A simple model of the Fe/Cu (001) reconstruction

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A simple model of the Fe/Cu (001) reconstruction

Š. Pick

To cite this version:

Š. Pick. A simple model of the Fe/Cu (001) reconstruction. Journal de Physique, 1989, 50 (13),

pp.1583-1586. �10.1051/jphys:0198900500130158300�. �jpa-00211018�

1583

LE JOURNAL DE PHYSIQUE

Short Communication

A simple model of the Fe/Cu (001) reconstruction 0160. Pick

J. Heyrovsky Institute of Physical Chemistry ^and Electrochemistry, Czechoslovak Academy ^{of Sci-}

ences, Dolej0161kova 3, 182 ²³ Prague 8, Czechoslovakia

(Reçu ^{le 24 mars 1989,} accepté ^sous forme définitive ^{le 3 mai} 1989)

Résumé.

Afin d’étudier la reconstruction displacive d’un film bicouche de Fe ^sur du Cu(001),

récemment observée, nous avons analysé ^un hamiltonien hautement simplifié de liaison forte par la méthode de récurrence. La reconstruction de surface ^se développe vraisemblablement dans ^une gamme étroite d’aimantation (probablement ~ ^par atome).

Abstract

To study ^the recently ^observed displacive reconstruction of the two-layer ^{Fe film on} Cu(001), ^a highly simplified tight-binding Hamiltonian is analyzed by the recursion method. In ^a

narrow range ^of magnetization (probably ~ ^{per atom), the} surface reconstruction is likely ^to develop.

J. Phys. ^{France 50} (1989) ^{1583-1586 1er JUILLET 1989,}

Classification

Physics ^Abstracts

68.35 - 73.60 - 75.70

1. Introduction.

Because of their interesting properties, ^thin layers ^of ferromagnetic transition metals have been studied extensively during ^{the last} years, ^both experimentally ^and theoretically. Particularly, ^much

attention has been paid ^{to iron grown on the} (001) face of noble and f.c.c. transition metals [1-6].

For very ^thin Fe-films, ^the following properties ^{seem to} be established [4-6] : a) ^the electronic d- states of Fe do not hybridize strongly ^{with the} electronic states of the substrate, ^and b) ^{the iron} magnetization is enhanced with respect ^{to the} bulk, amounting ^{to about 3} YB per ^atom.

Recently, ^a structural transition from (1 ^x 1) ^to (2 ^x 1) phase ^{has been} observed for two

Fe-monolayers deposited ^{on Cu} (001) when cooled to temperatures below 300 K [2]. ^{For films}

4 or 5 monolayers ^{thick, a related} phonon softening is detected. Such behaviour may indicate the repulsive ^stress within the surface (lst) Fe-layer, causing ^an important change of the surface force constants [2]. ^{It is} interesting ^that softening ^{of an} analogous (xi) ^{mode has been predicted}

theoretically [7] for the unstable (1 ^x 1) ^W (001) ^surface. According ^to [8], ^the repulsion ^between

the surface tungsten ^atoms results from the indirect interaction mediated by ^the subsurface (2nd layer) ^{atoms. T’he} explanation ^of the latter effect is based on the analysis ^{of the 4th-order moment}

of the electronic Hamiltonian. However, ^the generalization ^{of this} picture ^{to the Fe/Cu} (001)

system ^{is not} straightfonvard.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500130158300

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2. The model.

Below, ^an attempt ^{is made to} study ^{the Fe/Cu} (001) instability in the framework of a highly ^sim- plified one-electron tight-binding approach. Taking ^the properties mentioned in the introduction into account, ^we consider the isolated two-layer ^f.c.c. iron film. As pointed ^out by ^the referee, ^this assumption ^{is less} justified ^{for the} Cu-substrate than e.g. ^{in the} Ag ^{case. The reason} is the smaller interatomic distance and the position ^{of the} copper d-bands closer to the Fermi energy. ^{In the}

light ^{of recent results} [6], ^the very behaviour of thin Fe-films on the (001) faces of noble metals, ^Pd

and Pt seems to be universal and we believe that the gross qualitative ^{features are not} completely

lost in our modeL Besides that, ^{the mean field} (rigid ^band splitting) approximation [9]

is introduced for the electronic energy ^band dispersion ^Ea (k). ^{Above, the} index "+" ("-") ^stands

for the majority (minority) spin orientation. The existing theoretical results on ferromagnetic ^films

seem to support ^the approximation (1), especially ^{as the flat} t2g-bands ^{are concerned.}

