# Singling out SO(10) GUT models using recent PTA results

Stefan Antusch<EMAIL_ADDRESS>Kevin Hinze<EMAIL_ADDRESS>Shaikh Saad<EMAIL_ADDRESS>Jonathan Steiner
<EMAIL_ADDRESS>Department of Physics, University of Basel,
Klingelbergstrasse 82, CH-4056 Basel, Switzerland

###### Abstract

In this work, we construct promising model building routes towards SO(10) GUT
inflation and examine their ability to explain the recent PTA results hinting
at a stochastic gravitational wave (GW) background at nanohertz frequencies.
We consider a supersymmetric framework within which the so-called doublet-
triplet splitting problem is solved without introducing fine-tuning.
Additionally, realistic fermion masses and mixings, gauge coupling
unification, and cosmic inflation are incorporated by utilizing superfields
with representations no higher than the adjoint representation. Among the
three possible scenarios, two of these cases require a single adjoint Higgs
field, and do not lead to cosmic strings. In contrast, the third scenario
featuring two adjoints, can lead to a network of metastable cosmic strings
that generates a GW background contribution compatible with the recent PTA
findings and testable by various ongoing and upcoming GW observatories.

Introduction:– Global collaboration among pulsar timing arrays (PTAs)
(NANOGrav NANOGrav:2020bcs , PPTA Goncharov:2021oub , EPTA Chen:2021rqp , and
IPTA Antoniadis:2022pcn ) previously revealed evidence of common-spectrum
noise at nanohertz frequencies. Recent analysis, including CPTA Xu:2023wog ,
EPTA Antoniadis:2023ott , NANOGrav NANOGrav:2023gor , and PPTA Reardon:2023gzh
, identified spatial correlations (Hellings-Downs effect Hellings:1983fr ),
providing strong support for a stochastic gravitational-wave background
(SGWB). Although the mergers of supermassive black hole binaries (SMBHBs) are
natural astrophysical sources of the SGWB at nanohertz frequencies, the new
data somewhat disfavors SMBHBs in explaining the observed PTA SGWB signal
NANOGrav:2023gor . Therefore, the SGWB likely points toward new physics beyond
the Standard Model (SM). One of the explanations that fits well with the data
is a metastable cosmic string network (CSN) NANOGrav:2023hvm . Since such
cosmic strings (CSs) can arise from the multi-step spontaneous breaking of the
symmetry group of a Grand Unified Theory (GUT) after cosmic inflation, this
raises the question of what can be learned about GUTs from this finding.

GUTs Pati:1973rp ; Pati:1974yy ; Georgi:1974sy ; Georgi:1974yf ; Georgi:1974my
; Fritzsch:1974nn , combined with SUSY, offer an appealing framework for a
more fundamental theory beyond the SM of elementary particles. GUTs unify the
three fundamental forces of the SM, while SUSY provides a natural solution to
the gauge hierarchy problem and a potential weakly interacting dark matter
candidate when R-parity or matter-parity ensures its stability. SO(10)-based
GUTs are particularly interesting as they unify all SM fermions of each family
into a single irreducible 16-dimensional representation. This 16-dimensional
representation also includes a SM singlet right-handed neutrino, which,
through the type-I seesaw mechanism Minkowski:1977sc ; Yanagida:1979as ;
Glashow:1979nm ; Gell-Mann:1979vob ; Mohapatra:1979ia , generates tiny masses
for the SM neutrinos.

Promising GUT models must satisfy proton decay bounds and achieve successful
gauge coupling unification. In SUSY GUT models, the $d=5$ proton decay
operators are induced by color-triplet exchange, necessitating the superheavy
nature of color-triplet states compared to their doublet partners, known as
the doublet-triplet splitting (DTS) problem Randall:1995sh ; Yamashita:2011an
. A desirable GUT model should solve the DTS problem without fine-tuning
parameters. Since GUTs generate the Yukawa matrices out of joint GUT
operators, leading to constraints on the flavor structure, a further challenge
consists in realizing viable fermion masses and mixings.

Cosmic inflation Guth:1980zm ; Albrecht:1982wi ; Linde:1981mu ; Linde:1983gd
that solves the horizon and flatness problems of the standard Big Bang
cosmology, and explains the origin of structure formation of the observable
Universe, could have a deep connection to SUSY GUT models. In addition to the
similarity of the scales of inflation and gauge coupling unification,
inflation is also crucial to dilute away unwanted topological defects
Kibble:1976sj ; Linde:1981mu  like monopoles which generically form at some
stage of GUT symmetry breaking. Furthermore, supersymmetric theories typically
possess many flat directions, providing an attractive framework for realizing
inflation. While monopoles have to be diluted by inflation, other topological
defects, like (metastable) CSs Hindmarsh:1994re  that form after inflation can
leave an observable signature in the SGWB.

In this work, we explore supersymmetric SO(10) GUTs that naturally solve the
DTS problem, generate realistic fermion masses, and achieve successful gauge
coupling unification and inflation. We focus on lower-dimensional field
representations and investigate scenarios with Higgs fields no higher than the
adjoint representation. Three promising routes for SO(10) GUT model building
are identified: two cases use a single adjoint Higgs field, while the third
scenario requires two copies. In the latter case, the intermediate symmetry
contains two Abelian factors crucial for CSN formation. For the first time, we
construct a realistic SUSY SO(10) GUT scenario (particularly the third
scenario), satisfying the mentioned criteria and leads to metastable CSs
capable of explaining the recent PTA results for a stochastic GW background at
nanohertz frequencies.

SO(10) model building:– Two major guiding principles in building realistic
models in our framework are the natural DTS Dimopoulos:1981xm ;
Srednicki:1982aj  (see also Babu:1993we ; Babu:1994kb ; Berezhiani:1996bv ;
Barr:1997hq ; Chacko:1998jz ; Babu:1998wi ; Babu:2002fsa ; Kyae:2005vg ;
Babu:2010ej ; Wan:2022glq ) and employing smaller dimensional representations.
In achieving this, we utilize $45_{H}$ and $16_{H}+\overline{16}_{H}$ Higgs
representations to break the GUT symmetry down to the SM, which is
subsequently broken by $10_{H}$ (and possibly by $16_{H}+\overline{16}_{H}$).
The fundamental representation contains weak-doublet and color-triplet states,

$\displaystyle 10_{H}$
$\displaystyle=(2_{H}+3_{H})+(\overline{2}_{H}+\overline{3}_{H})$
$\displaystyle=(1,2,1/2)+(3,1,-1/3)+c.c..$

The VEV of the adjoint, $\langle 45_{H}\rangle\propto
i\tau_{2}\otimes\mathrm{diag}(a_{1},a_{2},a_{3},a_{4},a_{5})$ that breaks the
GUT symmetry is expected to provide superheavy masses to both these
components. With this setup, one can construct three classes of models:

  1. (i)

a single adjoint Higgs with $\langle 45_{H}\rangle\propto B-L$ generator,

  2. (ii)

a single adjoint Higgs with $\langle 45_{H}\rangle\propto I_{3R}$ generator,

  3. (iii)

two adjoint Higgses, one with $\langle 45_{H}\rangle\propto B-L$ generator and
another with $\langle 45_{H}^{\prime}\rangle\propto I_{3R}$ generator.

For each model, the superpotential takes the form,

$\displaystyle W=$ $\displaystyle W_{\mathrm{GUT-
breaking}}+\underbrace{W_{\mathrm{Inflation}}+W_{\mathrm{Mixed}}}_{W_{\mathrm{Intermedite-
breaking}}}$ $\displaystyle+W_{\mathrm{DTS}}+W_{\mathrm{Yukawa}},$

where terms in $W_{\mathrm{GUT-breaking}}$ and $W_{\mathrm{Intermedite-
breaking}}$ lead to a consistent symmetry breaking of the GUT symmetry down to
the SM gauge group. Terms in $W_{\mathrm{DTS}}$ realize DTS without fine-
tuning, and the $W_{\mathrm{Inflation}}$ part of the superpotential leads to
an inflationary period.

$\bullet$ $B-L$-case: The symmetry breaking chain in this scenario is given by

$\displaystyle SO(10)$
$\displaystyle\xrightarrow[45_{H}]{M_{\mathrm{GUT}}}SU(3)_{C}\times
SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$
$\displaystyle\xrightarrow[16_{H}+\overline{16}_{H}]{M_{I}}SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y}\;.$

The GUT scale symmetry breaking is achieved via

$\displaystyle W_{\mathrm{GUT-breaking}}$
$\displaystyle\supset\frac{m_{45}}{2}\text{Tr}[45_{H}^{2}]+\frac{\lambda}{4\Lambda}\text{Tr}[45_{H}^{4}],$
(1)

with the VEV $\langle 45_{H}\rangle\propto
i\tau_{2}\otimes\mathrm{diag}(a,a,a,0,0)$.

Note that breaking the GUT symmetry gives rise to superheavy monopoles that
must be inflated away. Therefore inflation must take place after the formation
of the monopoles. A straightforward option is to utilize hybrid Linde:1993cn ;
linde1991axions ; Dvali:1994ms  inflation (an alternative option is tribrid
inflation Antusch:2004hd ; Antusch:2010va ) at the last intermediate symmetry
breaking stage, which we achieve via employing $16_{H}+\overline{16}_{H}$ that
acquire VEVs 111As a result, the appearance of automatic R-parity from within
the SO(10) group is no longer possible. However a discrete symmetry, such as a
$Z_{2}$ symmetry (matter parity), can readily be imposed. in the right-handed
neutrino direction. Then the relevant superpotential term contributing to
inflation takes the following form,

$\displaystyle W_{\mathrm{Inflation}}\supset\kappa
S(\overline{16}_{H}16_{H}-m_{16}^{2}),$ (2)

which fixes the magnitude of the VEVs $\langle
16_{H}\overline{16}_{H}\rangle=m^{2}_{16}$. Here, $S$ is a GUT singlet
superfield, the scalar component of which plays the role of the inflaton.

Since $45_{H}$ and $16_{H}+\overline{16}_{H}$ have component fields that share
the same quantum numbers,

$\displaystyle
45_{H},16_{H},\overline{16}_{H}\supset(1,1,1)+(3,2,1/6)+(\overline{3},1,-2/3)+c.c.,$

to avoid additional would-be Goldstone bosons, which would ruin gauge coupling
unification, these fields must have non-trivial mixing terms. The simplest
possible interaction term, $\overline{16}_{H}45_{H}16_{H}$, is not welcome
since it would destabilize the VEV of $45_{H}$ from the desired “Dimopoulos-
Wilczek form”.

To circumvent this issue, we introduce a second copy of spinorial
representations, $16_{H}^{\prime}+\overline{16}_{H}^{\prime}$, which do not
acquire a VEV in the right-handed neutrino direction. Then a consistent
symmetry breaking without additional would-be Goldstone bosons can be achieved
via the addition of the following terms in the superpotential:

$\displaystyle W_{\mathrm{Mixed}}\supset$ (3)
$\displaystyle\overline{16}_{H}(\lambda_{1}45_{H}+\lambda^{\prime}_{1}1_{H})16_{H}^{\prime}+\overline{16}_{H}^{\prime}(\lambda_{2}45_{H}+\lambda^{\prime}_{2}1^{\prime}_{H})16_{H}.$

Here, we introduced the “sliding singlets” $1^{(\prime)}_{H}$, which are
assumed to have no other terms in the superpotential that could fix their
VEVs. They are needed to allow for vanishing $F$-terms corresponding to
$16_{H}^{\prime},\overline{16}_{H}^{\prime}$.

Concerning DTS, remarkably, the specific VEV structure of the $45_{H}$
provides masses to only the color-triplets, while the weak-doublets remain
massless, schematically

$\displaystyle 10_{1H}\langle 45_{H}\rangle 10_{2H}=$
$\displaystyle\cancelto{0}{\overline{2}_{1H}2_{2H}}+\cancelto{0}{\overline{2}_{2H}2_{1H}}+\overline{3}_{1H}3_{2H}+\overline{3}_{2H}3_{1H}.$
(4)

However, if only the above term is added to the superpotential, then the low
energy spectrum would contain four light doublets instead of the usual two
doublets of the MSSM. This would spoil the successful gauge coupling
unification of the MSSM. To avoid extra light states, we allow a direct mass
term for $10_{2H}$, i.e.,

$\displaystyle
10_{2H}10_{2H}=\overline{2}_{2H}2_{2H}+\overline{3}_{2H}3_{2H}.$ (5)

Then, the terms in the superpotential relevant for providing the masses of the
doublets and triplets and naturally realizing their splittings are

$\displaystyle W_{\mathrm{DTS}}\supset\gamma
10_{1H}45_{H}10_{2H}+m_{10}10_{2H}10_{2H}.$ (6)

A crucial remark is in order. Assuming that only $10_{1H}$ couples to the
fermions, the term in Eq. (Singling out SO(10) GUT models using recent PTA
results) by itself does not induce proton decay. Once the term in Eq. (5) is
also introduced, together they allow the proton to decay via color-triplet
Higgses, since now an effective mass term linking $\overline{3}_{1H}$ and
$3_{1H}$ can be written down after integrating out $\overline{3}_{2H}$ and
$3_{2H}$. This can be understood schematically as follows:

$\overline{3}_{1H}\langle 45_{H}\rangle{\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}3_{2H}}\hskip
20.0pt{\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\overline{3}_{2H}}m_{10}{\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}3_{2H}}\hskip
20.0pt{\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\overline{3}_{2H}}\langle
45_{H}\rangle 3_{1H}\;.$

With a sufficiently large effective triplet mass $\sim
M^{2}_{\mathrm{GUT}}/m_{10}$, the $d=5$ proton decay is suppressed.

