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ZnSe(001)
Journal of Magnetism and Magnetic Materials 78 (1989) 195-202 North-Holland, Amsterdam
195
STRUCTURAL, ELECTRONIC AND MAGNETIC PROPERTIES OF METAL/SEMICONDUCTOR SUPERLATI'ICES: Fe/ZnSe(001) A. CONTINENZA, S. MASSIDDA and A.J. FREEMAN Materials Research Center, and Department of Physics and Astronomy, Northwestern University, Evanston, 1L 60208, USA Received 16 August 1988
The structural, electronic and magnetic properties of a 1 x I (monolayer) Fe/ZnSe (001) superlattice are investigated using the highly precise all-electron spin-polarized full-potential linearized augmented plane wave (FLAPW) method. The equilibrium distance between Fe and Se is determined using a total energy approach. Band structure, density of states, and charge and spin density distributions are analyzed and compared with the calculated results for ZnSe bulk and an Fe free film surface. Both the magnetic hyperfine field and the magnetic moment on the Fe sites are studied as a function of Fe-Se distance. The magnetic properties of Fe are found to be remarkably enhanced with respect to the bulk values.
1. Introduction Metal-semiconductor interfaces and superlattices have recently captured the interest of both the scientific and technological world: New properties and the possibility of realizing novel microelectronic devices with unusual optical and electronic applications is now the focus of many research efforts. In this regard, the successful growth of single crystals of ferromagnetic materials, such as Fe, Co and only very recently Ni on semiconducting substrates (in particular ZnSe and GaAs) - achieved via molecular beam epitaxy (MBE) by Prinz and collaborators [1-4] - represents an even more tempting challenge to investigate and optimize the appealing properties that these new materials might offer for device applications. Although the possibility of realizing a material which is at the same time magnetic and semiconducting is not completely new (since very extensive theoretical and experimental studies have been performed on dilute magnetic semiconductor systems (DMS) such as CdTe and ZnSe "doped" with Mn), the "peculiarities" shown by these new magnetic-semiconductors seem to be completely different. In particular, experimental measurements of the magnetic properties of Fe on ZnSe substrates (magnetization hysteresis, ferromag-
netic resonance [5,6]) show an in-plane localization of the magnetization and a rather low value of the coercive field which might turn out to be very useful for obtaining planar microelectronic magnetic configurations. In view of all this interest, and because only few theoretical studies of these materials are available to help understand their electronic structure and properties, we performed a highly precise all-electron self-consistent full-potential linearized augmented plane wave (FLAPW) calculation of the electronic and magnetic properties of the 1 x 1 superlattice of Fe on ZnSe. Following a description of some details of the calculation in section 2, we present results on the structural, electronic and magnetic properties in section 3, and conclusions in section 4.
2. Calculational details The structural and electronic properties of the Fe/ZnSe superlattice were calculated by using the full-potential linearized augmented plane wave (FLAPW) method within the local spin density approximation (LSDA) and the exchange-correlation potential as parametrized by Von Barth and
0304-8853/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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A. Continenra
et al. / Fe/ZnSe
Hedin [7]. The core states were treated fully-relativistically and updated at each iteration whereas the valence states (which included the 3d states of the Zn atoms) were treated semi-relativistically. Furthermore, we used a total energy approach in order to determine the equilibrium distance between the Fe and the Se layers since experimental data are not available regarding the formation distance between the ZnSe substrate and the Fe epilayers. In performing the total energy calculations, we kept the cut-off parameter K,,, = 3.2 a.u. constant over the entire distance range considered in order to obtain the same degree of convergence; this choice results in about 340 basis functions (depending on the lattice parameter and on the particular k-point). We used about 3600 plane waves for the expansion of the charge density and the potential in the interstitial region and spherical harmonics up to I,,, = 8 for the expansion of both potential and charge density inside the muffin-tin spheres; the same expansion was used for the wave function. To obtain the output charge density, the integration over the irreducible Brillouin zone (IBZ) wedge was performed by using a Gaussian smearing procedure and 18 K-points chosen according to the special k-points technique
[81.
In order to check the convergence of the calculations, we performed several tests - increasing the number of k-points up to 40 and the wave function cut-off up to K,,, = 3.5 a.u. As a result, we found that the magnetic moment value and the hyperfine field were quite stable with respect to the variation of both parameters; we observed a 0.4% variation for the magnetic moment and a 2.6% variation for the conduction electron contribution to the hyperfine field whereas the variation was much smaller (about 0.3%) for the core contribution to the hyperfine field. The geometry considered is shown in fig. 1. The very small mismatch (only 1.1%) between twice the bee-Fe lattice constant (are = 2.866 A) and the ZnSe equilibrium lattice constant (a,,,, = 5.6676 A) allows the replacement of one layer of Zn with one layer of Fe and causes only a small strain on the ZnSe lattice; this can be done by just occupying both the Zn positions (Fel) and the sites at the edges of the structure (Fe2). The two
(001) superlattice
0 Fe
OSe
Fig. 1. Unit cell of the Fe/ZnSe@Ol)
@Zn superlattice.
Fe sites differ in that Fe1 points toward the Se sp3 hybrids, while Fe2 is located in the “antibonding” interstitial positions of the zincblende structure. The structure considered has a simple tetragonal unit cell and space group D$; in the (001) plane, the atoms form a square with lattice parameters equal to the ZnSe calculatedequilibrium structure. In performing our total energy calculations, we kept the distance between the Zn and Se layers constant and equal to the calculated equilibrium value in bulk ZnSe obtained in a previous total energy calculation [9]. The use of the calculated value (a = 5.6357 A) instead of the experimental value (a = 5.6676 A) does not change the physics very much since the calculated value has only a 0.6% deviation from experiment.
3. Results 3.1. Structural properties The results of the total energy calculations are shown in fig. 2 as a function of the Fe-Se distance. The calculated data agree very well with a parabola fitting whose root mean square deviation is 0.1 X lOA Ry. The equilibrium distance, given by the position corresponding to the energy minimum, is dFe_* = 2.4628 A which implies c = 5.7128 A as tetragonal parameter. This c value represents an increase of about 1.4% and 0.8% when compared, respectively, with the calculated and experimental lattice spacing of bulk ZnSe. Although there are no experimental values of
197
A, Continenza et al. / Fe / ZnSe (001) superlattice -0.893
/
-0.896
l
energy scale: this suggests that the material may be rather soft and easy to deform - at least along the F e - S e bond.
