Surface stacking faults in close-packed transition metals

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Surface stacking faults in close-packed transition metals

B. Piveteau, M. Desjonquères, D. Spanjaard

To cite this version:

B. Piveteau, M. Desjonquères, D. Spanjaard. Surface stacking faults in close-packed transition metals.

Journal de Physique I, EDP Sciences, 1992, 2 (8), pp.1677-1690. �10.1051/jp1:1992235�. �jpa-00246648�

Classification

Physics

Abstracts

68.20 68.55 68.65

Surface stacking faults in close-packed transition metals

B. Piveteau

(I),

M. C.

Desjonqu~res (')

and D.

Spanjaard (2)

(')

C-E-A-, C-E-N-

Saclay,

DSM, DRECAM, SRSIM, 91191 Gif-sur-Yvette Cedex, France (2) Universit6 Paris-Sud, Laboratoire de

Physique

des Solides, B£tirnent 510, 91405

Orsay

Cedex, France

(Received 7 February J992,

accepted

in

final form

J7 April J992)

Rksumk. Une monocouche

m6tallique d6pos6e

sur une surface dense d'un metal CFC _ou HC

peut se mettre en faute

d'empilement. L'6nergie

de _ces fautes

d'empilement

_sur des surfaces

CFC (i II) et HC (0001) de m6taux de transition est 6valude ~ 0 K par une m6thode de liaisons fortes _{et en} utilisant une

technique

de fraction continue. Los

potentiels

correcteurs au

voisinage

de la surface sont calcul£s _{avec une}

approximation

de

charge

nulle. Les _cas de l'homo _et de

l'h6t£ro-6pitaxie

sont

envisages.

Pour

l'homo-6pitaxie

nous avons r£alis6 _une £rude

syst6matique

en fonction du

remplissage

de la bande d du substrat. Nous montrons

qu'une

surface CFC (I II)

est

toujours

_en

£pitaxie parfaite

avec le substrat h 0 K. Ce mEme comportement ^est encore valable pour la surface HC (0001) sauf dons _une garnme de

remplissage

de la bande d

correspondant

h peu

prbs

_aux Elements de la colonne IIIB. Les

Energies

des fautes

d'empilement

sont du mEme

orate de grandeur en surface _et dans le volume. Pour

l'h£t£ro-dpitaxie,

_nous discutons, _{pour un} substrat donn£, la

possibilit6 d'apparition

^{d'une faute}

d'empilement

_en surface _en fonction du

remplissage

de la bande d de la monocouche. Nos conclusions _{sont en} accord _avec les donn6es

exp6rimentales

existantes.

Abstract. The

deposition

of _a metallic

monolayer

_{on a}

close-packed

surface of FCC and HCP metals _can induce _a surface

stacking

fault. The energy of such

stacking

faults _on FCC (I ii) and HCP (0001) surfaces of transition metals is evaluated at 0 K with _a

tight-binding

scheme

using

_a

continued fraction

technique.

Perturbative

potentials

near the surface _are calculated

using

a zero

charge approximation.

Both homo and heteroepitaxy are considered. In the _case of

homoepitaxy

a

systematic study

with the d-band

filling

of the substrate is carried out. It is

proved

that _a FCC (I II ) surface is

always

in

perfect registry

with the substrate at 0 K. The _same trend is also

found for the HCP (0001) surface except ^for a range of d-band

filling

which could

correspond

to the elements of the IIIB column. Surface

stacking

fault

energies

_are of the _same order of magnitude as in the bulk. In the _case of

heteroepitaxy,

the

possibility

of _occurrence of surface

stacking

fault is discussed _{as a} function of the d-band

filling

of the

overlayer

for _a

given

substrate.

Our conclusions _are in agreement with

existing experimental

data.

I. Introduction.

The three series of transition metals exhibit the _same sequence of structural

phase

transitions _: HCP-BCC-HCP-FCC. This

phenomenon

has been _a great

pole

of interest for

metallurgists,

chemists _as well _as

physicists during

_{many years} and can be

explained

with the

progressive filling

of the d-band

[ii.

However,

the

problem

of structural

phase stability

in transition metals still remains of

large

interest in surface science.

By

_now it is

possible,

under _some

specific experimental conditions,

to obtain _a

layer-by-layer growth

of _a metal _{on a} metal substrate

[2],

_or to

deposit

_a

single

atom _on a metallic surface

[3, 4]

and to observe it with _a field-ion

microscope.

In the

bulk,

the

stacking

fault

energies

of

close-packed (HCP

_or

FCC)

metals _are small

[5],

of the order of _a few

percent

^{of the} cohesive energy. If _we consider _now the

deposition

of an

overlayer

of a

transition metal M

(HCP

_or

FCC)

_{on a}

close-packed

surface of _a transition metal

M'

(HCP

or

FCC)

two types ^{of three-fold}

adsorption

sites _are

present

ⁱⁿ

equal

numbers _: normal sites which continue the bulk

regular stacking

or the fault sites which introduce _a

stacking

fault.

