mathematics Student (85), pp. 3-4 (2016).
  * [2] S. Fu, T. Hua, “On the Slack Euler Pair for Vector Partition,” Integers (18), pp. 1-7 (2018).
  * [3] J.W.L. Glaisher, “A Theorem in Partitions,” Messenger of Math (12), pp. 158-170 (1883).
  * [4] W.P. Johnson, An Introduction to $q$-Analysis, American Mathematical Society, Providence, R.I. (2020).
  * [5] D. Nyirenda, “A Note on a Finite Version of Euler’s Partition Identity,” Australasian Journal of Combinatorics (71)(2), pp. 241-245 (2018).
  * [6] “A Generalized Bijection for Partitions,” Solutions to Problem 11908. G.A. Edgar, D.H. Ullman, D.B. West, Problems and Solutions, American Mathematical Monthly 125 (3) (2018).


# Reversal of nanomagnets by propagating magnons in ferrimagnetic yttrium iron
garnet enabling nonvolatile magnon memory

Korbinian Baumgaertl  Laboratory of Nanoscale Magnetic Materials and
Magnonics, Institute of Materials (IMX), École Polytechnique Fédérale de
Lausanne (EPFL), 1015 Lausanne, Switzerland  Dirk Grundler
<EMAIL_ADDRESS>Laboratory of Nanoscale Magnetic Materials and
Magnonics, Institute of Materials (IMX), École Polytechnique Fédérale de
Lausanne (EPFL), 1015 Lausanne, Switzerland  Institute of Electrical and Micro
Engineering (IEM), École Polytechnique Fédérale de Lausanne (EPFL), 1015
Lausanne, Switzerland

###### Abstract

Despite the unprecedented downscaling of CMOS integrated circuits, memory-
intensive machine learning and artificial intelligence applications are
limited by data conversion between memory and processor. There is a
challenging quest for novel approaches to overcome this so-called von Neumann
bottleneck. Magnons are the quanta of spin waves and transport angular momenta
through magnets. They enable power-efficient computation without charge flow
and would solve the conversion problem if spin wave amplitudes could be stored
directly in a magnetic memory cell. Here, we report the reversal of
ferromagnetic nanostripes by spin waves which propagate through an underlying
spin-wave bus made from yttrium iron garnet. Thereby, the charge-free angular
momentum flow is stored after transmission over a macroscopic distance. We
show that spin waves can reverse large arrays of ferromagnetic stripes at a
strikingly small power level of nW. Combined with the already existing wave
logic, our discovery is path-breaking for the new era of magnonics-based in-
memory computation and beyond von Neumann computer architectures.

Charge-based electronics exploiting semiconductors and metals have led to the
digital age and given rise to ever increasing data-processing power during the
last decades. The significant development slows down however due to
overheating of processors at increased clock frequencies as well as the data-
transfer bottleneck between charge-based processors and storage devices [1,
2]. So far, storage devices exploit reprogrammable states in materials classes
which differ from the semiconductors and therefore require signal conversion.
Waves for data processing like photons in optical computers already avoid,
both, charge flow and the excessive Joule heating at high clock frequencies.
However, even for optical computers the currently implemented von-Neumann
device architecture still provokes the signal-conversion bottleneck,
particularly in case of memory-intensive machine learning applications.
Magnonics which is based on charge-free angular momentum flow via spin waves
(magnons) promises to overcome the dilemmas. Making use of the bosonic
quasiparticles [3], it offers low-power consuming wave-based computing with
microwave signals up to THz frequencies [4, 5, 6]. A technology platform for
disruptive information technologies is foreseen if the so far elusive
nanomagnet switching by propagating spin waves would be experimentally
demonstrated. This switching process in magnonic circuits would be path-
breaking for both non-volatile storage of wave signals without conversion
losses and in-memory computation [7, 8, 2]. Experiments presented in Ref. [9]
are encouraging in that spin waves in an antiferromagnet excited by incoherent
spin torques reversed an integrated micromagnet. However, such spin waves can
not be used for wave logic. In Ref. [10], spin waves coherently excited in a
magnon conduit by a microwave antenna induced the modification of magnetic
domains. Here, the required power was however in the mW regime. We follow a
different approach in that we excite spin waves in the ferrimagnetic insulator
yttrium iron garnet ${\textrm{Y}}_{3}{\textrm{Fe}}_{5}{\textrm{O}}_{12}$ (YIG)
by radiofrequency (RF) signals and observe the magnon-induced reversal of
ferromagnetic nanostripes integrated on its top surface (Fig. 1a). The shape
anisotropy of the long and narrow permalloy (Py) stripes (green) allows for
bistable magnetic states. Thereby, they represent magnetic bits with states
”0” and ”1”, and enable the nonvolatile storage of information (Fig. 1b, I and
II). The underlying YIG (blue) allows us to transmit coherent magnons over
macroscopic distances as their decay lengths in thin YIG are on the few 10 to
100 $\mu$m length scale [11, 12]. They are far larger than electron scattering
lengths in semiconductors and metals. We note that samples similar to the one
shown in Fig. 1a have thoroughly been investigated recently in view of the so-
called grating coupler effect. Different groups explored nanostripe arrays on
YIG in their saturated states for the emission and detection of short-wave
magnons [13, 14, 15, 16]. The magnon modes were transmitted over long
distances [14] and imaged in real space by means of x-ray based magnetic
microscopy [16]. However, the groups did not investigate if states of
ferromagnetic nanostripes were different before and after spin-wave excitation
at different power levels. Here, we report experiments that we performed on
YIG spin-wave buses (delay lines) with integrated grating couplers (Fig. 1a)
and led to the discovery of irreversible switching of the nanostripes by
propagating magnons. By combining power-dependent spin-wave spectroscopy and
magnetic force microscopy we evidence the storage of magnon signals in the
YIG-based magnonic circuit containing ferromagnetic stripes. Importantly, the
magnon-induced reversal reported here is induced by magnons which are
transported over a macroscopic distance ($>20~{}\mu$m). This goes beyond Refs.
[9, 17, 18, 19, 20] which considered distances on the nm length scale. Our
finding is key for implementing data storage in wavelogic or neuromorphic
computing architectures in magnonics [21]. Spin waves in YIG have already been
shown to read out different magnetic states of integrated ferromagnetic
stripes (Fig. 1b, III) [14, 22]. Combined with our discovery, in-memory
computing and a complete read-and-write process of non-volatile magnetic bits
via magnons in YIG circuits are now possible.