3.2. Energy band structure and density of states
vn- - 0 . 8 9 9 ILl -0.902"
-0.905 2.38
2.42
2.46
2.50
2.54
dFe-Se( A ) Fig. 2. Total energy (in Ry) versus Fe-Se distances (in ,~). The zero of the energy scale has been shifted to - 18376.0 Ry.
dFe_Se, comparing our result with the nearest neighbor distance [10] (dFe_F~ = 2.48 A.) and the volume per atom ( V = 11.77 ,~3) for bulk bcc-Fe shows that an expansion of the structure is very likely to occur. A more significant comparison might be obtained with the experimental work of Jonker et al. [11] regarding the dilute magnetic semiconductor Zn 1_~Fe~Se. In particular they observed, via X-ray diffraction, that the lattice parameter of the compound in the direction perpendicular to the MBE growth direction increases linearly with Fe composition (x). From a least-squares fit they obtained the following empirical law:
The calculated self-consistent band structures for the majority and minority spin components are shown in fig. 3 along the main symmetry lines of the irreducible Brillouin zone. Most of the main features which appear for the majority spin component are very similar to those appearing for minority spin; the main difference is due to the exchange interaction which lifts toward higher energies the bands originating from the minority Fe d-orbitals. The five-fold degenerate d-levels of the Fe are expected to be split at (F) by the tetragonal symmetry, into three singlets (dxy , dx:_y2, d3z2_r2) ' a n d one doublet (d~z , dyz); their identification, however, is not obvious due to the bonding between Fe and Se and the resulting superposition of their orbitals. In particular, the doubly degenerate bands at F do not have pure d-character; instead, they are a superposition of the Fe d~z-, dyz-orbitals and Se Px-, py-orbitals.
o.o
EF
a ( x ) = (5.6673 ± 0.0002) + (0.091 ± 0.002)x.
(1) If we extrapolate their results for an Fe composition of x = 0.5 we get a = 5.714 ,~ which is very close to our calculated result. It must be pointed out, however, that while the structure we are considered is not a dilute alloy and that the Fe composition in our case is much higher, this comparison shows, nevertheless, that the presence of Fe forces the structure to expand. The structure expansion is also a good indication, confirmed by experiment [6], that the diffusion of Fe atoms into the ZnSe lattice (which seems to occur in the GaAs case) is quite unlikely and that the formation of intermixed compounds (such as FeSe) might be inhibited. Another feature that should be noticed from fig. 2, is the small range in the
-8.o
r
x
M
r
a
z
4.0
0.0 ~'~ ~'~--~
~
____----m------.~__ -8.0
F
X
.
~ M
~ F
b Z
Fig. 3. Self-consistent band structure for (a) majority and (b) minority spin components at the equilibrium Fe-Se distance.
198
A. Continenza et aL / F e / ZnSe (001) superlattice
The lowest band shown in fig. 3 is an s-like band with (F]) symmetry at F resulting from the hybridization of the Zn s- and d-orbitals and the Se p-orbitals; due to the small contribution coming from the Fe d-orbitals, the exchange splitting of this state is rather small (less than 0.3 eV). The five higher bands are grouped together and are separated into two doublets and one singlet at F. The two doublets, which move in parallel across the IBZ, are hybrids of Fel d and Fe2 d with Se p, respectively, whereas the singlet is a combination of only the dy~ Fel and Fe2 orbitals. It is worthwhile pointing out that the levels with an Fe2 component are always at higher energies with respect to the corresponding levels with an Fel component - which stresses the less favorable bonding position of Fe2. The bands between - 3.2 eV ( - 1.0 eV) and the Fermi level ( + 1.6 eV) for majority (minority) spin are mostly due to the d-levels of both Fel and Fe2. A larger hybridization between the Se p- and Fe d-levels is present in the bands which cross the Fermi level for majority spin (the corresponding bands for minority spin are well above the Fermi level). Some of these bands have a very small dispersion in the F - M - X directions and are responsible for the peaks in the density of states (see fig. 4) at - 0 . 8 eV ( - 0 . 8 eV) and - 0 . 4 eV (1.6 eV) for majority (minority) spin. For these bands the Se contribution is about 17% whereas the two Fe atoms contribute about 44% of the total. The strong hybridization between the Se p- and Fe d-levels is also very clear from the density of states (DOS) plots, shown in fig. 4 and in more detail from the projected density of states per atomic site (PDOS) (figs. 5 and 6). At very low energies below E v there are states mainly due to Z n - S e bonds and in particular to the hybridization betweeen Zn s-levels and Se p-levels. Nevertheless, a substantial contribution (about 10%) from the d-states of the Fe (with some contribution from the s-states) is also present. At higher energies, but still below E v, the Se p-, Fe dhybridization is much more evident, resulting in quite a wide band between - 5.0 and - 3.2 eV; in this case the contribution due t o the two Fe sites adds up to more than 50% (in some cases 70%) of the total density of states. These states can be
10.0 EF
" ~ o.o
-10.C -8.0
-40
0.0
4.0
E (eV) Fig. 4. Total density of states (DOS), in states/eV-spin, for majority and minority (negative values) spin components at the equilibrium F e - S e distance.
recognized as those which form the valence band of bulk ZnSe. While a very small polarization can be observed on the Zn site, the polarization on the Se site is much more remarkable: the p-band of the Se shows an energy shift of about 0.7 eV between majority and minority spin bands. As previously discussed, the wide peak right below
4.0 EF
Fe, I
0 .m C
ID
~
0.0
-4.0 4.0 Fe2
'~
0.0
C
ID Q -4.0 -8.0
-4.0
0.0
4.0
E (eV)
Fig. 5. Partial density of states per atomic site: Fel and Fe2 contribution in units of states/eV-atoms-spin for majority (plotted positive) and minority (negative scale) spin.
