#  The Mass Gap of the Space-time and its Shape

Ahmed Farag Ali△∇<EMAIL_ADDRESS>△Essex County College, 303 University Ave,
Newark, NJ 07102, United States. ∇Department of Physics, Faculty of Science,
Benha University, Benha, 13518, Egypt.

###### Abstract

Snyder’s quantum space-time which is Lorentz invariant is investigated. It is
found that the quanta of space-time have a positive mass that is interpreted
as a positive real mass gap of space-time. This mass gap is related to the
minimal length of measurement which is provided by Snyder’s algebra. Several
reasons to consider the space-time quanta as a 24-cell are discussed.
Geometric reasons include its self-duality property and its 24 vertices that
may represent the standard model of elementary particles. The 24-cell symmetry
group is the Weyl/Coxeter group of the $F_{4}$ group which was found recently
to generate the gauge group of the standard model. It is found that 24-cell
may provide a geometric interpretation of the mass generation, Avogadro
number, color confinement, and the flatness of the observable universe. The
phenomenology and consistency with measurements is discussed.

“The knowledge at which geometry aims is knowledge of the eternal”— Plato.

###### Contents

  1. I Introduction
  2. II space-time quanta and Bekensetin universal bound
  3. III Shape of space-time quanta
  4. IV Symmetry of space-time quanta
  5. V space-time quanta and Spectral mass gap
  6. VI Phenomenological implications
  7. VII conclusion

##  I Introduction

In 1947, Snyder established a remarkable step that reconciles the minimal
length of measurement with Lorentz symmetry by constructing quantum Lorentzian
space-time Snyder (1947). The price was introducing non-commutative geometry
and the generalized uncertainty principle (GUP) in Snyder’s algebra. For Non-
commutative geometry part, it is found to emerge naturally at limits of
M/string theory Connes  _et al._ (1998) as higher dimensional corrections of
ordinary Yang-Mills theory Seiberg and Witten (1999). Several implications of
non-commutative geometry were investigated in quantum field theory and
condensed matter systems Douglas and Nekrasov (2001); Szabo (2003). For the
GUP part, it emerged in several approaches to quantum gravity such as string
theory, loop quantum gravity, and quantum geometry Amati  _et al._ (1989);
Garay (1995); Scardigli (1999); Konishi  _et al._ (1990); Kempf  _et al._
(1995); Maggiore (1993); Capozziello  _et al._ (2000). Phenomenological and
experimental implications of the GUP have been investigated in low and high-
energy systems Das and Vagenas (2008); Pikovski  _et al._ (2012); Marin  _et
al._ (2013); Petruzziello  _et al._ (2021); Kumar  _et al._ (2020); Moradpour
_et al._ (2019); Gao and Zhan (2016); Bawaj  _et al._ (2015); Girdhar  _et
al._ (2020); Ashoorioon  _et al._ (2005); Easther  _et al._ (2003); Dabrowski
and Wagner (2019); Easther  _et al._ (2001). Useful reviews on quantum space-
time and GUP can be found in Addazi  _et al._ (2021); Hossenfelder (2013);
Mignemi (2019). Snyder’s algebra is generated by three main generators which
are position ${x_{\mu}}$, momentum ${p_{\mu}}$ and Lorentz generators
${J_{\mu\nu}=x_{\mu}p_{\nu}-x_{\nu}p_{\mu}}$. They satisfy the Poincaré
commutation relations and suggest new commutation relations that provide a
quantum/minimal length as follows:

$[x_{\mu},x_{\nu}]=i\hbar~{}(\frac{\kappa~{}\ell_{Pl}}{\hbar
c})^{2}~{}J_{\mu\nu},$ (1)
$\qquad[x_{\mu},p_{\nu}]=i\hbar~{}(\eta_{\mu\nu}+(\frac{\kappa~{}\ell_{Pl}}{\hbar
c})^{2}~{}p_{\mu}p_{\nu}),\text{with}~{}~{}~{}~{}~{}~{}~{}~{}\mu,\nu=0,1,2,3$
(2)

where ${\ell_{Pl}}$ is a Planck length, $\kappa$ is a dimensionless parameter
that identifies the minimal measurable length, and
$\eta_{\mu\nu}=\left(-1,1,1,1\right)$. Eq. (1) introduces the non-commutative
geometry and Eq. (2) introduces a GUP. Both equations are invariant under
Lorentz symmetry Snyder (1947).

##  II space-time quanta and Bekensetin universal bound

In this section, we investigate the physical properties of space-time quanta
implied by Snyder’s algebra. It is clear that Eq. (1) only vanishes if there
is no fundamental minimal/quantum length (i.e $\kappa\ell_{Pl}=0$). This means
non-commutative geometry would vanish if there is no minimal/quantum length.
On the contrary, we find that the GUP commutation relation in Eq. (2)
vanishes. The time-energy commutation relation of Eq. (2) vanishes when:

$\displaystyle E$ $\displaystyle=$ $\displaystyle
E_{\kappa}=\pm~{}\frac{\hbar~{}c}{\kappa~{}\ell_{\text{Pl}}},$ (3)

where $E=p_{0}$ and $E_{\kappa}$ represents the maximum bound on energy. The
position-momentum commutation relation Eq. (2) vanishes when:

$\displaystyle p$ $\displaystyle=$ $\displaystyle
p_{\kappa}=\pm~{}i~{}\sqrt{3}~{}\frac{\hbar}{\kappa~{}\ell_{\text{Pl}}}$ (4)

where $p=\sqrt{p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}$, and $p_{\kappa}$ represents
momentum bound that are implied by minimal length
($\kappa~{}\ell_{\text{Pl}}$). This implies “real” solutions for energy and
“imaginary” solution for momentum. To put it another way, time and energy can
be known with certainty, implying a complete/perfect description of the space-
time quanta in terms of time and energy that is invariant under Lorentz
transformation which is satisfied by Snyder algebra. The quanta of space-time
in that sense behave like perfect classical physics at the minimum measurable
length scale in terms of energy and time. This may be consistent with the
asymptotic freedom property of Yang–Mills theory in which quantum fields
behave at a short distance in a very similar way to classical fields Gross and
Wilczek (1973); Politzer (1973). Vanishing time-energy uncertainty may shed
light on the nature of the wavefunction collapse to form the quanta of space-
time. Particle mass has been interpreted as a result of wavefunction collapse
Ellis  _et al._ (1989). We found recently that vanishing uncertainty could
resolve the EPR paradox Einstein  _et al._ (1935) and explain the radii values
of Hydrogen atom/nuclei Ali (2022a, b).
On another side, Bekenstein found a universal bound Bekenstein (1981, 2005,
2000) that defines the maximal amount of information that is necessary to
_perfectly_ and completely describes a physical object up to the quantum
level. Bekenstein universal bound is given by:

$\displaystyle H\leq\frac{2\pi~{}R~{}E}{\hbar~{}c}$ (5)

where $H$ is related to thermodynamic entropy as $S=k_{B}H$, where $k_{B}$ is
the Boltzmann constant. $H$ gives the number of bits contained in the quantum
states in the sphere with radius $R$ that encloses the physical system, and
$E$ is the energy of the physical system. According to Snyder’s algebra in Eq.
(1) and Eq. (2), the sphere that encloses the space-time quanta is identified
by a radius $R=\kappa~{}\ell_{\text{Pl}}$. Since the space-time quanta implies
bound on energy given by Eq. (3), the inequality in Eq. (5) turns to be
equality for space-time quanta that relates the maximum amount of information
$H_{\kappa}$ to the maximum measurable energy $E_{\kappa}$:

$\displaystyle
E_{\kappa}=\frac{\hbar~{}c}{\kappa~{}\ell_{\text{Pl}}}~{}\frac{H_{\kappa}}{2\pi}$
(6)

When we compare Eq. (3) with Eq. (6), we get:

$\displaystyle H_{\kappa}=2~{}\pi$ (7)

that completely describes the quanta of space-time. Notice here that
$H_{\kappa}$ depends only on $\pi$ and is independent of $\kappa$ and nature
constants.