### Reversal of ferromagnetic stripes on YIG observed by power-dependent spin-
wave spectroscopy

We studied different YIG-based magnonic circuits with integrated ferromagnetic
nanostripes which showed magnon-induced reversal of nanomagnets reported here.
We explored ferromagnetic nanostripes with widths between 50 and 200 nm,
lengths up to 27 $\mu$m, edge-to-edge separations down to 50 nm and different
orientations with respect to spin-wave emitters. In this publication we focus
on the magnonic circuit shown in Fig. 1a consisting of two nanostripe arrays
which form grating couplers [23] on YIG underneath coplanar waveguides (CPWs).
The insulating YIG with magnetization $M_{\rm YIG}$ is 100 nm thick. The
emitter (left) and detector (right) CPWs are connected to a vector network
analyzer (VNA) which provides electromagnetic (em) waves at GHz frequencies at
port 1. In such magnonic circuits information can be encoded in either the SW
amplitudes or phases [24, 25, 4, 26].

Figure 1: Magnonic memory effect–Reading and writing of magnetic bits by spin
waves. a Two lattices of bistable magnetic bits (Py nanostripes) underneath
CPWs on an insulating YIG film. b Depending on the SW amplitude bit writing (I
and II), reading (III) and data replication (IV) are achieved without charge
flow. c-e Transmission signals Mag(S21) taken at three different power levels
$P_{\mathrm{irr}}$. Analyzing signal strengths of mode branches (blue and
orange dashed lines), fields $H_{C1}$ and $H_{C2}$ are extracted reflecting
bit reversal. The red dashed lines indicate $+14~{}\mathrm{mT}$. Yield of bit
writing (dark) underneath f CPW1 and g CPW2 by propagating SWs at
$\mu_{0}H_{\mathrm{B}}=+14~{}$mT. The 50 % transition power levels
$P_{\mathrm{C1}}$ and $P_{\mathrm{C2}}$, respectively, are marked as black
dots. The error bars indicate the 70 % and 30 % transitions (Methods). h
Mag(S11) at $P_{\rm sens}$ and i Mag(S21) taken at +14 mT. In i, bits at CPW1
and CPW2 were magnetized to state ”1”.

The magnetic component of the em wave, $\mathbf{h}_{\rm rf}$ (field lines
sketched in Fig. 1a), is converted into propagating spin-wave (SW) signals at
the CPW via the torque $-\mid\gamma\mid\mu_{0}\mathbf{M}_{\rm
YIG}\times\mathbf{h}_{\rm rf}$, where $\gamma$ is the gyromagnetic ratio and
$\mu_{0}$ the vacuum permeability. We read out the SW signals both in
reflection configuration at the emitter CPW1 and, after transmission through
the YIG circuit, at the detector CPW2. The measured scattering parameters S11
and S21, respectively, are phase-resolved voltage signals induced by spin-
precessional motion. The VNA offers a frequency resolution of 1 Hz and
precisely adjusted power levels. In Fig. 1a, 25-$\upmu$m- and 27-$\upmu$m-long
ferromagnetic nanostripes are integrated between CPWs and YIG (see Methods and
Supplementary Fig. 1). These are bistable nanomagnets made from the
ferromagnetic metal $\mathrm{Ni_{81}Fe_{19}}$ (permalloy, Py). In earlier
works such bistable nanomagnets were investigated in the linear regime and
gave rise to the grating coupler (GC) effect [23, 13, 14, 16]. Here, we report
a nonlinear phenomenon in that GC stripes are reversed by spin waves which
propagate through YIG. We show that thereby the storage of SW signals occur.
To highlight the additional functionality of the ferromagnetic nanostripes we
consider them to represent magnetic bits with either state ”0” (magnetization
vector $\mathbf{M}_{\rm Py}$ pointing along $-y-$direction) or state ”1”
($\mathbf{M}_{\rm Py}$ along $+y-$direction). Our findings hence demonstrate
that propagating SWs in YIG can not only sense (”read”) the state of such
magnetic bits (Fig. 1 b) as reported before [14, 22], but also reverse
(”write”) magnetic states at appropriate power levels which can be still in
the linear SW excitation regime. The 100-nm-wide stripes considered here are
arranged in separated periodic arrays with a common lattice constant
$a=200~{}$nm. For the read-out (sensing) of ”0” and ”1” states we apply a
small power level $P_{\rm sens}$ of only $-25~{}$dBm at port 1 (or smaller)
and make use of the GC effect in S11 and S21 spectra. It has already been
shown that the signal strength and precise frequency of GC modes depended
decisively on the field direction and relative magnetization directions of
nanostripes and YIG [14, 15]. When the configuration of emitter and detector
grating couplers were not symmetric the detection of emitted GC modes was not
possible [27]. Using an in-plane magnetic field $H$ we reset the magnetic
states of stripes to a well-defined initial state (”erase cycle”) before
performing subsequent magnon-induced reversal experiments.
First, we show that the nanostripes in the magnonic circuit are switched from
”0” to ”1” by SWs exhibiting a certain amplitude. Figure 1 c-e displays the
magnitude Mag(S21) of the frequency-dependent voltages at CPW2 as a function
of field when measured for three different VNA power levels $P_{\mathrm{irr}}$
at CPW1. Dark branches indicate SW propagation. Initially, all nanostripes
were magnetized along the $-y-$direction (”erased”) by applying
$\mu_{0}H=-90~{}\mathrm{mT}$. Then $\mu_{0}H$ was increased in 1 mT steps to
the $+y-$direction. For all powers $P_{\mathrm{irr}}$, Mag(S21) contained a
prominent and continuous low-frequency branch (marked by green dashed lines)
attributed to the $k_{1}$-mode of SWs excited by CPW1 directly in the YIG film
[16]. The $k_{1}$-branch changed slope close to $\mu_{0}H\simeq
0~{}\mathrm{mT}$, indicating that the magnetization $M_{\rm YIG}$ of the soft
magnetic YIG film was aligned with the external magnetic field for values of
+1 mT and above. High frequency branches between roughly 7.5 and 12.5 GHz are
attributed to the GC effect. They showed distinct transitions (jumps) in
resonance frequencies. These frequency variations are known to indicate
irreversible switching of Py nanostripes [14, 15]. Following Refs. [28, 29],
the jumps occur at the switching fields at which the magnetization direction
of nanostripes reverse. We focus on the two prominent branches marked by blue
and orange dashed lines, which, in the following, we denote antiparallel (AP)
and parallel (P) mode, respectively (Supplementary Methods). $H_{C1}$
($H_{C2}$) defined the critical field value for which the signal strength of
the AP-mode (P-mode) decreased (increased) to 50 % of its maximum value (see
Methods and Supplementary Fig. 2). From Fig. 1 c to e, $H_{C1}$ and $H_{C2}$
diminished significantly with increasing $P_{\mathrm{irr}}$ indicating a
nonlinear process and the reversal of nanostripes. Before presenting the
magnon-induced reversal by means of frequency-resolved switching-yield maps
introduced below, it is instructive to present magnetic force microscopy (MFM)
data. We perform MFM at remanence to demonstrate that the magnon-induced
reversal of ferromagnetic nanostripes is non-volatile.

