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PHYSICAL REVIEW B 68, 104436 共2003兲

First-principles relativistic study of spin waves in thin magnetic films

L. Udvardi, 1,2 L. Szunyogh, 1,2 K. Palotás, 2 and P. Weinberger 2

1 Department of Theoretical Physics and Center for Applied Mathematics and Computational Physics, Budapest University of Technology

and Economics, Budafoki út 8, H-1111, Budapest, Hungary 2 Center for Computational Materials Science, Technical University Vienna, Gumpendorfer Str. 1a, A-1060 Vienna, Austria

共Received 9 April 2003; revised manuscript received 25 June 2003; published 30 September 2003兲

In order to study spin-wave excitations of itinerant ferromagnets a relativistic first-principles method based on the adiabatic approach is presented. The derivatives of the free energy up to second order with respect of the polar and azimuthal angles are derived within the framework of the magnetic force theorem and the fully relativistic Korringa-Kohn-Rostoker method. Exchange and spin-orbit coupling are thus incorporated on equal footing in the Hamiltonian. Furthermore, a detailed comparison to classical spin Hamiltonians is given and it is shown that the magnetocrystalline anisotropy energy contains contributions from both the on-site anisotropy and the off-site exchange coupling terms. The method is applied to an Fe monolayer on Cu共001兲 and Au共001兲 surfaces and for a Co monolayer on Cu共001兲. The calculations provide with the gap at zero wave number due to the spin-orbit coupling and uniaxial anisotropy energies in good agreement with the results of the band energy difference method. It is pointed out that the terms in the spin-wave Hamiltonian related to the mixed partial derivatives of the free energy, absent within a nonrelativistic description, introduce an asymmetry in the magnon spectrum with respect to two in-plane easy axes. Moreover, in the case of an in-plane magnetized system the long-wavelength magnons are elliptically polarized due to the difference of the second-order uniaxial and fourth-order in-plane magnetic anisotropy.

DOI: 10.1103/PhysRevB.68.104436 PACS number共s兲: 75.10.Jm, 75.30.Ds, 75.30.Gw, 75.70.Ak

I. INTRODUCTION

In recent years the magnetic properties of thin films have attracted much interest both experimentally as well as theoretically mainly due to possible applications in magnetic storage technology. In thin magnetic films relativistic effects such as magnetic anisotropy are more pronounced than in bulk materials. The spin-orbit coupling plays a key role in low-energy magnetic excitations of such materials as it opens a gap at zero wave number which in turn is necessary for the formation of long-range magnetic order. First-principles calculations of the spin-wave excitations can help a lot in understanding the details of the underlying physics and in tailoring the properties of different ultrathin magnetic films.

Spin-wave excitation spectra can be determined by calculating the dynamical spin susceptibilities using linear response theory 1 or by using the adiabatic approach. 2– 6 Within the adiabatic approach the fast motion of itinerant electrons is decoupled from the slow motion of the spins. Obviously, the condition for the validity of the adiabatic approximation is that the time scale of the precession of the magnetic moments should be sufficiently larger than the characteristic 共hopping兲 time of the motion of electrons. This condition corresponds to the criterion that the spin-wave energy must be small as compared to the bandwidth and to the exchange splitting. It has been shown that in terms of the rigid-spin approximation; i.e., by choosing a uniquely defined spinquantization axis within each cell, the general equation of motion for the spin density reduces to the semiclassical Landau-Lifshitz equation, 2,6 describing the orientational motion of the spin moments, whereas the magnitude of the spin magnetization can be evaluated self-consistently within the framework of a constrained density functional theory. 7 A non-self-consistent theory such as employed in the present 0163-1829/2003/68共10兲/104436共11兲/$20.00 work allows one to trace the orientational motion of the magnetic moments, describing thus transverse spin-wave excitations only. It should be mentioned that a conceptually clear description of spin dynamics within a time-dependent density functional theory is matter of ongoing investigation 8 and yet is not in the state of being used in practical terms.

In the nonrelativistic case the adiabatic approach leads within the harmonic approximation to a classical Heisenberg Hamiltonian,

H 1 2 ij Ji j i j , 1

where ␴ i is a classical unit vector parallel to the magnetization at site i and J i j is the exchange interaction energy between sites i and j. Basically two methods are used to determine the exchange interaction energy J i j . One of these is the so-called frozen magnon approximation, 4,9 in which the spin configuration is constrained to a spin wave of wave vector q and the energy of this spin wave is calculated by employing the generalized Bloch theorem for a spin-spiral configuration. 10 Since within the particular model described by Eq. 共1兲 the energy of a spin wave with respect to a ferromagnetic ground state can be expressed as

q j0 J 0 j e iq•R0 j 1 ,

共2兲 where R i j is the relative position vector connecting sites i and j, and q is the wave vector of the spin wave, the exchange coupling parameters J i j are obtained from the inverse Fourier transform of the spin-wave dispersion law.

