# Dzyaloshinskii-Moriya interaction induced by an ultrashort electromagnetic
pulse:
Application to coherent (anti)ferromagnetic skyrmion nucleation

L. Desplat<EMAIL_ADDRESS>Université de Strasbourg, CNRS,
Institut de Physique et Chimie des Matériaux de Strasbourg, UMR 7504, F-67000
Strasbourg, France  S. Meyer  Nanomat/Q-mat/CESAM, Université de Liège, B-4000
Sart Tilman, Belgium  J. Bouaziz  Peter Grünberg Institut and Institute for
Advanced Simulation, Forschungszentrum Jülich and JARA, D-52425 Jülich,
Germany  P. M. Buhl  Institute of Physics, Johannes Gutenberg University
Mainz, 55099 Mainz, Germany  S. Lounis  Peter Grünberg Institut and Institute
for Advanced Simulation, Forschungszentrum Jülich and JARA, D-52425 Jülich,
Germany  B. Dupé  Fonds de la Recherche Scientifique (FRS - FNRS), B-1000
Brussels, Belgium  Nanomat/Q-mat/CESAM, Université de Liège, B-4000 Sart
Tilman, Belgium  Université de Strasbourg, CNRS, Institut de Physique et
Chimie des Matériaux de Strasbourg, UMR 7504, F-67000 Strasbourg, France
P.-A. Hervieux<EMAIL_ADDRESS>Université de
Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg,
UMR 7504, F-67000 Strasbourg, France

###### Abstract

We show how a Dzyaloshinskii-Moriya interaction can be generated in an
ultrathin metal film from the induced internal electric field created by an
ultrashort electromagnetic pulse. This interaction does not require structural
inversion-symmetry breaking, and its amplitude can be tuned depending on the
amplitude of the field. We perform first-principles calculations to estimate
the strength of the field-induced magnetoelectric coupling for ferromagnetic
Fe, Co, and Ni, and antiferromagnetic Mn, as well as FePt alloys. Last, using
atomistic simulations, we demonstrate how an isolated antiferromagnetic
skyrmion can be coherently nucleated from the collinear background by an
ultrashort pulse in electric field on a 100-fs timescale.

The study of the ultrafast optical manipulation of magnetism constitutes a
fascinating and very active research topic [1]. A breakthrough in this domain
could provide the ingredients for the elaboration of a new generation of
spintronic devices and lead to computers an order of magnitude faster. A
remarkable result, first observed by Beaurepaire and Bigot in Nickel (Ni) thin
films [2], is the ultrafast loss of magnetization occurring within the first
100 fs following a laser pulse. More than 20 years after its discovery, there
is no general consensus about the underlying mechanisms of this ultrafast
demagnetization, which was attributed to various mechanisms, including, more
recently, the spin-orbit interaction [3, 4], or the superdiffusive electron
transport induced by the laser field [5]. This phenomenon is usually referred
to as ultrafast thermal demagnetization.

Another mechanism, not thermally induced, and suggested by Bigot et al. [6] is
a laser-induced coherent magnetization dynamics–also in Ni thin films, which
could be related to a direct coupling between photons and spins, i.e., between
the external electric field of the laser and the spins of the electrons in the
material. So far, this hypothesis has not been confirmed nor refuted, and is
the subject of intense research efforts in the field of ultrafast magneto-
optical spectroscopy [7, 1].

In Condensed Matter, the interplay between a static external electric field
and the magnetization is usually called the magnetoelectic (ME) effect [8]. It
allows the manipulation of the magnetocrystalline anisotropy [9] or that of
the Dzyaloshinskii-Moryia interaction (DMI) [10, 11, 12] with minimal energy
consumption, but Coulomb screening limits this effect to a few monolayers
[13].

