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On the feasibility of studying surface magnetism by spin-polarized low-energy electron diffraction
285
Surface Science 117 (1982) 285-293 North-Holland Publishing Company
ON THE FEASIBILITY OF STUDYING SURFACE MAGNETISM SPIN-POLARIZED LOW-ENERGY ELECTRON DIFFRACTION R. FEDER
and H. PLEYER
Theoretische Festkdrperphysik, Fed. Rep. of Germun~ Received
BY
14 September
FB IO, Universitiit
1981; accepted
Duishurg GH, D-4100 Dtrishurg,
for publication
21 October
198 I
In the framework of a dynamical theory of spin-polarized low-energy electron diffraction from ferromagnetic surfaces, a linear approximation scheme is presented. The large amount of computing time required for the analysis of magnetic structures involving a layer-dependent magnetization can thereby be reduced by a factor of at least 24. The validity of the scheme is verified by numerical calculations for various magnetization models of the Ni(OO1) and Fe( I IO) surfaces. Our polarization results further show a strong sensitivity to the surface magnetization partially decoupled from a strong response to a spin dependence of the mean free path. This implies the possibility of determining both physical properties. For special cases, the polarization is proportional to the surface magnetization, which suggests direct experimental access to the critical behaviour of the latter.
1. Introduction Recent theoretical and experimental progress suggests that spin-polarized low-energy electron diffraction (SPLEED) could become an excellent method for studying magnetic properties in the surface region of ferromagnets [ 11. In particular, the layer, temperature and field dependence of the magnetization, M,( T, H), is expected to emerge from comparison of experimental polarization versus energy and angle profiles with their theoretical counterparts calculated for assumed magnetic model structures. However, straightforward dynamical calculations for a variety of trial structures, which involve different magnetizations and therefore different S-matrices for each monoatomic layer (parallel to the surface), require an amount of computing time, which seems prohibitively large. In the present work, we propose a linear approximation scheme, which reduces, for a primary beam of given energy and orientation incident on a given surface, the numerical labour to essentially only the calculation of one “canonical” single-layer S-matrix. The validity of the scheme is verified by numerical calculations for two typical ferromagnetic surfaces: Ni(OO1) and Fe(ll0). The new method is applied to studies of the relative sensitivity of polarization profiles to changes of the surface magnetization and of ingredients 0039-6028/82/0000-0000/$02.75
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of the scattering potential model. Further, we point out conditions under which the polarization is proportional to the surface magnetization, i.e. under which the temperature dependence of the latter can be directly measured.
2. Linear approximation scheme The dynamical theory of SPLEED from magnetic surfaces is briefly characterized as follows [ 11. An electron with spin parallel ( +) or antiparallel (- ) to the crystal magnetization axis experiences an effective potential l’ + or I/ -~. As a consequence of the exchange interaction with the ground state electrons. I/+ is different from I/-. Two separate one-component “ordinary” LEED calculations for V” (u = *) yield intensities Z,” for the gth diffracted beam, from which an asymmetry A, or spin polarization % is defined as Ag=PB=(J+
g -zg
)/(I,+
+-I,),
(1)
where P, is the spin polarization obtained for the case of an unpolarized primary beam. (In the following, we drop the diffracted beam index g.) Each of the two LEED calculations proceeds in three steps: (a) scattering phase shifts 8: from a single “crystal atom” V”(r), (b) intra-layer multiple scattering leading to single-layer scattering matrices S;, where n enumerates the relevant layers, and (c) inter-layer multiple scattering to yield the intensities I”. In addition to the spin dependence of the real atomic potential V’(r), there is possibly some spin dependence of the mean free path, i.e. of the uniform absorptive potential V&,, which enters in steps (b) and (c) [I]. Magnetic structure analysis - at a given temperature T - by dynamical theory is based on comparing experimental P versus energy or angle profiles with the corresponding P-profiles calculated for a number t of trial magnetic structural models characterized by layer-dependent magnetizations M,(T). If inter-layer multiple scattering is treated by means of the renormalized-forward scattering method [2], more than 95% of the total computing time is required for the intra-layer step (b). For L layers contributing to the scattering, straightforward application of the above dynamical method thus raises the computing time by a factor of about LX r compared to the time for one calculation for a stack of identical layers. Since L is typically at least 6 and 1 can be expected to be not less than 4, L X t will be at least 24. It seems therefore very worthwhile to investigate whether expansions of the relevant quantities Q0 to first order in the magnetization M Q”(M)=Q”(M=O)+a(~Q+,‘~M),=,M,
a=+,
-1
(2)
for any of the spin-dependent quantities V”(r), to calculate Q’( MO) only for a single value MO. For any other value M, Q”(M) is then simply obtained as
