PHYSICAL REVIEW LETTERS 127, 077202 (2021)

Ultrafast Amplification and Nonlinear Magnetoelastic Coupling of Coherent Magnon
Modes in an Antiferromagnet

1

D. Bossini ,1,* M. Pancaldi,2 L. Soumah,2 M. Basini,2 F. Mertens ,3 M. Cinchetti ,3T. Satoh ,4 O. Gomonay ,5 and S. Bonetti 2,6

Department of Physics and Center for Applied Photonics, University of Konstanz, D-78457 Konstanz, Germany
2
Department of Physics, Stockholm University, 106 91 Stockholm, Sweden
3
Experimentelle Physik VI, Technische Universität Dortmund, Otto-Hahn Straße 4, 44227 Dortmund, Germany
4
Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan
5
Institut für Physik, Johannes Gutenberg Universität Mainz, D-55099 Mainz, Germany
6
Department of Molecular Sciences and Nanosystems, Ca’ Foscari University of Venice, 30172 Venezia-Mestre, Italy(Received 5 November 2020; revised 22 May 2021; accepted 24 June 2021; published 9 August 2021)
We investigate the role of domain walls in the ultrafast magnon dynamics of an antiferromagnetic NiO
single crystal in a pump-probe experiment with variable pump photon energy. Analyzing the amplitude ofthe energy-dependent photoinduced ultrafast spin dynamics, we detect a yet unreported coupling between
the material’s characteristic terahertz- and gigahertz-magnon modes. We explain this unexpected coupling
between two orthogonal eigenstates of the corresponding Hamiltonian by modeling the magnetoelastic
interaction between spins in different domains. We find that such interaction, in the nonlinear regime,
couples the two different magnon modes via the domain walls and it can be optically exploited via theexciton-magnon resonance.
DOI: 10.1103/PhysRevLett.127.077202

Antiferromagnets (AFMs) have recently surged as candidates for a novel paradigm of spintronics devices able tooutperform ferromagnetic and ferrimagnetic materials interms of operational frequency, storage density, and resilience to external fields [1–4]. Intrinsically the long-range
antiferromagnetic order presents domains, which canhardly be manipulated. This magnetic texture and themagnetoelastic coupling—which is intimately interconnected to the domain structure—have been very recently
shown to play a major role in the mechanism allowing
electric manipulations of the Néel vector [5–7]. The quest
for an ever faster and more energy efficient control of
AFMs motivates the use of ultrashort light pulses asstimulus to drive (sub)picosecond spin dynamics [8–17].
However, the role of domain walls in magnetoelastic AFMs
on the ultrafast Néel vector dynamics has been hitherto notaddressed, although being a crucial issue, since the overwhelming majority of AFMs in nature displays a multidomain magnetoelastic ground state.
Here, we demonstrate that the domain walls can activate
a novel functionality in an antiferromagnetic crystal,
namely a nonlinear magnetoelastic domain-walls-mediatedcoupling between coherent spin wave modes belonging todifferent branches of the magnon dispersion, affecting theultrafast dynamics of the Néel vector. We realize experimentally the tailored amplification of coherent terahertz
oscillations of the Néel vector by pumping a magnon modein an antiferromagnetic NiO crystal. This process is
triggered by driving a combined electronic and magnetic
0031-9007=21=127(7)=077202(6)transition and results even in the amplification of a different
gigahertz magnon mode via the aforementioned coupling.
Finally, we formulate a macroscopic phenomenologicalmodel able to explain the observations by taking into
account the role of the domain walls in the ultrafast
dynamics of the Néel vector.
Our specimen is a 100 μm-thick freestanding single
crystal of NiO, cut along the h111i direction and has amultidomain structure. A specimen in a multidomain state
can be described invoking as many antiferromagnetic
vectors (defined as n ≡ M ⇑ − M ⇓ , where M ⇑;⇓ represent
the magnetization of the two sublattices), each belonging toa T domain [Fig. 1(a)].
The domain structure of NiO, comprising spin (S) andtwin (T) domains, is tightly connected with the magnetoelastic coupling, since when the crystal enters the magnetic
phase strained magnetic domains are formed [18,19]. The
magnetoelastic energy is also the major contribution to theanisotropy gap in the magnon dispersion [20].
As a matter of fact, the magnon dispersion of NiO can be
described in the first approximation in terms of two
branches, so that at the center of the Brillouin zone twomodes are active [Figs. 1(b) and 1(c)]. A 1.07 THz mode,
which we are going to refer to as the high-frequency (hf)mode, has already been excited by means of two different
approaches: resonant terahertz excitation [10,11] andnonresonant impulsive stimulated Raman scattering[9,13,15,16,21] (ISRS) mainly by means of optical laser
pulses. The ISRS mechanism succeeded also in inducing a

