- Email: [email protected]
Surface magnetism of Fe(110) from polarized electron scattering
Solid State Communications, Vol. 44, No. 7, pp. 1101-l 104, 1982. Printed in Great Britain. SURFACE MAGNETISM OF Fe(ll0)
0038-1098/82/431101-04$03.00/O Pergamon Press Ltd.
FROM POLARIZED ELECTRON SCATTERING
E. Tamura and R. Feder Theoretische Festkorperphysik, FB 10, Universitat Duisburg GH, D-4100 Duisburg, West Germany (Received 12 July 1982 by P.H. Dederichs)
Exchange- and spin-orbit-induced scattering asymmetry spectra of polarized slow electrons from the ferromagnetic Fe( 110) surface have been calculated by dynamical theory and found to agree with recent experimental data taken at room temperature. Comparison of exchange asymmetry spectra, obtained for various interaction and layer-dependent magnetization models, with the data implies firstly an enhancement of the surface magnetization by about 30% with respect to the bulk, and secondly the importance of spin-dependent localized inelastic electronelectron scattering processes.SPIN-POLARIZED low-energy electron diffraction (SPLEED) is currently establishing itself as a practicable method for studying magnetic properties of ferromagnetic crystalline surfaces. (The underlying physical concepts, the theoretical formalism and the principle of magnetic structure determination are described in a recent review article [ 11, which also gives extensive references to earlier theoretical and experimental work.) Recent experimental and theoretical efforts concentrated on the controversial Ni(0 0 1) surface, for which the temperature dependence of the surface magnetization near the critical temperature [2] and the layerdependent magnetization in the surface region at lower temperatures [3] have been determined, Most recently, Waller and Gradmann [4] have performed the first SPLEED experiments on a ferromagnetic Fe surface, a (110) face epitaxially grown on a W(110) substrate. In the present note, we report corresponding calculations for the exchange-induced scattering asymmetry A,,, the spin-orbit-induced scattering asymmetry A,, and the spin-averaged intensity I at T = 300 K. By comparing theoretical A, profiles, calculated for various magnetic model assumptions, with the experimental data of [4], we obtain information on the layer-dependent surface magnetization of Fe (110) and on spin-dependent inelastic scattering processes in crystalline ferromagnetic Fe. In the experimental set-up of [4], the spin polarization vector of the incident beam, produced by a GaAs photoemission source, is (anti)-parallel to the majority spin direction and normal to the scattering plane, which is a (0 0 1) mirror plane. As mentioned in [2] and substantiated in [5], this geometry allows, in good approximation, the separate determination of the exchangeinduced scattering asymmetry A,, which would arise in the absence of spin-orbit coupling, and of the
spin-orbit-induced asymmetry A,, arising in the absence of ferromagnetic long-range order. To make contact with the experimental data, we have calculated A,, by means of a relativistic multiple scattering formalism (cf. [l] and references therein) using a spinindependent effective scattering potential Y [6], and A, by means of a Schrbdinger-equation-based scattering formalism using successively an effective scattering potential V+ for spin alignment parallel to the ferromagnetic majority spin axis and I/- for spin antiparallel (cf. [l] and references therein). It is useful to decompose the spin-dependent potentials Y’ in the form Y’ = v:+Av;
(1)
within each monoatomic layer of magnetization M,, the spin-averaged value 7 is independent of M, and equal to the spin-independent V, and the spin splitting A V is in good approximation proportional to M,, (cf. [7]). In the usual muffin tin approximation, I/(‘) consists of spherically symmetric potentials Y(‘)(r) around the lattice sites and a spatially uniform inner potential V$“, which in turn can be decomposed as in equation (1). Accordingly, the model assumptions entering our calculations fall into two categories: non-magnetic and magnetic. The former determine A,, and the associated intensity; addition of the latter determines A,. The real part of the inner potential was assumed as spin-independent and as energy-dependent in a form found by LEED intensity studies from Ni [8,9]. The real ion-core scattering potentials were constructed as earlier (cf. [7]) from self-consistent nonmagnetic and ferromagnetic bulk charge densities assuming an energydependent exchange-correlation approximation. The resulting phase shifts were corrected for thermal lattice vibrations at room temperature using the bulk Debye temperature of 420 K. As shownin [7], the ferromagnetic
