Ultrafast magnetization dynamics in pure and doped Heusler and inverse Heusler alloys
R. Chimata, E. K. Delczeg-Czirjak, J. Chico, M. Pereiro, B. Sanyal, O. Eriksson, D. Thonig
UUltrafast magnetization dynamics in pure and doped Heusler and inverse Heusleralloys
R. Chimata,
1, 2
E. K. Delczeg-Czirjak, J. Chico, M. Pereiro, B. Sanyal, O. Eriksson,
2, 4 and D. Thonig Argonne National Laboratory, Lemont, IL 60439, United States Department of Physics and Astronomy, Materials Theory, University Uppsala, SE-75120 Uppsala, Sweden Peter Gr¨unberg Institut and Institute for Advanced Simulation,Forschungszentrum J¨ulich & JARA, D-52425 J¨ulich, Germany School of Science and Technology, ¨Orebro University, SE-701 82 ¨Orebro, Sweden (Dated: February 21, 2018)By using a multiscale approach based on first-principles density functional theory combined withatomistic spin dynamics, we investigate the electronic structure and magnetization dynamics of aninverse Heusler and a Heusler compound and their alloys, i. e. Mn − x Z x CoAl and Mn − x Z x VAl,where Z = Mo, W, Os and Ru, respectively. A signature of the ferrimagnetic ordering of Mn CoAland Mn VAl Heusler alloys is reflected in the calculated Heisenberg exchange constants. They decayvery rapidly with the interatomic distance and have short range, which is a consequence of theexistence of the finite gap in the minority spin band. The calculated Gilbert damping parameterof both Mn CoAl and Mn VAl is high compared to other half-metals, but interestingly in theparticular case of the inverse Mn CoAl alloys and due to the spin-gapless semiconducting property,the damping parameters decrease with the doping concentration in clear contradiction to the generaltrend. Atomistic spin dynamics simulations predict ultrafast magnetisation switching in Mn CoAland Mn VAl under the influence of an external magnetic field, starting from a threshold field of2 T. Our overall finding extends with Heusler and inverse Heusler alloys, the class of materials thatexhibits laser induced magnetic switching.
I. INTRODUCTION
The field of the ultrafast magnetization dynamics hasbecome one of the most important topics in magnetism,starting from the pioneering experiment on ferromag-netic nickel from Beaurepaire et al. in 1996. Sincethen, numerous experiments were carried out on 3 d (Fe , Co , Ni ), 4 f (Tb and Gd ) ferromagnets, aswell as on several alloys (GdFeCo , TbCo , CoPt )and half metallic systems (CrO , Co Cr . Fe . Al ,Co FeSi, Co MnGe, Co FeAl , and Co Fe x Mn − x Si and Co MnSi ) aiming to find faster ways of manip-ulating spins in a controllable way, opening a new fieldin the advanced information/data storage and data pro-cessing technologies.Experimental observations revealed that the charac-teristic demagnetization times of 3 d elements are withinthe 100 fs time scale, much faster than that of the 4 f -ferromagnets, which show more complex behavior in-volving a two-step demagnetization process in 10 ps timescale. Surprisingly, recent pump-probe experiments onhalf-metallic Heusler alloys measured distinguished andtypically larger all-optical switching times when com-pared to 3 d -ferromagnets . In these materials, one ofthe spin channels is completely or partially unoccupiedaround the Fermi energy, consecutively the magneto op-tical excitations from one channel to another channel areforbidden.Attempts to understand the momentum transfer be-tween the electrons, spins and phonons after a shortlaser pulse have opened a new debate in the field. Sev-eral quantitative models had been proposed to describethe mechanism of the ultrafast demagnetization such as the microscopic three-temperature model , stochas-tic atomistic descriptions , models using the stochas-tic Landau-Lifshitz-Bloch equation and models sug-gesting the presence of diffusive or superdiffusive spincurrents . The first three models relate the spin-scattering to the Gilbert damping parameter, α , that de-scribes the energy dissipation in a magnetic system viaelementary spin-flip processes . Here, we combine the ab initio description of the magnetic exchange interactionand Gilbert damping parameter with the Landau-Lifshitz-Gilbert equation to investigate the demagneti-zation process in half-metallic ferrimagnetic Heusler andinverse Heusler alloys. a AlVMn a AlCoMn a) b)
FIG. 1. (Color online) Schematic crystal structures of a)the Heusler alloy Mn VAl and b) the inverse Heusler alloyMn CoAl. Different atom types are represented by differentcolours. Solid and dashed lines indicate the bond between theatoms and are added to guide the eye. The lattice constant a is also indicated. a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b Heusler and inverse Heusler alloys are defined asternary intermetallic compounds with a composition of X Y T (cf. Fig. 1). Heusler alloys crystallize in the L2 structure (space group Fm¯3m, 225), with the 4 a (0, 0, 0),4 b ( , , ) and 8 c ( , , ) Wyckoff positions. X and Y are transition metals occupying the 8 c and 4 a positions,respectively, and T is a main group III, IV or V elementsitting in the 4 b position. Inverse Heusler alloys adopt theHg CuTi prototype structure (space group F¯43m, 216),with 4 a (0,0,0), 4 b ( , , ), 4 c ( , , ) and 4 d ( , , ) positions. In this case, X and Y are transition metals, X occupying the 4 a and 4 d positions while Y is the 4 c po-sition. The main element T sits in the 4 b position. Bothstructures may be regarded as a cubic unit cell, whichconsists of four interpenetrating fcc sublattices. Thereare four atoms in the diagonal of the cube following the X - Y - X - T sequence for Heusler alloys and X - X - Y - T forthe inverse Heuslers.Here, we study the demagnetization dynamics of aHeusler and an inverse Heusler compound and their al-loys, i.e. Mn − x Z x VAl and Mn − x Z x CoAl, where Z =Mo, W, Os and Ru. Mn VAl is a well known half-metallicferrimagnetic Heusler compound where the minorityspin channel is the conducting one . Mn CoAl adoptsthe inverse Heusler structure and it is predicted andconfirmed to be a spin gapless magnetic semiconductor.These peculiarities of the band structure are reflected inthe Gilbert damping parameter and affect the magneti-sation dynamics under the influence of a laser pulse, aswill be described below.The paper is divided as follows: In Section II we intro-duce our numerical technique to study materials proper-ties and magnetization dynamics in Heusler alloys. Elec-tronic and magnetic properties of the parent Heusler al-loys Mn CoAl and Mn VAl as well as doping of thesematerials with Os, Ru, W, and Mo is discussed in Sec-tion III A. Demagnetisation studies of these alloys causedby a femtosecond laser are described in Section III E. Fi-nally, the article concludes in Section IV with an outlook.
