B. DISORDERED METALLIC ALLOYSTHE OPTICAL PROPERTIES OF DISORDERED METALS

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B. DISORDERED METALLIC ALLOYSTHE

OPTICAL PROPERTIES OF DISORDERED METALS

H. Myers

To cite this version:

H. Myers. B. DISORDERED METALLIC ALLOYSTHE OPTICAL PROPERTIES OF DISORDERED METALS. Journal de Physique Colloques, 1974, 35 (C4), pp.C4-31-C4-49.

�10.1051/jphyscol:1974406�. �jpa-00215597�

THE OPTICAL PROPERTIES OF DISORDERED METALS

H. P. MYERS

Chalmers University of Technology, Gothenburg, Sweden

Rksumk.

- Nous donnons des resultats de mesures optiques et de photo6mission pour differents alliages tels que LiMg, CuGa, AgPd, AgAu. Ces donnkes nous permettent de mieux cornprendre la structure electronique de ces alliages. Dans le cas de mktaux purs trks desordonnes, les propriktes optiques presentent des anomalies qui peuvent btre attribuees

a

la large fraction du volume occupee par des joints de grains.

Abstract.

- Representative optical and photoemission data for different alloys characterised by the following specific systems LiMg, CuGa, AgPd, AgAu are presented and the significance of the data for our understanding the electronic structure of alloys demonstrated. Even pure metals may be obtained in highly disordered form and they then show anomalous optical proper- ties attributed to the large volume fraction occupied by grain boundaries.

1. Introduction.

-

Clearly we need to define from the beginning what we mean by the terms

⁽⁽

optical properties

⁾⁾

and

⁽⁽

disordered metals

^D.

1 .1 DISORDERED

^METAL.

-Let us start by posing the question what is a disordered metal

?

It is in fact easier to describe the opposite concept namely, the perfectly ordered metal corresponding to an ideally pure infinitely large single crystal of a metallic element at 0 K. Usually we can accept the limitations of finite size and even the polycrystalline form. Thus any ordinary polycrystalline specimen of a pure metal at 0 K is in practice a highly ordered sample. Further- more the effect of temperature in disturbing this order is on the whole rather limited and most pure metals represent ordered structures at RT. We can illustrate this by comparing the optical absorption spectrum of aluminium when taken at 20 K and RT (Fig. la, b).

Admittedly the spectrum at 20 K is sharper than that typical of RT but the difference is of limited significance and both spectra can be associated with a regular arrangement of aluminium atoms. We shall see later that the prominent absorption band at 1.5 eV arises on account of the Fermi surface cutting the 200 Bril- louin zone boundaries and is therefore a direct result of the periodicity in the 200 direction. The liquid metal of course exhibits no periodic structural order and therefore is an example of a disordered metal.

On this account there can be no zone boundaries and liquid aluminium has an optical spectrum devoid of any absorption band at 1.5 eV (Fig. lc).

In principle it is possible to imagine the formation of a completely disordered pure solid metal by the rapid

FIG. 1. -

a) Optical absorption of

A1

at

20 K.

b)

Optical absorption of

A1

at

RT.

c)

Optical absorption

of liquid Al

(after Miller

[22]).

supercooling and subsequent freezing of the liquid structure. We would then describe the sample as amorphous. An approximation to this condition can be achieved if a sample is prepared by evaporating the metal onto a very cold substrate. It is however difficult to arrange amorphous samples of pure metals such as aluminium in this manner but the addition of a small amount of a second element with higher melting point can often stabilise the amorphous state. Thus amor- phous solid solutions, e. g. AgCu, AlCu are readily formed. Since in the amorphous state the concept of the Brillouin zone becomes inapplicable we expect the conduction electrons to occupy a simple featureless energy band and the band gaps characterising the crystalline state to be absent. The situation is analo- gous to that of the liquid, however such amorphous metals are usually studied at low temperatures. Amor-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974406

C4-32 H. P. MYERS

phous metals are clearly disordered substances having no connection with any crystalline basis. It is possible, in a restricted sense, to envisage a continuous variation from the crystalline to the amorphous state. Earlier we said that a polycrystalline pure metal could be considered as an ordered structure, the implication being that the grain size was sufficiently large that the material within the grain boundaries represented a negligible proportion of the sample volume. But how wide is a grain boundary

?