In the second approach which is frequently referred to as the torque method or the method of infinitesimal rotations, 11 68 104436-1 ©2003 The American Physical Society PHYSICAL REVIEW B 68, 104436 共2003兲 L. UDVARDI, L. SZUNYOGH, K. PALOTÁS, AND P. WEINBERGER the coupling constants J i j are calculated directly from the change of the energy associated with constrained rotations of the spin-polarization axes at sites i and j. Using the magnetic force theorem 11,12 the change of the total energy is approximated by the corresponding change of the one-particle energies, remarkably simplifying the calculations. A renormalization of this approach was recently proposed by Bruno 13 and transverse magnons 共see Introduction兲, the angles ␽ ri and ␸ ri depend on time, whereas, by supposing two-dimensional translational invariance for the ground state, the timeindependent magnitudes M ri depend only on the layer index; i.e., M ri ⫽M r for all sites i in a particular layer r.

Rewriting Eq. 共3兲 into spherical coordinates, the equations of motion for the angles ␽ ri and ␸ ri are given by

tion of this approach was recently proposed by Bruno and Antropov1 4 which considerably improved the calculated spectra in the region of shorter wavelengths. For magnetic systems with noncubic symmetry such as hcp Co or for layered systems the importance of relativistic effects is enhanced. In a relativistic formalism the effective Hamiltonian, Eq. 1, no longer applies since the invariance of the Hamiltonian against a global rotation of the spins is lost. So far, when calculating spin-wave spectra, relativistic effects such as the spin-orbit coupling and the magnetic dipole-dipole interaction were taken into account as additive terms in the Hamiltonian 1, while the exchange coupling parameters J i j were calculated nonrelativistically or were es- timated from experiments.1 5–19 The aim of the present paper is to provide with a relativ- istic description of spin waves in itinerant, transition-metal ferromagnets from first principles. Our method relies on the adiabatic decoupling of the local spin-density functional theory and the rigid-spin approximation, whereby a fully relativistic treatment of the electrons in terms of the Kohn- Sham-Dirac equation is used. In the next section we describe the main points of our theory based on the Landau-Lifshitz equation and a harmonic approximation to the free energy. It will be shown that due to the relativistic treatment the spin- wave Hamiltonian contains off-diagonal elements. Closed formulas for the derivatives of the free energy up to second order with respect of the polar and azimuthal angles are de- rived by means of the fully relativistic Korringa-Kohn- Rostoker KKR method.2 0 The relationship of our theory to classical spin Hamiltonians will also be given. Applications are then presented to typical ultrathin magnetic films with out-of-plane, Fe1 A u(001), and in-plane, Co1 C u(001), ground-state magnetic orientation. In the case of Fe1 C u(001) the effect of the magnetic dipole-dipole interaction to the spin-wave spectrum is emphasized. II. THEORY A. General formalism As shown in Refs. 2 and 6, within the rigid-spin approxi- mation the adiabatic dynamics of local spin moments is de- scribed by the Landau-Lifshitz equation M ri ˙r i 2 B F r i r i , 3 where M ri is the magnitude of the spin moment and r i is a of motion for the angles ri and ri are given by M r ˙ ris in ri 2 B F ri , 5 M r ˙ ris in ri 2 B F ri . 6 Choosing the polar z axis of the reference system to be perpendicular to the magnetization in the ferromagnetic ground state, Eqs. 5 and 6 can easily be linearized, M r ˙ ri 2 B F ri / 2, 0 , 7 M r ˙ ri 2 B F ri / 2, 0 , 8 where the constraint /2, 0 denotes that the partial derivatives have to be taken at ri /2, and ri 0 for all r and i. The linearized version of the Landau-Lifshitz equa- tions, Eqs. 7 and 8, are the canonical equations for the generalized coordinates q ri (M r / B) 1/2 ri and momenta p ri (M r / B) 1/2 ri . Adopting the harmonic approximation—i.e., expanding the free energy up to second order in the angular variables—the corresponding Hamilton function can be written as H 1 ri,s j q riA ri,s j q s j q riB ri,s j p s j p riB s j,riq s j p riC ri,s j p s j , 9 with A ri,s j M r / B 1/2 2 F ri s j / 2, 0 M s / B 1/2 , 10 B ri,s j M r / B 1/2 2 F ri s j / 2, 0 M s / B 1/2 , 11 C ri,s j M r / B 1/2 2 F ri s j / 2, 0 M s / B 1/2 . 12