are adequate,
where QU stands
S,Y, VP,, Sz, I”. If eq. (2) holds, it suffices
arbitrary
Q”(M)=t[Q+(M,)+Q-(M,)]+at[Q+(M,)-Q-
(3)
R. Fe&r, H. Pleyer / Feusihility of stujving surfuce mugnetism
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In the case of homogeneous magnetization, i.e. M, = A4 for all L layers, the most desirable situation would be the validity of eq. (2) for Qs = 1”. Eqs. (1) and (3) then yield P(M)
=p(&J
M/M),
(4)
i.e. the polarization is proportional to the magnetization M. Such linearity holds in the first Born approximation for 3d ferromagnets [3]. but it is not clear a priori whether partial wave analysis and multiple scattering do not add higher order odd terms. For a wider class of magnetization models characterized by layer-dependent M,, such that the M, for all n, which are relevant for scattering, scale by a common factor, linear expansion of I”(M,, .. .. ML) with respect to the M,, leads to a generalization of eq. (4): P( M,, .... M,)
=P(M,,,,
...>Mod
M/M,,,
(5)
i.e. the polarization is still proportional to the surface magnetization M, = Ms. The above means that P is approximately a first-order homogeneous function of M,, .... ML. For the general case of different M,, for the individual layers, eq. (5) is no longer possible. It would however, be almost as favourable, if eq. (2) held for the single-layer scattering matrices S:, which require most of the computing time. It would then suffice to calculate two “canonical” matrices S:( M,), u = k, instead of more than 24 different pairs. The validity of eq. (2) for Sl or of eqs. (4) and (5) for P can, for a given magnetic system, be only established by comparison of the approximate numerical results with the exact ones. It appears, for the 3d ferromagnets, however, likely from a consideration of the more basic quantities V”(r), ST, and V&. In a local density approximation (cf. refs. [4-61 and references therein), the zero-temperature exchange contribution V,“,(r) to the crystal atom potential V”(r) is proportional to [p”(r)]1/3, where p”(r) is the ground state charge density for spin u. Defining Ap=+(p+-p-) and p=i(p’+p-), Ap/p is very small for most r values. Thus AVex = i( VeT - V& ) is proportional to Ap. For T> 0, AV,, is multiplied by M(T)/M(T= 0), i.e. it is clearly proportional to M(T), and eq. (2) holds for V”(r). Since AV,, K V( M = 0), an integral representation of the phase shifts suggests AS, to be proportional to AV,, and thereby to M; eq. (2) thus also holds for 8:. For the (energy-dependent) absorptive potential V,“,(E), a simple model (cf. ref. [l]), which provides an upper limit for the absolute value of the spin splitting [7], gives V-P,(E)=
V,,(E;
M=O)
-d$(E;
M=O)
M,
u=
+,
-,
(6)
where Vi,(E; M = 0) is the total absorptive potential (accounting for the excitation of plasmons and electron-hole pairs) for the zero magnetization case, and V:i( E; M = 0) is its electron-hole part. Eq. (6) is already of the form of eq. (2), and more realistic models [8,9] yielding a substantially smaller spin splitting can a fortiori be expected to obey eq. (2). The validity of eq. (2) for V/“(r), 8: and vi”, together with the small splitting of these quantities suggests
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that it might hold also for the single-layer scattering matrices S; and that eqs. (4) and (5) might hold for P. We investigate this in the following with the aid of numerical calculations.
3. Numerical results for Ni(OO1) and Fe(ll0) Dynamical calculations at T= 0 have been performed for the typical ferromagnetic surfaces Ni(OO1) and Fe(l10). Atomic scattering potentials V”(r) were constructed from self-consistently calculated bulk charge densities p”(r) [lo] using an energy-dependent approximation for the exchange contribution [5], which was found superior to energy-independent approximations in SPLEED from W(OO1) [ 1 l] and Pt(ll1) [12]. Changes in the magnetization of the top layer or the entire crystal were introduced by modifying the intial p”( r ) according to eq. (3). The zero-magnetization (i.e. spin-averaged) imaginary potential V,,(E) was taken from an intensity-LEED study [13]. Various tentative spin splittings of Vi,< E ) were used. The surface potential barrier was chosen as an exponential-type function. Intra-layer scattering was treated exactly and inter-layer multiple scattering in the renormalized-forward scattering approximation (cf. ref. [2]). Calculations of the phase shifts 8: for magnetizations M ranging between 0 and 2M,,, where M, corresponds to the original ground state charge densities p”(r) [lo], reveal the validity of eq. (2) to within less than 1%. Fig. 1 shows the phase shifts obtained for MO (with the spin splitting magnified by two). The spin splitting is rather small, even for the ferromagnet Fe and about a factor 3.7 (obtained from the numerical data) smaller for Ni, i.e. it scales linearly with the magnetic moments of Fe (2.2 pa) and Ni (0.6 pn) (rato 3.7). In contrast to the spin-orbit interaction, the exchange interaction is seen to produce a splitting also of 8,. At the level of the single-layer scattering matrices 9’. the linear relation eq. (2) has also been confirmed for Ni and Fe to within about 1%. General results of our calculations for various angles of incidence for Ni(OO1) and Fe(ll0) are illustrated