II. METHODSA. Electronic structure calculation
The electronic and magnetic properties of the studiedmaterials are obtained from first principle calculationsby applying full-relativistic multiple scattering theoryas formulated in the Korringa-Kohn-Rostocker (KKR)approach . This method is implemented in the SPR-KKR package . Solving the Dirac equation, relativis-tic effects are fully accounted for, especially the spin-orbitinteraction which is essential for heavy elements such asthe here considered dopants Os, W, Ru, and Mo. Thepotentials are treated by the atomic sphere approxima-tion (ASA) and obtained by self-consistently solving theKohn-Sham density functional theory (DFT) equationwithin the local density (LDA) or generalized gradient approximation (PBE) as devised by Perdew, Burke andErnzerhof . Note that we applied the PBE functionalif not further specified. The irreducible Brillouin zone issampled by ≈
500 k-points. To describe substitutionaldisorder in the sub-lattices of the alloys we make useof the coherent potential approximation (CPA) . Thespin-polarized scalar relativistic full-potential (SR-FP)mode of the KKR approach is used to calculate the to-tal energies as a function of volume [ E ( V )], which givesan estimate of the lattice constant a . B. Calculation of Heisenberg exchange and Gilbertdamping
The angular momentum transfer in terms of Heisen-berg exchange interactions J ij and energy dissipationrelated to the Gilbert damping parameter α is deter-mined by an ab-initio method with the aim to ad-dress the magnetic ground state and also the dynami-cal properties by using the Landau-Lifshitz-Gilbert equa-tion. The interatomic exchange interactions, J ij , werecalculated via the Liechtenstein-Katsnelson-Antropov-Gubanov (LKAG) formalism J ij = 1 π (cid:90) ε F −∞ Im Tr (cid:16) ∆ i τ ↑ ij ∆ j τ ↓ ji (cid:17) d ε. (1)where ∆ i = t − i, ↑ − t − i, ↓ is the spin-resolved difference of thesingle-site scattering matrix t i at site i and τ ij is the scat-tering path operator, describing the propagation of theelectrons between two sites i and j . The Fermi energyis denoted by ε F . Note that in CPA, the multiple scat-tering matrix is replaced by the scattering properties ofthe effective medium ˆ τ iµ,jν = X iµ τ CP Aij X jν constructedfrom a defect of type µ, ν at site i, j , respectively. Thedefects are taken into account by the defect matrix X iµ .From the calculated exchange interactions, it is possibleto obtain the spin wave stiffness, D , which is expressedas: D = lim η → (cid:88) j e − η | r j | a J j | r ij | (2)by using super cell calculation with random configura-tions of the dopants in 12 ensembles and starting froma reference site i = 0. The distance between site i and j is given by r ij and the parameter η is introduced toguarantee convergence within a Pade interpolation ap-proximation.The Gilbert damping parameter is identified on thebasis of the linear response theory by means of themultiple scattering formalism . The diagonal elements µ = x, y, Z of the Gilbert damping tensor can be writtenas : α µµ = gπm tot (cid:88) j Tr (cid:10) T µ ˜ τ j T µj ˜ τ j (cid:11) c , (3)where the effective g-factor g = 2(1 + m orb /m spin ) andtotal magnetic moment m tot = m spin + m orb are given bythe spin and orbital moments, m spin and m orb , respec-tively, ascribed to a unit cell. Equation (3) gives α µµ forthe atomic cell at lattice site 0 and implies a summationover contributions from all sites indexed by j , including j = 0. Moreover, ˜ τ ij is related to the imaginary part ofthe multiple scattering operator that is evaluated only atthe Fermi energy ε F . Finally, T µi represents the matrixelements of the torque operator ˆ T µ = βσ µ B xc ( r ). Thenotation (cid:104) . . . (cid:105) c represents the configurational average, in-cluding vertex corrections derived by Butler and ac-counting for finite temperature using the alloy analogymodel within CPA . C. Atomistic spin dynamics
The evolution of atomistic spins in a thermal bath isdescribed by the Landau-Lifshitz-Gilbert (LLG) equa-tion , where the dynamics of a magnetic moment isexpressed in terms of precession and damping: d m i ( t ) dt = − γ (1 + α ) (cid:18) m i ( t ) × B i ( t )+ α m i m i ( t ) × ( m i ( t ) × B i ( t )) (cid:19) . (4)Here γ is the gyromagnetic ratio, α represents the di-mensionless Gilbert damping constant, and m i = m i e i is an individual atomic moment on site i . The effectivemagnetic field is given by B i = − ∂ H ∂ m i + b i , where H = − (cid:80) i (cid:54) = j J ij e i · e j and b i is an stochastic field. The latterdescribes white noise ( (cid:104) b i ( t ) · b j ( t (cid:48) ) (cid:105) = 2 Dδ ij δ ( t − t (cid:48) )),where the fluctuation width is D = αk B T s / γm . Thus, thespin temperature T s directly passes into LLG equationvia the stochastic magnetic field b i and is obtained fromsolving the two-temperature (2T) model . The analyti-cal expression of this two temperature model reads, T s = T + (5)( T P − T ) × (1 − exp ( − t/τ initial ) ) × exp ( − t/τ final ) +( T F − T ) × (1 − exp ( − t/τ final ) )where T is the initial temperature of the system, T P is the peak temperature after the laser pulse is appliedand T F is the final temperature. Both the initial andfinal temperature are set to 300 K, where the peak tem-perature is a parameter in the simulations. The time-dependent parameters τ initial and τ final are exponentialparameters, fixed by τ initial = 10 fs and τ final = 20 ps fromRef. [58]. Note that both relaxation times are materialsspecific and k B is the Boltzmann constant. III. RESULTS AND DISCUSSION
This current section is divided in five parts. In the firstand second part we discuss the electronic structure andthe magnetic moments, respectively, of pure and dopedHeusler and inverse Heusler materials based on DFT-optimized lattice constants. The third part deals withthe Heisenberg interaction, spin wave stiffness, as wellas the ordering temperature. The Gilbert damping isdiscussed in part four. The last part focuses on the de-magnetisation and reliable switching in Heusler materialsbased on the LLG equation.
A. Electronic structure calculations
Lattice parameters are estimated from total energy cal-culations, compared to Refs. [38] and [59], and listed inTable I. For undoped Mn CoAl and Mn VAl, we im-proved the theoretically predicted values used in Ref. [59]by 10% and they are closer to the experimentally mea-sured lattice constant. The improvement comes from tak-ing into account the full-potential, which is known to im-prove lattice constants . By doping Mn with 4d and5d metals Mo, Ru, W, and Os, we observe an expectedincrease of the lattice constant with the concentrationof the dopants, since the atomic radius of the dopant islarger than the one of Mn. For Mn VAl, the increaseof the lattice constant is substantially bigger ( ≈
1% for x = 1% doping) than for Mn CoAl ( ≈ .