### Evidence of non-volatile state reversal after magnon transport in YIG

Figure 2 a displays an MFM image taken in the remnant configuration after we
performed a broadband VNA measurement with $P_{\mathrm{irr}}=-25$ dBm at
$\mu_{0}H_{\mathrm{B}}=+14~{}$mT on the sample with initially ”reset” magnetic
stripes.

Figure 2: Non-volatile storage of spin-wave signals over macroscopic distance.
MFM measurements taken after the sample with erased bits (state ”0”, shown as
white end) was irradiated at $\upmu_{0}H_{\mathrm{B}}=+14~{}\mathrm{mT}$ with
different powers a $P_{\mathrm{irr}}=-25$ dBm, b $-15$ dBm, and c $-5$ dBm in
the frequency range 0.1 GHz to 12.5 GHz. In this range, spin waves are excited
in YIG. Reversed magnetic bits (highlighted by broken lines) appear with a
black end. In c, beyond $x=37~{}\upmu\mathrm{m}$ (marked by a yellow arrow)
stripes remained magnetized in state ”0” (white end), consistent with the
decay of the SW amplitude when propagating away from the emitter CPW1.

The MFM data show nanostripes at CPW1 and CPW2 which are magnetized still
along $-y-$direction and opposite to $\mathbf{M}_{\rm YIG}$, consistent with
our definition of the AP branch (cf. Fig. 1 c). Note that
$\mu_{0}H_{\mathrm{B}}$ was intentionally below the evaluated onset field of
quasistatistic reversal of magnetic stripes (Supplementary Fig. 3) in order to
reveal SW-induced reversal processes. Figure 2 b shows the MFM data after
applying an increased power level of $P_{\mathrm{irr}}=-15$ dBm (31.6
$\upmu\mathrm{W}$) at $\mu_{0}H_{\mathrm{B}}=+14~{}$mT (cf. Fig. 1 d). The
remnant MFM image shows nanostripes which on the right side of the signal line
of CPW1 are reversed to the $+y-$direction (state ”1”). On the left side with
$x<-1.1~{}\upmu\mathrm{m}$ they are still magnetized along $-y-$direction.
Fallarino _et al._ [30] showed that SWs were excited nonreciprocally by a CPW
for the same field geometry as used in our experiments. A large (small) SW
amplitude was reported for the right (left) side of the signal line with $x>0$
($x<0$). We hence attribute the spatially inhomogeneous switching of
nanostripes to the non-uniform (nonreciprocal) amplitude distribution of SWs
excited in YIG. Stripes at CPW2 were still magnetized in $-y-$direction for
$P_{\mathrm{irr}}=-15$ dBm, with the exception of the two stripes on the left
edge. For this experiment with an intermediate power level, a large number of
stripes of the two gratings was magnetized in opposite directions. This
magnetic disorder explained the vanishingly small signal amplitude for GC
modes in Fig. 1 d at 14 mT [27]. After a VNA measurement on again reset
magnetic stripes with a high power of $P_{\mathrm{irr}}=-5$ dBm (c.f. Fig. 1
e) the MFM data show that the majority of stripes below the emitter CPW1 was
switched (Fig. 2 c). Below CPW2, about two thirds of the magnetic stripes were
switched to ”1” as well. Only the stripes farthest away from CPW1 showed still
the initial state ”0” (yellow arrow). The spatial variation agrees with a
magnon-induced reversal process which requires a threshold amplitude of the
SWs propagating under CPW2. As the amplitude of the propagating SWs decayed in
positive $x$-direction, the required amplitude was not available for a large
propagation distance from CPW1. The MFM-detected magnetic states hence
represent a non-volatile memory which records whether a certain threshold SW
amplitude has locally been present (state ”1”) or not (state ”0”). Thereby,
wave-based computational results can be stored without signal conversion. The
reestablished magnetic order underneath both CPWs by high VNA power in Fig. 2
c explains the observation of the branch P at +14 mT in Fig. 1 e. We note that
the significant asymmetry of switched nanostripes observed in Fig. 2 b and the
subsequently presented experimental data allow us to exclude microwave-induced
heating as the trigger for reversal. Heating in our continuous wave
experiments would be most likely mirror-symmetric with respect to $x=0$ of the
emitter CPW. The direct electromagnetic cross talk between CPW 1 and CPW2 is
-50 dB ($10^{-5}$), i.e., vanishingly small.

### Power-dependent reversal field reduction

Figure 3 a summarizes how critical fields $H_{C1}$ (blue line with dots) and
$H_{C2}$ (orange line with dots) extracted from VNA data depended on the power
used in VNA spectroscopy. We find that $H_{C1}$ ($H_{C2}$) decreases from 16.8
mT (33.4 mT) at $P_{\mathrm{irr}}=1~{}\mu\mathrm{W}$ to 2.7 mT (12.9 mT) at
$200~{}\mu\mathrm{W}$. This corresponds to a reduction of $H_{C1}$ by 84 % and
$H_{C2}$ by 61 %. Surprisingly, for large $P_{\mathrm{irr}}$ we find again an
increase of $H_{C1}$ and $H_{C2}$. We attribute this observation to the onset
of nonlinear SW scattering (Supplementary Fig. 4), which limits the peak
amplitude of SWs emitted from CPW1 [31]. For $P_{\mathrm{irr}}$ above -9 dBm
(shaded with gray in Fig. 3 a) we found a red-shift of the $k_{1}$-mode
resonance frequency (Supplementary Fig. 4), which is an indication for
nonlinear SW excitation [32]. The reoccurring increase of critical fields is
counterintuitive if heating of Py nanostripes by high-power microwaves would
be at the origin of the observed reversal phenomenon. If we plot $H_{C1}$ and
$H_{C2}$ versus
$\upmu_{0}h_{\mathrm{rf,x}}=\upmu_{0}\sqrt{P_{\mathrm{irr.}}/(2Z_{0}w_{\mathrm{L}}^{2})}$
[33] (Fig. 3 b), the slopes (straight lines) show that a small in-plane RF
field amplitude of only 0.35 mT provokes a critical field reduction by about
10 mT ($Z_{0}=50~{}\Omega$ and $w_{\mathrm{L}}=2.1~{}\mu$m, see Supplementary
figures). Hence, the RF-field-excited SWs in YIG reduce the stripes’ switching
fields about $30$ times more efficiently than a static Oersted field applied
oppositely to $\mathbf{M}_{\rm Py}$.