by the Ni(OO1) polarization versus energy profiles shown in fig. 2. Firstly, for MS # M,, where MS denotes the top layer magnetization and M, the bulk magnetization assumed for all other layers, the profiles obtained by using the linear scaling approximation (eq. (3)) to the top layer S-matrix S, (cf. fig. 2d) are practically identical to those calculated with the aid of the exact S, corresponding to MS (cf. fig. 2~). The validity of our time-saving linear approximation scheme is thus clearly established [ 141. Secondly, for MS = M, = : M, the polarization profiles scale, in good approximation, linearly with M (cf. figs. 2a and 2b as anticipated by eq. (4). The proportionality of P and M, which holds in the first Born approximation (cf. ref. [3]), is thus only very weakly affected by using the exact atomic scattering amplitudes and by taking into account multiple scattering. Calculations further confirm, to within about l%, the proportionality of P to MS (eq. (5)) for
R. Fe&r, H. Pleyer / Fetisihili[v of stu&ing surface mugnetisnz
0
0
20
40
60 80 100 120 llr0 ENERGY [A'1
289
ENERGYLeVI
Fig I. Phase shifts describing the scattering of polarized electrons from magnetic “crystal atoms” and Ni (- - -). For each material, the higher (lower) phase shift corresponds ofFe( -----) to electron spin parallel (antiparallel) to the magnetic majority spin axis. For the purpose of illustration. the actual spin splitting has been magnified by a factor two. Fig. 2. Spin polarization of the specular beam from Ni(OO1) models at T=O for y,’ where MO is the actual bulk magnetization (0.6 /~a); (b) MS= Mb = M,=M,=M,. “exact” calculation, using appropriate top layer phase shifts and M,=I.SM,. M,=M,. S,; (d) M, = 1.5M,, M, = MO, calculation employing linear approximation (eqs. (2) and
= Vii: (a) l.5Mo; (c) S-matrices (3)) to 5,.
homogeneous layer-dependent magnetizations, M,. Thirdly, the P profiles for MS # M, are in general not proportional to some average magnetization. This is clearly illustrated by the P features near 25, 47, 94 and 96 eV in fig. 2c. which are not intermediate between the corresponding features in the homogeneous magnetization cases figs. 2a and 2b. The second and third of the above findings, although obtained at T = 0, have implications for the connection between P and MS at finite temperatures, at which each atomic layer has a characteristic magnetization M,(T) ranging between M,(T) -CMb(T) and Mb(T) (cf. ref. [ 151). According to our model results, P is generally not proportional to MS or some average surface magnetization. If, however, the magnetic coherence length is large compared to the total thickness of the atomic layers that contribute to scattering, the relevant magnetizations M,(T) scale like M,(T).This case is realized in the vicinity of
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the Curie temperature. Our proportionality result (eq. (5)) thus suggests that the measurement of P(T) provides a direct avenue to the.critical behaviour of the surface magnetization M,( T ). The sensitivity of P versus E profiles to a deviaton of the top layer magnetization MS from the bulk magnetization M, is illustrated in more detail for Fe(ll0) in fig. 3. Focusing first on the result from a model involving no spin dependence of the mean free path (i.e. vi,’ = Vi; ) (fig. 3a), numerous polarization features are seen to respond strongly to a change in A4,. In most cases, the absolute value of P increases with increasing MS. as one would naively expect. There are however, “anomalous” features, which behave in the opposite way (e.g. near 48 and 84 eV) or even change sign (near 28, 46 and 58 eV). We interpret this as due to interference of bulk and surface scattering amplitudes. Fig. 3a also demonstrates the absence of a proportionality between P and MS, which we already pointed out before. The introduction of a spin dependence of the absorptive potential (cf. eq. (6)) is found (see fig. 3b) to modify the P profiles in a rather selective way. While some features (e.g. near 24, 40 and 44 eV) are almost unaltered, other (e.g. near 29 and 54 eV) change appreciably, and some prominent new ones (near 57, 62 and 90 eV) appear.
1’
” 1
20
I
40
1
I
60 ENERGY IA’1
I
I
80
j
-40 100
Fig. 3. Specular beam from Fe( 110) models at T=O calculated by linear approximation to S,. Spin polarization for M, =A!,, and M,=0.5M, (--_), M,=M, (--) and M,=l.5M, (. - .- .). Intensity for all three cases (. .). (a) V,,+ =l’,,: (b) VP,= V,,(M=O)-UV~~~~O.~M,,~ (cf. eq. (6); we used only half of the maximal spin splitting).
R. Feder. H. Plqver / Ferrsihili!v of stu&ing surface mugneiism
This partial
decoupling
A%,(E)=[Y,‘(E)-
of the influence
291
of A4, and
Y,(E)]/2
is very valuable: via comparison with experimental data, features of the first kind permit the determination of the surface magnetization and those of the second and third kind promise to resolve the question of the spin dependence of the mean free path in ferromagnets [7-9,161. In view of the latter question, we would like to mention a further linear relation, which we have established. Denoting, for a model involving M, # M,, by P,( MS, Mb) and Pi( MS, Mb) the polarization produced by only V”(R) and V,> (eq. (5)), respectively, and replacing in eq. (6) M by cuM ((Y constant with (aI< I), the total polarization P,,,(M,, Mb) produced by V”(r) and I$,( (Y) together is [ 171 ‘t~,(~s?