1% for x = 1%doping).Thermal switching within our classical atomistic modelis completely determined by the Heisenberg exchangeand the Gilbert damping of the system , which are inturn identified by the scattering-path matrices and thesingle-site scattering matrices of the Kohn-Sham prob-lem in Eqs. (1) and (3). Hence, we first have to ad-dress the electronic structure by means of the density ofstates (DOS; Fig. 2). The here studied inverse HeuslerMn CoAl is known to be a spin gapless semiconductor,where an almost zero-width energy gap at the Fermi levelexists in the majority-spin channel (the majority statesare plotted with positive values and the minority spinstates with negative values) but a regular energy gap oc-curs in the minority spin-channel (see inset in the bottompanel of Fig. 2). This was already reported, for example,in Ref. [59]. The density of states and, consequently,the gap are sensitive to the applied exchange correlationfunctional. Using local density approximation (LDA),states are shifted up in energy (not shown here) com-pared to the PBE by about 10 meV and, consequently,no gap at the Fermi energy is observed. Note that theoffset of the energy from the real axis in Fig. 2 (thespectral width of the electron bands) is small and about1 meV, which causes sharp features in the DOS. A finitespectral width also gives rise to an overlap of the statesaround the Fermi level and ‘hide’ the zero-width energygap; a finite density of states at ε F is observed. Bands -120-80-4004080120 D e n s i t y o f s t a t e s ( R y d ) -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 E − E F (Ryd) -120-80-4004080120-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1-120-80-4004080120-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1-120-80-4004080120-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1-120-80-4004080120-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 Mn CoAl D O S -0.02 0.0 0.02 E − E F (Ryd) -0.02 0.0 0.02-0.02 0.0 0.02-0.02 0.0 0.02-0.02 0.0 0.02 -80-4004080 D e n s i t y o f s t a t e s ( R y d ) -80-4004080-80-4004080-80-4004080-80-4004080 WRuMoOs Mn VAl D O S E − E F (Ryd) FIG. 2. (Color online) Density of states (DOS) for Mn CoAl(lower panel) and Mn VAl (upper panel) without doping(graybackground) and with doping of W (red lines), Ru (blue lines),Mo (green lines), and Os (orange lines). Positive (negative)DOS values correspond to the majority-(minority-) spin elec-trons and are indicated by bold up-(down-) arrays. The insetis a magnification of the DOS around the Fermi level. that cross the Fermi level, are mainly allocated to Mn and Co (band structure is not shown here, but it can befound elsewhere ). Note that the superscripts 1 and 2between the two Mn atoms. In contrast to Ref. [59], theFermi energy is not located at the centre of the minorityband gap, which will affect the coupling between the col-lective and single-electron excitations, i.e. the exchangeinteractions.The chemical compound Mn VAl, however, is half-metallic (cf. Fig. 2) with a gap in the majority spin-channel. The width of the majority band gap (0 . CoAl (0 . − x Z x CoAl and Mn − x Z x VAl is destroyedby replacing some of the Mn atoms with heavy metals, Z = Mo, W, Os, Ru of a given concentration x = 0 .
05 and0 .
1. Comparing total energies (not shown here) allows usto conclude that for the inverse Heusler Mn − x Z x CoAldoping at both Mn-sites (Mn -Mn ) has the lowest en-ergy. We obtained a maximal energy difference of ∆ E ≈
40 meV when doping at Mn - Y , Mn - Y , or Y with 1%of the dopants W, Ru, Mo, Os. There is no major vari-ation found in ∆ E between the different dopands. Notethat we used here the same lattice constant as shown inTable I, but in principle it will vary when doping at Mn -Mn , Mn - Y , Mn - Y , or Y . However, Mn − x Z x VAl hasthe lowest energy when doping only the V atom, but totreat both material on the same footing, we consider alsoMn − x Z x VAl to be doped at the Mn -Mn atoms.In the case of Mn CoAl, W and Mo generate statesat the spin-gap majority states at the Fermi level, whereon the other hand the gap in the minority spin chan-nel survives. In terms of a rigid band model, W- andMo-doping decreases the Fermi energy, which relocatesthe DOS to higher energies. The dopants Os and Ruhave one electron more than Mn in the valence bandand, consequently, affect the density of states in the op-posite way: Minority states are added and become occu-pied. The Fermi energy increases, which shifts the den-sity of states to smaller energies. For Mn VAl, dopingwith Ru and Os preserves the half-metallic behaviour; itadd states below the Fermi energy and typically at theenergy ε = − .
025 Ryd. Doping with Mo and W reducesthe width of the band-gap and shifts it above the Fermienergy. Related to the alloying, the density of statessmears out in the whole energy range.
B. Magnetic moments
The exchange splitting in the DOS and, consequently,the total magnetic moment is affected by doping (seeFig. 3). Both Heusler materials are ferrimagnetic.An antiferromagnetic coupling between the Mn atomswas observed for the inverse Heusler alloy Mn CoAl(cf. Table I), caused by the inequivalence of the twoMn atoms. These results are in good agreement withexperiments and existing theoretical predictions .According to the Bethe-Slater curve , transition-metalatoms such as Mn tend to have an antiferromagneticspin moment when they are close to each other. InMn − x Z x VAl, the Mn atoms are equivalent and, thus,have the same magnetic moment that couple ferromag-netically. The V atom, however, is antiferromagneticwith respect to the Mn atoms and has a strong inducedmagnetic moment of 0 . µ B . Opposite to the total mag-netic moment, the size of the element resolved magneticmoments is sensitive to the lattice constant of the system m ( µ B ) x WRuMoOsMn − x Z x VAlMn − x Z x CoAl
FIG. 3. (Color online) Total magnetic moments ofMn − x Z x CoAl (triangles) and Mn − x Z x VAl (circles) as afunction of dopant concentration x . The symbol Z representsMo (green lines and symboles), Os (orange lines and symbols),Ru (blue lines and symbols), and W (red lines and symbols). and moments can vary up to 13 %, which was also foundin Ref. 59.The size but not the sign of the elemental magneticmoments changes by doping the Heusler materials with4d and 5d heavy metals, and, thus, also the total mag-netic moment. Typically, the induced magnetic momentsof dopants are parallel to the magnetic moment of Mnatoms and they become larger if the magnetic momentof the Mn atom is smaller. In the case of Mn CoAl, thedopants W, Ru, Mo, and Os cause a decay of the to-tal magnetic moment of about 0 . − . µ B for x = 1%,while in the case of Mn VAl, only the dopants Ru andMo decrease the magnetic moment. This is caused by asignificant change of the Mn magnetic moments of about∆ m ≈ . − . µ B , but also for Co atoms the momentvariation is about ∆ m ≈ . µ B . C. Heisenberg exchange parameter and Curietemperatures
Based on our electronic structure analysis in the Sec-tion III B, we calculated the Heisenberg exchange param-eter J ij (see Fig. 4). The already revealed ferrimag-netic behaviour is reflected also in the exchange con-stants J . The magnetic exchange parameters decay veryrapidly with the interatomic distance, r ij , which is as-cribed to the existence of the finite spin gap in theminority-channel . Our results for Mn CoAl are sim-ilar to the ones already reported in Refs. [62,59]. Notethe factor of 2 in Ref. [59] may be caused by a differentdouble-counting convention of the Heisenberg Hamilto-nian. For the compound Mn CoAl, the antiferromagneticinteraction between Mn and Mn dominates the ferri-magnetism, whereas the Mn -Co interatomic exchangeinteraction is ferromagnetic. In Mn VAl, the situation isthe opposite: the Mn to V interaction is dominating andantiferromagnetic, where only the Mn -Mn contributes -2.5-2.0-1.5-1.0-0.50.00.51.01.5 J i j ( m R y d ) r ij · a −
10 Mn -Mn Mn -Mn Mn -CoMn -Mn Mn -Co -0.8-0.6-0.4-0.20.00.20.4 J i j ( m R y d ) r ij · a −
10 Mn -Mn Mn -Mn Mn -VMn -Mn Mn -V FIG. 4. (Color online) Intersublattice Heisenberg exchangeparameter as a function of renormalized interatomic distancefor a) Mn VAl and b) Mn CoAl. Different colours representsthe coupling between Mn -Mn (red dotes), Mn -Mn (bluedotes), Mn -Co or Mn -V (green dotes), Mn -Mn (orangedotes) and Mn -Co or Mn -V (cyan dotes). with a ferromagnetic coupling but with half the strengthof the Mn-V interaction. The coupling between equiv-alent Mn atoms in Mn VAl (Mn -Mn and Mn -Mn )is small and negligible. The calculated interactions de-pend to some extent on the details of the calculations.In particular, the J Mn-Co and J Mn-V interactions dependstrongly on the applied exchange-correlation functional,but also on the lattice constant of the system. Notice thatfor J Mn-Co and J Mn-V in LDA we obtain twice the size ofthe J ’s from PBE (not shown here). The other couplings(e.g. J Mn-Al , J Co-Al , J V-Al ) turned out to be negligible,primarily caused by a vanishing magnetic moment on theAl atom.As shown in Fig. 5, doping with 4d and 5d elementsreduces nearest-neighbour interactions and the correla-tion length between magnetic moments, which is a directconsequence of the disorder and the coherent potentialapproximation . Nearest neighbour interactions are af-fected mostly by the doping. In general, the exchange Compound a (˚A) m [ Mn ] m [ Z ] m [ Mn ] m [ Z ] m [ Y ] Mn CoAl 5 .