The starting point ^{of our} computation ^{is the} d-electron nearest-neighbour tight-binding

Hamiltonian based on the canonical f.c.c. parameters of Pettifor taken from the reference [10]

(ddcr = - ^0.027784 Ry, ^{dd7r = 0.012535} Ry, ddc5 = - 0.001554 Ry). ^{In the} paramagnetic phase (8 ⁼ 0), ^the energy spectrum ^{of the} two-layer film is confined between the x2 - y2-like energy levels at -0.106868 Ry ^{and 0.084906} Ry, situated in the surface Brillouin zone points ^{r and} M, respectively. ^Th bring ^the tight-binding parameters ^{closer to those} predicted by ^Harrison [11] ^for

iron with the Cu-lattice spacing, ^{we have} multiplied ^the parameters of Pettifor by the factor 1.6.

Any ^other scaling, ^{as well as a} différent choice of the parameter Q introduced below would lead

simply ^{to the linear} rescaling ^{of the} energy difference curve given ⁱⁿ figure ^1.

1 If

Fig. ^1.- The electronic energy change per ^atom 6.Eel associated with the deformation with the amplitude

O.Ola. 1be deformation considered is shown in the inset. (The lst and 2nd layer ^{atoms are} represented by large ^{and small} circles, respectively).

On the ideal two layer ^sium, the deforinatioii suggested ⁱⁿ [2] ^{with the} amplitude O.Ola is im-

posed (see Fig. 1 ; ^the amplitude ^{on the 2n atomic} layer ^{is zéro from} symmetry reasons). ^The

corresponding electronic (band) energy change [12, 13] ^reads

Above, the indices i, ^{u refer to the} type of the orbital and to the spin, respectively, ^{hiiu =} 1=8,

and ni, (E) ^{is the} partial ^local density of electronic states. Contribution from both film layers ^is

included. Due to the approximation (1), ^the calculation for majority ^and minority spins ^{can be} performed separately. ^The density ^of states ni, is evaluated by the recursion method [10,12,13]

including ^the Hamiltonian moments up ^{to the 10th} order, ^and terminating the continued fraction

by ^the quadratic terminator fitted to the band limits specified ^above.

Similar models suffer from the uncertainty in the choice of the parameters [12,13], ^{and the}

situation is yet ^more unfavourable for the pseudomorphic ferromagnetic ^film, properties ^{of which}

are not known in the detaiL Hence, ^{it is more} promising ^to introduce further simplifications ^into

our model retaining ^{the most essential} aspects ^{of the} problem ^{at the same time,} rather than to insist on a careful fitting procedure. ^{First, we write the} perturbation ^{due to} the deformation as

AH = Hl = (grad H, ’1) , where q is the deformation. It is this term which is decisive in the stability analysis [14]. On the other hand, ^we disregard the contribution quadratic ⁱⁿ q ^and containing ^{the 2nd} derivatives of H. This term contributes to the repulsive (restoring) ^{forces and}

is not sensitive to the détails of the deformation [14]. Hence, ^{it is more} convenient to include it into the repulsive ^{force term} [12, 13] ^not considered here. Assuming ^the r- Q -distance-dependence ^of

the Hamiltonian matrix elements, ^the change of the matrix block HAB describing ^the interaction between atoms A, B is of the form

In equation (3), ^Aras is the bond length change and w is the infinitesimal rotation generator

associated with the A - B bond rotation by ^the angle Ao. The choice of Q represents ^{a serious} problem : ^{Q = 5} according ^to canonical theories [11-13], although ^{the recent} calculations [12,13j

suggest ^{a reduced value.} Here, ^we adopt ^{the value} Q ^{= 3.5. In} any case, the first ^term in the right-

hand side of equation (3) ^{is more} important than the second one. Th avoid complications arising

from the Q-dependent interplay ^{between the two terms, we} neglect ^{the rotation term} completely.