A. Continenza et a L / Fe / ZnSe (001) superlattice 1.0 Se
EF
.,= 0.0 C ffl ,~-1.0 1.0
~
Zn
o.o c a -1.o -8.0
-4.0
0.0
4.0
E (eV)
Fig. 6. Partial density of states per atomic site: Se and Zn contribution in units of states/eV per atom for majority and minority spin. (across) EF in the majority (minority) spin component originates mainly from the Fe d-bands; the contribution due to Zn and Se is in this case always less than 2%. At the Fermi level, finally, the contribution coming from the Se sites is comparable with the contribution coming from the majority states of the Fe sites, whereas the Zn contribution is much lower. The densities of states for Fel and Fe2, projected per atomic contribution, have roughly the same main features showing that the two sites play a very similar role independent of the difference in their geometrical configuration. How-
199
ever, a slight difference in the exchange splitting and the resulting different occupation of the majority and minority spin states implies also that a difference in the magnetic moment should be expected. The characteristic gap, due to the bonding-antibonding repulsion between d-states, which is present in the DOS of bulk Fe, is in this case suppressed by the narrowing of the Fe d-band localized in the F e - Z n S e interfacial region. The width of the d-band ( - 3.4 eV) in the present case is greatly reduced with respect to the bulk bcc-Fe density of states [13] ( - 6 . 2 eV) and resembles much more closely the shape calculated [12] for the free surface layer of a 7-layer slab. This agrees with the intuitive idea that a lower nearest neighbour coordination should result in a narrowing of the d-band and it is a further indication that the monolayer of Fe in this superlattice has a surface-like character. The states corresponding to the Fe d-orbitals and their high polarization (about a 1.2 eV exchange splitting) are responsible for the magnetic moment enhancement on the Fe sites to be described later.
3.3. Charge decomposition Further insights regarding the bonding nature of this structure can be found by looking at the decomposition into angular momentum contributions of the charge inside each muffin tin sphere shown in tables 1 and 2. These data have to be compared with the results obtained by similar (FLAPW) calculations for the Fe-film (free surface) [12] and bulk ZnSe [9], but noticing that these quantities are strongly dependent on the
Table 1 Charge decomposition (in electrons/unit celt) into different angular momentum components inside the muffin tin spheres and muffin tin radius values RMT (in a.u.) used Majority component
Minority component
R MT
Qs
Qp
Qd
QMT
Qs
Qp
Qd
QMT
0.151 0.145 0.686 0.296
0.163 0.151 1.127 0.225
4.445 4.438 0.036 4.835
4.771 4.746 1.855 5.417
0.132 0.134 0.678 0.286
0.139 0.138 1.064 0.205
1.730 1.806 0.033 4.878
2.010 2.089 1.779 5.378
Fe/ZnSe Fel Fe2 Se Zn
2.3 2.3 2.09 2.3
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A. Continenza et al. / F e / ZnSe (001) superlattice
Table 2 Total charge decomposition (in electrons/unit cell) into different angular m o m e n t u m components for Se and Zn compared with the bulk results [9]. The muffin tin radius values are the same as those used in the bulk calculation [9] Qs
Qp
Qd
QMT
Fe/ZnSe bulk ZnSe
Se Se
1.338 1.365
2.191 2.223
0.069 0.026
3.634 3.619
Fe/ZnSe bulk ZnSe
Zn Zn
0.582 0.539
0.430 0.453
9.713 9.758
10.795 10.768
muffin tin radii used. First of all, we should note that the total charge (table 2) for Zn and Se is very close to their bulk value. However, some differences occur, such as the charge increase into the Se d-orbitals (more than twice the bulk value), the corresponding decrease of the p-character charge and, only to a minor extent, the same kind of phenomena for the Zn case. As discussed above, these features are all related to the hybridization with the Fe d-orbitals. As far as the Fe charge decomposition is concerned, we find a much lower value for the l = 0 component with respect to the results for the Fe surface layer [12]. This is in agreement with what we should expect recalling that in the equilibrium structure of the F e / Z n S e superlattice the Fe is compressed with respect to its natural structure (Fe-bulk or Fe-film); as a consequence of the pressure resulting on the Fe atoms, the delocalized s states are lifted to higher energy and are therefore depopulated. The p-character charge is increased with respect to the film free surface layer but is lower than the bulk layer. The loss of electrons from the p-orbitals is a feature that also has been found by film calculations for several metallic surface systems and it reflects the tendency of these electrons to transfer into the vacuum region. We should point out that in this case the interface with the semiconductor plays a role somehow similar to the vacuum in reducing the Fe-site coordination. Actually, from a purely geometrical argument, the environmental situation of both Fel and Fe2 in this supercell is exactly equal to what it is for the surface layer of the slab: the number of its nearest neighbors (nn) is, in fact,
reduced from 8 as in the bulk case to 4 since Fe replaces the tetrahedrally coordinated Zn sites; moreover in this case the nn sites are occupied by Se atoms instead of Fe. As discussed before, the reduced coordination is responsible for the narrowing of the Fe d-band and for the characteristic surface-like behavior of the Fe (magnetic moment enhancement, loss of charge from p-orbitals, hyperfine fields). Of course the deviation from the free surface is to be attributed to the fact that the Se replaces Fe in all of its nearest neighbour sites and that the resulting bonding is more ionic (Fe-Se) than purely metallic (Fe-Fe).
3.4. Magnetic properties By looking at the difference between the total charge with spin up and the total charge with spin down at each site, it is possible to derive the magnetic moment of that particular site. In fig. 7 we show the calculated magnetic moment as a function of the F e - S e distance for the two atomic Fe sites (Fel, Fe2). As expected, the magnetic moment increases with distance ranging between 2.3 and 2.85/t B for Fel, and 2.1 and 2.7/~ for Fe2. The difference between the magnetic moment on the two sites (about a 4% constant deviation over the range of the F e - S e distance considered) indicates the influence of the site coordination on the magnetic behaviour. Even though that dif-
3.00 '
%
2.80-
~
2.60-
Fel •
,q
• • •
• Fe2
E o IE .~ 2.40 ~
2.20
2.00
2.38
l
2. 2
2.46 2. 0
2.54
d F e - S e ( '~ ) Fig. 7. Magnetic m o m e n t (/~e) on Fel and Fe2 sites as a function of the F e - S e distance (,~).