73 [ 59] − .
64 2 .
77 0 . . W . CoAl − . − .
52 2 .
75 0 .
26 0 . . Ru . CoAl − . − .
10 2 .
76 0 .
06 0 . . Mo . CoAl − . − .
56 2 .
75 0 .
33 0 . . Os . CoAl − . − .
12 2 .
76 0 .
18 0 . VAl 5 .
69 [ 38] 1 .
32 1 . − . . W . VAl 1 .
32 0 .
19 1 .
32 0 . − . . Ru . VAl 1 .
31 0 .
08 1 .
31 0 . − . . Mo . VAl 1 .
36 0 .
25 1 .
36 0 . − . . Os . VAl 1 .
31 0 .
09 1 .
31 0 . − . CoAl 5 .
79 [5 .
84 exp] − .
81 2 .
91 0 . . W . CoAl 5 . − . − .
46 2 .
62 0 .
22 0 . . Ru . CoAl 5 . − . − .
10 2 .
90 0 .
07 0 . . Mo . CoAl 5 . − . − .
60 2 .
89 0 .
39 0 . . Os . CoAl 5 . − . − .
12 2 .
92 0 .
19 0 . VAl 5 .
84 [5 .
88 exp] 1 .
47 1 . − . . W . VAl 5 .
92 1 .
73 0 .
37 1 .
73 0 . − . . Ru . VAl 5 .
86 1 .
50 0 .
05 1 .
50 0 . − . . Mo . VAl 5 .
91 1 .
72 0 .
45 1 .
72 0 . − . . Os . VAl 5 .
92 1 .
52 0 .
06 1 .
52 0 . − . µ B ) of the host Mn CoAl and Mn VAl . The upper panelshows results for a fixed lattice constant obtained from literature, where the lower panel is for lattice constants calculated fromtotal energy minimization. The superscripts 1 and 2 distinguish between the two Mn atoms. The symbol Y represents eitherCo or V. The magnetic moment of Al is negligibly small. couplings diminish with doping concentration x up to0 . x = 0 .
1. For Os and Ru doping,there is a slight increase of the exchange coupling (about0 .
03 mRyd).With knowledge about the trends in the exchange cou-plings { J } , one can estimate the spin-wave stiffness D and the phase transition temperature from both meanfield theory via k B T MFC = / (cid:80) j J j or from MonteCarlo simulations. The results are shown in Fig. 6. Thespin-wave stiffness (upper panel in Fig. 6) for Mn − x CoAlis in good agreement with Ref. [59], while for Mn − x VAlwe reproduce the spin wave stiffness constant D alreadyreported in Ref. [66] ( D = 324 meV˚A ), but not the ex-perimentally measured stiffness ( D = 534 meV˚A ).For the Co based Heusler compounds we obtain a hard-ening of the spin-waves after an initial softening, wherefor the V based Heusler compound, only hardening ofthe spin-waves with doping is observed. The phase tran-sition temperature T C , which turns out to be inverselyproportional to D , decreases with doping concentration x for two reasons, namely: i) reduction of the magneticmoment due to doping and, consequently, stronger fluc- tuations at a given temperature as well as ii) reduction ofcorrelation. The critical temperature T C is obtained fromMonte Carlo simulations on the Metropolis algorithm ,from Binder’s fourth cumulant for different simulatedsystem sizes but also from the spin susceptibility χ . Notethat the first method could fail for antiferro- and ferri-magnets. Thus, we obtain a systematic error of about ± CoAl and 475 K for Mn VAl) underestimate thetransition temperature observed from experiment (720 Kfor Mn CoAl and 768 K for Mn VAl). This discrep-ancy that is most notable for Mn − x VAl was reportedearlier and could have multiple reasons. First, magneticproperties in Heusler alloys are sensitive to the intersti-tial region spanned by the muffin tin potential. Thus,full-potential simulations are required as it was shown inRefs. [41, 42, and 69]. Also the results depend crucially onthe choice of the exchange-correlation functional and onelectron correlations e.g. addressed by including a Hub-bard U . Second, the Heisenberg exchange is calculatedfor a collinear ferrimagnetic state but when the magnetic -2.0-1.5-1.0-0.50.00.51.0 J i j ( m R y d ) r ij · a −
10 W Mn -Mn Mn -Mn Mn -CoMn -Mn Mn -Co r ij · a −
10 Ru -2.0-1.5-1.0-0.50.00.51.0 J i j ( m R y d ) Mo Os -1.0-0.8-0.6-0.4-0.20.00.2 J i j ( m R y d ) r ij · a −
10 W Mn -Mn ,Mn -Mn Mn -Mn Mn -V,Mn -VMn -X Mn -X r ij · a −
10 Ru -1.0-0.8-0.6-0.4-0.20.00.20.4 J i j ( m R y d ) Mo Os
FIG. 5. (Color online) Intersublattice Heisenberg exchangeparameter as a function of renormalized interatomic distancefor a) Mn − x Z x VAl and b) Mn − x Z x CoAl, where the differ-ent subpanels show the dopants W (bottom left), Ru (bottomright), Mo (top left), and Os (top right). Different coloursrepresents the coupling between Mn -Mn (red dotes), Mn -Mn (blue dotes), Mn -Co or Mn -V (green dotes), Mn -Mn (orange dotes) and Mn -Co or Mn -V (cyan dotes). disorder is taken into account in the electronic structure,usually the exchange interaction is biased . Based onthe alloy analogy model , we modelled also the temper-ature stability of the magnetic properties (magnetic mo-ments and magnetic exchange) coming from electronicstructure by the partial disordered local moment (DLM) T M F C ( K ) x WRuMoOs Mn − x X x VAlMn − x X x CoAl T C ( K ) D ( m e V˚A ) FIG. 6. (Color online) Spin wave stiffness D , critical tem-peratures T C , and mean field critical temperatures T MFC ofMn − x Z x CoAl (triangles) and Mn − x Z x VAl (circles) as afunction of dopand concentration x . Dopands are Mn (blackcircles), Os (red squares), Ru (green diamonds), and W (or-ange triangles). approximation within the Ising model . DLM approachis believed to accurately describe ‘spin temperature’ inthe electronic structure . However, it turned out thatthe disordered local moment theory can not be appliedto both Heusler and inverse Heusler for similar reasons asfor Ni : the magnetic moments in Al and Co/V disap-pear. For Mn CoAl, our simulations show furthermorethat the magnetic moment of the Mn atom is zero inthe paramagnetic phase and, consequently, the magneticexchange and the phase transition temperature are zero.This result is independent of the doping with 4d and 5delements. These results indicate the inconsistency of theDLM model for Heusler materials. It is still an openquestion, if results get improved by applying relativis-tic DLM theory . Third, we consider only a simplifiedapproach for electron correlation in the LDA and GGAdensity functional. However, it is known that improvedmodels for electron correlation have the trend to increaseslightly the phase transition temperature. D. Gilbert damping