##  III Shape of space-time quanta

A natural question arises, what is the geometric shape of the space-time
quanta? To answer this question, we need first to apply the energy-momentum
relation that is satisfied by Snyder’s algebra Lu and Stern (2012):

$E^{2}-p^{2}c^{2}=m^{2}c^{4}$ (8)

where $m$ is the mass of the physical object. Substituting Eq. (3) and Eq. (4)
into Eq. (8), we get:

$m_{\kappa}=\frac{2~{}}{~{}{\kappa}~{}\ell_{\text{Pl}}}\frac{\hbar}{c}$ (9)

This means that the mass of the space-time quanta is completely determined by
its unique length and it is strictly a “positive real value”. It is worth
mentioning that generalized forms of Snyder algebra with de Sitter background
imply modification of dispersion relation Banerjee  _et al._ (2011) that may
introduce corrections to the mass of space-time quanta obtained in Eq.(9). The
flatness of quantum Lorentzian space-time suggests that the space-time quanta
be described by 4-polytope geometry Coxeter (1973). We use elementary particle
physics as a guide to building the quanta of space-time. The standard model of
particle physics has 25 fundamental particles that include 12 fermions (quarks
and leptons), 4 gauge bosons that carry electromagnetic force and weak nuclear
force, 8 gluons that carry strong nuclear force, and 1 Higgs scalar field. The
quanta of space-time must carry a signature of information from all these
fundamental particles that constitute the fundamental structure of nature. The
space-time quanta must be “self-dual” as well to preserve its uniqueness.
Therefore, we look for a highly symmetric 4-dimensional geometric object that
is self-dual and identified by only unique length which could represent the
fundamental particles of nature on its vertices. This could be a uniform
4-polytope which is a 4-dimensional object with flat sides/faces and is
vertex-transitive symmetric which means an isometric map of any vertex onto
any other. A regular 4-polytope has the highest degree of symmetry as its
faces/cells are regular polytopes and transitive on the symmetries of the
polytope. More symmetries are found in regular 4-polytopes that define the
convex region as a subset that intersects every line into a single line
segment. There are two self-dual convex regular polytopes that are 5-cell,
which has 5 vertices, and 24-cell which has 24 vertices. We choose the 24-cell
because it has enough vertices to assign with elementary particles. A
representation in a plane of a regular 24-cell is given in Fig. (1):

Figure 1:

24-cell diagram: The vertices of 24-cell are divided into three subsets of
three colors, each is a regular 16-cell. Image copied from this  link with CC-
BY-SA-4.0 license

The 24-cell geometric properties can be summarized as follows:

  * •

Its boundary in 3-dimensions forms 24 octahedral cells with six meeting at
each vertex, and three at each edge

  * •

It has 96 triangular faces, 96 edges, and 24 vertices. The vertex figure is a
cube

  * •

self-dual Wenninger (2003)

  * •

identified by one length where edge length equals the distance between the
center and vertex (radius)

The 24-cell exists in 4- dimensional Euclidean geometry, but Minkowski space-
time is 4- dimensional “pseudo-Euclidean”. This can be simply resolved if time
is represented as an imaginary spatial dimension which is well-established in
quantum field theory Wick (1954) and in Euclidean quantum gravity Gibbons and
Hawking (1993). To put it another way, the space-time quanta is represented by
a 24-cell with considering time as an imaginary spatial dimension. The
covariance principle requires that the space-time quanta should represent the
elementary particles of the standard model Weinberg (1967). In the next
section, we show how to do this representation.

##  IV Symmetry of space-time quanta

The symmetry group of 24-cell is the Weyl/Coxeter group of $F_{4}$ group
Coxeter (1973). The 48 root vectors of $F_{4}$ represent the vertices of the
24-cell in two dual configurations. $F_{4}$ group is one of five exceptional
simple Lie groups that are non-abelian and do not have nontrivial connected
normal subgroups. The remarkable importance of the $F_{4}$ group has been
realized recently in Todorov and Dubois-Violette (2018) to explain the gauge
symmetry of the standard model of particle physics. It is proved that $F_{4}$
has two stabilizer groups that are $H_{1}=(SU(3)\times SU(3))/\mathbb{Z}_{3}$
and $H_{2}=Spin(9)$ that their intersection $H_{1}\cap H_{2}$ generates the
standard model gauge symmetry $(SU(3)×\times SU(2)\times
U(1))/\mathbb{Z}_{6}$. Further details and implications can be found in
Bernardoni  _et al._ (2008); Baez (2018); Boyle (2020); Dubois-Violette
(2016); Boyle and Farnsworth (2020); Todorov (2019); Bhatt  _et al._ (2022);
Vaibhav and Singh (2021); Todorov and Drenska (2018); Furey and Hughes (2022);
Krasnov (2021); Dubois-Violette and Todorov (2019); Furey (2018). This
supports our postulate of identifying the space-time quanta as the 24-cell. In
addition, a geometric connection between 24-cell and Calabi-Yau Threefolds
with Hodge Numbers (1,1) is realized in Braun (2012) that was useful in
determining the mass spectrum of type-IIB flux vacua Blanco-Pillado  _et al._
(2021). It is worth mentioning that 24-cell was crystallized to represent the
spinfoam topology by trivalent spin network Aschheim (2012).
Let us now have a close look at Fig. (1) which shows a representation in a
plane of 24-cell, where the vertices are divided into three subsets of three
colors, each being a regular 16-cell. 24-cell vertices can be grouped into
three different sets of eight vertices each one defining 16-cells with the
rest defining the dual tesseract that has sixteen vertices Coxeter (1973).
These three different sets could be used to represent the three colors in
quantum chromodynamics. Each set of eight vertices is represented by a 16-cell
which is a regular convex 4-polytope that has eight vertices that can
represent the eight gluons in the 24-cell. The other sixteen vertices of
tesseract could be used to represent the twelve fermions and four gauge
bosons. The tesseract is the four-dimensional analog of the cube. In that
sense, the 24-cell can be formed by the 16-cell which could be assigned with
eight gluons, and the tesseract which could be assigned with twelve fermions
and four gauge bosons. Since the Higgs field couples only with the twelve
fermions and four gauge bosons, therefore the tesseract as a whole could
represent the shape of the Higgs particle. In addition, this property of
24-cell may explain the geometric meaning of preserving the gluons symmetry
which may explain the color confinement. Besides, it explains why gluons could
have three colors by grouping 24-cell vertices into three different sets of
16-cell as shown in Fig. (1). It also explains the flatness of the observable
universe which is composed of 16 observable particles by the tesseract that
has sixteen vertices and which is flat. We can represent the particles by
orthogonal projections of 24-cell, 16-cell, and tesseract in the Coxeter plane
in two dimensions as in Fig. (2).

(a) 24-cell Coxeter plane that represents elementary particles of the standard
model

(b) Tesseract Coxeter plane- that act like Higgs particle with 16 vertices
assigned to 12 fermions and 4 gauge bosons

(c) 16-cell Coxeter plane that acts like eight gluons that may explain strong
force confinement

Figure 2: representation of fundamental particles on vertices of 24-cell,
16-cell and Tesseract

##  V space-time quanta and Spectral mass gap

The solution of the mass gap problem as described in Jaffe and Witten (2006)
requires proving that Yang-Mills theory exists and that the mass of all
particles predicted by the theory is strictly positive. Both conditions are
satisfied by the space-time quanta. According to Eq. (9), we find that the
mass of space-time quanta is a real positive value. The space-time quanta is
quatified by the parameter $\kappa$ in Snyder algebra that was found to
generate both non-commutative geometry and GUP at the same time. For Non-
commutative geometry part, it is found to emerge naturally at limits of
M/string theory Connes  _et al._ (1998) as higher dimensional corrections of
ordinary Yang-Mills theory Seiberg and Witten (1999). Since the space-time is
locally flat, the space-time quanta must be described by a 4-polytope. We
introduce a geometric and symmetric reason to consider the 24-cell as the
space-time quanta. The most important reason is that 24-cell is the
Weyl/Coxeter group of $F_{4}$ group that can generate the standard model gauge
symmetry as shown in recent studies Todorov and Dubois-Violette (2018).
Therefore, Yang-Mill’s theory exists as a $F_{4}$ group and is explained by
the space-time quanta from the first geometric principles as 24-cell. The
gluon masses should be related to the 16-cell which is related to the 24-cell
we explained in previous sections. We conclude that the space-time quanta
introduces a geometric origin of the spectral mass gap Jaffe and Witten
(2006). The spectral mass gap is entirely determined by the length/radius of
24-cell according to Eq. (9). Recently, it was shown that spectral gaps exist
in Hamiltonian with quasicrystalline order Hege  _et al._ (2022). Quasicrystal
considerations in Holography, the basic structure of nature, and cosmology are
discussed in Boyle  _et al._ (2020); Aschheim  _et al._ (2018); Irwin (2020a);
Fang  _et al._ (2018); Amaral  _et al._ (2022); Fang  _et al._ (2019); Irwin
_et al._ (2016); Irwin (2020b). We think the quantum space-time may be a
quasicrystal with a fundamental structure of a 24-cell. Experimental
observations of quantum time quasicrystal are reported in Autti  _et al._
(2018). This quasicrystal order is expected to follow from simulating Snyder’s
algebra with considering 24-cell as its fundamental structure. This needs
further investigation.