$\bullet$ $I_{3R}$-case: The symmetry breaking chain in this scenario is given
by

$\displaystyle SO(10)$
$\displaystyle\xrightarrow[45_{H}]{M_{\mathrm{GUT}}}SU(4)_{C}\times
SU(2)_{L}\times U(1)_{R}$
$\displaystyle\xrightarrow[16_{H}+\overline{16}_{H}]{M_{I}}SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y}\;,$

which is obtained by $\langle 45_{H}\rangle\propto
i\tau_{2}\otimes\mathrm{diag}(0,0,0,b,b)$. Although the $W_{\mathrm{GUT-
breaking}}$ and $W_{\mathrm{Intermedite-breaking}}$ parts of the
superpotential are identical to the $B-L$ case, $W_{\mathrm{DTS}}$ takes a
different form, which we discuss in the following.

Due to $\langle 45_{H}\rangle\propto I_{3R}$, we now have the opposite
situation compared to the previous case, namely

$\displaystyle 10_{1H}\langle 45_{H}\rangle 10_{2H}=$
$\displaystyle\overline{2}_{1H}2_{2H}+\overline{2}_{2H}2_{1H}+\cancelto{0}{\overline{3}_{1H}3_{2H}}+\cancelto{0}{\overline{3}_{2H}3_{1H}}.$
(7)

Therefore, a different strategy must be implemented to obtain light doublets
and superheavy color-triplets. By noting that
$16_{H}^{\prime}\supset\overline{2}_{H}^{\prime}$ is a $SU(2)_{R}$ singlet,
and, on the contrary, $16_{H}^{\prime}\supset\overline{3}_{H}^{\prime}$
resides in a $SU(2)_{R}$ doublet, one obtains a mass only for the color-
triplet, and not for the weak doublet, i.e.,

$\displaystyle\overline{16}_{H}^{\prime}\langle 45_{H}\rangle
16_{H}^{\prime}=\cancelto{0}{\overline{2}_{H}^{\prime}2_{H}^{\prime}}+\overline{3}_{H}^{\prime}3_{H}^{\prime}\;.$
(8)

If only the above term is included in the superpotential, then a pair of
triplets will remain massless in addition to one pair of doublets. To provide
large masses to all the color-triplets, we add two more terms

$\displaystyle W_{\mathrm{DTS}}\supset$ (9)
$\displaystyle\lambda_{3}\overline{16}_{H}^{\prime}45_{H}16^{\prime}_{H}+\lambda_{4}10_{H}16_{H}16_{H}+\lambda_{5}10_{H}\overline{16}_{H}\overline{16}_{H}\;.$

As for the $d=5$ proton decay, assuming the SM fermion masses are coming from
their coupling to the $10_{H}$ (i.e. neglecting all contributions from the
$16_{H}$), the effective triplet mass $m_{T}$ is approximately given by

$\displaystyle m_{T}=-\frac{\lambda_{3}\lambda_{4}\lambda_{5}\langle
16_{H}\rangle\langle\overline{16}_{H}\rangle}{2\lambda_{1}\lambda_{2}\langle
45_{H}\rangle}.$ (10)

Choosing somewhat small $\lambda_{1},\lambda_{2}$ allows having $m_{T}\gtrsim
10^{19}$ GeV, which is required by proton decay constraints.

$\bullet$ $B-L\;\&\;I_{3R}$-case: Depending on the values of the VEVs of the
two adjoints, various symmetry breaking chains may arise in this scenario,
examples of which are (a) $\langle 45_{H}\rangle>\langle
45_{H}^{\prime}\rangle>\langle 16_{H}\rangle,\langle\overline{16}_{H}\rangle$:

$\displaystyle SO(10)$
$\displaystyle\xrightarrow[45_{H}]{M_{\mathrm{GUT}}}SU(3)_{C}\times
SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$
$\displaystyle\xrightarrow[45_{H}^{\prime}]{M_{I}}SU(3)_{C}\times
SU(2)_{L}\times U(1)_{R}\times U(1)_{B-L}$
$\displaystyle\xrightarrow[16_{H}+\overline{16}_{H}]{M_{II}}SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y}\;,$

(b) $\langle 45_{H}^{\prime}\rangle>\langle 45_{H}\rangle>\langle
16_{H}\rangle,\langle\overline{16}_{H}\rangle$:

$\displaystyle SO(10)$
$\displaystyle\xrightarrow[45_{H}^{\prime}]{M_{\mathrm{GUT}}}SU(4)_{C}\times
SU(2)_{L}\times U(1)_{R}$
$\displaystyle\xrightarrow[45_{H}]{M_{I}}SU(3)_{C}\times SU(2)_{L}\times
U(1)_{R}\times U(1)_{B-L}$
$\displaystyle\xrightarrow[16_{H}+\overline{16}_{H}]{M_{II}}SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y}\;,$

(c) $\langle 45_{H}\rangle=\langle 45_{H}^{\prime}\rangle>\langle
16_{H}\rangle,\langle\overline{16}_{H}\rangle$:

$\displaystyle SO(10)$
$\displaystyle\xrightarrow[45_{H}+45_{H}^{\prime}]{M_{\mathrm{GUT}}}SU(3)_{C}\times
SU(2)_{L}\times U(1)_{R}\times U(1)_{B-L}$
$\displaystyle\xrightarrow[16_{H}+\overline{16}_{H}]{M_{I}}SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y}\;.$

In this scenario, for each of the adjoints, the GUT symmetry breaking
superpotential consists of the terms given in Eq. (1). Since $\langle
45_{H}\rangle$ and $\langle 45_{H}^{\prime}\rangle$ break SO(10) to the left-
right symmetry and quark-lepton symmetry, respectively, the first and the
second break the generators in $(3,2,+1/6)+(3,2,-5/6)+(3,1,2/3)+c.c$ and
$(3,2,+1/6)+(3,2,-5/6)+(1,1,+1)+c.c$, respectively. Consequently, there would
be additional massless states. To avoid such massless states, we add the
following mixing term in the superpotential,

$\displaystyle W_{\mathrm{GUT-breaking}}$
$\displaystyle\supset\frac{\eta}{\Lambda}Tr[45_{H}.45_{H}.45_{H}^{\prime}.45_{H}^{\prime}].$
(11)

As before, one requires non-trivial interactions between the spinorial
representations and the adjoints to give masses to the would-be Goldstones.
For the two adjoints, we now introduce two sets of additional spinorial
representations, $16_{H}^{\prime}+\overline{16}_{H}^{\prime}$ and
$16_{H}^{\prime\prime}+\overline{16}_{H}^{\prime\prime}$, and add the
following terms, such that the VEVs of the adjoints are not destabilized:

$\displaystyle W_{\mathrm{Mixed}}\supset$ (12)
$\displaystyle\overline{16}_{H}(\lambda_{1}45_{H}+\lambda_{1}^{\prime}1_{H})16_{H}^{\prime}+\overline{16}_{H}^{\prime}(\lambda_{2}45_{H}+\lambda_{2}^{\prime}1_{H}^{\prime})16_{H}$
$\displaystyle+$
$\displaystyle\overline{16}_{H}(\lambda_{3}45_{H}^{\prime}+\lambda_{3}^{\prime}1_{H}^{\prime\prime})16_{H}^{\prime\prime}+\overline{16}_{H}^{\prime\prime}(\lambda_{4}45_{H}^{\prime}+\lambda_{4}^{\prime}1_{H}^{\prime\prime\prime})16_{H}.$

For the DTS, we include the term $10_{1H}45_{H}10_{2H}$. However, here we can
construct an example model which does not lead to proton decay at leading
order via $d=5$ operators. To this end, we forbid the direct mass term
$10_{2H}10_{2H}$. Instead, we include a higher dimensional operator,
$10_{2H}.45^{\prime 2}.10_{2H}$, such that an effective triplet mass for
$3_{1H}$ and $\overline{3}_{1H}$ cannot be written down, since,

$\displaystyle 10_{2H}45_{H}^{\prime 2}10_{2H}=$
$\displaystyle\overline{2}_{2H}2_{2H}+\overline{2}_{2H}2_{2H}+\cancelto{0}{\overline{3}_{2H}3_{2H}}+\cancelto{0}{\overline{3}_{2H}3_{2H}}.$
(13)

With the inclusion of the above two terms, still one pair of color-triplets
and an additional pair of weak doublets remain massless. We cure this by
adding a term of the form $\overline{16}^{\prime\prime}_{H}16^{\prime}_{H}$ to
the superpotential,

$\displaystyle W_{\mathrm{DTS}}\supset$ (14)
$\displaystyle\gamma_{1}10_{1H}45_{H}10_{2H}+\frac{\gamma_{2}}{\Lambda}10_{2H}45_{H}^{\prime
2}10_{2H}+\omega_{16}\overline{16}^{\prime\prime}_{H}16^{\prime}_{H},$

that leads to a single pair of light doublets, as desired.

It is important to note that all the scenarios discussed above can
successfully reproduce correct charged fermion masses and mixings by
incorporating suitable higher-dimensional operators. The light neutrinos
acquire masses through the standard type-I seesaw mechanism. The Majorana
masses for the right-handed neutrinos are generated by the following higher-
dimensional operator:

$\displaystyle W_{\mathrm{Yukawa}}\supset
Y_{R}16_{i}16_{j}\frac{\overline{16}_{H}\overline{16}_{H}}{\Lambda}\sim
Y_{R}\frac{v^{2}_{R}}{\Lambda}\nu^{c}\nu^{c}\>.$ (15)

Gravitational wave signals:– In some of the models we consider, breaking e.g.
a simple group into a subgroup that contains a $U(1)$ factor leads to monopole
creation. To prevent overclosing the universe, inflation must get rid of the
monopoles. At some later stage, once the left-over Abelian symmetry is broken,
strings appear (we assume the ideal Nambu-Goto string approximation, where the
dominant radiation emission of CSs is into GWs Vachaspati:1984gt ). If these
two scales are very close, Schwinger nucleation of monopole-antimonopole pairs
Langacker:1980kd ; Lazarides:1981fv ; Vilenkin:1982hm  on the string cuts it
into pieces and makes it decay. How quickly these metastable strings decay
depends on a parameter $\kappa_{m}$ Leblond:2009fq ,

$\displaystyle\kappa_{m}=\frac{m^{2}}{\mu}\sim\frac{8\pi}{g^{2}}\left(\frac{v_{m}}{v_{R}}\right)^{2},$
(16)

where $m$ is the mass of the monopole and $v_{m}$ ($v_{R}$) is the monopole
(string) creation scale. The network behaves like a stable-string network for
$\kappa_{m}^{1/2}\gg 10$.

Metastable CSNs provide an intriguing explanation for the newly released PTA
data NANOGrav:2023hvm . The data indicates string tension ($\mu$) values in
the range $G\mu\sim 10^{-8}-10^{-5}$ for $\kappa_{m}^{1/2}\sim 7.7-8.3$ (with
a strong correlation, cf. Fig. 10 of NANOGrav:2023hvm ), consistent with CMB
bounds. Notably, the $68\%$ credible region in the $G\mu-\kappa_{m}^{1/2}$
parameter plane overlaps with the third advanced LIGO–Virgo–KAGRA (LVK) bound,
while major parts of the $95\%$ credible region are compatible, preferring
$G\mu\lesssim 10^{-7}$ and $\kappa_{m}^{1/2}\sim 8$ NANOGrav:2023hvm , as
shown in Fig.1. However, it should be remarked that the computation of the GW
spectrum from metastable CSs carries significant uncertainty Auclair:2019wcv .
Furthermore, various possible effects are not included in the above shown GW
spectrum, for instance, an extended matter domination phase after inflation
Cui:2018rwi ; Auclair:2019wcv ; Blasi:2020wpy  or the change of degrees of
freedom below the SUSY breaking scale Cui:2018rwi . Nevertheless, observing a
higher frequency SGWB signal in the next LIGO–Virgo–KAGRA rounds would be a
fascinating confirmation of the scenario.

Interestingly, $G\mu\sim 10^{-7}$ corresponds roughly to $v_{R}\sim 10^{15}$
GeV, which is fully consistent with the type-I seesaw contribution to neutrino
masses and corresponds to the right scale for inflation. On the other hand,
stable CSs are disfavored by the recent PTA data222Stable cosmic strings,
however, were consistent with the previous PTA data. For works on GWs, in
light of NANOGrav12.5 data, arising from cosmic strings within GUTs, c.f.,
Buchmuller:2019gfy ; King:2020hyd ; King:2021gmj ; Lazarides:2022jgr ;
Fu:2022lrn ; Saad:2022mzu ; Lazarides:2023iim ; Maji:2023fba ; Madge:2023cak
..