Previous studies have shown that Gilbert damping isa crucial parameter in the ultrafast switching procedureand, thus, call for ab-initio footing. Figure 7 shows theGilbert damping α as a function x at T = 300 K. Notethat for these calculations both lattice and magnetic fluc-tuations terms are considered, where the magnetic fluctu-ations are assumed from a linear correlation between themagnetization and the temperature. This could result inerrors, in particular at high temperatures. α · − n ( E F )( e V ) x Mn − x X x CoAl α · − n ( E F )( e V ) Mn − x X x VAl
FIG. 7. (Color online) Gilbert damping parameters α (solid lines) and density of states at the Fermi level n ( E F )(dotted lines) of Mn − x Z x CoAl (triangular symboles) andMn − x Z x VAl (circle symboles) vs dopand concentration x .Dopands are W (red color), Ru (blue color), Mo (green color),and Os (orange color). The Gilbert damping of both undoped Heusler materi-als (Mn CoAl: α = 0 . VAl: α = 0 . . Co . . The trends ofthe Gilbert damping parameters with dopant concentra-tion are different for Heusler and inverse Heusler materi-als. In Mn − x Z x VAl, doping leads to an increase of thedamping with x , except for the case of Ru. The slopeof α versus concentration x follows the general increaseof the total density of states at the Fermi level as it isproposed in Refs. [33, 73, and 74], but not linear to it.This non-linearity was already observed for Heusler ma-terials in Ref. [66] or doped permalloy with the heavy4d and 5d elements used here . The observed damping α is different from zero, however, small. This is in linewith the theory proposed in Ref. [74], in which dampingis proportional to the product of the spin-polarised DOSand, consequently α ≈
0. The increase of damping can bealso understood in terms of the Kambersk´y model :Alloying broadens the electron bands and more spin-fliptransitions between the electron states occur. This istrue only, if interband transitions are already dominat-ing. In the inverse Heusler material Mn CoAl we evenfind a decrease with x . This is due to the spin-gapless semiconducting behaviour (cf. Fig. 2): Only a low num-ber of states exist at the Fermi energy, making interbandtransitions unlikely. The damping is dominated by intra-band transitions, that tend to decrease with very small x . With increasing x , however, states appear within thegap and interband transition are preferred. Thus, a smallincrease with even higher concentration is expected andobserved. However, not only the number of states at theFermi energy and the spectral width of the states con-tribute to the damping, but also the spin-orbit coupling(SOC), the Land´e factor, and the saturation magnetiza-tion affect the damping parameter. Since we dope withrather heavy elements W, Mo, Ru, and Os, spin orbitcoupling strongly contributes to the variation of damp-ing with concentration x : the higher the ‘mass’ of thedopant atom (W and Os compared to Ru and Mo) is,the higher is the damping parameter.After we addressed all relevant parameters for the sim-ulation based on the Landau-Lifshitz-Gilbert equation,we are able to perform ultrafast switching calculations. E. Ultrafast switching -1.0-0.50.00.51.0 M / M s t (ps) VMn Mn -1.0-0.50.00.51.0 M / M s CoMn Mn FIG. 8. (Color online) Ultrafast switching behaviour ofMn CoAl (upper panel) and Mn VAl (lower panel). The de-magnetization is shown element resolved (blue and green lines- Mn atoms, red line - Co/ V atom). The peak temperature is600 K for Mn VAl and 900 K for Mn CoAl. The external mag-netic field is B = 2 . α = 0 . T P ( K ) B (T) α = 0 . B (T) α = 0 .
009 0.02.04.06.08.0 t (ps) T P ( K ) B (T) α = 0 . B (T) α = 0 .
009 0.02.04.06.08.0 t (ps) b)a) FIG. 9. (Color online) Thermal switching phase diagram fordifferent damping parameter (0.006 and 0.009) in a) Mn CoAland b) Mn VAl. The peak temperature is represented versusthe strength of the external magnetic field. The colour scale(fast switching - blue colour, slow switching - red colour) rep-resents the time in units of ps where the switching (indicatedby an arrow in Fig. (8)) takes place. No switching is repre-sented by the black background.
In order to study the ultrafast switching process inHeusler alloys we combined the two temperature modelwith an atomistic spin dynamics code . Here, we consid-ered a very long thermal pulse of 20 ps with different peaktemperatures T P . Typical timescales of the ultrafast de-magnetization and remagnetization process for Mn VAland Mn CoAl are in the orders of picoseconds (1 − α , which is varied in our studies between0 . . . . As demonstrated above, these dampingvalues are achievable by doping the ‘pure’ Heusler mate-rials. There is only a slight shift observable in the demag-netization time of each individual element in Mn CoAl,where for Mn VAl, it is not. After demagnetization, theHeusler material undergoes reliable switching only whenan external magnetic field induced by the pump-pulse ispresent. Thus, three parameters — damping, peak tem-perature and pulse induced external magnetic field —span a phase space for observing reliable switching, asshown in Fig. 9.We did not observe any magnetic switching for bothHeusler materials with α = 0 .