The situation for time-dependent external electric fields is quite different.
In the terahertz (THz) spectral region, the skin depth can, for instance, be
as large as a few tens of nanometers [14]. Moreover, it is nowadays possible
to produce powerful ultrashort sub-cycle THz pulses, leading to electric field
amplitudes above $10^{11}$ V.m-1 [15]. Under such conditions, an intense
electric field can penetrate inside the material, which may break the
inversion symmetry and induce a DMI, resulting in the presence of chiral
magnetic states such as skyrmions.

Within the search for novel spintronics devices, skyrmion-based designs appear
very promising, e.g., as racetrack memories and logic gates [16, 17], for
reservoir computing [18], or as reshufflers for probabilistic computing [19].
Magnetic skyrmions [20, 21] are two-dimensional, non-collinear solitonic spin
textures with a nontrivial topology. In ultrathin films and multilayers, small
skyrmions are stabilized by an interfacial form of the DMI, which arises under
a combination of strong spin-orbit coupling (SOC) and inversion-symmetry
breaking at the surface or interface [22, 23, 24]. In particular,
antiferromagnetic (AFM) skyrmions have been recently observed in a synthetic
antiferromagnet [25] and possess attractive properties for applications, i.e.,
fast internal modes in the THz range, vanishing dipolar fields, and immunity
to the skyrmion Hall effect [26]. As many antiferromagnets are insulating, the
optical manipulation of the magnetization in these materials is even more
compelling. Recently, the optical trapping of skyrmions was demonstrated in
spin-driven chiral multiferroics, where the coupling of an external electric
field to the ferroelectric polarization due to the large intrinsic ME effect
creates a DMI-like term, which allows the manipulation of skyrmions [27]. The
same effect was exploited to coherently switch the polarity and chirality in a
magnetic vortex by applying ultrashort electric field pulses [28].

In this Letter, we show that coherent magnetization dynamics can be induced by
a magnetoelectric interaction created by an ultrashort electromagnetic pulse.
This mechanism originates from the spin-orbit coupling between the electric
field of the pulse and the spins of the delocalized electrons, and reduces to
a Rashba SOC. By describing the sample as an ensemble of localized atomic
spins embedded in a two-dimensional electron gas (2DEG), and using the RKKY
model of indirect exchange, we express the magnetoelectric coupling as a DMI-
like term. Based on this model, we then show from density functional theory
(DFT) calculations that an external electric field is sufficient to create a
significant DMI without the need for a structural breaking of the inversion
symmetry. Last, we explore the creation of antiferromagnetic skyrmions in
metallic thin films, and demonstrate through atomistic simulations that
coherent nucleation is possible at the 100-fs timescale, i.e., an order of
magnitude faster than thermally-driven skyrmion nucleation processes [29, 30,
31].

In the following, we present a simple model to show that the interaction of an
external, time-dependent electric field with a magnetic thin metal film leads
to the creation of a magnetoelectric effect. A very simplified description of
the electronic properties of the material sample is based on the distinction
between itinerant magnetism carried by the conduction electrons, and localized
magnetism carried by the fixed ions. The $s$ electrons are assumed to be at
the origin of itinerant magnetism, whereas the $p-d$ electrons are localized
around their nuclei to form ionic spins that are responsible for localized
magnetism. We assume that the delocalized electrons can be modeled by a 2DEG
(ultrathin film geometry), and the light-matter interaction is described by a
semi-relativistic expansion of the Dirac-Maxwell mean-field model [32]. As
detailed in the Supplemental Material (SM) [33], and in Ref. 32, at the lowest
order in powers of $1/c$, the spin-light interaction Hamiltonian reads,

