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Surface energy and magnetism of the 3d metals
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Surface Science 315 (1994) 157-172
ELSEVIER
Surface energy and magnetism of the 3d metals M. AId& *7a,H.L. &river b, S. Mirbt a, B. Johansson a a Covered
Matter Theov Group, Physics Department, UppsaIa Uniuersi~, S-El21 UppsaIa, Sweden b Technical University of Denmark, DK-2800 Lyngby, Denmark Received 10 February 1994; accepted for publication 14 April 1994
Abstract We present an ab initio study of surface energies, surface magnetism and work functions of the 3d transition metals. The calculations are performed by means of a spin-polarized Green’s function technique based on the tight-binding linear muffin-tin orbitals method within the atomic sphere approximation. In addition to the conventional paramagnetic and spin-polarized calculations we use the fixed spin-moment method to clarify the effect of magnetism on the surface energies. The results are shown to be consistent with a Friedel model of d-electron bonding combined with spin-split state densities as well as with a Stoner-type description. It is established that the anomaly in the surface energy of the 3d metals deduced from surface tension measurements is purely a magnetic solid state effect. In addition, it is found that magnetism reduces the surface energy of open surfaces, e.g. the @Ol) crystal faces, to the extent that the usual anisotropy of the surface energy is reversed. Thus a complete realization of the surface energy anomaly only takes place in the less close-packed surface facets.
1. Introduction
The electronic properties of the surfaces of the ferromagnetic 3d transition metals Fe, Co, and Ni have been studied extensively by first-principles local spin-density calculations during the past decade D-71. Thereby one has established that the magnetization at the surface of a 3d metal generally is enhanced by up to 300%. This effect is commonly explained in terms of the decrease in coordination number and the increase in localization of the d-states at the surface. Direct support for the enhanced ma~etism from experiment exists but is rather limited [l], although it is
* Corresponding
author.
generally reported that magnetism is not lost in the surface layer. For a time Ni was in this respect experimentally a subject of controversy &91. In this situation it is of great interest to calculate the effect of magnetism on the surface energy of the 3d metals and to compare with available experimental information. This is particularly true because the surface energies of the 3d metals as deduced from the surface tension of the liquid metals by de Boer et al. [lo] are anomalously low when compared to the corresponding non-ma~etic metals in the 4d and 5d series. Hence, one suspects magnetism to play a role and the degree with which we are able to reproduce the anomaly by direct calculation will lend strong indirect support for the calculated enhanced surface magnetic moments.
~39-~28/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSL?lOO39-4028(94)00239-6
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M. Ald& et al. /Surface
The surface energy is defined as the energy required to transform a bulk atom into a surface atom with a corresponding decrease of the bulk volume and increase of the surface area, or equivalently, as half the energy needed to cut the crystal into two semi-infinite pieces. Similarly, the cohesive energy of a metal is defined as the energy gained in the transformation of a free atom into a bulk atom. It was recognized early on that the surface energy Es was approximately proportional to the cohesive energy EC,,,,and that the relationship
(1) was obeyed at the same level of accuracy by the 3d as well as by the 4d and 5d transition metals. The most common support for this relationship comes from the bond-cutting model, e.g. see Ref. [ll], which in its simplest form states that the total energy is proportional to the number of nearest neighbour atoms in the crystal. This type of model has proven successful, especially when modified into a form with a square-root dependence on the coordination number [ll], but it does not incorporate the effects of magnetism in a natural way. The parabolic variation with d-occupation number, which follows from d-band filling arguments, is common to both the surface energy E, and the cohesive energy Ecoh and consistent with the proportionality (1). In the 3d-transition series the expected parabolic behaviour is broken when the d band is approximately half full and instead one finds a pronounced minimum at the central element manganese. Since the anomaly occurs simultaneously in the surface energy and in the cohesive energy and since (1) remains valid even for the 3d series, it is common to assume that the two anomalies have the same physical origin. The loss of cohesive energy in the elements Cr-Co is well known to be caused by the large free atomic spin-pairing energy found in elements with approximately half-filled d-shells [12,13] and the cohesive energy anomaly is therefore an atomic effect. In contrast to the cohesive energy the surface energy is per definition not related to any atomic state and the anomaly in the surface energy of the 3d metals has remained a puzzle
Science 315 (1994) 157-I 72
although magnetic effects of course have been expected to play a role. We were recently able to verify the anomaly through first-principles, spinpolarized surface calculations [14] and to present a simple picture of its physical origin. The main issue of the present work is to address the longstanding problem of the origin of the surface energy anomaly by direct calculations of the quantities involved, and to present a full report on surface energies, surface magnetism, and work functions for the 3d metals from first-principles calculations. Ab initio calculations of the surface energies, surface relaxations, and work functions of the 4d metals have recently been reported by Methfessel et al. [ll] who used a full-potential, all-electron, slab-supercell linear-muffin-tin-orbital (LMTO) method. At the same time Skriver and Rosengaard 1151 presented a comprehensive study of the paramagnetic surface energies and work functions of all elemental metals by means of a Green’s function technique based on the tightbinding LMTO representation and the atomic sphere approximation. The spin-polarized analogue of this technique was subsequently used by Alden et al. [7], demonstrating its quality and accuracy by comparing calculated magnetic moments and work functions for Fe, Co, and Ni with those of previous full-potential, all-electron, slab-supercell calculations [3-61. Here, we present a theoretical study of the surface spin magnetism in Cr, Mn, Fe, Co, and Ni as well as of the general effect of magnetism on the surface energy and work function of the 3d elements based on the Green’s function technique. Results by the present method for the most densely packed surfaces of Cr and Mn with antiferromagnetic order within the surface layer are presented here for the first time. In addition to the unconstrained paramagnetic and spin-polarized calculations we employ the fixed-spin-moment (FSM) procedure to gain further insight into the effect of magnetism on surface energies. The results are discussed in terms of the Friedel model of cohesion and the Stoner model of itinerant magnetism. Both models are consistent with the unusual anisotropy of the surface energies found in Cr, Mn, and Fe and
M. Aldin
lead in conjunction with the complete calculations to a physically transparent picture of the anomaly in the surface energy of the 3d metals. The remaining part of the paper is organized as follows. In Section 2 we describe the computational details and approximations used throughout this work. In Section 3 we present our calculated surface energies, magnetic moments, and work functions for a number of crystal faces of the 3d metals SC-Cu. In Section 4 we review conventional models and predictions for the anisotropy of the surface energy of a (paramagnetic) metal. We discuss the Friedel model of (paramagnetic) d-bonding for surface energies and extend it to model the cases of half- and fully-saturated d-band ferromagnetism. The magnetic equation of state for ferromagnetic systems is illustrated in terms of the Stoner model for band magnetism. Finally, a summary with conclusions is given in Section 5.
2. Computational
method
2.1. TB-LMTO Greens function
159
et al. /Surface Science 315 (1994) 157-172
technique
The tight-binding linear-muffin-tin-orbitals (TB-LMTO) Green’s function technique for surfaces and interfaces, implemented by Skriver and Rosengaard [16], is based on the work of Andersen and co-workers [17-231. Shorter reviews of the method have recently been presented elsewhere [7,15]. In the present implementation we work within the frozen core approximation and use a minimal (spd) basis set and the atomic sphere approximation (ASA) for the charge densities as well as for the one-electron potentials except for additional electrostatic dipole contributions from neighbouring spheres. We use linear response theory and a linearized version of the Dyson equation to improve convergence speed. Depending on convergence tests the surface region consisted of 4-6 layers of metal plus two layers of empty spheres simulating the vacuum. Mainly due to the limitations of the computational approach, the relaxation of atomic positions are neglected. Since, however, the main perturbation at the surface is the change in coor-
dination number, we expect that the effect of relaxation on work functions, surface energies and magnetic moments is to be only a minor correction. The Green’s function matrices were sampled at 16 z-points on a complex energy contour and at 36-64 k,-points in the irreducible part of the 2D Brillouin zone, depending on surface facet. The underlying bulk calculations were performed with the same choice of kll-points in order to minimize sampling errors. State densities, presented in Section 4.3, were obtained by a final z-sampling of the Green’s functions on a fine mesh close to the real axis. Exchange and correlation were treated within the local spin density approximation (LSDA) using the Vosko-WilkNusair parametrization [24] of the data by Ceperley and Alder [25]. A systematic comparison 17,151 between the results obtained in recent calculations by the present method and those derived from full-potential, all-electron slab-supercell calculations [3-61 shows that key quantities such a surface energy or magnetic moments typically differ by much less than 10%. This suggests that the present approximations including the neglect of relaxation of the atomic positions are well-founded and indicates the efficiency of the Green’s function technique. 2.2. The FSM method The fixed-spin-moment (FSM) method [26,27] has been extensively used within density functional theory to deduce the total energy E as a function of magnetization M, thus enabling studies of, e.g. metamagnetism [28]. In a system with a fiied number of particles ilr under the additional constraints of fixed spin magnetic moments Mp at each site Q, the problem is formally to minimize the functional
F[n(r),
m(r)]
=E[n(r),
-
m(r)]
Chp( _/&W dr -4~)~ Q
(2) where P and h,
are Lagrange multipliers, with
160
M. Aldin et al. /Surface
respect to the charge density n(r) and the total and partial magnetization densities, m(r) and m,(r), respectively. By minimizing (2) with respect to the spin-up, n+(r), and spin-down, n-(r), densities n”(r)
=+2(r)
i-m(r)],
(3)
n_(r)
=+2(r)
-m(r)],
(4)
for the case of no Q-dependence, principle yields (he = h),
the variational
(5)
(6) The conventional procedure of the FSM method is to identify the right-hand sides of Eqs. (5) and (6), p f h, as chemical potentials for the two di~erent spins. This corresponds to using two different Fermi energies and to fix the number of particles of the two spins separately. A completely equivalent approach is to work with a single Fermi level whilst shifting the spin-dependent one-electron potentials, I$&@) = V,“(r)
3: ho,
(7)
where h, is to be determined self-consistently and forms the site-dependent external magnetic field, Ho = ho/pn. We have chosen this latter scheme since it is better suited to our present implementation of the Green’s function technique where boundary conditions and Fermi levei are set by the underlying bulk calculation.
