- Email: [email protected]
Ferromagnetic model system with spin-orbit coupling: Dynamical gap and effective spin-flip scattering
Journal of Magnetism and Magnetic Materials 471 (2019) 482–485
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Research articles
Ferromagnetic model system with spin-orbit coupling: Dynamical gap and effective spin-flip scattering Kai Leckron, Hans Christian Schneider
T
⁎
Department of Physics and Research Center OPTIMAS, TU Kaiserslautern, Kaiserslautern, Germany
A R T I C LE I N FO
A B S T R A C T
Keywords: Ultrafast magnetization dynamics Ultrafast demagnetization Ultrafast spin dynamics Dynamic band structure Elliott-Yafet scattering
We investigate ultrafast magnetization dynamics due to electron-phonon interaction in a ferromagnetic Rashba model, which includes spin-orbit coupling and a Stoner mean-field splitting. By computing the reduced spindensity matrix including explicitly spin-independent electron-phonon scattering at the level of Boltzmann-type scattering integrals, we investigate the influence of a time-dependent Stoner mean field on the magnetization dynamics. We find that the dynamical gap increases the magnetization quenching on short timescales and slows down the remagnetization process. We further show how the noncollinear dynamics of electronic spins in the band structure with internal spin orbit fields can be approximated by effective spin-flip scattering matrix elements.
1. Introduction The measurement of ultrafast (sub-ps) demagnetization in ferromagnetic materials after ultrashort laser pulses [1] has led to a new field of research with very promising prospects for applications in data storage. Different models are used to describe these ultrafast magnetization dynamics, one of the more prominent being the phenomenological three temperature model (3TM) that separates the three systems “spin”, “electrons” and “lattice” and couples them with different relaxation times to each other. Extensions to this model (e.g. the μ3TM [2]) or completely different approaches like the Langevin theory with stochastic fields [3], time-dependent density functional theory [4], or hot-electron transport [5] consider additional/different aspects, but there is yet no complete microscopic theory that can explain the observed ultrafast demagnetization and remagnetization behavior from first principles. This is mainly due to the large variety of effects that possibly play a role in these ultrafast dynamics (coherent interactions with the photons of the laser, redistribution of spin and orbital angular momentum, incoherent processes like scattering between electrons, phonons, magnons, etc.), and it is difficult to disentangle the different contributions. Here, we analyze a simplified ferromagnetic model system with spin-orbit coupling. In this way, we can include the band structure of the model, the laser/heat-induced dynamics and the incoherent scattering with phonons in a unified framework, i.e., at the level of a quantum mechanical hamiltonian. We focus on incoherent electron-
⁎
Corresponding author. E-mail address: [email protected] (H.C. Schneider).
https://doi.org/10.1016/j.jmmm.2018.09.105 Received 30 August 2018; Accepted 27 September 2018 Available online 28 September 2018 0304-8853/ © 2018 Elsevier B.V. All rights reserved.
phonon scattering as the mechanism responsible for the incoherent electronic dynamics, as this is widely believed to be one of the most important contributions to demagnetization dynamics. A commonly used picture for magnetization dynamics resulting from electron–phonon scattering is the Elliot-Yafet mechanism, i.e., the change of the ensemble spin via microscopic spin-flip transitions, but as we have shown recently, this description is not always valid and depends on the internal effective fields and the momentum scattering time [6]. In Ref. [6] we also found that because of the large exchange splitting in ferromagnets, the precessional (noncollinear) electronic spin dynamics around internal fields can be approximated by a collinear spin-flip scattering process. The purpose of this paper is twofold. First, we want to further investigate the approximation of the noncollinear electronic dynamics by spin-flip scattering. Second, we want to focus on the effect of a timedependent Stoner mean-field splitting on the electronic spin dynamics. While the electron–phonon interaction is too weak to be responsible for ultrafast demagnetization in a fixed band structure [7,8], it has already been shown that the demagnetization can be accelerated and enhanced in a collinear dynamical calculation [9]. We go beyond this earlier work by including the internal effective magnetic fields and by investigating the effect of the dynamical Stoner gap in such a noncollinear calculation. The paper is organized as follows. In Section 2 we introduce our model system and the dynamical equations for the reduced spin-density matrix. In Section 3 we present the excitation conditions for the
Journal of Magnetism and Magnetic Materials 471 (2019) 482–485
K. Leckron, H.C. Schneider
elevated temperature Te ≫ Teq by adjusting the distribution functions to the higher temperature while maintaining the electron density in each band. This leads to a small change of the ensemble spin due to the different projections of the effective fields on the z-direction for different k, a difference of the chemical potentials between the + / −-bands and to a non-equilibrium between electronic system and phonon bath. Starting from an instantaneous heating of the self-consistently determined equilibrium state, we solve the equations of motion (3). From the time-dependent reduced spin-density matrix, we compute three quantities to characterize the dynamics of the electronic system, the ensemble spin S, the effective temperature T and the chemical potentials μC± for each band. The ensemble spin is given by 1 1 S = 2 〈σ^z 〉 = 2 ∑k, νν′ 〈k, ν σ^z k, ν′〉 ρkνν′. Note that our system is isotropic in the x-y-plane which is why the in-plane components cancel each other out. The effective temperature is defined by the following procedure. We calculate the total energy E of the system and then adjust the effective temperature T of a fictitious Fermi–Dirac equilibrium distribution so that its total energy matches the current energy of the system. The temperature determined in this way is a quantity suitable to describe energy relaxation for short times, where a strong nonequilibrium is present. The chemical potentials are calculated using the current temperature T of the system by fitting equilibrium distributions to the electron densities for each band separately. The chemical potentials are therefore a means to measure the extent of the non-equilibrium via the discrepancy between μC+ and μC−.