$\mathcal{H}_{\mathrm{SOC}}=\frac{e\hbar}{4m^{2}c^{2}}\mbox{\boldmath$\sigma$}\cdot(\mathbf{E}_{\mathrm{ext}}+\mathbf{E}_{\mathrm{int}})\wedge\mathbf{p},$
(1)

where $m$ is the effective electron mass, $\mathbf{p}$ is the momentum of the
electron, $c$ is the speed of light in vacuum, $\sigma$ is the vector of the
Pauli matrices, $\mathbf{E}_{\mathrm{ext}}$ the external electric field of the
electromagnetic pulse, and $\mathbf{E}_{\mathrm{int}}$ is the internal
electric field solution to the Poisson equation. Thus, the many-electron
problem reduces to a one-electron problem, in which the electron experiences
SOC with the electric field within the material. In the limit of a weak
external electromagnetic field–used to demonstrate the linearity of the
magnetoelectric effect at low field, the internal field
$\mathbf{E}_{\mathrm{int}}$ is proportional to the external field
$\mathbf{E}_{\mathrm{ext}}$, therefore
$\mathbf{E}_{\mathrm{int}}+\mathbf{E}_{\mathrm{ext}}\propto\mathbf{E}_{\mathrm{ext}}$.
The spin-orbit Hamiltonian $\mathcal{H}_{\mathrm{SOC}}$ may be rewritten as a
Rashba SOC proportional to
$\left(\mathbf{k}\wedge\mathbf{E}_{\mathrm{ext}}\right)\cdot\mbox{\boldmath$\sigma$}$,
where $\mathbf{k}$ is the electron wave vector. The effect of the Rashba SOC
on a 2DEG has been studied in Refs. 34, 35. When a pair of atomic spins
$\mathbf{S}_{\mathrm{i}}$ and $\mathbf{S}_{\mathrm{j}}$ are separated by a
distance $r_{\mathrm{ij}}$, the field-induced DMI reads

$|\mathbf{D}^{\mathrm{E}}_{\mathrm{ij}}|=\alpha_{\mathrm{ME}}E_{\mathrm{ext}},$
(2)

where
$\alpha_{\mathrm{ME}}=\frac{e\hbar}{4m^{2}c^{2}}\mathcal{F}(k_{\mathrm{F}},r_{\mathrm{ij}})$
is the antisymmetric magnetoelectric coupling, in which $k_{\mathrm{F}}$ is
the Fermi wave vector, and $\mathcal{F}$ is defined in the SM [33]. In
particular, for an external electric field applied along $z$, the DMI vector
has the same symmetry as that of interfacial DMI. Let us stress that in the
absence of an electric field, $|\mathbf{D}^{\mathrm{E}}_{\mathrm{ij}}|=0$.

Next, we perform density functional theory calculations in order to estimate
the strength of the DMI created by an external electric field. To this extent,
we apply the full-potential linearized augmented planewave (FLAPW) method, as
implemented in the FLEUR code [36]. We consider model systems in the form of
3d transition metal unsupported monolayers (UMLs), in which the d-band is
progressively filled from Mn to Fe, Co, and Ni, as well as unsupported
trilayers of Fe/Pt/Fe, Pt/Fe/Pt, and Pt/Mn/Pt. We use the lattice parameters
and crystal structures of bulk Mn, Fe, Co, Ni, and L10 PtFe or PtMn binary
alloys [37]. A static electric field is applied perpendicular to the film
plane, which allows us to circumvent the calculation of the long-range
Coulomb-type interaction present in bulk systems. This drastic approximation
thus only holds at ultrafast timescales, at which the mobility of the
electrons is mainly influenced by their effective mass, rather than by
scattering events.

Figure 1: Charge density and potential in an Fe unsupported monolayer (UML)
along the $z$-direction perpendicular to the film. (a) Charge density for spin
up ($\uparrow$) and spin down ($\downarrow$) without applying an external
electric field. $z_{0}$ denotes the position of the atomic center whereas the
$y$ position lies in the middle of the unit cell. (b) Potential difference for
both spins under an electric field of $E_{\textrm{ext}}=+6.66\times
10^{8}$V.m-1 with charged sheets at position $\pm 6\,\textrm{\AA}$. (c)
Difference of charge density after applying the electric field.

The calculations can be understood based on the example of a Fe UML, as shown
in Fig. 1. In the absence of an external electric field (Fig. 1(a)), the
system is centrosymmetric, so the charge density for both spin up ($\uparrow$,
orange line) and spin down ($\downarrow$, dashed black line) as a function of
the vertical distance from the center of an atom is symmetric, and no internal
field is created ($E_{\textrm{int}}=0$). In that case, there is no DMI. This
behavior holds for all centrosymmetric, unsupported monolayers.