Figure 1: Examples of GW signals (which we estimate following
Buchmuller:2021mbb ) from metastable cosmic strings explaining the recent PTA
result NANOGrav:2023hvm  while being at the edge of the sensitivity reach of
LVK LIGOScientific:2021nrg ; KAGRA:2021kbb . The plot also shows examples of
sensitivities of possible future observatories (SKA Janssen:2014dka , THEIA
Theia:2017xtk , $\mu$Ares Sesana:2019vho , LISA Audley:2017drz , Taiji
Guo:2018npi , TianQin Luo:2015ght , BBO Corbin:2005ny , DECIGO Seto:2001qf ,
ET Sathyaprakash:2012jk  and CE Evans:2016mbw ) which can test the signal at
various frequencies. Moreover, the recent NANOGrav 15 yrs result is also shown
with grey lines. $f$ is the GW frequency observed today.

The first (and second) model studied, the $B-L$\- (and $I_{3R}$-) case, leads
to embedded strings, which are generally unstable James:1992zp ; James:1992wb
; Goodband:1995he . Interestingly, all three models in the
$B-L\;\&\;I_{3R}$-case have the potential to produce metastable strings for
nearly degenerate monopole and string formation scales: $M_{I}\sim M_{II}$ for
cases (a) and (b), and $M_{\mathrm{GUT}}\sim M_{I}$ for case (c). However, in
case (c), a lower GUT scale $\sim 10^{15}$ GeV would have to be arranged that
requires suppression of $d=6$ proton decay utilizing the freedom in the Yukawa
sector, which makes this case somewhat less appealing. We like to point out
that the class of promising SO(10) models we considered in this work may or
may not lead to the formation of CSs, contrary to the class of models
considered in Jeannerot:2003qv , where the appearance of CSs is unavoidable.

Before concluding, we discuss the gauge coupling unification for an example
scenario that leads to metastable CSs (specifically, we choose case (a) within
$B-L\;\&\;I_{3R}$). To achieve metastable strings, the monopole and string
formation scales must nearly coincide. Therefore, we effectively have three
scales: the GUT scale, the monopole/string formation scale, and the SUSY
breaking scale (fixed at 3 TeV). To simplify the analysis, we assume that the
fields breaking a symmetry are degenerate with the corresponding scale, while
the remaining states have GUT scale masses. This minimal number of free
parameters allows us to find a wide range for the monopole/string formation
scale, approximately $M_{I}\sim M_{II}\sim[10^{9}-10^{17}]$ GeV (with
$10^{16}$ GeV $\leq M_{\mathrm{GUT}}\leq 10^{18}$ GeV and
$M_{\mathrm{GUT}}>M_{I}$), while still being consistent with gauge coupling
unification. Our analysis considers a $1\%$ uncertainty on the measured values
of the gauge couplings to account for GUT threshold uncertainties.

A comprehensive analysis encompassing gauge coupling unification, fermion
masses and mixings, proton decay, GW signal, and the mass spectrum of the
component fields from the superpotential terms will be presented in a
forthcoming publication.

Conclusions:– We explored promising model-building routes for SO(10) GUT
inflation in light of the recent PTA results suggesting the presence of a SGWB
at nanohertz frequencies. Our investigation focused on a supersymmetric SO(10)
framework with small dimensional representations, effectively solving the
doublet-triplet splitting problem without fine-tuning. This approach enables
realistic fermion masses, gauge coupling unification, and simple options for
embedding cosmic inflation. Among the three model classes studied, one
involves two adjoint fields capable of generating a network of metastable
cosmic strings. This network generates a SGWB background contribution that can
explain the recent PTA data, and will be tested by various upcoming GW
observatories.

Note added: As we were completing this work, several papers appeared that also
discussed the impact of the new PTA results on new physics scenarios
Zu:2023olm ; Vagnozzi:2023lwo ; Han:2023olf ; Lambiase:2023pxd ; Ellis:2023dgf
; Guo:2023hyp ; Megias:2023kiy ; Fujikura:2023lkn ; Yang:2023aak ;
Franciolini:2023wjm ; Shen:2023pan ; Kitajima:2023cek ; Ellis:2023tsl ;
Franciolini:2023pbf ; Ghoshal:2023fhh ; Bai:2023cqj ; Addazi:2023jvg ;
Athron:2023mer ; Kitajima:2023vre ; Lazarides:2023ksx ; Broadhurst:2023tus ;
Cai:2023dls ; Yang:2023qlf ; Blasi:2023sej ; Inomata:2023zup ; Depta:2023qst ;
Gouttenoire:2023ftk ; Borah:2023sbc ; Murai:2023gkv ; Datta:2023vbs ;
Barman:2023fad ; Bi:2023tib ; Lu:2023mcz ; Xiao:2023dbb ; Li:2023bxy ;
Anchordoqui:2023tln ; Niu:2023bsr ; Ebadi:2023xhq ; Gouttenoire:2023nzr ;
Abe:2023yrw ; Ghosh:2023aum ; Unal:2023srk ; Bian:2023dnv ; Figueroa:2023zhu ;
Du:2023qvj ; Servant:2023mwt ; Li:2023tdx ; DiBari:2023upq ; Geller:2023shn ;
Liu:2023ymk .