Mb) =pr(Ms*
Mb) + (ypi(Ms,
Mb).
(7)
Instead of calculating P,,, for a number of trial values of LYit thus suffices to calculate, in addition to P,, only one Pi (corresponding to vP,( a = 1)). As is also illustrated by fig. 3, the (spin-averaged) intensity profiles obtained for the same VU(r) are practically identical for different A4, and AK,,, models, which corroborates an earlier statement (cf. ref. [ 11) that this magnetic information is exclusively coded in the spin polarization. In the above, we have focused on the feasibility of determining J4, and AC, while assuming the ion-core potential P(r) as known. In order to discriminate between different models of VO(r), one can extend the above idea of selective senstivity of the exchange-induced spin polarization. Calculations using an energy-independent exchange approximation actually demonstrate the existence of features, which respond predominantly to P(r). For example, the height of the positive peak near 90 eV in fig. 3, which is only very weakly affected by changes in iL4, and AV,,, is very sensitive to V”(r). Since, however, the principle of construction of V”(r) is closely related to that of the nonmagnetic V(r), changes in Vu(r) are associated with changes in V(r) and therefore manifest themselves also in the intensity profile. Comparison between calculated and measured intensity or spin-orbit-induced polarization profiles thus already provides at least some discrimination between Vu(r) models.
4. Conclusions The proportionality of the spin polarization to the surface magnetization for the case of homogeneously layer-dependent magnetization (eq. (5)), which we have found to hold for Ni(OO1) and Fe( 1 IO) in very good approximation, suggests the feasibility of a direct experimental study of the behaviour of M,(T) near the critical temperature. The determination of the layer depen-
292
R. Feder. H. Pleyr
/
Feasibility
of studyingsurfucemugnetism
dence of M, i.e. magnetic structure analysis, requires dynamical SPLEED calculations. These are made computationally feasible by means of a linear approximation sheme for the single-layer S-matrices, which reduces the computing time by at least a factor of 24 with respect to straightforward calculation. A further prerequisite for magnetic structure analysis, the existence of polarization features which are strongly sensitive to the surface magnetization, while responding only weakly to a possible spin splitting AVim of the imaginary potential, is verified by numerical calculations for Ni(OO1) and Fe(ll0). The finding of other features, which are produced by AV,, and are very sensitive to it, provides a basis for determining the actual spin dependence of the mean free path. In fact, a joint experimental and theoretical study currently in progress [ 181 already indicates such information on the ferromagnetic Ni(OO1) surface.
Acknowledgements We would like to thank Dr. S. Alvarado for topical Professor G. Eilenberger for his interest and kind hospitality.
discussions
and
References [I] Details and references to the original literature may be found in a recent review article: R. Feder, J. Phys. Cl4 (1981) 2049. [2] J.B. Pendry, Low Energy Electron Diffraction (Academic Press, London, 1974). and references therein. [31 L.A. Vredevoe and R.E. de Wames, Phys. Rev. 176 (1968) 684: X.1. Saldana and J.S. Helman, Phys. Rev. Bl6 (1977) 4978. [41 W. Kohn and L.J. Sham, Phys. Rev. 140 (1965) Al 133. [51 J.C. Slater. T.M. Wilson and J.W. Wood, Phys. Rev. 179 (1969) 28. [61 A.K. Rajagopal, Advan. Chem. Phys. 41 (I 980) 59. model calcultion of the spin splitting AI’,, [8] for Fe and Ni produces 171 A more sophisticated values which are by about an order of magnitude smaller. more strongly energy dependent, and even have the opposite sign over wide ranges of energy. In ontrast, a most recent microscopic model calculation for Fe [9] confirms the sign of AI’,, of the simple model (cf. ref. [l] and eq. (6)) and predicts its value to be reduced by an energy-dependent factor ranging between 6 and 10. As will become apparent in the context of fig. 3, knowledge of V,, is, however, not only not required for magnetic structure analysis by SPLEED. but can be gained by the analysis. PI R.W. Rendell and D.R. Penn, Phys. Rev. Letters 45 (1980) 2057. 191 Soe Yin and E. Tosatti. to be published. [lOI D. Gloetzel, private communication. [Ill R. Feder and J. Kirschner, Surface Sci. 102 (1981) 75. [I21 R. Feder. H. Pleyer. P. Bauer and N. Mtiller, Surface Sci. 109 (1981) 419. iI31 J.E. Demuth. P.M. Marcus and D.W. Jepsen, Phys. Rev. Bll (1975) 1460. 1141 While the results of fig. 2 were obtained for models with V,z = Vi,, the linear approximation scheme is also valid for models involving
a spin-dependent
mean free path.
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[ 151 S.F. Alvarado, Z. Physik 33 ( 1979) 5 1. and references therein. [ 161 The energy dependence of V&(E) does not prohibit its determination. It only implies that for a particular beam and diffraction geometry one obtains values of ViK,for a number of selected E points. and that different spectra may have to be studied to get a sufficiently dense mesh. [ 171 For the special case of MS = M,, such a relation was found previously (cf. ref. [I]). [18] R. Feder. SF. Alvarado, H. Hopster and E. Tamura, to be published.