hcp Co or for layered systems the importance of relativistic effects is enhanced. In a relativistic formalism the effective Hamiltonian, Eq. 1, no longer applies since the invariance of the Hamiltonian against a global rotation of the spins is lost. So far, when calculating spin-wave spectra, relativistic effects such as the spin-orbit coupling and the magnetic dipole-dipole interaction were taken into account as additive terms in the Hamiltonian 1, while the exchange coupling parameters J i j were calculated nonrelativistically or were es- timated from experiments.1 5–19 The aim of the present paper is to provide with a relativ- istic description of spin waves in itinerant, transition-metal ferromagnets from first principles. Our method relies on the adiabatic decoupling of the local spin-density functional theory and the rigid-spin approximation, whereby a fully relativistic treatment of the electrons in terms of the Kohn- Sham-Dirac equation is used. In the next section we describe the main points of our theory based on the Landau-Lifshitz equation and a harmonic approximation to the free energy. It will be shown that due to the relativistic treatment the spin- wave Hamiltonian contains off-diagonal elements. Closed formulas for the derivatives of the free energy up to second order with respect of the polar and azimuthal angles are de- rived by means of the fully relativistic Korringa-Kohn- Rostoker KKR method.2 0 The relationship of our theory to classical spin Hamiltonians will also be given. Applications are then presented to typical ultrathin magnetic films with out-of-plane, Fe1 A u(001), and in-plane, Co1 C u(001), ground-state magnetic orientation. In the case of Fe1 C u(001) the effect of the magnetic dipole-dipole interaction to the spin-wave spectrum is emphasized. II. THEORY A. General formalism As shown in Refs. 2 and 6, within the rigid-spin approxi- mation the adiabatic dynamics of local spin moments is de- scribed by the Landau-Lifshitz equation M ri ˙r i 2 B F r i r i , 3 where M ri is the magnitude of the spin moment and r i is a unit vector pointing along the spin-quantization axis in the atomic cell at the site i of layer r, r i sin ric os ri ,sin ris in ri ,cos ri , 4 M r ˙ ris in ri 2 B F ri . 6 Choosing the polar z axis of the reference system to be perpendicular to the magnetization in the ferromagnetic ground state, Eqs. 5 and 6 can easily be linearized, M r ˙ ri 2 B F ri / 2, 0 , 7 M r ˙ ri 2 B F ri / 2, 0 , 8 where the constraint /2, 0 denotes that the partial derivatives have to be taken at ri /2, and ri 0 for all r and i. The linearized version of the Landau-Lifshitz equa- tions, Eqs. 7 and 8, are the canonical equations for the generalized coordinates q ri (M r / B) 1/2 ri and momenta p ri (M r / B) 1/2 ri . Adopting the harmonic approximation—i.e., expanding the free energy up to second order in the angular variables—the corresponding Hamilton function can be written as H 1 ri,s j q riA ri,s j q s j q riB ri,s j p s j p riB s j,riq s j p riC ri,s j p s j , 9 with A ri,s j M r / B 1/2 2 F ri s j / 2, 0 M s / B 1/2 , 10 B ri,s j M r / B 1/2 2 F ri s j / 2, 0 M s / B 1/2 , 11 C ri,s j M r / B 1/2 2 F ri s j / 2, 0 M s / B 1/2 . 12 It is straightforward now to quantize the system using the usual commutation rules q ri ,p s j (/i) i j rs for the gen- eralized coordinates and momenta. By introducing bosonic annihilation an d creation operators a ri (1/ 2 )(q ri i p ri) †

eralized coordinates and momenta. By introducing bosonic annihilation an d creation operators a ri (1/ 2 )(q ri i p ri) and a ri † (1/ 2 )(q ri ip ri) , the Hamilton operator of mag-

netic excitations can be written as 104436-2 PHYSICAL REVIEW B 68, 104436 共2003兲 FIRST-PRINCIPLES RELATIVISTIC STUDY OF SPIN . . .

H 1 2 rs,i j A ri,s j iB ri,s j iB s j,ri C ri,s j a ri † a s j H.c., 1 2 rs,i j A ri,s j iB ri,s j iB s j,ri C ri, js a ri † a s j † H.c. 13

By utilizing the two-dimensional translational symmetry of the coefficients A ri,s j , B ri,s j , and C ri,s j one can rewrite the above Hamiltonian as

H 1 2 q,rs h rs q a r † q a s q a s q a r † q 1 2 q,rs w rs q a r † q a s † q w sr* q a r q a s q , 14

Quite clearly, the diagonal matrix ␧ (q) contains the excitation energies of the magnons.

So far we have not made any distinction between a relativistic and a nonrelativistic theory. Relativistic effects enter through the parameters A ri,s j , B ri,s j , and C ri,s j defined by Eqs. 共10兲, 共11兲, and 共12兲, respectively. However, it is easy to see that in a nonrelativistic case the mixed partial derivatives of the free energy, B ri,s j , vanish and there is no difference between the second derivatives with respect to the azimuthal and polar angles, i.e., A ri,s j ⫽C ri,s j . Consequently, the second term in Eq. 共14兲 disappears and the Hamiltonian is of the form of a simple harmonic oscillator. In a relativistic treatment the second term in Eq. 共14兲 appears to be nonzero due to the magnetic dipole-dipole interaction and spin-orbit coupling. The effect of the magnetic dipole-dipole interaction is described in detail in Refs. 15–17, but according to our knowledge the effect of the spin-orbit coupling on the spinwave spectra has not yet been studied from first principles.