A. Continenza et al. / Fe / ZnSe (001) superlattice
Zn i',
]'
Fe2
Fe2
Fel
Fel Zn
Fig. 8. Spin density d i s t r i b u t i o n in the (101) p l a n e at the e q u i l i b r i u m F e - S e distance. The d a s h e d lines i n d i c a t e the c o n t o u r s of negative spin d e n s i t y in units of 10 - 4 e / a . u . 3. Successive c o n t o u r s h o w n differ b y a factor of 2.
ference is rather small, it is a direct consequence of the spin density distribution plotted in fig. 8 which shows a localization of the negative spin polarization in the region between nearest neighbour Fe but, most of all, in the zinc blende interstitial regions. The direct exchange interaction present between Fe sites and, also, between Fe and the positively polarized Se, repels the electrons with negative polarization away from the F e - S e and F e - F e bonds in the region where, because of the zinc blende crystal structure, there is a charge depletion. It is this enhanced spin down polarization localized close to the Fe2 atom which lowers the magnetic moment on that site. In that sense, the role played by the Se atom is, therefore, to favor the direct polarization of the outermost Fe electrons and to further enhance the total magnetic moment on the Fel sites. Moreover, in agreement with experiment [6], the spin density distribution indicates that the overall superlattice magnetization is anisotropic. The magnetic moment at the equilibrium position (2.71#B for Fel and 2.67#B for Fe2) is only about 8% lower than the value calculated for the free surface (2.98~B) and about 15% lower than the monolayer; it is remarkably enhanced (by about 30%) when compared with the experimental bulk value (2.12/~a). It is striking that the Fe moment is enhanced relative to bulk by almost the same amount as the Fe moment at the free surface or even for the extreme case of a free monolayer.
201
In order to further investigate the magnetic properties of the superlattice, we calculated the Fermi contact contribution to the magnetic hyperfine field (table 3). As expected, the core contribution is strongly negative due to the direct exchange interaction between the core s-orbitals and the magnetic d-electrons; it is comparable with the free surface value [12] and, like the other values shown, scales with the magnetic moment all over the F e - S e distance range considered. On the other hand, the conduction electron contribution (CE) is positive (as for the free atom) and even larger than at the clean surface indicating that the direct coupling of the CE to the unpaired d-electrons prevails over the indirect (covalent) polarisation. We also found that the CE contribution decreases continuously as the F e - S e distance is increased and to approach the clean surface value. The observed overall variation at the extreme value of the dve_s~ interval considered is about 8% for Fel and 25% for Fe2. This might indicate that the slightly polarized Se favors the direct polarization of the outermost s-like Fe electrons which are not involved in the Se p - F e d hybridization and in the F e - S e bonding. This is also in agreement with the fact that the CE contribution is not dependent on the magnetic moment (a property mainly due to the localized d-electrons), but unlike the core contribution, is strongly affected by the environment surrounding the conduction electron charge.
Table 3 M a g n e t i c m o m e n t s (in #B) a n d F e r m i c o n t a c t h y p e r f i n e fields (in k G ) b r o k e n d o w n i n t o core a n d c o n d u c t i o n electron (CE) contributions Magnetic moments
H y p e r f i n e field core
CE
total
2.76 2.67
- 386 - 371
+ 218 + 119
- 168 - 252
surface layer center layer
2.98 2.25
- 398 - 291
+ 143 - 75
- 255 - 366
F e - m o n o l a y e r b) e x p e r i m e n t c)
3.18 2.12
-- 424
+ 323
-- 101 - 340
Fe/ZnSe Fel Fe2 Fe-film a)
a) Ref. [12]. b) Ref. [14]. c) Ref. [10].
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A. Continenza et al. / F e / ZnSe (001) superlattice
As a result of such a large CE, contribution, the total hyperfine field H c is reduced in magnitude as found for the free-Fe surface, despite the considerable enhancement of the magnetic moment and is therefore not proportional to the magnetic moment.
4. Conclusion We have presented results of self-consistent all-electron local spin density functional calculations for a 1 × 1 F e / Z n S e (001) superlattice. Our total energy approach determined the equilibrium distance between the Fe layer and the ZnSe substrate to be d = 2.4628 ,~ - in good agreement with experimental evidence [11] available to date. The analysis of the calculated electronic and magnetic properties (magnetic moment and hyperfine contact field) lead us to the conclusion that an important factor in determining the Fe properties might be the geometrical coordination of the Fe sites. The Fe magnetic moment is found to be remarkably enhanced (by 30%) with respect to the bulk value; the magnitude of the total Fe hyperfine field is reduced relative to the bulk value, however, because of a large positive contribution from the 4s-like conduction electrons. Finally, the strong hybridization between the Fe 3d- and the highly directional Se 3p-electrons results in a weak polarization on the Se sites and in lowering the value of the Fe magnetic moment with respect to the free surface.
Acknowledgements We are grateful to G. Prinz for suggesting this study and for critical comments. Work supported
by the National Science Foundation [through the Northwestern University Material Center, Grant No. DMR85-20280 and by a grant of CRAY supercomputing time from its D.ivision of Advanced Scientific Computing at the National Center for Supercomputer Applications at the University of Illinois, U r b a n a / C h a m p a i g n ] and by the Office of Naval Research (Grant No. N00014-81-K-0438). One of us (A.J.F.) thanks L.J. Varnerin for handling the editorial aspects of this ms. (including the anonymous refereeing).
References [1] G.A. Prinz, B.T. Jonker, J.J. Krebs, J.M. Ferrari and F. Kovanic, Appl. Phys. Lett. 48 (1986) 1756. [2] G.A. Prinz and J.J. Krebs, Appl. Phys. Lett. 39 (1981) 397. [3] G.A. Prinz, Phys. Rev. Lett. 54 (1985) 1051. [4] G.A. Prinz, B.T. Jonker and J.J. Krebs, Bull. Am. Phys. Soc. 33 (1988) 562. [5] J.J. Krebs, B.T. Jonker and G.A. Prinz, J. Appl. Phys. 61 (1987) 3744. [6] B.T. Jonker, J.J. Krebs, G.A. Prinz and S.B. Qadri, J. Crystal Growth 81 (1987) 524. [7] L. Hedin and B.I. Lundqvist, J. Phys. C4 (1971) 2064. U. von Barth and L. Hedin, ibid. 5 (1972) 1629. [8] A. Baldereschi, Phys. Rev. B7 (1973) 5212. D.J. Chadi and M.L. Cohen, Phys. Rev. B8 (1973) 5747. H.J. Monkhorst and J.D. Peck, Phys. Rev. B13 (1976) 5188. [9] A. Continenza, S. Massidda and A.J. Freeman, Phys. Rev. B38 (1988) 12996. [10] American Inst. of Physics Handbook (McGraw-Hill, New York, 1987). [11] B.T. Jonker, J.J. Krebs, S.B. Qadri and G.A. Prinz, Appl. Phys. Lett. 50 (1987) 848. [12] S. Ohnishi, A.J. Freeman and M. Weinert, Phys. Rev. B28 (1983) 6741. [13] K.B. Hathaway, H.J.F. Jansen and A.J. Freeman, Phys. Rev. B31 (1985) 7603. [14] S. Ohnishi, M. Weinert and A.J. Freeman, Phys. Rev. B30 (1984) 36.