We then look at the effect of an external electric field, which can be applied
by adding two oppositely charged sheets below and above the film. Here,
positive fields are related to a positively charged sheet below, and a
negatively charged sheet above the film. The potential difference created by
an applied field of $6.66\times 10^{8}$ V.m-1 is shown in Fig. 1(b), for which
the charged sheets are placed at $\pm$6 Å from the film surface. The sheets
create a linear potential along $z$, which vanishes in the film region due to
Coulomb screening. The effect of the latter is visible in the difference in
the charge density compared to the zero-field scenario, as shown in Fig. 1(c).
In the presence of the external field, the mirror symmetry of the charge
density is broken, and the charges recombine depending on their spin channels.
The charge density differences exhibit two peaks: A positive one for $z<0$,
and a negative one for $z>0$, which respectively translates an accumulation or
a depletion of charges in the film. This charge imbalance creates an internal
electric field $E_{\mathrm{int}}$, which can act as a spin-orbit coupling
contribution.

To confirm this hypothesis, we calculate the SOC contribution in the presence
of such an internal electric field. To do so, we perform self-consistent spin-
spiral calculations [38], where SOC is added to first order in perturbation
theory [39]. The strength of the field-induced SOC, which we interpret as a
DMI term $D_{\textrm{eff}}^{E}$, is then determined in the limit of
$q\rightarrow 0$ in the vicinity of the magnetic ground-state, as a function
of the external electric field. As expected, the DMI depends linearly on the
electric field [33], which allows the extraction of the linear ME coefficient,
$\alpha_{\mathrm{ME}}$. Figure 2 shows the value of $\alpha_{\mathrm{ME}}$ for
different $d$-band fillings of the 3$d$ transition-metal monolayers. For Mn
and Fe, the DMI favors a right-rotating spin spiral, and
$\alpha_{\mathrm{ME}}$ goes from $4\times 10^{-15}$ eV.m.V-1 for Mn, to
$2\times 10^{-15}$ eV.m.V-1 for Fe. As the $d$-band filling increases, i.e.,
for Co and Ni, the DMI changes sign, and $\alpha_{\mathrm{ME}}$ reaches
$-4\times 10^{-15}$ eV.m.V-1 for Ni.

Figure 2: DFT-calculated values of the magnetoelectric coupling,
$\alpha_{\textrm{ME}}$. Positive (negative) values correspond to increasing
strength of clockwise- (counterclockwise-) preferring DMI with the applied
electric field.

To increase the value of $\alpha_{\mathrm{ME}}$, we associate 5$d$ elements to
the 3$d$ ultrathin films. In bulk, these combinations could be achieved by
exploring binary alloys such as FePt, CoPt or MnPt. In this case,
$\alpha_{\mathrm{ME}}$ can reach values up to $-17\times 10^{-15}$ eV.m.V-1
for Pt/Fe/Pt. This corresponds to a DMI of 17 meV/magnetic atom for an
electric field amplitude of $10^{12}$ V.m-1, which, as we show in the
following, is sufficient to nucleate isolated skyrmions at the 100-fs
timescale.

Finally, we perform atomistic simulations to demonstrate how this new
mechanism can be used to nucleate an AFM skyrmion coherently, by applying an
ultrafast electromagnetic pulse. To that extent, we simulate an ultrathin
magnetic film consisting of $N$ magnetic moments
$\\{\mathbf{\hat{m}}_{\mathrm{i}}\\}$, $i=1\ldots N$, of norm unity, on a two-
dimensional simple square lattice. The grid size is $50\times 50$ with
periodic boundary conditions at the edges. The pulse is assumed to be linearly
polarized, and in an oblique incidence configuration, which produces a
longitudinal component $E_{z}$ inside the film. We give the effective
Heisenberg Hamiltonian,