Let us arbitrarily assume the width to be 10 A (roughly three atom diameters for say Al). For a sample with average grain diameter 100 then some 30 % of the sample is in the grain boundaries and it is clear that one can easily imagine smaller grain sizes and correspondingly larger pro- portions of the sample in the form of boundaries. If we avoid the particular cases of low angle and twin boundaries then we expect an average grain boundary to have a structure very different from that of a crystallite. In particular there should be little or no periodicity in the plane of the boundary and most probably a lower packing density for the atoms. Quite simply the grain boundary represents another phase of the pure metal albeit a poorly defined one. When pure metals are condensed onto cooled substrates the grain size becomes smaller the lower the temperature of the substrate. Eventually the grain size can become so small that the sample, with regard to X-ray diffrac- tion, is indistinguishable from an amorphous structure.

Before this limit is reached the metal may be considered as a two phase mixture of crystallites and grain boundaries the latter occurring as thin plates and on this account having a very significant effect on the optical properties of the metal as we shall see later.

When we turn to alloys then it is possible to have disordered systems even though the individual atoms may be located on a well defined lattice. Thus a solid solution of one metal in another represents a disor- dered system because the perfect translational periodi- city of the solvent is broken by the addition of the solute atoms. At certain simple proportions of the components superlattices may arise and, when well developed, we again obtain an ordered system, one which might be considered the simplest alloy from the theoretical point of view. The equilibrium crystalline form of an alloy is that which has received most attention but from what has been said earlier it is always possible to study metastable amorphous alloys too (see for example the contributions of Granquist and Claeson and that of Nilsson and Forsell to this conference). Our attention will be directed primarily to the crystalline alloys and for the present purpose we can divide them into three groups

:

I) Pseudoatom alloys examples KRb

Mg Cd LiMg PbTl.

11) Alloys based on the noble metals

a) noble metals as solvents for polyvalent metals

examples CuZn CuGa AgIn

b) noble metals as solvents for transition metals

examples CuNi AgPd Ag Mn

c)

mutual alloys examples CuAu

AgAu.

111) Other alloys

a) intermetallic compounds examples P brass CuZn

Li Ag

b) alloys between transition metals examples CrV

CrMn.

Without any doubt most attention both experi- mental and theoretical has so far been directed to alloys categorised as group I1 above. This is due to several factors among which we may mention

:

a) the ease with which measurements can be made on the pure elements Cu, Ag, Au as well as their alloys,

b) the strong spectral features which can be asso- ciated partly with the conduction electrons partly with transitions from d states,

c) the good solubilities of Cu, Ag, Au for many other metals,

d)

the fact that the d states are involved.

There has long been a strong interest in the descrip- tion of the d states in metals and alloys. The problem is still somewhat controversial (see for example Friedel [I]). In recent years calculations of d band structure have become more detailed and more accu- rate with regard to structure and breadth of the d bands but it is still difficult to forecast their exact position in energy. Likewise the question of what happens to the d bands on alloying is one with a long history and for which the solution is becoming clearer as a result of both experimental and theoretical work.

Thus alloys of group I1 above tell us primarily about certain aspects of d state behaviour

;

the latter are so dominant that the behaviour of the conduction electrons (at least with regard to interband transitions) is often more difficult to assess.

On the other hand the properties of the alkali and

polyvalent metals which form the alloys categorised

as group I are determined solely by the conduction

electrons. The pure metals have been much studied

with regard to transport and Fermi surface properties.

They are well adapted to the pseudopotential approach particularly so in the case of the lighter elements where spin orbit splitting is negligible. The optical properties of such metals have been actively studied in recent years and the polyvalent metals show strong spectral structure as a direct result of the Fermi surface cutting the zone boundaries

;

we shall see that this effect allows one to measure energy gaps with good accuracy. Little, however, has been done concern- ing the optical properties of

⁽⁽

pseudoatom

⁾⁾

alloys.