1101
1102
SURFACE MAGNETISM OF Fe(ll0)
spin splitting A8r is, in the nth layer of magnetization M,,, in good approximation proportional to M,. The values M, were viewed as parameters to be chosen such as to optimize the agreement of calculated exchange scattering asymmetry profiles A,, with their experimental counterparts. The influence of inelastic processes on the elastic scattering channel is, in the spirit of an optical potential, described by imaginary contributions to the effective scattering potential. In the frame of the muffin-tin approximation, both the spatially uniform ‘inner potential” and the spherically symmetric “ion-core potential” may thus have an imaginary part. The imaginary inner potential V&,typically of the order of 4 eV, is an important ingredient of all present-day LEED calculations (cf. [l] and references therein). A spherical imaginary contribution V,(r), which corresponds to a localization of inelastic events and leads to imaginary parts 6r1of the phase shifts 6r (in addition to the imaginary contributions associated with thermal lattice vibrations), has only recently been found to be of importance in a spin-polarized LEED analysis of W(0 0 1) [9]. In the present calculations from Fe(1 lo), we assume a uniform Vorincreasing with energy in a manner recently found for Ni [8]. In addition, we investigate the influence of adding imaginary 6r1as adjustable parameters. While both Vorand 6,, are spinindependent in calculating the spin-orbit asymmetry A,, in the absence of magnetism, magnetic order may cause a spin dependence of inelastic scattering crosssections and thus of V& and Sri. For collisions with the “magnetic” 3d electrons of Fe, spin dependence has been proposed on the grounds of various theoretical models [ 11-141, with partly controversial results. Collisions exciting 3p electrons into the empty, i.e. predominantly spin-down, part of the 3d band should generally have a larger cross-section for spin-down incident electrons as a consequence of exchange scattering (cf. [ 151). Since 3p excitation is more localized, it is plausible that, above the threshold of 50 eV [ 161, this process should manifest itself predominantly in a spin splitting 8, > i$, while 3d excitation would seem more likely to also induce a spin splitting of V&. In the present work, we abstain from first-principles microscopic calculations of AV,, and A&, but rather consider both spin splittings as parameters to be determined by comparing calculated and measured & profiles. From Vif, and Si a total spin-dependent mean free path h* resulting from both uniform and localized absorption is obtained as h;’ = 2V&/2E x [l - exp (-
+ N JtE -
#dl,
f: (21+ 1)
(2)
Vol. 44, No. 7
where E = Ekin + Vo, and N is the number of atoms per unit cell. As a last potential ingredient, we assumed an exponential-type surface barrier with image asymptotics. The geometry of the Fe(ll0) surface was taken from LEED intensity analyses [ 17, 181 as a simple truncation of the bulk with no relaxation of the topmost atomic layer. Comparison of our semi-infinite crystal calculations with the finite-thickness film data of Waller and Gradmann [4] is legitimate, since the data were found to be converged beyond 34 mono-layers [4], which is in accordance with theoretical LEED knowledge ([l] and references therein). As a check of our non-magnetic model assumptions and of the quality of the experimental data [4], we compare in Fig. 1 calculated and measured intensity I and spin-orbit asymmetry A,, vs. energy profiles. Good agreement is found with respect to existence and position of features and to general lineshape. The negative peak in A,, near 73 eV, which is very weak in the data, is strong only in the profile calculated with a reflecting barrier model. This identifies it as a surface resonance, which is very sensitive to details of the barrier model. We did not attempt to optimize this model, since most