Surface magnetic structure analysis of Ni(001) by polarized electron diffraction

83

Surface Science 127 (1983) 83-107 North-Holland Publishing Company

SURFACE MAGNETIC STRUCTURE ANALYSIS POLARIZED ELECTRON DIFFRACTION

OF Ni(O01) BY

R. FEDER Theoretische Fesfktirperphysik, Rep. of Germany

Fachbereich

IO, Unioersitiit Duisburg

GH, D - 4100 Duisburg, Fed.

S.F. ALVARADO Institut fiir Festkiirperforschung, Rep. of Germany

Kernforschungsanlage

Jiilich, Postfach 1913, D -5170 Jiilich, Fed.

E. TAMURA Theoretische Festkijrperphysik, Rep. of Germany

Fachbereich

IO, Uniuersitiit Duisburg GH, D -4100 Duisburg, Fed.

and E. KISKER Institut fiir Festkijrperforschung Rep. of Germany Received

25 November

Kemforschungsanlage

Jiilich, Postfach 1913, D - 5170 Jiilich, Fed.

1982

For spin-polarized low-energy electrons diffracted from ferromagnetic Ni(OOl), the exchangeand the spin-orbit-induced scattering asymmetries A,, and A,, of two diffracted beams have been simultaneously measured at T = 300 K and at T = 520 K for several constant energies as functions of the polar angle of incidence. Corresponding “rocking curves” calculated by dynamical theory are in generally good agreement with the data. Analysis of A,, shows firstly that an energy-dependent exchange approximation is adequate, secondly that the topmost interlayer spacing is essentially bulk-like with a possible slight outward relaxation (l%), and thirdly that 3p + 3d excitation is significant. From A,, the following magnetic information is obtained: The surface magnetization for T = 300 K and 520 K is such as to imply, via extrapolation by mean field theory, a T = 0 value slightly enhanced (by + 5% f 5X) with respect to the bulk magnetization. The spin asymmetries of localized and uniform inelastic processes are very small, leading, for incident electron energy 69 and 89 eV, to a combined inverse mean free path asymmetry of less than 0.3%.

1.Introduction Magnetic properties of the surface region of 3d ferromagnets are in the focus of great current experimental and theoretical interest (cf. refs. [l-7]). In

0039-6028/83/0000-0000/$03.00 0 1983 North-Holland

particular, there is a still unresolved controversy about ferromagnetic Ni(OO1): recent self-consistent electronic structure calculations agree qualitatively regarding the non-existence of a magnetically “dead” layer, but contradict each other in yielding an enhancement 14-61 and a reduction [7] of the spin magnetic moment of the outermost layer compared to that of the bulk. Since the exchange part of the interaction of an incident electron with a ferromagnet depends on the relative orientation of the electron spin and the magnetization, spin-polarized low-energy electron diffraction (SPLEED) recommends itself as a method for studying surface magnetism (cf. review articles [8,9] and references therein). Most recently, this method has revealed the critical behaviour of the surface magnetization of Ni(OO1) [IO]. Also, it has been suggested to be practicable for determining the layer dependence of the surface magnetization at any fixed temperature (below T,) fl1,12]. In this paper, we report for Ni(OOl) - at room temperature and at 520 K the first analysis of surface magnetic moments based on comparing experimental exchange-scattering asymmetry versus polar angle profiles with their theoretical counterparts calculated for various surface magnetic moments and for several models of the spin dependence of electron-electron scattering processes.

2. Experiment The apparatus used in this work has been described in detail elsewhere [ 10,131. It essentially consists of an ultra-high-vacuum multiple-chamber system and a negative-electron-affinity AlGaAs( 100) source of spin-polarized electrons [ 141. The target (cf. ref. [lo]), cut in a window-frame shape from a single crystal, forms a closed magnetic circuit, which minimizes stray fields near the surface [IS]. The high-temperature measurements were performed using the experimental set-up shown in fig. 1 of ref. [lo]. The temperature of the target is stabilized by indirect pulse heating with a tungsten filament. The sample is magnetized after each heating pulse in order to account for the demagnetization induced by the heating-current pulse. The data collection is switched off during the heating and ma~eti~ng current pulses and is resumed after a convenient time delay. For this particular measuement the temperature fluctuations were kept at about - 3 K. The set-up is such that the polarization vector PO of the incident beam as well as the ma~etization M are normal to the scattering plane, i.e. parallel or antiparallel to n = (k, x k*)/(lk,

X&l),

where k, and k, denote the momentum of the incident and the scattered beam, respectively; n is oriented along a (110) direction. Experimentally we determine the scattering asymmetry: A’=

(l/]P,])(lyc-1!!)/(1:+

I!$

(1)

R. Feder et al. / Surjace magnetic structure analysis of Ni(001)

85

where P, is the degree of spin polarization of the incident electron beam and 1,” is the diffraction intensity for PO parallel (a = +) and antiparallel (a = -) and for M antiparallel (p = + ) and parallel (1_1= - ) to n. Thus u = p( - EL) corresponds to a spin orientation of the incident electron parallel (antiparallel~ to the majority spin direction in the target. For a given magnetization state of the sample, A’ is determined by switching the spin polarization of the incident electrons with a frequency of - 7 Hz and synchronously measuring the scattered intensities 1: by means of a desk computer which also calculates the scattering asymmetries. For Ni, the experimental quantities A,,=(A+-A-)/2

and

A,,=(A++A-)/2

(2)

represent in a very good approximation (within 1% of the measured value) the exchange-induced asymmetry in the absence of spin-orbit coupling and the spin-orbit-induced asymmetry in the absence of magnetism (cf. ref. [ 121).

3. Theory In view of the above decoupling of spin-orbit and magnetic effects there is no need for developing a multiple-scattering formalism including both interactions simultaneously, which would be complicated due to a loss of symmetry of the effective interaction potential. Instead, the two scattering asymmetries A,, can be calculated separately: A,, by means of a relativistic LEED and A,, formalism applied to the paramagnetic crystal, and A,, by means of a non-relativistic scalar formalism (neglecting spin-orbit coupling) applied to the corresponding ferromagnetic crystal. The two formalisms are reviewed in detail in ref. [S]. In the magnetic calculations for different layer-dependent magnetizations M, - with layer index n - the computational effort is greatly reduced by means of a linear approximation scheme at the level of the layer transfer matrices, which was introduced and validated in ref. [ 1 I]. The present calculations of A,, and A,, for Ni(OO1) share the following non-magnetic model characteristics. The exchange part of the ion-core potential is approximated firstly independent of the electron energy as in paramagnetic and ferroma~etic self-consistent band structure calculations [ 161and secondly, starting from the band structure ground state charge densities (p for way [17]. Thermal lattice 4, and P’, p1 for A,,), in an energy-dependent motion at room temperature is taken into account via imaginary parts of the phase shifts, assuming a Debye temperature of 335 K for both bulk and surface atoms (cf. ref. [18]). We also study the influence of using a smaller Debye temperature for the surface atoms (cf. refs. [19,20]. The real and imaginary parts of the inner potential are both taken as energy-dependent from earlier LEED intensity studies [ 18,211. Going beyond the uniform absorption contained in the inner potential, we