where

B. Evaluation in terms of the Korringa-Kohn-Rostoker

method

a r q 1 N i e iq•Ti a ri 15

and

h rs q i A r0,s j iB r0,s j iB s j,r0 C r0,s j e iq(Tj Cs Cr ) , 16 w rs q i A r0,s j iB r0,s j iB s j,r0 C r0,s j e iq(Tj Cs Cr ) ,

i 16 w rs q i A r0,s j iB r0,s j iB s j,r0 C r0,s j e iq(Tj Cs Cr ) , 17 with the two-dimensional translational vectors Ti and the layer generating vectors Cr (Rr i Cr Ti ) . The same structure of the Hamiltonian has been obtained by Erickson and Mills1 6 in terms of the Holstein-Primakoff transformation of a Heisenberg Hamiltonian with additional magnetic dipole-dipole interactions. Following their proce- dure, in order to diagonalize the Hamiltonian in Eq. 14 we introduce a new set of creation and annihilation operators s † (q) and s( q), respectively, defined by the transforma- tions a r q s S rs q s q T rs* q s † q , 18 a r † q s S rs * q s † q T rs q s q , 19 where the transformation matrices S(q) and T(q) satisfy the eigenvalue equation h q w q w† q h† q S q T* q T q S* q S q T* q T q S* q q q . 20 perature is approximated by F EF d E F n EF dN , 21 where E F denotes the Fermi energy of the system, n() is the density of states DOS, and N() is the integrated DOS. Employing Lloyd’s formula,2 1 apart from a constant term corresponding to the potential-free system, the free energy can further be written as F 1 Im EF d Tr ln , 22 where the site representation of the scattering path operator ( ) i j () can be expressed in terms of a site-diagonal single-site scattering matrix t() t i ( ) i j and the struc- ture constants G0 ( ) G 0,i j ()(1 i j ) as t 1 G0 1 . 23 In here, i j (), t i( ), and G i j () are matrices in angular momentum space. The change of the free energy has to be expressed up to second order with respect to a small deviation of the orien- tation of the magnetizations at sites i and j relative to the ferromagnetic ground-state orientation. As it is explained in Ref. 20 the orientational dependence of the single-site t ma- trix corresponds to a similarity transformation that rotates the z axis of the reference system to the desired orientation given by the angles i and i , t i 1 m i R i , i m i 0 R † i , i , 24

PHYSICAL REVIEW B 68, 104436 共2003兲 L. UDVARDI, L. SZUNYOGH, K. PALOTÁS, AND P. WEINBERGER

where m 0 i denotes the inverse of the t matrix at site i in a local frame in which the z axis coincides with the axis of the

local frame in which the z axis coincides with the axis of the spin quantization magnetization. Note that for brevity we

have dropped the energy argument from the corresponding

matrices. The change of m i up to second order in i and i , m i (1) m i d i m i d i , 25 m i (2) 1 2 m i d id i m i d id i 1 2 m i d id i , 26 can easily be expressed by means of the derivatives of the rotation matrices R( i , i) , m i m i i R i i m i 0 R i † R im i 0 R i † i , 27 m i m i i R i i m i 0 R i † R im i 0 R i † i , 28 m i 2 m i i i 2 R i i 2 m i 0 R i † R im i 0 2 R i † i 2 2 R i i m i 0 R i † i , 29 m i 2 m i i i 2 R i i i m i 0 R i † R i i m i 0 R i † i R i i m i 0 R i † i R im i 0 2 R i † i i , 30 m i 2 m i i i 2 R i i 2 m i 0 R i † R im i 0 2 R i † i 2 2 R i i m i 0 R i † i , 31 where for simplicity we used the abbreviation R i R( i , i) . First we evaluate the diagonal terms of the second deriva- tive tensor of the free energy by introducing a change of the orientation of the magnetization only on site i. The logarithm of the scattering path operator appearing in Eq. 22 can now be written as ln ln mmi G0 1 ln ln 1 mi . 32 Quite clearly, the site-diagonal matrix mi has only the block corresponding to site i with nonzero elements. Expand- ing the logarithm in Eq. 32 and keeping the terms up to second order one obtains ln ln mi( 1) mi( 2) 1 2 mi( 1) mi( 1) . 33 ln ln mmi mj G0 1 ln 1 mi mj , 34 which can be rewritten into the form ln ln ln 1 mi ln 1 mj ln 1 i j , 35 where we introduced the notation i mi ( 1 mi ) 1 . Expanding Eq. 35 up to second order gives ln ln mi( 1) mj( 1) mi( 1) mj( 1) . 36 Similar to the site-diagonal case the second derivatives can be deduced from the second term of the right-hand side RHS of Eq. 36. In the following a layered ferromagnetic order is as- sumed; i.e., the orientation of the magnetization is the same within each layer but the magnetic moment can be different in different layers. With these conditions the single-site t matrix depends only on the layer index. Substituting Eqs. 25–31 into Eqs. 33 and 36 and applying Eq. 22, the second derivative tensor of the free energy with respect to the site-dependent orientations of the magnetization can be written as follows: diagonal terms: 2 F ri ri 1 Im EF d Tr r0,r0 m r r0,r0 m r r0,r0 m r , 37 2 F ri ri 1 Im EF d Tr r0,r0 m r r0,r0 m r r0,r0 m r , 38 2 F ri ri 1 Im EF d Tr r0,r0 m r r0,r0 m r r0,r0 m r ; 39 off-diagonal terms: 2 F ri s j

The first term on the right-hand side of the above equation contributes to the gradient of the free energy, while the sitediagonal elements of its second derivative tensor are related to the second and third terms of Eq. 共33兲.