$\begin{split}&\mathcal{H}=-J_{\mathrm{eff}}\sum_{\langle
ij\rangle}\mathbf{\hat{m}}_{\mathrm{i}}\cdot\mathbf{\hat{m}}_{\mathrm{j}}-K\sum_{\mathrm{i}}m_{z,\mathrm{i}}^{2}\\\
&-\sum_{\langle
ij\rangle}\left(D_{\mathrm{eff}}^{\mathrm{interf}}+\alpha_{\mathrm{ME}}E_{z,\mathrm{i}}\right)\left(\mathbf{\hat{z}}\times\mathbf{\hat{r}}_{\mathrm{ij}}\right)\cdot\left(\mathbf{\hat{m}}_{\mathrm{i}}\times\mathbf{\hat{m}}_{\mathrm{j}}\right),\end{split}$
(3)

where the double summations are carried out over all pairs of first nearest
neighbors $\langle\mathrm{ij}\rangle$, $J_{\mathrm{eff}}<0$ is the effective
AFM Heisenberg exchange coupling constant, $K$ is the perpendicular magnetic
anisotropy constant, $D_{\mathrm{eff}}^{\mathrm{interf}}$ is the effective
interfacial DMI coupling at zero electric field, $E_{z,\mathrm{i}}$ is the
perpendicular electric field at site i such that
$D_{\mathrm{eff},\mathrm{i}}^{E}=\alpha_{\mathrm{ME}}E_{z,\mathrm{i}}$, and
$\mathbf{\hat{r}}_{\mathrm{ij}}$ is the unit displacement vector between first
neighbors. We furthermore define reduced parameters as
$d=D_{\mathrm{eff}}^{\mathrm{interf}}/|J_{\mathrm{eff}}|$,
$k=K/|J_{\mathrm{eff}}|$, and
$d_{E}=\alpha_{\mathrm{ME}}E_{z}/|J_{\mathrm{eff}}|$. Simulations are
performed at zero temperature by solving the Laudau-Lifshitz-Gilbert equation
[40] with a Gilbert damping of $\lambda=0.3$, and $J_{\mathrm{eff}}=-11$ meV.
The magnetization is initialized in the AFM state with a single in-plane
defect in the center, such that an electric field exerts a nonzero torque on
the magnetic moments. At $t=0$, the electric field is applied in the lattice
center, with a spot diameter of ten lattice sites, and a time dependence taken
as a Heaviside step function. While electromagnetic sources such as lasers
typically produce mm-$\mu$m-wide spots [41], here, no new physics would emerge
by simulating a wider spot since the DMI exerts no torque on the collinear
state at zero temperature. Last, the total topological charge $Q$ of the
vector of the AFM order, $\mathbf{L}=(\mathbf{M}_{1}-\mathbf{M}_{2})/2$, where
$\mathbf{M}_{1,2}$ correspond to the AFM sublattices, is computed using a
discretized description [42, 43] and used to track the skyrmion nucleation.

Figure 3: AFM skyrmion creation under a pulse in electric field. The graph
shows the time evolution of the topological charge (left, dotted line) and of
the mean energy per site (right, solid line). The blue shaded area indicates
the duration of the electric field. The spin maps show snapshots of the
magnetization around the lattice center at different times of the simulation.
The beam spot is shown as an overlayed white disk. The parameters of the
simulations are $k=0.5$, $d=0.44$, $d_{E}=1.13$.