Figure 3: Efficiency analysis of reversal field reduction by linear and
nonlinear spin waves. a Dependence of critical fields $H_{C1}$ (solid blue
line) and $H_{C2}$ (solid orange line) extracted from VNA spectra on
$P_{\mathrm{irr}}$ applied in a broad frequency range from 0.1 GHz to 12.5
GHz. The power region attributed to nonlinear effects in the SW modes is
shaded in light gray. b $H_{C1}$ and $H_{C2}$ as a function of the evaluated
in-plane dynamic field amplitude $\upmu_{0}h_{\mathrm{rf,x}}$ in the linear SW
regime. The error bars represent the 30 % and 70 % switching field values. The
straight lines are guides to the eye.

### Magnon-induced reversal of magnetic nanostripes at low power level

To evaluate quantitatively the critical powers $P_{\mathrm{C}}$ needed to
reverse ferromagnetic nanostripes (”switching-yield map”), we excited SWs at
well-defined frequencies $f_{\mathrm{irr}}$, varied the excitation power
$P_{\mathrm{irr}}$ and explored after each power step whether the magnetic
configuration of nanostripes changed (Supplementary Fig. 2). We defined
$P_{\mathrm{C1}}$ ($P_{\mathrm{C2}}$) as the critical power value at which
mode branches exhibited 50 % of their maximum signal strength, indicating that
a considerable number of Py nanostripes under CPW1 (CPW2) were switched; the
results for an applied field $\mu_{0}H_{\mathrm{B}}=+14$ mT are summarized by
black dots in Fig. 1 f (g) (datasets for a separate sample are shown in
Supplementary Fig. 5). We considered further
$H_{\mathrm{B}}<H_{\mathrm{C1}}(-25~{}$dBm$)=16.8~{}$mT and obtained
consistent diagrams (not shown). We find the minimum $P_{\mathrm{C1}}$ of 12.6
$\upmu$W when applying an RF signal to CPW1 in the frequency window
$f_{\mathrm{irr}}=1.5$ GHz to 1.75 GHz. This window includes the peak
frequency of the $k_{1}$-mode (1.70 GHz) with a wavelength of 7.4 $\mu$m
(Methods), as evidenced by the SW absorption spectrum Mag(S11) (Fig. 1 h). We
note that this wavelength is much longer than both the width of stripes and
the period $a$ of arrays. The wavelength is not commensurable with the stripe
period $a$. The second lowest $P_{\mathrm{C1}}$ of 68.9 $\upmu$W occurs for
$f_{\mathrm{irr}}=2.75$ GHz to 3.0 GHz. This frequency range coincides with
the frequency of the GC mode named $k_{1}+1G$. This mode exhibits a wavelength
of 195 nm which is a value close to period $a$ (for mode identification see
Supplementary Fig. 6 and Ref. [16] for the P-mode). The microwave power
$P_{\mathrm{prec}}$ which is transferred into magnetization precession is
evaluated by
$P_{\mathrm{prec}}(f_{\mathrm{irr}})=P_{\mathrm{irr}}(f_{\mathrm{irr}})\cdot[\mathrm{Mag(S11)}(f=f_{\mathrm{irr}})]^{2}$.
We obtain critical power values $P_{\mathrm{C1,prec}}=36~{}\mathrm{nW}$ and
$P_{\mathrm{C1,prec}}=3.4~{}\mathrm{nW}$ for magnetic stripe reversal by means
of the $k_{1}$ mode and the short-wave magnon labelled $k_{1}+1G$,
respectively. The switching-yield map features three further dips at high
frequency marked as A, B, and C. They occur at relatively large power values
$P_{\mathrm{C1}}$ of 0.69 mW, 1.91 mW and 1.13 mW for $f_{\mathrm{irr}}$ near
9.25 GHz, 10.75 GHz and 11.75 GHz, respectively. These dips agree with
eigenresonances of the Py stripes. At such frequencies, the concomitant
reversal of Py stripes underneath CPW1 is attributed to the conventional
microwave-assisted magnetization reversal (MAMR). MAMR of individual
nanomagnets was pioneered in 2003 [34] and then applied to mesoscopic Py
magnets in e.g. Refs. [28, 29, 35, 33]. These micromagnets were not exposed to
a propagating spin wave. Instead, they were irradiated directly by an RF
signal. Its frequency hit exactly the eigenfrequency of the Py. We measured
the direct electromagnetic crosstalk between the CPWs. At CPW2, it amounted to
-50 dB. In other words, the directly irradiated microwave power was five (!)
orders of magnitude lower and too small for MAMR at CPW2. The dips in Fig. 1f
that occur near 8 GHz will not be discussed in the following. The important
features are the ones at low frequency in Fig. 1f and g. They go beyond MAMR
in that (1) we excite spin precession in YIG and not in the Py nanostripes,
(2) different spin-wave frequencies in the YIG induce switching of Py
nanostripes and (3), nanostripes more than 25 micrometres away from the
emitter CPW reverse (Fig. 1g).
$P_{\mathrm{C2}}$ (Fig. 1g) is the critical power for magnetic stripe reversal
under CPW2. We find $P_{\mathrm{C2}}=58.4~{}\upmu$W when exciting the
$k_{1}$-mode in YIG (Fig. 1 i). Its eigenfrequency was slightly blue-shifted
compared to spectra taken for $P_{\mathrm{irr}}<P_{\mathrm{C1}}$ evidencing
the concomitant magnetic stripe reversal by the $k_{1}$-mode at CPW1. Hence, a
SW mode near 2 GHz can ”write” states ”1” at CPW1 and perform a real-time
replication of these bit-states ”1” about 25 $\mu$m away [Fig. 1(b), panel
IV]. Similar to Ref. [22] we observed that reversed stripes changed the
transmitted SW amplitude. Such a signal variation was recently functionalized
in a neuronal network based on propagating magnons [21]. The maximum distance
over which reversal is possible is determined by the intermediate damping as
will be discussed in the following.
The minimum $P_{\mathrm{C2}}$ at CPW2 is about 4.5 times larger than the
minimum $P_{\mathrm{C1}}$. We attribute this discrepancy to the damping of the
propagating $k_{1}$ SW mode. Assuming $I(s)=I(x=0)\exp{(-s/l_{\rm d})}$ and a
decay length $l_{\rm d}$ of $23~{}\upmu\mathrm{m}$, the intensity $I$ of a SW
would decay by 1/4.5 over a distance $x=s=35~{}\upmu\mathrm{m}$, i.e., the
center-to-center distance between CPW1 and CPW2. Based on the intrinsic
Gilbert damping $\alpha_{\rm i}=9\times 10^{-4}$ of the YIG film, we calculate
a decay length of $99~{}\upmu\mathrm{m}$ for SWs with a wave vector $k_{1}$ in
unpatterned YIG (Supplementary Fig. 7). It is reasonable to assume that the
effective damping parameter in the magnonic circuit of Fig. 1 was larger due
to additional magnon scattering and radiative losses [36]. We hence consider
$l_{\rm d}=23~{}\mu$m to be a reasonable value. The mode $k_{1}+1G$ has a
smaller velocity than mode $k_{1}$ (Supplementary Fig. 7b) and thereby a
further decreased decay length. This consideration explains why the maximum
available power of the VNA was not sufficient for nanostripe reversal via mode
$k_{1}+1G$ underneath CPW2.