003 (data not shown here).Typically for certain threshold peak temperatures T P above the magnetic phase transition temperature ( T C =700 K for Mn CoAl and T C = 475 K for Mn VAl) switch-ing occurs. The peak temperature can be tuned by thelaser intensity and the pulse duration. The presence ofan effective magnetic field during pumping is discussed in literature . It was argued that the electric field ofthe pump pulse induces a strong material specific mag-netic field of 10 −
100 T. Even below but above certainminimum magnetic field of 1 − − CoAl and Mn VAl oc-curs.Nevertheless, our approach has certain limitations. Forinstance, we explicitly neglect the electronic motion andeffects like super diffusion or spin-flip scattering, as dis-cussed in Ref. . We also assume the damping to be ‘spin-and phonon-temperature’ independent. This is a roughapproximation, in particular, due to the important role ofphonons in the demagnetization process (e.g. Ref. [81])and for energy dissipation in magnetic systems . Fur-thermore, we neglect the change of the magnetic ex-change interaction with temperature, although magneticmoments of Co and V atoms vanish in the DLM approx-imation. This behaviour in the disordered local momenttheory is well studied and occurs also for Ni atoms. Butwe have shown elsewhere but also others , that ourmethodology is applicable for demagnetization in bulkbcc Fe and hcp Co compounds and, likely, for the Heuslermaterials studied here. We also neglect possible struc-tural phase transition to A2 or B IV. CONCLUSION
We have demonstrated thermal switching in Heuslerand inverse Heusler materials making use of magneticfield pulse induced by the pump-pulse. We found a sensi-tive dependence of the possible switching and the switch-ing time on the magnetic field pulse strength, the peaktemperature in the effective two-temperature model aswell as intrinsic materials properties, say the Heisenbergexchange and the Gilbert damping parameter. We haveshown that the latter can be tuned by doping heavy ele-ments, say W, Mo, Ru, Os, to both, higher and lowerdamping values, especially in the case of spin-gaplesssemiconductor. This calls for further investigations onother spin-gapless semiconductor , aiming for tuning theGilbert damping to very low values, which may enable in-teresting spintronic and magnonic applications . Withinour methodology, we could reproduce exchange parame-ter and, consequently, phase transition temperatures re-ported in literature . Our overall finding extends withHeusler and inverse Heusler alloys the class of materials0that exhibits laser induced magnetic switching and callsfor future theoretical and experimental studies. V. ACKNOWLEDGEMENT
We acknowledge financial support from the SwedishResearch Council. O.E. and and E.K.D.-Cz. ac-knowledged support from KAW (projects 2013.0020and 2012.0031) as well as acknowledges eSSENCE andSTandUP. The calculations were performed at NSC(Link¨oping University, Sweden) under a SNAC project. E. Beaurepaire, J. C. Merle, A. Daunois, and J. Y. Bigot,Phys. Rev. Lett. , 4250 (1996), URL https://link.aps.org/doi/10.1103/PhysRevLett.76.4250 . E. Carpene, E. Mancini, C. Dallera, M. Brenna, E. Pup-pin, and S. De Silvestri, Phys. Rev. B , 174422 (2008),URL https://link.aps.org/doi/10.1103/PhysRevB.78.174422 . M. Cinchetti, M. S. Albaneda, D. Hoffmann, T. Roth,J. P. W¨ustenberg, M. Krauß, O. Andreyev, H. C. Schnei-der, M. Bauer, and M. Aeschlimann, Phys. Rev. Lett. , 177201 (2006), URL https://link.aps.org/doi/10.1103/PhysRevLett.97.177201 . C. Stamm, T. Kachel, N. Pontius, R. Mitzner, T. Quast,K. Holldack, S. Khan, C. Lupulescu, E. F. Aziz, M. Wi-etstruk, et al., Nature Materials , 740 (2007), URL http://adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2007NatMa...6..740S&link_type=EJOURNAL . H. S. Rhie, H. A. D¨urr, and W. Eberhardt, Phys. Rev. Lett. , 247201 (2003), URL https://link.aps.org/doi/10.1103/PhysRevLett.90.247201 . M. Wietstruk, A. Melnikov, C. Stamm, T. Kachel, N. Pon-tius, M. Sultan, C. Gahl, M. Weinelt, H. A. D¨urr, andU. Bovensiepen, Phys. Rev. Lett. , 127401 (2011),URL https://link.aps.org/doi/10.1103/PhysRevLett.106.127401 . C. D. Stanciu, A. V. Kimel, F. Hansteen, A. Tsukamoto,A. Itoh, A. Kirilyuk, and T. Rasing, Phys. Rev. B , 220402 (2006), URL https://link.aps.org/doi/10.1103/PhysRevB.73.220402 . K. Vahaplar, A. M. Kalashnikova, A. V. Kimel, D. Hinzke,U. Nowak, R. Chantrell, A. Tsukamoto, A. Itoh, A. Kiri-lyuk, and T. Rasing, Phys. Rev. Lett. , 117201 (2009),URL https://link.aps.org/doi/10.1103/PhysRevLett.103.117201 . C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk,A. Tsukamoto, A. Itoh, and T. Rasing, Phys. Rev. Lett. , 047601 (2007), URL https://link.aps.org/doi/10.1103/PhysRevLett.99.047601 . D. Steil, S. Alebrand, A. Hassdenteufel, M. Cinchetti,and M. Aeschlimann, Phys. Rev. B , 224408 (2011),URL https://link.aps.org/doi/10.1103/PhysRevB.84.224408 . T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell,U. Atxitia, O. Chubykalo-Fesenko, S. El Moussaoui,L. Le Guyader, E. Mengotti, L. J. Heyderman, et al., NatComms , 666 (2012), URL . J. H. Mentink, J. Hellsvik, D. V. Afanasiev, B. A. Ivanov,A. Kirilyuk, A. V. Kimel, O. Eriksson, M. I. Katsnel-son, and T. Rasing, Phys. Rev. Lett. , 057202 (2012),URL https://link.aps.org/doi/10.1103/PhysRevLett. 108.057202 . R. Chimata, L. Isaeva, K. K´adas, A. Bergman, B. Sanyal,J. H. Mentink, M. I. Katsnelson, T. Rasing, A. Kiri-lyuk, A. Kimel, et al., Phys. Rev. B , 094411 (2015),URL https://link.aps.org/doi/10.1103/PhysRevB.92.094411 . S. Alebrand, U. Bierbrauer, M. Hehn, M. Gottwald,O. Schmitt, D. Steil, E. E. Fullerton, S. Mangin,M. Cinchetti, and M. Aeschlimann, Phys. Rev. B , 144404 (2014), URL https://link.aps.org/doi/10.1103/PhysRevB.89.144404 . J.