3. The calculations The 3d metals form in a number of different crystal lattices and exhibit several magnetic structures some of which are rather complicated, especially when surface magnetism is included. We shall therefore explain which crysta1 and magnetic structure were used in the present calculations. bee Fe, hcp Co, and fee Ni order ferromagnetically with one (Fe, Ni) or two (Co) equivalent atoms per primitive cell and were treated within
Science 315 (1994) 157-172
these structures. The true ground state of bee Cr is that of an incommensurable antiferromagnet with an average magnetic moment of 0.46~~ per atom [32]. Here, we have assumed a commensurable antiferromagnetic state as was done in previous ab initio bulk calculations 133,341 and treated Cr in the CsCl structure. Mn orders magnetically in a complicated bee structure with 58 atoms per unit cell, which to our knowledge has not been treated by ab initio methods. Theoretical studies of the magneto-volume properties of both one-atomic bee Mn [35-371 and two-atomic CsCl Mn [38] predict the ground state of Mn to be a CsCl structure ferrimagnet at the normal volume. In the present calculations we have treated Mn in both the ferromagnetic (bee) and the ferrimagnetic (CsCl) state. Finally, fee Mn and fee Fe were constrained antiferromagnetitally in the CuAuI structure with the sign of the moments alternating along the (001) axis, as recommended in ab initio magneto-volume studies E391. In the following we shall refer to the most closely packed surface of each crystal structure as a type P surface (“packed”), and to the second most closely packed surface as a type 0 surface (“open”). This is mainly for convenience but will also benefit the discussion later on. Thus f&111), h~(OOl), and bcc(ll0) are the type P facets and fcc(OO1) and bcc(001) are the type 0 facets. In the cases of the CsCl and the AuCuI structures which are used in the calculation of antiferromagnetic bcc(ll0) Cr, fcc(ll1) Mn, and fcc(lll) Fe, the type P surfaces have two sites in each layer which differ in the direction of the spin while the type 0 surfaces have only a single site per layer. The so-called inside-the-su~ace 42 x 2) antiferromagnetism investigated for Cr(OO1) by Bhigel et al. [31] was not considered for the type 0 facets. The calculated paramagnetic surface energies for a number of surface facets of the 3d metals are presented in Table 1 together with the values listed by de Boer et al. [lo]. Since the Iatter values are partially derived from measurements of the surface tension of liquid metals they may not be attributed to any particular crystal surface and are best interpreted as weighted averages. For this reason we compare in Fig. 1 these exper-
M. Ati&
et al. /Surface
imental values with the surface energies obtained for a common close-packed surface facet, i.e. fcc(l11). The calculated surface energies in the figure exhibit a parabolic variation with the number of d-electrons which is in accord with the simple picture of d-electron contribution to the surface energy suggested by Friedel [42]. In contrast to this, the parabolic behaviour of the experimental data is broken in the middle of the series leading to a large discrepancy with the calculated paramagnetic values. Clearly, one expects this
Science
315
(1994)
i 1.0
161
157-172
1
0/
,n’
fcc(lll)
0.5 1 Table 1 Paramagnetic surface energies for the 3d metals and experimentally observed equilibrium Wigner-Seitz radii, S, used in the calculations Metal
SC
Ti
S (Bohr)
Surface (a.u.1
(eV)
(J/m*)
(J/m*)
3.427
hcp(001) fcdlll)
0.48 0.44
0.82 0.76
1.28
hcp(OW
0.90 0.12
1.95 1.56
2.10
fcc(ll1)
3.052
Surface energy Theory
Exp. a
V
2.818
bcc(l10) bcdO01) fcdlll)
0.82 1.06 0.99
2.02 1.76 2.55
2.55
Cr
2.684
be&IO) bcc(001) fcdlll)
1.33 1.74 1.10
3.63 3.36 3.09
2.30
Mn
2.699
fcc(lll) fcc(OO1) bcc(l10) bcd001)
1.17 1.40 1.27 1.60
3.24 3.35 3.43 3.05
1.60
Fe
2.662
bcctll0) bcc(OO1) fdlll) fcdool)
1.12 1.57 1.15 1.43
3.09 3.08 3.28 3.48
2.48
co
2.621
hcp@Ql) fcc(ll1)
3.18 3.23 3.40 2.78 2.83
2.55
bcdOO1)
1.08 1.10 1.33 0.98 1.41
Ni
2.602
fccflll) fcc(OO1)
0.88 1.03
2.63 2.67
2.45
cn
2.669
fcdlll) fcc(OO1)
0.69 0.85
1.96 2.09
1.83
2.631
a See Ref. [lo].
fccml) bcctl10)
1 i
SC Ti V Cr Mn Fe Co Ni Cu L,l_li’,I’J~~~J~J’ 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 Number of d-electrons Fig. 1. The calculated surface energy for paramagnetic fcdlll) surfaces of the 3d metals quoted from Skriver and Rosengaard [15] compared with the experimental values listed by de Boer et al. [lo].
difference to be due mainly to the appearance of magnetic ground states in the elemental metals and hence to the neglect of spin-polarization in the calculations. The results of the spin-polarized calculations are summarized in Table 2 and in Figs. 2 and 3 we present surface energies for the type J? and 0 facets of the magnetic 3d metals. As explained above all the calculations are performed for the experimentally observed magnetic structures except for Mn which is treated in the magnetic fee and bee structures of lowest energy. Since Mn transforms into the bee structure before melting it is the results of the magnetic bee calculations which are shown in the figures. It is seen that spin-polarization generally reduces the surface energy and thereby brings the calculation into closer agreement with the experimentally observed anomaly. It may also be observed that owing to lower coordination numbers and thereby increased localization effects surface magnetic moments are systematically larger at the type 0 surfaces than at the type P surfaces. As a result the surface-energy anomaly is only completely realized for the type 0 surfaces, Fig. 3, of Cr, Mn, and Fe where the magnetization is driven closer to saturation.
M. AEd& et al. /Surface Science315 (1994) 157-I 12
162
The degree of magnetic saturation, as e.g. defined in the following section, is known to increase when going from Cr to Ni and is found to be especially small for Cr and Mn. At the surface, the magnetic moments in Cr, Mn, and Fe are driven closer to full saturation, whereas for Co and Ni the moment enhancement is small due to the already large saturation in the bulk. For Cr the surface magnetic moment is found to be strongly dependent on the surface facet and the magnitudes of the moments for the bee type P and 0 facets differ by a factor of 2, see Table 2. As a result the magnetic reduction of the surface energy in Cr is very large for the type 0 facet, but essentially nonexistent for the type P facet which
in addition has antiferromagnetic order within each layer. In Mn the surface magnetic moment is already sufficiently large for the type P facet to lower the surface energy substantially with respect to the paramagnetic value, but also in this case is comptete agreement with the experimental surface energy only realized for the type 0 facet. It is interesting to note that the increased magnetic saturation in the type 0 surfaces reduces the surface energy to the extent that these surfaces become more stable than the type P surfaces. This is in contrast to the anisotropy commonly found in paramagnetic systems where the close-packed surfaces as a rule have the lowest energies.
Table 2 Calculated surface energies and spin moments in the bulk and in the top-most surface layers for the magnetic 3d metals; the second row of ferrimagnetic bcc(ll0) Mn refers to the second site in the layer; the surface layer is denoted by S Metal
Surface
Spin magnetic moments &a)
Surface energy
Bulk
s-3
s-2
S-l
S
(eV)
(J/m’)
Exp. g (J/m2)
Cr
bcdll0) a,b bcc(001) b
f 0.21 f 0.23
f 0.42 - 0.57
+ 0.42 0.74
f 0.65 - 1.16
+ 1.39 2.61
1.31 1.16
3.58 2.23
2.30
Mn
bcc(ll0) = bcc(ll0) d
1.13 2.23 - 0.24 - 0.57 - 0.29 * 1.75 - 1.70
1.07 2.23 - 0.32 2.50 2.42 + 1.68 1.74
0.94 2.41 -0.14 - 0.61 -0.70 f 1.72 - 2.01
1.51 2.96 - 2.95 3.66 3.64 + 2.58 2.84
1.22 0.90
3.28 2.42
1.60
bc~~l) ’ bcdOO1) d fcc(l11) +Gf fcdOO1) f
1.03 2.26 - 0.29 0.99 2.28, - 0.31 + 1.72 f 1.68
0.82 0.73 1.01 0.88
1.57 1.38 2.80 2.12
Fe
bcc(ll0) bcc@Ol) fcc(ll1) Gf fcc(OO1)f
2.24 2.24 + 1.76 kl.78
2.24 2.25 f 1.79 - 1.38
2.25 2.37 f 1.76 1.99
2.35 2.30 + 1.76 - 1.40
2.57 2.97 + 2.30 2.59
0.96 1.09 1.02 0.93
2.66 2.18 2.88 2.25
2.48
CO
hc~~l) fcc(lll)
1.61 1.64 1.64 1.74 1.74
1.60 1.63 1.66 1.73 1.75
1.65 1.67 1.63 1.75 1.72
1.70 1.72 1.84 1.78 1.94
0.94 0.91 1.09 0.99 1.26
2.74 2.70 2.78 2.79 2.52
2.55
bcc(ll0) bcc(001)
1.61 1.64 1.64 1.74 1.74
fcdlll) fcc(OO1)
0.63 0.64
0.63 0.64
0.65 0.66
0.67 0.64
0.62 0.69
0.90 1.07
2.69 2.77
2.45
fcc@Ol 1
Ni
a ~tiferromagnetic within layers in the C&Cl structure. b Underlying antiferromagnetic bulk calculation in the CsCl structure. ’ Underlying ferromagnetic bulk calculation, one-atomic bee unit cell. ’ Ferrimagnetic in the CsCl structure. ’ Antiferromagnetic within layers in the CuAu structure. ’ Underlying antiferromagnetic bulk calculation in the CuAu structure. g See Ref. [lo].
M. Ald&
2.0
3.0
4.0
5.0
6.0
7.0
8.0
163
et nl. /Surface Science 315 (1994) 1.57-I 72
9.0
Number
of d-electrons Fig. 2, Comparison between the calculated paramagnetic (Para.) and spin-~larized (Mag.) surface energy for the type P facets of the 36 metals. Paramagnetic rest&s are taken from &river and Rosengaard [15]. Also illustrated are the experimental values listed by de Boer et al. (Exp.) DO].
To gain a deeper understanding of the magnetic contribution to the surface energy and in particular of the role played by the degree of saturation, we have performed fixed-spin calculations for the 3d metals Sc-Ni. In these calculations the d-magnetization is fixed to be uniform in the bulk as well as in the surface layers, and we define the d-saturation to be md/nd if nd < 5 or m,/(lO - n,) if n4 > 5, respectively, where md is the d-moment and nd is the number of d-electrons. Thus md is defined by choosing a certain constant amount of saturation through the series. The results for O%, SO%, and 80% d-magnetization for the fcc(ll1) facet are shown in Fig. 4, where it may be seen that the minimum of the anomaly has been shifted to Cr. Within the Friedel model to be discussed later this fallows from the fact that Cr is effectively closer to having a hdf-filled d-shell than Mn. On the other hand, the surface magnetic moment (Table 2) and the saturation is larger for the type 0 surfaces of Mn than they are for Cr, and therefore the minima in the unconstrained calculation occ!ur for Mn.