## References

  * (1) NANOGrav Collaboration, Z. Arzoumanian et al., “The NANOGrav 12.5 yr Data Set: Search for an Isotropic Stochastic Gravitational-wave Background,” Astrophys. J. Lett. 905 no. 2, (2020) L34, arXiv:2009.04496 [astro-ph.HE].
  * (2) B. Goncharov et al., “On the Evidence for a Common-spectrum Process in the Search for the Nanohertz Gravitational-wave Background with the Parkes Pulsar Timing Array,” Astrophys. J. Lett. 917 no. 2, (2021) L19, arXiv:2107.12112 [astro-ph.HE].
  * (3) S. Chen et al., “Common-red-signal analysis with 24-yr high-precision timing of the European Pulsar Timing Array: inferences in the stochastic gravitational-wave background search,” Mon. Not. Roy. Astron. Soc. 508 no. 4, (2021) 4970–4993, arXiv:2110.13184 [astro-ph.HE].
  * (4) J. Antoniadis et al., “The International Pulsar Timing Array second data release: Search for an isotropic gravitational wave background,” Mon. Not. Roy. Astron. Soc. 510 no. 4, (2022) 4873–4887, arXiv:2201.03980 [astro-ph.HE].
  * (5) H. Xu et al., “Searching for the Nano-Hertz Stochastic Gravitational Wave Background with the Chinese Pulsar Timing Array Data Release I,” Res. Astron. Astrophys. 23 no. 7, (2023) 075024, arXiv:2306.16216 [astro-ph.HE].
  * (6) J. Antoniadis et al., “The second data release from the European Pulsar Timing Array III. Search for gravitational wave signals,” arXiv:2306.16214 [astro-ph.HE].
  * (7) NANOGrav Collaboration, G. Agazie et al., “The NANOGrav 15 yr Data Set: Evidence for a Gravitational-wave Background,” Astrophys. J. Lett. 951 no. 1, (2023) L8, arXiv:2306.16213 [astro-ph.HE].
  * (8) D. J. Reardon et al., “Search for an Isotropic Gravitational-wave Background with the Parkes Pulsar Timing Array,” Astrophys. J. Lett. 951 no. 1, (2023) L6, arXiv:2306.16215 [astro-ph.HE].
  * (9) R. w. Hellings and G. s. Downs, “Upper Limits on the Isotropic Gravitational Radition Background from Pulsar Timing Analysis,” Astrophys. J. Lett. 265 (1983) L39–L42.
  * (10) NANOGrav Collaboration, A. Afzal et al., “The NANOGrav 15 yr Data Set: Search for Signals from New Physics,” Astrophys. J. Lett. 951 no. 1, (2023) L11, arXiv:2306.16219 [astro-ph.HE].
  * (11) J. C. Pati and A. Salam, “Is Baryon Number Conserved?,” Phys. Rev. Lett. 31 (1973) 661–664.
  * (12) J. C. Pati and A. Salam, “Lepton Number as the Fourth Color,” Phys. Rev. D 10 (1974) 275–289. [Erratum: Phys.Rev.D 11, 703–703 (1975)].
  * (13) H. Georgi and S. L. Glashow, “Unity of All Elementary Particle Forces,” Phys. Rev. Lett. 32 (1974) 438–441.
  * (14) H. Georgi, H. R. Quinn, and S. Weinberg, “Hierarchy of Interactions in Unified Gauge Theories,” Phys. Rev. Lett. 33 (1974) 451–454.
  * (15) H. Georgi, “The State of the Art—Gauge Theories,” AIP Conf. Proc. 23 (1975) 575–582.
  * (16) H. Fritzsch and P. Minkowski, “Unified Interactions of Leptons and Hadrons,” Annals Phys. 93 (1975) 193–266.
  * (17) P. Minkowski, “$\mu\to e\gamma$ at a Rate of One Out of $10^{9}$ Muon Decays?,” Phys. Lett. 67B (1977) 421–428.
  * (18) T. Yanagida, “Horizontal gauge symmetry and masses of neutrinos,” Conf. Proc. C7902131 (1979) 95–99.
  * (19) S. Glashow, “The Future of Elementary Particle Physics,” NATO Sci. Ser. B 61 (1980) 687.
  * (20) M. Gell-Mann, P. Ramond, and R. Slansky, “Complex Spinors and Unified Theories,” Conf. Proc. C 790927 (1979) 315–321, arXiv:1306.4669 [hep-th].
  * (21) R. N. Mohapatra and G. Senjanovic, “Neutrino Mass and Spontaneous Parity Nonconservation,” Phys. Rev. Lett. 44 (1980) 912.
  * (22) L. Randall and C. Csaki, “The Doublet - triplet splitting problem and Higgses as pseudoGoldstone bosons,” in International Workshop on Supersymmetry and Unification of Fundamental Interactions (SUSY 95), pp. 99–109.  3, 1995.  arXiv:hep-ph/9508208.
  * (23) T. Yamashita, “Doublet-Triplet Splitting in an SU(5) Grand Unification,” Phys. Rev. D 84 (2011) 115016, arXiv:1106.3229 [hep-ph].
  * (24) A. H. Guth, “The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems,” Phys. Rev. D 23 (1981) 347–356.
  * (25) A. Albrecht and P. J. Steinhardt, “Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking,” Phys. Rev. Lett. 48 (1982) 1220–1223.
  * (26) A. D. Linde, “A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems,” Phys. Lett. 108B (1982) 389–393. [Adv. Ser. Astrophys. Cosmol.3,149(1987)].
  * (27) A. D. Linde, “Chaotic Inflation,” Phys. Lett. B 129 (1983) 177–181.
  * (28) T. W. B. Kibble, “Topology of Cosmic Domains and Strings,” J. Phys. A 9 (1976) 1387–1398.
  * (29) M. B. Hindmarsh and T. W. B. Kibble, “Cosmic strings,” Rept. Prog. Phys. 58 (1995) 477–562, arXiv:hep-ph/9411342.
  * (30) S. Dimopoulos and F. Wilczek, “Incomplete Multiplets in Supersymmetric Unified Models,”.
  * (31) M. Srednicki, “Supersymmetric Grand Unified Theories and the Early Universe,” Nucl. Phys. B 202 (1982) 327–335.
  * (32) K. S. Babu and S. M. Barr, “Natural suppression of Higgsino mediated proton decay in supersymmetric SO(10),” Phys. Rev. D 48 (1993) 5354–5364, arXiv:hep-ph/9306242.
  * (33) K. S. Babu and R. N. Mohapatra, “Mass matrix textures from superstring inspired SO(10) models,” Phys. Rev. Lett. 74 (1995) 2418–2421, arXiv:hep-ph/9410326.
  * (34) Z. Berezhiani and Z. Tavartkiladze, “More missing VEV mechanism in supersymmetric SO(10) model,” Phys. Lett. B 409 (1997) 220–228, arXiv:hep-ph/9612232.
  * (35) S. M. Barr and S. Raby, “Minimal SO(10) unification,” Phys. Rev. Lett. 79 (1997) 4748–4751, arXiv:hep-ph/9705366.
  * (36) Z. Chacko and R. N. Mohapatra, “Economical doublet triplet splitting and strong suppression of proton decay in SO(10),” Phys. Rev. D 59 (1999) 011702, arXiv:hep-ph/9808458.
  * (37) K. S. Babu, J. C. Pati, and F. Wilczek, “Fermion masses, neutrino oscillations, and proton decay in the light of Super-Kamiokande,” Nucl. Phys. B 566 (2000) 33–91, arXiv:hep-ph/9812538.
  * (38) K. S. Babu and S. M. Barr, “Eliminating the d = 5 proton decay operators from SUSY GUTs,” Phys. Rev. D 65 (2002) 095009, arXiv:hep-ph/0201130.
  * (39) B. Kyae and Q. Shafi, “Inflation with realistic supersymmetric SO(10),” Phys. Rev. D 72 (2005) 063515, arXiv:hep-ph/0504044.
  * (40) K. S. Babu, J. C. Pati, and Z. Tavartkiladze, “Constraining Proton Lifetime in SO(10) with Stabilized Doublet-Triplet Splitting,” JHEP 06 (2010) 084, arXiv:1003.2625 [hep-ph].
  * (41) Q. Wan and D.-X. Zhang, “An extended study on the supersymmetric SO(10) models with natural doublet-triplet splitting,” arXiv:2210.01589 [hep-ph].
  * (42) A. D. Linde, “Hybrid inflation,” Phys. Rev. D 49 (1994) 748–754, arXiv:astro-ph/9307002.
  * (43) A. Linde, “Axions in inflationary cosmology,” Physics Letters B 259 no. 1-2, (1991) 38–47.
  * (44) G. R. Dvali, Q. Shafi, and R. K. Schaefer, “Large scale structure and supersymmetric inflation without fine tuning,” Phys. Rev. Lett. 73 (1994) 1886–1889, arXiv:hep-ph/9406319.
  * (45) S. Antusch, M. Bastero-Gil, S. F. King, and Q. Shafi, “Sneutrino hybrid inflation in supergravity,” Phys. Rev. D 71 (2005) 083519, arXiv:hep-ph/0411298.
  * (46) S. Antusch, M. Bastero-Gil, J. P. Baumann, K. Dutta, S. F. King, and P. M. Kostka, “Gauge Non-Singlet Inflation in SUSY GUTs,” JHEP 08 (2010) 100, arXiv:1003.3233 [hep-ph].
  * (47) As a result, the appearance of automatic R-parity from within the SO(10) group is no longer possible. However a discrete symmetry, such as a $Z_{2}$ symmetry (matter parity), can readily be imposed.
  * (48) T. Vachaspati and A. Vilenkin, “Gravitational Radiation from Cosmic Strings,” Phys. Rev. D 31 (1985) 3052.
  * (49) P. Langacker and S.-Y. Pi, “Magnetic Monopoles in Grand Unified Theories,” Phys. Rev. Lett. 45 (1980) 1.
  * (50) G. Lazarides, Q. Shafi, and T. F. Walsh, “Cosmic Strings and Domains in Unified Theories,” Nucl. Phys. B 195 (1982) 157–172.
  * (51) A. Vilenkin, “Cosmological Evolution og Monopoles Connected by Strings,” Nucl. Phys. B 196 (1982) 240–258.
  * (52) L. Leblond, B. Shlaer, and X. Siemens, “Gravitational Waves from Broken Cosmic Strings: The Bursts and the Beads,” Phys. Rev. D 79 (2009) 123519, arXiv:0903.4686 [astro-ph.CO].
  * (53) P. Auclair et al., “Probing the gravitational wave background from cosmic strings with LISA,” JCAP 04 (2020) 034, arXiv:1909.00819 [astro-ph.CO].
  * (54) Y. Cui, M. Lewicki, D. E. Morrissey, and J. D. Wells, “Probing the pre-BBN universe with gravitational waves from cosmic strings,” JHEP 01 (2019) 081, arXiv:1808.08968 [hep-ph].
  * (55) S. Blasi, V. Brdar, and K. Schmitz, “Fingerprint of low-scale leptogenesis in the primordial gravitational-wave spectrum,” Phys. Rev. Res. 2 no. 4, (2020) 043321, arXiv:2004.02889 [hep-ph].
  * (56) Stable cosmic strings, however, were consistent with the previous PTA data. For works on GWs, in light of NANOGrav12.5 data, arising from cosmic strings within GUTs, c.f., Buchmuller:2019gfy ; King:2020hyd ; King:2021gmj ; Lazarides:2022jgr ; Fu:2022lrn ; Saad:2022mzu ; Lazarides:2023iim ; Maji:2023fba ; Madge:2023cak .
  * (57) W. Buchmuller, V. Domcke, and K. Schmitz, “Stochastic gravitational-wave background from metastable cosmic strings,” JCAP 12 no. 12, (2021) 006, arXiv:2107.04578 [hep-ph].
  * (58) LIGO Scientific, Virgo, KAGRA Collaboration, R. Abbott et al., “Constraints on Cosmic Strings Using Data from the Third Advanced LIGO–Virgo Observing Run,” Phys. Rev. Lett. 126 no. 24, (2021) 241102, arXiv:2101.12248 [gr-qc].
  * (59) KAGRA, Virgo, LIGO Scientific Collaboration, R. Abbott et al., “Upper limits on the isotropic gravitational-wave background from Advanced LIGO and Advanced Virgo’s third observing run,” Phys. Rev. D 104 no. 2, (2021) 022004, arXiv:2101.12130 [gr-qc].
  * (60) G. Janssen et al., “Gravitational wave astronomy with the SKA,” PoS AASKA14 (2015) 037, arXiv:1501.00127 [astro-ph.IM].
  * (61) Theia Collaboration, C. Boehm et al., “Theia: Faint objects in motion or the new astrometry frontier,” arXiv:1707.01348 [astro-ph.IM].
  * (62) A. Sesana et al., “Unveiling the gravitational universe at $\mu$-Hz frequencies,” Exper. Astron. 51 no. 3, (2021) 1333–1383, arXiv:1908.11391 [astro-ph.IM].
  * (63) LISA Collaboration, P. Amaro-Seoane et al., “Laser Interferometer Space Antenna,” arXiv:1702.00786 [astro-ph.IM].
  * (64) W.-H. Ruan, Z.-K. Guo, R.-G. Cai, and Y.-Z. Zhang, “Taiji program: Gravitational-wave sources,” Int. J. Mod. Phys. A 35 no. 17, (2020) 2050075, arXiv:1807.09495 [gr-qc].
  * (65) TianQin Collaboration, J. Luo et al., “TianQin: a space-borne gravitational wave detector,” Class. Quant. Grav. 33 no. 3, (2016) 035010, arXiv:1512.02076 [astro-ph.IM].
  * (66) V. Corbin and N. J. Cornish, “Detecting the cosmic gravitational wave background with the big bang observer,” Class. Quant. Grav. 23 (2006) 2435–2446, arXiv:gr-qc/0512039.
  * (67) N. Seto, S. Kawamura, and T. Nakamura, “Possibility of direct measurement of the acceleration of the universe using 0.1-Hz band laser interferometer gravitational wave antenna in space,” Phys. Rev. Lett. 87 (2001) 221103, arXiv:astro-ph/0108011.
  * (68) B. Sathyaprakash et al., “Scientific Objectives of Einstein Telescope,” Class. Quant. Grav. 29 (2012) 124013, arXiv:1206.0331 [gr-qc]. [Erratum: Class.Quant.Grav. 30, 079501 (2013)].
  * (69) LIGO Scientific Collaboration, B. P. Abbott et al., “Exploring the Sensitivity of Next Generation Gravitational Wave Detectors,” Class. Quant. Grav. 34 no. 4, (2017) 044001, arXiv:1607.08697 [astro-ph.IM].
  * (70) M. James, L. Perivolaropoulos, and T. Vachaspati, “Stability of electroweak strings,” Phys. Rev. D 46 (1992) R5232–R5235.
  * (71) M. James, L. Perivolaropoulos, and T. Vachaspati, “Detailed stability analysis of electroweak strings,” Nucl. Phys. B 395 (1993) 534–546, arXiv:hep-ph/9212301.
  * (72) M. Goodband and M. Hindmarsh, “Instabilities of electroweak strings,” Phys. Lett. B 363 (1995) 58–64, arXiv:hep-ph/9505357.
  * (73) R. Jeannerot, J. Rocher, and M. Sakellariadou, “How generic is cosmic string formation in SUSY GUTs,” Phys. Rev. D 68 (2003) 103514, arXiv:hep-ph/0308134.
  * (74) L. Zu, C. Zhang, Y.-Y. Li, Y.-C. Gu, Y.-L. S. Tsai, and Y.-Z. Fan, “Mirror QCD phase transition as the origin of the nanohertz Stochastic Gravitational-Wave Background detected by the Pulsar Timing Arrays,” arXiv:2306.16769 [astro-ph.HE].
  * (75) S. Vagnozzi, “Inflationary interpretation of the stochastic gravitational wave background signal detected by pulsar timing array experiments,” arXiv:2306.16912 [astro-ph.CO].
  * (76) C. Han, K.-P. Xie, J. M. Yang, and M. Zhang, “Self-interacting dark matter implied by nano-Hertz gravitational waves,” arXiv:2306.16966 [hep-ph].
  * (77) G. Lambiase, L. Mastrototaro, and L. Visinelli, “Astrophysical neutrino oscillations after pulsar timing array analyses,” arXiv:2306.16977 [astro-ph.HE].
  * (78) J. Ellis, M. Fairbairn, G. Hütsi, J. Raidal, J. Urrutia, V. Vaskonen, and H. Veermäe, “Gravitational Waves from SMBH Binaries in Light of the NANOGrav 15-Year Data,” arXiv:2306.17021 [astro-ph.CO].
  * (79) S.-Y. Guo, M. Khlopov, X. Liu, L. Wu, Y. Wu, and B. Zhu, “Footprints of Axion-Like Particle in Pulsar Timing Array Data and JWST Observations,” arXiv:2306.17022 [hep-ph].
  * (80) E. Megias, G. Nardini, and M. Quiros, “Pulsar Timing Array Stochastic Background from light Kaluza-Klein resonances,” arXiv:2306.17071 [hep-ph].
  * (81) K. Fujikura, S. Girmohanta, Y. Nakai, and M. Suzuki, “NANOGrav Signal from a Dark Conformal Phase Transition,” arXiv:2306.17086 [hep-ph].
  * (82) J. Yang, N. Xie, and F. P. Huang, “Nano-Hertz stochastic gravitational wave background as hints of ultralight axion particles,” arXiv:2306.17113 [hep-ph].
  * (83) G. Franciolini, D. Racco, and F. Rompineve, “Footprints of the QCD Crossover on Cosmological Gravitational Waves at Pulsar Timing Arrays,” arXiv:2306.17136 [astro-ph.CO].
  * (84) Z.-Q. Shen, G.-W. Yuan, Y.-Y. Wang, and Y.-Z. Wang, “Dark Matter Spike surrounding Supermassive Black Holes Binary and the nanohertz Stochastic Gravitational Wave Background,” arXiv:2306.17143 [astro-ph.HE].
  * (85) N. Kitajima, J. Lee, K. Murai, F. Takahashi, and W. Yin, “Nanohertz Gravitational Waves from Axion Domain Walls Coupled to QCD,” arXiv:2306.17146 [hep-ph].
  * (86) J. Ellis, M. Lewicki, C. Lin, and V. Vaskonen, “Cosmic Superstrings Revisited in Light of NANOGrav 15-Year Data,” arXiv:2306.17147 [astro-ph.CO].
  * (87) G. Franciolini, A. Iovino, Junior., V. Vaskonen, and H. Veermae, “The recent gravitational wave observation by pulsar timing arrays and primordial black holes: the importance of non-gaussianities,” arXiv:2306.17149 [astro-ph.CO].
  * (88) A. Ghoshal and A. Strumia, “Probing the Dark Matter density with gravitational waves from super-massive binary black holes,” arXiv:2306.17158 [astro-ph.CO].
  * (89) Y. Bai, T.-K. Chen, and M. Korwar, “QCD-Collapsed Domain Walls: QCD Phase Transition and Gravitational Wave Spectroscopy,” arXiv:2306.17160 [hep-ph].
  * (90) A. Addazi, Y.-F. Cai, A. Marciano, and L. Visinelli, “Have pulsar timing array methods detected a cosmological phase transition?,” arXiv:2306.17205 [astro-ph.CO].
  * (91) P. Athron, A. Fowlie, C.-T. Lu, L. Morris, L. Wu, Y. Wu, and Z. Xu, “Can Supercooled Phase Transitions explain the Gravitational Wave Background observed by Pulsar Timing Arrays?,” arXiv:2306.17239 [hep-ph].
  * (92) N. Kitajima and K. Nakayama, “Nanohertz gravitational waves from cosmic strings and dark photon dark matter,” arXiv:2306.17390 [hep-ph].
  * (93) G. Lazarides, R. Maji, and Q. Shafi, “Superheavy quasi-stable strings and walls bounded by strings in the light of NANOGrav 15 year data,” arXiv:2306.17788 [hep-ph].
  * (94) T. Broadhurst, C. Chen, T. Liu, and K.-F. Zheng, “Binary Supermassive Black Holes Orbiting Dark Matter Solitons: From the Dual AGN in UGC4211 to NanoHertz Gravitational Waves,” arXiv:2306.17821 [astro-ph.HE].
  * (95) Y.-F. Cai, X.-C. He, X. Ma, S.-F. Yan, and G.-W. Yuan, “Limits on scalar-induced gravitational waves from the stochastic background by pulsar timing array observations,” arXiv:2306.17822 [gr-qc].
  * (96) A. Yang, J. Ma, S. Jiang, and F. P. Huang, “Implication of nano-Hertz stochastic gravitational wave on dynamical dark matter through a first-order phase transition,” arXiv:2306.17827 [hep-ph].
  * (97) S. Blasi, A. Mariotti, A. Rase, and A. Sevrin, “Axionic domain walls at Pulsar Timing Arrays: QCD bias and particle friction,” arXiv:2306.17830 [hep-ph].
  * (98) K. Inomata, K. Kohri, and T. Terada, “The Detected Stochastic Gravitational Waves and Sub-Solar Primordial Black Holes,” arXiv:2306.17834 [astro-ph.CO].
  * (99) P. F. Depta, K. Schmidt-Hoberg, and C. Tasillo, “Do pulsar timing arrays observe merging primordial black holes?,” arXiv:2306.17836 [astro-ph.CO].
  * (100) Y. Gouttenoire and E. Vitagliano, “Domain wall interpretation of the PTA signal confronting black hole overproduction,” arXiv:2306.17841 [gr-qc].
  * (101) D. Borah, S. Jyoti Das, and R. Samanta, “Inflationary origin of gravitational waves with \textitMiracle-less WIMP dark matter in the light of recent PTA results,” arXiv:2307.00537 [hep-ph].
  * (102) K. Murai and W. Yin, “A Novel Probe of Supersymmetry in Light of Nanohertz Gravitational Waves,” arXiv:2307.00628 [hep-ph].
  * (103) S. Datta, “Inflationary gravitational waves, pulsar timing data and low-scale-leptogenesis,” arXiv:2307.00646 [hep-ph].
  * (104) B. Barman, D. Borah, S. Jyoti Das, and I. Saha, “Scale of Dirac leptogenesis and left-right symmetry in the light of recent PTA results,” arXiv:2307.00656 [hep-ph].
  * (105) Y.-C. Bi, Y.-M. Wu, Z.-C. Chen, and Q.-G. Huang, “Implications for the Supermassive Black Hole Binaries from the NANOGrav 15-year Data Set,” arXiv:2307.00722 [astro-ph.CO].
  * (106) B.-Q. Lu and C.-W. Chiang, “Nano-Hertz stochastic gravitational wave background from domain wall annihilation,” arXiv:2307.00746 [hep-ph].
  * (107) Y. Xiao, J. M. Yang, and Y. Zhang, “Implications of Nano-Hertz Gravitational Waves on Electroweak Phase Transition in the Singlet Dark Matter Model,” arXiv:2307.01072 [hep-ph].
  * (108) S.-P. Li and K.-P. Xie, “A collider test of nano-Hertz gravitational waves from pulsar timing arrays,” arXiv:2307.01086 [hep-ph].
  * (109) L. A. Anchordoqui, I. Antoniadis, and D. Lust, “Fuzzy Dark Matter, the Dark Dimension, and the Pulsar Timing Array Signal,” arXiv:2307.01100 [hep-ph].
  * (110) X. Niu and M. H. Rahat, “NANOGrav signal from axion inflation,” arXiv:2307.01192 [hep-ph].
  * (111) R. Ebadi, S. Kumar, A. McCune, H. Tai, and L.-T. Wang, “Gravitational Waves from Stochastic Scalar Fluctuations,” arXiv:2307.01248 [astro-ph.CO].
  * (112) Y. Gouttenoire, S. Trifinopoulos, G. Valogiannis, and M. Vanvlasselaer, “Scrutinizing the Primordial Black Holes Interpretation of PTA Gravitational Waves and JWST Early Galaxies,” arXiv:2307.01457 [astro-ph.CO].
  * (113) K. T. Abe and Y. Tada, “Translating nano-Hertz gravitational wave background into primordial perturbations taking account of the cosmological QCD phase transition,” arXiv:2307.01653 [astro-ph.CO].
  * (114) T. Ghosh, A. Ghoshal, H.-K. Guo, F. Hajkarim, S. F. King, K. Sinha, X. Wang, and G. White, “Did we hear the sound of the Universe boiling? Analysis using the full fluid velocity profiles and NANOGrav 15-year data,” arXiv:2307.02259 [astro-ph.HE].
  * (115) C. Unal, A. Papageorgiou, and I. Obata, “Axion-Gauge Dynamics During Inflation as the Origin of Pulsar Timing Array Signals and Primordial Black Holes,” arXiv:2307.02322 [astro-ph.CO].
  * (116) L. Bian, S. Ge, J. Shu, B. Wang, X.-Y. Yang, and J. Zong, “Gravitational wave sources for Pulsar Timing Arrays,” arXiv:2307.02376 [astro-ph.HE].
  * (117) D. G. Figueroa, M. Pieroni, A. Ricciardone, and P. Simakachorn, “Cosmological Background Interpretation of Pulsar Timing Array Data,” arXiv:2307.02399 [astro-ph.CO].
  * (118) X. K. Du, M. X. Huang, F. Wang, and Y. K. Zhang, “Did the nHZ Gravitational Waves Signatures Observed By NANOGrav Indicate Multiple Sector SUSY Breaking?,” arXiv:2307.02938 [hep-ph].
  * (119) G. Servant and P. Simakachorn, “Constraining Post-Inflationary Axions with Pulsar Timing Arrays,” arXiv:2307.03121 [hep-ph].
  * (120) X.-F. Li, “Probing the high temperature symmetry breaking with gravitational waves from domain walls,” arXiv:2307.03163 [hep-ph].
  * (121) P. Di Bari and M. H. Rahat, “The split majoron model confronts the NANOGrav signal,” arXiv:2307.03184 [hep-ph].
  * (122) M. Geller, S. Ghosh, S. Lu, and Y. Tsai, “Challenges in Interpreting the NANOGrav 15-Year Data Set as Early Universe Gravitational Waves Produced by ALP Induced Instability,” arXiv:2307.03724 [hep-ph].
  * (123) L. Liu, Z.-C. Chen, and Q.-G. Huang, “Implications for the non-Gaussianity of curvature perturbation from pulsar timing arrays,” arXiv:2307.01102 [astro-ph.CO].
  * (124) W. Buchmüller, V. Domcke, H. Murayama, and K. Schmitz, “Probing the scale of grand unification with gravitational waves,” Phys. Lett. B 809 (2020) 135764, arXiv:1912.03695 [hep-ph].
  * (125) S. F. King, S. Pascoli, J. Turner, and Y.-L. Zhou, “Gravitational Waves and Proton Decay: Complementary Windows into Grand Unified Theories,” Phys. Rev. Lett. 126 no. 2, (2021) 021802, arXiv:2005.13549 [hep-ph].
  * (126) S. F. King, S. Pascoli, J. Turner, and Y.-L. Zhou, “Confronting SO(10) GUTs with proton decay and gravitational waves,” JHEP 10 (2021) 225, arXiv:2106.15634 [hep-ph].
  * (127) G. Lazarides, R. Maji, and Q. Shafi, “Gravitational waves from quasi-stable strings,” JCAP 08 no. 08, (2022) 042, arXiv:2203.11204 [hep-ph].
  * (128) B. Fu, S. F. King, L. Marsili, S. Pascoli, J. Turner, and Y.-L. Zhou, “A predictive and testable unified theory of fermion masses, mixing and leptogenesis,” JHEP 11 (2022) 072, arXiv:2209.00021 [hep-ph].
  * (129) S. Saad, “Probing minimal grand unification through gravitational waves, proton decay, and fermion masses,” JHEP 04 (2023) 058, arXiv:2212.05291 [hep-ph].
  * (130) G. Lazarides, Q. Shafi, and A. Tiwari, “Composite topological structures in SO(10),” JHEP 05 (2023) 119, arXiv:2303.15159 [hep-ph].
  * (131) R. Maji, W.-I. Park, and Q. Shafi, “Gravitational waves from walls bounded by strings in $SO(10)$ model of pseudo-Goldstone dark matter,” arXiv:2305.11775 [hep-ph].
  * (132) E. Madge, E. Morgante, C. Puchades-Ibáñez, N. Ramberg, W. Ratzinger, S. Schenk, and P. Schwaller, “Primordial gravitational waves in the nano-Hertz regime and PTA data – towards solving the GW inverse problem,” arXiv:2306.14856 [hep-ph].