Surface Science 117 (1982) 285-293 North-Holland Publishing Company
ON THE FEASIBILITY OF STUDYING SURFACE MAGNETISM SPIN-POLARIZED LOW-ENERGY ELECTRON DIFFRACTION R. FEDER
and H. PLEYER
Theoretische Festkdrperphysik, Fed. Rep. of Germun~ Received
BY
14 September
FB IO, Universitiit
1981; accepted
Duishurg GH, D-4100 Dtrishurg,
for publication
21 October
198 I
In the framework of a dynamical theory of spin-polarized low-energy electron diffraction from ferromagnetic surfaces, a linear approximation scheme is presented. The large amount of computing time required for the analysis of magnetic structures involving a layer-dependent magnetization can thereby be reduced by a factor of at least 24. The validity of the scheme is verified by numerical calculations for various magnetization models of the Ni(OO1) and Fe( I IO) surfaces. Our polarization results further show a strong sensitivity to the surface magnetization partially decoupled from a strong response to a spin dependence of the mean free path. This implies the possibility of determining both physical properties. For special cases, the polarization is proportional to the surface magnetization, which suggests direct experimental access to the critical behaviour of the latter.
1. Introduction Recent theoretical and experimental progress suggests that spin-polarized low-energy electron diffraction (SPLEED) could become an excellent method for studying magnetic properties in the surface region of ferromagnets [ 11. In particular, the layer, temperature and field dependence of the magnetization, M,( T, H), is expected to emerge from comparison of experimental polarization versus energy and angle profiles with their theoretical counterparts calculated for assumed magnetic model structures. However, straightforward dynamical calculations for a variety of trial structures, which involve different magnetizations and therefore different S-matrices for each monoatomic layer (parallel to the surface), require an amount of computing time, which seems prohibitively large. In the present work, we propose a linear approximation scheme, which reduces, for a primary beam of given energy and orientation incident on a given surface, the numerical labour to essentially only the calculation of one “canonical” single-layer S-matrix. The validity of the scheme is verified by numerical calculations for two typical ferromagnetic surfaces: Ni(OO1) and Fe(ll0). The new method is applied to studies of the relative sensitivity of polarization profiles to changes of the surface magnetization and of ingredients 0039-6028/82/0000-0000/$02.75
0 1982 North-Holland
286
R. Feder. H. Pleyr
/ Feusihlity
of stu+ng
surfuce nugnerisnl
of the scattering potential model. Further, we point out conditions under which the polarization is proportional to the surface magnetization, i.e. under which the temperature dependence of the latter can be directly measured.
2. Linear approximation scheme The dynamical theory of SPLEED from magnetic surfaces is briefly characterized as follows [ 11. An electron with spin parallel ( +) or antiparallel (- ) to the crystal magnetization axis experiences an effective potential l’ + or I/ -~. As a consequence of the exchange interaction with the ground state electrons. I/+ is different from I/-. Two separate one-component “ordinary” LEED calculations for V” (u = *) yield intensities Z,” for the gth diffracted beam, from which an asymmetry A, or spin polarization % is defined as Ag=PB=(J+
g -zg
)/(I,+
+-I,),
(1)
where P, is the spin polarization obtained for the case of an unpolarized primary beam. (In the following, we drop the diffracted beam index g.) Each of the two LEED calculations proceeds in three steps: (a) scattering phase shifts 8: from a single “crystal atom” V”(r), (b) intra-layer multiple scattering leading to single-layer scattering matrices S;, where n enumerates the relevant layers, and (c) inter-layer multiple scattering to yield the intensities I”. In addition to the spin dependence of the real atomic potential V’(r), there is possibly some spin dependence of the mean free path, i.e. of the uniform absorptive potential V&,, which enters in steps (b) and (c) [I]. Magnetic structure analysis - at a given temperature T - by dynamical theory is based on comparing experimental P versus energy or angle profiles with the corresponding P-profiles calculated for a number t of trial magnetic structural models characterized by layer-dependent magnetizations M,(T). If inter-layer multiple scattering is treated by means of the renormalized-forward scattering method [2], more than 95% of the total computing time is required for the intra-layer step (b). For L layers contributing to the scattering, straightforward application of the above dynamical method thus raises the computing time by a factor of about LX r compared to the time for one calculation for a stack of identical layers. Since L is typically at least 6 and 1 can be expected to be not less than 4, L X t will be at least 24. It seems therefore very worthwhile to investigate whether expansions of the relevant quantities Q0 to first order in the magnetization M Q”(M)=Q”(M=O)+a(~Q+,‘~M),=,M,
a=+,
-1
(2)
for any of the spin-dependent quantities V”(r), to calculate Q’( MO) only for a single value MO. For any other value M, Q”(M) is then simply obtained as
are adequate,
where QU stands
S,Y, VP,, Sz, I”. If eq. (2) holds, it suffices
arbitrary