86

R. Feder et al. / Surface magnetic structure analysis of Ni(iW1)

allow for the more localized nature of inelastic processes like the excitation of 3d and 3p electrons - by positive imaginary contributions 6,i to the phase shifts (in addition to those describing lattice vibrations~. Such localized absorption was found to have a significant influence in a SPLEED analysis of W(OO1) (221. In the absence of a microscopic calculation of the associated optical potential, we consider the 6,i as adjustable parameters to be estimated from comparison of calculated and measured asymmetry profiles. The extent to which the 6,i physically correspond to 3d or 3p excitation can be assessed by comparing their values at energies well below and above the threshold energy for 3p + 3d excitation (for Ni: Ekin = 61 eV 1231). In view of determining the local-absorption-induced aIi, it seems relevant to consider a third kind of imaginary phase shift contributions, ali, which arise from treating the ion-core scattering as embedded in a uniform electron gas. Formally, this can be done by replacing, in the Dyson equation for the single-site T matrix, the free electron propagator by a self-energy-corrected propagator (cf. ref. [24J and references therein). Denoting by V, and Vi the negative real and imaginary parts of the inner potential (self-energy) and by S,(E - V, - V;) the total phase shift, a good approximation is obtained from first-order expansion as [25] ~~i(E-

K(E)-iv,(E))=

- [d&,,(E-

Y,(E))/d(E-

I/,(E))]

y(E).

(3)

In contrast to the thermal and excitation parts, gli is seen to be either positive or negative depending on whether the slope of a,, is positive or negative. To our knowledge, this correction has been neglected in all recent LEED analysis with the exception of a study on Al(001) [26], according to which c?,~ is significant for small I below about 60 eV. The surface potential barrier is assumed to be of an exponential form near the surface and of the image form asymptotically (cf. ref. [27]). The relaxation a,, of the topmost atomic layer is varied between - 2.5% and + 2.5% of the bulk interlayer spacing with the aim of determining it by comparison with experimental data. The fundamental magnetic property, to which A,, owes its existence and its detailed behaviour as a function of incident energy and scattering geometry, is the temperatureand layer-dependent thermal-average spin magnetic moment M,,(T). This quantity can, for any given T, be determined by comparing A,, profiles, which are obtained by calculations for assumed values of M,, (in general M, * M2 * M3 etc.), with the corresponding experimental A,, profiles measured at T. In order to reduce the number of trial models, we specify only M, (T = 0) = M,” and therefrom calculate M,(T) as described below. Beyond the obvious “homogeneous model” (Mp = Mf = . . . = Ml, where Mt denotes the bulk magnetic moment at T = 0) and a prototype “dead-layer model” Mt), it seems most interesting to use models obtained (M;=O, M;=... recently from first principles by self-consistent spin-dependent electronic structure calculations for thin films consisting of a finite number of atomic layers 14-71. The results for a 9-layer film [7] and a 5-layer film [5], the central planes

R. Feder et al. / Surface magnetic structure analysis of Ni(OOl)

0.21

2

L

6

2 LAYER

I NUMBER

87

6 n

Fig. 1. Spin magnetic moments M, (relative to the zero temperature bulk magnetic moment Ma) of the topmost six monatomix layers of ferromagnetic Ni(OO1). At T = 0: homogeneous model (0) predictions by Jepsen et al. [5a) (i) and by Wang and Freeman [7] ( x), taking the central layer moment as the bulk moment. The corresponding results at 7’= 300 K and 520 K were therefrom obtained by means of eqs. (5).

of which have a magnetization very close to the bulk magnetization M,, = 0.6~~ [28], are shown - normalized to M, - in fig. la [29]. While both calculations are seen to predict deviations from the bulk value for the outermost layer, there is the striking contrast of an enhancement of about 5% according to ref. [5] as opposed to a reduction by about 20% according to ref. [7]. For deeper layers, both calculations predict an oscillation of opposite sign; according to ref. IS], this surface-induced perturbation is significant only for the second layer, whereas it still strongly affects the third according to ref. 171. Results for a 5-layer film reported in ref. f4] show the same relative enhancement for the top layer as ref. [5] and a somewhat smaller relative reduction for the second layer. A non-self-consistent extension of the 5-layer result [5] to 13 layers [5] also yields such enhancement. In view of comparing with experiment, which was done at room temperature and at T= 520 K, we require M,(300 K) and M,(520 K). Since realistic first-principles calculations of M,,(T) at arbitrary finite T for semi-infinite 3d ferromagnets are not yet available, we adopt the following semi-empirical approach based on a Heisenberg model in the mean field approximation. Restricting ourselves to nearest-neighbour exchange, we choose the exchange constant (assumed to be the same for bulk and surface) such that the experimental Curie temperature (T, = 631 K for Ni) results for all layers [30]. For the effective spin S, in the n th layer we take S, = 0.5 in the bulk, since this value was found to reasonably reproduce the experimental temperature dependence of the bulk Ni magnetization at intermediate temperatures [31], and $J%

= M?Y@

>

(4)

R. Feder et al. / Surface magnetic structure analysis of Ni(OO1)

88

where the magnetic moments are those at T = 0. Defining the reduced temperature t = T/T, and the reduced layer-dependent magnetic moment m, = M,(T)/Mf, one obtains (cf. refs. [32-341) for a semi-infinite stack of layers the set of mean field equations

=4,[~,(zllml-z,m2>/(zt)l, mn=Bs [~n(zIImn+zJmn-I +m,+d)/(zd], mN=BS,.,[~NmN/t]~ ml

n

(5c)

where si = 3Si/(Si + 1) for i = 1, n, N; z is the total number of nearest neighbours, and z,, and zI give the nearest neighbours in an atomic layer and in the adjacent layer, respectively. For the present case of an fee (001) surface we have z,, = zI = 4. B,, . . ] is the usual Brillouin function for spin S,. In eq. (4b), n = 2,. . . , N - 1, where N is chosen large enough to represent a typical bulk layer as described by eq. (4~). Eqs. (4) are, for given t, solved numerically by iteration starting from an arbitrary set of initial values {m,). For room temperature, starting from mi = 0.8, convergence to within lo-’ was reached in 12 iterative steps. We note that eqs. (4) are somewhat more general than the analogous equations in refs. [32-341, as they allow for arbitrary zero-temperature values M,” different from the bulk value Mt. The T = 300 K and 520 K results thus obtained for the three M,” models of fig. la are, after renormalization to Mz, shown in figs. 1b and lc. We firstly notice the well-known enhanced reduction of the surface magnetic moment (cf. refs. [1,2,35] and references therein). Secondly, the similarity between the homogeneous case and the results from ref. [5], and the comparatively strong deviation of the results from ref. [7], found at T = 0, are seen to persist at elevated temperatures. A particularly interesting question concerning the interaction between slow electrons with a ferromagnet is the spin dependence of the inelastic channel of electron-hole pair excitation, which entails a spin dependence of the inelastic mean free path:

[.

x;‘(E)=

[l +/LA(E)]

h-‘(E),

(6)

where p = + ( - ) refers to spin parallel (antiparallel) to the majority spin axis, E denotes the kinetic energy, X- '(E) the inverse mean free path for zero magnetization and A(E) its magnetization-induced asymmetry. Such spin dependence was proposed in refs. [36,37] and described by a simple model, according to which singlet scattering should dominate, leading to A; ’ < XI I, i.e. A < 0, with A = -0.05 for Ni below the threshold for (spin-independent) at about 80 eV. More sophisticated plasmon excitation and, e.g., -0.035 microscopic model calculations for Fe and Ni [38] and Fe [39] confirmed the effect, but reduced in magnitude by a factor ranging between 6 and 10. Ref. [38], however, even predicts the opposite sign over wide ranges of energy. In

R. Feder et al. / Surface magnetic structure analysis of Ni(OO1)

89

SPLEED theory, spin-dependent pair production can be accounted for by a spin-dependent uniform imaginary potential r/i”, if nearly free electrons are excited, and/or by spin-dependent imaginary contributions Sfi to the ion-core potential [40]. The spin splitting of yip can be characterized by an asymmetry Ai such that KP=(l

+j.UQIJi:,,

/_4=+,

-.

(7)

In the present study, we have calculated A,, profiles assuming various spin splittings of Vi and of the Sli in order to determine the actual splitting values via comparison with experiment. The inverse mean free path “(1’ can then easily be obtained from V,fi and the total inelastic scattering cross section associated with the 8; as 1) [1 -exp(-4601,

x,‘=2Y,~/~+N(a/2E)C(22+ I

(8)

where E = Ekin + V, and N is the number of atoms per unit cell.

4. Results and discussion Although the magnetic surface information is exclusively coded in the exchange asymmetry A,, , an accompanying analysis of the spin-orbit asymmetry A,* is valuable firstly for establishing the relevant non-magnetic model quantities and secondly for verifying the decoupling of exchange and spin-orbit interaction in our geometry. Measurements and calculations have been performed for the specular beam in the (110) azimuth and for the (10) beam (which has the same scattering plane as the specular beam) at constant energies as functions of the polar angle 9 (“rocking curves”). This mode, which has been found preferable for both theoretical and experimental reasons in a recent SPLEED analysis of Pt( 11 I) 141,421, also allows more definite conclusions in the present work. 4.1. Spin-orbit asymmetry and non-magnetic

information

The influence of a variation a,, of the topmost interlayer spacing and of a variation of the local absorption contributions 6,; to the ion-core phase shifts is - for two ion-core potentials V, and V, constructed with exchange dependent and independent of energy, respectively (see section 3) - demonstrated in fig. 2 for (10) and (00) beam rocking curves at energy 89 eV. In order to determine via comparison with the experimental data - the optimal choice of these three model characteristics, it is desirable to find features in the A,, profiles, which are selectively sensitive to variation of only one or two of the model cnaracteristics. In order to discriminate between V, and V,, let us first focus on the

90

R. Feder et al. / Surface magnetic structure analysis of Ni(001)

01

’ me’4 T=300K

IO BEAM

2

-6 2

0

10 POLAR

20

ANGLE 3 (deg)

-6 0

10 POLAR

POLAR

ANGLE

9

20 ANGLE

3 (dog )

(degl

Fig. 2. Spin-orbit-induced scattering asymmetries A,, of the 00 beam and the 10 beam from Ni(OO1) in a (110) scattering plane at kinetic energy E = 89 eV and temperature T= 300 K as functions of the polar angle of incidence 9. The theoretical results have been calculated for top

R. Feder et al. / Surface magnetic structure analysis

20

of Ni(001)

30

POLAR ANGLE S ldeg

20

30

WLAR

1

LO

ANGLE 3 (deg )

p -2

20

30 POLAR

ANGLE

LO 9 (degl

and + 2.5% (- - -), imaginary phase shift layer relaxation&,,= -2.5% (.-.-.). 0% ( -) contributions a,, = 0.0, 0.1, 0.2 and 0.3 (for I = 0, 1, 2) as indicated in each panel; parts (a), (c), (d) and (f) were obtained using an energy-dependent potential Va [17], and parts (b) and (e) using a band structure potential Va [ 161; the bulk and surface layer Debye temperatures Tg and T& were both 335 K for parts (a), (b), (d), (e) and 335 K and 210 K, respectively, for parts (c) and (f). Vertical bars: experimental A,, data.

92

R. Feder ef al. / Surfuce magnetic structure analysis of Ni(OO1)

Table I Imaginary contributions to ion-core and 89 eV; 6z(Tt$) and Sl(TA) T= 300 K (upper rows) and 520 K and 210 K, respectively; gfi (cf. eq. and an is associated with localized

scattering phase shifts for I = 0, 1, 2 at kinetic energies 49, 69 designate the contribution due to thermal lattice motion at (lower rows, values in brackets) for Debye temperatures 335 K (3)) arises from the embedding in a homogeneous electron gas absorptive processes

Energy 49

0

69

2 0

0.067 0.026 0.015

0.154 0.061 0.034 0.229

- 0.035 - 0.025 -0.001 -0.031

0.0 0.0 0.0 0.2

(0.061)

(0.349) 0.111

-0.023

0.2

(0.078) 0.036

(0.176) 0.081

- 0.003

0.2

(0.058) 0.129

(0.122) 0.306

- 0.028

0.1

(0.215) 0.073

(0.406) 0.174

- 0.02 1

0.1

(0.122) 0.066

(0.274) 0.146

- 0.004

0.1

(0.107)

(0.246)