In Fig. 5 we present the results of constraining the surface magnetic moments for the type 0 facets to their relaxed b&k values, When these results are compared with those of different saturation in Fig. 4 as well as with those obtained in the relaxed ab initio calculations, Fig. 5, it is seen that the introduction of uniform magnetic moment reduces the surface energy relative to the paramagnetic case, except if the d-band is either empty or filled. However, the most important contribution to the anomaly results from the enhanced saturation caused by low surface coordination numbers. The work functions obtained from the paramagnetic and the spin-polarized calculations are given in Table 3. The trend for the type P 3d surfaces and the effects of magnetism are illustrated in Fig. 6, Except for the various type 0 facets of Cr and Mn, there is a subst~tial reduction in the work function when spin-~lari~ation is included improving the agreement with the experimental values [40]. Flowever, even includ-
3.0
2.5 $ 3
2.0
ui” 1.5 1.0
0.5
Sc
1
2.0
Ti
V
a./,I,I,I,j.,, 3.0
4.0
Cr
Mn 5.0
Number
6.0
Fe 7.0
Co 8.0
Ni
Cc
9.0
ofd-electrons
Fig. 3. Comparison between the calculated par~a~et~c fFara.f and spin-polarized fMag.) surface energy for the type 0 facets of the 5d metals. Paramagnetic results are taken from Skriver and Rosengaard [X5]. The calculations were performed far the experimentally observed crystal structures, except for Mn and Co which were treated as bc4001) and fcc@Ol), respectively. Experimental values are those listed by de Boer et al. (Exp.) [lo].
r-e-O%
--a-50%
ing the magnetic reduction the calculations provide an overestimate of thle work functions by 0.5-l eV. This systematic difference between the calculated and experimental work functions may partly be attributed to a failure of the local spin density a~pro~mation, since for instance the calculated values have been shown E16,7f to depend quite sensitively on the actual parametrization used for the LSDA functional. Since full-potential, layer-reIaxed results compare favourably [15] with the present type of calcufations, it seems
--+80%
SC
Ti
AL___,_,
v
cr 3
ml 1
Fe .a
co
Ni
.-_l”“_““i----j---C
2.0 3.0 4.0 5.0 6.0 7.0 8.0 Number of d-electrons Fig. 4. Fixed-spin calculations of surface energies for the fcc(Ill) surface of the 3d metals. TIte top curve shows paramagnetic rest&s, whereas the middle and bottom curves correspond to 5tI% and 80% saturation, respectiveiy, of the d magnetic moments
Tabfe 3 Calculated and experimental work functions for the 3d metals; P and F denote the pammagnetic and spin-polarized results, respectively Metal
Surface
Theory
Exp.a
hcp(001) fcc(ll1)
4.59 4.63
(4.333
b&I 10) bcciOOlt iccQ11t
5.12 4.57 4.88
(4.3)
5.45 4.58 5.27
5.30
5.45 5.37 5.69 4.90
5.18 5.76 5.34 5.31
(4.1)
5.78 5,t-S 5.54 5.55
5.21 4.S 5.50 6.12
@.5)
hcp(001) fcc(ll1) fcc(OO1) bcc(llO~ bcc@Ol)
5.81 5.76 5.83 5.74 4.91
5.53 5.55 5.52 5.66 5.15
CS.0)
fc&Ii~ fccioolf
5:77 584
5.70 5.75
5.35 5.22
fcc(lll) fec(OO1)
5.30 5.26
fcctlllf fcc@Ol) bcdll0) bcc@Ol~
4.0
7.0 5.0 6.0 Number of d-electrons
8.0
Fig. 5. Fixed-spin c&ulations of the surface energy assuming a uniform m~eti~~tion equal fo the setf-consistent butk vatue ~n~f~~rn~ compared v&h the diamagnetic restrfts (Pam.) and those of the uncoustrained, ~~j~-p~j~iz~~ c&a&tions (Mag.1. Except for Mn and Co which are treated as bcc@Ol) and fcc@Ol), respectively, the calculations are performed by type 0 facets of the experimentally observed crystal structures.
Work function (eV)
a SeeRef. [401.
f4.5)
4.94
4.94 4.59
M. Ak& et al. /&face
Science 31.5 (1994) 157-l 72
respectively, becomes Es = IG
-0
Exn.
Element Fig. 6. Work functions for the 3d metal. The calculated results are obtained for the close-packed type P surfaces assuming either a paramagnetic (Para.) or a spin-polarized (Msg.) ground state. The experimental values [40] (Exp.) for Se-Co are obtained on polycrystalline samples while those for Ni and Cu are obtained on the type P facets.
that the ASA and the omittance of surface layer relaxations is less responsible for the discrepancy. Finally, one should also remember that a closepacked single-crystal surface usually experiences a larger work function than a polyc~stalline sample, and that this difference may be of the order 0.5 eV [HI. Of the cited experiments in Fig. 6 only the measurements on Ni and Cu are performed for a single surface-facet while those for SC-CO are performed on polycrystalline samples.
4. Theory of surface energies 4.1. Bond-cutting and anisotropy The surface energy, and in particular its relation to the cohesive energy, Eq. (l), has often been discussed in terms of bond-cutting models and nearest neighbour interactions. The primary assumption is that the energy for cutting a single nearest-neighbour-bond is a well-defined quantity, and that the total energy therefore scales to the number, C, of nearest neighbours. If these are C, and C, for the bulk and surface atom,
165
the relation between - Cs)/CnlK0h*
Es and Emh (8)
This approach may be acceptable in the limit of strongly covalent bonding, whereas for a metal one expects that the bond saturates when the coordination number is increased. Direct determination of the metallic bond strength by means of ab initio total-energy calculations have recently been presented for Al and for the 4d transition metals by Methfessel and co-workers [ll]. In their work the total energy was fitted to the sum of a dominating bonding term (proportional to the square-root of the number of nearest neighbours, fi) and a small repulsive term (linear in C) E(C) =E,-A@
+BC,
(9)
reducing the “effective” nearest-neighbour bonding to half of that of the simple linear model. By here letting B scale with Ecoi,, B = f&E,, as suggested by Methfessel et al. [ill, one obtains Es= [1+&r,-
~(I+B,,CB)]E_,b.
(10)
Within the framework reviewed here, the bondcutting model supports a proportionali~ k between Es and Ecoh at 3 different levels of approximation. Having k = &/Ecoh, the quasi-chemicai, linear k is given directly through Eq. (8). Neglecting the linear term in (9) gives k = CJc, - G>/G whereas the full k follows from Eq. (10). Numerical values of k for these different cases and for 5 common crystal planes have been tabulated in Table 4. From this it follows that the square-root description using Eq. (10) gives the best overall agreement with experiment for the 3d and 4d transition metals. It is difficult to obtain a good estimate of k directly from calculations of E,, and Es, since E _,, is well known to be overestimated in the LSDA. The error is often stated to lie in the atomic c~c~ation, whereas the bulk total energy is expected to be given more accurately by the LSDA. In the context of bond-cutting models it is important to use non-magnetic quantities, and to carefully consider the dependence of the valence
M. Aldh
166
et al. /Surface
Table 4 Study of the proportionality constant k, in the relation Es = models; Cn and Cs are the kr&oh for various bond-cutting coordination numbers for a bulk and surface atom, respectively; k, = (C, - C,)/C,, k, = C& - &I/&, and k, = (1+ B&)(1 the text; (k,,) proportionalities respectively Surface fcc(ll1) fcc@Ol) fcc(l10) bcc(l10) bcc(001)
with B, = 0.03, as explained in - JW) and (k,,) refers to averages of experimental [lo] for the 3d and 4d transition metals, C, 12 12 12 8 8
(k,)
C,
k,
k,
k,
9 8 6 6 4
0.25 0.33 0.50 0.25 0.50 0.37
0.13 0.18 0.29 0.13 0.29 0.20
0.09 0.13 0.22 0.11 0.22 0.16
c&3,>
0.156
&j>
0.149
configuration in the free atom [11,13]. Most important is the atomic spin-polarization (SP) energy, which lowers the energy of the atom and therefore reduces the cohesive energy. This SPenergy is especially large for elements in the middle of a d-transition series. The effect of the bulk magnetization on the energy is one order of magnitude less, since in the solid this polarization energy has to be balanced against the band (kinetic) energies. In this light, it is surprising that the empirical relation (1) holds just as well for the 3d series, where magnetic effects are especially large, as it does for the 4d and 5d series. Denoting the parumagnetic component of the surface and cohe-
Science 315 (1994) 157-172
sive energies Er and E:tr, respectively, this (accidental) empirical fact may be expressed as Er+As-
bee bee bee bee fee fee
V Cr Mn Fe co Ni
ESF
1 .Ol 1.74 1.60 1.57 1.33 1.03
_ 1.16 0.73 1.12 1.09 1.07
(11)
Here, A, and A, are the (negative) energy contributions due to magnetism in the atom and bulk, respectively, and A, is the total magnetic contribution to Es (A, = Es-Erra>. For the 4d and 5d series one expects Erra and EFlr to be related by the original relation (11, and this should also hold for the 3d elements. Therefore, the magnetic contributions appear, empirically, to behave as A, = +(A,4 -Ad.
(12)
In order to test this relation we considered our calculated values for A, (type 0 facets) and A,, and performed first-principles calculations of the atomic SP-energies (A,) for Cr-Ni. The results, given in Table 5 and illustrated in Fig. 7, show that the empirical relation (12) is approximately obeyed and consistent with ab initio theory. However, for e.g. the 4d transition metals magnetic effects in the surface energies are nonexistent [lo] whereby relation (12) becomes meaningless. We conclude that Eq. (1) is not well-founded for magnetic systems, but only fortuitously obeyed for the 3d metals. The anisotropy of the surface energy for a parumagnetic metal is predicted by the bond-cutting models (Table 4) and by tight-binding studies [41]. One finds for the fee crystal that E S(111)
Table 5 Calculated surface and magnetization energies (in eV) of V-Ni for structures; El and E: are paramagnetic and spin-polarized surface surface energies assuming a uniform d-magnetization in the bulk and unconstrained bulk value), Ez”% (50% d-saturation) and Es80% (80% atom (A,) and bulk (A,) E,P
;(~,P,ah’“+A~-d~).
<-&(001)
(13)
the type 0 facets of the experimentally observed crystal energies, respectively, quoted from Tables 1-4; fixed-spin at the surface are denoted by EC (d-moments equal to the d-saturation); magnetization energies are given for the free
50%
ESB
Es
EF%
A,
AB
1.746 1.522 1.455 1.168 1.09
0.677 0.538 0.994 1.500 1.211 1.138
0.412 0.047 0.400 1.063 1.060
1.64 5.02 5.07 3.40 1.99 0.91
0.01 0.09 0.53 0.19 0.05
M. Aldth et al. /Surface
167
Science 315 (1994) 157-I 72
energy Es for a paramagnetic approximately [42] E, = $q, (
Cr
Mn
Fe
Co
Ni
Element Fig. 7. Study of the empirical relation (121, by means of comparing ab initio results for A, and i(A, - A,). Quantities are as defined in the text.
and for the bee crystal J&O) < &01) <
-%(,11,7
(14)
i.e. the surface energy is expected to increase as the surface becomes more open. Whether these inequalities will hold for a magnetic system or not depend upon the magnetic energy contribution to the surface energy. Clearly, they will be violated if the magnetic effects are sufficiently large and surface sensitive. In our spin-polarized calculations for the type P and 0 facets, this is indeed seen to occur for Cr, Mn, and Fe. 4.2. The Friedel model In the band-picture of d-electron bonding as suggested by Friedel [42], the average electronic energy per atom is (E)
=
/
+D( E)E
dE,
1- $
(16)
6W, 1
where n,, is the number of d-electrons and the 6W is the reduction in band width at the surface relative to the bulk. Thus, through a d-series of elements a parabolic-like behaviour is expected for the surface energy. Experiment [lo] and recent calculations [15,11] support this parabolic behaviour for the 4d and 5d metals. Support is also found from the present paramagnetic calculations for the 3d metals, as can be seen in Fig. 1. This is in contrast to the experimental results which exhibit a marked dip in the middle of the series. This bears, as already mentioned, a close resemblance to the the mid-series cusp in the 3d cohesive energies, caused by atomic spin-pairing effects. We incorporate band magnetism in the Friedel model by first regarding the case of a saturated itinerant ferromagnet. Here, the spin-up and spin-down d-bands in the bulk and at the surface, respectively, are completely separated. Hence for a less than half-filled d-shell (nd < 5) the narrowing of the unoccupied spin-band has no effect on the surface energy. The surface energy of such a saturated ferromagnet becomes Es=&,
(
l-
$
)
(17)
6w,
and for a saturated magnet with rzd > 5 the same expression is valid if nd is replaced by (nd - 5). Thus the parabolic behaviour of the surface energy for the paramagnetic state is replaced by two parabolas, with maximum for n,, = 2.5 and 7.5, respectively. For a non-saturated magnet there is a continuum of solutions, but if the bulk magnetic moment md is maintained up to the surface one finds the surface energy to be
(15)
i.e. the one-electron term in the independentparticle model. Here or is the Fermi level and D(E) is the d-state density at energy E. Assuming a rectangular state density in the bulk and at the surface, the d-electron contribution to the surface
metal becomes
6W.