magnetization dynamics and we investigate the effects of a time-dependent band structure on the electron spin dynamics in Section 4. Finally, in Section 5 we look at different simplifications/approximations to the dynamical equations that describe the spin dynamics in our model system and study their influence on the dynamics. 2. Model We use a Rashba model with ferromagnetism at the level of a Stoner mean-field that has been discussed in Ref. [6]. It describes itinerant electrons in a thin ferromagnetic film with out-of-plane magnetization and Bychkov-Rashba spin-orbit coupling. It has a two-dimensional kspace and the electronic states are described by a model Hamiltonian that consists of three terms
e = H kin + H Stoner + H Rashba. H
(1)
In the spin |↑, ↓〉-basis this Hamiltonian is given by −iφ k ⎞, e = ⎜⎛ Ekin + Δz − iαke H ⎟ iφ k iαke E − Δ kin z ⎝ ⎠
(2)
where the φ k and k correspond to the angle and the value of k in the xy-plane respectively, α is the Bychkov-Rashba parameter and Δz the splitting due to the ferromagnetic Stoner contribution. Diagonalizing the hamiltonian (2) yields two bands (labeled “ν = +” and “ν = −”), which are separated by the Stoner (exchange) splitting. For more details regarding the different contributions to the Hamiltonian as well as its analytically known eigenstates and -energies refer to e.g. Refs. [6,10]. The important point for the following section is the spin-depen2 dence of the Stoner splitting, i.e., Δz = − 3 Ueff 〈σẑ 〉 where Ueff is the Stoner parameter and the Pauli matrix in z-direction σẑ is directly connected to the dimensionless ensemble spin-expectation value 1 S = 2 〈σ^z〉. We derive the dynamics of the electronic system due to electron–phonon interaction for the reduced spin density matrix ρkνν′ = 〈ck†ν c kν〉 where ck†ν and c kν , respectively, create and annihilate an electron in the single-particle state k, ν , which are the eigenstates of e and where ν ∈ { +,−} enumerates the bands. The equation of motion H (EOM) for the reduced spin density matrix
∂ νν′ ∂ ρ = ρkνν′ ∂t k ∂t
+ coh
∂ νν′ ρ ∂t k
scat
4. Dynamical band structure In Refs. [6,10], we determined the electronic band structure during the self consistent search for the ground state and kept it fixed throughout all calculations to this equilibrium configuration. In Fig. 1(a) we plot the dynamics, i.e., ensemble spin, chemical potentials and effective temperature for the dynamics computed including all terms of (3) and the excitation conditions described in the last section for this fixed band structure. In Fig. 1(b) we plot the same quantities but now for a calculation with a time-dependent Stoner mean-field. This means that we use the ensemble spin S at each time step—modulo 1 —to determine the Stoner2 splitting part Δz of the Hamiltonian (2). This changes the eigenenergies and states and thus the quasiparticle band structure. Because of this change of the quasiparticle energies and states we transform the spin density matrix at each time step to the instantaneous basis during the calculation. There are several similarities and differences regarding these two scenarios. The ensemble spin S reveals the most obvious difference: The demagnetization for the dynamic band structure is faster and more pronounced while the remagnetization takes about one order of magnitude longer. We can explain the stronger and faster demagnetization by an expanding scattering phase space during the beginning of the dynamics. The dominant contribution to the spin dynamics is the interband scattering. Due to the shrinking exchange (Stoner) splitting, hot electrons in the high energy part of the lower (majority electron) band at larger k have more time to scatter to the low energy part of the upper (minority electron) band at smaller k and thus contribute to a reduction of spin polarization and thus demagnetization. In the case of the constant exchange splitting these states would already lie under the bottom of the upper band. The longer remagnetization time can be explained with the same argument: The nearly equilibrated electrons in the upper (minority electron) band have less phase space to scatter into the lower (majority electron) band, since the possible final states are below the chemical potential resulting in a Pauli-blocking. The dynamics of the temperature and thus the energy relaxation are very similar for the case without and with dynamic exchange splitting, so that the cooling effect of the electron–phonon scattering, which occurs dominantly by intraband scattering, is only weakly affected.
(3)
consists of two contributions, the first term describes a coherent precession around k-local effective fields, the second term describes spinconserving electron–phonon scattering terms similar to Boltzmannscattering integrals. For more details on the EOM as well as consequences arising from the spin-conserving nature of the scattering contribution see again Refs. [6,10]. As stated above, we focus here on the dynamic adjustment of the single-particle states throughout the dynamics due to the changing mean-field contribution from the ensemble spin expectation value S in the Stoner splitting Δz and compare them to earlier calculations done for fixed states. 3. Excitation conditions, effective temperatures, etc. Here we describe the initial conditions for the dynamics and introduce quantities of interest that characterize the electronic dynamics. All calculations below use our standard parameters Ueff = 720 meV and α = 30 meV nm , an electron density ne = 1 nm−2 and an equilibrium temperature Teq = 70 K , which is also the temperature of the phonon bath and lies well below the Curie-temperature TC = 514 K. We self-consistently calculate the equilibrium configuration of the system for our standard parameters by fitting the chemical potential μC so that the correct electron density is achieved. The laser excitation is modeled by an instantaneous heating of the electronic system to an 483
Journal of Magnetism and Magnetic Materials 471 (2019) 482–485
K. Leckron, H.C. Schneider
Fig. 1. Dynamics of the ensemble Spin S, the chemical potentials μC± and the temperature T using all terms of (3) for our standard parameters (see text) and an excitation temperature of Te = 1000 K (a) for a fixed band structure and (b) for a band structure that is dynamically adjusted to the current ensemble spin. Equilibrium values are indicated by the gray dotted horizontal lines.