.0.097

1 0.047

89

0

2

negative 00 beam A,, maximum between 31” and 32”, which responds significantly more weakly to changing S,, and 6,i than to going from V, (fig. 2d) to V, (fig. 2e). For V,, this feature is below - 1% for all S,, and Sfi, while it is always above - 1% for V,. Comparison with experiment therefore favours V,. This finding is confirmed by the 10 beam profiles (figs. 2a and 2b) between 8” and 16’, which are consistently too negative for V, (fig. lb). (An increase of 6,i to 0.2 or beyond for V,, while bringing A,, closer to the data for 8’ < 8 < 14’, increases the discrepancy around 16’.) The parameters S,, (top layer relaxation), 6,i (imaginary phase shift part due to localized inelatic scattering) and TA (surface Debye temperature) have therefore to be determined from the A,, profiles calculated for V,. Since a reduction of T,$ with respect to the bulk value Yf’i entails an increase of the total imaginary phase shifts (cf. table l), we anticipate a linkage of the influence of varying 7’; and Sli for A,, features mainly produced by the top layer. In the 00 beam profiles obtained with T; = 335 K (fig. 2d), the range below 16’ suggests a value of & between 0.1 and about 0.15. This choice lies within the larger range 0.0-0.2 indicated by the 10 beam profiles between 9 = 10” and 16” (fig. 2a). Lowering the surface Debye temperature T& to the extreme perpendicular-vibration value 210 K (inferred from ref. [20] and close to the value 225 K given in ref. [ 191, produces in the 00 profile for each fixed Sji (cf. fig. 2f) a substantial enhancement both

R. Feder et al. / Surface magnetic structure analysis of Ni(OO1)

93

of the negative peak near 15” and of the positive peak near 24”. Comparison with the data favours, for each S,i, the TA = 335 K results (fig. 2d). For the 10 profiles (fig. 2c), TG = 210 K for S,i = 0.1-0.2 improves the agreement with the data, while worsening it between 2” and 10”. We can therefore maintain our conclusion concerning 6,i at 89 eV but have to seek further information on Th from other profiles. The influence of including the embedding correction sri (cf. eq. (3)) is almost negligible at 89 eV and therefore not shown in fig. 2. For the relaxation S ,2, the 00 and 10 profiles at E = 89 eV only allow the conclusion that a value between -2.5% and +2.5% seems reasonable. A more precise value will be seen to emerge from the following analysis of A,, rocking curves at E = 69 eV (fig. 3). The preference for the energy-dependent ion-core potential I’,, which we deduced above from the 89 eV rocking curves, is confirmed by results at 69 eV. To avoid redundance, we therefore show in fig. 3 only results for I’,. Also, the effect of bIi (eq. (3)) is still very small at 69 eV. Since inelastic scattering cross sections are generally energy-dependent, the influence of 6,i and the partly concurrent influence of varying TA have to be studied again at 69 eV. The 00 beam A,, profiles calculated with TA = T; = 335 K are seen (fig. 3c) to respond, for 22” < 9 < 26O, particularly strongly in absolute height and line shape to increasing S,i from 0.0 to 0.3. For the reduced TA = 210 K (fig. 3d), similar behaviour occurs starting from smaller A,, values at 6,i = 0.0. Best agreement with the experimental data is reached for TA = 335 K and 6,i = 0.2 (fig. 3~). For TA = 210 K (fig. 3d), the relatively best agreement is produced by S,i = 0.1, but the discrepancies near 8 = lo, 25” and 35’-40” render it inferior to that in the TA = 335 K case. The 10 beam profiles at 69 eV (figs. 3a and 3b) support values of Tf,closer to 335 K and of S,i around 0.2 on the basis of a “minimal misfit” in the very sensitive range 6” < 6 < 10’. In particular, we note that, for any value of the relaxation a,, (between - 2.5% and + 2.5%), the calculated A,, maximum near 6 = 7’ is raised by lowering T& i.e. moves away from the experimental data. From the strong sensitivity to a,, in the range 6” < 9 < 10” in fig. 3a, a contraction by more than about 0.01% is clearly ruled out. This leaves a,, in the expansion range O-2.5%. Turning to the 00 beam (fig. 3c), the S,,-sensitive range 18” < 6 < 24” permits a further specification of S,, to a value between 0% and about 1%. For A,, rocking curves at a third energy, 49 eV, the potential V, and a surface Debye temperature of about 335 K are also found to be adequate. From the variation of the 00 and 10 beam curves with S,i and a,, (see fig. 4) the value a,, = 0 is suggested, expecially by the 10 beam. Inclusion of the embedding correction d,i (eq. (3)) leads to a mild overall improvement for the 10 beam (fig. 4a) and to a substantial improvement for the 00 beam (fig. 4b) in the range below 18’. As for the relaxation S,,, the message is less clear. While the 00 curve at 6 = 14” implies S,, = 0% and the 10 curve for 0 5 6 I 4” points to a value between 0 and + 2.5’%, the 10 curve for 8 > 6” is best reproduced by

R. Feder et al. / Surface magnetic structure analysis of Ni(OO1)

94

= - 2.5%. The latter finding is, however, weakened by the fact that for 9 2 10” large exit angles of the 10 beam at E = 49 eV render the measurement more difficult. In view of further examining F’g and of strengthening the magnetic A,, analysis (see below), we extended our study to an elevated temperature (520 K), which is still below the Curie temperature (631 K). A selection of corresponding A,, rocking curves is shown in fig. 5. We firstly notice, by comparing with figs. 2 and 3, that all the calculated results are very close to those obtained at 300 K. This is in accordance with earlier spin-orbit SPLEED results (cf. ref. [8], and references therein), which have shown that, for increasing temperature, spin polarization (or, equivalently, A,,) can - in contrast to the well-known decrease of LEED intensities - remain almost unaltered or even increase. Most of the experimental A,, data at 520 K are also close to their 300 K counterparts. But there are some still unexplained S,,

I.

2

0

-2

I

2

0

-2

-L

L

2

0

-2

Y 0

II

I

I

10

20

POLAR

ANGLE

I

-L 0

10

J idegl FOLAR

ANGLE

20 3 (deg 1

--+

95

R. Feder et al. / Surface magnetic structure analysis of Ni(OO1)

20 POLAR

I..

1

.

20 ANGLE

3

I.0 ldeg

1

I

I,,

30 POLAR

30 ANGLE

LO J (dcg

I

Fig. 3. As fig. 2 except: E = 69 eV, ion-core potential VE throughout; Debye T& = T,$ = 335 K in parts (a) and (c). and Ti = 335 K in parts (b) and (d).

temperatures

deviations, notably for the 00 beam at 89 eV for B > 32’. In fig. 5a, the 00 curves clearly favour TA = 335 K, while the 10 curves do so more weakly. For the relaxation a,,, comparison between experiment and theory suggests an expansion between 0% and 2.5%. For all the ten different experimental A,, rocking curves, we have thus reached consistently good to fair agreement with theoretical results calculated with specific model characteristics. Firstly, the ion-core potential V, is appropriate, which involves a local-density exchange contribution decreasing with increasing energy. This determination of a suitable exchange approximation already from A,, is very useful in view of the magnetic analysis, where the exchange interaction is of fundamental importance. Secondly, we have found a possible slight outward relaxation of the topmost layer between 0% and 1% at T = 300 K and between 0% and 2.5% at T = 520 K. This is consistent with results from LEED intensity analyses ( + 1.1% [ 181 and 0% [43]) and from a high-temperature LEED spin polarization analysis (+2.5% [27]), the latter value being possibly due to an enhanced thermal lattice expansion coefficient. In contrast to this consensus between various different LEED analyses, an ion-scattering study [20] reports an appreciable contraction S,, = - 3.9 k 0.5%, which is a rather novel result for an fee (001) surface.