(18)
The Friedel models including (18) for a halfsaturated magnet, i.e. md = n,/2 for n,, < 5 and m,, = (10 - n,)/2 for nd > 5 are illustrated in Fig. 8. From this it is now evident that the dip in the
M. Aldh
168
et al./Surface
Science 31.5 (1994) 157-172
4.3. The Stoner model PARAUAGNETIC
t :
0
2
4
6
8
10
Number of d-ekctrons
Fig. 8. Model calculation of the surface energy of the 3d transition metals based on a rectangular d-state density. The three curves correspond to the paramagnetic, the halfsaturated Cm, = n,, /2 for nd < 5 and m,, = (lo- nd)/2 for nd > 5), and the saturated case, respectively.
middle of the series may originate from simple band-filling effects. The Friedel model predicts that the mere introduction of a uniform magnetization will lower the surface energy of a metallic crystal. Hence, if the magnetism is constrained to be uniform, as in the fixed-spin calculation presented in Section 3, with moments equal to the unconstrained bulk (nonzero) values, the surface energy should decrease relative to that of the paramagnetic state, as is indeed found to be the case. In general, the magnitude of the surface magnetic moment is never lower than that in the bulk, and relaxing the spin-moment at the surface automatically gives rise to a further reduction of the surface energy. The size of this reduction will depend on the degree of magnetic saturation in the bulk relative to the surface, so that for Co and Ni the reduction is small while it is substantial for Cr and Mn.
The existence of ferromagnetism in metallic systems has often been explained in terms of properties of the paramagnetic ground state within the so-called Stoner model [43-471. The approximations of this model are rigid band splittings and a Stoner parameter I independent of magnetization and they lead to the simple and well-known Stoner criterion for the onset of ferromagnetism as well as to the equations of state and the susceptibili~. Before the advent of accurate surface calculations, Stoner analysis has been applied also to the question of surface magnetism [48,49]. The results indicate that the magnetic moments will be maintained at the surface. In the context of surface energies and the effect of the onset of spin-polarization in the surface layer, it is interesting to consider the following Stoner equation of state E(M,
I’) = +j”A(M’,
V) dM’-
$4’,
(19)
0
where E(M, V) represents the change in total energy due to magnetization M at volume L’, A&W, V> is the (constant) exchange-splitting of the d-states and I the Stoner parameter, taken to be an intrinsic atomic quantity independent of the crystal surroundings. The first term in (19) describes the loss in kinetic energy due to occupancy of less bonding d-states, as discussed in the last section, whereas the second term represents the gain in exchange energy. Within the rigid-band ansatz and neglecting the volume dependence, the exchange-splitting A(M) may be obtained from M=/
‘“+4%(E)
dE = [;_4
D(E)
dE,
(20)
6F
A(M)
=A, +A,,
(21)
where the spin-up and spin-down band shifts, A, and A,, are given in terms of the magnetization M and the paramagnetic d-state density D(E). The onset of ferromagnetism is given by the condition that for A4> 0, A(M)
=ZM,
(22)
M. Aldhz et al. /Surface Science 315 (1994) 157-172
which may be solved graphically using (20) and (21). In Figs. 9a-9c we have plotted A(M), derived from the calculated paramagnetic state densities for bulk bee Fe, fee Co, and fee Ni and their respective type P and 0 facets. If the criterion (22) is obeyed, the energy gain per atom due to spin-polarization (19) may be obtained as half the area enclosed by the straight line and the A-curve.
169
The Stoner estimates of the magnetization energies for Fe, Co, and Ni are compared with our calculated ab initio results in Fig. 10. The qualitative agreement found in the figure supports the Stoner approach in the case of ferromagnetism. For Fe the large energy difference between the two facets is a result of the surface-dependence of A, (Section 4.1) and leads to an anisotropy
0-4
(4
1.6
>^ s
1.2
a 0.8
1
0.8
M
(cl,)
M
1.2
1.6
(ClB)
0.6
0.2
0.4 M
01,)
Fig. 9. The rigid exchange splittings A(M) as defined within the Stoner model and obtained from the paramagnetic densities: (a) bee Fe; (b) fee Co; (cl fee Ni.
d-state
170
M. Aldh
et al. /Surface
which is opposite to that usually found for surface energies (14). For Co and Ni, the magnetic energy differences and hence surface energy anisotropy are small. Clearly, already the simple Stoner model gives this result, although the effect is overestimated for Fe. Detailed theoretical investigations of the magnetic properties of the bcc(001) surface of Cr have been carried out by Bliigel and co-workers 131,501, and on the experimental side this surface was found to order ferromagnetically within the layer [30]. In the p(1 x 1) structure with insidethe-surface ferromagnetism, calculations, such as our present ones, usually give magnetic moments that are enhanced N 300% as compared to the bulk moment. Nevertheless, here the Stoner criterion is never met, which is in sharp contrast to an early calculation within the tight-binding approximation [48]. This is because the state densities obtained from the tight-binding scheme differ substantially from those of our self-consistent calculation, which are displayed in Fig. 11. This simple Stoner analysis for the bcc(001) surface of Cr shows that its ferromagnetism is not explicitly due to a Stoner instability, but only supported by the antiferromagnetic coupling to the underlying
Science 315 (1994) 157-172
-3 t
Cr bulk
1
I -4
0
-2 E
2
4
(eV)
Fig. 11. Calculated paramagnetic (total) d-state density, D(E), for bulk and for the (001) surface layer of bee Cr. Notice that IL&,)< 2 for the surface layer, implying that a surface Stoner criterion for ferromagnetism is not obeyed here. Stoner parameter for Cr: I = 0.82 eV [341. The Fermi level is marked with a vertical line at zero energy.
layers and enhanced fects.
due to band narrowing ef-
5. Summary
Type 0
\
Fe Co Ni
TypeP
71
Fe Co Ni
Fe Co Ni
Fig. 10. Bulk and surface layer magnetization energies for Fe, Co, and Ni obtained from the Stoner equation of state compared with the ab initio results, resolved for the surface layer.
We have calculated the surface energy, surface magnetic moments, and work function of the elements Cr, Mn, Fe, Co, and Ni. Our main objective has been to study the effect of magnetism on the surface energy of the 3d metals with a view to clarify the anomaly in the surface energy as deduced from surface tension measurements. With this objective in mind we have introduced two models in order to obtain an understanding of the calculated results and to isolate the magnetic contributions to the surface energy. The Friedel model accounts for the fact that metallic d-bonding is affected when the d-bands are spin-split, whereas the Stoner model gives a good estimate of the gain in total energy when ferromagnetism is introduced at a paramagnetic surface. The Friedel model is fully consistent with
M. AId& et al. /Surface Science 315 (1994) 157-172
the anomaly in the 3d surface energies as deduced from experiment and explains the dip in the middle of the series in terms spin-split d-state densities. With respect to our calculated surface energies the spin-polar~ation substantially improves the agreement with experiment but the full surface energy anomaly is only realized for the more open surfaces such as the (001) crystal facets where the calculated magnetic moments are found to be much larger than those of the closely packed facets, especially for Cr and Mn. Our calculations suggest that the anisotropy in the surface energy as obeyed by parama~etic surfaces, i.e. the most densely packed facet exhibits the lowest surface energy, may be reversed if magnetic effects are sufficiently large. Finally, we conclude that the anomalies in the surface and cohesive energies of the 3d metals are of different origin, and that the observed proportionality Es = iEm,, between the two quantities therefore must be fortuitous. Acknowledgements The Uppsala group is grateful to The G&an Gustafsson Foundation and The Swedish Natural Science Research Council for financial support. The work of HIS was supported by grants from the Novo Nordisk Foundation and from the Danish Research Councils through the Center for Surface Reactivity. We would also like to thank Dr. 0. Eriksson for assisting us with the spinpolarized atomic calculations. References [II For a review, see e.g., J. Mathon, Rep. Prop. Phys. 51 (1988) 1.
@I 0. Jepsen, J. Madsen and O.K. Andersen, J. Magn. Magn. Mater. 15-18 (1980) 867; Phys. Rev. B 26 (1982) 2790. DI S. Ohnishi, A.J. Freeman and M. Weinert, Phys. Rev. B 28 (1983) 6741. 141 E. Wimmer, A.J. Freeman and H. Krakauer, Phys. Rev. B 30 (1984) 3113. 151C. Li and A.J. Freeman, J. Magn. Magn. Mater. 75 (1988) 53.
171
[6] 0. Eriksson, A.M. Boring, R.C. Albers, G.W. Fernando and B.R. Cooper, Phys. Rev. B 45 (1992) 2868. [7] M. Alden, S. Mirbt, H.L. Skriver, N.M. Rosengaard and B. Johansson, Phys. Rev. B 46 (199216303. [8] L. Liebermann, J. Clinton, D.M. Edwards and J. Mathon, Phys. Rev. Lett. 25 (1970) 232. [9] M. Landolt and M. Campagna, Phys. Rev. Lett. 38 (19771 663. [lo] F.R. de Boer, R. Boom, W.C.M. Mattens, A.R. Miedema and A.K. Niessen, in: Cohesion in Metals, Vol. I, Eds. F.R. de Boer and D.G. Pettifor (North-Holland, Amsterdam, 1988) p. 676. [II] M. Methfessel, D. Hennig and M. Scheffler, Phys. Rev. B 46 (1992) 4816. [12] V.L. Momzzi, A.R. Williams and J.F. Janak, Phys. Rev. B 15 (1977) 2854. 1131 M.S.S. Brooks and B. Johansson, J. Phys. F 13 (1983) L 197. [14] M. Alden, H.L. Slcriver, S. Mirbt and B. Johansson, Phys. Rev. Lett. 69 (1992) 2296. [15] H.L. Skriver and N.M. Rosengaard, Phys. Rev. B 46 (1992) 7157. [16] H.L. &river and N.M. Rosengaard, Phys. Rev. B 43 (199119538. 1173 O.K. Andersen, Phys. Rev. B 12 (1975) 3060. 1181 0. Gunnarsson, 0. Jepsen and O.K. Andersen, Phys. Rev. B 27 (1983) 7144. [19] H.L. Skriver, The LMTO Method (Springer, Berlin, 1984). [20] O.K. Andersen and 0. Jepsen, Phys. Rev. Lett. 53 (19841 2571. [21] O.K. Andersen, 0. Jepsen and D. Glotzel, in: Highlights of Conden~d-Matter Theory, Eds. F. Bassani, F. Fumi and M.P. Tosi ~No~h-Holland, New York, 1985). 1221 O.K. Andersen, Z. Pawlowska and 0. Jepsen, Phys. Rev. B 34 (1986) 5253. [23] W.R.L. Lambrecht and O.K. Andersen, Surf. Sci. 178 (1986) 256. [24] S.H. Vosko, L. Wilk and M. Nusair, Can. J. Phys. 58 (1980) 1200. [25] D.M. Ceperley and B.J. Alder, Phys. Rev. Lett. 45 (1980) 566. [26] K. Schwarz and P. Mohn, J. Phys. F 14 (1984) L129. 1271A.R. Williams, V.L. Moruzzi, J. Kiibler and K. Schwarz, Bull. Am. Phys. Sot. 29 (1984) 278. [28] See for instance, L. Nordstrom, B. Johansson, 0. Eriksson and MS.8 Brooks, Phys. Rev. B 42 (1990) 8367. 1291CL. Fu and A.J. Freeman, Phys. Rev. B 33 (1986) 1755. 1301L.E. Klebanoff, SW. Robey, G. Liu and D.A. Shirley, Phys. Rev. B 30 (1984) 1048. 1311 S. B&gel, D, Pescia and P.H. Dederichs, Phys. Rev. B 39 (1989) 1392. [32] G. Shirane and W.J. Takei, J. Phys. Sot. Jpn. Suppl. 17 (1962) 35. [33] J. Klibler, J. Magn. Magn. Mater. 20 (1980) 277. [34] H.L. Skriver, J. Phys. F 11 (1981) 97.
