Magnons in complex systems from TDDFT
L. Sandratskii, P. Buczek, A. Ernst1 Introduction
The properties of excited magnetic states are of great importance in the
fundamental and applied magnetism. Their spectrum determines the thermody
namical properties of magnets, including the Curie temperature
^{[11]
}
The excitations contribute to the electronic specific heat
^{
[5]
}
and the electrical and thermal conductivity, couple to char
ge degrees of freedom
^{
[14]
}
,
and can provide a coupling mec
hanism in high temperature superconductors alternative to phonons
^{
[10]
}
The control of spin dynamics and its attenuation are the centr
al problems in the rapidly growing field of spintronics
^{
[9]
}
Until now the main body of theoretical studies on magnetic excitations ha
s been based on the adiabatic treatment of magnetic degrees of freedom
^{
[8]
}
The approach describes correctly long wavelength
spinwaves (magnons), i.e. collective
modes associated with the coherent precession of atomic moments.
It captures most of the physics of nonconducting materials, but is of limited
validity in the metallic magnets, because it neglects the presence of particlehole states with
opposite spin called
Stoner excitations. The Stoner states are p
ronounced mostly at the energies corresponding roughly to the exchange sp
litting of electron states, but they can have nonvanishing contribution
also in the lower energy region where the magnons appear. The resulting h
ybridization leads to the attenuation of spinwaves. The effect is called
Landau damping, influences qualitatively the spin dynamics of m
etals and cannot be described in the adiabatic theories.
The damping can be captured in calculations of wavevector and frequency
dependent transverse magnetic susceptibility χ(q,ω),
where spinwaves and Stoner states are treated on an equal footing. The dynam
ic method became particularly powerful after the parameter free linear re
sponse density functional theory (LRDFT)
^{
[7]
}
was formulated.
Such calculations are, however, very demanding both from the point of vi
ew of algorithmic complexity and computer resources and for a long time t
hey were restricted to simple bulk systems.
^{
[13],
[15]
}
Recently, we have developed a novel efficient numerical scheme allowi
ng to evaluate the spin susceptibility of complex magnets and applied it
to study energies and lifetimes of magnons in complex bulk phases
^{
[1],
[4]
}
and ultrathin films
^{
[2] ,
[3]
}
.
The spinwave attenuation is determined by fine properties of Stoner cont
inuum. The first principle approaches based on the calculation of transve
rse magnetic susceptibility are indispensable in the consistent descripti
on of spin dynamics in real materials.
2 Methods
Linear response time dependent density functional theory allows
to compute the generalized susceptibility in the following two step procedure.
^{
[7] ,
[12]
}
We focus on the transverse magnetization
dynamics. First, one considers the KohnSham susceptibility
giving the retarded response of the formally noninteracting KohnSham system.
φ_{j}(xα)'s and ε_{j}'s denote
respectively KS eigenfunctions and corresponding eigenenergies.
f_{j} ≡ f_{T}(ε_{j}),
where f_{T}(&epsilon)
is the FermiDirac distribution function. T
he induced magnetization densities described by the KohnSham susceptibil
ity modify the exchangecorrelation potential, giving rise to a selfcons
istent problem: the induced densities contribute to the effective fields
and are, simultaneously, induced by it. The selfconsistency is reflected
in the second step of the formalism
The last equation is referred to as "susceptibility Dyson equation", be
cause of its characteristic form. χ is the physical (enhanced) susce
ptibility of the system. The exchangecorrelation kernel, K_{xc},
is defined as a functional derivative of exchangecorrelation potential
with respect to the density
evaluated at the ground state values of electronic and magnetic densities.
3 Examples
3.1 Spinwaves in halfmetals
The halfmetalls are very attractive materials for spintronic ap
plications. We studied the relation between the halfmetallicity and life
time of the spinwaves in a series of Heusler alloys
^{
[1]
}
We demonstrated that the acoustic spin wave mode remains practically undamped
for spinwave momenta in the entire Brillouin zone. On the contrary
the optical modes feature a finite lifetime changing strongly and nonmo
notonously with the momentum, cf. Fig. 1.
 Figure 1: Spinwaves in Co_{2}MnSi for different wavevectors in the first Brillouin zone. Panel a) presents energies of the spin waves. Panel
 b) shows the halfwidth at halfmaximum (HWHM) of the spinwave peak (inversely proportional to the state's lifetime) in the spectral density. The HWHM of the acoustic (EV 1) mode in this system is less than and is not shown.
The knowledge of the lifetime is however essential for applications. In
the storage devices the excited states should decay as soon as possible,
leaving a bit after a readin or readout in a steady state. On the contr
ary, in the interchip communication, the spinwave should live as long,
as it is necessary for signal to travel undistorted between emitter and antenna.
3.2 Controlling terahertz magnetization dynamics
Spintronics utilizes the spin degree of freedom to process and s
tore information. Typical spintronic devices are operated at frequencies
of the Larmor precession in magnetic anisotropy fields, which corresponds
roughly to the GHz band. Only recently their spatial sizes has
been reduced to the subμmeter regime. Recently, we focused on t
he possibility of controlling the magnetization dynamics in the THz
range and on the scale of single nanometers
^{
[2]
}
We s
uggested that spatially confined (between the surface and the interface o
f the film) exchange driven spin waves could be utilized in a ne
w generation of spintronic devices to scale down their sizes and to accel
erate speed.
3.3 The impact of the substrate on the Landau damping in ultrathin films
We considerably advanced the understanding of the way a nonmagnetic subs
trate influences the properties of the spin waves in thin magnetic films.
To get insight into the properties of magnetic excitations formed by the
combination of the 2D magnetic film and 3D nonmagnetic substrate we inve
nted socalled Landau maps that vizualize the kresolved
intensity of the Stoner continuum and allow to determine the states resp
onsible for the decay of the spin waves.