R. Feder ef al. / Surface magnefic structure analysis of Ni(oO1)

F‘OLAR ANGLE J (deg ,

POLAR ANGLE 3 (dql

Fig. 4. As fig. 2 except: E = 49 eV, potential VE and T& = T; = 335 K only. The dotted curves in the top panels parts (a) (10) beam) and (b) (MI beam) demonstrate (for a,, = 0%) the effect of including b,i (eq. (3)).

A synopsis of the imaginary phase shift contributions involved in our calculations is given in table 1. The embedding corrections & (cf. eq. (3)) are seen to have a sign opposite to the lattice vibration corrections SE and a magnitude reduced (except for I = 1 at 49 eV) by a factor of a least two. This explains their significant beneficial effect on some features at 49 eV and their relative unimportance at 69 and 89 eV. The present A,, analysis thus indicates - in accordance with the intensity analysis on Al by Le Bosse [26] - that (T,i should be taken into account below about 60 eV. The local-absorption S,i was found to be zero at 49 eV and non-va~s~ng at 69 and 89 eV. The values q,, = 0.2 at 69 eV and 0.1 at 89 eV imply inelastic mean free paths of 15 and 31 A, respectively. By comparison, the uniform absorption mean free path (corresponding to our Vi) is about 5 A at these two energies. Recalling that the threshold for inelastic scattering by 3p electrons is 61 eV 1231, our &ri values

R. Feder et al. / Surface magnetic structure analysis of Ni(OO1) bl

97

I I I I I I I IT

4-

10

/I 1I

2-

BEAM

I I mev T.520K _ r; = 335K

1 I

6,,=+O.2 0

10

_

20

61,= +0.2 -4

I

I

20

30

POLAR

ANGLE

3

’

’

’ ’ ’ ’ ’ ’ ’ ’ 30 20 8 (deg POLAR ANGLE

’

’

40

1

LO (deg)

Fig. 5. As fig. 2 except: T = 520 K, potential VE only. (a) T& = 335 K (upper panel for each beam) and TS, = 210 K (lower panel, E = 89 eV; (b) T;J = 335 K, E = 69 eV.

suggest the relevance of this loss process for elastic SPLEED analyses. It therefore seems worthwhile to replace the present crude parameterization (Sli as an I-independent fit parameter for I = 0, 1, 2 and 6,i = 0 for 12 3) by a first-principles theory. As for the Debye temperature TA of the topmost layer, which enters isotropically into our calculations via imaginary phase shift parts (see table l), all the above A,, results - with the exception of a single feature in one profile - suggest that 335 K is more appropriate than 210 K. To assess the

9x

R. Feder et al. / Surface magnetic structure analysis of Ni(001)

range of TA compatible with our data, we have performed some further calculations. Using Ti = 375 K (the bulk value quoted in ref. f44]) and Tfs = 235 K (the corresponding normal-vibration surface value inferred from ref. f20]), the A,, profiles are quite close to those obtained for the above combination (335 and 210 K) and therefore also discriminated against by our data. We estimate that T& should be above about 300 K. In comparing this with the surface-perpendicular value 225 K predicted by ref. [ 191, we note that the average from these values, 312 K, is compatible with our result. A value of T& close to the bulk value was, incidentally, also found in a recent SPLEED analysis for Pt( 111) [41]. We conclude that, for the diffraction conditions used, multiple scattering generally mixes the effects of perpendicular and parallel surface lattice vibrations such that the resulting average T; is appropriate. We have thus, via A,,, determined the relevant non-magnetic model characteristics. Using these as a fixed input for calculating the exchange-induced scattering asymmet~ A,, , we can now proceed towards our chief goal, the magnetic structure analysis via A,,. 4.2. Exchange asymmetry

and magnetic information

In the following, we present the A,, rocking curves measured simultaneously (see eqs. (1 and 2)) with the above A,, curves, and compare them to their theoretical counterparts obtained for the magnetic surface structure models, which were introduced in section 3 (see in particular fig. l), and for typical magnetic splittings of the imaginary phase shift parts 6,i and of the uniform imaginary potential Vi. For the 10 beam at 300 K and 89 eV (fig. 6) we notice that the theoretical A,, curves in the range 0 I 9 < 12” are affected appreciably by a spin splitting AS,, = (S,; - a;)/2 = 0.001 as well as by an imaginary potential asymmetry Ai = kO.0044 (cf. eq.(7)), whereas the change in going from the magnetization model of Jepsen, Madsen and Andersen (JMA) [5] to that of Wang and Freeman (WF) ]7] (see also fig. 1) is comparatively small. Such selective sensitivity to different types of magnetic ingredient is, as was emphasized in ref. [ll], very valuable for their determination. Comparison with the experimental data shows that both AS,, = 0.001 and Ai > 0 are ruled out, while in ref. Ai = - 0.0022 appears favoured. (As was shown by explicit calculations [ 1 l] and is also apparent from the present results for Ai = -0.0044, 0 and + 0.0044, the exchange asymmetry A,, scales, for small Ai, in good approximation linearly with A,. We can therefore, from A,. for Ai = 0 and -0.0044, interpolate to A,, for A, = -0.0022.) The positive A,, peak at 8 2: 17.5O is seen to increase strongly when going from the JMA model to the WF model and further to the dead layer model, i.e. when decreasing the magnetization of the topmost layer. This is, by the way, a strong demonstration that in multiple scattering theory, and consequently in reality, A,, is in general not propor-

R. Feder et a/. / Surface magnetic strucfure analysis of Ni(O01)

I:,

0

1

IO POLAR ANGLE

99

I

‘1’

20 8 (deg

1

in a (110) Fig. 6. ~change-indu~ scattering asymmetry A,, of the IO beam from Ni(~l) scattering plane at kinetic energy 89 eV and temperature T = 300 K. Panel A: theoretical results obtained for (temperature-extrapolated) surface magnetization model WF ([7] and fig. I) with S,, = OS, 6,i = 0.1 and with imaginary-phase-shift spin splitting AS,, = 0 and imaginary potential A8,i=0.001 and A,=0 (.s....), AS,i=O and Ai= +0.0044 asymmetry A,=0 ( -), Panel B: as panel A, but magnetization model (- - -), AS,, = 0 and Ai = -0.0044 ( .-.--)_ model WF JMA (ref. [5] and fig. 1). Panel C: a,, = 0.1, AS,, =i 0, Ai = 0 for magnetization and model with p~ama~etic top layer (“dead layer model”) (.-. -.f. (- - -), JMA ( -) Panel D: as C, but 6,i = 0. Vertical bars: experimental A,, data.

