Theory Department
Max Planck Institute of Microstructure Physics

Theory Department > Research > Magnons in complex materials

Magnons in complex systems from TDDFT

L. Sandratskii, P. Buczek, A. Ernst

1 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 spin-waves (magnons), i.e. collective modes associated with the coherent precession of atomic moments. It captures most of the physics of non-conducting materials, but is of limited validity in the metallic magnets, because it neglects the presence of particle-hole 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 non-vanishing contribution also in the lower energy region where the magnons appear. The resulting h ybridization leads to the attenuation of spin-waves. 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 wave-vector and frequency dependent transverse magnetic susceptibility χ(q,ω), where spin-waves 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 life-times of magnons in complex bulk phases ^{[1],
[4]} and ultrathin films ^{[2] ,
[3]} .

The spin-wave 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 Kohn-Sham susceptibility

(1)

giving the retarded response of the formally non-interacting Kohn-Sham 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 Fermi-Dirac distribution function. T he induced magnetization densities described by the Kohn-Sham susceptibil ity modify the exchange-correlation potential, giving rise to a self-cons istent problem: the induced densities contribute to the effective fields and are, simultaneously, induced by it. The self-consistency is reflected in the second step of the formalism

(2)

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 exchange-correlation kernel, K_xc, is defined as a functional derivative of exchange-correlation potential with respect to the density

(3)

evaluated at the ground state values of electronic and magnetic densities.

3 Examples

3.1 Spin-waves in half-metals

The half-metalls are very attractive materials for spintronic ap plications. We studied the relation between the half-metallicity and life -time of the spin-waves in a series of Heusler alloys ^[1] We demonstrated that the acoustic spin wave mode remains practically undamped for spin-wave momenta in the entire Brillouin zone. On the contrary the optical modes feature a finite life-time changing strongly and non-mo notonously with the momentum, cf. Fig. 1.

: Figure 1: Spin-waves in Co₂MnSi for different wave-vectors in the first Brillouin zone. Panel a) presents energies of the spin waves. Panel

: b) shows the half-width at half-maximum (HWHM) of the spin-wave peak (inversely proportional to the state's life-time) 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 life-time is however essential for applications. In the storage devices the excited states should decay as soon as possible, leaving a bit after a read-in or read-out in a steady state. On the contr ary, in the inter-chip communication, the spin-wave 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 sub-μmeter 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 non-magnetic 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 so-called Landau maps that vizualize the k-resolved 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 bulk-like 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 in-plane wave vector. Six Lorentzian peaks can be distinguished. They correponds to the standing spin-waves of the 6 ML Co/Cu(100) system. (b) Corresponding mag

: Figure 3: Spectral intesity of spin-flip excitations in 3 ML Fe/Cu(100) system. Three branches of weakly damped standing spin-waves are clearly discernable.

Surprisingly, the damping of Fe/W(110) is still dominated by hot-spots (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, ω₀(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 free-standing 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, spin-waves in non-collinear systems, magnetic ex citations of clusters on metallic and non-metallic substrates.

Figure 5: Intensity of Stoner transitions with momentum q₀ and energy ω₀ in Fe layer resolved for different final k-vectors in the first Brillouin zone. The Stoner states cause the damping of magnons indicated in Fig. 4.

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3.4 Chirality-dependent magnon lifetimes in a half-metallic 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 half-Heusler CrMnSb, the prototype of half-metallic antiferromagnets (HM-AFM) ^[18]. The unique combination of half-metallicity 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 spin-wave 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 chirality-dependent 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: Spin-polarized density of states of half-metallic CrMnSb, a fully compensated ferrimagnet (details around the Fermi level are depicted in the inset). The gap in the spin-up 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 spin-wave (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 ab-initio 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 spin-fluctuation spectrum takes a form of well-defined brunches of paramagnon excitations. In contrast to the spin-waves 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 Goldstone-type excitations and the fluctuations in the PM phase.

Our finding of the well-defined high-spectral-density paramagnon excitations in FeSe supports the scenario of the unconventional spin-fluctuations mediated super-conductivity in Fe-based 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 .

References

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