### Discussion

The magnon-induced reversal of bistable ferromagnetic nanoelements via
different propagating spin wave modes such as $k_{1}$ and $k_{1}+1G$ in YIG is
a key asset when aiming at in-memory computation and beyond von-Neumann device
architectures based on multi-frequency magnonic circuits. Further studies are
needed to understand the microscopic mechanism behind the magnon-induced
reversal reported here and thereby predict the theoretical limit of minimum
power consumption. But already at this stage several considerations can be
made. CPWs and grating couplers emit spin waves at specifically designed wave
vectors and due to the spin wave dispersion relation in YIG a specific
frequency band is thereby realized. Importantly we observe low-power magnon-
induced reversal for both a pure CPW mode (mode $k_{1}$) and a GC mode
($k_{1}+1G$). The nature of the mode does not seem to play a decisive role.
The observed frequency selectivity is so far attributed to the microwave-to-
magnon transducer used and not the reversal process itself. A decade ago, J.
C. Slonczewski pointed out that magnons in a ferrimagnet/ferromagnet hybrid
structure have the potential to provide spin-transfer torque for nanomagnet
reversal more efficiently than charge flow [17]. His theoretical predictions
reflected that for a given amount of energy more bosonic magnons than
fermionic electrons could be generated. If transferred to an adjacent
nanomagnet, each of these magnons provided an angular momentum of $\hbar$ for
reversal. Beyond this so-called magnon torque, spin precessional motion in YIG
was found to inject a spin-polarized current $\mathbf{j}_{\rm
s}\propto\mathbf{M}_{\rm YIG}\times d\mathbf{M}_{\rm YIG}/dt$ into Py [37].
The spin current $\mathbf{j}_{\rm s}$ displayed a nonreciprocal behavior [38]
and its amplitude increased with the SW wave vector in YIG [38, 39]. The
spatial symmetry of our MFM data of reversed stripes is consistent with both
the predicted magnon torque or $\mathbf{j}_{\rm s}$. Furthermore, the $k_{1}$
and $k_{1}+1G$ modes have indeed shown a characteristically different power
level for reversal. One might hence speculate that magnon torque [10, 9, 19,
20] and/or spin current [18, 38] could play a role for the switching of Py
stripes. We note however that we observe similar reversal also in hybrid
structures with a 5 nm thick intermediate layer between YIG and Py consisting
of either the insulator SiO2 or metal Cu [40]. These further experiments and
results will be presented elsewhere, but render the so far discussed scenarios
unlikely as their efficiency should vary significantly with the different
intermediate layers. We speculate that the dipolar magnetic fields of
propagating spin waves introduce torques on Py magnetic moments. As the
frequencies of SW modes $k_{1}$ and $k1+1G$ in YIG are well below the
eigenfrequencies of the Py nanostripes (Fig. 1f and g and Supplementary Fig.
5), they do not fulfill the condition for resonant chiral absorption discussed
theoretically in Ref. [41]. A possible scenario might instead consist of SW-
induced nucleation of a domain wall followed by its propagation due to the
applied static magnetic field. We now compare the power levels reported here
with the pioneering spin-transfer torque experiment in Ref. [42] performed in
zero and non-zero applied fields $H$. In Ref. [42], a directly injected
electrical (el) current was used to switch one single Co nanomagnet of a small
volume $(60\times 130\times 2.5)~{}{\rm nm^{3}}$. A minimum power $P^{*}_{\rm
el}\approx 6~{}\mu{\rm W}$ was required. In our study, we evaluated a critical
power of $P_{\mathrm{C1,prec}}=3.4~{}\mathrm{nW}$ when reversing a few 10 Py
nanostripes of a much larger volume $(100\times 26000\times 20)~{}{\rm
nm}^{3}$ each. In our so-far non-optimized magnonic devices, the required
power levels are also orders of magnitude below the ones recently reported for
domain-wall manipulation by SWs in Py stripes and the torques produced by
incoherent magnons in a thin antiferromagnetic barrier [10, 9]. Stimulated by
Ref. [17], we argue that the bosonic nature of magnons is in general
advantageous to initiate magnet reversal. In our experiments, the magnons in
YIG had frequencies $f$ around 2 GHz. They were hence of low energy $hf$ where
$h$ is the Planck constant. Consequently, a small power can generate a large
number of identical low-energy magnons in a short time which then produce
torques for reversal via different scenarios outlined above. We expect further
reduced critical power levels for magnetic bit reversal and smaller
$P_{\mathrm{C,prec}}$ by optimizing microwave-to-magnon transducers, modifying
the shape or volume of nanomagnets and engineering the interface between
magnetic bit and magnon conduit via e.g. an antiferromagnetic layer.