[1,2]Lauren J. Riddiford These authors contributed equally to this work.

[1,2]Jeffrey A. Brock These authors contributed equally to this work.

[1,2]Aleš Hrabec

1]Laboratory for Mesoscopic Systems, Department of Materials, ETH Zurich,
8093, Zurich, Switzerland

2]Laboratory for Multiscale Materials Experiments, Paul Scherrer Institute,
5232, Villigen PSI, Switzerland

# Grayscale control of local magnetic properties with direct-write laser
annealing

<EMAIL_ADDRESS><EMAIL_ADDRESS>Katarzyna Murawska
<EMAIL_ADDRESS>Laura J. Heyderman  [  [

###### Abstract

Across the fields of magnetism, microelectronics, optics, and others,
engineered local variations in physical properties can yield groundbreaking
functionalities that play a crucial role in enabling future technologies [1,
2, 3, 4, 5, 6]. Beyond binary modifications, 1D lateral gradients in material
properties (achieved by gradients in thickness, stoichiometry, temperature, or
strain) give rise to a plethora of new effects in thin film magnetic systems
[7, 8, 9, 10, 11]. However, extending such gradient-induced behaviors to 2D is
challenging to realize with existing methods, which are plagued by slow
processing speeds, dose instabilities, or limitation to variation along one
dimension [12, 13, 14]. Here, we show for the first time how commonplace
direct-write laser exposure techniques, initially developed for grayscale
patterning of photoresist surfaces [15], can be repurposed to perform
grayscale direct-write laser annealing. With this technique, we demonstrate
the ease with which two-dimensional, continuous variations in magnetic
properties can be created at the mesoscopic scale in numerous application-
relevant materials, including ferromagnetic, ferrimagnetic, and synthetic
antiferromagnetic thin-film systems. The speed, versatility, and new
possibilities to create complex magnetic energy landscapes offered by direct-
write laser annealing opens the door to the lateral modification of the
magnetic, electronic, and structural properties of a variety of thin films
with an abundance of applications.

Laser annealing, in which thermal energy is imparted to a material using a
high-powered laser, has been used in magnetism to modify the saturation
magnetization, exchange bias, and magnetic easy axis orientation of thin films
[16, 17, 18]. However, the instruments previously used have been custom setups
or repurposed lasers with a spot size of $\sim 10-20$ $\mu$m and without
spatial patterning capabilities beyond that provided by a simple Gaussian
intensity profile [19]. While local control of magnetic anisotropy through
electron beam exposure has been demonstrated, the technique has been applied
sparingly [20, 21].

Here, we demonstrate the implementation of direct-write laser annealing (DWLA)
to create arbitrarily-shaped magnetic energy landscapes in numerous important
magnetic thin-film systems. We locally engineer continuous variations in the
magnetic anisotropy in ferromagnetic and synthetic antiferromagnetic
materials, the compensation temperature of ferrimagnets, and the Ruderman-
Kittel-Kasuya-Yosida (RKKY) coupling strength of synthetic antiferromagnets,
exemplifying the broad utility of DWLA. Crucially, as DWLA does not require
patterned resist layers or ultrahigh vacuum environments, the process is
significantly streamlined compared to other approaches. In addition, the
applicability of this technique to a wide variety of material systems opens
the door to creating novel and precise potential energy landscapes for
spintronic and magnonic applications, and beyond.

The principle of DWLA is summarized in Fig. 1a. A 3D CAD structure (input) is
used to create gradients in the film plane, with the local laser fluence
determined by the height of the 3D structure. The photolithography software
interprets this 3D object as a grayscale image, where white indicates maximum
laser power and black indicates no exposure. The laser writer is then used to
expose the pattern onto the film by rastering the laser across the surface and
modulating the laser power according to the design. The resulting magnetic
texture is observed with a Kerr microscope (output), seen in Fig. 1b. In the
following sections, we highlight four material systems that illustrate four
mechanisms of thermally-induced material transformation that DWLA offers:
crystallization, oxidation, interfacial alloying, and diffusion in
multilayers.

##  1 Interface crystallization in ferromagnetic heterostructures

CoFeB/MgO-based heterostructures have been extensively investigated for
implementation in magnetic tunnel junctions (MTJs) due to the high tunneling
magnetoresistance of CoFeB/MgO/CoFeB. Furthermore, strong perpendicular
magnetic anisotropy (PMA) can be achieved in these heterostructures through
thermal annealing, where crystallization of the CoFeB/MgO layers yields
interfacial PMA ($K_{\text{i}}$) due to hybridization of the Fe-O orbitals.
The effective magnetic anisotropy energy $K_{\text{eff}}$ is given by:

$K_{\text{eff}}=K_{\text{u}}-\frac{1}{2}\mu_{0}M_{\text{s}}^{2}+\frac{K_{\text{i}}}{t_{\text{CFB}}}$
(1)

where $K_{\text{u}}$ is the (negligible) magnetocrystalline anisotropy,
$\mu_{0}$ is the vacuum permeability, $\mu_{0}M_{\text{s}}$ is the saturation
magnetization, $\frac{1}{2}\mu_{0}M_{\text{s}}^{2}$ is the shape anisotropy,
and $t_{\text{CFB}}$ is the CoFeB thickness [22]. By tuning the magnitude of
$K_{\text{i}}$, the effective anisotropy can be modified from in-plane
($K_{\text{eff}}<0$) to out-of-plane ($K_{\text{eff}}>0$). In particular, when
$t_{\text{CFB}}\leq 1.4$ nm, interfacial PMA can overcome the shape
anisotropy. With DWLA, thermal energy is deposited in small regions of CoFeB
with high grayscale precision, allowing for continuous tuning of
$K_{\text{eff}}$ over a broad range. In this materials system, we observe that
DWLA increases the ordering of the film, providing a process window where the
properties are transformed through crystallization rather than film damage.