077202-1© 2021 American Physical Society

PHYSICAL REVIEW LETTERS 127, 077202 (2021)
(a)

(a)

T2
Domain
D
main

1600

Absorption (1/cm)

Domain Wall

T1
Domain

0

(c)

(b)

hf modelf modeExcitonpeak

Magnonsideband

1200

T=5K
T = 40 K
800

T = 60 K
T = 77 K
400n

T = 90 Kn

T = 110 K
0
0.96

0.97

0.975

0.98

Energy (eV)FIG. 1. (a) Two of the four possible T domains and thecorresponding orientation of the Néel vector (green arrows). The
magenta area represents the wall; the order parameter (dark arrows)
rotates in this region.At the center of thewall (ξ ≈ 0) the orientationsof n in the T1 and T2 domain are parallel with each other. (b) In- and(c) out-of-plane dynamics of the order parameter induced by thelow- and high-frequency magnon mode.

(b)
1.4

5.4

E (meV)

1.3
5.0
1.2
4.6

1.1

Frequency (THz)lower frequency magnon mode, which will be labeled as
low frequency (lf), with a frequency on the order of
130 GHz [9,13]. The previous time-resolved investigations
reporting the photoactivation of both modes, mostly performed focusing the pump and probe beams into a single T
domain, do not show any form of coupling or interaction
between the two magnetic eigenmodes.
An unexplored pathway to the femtosecond opticalgeneration of the hf mode relies on the exciton-magnon(X-M) transition [22,23]. This process consists in thesimultaneous excitation of a spin-forbidden (i.e.,
ΔS ¼ 1) electronic transition and of a magnon (i.e.,
ΔS ¼ −1), restoring the overall conservation of spin as
required for electric dipole transitions [22,23]. We thus
measured the absorption spectrum of our sample as afunction of temperature (for details, see [24]). The spectra
obtained for T < 100 K display a peak centered at approximately 0.97 eV and a sideband at higher energy [Fig. 2(a)].
The position of the sideband is temperature dependent, andthe energy shift between the two spectral features(≈4 meV) is consistent with the energy of the hf mode
[Fig. 2(b)] observed by Raman spectroscopy [36]. Our
observations are in excellent agreement with the literature[25,36], so resonantly pumping our sample in the 0.97 eV
spectral range is expected to result in the generation of the
hf mode. We aim at answering two open scientific questions. First, whether the X-M transition can actuallyresonantly induce coherent magnons on the femtosecond
timescale. Second, whether the domain walls play a role in
the ultrafast spin dynamics of a multidomain AFM and, incase they do, what this role is.
We tackle these questions in a magneto-optical pumpprobe experiment, in which the pump photon energy can be

0.965

1.0

4.2
20

40

60

80

100

T (K)

FIG. 2. (a) Absorption coefficient of NiO around the excitonmagnon resonance detected at different values of temperature.
Traces are displaced vertically for the sake of clarity. (b) Bluesymbols: temperature-dependent energy shift between the electronic peak and the sideband [24]. Green symbols: reported
values of the temperature dependence of the 1 THz magnon
mode, detected with Raman spectroscopy [36].

tuned in the 0.92–1.07 eV spectral range [Fig. 3(a)],allowing us to compare the spin dynamics triggered by aresonant pumping of the X-M with the signal detected byexciting NiO nonresonantly (setup described in [24]). The
rotation of the polarization detected in every trace shown in
Fig. 3(b) reveals oscillations at the frequency of approximately 110 GHz. Considering the value of the frequency,
we ascribe this harmonic component of the signal to the lf
mode [9,13]. The slight deviation of the frequency from thereported value is due to magnetostriction induced by bothinternal and external strains of the sample, which cansignificantly affect the magnon frequency in antiferromagnets [37]. Additionally, some of the traces in Fig. 3(b)display also a faster oscillatory component, whose frequency matches the reported 1.07 THz value of the hf
mode. In the inset of Fig. 3(b) the power spectra (i.e.,square modulus of the Fourier transform) of the 0.92 and0.97 eV time traces are shown, displaying the presence andabsence of the 1 THz magnon. The discussion of the077202-2