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[35] J.L. Fry, Y.Z. Zhao,
[36] [37] [38] [39] [40] [41] [42] [43]
N.E. Brener, G. Fuster and J. Callaway, Phys. Rev. B 36 (1987) 868. V.L. Moruzzi, P.M. Marcus and P.C. Pattnaik, Phys. Rev. B 37 (1988) 8003. V.L. Moruzzi and P.M. Marcus, Phys. Rev. B 38 (1988) 1613. V.L. Moruzzi and P.M. Marcus, Solid State Commun. 71 (1989) 203. V.L. Moruzzi, P.M. Marcus and J. Kiibler, Phys. Rev. B 39 (1989) 6957. H.B. Michaelson, J. Appl. Phys. 48 (1977) 4729. M.C. Desjonqueres and F. Cyrot-Lackman, Surf. Sci. 50 (1975) 257; 48 (1977) 4729. J. Friedel, Ann. Phys. 1 (1976) 257. E.C. Stoner, Proc. R. Sot. London Ser. A 154 (1936) 656; 169 (1939) 339;
[44] [45] [46]
[47] [48] [49] [50]
E.P. Wolfarth, Proc. R. Sot. London Ser. A 195 (1949) 434. 0. Gunnarsson, J. Phys. F 6 (1976) 593. D.G. Pettifor, J. Magn. Magn. Mater. 15-18 (1980) 847. R. Coehoorn, in: Supermagnets, Hard Magnetic Materials, Eds. G.J. Lang and F. Grandjean (Kluwer, Dordrecht, 1991) p. 133. 0. Eriksson, Uppsala University, Thesis, 1989. G. Ahan, Surf. Sci. Rep. 1 (1981) 121. A. Zangwill, Physics at Surfaces (Cambridge University Press, Cambridge, 1988) p. 125. S. Bliigel, M. Weinert and P.H. Dederichs, Phys. Rev. Lett. 60 (1988) 1077.
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Surface Science 315 (1994) 157-172
ELSEVIER
Surface energy and magnetism of the 3d metals M. AId& *7a,H.L. &river b, S. Mirbt a, B. Johansson a a Covered
Matter Theov Group, Physics Department, UppsaIa Uniuersi~, S-El21 UppsaIa, Sweden b Technical University of Denmark, DK-2800 Lyngby, Denmark Received 10 February 1994; accepted for publication 14 April 1994
Abstract We present an ab initio study of surface energies, surface magnetism and work functions of the 3d transition metals. The calculations are performed by means of a spin-polarized Green’s function technique based on the tight-binding linear muffin-tin orbitals method within the atomic sphere approximation. In addition to the conventional paramagnetic and spin-polarized calculations we use the fixed spin-moment method to clarify the effect of magnetism on the surface energies. The results are shown to be consistent with a Friedel model of d-electron bonding combined with spin-split state densities as well as with a Stoner-type description. It is established that the anomaly in the surface energy of the 3d metals deduced from surface tension measurements is purely a magnetic solid state effect. In addition, it is found that magnetism reduces the surface energy of open surfaces, e.g. the @Ol) crystal faces, to the extent that the usual anisotropy of the surface energy is reversed. Thus a complete realization of the surface energy anomaly only takes place in the less close-packed surface facets.
1. Introduction
The electronic properties of the surfaces of the ferromagnetic 3d transition metals Fe, Co, and Ni have been studied extensively by first-principles local spin-density calculations during the past decade D-71. Thereby one has established that the magnetization at the surface of a 3d metal generally is enhanced by up to 300%. This effect is commonly explained in terms of the decrease in coordination number and the increase in localization of the d-states at the surface. Direct support for the enhanced ma~etism from experiment exists but is rather limited [l], although it is
* Corresponding
author.
generally reported that magnetism is not lost in the surface layer. For a time Ni was in this respect experimentally a subject of controversy &91. In this situation it is of great interest to calculate the effect of magnetism on the surface energy of the 3d metals and to compare with available experimental information. This is particularly true because the surface energies of the 3d metals as deduced from the surface tension of the liquid metals by de Boer et al. [lo] are anomalously low when compared to the corresponding non-ma~etic metals in the 4d and 5d series. Hence, one suspects magnetism to play a role and the degree with which we are able to reproduce the anomaly by direct calculation will lend strong indirect support for the calculated enhanced surface magnetic moments.
~39-~28/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSL?lOO39-4028(94)00239-6
158
M. Ald& et al. /Surface
The surface energy is defined as the energy required to transform a bulk atom into a surface atom with a corresponding decrease of the bulk volume and increase of the surface area, or equivalently, as half the energy needed to cut the crystal into two semi-infinite pieces. Similarly, the cohesive energy of a metal is defined as the energy gained in the transformation of a free atom into a bulk atom. It was recognized early on that the surface energy Es was approximately proportional to the cohesive energy EC,,,,and that the relationship
(1) was obeyed at the same level of accuracy by the 3d as well as by the 4d and 5d transition metals. The most common support for this relationship comes from the bond-cutting model, e.g. see Ref. [ll], which in its simplest form states that the total energy is proportional to the number of nearest neighbour atoms in the crystal. This type of model has proven successful, especially when modified into a form with a square-root dependence on the coordination number [ll], but it does not incorporate the effects of magnetism in a natural way. The parabolic variation with d-occupation number, which follows from d-band filling arguments, is common to both the surface energy E, and the cohesive energy Ecoh and consistent with the proportionality (1). In the 3d-transition series the expected parabolic behaviour is broken when the d band is approximately half full and instead one finds a pronounced minimum at the central element manganese. Since the anomaly occurs simultaneously in the surface energy and in the cohesive energy and since (1) remains valid even for the 3d series, it is common to assume that the two anomalies have the same physical origin. The loss of cohesive energy in the elements Cr-Co is well known to be caused by the large free atomic spin-pairing energy found in elements with approximately half-filled d-shells [12,13] and the cohesive energy anomaly is therefore an atomic effect. In contrast to the cohesive energy the surface energy is per definition not related to any atomic state and the anomaly in the surface energy of the 3d metals has remained a puzzle
Science 315 (1994) 157-I 72
although magnetic effects of course have been expected to play a role. We were recently able to verify the anomaly through first-principles, spinpolarized surface calculations [14] and to present a simple picture of its physical origin. The main issue of the present work is to address the longstanding problem of the origin of the surface energy anomaly by direct calculations of the quantities involved, and to present a full report on surface energies, surface magnetism, and work functions for the 3d metals from first-principles calculations. Ab initio calculations of the surface energies, surface relaxations, and work functions of the 4d metals have recently been reported by Methfessel et al. [ll] who used a full-potential, all-electron, slab-supercell linear-muffin-tin-orbital (LMTO) method. At the same time Skriver and Rosengaard 1151 presented a comprehensive study of the paramagnetic surface energies and work functions of all elemental metals by means of a Green’s function technique based on the tightbinding LMTO representation and the atomic sphere approximation. The spin-polarized analogue of this technique was subsequently used by Alden et al. [7], demonstrating its quality and accuracy by comparing calculated magnetic moments and work functions for Fe, Co, and Ni with those of previous full-potential, all-electron, slab-supercell calculations [3-61. Here, we present a theoretical study of the surface spin magnetism in Cr, Mn, Fe, Co, and Ni as well as of the general effect of magnetism on the surface energy and work function of the 3d elements based on the Green’s function technique. Results by the present method for the most densely packed surfaces of Cr and Mn with antiferromagnetic order within the surface layer are presented here for the first time. In addition to the unconstrained paramagnetic and spin-polarized calculations we employ the fixed-spin-moment (FSM) procedure to gain further insight into the effect of magnetism on surface energies. The results are discussed in terms of the Friedel model of cohesion and the Stoner model of itinerant magnetism. Both models are consistent with the unusual anisotropy of the surface energies found in Cr, Mn, and Fe and
M. Aldin
lead in conjunction with the complete calculations to a physically transparent picture of the anomaly in the surface energy of the 3d metals. The remaining part of the paper is organized as follows. In Section 2 we describe the computational details and approximations used throughout this work. In Section 3 we present our calculated surface energies, magnetic moments, and work functions for a number of crystal faces of the 3d metals SC-Cu. In Section 4 we review conventional models and predictions for the anisotropy of the surface energy of a (paramagnetic) metal. We discuss the Friedel model of (paramagnetic) d-bonding for surface energies and extend it to model the cases of half- and fully-saturated d-band ferromagnetism. The magnetic equation of state for ferromagnetic systems is illustrated in terms of the Stoner model for band magnetism. Finally, a summary with conclusions is given in Section 5.
2. Computational
method
2.1. TB-LMTO Greens function
159
et al. /Surface Science 315 (1994) 157-172
technique
The tight-binding linear-muffin-tin-orbitals (TB-LMTO) Green’s function technique for surfaces and interfaces, implemented by Skriver and Rosengaard [16], is based on the work of Andersen and co-workers [17-231. Shorter reviews of the method have recently been presented elsewhere [7,15]. In the present implementation we work within the frozen core approximation and use a minimal (spd) basis set and the atomic sphere approximation (ASA) for the charge densities as well as for the one-electron potentials except for additional electrostatic dipole contributions from neighbouring spheres. We use linear response theory and a linearized version of the Dyson equation to improve convergence speed. Depending on convergence tests the surface region consisted of 4-6 layers of metal plus two layers of empty spheres simulating the vacuum. Mainly due to the limitations of the computational approach, the relaxation of atomic positions are neglected. Since, however, the main perturbation at the surface is the change in coor-
dination number, we expect that the effect of relaxation on work functions, surface energies and magnetic moments is to be only a minor correction. The Green’s function matrices were sampled at 16 z-points on a complex energy contour and at 36-64 k,-points in the irreducible part of the 2D Brillouin zone, depending on surface facet. The underlying bulk calculations were performed with the same choice of kll-points in order to minimize sampling errors. State densities, presented in Section 4.3, were obtained by a final z-sampling of the Green’s functions on a fine mesh close to the real axis. Exchange and correlation were treated within the local spin density approximation (LSDA) using the Vosko-WilkNusair parametrization [24] of the data by Ceperley and Alder [25]. A systematic comparison 17,151 between the results obtained in recent calculations by the present method and those derived from full-potential, all-electron slab-supercell calculations [3-61 shows that key quantities such a surface energy or magnetic moments typically differ by much less than 10%. This suggests that the present approximations including the neglect of relaxation of the atomic positions are well-founded and indicates the efficiency of the Green’s function technique. 2.2. The FSM method The fixed-spin-moment (FSM) method [26,27] has been extensively used within density functional theory to deduce the total energy E as a function of magnetization M, thus enabling studies of, e.g. metamagnetism [28]. In a system with a fiied number of particles ilr under the additional constraints of fixed spin magnetic moments Mp at each site Q, the problem is formally to minimize the functional
F[n(r),
m(r)]
=E[n(r),
-
m(r)]
Chp( _/&W dr -4~)~ Q
(2) where P and h,
are Lagrange multipliers, with
160
M. Aldin et al. /Surface
respect to the charge density n(r) and the total and partial magnetization densities, m(r) and m,(r), respectively. By minimizing (2) with respect to the spin-up, n+(r), and spin-down, n-(r), densities n”(r)
=+2(r)
i-m(r)],
(3)
n_(r)
=+2(r)
-m(r)],
(4)
for the case of no Q-dependence, principle yields (he = h),
the variational
(5)
(6) The conventional procedure of the FSM method is to identify the right-hand sides of Eqs. (5) and (6), p f h, as chemical potentials for the two di~erent spins. This corresponds to using two different Fermi energies and to fix the number of particles of the two spins separately. A completely equivalent approach is to work with a single Fermi level whilst shifting the spin-dependent one-electron potentials, I$&@) = V,“(r)
3: ho,
(7)
where h, is to be determined self-consistently and forms the site-dependent external magnetic field, Ho = ho/pn. We have chosen this latter scheme since it is better suited to our present implementation of the Green’s function technique where boundary conditions and Fermi levei are set by the underlying bulk calculation.