Figure 1: DWLA and local anisotropy tuning of ferromagnetic thin films. a,
Procedure for local grayscale laser annealing. 2D grayscale intensity in the
pattern is interpreted as a laser fluence adjustment, which is executed by the
modulator. The deflector rasters the laser on the film as the stage
simultaneously moves to write a 2D pattern. b, A background-subtracted Kerr
micrograph of the written pattern on a CoFeB thin film (scale bar = 10
$\mu$m). The magnetic domain size increases as laser fluence increases,
indicating stronger PMA. c, Changes in the pMOKE hysteresis loops for regions
of 1.3-nm-thick CoFeB exposed to increasing laser fluence. PMA is achieved in
a fluence range of 1.25-1.55 J/cm2, while fluences >1.55 J/cm2 disrupt PMA.
The standard error of the measurements shown in c are smaller than the lines
used.

Ta/CoFeB/MgO films are grown with in-plane magnetic anisotropy. As regions of
the film are laser annealed with increasing fluence, hysteresis loops recorded
with polar magneto optic Kerr effect (pMOKE) magnetometry become increasingly
square. As shown in Fig. 1c, sharp switching steps indicative of PMA are
observed by a fluence of 1.25 J/cm2. When the laser fluence is further
increased above a threshold value of 1.55 J/cm2, the hysteresis loops become
more slanted, indicating a loss of PMA. These results are consistent with
those obtained by furnace annealing CoFeB films, where there is an optimal
temperature range of $\sim 200-350$ ∘C that promotes strong PMA [22]. When
furnace annealing above $\sim 350$ ∘C, Ta diffusion degrades the interfacial
anisotropy energy, and the magnetization easy axis falls in-plane. Comparing
the effects of DWLA to furnace annealing, we estimate that DWLA in the optimal
fluence range is equivalent to 1 hour in a 300 ∘C furnace (see Extended Data
Fig. 1). Meanwhile, DWLA accomplishes the same effect for a $100\times 100$
$\mu$m2 square in 30 seconds. An anisotropy energy range of up to $\Delta
K_{\text{eff}}\approx 3\times 10^{5}$ J/m3 can be created in arbitrary
patterns in the film plane (Fig. 1b). This energy difference is sufficient to
dramatically influence propagation of spin waves [23]. Furthermore, DWLA could
be used to selectively anneal a series of MTJs to different PMA strengths for
the construction of sophisticated neural networks [24].

##  2 Control of ferrimagnetic compensation temperature by oxidation

Near the magnetic compensation temperature (T${}_{\text{m}}$), it is well-
known that transition metal-rare earth (TM-RE) ferrimagnetic films exhibit
novel, application-relevant magnetization dynamics, such as extraordinarily
efficient current-induced domain wall motion [25, 26, 27] and ultrafast laser-
induced magnetization switching [28]. More recently, it has been demonstrated
that the handedness of spin waves in TM-RE ferrimagnets depends on whether the
sample temperature is above or below T${}_{\text{m}}$ [29]. In addition to the
demonstration of localized, binary control of T${}_{\text{m}}$ using focused
ion irradiation and selective oxidation techniques [30, 31, 32], the symmetry
breaking originating from 1D lateral gradients in T${}_{\text{m}}$ has
recently been shown to be a valuable means of achieving magnetic field-free,
all-electrical magnetization switching [31]. As all-electrical control of
magnetization is necessary to realize energy-efficient, high-density
spintronic computing architectures, there is a crucial need to develop
scalable techniques by which these functionalities can be extended to 2D [33].

Here, we demonstrate that DWLA can be applied to TM-RE ferrimagnets to impart
two-dimensional, continuous gradients in T${}_{\text{m}}$ through local
oxidation. For this, we exposed $100\times 100$ $\mu$m2 regions of a CoGd film
(see Methods for the sample composition) using varying laser fluences, and
collected hysteresis loops using pMOKE (Fig. 2a). The positive polarity of the
room temperature pMOKE hysteresis loop of a virgin CoGd film (orange loop in
Fig. 2a) indicates the as-deposited film is RE dominant in its magnetic
properties at room temperature [34]. For laser fluences less than 1.3 J/cm2,
the coercivity increases with increasing fluence (light purple loop in Fig.
2a). For fluences greater than or equal to 1.3 J/cm2, the polarity of the
pMOKE loop is reversed relative to the virgin film (dark purple loop in Fig.
2a), and the coercivity decays with increasing fluence (black loop in Fig.
2a). The change in the pMOKE loop polarity, coupled with the non-monotonic,
divergent trend in coercivity as a function of fluence shown in Fig. 2b
demonstrates how DWLA permits tuning of magnetic compensation in ferrimagnetic
films [30, 25]. This scenario is further confirmed by magnetometry
measurements of continuous films uniformly exposed using various laser
fluences (see Extended Data Figs. 2b and 2c).

Figure 2: Tuning the ferrimagnetic compensation temperature in CoGd films with
DWLA. a, pMOKE hysteresis loops for $100\times 100$ $\mu$m2 regions that have
been subjected to DWLA using different laser fluences. b, Coercive field
$\mu_{0}H_{\text{c}}$ as a function of laser fluence extracted from pMOKE
loops. c, The 2D fluence profile used to create the ferrimagnetic
“compensation surface” demonstrated in d (scale bar = 10 $\mu$m). d, pMOKE
image of the compensation surface created using the fluence profile shown in c
and schematic representations of the magnetic Co (red) and Gd (blue)
sublattice configurations of the RE-dominant (top), compensated (middle), and
TM-dominant (bottom) regions that define the compensation surface (scale bar =
10 $\mu$m). The measurement protocol used to generate d is provided in the
Methods section. See Extended Data Fig. 2a for the detailed pMOKE
characterizations used to generate b. The standard error of the measurements
shown in a and b are smaller than the symbols or lines used.

Unlike past approaches, DWLA allows for continuous 2D patterning such as the
radial fluence gradient demonstrated in Fig. 2c. In Fig. 2d, we show how the
fluence profile of Fig. 2c results in the formation of a two-dimensional
“compensation surface” at remanence after the sample was exposed to a +500 mT
perpendicular magnetic field, characterized by the presence of three distinct
magnetic regions whose configurations are illustrated in the insets of Fig.
2d. Separating the TM (in black) and RE (in white) dominant regions, there
exists a shell of intermediate gray contrast, corresponding to an area whose
magnetization has not switched relative to the subtracted reference image (see
Methods). Given that the coercivity diverges near magnetic compensation, this
inability to switch the magnetic orientation in a large magnetic field
indicates that these areas are close to magnetic compensation at room
temperature. While linear composition gradients have previously been used to
obtain 1D compensation walls [35], DWLA permits this first-ever demonstration
of a 2D “compensation surface.”

##  3 Modifying RKKY coupling fields in synthetic antiferromagnets through
interfacial alloying

Thin-film systems in which individual ferromagnetic layers are
antiferromagnetically coupled to each other through a spacer layer that
fosters indirect RKKY coupling (so-called synthetic antiferromagnets, or SAFs)
have long enjoyed significant interest. Besides playing a crucial role in
modern magnetic sensing technologies that operate on the principle of giant
magnetoresistance [36, 37, 38], there has been a renewed enthusiasm for SAFs
in the past decade due to their extraordinarily fast and efficient domain wall
motion in response to a spin-orbit torque [39]. More recently, the utility of
creating local gradients in the RKKY coupling field in SAFs has been
highlighted by works showing that such gradients permit field-free, all-
electrical magnetization switching using spin-orbit torques [40]. As such, the
development of new approaches by which the RKKY coupling strength of SAFs can
be locally controlled is of paramount technological interest in the
development of magnetic components for integration in spin-based computing and
memory schemes [33, 41, 42].

To demonstrate the unique capability of DWLA to pattern the local RKKY
coupling in SAFs by means of interfacial alloying, we consider a Co/Cr/Co
multilayer (see Methods for the full sample structure). The as-deposited
Co/Cr/Co sample exhibits a pMOKE hysteresis loop typical of SAFs with PMA (see
Extended Data Fig. 3a). Defining the RKKY coupling field
$\mu_{0}H_{\text{RKKY}}$ as the field about which the spin-flip transitions
are centered, the as-deposited sample possesses a $\mu_{0}H_{\text{RKKY}}$ of
360 mT. Next, we patterned a spiral-shaped exposure gradient, as shown in Fig.
3a. Starting from zero fluence, the fluence was continuously increased along
the spiral to a maximum of 4.5 J/cm2. From pMOKE hysteresis loops collected at
selected locations along the spiral, we have extracted
$\mu_{0}H_{\text{RKKY}}$ as a function of laser fluence, as shown in Fig. 3b
(see Extended Data Fig. 3a for the full pMOKE characterization). From Fig. 3b,
it can be seen that an increase in laser fluence causes a decrease in
$\mu_{0}H_{\text{RKKY}}$. When Co/Cr/Co is exposed using laser fluences
greater than approximately 4 J/cm2, a spin-flip transition is no longer
apparent in the pMOKE hysteresis loops (see Extended Data Fig. 3a), suggesting
that the two FM layers comprising the SAF structure are no longer
antiferromagnetically coupled to one another, to which we assign a
$\mu_{0}H_{\text{RKKY}}$ value of zero.

Figure 3: Continuous reduction of the RKKY coupling field in Co/Cr/Co SAF
structures with DWLA. a, Optical micrograph of the spiral track structure with
an overlay of the 2D fluence profile imparted to the structure (scale bar = 10
$\mu$m). b, RKKY coupling field ($\mu_{0}H_{\text{RKKY}}$) as a function of
laser fluence, measured along the spiral structure shown in a. The inset shows
the saturation magnetization $M_{\text{s}}$ as a function of laser fluence,
determined from SQUID - VSM measurements. See Extended Data Fig. 3 for the
full pMOKE and SQUID - VSM measurements used to generate Figs. 3b. The
standard error of the measurements shown in b are smaller than the symbols or
lines used.

To understand the relationship between laser fluence and the measured
$\mu_{0}H_{\text{RKKY}}$, we consider the simple algebraic relationship:

$J_{\text{RKKY}}=t_{\text{m}}M_{\text{s}}\mu_{0}H_{\text{RKKY}}$ (2)

t${}_{\text{m}}$ is the total thickness of magnetic material in the SAF, and
J${}_{\text{RKKY}}$ is the interlayer coupling strength (set by the thickness
of the spacer layer). Therefore, possible explanations for the observed
reduction in $\mu_{0}H_{\text{RKKY}}$ with increasing laser fluence could be a
reduction in $J_{\text{RKKY}}$ (due to a change in the effective thickness or
stoichiometry of the RKKY spacer layer) or $M_{\text{s}}$ (resulting from the
oxidation of Co). To delineate between these two mechanisms, we determined the
$M_{\text{s}}$ of continuous Co/Cr/Co films uniformly exposed using various
laser fluences (see the inset of Fig. 3b and Extended Data Fig. 3b). Over the
range of fluences that modify the magnetic properties of our Co/Cr/Co SAFs (0
- 5.5 J/cm2), DWLA reduces $M_{\text{s}}$ by approximately 15% relative to the
as-deposited sample. However, within the context of Eq. 2, a 15% reduction in
$M_{\text{s}}$ cannot be solely responsible for the extensive changes in RKKY
coupling field evident in Fig. 3b. Thus, while a reduction in $M_{\text{s}}$
due to heating-induced oxidation of Co may occur, a change in the effective
thickness of the Cr spacer layer as a result of the alloying of Cr and Co must
also be responsible for the observed variation in both the strength and nature
(i.e., ferromagnetic or antiferromagnetic) of the interlayer RKKY coupling.

##  4 Diffusion-induced anisotropy modification of synthetic antiferromagnets

In addition to local control of RKKY coupling, choice of different materials
can allow us to instead modify the magnetic anisotropy of a SAF. Two CoFeB/Pt
magnetic multilayers coupled by a Ru spacer are synthesized for this purpose.
In previous studies of CoFeB/Pt multilayers, it was found that PMA is lost
when the structure was annealed above $\sim 300$ ∘C [43]. With DWLA,
thermally-induced diffusion within each magnetic layer leads to a coherent
change in magnetic anisotropy of a SAF as the laser fluence increases – from
strongly out-of-plane, to weakly out-of-plane, to in-plane, without disrupting
the RKKY coupling. This ability to modify anisotropy is of particular
technological relevance for domain wall devices, as domain walls are expected
to react more strongly to an anisotropy gradient in an antiferromagnet, and
therefore in SAFs, than a ferromagnet [44].