PHYSICAL REVIEW LETTERS 127, 077202 (2021)
1

NiO

Max
absorption

Normalized amplitude

0.8

111

(b)

E = 1.07 eV
E = 1.01 eV

-0.02

-0.04

-0.06

-0.08

Bandwidth
pulses
PS2

0.4

PS1

Absorption

0.2

E = 0.92 eV

X-M

Amplitude (arb. units)

Rotation (deg)

E = 0.97 eV

0.6

Lowfrequency
mode

0.97 eV

0.0

0.92 eV

0.92 0.94
1.00

1.10

High-frequency
mode

0.96

0.98 1.00 1.02
Energy (eV)

1.04

1.06

1.20

Frequency (THz)

0

5

10

15

20

25

30

Delay (ps)FIG. 3. (a) Schematic representation of the setup. (b) Selectedpump-probe traces for different pump-photon energies. The
probe photon energy is 1.55 eV in each dataset. The temperatureof the sample was set to 77 K. The polarization of the pump beam
is linear and parallel to the ½112̄ direction, while the probe beam
was linearly polarized 45° away from the ½112̄ axis. The pumpfluence was kept constant to ≈10 mJ=cm2 . We have not performed fluence-dependent measurements, as increasing thefluence results in damaging the sample and lower fluence value
implied an unfavorable signal-to-noise ratio. The incoherent
background is ascribed to heating of both the lattice and themagnetic system, consistently with the literature [26,38].

magneto-optical effects involved in our experiments isreported in [24].
Hence, we analyzed the spectral dependence of theamplitude of both magnon modes [24]. The results ofthe data processing are shown in Fig. 4. We first discuss thetrend of the hf mode. The off-resonant photoexcitation (I)does not induce terahertz magnons, implying that an
impulsive stimulated Raman generation of the hf mode
is not observed. The amplitude of the hf mode increases
steeply as the spectral range containing the X-M (II) is
covered by the spectrum of the pump pulses. A comparisonof the spectral dependence of the hf mode with theabsorption spectrum of NiO (both plotted in Fig. 4) reveals
the amplification of the magnon mode to occur in a broader
spectral range than the X-M itself. This behavior is due tothe bandwidth of the pump pulses, which being ultrashort(≈50 fs) are intrinsically broadband (≈40 meV). As thepump photon energy is further increased (III), so that
the X-M is not directly induced anymore, the amplitude of
the hf mode is reduced. Two other transitions (PS1 andPS2) are photoinduced in spectral region III: they are

FIG. 4. Amplification and nonlinear coupling of the two
magnon modes. The amplitudes are normalized on their maximum values. The experimentally accessed spectral range isdivided into four regions, as described in detail in the main text.

phonon sidebands of the excitonic peak [25]. Therefore aresonant pumping of such sidebands results again in
inducing both the excitonic peak and the X-M, as recently
experimentally demonstrated [39]. However, the phonon
sidebands implies a stronger optical absorption, so that theoverall amplitude of the hf mode is reduced in comparison
with spectral range II, since in region III a portion of thepump photons are absorbed by the lattice. Finally, in region
IV the X-M is not induced anymore and, accordingly, theamplitude of the hf mode decreases to an almost vanishingvalue. We conclude that the pumping of the X-M unambiguously amplifies the hf mode. The mechanism is aresonant drive but it is not purely dissipative, implying thatpumping the material with photon energy corresponding to
the maximum absorption (red dot in Fig. 4) does not deliver
the most intense magnonic oscillations. This result isachieved if the X-M is resonantly driven.
We now turn the discussion to the lf mode. Under
nonresonant pumping (I and IV) the amplitude of the lf
mode is not negligible, consistently with the literature [13].
The observation of coherent oscillations in a multidomainstate of the sample is ascribed to the different amplitudes ofthe lf oscillations photoinduced in different T domains. The
ISRS excitation is conventionally described in terms of alight-induced effective field, which torques the spins
triggering spin precession [40,41]. The torque is maximumin T domains in which spins lie along a directional
orthogonal to the effective field, implying that different
T domains differently contribute to the overall detectedmagneto-optical response, which is hence not averaged out.
Unexpectedly, also the lf mode experiences an