## Methods

### Sample Fabrication

The samples were prepared on a 100 nm thick yttrium iron garnet (YIG) film,
which was grown by liquid phase epitaxy and purchased from the Matesy GmbH in
Jena, Germany. On top of the insulating YIG, a 20 nm thick metallic Py
($\mathrm{Ni_{81}Fe_{19}}$) film was deposited via electron beam evaporation.
The grating pattern was defined via electron beam lithography (EBL) using
negative hydrogen silsesquioxane resist and transferred into the Py film by
$\mathrm{Ar^{+}}$ ion beam etching. The etching time was optimized to ensure
minimal overetching into the YIG film. Coplanar waveguides were prepared by
EBL lift-off processing using a PMMA/MMA double layer positive resist and
evaporation of Ti/Cu (5 nm/110 nm).

### Spin wave spectroscopy with a vector network analyzer

SW spectroscopy measurements were conducted with a VNA (Keysight PNA N5222A)
in a microwave probe station with integrated in-plane magnetic field control.
The magnetic field was created by electromagnets and stabilized via a Hall-
sensor controlled feedback loop. High-frequency coaxial cables with
$50~{}\mathrm{\Omega}$ impedance and microwave probes were used to connect
port 1 and port 2 of the VNA to CPW1 and CPW2, respectively. For all conducted
measurements, microwave power $P_{\mathrm{irr}}$ (or $P_{\mathrm{sens}}$) was
only applied to port 1, i.e. only the scattering parameters S11 and S21 were
measured. The front panel jumpers of the PNA N5222A were adjusted at port 2 to
provide increased forward (S21) dynamic range. For measurement of the
microwave spectra at different power levels (cf. Fig. 1 c-e and raw data for
Fig. 3) the frequency was scanned from 10 MHz to 12.5 GHz in 3.1 MHz steps. An
IF bandwidth of 1 kHz was used. For the switching-yield measurements (cf. Fig.
1 f and g) $f_{\mathrm{irr}}$ was scanned in 2.5 MHz steps. An IF bandwidth of
1 kHz and a dwell time of 50 ms was used. The magnitudes of the reflection and
transmission scattering parameters Mag(S11) and Mag(S21) were calculated as
the square‐root of the sum of the respective squared real and squared
imaginary parts, after removing nonmagnetic background signals, which did not
depend on the applied external magnetic field. $\mathrm{Mag(S11)}^{2}$ is the
measure of the relative absorbed power and used to calculate the power
$P_{\mathrm{prec}}=P_{\mathrm{irr}}\cdot\mathrm{Mag(S11)}^{2}$ transferred
from the CPW into the spin-precessional (prec) motion in the magnonic circuit.
The most prominent SW excitation of CPW1 occurred at a wave vector
$k_{1}=0.85~{}{\rm rad}~{}\mu$m-1 [16]. The wavelength of the corresponding
$k_{1}$ mode in YIG amounted to about 7.4 $\mu$m. The periodic array of Py
nanostripes gave rise to a reciprocal lattice vector $1G$ and enabled the
excitation of short-wave magnons by the grating coupler effect [23]. Its wave
vector amounted to $k_{1G}=2\pi/a=2\pi/(0.2~{}\mu$m) $=31.4~{}{\rm
rad}~{}\mu$m-1. The absorption spectrum Mag(S11) in Fig. 1 h showed a double
peak structure for the resonant feature labelled with 1G. We attributed the
smaller peak on the low-frequency side of 1G to a grating coupler mode with
$-k_{1}+1G$ and the larger high-frequency peak to $k_{1}+1G$. The mode
$k_{1}+1G$ exhibited an absolute wave vector of $k_{k1+1G}=k_{1}+k_{1G}$. Its
wavelength amounted to $2\pi/k_{k1+1G}=195$ nm under the stripe array [16].
This value was close to $a$ due to $k_{1}\ll k_{1G}$. The GC mode $-k_{1}+1G$
corresponded to a wavelength of 206 nm under CPW1. Considering its larger
absorption strength we assumed mode $k_{1}+1G$ to induce mainly the reversal
of nanostripes in the text.

### Analysis of Switching Fields

To determine the power and frequency dependence of the discovered switching
process, the following measurement routine was conducted: The stripes were
first magnetized downwards with $\mu_{0}H=-90~{}\mathrm{mT}$ and then the
field was increased slowly to a positive bias field of
$\mu_{0}H_{\mathrm{B}}=14~{}\mathrm{mT}$. This field value was below
$\mu_{0}H_{\rm C1}$ at -25 dBm, i.e., below the evaluated minimum field of the
switching field distribution (Supplementary Fig. 3). We monitored Mag(S21)
with a low sensing power $P_{\mathrm{sens}}=-25$ dBm. This power level was
chosen such that $P_{\mathrm{sens}}$ itself was insufficient to induce
switching. While staying at $\mu_{0}H_{\mathrm{B}}=14~{}$mT, we applied
$P_{\mathrm{irr}}$ in a small frequency window $f_{\mathrm{irr}}$ with a range
of 0.25 GHz. $P_{\mathrm{irr}}$ was increased from -30 dBm to 6 dBm in 1 dBm
steps. After each power step, Mag(S21) was measured with $P_{\mathrm{sens}}$
in a frequency window from 7.5 GHz to 12.5 GHz. Thereby we covered the
branches indicating the magnetic configuration of the stripes. After each such
sequence the sample was reset to the initial state (bits were erased) for the
next frequency $f_{\rm irr}$. Supplementary Fig. 2a shows exemplarily Mag(S21)
measured as a function of $P_{\mathrm{irr}}$ with $1.5~{}\mathrm{GHz}\leq
f_{\mathrm{irr}}\leq 1.75~{}\mathrm{GHz}$. With increasing $P_{\mathrm{irr}}$
the AP-mode vanishes and the P-mode appears. We evaluate mode strengths as a
function of $P_{\mathrm{irr}}$ (Supplementary Fig. 2b) by integrating Mag(S21)
in the vicinity of the AP-mode (enclosed by blue dashed lines) and P-mode
(enclosed by orange dashed lines). We define $P_{\mathrm{C1}}$
($P_{\mathrm{C2}}$) as the critical power value for which the signal strength
of the AP-mode (P-mode) is at 50 % of its maximum value (Supplementary Fig.
2b), indicating that 50 % of the Py nanostripes in the grating under CPW1
(CPW2) were switched. Further we extract the values for which 30 % and 70 % of
the signal strength are reached as a measure of the switching power
distribution (error bar). The measurements were repeated for different
$f_{\mathrm{irr}}$ altered from 1 GHz to 13.25 GHz in 0.25 GHz wide windows.
We quantified the relative numbers of reversed nanostripes by integrating the
signal strengths in Mag(S21) spectra in the vicinity of the AP- and the P-mode
(Methods). At the low power level of $P_{\rm sens}=-25~{}$dBm we analyzed the
quasistatic switching field (SF) distribution of Py nanostripes under CPW1
(Supplementary Fig. 3a and b) and found a full width half maximum
$\mathrm{SF_{S11,f}}-\mathrm{SF_{S11,i}}$ of roughly 13 mT centered around a
switching field value of
$(\mathrm{SF_{S11,f}}+\mathrm{SF_{S11,i}})/2=21.5~{}$mT (Supplementary Fig.
3c). With increasing $P_{\rm irr}$ we observed a decrease in SF values and a
narrower distribution (Supplementary Fig. 3c) consistent with the writing of
bits reported in the manuscript.