The curves of the chemical potentials may look similar at first glance, but reveal some interesting details. It is important to know that our zero point energy is located in the middle between the two bands at k = 0 , so that it does not change even though the band structure is dynamically adjusted. While μC+ < μC− holds for all times for the fixed exchange splitting, in the case of the dynamic band structure, the chemical potential of the upper (minority-electron) band overtakes μC− at the time of the strongest demagnetization, slightly before 0.5 ps, and approaches the equilibrium curve on the time scale of the remagnetization. This is related to the dynamics of the ensemble spin, since the maximum quenching of the spin is approximately reached at the time when the electron density in the upper band is maximized. The electrons then have little phase space left for interband scattering and equilibrate on a ps-time scale in their respective bands due to intraband scattering. The deviation from the equilibrium-electron-density distribution is reflected in the elevated μC+ for times between 0.5 ps and about 10 ps. These results show that dynamically adjusting the exchange splitting increases the quenching of the magnetization due to microscopic electron–phonon scattering and brings it closer to values observed in experiments on 3d-ferromagnets [11]. In these real materials, other processes may still dominate over the electron–phonon interaction during the demagnetization, but the cooling (and thus the remagnetization) can be mainly attributed to the scattering of electrons and phonons. Thus the behavior of the band structure on longer times is also crucial for an understanding of the spin dynamics involved in demagnetization experiments. Our result for the dynamic splitting is in agreement with earlier investigations on the influence of a dynamic exchange splitting. [9]. However, in our system, there is the additional effect of the precession due to the coherences, which makes the feedback effect between band and magnetization dynamics even more pronounced.
the correct eigenenergies and eigenstates, we use |↑/↓〉-states, so that the influence of the SOC is neglected, i.e., α = 0 , everywhere except for the electron–phonon matrix elements gkν, k′ν′ = Deff k′−k k, ν k′, ν′ , which is proportional to an effective deformation potential constant Deff [12] and can effectively flip the spin. That goes along with using distribution functions nkν ≡ ρkνν instead of the full spin density matrix, since there is never any overlap of ↑ and ↓ states, so that coherences (off-diagonal elements of the spin-density matrix) will not occur. We will refer to this as approximation I. The resulting dynamics are shown as red solid lines in Fig. 2. Since the ensemble spin in equilibrium Seq differs for this case and the full calculation, which is drawn as black lines in Fig. 2 (remember that the k-local effective fields are k-depenS − Seq dent with SOC), we show here the relative change ΔS ≡ S instead of
5. Approximations
n eq−nkν ∂ nkν = kν . τ ∂t
eq
the absolute value of the ensemble spin S as in Fig. 1(a). We observe a negligibly small difference in the temperature dynamics and thus the energy relaxation, as was expected, since the electron–phonon scattering is nearly unaffected by the approximation in the electronic energies in the denominator. The difference for the spin dynamics is bigger, but still not drastic, and mainly stems from the evaluation of the spin expectation value for the different states: The difference between a pure transition |↓〉⟶ |↑〉 is simply higher than that of a (k-dependent) spin-mixed transition k, −〉⟶ k′ , +〉. Note that this approximation uses the same parameters as the full calculation without any fitting procedure. Approximation is similar to Elliott-Yafet scattering as used in Ref. [7]. We now consider a further simplification: We calculate the scattering for pure ↑/↓-bands (this time also for the electron–phonon matrix element), so that—as the scattering is spin conserving and thus no interband scattering occurs—there is only intraband scattering, which only provides cooling. To achieve spin dynamics, we introduce a time constant τ that is used to relax the distribution functions towards a Fermi-distributed equilibrium configuration nkeqν ≡ ρkνν (eq) with the current temperature T of the form
The calculation of the full dynamics including the precession of the electronic spins around internal effective fields is numerically demanding already for model systems and not practical for complicated spin resolved band structures. We therefore introduce three approximations in this section. (I) We replace the spin-mixed bands by pure up-spin (↑) and down-spin (↓) bands. For pure ↑/ ↓ states, we introduce additional simplifications by fitting procedures to approximation (I): (II) a relaxation time, and (III) a k-independent electron–phonon matrix element. We begin with a seemingly drastic approximation. Instead of using
(4)
This relaxation leads to a redistribution of electron density between the ↑- and ↓-bands resulting in a spin change without cooling effects. From the viewpoint of our model this is not a spin-flip time as in the 3-temperature model, but a relaxation time. The relaxation-time approximation cannot completely reproduce the full magnetization dynamics in slope and quenching. We can only achieve a fit with a relaxation time for an effective intraband scattering deformation potential Deff , which differs from the real one. In Fig. 2 we show the ensemble spin and temperature dynamics for the relaxation-time calculation as blue 484