Alloys of the alkali metals (e. g. KRb, KCs, CsRb) would be good objects for study but they present severe difficulties not so much on account of their chemical reactivity as for their very low absorptivities which makes it difficult to obtain high accuracy.

Alloy studies are however hampered by the paucity of binary pseudo atom alloys with extensive ranges of solid solubility (witness the very poor solubility of aluminium for other elements). As yet only one alloy system has been studied in any detail namely LiMg

;

it will be of value in illustrating the usefulness of work on pseudoatom alloys and as an indication of how compositional disorder affects conduction band structure.

Our group I11 above contains alloys between tran- sition metals (including the rare earths and actinides).

Very little optical work has been done on such alloys on account of experimental difficulties and problems of interpretation. The influence of alloying on the anti- ferromagnetic spin wave superstructure in chromium has received some attention (Lynch et al. [2, 31). We shall not consider such alloys here and centre our interest on the alloys of groups I and I1 together with the disordered pure metals. First we need to clarify what we mean by the optical properties of metals.

1 . 2 OPTICAL PROPERTIES.

-

Here we mean the response of the metal to incident monochromatic light as determined by the reflectivity, transmission or say ellipsometry

;

furthermore we include the ana- lysis of the electrons photoemitted from the metal by ultra violet light (hw 5 40 eV).

The optical behaviour of the metal is governed by two real parameters, the frequency dependent conduc- tivity a and the frequency dependent dielectric constant

^E.

The electric field vector of the light produces an electron current within the metal which via Joule heating causes a power absorption,

o

is a direct measure of the power absorbed per unit volume and unit field strength. The optical absorption as measured by the conductivity is expressed in s-I and in the visible region of the spectrum lies in the interval 1014-1016

^{S - l .}

(The static conductivity of copper corresponds to -- 1017s-I.) Sometimes one chooses to describe a metal as a lossy dielectric and defines a complex dielectric constant,

A ^{E = E,}

+ i~,,

^{8 ,}

being the conventional real dielectric constant and

E,

a parame- ter related to a (in fact

a =

we2/4 n), it is then conven-

tional to describe the optical absorption by &,/A and the unit becomes pm-l. Some people prefer to des- cribe the optical properties in terms of a complex refractive index n + ik

;

we then have

=

(n + ^ik)'

and el

=

n2 - k2,

^E~ =

2 nk

;

the optical absorption then becomes 2 nk/A. In the figures presented later all three ways of expressing the absorption are used.

The current induced in the metal by the incident light may be carried wholly by electrons within (and remaining within) the conduction band. This gives rise to what is known as the intraband absorption.

This, in its simplest form, approximates the behaviour of a free electron gas, which is characterised by the Drude equations

The symbols have their usual meanings

^z

being a relaxation time, no the volume density of electrons, mo an optical effective electron mass and o, the plasma frequency wi

=

4 nn, e2/mo.

On the other hand the incident photons may have sufficient energy to excite electrons from filled bands below the Fermi level to empty states in higher lying bands. We then speak of an interband absorption and associate a specific component of the conductivity, a I B with this process. In pure metals such transitions demand

k

conservation and within the reduced zone scheme correspond to vertical transitions.

The interband conductivity may be expressed

We have excluded the Fermi factors it being assumed that the initial state I i > is occupied and the final state < f 1 originally empty. If the matrix ele- ments may be considered constant then (3) may be more simply written

Any evaluation of eq. (4) requires knowledge of the

matrix elements M and the joint density of states

(JDS) evaluated throughout the whole zone. A priori

calculations even for simple pure metals such as the

alkalis have yielded rather poor agreement with expe-

riment and most discussion of the interband optical

spectra of metals is qualitative. The earlier association

of spectral features with critical points at symmetry

positions in the zone appears now to have been

overemphasized and spectral detail may well arise due

to the juxtaposition of several transitions arising at

many isolated points distributed throughout the zone

(Christensen [4]). For the present purpose the poor

C4-34 H. P. MYERS

quantitative agreement between experiment and

theory will not inhibit useful qualitative discussion.