-Y. Bigot, M. Vomir, and E. Beaurepaire, Nat Phys , 515 (2009), URL . G. M. M¨uller, J. Walowski, M. Djordjevic, G.-X. Miao,A. Gupta, A. V. Ramos, K. Gehrke, V. Moshnyaga,K. Samwer, J. Schmalhorst, et al., Nature Publish-ing Group , 56 (2008), URL . J. P. W¨ustenberg, D. Steil, S. Alebrand, T. Roth,M. Aeschlimann, and M. Cinchetti, Phys. Status SolidiB , 2330 (2011), URL http://onlinelibrary.wiley.com/doi/10.1002/pssb.201147087/full . A. Mann, J. Walowski, M. M¨unzenberg, S. Maat, M. J.Carey, J. R. Childress, C. Mewes, D. Ebke, V. Drewello,G. Reiss, et al., Phys. Rev. X , 041008 (2012),URL https://link.aps.org/doi/10.1103/PhysRevX.2.041008 . D. Steil, S. Alebrand, T. Roth, M. Krauß, T. Kubota,M. Oogane, Y. Ando, H. C. Schneider, M. Aeschlimann,and M. Cinchetti, Phys. Rev. Lett. , 217202 (2010),URL https://link.aps.org/doi/10.1103/PhysRevLett.105.217202 . Y. Liu, L. R. Shelford, V. V. Kruglyak, R. J. Hicken,Y. Sakuraba, M. Oogane, and Y. Ando, Phys. Rev. B , 094402 (2010), URL https://link.aps.org/doi/10.1103/PhysRevB.81.094402 . B. Koopmans, G. Malinowski, F. D. Longa, D. Steiauf,M. F¨ahnle, T. Roth, M. Cinchetti, and M. Aeschlimann,Nature Publishing Group , 259 (2009), URL . I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius,H. A. D¨urr, T. A. Ostler, J. Barker, R. F. L. Evans, R. W.Chantrell, et al., Nature , 205 (2011), URL . N. Kazantseva, U. Nowak, R. W. Chantrell,J. Hohlfeld, and A. Rebei, EPL , 27004 (2008),URL http://iopscience.iop.org/article/10.1209/0295-5075/81/27004 . U. Atxitia, O. Chubykalo-Fesenko, N. Kazantseva,D. Hinzke, U. Nowak, and R. W. Chantrell, Applied Physics Letters , 232507 (2007), URL http://aip.scitation.org/doi/10.1063/1.2822807 . M. Battiato, K. Carva, and P. M. Oppeneer, Phys. Rev.Lett. , 027203 (2010), URL https://link.aps.org/doi/10.1103/PhysRevLett.105.027203 . A. Melnikov, I. Razdolski, T. O. Wehling, E. T. Pa-paioannou, V. Roddatis, P. Fumagalli, O. Aktsipetrov,A. I. Lichtenstein, and U. Bovensiepen, Phys. Rev. Lett. , 076601 (2011), URL https://link.aps.org/doi/10.1103/PhysRevLett.107.076601 . K. Carva, M. Battiato, and P. M. Oppeneer, Phys. Rev.Lett. , 207201 (2011), URL https://link.aps.org/doi/10.1103/PhysRevLett.107.207201 . S. Essert and H. C. Schneider, Phys. Rev. B ,224405 (2011), URL https://link.aps.org/doi/10.1103/PhysRevB.84.224405 . U. Atxitia and O. Chubykalo-Fesenko, Phys. Rev. B , 144414 (2011), URL https://link.aps.org/doi/10.1103/PhysRevB.84.144414 . M. F¨ahnle and C. Illg, J. Phys.: Condens. Matter , 493201 (2011), URL http://iopscience.iop.org/article/10.1088/0953-8984/23/49/493201 . K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev.Lett. , 027204 (2007), URL https://link.aps.org/doi/10.1103/PhysRevLett.99.027204 . A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys.Rev. Lett. , 037207 (2008), URL https://link.aps.org/doi/10.1103/PhysRevLett.101.037207 . H. Ebert, S. Mankovsky, D. K¨odderitzsch, and P. J. Kelly,Phys. Rev. Lett. , 066603 (2011), URL http://link.aps.org/doi/10.1103/PhysRevLett.107.066603 . H. Itoh, T. Nakamichi, Y. Yamaguchi, and N. Kazama,Transactions of the Japan Institute of Metals , 265(1983), URL . Y. Yutaka, K. Masayuki, and N. Takuro, J. Phys. Soc. Jpn. , 2203 (2013), URL http://journals.jps.jp/doi/10.1143/JPSJ.50.2203 . C. Jiang, M. Venkatesan, and J. M. D. Coey,Solid State Communications , 513 (2001), URL http://linkinghub.elsevier.com/retrieve/pii/S003810980100151X . I. Shoji, A. Setsuro, and I. Junji, J. Phys. Soc. Jpn. , 2718 (2013), URL http://journals.jps.jp/doi/10.1143/JPSJ.53.2718 . K. ¨Ozdogan, I. Galanakis, E. S¸a¸sioglu, and B. Akta¸s,J. Phys.: Condens. Matter , 2905 (2006), URL http://iopscience.iop.org/article/10.1088/0953-8984/18/10/013 . E. S¸a¸sioglu, L. M. Sandratskii, and P. Bruno, J.Phys.: Condens. Matter , 995 (2005), URL http://iopscience.iop.org/article/10.1088/0953-8984/17/6/017 . R. Weht and W. E. Pickett, Phys. Rev. B , 13006 (1999),URL https://link.aps.org/doi/10.1103/PhysRevB.60.13006 . G. D. Liu, X. F. Dai, H. Y. Liu, J. L. Chen, Y. X. Li,G. Xiao, and G. H. Wu, Phys. Rev. B , 014424 (2008),URL https://link.aps.org/doi/10.1103/PhysRevB.77.014424 . S. Ouardi, G. H. Fecher, C. Felser, and J. K¨ubler, Phys.Rev. Lett. , 100401 (2013), URL https://link.aps.org/doi/10.1103/PhysRevLett.110.100401 . J. Zabloudil, L. Szunyogh, R. Hammerling, and P. Wein-berger,
Electron Scattering in Solid Matter , A Theoreticaland Computational Treatise (2006), URL http://bookzz.org/md5/190C5DE184B30E2B6898DE499DFB7D78 . H. Ebert, D. K¨odderitzsch, and J. Min´ar, Rep. Prog.Phys. , 096501 (2011), URL http://iopscience.iop.org/article/10.1088/0034-4885/74/9/096501 . H. Ebert,
The Munich SPR-KKR package, version6.3, (2012), URL http://ebert.cup.uni-muenchen.de/SPRKKR . E. K. U. Gross and R. M. Dreizler,
Density FunctionalTheory (Springer Science & Business Media, 2013), ISBN1475799756, URL http://books.google.se/books?id=aG4ECAAAQBAJ&pg=PR4&dq=10.1007/978-1-4757-9975-0&hl=&cd=1&source=gbs_api . J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.Lett. , 3865 (1996), URL https://link.aps.org/doi/10.1103/PhysRevLett.77.3865 . B. L. Gyorffy, Phys. Rev. B , 2382 (1972), URL https://link.aps.org/doi/10.1103/PhysRevB.5.2382 . T. Huhne, C. Zecha, H. Ebert, P. H. Dederichs,and R. Zeller, Physical Review B (Condensed Mat-ter and Materials Physics) , 10236 (1998), URL http://adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1998PhRvB..5810236H&link_type=EJOURNAL . A. I. Liechtenstein, M. I. Katsnelson, and V. A.Gubanov, J. Phys. F: Met. Phys. , L125 (1984),URL http://iopscience.iop.org/article/10.1088/0305-4608/14/7/007 . M. Pajda, J. Kudrnovsk´y, I. Turek, V. Drchal, andP. Bruno, Phys. Rev. B , 174402 (2001), URL https://link.aps.org/doi/10.1103/PhysRevB.64.174402 . S. Mankovsky, D. K¨odderitzsch, G. Woltersdorf, andH. Ebert, Phys. Rev. B , 014430 (2013), URL http://link.aps.org/doi/10.1103/PhysRevB.87.014430 . W. H. Butler, Phys. Rev. B , 3260 (1985), URL https://link.aps.org/doi/10.1103/PhysRevB.31.3260 . H. Ebert, S. Mankovsky, K. Chadova, S. Polesya, J. Min´ar,and D. K¨odderitzsch, Phys. Rev. B , 165132 (2015),URL https://link.aps.org/doi/10.1103/PhysRevB.91.165132 . V. P. Antropov, M. I. Katsnelson, B. N. Harmon,M. van Schilfgaarde, and D. Kusnezov, Phys. Rev. B ,1019 (1996), URL http://link.aps.org/doi/10.1103/PhysRevB.54.1019 . O. Eriksson, A. Bergman, L. Bergqvist, andJ. Hellsvik,