104436-4

ri s j 1 Im EF d Tr s j,ri m r ri,s j m s , 40

PHYSICAL REVIEW B 68, 104436 共2003兲 FIRST-PRINCIPLES RELATIVISTIC STUDY OF SPIN . . .

2 F ri s j 1 Im EF d Tr s j,ri m r ri,s j m s ,

where the first and second terms on the RHS are the isotropic and the symmetric anisotropic exchange interactions, respec-

ri s j 1 Im EF d Tr s j,ri m r ri,s j m s , 41 2 F ri s j 1 Im EF d Tr s j,ri m r ri,s j m s , 42 tively, while the third term represents the Dzyaloshinsky- Moriya DM interaction,2 2,23 the vector Di j being defined as D i j x 1 2 J i j yz J i j zy , D i j y 1 2 J ij xz J i j zx , D i j z 1 2 J ij xy J i j yx . 49 Since the on-site term in Eq. 43, i J i i i , contains only symmetric contributions, it can readily be included as a second-order term to the magnetic anisotropy K( i ) . Conse- quently, the summation in the second term on the RHS of Eq.

2 F ri s j 1 Im EF d Tr s j,ri m r ri,s j m s , 42 where ri,s j is the block of the real-space scattering path operator that corresponds to site i of layer r and site j of layer s. In Eqs. 37–42 the trace has to be performed in angular momentum space. C. Relation to classical spin Hamiltonians In order to study, e.g., magnetic properties at finite tem- peratures the energy of an itinerant electron system is often mapped onto a simple classical spin Hamiltonian. Using a quadratic approximation of the effective spin-spin interac- tion, such a Hamiltonian can be written in the most general form as H i K i 1 2 ij i J i j j , 43 where the first term stands for the on-site anisotropy energy and the Ji j in the second term are 33 matrices. It is evi- dent that the matrices Ji j (i j) can be chosen such that Ji j J ji t , where t refers to the transposed matrix. In order to relate the Hamiltonian 43 to physically significant interac- tions, Ji j has to be decomposed as Ji j J i j IJ i j S J i j A , 44 where I is the unit matrix, J i j 1 3 Tr Ji j , 45 and the symmetric, traceless part of Ji j , J i j S , is defined by J i j S 1 2 Ji j J i j t J i j I, 46 while the antisymmetric part of Ji j , J i j A , by J i j A 1 2 Ji j J i j t . 47 Clearly, therefore, a typical intersite interaction consists of the three terms2 2 i J i j j J i j i • j i J i j S j Di j i j , 48 D i j z 1 2 J ij xy J i j yx . 49 Since the on-site term in Eq. 43, i J i i i , contains only symmetric contributions, it can readily be included as a second-order term to the magnetic anisotropy K( i ) . Conse- quently, the summation in the second term on the RHS of Eq. 43 has to run over i j only. In what follows we show how the parameters of the Hamiltonian 9 can be related to those in Eq. 43, the magnetocrystalline anisotropy and the gen- eralized exchange interactions occurring in Eq. 48 being thus automatically included in our Hamiltonian for trans- verse magnons. We start by calculating the second derivatives of the Hamiltonian in Eq. 43 with respect to the polar and azi- muthal angles, 2 H i k ikK i ik j(i) i Ji j j 1 ik i Ji k k , 50 where and can be either or and K i 2 K i i i , i i i , i 2 i i i . Note that, according to Eqs. 7 and 8, all the derivatives have to be taken at i /2 and i 0 in a coordinate sys- tem with the z axis normal to the reference orientation of the magnetization. If the magnetization points along the x axis of the global coordinate system, i.e., to which the matrices Ji j are related, Eq. 4 yields, i 1 0 0 i 0 1 0 i 0 0 1 i 1 0 0 i 1 0 0 i 0 0 0 , 51 which, when substituted into Eq. 50, leads to 2 H i k ikK i ik j(i) J i j xx 1 ik J ik yy , 52 2 H i k ikK i ik j(i) J i j xx 1 ik J ik zz , 53