Q”(M)=t[Q+(M,)+Q-(M,)]+at[Q+(M,)-Q-
(3)
R. Fe&r, H. Pleyer / Feusihility of stujving surfuce mugnetism
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In the case of homogeneous magnetization, i.e. M, = A4 for all L layers, the most desirable situation would be the validity of eq. (2) for Qs = 1”. Eqs. (1) and (3) then yield P(M)
=p(&J
M/M),
(4)
i.e. the polarization is proportional to the magnetization M. Such linearity holds in the first Born approximation for 3d ferromagnets [3]. but it is not clear a priori whether partial wave analysis and multiple scattering do not add higher order odd terms. For a wider class of magnetization models characterized by layer-dependent M,, such that the M, for all n, which are relevant for scattering, scale by a common factor, linear expansion of I”(M,, .. .. ML) with respect to the M,, leads to a generalization of eq. (4): P( M,, .... M,)
=P(M,,,,
...>Mod
M/M,,,
(5)
i.e. the polarization is still proportional to the surface magnetization M, = Ms. The above means that P is approximately a first-order homogeneous function of M,, .... ML. For the general case of different M,, for the individual layers, eq. (5) is no longer possible. It would however, be almost as favourable, if eq. (2) held for the single-layer scattering matrices S:, which require most of the computing time. It would then suffice to calculate two “canonical” matrices S:( M,), u = k, instead of more than 24 different pairs. The validity of eq. (2) for Sl or of eqs. (4) and (5) for P can, for a given magnetic system, be only established by comparison of the approximate numerical results with the exact ones. It appears, for the 3d ferromagnets, however, likely from a consideration of the more basic quantities V”(r), ST, and V&. In a local density approximation (cf. refs. [4-61 and references therein), the zero-temperature exchange contribution V,“,(r) to the crystal atom potential V”(r) is proportional to [p”(r)]1/3, where p”(r) is the ground state charge density for spin u. Defining Ap=+(p+-p-) and p=i(p’+p-), Ap/p is very small for most r values. Thus AVex = i( VeT - V& ) is proportional to Ap. For T> 0, AV,, is multiplied by M(T)/M(T= 0), i.e. it is clearly proportional to M(T), and eq. (2) holds for V”(r). Since AV,, K V( M = 0), an integral representation of the phase shifts suggests AS, to be proportional to AV,, and thereby to M; eq. (2) thus also holds for 8:. For the (energy-dependent) absorptive potential V,“,(E), a simple model (cf. ref. [l]), which provides an upper limit for the absolute value of the spin splitting [7], gives V-P,(E)=
V,,(E;
M=O)
-d$(E;
M=O)
M,
u=
+,
-,
(6)
where Vi,(E; M = 0) is the total absorptive potential (accounting for the excitation of plasmons and electron-hole pairs) for the zero magnetization case, and V:i( E; M = 0) is its electron-hole part. Eq. (6) is already of the form of eq. (2), and more realistic models [8,9] yielding a substantially smaller spin splitting can a fortiori be expected to obey eq. (2). The validity of eq. (2) for V/“(r), 8: and vi”, together with the small splitting of these quantities suggests
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that it might hold also for the single-layer scattering matrices S; and that eqs. (4) and (5) might hold for P. We investigate this in the following with the aid of numerical calculations.
3. Numerical results for Ni(OO1) and Fe(ll0) Dynamical calculations at T= 0 have been performed for the typical ferromagnetic surfaces Ni(OO1) and Fe(l10). Atomic scattering potentials V”(r) were constructed from self-consistently calculated bulk charge densities p”(r) [lo] using an energy-dependent approximation for the exchange contribution [5], which was found superior to energy-independent approximations in SPLEED from W(OO1) [ 1 l] and Pt(ll1) [12]. Changes in the magnetization of the top layer or the entire crystal were introduced by modifying the intial p”( r ) according to eq. (3). The zero-magnetization (i.e. spin-averaged) imaginary potential V,,(E) was taken from an intensity-LEED study [13]. Various tentative spin splittings of Vi,< E ) were used. The surface potential barrier was chosen as an exponential-type function. Intra-layer scattering was treated exactly and inter-layer multiple scattering in the renormalized-forward scattering approximation (cf. ref. [2]). Calculations of the phase shifts 8: for magnetizations M ranging between 0 and 2M,,, where M, corresponds to the original ground state charge densities p”(r) [lo], reveal the validity of eq. (2) to within less than 1%. Fig. 1 shows the phase shifts obtained for MO (with the spin splitting magnified by two). The spin splitting is rather small, even for the ferromagnet Fe and about a factor 3.7 (obtained from the numerical data) smaller for Ni, i.e. it scales linearly with the magnetic moments of Fe (2.2 pa) and Ni (0.6 pn) (rato 3.7). In contrast to the spin-orbit interaction, the exchange interaction is seen to produce a splitting also of 8,. At the level of the single-layer scattering matrices 9’. the linear relation eq. (2) has also been confirmed for Ni and Fe to within about 1%. General results of our calculations for various angles of incidence for Ni(OO1) and Fe(ll0) are illustrated by the Ni(OO1) polarization versus energy profiles shown in fig. 2. Firstly, for MS # M,, where MS denotes the top layer magnetization and M, the bulk magnetization assumed for all other layers, the profiles obtained by using the linear scaling approximation (eq. (3)) to the top layer S-matrix S, (cf. fig. 2d) are practically identical to those calculated with the aid of the exact S, corresponding to MS (cf. fig. 2~). The validity of our time-saving linear approximation scheme is thus clearly established [ 141. Secondly, for MS = M, = : M, the polarization profiles scale, in good approximation, linearly with M (cf. figs. 2a and 2b as anticipated by eq. (4). The proportionality of P and M, which holds in the first Born approximation (cf. ref. [3]), is thus only very weakly affected by using the exact atomic scattering amplitudes and by taking into account multiple scattering. Calculations further confirm, to within about l%, the proportionality of P to MS (eq. (5)) for
