MAGNETIC PROPERTIES OF SMALL 3d-TRANSITION METAL CLUSTERS

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MAGNETIC PROPERTIES OF SMALL

3d-TRANSITION METAL CLUSTERS

G. Pastor, J. Dorantes-Dávila, K. Bennemann

To cite this version:

G. Pastor, J. Dorantes-Dávila, K. Bennemann.

MAGNETIC PROPERTIES OF SMALL

JOURNAL DE PHYSIQUE

Colloque C8, Supplkment au no 12, Tome 49, d4cembre 1988

MAGNETIC PROPERTIES OF SMALL 3d-TRANSITION METAL CLUSTERS

G.

M.

Pastor, J. Dorantes-Divila and K. H. Bennemann

Freie Universitat Berlin, Institut fur Theoretische Physik, Arnimallee 14, D-1000 Berlin 33, F.R.G.

Abstract. - We determine the size and structural dependence of magnetic properties of small Cr,, Fen and Ni, clusters

by using a tight-binding Hubbard Hamiltonian in the unrestricted Hartree-Fock approximation. The role of magnetism for the structural stability of these clusters is also discussed.

Recently, significant interest in the electronic properties of transition metal clusters has developed [I-41. In the following we determine the size and struc- tural dependence of magnetic properties of small Cr,, Fen and Ni, clusters and compare our results with ex- periment and previous calculations.

We consider the Hubbard Hamiltonian for, d- electrons:

where c&, (c;,,) refers to the creation (destruc- tion) operator of an electron on site i , orbital a and spin a , and t~~ to the hopping integrals for which we use the canonical values [5] dd

( a ,

n,

6) =

(-6,4, -1) (Wd12.5) ( ~ l d ) ~ . Here, Wd stands for the bulk d-band width, S for the Wigner-Seitz radius, and d for the interatomic distance. The interaction Hamil- tonian HI in the unrestricted Hartree-Fock approxima- tion is given by:

2

a J

= E:

+

UAn (i)

-

- p (i)

.

2 (2)

Here, fi;, = c~,c;,, refers to d-electron number operator and Edc to the correction due to dou- ble counting. The direct and exchange intra-atomic Coulomb integrals are denoted by U and

J,

respec- tively. The number of d-electrons n (i) = (nit)

+

(nil) (An (i) = n (i) - nd, nd = ( l l n ) X;n (i)) and the magnetic moment p (i) = (niT)

-

(nil) on site i

are calculated self-consistently by requiring:

Here, Ni, ( E ) refers to the local-electronic density of

states of spin a on site i and is calculated using the recursion method [6].

The cohesive energy per atom is calculated from

Ecoh (n) = Eb (1) - Eb (n)

-

E R (n)

,

where

E (n) = ( 1i ~ N i u ( E ) d~ - Ed=

is the electronic d-band contribution and

ER

(n) = ( l l n ) C i z i A exp { - p (dl& - 1))

the Born-Mayer repulsive energy. The parameters A,

p are fitted to the bulk compressibility and equilib- rium condition at the bulk interatomic distance db.zi refers to the local coordination number. Finally, for each assumed structure, we minimize the total energy with respect to d (uniform relaxation) and obtain p (i)

,

pn = ( l l n ) Cip (i) and Ecoh (n) for the relaxed geome- tries.

The parameters used for the calculations are listed in table I. nd is taken t o be independent of cluster size and such that the number of s-electrons n, = 1[7]. J is fitted to the bulk magnetic moment pb, U is estimated

from atomic spectroscopic data and Wd is taken from band calculations [8].

Table I. - Parameters used for the calculations (see text). Energies in eV and nd in electrons per atom. The calculated bulk-magnetic moment pb is given in units of p ~ .

Results for the average magnetization p, at T = 0

and for the local magnetic moments on different clus- ter shells p (i) of small Fe,, Cr,, and Ni, clusters are given in table 11. &r all studied clusters we obtain larger magnetic moments than for bulk material [9]. Note, that the increase of p (i) becomes more and more important as we go from Ni to Cr, i.e. as we approach half-band-filling, and that Cr, shows even larger local magnetic moments than Fen in contrast to bulk ma- terial [9]. Upon relaxation the interatomic distance d reduces typically 2-4 % (9 % for bcc-Fes). The mag- netic moments p ( i ) also decrease for Cr, and Fen since the band width increases. In Nils u, (1)

>

p (2) due to charge transfer from the central atom to the surface of

C8 - 1816 JOURNAL DE PHYSIQUE

Table 11. - Results for the size and structural depen- (nd = 9) charge transfer effects, resulting mainly from dence of the magnetic moments p (i) and average mag- the different effective band-width a t outer and inner netization fin (in units of p ~ ) of Cr,, Fen and Ni, atoms, become increasingly important. Magnetism clusters. DzBerent shells i are orderd by increasing dis- and charge redistribution also play an important role tance to the cluster center. I n brackets results for the in the structural stability of Ni-clusters. In fact, if unrelaxed geometries are given. we set U = 0 and J = 0, and thus disregard ef-

the cluster. Note, p (1)

>

1 can be only obtained if n (1)

<

nd = 9. Upon contraction the charge transfer

Feg Fels Fel3 Fe15 Fe19 Crg Cr15 Nil3 Nil3

increases slightly and thus p. (1) increases.