195
STRUCTURAL, ELECTRONIC AND MAGNETIC PROPERTIES OF METAL/SEMICONDUCTOR SUPERLATI'ICES: Fe/ZnSe(001) A. CONTINENZA, S. MASSIDDA and A.J. FREEMAN Materials Research Center, and Department of Physics and Astronomy, Northwestern University, Evanston, 1L 60208, USA Received 16 August 1988
The structural, electronic and magnetic properties of a 1 x I (monolayer) Fe/ZnSe (001) superlattice are investigated using the highly precise all-electron spin-polarized full-potential linearized augmented plane wave (FLAPW) method. The equilibrium distance between Fe and Se is determined using a total energy approach. Band structure, density of states, and charge and spin density distributions are analyzed and compared with the calculated results for ZnSe bulk and an Fe free film surface. Both the magnetic hyperfine field and the magnetic moment on the Fe sites are studied as a function of Fe-Se distance. The magnetic properties of Fe are found to be remarkably enhanced with respect to the bulk values.
1. Introduction Metal-semiconductor interfaces and superlattices have recently captured the interest of both the scientific and technological world: New properties and the possibility of realizing novel microelectronic devices with unusual optical and electronic applications is now the focus of many research efforts. In this regard, the successful growth of single crystals of ferromagnetic materials, such as Fe, Co and only very recently Ni on semiconducting substrates (in particular ZnSe and GaAs) - achieved via molecular beam epitaxy (MBE) by Prinz and collaborators [1-4] - represents an even more tempting challenge to investigate and optimize the appealing properties that these new materials might offer for device applications. Although the possibility of realizing a material which is at the same time magnetic and semiconducting is not completely new (since very extensive theoretical and experimental studies have been performed on dilute magnetic semiconductor systems (DMS) such as CdTe and ZnSe "doped" with Mn), the "peculiarities" shown by these new magnetic-semiconductors seem to be completely different. In particular, experimental measurements of the magnetic properties of Fe on ZnSe substrates (magnetization hysteresis, ferromag-
netic resonance [5,6]) show an in-plane localization of the magnetization and a rather low value of the coercive field which might turn out to be very useful for obtaining planar microelectronic magnetic configurations. In view of all this interest, and because only few theoretical studies of these materials are available to help understand their electronic structure and properties, we performed a highly precise all-electron self-consistent full-potential linearized augmented plane wave (FLAPW) calculation of the electronic and magnetic properties of the 1 x 1 superlattice of Fe on ZnSe. Following a description of some details of the calculation in section 2, we present results on the structural, electronic and magnetic properties in section 3, and conclusions in section 4.
2. Calculational details The structural and electronic properties of the Fe/ZnSe superlattice were calculated by using the full-potential linearized augmented plane wave (FLAPW) method within the local spin density approximation (LSDA) and the exchange-correlation potential as parametrized by Von Barth and
0304-8853/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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A. Continenra
et al. / Fe/ZnSe
Hedin [7]. The core states were treated fully-relativistically and updated at each iteration whereas the valence states (which included the 3d states of the Zn atoms) were treated semi-relativistically. Furthermore, we used a total energy approach in order to determine the equilibrium distance between the Fe and the Se layers since experimental data are not available regarding the formation distance between the ZnSe substrate and the Fe epilayers. In performing the total energy calculations, we kept the cut-off parameter K,,, = 3.2 a.u. constant over the entire distance range considered in order to obtain the same degree of convergence; this choice results in about 340 basis functions (depending on the lattice parameter and on the particular k-point). We used about 3600 plane waves for the expansion of the charge density and the potential in the interstitial region and spherical harmonics up to I,,, = 8 for the expansion of both potential and charge density inside the muffin-tin spheres; the same expansion was used for the wave function. To obtain the output charge density, the integration over the irreducible Brillouin zone (IBZ) wedge was performed by using a Gaussian smearing procedure and 18 K-points chosen according to the special k-points technique
[81.
In order to check the convergence of the calculations, we performed several tests - increasing the number of k-points up to 40 and the wave function cut-off up to K,,, = 3.5 a.u. As a result, we found that the magnetic moment value and the hyperfine field were quite stable with respect to the variation of both parameters; we observed a 0.4% variation for the magnetic moment and a 2.6% variation for the conduction electron contribution to the hyperfine field whereas the variation was much smaller (about 0.3%) for the core contribution to the hyperfine field. The geometry considered is shown in fig. 1. The very small mismatch (only 1.1%) between twice the bee-Fe lattice constant (are = 2.866 A) and the ZnSe equilibrium lattice constant (a,,,, = 5.6676 A) allows the replacement of one layer of Zn with one layer of Fe and causes only a small strain on the ZnSe lattice; this can be done by just occupying both the Zn positions (Fel) and the sites at the edges of the structure (Fe2). The two
(001) superlattice
0 Fe
OSe
Fig. 1. Unit cell of the Fe/ZnSe@Ol)
@Zn superlattice.
Fe sites differ in that Fe1 points toward the Se sp3 hybrids, while Fe2 is located in the “antibonding” interstitial positions of the zincblende structure. The structure considered has a simple tetragonal unit cell and space group D$; in the (001) plane, the atoms form a square with lattice parameters equal to the ZnSe calculatedequilibrium structure. In performing our total energy calculations, we kept the distance between the Zn and Se layers constant and equal to the calculated equilibrium value in bulk ZnSe obtained in a previous total energy calculation [9]. The use of the calculated value (a = 5.6357 A) instead of the experimental value (a = 5.6676 A) does not change the physics very much since the calculated value has only a 0.6% deviation from experiment.
3. Results 3.1. Structural properties The results of the total energy calculations are shown in fig. 2 as a function of the Fe-Se distance. The calculated data agree very well with a parabola fitting whose root mean square deviation is 0.1 X lOA Ry. The equilibrium distance, given by the position corresponding to the energy minimum, is dFe_* = 2.4628 A which implies c = 5.7128 A as tetragonal parameter. This c value represents an increase of about 1.4% and 0.8% when compared, respectively, with the calculated and experimental lattice spacing of bulk ZnSe. Although there are no experimental values of
197
A, Continenza et al. / Fe / ZnSe (001) superlattice -0.893
/
-0.896
l
energy scale: this suggests that the material may be rather soft and easy to deform - at least along the F e - S e bond.