other features respond to it only very weakly. Anticipating the importance of imaginary phase shift parts 6ri for the exchange asymmetry (see below and Fig. 2), we display their influence on I and A,, in Fig. 1 (curves 2). Both are generally reduced and somewhat smeared out. Since the absorption cross-section corresponding to constant 6r1 [cf. equation (2)] decreases with increasing energy, features at lower energies are more strongly affected. Figure 1 shows that the absolute heights of the negative A,, peaks below 80 eV move closer to the experimental values; so do the relative peak heights in the intensity spectrum. In Fig. 2 we illustrate, for the typical example of the specular beam for an angle of incidence of 45’ with respect to the surface normal, how magnetic surface information is obtained by comparing exchange asymmetry A, vs. energy profiles, which have been calculated for various magnetic model assumptions, with their experimental counterpart. For the simplest assumptions - homogeneous magnetization Ml = M2 = . . . = Mb, spin-dependent uniform absorptive potential Vii = Vii and no localized absorption - the A, profile (curve 1 in Fig. 2) already shows some correspondence with experiment (curve 0), but there are several discrepancies. In particular, the calculated A, has a negative peak around 70 eV instead of a positive one, a peak at 74 eV instead of a dip and generally too low (even negative) values above 80 eV. Using for spin up and for spin down the imaginary phase shifts 6r1= 0.085, which were introduced in calculating I and A,, (see Fig. l), we find no improvement (see dashed curve 1 in Fig. 2). Assuming
Vol. 44, No. 7
1103
SURFACE MAGNETISM OF Fe(1 IO)
30 20 7 & -10
10
5 5 5
0
-
1.0
0 -10 20
-0
10
z
0
5 3 5 50
100 ENERGY [evl
z 0
-10
& t; F
10
Y
-10
0
kg 20 z
10 0 -10 10 0
ENERGY Id]
-6
I
so
II
1
I
II 100
I
ENERGY IeVl
Fig. 1. Intensity (part a) and spin-orbit asymmetry A,, @art b) of the 0 0 beam from Fe(ll0) for incidence at a = 45” in the (0 0 1) plane. Curves 3 : experiment [4]; curves 1 and 2: theory using imaginary phase shift parts Sri = 0 and S,, = 0.085 (for 1= 0, 1,2), respectively; with continuous surface barrier (full time) and non-reflecting barrier (dashed line). (It should be noted that the absolute values of the experimental A,, may be systematically larger by a common scaling factor, since the polarization of the electron gun was not exactly known [4].) I’&> v$ (e.g. with a splitting asymmetry of 15% as in curve 2) improves the situation above 80 eV and at 70 eV. However, there is a large positive peak at 74 eV, which is absent in the data. The assumption of localized
Fig. 2. Exchange asymmetry A,, of the 0 0 beam from Fe(ll0) for incidence at a = 45” in the (0 0 1) plane. Experiment [4] : curves 0. The theoretical profiles (curves l-5) have been obtained using the following magnetic model parameters: (1) M, = Mb for all IZ (homogeneous model), A Vi0= 0; S,i = 0 (full line) and 6~~=0~085for1=0,1,2;(2)asinfulllineof(l)except spin splitting of I$, by 15%; (3) as in (1) except 6, = 0.1 and 8fi + 0.07 for I = 0, 1,2; (4) AI& = 0, 6, and 6; as in (3), M1 = O.BM, (full line) and Mr = 0 (dead layer model) (dashed line); (5) as in (4) except Mr = 1.3Mr,. (For scaling of experimental A, cf. note on A,, in caption to Fig. 1.) absorption with 6; > SY,(see curve 3 for our present “best choice” of S$ also leads to reasonable agreement at 70 eV and in the energy range above 80 eV. The objectionable 74 eV peak is still produced, but only about half as high. It turns out that this peak is highly sensitive to changes in the surface magnetization Ml, while the range above 80 eV is fairly insensitive. As is shown in curves 4 in Fig. 2, the 74 eV peak increases with a reduction of Ml, and comparison with the data
1104
Vol. 44, No. 7
SURFACE MAGNETISM OF Fe( 110)