Nor will the disorder caused by alloying create diffi- culties by the relaxation of the condition Ak

=

0.

This is because much of our attention is directed to the uppermost filled d levels and these are already strongly localised in real space

;

they therefore already have extensive spread in k and very little dispersion.

Clearly if we can separate the intra and interband components of the optical behaviour then the former will provide us with values of n,/m, and

z ;

when the volume density of electrons is known this also gives a direct measure of the optical mass which is directly related to the average band mass for the electrons.

This is often possible for the noble metals and their alloys since interband transitions start at sharp and relatively high energy thresholds. For the polyvalent metals (e. g. Al, Mg, etc.) this is not the case and interband transitions persist to very low energies well into the infra red region.

Light incident on the metal penetrates a distance 100-500 A dependent upon the wavelength

;

electrons excited by the absorption of a photon can exist to this depth under the surface where they form a current of electrons occupying a range of final energy states Ef.

The current of excited electrons can travel to the sur- face and if the final energy state lies above the vacuum level then electrons may escape and give rise to an external photocurrent. This current can be collected and analysed. In practice many electrons are strongly scattered before reaching the surface, furthermore only those with velocities in excess of a critical value can escape. Thus the emerging photocurrent is strongly modified and it is thought that only those electrons excited within two or three atom layers of the surface contribute significantly to the external current. It would appear unlikely that they could convey infor- mation relating to the bulk properties. Experimentally the evidence is very convincing that the photocurrent reflects the bulk properties. Neglecting all these drawbacks and difficulties we expect the photo- current to provide a measure of the distribution of electrons in the final excited state and this is propor- tional to the quantity

A proper interpretation of photoemission experi- ments requires data obtained with a wide range of incident photon energies and access to a reasonably accurate and complete band structure diagram. Just as in the case of the ordinary optical transition, the wave vector is conserved but the inclusion of the term 6(E

-

Ef) in eq. (5) offers, in principle at least, a certain measure of simplification since the photo current at a particular energy provides information on the probability of excitation to a specific final state Ef whereas the optical conductivity gives a

measure of the transitions to all final states subject only to the condition

tzo=

Eif.

In studying transitions from initial states that are strongly localised in real space or in the case of disordered metals the condition Ak

=

0 becomes less relevant and for the alloy systems so far studied (as opposed to the case of the pure metal) it is often assumed that the energy distribution of the photo- electrons is proportional to the product of the initial and final energy state densities. Our experience is that on the assumption of constant or slowly varying matrix elements the energy distribution reflects the density of initial states. Under these conditions of limi- ted generality but particular to disordered systems and narrow initial energy bands we expect a photoemission experiment to give a measure of the work function, the position of the Fermi level, the top of the d band and possibly the position of any localised level or conduction band transition not masked by the d states of the matrix. If the incident energy is suffi- ciently large then we can also determine d band breadths Eastman

[ 5 ] .

This can also be done via X-ray photoemission spectroscopy but the latter has an inherently poorer resolution than the UV technique.

As yet, however, d band breadths in alloy systems have not been measured using UV photoemission but such measurements would be worth while particu- larly in connexion with the deeper lying d levels associated with Zn, In and similar polyvalent post transition metals. An important advantage of photo- emission spectroscopy is that it provides information about the states both at and below the Fermi level whereas

in

ordinary optical experiments the low energy conductivity is dominated by the intraband terms and any low energy transitions are usually hidden in the large Drude background. A new technique known as appearance potential spectroscopy allows the study of the empty levels at and above the Fermi level. As yet it has had limited application although data for CuNi alloys are available 161.

We shaII not consider this development here.

Anyone wishing to penetrate deeper into the subject of optical properties may consult books by Hodg- son 171, Wooten [8] and the articles by Abelhs, Harbeke and Spicer in the recent omnibus presentation of the optical properties of solids, Abelb

[9].