##  VI Phenomenological implications

Phenomenological studies expect to prop information about dark matter at ILC
Fujii  _et al._ (2017). We intuit that the fundamental structure of dark
matter/energy is the space-time quanta. Previous studies suggested that a
minimal length of measurements forms a dark matter candidate Adler  _et al._
(2001). There is an interesting observation about the 24-cell. All
permutations of vertices in 24-cell are given by factorial $24!=6.2044\times
10^{23}$ which is quite close to the Avogadro number ($6.0221\times 10^{23}$)
which determines the “approximate” number of nucleons per gram of matter based
on thermodynamical and statistical computations. The relative error between
the two values is around $2.9\%$. This may shed light on the geometric meaning
of the mole unit. The number of observable stars in the universe is between
$10^{22}$ to $10^{23}$! Rudnick  _et al._ (2003); Zyla  _et al._ (2020) which
is again close to the number predicted by 24-cell. According to the
interpretation of creating a mass of space-time quanta by vanishing
uncertainty, the minimal length is expected to correspond to the electroweak
length scale at which masses are created. Therefore, the minimal length is
determined from the measured Higgs mass ($\approx 125.35~{}GeV/c^{2}$) Zyla
_et al._ (2020) and substituting it in Eq. (9) that implies a minimal length
as follows:

$\kappa~{}\ell_{\text{Pl}}\approx 10^{-18}~{}~{}\text{m}$ (10)

This would give the length scale at which uncertainty vanishes to form a mass
of space-time quanta. In that sense, we reinterpret the Higgs mechanism as a
state in which the uncertainty vanishes to form the mass of space-time quanta.
The electroweak length scale determines the unique length of the 24-cell. The
effective minimal length has been interpreted recently as a charge radius of
scattering for every physical particle Ali  _et al._ (2022). Recent studies
show that the minimum measurable charge radius measured so far is around
$10^{-18}$ for the neutrinos, Higgs, W-bosons, and Z-boson, Hutauruk  _et al._
(2022); Lehnert (2014); Zyla  _et al._ (2020) which is consistent with our
interpretations.

One notice that the permutations of the 16-cell that forms the gluons part of
the 24-cell are equal to factorial $8!$ and the permutation of tesseract that
represents Higgs, photon, W, Z, quarks, and leptons is given by factorial
$16!$. Based on our model, the 16-cell represents the Planck scale and the
tesseract represents the electroweak scale. The multiplications of their
permutations ($8!16!=8.4360689\times 10^{17}$) may explain the scale ratio
between the Planck scale and the electroweak scale. The measured scale ratio
between the Planck scale and the electroweak scale is approximately $10^{17}$.
This may imply a quantum computational explanation of the hierarchy in nature.

##  VII conclusion

We found that the GUP implied by Snyder’s algebra vanishes at a specific
energy scale. We define this energy scale as the scale of space-time quanta at
which wavefunction collapses to form a mass. The mass of space-time quanta
forms a mass gap of space-time. The covariance principle requires the space-
time quanta to be a 4-dimensional object and to represent the elementary
particles. Based on the geometric and symmetric analysis, we propose that the
space-time quanta be represented by the 24-cell. First, it is highly-symmetric
convex regular 4-polytope and self-dual. Second, The symmetry group of 24-cell
is the Weyl/Coxeter group of $F_{4}$ group that generates the gauge group of
the standard model by the intersection of its two stabilizer groups. In
addition, the 24-cell has a beautiful geometric property in which its vertices
can be grouped into 3 different sets of eight vertices, each defining 16-cell
with the rest defining the dual tesseract with 16 vertices. Therefore, we
represent 8 vertices of 16-cell with the 8 gluons that may give a geometric
interpretation of the color confinement. We represent 16 vertices of tesseract
with the 12 fermions and 4 gauge bosons which may explain the flatness of the
observable universe. The Higgs particle is represented as the tesseract which
explains why Higgs only couple with 12 fermions and 4 gauge bosons and does
not couple with 8 gluons. Vanishing uncertainty implies the creation of a mass
of space-time quanta, so the length of 24-cell is identified by the
electroweak length scale $10^{-18}$ m of mass creation and which is consistent
with experimental measurements of the smallest measured value of charge radius
of scattering for neutrinos, Higgs, Z-boson, and W boson. We think that a
24-cell symmetry group which is a solvable group of order “1152” could be
useful in quantum computing. We hope to report on this application in the
future.

## Acknowledgments

I thank the Editor and referee for their meticulous review of this manuscript
and express gratitude towards Klee Irwin, Raymond Aschheim, Fang Fang, Richard
Clawson, and Dugan Hammock for their enlightening discussions on Polytopes and
Quasicrystals. Special thanks to Ahmed Almheiri for his engaging lecture on
the Path integral for chords.

This manuscript is a tribute to my father, Farag Mohamed Ali, whose legacy
illuminates my journey, and in memory of Mustafa Shalaby, whose teachings have
profoundly inspired my exploration of physics’ core principles.