Figure 4: Magnetic domain wall motion along an anisotropy gradient in a SAF.
a, As-grown SAFs display a spin-flip transition (black solid line), where the
two magnetic layers abruptly switch from antiferromagnetic to ferromagnetic
alignment (at $\sim 10$ mT). After laser exposure at a fluence of 2.9 J/cm2,
the SAF instead has a more gradual spin-flop transition (black dashed line).
The standard error of the measurements shown in a are smaller than the lines
used. b, The circular anisotropy gradient written into a continuous CoFeB-
based SAF, giving a radial domain wall ratchet. c-f, Background-subtracted
Kerr microscope images of the device (scale bar = 10 $\mu$m). The background
was taken at remanence after application of a -30 mT magnetic field. The red
arrows indicate the direction of domain wall motion. As the Kerr contrast of
SAFs at remanence is small, the contrast was enhanced with post-processing.
Unaltered images are shown in Extended Data Fig. 4. c, The device just at the
moment a +4.5 mT magnetic field is applied. A domain nucleates at the lowest
anisotropy region (labeled with a white dumbbell), with the domain wall
configuration shown on the right. Here, m${}_{\text{L}}$ refers to the
magnetization of the lower magnetic layer in the SAF, which determines the
Kerr contrast. d, The device shortly after applying a +4.5 mT magnetic field
pulse. DW1 travels along the gradient, while DW2 remains pinned at the barrier
between low and high anisotropy. e, 45 s after the applied magnetic field
returns to zero, DW1 has started retreating back down the anisotropy gradient.
DW2 moves counterclockwise slightly, possibly due to sample inhomogeneity
which leads to a small region of reduced remanent magnetization. f, At 86
seconds after field pulse, DW1 has continued to move spontaneously down the
anisotropy gradient towards DW2, where the domain nucleated.

To monitor tuning of magnetic anisotropy in the CoFeB-based SAFs due to DWLA,
we record pMOKE hysteresis loops of regions annealed with varying laser
fluences. We observe a shift in the pMOKE loop from a spin-flip SAF, where
there is a sharp switching event from antiferromagnetic to ferromagnetic
alignment at $\mu_{0}H_{\text{RKKY}}$, to a spin-flop SAF, where switching
from antiferromagnetic to ferromagnetic alignment occurs gradually, at a
fluence of 2.9 J/cm2(seen in Fig. 4a). If laser annealing modified the
anisotropy of the top magnetic layer only, an increase of the remanent
magnetization would be expected. However, the remanent magnetization is
unaffected, suggesting the anisotropy of both magnetic layers are reduced
simultaneously (see Extended Data Fig. 5a). A further increase in laser
fluence (to $\geq 3.4$ J/cm2) leads to a loss of PMA.

The grayscale and complex design capabilities of DWLA provide an opportunity
to make devices with much greater freedom than with existing thin film
modification methods. Specifically, gradients along curved paths and around
corners can be created. As a proof of concept, we fabricate a circular domain
wall track where the laser fluence linearly decreases along the track such
that the anisotropy gradient is always parallel to the track (seen in Fig.
4b). Such a structure is impossible to fabricate with most other methods
except ion irradiation, which suffers from difficulty in maintaining constant
doses radially [45]. The linear increase in laser fluence leads to a linear
decrease in the PMA of the SAF, with $\Delta K_{\text{eff}}=1.5\times 10^{5}$
J/m3 (see Extended Data Fig. 5b). The device is initialized with a -30 mT
magnetic field to a down-up state (referring to the magnetization direction in
the bottom and top magnetic layer, respectively). When a positive magnetic
field is applied, an up-down domain nucleates in the lowest anisotropy region
(indicated by the white dumbbell in Fig. 4c), and one domain wall (DW1)
propagates counterclockwise along the circular anisotropy gradient. Due to the
anisotropy energy barrier, the other domain wall (DW2) is prevented from
moving clockwise (see Fig. 4d). When the field is removed, DW1 retreats to the
area with the lowest anisotropy, moving spontaneously down the anisotropy
gradient, seen in Fig. 4e,f. Such “resetting functionality” is promising for
providing a feedback mechanism for domain wall logic, which is essential for
the development of in-memory computing, or as a resetting mechanism for
neuromorphic computing [46, 47].

##  5 Discussion

There are five key advantages that direct-write laser annealing offers over
pre-existing methods for local modification of materials properties:

  1. 1.

Ease of use: DWLA is performed using commercial equipment designed for 2D and
2.5D photolithography. The desired exposure profile is straightforwardly
created using CAD software and interpreted by the tool.

  2. 2.

Grayscale capability: Unlike magnetic property gradients produced by
composition or thickness wedges, gradients produced by DWLA can have arbitrary
2D shapes and energy profiles. While light ion irradiation offers capabilities
similar to DWLA, beam instability may impact exposure precision over large
areas, and the required instrumentation is less readily available. Hence, DWLA
significantly expands the possibilities for creating mesoscopic, arbitrary
profiles in magnetic potential energy that extend over large areas. This
flexibility makes DWLA ideal to couple with computational approaches to
control magnon, phonon, or photon propagation [48, 49].

  3. 3.

Modification through the whole heterostructure: Unlike selective oxidation by
oxygen plasma and thermal scanning probe lithography, thermal energy is
deposited throughout the heterostructure, allowing alteration of magnetic
properties through the entire thickness of a film rather than just at the
surface.

  4. 4.

Speed: For 180 nm feature sizes (see Extended Data Fig. 6), a $100\times 100$
$\mu$m2 area can be exposed in under one minute. This is an important
practical advantage over techniques like thermal scanning probe lithography,
which typically requires more processing time [19]. The high speed of DWLA
also means that large areas (mm to cm scale) can be exposed with high
uniformity. It is further anticipated that using multiple laser sources in
parallel could facilitate even faster exposure speeds.

  5. 5.

Generality: Because DWLA can be implemented in any thin film modified by heat-
induced crystallization, oxidation, alloying, or diffusion, it is relevant to
a wide variety of material systems beyond magnetic materials. For example,
since non-uniform charge currents at corners of heavy metal conduits can
induce field-free spin-orbit torque switching [50], a precise control of
switching characteristics could be envisioned from a spatial tuning of
resistivity. The possibility to create nanoscale gradients in the refractive
index of a material extends the applicability of DWLA to the field of photonic
metamaterials [51]. Very recently, DWLA was used to create photonic circuits
by modifying the crystal structure of a thin film in a binary manner [52]. Our
grayscale technique dramatically expands the possibilities for photonic
devices, and can be extended to any photonic material whose properties can be
continuously tuned by laser irradiation. While surface roughness gradients
created using multiple chemical and microfluidic patterning steps have been
used to control the properties of cultured cells [53], DWLA offers a path to
obtain similar effects with a substantially simpler process.

##  6 Conclusions

We have demonstrated how intricate magnetic energy landscapes can be created
on demand using direct-write laser annealing. This opens up the possibility of
realizing theoretically proposed device designs that could not be fabricated
using existing methods. Such fine control of the energy landscape, both in
terms of pattern and energy scale, will make it possible to characterize with
high precision how spin waves and magnetic domain walls interact with
variations in magnetic properties. This will lead to new schemes for in-memory
computation, data storage, and sensing. Beyond the field of magnetism, direct-
write laser annealing provides a path to locally control and exploit the
properties of any thin film that transforms in response to heat, ushering in
further applications across many fields.

##  7 Methods

###  7.1 Synthesis

All films were deposited on Si substrates with a 300 nm-thick thermal oxide
coating using magnetron sputtering in a 3 mTorr Ar partial pressure. The ATC
Orion sputtering tool (AJA International, Inc.) used had a base pressure of
$4\times 10^{-8}$ Torr. MgO was deposited with rf sputtering; all other layers
were deposited with dc sputtering. The full composition of the films are
detailed below, where all thicknesses are in nm:

  * •

CoFeB films, with the structure Ta(5)/ Co40Fe40B20(1.3)/ MgO(1.5)/ Ta(5) were
sputtered at powers of 80 W, 40 W, and 150 W, respectively.

  * •

CoGd films of the structure Ta(2)/ Pt (5)/ Co70%Gd30% (5.5)/ Ta (3) were
deposited using a co-sputtering technique, in which the Co (Gd) was deposited
using a sputtering power of 50 W (24 W).

  * •

The Co/Cr/Co SAFs have the composition Ta(2)/ Pt(3)/ Co(0.7)/ Cr(1)/ [Co(0.7)/
Pt(1)]3/ Pt(5). All layers were deposited using a sputtering power of 100 W.

  * •

The CoFeB/Pt-based SAFs have the structure Ta(5)/ [Pt(1.5)/ CoFeB(0.4)]3/
Ru(1.1)/ [CoFeB(0.4)/ Pt(1.5)]2. Ta and Ru were deposited at 80 W, CoFeB was
deposited at 40 W, and Pt was deposited at 100 W.

After treatment with DWLA, the Co/Cr/Co sample was patterned to a 2
$\mu$m-wide spiral feature (as shown in Fig. 3a) using conventional UV
photolithography and ion beam etching techniques.

###  7.2 Direct-write laser annealing

A Heidelberg DWL66+ direct-write photolithography system with 405 nm CW
illumination was used to perform DWLA. In the “Advanced Grayscale”
configuration, this platform provides 256 levels of grayscale exposure
intensity, which allows for quasi-continuous variations in magnetic
properties. Design files made using standard CAD programs modulate the power
of the laser as it is raster-scanned over the film. The “Hi-Res” write head
used throughout this work offers a 50 nm address grid with feature sizes of
180 nm (see Extended Data Fig. 6). The maximum laser fluence possible is $\sim
31$ kJ/cm2. Based on the published write speed of 3 mm2/minute, we estimate
that each pixel in the address grid is exposed to the laser for 50 ns.

The substrate on which a film is grown can dramatically impact the laser
fluence required to induce transformations in the physical properties. As the
laser annealing process is thermal in nature, this variance can be explained
by the differing thermal conductivities or optical absorptivities of the
substrates. Silicon with a 300 nm-thick thermal oxide coating is used for all
experiments in the paper, as the low thermal conductivity of SiO2 (1.3
W/m$\cdot$K) reduces the laser fluence needed to obtain significant
temperature increases. Similar modifications in magnetic properties can be
achieved for films deposited on Si3N4 (55 W/m$\cdot$K)[54] and Si (140
W/m$\cdot$K)[55] substrates, but much higher fluences must be used. We also
found that different SiO2 wafers required slightly different fluences to
achieve the same effects, indicating a strong sensitivity to the thermal
properties of the substrate. As the maximum fluence ($\sim 31$ kJ/cm2) was not
required to modify films grown on any of the substrates mentioned above, DWLA
is suitable for materials with higher thermal conductivities or lower optical
absorptivities.

###  7.3 Magneto optic Kerr effect measurements

Magneto optic Kerr effect (MOKE) hysteresis loops were measured using a
NanoMOKE magnetometry system manufactured by Durham Magneto Optics, Ltd.,
which employs a 660 nm laser. The pMOKE hysteresis loops were collected from
areas that were direct-write laser annealed using the indicated laser fluence.
Imaging of magnetic domain structures was performed using a MOKE microscopy
platform with wide-spectrum LED illumination manufactured by Evico Magnetics
GmbH. As is common in MOKE microscopy, the contrast of MOKE images was
enhanced using a background subtraction procedure in which an image collected
during exposure to a saturating magnetic field was subtracted from the
recorded image.

###  7.4 Magnetic characterization

A Quantum Design magnetic property measurement system (MPMS) equipped with
superconducting quantum interference device - vibrating sample magnetometry
(SQUID - VSM) was used to measure the saturation magnetization of selected
continuous films subjected to uniform laser exposure.

##  8 End Notes

###  8.1 Acknowledgements

JAB acknowledges funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement No
884104 (PSI-FELLOW-III-3i). LJR acknowledges support from the ETH Zurich
Postdoctoral Fellowship Program 22-2 FEL-006. We thank J. Moritz Bosse for
assistance with furnace annealing experiments. We thank the staff of the
cleanroom facilities at the Laboratory for Nano and Quantum Technologies at
the Paul Scherrer Institute for technical support.

###  8.2 Author contributions

JAB and LJR synthesized samples, carried out experiments, and performed
analysis. KM synthesized SAF samples and carried out characterization
experiments. JAB, LJR, AH, and LJH prepared the manuscript.