3. The calculations The 3d metals form in a number of different crystal lattices and exhibit several magnetic structures some of which are rather complicated, especially when surface magnetism is included. We shall therefore explain which crysta1 and magnetic structure were used in the present calculations. bee Fe, hcp Co, and fee Ni order ferromagnetically with one (Fe, Ni) or two (Co) equivalent atoms per primitive cell and were treated within
Science 315 (1994) 157-172
these structures. The true ground state of bee Cr is that of an incommensurable antiferromagnet with an average magnetic moment of 0.46~~ per atom [32]. Here, we have assumed a commensurable antiferromagnetic state as was done in previous ab initio bulk calculations 133,341 and treated Cr in the CsCl structure. Mn orders magnetically in a complicated bee structure with 58 atoms per unit cell, which to our knowledge has not been treated by ab initio methods. Theoretical studies of the magneto-volume properties of both one-atomic bee Mn [35-371 and two-atomic CsCl Mn [38] predict the ground state of Mn to be a CsCl structure ferrimagnet at the normal volume. In the present calculations we have treated Mn in both the ferromagnetic (bee) and the ferrimagnetic (CsCl) state. Finally, fee Mn and fee Fe were constrained antiferromagnetitally in the CuAuI structure with the sign of the moments alternating along the (001) axis, as recommended in ab initio magneto-volume studies E391. In the following we shall refer to the most closely packed surface of each crystal structure as a type P surface (“packed”), and to the second most closely packed surface as a type 0 surface (“open”). This is mainly for convenience but will also benefit the discussion later on. Thus f&111), h~(OOl), and bcc(ll0) are the type P facets and fcc(OO1) and bcc(001) are the type 0 facets. In the cases of the CsCl and the AuCuI structures which are used in the calculation of antiferromagnetic bcc(ll0) Cr, fcc(ll1) Mn, and fcc(lll) Fe, the type P surfaces have two sites in each layer which differ in the direction of the spin while the type 0 surfaces have only a single site per layer. The so-called inside-the-su~ace 42 x 2) antiferromagnetism investigated for Cr(OO1) by Bhigel et al. [31] was not considered for the type 0 facets. The calculated paramagnetic surface energies for a number of surface facets of the 3d metals are presented in Table 1 together with the values listed by de Boer et al. [lo]. Since the Iatter values are partially derived from measurements of the surface tension of liquid metals they may not be attributed to any particular crystal surface and are best interpreted as weighted averages. For this reason we compare in Fig. 1 these exper-
M. Ati&
et al. /Surface
imental values with the surface energies obtained for a common close-packed surface facet, i.e. fcc(l11). The calculated surface energies in the figure exhibit a parabolic variation with the number of d-electrons which is in accord with the simple picture of d-electron contribution to the surface energy suggested by Friedel [42]. In contrast to this, the parabolic behaviour of the experimental data is broken in the middle of the series leading to a large discrepancy with the calculated paramagnetic values. Clearly, one expects this
Science
315
(1994)
i 1.0
161
157-172
1
0/
,n’
fcc(lll)
0.5 1 Table 1 Paramagnetic surface energies for the 3d metals and experimentally observed equilibrium Wigner-Seitz radii, S, used in the calculations Metal
SC
Ti
S (Bohr)
Surface (a.u.1
(eV)
(J/m*)
(J/m*)
3.427
hcp(001) fcdlll)
0.48 0.44
0.82 0.76
1.28
hcp(OW
0.90 0.12
1.95 1.56
2.10
fcc(ll1)
3.052
Surface energy Theory
Exp. a
V
2.818
bcc(l10) bcdO01) fcdlll)
0.82 1.06 0.99
2.02 1.76 2.55
2.55
Cr
2.684
be&IO) bcc(001) fcdlll)
1.33 1.74 1.10
3.63 3.36 3.09
2.30
Mn
2.699
fcc(lll) fcc(OO1) bcc(l10) bcd001)
1.17 1.40 1.27 1.60
3.24 3.35 3.43 3.05
1.60
Fe
2.662
bcctll0) bcc(OO1) fdlll) fcdool)
1.12 1.57 1.15 1.43
3.09 3.08 3.28 3.48
2.48
co
2.621
hcp@Ql) fcc(ll1)
3.18 3.23 3.40 2.78 2.83
2.55
bcdOO1)
1.08 1.10 1.33 0.98 1.41
Ni
2.602
fccflll) fcc(OO1)
0.88 1.03
2.63 2.67
2.45
cn
2.669
fcdlll) fcc(OO1)
0.69 0.85
1.96 2.09
1.83
2.631
a See Ref. [lo].
fccml) bcctl10)
1 i
SC Ti V Cr Mn Fe Co Ni Cu L,l_li’,I’J~~~J~J’ 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 Number of d-electrons Fig. 1. The calculated surface energy for paramagnetic fcdlll) surfaces of the 3d metals quoted from Skriver and Rosengaard [15] compared with the experimental values listed by de Boer et al. [lo].
difference to be due mainly to the appearance of magnetic ground states in the elemental metals and hence to the neglect of spin-polarization in the calculations. The results of the spin-polarized calculations are summarized in Table 2 and in Figs. 2 and 3 we present surface energies for the type J? and 0 facets of the magnetic 3d metals. As explained above all the calculations are performed for the experimentally observed magnetic structures except for Mn which is treated in the magnetic fee and bee structures of lowest energy. Since Mn transforms into the bee structure before melting it is the results of the magnetic bee calculations which are shown in the figures. It is seen that spin-polarization generally reduces the surface energy and thereby brings the calculation into closer agreement with the experimentally observed anomaly. It may also be observed that owing to lower coordination numbers and thereby increased localization effects surface magnetic moments are systematically larger at the type 0 surfaces than at the type P surfaces. As a result the surface-energy anomaly is only completely realized for the type 0 surfaces, Fig. 3, of Cr, Mn, and Fe where the magnetization is driven closer to saturation.
M. AEd& et al. /Surface Science315 (1994) 157-I 12
162
The degree of magnetic saturation, as e.g. defined in the following section, is known to increase when going from Cr to Ni and is found to be especially small for Cr and Mn. At the surface, the magnetic moments in Cr, Mn, and Fe are driven closer to full saturation, whereas for Co and Ni the moment enhancement is small due to the already large saturation in the bulk. For Cr the surface magnetic moment is found to be strongly dependent on the surface facet and the magnitudes of the moments for the bee type P and 0 facets differ by a factor of 2, see Table 2. As a result the magnetic reduction of the surface energy in Cr is very large for the type 0 facet, but essentially nonexistent for the type P facet which
in addition has antiferromagnetic order within each layer. In Mn the surface magnetic moment is already sufficiently large for the type P facet to lower the surface energy substantially with respect to the paramagnetic value, but also in this case is comptete agreement with the experimental surface energy only realized for the type 0 facet. It is interesting to note that the increased magnetic saturation in the type 0 surfaces reduces the surface energy to the extent that these surfaces become more stable than the type P surfaces. This is in contrast to the anisotropy commonly found in paramagnetic systems where the close-packed surfaces as a rule have the lowest energies.
Table 2 Calculated surface energies and spin moments in the bulk and in the top-most surface layers for the magnetic 3d metals; the second row of ferrimagnetic bcc(ll0) Mn refers to the second site in the layer; the surface layer is denoted by S Metal
Surface
Spin magnetic moments &a)
Surface energy
Bulk
s-3
s-2
S-l
S
(eV)
(J/m’)
Exp. g (J/m2)
Cr
bcdll0) a,b bcc(001) b
f 0.21 f 0.23
f 0.42 - 0.57
+ 0.42 0.74
f 0.65 - 1.16
+ 1.39 2.61
1.31 1.16
3.58 2.23
2.30
Mn
bcc(ll0) = bcc(ll0) d
1.13 2.23 - 0.24 - 0.57 - 0.29 * 1.75 - 1.70
1.07 2.23 - 0.32 2.50 2.42 + 1.68 1.74
0.94 2.41 -0.14 - 0.61 -0.70 f 1.72 - 2.01
1.51 2.96 - 2.95 3.66 3.64 + 2.58 2.84
1.22 0.90
3.28 2.42
1.60
bc~~l) ’ bcdOO1) d fcc(l11) +Gf fcdOO1) f
1.03 2.26 - 0.29 0.99 2.28, - 0.31 + 1.72 f 1.68
0.82 0.73 1.01 0.88
1.57 1.38 2.80 2.12
Fe
bcc(ll0) bcc@Ol) fcc(ll1) Gf fcc(OO1)f
2.24 2.24 + 1.76 kl.78
2.24 2.25 f 1.79 - 1.38
2.25 2.37 f 1.76 1.99
2.35 2.30 + 1.76 - 1.40
2.57 2.97 + 2.30 2.59
0.96 1.09 1.02 0.93
2.66 2.18 2.88 2.25
2.48
CO
hc~~l) fcc(lll)
1.61 1.64 1.64 1.74 1.74
1.60 1.63 1.66 1.73 1.75
1.65 1.67 1.63 1.75 1.72
1.70 1.72 1.84 1.78 1.94
0.94 0.91 1.09 0.99 1.26
2.74 2.70 2.78 2.79 2.52
2.55
bcc(ll0) bcc(001)
1.61 1.64 1.64 1.74 1.74
fcdlll) fcc(OO1)
0.63 0.64
0.63 0.64
0.65 0.66
0.67 0.64
0.62 0.69
0.90 1.07
2.69 2.77
2.45
fcc@Ol 1
Ni
a ~tiferromagnetic within layers in the C&Cl structure. b Underlying antiferromagnetic bulk calculation in the CsCl structure. ’ Underlying ferromagnetic bulk calculation, one-atomic bee unit cell. ’ Ferrimagnetic in the CsCl structure. ’ Antiferromagnetic within layers in the CuAu structure. ’ Underlying antiferromagnetic bulk calculation in the CuAu structure. g See Ref. [lo].