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PHYSICAL REVIEW LETTERS 127, 077202 (2021)
amplification in spectral regions II and III, although not as
pronounced as the THz mode. The lf mode is not reported
to be involved in the X-M transition [25]. We stress that theobserved behavior is strongly surprising and atypical.
Experiments have demonstrated that the amplitude ofmagnonic oscillations induced by ISRS in a single-domain
antiferromagnet, are not affected by a modification of thepump photon-energy in a spectral range lying below theband gap [26].
A key characteristic of our NiO samples is that they arein a multidomain state. Since S domains can be neglected[13,24] the T domains have to be taken into account.
Hence, we formulate a macroscopic model. In particular,
we consider two T domains (T1 and T2) and a wall
separating them, and thus the unit vectors representing thetwo modes (ehf and elf ) have different nonorthogonal
orientations in the T1 and T2 domains. In the presenceof such a domain wall the two modes can couple, due to theexchange interaction between the Néel vectors of the twodifferent domains. In addition, since the wall breaks thetranslational symmetry, the eigenstates in one domain are
not eigenstates in the other one. In a single T domain and in
the linear regime the two modes cannot couple, because
they have different frequencies and are two different
eigenstates of the Hamiltonian of the magnetic system.
As a starting point, we assume that the photoexcitationinduces the hf mode in a T1 [ξ < 0, where ξ is introduced inFig. 1(a)] domain: n ¼ aehf ðT1Þ exp ðiωhf tÞ, where a is theamplitude of the photoinduced magnons and depends on
the intensity of the pump beam and on the cross section of
the X-M process. We model then how the hf magnons in T1
can interact with the lf-magnon mode in T2 (ξ ≥ 0), whose
amplitude is represented in the following by b, via thedomain wall. Considering the standard equations for the
dynamics of n [42], we obtainn × ð̈n − c2 Δn þ γ 2 H ex Han Þ¼ ω2hf an × ehf ðT1Þ exp ðiωhf tÞδðξÞ;

ð1Þ

where c is the limiting propagation velocity of magnons,
γ is the gyromagnetic ratio, H ex is the exchange field, Han isthe effective anisotropy field given as usual by
Han ¼ −∂wan =∂n, wan being the magnetic anisotropy
energy [24].
Equation (1) can be solved in the stationary case in theabsence of pump pulses, giving the spatial orientation of theNéel vector from the T1 to the T2 domain across the wall,n0 ðξÞ ¼ cos φ0 e0 ðξÞ þ sin φ0 elf ðξÞ, parametrized with thefunction φ0 ðξÞ and space-dependent vectors [24]. The
stationary solution is represented in Fig. 1(a), whichdisplays how the orientation of the Néel vector changes
in the domain wall. In particular, at the center of the wall(ξ ≈ 0) the n vectors in the two domains are oriented
parallel to each other. We calculate [24] the magnon spectra
in a two-domain sample and find that they exhibit not only

propagating modes, but also localized modes that correspond to oscillations of the domain wall. The frequencies ofthe latter modes depend on both the magnetoelastic
coupling and on the exchange interaction between theNéel vectors in the two T domains. The frequency of one of
the localized modes, ωDW , can be approximated as ωDW ≈ωhf þ ωlf [24].
The experimentally observed coupling between magnon
modes with different frequencies is an intrinsically nonlinear process, as an input signal at a given frequency (ωhf )is converted into an output with a different frequency (ωlf ).
We thus expand n in series of powers of small deflections
δn, around the stationary solution n0 ðξÞ, so that the lowestorder nonlinearity is achieved. We represent further δn as asum of three eigenmodes: the localized domain wall mode,
the lf mode and the hf mode. Relying on the initialconditions, in which the domain wall is not in motion
prior to the photoexcitation, the equations for the corresponding time-dependent amplitudes bDW , blf , and bhf in
the T2 domain are [24]1b̈ DW þ b_ DW þ ω2DW bDW ¼ ω2hf a exp ðiωhf tÞ; ð2aÞτ
1
1b̈ hf þ b_ hf þ ω2hf bhf ¼ ω2hf a exp ðiωhf tÞ;
τ
3