Then,

the

following questions

arise _: if M and M' _are of the _same chemical

species,

is the _«

stacking

fault _» favoured at the surface ? _or is the surface

stacking

^fault _energy

comparable

to the bulk _one ? If M and M' _{are not} of the _same chemical

species

and have different bulk

crystal

structure

(HCP

_or

FCC)

which type ^{of sites will be}

preferred by

the M atoms ?

In this

study,

_{we try} to _answer these

questions by evaluating

the energy of surface

stacking

faults _on

close-packed

transition metals. The method is _a

tight-binding

scheme and is

briefly

summarized in section 2. Section 3

presents

an evaluation of the energy difference between

the HCP and FCC _structures in the bulk.

Then,

in section

4,

_we tum to the

study

of surface

stacking

faults.

Heteroepitaxy

is

briefly investigated

in section 5.

2. The method.

Many

times in the past, ^{it has been}

proved

that

general

trends

conceming

the cohesive

properties

of transition metals _can be derived from the

study

of the d-band electrons. In _a first

step,

^{the valence} sp electrons _can be

neglected

in _a

systematic study

of

properties

of transition

metals. These electrons should

only

have _some effects for metals with _a

nearly

_{empty or} filled d-band. More detailed

arguments

will be

given

in sections 3 and 4 to

justify

that their influence _can be

disregarded

in the

particular

_case of structural energy difference. Let _us start

by summarizing

the method to describe the d-valence band.

Since this d-band is rather _narrow

(5-10

eV _or 0.4-0.7

Ryd)

^{and the extension} of d-orbitals is small

compared

with interatomic

distances,

it is well

justified

to describe it in _a

tight-binding

approximation.

In this

approximation,

the _one electron wave-function is written _{as a} linear combination of atomic orbitals centered _on all sites _:

4l~

₌

£ a;~(E~) 4~(r R;), (1)

where the

following

notations _are used _:

4~ (r R;)

is _an atomic d-orbital centered _on site I _; A denotes the

degeneracy

^{and varies from} I to 5 and

E~

^is the energy of

4l~.

In the

following 4~ (r R;)

will be denoted

(iA ).

The energy of the five atomic d-orbitals is chosen _as the

origin

of

energies. 4l~

must

obey

the

Schr6dinger equation

_:

H4l~

₌

E~ 4l~, (2)

where H is the one-electron Hamiltonian of the system ^which can be written _:

H=T+

£V,, (3)

T

being

the kinetic energy operator ^and

V;

the atomic

potential

centered _on site I.

The

projection

of the

Schr6dinger equation

on the basis set

ii

A

)

needs the

knowledge

of the matrix elements of the Hamiltonian _on this basis. As

usually

done in the

tight-binding

approximation,

three _center

integrals

_are

neglected, I-e-,

_we restrict ourselves _to matrix

elements of the form

(iA [V~ jM)

with I #

j

and k

= I _or k

=

j.

These

hopping integrals

_can be

expressed simply

_{as a} function of the Slater-Koster

parameters

which _are assumed to decrease

exponentially

with the distance

R;j

=

[R; R;(

[6]. Moreover,

in the FCC and HCP structures, the second _nearest

neighbour

distance

R~

is much

larger

than that of the first nearest

neighbour Rj (R~

=

/Ri). Consequently, hopping integrals beyond

the first nearest

neighbours

_are

neglected.

In the _case of

d-electrons,

there _are

only

three Slater-Koster parameters

dd«, ddw,

dd8

[6].

All the transition

metals,

which have the _same

crystalline

structure, have

roughly

the _same

band structure except ^for a

scaling

factor determined

by

the bandwidth W.

Hence,

all

computations

have been carried _out with the _same set of Slater-Koster parameters ^and

energies

_are

given

in units of bandwidth. The

following

ratios of these

parameters

^{have been}

used

throughout

:

iii

_o.45

m

-

o.056 _~~~

With these

assumptions

_{we compute} the local

density

^of states

(LDOS) niA(E)

at site I

projected

_{on a}

given

orbital A. The LDOS is

simply

related _to the Green operator ^G

by

the

following

relation _:

G

=

(z

^H

)-1

,

(5)

and

n;~

(E )

= lim Im G

(

^~

_(E

₊ I_e

)

,

(6)

ar

~

where

G(~(E+ _is)

denotes _a

diagonal

matrix element of G _on the basis

(iA).

G;(~(z)

is written _{as a} continued fraction

[7-10]

:

G(~(z)

= ~~

(7)

iA

~l

~ ~'

iA ~iA

Z ~2 ₂

z-a(~-bji~lY(z)

The coefficients

a(~, b(~,

_are

computed by

a recursion method

[7]

from the Hamiltonian

matrix elements _on the basis

(iA).

The

larger

the number of _exact coefficients

a(~,

b(~,

the _more accurate is the

computation

_{of n~~}

(E ).

The coefficients

a~, b~

which appear in the terminator

lY(z)

_are

simply

related _to the lowest and

highest

limits of the d-band

energy

[8].

Once the LDOS is

computed,

a

simple integration gives

the electronic energy of _{an atom} in the metal. In

particular,

in all this

work,

_{we are} interested in

evaluating

small energy

differences

(about

10 ^~

Ryd), hence,

a

good

_{accuracy on} the LDOS is

required. Thus,

14

pairs

of coefficients

(a(~, b(~)

have been used.