## Acknowledgments

The authors acknowledge discussions with Andrea Mucchietto, Shreyas Joglekar,
Michal Mruczkiewicz, and Anna Fontcuberta i Morral.

### Funding

The research was supported by the SNSF via grant number 163016.

### Author contributions

D.G. and K.B. planned the experiments and designed the samples. K.B. prepared
the samples and performed the experiments. K.B. and D.G. analyzed and
interpreted the data. K.B. and D.G. wrote the manuscript.

### Competing interests

The authors declare that they have no competing interests.

### Data availability

The datasets generated and/or analysed during the current study are available
from the corresponding author on reasonable request.

## Additional information

Supplementary Information is available for this paper.
Correspondence and requests for materials should be addressed to D.G. (email:
dirk.grundler@epfl.ch).
Reprints and permissions information is available at www.nature.com/reprints.

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### SUPPLEMENTARY INFORMATION

Supplementary Fig. 1: $\mid$ Layout and scanning electron microscopy images of
the sample. a Sample layout. Two CPWs (blue color) with a total length of
$121.6~{}\upmu\mathrm{m}$ are separated by a center-to-center distance of
$35~{}\upmu\mathrm{m}$. Below each CPW three $25~{}\upmu\mathrm{m}$ long Py
grating couplers (GCs) are arranged. Every second stripe of the GCs is
prolonged on both sides by $1~{}\upmu\mathrm{m}$ and then its length amounted
to 27 $\mu$m. The edge-to-edge separation of neighboring GCs underneath one
and the same CPW amounts to $15~{}\upmu\mathrm{m}$ to avoid dipolar
interaction. b Scanning electron microscopy image showing both CPWs with GCs.
c Zoom in on the end of one of the GCs. Supplementary Fig. 2: $\mid$
Evaluation of critical power levels $P_{\mathrm{C1}}$ and $P_{\mathrm{C2}}$. a
Stripes were magnetized along $-y-$direction and a small power of
$P_{\mathrm{sens}}=-25$ dBm was used to measure Mag(S21). Staying at
$\mu_{0}H_{\mathrm{B}}=+14~{}\mathrm{mT}$ a microwave signal of power
$P_{\mathrm{irr}}$ was applied in 0.25 GHz wide windows exemplary plotted for
$f_{\mathrm{irr}}=1.5$ GHz to 1.75 GHz and increased in a stepwise manner.
After each increment of $P_{\mathrm{irr}}$, Mag(S21) was measured with the
small power $P_{\mathrm{sens}}$ between 7.5 GHz to 12.5 GHz. A transition from
the AP-mode (enclosed by blue dashed lines) to the P-mode (enclosed by orange
dashed lines) as function of $P_{\mathrm{irr}}$ was observed. b We quantify
critical power levels $P_{\mathrm{C1}}$ and $P_{\mathrm{C2}}$ by evaluating
the 50 % signal strength transitions of AP- (blue) and P-mode (orange),
respectively. We argue that the decay of SW amplitudes below CPW1 and CPW2 and
the inhomogeneous stripe reversal in the grating couplers observed by MFM
explain why AP- and P-mode transitions appear gradual and are not sharp step
functions.  Supplementary Fig. 3: $\mid$ Power-dependent switching field
distribution of Py nanostripes underneath CPW1. a Exemplary Mag(S11) for
$P_{\mathrm{irr}}=-25$ dBm. We track a characteristic mode of the AP-
configuration (blue dots) and the P-configuration (orange dots). b We quantify
the switching field $\mathrm{SF_{S11,i}}$ ($\mathrm{SF_{S11,f}}$) by
evaluating the decay (rise) of the signal strength to 50 % of its maximum
value with respect to the noise floor (estimated from the region between the
gray dashed lines in a). c $\mathrm{SF_{S11,f}}$ diminished more significantly
than $\mathrm{SF_{S11,i}}$ with $P_{\mathrm{irr}}$. Above
$P_{\mathrm{irr}}\geq-15$ dBm, $\mathrm{SF_{S11,i}}$ and $\mathrm{SF_{S11,f}}$
were separated by less than 1 mT and automatic evaluation with the previously
introduced methodology (cf. a, b) was not performed.  Supplementary Fig. 4:
$\mid$ Onset power level of nonlinear spin-wave regime. Power dependence of
resonance frequency $f_{\mathrm{res}}$ (red color) and full width half maximum
(FWHM) of the $k_{1}$ mode resonance in Mag(S11) extracted by fitting with a
Lorentzian function at $\upmu_{0}H=-10$ mT. Above $P_{\mathrm{irr}}=-9$ dBm
(indicated by dashed arrow) $f_{\mathrm{res}}$ starts to decrease
significantly with increasing power. Around the same $P_{\mathrm{irr}}$ the
FWHM starts to increase significantly. These two observations are attributed
to the onset of the nonlinear regime of SW excitation.  Supplementary Fig. 5:
$\mid$ Switching-yield maps at $\mu_{0}H_{\mathrm{B}}=14~{}\mathrm{mT}$ for an
independently prepared second sample with nominally identical Py nanostripe
arrays. The YIG film was taken from the same wafer, but, evaporation of the Py
film and subsequent process steps were conducted separately from the sample
presented in the main text. The yields for magnetic bit writing of this second
sample out of a different fabrication batch agree well with the findings
presented in Fig. 1 of the main text. The agreement is true for writing
magnetic bits, both, under a CPW1 and b CPW2. Supplementary Fig. 6: $\mid$