Journal of Magnetism and Magnetic Materials 471 (2019) 482–485
K. Leckron, H.C. Schneider
Fig. 2. Relative change of the ensemble Spin ΔS and the temperature T using all terms of (3) for our standard parameters (see text) and an excitation temperature of Te = 1000 K for the fixed band structure as in Fig. 1 (a) (solid black lines), a fixed band structure with neglected SOC except for the electron–phonon-matrix elements (solid red lines) and ↑/↓-bands with a relaxation time τ (dashed dark blue lines) as well as ↑/↓-bands with a constant spin-flip matrix element (dash-dotted light blue lines). The equilibrium temperature is indicated by a gray dotted horizontal line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Acknowledgement
dashed lines. The effective parameters are a time constant τ = 162.5 fs and a scattering strength Deff of 1.87 times the scattering strength of the full calculation. Thus using the relaxation-time we may obtain a good fit for the magnetization dynamics only. We cannot draw a conclusion as to the value of the deformation potential from such a fit. This is also reflected in the temperature dynamics, which come out too fast, since the scattering strength is nearly doubled. Last, we use pure spin ↑/↓-bands and assume a k-independent electron–phonon matrix element gkν, k′ν′ ≡ Deff b for ν ≠ ν′ instead of calculating it from the states k, ν . The fit procedure yields a Deff that is 0.956 times the scattering strength of the full calculation and an overlap of b = 0.257 . The result of this approximation is shown as light blue dash-dotted curves in Fig. 2. The agreement with the red curve representing the dynamics from approximation I is quite good for both spin and temperature dynamics. We conclude that the k-dependence of the electron–phonon matrix element, at least for weak SOC as in the present case, has a nearly negligible influence on the dynamics and that this approximation seems to be reasonable for efficient computations.
We acknowledge support by the Deutsche Forschungsgemeinschaft (DFG), Germany via SFB/TRR 173 (Spin+X). References [1] E. Beaurepaire, J.C. Merle, A. Daunois, J.-Y. Bigot, Ultrafast spin dynamics in ferromagnetic nickel, Phys. Rev. Lett. 76 (1996) 4250. [2] B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf, M. Fähnle, T. Roth, M. Cinchetti, M. Aeschlimann, Explaining the paradoxical diversity of ultrafast laser-induced demagnetization, Nat. Mater. 9 (2010) 259. [3] S. Wienholdt, D. Hinzke, K. Carva, P.M. Oppeneer, U. Nowak, Orbital-resolved spin model for thermal magnetization switching in rare-earth-based ferrimagnets, Phys. Rev. B 88 (2013) 020406. [4] V. Shokeen, M. Sanchez Piaia, J.-Y. Bigot, T. Müller, P. Elliott, J.K. Dewhurst, S. Sharma, E.K.U. Gross, Spin flips versus spin transport in nonthermal electrons excited by ultrashort optical pulses in transition metals, Phys. Rev. Lett. 119 (2017) 107203. [5] M. Battiato, K. Carva, P.M. Oppeneer, Superdiffusive spin transport as a mechanism of ultrafast demagnetization, Phys. Rev. Lett. 105 (2) (2010) 027203. [6] K. Leckron, S. Vollmar, H.C. Schneider, Ultrafast spin-lattice relaxation in ferromagnets including spin-orbit fields, Phys. Rev. B 96 (14) (2017) 140408. [7] S. Essert, H.C. Schneider, Electron-phonon scattering dynamics in ferromagnetic metals and their influence on ultrafast demagnetization processes, Phys. Rev. B 84 (22) (2011) 224405. [8] K. Carva, M. Battiato, P.M. Oppeneer, Ab-initio investigation of the elliott-yafet electron-phonon mechanism in laser-induced ultrafast demagnetization, Phys. Rev. Lett. 107 (20) (2011) 207201. [9] B.Y. Mueller, A. Baral, S. Vollmar, M. Cinchetti, M. Aeschlimann, H.C. Schneider, B. Rethfeld, Feedback effect during ultrafast demagnetization dynamics in ferromagnets, Phys. Rev. Lett. 111 (2013) 167204. [10] K. Leckron, H.C. Schneider, Spin-dependent electronic scattering dynamics in a ferromagnetic model system: numerical aspects, Proc. SPIE 10638 (2018) 10638–10638 – 12. [11] M. Krauß, T. Roth, S. Alebrand, D. Steil, M. Cinchetti, M. Aeschlimann, H.C. Schneider, Ultrafast demagnetization of ferromagnetic transition metals: the role of the Coulomb interaction, Phys. Rev. B 80 (2009) 180407. [12] A. Baral, S. Vollmar, S. Kaltenborn, H.C. Schneider, Re-examination of the ElliottYafet spin-relaxation mechanism, New J. Phys. 18 (2) (2016) 023012.
6. Conclusion We showed that a dynamically adjusted band gap strongly influences the magnetization in a ferromagnetic model system, for which we calculated the spin dynamics including precessional dynamics and electron–phonon scattering. We found a more pronounced initial quenching in the demagnetization and a slow down of the remagnetization. We then showed three approximations that avoid the complication of the k-dependent precessional dynamics due to spinorbit fields, and work in ↑/↓-bands: One with the correct electron–phonon matrix elements (I), the other with ↑/↓ matrix elements and an additional relaxation time (II) and the last with a constant electron–phonon matrix element (III). We saw that (I) and (III) lead to similar dynamics and that for (II) to reproduce the spin dynamics of (I), the scattering strength has to be nearly doubled.