Moreover,

in the _case of FCC

metals,

the LDOS varies rather

abruptly

with E and _a numerical

integration

_can lead to some error. In order to achieve _an accuracy of the order of 10~^~ 9b _on the

filling

of the

d-band,

_an

analytical

integration

has been carried _out

[I II-

Note that _a

repulsive

_energy term should be added to obtain the total energy ^{of the}

system.

This term, which _ensures the

stability

of the

crystal,

is assumed to be

pairwise

and

depends

only

_on the interatomic distances.

Consequently,

it does not contribute to the energy

difference between _two atomic structures

which,

_as in the

present study,

have the _same coordination numbers and interatomic distances.

Finally,

electronic correlations and

spin-orbit coupling

have been

neglected.

These contributions to the cohesive energy of transition metals _are small and we

expect

that

they

do not

play

_a

leading

role in the energy difference between two

phases. Nevertheless,

this method must be

slightly

modified since it is _not self-consistent when

applied

to surface

electronic bands. This correction will be

presented

ⁱⁿ section 4. The _next section

(Sect. 3) only

deals with bulk band calculations.

3. Relative

phase stability

of the FCC and the HCP structures,

The three transition

series, excluding

the four

magnetic

3d

metals,

exhibit the _same HCP- BCC-HCP-FCC _sequence. It is _now

generally accepted

that this

crystal

structure sequence is related to the

change

in the d-band

occupation

which takes

place along

a transition series

[I].

Assuming

that the interatomic distances _are the _same in the FCC and HCP

phases,

there is _no variation in the

repulsive

_energy between the two structures.

Thus,

the relative

stability

of the FCC and HCP

phases

_can be deduced from the

computation

of the d-band _energy difference.

We have

computed

these electronic

energies

with the method described in section 2 and derived the _curve shown in

figure

I. The energy difference AE oscillates with the d-band

filling N~

and cancels for four values of

N~ (Tab. I).

This _curve accounts

qualitatively

for the

crystal

structures of the _non

magnetic

transition metals _:

when _~

N~

~

3.5,

the HCP structure is stable _;

when

N~

_~ ⁶ a HCP-FCC

phase

transition _occurs around

N~

= 7.5.

FCC stable

o

o

~ 0

x

o

o

~ i

~ 1

~

0

~

Fig.

number ^N~) d-valence electrons. AE is given in units of -bandwidth (W) : _{W~cc for} the _FCC

structure, W~cp ^{for the HCP tructure.}

Table I.

Special

values

of N~ (number of

d-valence

electrons) for

which the energy

difference

AE

(Fig. I)

_or the

stacking fault

_energy

(Figs.

3 and

5)

cancels. For the notations

of

bulk

stacking faults

in FCC and in HCP structures, see

figure

5.

Figure

I AE 1.0 4.1 6.0 7.5

Figure

3a

AEs~

3. 5.0 7.4

nV 0.8 3.4 4.7 7.4

Figure

5a I2V

(intrinsic)

3.7 4.7 7.4

I2V

(extrinsic)

3.4 4.7 .2

Figure

3b

AEs~

4.3 6.0 7.65

nV 1.3 4.5 6.0 7.8

Figure

5b I2V 1.2 4.5 6.0 7.7

I3V 1.2 4.3 5.9 7.5

At intermediate

N~ (3.5 ~N~~6),

this _curve is

meaningless

_as the stable structure is

BCC. Noble metals _are excluded from _our

study

since _our

description, neglecting

the sp band

contribution,

leads _{to a zero} energy difference

owing

to the _zero value of the cohesive energy for _a filled d-band.

Choosing

_a

typical

d-bandwidth of 0.5

Ryd (6.8 eV)

leads to an energy difference which

does not exceed AE

=

6

mRyd (80meV).

This is in accordance with the fact that _some

transition metals

undergo

a HCP-FCC

phase

transition at

temperatures

^of a few

10~ K.

Yet,

it is _not the first time that this d-band energy difference has been evaluated

[1, 5, 12, 13]. Irrespective

of the

method,

the sequence of structural

stability

is rather well described.

Previous calculations in the

tight-binding approximation [1, 5]

^lead to the _same results and have shown that the

sp-d hybridization only brings

small corrections. It is

interesting

to stress here that

figure

I is very close to that obtained

by treating

the d and sp bands in _a

tight- binding

and a

nearly

free electron

picture, respectively [il.

Note that in reference

[II,

_two

curves for the d-band energy difference between the HCP and the FCC _{structures are}

presented

we must compare

figure

I

only

with the _curve obtained _N~ith similar

parameters

: d bandwidth and direct

hopping

between first

neighbours.

More

so~,histicated computations using

_a Linear Muffin-Tin Orbital

(LMTO)

method

[12]

_{or a} Linear

~ugmented-Slater-Type

Orbital

(LASTO)

scheme

[13]

have also been carried out.

However,

it is difficult to make _a

comparison

^between the values of AE obtained with these different,~nethods _as the d-band

filling

^for a same metal _can vary

significantly

with the

approximation

^{used. The} values of AE which _are

given

in

[12]

and

[13]

_seem rather

large

and inconsistent with _a

phase

transition

induced

by

a

temperature

^of a few 102 K.