## References

  * Snyder (1947) H. S. Snyder, Phys. Rev. 71, 38 (1947).
  * Connes  _et al._ (1998) A. Connes, M. R. Douglas, and A. S. Schwarz, JHEP 02, 003 (1998), arXiv:hep-th/9711162 .
  * Seiberg and Witten (1999) N. Seiberg and E. Witten, JHEP 09, 032 (1999), arXiv:hep-th/9908142 .
  * Douglas and Nekrasov (2001) M. R. Douglas and N. A. Nekrasov, Rev. Mod. Phys. 73, 977 (2001), arXiv:hep-th/0106048 .
  * Szabo (2003) R. J. Szabo, Phys. Rept. 378, 207 (2003), arXiv:hep-th/0109162 .
  * Amati  _et al._ (1989) D. Amati, M. Ciafaloni, and G. Veneziano, Phys. Lett. B 216, 41 (1989).
  * Garay (1995) L. J. Garay, Int. J. Mod. Phys. A 10, 145 (1995), arXiv:gr-qc/9403008 .
  * Scardigli (1999) F. Scardigli, Phys. Lett. B 452, 39 (1999), arXiv:hep-th/9904025 .
  * Konishi  _et al._ (1990) K. Konishi, G. Paffuti, and P. Provero, Phys. Lett. B 234, 276 (1990).
  * Kempf  _et al._ (1995) A. Kempf _et al._ , Phys. Rev. D 52, 1108 (1995), arXiv:hep-th/9412167 .
  * Maggiore (1993) M. Maggiore, Phys. Lett. B 304, 65 (1993), arXiv:hep-th/9301067 .
  * Capozziello  _et al._ (2000) S. Capozziello _et al._ , Int. J. Theor. Phys. 39, 15 (2000), arXiv:gr-qc/9910017 .
  * Das and Vagenas (2008) S. Das and E. C. Vagenas, Phys. Rev. Lett. 101, 221301 (2008), arXiv:0810.5333 [hep-th] .
  * Pikovski  _et al._ (2012) I. Pikovski _et al._ , Nature Phys. 8, 393 (2012), arXiv:1111.1979 [quant-ph] .
  * Marin  _et al._ (2013) F. Marin _et al._ , Nature Phys. 9, 71 (2013).
  * Petruzziello  _et al._ (2021) L. Petruzziello _et al._ , Nature Commun. 12, 4449 (2021), arXiv:2011.01255 [gr-qc] .
  * Kumar  _et al._ (2020) S. P. Kumar _et al._ , Nature Commun. 11, 3900 (2020), arXiv:1908.11164 [quant-ph] .
  * Moradpour  _et al._ (2019) H. Moradpour _et al._ , Mon. Not. Roy. Astron. Soc. 488, L69 (2019), arXiv:1907.12940 [gr-qc] .
  * Gao and Zhan (2016) D. Gao and M. Zhan, Physical Review A 94, 013607 (2016).
  * Bawaj  _et al._ (2015) M. Bawaj _et al._ , Nature Commun. 6, 7503 (2015), arXiv:1411.6410 [gr-qc] .
  * Girdhar  _et al._ (2020) P. Girdhar _et al._ , New J. Phys. 22, 093073 (2020), arXiv:2005.08984 [quant-ph] .
  * Ashoorioon  _et al._ (2005) A. Ashoorioon _et al._ , Phys. Rev. D 71, 023503 (2005), arXiv:astro-ph/0410139 .
  * Easther  _et al._ (2003) R. Easther _et al._ , Phys. Rev. D 67, 063508 (2003), arXiv:hep-th/0110226 .
  * Dabrowski and Wagner (2019) M. P. Dabrowski and F. Wagner, The European Physical Journal C 79, 1 (2019).
  * Easther  _et al._ (2001) R. Easther _et al._ , Phys. Rev. D 64, 103502 (2001), arXiv:hep-th/0104102 .
  * Addazi  _et al._ (2021) A. Addazi _et al._ , (2021), arXiv:2111.05659 [hep-ph] .
  * Hossenfelder (2013) S. Hossenfelder, Living Rev. Rel. 16, 2 (2013), arXiv:1203.6191 [gr-qc] .
  * Mignemi (2019) S. Mignemi, Ukr. J. Phys. 64, 991 (2019), arXiv:1911.06127 [hep-th] .
  * Gross and Wilczek (1973) D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973).
  * Politzer (1973) H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973).
  * Ellis  _et al._ (1989) J. R. Ellis, S. Mohanty, and D. V. Nanopoulos, Phys. Lett. B 221, 113 (1989).
  * Einstein  _et al._ (1935) A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).
  * Ali (2022a) A. F. Ali, (2022a), arXiv:2210.13974 [quant-ph] .
  * Ali (2022b) A. F. Ali,  (2022b), 10.2139/ssrn.4292472.
  * Bekenstein (1981) J. D. Bekenstein, Phys. Rev. D 23, 287 (1981).
  * Bekenstein (2005) J. D. Bekenstein, Found. Phys. 35, 1805 (2005), arXiv:quant-ph/0404042 .
  * Bekenstein (2000) J. D. Bekenstein, Phys. Lett. B 481, 339 (2000), arXiv:hep-th/0003058 .
  * Lu and Stern (2012) L. Lu and A. Stern, Nucl. Phys. B 860, 186 (2012), arXiv:1110.4112 [hep-th] .
  * Banerjee  _et al._ (2011) R. Banerjee, K. Kumar, and D. Roychowdhury, JHEP 03, 060 (2011), arXiv:1101.2021 [hep-th] .
  * Coxeter (1973) H. S. M. Coxeter,  _Regular polytopes_ (Courier Corporation, 1973).
  * Wenninger (2003) M. J. Wenninger,  _Dual models_ (Cambridge University Press, 2003).
  * Wick (1954) G. C. Wick, Phys. Rev. 96, 1124 (1954).
  * Gibbons and Hawking (1993) G. W. Gibbons and S. W. Hawking,  _Euclidean quantum gravity_ (World Scientific, 1993).
  * Weinberg (1967) S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967).
  * Todorov and Dubois-Violette (2018) I. Todorov and M. Dubois-Violette, Int. J. Mod. Phys. A 33, 1850118 (2018), arXiv:1806.09450 [hep-th] .
  * Bernardoni  _et al._ (2008) F. Bernardoni, S. L. Cacciatori, B. L. Cerchiai, and A. Scotti, Adv. Theor. Math. Phys. 12, 889 (2008), arXiv:0705.3978 [math-ph] .
  * Baez (2018) J. C. Baez, The n-Category Cafe 27 (2018).
  * Boyle (2020) L. Boyle, (2020), arXiv:2006.16265 [hep-th] .
  * Dubois-Violette (2016) M. Dubois-Violette, Nucl. Phys. B 912, 426 (2016), arXiv:1604.01247 [math.QA] .
  * Boyle and Farnsworth (2020) L. Boyle and S. Farnsworth, New J. Phys. 22, 073023 (2020), arXiv:1910.11888 [hep-th] .
  * Todorov (2019) I. Todorov, Springer Proc. Math. Stat. 335, 29 (2019), arXiv:1911.13124 [hep-th] .
  * Bhatt  _et al._ (2022) V. Bhatt, R. Mondal, V. Vaibhav, and T. P. Singh, J. Phys. G 49, 045007 (2022), arXiv:2108.05787 [hep-ph] .
  * Vaibhav and Singh (2021) V. Vaibhav and T. P. Singh, (2021), arXiv:2108.01858 [hep-ph] .
  * Todorov and Drenska (2018) I. Todorov and S. Drenska, Adv. Appl. Clifford Algebras 28, 82 (2018), arXiv:1805.06739 [hep-th] .
  * Furey and Hughes (2022) N. Furey and M. J. Hughes, Phys. Lett. B 831, 137186 (2022), arXiv:2210.10126 [hep-ph] .
  * Krasnov (2021) K. Krasnov, J. Math. Phys. 62, 021703 (2021), arXiv:1912.11282 [hep-th] .
  * Dubois-Violette and Todorov (2019) M. Dubois-Violette and I. Todorov, Nucl. Phys. B 938, 751 (2019), arXiv:1808.08110 [hep-th] .
  * Furey (2018) N. Furey, Phys. Lett. B 785, 84 (2018), arXiv:1910.08395 [hep-th] .
  * Braun (2012) V. Braun, JHEP 05, 101 (2012), arXiv:1102.4880 [hep-th] .
  * Blanco-Pillado  _et al._ (2021) J. J. Blanco-Pillado, K. Sousa, M. A. Urkiola, and J. M. Wachter, JHEP 04, 149 (2021), arXiv:2007.10381 [hep-th] .
  * Aschheim (2012) R. Aschheim, in  _Loops 11: International Conference on Quantum Gravity_ (2012) arXiv:1212.5473 [cs.IT] .
  * Jaffe and Witten (2006) A. Jaffe and E. Witten, The millennium prize problems 1, 129 (2006).
  * Hege  _et al._ (2022) P. Hege, M. Moscolari, and S. Teufel, Phys. Rev. B 106, 155140 (2022), arXiv:2205.10622 [quant-ph] .
  * Boyle  _et al._ (2020) L. Boyle, M. Dickens, and F. Flicker, Phys. Rev. X 10, 011009 (2020), arXiv:1805.02665 [hep-th] .
  * Aschheim  _et al._ (2018) R. Aschheim, L. Bubuianu, F. Fang, K. Irwin, V. Ruchin, and S. I. Vacaru, Annals of Physics 394, 120 (2018).
  * Irwin (2020a) K. Irwin, Activitas Nervosa Superior 62, 48 (2020a).
  * Fang  _et al._ (2018) F. Fang, S. Paduroiu, D. Hammock, and K. Irwin, Crystals 8, 416 (2018).
  * Amaral  _et al._ (2022) M. Amaral, D. Chester, F. Fang, and K. Irwin, Symmetry 14, 1780 (2022).
  * Fang  _et al._ (2019) F. Fang, S. Paduroiu, D. Hammock, and K. Irwin, in  _Electron Crystallography_ (IntechOpen, 2019) p. 27.
  * Irwin  _et al._ (2016) K. Irwin, M. Amaral, R. Aschleim, and F. Fang, in  _Proceedings of the Fourth International Conference on the Nature and Ontology of Spacetime, Varna, Bulgaria_ , Vol. 30 (2016) pp. 117–160.
  * Irwin (2020b) K. Irwin, “Simple programs - qc cycle clocks,”  (2020b).
  * Autti  _et al._ (2018) S. Autti, V. B. Eltsov, and G. E. Volovik, Phys. Rev. Lett. 120, 215301 (2018), arXiv:1712.06877 [cond-mat.other] .
  * Fujii  _et al._ (2017) K. Fujii _et al._ , (2017), arXiv:1710.07621 [hep-ex] .
  * Adler  _et al._ (2001) R. J. Adler, P. Chen, and D. I. Santiago, Gen. Rel. Grav. 33, 2101 (2001), arXiv:gr-qc/0106080 .
  * Rudnick  _et al._ (2003) G. Rudnick, H.-W. Rix, M. Franx, I. Labbé, M. Blanton, E. Daddi, N. M. F. Schreiber, A. Moorwood, H. Röttgering, I. Trujillo, _et al._ , The Astrophysical Journal 599, 847 (2003).
  * Zyla  _et al._ (2020) P. A. Zyla _et al._ (Particle Data Group), PTEP 2020, 083C01 (2020).
  * Ali  _et al._ (2022) A. F. Ali, I. Elmashad, and J. Mureika, Phys. Lett. B 831, 137182 (2022), arXiv:2205.14009 [physics.gen-ph] .
  * Hutauruk  _et al._ (2022) P. T. P. Hutauruk, A. Sulaksono, and K. Tsushima, Nucl. Phys. A 1017, 122356 (2022), arXiv:2009.08781 [hep-ph] .
  * Lehnert (2014) B. Lehnert, Progress in Physics 10, 5 (2014).