M. Ald&
2.0
3.0
4.0
5.0
6.0
7.0
8.0
163
et nl. /Surface Science 315 (1994) 1.57-I 72
9.0
Number
of d-electrons Fig. 2, Comparison between the calculated paramagnetic (Para.) and spin-~larized (Mag.) surface energy for the type P facets of the 36 metals. Paramagnetic rest&s are taken from &river and Rosengaard [15]. Also illustrated are the experimental values listed by de Boer et al. (Exp.) DO].
To gain a deeper understanding of the magnetic contribution to the surface energy and in particular of the role played by the degree of saturation, we have performed fixed-spin calculations for the 3d metals Sc-Ni. In these calculations the d-magnetization is fixed to be uniform in the bulk as well as in the surface layers, and we define the d-saturation to be md/nd if nd < 5 or m,/(lO - n,) if n4 > 5, respectively, where md is the d-moment and nd is the number of d-electrons. Thus md is defined by choosing a certain constant amount of saturation through the series. The results for O%, SO%, and 80% d-magnetization for the fcc(ll1) facet are shown in Fig. 4, where it may be seen that the minimum of the anomaly has been shifted to Cr. Within the Friedel model to be discussed later this fallows from the fact that Cr is effectively closer to having a hdf-filled d-shell than Mn. On the other hand, the surface magnetic moment (Table 2) and the saturation is larger for the type 0 surfaces of Mn than they are for Cr, and therefore the minima in the unconstrained calculation occ!ur for Mn.
In Fig. 5 we present the results of constraining the surface magnetic moments for the type 0 facets to their relaxed b&k values, When these results are compared with those of different saturation in Fig. 4 as well as with those obtained in the relaxed ab initio calculations, Fig. 5, it is seen that the introduction of uniform magnetic moment reduces the surface energy relative to the paramagnetic case, except if the d-band is either empty or filled. However, the most important contribution to the anomaly results from the enhanced saturation caused by low surface coordination numbers. The work functions obtained from the paramagnetic and the spin-polarized calculations are given in Table 3. The trend for the type P 3d surfaces and the effects of magnetism are illustrated in Fig. 6, Except for the various type 0 facets of Cr and Mn, there is a subst~tial reduction in the work function when spin-~lari~ation is included improving the agreement with the experimental values [40]. Flowever, even includ-
3.0
2.5 $ 3
2.0
ui” 1.5 1.0
0.5
Sc
1
2.0
Ti
V
a./,I,I,I,j.,, 3.0
4.0
Cr
Mn 5.0
Number
6.0
Fe 7.0
Co 8.0
Ni
Cc
9.0
ofd-electrons
Fig. 3. Comparison between the calculated par~a~et~c fFara.f and spin-polarized fMag.) surface energy for the type 0 facets of the 5d metals. Paramagnetic results are taken from Skriver and Rosengaard [X5]. The calculations were performed far the experimentally observed crystal structures, except for Mn and Co which were treated as bc4001) and fcc@Ol), respectively. Experimental values are those listed by de Boer et al. (Exp.) [lo].
r-e-O%
--a-50%
ing the magnetic reduction the calculations provide an overestimate of thle work functions by 0.5-l eV. This systematic difference between the calculated and experimental work functions may partly be attributed to a failure of the local spin density a~pro~mation, since for instance the calculated values have been shown E16,7f to depend quite sensitively on the actual parametrization used for the LSDA functional. Since full-potential, layer-reIaxed results compare favourably [15] with the present type of calcufations, it seems
--+80%
SC
Ti
AL___,_,
v
cr 3
ml 1
Fe .a
co
Ni
.-_l”“_““i----j---C
2.0 3.0 4.0 5.0 6.0 7.0 8.0 Number of d-electrons Fig. 4. Fixed-spin calculations of surface energies for the fcc(Ill) surface of the 3d metals. TIte top curve shows paramagnetic rest&s, whereas the middle and bottom curves correspond to 5tI% and 80% saturation, respectiveiy, of the d magnetic moments
Tabfe 3 Calculated and experimental work functions for the 3d metals; P and F denote the pammagnetic and spin-polarized results, respectively Metal
Surface
Theory
Exp.a
hcp(001) fcc(ll1)
4.59 4.63
(4.333
b&I 10) bcciOOlt iccQ11t
5.12 4.57 4.88
(4.3)
5.45 4.58 5.27
5.30
5.45 5.37 5.69 4.90
5.18 5.76 5.34 5.31
(4.1)
5.78 5,t-S 5.54 5.55
5.21 4.S 5.50 6.12
@.5)
hcp(001) fcc(ll1) fcc(OO1) bcc(llO~ bcc@Ol)
5.81 5.76 5.83 5.74 4.91
5.53 5.55 5.52 5.66 5.15
CS.0)
fc&Ii~ fccioolf
5:77 584
5.70 5.75
5.35 5.22
fcc(lll) fec(OO1)
5.30 5.26
fcctlllf fcc@Ol) bcdll0) bcc@Ol~
4.0
7.0 5.0 6.0 Number of d-electrons
8.0
Fig. 5. Fixed-spin c&ulations of the surface energy assuming a uniform m~eti~~tion equal fo the setf-consistent butk vatue ~n~f~~rn~ compared v&h the diamagnetic restrfts (Pam.) and those of the uncoustrained, ~~j~-p~j~iz~~ c&a&tions (Mag.1. Except for Mn and Co which are treated as bcc@Ol) and fcc@Ol), respectively, the calculations are performed by type 0 facets of the experimentally observed crystal structures.
Work function (eV)
a SeeRef. [401.
f4.5)
4.94
4.94 4.59
M. Ak& et al. /&face
Science 31.5 (1994) 157-l 72
respectively, becomes Es = IG
-0
Exn.
Element Fig. 6. Work functions for the 3d metal. The calculated results are obtained for the close-packed type P surfaces assuming either a paramagnetic (Para.) or a spin-polarized (Msg.) ground state. The experimental values [40] (Exp.) for Se-Co are obtained on polycrystalline samples while those for Ni and Cu are obtained on the type P facets.
that the ASA and the omittance of surface layer relaxations is less responsible for the discrepancy. Finally, one should also remember that a closepacked single-crystal surface usually experiences a larger work function than a polyc~stalline sample, and that this difference may be of the order 0.5 eV [HI. Of the cited experiments in Fig. 6 only the measurements on Ni and Cu are performed for a single surface-facet while those for SC-CO are performed on polycrystalline samples.
4. Theory of surface energies 4.1. Bond-cutting and anisotropy The surface energy, and in particular its relation to the cohesive energy, Eq. (l), has often been discussed in terms of bond-cutting models and nearest neighbour interactions. The primary assumption is that the energy for cutting a single nearest-neighbour-bond is a well-defined quantity, and that the total energy therefore scales to the number, C, of nearest neighbours. If these are C, and C, for the bulk and surface atom,
165
the relation between - Cs)/CnlK0h*
Es and Emh (8)
This approach may be acceptable in the limit of strongly covalent bonding, whereas for a metal one expects that the bond saturates when the coordination number is increased. Direct determination of the metallic bond strength by means of ab initio total-energy calculations have recently been presented for Al and for the 4d transition metals by Methfessel and co-workers [ll]. In their work the total energy was fitted to the sum of a dominating bonding term (proportional to the square-root of the number of nearest neighbours, fi) and a small repulsive term (linear in C) E(C) =E,-A@
+BC,
(9)
reducing the “effective” nearest-neighbour bonding to half of that of the simple linear model. By here letting B scale with Ecoi,, B = f&E,, as suggested by Methfessel et al. [ill, one obtains Es= [1+&r,-
~(I+B,,CB)]E_,b.
(10)
Within the framework reviewed here, the bondcutting model supports a proportionali~ k between Es and Ecoh at 3 different levels of approximation. Having k = &/Ecoh, the quasi-chemicai, linear k is given directly through Eq. (8). Neglecting the linear term in (9) gives k = CJc, - G>/G whereas the full k follows from Eq. (10). Numerical values of k for these different cases and for 5 common crystal planes have been tabulated in Table 4. From this it follows that the square-root description using Eq. (10) gives the best overall agreement with experiment for the 3d and 4d transition metals. It is difficult to obtain a good estimate of k directly from calculations of E,, and Es, since E _,, is well known to be overestimated in the LSDA. The error is often stated to lie in the atomic c~c~ation, whereas the bulk total energy is expected to be given more accurately by the LSDA. In the context of bond-cutting models it is important to use non-magnetic quantities, and to carefully consider the dependence of the valence
M. Aldh
166
et al. /Surface
Table 4 Study of the proportionality constant k, in the relation Es = models; Cn and Cs are the kr&oh for various bond-cutting coordination numbers for a bulk and surface atom, respectively; k, = (C, - C,)/C,, k, = C& - &I/&, and k, = (1+ B&)(1 the text; (k,,) proportionalities respectively Surface fcc(ll1) fcc@Ol) fcc(l10) bcc(l10) bcc(001)
with B, = 0.03, as explained in - JW) and (k,,) refers to averages of experimental [lo] for the 3d and 4d transition metals, C, 12 12 12 8 8
(k,)
C,
k,
k,
k,
9 8 6 6 4
0.25 0.33 0.50 0.25 0.50 0.37
0.13 0.18 0.29 0.13 0.29 0.20
0.09 0.13 0.22 0.11 0.22 0.16
c&3,>
0.156
&j>
0.149
configuration in the free atom [11,13]. Most important is the atomic spin-polarization (SP) energy, which lowers the energy of the atom and therefore reduces the cohesive energy. This SPenergy is especially large for elements in the middle of a d-transition series. The effect of the bulk magnetization on the energy is one order of magnitude less, since in the solid this polarization energy has to be balanced against the band (kinetic) energies. In this light, it is surprising that the empirical relation (1) holds just as well for the 3d series, where magnetic effects are especially large, as it does for the 4d and 5d series. Denoting the parumagnetic component of the surface and cohe-
Science 315 (1994) 157-172
sive energies Er and E:tr, respectively, this (accidental) empirical fact may be expressed as Er+As-
bee bee bee bee fee fee
V Cr Mn Fe co Ni
ESF
1 .Ol 1.74 1.60 1.57 1.33 1.03
_ 1.16 0.73 1.12 1.09 1.07
(11)
Here, A, and A, are the (negative) energy contributions due to magnetism in the atom and bulk, respectively, and A, is the total magnetic contribution to Es (A, = Es-Erra>. For the 4d and 5d series one expects Erra and EFlr to be related by the original relation (11, and this should also hold for the 3d elements. Therefore, the magnetic contributions appear, empirically, to behave as A, = +(A,4 -Ad.
(12)
In order to test this relation we considered our calculated values for A, (type 0 facets) and A,, and performed first-principles calculations of the atomic SP-energies (A,) for Cr-Ni. The results, given in Table 5 and illustrated in Fig. 7, show that the empirical relation (12) is approximately obeyed and consistent with ab initio theory. However, for e.g. the 4d transition metals magnetic effects in the surface energies are nonexistent [lo] whereby relation (12) becomes meaningless. We conclude that Eq. (1) is not well-founded for magnetic systems, but only fortuitously obeyed for the 3d metals. The anisotropy of the surface energy for a parumagnetic metal is predicted by the bond-cutting models (Table 4) and by tight-binding studies [41]. One finds for the fee crystal that E S(111)
Table 5 Calculated surface and magnetization energies (in eV) of V-Ni for structures; El and E: are paramagnetic and spin-polarized surface surface energies assuming a uniform d-magnetization in the bulk and unconstrained bulk value), Ez”% (50% d-saturation) and Es80% (80% atom (A,) and bulk (A,) E,P
;(~,P,ah’“+A~-d~).