485
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Research articles
Ferromagnetic model system with spin-orbit coupling: Dynamical gap and effective spin-flip scattering Kai Leckron, Hans Christian Schneider
T
⁎
Department of Physics and Research Center OPTIMAS, TU Kaiserslautern, Kaiserslautern, Germany
A R T I C LE I N FO
A B S T R A C T
Keywords: Ultrafast magnetization dynamics Ultrafast demagnetization Ultrafast spin dynamics Dynamic band structure Elliott-Yafet scattering
We investigate ultrafast magnetization dynamics due to electron-phonon interaction in a ferromagnetic Rashba model, which includes spin-orbit coupling and a Stoner mean-field splitting. By computing the reduced spindensity matrix including explicitly spin-independent electron-phonon scattering at the level of Boltzmann-type scattering integrals, we investigate the influence of a time-dependent Stoner mean field on the magnetization dynamics. We find that the dynamical gap increases the magnetization quenching on short timescales and slows down the remagnetization process. We further show how the noncollinear dynamics of electronic spins in the band structure with internal spin orbit fields can be approximated by effective spin-flip scattering matrix elements.
1. Introduction The measurement of ultrafast (sub-ps) demagnetization in ferromagnetic materials after ultrashort laser pulses [1] has led to a new field of research with very promising prospects for applications in data storage. Different models are used to describe these ultrafast magnetization dynamics, one of the more prominent being the phenomenological three temperature model (3TM) that separates the three systems “spin”, “electrons” and “lattice” and couples them with different relaxation times to each other. Extensions to this model (e.g. the μ3TM [2]) or completely different approaches like the Langevin theory with stochastic fields [3], time-dependent density functional theory [4], or hot-electron transport [5] consider additional/different aspects, but there is yet no complete microscopic theory that can explain the observed ultrafast demagnetization and remagnetization behavior from first principles. This is mainly due to the large variety of effects that possibly play a role in these ultrafast dynamics (coherent interactions with the photons of the laser, redistribution of spin and orbital angular momentum, incoherent processes like scattering between electrons, phonons, magnons, etc.), and it is difficult to disentangle the different contributions. Here, we analyze a simplified ferromagnetic model system with spin-orbit coupling. In this way, we can include the band structure of the model, the laser/heat-induced dynamics and the incoherent scattering with phonons in a unified framework, i.e., at the level of a quantum mechanical hamiltonian. We focus on incoherent electron-
⁎
Corresponding author. E-mail address: [email protected] (H.C. Schneider).
https://doi.org/10.1016/j.jmmm.2018.09.105 Received 30 August 2018; Accepted 27 September 2018 Available online 28 September 2018 0304-8853/ © 2018 Elsevier B.V. All rights reserved.
phonon scattering as the mechanism responsible for the incoherent electronic dynamics, as this is widely believed to be one of the most important contributions to demagnetization dynamics. A commonly used picture for magnetization dynamics resulting from electron–phonon scattering is the Elliot-Yafet mechanism, i.e., the change of the ensemble spin via microscopic spin-flip transitions, but as we have shown recently, this description is not always valid and depends on the internal effective fields and the momentum scattering time [6]. In Ref. [6] we also found that because of the large exchange splitting in ferromagnets, the precessional (noncollinear) electronic spin dynamics around internal fields can be approximated by a collinear spin-flip scattering process. The purpose of this paper is twofold. First, we want to further investigate the approximation of the noncollinear electronic dynamics by spin-flip scattering. Second, we want to focus on the effect of a timedependent Stoner mean-field splitting on the electronic spin dynamics. While the electron–phonon interaction is too weak to be responsible for ultrafast demagnetization in a fixed band structure [7,8], it has already been shown that the demagnetization can be accelerated and enhanced in a collinear dynamical calculation [9]. We go beyond this earlier work by including the internal effective magnetic fields and by investigating the effect of the dynamical Stoner gap in such a noncollinear calculation. The paper is organized as follows. In Section 2 we introduce our model system and the dynamical equations for the reduced spin-density matrix. In Section 3 we present the excitation conditions for the
Journal of Magnetism and Magnetic Materials 471 (2019) 482–485
K. Leckron, H.C. Schneider
elevated temperature Te ≫ Teq by adjusting the distribution functions to the higher temperature while maintaining the electron density in each band. This leads to a small change of the ensemble spin due to the different projections of the effective fields on the z-direction for different k, a difference of the chemical potentials between the + / −-bands and to a non-equilibrium between electronic system and phonon bath. Starting from an instantaneous heating of the self-consistently determined equilibrium state, we solve the equations of motion (3). From the time-dependent reduced spin-density matrix, we compute three quantities to characterize the dynamics of the electronic system, the ensemble spin S, the effective temperature T and the chemical potentials μC± for each band. The ensemble spin is given by 1 1 S = 2 〈σ^z 〉 = 2 ∑k, νν′ 〈k, ν σ^z k, ν′〉 ρkνν′. Note that our system is isotropic in the x-y-plane which is why the in-plane components cancel each other out. The effective temperature is defined by the following procedure. We calculate the total energy E of the system and then adjust the effective temperature T of a fictitious Fermi–Dirac equilibrium distribution so that its total energy matches the current energy of the system. The temperature determined in this way is a quantity suitable to describe energy relaxation for short times, where a strong nonequilibrium is present. The chemical potentials are calculated using the current temperature T of the system by fitting equilibrium distributions to the electron densities for each band separately. The chemical potentials are therefore a means to measure the extent of the non-equilibrium via the discrepancy between μC+ and μC−.