## References

  *   * [1] Choe, H. S. _et al._ Ion write microthermotics: Programing thermal metamaterials at the microscale.  _Nano Letters_ 19, 3830 (2019).
  * [2] Ghosh, S. _et al._ On-demand local modification of high-Tc superconductivity in few unit-cell thick Bi2Sr2CaCu2O8+δ.  _Advanced Materials_ 32, 2002220 (2020).
  * [3] Bonfadini, S. _et al._ Femtosecond laser-induced refractive index patterning in inorganic/organic hybrid films.  _Advanced Photonics Research_ 3, 2100257 (2022).
  * [4] Sarcan, F. _et al._ Understanding the impact of heavy ions and tailoring the optical properties of large-area monolayer WS2 using focused ion beam.  _npj 2D Materials and Applications_ 7, 23 (2023).
  * [5] Chitnis, G., Ding, Z., Chang, C.-L., Savran, C. A. & Ziaie, B.  Laser-treated hydrophobic paper: an inexpensive microfluidic platform.  _Lab on a Chip_ 11, 1161 (2011).
  * [6] Kiechle, M. _et al._ Spin-wave optics in YIG realized by ion-beam irradiation.  _Small_ 19, 2207293 (2023).
  * [7] Yamahara, H. _et al._ Flexoelectric nanodomains in rare-earth iron garnet thin films under strain gradient.  _Communications Materials_ 2, 95 (2021).
  * [8] Kim, K. _et al._ Chiral-phonon-activated spin seebeck effect.  _Nature Materials_ 22, 322 (2023).
  * [9] Hou, D. _et al._ Observation of temperature-gradient-induced magnetization.  _Nature Communications_ 7, 12265 (2016).
  * [10] Uchida, K.-I. _et al._ Observation of the spin seebeck effect.  _Nature_ 455, 778 (2008).
  * [11] Yu, G. _et al._ Switching of perpendicular magnetization by spin–orbit torques in the absence of external magnetic fields.  _Nature Nanotechnology_ 9, 548 (2014).
  * [12] Gigax, J. G. _et al._ The influence of ion beam rastering on the swelling of self-ion irradiated pure iron at 450 C.  _Journal of Nuclear Materials_ 465, 343 (2015).
  * [13] Schmidt, M., Byrne, J. M. & Maasilta, I. J.  Bio-imaging with the helium-ion microscope: A review.  _Beilstein Journal of Nanotechnology_ 12, 1 (2021).
  * [14] McGinn, P. J.  Thin-film processing routes for combinatorial materials investigations—a review.  _ACS Combinatorial Science_ 21, 501 (2019).
  * [15] Grushina, A.  Direct-write grayscale lithography.  _Advanced Optical Technologies_ 8, 163 (2019).
  * [16] Ando, K., Yokoyama, Y., Okuda, T. & Koshizuka, N.  Localised modification of magnetic anisotropy in LPE iron garnet films by laser annealing.  _Journal of Magnetism and Magnetic Materials_ 35, 350 (1983).
  * [17] Berthold, I., Löschner, U., Schille, J., Ebert, R. & Exner, H.  Exchange bias realignment using a laser-based direct-write technique.  _Physics Procedia_ 56, 1136 (2014).
  * [18] Sharma, A. _et al._ Exchange bias and diffusion processes in laser annealed CoFeB/IrMn thin films.  _Journal of Magnetism and Magnetic Materials_ 489, 165390 (2019).
  * [19] Levati, V. _et al._ Phase nanoengineering via thermal scanning probe lithography and direct laser writing.  _Advanced Materials Technologies_ 8, 2300166 (2023).
  * [20] Allenspach, R., Bischof, A., Dürig, U. & Grütter, P.  Local modification of magnetic properties by an electron beam.  _Applied Physics Letters_ 73, 3598 (1998).
  * [21] Guang, Y. _et al._ Electron beam lithography of magnetic skyrmions.  _Advanced Materials_ 32, 2003003 (2020).
  * [22] Liu, Y., Hao, L. & Cao, J.  Effect of annealing conditions on the perpendicular magnetic anisotropy of Ta/CoFeB/MgO multilayers.  _AIP Advances_ 6, 045008 (2016).
  * [23] Yu, H. _et al._ Magnon-bandgap controllable artificial domain wall waveguide.  _Applied Physics Letters_ 122, 242403 (2023).
  * [24] Rzeszut, P. _et al._ Multi-state MRAM cells for hardware neuromorphic computing.  _Scientific Reports_ 12, 7178 (2022).
  * [25] Caretta, L. _et al._ Fast current-driven domain walls and small skyrmions in a compensated ferrimagnet.  _Nature Nanotechnology_ 13, 1154 (2018).
  * [26] Kim, K.-J. _et al._ Fast domain wall motion in the vicinity of the angular momentum compensation temperature of ferrimagnets.  _Nature Materials_ 16, 1187 (2017).
  * [27] Bläsing, R. _et al._ Exchange coupling torque in ferrimagnetic Co/Gd bilayer maximized near angular momentum compensation temperature.  _Nature Communications_ 9, 4984 (2018).
  * [28] Stanciu, C. D. _et al._ All-optical magnetic recording with circularly polarized light.  _Physical Review Letters_ 99, 047601 (2007).
  * [29] Kim, C. _et al._ Distinct handedness of spin wave across the compensation temperatures of ferrimagnets.  _Nature Materials_ 19, 980 (2020).
  * [30] Liu, Z. _et al._ Strong lateral exchange coupling and current-induced switching in single-layer ferrimagnetic films with patterned compensation temperature.  _Physical Review B_ 107, L100412 (2023).
  * [31] Lee, T. _et al._ Field-free spin-orbit torque switching of GdCo ferrimagnet with broken lateral symmetry by He ion irradiation.  _Acta Materialia_ 246, 118705 (2023).
  * [32] Frąckowiak, Ł. _et al._ Magnetic domains without domain walls: A unique effect of He+ ion bombardment in ferrimagnetic Tb/Co films.  _Physical Review Letters_ 124, 047203 (2020).
  * [33] Lin, X., Yang, W., Wang, K. L. & Zhao, W.  Two-dimensional spintronics for low-power electronics.  _Nature Electronics_ 2, 274 (2019).
  * [34] Khorsand, A. _et al._ Element-specific probing of ultrafast spin dynamics in multisublattice magnets with visible light.  _Physical Review Letters_ 110, 107205 (2013).
  * [35] Hrabec, A., Nam, N., Pizzini, S. & Ranno, L.  Magnetization reversal in composition-controlled Gd1-xCox ferrimagnetic films close to compensation composition.  _Applied Physics Letters_ 99, 052507 (2011).
  * [36] Grünberg, P., Schreiber, R., Pang, Y., Brodsky, M. & Sowers, H.  Layered magnetic structures: Evidence for antiferromagnetic coupling of Fe layers across Cr interlayers.  _Physical Review Letters_ 57, 2442 (1986).
  * [37] Baibich, M. N. _et al._ Giant magnetoresistance of (001) Fe/(001) Cr magnetic superlattices.  _Physical Review Letters_ 61, 2472 (1988).
  * [38] Parkin, S. S. P.  Systematic variation of the strength and oscillation period of indirect magnetic exchange coupling through the 3d, 4d, and 5d transition metals.  _Physical Review Letters_ 67, 3598 (1991).
  * [39] Yang, S.-H., Ryu, K.-S. & Parkin, S.  Domain-wall velocities of up to 750 m s-1 driven by exchange-coupling torque in synthetic antiferromagnets.  _Nature Nanotechnology_ 10, 221 (2015).
  * [40] Wang, Z. _et al._ Field-free spin-orbit torque switching of synthetic antiferromagnet through interlayer Dzyaloshinskii-Moriya interactions.  _Cell Reports Physical Science_ 4, 101334 (2023).
  * [41] Mitin, D. _et al._ Magnetic properties of artificially designed magnetic stray field landscapes in laterally confined exchange-bias layers.  _Nanotechnology_ 29, 355708 (2018).
  * [42] Koch, L., Samad, F., Lenz, M. & Hellwig, O.  Manipulating the energy balance of perpendicular-anisotropy synthetic antiferromagnets by He+-ion irradiation.  _Physical Review Applied_ 13, 024029 (2020).
  * [43] Zhu, Y., Zhang, Z., Ma, B. & Jin, Q. Y.  Thermal stability of CoFeB/Pt multilayers with perpendicular magnetic anisotropy.  _Journal of Applied Physics_ 111, 07C106 (2012).
  * [44] Wen, D. _et al._ Ultralow-loss domain wall motion driven by a magnetocrystalline anisotropy gradient in an antiferromagnetic nanowire.  _Physical Review Research_ 2, 013166 (2020).
  * [45] Franken, J., Swagten, H. & Koopmans, B.  Shift registers based on magnetic domain wall ratchets with perpendicular anisotropy.  _Nature Nanotechnology_ 7, 499 (2012).
  * [46] Luo, Z. _et al._ Current-driven magnetic domain-wall logic.  _Nature_ 579, 214 (2020).
  * [47] Mah, W. L. W., Kumar, D., Jin, T. & Piramanayagam, S. N.  Domain wall dynamics in (Co/Ni)${}_{\text{n}}$ nanowire with anisotropy energy gradient for neuromorphic computing applications.  _Journal of Magnetism and Magnetic Materials_ 537, 168131 (2021).
  * [48] Wang, Q., Chumak, A. V. & Pirro, P.  Inverse-design magnonic devices.  _Nature Communications_ 12, 2636 (2021).
  * [49] Dorn, C. & Kochmann, D. M.  Inverse design of graded phononic materials via ray tracing.  _Journal of Applied Physics_ 134, 195103 (2023).
  * [50] Kateel, V. _et al._ Field-free spin–orbit torque driven switching of perpendicular magnetic tunnel junction through bending current.  _Nano Letters_ 23, 5482 (2023).
  * [51] Jin, Y., Djafari-Rouhani, B. & Torrent, D.  Gradient index phononic crystals and metamaterials.  _Nanophotonics_ 8, 685 (2019).
  * [52] Wu, C. _et al._ Freeform direct-write and rewritable photonic integrated circuits in phase-change thin films.  _Science Advances_ 10, eadk1361 (2024).
  * [53] Zhou, B. _et al._ Functionalized PDMS with versatile and scalable surface roughness gradients for cell culture.  _ACS Applied Materials & Interfaces_ 7, 17181 (2015).
  * [54] Dow, H. S., Kim, W. S. & Lee, J. W.  Thermal and electrical properties of silicon nitride substrates.  _AIP Advances_ 7, 095022 (2017).
  * [55] Li, X. _et al._ Thermal conduction across a boron nitride and SiO2 interface.  _Journal of Physics D: Applied Physics_ 50, 104002 (2017).

##  9 Extended Data

Extended Data Fig.1 Comparison of CoFeB magnetic properties for laser
annealing and furnace annealing. A laser annealed CoFeB film is compared to an
identical furnace annealed sample (annealed in a tube furnace at atmosphere
for 1 hour at 300 ∘C with a temperature ramp rate of 20 ∘C/minute). a, Polar
MOKE measurements of the two samples reveals similarly square hysteresis
loops, with only a small difference in the coercive field. b, Out-of-plane
SQUID - VSM measurements for both a laser annealed and a furnace annealed
sample. The saturation magnetization is the same for both samples, indicating
that the two techniques give an equivalent transformation of magnetic
properties in CoFeB. The standard error of the measurements shown are smaller
than the lines used.

Extended Data Fig.2 Magnetic characterization of CoGd films. a, Room
temperature pMOKE hysteresis loops for $100\times 100$ $\mu$m2 regions of a
CoGd film exposed using various laser fluences. b, Room temperature, in-plane
SQUID - VSM data for continuous CoGd films that were uniformly exposed using
the indicated laser fluences. c, The saturation magnetization as a function of
laser fluence extracted from b. The standard error of the measurements shown
are smaller than the symbols or lines used.

Extended Data Fig.3 Magnetic characterization of Co/Cr/Co films. a, Room
temperature, pMOKE hysteresis loops for $100\times 100$ $\mu$m2 regions of a
Co/Cr/Co film exposed using the indicated laser fluences. b, Room temperature,
in-plane SQUID - VSM measurements of continuous Co/Cr/Co films that were
uniformly exposed using the indicated laser fluences. The standard error of
the measurements shown are smaller than the lines used.

Extended Data Fig.4 Kerr microscope images of domain wall motion around a
loop. The original unprocessed Kerr microscope images seen in Fig. 4c-f (scale
bar = 10 $\mu$m). For images in the main text, a Gaussian blur was applied and
contrast was enhanced.

Extended Data Fig.5 Magnetometry of CoFeB/Pt-SAF films. Room temperature
SQUID-VSM measurements of two identical SAF films, one of which has been laser
annealed at a fluence of 2.9 J/cm2. a, Out-of-plane magnetometry reveals that
the saturation magnetization of the laser annealed sample is $\sim 10$ % lower
than that of the unannealed film. Inset: the remanent magnetization of the
laser annealed film is also slightly lower than that of the unannealed film.
If only the anisotropy of the top layer was reduced by DWLA, an increase of
the remanent magnetization would be expected, as a canted moment in the top
magnetic layer would not compensate as much of the moment from the bottom
magnetic layer. The nearly identical remanent magnetization indicates that the
thermal energy is being evenly deposited in both the top and bottom magnetic
layers. b, In-plane magnetometry of the laser annealed and unannealed SAF
films. The effective anisotropy field and saturation magnetization is slightly
lower for the annealed sample. The standard error of the measurements shown
are smaller than the lines used.

Extended Data Fig.6 Determination of DWLA feature size using magnetic force
microscopy. Line profile (solid blue line) and Gaussian fit (dashed red line)
of magnetic force microscopy (MFM) contrast for a Co/Cr/Co SAF film in which
DWLA (with a laser fluence of 6.5 J/cm2) was used to create a
ferromagnetically coupled region (dark region in the inset image), which
exhibits a full-width half-minimum of approximately 180 nm. The profile was
extracted along the blue line shown in the inset MFM image (scale bar = 500
nm). The nominal width of the exposed area was 50 nm in the design.