<-&(001)
(13)
the type 0 facets of the experimentally observed crystal energies, respectively, quoted from Tables 1-4; fixed-spin at the surface are denoted by EC (d-moments equal to the d-saturation); magnetization energies are given for the free
50%
ESB
Es
EF%
A,
AB
1.746 1.522 1.455 1.168 1.09
0.677 0.538 0.994 1.500 1.211 1.138
0.412 0.047 0.400 1.063 1.060
1.64 5.02 5.07 3.40 1.99 0.91
0.01 0.09 0.53 0.19 0.05
M. Aldth et al. /Surface
167
Science 315 (1994) 157-I 72
energy Es for a paramagnetic approximately [42] E, = $q, (
Cr
Mn
Fe
Co
Ni
Element Fig. 7. Study of the empirical relation (121, by means of comparing ab initio results for A, and i(A, - A,). Quantities are as defined in the text.
and for the bee crystal J&O) < &01) <
-%(,11,7
(14)
i.e. the surface energy is expected to increase as the surface becomes more open. Whether these inequalities will hold for a magnetic system or not depend upon the magnetic energy contribution to the surface energy. Clearly, they will be violated if the magnetic effects are sufficiently large and surface sensitive. In our spin-polarized calculations for the type P and 0 facets, this is indeed seen to occur for Cr, Mn, and Fe. 4.2. The Friedel model In the band-picture of d-electron bonding as suggested by Friedel [42], the average electronic energy per atom is (E)
=
/
+D( E)E
dE,
1- $
(16)
6W, 1
where n,, is the number of d-electrons and the 6W is the reduction in band width at the surface relative to the bulk. Thus, through a d-series of elements a parabolic-like behaviour is expected for the surface energy. Experiment [lo] and recent calculations [15,11] support this parabolic behaviour for the 4d and 5d metals. Support is also found from the present paramagnetic calculations for the 3d metals, as can be seen in Fig. 1. This is in contrast to the experimental results which exhibit a marked dip in the middle of the series. This bears, as already mentioned, a close resemblance to the the mid-series cusp in the 3d cohesive energies, caused by atomic spin-pairing effects. We incorporate band magnetism in the Friedel model by first regarding the case of a saturated itinerant ferromagnet. Here, the spin-up and spin-down d-bands in the bulk and at the surface, respectively, are completely separated. Hence for a less than half-filled d-shell (nd < 5) the narrowing of the unoccupied spin-band has no effect on the surface energy. The surface energy of such a saturated ferromagnet becomes Es=&,
(
l-
$
)
(17)
6w,
and for a saturated magnet with rzd > 5 the same expression is valid if nd is replaced by (nd - 5). Thus the parabolic behaviour of the surface energy for the paramagnetic state is replaced by two parabolas, with maximum for n,, = 2.5 and 7.5, respectively. For a non-saturated magnet there is a continuum of solutions, but if the bulk magnetic moment md is maintained up to the surface one finds the surface energy to be
(15)
i.e. the one-electron term in the independentparticle model. Here or is the Fermi level and D(E) is the d-state density at energy E. Assuming a rectangular state density in the bulk and at the surface, the d-electron contribution to the surface
metal becomes
6W.
(18)
The Friedel models including (18) for a halfsaturated magnet, i.e. md = n,/2 for n,, < 5 and m,, = (10 - n,)/2 for nd > 5 are illustrated in Fig. 8. From this it is now evident that the dip in the
M. Aldh
168
et al./Surface
Science 31.5 (1994) 157-172
4.3. The Stoner model PARAUAGNETIC
t :
0
2
4
6
8
10
Number of d-ekctrons
Fig. 8. Model calculation of the surface energy of the 3d transition metals based on a rectangular d-state density. The three curves correspond to the paramagnetic, the halfsaturated Cm, = n,, /2 for nd < 5 and m,, = (lo- nd)/2 for nd > 5), and the saturated case, respectively.
middle of the series may originate from simple band-filling effects. The Friedel model predicts that the mere introduction of a uniform magnetization will lower the surface energy of a metallic crystal. Hence, if the magnetism is constrained to be uniform, as in the fixed-spin calculation presented in Section 3, with moments equal to the unconstrained bulk (nonzero) values, the surface energy should decrease relative to that of the paramagnetic state, as is indeed found to be the case. In general, the magnitude of the surface magnetic moment is never lower than that in the bulk, and relaxing the spin-moment at the surface automatically gives rise to a further reduction of the surface energy. The size of this reduction will depend on the degree of magnetic saturation in the bulk relative to the surface, so that for Co and Ni the reduction is small while it is substantial for Cr and Mn.
The existence of ferromagnetism in metallic systems has often been explained in terms of properties of the paramagnetic ground state within the so-called Stoner model [43-471. The approximations of this model are rigid band splittings and a Stoner parameter I independent of magnetization and they lead to the simple and well-known Stoner criterion for the onset of ferromagnetism as well as to the equations of state and the susceptibili~. Before the advent of accurate surface calculations, Stoner analysis has been applied also to the question of surface magnetism [48,49]. The results indicate that the magnetic moments will be maintained at the surface. In the context of surface energies and the effect of the onset of spin-polarization in the surface layer, it is interesting to consider the following Stoner equation of state E(M,
I’) = +j”A(M’,
V) dM’-
$4’,
(19)
0
where E(M, V) represents the change in total energy due to magnetization M at volume L’, A&W, V> is the (constant) exchange-splitting of the d-states and I the Stoner parameter, taken to be an intrinsic atomic quantity independent of the crystal surroundings. The first term in (19) describes the loss in kinetic energy due to occupancy of less bonding d-states, as discussed in the last section, whereas the second term represents the gain in exchange energy. Within the rigid-band ansatz and neglecting the volume dependence, the exchange-splitting A(M) may be obtained from M=/
‘“+4%(E)
dE = [;_4
D(E)
dE,
(20)
6F
A(M)
=A, +A,,
(21)
where the spin-up and spin-down band shifts, A, and A,, are given in terms of the magnetization M and the paramagnetic d-state density D(E). The onset of ferromagnetism is given by the condition that for A4> 0, A(M)
=ZM,
(22)
M. Aldhz et al. /Surface Science 315 (1994) 157-172
which may be solved graphically using (20) and (21). In Figs. 9a-9c we have plotted A(M), derived from the calculated paramagnetic state densities for bulk bee Fe, fee Co, and fee Ni and their respective type P and 0 facets. If the criterion (22) is obeyed, the energy gain per atom due to spin-polarization (19) may be obtained as half the area enclosed by the straight line and the A-curve.
169
The Stoner estimates of the magnetization energies for Fe, Co, and Ni are compared with our calculated ab initio results in Fig. 10. The qualitative agreement found in the figure supports the Stoner approach in the case of ferromagnetism. For Fe the large energy difference between the two facets is a result of the surface-dependence of A, (Section 4.1) and leads to an anisotropy
0-4
(4
1.6
>^ s
1.2
a 0.8
1
0.8
M
(cl,)
M
1.2
1.6
(ClB)
0.6
0.2
0.4 M
01,)
Fig. 9. The rigid exchange splittings A(M) as defined within the Stoner model and obtained from the paramagnetic densities: (a) bee Fe; (b) fee Co; (cl fee Ni.
d-state
170
M. Aldh
et al. /Surface
which is opposite to that usually found for surface energies (14). For Co and Ni, the magnetic energy differences and hence surface energy anisotropy are small. Clearly, already the simple Stoner model gives this result, although the effect is overestimated for Fe. Detailed theoretical investigations of the magnetic properties of the bcc(001) surface of Cr have been carried out by Bliigel and co-workers 131,501, and on the experimental side this surface was found to order ferromagnetically within the layer [30]. In the p(1 x 1) structure with insidethe-surface ferromagnetism, calculations, such as our present ones, usually give magnetic moments that are enhanced N 300% as compared to the bulk moment. Nevertheless, here the Stoner criterion is never met, which is in sharp contrast to an early calculation within the tight-binding approximation [48]. This is because the state densities obtained from the tight-binding scheme differ substantially from those of our self-consistent calculation, which are displayed in Fig. 11. This simple Stoner analysis for the bcc(001) surface of Cr shows that its ferromagnetism is not explicitly due to a Stoner instability, but only supported by the antiferromagnetic coupling to the underlying
Science 315 (1994) 157-172
-3 t
Cr bulk
1
I -4
0
-2 E
2
4
(eV)
Fig. 11. Calculated paramagnetic (total) d-state density, D(E), for bulk and for the (001) surface layer of bee Cr. Notice that IL&,)< 2 for the surface layer, implying that a surface Stoner criterion for ferromagnetism is not obeyed here. Stoner parameter for Cr: I = 0.82 eV [341. The Fermi level is marked with a vertical line at zero energy.
layers and enhanced fects.
due to band narrowing ef-
5. Summary
Type 0
\
Fe Co Ni
TypeP
71
Fe Co Ni
Fe Co Ni
Fig. 10. Bulk and surface layer magnetization energies for Fe, Co, and Ni obtained from the Stoner equation of state compared with the ab initio results, resolved for the surface layer.
We have calculated the surface energy, surface magnetic moments, and work function of the elements Cr, Mn, Fe, Co, and Ni. Our main objective has been to study the effect of magnetism on the surface energy of the 3d metals with a view to clarify the anomaly in the surface energy as deduced from surface tension measurements. With this objective in mind we have introduced two models in order to obtain an understanding of the calculated results and to isolate the magnetic contributions to the surface energy. The Friedel model accounts for the fact that metallic d-bonding is affected when the d-bands are spin-split, whereas the Stoner model gives a good estimate of the gain in total energy when ferromagnetism is introduced at a paramagnetic surface. The Friedel model is fully consistent with
M. AId& et al. /Surface Science 315 (1994) 157-172
the anomaly in the 3d surface energies as deduced from experiment and explains the dip in the middle of the series in terms spin-split d-state densities. With respect to our calculated surface energies the spin-polar~ation substantially improves the agreement with experiment but the full surface energy anomaly is only realized for the more open surfaces such as the (001) crystal facets where the calculated magnetic moments are found to be much larger than those of the closely packed facets, especially for Cr and Mn. Our calculations suggest that the anisotropy in the surface energy as obeyed by parama~etic surfaces, i.e. the most densely packed facet exhibits the lowest surface energy, may be reversed if magnetic effects are sufficiently large. Finally, we conclude that the anomalies in the surface and cohesive energies of the 3d metals are of different origin, and that the observed proportionality Es = iEm,, between the two quantities therefore must be fortuitous. Acknowledgements The Uppsala group is grateful to The G&an Gustafsson Foundation and The Swedish Natural Science Research Council for financial support. The work of HIS was supported by grants from the Novo Nordisk Foundation and from the Danish Research Councils through the Center for Surface Reactivity. We would also like to thank Dr. 0. Eriksson for assisting us with the spinpolarized atomic calculations. References [II For a review, see e.g., J. Mathon, Rep. Prop. Phys. 51 (1988) 1.
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