# Accelerating the laser-induced phase transition in nanostructured FeRh via
plasmonic absorption

M. Mattern  Institut für Physik und Astronomie, Universität Potsdam, 14476
Potsdam, Germany  J.-E. Pudell  Institut für Physik und Astronomie,
Universität Potsdam, 14476 Potsdam, Germany  Helmholtz-Zentrum Berlin für
Materialien und Energie GmbH, Wilhelm-Conrad-Röntgen Campus, BESSY II, 12489
Berlin, Germany  European XFEL, 22869 Schenefeld, Germany  J. A. Arregi
<EMAIL_ADDRESS>CEITEC BUT, Brno University of Technology, 61200
Brno, Czech Republic  J. Zlámal  CEITEC BUT, Brno University of Technology,
61200 Brno, Czech Republic  Institute of Physical Engineering, Brno University
of Technology, 61669 Brno, Czech Republic  R. Kalousek  Institute of Physical
Engineering, Brno University of Technology, 61669 Brno, Czech Republic  V.
Uhlíř  CEITEC BUT, Brno University of Technology, 61200 Brno, Czech Republic
Institute of Physical Engineering, Brno University of Technology, 61669 Brno,
Czech Republic  M. Rössle  Helmholtz-Zentrum Berlin für Materialien und
Energie GmbH, Wilhelm-Conrad-Röntgen Campus, BESSY II, 12489 Berlin, Germany
M. Bargheer<EMAIL_ADDRESS>Institut für Physik und Astronomie,
Universität Potsdam, 14476 Potsdam, Germany  Helmholtz-Zentrum Berlin für
Materialien und Energie GmbH, Wilhelm-Conrad-Röntgen Campus, BESSY II, 12489
Berlin, Germany

###### Abstract

By ultrafast x-ray diffraction we show that the laser-induced
magnetostructural phase transition in FeRh nanoislands proceeds faster and
more complete than in continuous films. We observe an intrinsic $8\,\text{ps}$
timescale for nucleation of ferromagnetic (FM) domains in both types of
samples. For the continuous film, the substrate-near regions, which are not
directly exposed to light, are only slowly transformed to the FM state by
domain wall motion following heat transport. In contrast, numerical modeling
of the plasmonic absorption in the investigated nanostructure reveals a strong
contribution near the FeRh/MgO interface. On average, the absorption is larger
and more homogeneous in the nanoislands, enabling the phase transition
throughout the entire volume at the intrinsic nucleation timescale.

Reducing the structure-size of metallic ferromagnets to the nanoscale not only
helps increasing the information storage density but also enables direct
plasmonic coupling of light to the magnetic nano-bit for magnetoplasmonic
control and readout[1, 2]. This is a particularly exciting perspective in the
context of femtosecond optomagnetism[3] with ultrafast optical manipulation of
magnetic properties such as a polarization control of two magnetic nanolayers
mediated by plasmon-polaritons [4] and plasmonic enhanced all-optical
switching in magnetic nanodisks[5]. Heat assisted magnetic recording (HAMR)[6,
7] already uses optical near fields to confine the magnetic switching in the
new generation of magnetic hard drives to a few nanometers. Resonant magnetic
x-ray scattering studies confirmed plasmonically enhanced ultrafast switching
for nano-granular FePt thin films, which constitute the classical HAMR
material[8].

The potential consequences of nanostructuring FeRh go well beyond plasmonic
coupling. Lateral nanostructuring limits the number of independent nucleation
sites through the antiferromagnetic-to-ferromagnetic (AF-FM) phase transition
around $370\,\text{K}$, which changes the nature of magnetization reversal
from multi-domain to single-domain and results in discrete avalanche-like
jumps of the order parameter upon cooling [9, 10]. In thermal equilibrium, the
phase transition that is accompanied by a $1\%$ volume expansion, crucially
depends on the lattice structure. The tetragonal distortion of the unit cell
originating from an in-plane substrate-induced compression enhances the
transition temperature [10, 11, 12]. In FeRh nanoislands, the partial
relaxation of this tetragonal distortion reduces the transition temperature
[10, 13]. Generally, in-plane nano-structuring unlocks in-plane expansion on
the picosecond timescale in contrast to the exclusive out-of-plane expansion
of laser-excited continuous thin films [14]. The three-dimensional nature of
the picosecond strain response of nanoislands preserves bulk-like material-
specific expansion properties and results in a complex strain response due to
cross-talk of in- and out-of-plane expansion via the Poisson effect [15, 16,
17, 18].

Previous experiments studied the laser-induced phase transition in FeRh by the
emergence of magnetization[19, 20, 21, 22], changes in the electronic
structure[23] and the rise of the larger FM lattice constant[24, 25, 26].
Probing the structural order parameter by ultrafast x-ray diffraction (UXRD),
we recently disentangled FM domain nucleation and growth in inhomogeneously
excited FeRh continuous films, whose thickness exceed the optical penetration
depth[26]. We identified a universal $8\,\text{ps}$ nucleation timescale in
FeRh, which does not depend on the film thickness and temperature nor on
applied laser fluence and magnetic field [26]. The effects of nanostructuring
on the coupled ultrafast dynamics of demagnetization[17], remagnetization [27]
and strain[15, 16, 17] have been thoroughly studied for FePt. Ultrafast
experiments on FeRh nanoislands that study the influence of the in-plane
expansion, reduced number of nucleation sites and plasmonic excitation are
lacking up to now.

In this letter, we explore for the first time the kinetics of the laser-driven
phase transition in FeRh nanoislands by probing the rise of a larger lattice
constant parameterizing the FM phase as structural order parameter via UXRD.
In order to access the effect of finite lateral dimensions, we compare the
results to a similarly thick continuous FeRh film as reference. In the
nanoislands, the AF-FM phase transition drives a partial in-plane expansion of
the nanoislands both in equilibrium and on ultrafast timescales. Upon laser
excitation, we observe the same $8\,\text{ps}$ nucleation timescale in both
samples indicating an intrinsic property irrespective of the sample
morphology. However, while we observe a relatively slow heat transport-driven
domain growth in the thin film, the phase transition of the nanostructured
film occurs precisely on the intrinsic timescale of domain nucleation. By
modeling the absorption of the nanostructures, we relate this acceleration of
the phase transition to a homogeneous optical excitation due to plasmonic
effects enabled by the size of the metal islands below the excitation
wavelength.

Figures 1(a) and (b) sketch the continuous and nanostructured FeRh film grown
on MgO(001) substrates. The continuous $55\,\text{nm}$ thick FeRh(001) film
was grown by magnetron sputtering from an equiatomic FeRh target [11] and
capped by $2\,\text{nm}$ of Pt. The nanostructured sample is composed of
epitaxial FeRh(001) nanoislands formed by solid state dewetting of an
epitaxial FeRh(001) film via self-assembly resulting in maze-like
structures[13] with a mean height of $52\,\text{nm}$.

Figure 1:  Morphology-dependent phase transition in thermal equilibrium:
Sketch of the UXRD experiment mapping the reciprocal space via
$\theta-2\theta$-scans and the sample structure of the continuous film (a) and
the nanoislands (b). Panels (c–e) and (f–h) characterize the equilibrium AF-FM
phase transition in the continuous film and the nanostructures, respectively.
(c, f) The diffracted intensity (grey symbols) is the superposition of an AF
and an arising FM Bragg peak at a larger out-of-plane lattice constant during
heating above $T_{\text{T}}$. (d, g) Temperature-dependent ferromagnetic
volume fraction $V_{\text{FM}}$ determined by the relative integrated
intensity of the FM Bragg peak (symbols) as structural order parameter and the
magnetization normalized to its maximum value as magnetic order parameter
(solid lines). (e, h) Temperature-dependent lattice constant (symbols) modeled
by Eq. 1 using bulk expansion parameters (solid lines).

Static and time-resolved reciprocal space maps around the (002) FeRh Bragg
peak are recorded at the KMC-3 XPP endstation at BESSY II in the low-alpha
operation mode[28] with monochromized $9\,\text{keV}$ hard x-ray photons. The
diffracted intensity in Figs. 1(c) and (f) displays the emergence of an
additional Bragg peak at lower values of the out-of-plane reciprocal lattice
coordinate $q_{z}$ when the temperature approaches the mean transition
temperature of the thin film ($370\,\text{K}$) and the nanoislands
($350\,\text{K}$), respectively. The integrated intensity of this Bragg peak
is directly proportional to the volume of FeRh exhibiting the FM phase[24] and
thus parameterizes the FM phase during the temperature-driven AF-FM phase
transition. The proportion of the FM Bragg peak in the total intensity yields
the temperature-dependent FM volume fraction $V_{\text{FM}}$. Figures 1(d) and
(g) compare this structural parameterization of the phase transition (symbols)
to the macroscopic magnetization normalized to its maximum (solid lines)
serving as complementary order parameter of the FM phase. The magnetization is
measured via vibrating sample magnetometry (VSM) using a QuantumDesign
VersaLab magnetometer, which results in a broadening of the hysteresis by the
heterogeneous transition temperature at different sample sites in contrast to
the narrow hysteresis of the structural order parameter locally ($300\times
300\,\text{\textmu m}^{2}$) measured via x-ray diffraction.