magnetization dynamics and we investigate the effects of a time-dependent band structure on the electron spin dynamics in Section 4. Finally, in Section 5 we look at different simplifications/approximations to the dynamical equations that describe the spin dynamics in our model system and study their influence on the dynamics. 2. Model We use a Rashba model with ferromagnetism at the level of a Stoner mean-field that has been discussed in Ref. [6]. It describes itinerant electrons in a thin ferromagnetic film with out-of-plane magnetization and Bychkov-Rashba spin-orbit coupling. It has a two-dimensional kspace and the electronic states are described by a model Hamiltonian that consists of three terms
e = H kin + H Stoner + H Rashba. H
(1)
In the spin |↑, ↓〉-basis this Hamiltonian is given by −iφ k ⎞, e = ⎜⎛ Ekin + Δz − iαke H ⎟ iφ k iαke E − Δ kin z ⎝ ⎠
(2)
where the φ k and k correspond to the angle and the value of k in the xy-plane respectively, α is the Bychkov-Rashba parameter and Δz the splitting due to the ferromagnetic Stoner contribution. Diagonalizing the hamiltonian (2) yields two bands (labeled “ν = +” and “ν = −”), which are separated by the Stoner (exchange) splitting. For more details regarding the different contributions to the Hamiltonian as well as its analytically known eigenstates and -energies refer to e.g. Refs. [6,10]. The important point for the following section is the spin-depen2 dence of the Stoner splitting, i.e., Δz = − 3 Ueff 〈σẑ 〉 where Ueff is the Stoner parameter and the Pauli matrix in z-direction σẑ is directly connected to the dimensionless ensemble spin-expectation value 1 S = 2 〈σ^z〉. We derive the dynamics of the electronic system due to electron–phonon interaction for the reduced spin density matrix ρkνν′ = 〈ck†ν c kν〉 where ck†ν and c kν , respectively, create and annihilate an electron in the single-particle state k, ν , which are the eigenstates of e and where ν ∈ { +,−} enumerates the bands. The equation of motion H (EOM) for the reduced spin density matrix
∂ νν′ ∂ ρ = ρkνν′ ∂t k ∂t
+ coh
∂ νν′ ρ ∂t k
scat
4. Dynamical band structure In Refs. [6,10], we determined the electronic band structure during the self consistent search for the ground state and kept it fixed throughout all calculations to this equilibrium configuration. In Fig. 1(a) we plot the dynamics, i.e., ensemble spin, chemical potentials and effective temperature for the dynamics computed including all terms of (3) and the excitation conditions described in the last section for this fixed band structure. In Fig. 1(b) we plot the same quantities but now for a calculation with a time-dependent Stoner mean-field. This means that we use the ensemble spin S at each time step—modulo 1 —to determine the Stoner2 splitting part Δz of the Hamiltonian (2). This changes the eigenenergies and states and thus the quasiparticle band structure. Because of this change of the quasiparticle energies and states we transform the spin density matrix at each time step to the instantaneous basis during the calculation. There are several similarities and differences regarding these two scenarios. The ensemble spin S reveals the most obvious difference: The demagnetization for the dynamic band structure is faster and more pronounced while the remagnetization takes about one order of magnitude longer. We can explain the stronger and faster demagnetization by an expanding scattering phase space during the beginning of the dynamics. The dominant contribution to the spin dynamics is the interband scattering. Due to the shrinking exchange (Stoner) splitting, hot electrons in the high energy part of the lower (majority electron) band at larger k have more time to scatter to the low energy part of the upper (minority electron) band at smaller k and thus contribute to a reduction of spin polarization and thus demagnetization. In the case of the constant exchange splitting these states would already lie under the bottom of the upper band. The longer remagnetization time can be explained with the same argument: The nearly equilibrated electrons in the upper (minority electron) band have less phase space to scatter into the lower (majority electron) band, since the possible final states are below the chemical potential resulting in a Pauli-blocking. The dynamics of the temperature and thus the energy relaxation are very similar for the case without and with dynamic exchange splitting, so that the cooling effect of the electron–phonon scattering, which occurs dominantly by intraband scattering, is only weakly affected.
(3)
consists of two contributions, the first term describes a coherent precession around k-local effective fields, the second term describes spinconserving electron–phonon scattering terms similar to Boltzmannscattering integrals. For more details on the EOM as well as consequences arising from the spin-conserving nature of the scattering contribution see again Refs. [6,10]. As stated above, we focus here on the dynamic adjustment of the single-particle states throughout the dynamics due to the changing mean-field contribution from the ensemble spin expectation value S in the Stoner splitting Δz and compare them to earlier calculations done for fixed states. 3. Excitation conditions, effective temperatures, etc. Here we describe the initial conditions for the dynamics and introduce quantities of interest that characterize the electronic dynamics. All calculations below use our standard parameters Ueff = 720 meV and α = 30 meV nm , an electron density ne = 1 nm−2 and an equilibrium temperature Teq = 70 K , which is also the temperature of the phonon bath and lies well below the Curie-temperature TC = 514 K. We self-consistently calculate the equilibrium configuration of the system for our standard parameters by fitting the chemical potential μC so that the correct electron density is achieved. The laser excitation is modeled by an instantaneous heating of the electronic system to an 483
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Fig. 1. Dynamics of the ensemble Spin S, the chemical potentials μC± and the temperature T using all terms of (3) for our standard parameters (see text) and an excitation temperature of Te = 1000 K (a) for a fixed band structure and (b) for a band structure that is dynamically adjusted to the current ensemble spin. Equilibrium values are indicated by the gray dotted horizontal lines.