clearly rules out models involving reduced surface magnetization (in particular the dead layer model). By contrast , increasing Mr leads to significant improvement, best agreement resulting for M, = 1.3M, (see curve 5). Assuming, in addition, a reduced magnetization Mz = 0.8 Me for the second layer slightly worsens the overall agreement. This suggests that there should be at most a rather weak Friedel oscillation in the layer dependence of the magnetic moment. We note that removal of the 74 eV peak in curve 3 (for Vi > F,,) (without assuming S, > 8;) would either require an unphysically huge enhancement of the surface magnetization or a smaller spin splitting of P&. In the latter case, however, a negative peak remains near 70 eV in contrast to the data. Assuming S, > 8; is seen to also improve the agreement below 45 eV, i.e. below the 3p excitation threshold (near 50 eV [ 16]), which suggests localization and spin dependence of 3d excitation processes. As for the discrepancies in the absolute heights of the peaks near 49 and 55 eV, we note that these energies are close to the 3p-3d excitation threshold and that our present ad hoc Sg are independent of 1 and of energy. In fact, by assuming a set of Sf, decreasing with increasing I, which corresponds to a strong localization of the excitation process, we succeeded in reproducing the measured height of the negative peak near 55 eV. It is, however, unsatisfactory to inflate the number of adjustable parameters by trying arbitrary SF,for each I and energy. Rather, it is desirable to incorporate the detailed local excitation mechanism (cf. [ 151) in an optical potential I/:(r) and therefrom calculate the Sit. The above findings at a= 45’ are confirmed by comparing exchange asymmetry profiles calculated and measured at other angles. From the spin-independent Vel and the spin-dependent Si (which lead to curves 3 and 5 in Fig. 2), we have calculated the total mean free paths according to equation (2). The resulting spin asymmetry Ah = (A_ - A+)/@_ + h,) changes steadily from - 0.04 at 30 eV to - 0.02 at 130 eV. It is interesting to note that the present A, has the same sign, although smaller magnitude, as the handwaving prediction of [ 111, and agrees even in magnitude with the results of a more sophisticated microscopic calculation [14]. By contrast, the calculation of [ 121 yields much smaller magnitudes for Ah and, below about 80 eV, even the wrong sign. Our finding of an enhancement of the surface magnetization (at T = 300 K, i.e. well below T,) by about 30% is, to our knowledge, the first prediction of this
kind for the Fe(ll0) surface. It is, however, interesting to note that self-consistent electronic structure calculations for a seven-layer b.c.c. Fe(0 0 1) film [ 191 and for a single Fe(0 0 1) layer [20] also yield large enhancements by about 50% and 66%, respectively [21]. Acknowledgements - We appreciate fruitful discussions
with Prof. U. Gradmann and thank him and G. Waller for making their data available to us prior to publication. The sponsoring of this work by the Deutsche Forschungsgemeinschaft is gratefully acknowledged. REFERENCES ::
3. 4. 5. 6.
7. 8. 9. 10. 11. 12. ::: ::: ::: 19. 20. 21.
R. Feder, J. Phys. C14,2049 (1982). S. Alvarado, M. Campagna & H. Hopster,Phys. Rev. Lett. 48,51 (1982). R. Feder, S.F. Alvarado, E. Tamura, H. Hopster & E. Kisker (to be published). Fret? & U. Gradmann, Phys. B (1982) (in S.F. Alvarado, R. Feder, H. Hopster, F. Ciccacci & H. Pleyer 2. Physik (in press). We recall that spin-orbit coupling is inherent in the four-component Dirac equation and appears as a separate term only in two-component approximations. R. Feder & H. Pleyer, Surf: Sci. 117,285 (1982). J.E. Demuth, P.M. Marcus & D.W. Jepsen, Phys. Rev. Bll, 1460 (1975). P.J. Jennings & S.M. Thurgate, Surf: Sci. 104, L210 (1981). R. Feder &J. Kirschner, SurjI Sci. 103,75 (1981). A. Bringer, M. Campagna, R. Feder, W. Gudat, E. Kisker & E. Kuhhnann, Phys. Rev. Lett. 42,1705 (1979). R.W. Rendell & D.R. Penn, Phys. Rev. Lett. 45, 2057 (1980). J.A.D. Matthew, Phys. Rev. B25,3326 (1982). Soe Yin & E. Tosatti (to be published). R.K. Nesbet (to be published). B. Egert &G. Panzner, J. Phys. Fll, L233 (1981). R. Feder & G. Gafner, Surf: Sci. 57,45 (1976). H.D. Shlh, F. Jona, U. Bardi & P.M. Marcus, J. Phys. C13,3801 (1980). C.S. Wang & A. J. Freeman, Phys. Rev. B24,4363 ?%!f!ke
& L. Fritsche J Phys Cl4 89 (1981). The reduction of the s&face magnetiiation by about 30%, which was inferred for non-crystalline and polycrystalline Fe films from spin-polarized photoemission data by S.F. Alvarado [Z. Physik B33,51 (1979)] need not be in conflict with the enhancement for low-index surfaces of a single crystal, since ferromagnetism is sensitive to the geometrical structure (cf. the absence of ferromagnetism for f.c.c. Fe).