In what follows we shall discuss particular experi- ments and present the experimental data demonstrat- ing the usefulness of optical studies in establishing the electronic structure of disordered metals. This review has therefore a limited objective and it is not meant in any way to be a comprehensive survey of the literature and a complete bibliography is not provided.

In addition to presenting representative optical data we shall attempt a qualitative interpretation and this requires that we have some idea of what happens to the electron states when two metalsare alloyed together.

This is a complicated theoretical problem but we

shall adopt a primitive but useful approach in which

it is assumed that the constituents of an alloy remain electrostatically neutral, i. e. they are perfectly screened. This will lead us to the idea of local conduc- tion electron densities and local band breadths. It will allow a coarse understanding of alloy properties and neglects completely the possibility of charge transfer between alloy components. (But it is also true that optical data do not give any direct informa- tion concerning charge transfer.) However, to go beyond our simple approach demands a degree of theoretical sophistication beyond that of the experi- mentalist. Needless to say our appreciation of the electron structure of metals and alloys remains completely within the one electron approximation.

2. Pseudo-atom alloys. LiMg. - The simple metals i. e. the alkalis and polyvalent metals such as Al, Mg, etc. have very nearly spherical Fermi surfaces.

The band gaps at zone boundaries are small compared to the Fermi energy and the distortion of the Fermi sphere is confined to the intersection with the zone boundaries. Consider a specific direction of transla- tional symmetry [hkl]. In the reciprocal lattice we expect the energy band to be approximately parabolic and appear as in figure 2. The energy contours are

FIG. 2. - The origin of the characteristic parallel band absorption in ct nearly free electron like ^D polyvalent metals.

The parallel energy bands separated by the band gap energy lie in the plane of the zone face. Different possible Fermi levels are shown. Only when the Fermi level cuts the band gap does

the parallel band absorption arise.

spherical except where they contact a zone boundary and where cylindrical necks form. Thus on the zone boundary contours of constant energy are circles and there will be two sets, one for the first zone and one for the second

;

these two sets will be separated by the energy gap characteristic of the particular zone face i. e. 2 U(G,,,) where U(G,,,) is the pseudopotential appropriate to wave vector G,,,. This is most readily seen by plotting E as a function not only of

k

parallel to G but in the direction perpendicular to G. The figure 2 shows that in the plane of the zone boundary the 1st and 2nd bands are parallel parabolas with a

constant energy separation equal to the band gap, Harrison [lo], Ashcroft and Sturm [Ill. If the Fermi surface contacts the zone boundary then this means that a strong interband absorption can arise as is evident from figure 2. The absorption appears as a pronounced asymmetric peak (Fig. la), were it not for the presence of broadening mechanisms it would be an infinite edge. Zone planes characterised by different G values have different energy gaps

;

thus for every zone plane cut by the Fermi surface there will be such an absorption band. For A1 we expect bands due to the G(200) and G(110) but normally only the former is seen since the latter lies at energies below the usual range of measurement (it has however been observed Bos and Lynch 1121). In the case of hexagonal Mg there are three different classes of zone plane (0002), (1100) and (1101) but again only optical absorption due to the (1i01) plane is normally observed at approximately 0.7 eV, figure 3. Now Mg is soluble in

- ^15-

-

I mm

-0 w Y

b l o -

5

-

FIG. 3. - The optical absorption for pure Li and pure Mg at RT [13]. The sharp peak for Mg at 0.7 eV is due to parallel

band transitions in the 3101 zone faces.

Li to about 70 %. The alloys are in the b. c. c. form and the zone structure is particularly simple being a dodecahedron bounded only by (110) planes. If we calculate the vector G(110) for an assumed b. c. c.

Mg (the atomic volume of Mg being assumed inva- riant and given by data for the h. c. p. form) then we find that G(110)

=

G(lTO1) implying that a b. c. c.

Mg would in fact have an optical spectrum identical

H. P. MYERS

FIG. 4.