Micromagnetic simulations of GC modes in P- and AP- scenario with and without
interlayer coupling. We used Mumax3.10 [43] to simulate the microwave response
of the investigated GC sample assuming different interlayer exchange coupling
strengths. a The simulated geometry consisted of a 100 nm thick YIG film
(shown in blue) with a total width of $41~{}\upmu\mathrm{m}$. On top of the
YIG film was a periodic grating ($a=200~{}\mathrm{nm}$) consisting of 51 Py
stripes (shown in red) with a thickness of 20 nm and width of 100 nm. We
simulated a slice of the GC sample in the $x-z$ plane. In $y$-direction
periodic boundary condition was used (with 2048 repetitions in both $+y-$ and
$-y-$direction). The grid size was $5~{}\mathrm{nm}\times
5~{}\mathrm{nm}\times 5~{}\mathrm{nm}$. As material parameters we used
$\upmu_{0}M_{\mathrm{S}}=176$ mT (1005 mT),
$A_{\mathrm{ex}}=3.75~{}\mathrm{pJ/m}$ ($13~{}\mathrm{pJ/m}$) and
$\alpha=0.0009$ (0.01) for the YIG film (Py grating). b For SW excitation we
applied a small $\bf{h_{\mathrm{rf}}}$. The spatial profile of
$\bf{h_{\mathrm{rf}}}$ was simulated in COMSOL Multiphysics for the nominal
CPW dimensions. Following Ref. [44] $\bf{h_{\mathrm{rf}}}$ was modulated in
time by multiplication with a Sinc-function; the SW spectra were subsequently
computed from the Fourier amplitudes of the dynamic magnetization components.
We simulated the SW spectra for the (c, e) P- and (d, f) AP-scenario and for
the interlayer exchange between YIG and Py either (c, d) completely turned off
(by setting the scaling parameter $S$ in Mumax3 to zero) or (e, f) set to its
standard value $S=1$ for which Mumax3 assumes the harmonic mean of
$A_{\mathrm{ex}}$ between both layers [43]. In the latter case (e, f), the
simulations show a strong modification of the $k_{1}$ mode in the AP-scenario,
which was not observed in the measured spectra. We thus assume that in the
experimentally investigated sample the interlayer exchange coupling was only
small or completely absent. When the interlayer exchange was turned off in the
simulation, we found a good qualitative agreement between simulation and
measured data. In particular, the $k_{1}$ mode was not significantly shifted
between P- and AP-scenario, while the GC modes (see labels in (c, d)) were
red-shifted by several hundreds of MHz for the AP-scenario. We attribute this
frequency shift in the absence of interlayer-exchange coupling to a modified
dynamic demagnetization field. For the simulated GC modes we observed that in
the AP-configuration the dynamic out-of-plane magnetization component
$m_{\mathrm{z}}$ of the Py stripes was in-phase with $m_{\mathrm{z}}$ of the
underlying YIG film, while in the P-configuration there was an $\pi$ phase
shift between them, leading to an increased dynamic demagnetization field.
Supplementary Fig. 7: $\mid$ Estimation of the SW decay length
$l_{\mathrm{d}}$ of the $k_{1}$ mode at
$\mu_{0}H_{\mathrm{B}}=14~{}\mathrm{mT}$. a To extract the intrinsic Gilbert
damping $\alpha_{\mathrm{i}}$ of the YIG film, its ferromagnetic resonance
(FMR) linewidth was measured with perpendicular-to-plane magnetic fields up to
0.5 T. A CPW with a broad signal line of 20 $\upmu$m width was used, which
excited spin-precessional motion at $k^{\prime}_{1}\simeq
0~{}\mathrm{rad}\,\upmu\mathrm{m}^{-1}$. The extracted full width half maximum
(FWHM) of the FMR mode in Mag(S11) as a function of its resonance frequency is
fitted by a linear function with a slope of $m=0.0031$ (green line) in a.
Following Ref. [45] the linewidth broadening depended on $\alpha_{\mathrm{i}}$
according to $\delta f=\frac{|\gamma|}{2\pi}\upmu_{0}\Delta
H+2\alpha_{\mathrm{i}}f$ where $\Delta H$ was the inhomogeneous line
broadening. $\delta f$ was the linewidth of the imaginary part, which was
estimated by dividing the linewidth of the magnitude Mag(S11) by a factor of
$\sqrt{3}$ [46]. Accordingly we calculated
$\alpha_{\mathrm{i}}=m/(2\sqrt{3})=9\times 10^{-4}$. b Group velocity
$v_{\mathrm{g}}$ (line) of SWs in a 100 nm thick YIG film as calculated by the
Kalinikos-Slavin formalism [47]. $v_{\mathrm{g}}$ was calculated for
$\mu_{0}H=14~{}\mathrm{mT}$ in the Damon-Eshbach configuration
($\bf{k}\perp\bf{M}_{\rm YIG}$) using a saturation magnetization
$\upmu_{0}M_{\mathrm{YIG,S}}=176$ mT, an exchange constant
$A_{\mathrm{ex}}=3.7~{}\mathrm{pJ/m}$ and $\gamma=28.0$ GHz/T as material
parameters for the YIG film. At $k_{1}=0.85~{}\upmu\mathrm{m/rad}$ (marked by
the left black arrow), we extract $v_{\mathrm{g}}=1.9~{}\upmu$m/ns. The decay
length $l_{\mathrm{d}}$ for the SW intensity is calculated by
$l_{\mathrm{d}}=\tau\cdot v_{\mathrm{g}}/2$, where $\tau=\frac{1}{2\pi\alpha
f}$ is the relaxation time. Using $f=1.7$ GHz and $v_{\mathrm{g}}$ extracted
for the $k_{1}$ mode, we find $l_{\mathrm{d}}=99~{}\upmu$m. The wave vector at
1G (2G) reads $k_{1G}=2\pi/a=2\pi/(0.2~{}\mu$m) $=31.4~{}{\rm rad}~{}\mu$m-1
($62.8~{}{\rm rad}~{}\mu$m-1)