The comparison of the two samples reveals a dependence of the AF-FM phase
transition in thermal equilibrium on the sample morphology. The enhanced
surface-to-volume ratio of the nanoislands results in a noticeable residual FM
phase that persists below the transition temperature $T_{\text{T}}$ at the
symmetry breaking surface [29] and at the film-substrate interface [30]. In
addition, the small lateral extent of the islands partially relaxes the
substrate-induced tetragonal distortion of FeRh, which lowers the transition
temperature for the nanoislands [10, 11, 13]. This is indicated by the lower
mean out-of-plane lattice constant $d$ with respect to the continuous film
(see Figs. 1(e) and (h)) given by the center-of-mass (COM) of the diffracted
intensity via $d=4\pi/q_{z,\text{COM}}$. This applies in particular to the
out-of-plane expansion associated with the phase transition. While we find
$0.4\%$ expansion for the nanoislands close to the bulk value of
$0.3\,\%$[31], the substrate-induced clamping of the in-plane expansion
suppresses the Poisson effect[14] and results in an out-of-plane expansion of
$0.6\%$ for the thin film. Accounting for the different substrate-induced
constraints of the in-plane expansion, the out-of-plane expansion of the FeRh
samples is described by[14]:

$\displaystyle\alpha_{\text{eff}}$
$\displaystyle=\alpha_{\text{bulk}}(T)+2\chi\frac{c_{1133}}{c_{3333}}\left(\alpha_{\text{bulk}}(T)-\alpha_{\text{MgO}}\right)\;,$
(1)

where $\alpha_{\text{bulk}}(T)$ and $\alpha_{\text{MgO}}=10.5\cdot
10^{-6}\text{K}^{-1}$ denote the thermal expansion coefficients of bulk FeRh
and MgO, respectively. The expansion of FeRh $\alpha_{\text{bulk}}(T)$ is
given by the expansion coefficients $\alpha^{\text{FM}}=6.0\cdot
10^{-6}\text{K}^{-1}$ and $\alpha^{\text{AF}}=9.7\cdot 10^{-6}\text{K}^{-1}$
in the AF and FM phase[32] and the expansion of $0.3\,\%$ across the AF-FM
phase transition[31], considering the temperature-dependent volume fraction in
the FM phase $V_{\text{FM}}(T)$, which we derived from the integrated
intensity of the FM Bragg peak in Figs. 1(c) and (f). The elastic constants of
FeRh $c_{1133}$ and $c_{3333}$ quantify the effect of the in-plane expansion
on the out-of-plane expansion via the Poisson effect [14] and
$\alpha_{\text{eff}}$ denotes the modified expansion coefficient of the
samples depending on the parameter $\chi$. This parameter measures the epitaxy
to the substrate, where $\chi=0$ corresponds to pure bulk-like in-plane
expansion and $\chi=1$ to an in-plane expansion completely determined by the
MgO substrate.

Figure 2:  Reduced laser-induced out-of-plane expansion in FeRh
nanostructures: (a) Transient Bragg peaks of the thin film for an excitation
of $12.0\,\text{mJ/cm}^{2}$ at $340\,\text{K}$ dissected into the FM (green)
and AF (blue) Bragg peak that are well-separated in reciprocal space. (b) In
the nanoislands, the FM (pink) and AF (purple) Bragg peak are less separated
due to the partial in-plane expansion of the unit cell across the laser-
induced phase transition for $5.2\,\text{mJ/cm}^{2}$ at $230\,\text{K}$. The
data for different pump-probe delays are vertically off-set to improve
visibility.

Our modeling of the temperature-dependent lattice constant in Figs. 1(e) and
(h) (symbols) by Eq. 1 (solid line) yields excellent agreement for $\chi=1$
and $\chi=0.42$ for the continuous thin film and the nanoislands,
respectively. While the thin film purely follows the in-plane expansion of the
MgO substrate ($\chi=1$), the nanoislands behave partially bulk-like
($\chi=0.42$). This relaxation of the in-plane constraints is expected to
increase towards the surface and to depend on the in-plane dimensions of the
different nanoislands [33].

In the UXRD experiment, the FeRh samples are excited by a $600\,\text{fs}$
p-polarized pump pulse with a central wavelength of $1028\,\text{nm}$ that is
incident at $\approx 30^{\circ}$ with respect to the sample surface. As
sketched in Fig. 1(a), we probe the emergence of the FM Bragg peak that
parameterizes the laser-induced AF-FM phase transition by $17\,\text{ps}$
$9\,\text{keV}$ hard x-ray pulses [28] performing symmetric $\theta$-$2\theta$
scans around the (002) Bragg reflection at $28^{\circ}$.

Figure 3:  Nucleation-dominated phase transition in FeRh nanoislands: (a)
Transient FM volume fraction of the nanoislands at $T=190\,\text{K}$ for
various fluences $F$. (b) Same for $T=230\,\text{K}$, which increases the
conversion to the FM state at low fluence. (c) Temperature series at a
relatively low fluence of $F=8.6\,\text{mJ/cm}^{2}$ for the thin film. (d)
Same for $F=12.0\,\text{mJ/cm}^{2}$. The two-step rise of $V_{\text{FM}}$ in
the thin film (c and d) indicates a growth of the nucleated FM domains into
the depth of the layer driven by near-equilibrium heat transport. In all
panels, solid lines denote the kinetics of FM domain nucleation according to
Eq. 2 convoluted with the time-resolution given by the duration of the x-ray
pulse.

Figures 2(a) and (b) display the diffracted intensity along $q_{z}$ before and
after an excitation with $12.0\,\text{mJ/cm}^{2}$ at $340\,\text{K}$ for the
thin film and with $5.2\,\text{mJ/cm}^{2}$ at $230\,\text{K}$ for the
nanoislands, respectively. The emerging FM Bragg peaks indicate the optically
induced AF-FM phase transition for both samples. The AF and FM Bragg peaks are
well separated for the thin film. For the nanoislands, an ultrafast in-plane
expansion on short timescales is enabled by their small lateral extent[14].
The concomitant transient out-of-plane Poisson-contraction results in less
separated Bragg peaks (Fig. 2(b)) for the nanoislands. This indicates a
reduced tetragonal distortion of the unit cell upon laser-excitation and the
emergence of more cubic FM unit cells upon nucleation in the nanoislands as
already discussed for thermal equilibrium. In addition to the lattice constant
change across the phase transition, the dynamics of the laser-induced phase
transition also depends on the sample morphology. While the integrated
intensity of the FM Bragg peak barely changes between $40$ and
$240\,\text{ps}$ for the nanoislands, the FM Bragg peak increases after
$40\,\text{ps}$ for the thin film.

Figure 3 displays the resulting transient FM volume fraction $V_{\text{FM}}$
for both samples under various excitation conditions. The solid lines denote
the expected dynamics for nucleation of FM domains at independent sites
described by Eq. 2 with the previously identified universal nucleation
timescale $\tau=8\,\text{ps}$[26] convoluted with the $17\,\text{ps}$-long
x-ray pulse limiting the time-resolution. The final FM volume fraction
$V_{\text{FM}}^{*}$ is adjusted to the experimental value of
$V_{\text{FM}}(t=40\,\text{ps})$ for the respective measurement and we include
a finite residual FM phase in the nanoislands being present before excitation:

$\displaystyle V_{\text{FM}}(t)$
$\displaystyle=V_{\text{FM}}^{*}\cdot\left(1-e^{-t/\tau}\right)\;.$ (2)

For the nanoislands, the transient FM volume fraction in Figs. 3(a–b) is well
described by Eq. 2 indicating that nucleation dominates the laser-induced AF-
FM phase transition. With increasing fluence and initial sample temperature a
larger fraction of the nanoislands is excited above the critical threshold
characteristic for first-order phase transitions[20, 24], which results in an
enhanced $V_{\text{FM}}^{*}$. Figs. 3(c–d) show that within the first
$40\,\text{ps}$ the laser-induced phase transition in the continuous film is
equally well described by Eq. 2 (solid lines) with the same $8\,\text{ps}$
nucleation timescale. Thus the nucleation kinetics of the AF-FM phase
transition in FeRh is insensitive to the substrate-induced or transient
tetragonal distortion of the unit cell that is smaller in case of the
nanoislands as identified by the less separated AF and FM Bragg peaks shown in
Fig. 2.

Figure 4:  Optical excitation of nanostructured FeRh: Re-build of the
topography of the FeRh nanostructures characterized by AFM (a) in COMSOL (b).
(c) Local absorbed power per area of the FeRh nanostructures calculated in
COMSOL by solving the Maxwell equations. (d) Local optical penetration depth
determined from $P_{\text{abs}}$ as function of the depth relative to the
local height $h$. (e) Absorption of different nanoislands as function of $z$
at $y=3.2\,\text{\textmu m}$. (f) Integrated absorption of the nanoislands as
function of $z$ (blue symbols). The purple solid line displays the
$z$-dependent absorption of an hypothetical ensemble of continuous FeRh films
resembling the height distribution of the nanoislands and the grey line
denotes the absorption profile of the continuous $55\,\text{nm}$ thick FeRh
film.