3. Results and discussion.

The electronic energy change per atom, associated with ^the deformation is shown in figure ^{1 as}

a function of the electronic occupation ^{number N} (0 N 5) for électrons with particular (ma- jority ^or minority) spin orientation. In figure 1, negative ^{values of AEei are} favourable for the destabilization. The contribution from the surface and subsurface is comparable, and this is expe-

cially ^{true in the} vicinity of the marked minimum at about N ⁼ 4.7. The origin ^{of this} outstanding

feature can be traced to the flat t2g energy ^{bands. The three} t2g ^orbitals contribute most to the A Eel

curve, whereas the x2 - y2 ^{orbital is almost not involved. For} transition metals, ^{the electronic} energy gain for deformations with the amplitude ^{0.01a should be} several 10-5 Ry (10-4 eV ^[15]

to overcome the repulsive forces. That is why ^the position ^{of the} Fermi level EF ^{in the} region

close to N ⁼ 4.7 for majority spins ^{would be} very favourable for ^the surface softening. ^{We find an} essentially ^smaller energy gain ^{at N N 0.5 and also in the central} part ^{of the energy} spectrum.

In the latter _case, however, ^the part ^{of the} restoring ^{forces due to the} second derivatives of the Hamiltonian (which reduce the Hamiltonian second moment m2 and are not considered here)

are maximaL Indeed, according ^{to common,} although ^crude approximation, ^{JE’et is} proportional ^to m2, ^{with the} proportionality ^factor being maximal for roughly half occupied ^{bands. The result-} ing guess (which ^is by ^{no means} reliable) indicates that the energy gain is reduced by one ^{third in}

this energy region.

Similarly ^{as in} [12, 13], ^{the exact} d-electron count per ^{atom is not} known, although ^{it is un-}

1586

doubtedly higher ^{than 6} (cf. ^Ref. [16]). Accepting ^{the value} Nd+ ^{4.7 for} majority spin ^electrons,

the magnetic ^moment per ^{atom will} be > ^{3 PB,} in accord with thé values quoted ^above.

It would be nice to offer a simple picture of the surface destabilization, e.g. ^{in the} following

form : let us consider (x ^a y) z ^orbitals interacting predominantly along ^the atomic chains in the

[1, 1] direction. (The situation is the very ^same for (x - y) z orbitals and the [1, -1] direction).

Ileating ^{the chains as} one-dimensional arrays of alternating ^{lst and 2nd} layer ^atoms, the deforma- tion studied in the present paper ^has formally period ⁴ along ^{the chain.} (Note ^{that the} glide-plane symmetry operation, interchanging the lst and the 2nd layer, ^{behaves in most} respects ^{as a "com-}

mon" translation along ^the chain). Consequently, ^{two Peierls} gaps open ^{in the} upper ^{and in the} lower part ^{of the} density ^of states, respectively [17]. The minima at about N ⁼ 0.5 and N ⁼ 4.7 in

figure ^1, together ^{with the} proof ^{of the} importance ^of t2g ^{orbitals, may} give ^{some credit to the above}

idea. Nevertheless, the model of the quasi-one-dimensionai ^{chains is} probably ^too simplified ^{to be}

taken literally.

Tb summarize, ^our simple tight-binding model indicates that the two-layer ^{iron film on} Cu(001) ^can reconstruct providing ^{that the} Fermi level falls into a narrow range ^of energies ^cor- responding probably ^{to the} magnetic ^{moment value} > 3 _UB per ^{atom. The cause of the} instability

is the indirect repulsive interaction between surface atoms, ^mediated by ^{the 2nd} layer. ^{For this} interaction, t2g ^{orbitals are} important.

Acknowledgement.

The author is indebted to Professor Dr. Harald Ibach for drawing his attention to the problem.

References

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