The nucleation mechanism is illustrated in Fig. 3. We show the time evolution
of the total topological charge of the system, $Q$ (left), and of the mean
energy per site normalized by the exchange coupling constant,
$\big{(}\langle\mathcal{E}\rangle-\langle\mathcal{E}_{\mathrm{sk}}\rangle\big{)}/|J_{\mathrm{eff}}|$
(right), where $\langle\mathcal{E}_{\mathrm{sk}}\rangle$ is the mean energy
per site of the partially relaxed skyrmion at $t=1000$ fs. The blue shaded
area indicates the duration of the electric field, which is 100 fs in this
case. The spin maps correspond to snapshots of the magnetization around the
center of the simulated area at different times. When the field is switched on
($t=0$), the mean energy first decreases towards a local minimum corresponding
to a perturbed collinear state (snapshot 2, 29 fs) and increases again to a
maximum–the barrier top for the skyrmion nucleation in this scenario (snapshot
3, 64 fs), before dropping brutally as a unit of topological charge is
nucleated (snapshot 4, 90 fs). This behavior stems from the fact that the
increase in the DMI induced by the field momentarily changes the ground state
of the system in the beam cross-section, from the collinear to the spin spiral
(SS) state. It follows that the skyrmion state becomes more favorable that the
collinear state and is more easily nucleated during the relaxation process.
Snapshot 5 corresponds to the skyrmion state at the time when the beam is
turned off (100 fs). This results in a discontinuous jump in energy, at which
point the collinear ground state is restored, followed by the relaxation of
the newly formed metastable skyrmion (snapshot 6). We note that we used the
minimum value of $d_{E}$ that enables nucleation, and that, for a larger
value, the nucleation process is typically even faster.

In order for a skyrmion to survive at zero electric field, the material
parameters must allow skyrmions to be metastable. In Fig. 4, the area with a
color gradient indicates the existence of metastable skyrmion excitations on
the collinear background as a function of reduced DMI and anisotropy [44, 45,
46].

Figure 4: Minimum DMI induced by an electric field required for the nucleation
of a metastable AFM skyrmion. The area with the color gradient corresponds to
the region of existence of metastable skyrmions as a function of anisotropy
and effective interfacial DMI at zero electric field. The colorscale indicates
the minimum value of the field-induced DMI required for skyrmion nucleation
for $J_{\mathrm{eff}}=-11$ meV and a damping of $\lambda=0.3$.

The red dotted line marks the transition line between the spin spiral ground
state and the collinear ground state as $d_{c}=2\pi^{-1}\sqrt{k}$ [47, 48].
The colorscale indicates the minimum value of the reduced DMI induced by the
electric field, $d_{E}$, that is required to nucleate a skyrmion for each set
of $(d,k)$. We find the smallest value of $d_{E}=1.12$ in the large $(d,k)$
region, which also corresponds to the fastest topological charge nucleation in
90 fs, and a maximum value of $d_{E}=1.25$ at low $(k,d)$, at which the
topological charge takes a maximum of 210 fs to nucleate. With our choice of
parameters and the value of $\alpha_{\mathrm{ME}}$ for Pt/Mn/Pt, we obtain a
minimum applied field of $E_{z}=2\times 10^{12}$ V.m-1. In the SM, we show
similar results for the case of a ferromagnetic skyrmion nucleation [33], for
which the minimum $d_{E}$ is 1.52, yielding a minimum field $E_{z}=10^{12}$
V.m-1 for Pt/Fe/Pt. In both cases, the required field values are realistically
achievable [15].

In conclusion, we have shown that at ultrashort timescales, an external
electric field can induce an internal electric field in metals, which breaks
the inversion symmetry and creates a DMI. We have shown how this mechanism
allows a coherent nucleation of skyrmions on a 100-fs timescale. This result
provides a new handle for the ultrafast manipulation of the magnetization, and
opens the way to an optical control of skyrmions, e.g., via ultrafast laser
pulses in metallic thin films.

###### Acknowledgements.

We thank Melanie Dupé and Giovanni Manfredi for useful discussions. This work
used the ARCHER UK National Supercomputing Service (http://www.archer.ac.uk)
and was supported by the University of Strasbourg Institute for Advanced Study
(USIAS) via a Fellowship, within the French national program “Investment for
the Future” (IdEx-Unistra). B. Dupé and S. Meyer acknowledge support via DARPA
Grant No. HR0011727183-D18AP00010 (TEE program). B. Dupé and P. Buhl
acknowledge funding by the DFG under Grant No. DU 1489/3-1.

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