However, we observe a second slow contribution to the rise of $V_{\text{FM}}$
in the continuous film at initial sample temperatures above $300\,\text{K}$.
In a previous publication[26], we revealed this two-step rise of the FM phase
to originate from a nucleation of FM domains within the optically excited near
surface region and a subsequent growth of the domains into the depth of the
inhomogeneously excited layer driven by near-equilibrium heat transport. The
equilibration of the phonon temperature within the continuous film via heat
diffusion slowly heats the backside of the film and induces the phase
transition if the temperature surpasses $T_{\text{T}}$. For initial sample
temperatures only slightly below $T_{\text{T}}$ and higher excitation fluences
a larger fraction of the film is heated above $T_{\text{T}}$ and undergoes the
phase transition[26]. This results in an enhanced delayed contribution arising
from domain growth as displayed in Figs. 3(c–d).

To explain the absence of such a heat transport-driven domain growth in the
nanoislands we calculate their optical absorption for the experimental
excitation conditions in COMSOL utilizing the refractive index of MgO
$n_{\text{MgO}}=1.72$ [34] and of FeRh $n_{\text{FeRh}}=4.67+5.58i$. This was
measured via spectroscopic ellipsometry for a similar FeRh film, grown under
the same conditions. For this we reproduce the topography of the nanoislands
characterized by AFM (Fig. 4(a)) in COMSOL by ellipsoids utilizing an
algorithm for rapid contour detection [35] (see Fig. 4(b)). Figure 4(c)
displays the local power absorption $P_{\text{abs}}$ of the nanostructures
that reveals the existence of plasmonic hot-spots with drastically enhanced
absorption. By fitting an exponential decay function to the local
$z$-dependent absorption (FeRh-MgO interface corresponds to $z=0$), we find a
large spread of the optical penetration depth $\delta_{\text{p}}$. Figure 4(d)
shows this distribution relative to the semi-infinite medium value
$\delta_{\text{p},0}=14.7\,\text{nm}$. Yellow color-code indicates a locally
strongly enhanced optical penetration depth due to nanostructuring. The
exemplary lineout of the dissipated power at $y=3.2\,\text{\textmu m}$ in Fig.
4(e) depicts the $z$-dependent absorption as a function of the in-plane $x$
coordinate, with characteristic plasmonic enhancement near the FeRh-MgO
interface[36] that is responsible for the local absorption hot-spots shown in
Fig. 4(c). This increases the absorption at depth that in thin films receives
only a negligible exponentially damped amount of light energy.

Both the penetration depth enhancement and the optical power dissipation
build-up near the FeRh-MgO interface make the absorption of the pump-pulse in
the nanostructures more homogeneous with respect to the out-of-plane
$z$-coordinate than in a continuous thin film. The average total absorption in
the nanostructures as function of the distance from the FeRh-MgO interface
($z=0$) is displayed by symbols in Fig. 4(f). In addition, the grey solid line
denotes the $z$-dependent absorption of a $55\,\text{nm}$ FeRh film scaled by
the surface coverage of the nanoislands ($49\,\%$) in the studied sample. Its
thickness is comparable to the average nanoisland height of $52\,\text{nm}$.
Lastly, the purple solid line represents the integrated $z$-dependent optical
absorption from each pixel assuming its local profile is identical to that of
a continuous film of an equivalent thickness. The decrease of the absorption
for large $z$ values (purple line) agrees with the COMSOL simulation
additionally including plasmonic effects (symbols) and exclusively originates
from the decreasing number of nanoislands higher than the $z$ value. However,
the average absorption of nanostructures (symbols) for $z<30\,\text{nm}$ is
much larger than that predicted by the pixel-integrated absorption of
equivalent thin films, which neglects the plasmonic enhancement near the FeRh-
MgO interface. Comparing the area under the grey and dotted curves in Fig.
4(f), we find that the total $z$-integrated optical absorption of the
nanostructures amounts to $34\,\%$ of the incident power, which exceeds the
absorption of the $55\,\text{nm}$-thick continuous FeRh film with the same
value by a factor of $1.5$. In essence, the optical excitation of the
nanostructures is significantly more homogeneous along their vertical extent
than for the thin film case, where only the near surface region receives a
sizeable light intensity.

This stronger and almost homogeneous excitation of the complete volume of the
nanostructures supports a nucleation-driven phase transition throughout the
entire nanoislands. This suppresses the slow phase transition by domain wall
motion observed for the thin film (Figs. 3(c–d)), which is driven by near-
equilibrium heat-transport. This drastically accelerates the laser-induced
phase transition in FeRh nanostructures with small lateral extension that is
even more efficiently driven due to the overall enhanced absorption.

In summary, we studied the morphology dependence of the laser-induced AF-FM
phase transition by comparing a continuous and a nanostructured thin FeRh
film. We find an ultrafast in-plane expansion of the nanoislands, whereas the
thin FeRh film is pinned to the MgO. This results in a less tetragonal
distortion of the unit cell across the phase transition, however, it has no
influence on the nucleation timescale of the FM domains. Instead, only
plasmonic effects change the dynamics of the phase transition: By modelling
the spatially resolved optical absorption of the FeRh nanostructures, we
identified a plasmon-enhanced absorption near the FeRh-MgO interface and an
enhanced optical penetration depth. This results in a homogeneous excitation
of the nanoislands, which drives a nucleation of FM domains on an
$8\,\text{ps}$ timescale within the volume of the FeRh nanostructures and
makes slow heat-transport driven domain growth irrelevant. This accelerates
the phase transition in comparison with the thin film that exhibits nucleation
only within the optically excited near-surface region and shows a subsequent
slow growth of the FM phase into the depth of film at initial sample
temperatures slightly below the transition temperature.

We acknowledge the DFG for financial support via Project-No. 328545488 – TRR
227 project A10 and the BMBF for funding via 05K22IP1. Access to the CEITEC
Nano Research Infrastructure was supported by the Ministry of Education, Youth
and Sports (MEYS) of the Czech Republic under the project CzechNanoLab
(LM2023051). Measurements were carried out at the KMC3-XPP instrument at the
BESSY II electron storage ring operated by the Helmholtz-Zentrum Berlin für
Materialien und Energie.