tional to the top layer magnetization or some average surface magnetization (cf. also ref. [ 111). Comparison with the experimental data clearly favours the JMA model. In the range below i? = 12’, the calculations using 6,i = 0.1 (as established above from A,,) (see fig. 6 panel C) also favour the JMA model, but there remains some discrepancy with the data. As implied above, this

100

R. Feder et al. / Surface magnetic structure analysis of Ni(001) 2-

00

20 POLAR

ANGLE

BEAM

30 9 (deg

89 8” 3WK

1

Fig. 7. As panels C and D of fig. 6, but 00 beam.

discrepancy can be removed by choosing A,= -0.005. An alternative way of improving the agreement is to reduce S,i (as shown in fig. 6 panel D for 6,; = 0). This ambiguous situation stresses the need of going beyond the present crude

0

IO POLAR

20 ANGLE

9

i&q1

Fig. 8. As fig. 6 but 69 eV (instead of 89 ev), S,i = 0.2, and panel D replaced by a lower part added to C: WF (---) and JMA () model results for 6,i = 0.2 and top layer relaxation 13,~ = + 1.0% (instead of 6,, = 0).

R. Feder et al. / Surfore magnetic sfructure analysis of Ni(OO1)

20 POLAR

ANGLE

101

30 ?t tdeg I

Fig. 9. As panel C (upper part) of fig. 8, but 00 beam.

parameterization of 6,i. It does, however, not affect our conclusion concerning the magnetization model. For the 00 beam at 89 eV (fig. 7), the A,, data show generally much smaller absolute magnitude than the calculated results and no definite conclusion can be drawn. For the 10 beam at 300 K and 69 eV (fig. 8), the calculated A,, profiles respond comparably strongly to all the magnetic model ingredients. One therefore has to try to determine them simultaneously. This difficulty is somewhat alleviated by noting that in the range 0 I 9 I 4’ the phase shift splitting AC?,,= 0.002 is, for both the JMA and the WF model, clearly ruled out by comparison with the data. Since any AS,, > 0 moves the positive A,, peaks near 7’-8’ and 13” away from the data, and AS,i < 0 - as can be inferred by linear extrapolation - lowers A,, for 8 > 18” away from the data, it seems best to choose A6,i = 0. We then notice that, for any value of Ai, the prominent 7”-8” peak is much larger for the WF model than for the JMA model. Comparison with the data therfore implies that the JMA model is more adequate. For both WF and JMA, A, -c 0 increases the 7”-8” peak away from the data and is therefore ruled out. For A, > 0, A,, below about 6” is raised.

I

2-

10 ,?

BEAM

&9& 300 K

20 POLAR

ANGLE

30 9 (degl

Fig. 10. A,, of 10 and 00 beam at E = 49 eV and T = 300 K. Calculation using a,, = OX, 6,i = 0, Ai = 0 and embedding correction 6,i from eq. (3) (values see table 1) (lower part of each panel) and ), WF (- - -) and “dead layer” b,i = 0 (upper part of each panel); models JMA ((. - .- .). Vertical bars: experimental data.

102

R. Feder et al. / Surface magnetic structure analysis of Ni(001)

Ths means, for any Ai > 0, an increase of the misfit with the data in the WF case and, for Ai up to about 0.0022, an improvement in the JMA case. In summary, A,, below about 10” implies a preference for the JMA model and Ai = 0.0022. The lower part of panel C of fig. 8 reveals that a slight outward relaxation a,, = + l%, which is the approximate upper limit from our A,, analysis at 300 K, improves the agreement while leaving the magnetic conclusions unaltered. For the 00 beam at 69 eV (fig. 9) the scatter in the data and the lack of detailed agreement for any of the magnetic model preclude magnetic information. The 10 beam A,, profiles at 49 eV (fig. lo), for 9 < 9”, clearly eliminate the dead layer model. The discrimination between JMA and WF is more subtle. While the data at one angle (2”) favour the latter model, the data at three other angles (4’, 6’ and 8”) support JMA. The range 9 2 10” should not be taken into consideration, since A,, is measured simultaneously with A,, and the A,, analysis casts serious doubt on the data in this range. In accordance with our results from A,, at 49 eV, we further notice that the embedding correction ali (cf. eq. (3)) produces a mild improvement of the agreement, without affecting the magnetic conclusion. The same is seen to hold for the 00 results at 49 eV (fig. 10). While also discarding the dead layer model, they do, however, not permit to discriminate between WF and MJA. The A,, situation at the elevated temperature 520 K is first illustrated by the 10 beam at 89 eV (fig. 11). Comparing, for relaxation a,, = 0, with the corresponding 300 K results (fig. 6 panel C) we notice that over the wide range 0 I 9 I 14” the rise in temperature reduces both the experimental A,, data and the theoretical results for the individual magnetic models (cf. fig. 1) by about a factor 2. This behaviour of A,, is in marked contrast to the temperature-inas a consequence of decreasing sensitivity of A,,, but of course plausible magnetization. There is, however, an interesting exception from this expectation: the positive peak at 17” is unchanged both in experiment and in the JMA model calculation, slightly reduced for the WF model and appreciably reduced for the dead-layer model. A key to understanding this apparent anomaly is offered by noticing that both at 300 and 520 K, i.e. for fixed bulk magnetic moment, this peak increases - as a consequence of multiple scattering - with decreasing surface magnetic moment. Raising the temperature induces firstly a decrease of the bulk magnetic moment and secondly, for models with non-zero surface magnetization, a relatively strong decrease of the surface magnetic moment M,. Since a homogeneous decrease of all layer magnetic moments M, with temperature leads to a reduction of A,, (cf. ref. [ 1 l]), an insensitivity of the 17” peak to increasing T can be understood as a compensation of the reduction due to the homogeneous (bulk-like) decrease of M, by an enhancement due to the relatively stronger decrease of M,. For the dead-layer model (M, = 0 at all T), the latter is of course not possible and the 17” peak consequently gets reduced, as is verified by fig. 11. The finding that raising T leaves the 17” peak unchanged both in experiment and in the JMA model

R. Feder et al. / Surface magnetic structure analysis of Ni(OO1)

!s

F-2’

O

Y w2-, 8

20 POLAR

1

20 00

-2+------J

30 ANGLE

/ 10

9 (deg)

POLAR

ANGLE

t

I

BEAM

30 3 ldeg)

.CO

Fig. 11. A,, of 10 and 00 beam at E = 89 eV and T= 520 K. Calculation using relaxation i3,, = 0 (upper part of each panel) and 6,, = + 2.5% (lower part), IS,, = 0.1, AS,, = 0 and Ai = 0. Models JMA ( -), WF (- - -) and “dead layer” (.-_- .) as extrapolated to T= 520 K. Vertical bars: experimental data. Fig. 12. As fig. I1 except E = 69 eV and S,, = 0.2.