-

The optical absorption of LiMg alloys at RT (after Mathewson and Myers [13]).

to that measured for h. c. p. Mg. This is a happy accident. The spectrum of pure Li is typical of an alkali metal, the absorptivity is low and there is a shallow edge but no indication of parallel band absorption associated with zone boundary contact.

Note the very large difference in level of absorption between Mg and Li due to the intense parallel band absorption of the former. On alloying Li and Mg the parallel band absorption does not start until ca 30 % Mg has been added to Li (little or no change in lattice parameter occurs in this alloy system). For

50

% Mg the parallel band absorption is well deve- loped, figure 4. Note that although the bands are broader than in pure Mg they are still very well defined and one can readily obtain values of the energy gap on the (1 10) zone faces from these data [13].

The very presence of the parallel band absorption indicates the existence of well defined zone bounda- ries. Thus potential scattering in this pseudo-atom alloy is insufficient to destroy the zone structure. From their optical data Mathewson and Myers [13] deduced the variation of band gap throughout the b. c. c. LiMg system. This is shown in figure 5, together with the virtual crystal approximation. We see that the virtual crystal approximation is neither a very good nor very bad description of the measured band gaps.

A continuation of this kind of study using atoms with a larger difference in valence would be worth while. Similarly in principle one might expect quite

FIG. 5.

-

Derived values of the (1 10) band gap in LiMg alloys compared with the predictions of the virtuai crystal mode1 as shown by the dotted line (after Mathewson and Myers [13]).

large effects due to order-disorder transition (e. g. the

MgCd system). However as we said earlier the number

of metallurgically suitable systems is limited and not

all components have such large differences in pseudo-

potential as do Li and Mg. Nevertheless our conclusion

is that potential scattering of conduction electrons is

not likely to destroy the band structure and we can

maintain the concept of Brillouin zone even in a disor-

dered alloy. Furthermore the data give us no reason to

suppose that the condition

Ak = 0

is relaxed in this

particular system.

3. Alloys based on the coinage metals.

-

3.1 THE

NOBLE METALS AS SOLVENTS FOR METALS OF HIGHER VALENCE

Al, Zn, In, ETC. -In these alloys we neglect the presence of d electrons on the solute atoms

;

either there is none as for A1 or they lie deep below the Fermi level and out of action. Just what happens to them is a matter for later discussion. Since there is no d state density at the Fermi level there can be no s-d scattering and we have a situation similar to the case of LiMg, where potential scattering dominates

;

but it should be remembered that this scattering increases rapidly with difference in valency between the solvent and solute atoms. The added possibility of transitions from the filled d states to the conduction states produces a more complicated spectrum than is the case for the simpler pseudoatom metals.

The pure metals Cu, Ag and Au have attracted much attention, Spicer [14], Nilsson, Norris and WalldCn [I

51,

Thkye [16]. Copper has been a favourite subject from both experimental and theoretical viewpoints but the recent work of Christensen [17] has also been important in providing a basis for interpret- ing experimental work on Ag and Au. We shall of necessity confine our discussion primarily to Cu and certain alloys, referring particularly to the work of Pells and Shiga [18] and Pells and Montgomery 1191.

Band structure calculations for the noble metals show that whereas the breadth of the band can be calculated accurately the position of d band relative to the conduction bands is very sensitive to the assumed potential. This implies that we might expect the posi- tion of the d states to be very sensitive to added solute atoms perturbing the matrix but paradoxically this appears not to be the case and we must conclude that on account of their strong spatial localisation they do not feel the presence of a polyvalent solute atom (which furthermore is assumed to be very strongly screened). However the position of the L, level which has s like symmetrv is found to be much more sensitive to changes in the potential than the L, levels which have p type symmetry. These facts are of importance in appreciating the optical spectra of the pure noble metals and alloys based on them.