0038-1098/82/431101-04$03.00/O Pergamon Press Ltd.
FROM POLARIZED ELECTRON SCATTERING
E. Tamura and R. Feder Theoretische Festkorperphysik, FB 10, Universitat Duisburg GH, D-4100 Duisburg, West Germany (Received 12 July 1982 by P.H. Dederichs)
Exchange- and spin-orbit-induced scattering asymmetry spectra of polarized slow electrons from the ferromagnetic Fe( 110) surface have been calculated by dynamical theory and found to agree with recent experimental data taken at room temperature. Comparison of exchange asymmetry spectra, obtained for various interaction and layer-dependent magnetization models, with the data implies firstly an enhancement of the surface magnetization by about 30% with respect to the bulk, and secondly the importance of spin-dependent localized inelastic electronelectron scattering processes.SPIN-POLARIZED low-energy electron diffraction (SPLEED) is currently establishing itself as a practicable method for studying magnetic properties of ferromagnetic crystalline surfaces. (The underlying physical concepts, the theoretical formalism and the principle of magnetic structure determination are described in a recent review article [ 11, which also gives extensive references to earlier theoretical and experimental work.) Recent experimental and theoretical efforts concentrated on the controversial Ni(0 0 1) surface, for which the temperature dependence of the surface magnetization near the critical temperature [2] and the layerdependent magnetization in the surface region at lower temperatures [3] have been determined, Most recently, Waller and Gradmann [4] have performed the first SPLEED experiments on a ferromagnetic Fe surface, a (110) face epitaxially grown on a W(110) substrate. In the present note, we report corresponding calculations for the exchange-induced scattering asymmetry A,,, the spin-orbit-induced scattering asymmetry A,, and the spin-averaged intensity I at T = 300 K. By comparing theoretical A, profiles, calculated for various magnetic model assumptions, with the experimental data of [4], we obtain information on the layer-dependent surface magnetization of Fe (110) and on spin-dependent inelastic scattering processes in crystalline ferromagnetic Fe. In the experimental set-up of [4], the spin polarization vector of the incident beam, produced by a GaAs photoemission source, is (anti)-parallel to the majority spin direction and normal to the scattering plane, which is a (0 0 1) mirror plane. As mentioned in [2] and substantiated in [5], this geometry allows, in good approximation, the separate determination of the exchangeinduced scattering asymmetry A,, which would arise in the absence of spin-orbit coupling, and of the
spin-orbit-induced asymmetry A,, arising in the absence of ferromagnetic long-range order. To make contact with the experimental data, we have calculated A,, by means of a relativistic multiple scattering formalism (cf. [l] and references therein) using a spinindependent effective scattering potential Y [6], and A, by means of a Schrbdinger-equation-based scattering formalism using successively an effective scattering potential V+ for spin alignment parallel to the ferromagnetic majority spin axis and I/- for spin antiparallel (cf. [l] and references therein). It is useful to decompose the spin-dependent potentials Y’ in the form Y’ = v:+Av;
(1)
within each monoatomic layer of magnetization M,, the spin-averaged value 7 is independent of M, and equal to the spin-independent V, and the spin splitting A V is in good approximation proportional to M,, (cf. [7]). In the usual muffin tin approximation, I/(‘) consists of spherically symmetric potentials Y(‘)(r) around the lattice sites and a spatially uniform inner potential V$“, which in turn can be decomposed as in equation (1). Accordingly, the model assumptions entering our calculations fall into two categories: non-magnetic and magnetic. The former determine A,, and the associated intensity; addition of the latter determines A,. The real part of the inner potential was assumed as spin-independent and as energy-dependent in a form found by LEED intensity studies from Ni [8,9]. The real ion-core scattering potentials were constructed as earlier (cf. [7]) from self-consistent nonmagnetic and ferromagnetic bulk charge densities assuming an energydependent exchange-correlation approximation. The resulting phase shifts were corrected for thermal lattice vibrations at room temperature using the bulk Debye temperature of 420 K. As shownin [7], the ferromagnetic