For Fen the bcc-structure is more stable than the fcc- or icosahedral structure (E,"::

-

EZh 21 0.2 - 0.4 eV).

While for bcc-Fen the ferromagnetic-like order (all mo- ments pointing in the same direction) is favoured, the fcc-like Fen-clusters show an antiferromagnetic-like or- dering. The magnetic moment on the central atom points in the direction opposite t o that of the outer- most shells (Fels: tit, and Felg : t t l t t ) . This seems

Struct. bcc bcc fcc bcc fcc bcc bcc fcc icos.

t o be observed for small y-Fe particles and surfaces

[lo] and is probably related t o the antiferromagnetic ordering observed in bulk y-Fe [ll].

Pn 2.31 (2.98) 2.52 (2.96) 1.94 (2.04) 2.58 (2.72) 1.86 (191) (3.86) (0.33) 0.82 (0.82) 0.97 (0.96) p (1) 0.40 (2.91) 0.37 (2.89) -1.67 (-1.74) 0.48 (1.23) -1.07 (-1.2) (-2.63) (-2.13) 1.20 (1.17) 1.27 (1.23)

Crn-clusters with bcc-like structure show antiferromagnetic-like order with moments on atoms

p (2) 2.55 (2.98) 2.63 (2.98) 2.24 (2.36) 2.64 (2.80) 1.81 (1.88 (4.67) (3.23) 0.79 (0.79) 0.94 (0.94)

belonging to different sublattices of antiferromagnetic bulk-Cr pointing in opposite directions (Cr9:

tit and

Cr15 : 1t.lt-l). This is in agreement with results re-

p (3) 2.82 (2.94) 2.85 (2.70) 2.48 (2.52) (-3.13)

ported previously [3]. The average magnetization fin + 0 for increasing cluster size due t o the cance- lation of contributions with opposite sign. Already for Cr15 fi15 is much smaller than the local magnetic mo- ments ( ~ 1 5 = 0.33 p ~ < < Ip (surface)l 21 3 p ~ )

.

For Nils-clusters with icosahedral and fcc-like struc- ture we obtain ferromagnetic-like order in constrast t o fcc- or icosahedral-Fe13. For nearly filled d-band

fects due to magnetism and charge transfer, we ob- tain that icosahedral-Nils is more stable than fcc-Nil3

( E E ~ - E ~ ;

=

-

0.07 eV)

,

in agreement with simi- lar previous calculations [4]. If we take into account charge transfer effects but still keep J = 0 we ob- tain EFh - E z c = 0.02 eV indicating that fcc-Nil3 is slightly favoured. Now, using J as given in ta- ble I, we obtain that fcc-Nils is further stabilized with respect to icosahedral Nil3

(EE-,

-

E ~ Z

= 0.06 eV). This results illustrate the importance of calculating the spin-polarized charge distribution self-consistently when comparing structures with similar binding en- ergies. However, we cannot conclude safely how the most stable geometrical arrangement of Nil3 should look like since the approximations we made (e.g. ne-

glect of s-electrons and s-d hybridization) preclude the determination of AEcoh t o the accuracy required for

this case (N 0.01 eV) . Further work on this subject is currently in progress and will be published elsewhere.

[I] Cox, D.

M.,

Trevor, D. J., Whetten, R. L., Rohlf- ing, E. A. and Kaldor, A., Phys. Rev. B 32 (1985) 7290.

[2] Lee, K., Callaway,

J.

and Dhar, S., Phys. Rev. B

30 (1984) 1724.

[3] Salahub, D. R. and Messmer, R. P., Surf. Sci.

106 (1981) 415.

[4] Gordon, M. B., Cyrot-Lackman, F. and DesjonquBres, M. C., Surf. Sci. 80 (1979) 159. [5] Pettifor, D. G., J. Phys. F 7 (1977) 613.

[6] Haydock, R., Solid State Phys. 35 (1980) 216. [7] n,

--

1 for n

>

2, 3 seems reasonable due t o

maximal s-bonding and W s > > ~ d

-

E ~ . Small s- d charge transfer do not change qualitatively our results.

[8] Moruzzi, V. L., Janak, J. F. and Williams, A. R., Calculated Electronic Properties of Metals (Perg- amon Press, New York) 1978, p. 168.

[9] Victora, R. H., Falicov, L.

M.

andIshida, S., Phys. Rev. B 30 (1984) 3896.

[lo] Keune, W., Halbauer, R., Gonser, U., Lauer, J. and Williamson, D. L.,

J.

Magn. Magn. Muter.

6 (1978) 192;

Gonser, U., private communication.

[ l l ] Johanson, G. J., McGirr, M. B. and Wheeler, D. A., Phys. Rev. B 1 (1970) 3208;