3.2. Energy band structure and density of states
vn- - 0 . 8 9 9 ILl -0.902"
-0.905 2.38
2.42
2.46
2.50
2.54
dFe-Se( A ) Fig. 2. Total energy (in Ry) versus Fe-Se distances (in ,~). The zero of the energy scale has been shifted to - 18376.0 Ry.
dFe_Se, comparing our result with the nearest neighbor distance [10] (dFe_F~ = 2.48 A.) and the volume per atom ( V = 11.77 ,~3) for bulk bcc-Fe shows that an expansion of the structure is very likely to occur. A more significant comparison might be obtained with the experimental work of Jonker et al. [11] regarding the dilute magnetic semiconductor Zn 1_~Fe~Se. In particular they observed, via X-ray diffraction, that the lattice parameter of the compound in the direction perpendicular to the MBE growth direction increases linearly with Fe composition (x). From a least-squares fit they obtained the following empirical law:
The calculated self-consistent band structures for the majority and minority spin components are shown in fig. 3 along the main symmetry lines of the irreducible Brillouin zone. Most of the main features which appear for the majority spin component are very similar to those appearing for minority spin; the main difference is due to the exchange interaction which lifts toward higher energies the bands originating from the minority Fe d-orbitals. The five-fold degenerate d-levels of the Fe are expected to be split at (F) by the tetragonal symmetry, into three singlets (dxy , dx:_y2, d3z2_r2) ' a n d one doublet (d~z , dyz); their identification, however, is not obvious due to the bonding between Fe and Se and the resulting superposition of their orbitals. In particular, the doubly degenerate bands at F do not have pure d-character; instead, they are a superposition of the Fe d~z-, dyz-orbitals and Se Px-, py-orbitals.
o.o
EF
a ( x ) = (5.6673 ± 0.0002) + (0.091 ± 0.002)x.
(1) If we extrapolate their results for an Fe composition of x = 0.5 we get a = 5.714 ,~ which is very close to our calculated result. It must be pointed out, however, that while the structure we are considered is not a dilute alloy and that the Fe composition in our case is much higher, this comparison shows, nevertheless, that the presence of Fe forces the structure to expand. The structure expansion is also a good indication, confirmed by experiment [6], that the diffusion of Fe atoms into the ZnSe lattice (which seems to occur in the GaAs case) is quite unlikely and that the formation of intermixed compounds (such as FeSe) might be inhibited. Another feature that should be noticed from fig. 2, is the small range in the
-8.o
r
x
M
r
a
z
4.0
0.0 ~'~ ~'~--~
~
____----m------.~__ -8.0
F
X
.
~ M
~ F
b Z
Fig. 3. Self-consistent band structure for (a) majority and (b) minority spin components at the equilibrium Fe-Se distance.
198
A. Continenza et aL / F e / ZnSe (001) superlattice
The lowest band shown in fig. 3 is an s-like band with (F]) symmetry at F resulting from the hybridization of the Zn s- and d-orbitals and the Se p-orbitals; due to the small contribution coming from the Fe d-orbitals, the exchange splitting of this state is rather small (less than 0.3 eV). The five higher bands are grouped together and are separated into two doublets and one singlet at F. The two doublets, which move in parallel across the IBZ, are hybrids of Fel d and Fe2 d with Se p, respectively, whereas the singlet is a combination of only the dy~ Fel and Fe2 orbitals. It is worthwhile pointing out that the levels with an Fe2 component are always at higher energies with respect to the corresponding levels with an Fel component - which stresses the less favorable bonding position of Fe2. The bands between - 3.2 eV ( - 1.0 eV) and the Fermi level ( + 1.6 eV) for majority (minority) spin are mostly due to the d-levels of both Fel and Fe2. A larger hybridization between the Se p- and Fe d-levels is present in the bands which cross the Fermi level for majority spin (the corresponding bands for minority spin are well above the Fermi level). Some of these bands have a very small dispersion in the F - M - X directions and are responsible for the peaks in the density of states (see fig. 4) at - 0 . 8 eV ( - 0 . 8 eV) and - 0 . 4 eV (1.6 eV) for majority (minority) spin. For these bands the Se contribution is about 17% whereas the two Fe atoms contribute about 44% of the total. The strong hybridization between the Se p- and Fe d-levels is also very clear from the density of states (DOS) plots, shown in fig. 4 and in more detail from the projected density of states per atomic site (PDOS) (figs. 5 and 6). At very low energies below E v there are states mainly due to Z n - S e bonds and in particular to the hybridization betweeen Zn s-levels and Se p-levels. Nevertheless, a substantial contribution (about 10%) from the d-states of the Fe (with some contribution from the s-states) is also present. At higher energies, but still below E v, the Se p-, Fe dhybridization is much more evident, resulting in quite a wide band between - 5.0 and - 3.2 eV; in this case the contribution due t o the two Fe sites adds up to more than 50% (in some cases 70%) of the total density of states. These states can be
10.0 EF
" ~ o.o
-10.C -8.0
-40
0.0
4.0
E (eV) Fig. 4. Total density of states (DOS), in states/eV-spin, for majority and minority (negative values) spin components at the equilibrium F e - S e distance.
recognized as those which form the valence band of bulk ZnSe. While a very small polarization can be observed on the Zn site, the polarization on the Se site is much more remarkable: the p-band of the Se shows an energy shift of about 0.7 eV between majority and minority spin bands. As previously discussed, the wide peak right below
4.0 EF
Fe, I
0 .m C
ID
~
0.0
-4.0 4.0 Fe2
'~
0.0
C
ID Q -4.0 -8.0
-4.0
0.0
4.0
E (eV)
Fig. 5. Partial density of states per atomic site: Fel and Fe2 contribution in units of states/eV-atoms-spin for majority (plotted positive) and minority (negative scale) spin.