Atomistic Spin Dynamics , Foundationsand Applications (Oxford University Press, 2016),URL https://global.oup.com/academic/product/atomistic-spin-dynamics-9780198788669 . U. Bovensiepen, J. Phys.: Condens. Matter , 083201(2007), URL http://iopscience.iop.org/article/10.1088/0953-8984/19/8/083201 . R. Chimata, A. Bergman, L. Bergqvist, B. Sanyal,and O. Eriksson, Phys. Rev. Lett. , 157201 (2012),URL https://link.aps.org/doi/10.1103/PhysRevLett.109.157201 . A. Jakobsson, P. Mavropoulos, E. S¸a¸sıo˘glu, S. Bl¨ugel,M. Leˇzai´c, B. Sanyal, and I. Galanakis, Phys. Rev. B ,174439 (2015), URL http://link.aps.org/doi/10.1103/PhysRevB.91.174439 . K. Motizuki, H. Ido, T. Itoh, and M. Mori-fuji,
Electronic Structure and Magnetism of 3d-Transition Metal Pnictides (Springer Science & Business Media, 2009), ISBN 3642034209, URL http://books.google.se/books?id=g1wv4vHY58cC&printsec=frontcover&dq=intitle:Electronic+Structure+and+Magnetism+of+3d+Transition+Metal+Kazuko+Motizuki+Springer&hl=&cd=1&source=gbs_api . R. Chimata, E. K. Delczeg-Czirjak, A. Szilva, R. Car-dias, Y. O. Kvashnin, M. Pereiro, S. Mankovsky,H. Ebert, D. Thonig, B. Sanyal, et al., Phys. Rev. B ,214417 (2017), URL http://link.aps.org/doi/10.1103/PhysRevB.95.214417 . M. Meinert, J.-M. Schmalhorst, and G. Reiss, J.Phys.: Condens. Matter , 116005 (2011), URL http://iopscience.iop.org/article/10.1088/0953-8984/23/11/116005 . D. Jiles,
Introduction to Magnetism and MagneticMaterials, Third Edition (CRC Press, 2015), ISBN1482238888, URL http://books.google.se/books?id=2diYCgAAQBAJ&printsec=frontcover&dq=intitle:Introduction+to+Magnetism+and+Magnetic+Materials+Second+Edition&hl=&cd=1&source=gbs_api . J. Rusz, L. Bergqvist, J. Kudrnovsk´y, and I. Turek, Phys.Rev. B , 214412 (2006), URL https://link.aps.org/doi/10.1103/PhysRevB.73.214412 . D. B¨ottcher, A. Ernst, and J. Henk, Journal ofMagnetism and Magnetic Materials , 610 (2012),URL http://linkinghub.elsevier.com/retrieve/pii/S0304885311006299 . J. Chico, S. Keshavarz, Y. Kvashnin, M. Pereiro,I. Di Marco, C. Etz, O. Eriksson, A. Bergman, andL. Bergqvist, Phys. Rev. B , 214439 (2016), URL https://link.aps.org/doi/10.1103/PhysRevB.93.214439 . R. Y. Umetsu and T. Kanomata, Physics Procedia , 890 (2015), URL http://linkinghub.elsevier.com/retrieve/pii/S187538921501754X . K. Binder and D. Heermann,
Monte Carlo Simulation inStatistical Physics , vol. 5 of
An Introduction (Berlin Hei-delberg, 2010), springer-verlag ed., URL . K. Meinel, A. Beckmann, M. Klaua, and H. Bethge, phys-ica status solidi (a) , 521 (1995), URL http://doi.wiley.com/10.1002/pssa.2211500146 . J. Staunton,
Relativistic Effects and Disordered Local Mo-ments in Magnets (2013), URL . H. Akai and P. H. Dederichs, Phys. Rev. B , 8739 (1993),URL https://link.aps.org/doi/10.1103/PhysRevB.47.8739 . A. Buruzs, Ph.D. thesis, cms.tuwien.ac.at, Wien(2008), URL . M. A. W. Schoen, D. Thonig, M. L. Schneider, T. J. Silva,H. T. Nembach, O. Eriksson, O. Karis, and J. M. Shaw,Nat Phys , 839 (2016), URL . S. Lounis, M. dos Santos Dias, and B. Schweflinghaus,Phys. Rev. B , 104420 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.91.104420 . F. Pan, J. Chico, J. Hellsvik, A. Delin, A. Bergman, andL. Bergqvist, Phys. Rev. B , 214410 (2016), URL https://link.aps.org/doi/10.1103/PhysRevB.94.214410 . V. Kambersk´y, Czech J Phys , 1111 (1984), URL https://link.springer.com/article/10.1007/BF01590106 . V. Kambersk´y, Czech J Phys , 1366 (1976), URL http://link.springer.com/10.1007/BF01587621 . B. Skubic, J. Hellsvik, L. Nordstr¨om, and O. Eriks-son, J. Phys.: Condens. Matter , 315203 (2008),URL http://iopscience.iop.org/article/10.1088/0953-8984/20/31/315203 . M. Berritta, R. Mondal, K. Carva, and P. M. Oppeneer,Phys. Rev. Lett. , 137203 (2016), URL https://link.aps.org/doi/10.1103/PhysRevLett.117.137203 . R. John, M. Berritta, D. Hinzke, C. M¨uller, T. San-tos, H. Ulrichs, P. Nieves, J. Walowski, R. Mondal,O. Chubykalo-Fesenko, et al., Sci. Rep. , 4114 (2017),URL http://adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2017NatSR...7.4114J&link_type=EJOURNAL . C. Illg, M. Haag, and M. F¨ahnle, Phys. Rev. B , 214404 (2013), URL https://link.aps.org/doi/10.1103/PhysRevB.88.214404 . R. F. L. Evans, U. Atxitia, and R. W. Chantrell, Phys.Rev. B , 144425 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.91.144425 . D. Hinzke, U. Atxitia, K. Carva, P. Nieves, O. Chubykalo-Fesenko, P. M. Oppeneer, and U. Nowak, Phys. Rev.B , 259 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.92.054412 . U. Atxitia, D. Hinzke, O. Chubykalo-Fesenko, U. Nowak,H. Kachkachi, O. N. Mryasov, R. F. Evans, and R. W.Chantrell, Phys. Rev. B , 134440 (2010), URL https://link.aps.org/doi/10.1103/PhysRevB.82.134440 . C. J. Palmstrøm, Progress in Crystal Growth andCharacterization of Materials , 371 (2016), URL http://linkinghub.elsevier.com/retrieve/pii/S0960897416300237http://linkinghub.elsevier.com/retrieve/pii/S0960897416300237