1101
1102
SURFACE MAGNETISM OF Fe(ll0)
spin splitting A8r is, in the nth layer of magnetization M,,, in good approximation proportional to M,. The values M, were viewed as parameters to be chosen such as to optimize the agreement of calculated exchange scattering asymmetry profiles A,, with their experimental counterparts. The influence of inelastic processes on the elastic scattering channel is, in the spirit of an optical potential, described by imaginary contributions to the effective scattering potential. In the frame of the muffin-tin approximation, both the spatially uniform ‘inner potential” and the spherically symmetric “ion-core potential” may thus have an imaginary part. The imaginary inner potential V&,typically of the order of 4 eV, is an important ingredient of all present-day LEED calculations (cf. [l] and references therein). A spherical imaginary contribution V,(r), which corresponds to a localization of inelastic events and leads to imaginary parts 6r1of the phase shifts 6r (in addition to the imaginary contributions associated with thermal lattice vibrations), has only recently been found to be of importance in a spin-polarized LEED analysis of W(0 0 1) [9]. In the present calculations from Fe(1 lo), we assume a uniform Vorincreasing with energy in a manner recently found for Ni [8]. In addition, we investigate the influence of adding imaginary 6r1as adjustable parameters. While both Vorand 6,, are spinindependent in calculating the spin-orbit asymmetry A,, in the absence of magnetism, magnetic order may cause a spin dependence of inelastic scattering crosssections and thus of V& and Sri. For collisions with the “magnetic” 3d electrons of Fe, spin dependence has been proposed on the grounds of various theoretical models [ 11-141, with partly controversial results. Collisions exciting 3p electrons into the empty, i.e. predominantly spin-down, part of the 3d band should generally have a larger cross-section for spin-down incident electrons as a consequence of exchange scattering (cf. [ 151). Since 3p excitation is more localized, it is plausible that, above the threshold of 50 eV [ 161, this process should manifest itself predominantly in a spin splitting 8, > i$, while 3d excitation would seem more likely to also induce a spin splitting of V&. In the present work, we abstain from first-principles microscopic calculations of AV,, and A&, but rather consider both spin splittings as parameters to be determined by comparing calculated and measured & profiles. From Vif, and Si a total spin-dependent mean free path h* resulting from both uniform and localized absorption is obtained as h;’ = 2V&/2E x [l - exp (-
+ N JtE -
#dl,
f: (21+ 1)
(2)
Vol. 44, No. 7
where E = Ekin + Vo, and N is the number of atoms per unit cell. As a last potential ingredient, we assumed an exponential-type surface barrier with image asymptotics. The geometry of the Fe(ll0) surface was taken from LEED intensity analyses [ 17, 181 as a simple truncation of the bulk with no relaxation of the topmost atomic layer. Comparison of our semi-infinite crystal calculations with the finite-thickness film data of Waller and Gradmann [4] is legitimate, since the data were found to be converged beyond 34 mono-layers [4], which is in accordance with theoretical LEED knowledge ([l] and references therein). As a check of our non-magnetic model assumptions and of the quality of the experimental data [4], we compare in Fig. 1 calculated and measured