Pells and Shiga

[18]

studied the optical spectrum of pure copper as a function of temperature. Comparing their data with theoretical predictions [20, 211 showed that the sharp edge at 2 eV, which is due to transitions from the top of the d band to the Fermi level (L,-E,), was insensitive to changes in temperature apart from a slight broadening due to smearing of the Fermi surface (Fig. 6 and 7). Miller [22] has also demonstrated the persistence of the d band edge into the liquid state.

Thus we conclude that this transition is insensitive to volume changes. On the other hand the sensitivity of conduction band-conduction band transitions to temperature are clearly demonstrated in their data, figure 6. The principal conduction band transitions are L,,-Lx4.8 eV) and X,-X,, (4.0 eV) but the latter is expected to be weak (in contradiction to an earlier

FIG. 6. - Optical absorption of pure Cu at different tempe- ratures (after Pells and Shiga [18]).

FIG. 7. - Band structure of Cu according to Segall with probable optical transitions [20].

analysis of alloy data by Lettington [23]). In addition there is a transition from the bottom of the d band to the Fermi level L:-E, at 5.2 eV. At KT the L,,-L; and L:-E, are lumped into one broad peak around 5 eV but the expansion of the lattice caused by an increase in temperature moves the L,,-L>o lower energies and in so doing smothers the weaker X,-X,, peak which is best resolved at the lowest temperatures.

Note that all the above transitions are direct i. e.

Ak =

0. Thus the copper spectrum contains a lot of

C4-38 H. P. MYERS

information but without access to a good band structure calculation and an idea of what to expect it would be well nigh impossible to benefit from the experimental data. We conclude that the L, levels are insensitive to volume changes and that the conduction band gap decreases with increase in temperature partly as a result of volume increases partly due to Debye Waller factor in typical pseudopotential fashion.

The most recent and possibly most detailed study of 7

copper based alloys of the kind under discussion is

a

that of Pells and Montgomery [19] who investigated 2

'5

CuZn, CuGa, CuGe and CuAs alloys. We reproduce 2

their data for CuGa and CuAs in figures 8 and 9.

^N

The CuGa (as do the CuZn and CuGe) alloys show, in addition to a broadened and somewhat shifted d band edge, two peaks near 5 eV which separate progressively with increase in the solute concentration. The origin of the two peaks is associated with the separation of the L,.-L; and L:-E, transitions due to the pertur- bation of the L, levels by the solute. All these solutes cause an expansion of the Cu matrix.

hv (eV1

The principal edge at 2 eV, in contrast to the effect

of temperature, shifts to higher energies and at the

^{FIG. 9.}

-

Optical absorption in CuAs alloys (after Pells and

same time broadens somewhat. Part of this shift must

Montgomery [191). 0.2 % :

A,

4 % :

+,

6.5

%.

be attributed to a narrowing of the matrix d band due - to the reduced concentration of copper and the attendant reduction in interatomic d-d broadening.

Surprisingly the shift of this edge is much more pronounced for additions of Zn than for the other solutes Ga, Ge and As. On the other hand it is difficult to assess these shifts in the edge since there is significant tailing and in fact very little movement of the foot of the edge. A shift of the edge could arise from a change in the position of the top.of the Cu d band, a change

FIG. 8. - Optical absorption in CuGa alloys (after Pells and Montgomery [19]). 0.6 % :

A,

8 % :

+,

^{12 % : U,16}

^%.

in the Fermi level or both. The d band cuts the conduction band in two and one might expect any narrowing of the d band t o be accompanied by a movement of the L, level to lower energy. However such a trend will be offset by a reduction of the conduction band gap L2,-L>y the addition of solute with a weaker pseudopotential than Cu. Heine 1241, in contrast to Mueller

[25],

attributes the L2,-L; gap largely to s-p splitting in a conventional pseudo- potential (1

⁼ 0,l)

fashion. The contribution from any 1

=

2 potential being compensated by orthogo- nality terms. Now the solutes Zn, Ga, Ge, etc. have weak pseudopotentials compared with Cu and we can argue (just as for Mg in Li), that their addition will serve to decrease the L2*-L",ap. The L2, level is therefore affected in opposite manners by the process of d band narrowing and a reduction of the conduction band gap. We therefore suppose that the L,, level is essentially unaffected by alloying

-

as is the centre of gravity of the d states on account of their strong localisation to the parent copper atoms.