Of _course, there _are few

experimental

determinations of these energy differences _as

they

deal with metastable

phases.

Some values have been obtained from the

study

of

binary phase

diagrams [14]. They

_{are more} consistent with _our

computations.

As _a

conclusion,

_a

simple tight-binding approximation, dealing only

with valence d-

electrons,

accounts for the

experimental crystal1ilie

structure of the transition metals _: the

crystal

structure sequence is well

reproduced through

_a transition series. This

comforting

result allows _us to use the _same

approximation

in the

study

of surface

stacking

faults.

4. Surface

stacking

faults of FCC and HCP

crystals.

The

(I it) (resp. (0001))

surface in the FCC

(resp. HCP)

structure is

close-packed

and shows very small relaxation effects which _can be

neglected.

Each of these surfaces presents two types of sites in

equal

numbers _: FCC

(resp. HCP) sites,

_at which _an added _atom continues the

normal FCC

(resp. HCP)

structure and HCP

(resp. FCC) sites,

where continued

growth

introduces _a fault

plane.

We _are interested in the situation in which _a whole surface

plane

is faulted. The surface

stacking

fault energy,

AEs~,

is the energy difference between the _two

following

situations

(Fig. 2)

:

a « normal _»

(lll)

^FCC

(resp. (0001) HCP)

surface _{on a} FCC

(resp. HCP)

^bulk _;

a faulted

plane

_{on a} FCC

(resp. HCP)

bulk.

B <- _sudace layer -> C

A A

C ^C

B B

A A

a)

A B

§ /

C surface layer

,

C

A A

c c

A A

b)

Fig,

2. Nornlal and faulted

stockings

which have been studied in this work. a) At _a (I II) FCC surface. b) At _a (0001) ^HCP surface.

As the FCC and the HCP _structures have the _same interatomic distances and since we

neglect

normal relaxation

effects, AEs~

^is

simply given by

the electronic energy difference, A

rigorous

calculation of

AEs~

should take into account both _s and d electrons.

Nevertheless,

several

points

lead _us to _assume that the

major

contribution to

AEs~

_comes

from the d-band.

First,

_as it was stressed in section

3,

_a

simple tight-binding approximation dealing only

with valence d-electrons accounts for the

crystal

structure sequence

through

the transition series.

Second,

it is well known that the s-band is much broader than the

d-band,

so

that it _can be

thought

that s-electrons _are rather insensitive _to the

geometrical

structure. This

point

can be

justified

with the

help

of the effective medium

theory.

In this

theory,

the

binding

energy

EB

of the system ^{consists of three terms}

[15]

_:

~B

"

Z ~c,

I

(~i )

₊

~AS

+

~l

cl '

(8)

1

The first term

£ E~ I(fi; )

is _{a sum over} all atoms in the system ^{and is}

closely

related to

the

embedding

_energy of the _atom in _a

homogeneous

electron gas of

density

n~. ^{The second} term

(E~s)

is called the

atomic-sphere

^correction _energy and _comes from the electrostatic

repulsion

that _occurs when two neutral

spheres overlap. Finally,

the third term

(Ej

~j) ^{is the} difference in the one-electron energy

spectrum

^{between the} atom in the real host and in the

homogeneous jellium.

In the _case of _a

single stacking

fault between _{two structures} which have the _same

compacity, I-e-,

in which the coordination numbers and the interatomic distances

remain the _{same so} that

only

the directions of the first

nearest-neighbours

_are

modified,

the first and second _{terms are not} altered. Thus the

stacking

fault energy can come

only

from the variation of the third term between the _two

geometric configurations.

This term is

largely

dominated

by

the contribution of the d-electrons

[15].

The

sp-d hybridization

_can

modify

this contribution

only

at the very

beginning

_or end of the series.

Moreover several

experimental

data have shown that the contribution of s-electrons _to

stacking

fault

energies

is small and is _not very sensitive _to the number of s-electrons in the band

[16]

_:

the

stacking

fault _{energy per} unit _area

(ys~),

is much smaller in bulk Cu

(~100 ergs/cm2)

than in bulk Ni

(ys~

= 240

erg/cm~)

a

systematic study

of

stacking-fault

_energy has been carried out in

Pd-Ag

_systems. As the concentration of Silver

increases,

the number of d-holes in the d-band of Palladium

decreases and vanishes for _a Silver concentration of about 60 9b atoms.

Correlatively,

_a

sharp

decrease of ys~ is observed from ys~ =180

ergs/cm~ (pure Pd)

to ys~ = 50

ergs/cm~.

From 60 9b atoms of Silver _to pure

Ag,

the number of s-electrons increases and ys~ varies very little.

So,

_{we can} consider that the

general trends, conceming

the

stacking

faults in transition

metals,

_can be derived from _a

tight-binding

calculation

dealing only

with valence d-electrons.

The d-band energy differences _are evaluated

using

the method described in section 2.

However,

in the presence of the

surface,

there is _a redistribution of

charge modifying

the

potential and, then,

the

tight-binding parameters.

^{It is}

usually

assumed

that,

in _a first

approximation, only

the atomic d-levels _are modified

[17-19].