intensity I and spin-orbit asymmetry A,, vs. energy profiles. Good agreement is found with respect to existence and position of features and to general lineshape. The negative peak in A,, near 73 eV, which is very weak in the data, is strong only in the profile calculated with a reflecting barrier model. This identifies it as a surface resonance, which is very sensitive to details of the barrier model. We did not attempt to optimize this model, since most other features respond to it only very weakly. Anticipating the importance of imaginary phase shift parts 6ri for the exchange asymmetry (see below and Fig. 2), we display their influence on I and A,, in Fig. 1 (curves 2). Both are generally reduced and somewhat smeared out. Since the absorption cross-section corresponding to constant 6r1 [cf. equation (2)] decreases with increasing energy, features at lower energies are more strongly affected. Figure 1 shows that the absolute heights of the negative A,, peaks below 80 eV move closer to the experimental values; so do the relative peak heights in the intensity spectrum. In Fig. 2 we illustrate, for the typical example of the specular beam for an angle of incidence of 45’ with respect to the surface normal, how magnetic surface information is obtained by comparing exchange asymmetry A, vs. energy profiles, which have been calculated for various magnetic model assumptions, with their experimental counterpart. For the simplest assumptions - homogeneous magnetization Ml = M2 = . . . = Mb, spin-dependent uniform absorptive potential Vii = Vii and no localized absorption - the A, profile (curve 1 in Fig. 2) already shows some correspondence with experiment (curve 0), but there are several discrepancies. In particular, the calculated A, has a negative peak around 70 eV instead of a positive one, a peak at 74 eV instead of a dip and generally too low (even negative) values above 80 eV. Using for spin up and for spin down the imaginary phase shifts 6r1= 0.085, which were introduced in calculating I and A,, (see Fig. l), we find no improvement (see dashed curve 1 in Fig. 2). Assuming
Vol. 44, No. 7
1103
SURFACE MAGNETISM OF Fe(1 IO)
30 20 7 & -10
10
5 5 5
0
-
1.0
0 -10 20
-0
10
z
0
5 3 5 50
100 ENERGY [evl
z 0
-10
& t; F
10
Y
-10
0
kg 20 z
10 0 -10 10 0
ENERGY Id]
-6
I
so
II
1
I
II 100
I
ENERGY IeVl
Fig. 1. Intensity (part a) and spin-orbit asymmetry A,, @art b) of the 0 0 beam from Fe(ll0) for incidence at a = 45” in the (0 0 1) plane. Curves 3 : experiment [4]; curves 1 and 2: theory using imaginary phase shift parts Sri = 0 and S,, = 0.085 (for 1= 0, 1,2), respectively; with continuous surface barrier (full time) and non-reflecting barrier (dashed line). (It should be noted that the absolute values of the experimental A,, may be systematically larger by a common scaling factor, since the polarization of the electron gun was not exactly known [4].) I’&> v$ (e.g. with a splitting asymmetry of 15% as in curve 2) improves the situation above 80 eV and at 70 eV. However, there is a large positive peak at 74 eV, which is absent in the data. The assumption of localized
Fig. 2. Exchange asymmetry A,, of the 0 0 beam from Fe(ll0) for incidence at a = 45” in the (0 0 1) plane. Experiment [4] : curves 0. The theoretical profiles (curves l-5) have been obtained using the following magnetic model parameters: (1) M, = Mb for all IZ (homogeneous model), A Vi0= 0; S,i = 0 (full line) and 6~~=0~085for1=0,1,2;(2)asinfulllineof(l)except spin splitting of I$, by 15%; (3) as in (1) except 6, = 0.1 and 8fi + 0.07 for I = 0, 1,2; (4) AI& = 0, 6, and 6; as in (3), M1 = O.BM, (full line) and Mr = 0 (dead layer model) (dashed line); (5) as in (4) except Mr = 1.3Mr,. (For scaling of experimental A, cf. note on A,, in caption to Fig. 1.) absorption with 6; > SY,(see curve 3 for our present “best choice” of S$ also leads to reasonable agreement at 70 eV and in the energy range above 80 eV. The objectionable 74 eV peak is still produced, but only about half as high. It turns out that this peak is highly sensitive to changes in the surface magnetization Ml, while the range above 80 eV is fairly insensitive. As is shown in curves 4 in Fig. 2, the 74 eV peak increases with a reduction of Ml, and comparison with the data