ð2bÞ

1
1
b̈ lf þ b_ lf þ ω2lf ð1 − 4PbDW Þblf ¼ ω2hf a expðiωhf tÞ;
τ
3

ð2cÞ

where 1=τ ≪ ωlf , ωhf , ωDW is the relaxation time due to theGilbert damping and the coefficient P describes thecoupling between the lf- and domain wall mode. Let us
stress that the source terms in Eqs. (2) are photodriven hf
magnons in the T1 domain. The relevant quantity tocompute, for the sake of comparison with the experimentis thus blf :blf ¼

ω2hf a4a2 ω2lf ω2hf eλt
exp ðiωhf tÞ þ222
ωlf − ωhf
3ðωDW − ω2hf Þ2 ðω2lf − ω2hf Þ−ωlfzﬄﬄﬄﬄﬄﬄﬄﬄ}|ﬄﬄﬄﬄﬄﬄﬄﬄ{
× exp½iðωhf − ωDW Þt;

ð3Þ

pﬃﬃﬃ
where λ ¼ 2aPω2hf =3 − 1=τ is the instability parameterfor parametric down-conversion. The first term corresponds
to nonresonant excitations of lf magnons with wave vector
high enough, that they match the frequency of the hf mode
(i.e., 1.07 THz) [43]. These magnons may only contributeto the background signal, once they relax via scattering
events. The second contribution originates from the nonlinear term in Eq. (2c) and corresponds to the beating
between frequencies ωDW and ωhf . Crucially, we note thatquantitatively ωhf − ωDW matches ωlf . Moreover, sinceωlf ≈ 0.1ωhf ≪ ωhf , we can reformulate the second contribution as

077202-4

PHYSICAL REVIEW LETTERS 127, 077202 (2021)
blf ¼

a2 eλtexp ðiωlf tÞ:3

ð4Þ

In essence, our model describes an additional contribution to the lf mode in the T2 domain, originating from thenonlinear magnetoelastic interaction between the domain
wall mode and the photoinduced hf magnons in the T1
domain, activated by the hf magnons hitting on the wall.
This concept is consistent with the experimental observations reported in Fig. 2. The interpretation of our data interms of the mechanism described in the model above isfurther substantiated by measurements performed on a NiO
sample in a single-domain state, displaying amplification of
the hf mode only [24]. Further details of the model havebeen recently published [44].
We rule out the possibility that the lattice mediate thecoupling between the modes, as no phonon with therequired frequency for the frequency mixing is reported
in the dispersion of NiO [45].
Our results suggest that a proper sample engineering
may allow both an even more pronounced amplitude
amplification of the oscillations of n and a coupling evenamong propagating magnons. The former phenomenon isrelevant for achieving nonlinearities in the spin dynamics,
while the latter is a milestone toward the establishment ofcoherence transfer and energy transfer between magnonic
branches on the characteristic femtosecond time- and (sub)micrometer length scales of collective spin eigenmodes.
This work was supported by the Deutsche
Forschungsgemeinschaft through the International
Collaborative Research Centre TRR160 (Project B9), theDFG programme BO 5074/1-1, by the COST Action
MAGNETOFON (Grants No. CA17123, STSM
No. CA17123-44749). M. B. and L. S. acknowledge support from the Knut and Alice Wallenberg Foundation,
Grant No. 2017.0158. S. B. acknowledges support from theSwedish Research Council (VR), Grant No. 2018-04611.
O. G. acknowledges funding by the Deutsche
Forschungsgemeinschaft (DFG, German Research
Foundation)—TRR 173—268565370 (Project No. A11),
TRR 288—422213477 (Project No. A09), and Project
SHARP 397322108, the ERC Synergy Grant SC2(No. 610115), and partial support by the National
Science Foundation under Grant No. NSF PHY1748958. She also acknowledges discussions with Oleg
Tchernyshyov. T. S. ackowledges H. Ueda for providing
the single-domain sample.

*

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