c~culation, while slightly reducing it in the WF model calculation, implies that the JMA model is more appropriate. This conclusion is supported by the agreement, also at T = 520 K, in absolute height. These conclusions are reached if we assume, also for T = 520 K, the relaxation a,, = 0%. If we assumed the upper limit i3,2 = 2.5%, which was set by the A,, analysis at 520 IL, a discri~nation between JMA and WF would seem not possible (see fig. 11, lower panel of 10 results). However, for a,, = 2.5% neither the JMA nor the WF model calculation reproduces height and position of the 17” peak correctly. Further, since S,, = 2.5% is excluded for T = 300 IS, the calculated temperature dependence is then at variance with experiment. We can therefore conclude that the upper limit S,2 = 2.5% set by A,, is substantially smaller and the above deductions from the 6,, = 0% results are essentially realistic. For the 00 beam A,, profiles at 520 K for 89 eV (fig. ll} and 69 eV (fig. 12), comparison with the corresponding profiles at 300 K (figs. 7 and 9) shows, over wide angular ranges, a reduction of the calculated results by about a

104

R. Feder et al. / Surface magnetic structure analysis of Ni(OO1)

factor 2. In the measured data, such reduction is only weakly discernible, since the absolute values are for many angles of the order of the statistical uncertainty. For 00 at 520 K, the ranges 23’ < 9 < 27” at 89 eV (fig. 11) and 16” < 9 < 21’ at 69 eV (fig. 12) indicate a slight preference for the JMA magnetization model. The 10 profiles at 520 K (fig. 12) are again reduced, both in theory and experiment, with respect to 300 K, and the JMA model is clearly favoured. As for the relaxation a,,, fig, 12 apparently suggests +2/5%. Recalling, however, the concurrent influence of the absorption potential asymmetry Ai (cf. eq. (7)) shown in fig. 8, one sees that the combination Ai = 0.0022 and S,, EJ + 1% produces a comparable fit. Ths option has the advantage of being compatible with the value S,, close to 0 deduced above from the 10 profiles in fig. 11. In summary, the consistently good to fair experiment/theory agreement achieved for all the ten exchange asymmetry A,, rocking curves reveals the following magnetic information on Ni(OO1). The topmost monoatomic layer is not “magnetically dead”, which is in accordance with self-consistent theories [4-71 as well as with experimental evidence from spin-polarized field emission [45], spin-polarized photoemission [46] and electron capture by fast deuterons [47]. Moreover, the Wang-Freeman prediction [7] of a strongly reduced surface magnetic moment (at T = 0) is clearly distriminated against in favour of the Jepsen et al. (JMA) prediction [5a] of an enhancement of the surface magnetic moment by about 5%. Calculations using the homogeneous magnetization model (cf. fig. 1) gave results, which are so close to the JMA model results (and therefore not shown above) that the present levels of experimental and theoretical accuracy do not yet allow a distinction between these two models. We thus find the homogeneous ode1 value MS = M, as an approximate lower limit of the T = 0 surface magnetization. As an upper limit, we estimate MS = 1. lM,. Our analysis thus yields an enhancement of MS in the range from 0% to lo%, i.e. + 5% f 5%. This is also in very good accordance with a refined 5-layer film result ( + 6.5%) and a 13-layer film result (+ 6.8%) by Jepsen et al. [5b] as well as with the M, versus M3 enhancement (+ 5.8%) found in a 5-layer film calculation by Freeman et al. [4]. The result of a parameterized tight-binding calculation by Tersoff and Falicov [6] is, after subtracting the estimated exaggeration by about O.lpa, also within the above range. An enhancement of the surface magnetic moment relative to the bulk value seems physically plausible on the grounds of d-band narrowing and sp-d dehybridization [4,6]. This interpretation is supported by the strong further enhancements found by self-consistent calculations for an unsupported monolayer by Noffke and Fritsche (+43%) [3] and by Jepsen et al. (+61%) [5b]. As for the manifestation of localized 3p + 3d excitations in the elastic channel, we obtain - for E = 69 eV - a spin splitting AS,, = (6, - al)/2 of the imaginary phase shift parts in the range between 0 and 0.0004 (cf. the spin-averaged value S,i = 0.2). This indicates a weak preference for excitation

R. Feder et al. / Surface

magnetic

structure

analysis of Ni(001)

105

by spin-down electrons, in accordance with qualitative considerations by Nesbet [40]. At 89 eV, a splitting AS,i between -0.0005 and + 0.0005 is inferred. These results suggest a possible weak spin dependence, but we feel that in order to obtain a more definite realistic result a theoretical treatment beyond the present crude parameterization (in terms of Z-independent parameters ali and AS,i) is necessary. In the absence of a spin asymmetry in the uniform absorption (Ai = 0 in eq. (7)), the above AS,, imply, via eqs. (6 and S), an upper limit of 0.06% for the (absolute) spin asymmetry of the inverse total mean free path. For the uniform-absorption asymmetry Ai (eq. (7)), the value Ai = -0.22% from the 10 profile at 300 K and 69 eV has about the same magnitude but the opposite sign as the result from the 10 profile at 89 eV. The joint message from these two profiles is thus lAil < 0.3% for 69 and 89 eV, but further investigation of this quantity is required.

5. Conclusion From the consistent theory/experiment agreement for all the ten A,, rocking curves at 300 K and at 520 K we can firstly conclude that our mean field theory extrapolation of zero-temperature magnetization models to finite temperatures (see section 3) is a reasonable approximation. Secondly, this agreement corroborates the adequacy of density functional formalism in terms of Kohn-Sham single particle equations (with a local exchange-correlation potential approximation) for calculating ground state charge and spin density distributions. The main result of our A,, analysis, the slight enhancement (by 5% f 5%) of the spin magnetic moment at the Ni(OO1) surface (at T = 0) is in very good accordance with the first-principles theoretical results of Jepsen et al. [5a,b] and clearly rules out not only a dead layer model but also the competing first-principles theoretical result of Wang and Freeman [7] (i.e. a reduction of the surface magnetic moment by about 20%). From the present magnetic analysis is also becomes clear why our earlier A,, calculations [ 121 using the simplest magnetic model assumptions (homogeneous magnetization model and no spin dependence of the mean free path) already led to good agreement with experiment: indeed these simplest assumptions fall within the limits found in the present study. In conclusion, our results establish elastic SPLEED as a valuable tool for determining - via comparison of experimental and theoretical exchange-induced scattering asymmetry profiles - the layer- and temperature-dependence of the magnetization at ferromagentic surfaces. It therefore seems worthwhile to make efforts to increase the experimental accuracy and to improve upon the theoretical model ingredients, notably the description of localized inelastic scattering processes.