PHYSICAL REVIEW B 68, 104436 共2003兲 L. UDVARDI, L. SZUNYOGH, K. PALOTÁS, AND P. WEINBERGER

2 H i k ikK i 1 ik J ik yz , 54 2 H i k ikK i 1 ik J ik zy . 55

Assuming for matters of simplicity second-order uniaxial magnetic anisotropy

K i K 2,is in2 i , 56

the diagonal components of the second derivative tensor are given by

2 H i i j(i) J ij xx , 2 H i i 2K 2,i j(i) J ij xx , 2 H i i 2 H i i 0, 57

while for the off-diagonal components (i⫽k) we get

2 H i k J ik yy , 2 H i k J ik zz , 2 H i k J ik yz , 2 H i k J ik zy . 58

If the magnetization points parallel to the z axis of the global coordinate system, the corresponding derivatives can

global coordinate system, the corresponding derivati be obtained by applying t he unitary transformation U 0 0 1 0 1 0 1 0 0 ,

U 0 0 1 0 1 0 1 0 0 , 59 whereas also the uniaxial magnetic anisotropy, Eq. 56, has to be transformed to K i K 2,is in2 i cos2 i . 60 Thus, one ends up with the expressions for the on-site terms, 2 H i i 2 H i i 2K 2,i j(i) J ij zz , 2 H i i 2 H i i 0, 61 and for ik with 2 H i k J ik yy , 2 H i k J ik xx , 2 H i k J ik yx , 2 H i k J ik xy . 62 the magnetization, the energy of the system is invariant against a global rotation of all the magnetic moments, and the following condition has to be satisfied: 2 H i i j(i) 2 H i j 0 or , 63 which is equivalent to the condition for the exchange inter- action parameters,1 1 J i j(i) J ij . 64 In the relativistic case this sum rule no longer applies. Indeed, the expression on the LHS of Eq. 63 can be used to define an effective second-order magnetic anisotropy con- stant i . For the case of a magnetization parallel to the z axis of the global coordination frame, Eqs. 61 and 62 imply i 1 2 2 H i i j(i) 2 H i j K 2,i 1 2 j(i) J i j zz J i j xx . 65

Eq. 24 can be performed in spin space and the off-diagonal
terms 40 or 42 as well as the diagonal terms 37 or 39

and the elements of the matrices Ji j can be calculated from Eqs. 57, 58, 61, or 62 using Eqs. 37–42. The missing elements J i j xz and J i j zx can be derived by choosing the

The physical content of the above equation is obvious: beyond the on-site anisotropy term the magnetocrystalline anisotropy energy of the system contains contributions that arise from the anisotropy of the intersite exchange interactions.

Going back now to Eqs. 共39兲 and 共42兲 and using the notation used in Sec. II B, one obtains, for the layer-resolved anisotropies, 104436-6 PHYSICAL REVIEW B 68, 104436 共2003兲 FIRST-PRINCIPLES RELATIVISTIC STUDY OF SPIN . . .

r 1 2 Im EF d Tr r0,r0 m r r0,r0 m r r0,r0 m r sj 1 2 Im EF d Tr s j,r0 m r r0,s j m s , 66 which for practical purposes can be rewritten in terms of reciprocal-space summations as r 1 2 Im EF ddk T r rr ,k m r 1 2 s Im EF ddk T r sr ,k m r rs ,k m s , 67 E dip 1 c 2 ij R i j 2 Mi • Mj 3 Mi • Ri j Ri j •Mj R i j 5 . 68 2 0 FIG. 1. Spin-wave spectrum for Fe1 /Au(001). The almost dispersion-less bands belong to the two Au layers adjacent the Fe monolayer.

As has been shown by Szunyogh et al. 20 the orientation of the magnetization for an Fe monolayer on Au共001兲 is perpendicular to the surface due to the relatively strong band energy anisotropy. In the case of an Fe monolayer on Cu共001兲 the magnetic dipole-dipole interaction overrides the positive band energy contribution 共out-of-plane orientation兲, resulting in an in-plane ferromagnetic order. 28 For a cobalt monolayer on Cu共001兲 surface both the magnetic anisotropy and the magnetic dipole-dipole interaction favor the in-plane orientation of the magnetization. 29 The magnon spectra of the three systems should therefore account for these three different situations.

The calculated magnon spectra along the high-symmetry directions of the Brillouin zone are depicted in Figs. 1–3. The almost dispersion-less bands belong to the nonmagnetic nearest- and next-nearest-neighbor layers. There are anticrossings between the bands which are the most pronounced in the case of Co/Cu共001兲, indicating the largest interactions between the magnetic and nonmagnetic layers. For Fe/ Cu共001兲, the dispersion relation is very similar to that ob-

FIG. 2. Spin-wave spectrum for Fe1 /Cu(001) with an in-plane ground-state magnetization. Four additional Cu layers indicated by bands of very low dispersion have been taken into account. The almost identical solid and dashed lines represent the 100 and 010 directions of the magnetization, respectively.