Let _us

denote, 8V~,

the

perturbation

of the atomic level of _atom I relative to that of bulk atoms. One _can show

that,

in

a metal, these

8V~

are obtained with _a very

good approximation by assuming

_a local

charge neutrality [17-19].

In sections 4 and

5,

_{we compute} the

displacements

of the atomic levels

by imposing charge neutrality

with an accuracy of 10~^~ electron per

spin

orbital, Such a model is not

exactly

self-consistent _as the

potentials 8V,

are not calculated from the electronic

charges

but with _{a zero}

charge approximation. Nevertheless,

this

approxi,nation gives

results very

near from the self-consistent _ones in transition metals

[20].

In the

fcllowing,

_we will call the results obtained within this

approximation

the _« self-consistent _{» oni;s.}

A

positive (resp. negative) 8V,

_may increase the upper

(resp.

^{den-ease the}

lower)

limit of the energy of the

band, owing

to the existence of surface _states the en~.rgy of which is outside the bulk band

[9].

On

close-packed

surfaces these

«

self-consistency

_» effects _{are not} very

strong [9]. However,

_as it _was yet ^{stressed in section}

2,

_{we are} interested in

evaluating

small energy differences _so that these effects cannot be

neglected. Throughout

this

work,

the

broadening

of surface bands _was determined

by ensuring

a normalization of the

computed

LDOS with _an accuracy better than

10~~

_{9b, The}

largest

value reached

by 8V~ (displacement

of the atomic level of _{an atom at} the

surface)

is about 10~^~

Ryd.

This maximum value leads to

a

broadening

of the surface band of 3 9b

(resp.

0.5

9b)

when the bulk is FCC

(resp. HCP).

In the

present

case

8V~

and the LDOS have the _same values for all atoms of the _same

plane,

For each

plane

_r, the d-band energy is calculated

using

the

following

formula _:

E~

E~

E~

₌

En~(E )

dE N

~ 8V

~

Ei n~(E )

dE N

j

,

(9)

m m

in which

Ei

and

N~

_are,

respectively,

the Fermi energy and the number of d-valence electrons per atom in the metal.

8V~

is the

perturbation

of the atomic level of _an atom of the

plane

_{r m}

is the lower limit of the d-band energy.

Finally, n~(E )

is the total LDOS for _an atom I of the

plane

r

given by

:

n~(E )

= 2

£

_ni~

(E ) (10)

The second term in

(9)

arises from the Coulomb interactions which _are counted twice in the

sum of one-electron

energies.

The third term in

(9)

is _an additional term to the _« self-

consistency

_» correction.

Indeed,

it _must be noted that the _exact

charge neutrality

which is

imposed

here is _never

strictly

reached in the

computation. Hence,

there is _a lack

(or

_an

excess)

of

charge,

which _never exceeds

10-2

_{electron for the} whole d-band. In order to avoid any lack

(or excess)

of

charge

in the

d-band,

_{we put}

(or withdraw)

the

corresponding

number

of electrons _at

(from)

^the Fermi level. This

corresponds

to a

slight,

but

unphysical,

displacement

of the Fenni energy ; it _can be shown

that,

to first order, this correction is

equivalent

to the self-consistent _one

[20].

To evaluate the

stacking

fault energy per ^atom

AEs~,

_we must sum over several

layers

the energy differences between the situations shown in

figure

2.

Obviously,

the

major

contri-

bution to

AEs~

_comes from the surface

(r= I)

and the first

(two

_or

three) sublayers.

However,

in order to achieve _a

good

_{accuracy on}

AEs~,

and

especially

when

AEs~ changes sign,

we choose to _sum these energy differences _{over seven}

layers.

The _curve

AEs~(N~)

for _a

(I it)

FCC

(resp. (0001) HCP)

surface is shown in

figure

3a

(resp. 3b).

The

stacking

fault is

energetically

favoured when

AEs~

is

negative.

On each

figure

we compare ^{the results} without and with the

«

self-consistency

_» correction. As could be

expected,

neither the

shape

of the _{curve nor} the values of

AEs~

_are

dramatically changed by

this correction, Moreover there is a

striking similarity

between these _curves and the energy difference AE

(N~)

between the FCC and HCP structures

given

in

figure

I

(note

that _we must

compare

AEs~

to AE in the _case of _a

(0001)

HCP

surface),

Both

quantities AEs~

and AE

oscillate _as a function of N

~ with the _same altemations of

sign

and reach values of the order of

1/100

of the d-bandwidth per ^atom.

However,

_as stressed in table

I,

the

special

values of

N~

for which AE and

AEs~

cancel are not

exactly

the _same. Thus if the

stability

of _a surface

stacking

fault is to be

determined,

the results

given

in

figure

3a

(resp. 3b)

have _to be considered

only

in the

N~-domain

for which the FCC

(resp. HCP)

structure is stable.

For the FCC structuie,

only

the domain

N~~7.5

is relevant: for such values of

N~, AEs~

^is

positive. Herce,

a

stacking

fault

always

costs some energy on a

(I

I

I)

FCC surface.

For the HCP structu«. two domains _are to be considered _:

l~N~~3.5

and

6~N~~7.5.