The curves of the chemical potentials may look similar at first glance, but reveal some interesting details. It is important to know that our zero point energy is located in the middle between the two bands at k = 0 , so that it does not change even though the band structure is dynamically adjusted. While μC+ < μC− holds for all times for the fixed exchange splitting, in the case of the dynamic band structure, the chemical potential of the upper (minority-electron) band overtakes μC− at the time of the strongest demagnetization, slightly before 0.5 ps, and approaches the equilibrium curve on the time scale of the remagnetization. This is related to the dynamics of the ensemble spin, since the maximum quenching of the spin is approximately reached at the time when the electron density in the upper band is maximized. The electrons then have little phase space left for interband scattering and equilibrate on a ps-time scale in their respective bands due to intraband scattering. The deviation from the equilibrium-electron-density distribution is reflected in the elevated μC+ for times between 0.5 ps and about 10 ps. These results show that dynamically adjusting the exchange splitting increases the quenching of the magnetization due to microscopic electron–phonon scattering and brings it closer to values observed in experiments on 3d-ferromagnets [11]. In these real materials, other processes may still dominate over the electron–phonon interaction during the demagnetization, but the cooling (and thus the remagnetization) can be mainly attributed to the scattering of electrons and phonons. Thus the behavior of the band structure on longer times is also crucial for an understanding of the spin dynamics involved in demagnetization experiments. Our result for the dynamic splitting is in agreement with earlier investigations on the influence of a dynamic exchange splitting. [9]. However, in our system, there is the additional effect of the precession due to the coherences, which makes the feedback effect between band and magnetization dynamics even more pronounced.
the correct eigenenergies and eigenstates, we use |↑/↓〉-states, so that the influence of the SOC is neglected, i.e., α = 0 , everywhere except for the electron–phonon matrix elements gkν, k′ν′ = Deff k′−k k, ν k′, ν′ , which is proportional to an effective deformation potential constant Deff [12] and can effectively flip the spin. That goes along with using distribution functions nkν ≡ ρkνν instead of the full spin density matrix, since there is never any overlap of ↑ and ↓ states, so that coherences (off-diagonal elements of the spin-density matrix) will not occur. We will refer to this as approximation I. The resulting dynamics are shown as red solid lines in Fig. 2. Since the ensemble spin in equilibrium Seq differs for this case and the full calculation, which is drawn as black lines in Fig. 2 (remember that the k-local effective fields are k-depenS − Seq dent with SOC), we show here the relative change ΔS ≡ S instead of
5. Approximations
n eq−nkν ∂ nkν = kν . τ ∂t
eq
the absolute value of the ensemble spin S as in Fig. 1(a). We observe a negligibly small difference in the temperature dynamics and thus the energy relaxation, as was expected, since the electron–phonon scattering is nearly unaffected by the approximation in the electronic energies in the denominator. The difference for the spin dynamics is bigger, but still not drastic, and mainly stems from the evaluation of the spin expectation value for the different states: The difference between a pure transition |↓〉⟶ |↑〉 is simply higher than that of a (k-dependent) spin-mixed transition k, −〉⟶ k′ , +〉. Note that this approximation uses the same parameters as the full calculation without any fitting procedure. Approximation is similar to Elliott-Yafet scattering as used in Ref. [7]. We now consider a further simplification: We calculate the scattering for pure ↑/↓-bands (this time also for the electron–phonon matrix element), so that—as the scattering is spin conserving and thus no interband scattering occurs—there is only intraband scattering, which only provides cooling. To achieve spin dynamics, we introduce a time constant τ that is used to relax the distribution functions towards a Fermi-distributed equilibrium configuration nkeqν ≡ ρkνν (eq) with the current temperature T of the form
The calculation of the full dynamics including the precession of the electronic spins around internal effective fields is numerically demanding already for model systems and not practical for complicated spin resolved band structures. We therefore introduce three approximations in this section. (I) We replace the spin-mixed bands by pure up-spin (↑) and down-spin (↓) bands. For pure ↑/ ↓ states, we introduce additional simplifications by fitting procedures to approximation (I): (II) a relaxation time, and (III) a k-independent electron–phonon matrix element. We begin with a seemingly drastic approximation. Instead of using
(4)
This relaxation leads to a redistribution of electron density between the ↑- and ↓-bands resulting in a spin change without cooling effects. From the viewpoint of our model this is not a spin-flip time as in the 3-temperature model, but a relaxation time. The relaxation-time approximation cannot completely reproduce the full magnetization dynamics in slope and quenching. We can only achieve a fit with a relaxation time for an effective intraband scattering deformation potential Deff , which differs from the real one. In Fig. 2 we show the ensemble spin and temperature dynamics for the relaxation-time calculation as blue 484