106

R. Feder et al. / Surface magnetic structure analysis of Ni(OO1)

Acknowledgements

We appreciate the participation of Dr. H. Hopster in the initial stage of the experimental part of this work. We are also very grateful to Professor O.K. Andersen and Dr. 0. Jepsen for making their self-consistent layer magnetization results available to us prior to publication. Further, we appreciate stimulating discussions with Professor M. Campagna. The theoretical part of this work was financially supported by the Deutsche Forschungsgemeinschaft.

References [I] [2] [3] [4] [5]

[6] [7] [8] [9] [IO] [ 1l] [12] [ 131 [ 141 [IS]

[ 161 [17] [I81 [ 191 [20] [21] [22] [23] [24] [25] [26] [27] [28]

U. Gradmann, Appl. Phys. 3 (1974) 161; J. Magnetism Magnetic Mater. 6 (19770 173. SF. Alvarado, Z. Physik 33 (1979) 5 1. J. Noffke and L. Fritsche, J. Phys. Cl4 (1981) 89. A.J. Freeman, D.S. Wang and H. Krakauer, J. Vacuum Sci. Technol., in press. (a) 0. Jepsen, J. Madsen and O.K. Andersen, J. Magnetism Magnetic Mater. 15-18 (1980) 867; (b) 0. Jepsen, J. Madsen and O.K. Andersen, Phys. Rev. B26 (1982) 2790. J. Tersoff and L.M. Falicov, Phys. Rev. B26 (1982) 459. C.S. Wang and A.J. Freemann, J. Magnetism Magnetic Mater. 15-18 (1980) 869; Phys. Rev. B21 (1980) 4585. R. Feder, J. Phys. Cl4 (1981) 2049. D.T. Pierce and R.J. Celotta, Advan. Electron. Electron Phys. 56 (1981) 219. S. Alvarado, M. Campagna and H. Hopster, Phys. Rev. Letters 48 (1982) 51. R. Feder and H. Pleyer, Surface Sci. 117 (1982) 285. SF. Alvarado, R. Feder, H. Hopster, F. Ciccacci and H. Pleyer, Z. Physik B49 (1982) 129. S.F. Alvarado, F. Ciccacci and M. Campagna, Appl. Phys. Letters 39 (1981) 165. F. Ciccacci, S.F. Alvarado and S. Valeri, J. Appl. Phys. 53 (19820 4395. A window-frame crystal has previously been used in a spin-polarized photoemission experiment by R. Clauberg, W. Gudat, E. Kisker, E. Kuhlmann and G.M. Rothberg, Phys. Rev. Letters 47 (19810 1314. D. Gloetzel, private communication. J.C. Slater, T.M. Wilson and J.W. Wood, Phys. Rev. 179 (1969) 28. J.E. Demuth, P.M. Marcus and D.W. Jepsen, Phys. Rev. Bl 1 (1975) 1460. D.P. Jackson, Surface Sci. 43 (1974)43 1. J.W.M. Frenken, R.G. Smeenk, J.F. van der Veen and F.W. Saris, 5th European Conf. on Surface Science (ECOSS-5), Gent, 1982. P.J. Jennings and S.M. Thurgate, Surface Sci. 104 (1981) L210. R. Feder and J. Kirschner, Surface Sci. 102 (1981) 75. T. Jach and C.J. Powell, Solid State Commun. 40 (1981) 967, and references therein. C.B. Duke, Advan. Chem. Phys. 27 (1974) 1. J.B. Pendry, Low Energy Electron Diffraction (Academic Press, London, 1974). J.C. Le Boss&, These d’Etat, Lyon (1981). S.F. Alvarado, H. Hopster, R. Feder and H. Pleyer, Solid State Commun. 39 (1981) 1319. Due to spin-orbit coupling, the g-factor of the magnetic electrons in Ni is 2.2 instead of 2.0, i.e. a magneton number nn = 0.606 pe atom corresponds to an effective magnetic electron number ne = 0.551 (cf., e.g., C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1971) ch. 16). Spin-polarized electronic structure calculations primarily yield ne. This is also

R. Feder et al. / Surface magnetic structure analysis of Ni(001)

107

the quantity relevant for magnetic electron scattering, which is caused by exchange interaction. In contrast, for neutron scattering, in which the magnetic part of the interaction is of a dipole-dipole nature, n a is the primary quantity. to the bulk value, the magnetization rr~‘~a and the effective magnetic v91 When normalizing electron number nr) of the n th layer (cf. ref. [ZS]) need not be distinguished. LEED measurements [ 101 have shown the surface Tc to equal the bulk [301 Recent spin-polarized T, to within *4 K. to Solid State Physics (Wiley, New York, 19710 ch. 16, fig. 4. [311 C. Kittel, Introduction 1321 T. Wolfram, R.E. De Wames, W.F. Hall and P.W. Palmberg, Surface Sci. 28 91971) 45. [331 T. Wolfram and R.E. De Wames, Progr. Surface Sci. 2 91974) 233. J. Phys. Sot. Japan 40 (1975) 925. [341 T. Takeda and H. Fukuyama, IEEE Trans. Magnetic Mater. 12 (1976) 66. 1351 K. Binder and P.C. Hohenberg, R. Feder, W. Gudat, E. Kisker and E. Kuhlmann, Phys. Rev. 1361 A. Bringer, M. Campagna, Letters 42 (1979) 1705. [371 R. Feder, Solid State Commun. 31 (1979) 821. [381 R.W. Rendell and D.R. Penn, Phys. Rev. Letters 45 91980) 2057. International Centre for Theoretical Physics (Trieste) Report 1391 Soe Yin and E. Tosatti, IC/81/129, 1981. WI The spin dependence of inelastic electron scattering involving localized states has recently also been discussed by R.K. Nesbet, to be published. [411 R. Feder, H. Pleyer, P. Bauer and N. Mtiller, Surface Sci. 109 (1981) 419. constant energy mode, rotation diagrams at constant iI, which has similar 1421 The alternative advantages (cf. refs. [41,8], and references therein), cannot be employed in this study, firstly because our decoupling of A,, and A,, requires the scattering plane to coincide with a mirror plane of the crystal, and secondly because our geometry as well as magnetic anisotropy prevents a concomitant rotation of the direction of magnetization. 1431 R. Baudoing, private communication. New York, 1976). 1441 N.W. Ashcroft and N.D. Mermin, Solid State Physics (Holt-Saunders, Phys. Rev. Letters 38 (1977) 663. [451 M. Landolt and M. Campagna, [461 I.D. Moore and J.B. Pendry, J. Phys. Cl 1 (1978) 4615, and references therein. [47] C. Rau, Comments Solid State Phys. 9 (1980) 177, and references therein.