104436-7 L. UDVARDI, L. SZUNYOGH, K. PALOTÁS, AND P. WEINBERGER PHYSICAL REVIEW B 68, 104436 共2003兲

tained by Pajda et al.1 8 using the method of infinitesimal rotations combined with the nonrelativistic tight-binding lin- ear muffin-tin orbital TB-LMTO approach. A local mini- mum observed between the points X and M is an indication for a so-called Kohn anomaly4 which is the consequence of the long-range RKKY-like behavior of the exchange interac- tions. In Fig. 4 the spectra in the vicinity of the point are shown for Fe/Au001 as calculated for an in-plane and for an out-of-plane magnetic orientation, and also for the case of neglecting spin-orbit coupling. In the absence of spin-orbit coupling the dispersion curve starts from zero as has to be expected. Inclusion of spin-orbit coupling opens up a gap of 43 Ry for the normal-to-plane orientation. For an in- plane magnetization the relativistic calculation resulted in an imaginary magnon energy close to the point, indicating that the in-plane ferromagnetic order corresponds to a local maximum in the free energy and the easy axis is perpendicu- lar to the surface. The long-wavelength parts of the relativistic spectra of the Fe/Cu001 monolayer without and with inclusion of the magnetic dipole-dipole interaction are shown in Fig. 5. Con- sidering only the band-energy contributions the easy axis turned out to be perpendicular to the surface and a gap of 17.5 Ry appeared in the corresponding magnon spec- FIG. 3. Spin-wave spectrum for Co1 /Cu(001) with an in-plane ground-state magnetization. The four additional Cu layers considered in the calcu- lations are coupled relatively strongly to the Co monolayer as indicated by the noncrossing be- havior hybridization of the corresponding bands. The solid and dashed lines represent the 100 and 010 directions of the magnetization, respectively. The spectrum between the symme- try points X and as well as between M and X is shown on an enlarged scale in upper left and up- per right insets, respectively. FIG. 4. Spin-wave spectrum for Fe1 /Au(100) in the vicinity of the point. The solid lines correspond to the normal–to–plane orientation of the magnetization, the dotted line represents the spec- trum for an in-plane orientation, and the dashed line denotes the magnon spectrum with spin-orbit–coupling turned off. FIG. 5. Spin-wave spectrum in the vicinity of the point for Fe1 /Cu(100) neglecting the magnetic dipole–dipole interaction a and including the magnetic dipole–dipole interaction b. The solid and dashed lines represent the spectra corresponding to the normal– to–plane and to the in-plane magnetic orientations, respectively.

104436-8 PHYSICAL REVIEW B 68, 104436 共2003兲 FIRST-PRINCIPLES RELATIVISTIC STUDY OF SPIN . . .

trum, while, similar to the previous case of Fe/Au共001兲, there are imaginary excitation energies in the in-plane magnon spectrum at small wave numbers. Inclusion of the magnetic dipole-dipole term, Eq. 共68兲, reverses the situation: there is a tiny gap for the in-plane magnetized film and the magnon spectrum becomes negative for the out-of-plane directions.

In the case of a thin film with an easy axis perpendicular to the layers the gap is determined by the second-order uniaxial anisotropy. For a magnetic layer with an in-plane easy axis the lowest energy of the magnetic excitations—i.e., the gap at q⫽0—lies in the order of the in-plane anisotropy energy which for the present systems of c 4 v symmetry is at least by an order of magnitude smaller than the uniaxial one. Our calculation resulted in a gap of ⌬⫽0.3 ␮ Ry for the Fe/Cu共001兲 monolayer and the easy axis turned out to be parallel to the 共110兲 direction. We have to mention that such a small value is very close to the numerical accuracy of our calculations inherent mainly to the Brillouin zone sampling.

A similar drop of the gap has, however, been indicated in the experiments for a thermally deposited three-monolayer film of Fe on Cu共001兲. In contrast to the pulsed-laserdeposited 共PLD兲 samples having an in-plane ground-state magnetization 30,31 thermal deposition 共TD兲 produces Fe films on Cu共001兲 with an out-of-plane easy axis. 32 In their Brillouin light scattering experiment Dutcher et al. 32 studied the lowest-lying spin-wave mode in the presence of an external magnetic field parallel to the substrate. They found that the frequency of this mode has a minimum as the magnetic field passes a critical value where the spins become parallel to the film.

In the case of an in-plane magnetization the c 4 v symmetry of the fcc 共001兲 surface is lifted and the spin-wave spectra for the 共001兲 and 共010兲 directions of the magnetization become different as is shown in Fig. 3 for Co/Cu共001兲. A splitting of a few meV due to spin-orbit coupling can clearly be seen in the insets in Fig. 3. This splitting of the spectrum can also be interpreted in terms of the Dzyaloshinsky-Moriya interaction. 22,23 On the one hand, it is straightforward to show that transverse magnons are affected only by the component of the DM vector, D i j 关see Eq. 共49兲兴, parallel to the ground-state magnetization. On the other hand, in agreement with the symmetry rules set up by Moriya, 22 we observed noticeable parallel components of the DM vectors only if the vector R i j connecting the two interacting spins is aligned close to a direction perpendicular to the magnetization, while the DM interactions along a direction parallel to the groundstate magnetization turned out to be practically zero. Evidently, the above real-lattice asymmetry of the DM interactions shows up as a corresponding asymmetry in the spinwave spectrum for wave vectors parallel and normal to the magnetization.