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0
0
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60 80 100 120 llr0 ENERGY [A'1
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ENERGYLeVI
Fig I. Phase shifts describing the scattering of polarized electrons from magnetic “crystal atoms” and Ni (- - -). For each material, the higher (lower) phase shift corresponds ofFe( -----) to electron spin parallel (antiparallel) to the magnetic majority spin axis. For the purpose of illustration. the actual spin splitting has been magnified by a factor two. Fig. 2. Spin polarization of the specular beam from Ni(OO1) models at T=O for y,’ where MO is the actual bulk magnetization (0.6 /~a); (b) MS= Mb = M,=M,=M,. “exact” calculation, using appropriate top layer phase shifts and M,=I.SM,. M,=M,. S,; (d) M, = 1.5M,, M, = MO, calculation employing linear approximation (eqs. (2) and
= Vii: (a) l.5Mo; (c) S-matrices (3)) to 5,.
homogeneous layer-dependent magnetizations, M,. Thirdly, the P profiles for MS # M, are in general not proportional to some average magnetization. This is clearly illustrated by the P features near 25, 47, 94 and 96 eV in fig. 2c. which are not intermediate between the corresponding features in the homogeneous magnetization cases figs. 2a and 2b. The second and third of the above findings, although obtained at T = 0, have implications for the connection between P and MS at finite temperatures, at which each atomic layer has a characteristic magnetization M,(T) ranging between M,(T) -CMb(T) and Mb(T) (cf. ref. [ 151). According to our model results, P is generally not proportional to MS or some average surface magnetization. If, however, the magnetic coherence length is large compared to the total thickness of the atomic layers that contribute to scattering, the relevant magnetizations M,(T) scale like M,(T).This case is realized in the vicinity of
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the Curie temperature. Our proportionality result (eq. (5)) thus suggests that the measurement of P(T) provides a direct avenue to the.critical behaviour of the surface magnetization M,( T ). The sensitivity of P versus E profiles to a deviaton of the top layer magnetization MS from the bulk magnetization M, is illustrated in more detail for Fe(ll0) in fig. 3. Focusing first on the result from a model involving no spin dependence of the mean free path (i.e. vi,’ = Vi; ) (fig. 3a), numerous polarization features are seen to respond strongly to a change in A4,. In most cases, the absolute value of P increases with increasing MS. as one would naively expect. There are however, “anomalous” features, which behave in the opposite way (e.g. near 48 and 84 eV) or even change sign (near 28, 46 and 58 eV). We interpret this as due to interference of bulk and surface scattering amplitudes. Fig. 3a also demonstrates the absence of a proportionality between P and MS, which we already pointed out before. The introduction of a spin dependence of the absorptive potential (cf. eq. (6)) is found (see fig. 3b) to modify the P profiles in a rather selective way. While some features (e.g. near 24, 40 and 44 eV) are almost unaltered, other (e.g. near 29 and 54 eV) change appreciably, and some prominent new ones (near 57, 62 and 90 eV) appear.
1’
” 1
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I
40
1
I
60 ENERGY IA’1
I
I
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j
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Fig. 3. Specular beam from Fe( 110) models at T=O calculated by linear approximation to S,. Spin polarization for M, =A!,, and M,=0.5M, (--_), M,=M, (--) and M,=l.5M, (. - .- .). Intensity for all three cases (. .). (a) V,,+ =l’,,: (b) VP,= V,,(M=O)-UV~~~~O.~M,,~ (cf. eq. (6); we used only half of the maximal spin splitting).
R. Feder. H. Plqver / Ferrsihili!v of stu&ing surface mugneiism
This partial
decoupling
A%,(E)=[Y,‘(E)-
of the influence
291
of A4, and
Y,(E)]/2
is very valuable: via comparison with experimental data, features of the first kind permit the determination of the surface magnetization and those of the second and third kind promise to resolve the question of the spin dependence of the mean free path in ferromagnets [7-9,161. In view of the latter question, we would like to mention a further linear relation, which we have established. Denoting, for a model involving M, # M,, by P,( MS, Mb) and Pi( MS, Mb) the polarization produced by only V”(R) and V,> (eq. (5)), respectively, and replacing in eq. (6) M by cuM ((Y constant with (aI< I), the total polarization P,,,(M,, Mb) produced by V”(r) and I$,( (Y) together is [ 171 ‘t~,(~s?