A. Continenza et a L / Fe / ZnSe (001) superlattice 1.0 Se
EF
.,= 0.0 C ffl ,~-1.0 1.0
~
Zn
o.o c a -1.o -8.0
-4.0
0.0
4.0
E (eV)
Fig. 6. Partial density of states per atomic site: Se and Zn contribution in units of states/eV per atom for majority and minority spin. (across) EF in the majority (minority) spin component originates mainly from the Fe d-bands; the contribution due to Zn and Se is in this case always less than 2%. At the Fermi level, finally, the contribution coming from the Se sites is comparable with the contribution coming from the majority states of the Fe sites, whereas the Zn contribution is much lower. The densities of states for Fel and Fe2, projected per atomic contribution, have roughly the same main features showing that the two sites play a very similar role independent of the difference in their geometrical configuration. How-
199
ever, a slight difference in the exchange splitting and the resulting different occupation of the majority and minority spin states implies also that a difference in the magnetic moment should be expected. The characteristic gap, due to the bonding-antibonding repulsion between d-states, which is present in the DOS of bulk Fe, is in this case suppressed by the narrowing of the Fe d-band localized in the F e - Z n S e interfacial region. The width of the d-band ( - 3.4 eV) in the present case is greatly reduced with respect to the bulk bcc-Fe density of states [13] ( - 6 . 2 eV) and resembles much more closely the shape calculated [12] for the free surface layer of a 7-layer slab. This agrees with the intuitive idea that a lower nearest neighbour coordination should result in a narrowing of the d-band and it is a further indication that the monolayer of Fe in this superlattice has a surface-like character. The states corresponding to the Fe d-orbitals and their high polarization (about a 1.2 eV exchange splitting) are responsible for the magnetic moment enhancement on the Fe sites to be described later.
3.3. Charge decomposition Further insights regarding the bonding nature of this structure can be found by looking at the decomposition into angular momentum contributions of the charge inside each muffin tin sphere shown in tables 1 and 2. These data have to be compared with the results obtained by similar (FLAPW) calculations for the Fe-film (free surface) [12] and bulk ZnSe [9], but noticing that these quantities are strongly dependent on the
Table 1 Charge decomposition (in electrons/unit celt) into different angular momentum components inside the muffin tin spheres and muffin tin radius values RMT (in a.u.) used Majority component
Minority component
R MT
Qs
Qp
Qd
QMT
Qs
Qp
Qd
QMT
0.151 0.145 0.686 0.296
0.163 0.151 1.127 0.225
4.445 4.438 0.036 4.835
4.771 4.746 1.855 5.417
0.132 0.134 0.678 0.286
0.139 0.138 1.064 0.205
1.730 1.806 0.033 4.878
2.010 2.089 1.779 5.378
Fe/ZnSe Fel Fe2 Se Zn
2.3 2.3 2.09 2.3
200
A. Continenza et al. / F e / ZnSe (001) superlattice
Table 2 Total charge decomposition (in electrons/unit cell) into different angular m o m e n t u m components for Se and Zn compared with the bulk results [9]. The muffin tin radius values are the same as those used in the bulk calculation [9] Qs
Qp
Qd
QMT
Fe/ZnSe bulk ZnSe
Se Se
1.338 1.365
2.191 2.223
0.069 0.026
3.634 3.619
Fe/ZnSe bulk ZnSe
Zn Zn
0.582 0.539
0.430 0.453
9.713 9.758
10.795 10.768
muffin tin radii used. First of all, we should note that the total charge (table 2) for Zn and Se is very close to their bulk value. However, some differences occur, such as the charge increase into the Se d-orbitals (more than twice the bulk value), the corresponding decrease of the p-character charge and, only to a minor extent, the same kind of phenomena for the Zn case. As discussed above, these features are all related to the hybridization with the Fe d-orbitals. As far as the Fe charge decomposition is concerned, we find a much lower value for the l = 0 component with respect to the results for the Fe surface layer [12]. This is in agreement with what we should expect recalling that in the equilibrium structure of the F e / Z n S e superlattice the Fe is compressed with respect to its natural structure (Fe-bulk or Fe-film); as a consequence of the pressure resulting on the Fe atoms, the delocalized s states are lifted to higher energy and are therefore depopulated. The p-character charge is increased with respect to the film free surface layer but is lower than the bulk layer. The loss of electrons from the p-orbitals is a feature that also has been found by film calculations for several metallic surface systems and it reflects the tendency of these electrons to transfer into the vacuum region. We should point out that in this case the interface with the semiconductor plays a role somehow similar to the vacuum in reducing the Fe-site coordination. Actually, from a purely geometrical argument, the environmental situation of both Fel and Fe2 in this supercell is exactly equal to what it is for the surface layer of the slab: the number of its nearest neighbors (nn) is, in fact,
reduced from 8 as in the bulk case to 4 since Fe replaces the tetrahedrally coordinated Zn sites; moreover in this case the nn sites are occupied by Se atoms instead of Fe. As discussed before, the reduced coordination is responsible for the narrowing of the Fe d-band and for the characteristic surface-like behavior of the Fe (magnetic moment enhancement, loss of charge from p-orbitals, hyperfine fields). Of course the deviation from the free surface is to be attributed to the fact that the Se replaces Fe in all of its nearest neighbour sites and that the resulting bonding is more ionic (Fe-Se) than purely metallic (Fe-Fe).
3.4. Magnetic properties By looking at the difference between the total charge with spin up and the total charge with spin down at each site, it is possible to derive the magnetic moment of that particular site. In fig. 7 we show the calculated magnetic moment as a function of the F e - S e distance for the two atomic Fe sites (Fel, Fe2). As expected, the magnetic moment increases with distance ranging between 2.3 and 2.85/t B for Fel, and 2.1 and 2.7/~ for Fe2. The difference between the magnetic moment on the two sites (about a 4% constant deviation over the range of the F e - S e distance considered) indicates the influence of the site coordination on the magnetic behaviour. Even though that dif-
3.00 '
%
2.80-
~
2.60-
Fel •
,q
• • •
• Fe2
E o IE .~ 2.40 ~
2.20
2.00
2.38
l
2. 2
2.46 2. 0
2.54
d F e - S e ( '~ ) Fig. 7. Magnetic m o m e n t (/~e) on Fel and Fe2 sites as a function of the F e - S e distance (,~).
A. Continenza et al. / Fe / ZnSe (001) superlattice
Zn i',
]'
Fe2
Fe2
Fel
Fel Zn
Fig. 8. Spin density d i s t r i b u t i o n in the (101) p l a n e at the e q u i l i b r i u m F e - S e distance. The d a s h e d lines i n d i c a t e the c o n t o u r s of negative spin d e n s i t y in units of 10 - 4 e / a . u . 3. Successive c o n t o u r s h o w n differ b y a factor of 2.
ference is rather small, it is a direct consequence of the spin density distribution plotted in fig. 8 which shows a localization of the negative spin polarization in the region between nearest neighbour Fe but, most of all, in the zinc blende interstitial regions. The direct exchange interaction present between Fe sites and, also, between Fe and the positively polarized Se, repels the electrons with negative polarization away from the F e - S e and F e - F e bonds in the region where, because of the zinc blende crystal structure, there is a charge depletion. It is this enhanced spin down polarization localized close to the Fe2 atom which lowers the magnetic moment on that site. In that sense, the role played by the Se atom is, therefore, to favor the direct polarization of the outermost Fe electrons and to further enhance the total magnetic moment on the Fel sites. Moreover, in agreement with experiment [6], the spin density distribution indicates that the overall superlattice magnetization is anisotropic. The magnetic moment at the equilibrium position (2.71#B for Fel and 2.67#B for Fe2) is only about 8% lower than the value calculated for the free surface (2.98~B) and about 15% lower than the monolayer; it is remarkably enhanced (by about 30%) when compared with the experimental bulk value (2.12/~a). It is striking that the Fe moment is enhanced relative to bulk by almost the same amount as the Fe moment at the free surface or even for the extreme case of a free monolayer.