If the continuum of the substrate bulklike states were the decisive fact
or in the strong Landau damping of the supported monolayer, the correspon
ding Landau map of Fe/W(110) would show hardly any sharp features. Instead,
the Stoner transitions for the energy associated with the magnon would
be available for almost any k_{ } resulting in a relativ
ely uniform filling of the map.
 Figure 2: (a) Spectral power of spin excitations for zero inplane wave vector. Six Lorentzian peaks can be distinguished. They correponds to the standing spinwaves of the 6 ML Co/Cu(100) system. (b) Corresponding mag
 Figure 3: Spectral intesity of spinflip excitations in 3 ML Fe/Cu(100) system. Three branches of weakly damped standing spinwaves are clearly discernable.
Surprisingly, the damping of Fe/W(110) is still dominated by hotspots
(cf. Fig. 5b)
The hot spots are responsible for 70% of the damping. The analysis of these features
shows that they are formed by transition between so called interface
electron complexes, i.e. electronic states resulting from the hybridization
of the states of the Fe film and the surface states of the W(110)
^{
[6]
}
. Region marked with E originates from electron states
in the film hybridizing strongly with the continuum of bulk states in bot
h spin channels. Such Stoner pairs are of minor importance.
 Figure 4: Magnon spectra in iron films. Thick lines denote the dispersio relation, ω_{0}(q), and the width of the shaded area corresponds to the full width at half maximum on the energy axis. The Stoner spectrum contributing to the damping of marked magnons ( • ) is an alyzed in Fig. 5.
In contrast to the Fe/W(110) case, the electronic structure of the Fe/Cu(
100) differs weakly from its freestanding counterpart. Additionally, Cu(
100) does not feature surface states crossing Fermi level. As a result, t
he magnon spectrum is weakly affected by the substrate, cf. Fig. 4.
Our current and future research will include such topics as the study of
paramagnons excitations, spinwaves in noncollinear systems, magnetic ex
citations of clusters on metallic and nonmetallic substrates.
Figure 5: Intensity of Stoner transitions with momentum q_{0} and energy ω_{0} in Fe layer resolved for different final kvectors in the first Brillouin zone. The Stoner states cause the damping of magnons indicated in Fig. 4.
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3.4 Chiralitydependent magnon lifetimes in a halfmetallic antiferromagnet
By exploring different magnetic orderings one can enhance the functionality of magnonic devices. Recent ultrafast
experiments have shown the reversal of the polarity of AFM magnons via circularly polarized light
^{[16]}
, and the optical
directioning of the spin wave emission in a ferrimagnet
^{[17]
} .
We performed the study of the lifetimes of magnons in the halfHeusler CrMnSb, the prototype of halfmetallic
antiferromagnets (HMAFM) ^{
[18]}.
The unique combination of halfmetallicity and magnetic compensation of inequiv
alent sublattices leads to unexpected properties of the magnon excitations. Despite inequivalence of the sublattices
we obtained two acoustic magnon branches with linear dependence of the magnon energies on the wave vector of
the excitations and equal spinwave velocities. On the other hand, the damping properties of the two branches
are strongly different. We reveal the origin of the unusual magnon properties and discuss a possible way of the
engineering of the chiralitydependent attenuation of the magnons.
In Fig. 6 we present the density of states of CrMnSb and the results of the calculation of the energies and
attenuation of the magnons. The description of the presented data is given in the caption to the figure.
Figure 6: Spinpolarized density of states of halfmetallic CrMnSb, a fully compensated ferrimagnet (details around the Fermi level are depicted in the inset). The gap in the spinup channel directly affects the spin excitation properties. On the right side, we show the spin wave spectra from Heisenberg adiabatic dynamics (in blue), and from the dynamic transverse magnetic susceptibility (red dots) with the inverse lifetimes (error bars), evidencing the asymmetric spin wave damping. The inset presents the spectral densities of spinwave (solid orange line) and Stoner (green line) excitations. The spin wave branch of positive chirality is damped due to the presence of Stoner excitations.
3.5 Paramagnons in FeSe
FeSe is a member of the family of iron based superconductors.The critical
temperature at ambient pressure is moderate with 9 K, but it is strongly enhanced by
pressure. The application of pressure intensifies also the antiferromagnetic(AFM)
spin fluctuations
^{ [19]},indicating a connection between spin excitations and
superconductivity.The incorporation of the magnetic fluctuation in an abinitio theory for
superconductivity is also ongoing research
at the institute .
In our study we modeled the approaching to the point of the quantum phase
transition from the paramagnetic state to the antiferromagnetic state by the variation
of the position of the Se atoms, zSe . Figure 7 shows the spectral
density of the
collective spin excitations for three values of the parameter. For zSe = 0.662 Å the
system is far from the phase transition and the spectrum of collective excitations is
very weak. Approaching closer to the phase transition (zSe = 0.994 Å) is accompanied
by gradual increase of the spectral density of the fluctuations. Near the point of the phase
transition, the spinfluctuation spectrum takes a form of welldefined brunches of
paramagnon excitations. In contrast to the spinwaves in magnetically ordered systems,
in the paramagnon spectrum there is no acoustic mode characterized by zero energy at
zero wave vector. This demonstrates a principle difference between the magnons being
Goldstonetype excitations and the fluctuations in the PM phase.
Our finding of the welldefined highspectraldensity paramagnon excitations in FeSe
supports the scenario of the unconventional spinfluctuations mediated superconductivity in
Febased superconductors.
Figure 7: Spectral density along a path in inverse space for various zSe . The high synmmetry points are Γ = (0, 0, 0), X = ( π , 0, 0) and M = π , π , 0 .
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