In these domains there is

only

_a small AN

~ interval

(1

_~ N

~ ^~ l.35

)

in which

AEs~

is

negative, I-e-,

where such _a fault could be favoured on a

(0001)

HCP surface. If _{we note} that, ⁱⁿ the IIIB

column of the

periodic table,

Lanthanum

undergoes

a HCP

- FCC transition _at about 600 K, it _can be inferred that its d-band

filling N~ (La )

_may

belong

to the interval

AN~,

_even

though

2

AE /Vi

SF

(x loo)

I

o

+

-I

tL -2

0 2 4 N

~ 6 8 lo

a)

2

A E /Vi ^*

SF

(x 100)

0

+

-1

-2

0 b)

Fig.

electrons :

with the

« » correction (o) _and the « lf-consistent »

correction (+).

a) fault for _a (I _II ) FCC surface. b)

Stacking for _a (0001) HCP

nits

of

bandwidth(W~cc or ^~cp)of

he bulk. The ack%g

fault

is

energetically

when

AEs~ is negative.

this

nterval

may

good candidate for the appearance

of a

pontaneous

surface

stacking fault. This _same

possibility cannot be

excluded for ^ttrium and

Scandium

which

have

a band _filling

larger

than anthanum [21]. An experimental study of surface stacking aults on

these

elements would be

very

interesting.

In

figures

4a and

4b,

_we

present

^the contributions

AEs~,~

^of

layers

r

=1,

2 and 3 to

AEs~.

In both _cases

III lI)

FCC and

(0001) HCP)

the _sum of these three contributions is

nearly equal

to

AEs~. Moreover,

the contribution of

layer

_r

=

2, I-e-,

the subsurface

layer,

is rather different from the other _ones.

AEs~,

j and

AEs~,

~ are very similar _:

they

have about the

1-o

AE /Vi

SF,r

~~

~~~~

0.5

o-o

»

-1.0

0 d

a)

.o

A E /Vi

SF,r

(x 100)

_{o 5} ...

o-o

+

2 4

N~

^{6 8 10}

b)

Fig.

4. Contributions

AEs~_,

of the

layers

_r

= 1, 2 and 3 _to AEs~ versus the number of d-valence

electrons

(Nd). AEs~,,

^is given in units of d-bandwidth. (o):

AESF,ii

(+):

AEsF2, (A);

AEs~,_~. ^For AEs~,_~ no

guideline

is drawn _; it is very similar to

AEs~

j. a) Surface

stacking

fault _{on a} (II I) FCC surface.

b)

Surface

stacking

fault _{on a}

(0001)

HCP surface.

same values and show the _same

oscillatory

behaviour _as

AEs~. AEs~,

~ is very small in the

N~

interval

(4 ~N~ ~7)

and does _not cancel for the _same values of

N~

_as

AEs~,

j and

AEs~,

_~. ^{In the}

beginning

of the transition

series, AEs~,

~

changes sign

for a smaller value of

N~

than for

AEs~,

j and

AEs~,

_~ ^{the subsurface}

layer prefers

a local HCP

stacking.

At the end of the

series,

the

change

of

sign

of

AEs~,~

occurs at a smaller value of

N~

than for

AEs~,j

and

AEs~,~.

the subsurface

layer greatly

favours _a local FCC

stacking.

These comments may lead to the conclusion that _a

stacking

fault for two

layers

_{on a}

(COOT)

HCP would be

energetically

favourable for

N~ varying

between 6 and 7.5.

An

interesting comparison

can be made with

previous

calculations of twin and

stacking

fault

energies (r)

^{in the} FCC and HCP structures. In _a FCC _{or a} HCP structure several

stacking

faults _can be obtained

by modifying

one, two or three

stacking

_sequences

(the respective

faults _are denoted I

V,

2 V and 3

V).

These faults

easily

_appear under _a mechanical

or a thermal stress. Their

energies

have been evaluated with _a

tight-binding approximation [5].

The results _are shown in

figure

5a and 5b. The

similarity

of these _curves with _ours is

striking.

On the _one

hand,

the values of

N~

for which r cancels _are

nearly equal

to those for

AEs~ (Tab. I).

On the other

hand,

the values of rand

AEs~

are rather similar.

Thus,

it _seems that _a

stacking

fault costs about the _same energy in the bulk _{as on} the surface. It _must be

noted that surface _or bulk

stacking

fault

energies

for the HCP structure are greater ^{than those} for the FCC structure

by

about _a factor of 2.

However,

these

energies

_are

remarkably

small and _our

computation

_{assumes a}

temperature

^T ₌ 0 K. These faults could be stabilized

by

_a

slight

temperature increase.

Thus,

we conclude that _a

stacking

fault _{on a}

(I

it FCC surface _{or a}

(0001)

HCP surface is

never

energetically favoured,

_except

possibly

for the elements of the IIIB column. Orders of

magnitude

of

energies

_are the _{same as} in the bulk.

5. Surface

stacking

fault for _a metal M _{on a} substrate M'.

At the

present time,

the

evaporation

of _a metal M _{on a} metal M' is _an

exciting pole

of interest

as it _can lead _to the

building

of

superlattices,

which _are known to have

specific magnetic properties [22].

Here, _we tum to the

study

of the

crystalline

structure of such films. We have

especially

studied two _{cases :}

M' is _a metal which is FCC stable _:

N~(M')

₌ 9.05. M is _a metal which is HCP stable.

According

to the results shown in

figure I,

_we choose

N~(M) varying

between 6 and 7.5.

M' is _a metal which is HCP stable _:

N~(M')

=

7.00. M is _a metal which is FCC stable _: 7.5 _~

N~ (M)

_~ 8. 8.

The _two metals M' _were chosen in domains of

figure

^I where

(AE

is _{near a} maximum.

Thus,

the

sp-d hybridization

_may

only slightly

affect the numerical values without

changing

the trends

conceming

the relative

stability

of the

stacking

faults.

As in section

4,

_we

neglect

all normal relaxation effects. Moreover, we make the

approximation

that the Slater-Koster parameters ^{of the} two metals _are

equal.

This

approximation

leads to

simpler

calculations and is

justified

since the d-bandwidth varies rather

slowly

from _a transition metal to another. The Fermi energy

Ei

is fixed

by

the substrate M'. With these

approximations,

the different nature of metals M and M' is mimicked

by

the

perturbation

of the atomic

level, 8Vs,

of surface atoms which is

computed

to achieve a d-band

filling

of N

~(M)

at the surface. This

8Vs

is

approximately equal

to the difference between the

Fermi energy of bulk metals M and M' and is much

larger

than the values reached in

section 4. The surface

stacking

fault energy, ^at T

= 0

K,

is evaluated

by

the _same method _as

previously (Sect. 4).

For the FCC

bulk,

_a

stacking

fault

always

costs some energy

(Fig. 6).

For the HCP

bulk,

m

lf.c.c.I

I I

/

i

j i

j ^I

I '

X

~

,

~ '_

' i

I

~'

i

-2

0 5 N~ lo

a)

/~',

_lh.c.p.I

/ _',

"

' a

R

"~.,

/

,'

/

i / -3

i~/

0 5 N~ lo

b)

Fig.

5. a) Defect

energies

r _versus d-band

fillipg

in FCC structure (from [5]). The notations _are the

following

_: I v CABCBAC (--), indinsic 2 v _: CABCBCA ), extrinsic 2 v _: CABCBABC (---). b) Defect

energies

r _versus d-band

filling

in HCP structure (from [5]). The notations _are the

following

_: I v CABCBAC (--), 2 v _: CABCBCA ( ), 3 v _: CABCBABC (---).

two _cases must be considered

according

to the d-band

filling

of the metal M

(Fig. 6)

_:

7.5

~N~(M)

_~

8.3,

the

stacking

fault costs _some energy.

8.3 ~

N~(M)

_~

8.8,

the

stacking

fault is

energetically

favoured.

Hence, stacking

faults _are

likely

to appear for _some d-band

fillings.

2

A E*/lV (x loo)

i

o

5 6 7 8 9 0

N (M) d

Fig.

6.

Energy

AE* of _a surface

stacking

fault for _a surface

layer

different from the substrate. M denotes the surface

layer

^metal and is characterised

by

its number of d-valence electrons:

Nd(M).

AE* is

given

in units of d-bandwidth of the bulk and is

negative

when the

stacking

fault is favoured. (+)

Nd(M')

= 9.05 (bulk FCC), (o)

N~(M')

= 7.00 (bulk HCP).

Unfortunately

there _{are scarce}

experimental

studies _to which _our results _can be

compared.

Heteroepitaxy

is often realized _on BCC substrates _or with noble metals.

However,

_some

experiments

have been

performed

_on Ru-Ir

superlattices [23, 24]. They

have shown that thin Ir films

(s

10

ML)

_on the

(000I)

face of Ru have the HCP structure. For thicker Ir films

(a10 ML),

the FCC structure is observed. This

strongly suggests

that _an ideal

epitaxy

is realized at the interface.

According

to _our results this is

expected

for _a metal M with

N~(M)

_~ 8.3 _{on a} substrate M' with

N~(M')

= 7, a

band-filling

which is rather close _to that of Ru.

Thus,

the above mentioned

experimental

result is in agreement ^with our

theory

if _we assuHe

)hat thi number

of'd-electrons per ^atom in'Ir is §maller than 8.3. This latter

valui

seems an upper limit of the values

quoted

in the litterature

[21, 25, 26].

In

conclusion,

it would be very

interesting

to

perform systematic experimental

studies of the atomic structure of

overlayers

of _a metal M _{on a} metal

M',

M and M'

being

FCC _or HCP

transition metals.

6. Conclusion.

Using

a

tight-binding

method for

describing

the d-valence electrons in transition

metals,

_we have realised _a

systematic study

of surface

stacking

faults _versus the d-band

filling,

for the

FCC and the HCP structures. We have first checked that _our method _accounts for the

sequence of structural

phase

transitions

(HCP-BCC-HCP-FCC) along

a series of transition metals.

For the evaluation of surface

stacking

fault

energies,

the

charge rearrangement

near the surface _was taken into account,

by

_a

displacement

of atomic levels. We have shown that this

« self-consistent _» correction does not

seriously

affect the results. The surface

stacking

fault

energies

are small

(I/IOOC

of cohesive

energy)

and

comparable

to the bulk _ones. As _a

conclusion,

it _seems that such

stacking

faults may sometimes be favoured _{on a} HCP bulk. On the contrary, a FCC II

I)

surface is

always

in

epitaxy

with the

substrate,

at K.

Nevertheless,

energies

are

always

small _so that _a

slight

increase of

temperature

could induce the

stacking

fault.

We have also studied _{some cases} in which the surface

layer

is made of _a transition metal M different from the substrate M'. We have

found,

for

instance,

that _{on a} HCP substrate with

N~ (M')

=

7,

the

stacking

fault is

energetically

favoured when N

~(M)

~ 8.3. It would then be

interesting

to

study experimentally

the atomic structure of _an

overlayer

of _a transition metal of the end of the series _on HCP metals.

The

crystalline

structure of _an

overlayer

is

clearly

connected with the formation of the

overlayer

at the very

beginning.

It would

be, thus, instructive,

_{as a}

starting point,

to calculate the energy of

adsorption

of _a

single

atom on

(I II)

FCC and (COOT) ^HCP surfaces and the

corresponding

activation energy ^to

jump

from one site to another. Such _a

systematic study

is

currently

in progress.

References

[Ii PETTIFOR D. G., Calphad1 (1977) 305.

[2] KUNKEL R., POELSEMA B., VERHEUAND L. K., COMSA G.,

Phys.

Rev. Lett. 65

(1990)

733.

[3] WANG S. C., EHRLICH G., J. Chem.

Phys.

94 (1991) 4071.

[4] KELLOGG G. L.,

Phys.

Rev. Lett. 67 (1991) 216.

[5] PAPON A. M., SIMON J. P., GUYOT P., DESJONQUtRES M. C., Philos.

Mag.

39 (1979) 301.

[6] SLATER J. C., KOSTER G. F.,

Phys.

Rev. 94

(1954)

1498.

[7] HAYDOCK R., HEINE V., KELLY M. J., J.

Phys.

C 5 (1972) 2845.

[8] GASPARD J. P., CYROT-LACKMANN F., J.

Phys.

C6 (1973) 3077.

[9] DESJONQUtRES ^M. C., CYROT-LACKMANN F., J. Phys.. Met. Phys. 5 (1975) 1368.

[10] KELLY M. J., Solid State Phys. ³⁵ (1980) 296.

[I Ii ALLAN G., DESJONQUtRES M. C., SPANJAARD D., Solid State Commun. 50 (1984) 401.

[12] SKRIVER H. L.,

Phys.

Rev. B 31(1985) 1909.

[13] FERNANDO G. W., WATSON R. E., WEINERT M., WANG Y. J., DAVENPORT J. W.,

Phys.

Rev. B

41 (1990) l1813.

[14] SAUNDERS N., MIODOWNIK A. P., DINSDALE A. T.,

Calphad

12 (1988) 351.

[15] JACOBSEN K. W., Comments Cond. Mat.

Phys.

14

(1988)

129 _;

JACOBSEN K. W., NoRsKov J. K., PUSKA M. J.,

Phys.

Rev. B 35 (1987) 7423.

[16] HUME-ROTHERY W., SMALLMAN R. E., HAWORTH C. W., The Structure of Metals and

Alloys

(Institute of Metals, London, 1969).

[17] ALLAN G., Ann. Phys. 5 (1970) 169.

[18] ALLAN G., Handbook of surfaces and interfaces, L.

Dobrzynski,

Garland L. Ed. (STPM Press, New York London, 1978) _p. 299.

[19] DESJONQUtRES M. C., SPANJAARb D.,

Concepts

in Surface

Physics

(to be

published,

editor

Splinger Verlag).

[20] ALLAN G., LANNOO M., J.

Phys.

Chem. Solids 37

(1976)

699 PICK S., J.

Phys.

France 51 (1990) 2011.

[21] ANDERSEN O. K., JEPSEN O., GLnTzEL D.,

Highlights

of Condensed Matter

Theory (Proceedings

of the Intemational School of Physics _« Enrico Fernli _», 1983) 89 (1985) 59.

[22] MoscA D. H., BARTHELEMY A., PETROFF F., FERT A., SCHROEDER P. A., PRATT Jr. W. P.,

LALOEE R., CABANEL R., J. Magn. Magn. Mat. 94 (1991) 480

BARTHELEMY A., FERT A., BAIBICH M. N., HADJOUDJ S., PETROFF F., ETIENNE P., CABANEL R., LEQUIEN S., NGUYEN VAN DAU F., CREUzET G., J.

Appl. Phys.

67 (1990) 5908.

j23] CUNNINGHAM J. E., FLYNN C. P., J.

Phys..

Met.

Phys.

is (1985) L-221.

[24] CLARKE R., LAMELAS

I,

_{UHER C., FLYNN C.} P., CUNNINGHA3i J. E.,

Piys.

_Rev. B 34 (1986) 2022.

[25] BOURDIN J. P., DESJONQUtRES M. C., SPANJAARD D., FRIEDEL J., Suj Sci. 157 (1985) L-345.

[26] BARRETT N. T., GUILLOT C., VILLETTE B., TRtGLIA G., LEGRAND B.,

Suj

Sci. 251-252 (1991) 717.