## References

  * Temnov  _et al._ [2010] V. V. Temnov, G. Armelles, U. Woggon, D. Guzatov, A. Cebollada, A. Garcia-Martin, J.-M. Garcia-Martin, T. Thomay, A. Leitenstorfer, and R. Bratschitsch, “Active magneto-plasmonics in hybrid metal–ferromagnet structures,” Nature Photonics 4, 107–111 (2010).
  * Armelles  _et al._ [2013] G. Armelles, A. Cebollada, A. García-Martín, and M. U. González, “Magnetoplasmonics: Combining magnetic and plasmonic functionalities,” Advanced Optical Materials 1, 10–35 (2013).
  * Bossini  _et al._ [2016] D. Bossini, V. I. Belotelov, A. K. Zvezdin, A. N. Kalish, and A. V. Kimel, “Magnetoplasmonics and femtosecond optomagnetism at the nanoscale,” ACS Photonics 3, 1385–1400 (2016).
  * Ignatyeva  _et al._ [2019] D. Ignatyeva, C. Davies, D. Sylgacheva, A. Tsukamoto, H. Yoshikawa, P. Kapralov, A. Kirilyuk, V. Belotelov, and A. Kimel, “Plasmonic layer-selective all-optical switching of magnetization with nanometer resolution,” Nature communications 10, 4786 (2019).
  * Vergès  _et al._ [2023] M. Vergès, S. Perumbilavil, J. Hohlfeld, F. Freire-Fernández, Y. Le Guen, N. Kuznetsov, F. Montaigne, G. Malinowski, D. Lacour, M. Hehn, _et al._ , “Energy efficient single pulse switching of [co/gd/pt] n nanodisks using surface lattice resonances,” Advanced Science 10, 2204683 (2023).
  * Rottmayer  _et al._ [2006] R. Rottmayer, S. Batra, D. Buechel, W. Challener, J. Hohlfeld, Y. Kubota, L. Li, B. Lu, C. Mihalcea, K. Mountfield, K. Pelhos, C. Peng, T. Rausch, M. Seigler, D. Weller, and X.-M. Yang, “Heat-assisted magnetic recording,” IEEE Transactions on Magnetics 42, 2417–2421 (2006).
  * Weller  _et al._ [2016] D. Weller, G. Parker, O. Mosendz, A. Lyberatos, D. Mitin, N. Y. Safonova, and M. Albrecht, “Fept heat assisted magnetic recording media,” Journal of Vacuum Science & Technology B 34 (2016), 10.1116/1.4965980.
  * Granitzka  _et al._ [2017] P. W. Granitzka, E. Jal, L. Le Guyader, M. Savoini, D. J. Higley, T. Liu, Z. Chen, T. Chase, H. Ohldag, G. L. Dakovski, W. F. Schlotter, S. Carron, M. C. Hoffman, A. X. Gray, P. Shafer, E. Arenholz, O. Hellwig, V. Mehta, Y. K. Takahashi, J. Wang, E. E. Fullerton, J. Stöhr, A. H. Reid, and H. A. Dürr, “Magnetic switching in granular fept layers promoted by near-field laser enhancement,” Nano Letters 17, 2426–2432 (2017).
  * Uhlíř, Arregi, and Fullerton [2016] V. Uhlíř, J. A. Arregi, and E. E. Fullerton, “Colossal magnetic phase transition asymmetry in mesoscale ferh stripes,” Nature communications 7, 13113 (2016).
  * Arregi  _et al._ [2018] J. A. Arregi, M. Horký, K. Fabianová, R. Tolley, E. E. Fullerton, and V. Uhlíř, “Magnetization reversal and confinement effects across the metamagnetic phase transition in mesoscale ferh structures,” Journal of Physics D: Applied Physics 51, 105001 (2018).
  * Arregi, Caha, and Uhlíř [2020] J. A. Arregi, O. Caha, and V. Uhlíř, “Evolution of strain across the magnetostructural phase transition in epitaxial ferh films on different substrates,” Physical Review B 101, 174413 (2020).
  * Cherifi  _et al._ [2014] R. Cherifi, V. Ivanovskaya, L. Phillips, A. Zobelli, I. Infante, E. Jacquet, V. Garcia, S. Fusil, P. Briddon, N. Guiblin, _et al._ , “Electric-field control of magnetic order above room temperature,” Nature materials 13, 345–351 (2014).
  * Motyčková  _et al._ [2023] L. Motyčková, J. A. Arregi, M. Staňo, S. Průša, K. Částková, and V. Uhlíř, “Preserving metamagnetism in self-assembled ferh nanomagnets,” ACS Applied Materials & Interfaces 15, 8653–8665 (2023).
  * Mattern  _et al._ [2023a] M. Mattern, A. von Reppert, S. P. Zeuschner, M. Herzog, J.-E. Pudell, and M. Bargheer, “Concepts and use cases for picosecond ultrasonics with x-rays,” Photoacoustics , 100503 (2023a).
  * von Reppert  _et al._ [2018] A. von Reppert, L. Willig, J.-E. Pudell, M. Rössle, W. Leitenberger, M. Herzog, F. Ganss, O. Hellwig, and M. Bargheer, “Ultrafast laser generated strain in granular and continuous fept thin films,” Applied Physics Letters 113, 123101 (2018).
  * von Reppert  _et al._ [2020] A. von Reppert, L. Willig, J.-E. Pudell, S. Zeuschner, G. Sellge, F. Ganss, O. Hellwig, J. Arregi, V. Uhlíř, A. Crut, _et al._ , “Spin stress contribution to the lattice dynamics of fept,” Science advances 6, eaba1142 (2020).
  * Reid  _et al._ [2018] A. Reid, X. Shen, P. Maldonado, T. Chase, E. Jal, P. Granitzka, K. Carva, R. Li, J. Li, L. Wu, _et al._ , “Beyond a phenomenological description of magnetostriction,” Nature communications 9, 388 (2018).
  * Chang  _et al._ [2015] W.-S. Chang, F. Wen, D. Chakraborty, M.-N. Su, Y. Zhang, B. Shuang, P. Nordlander, J. E. Sader, N. J. Halas, and S. Link, “Tuning the acoustic frequency of a gold nanodisk through its adhesion layer,” Nature Communications 6, 7022 (2015).
  * Ju  _et al._ [2004] G. Ju, J. Hohlfeld, B. Bergman, R. J. van de Veerdonk, O. N. Mryasov, J.-Y. Kim, X. Wu, D. Weller, and B. Koopmans, “Ultrafast generation of ferromagnetic order via a laser-induced phase transformation in ferh thin films,” Physical review letters 93, 197403 (2004).
  * Bergman  _et al._ [2006] B. Bergman, G. Ju, J. Hohlfeld, R. J. van de Veerdonk, J.-Y. Kim, X. Wu, D. Weller, and B. Koopmans, “Identifying growth mechanisms for laser-induced magnetization in ferh,” Physical Review B 73, 060407 (2006).
  * Radu  _et al._ [2010] I. Radu, C. Stamm, N. Pontius, T. Kachel, P. Ramm, J.-U. Thiele, H. Dürr, and C. Back, “Laser-induced generation and quenching of magnetization on ferh studied with time-resolved x-ray magnetic circular dichroism,” Physical Review B 81, 104415 (2010).
  * Li  _et al._ [2022] G. Li, R. Medapalli, J. Mentink, R. Mikhaylovskiy, T. Blank, S. Patel, A. Zvezdin, T. Rasing, E. Fullerton, and A. Kimel, “Ultrafast kinetics of the antiferromagnetic-ferromagnetic phase transition in ferh,” Nature Communications 13, 2998 (2022).
  * Pressacco  _et al._ [2021] F. Pressacco, D. Sangalli, V. Uhlíř, D. Kutnyakhov, J. A. Arregi, S. Y. Agustsson, G. Brenner, H. Redlin, M. Heber, D. Vasilyev, _et al._ , “Subpicosecond metamagnetic phase transition in ferh driven by non-equilibrium electron dynamics,” Nature Communications 12, 5088 (2021).
  * Mariager  _et al._ [2012] S. O. Mariager, F. Pressacco, G. Ingold, A. Caviezel, E. Möhr-Vorobeva, P. Beaud, S. Johnson, C. Milne, E. Mancini, S. Moyerman, _et al._ , “Structural and magnetic dynamics of a laser induced phase transition in ferh,” Physical Review Letters 108, 087201 (2012).
  * Quirin  _et al._ [2012] F. Quirin, M. Vattilana, U. Shymanovich, A.-E. El-Kamhawy, A. Tarasevitch, J. Hohlfeld, D. von der Linde, and K. Sokolowski-Tinten, “Structural dynamics in ferh during a laser-induced metamagnetic phase transition,” Physical Review B 85, 020103 (2012).
  * Mattern  _et al._ [2023b] M. Mattern, J. Jarecki, J. A. Arregi, V. Uhlíř, M. Rössle, and M. Bargheer, “Disentangling nucleation and domain growth during a laser-induced phase transition,”  (2023b), 10.48550/arXiv.2305.02094.
  * Willig  _et al._ [2019] L. Willig, A. von Reppert, M. Deb, F. Ganss, O. Hellwig, and M. Bargheer, “Finite-size effects in ultrafast remagnetization dynamics of fept,” Physical Review B 100, 224408 (2019).
  * Rössle  _et al._ [2021] M. Rössle, W. Leitenberger, M. Reinhardt, A. Koç, J. Pudell, C. Kwamen, and M. Bargheer, “The time-resolved hard x-ray diffraction endstation kmc-3 xpp at bessy ii,” Journal of Synchrotron Radiation 28, 948–960 (2021).
  * Pressacco  _et al._ [2016] F. Pressacco, V. Uhlíř, M. Gatti, A. Bendounan, E. E. Fullerton, and F. Sirotti, “Stable room-temperature ferromagnetic phase at the ferh (100) surface,” Scientific reports 6, 22383 (2016).
  * Fan  _et al._ [2010] R. Fan, C. J. Kinane, T. Charlton, R. Dorner, M. Ali, M. De Vries, R. M. Brydson, C. H. Marrows, B. J. Hickey, D. A. Arena, _et al._ , “Ferromagnetism at the interfaces of antiferromagnetic ferh epilayers,” Physical Review B 82, 184418 (2010).
  * Zsoldos [1967] L. Zsoldos, “Lattice parameter change of ferh alloys due to antiferromagnetic-ferromagnetic transformation,” physica status solidi (b) 20, K25–K28 (1967).
  * Ibarra and Algarabel [1994] M. Ibarra and P. Algarabel, “Giant volume magnetostriction in the ferh alloy,” Physical Review B 50, 4196 (1994).
  * Laskin  _et al._ [2019] G. Laskin, H. Wang, H. Boschker, W. Braun, V. Srot, P. A. van Aken, and J. Mannhart, “Magnetic properties of epitaxially grown srruo3 nanodots,” Nano letters 19, 1131–1135 (2019).
  * Stephens and Malitson [1952] R. E. Stephens and I. H. Malitson, “Index of refraction of magnesium oxide,” Journal of Research of the National Bureau of Standards 49, 249–252 (1952).
  * Rasche [2018] C. Rasche, “Rapid contour detection for image classification,” IET Image Processing 12, 532–538 (2018).
  * Garcia-Vidal  _et al._ [2022] F. J. Garcia-Vidal, A. I. Fernández-Domínguez, L. Martin-Moreno, H. C. Zhang, W. Tang, R. Peng, and T. J. Cui, “Spoof surface plasmon photonics,” Reviews of Modern Physics 94, 025004 (2022).