201
In order to further investigate the magnetic properties of the superlattice, we calculated the Fermi contact contribution to the magnetic hyperfine field (table 3). As expected, the core contribution is strongly negative due to the direct exchange interaction between the core s-orbitals and the magnetic d-electrons; it is comparable with the free surface value [12] and, like the other values shown, scales with the magnetic moment all over the F e - S e distance range considered. On the other hand, the conduction electron contribution (CE) is positive (as for the free atom) and even larger than at the clean surface indicating that the direct coupling of the CE to the unpaired d-electrons prevails over the indirect (covalent) polarisation. We also found that the CE contribution decreases continuously as the F e - S e distance is increased and to approach the clean surface value. The observed overall variation at the extreme value of the dve_s~ interval considered is about 8% for Fel and 25% for Fe2. This might indicate that the slightly polarized Se favors the direct polarization of the outermost s-like Fe electrons which are not involved in the Se p - F e d hybridization and in the F e - S e bonding. This is also in agreement with the fact that the CE contribution is not dependent on the magnetic moment (a property mainly due to the localized d-electrons), but unlike the core contribution, is strongly affected by the environment surrounding the conduction electron charge.
Table 3 M a g n e t i c m o m e n t s (in #B) a n d F e r m i c o n t a c t h y p e r f i n e fields (in k G ) b r o k e n d o w n i n t o core a n d c o n d u c t i o n electron (CE) contributions Magnetic moments
H y p e r f i n e field core
CE
total
2.76 2.67
- 386 - 371
+ 218 + 119
- 168 - 252
surface layer center layer
2.98 2.25
- 398 - 291
+ 143 - 75
- 255 - 366
F e - m o n o l a y e r b) e x p e r i m e n t c)
3.18 2.12
-- 424
+ 323
-- 101 - 340
Fe/ZnSe Fel Fe2 Fe-film a)
a) Ref. [12]. b) Ref. [14]. c) Ref. [10].
202
A. Continenza et al. / F e / ZnSe (001) superlattice
As a result of such a large CE, contribution, the total hyperfine field H c is reduced in magnitude as found for the free-Fe surface, despite the considerable enhancement of the magnetic moment and is therefore not proportional to the magnetic moment.
4. Conclusion We have presented results of self-consistent all-electron local spin density functional calculations for a 1 × 1 F e / Z n S e (001) superlattice. Our total energy approach determined the equilibrium distance between the Fe layer and the ZnSe substrate to be d = 2.4628 ,~ - in good agreement with experimental evidence [11] available to date. The analysis of the calculated electronic and magnetic properties (magnetic moment and hyperfine contact field) lead us to the conclusion that an important factor in determining the Fe properties might be the geometrical coordination of the Fe sites. The Fe magnetic moment is found to be remarkably enhanced (by 30%) with respect to the bulk value; the magnitude of the total Fe hyperfine field is reduced relative to the bulk value, however, because of a large positive contribution from the 4s-like conduction electrons. Finally, the strong hybridization between the Fe 3d- and the highly directional Se 3p-electrons results in a weak polarization on the Se sites and in lowering the value of the Fe magnetic moment with respect to the free surface.
Acknowledgements We are grateful to G. Prinz for suggesting this study and for critical comments. Work supported
by the National Science Foundation [through the Northwestern University Material Center, Grant No. DMR85-20280 and by a grant of CRAY supercomputing time from its D.ivision of Advanced Scientific Computing at the National Center for Supercomputer Applications at the University of Illinois, U r b a n a / C h a m p a i g n ] and by the Office of Naval Research (Grant No. N00014-81-K-0438). One of us (A.J.F.) thanks L.J. Varnerin for handling the editorial aspects of this ms. (including the anonymous refereeing).
References [1] G.A. Prinz, B.T. Jonker, J.J. Krebs, J.M. Ferrari and F. Kovanic, Appl. Phys. Lett. 48 (1986) 1756. [2] G.A. Prinz and J.J. Krebs, Appl. Phys. Lett. 39 (1981) 397. [3] G.A. Prinz, Phys. Rev. Lett. 54 (1985) 1051. [4] G.A. Prinz, B.T. Jonker and J.J. Krebs, Bull. Am. Phys. Soc. 33 (1988) 562. [5] J.J. Krebs, B.T. Jonker and G.A. Prinz, J. Appl. Phys. 61 (1987) 3744. [6] B.T. Jonker, J.J. Krebs, G.A. Prinz and S.B. Qadri, J. Crystal Growth 81 (1987) 524. [7] L. Hedin and B.I. Lundqvist, J. Phys. C4 (1971) 2064. U. von Barth and L. Hedin, ibid. 5 (1972) 1629. [8] A. Baldereschi, Phys. Rev. B7 (1973) 5212. D.J. Chadi and M.L. Cohen, Phys. Rev. B8 (1973) 5747. H.J. Monkhorst and J.D. Peck, Phys. Rev. B13 (1976) 5188. [9] A. Continenza, S. Massidda and A.J. Freeman, Phys. Rev. B38 (1988) 12996. [10] American Inst. of Physics Handbook (McGraw-Hill, New York, 1987). [11] B.T. Jonker, J.J. Krebs, S.B. Qadri and G.A. Prinz, Appl. Phys. Lett. 50 (1987) 848. [12] S. Ohnishi, A.J. Freeman and M. Weinert, Phys. Rev. B28 (1983) 6741. [13] K.B. Hathaway, H.J.F. Jansen and A.J. Freeman, Phys. Rev. B31 (1985) 7603. [14] S. Ohnishi, M. Weinert and A.J. Freeman, Phys. Rev. B30 (1984) 36.