There is another interesting feature of the in-plane magnons. The long-wavelength part of the spin-wave spectrum is determined by the anisotropy fields which are considerably different for the out-of-plane and for the in-plane motion of the magnetic moments and, consequently, the spin waves are elliptically polarized. In our theoretical picture the out-ofplane and in-plane motions are assigned to the angles ␽ ri and ␸ ri , respectively. In order to characterize the ellipticity of a

magnon in a specific mode band , the difference of the mean-square deviation of the out-of-plane and in-plane com- ponents of the magnetization, FIG. 6. Spectral resolution of the ellipticity, r (q), for the mag- non band having the largest weight at the Co monolayer in Co1 /Cu(001). The solid and dashed lines refer to the 100 and 010 directions of the magnetization, respectively.

nts of the magnetization, M 2 M 2 ri M r 2 ri 2 ri 2 B ri q ri 2 p ri 2 , 69

is introduced, where the notation 具典 ␣ refers to a quantummechanical expectation value with respect to the eigenstate labeled by ␣ of the Hamiltonian 共14兲. The ellipticity 共69兲 can obviously be expressed as a sum in reciprocal space,

r,q r q , 70 r q a r † q a r † q a r q a r q , 71

while r (q) can easily be calculated using the matrices in Eq. 20, r q S r q T r q T r* q S r* q . 72

In Fig. 6, ␩ r ␣ (q) is shown for the band having the largest weight in the cobalt layer in the case of Co/Cu共001兲 along the 共001兲 and 共010兲 directions. As the wave number is increased the isotropic exchange interaction plays an increasingly important role in the formation of the magnetic excitations and the ellipticity rapidly tends to zero. The band crossing and the asymmetry are also reflected by the ellipticity.

Finally in Table I we summarize the uniaxial anisotropy constants provided by Eq. 共67兲 together with those obtained by taking the band-energy difference between the cases of an in-plane and a normal-to-plane magnetization. 20,28,29 The results of the two types of calculations are in very good agreement. It is worth mentioning that the present method naturally supplies layer-resolved anisotropy constants that can 104436-9

L. UDVARDI, L. SZUNYOGH, K. PALOTÁS, AND P. WEINBERGER

TABLE I. The gap at the ⌫ point for monolayers with out-ofplane magnetization and the uniaxial magnetic anisotropy constants given by the present method and by the band energy difference method.

System

Fe 1 /Au(001) Fe 1 /Cu(001) Co 1 /Cu(001)

aGap (q⫽0)

␭ Present method ␭ Band energy diff.

56.8 Ry 42.8 Ry 42.9 Ry 17.5 Ry a 12.1 Ry 12.2 Ry 30.6 Ry 31.3 Ry

Magnetic dipole–dipole interaction is not included.

differ from the projection of the band-energy anisotropies to the layers as applied so far in magnetic anisotropy calculations.

IV. SUMMARY

In order to study low-energy spin-wave excitations of itinerant ferromagnets we have developed a relativistic firstprinciples method based on the adiabatic decoupling and rigid-spin approximation. A spin-wave Hamiltonian has been constructed starting from the Landau-Lifshitz equation and using a harmonic approximation for the free energy. Closed formulas for the derivatives of the free energy up to second order with respect of the polar and azimuthal angles have been derived within the framework of the magnetic force theorem by means of the fully relativistic Korringa-KohnRostoker method. Exchange and spin-orbit couplings have thus been treated in equal terms. We discussed in details the relevance of classical spin models with respect to the Hamiltonian of transverse magnons. The method has been applied to an Fe monolayer on Cu共001兲 and Au共001兲 surfaces and for a Co monolayer on Cu共001兲. Our calculations reproduced the gap at zero wave number due to spin-orbit coupling and provided uniaxial anisotropy energies in good agreement with the results of the band-energy difference method. We

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PHYSICAL REVIEW B 68, 104436 共2003兲 pointed out that a relativistic treatment gives rise to an asymmetry in the magnon spectra with respect to magnetic orientations along two in-plane easy axes. We also emphasized that in the case of an in-plane magnetized system the longwavelength magnons are elliptically polarized due to the difference of the second-order uniaxial and fourth-order inplane magnetic anisotropy.

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ACKNOWLEDGMENT

This paper resulted from a collaboration partially funded by the RTN network ‘‘Computational Magnetoelectronics’’ 共Contract No. HPRN-CT-2000-00143兲 and by the Research and Technological Cooperation Project between Austria and Hungary 共Contract No. A-23/01兲. Financial support was also provided by the Center for Computational Materials Science 共Contract No. GZ 45.531兲, the Austrian Science Foundation 共Contract No. W004兲, and the Hungarian National Scientific Research Foundation 共Grant Nos. OTKA T038162 and OTKA T037856兲.

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