1104
Vol. 44, No. 7
SURFACE MAGNETISM OF Fe( 110)
clearly rules out models involving reduced surface magnetization (in particular the dead layer model). By contrast , increasing Mr leads to significant improvement, best agreement resulting for M, = 1.3M, (see curve 5). Assuming, in addition, a reduced magnetization Mz = 0.8 Me for the second layer slightly worsens the overall agreement. This suggests that there should be at most a rather weak Friedel oscillation in the layer dependence of the magnetic moment. We note that removal of the 74 eV peak in curve 3 (for Vi > F,,) (without assuming S, > 8;) would either require an unphysically huge enhancement of the surface magnetization or a smaller spin splitting of P&. In the latter case, however, a negative peak remains near 70 eV in contrast to the data. Assuming S, > 8; is seen to also improve the agreement below 45 eV, i.e. below the 3p excitation threshold (near 50 eV [ 16]), which suggests localization and spin dependence of 3d excitation processes. As for the discrepancies in the absolute heights of the peaks near 49 and 55 eV, we note that these energies are close to the 3p-3d excitation threshold and that our present ad hoc Sg are independent of 1 and of energy. In fact, by assuming a set of Sf, decreasing with increasing I, which corresponds to a strong localization of the excitation process, we succeeded in reproducing the measured height of the negative peak near 55 eV. It is, however, unsatisfactory to inflate the number of adjustable parameters by trying arbitrary SF,for each I and energy. Rather, it is desirable to incorporate the detailed local excitation mechanism (cf. [ 151) in an optical potential I/:(r) and therefrom calculate the Sit. The above findings at a= 45’ are confirmed by comparing exchange asymmetry profiles calculated and measured at other angles. From the spin-independent Vel and the spin-dependent Si (which lead to curves 3 and 5 in Fig. 2), we have calculated the total mean free paths according to equation (2). The resulting spin asymmetry Ah = (A_ - A+)/@_ + h,) changes steadily from - 0.04 at 30 eV to - 0.02 at 130 eV. It is interesting to note that the present A, has the same sign, although smaller magnitude, as the handwaving prediction of [ 111, and agrees even in magnitude with the results of a more sophisticated microscopic calculation [14]. By contrast, the calculation of [ 121 yields much smaller magnitudes for Ah and, below about 80 eV, even the wrong sign. Our finding of an enhancement of the surface magnetization (at T = 300 K, i.e. well below T,) by about 30% is, to our knowledge, the first prediction of this
kind for the Fe(ll0) surface. It is, however, interesting to note that self-consistent electronic structure calculations for a seven-layer b.c.c. Fe(0 0 1) film [ 191 and for a single Fe(0 0 1) layer [20] also yield large enhancements by about 50% and 66%, respectively [21]. Acknowledgements - We appreciate fruitful discussions
with Prof. U. Gradmann and thank him and G. Waller for making their data available to us prior to publication. The sponsoring of this work by the Deutsche Forschungsgemeinschaft is gratefully acknowledged. REFERENCES ::
3. 4. 5. 6.
7. 8. 9. 10. 11. 12. ::: ::: ::: 19. 20. 21.
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& L. Fritsche J Phys Cl4 89 (1981). The reduction of the s&face magnetiiation by about 30%, which was inferred for non-crystalline and polycrystalline Fe films from spin-polarized photoemission data by S.F. Alvarado [Z. Physik B33,51 (1979)] need not be in conflict with the enhancement for low-index surfaces of a single crystal, since ferromagnetism is sensitive to the geometrical structure (cf. the absence of ferromagnetism for f.c.c. Fe).