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Fig. 2. Relative change of the ensemble Spin ΔS and the temperature T using all terms of (3) for our standard parameters (see text) and an excitation temperature of Te = 1000 K for the fixed band structure as in Fig. 1 (a) (solid black lines), a fixed band structure with neglected SOC except for the electron–phonon-matrix elements (solid red lines) and ↑/↓-bands with a relaxation time τ (dashed dark blue lines) as well as ↑/↓-bands with a constant spin-flip matrix element (dash-dotted light blue lines). The equilibrium temperature is indicated by a gray dotted horizontal line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Acknowledgement
dashed lines. The effective parameters are a time constant τ = 162.5 fs and a scattering strength Deff of 1.87 times the scattering strength of the full calculation. Thus using the relaxation-time we may obtain a good fit for the magnetization dynamics only. We cannot draw a conclusion as to the value of the deformation potential from such a fit. This is also reflected in the temperature dynamics, which come out too fast, since the scattering strength is nearly doubled. Last, we use pure spin ↑/↓-bands and assume a k-independent electron–phonon matrix element gkν, k′ν′ ≡ Deff b for ν ≠ ν′ instead of calculating it from the states k, ν . The fit procedure yields a Deff that is 0.956 times the scattering strength of the full calculation and an overlap of b = 0.257 . The result of this approximation is shown as light blue dash-dotted curves in Fig. 2. The agreement with the red curve representing the dynamics from approximation I is quite good for both spin and temperature dynamics. We conclude that the k-dependence of the electron–phonon matrix element, at least for weak SOC as in the present case, has a nearly negligible influence on the dynamics and that this approximation seems to be reasonable for efficient computations.
We acknowledge support by the Deutsche Forschungsgemeinschaft (DFG), Germany via SFB/TRR 173 (Spin+X). References [1] E. Beaurepaire, J.C. Merle, A. Daunois, J.-Y. Bigot, Ultrafast spin dynamics in ferromagnetic nickel, Phys. Rev. Lett. 76 (1996) 4250. [2] B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf, M. Fähnle, T. Roth, M. Cinchetti, M. Aeschlimann, Explaining the paradoxical diversity of ultrafast laser-induced demagnetization, Nat. Mater. 9 (2010) 259. [3] S. Wienholdt, D. Hinzke, K. Carva, P.M. Oppeneer, U. Nowak, Orbital-resolved spin model for thermal magnetization switching in rare-earth-based ferrimagnets, Phys. Rev. B 88 (2013) 020406. [4] V. Shokeen, M. Sanchez Piaia, J.-Y. Bigot, T. Müller, P. Elliott, J.K. Dewhurst, S. Sharma, E.K.U. Gross, Spin flips versus spin transport in nonthermal electrons excited by ultrashort optical pulses in transition metals, Phys. Rev. Lett. 119 (2017) 107203. [5] M. Battiato, K. Carva, P.M. Oppeneer, Superdiffusive spin transport as a mechanism of ultrafast demagnetization, Phys. Rev. Lett. 105 (2) (2010) 027203. [6] K. Leckron, S. Vollmar, H.C. Schneider, Ultrafast spin-lattice relaxation in ferromagnets including spin-orbit fields, Phys. Rev. B 96 (14) (2017) 140408. [7] S. Essert, H.C. Schneider, Electron-phonon scattering dynamics in ferromagnetic metals and their influence on ultrafast demagnetization processes, Phys. Rev. B 84 (22) (2011) 224405. [8] K. Carva, M. Battiato, P.M. Oppeneer, Ab-initio investigation of the elliott-yafet electron-phonon mechanism in laser-induced ultrafast demagnetization, Phys. Rev. Lett. 107 (20) (2011) 207201. [9] B.Y. Mueller, A. Baral, S. Vollmar, M. Cinchetti, M. Aeschlimann, H.C. Schneider, B. Rethfeld, Feedback effect during ultrafast demagnetization dynamics in ferromagnets, Phys. Rev. Lett. 111 (2013) 167204. [10] K. Leckron, H.C. Schneider, Spin-dependent electronic scattering dynamics in a ferromagnetic model system: numerical aspects, Proc. SPIE 10638 (2018) 10638–10638 – 12. [11] M. Krauß, T. Roth, S. Alebrand, D. Steil, M. Cinchetti, M. Aeschlimann, H.C. Schneider, Ultrafast demagnetization of ferromagnetic transition metals: the role of the Coulomb interaction, Phys. Rev. B 80 (2009) 180407. [12] A. Baral, S. Vollmar, S. Kaltenborn, H.C. Schneider, Re-examination of the ElliottYafet spin-relaxation mechanism, New J. Phys. 18 (2) (2016) 023012.
6. Conclusion We showed that a dynamically adjusted band gap strongly influences the magnetization in a ferromagnetic model system, for which we calculated the spin dynamics including precessional dynamics and electron–phonon scattering. We found a more pronounced initial quenching in the demagnetization and a slow down of the remagnetization. We then showed three approximations that avoid the complication of the k-dependent precessional dynamics due to spinorbit fields, and work in ↑/↓-bands: One with the correct electron–phonon matrix elements (I), the other with ↑/↓ matrix elements and an additional relaxation time (II) and the last with a constant electron–phonon matrix element (III). We saw that (I) and (III) lead to similar dynamics and that for (II) to reproduce the spin dynamics of (I), the scattering strength has to be nearly doubled.
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