Mb) =pr(Ms*
Mb) + (ypi(Ms,
Mb).
(7)
Instead of calculating P,,, for a number of trial values of LYit thus suffices to calculate, in addition to P,, only one Pi (corresponding to vP,( a = 1)). As is also illustrated by fig. 3, the (spin-averaged) intensity profiles obtained for the same VU(r) are practically identical for different A4, and AK,,, models, which corroborates an earlier statement (cf. ref. [ 11) that this magnetic information is exclusively coded in the spin polarization. In the above, we have focused on the feasibility of determining J4, and AC, while assuming the ion-core potential P(r) as known. In order to discriminate between different models of VO(r), one can extend the above idea of selective senstivity of the exchange-induced spin polarization. Calculations using an energy-independent exchange approximation actually demonstrate the existence of features, which respond predominantly to P(r). For example, the height of the positive peak near 90 eV in fig. 3, which is only very weakly affected by changes in iL4, and AV,,, is very sensitive to V”(r). Since, however, the principle of construction of V”(r) is closely related to that of the nonmagnetic V(r), changes in Vu(r) are associated with changes in V(r) and therefore manifest themselves also in the intensity profile. Comparison between calculated and measured intensity or spin-orbit-induced polarization profiles thus already provides at least some discrimination between Vu(r) models.
4. Conclusions The proportionality of the spin polarization to the surface magnetization for the case of homogeneously layer-dependent magnetization (eq. (5)), which we have found to hold for Ni(OO1) and Fe( 1 IO) in very good approximation, suggests the feasibility of a direct experimental study of the behaviour of M,(T) near the critical temperature. The determination of the layer depen-
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dence of M, i.e. magnetic structure analysis, requires dynamical SPLEED calculations. These are made computationally feasible by means of a linear approximation sheme for the single-layer S-matrices, which reduces the computing time by at least a factor of 24 with respect to straightforward calculation. A further prerequisite for magnetic structure analysis, the existence of polarization features which are strongly sensitive to the surface magnetization, while responding only weakly to a possible spin splitting AVim of the imaginary potential, is verified by numerical calculations for Ni(OO1) and Fe(ll0). The finding of other features, which are produced by AV,, and are very sensitive to it, provides a basis for determining the actual spin dependence of the mean free path. In fact, a joint experimental and theoretical study currently in progress [ 181 already indicates such information on the ferromagnetic Ni(OO1) surface.
Acknowledgements We would like to thank Dr. S. Alvarado for topical Professor G. Eilenberger for his interest and kind hospitality.
discussions
and
References [I] Details and references to the original literature may be found in a recent review article: R. Feder, J. Phys. Cl4 (1981) 2049. [2] J.B. Pendry, Low Energy Electron Diffraction (Academic Press, London, 1974). and references therein. [31 L.A. Vredevoe and R.E. de Wames, Phys. Rev. 176 (1968) 684: X.1. Saldana and J.S. Helman, Phys. Rev. Bl6 (1977) 4978. [41 W. Kohn and L.J. Sham, Phys. Rev. 140 (1965) Al 133. [51 J.C. Slater. T.M. Wilson and J.W. Wood, Phys. Rev. 179 (1969) 28. [61 A.K. Rajagopal, Advan. Chem. Phys. 41 (I 980) 59. model calcultion of the spin splitting AI’,, [8] for Fe and Ni produces 171 A more sophisticated values which are by about an order of magnitude smaller. more strongly energy dependent, and even have the opposite sign over wide ranges of energy. In ontrast, a most recent microscopic model calculation for Fe [9] confirms the sign of AI’,, of the simple model (cf. ref. [l] and eq. (6)) and predicts its value to be reduced by an energy-dependent factor ranging between 6 and 10. As will become apparent in the context of fig. 3, knowledge of V,, is, however, not only not required for magnetic structure analysis by SPLEED. but can be gained by the analysis. PI R.W. Rendell and D.R. Penn, Phys. Rev. Letters 45 (1980) 2057. 191 Soe Yin and E. Tosatti. to be published. [lOI D. Gloetzel, private communication. [Ill R. Feder and J. Kirschner, Surface Sci. 102 (1981) 75. [I21 R. Feder. H. Pleyer. P. Bauer and N. Mtiller, Surface Sci. 109 (1981) 419. iI31 J.E. Demuth. P.M. Marcus and D.W. Jepsen, Phys. Rev. Bll (1975) 1460. 1141 While the results of fig. 2 were obtained for models with V,z = Vi,, the linear approximation scheme is also valid for models involving
a spin-dependent
mean free path.
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[ 151 S.F. Alvarado, Z. Physik 33 ( 1979) 5 1. and references therein. [ 161 The energy dependence of V&(E) does not prohibit its determination. It only implies that for a particular beam and diffraction geometry one obtains values of ViK,for a number of selected E points. and that different spectra may have to be studied to get a sufficiently dense mesh. [ 171 For the special case of MS = M,, such a relation was found previously (cf. ref. [I]). [18] R. Feder. SF. Alvarado, H. Hopster and E. Tamura, to be published.