The primary effect of alloying will therefore concern the L, levels and in particular we expect L>o move to lower energy and this is in agreement with the optical data.

How about the change in Fermi level and conduc-

tion band breadth

?

In this connexion it is important

to recognise that alloys are very inhomogeneous

substances. We shall as a naive but practical approxi-

mation consider the solvent and solute atoms to

exist in the alloy as neutral entities, thus each unit cell

whether it contain solvent or solute is considered

electrostatically neutral and the solute ion is compen-

sated by the presence of all its valency electrons. Thus the distribution of conduction electron charge in the alloy is very uneven. For a small addition of solute there can be no change in the Fermi level from that of the pure matrix, Friedel [26]. We assume insufficient solute concentration to affect the matrix at large and therefore the Fermi level and conduction band breadth remain unchanged. What then happens to the solute valency electrons

?

There are two possibilities. Either the extra valency electrons provided by the solute enter a bound state below the bottom of the matrix conduction band or they are squashed into the lower regions of this band. We attempt to illustrate this situation in figure 10. As yet there is no experimental evidence for the occurrence of the bound state and we shall therefore assume that when small amounts of solute are added the extra electrons entering the conduction electron gas are squeezed into a conduction band of unchanged breadth, Ingelsfield [27]. Thus any

change in the d band edge will be associated wholly with d band narrowing which will be very small for small solute concentrations, as is observed experimen- tally.

As more solute is added the above situation must prevail until the interaction between solute atoms allows itinerant states to form below the bottom of the matrix conduction band. The conduction band will then deepen but, on the assumption of perfect screening, only about the solute atoms. Thus we consider there to be local breadths to the conduction band dependent upon whether we are considering the solvent or the solute atoms. (See also Heine and Weaire 1281.) To a good first approximation the Fermi level measured with respect to the bottom of the conduction band appropriate to the matrix copper atoms will not change. Thus even at high concentra- tions of solute d band narrowing will be the major cause of the shift in the absorption edge at 2 eV.

E f E f

C u cell Cu cell

E f

G a cell G a cell

C U Ga

C) FIG. 10. - Stylised presentation of:

a) Possible energy levels for the solute valency electrons in very dilute CuGa alloys ; the choice lies between the formation of a bound state or their accommodation in a local conduction

CU cell

band of breadth determined by the matrix. Effects of any broadening process neglected.

b) Densities of states in dilute CuGa and more concentrated CuGa alloys. We attempt to demonstrate the average density of state per atom of alloy in the uppermost sketch and the density

I

of state per atom of Cu or Ga in the lower figures. As more Ga is

added to the Cu the local conduction band associated with the

1

Ga atoms will deepen. Neutral atoms are assumed. If the Ga

1 Ga cell

potential is felt by the Cu atoms then the Cu conduction band

I breadth will also increase and charge transfer will occur but this

p-

aspect is neglected.

c) Even in the absence of charge transfer interaction of the solvent and solute conduction bands will produce G tailing))

of the former in more concentrated alloys.

MYERS

This concept of local conduction electron densities and band breadths is supported by the soft X-ray emission spectra of concentrated alloys, but see figure 10c.

In practice the screening of the solute atom will not be perfect and this will lead to charge transfer between solute and solvent atoms with resultant shifts in the energy levels. However optical data do not give us any direct measure of such charge transfer and it is a very difficult problem to go from the naive qualitative appreciation described above to a more detailed quantitative microscopic description as a later discus- sion of AgAu alloys demonstrates. We again emphasize that conduction band features typical of the pure metal are recognisable in concentrated alloys and we again conclude that the Brillouin zone and Fermi surface maintain their essential significance in these alloys.

Turning now to the CuAs alloys we notice that the low energy peak is not so clearly marked as in the CuGa data (the same is also true of CuGe alloys).

Pells and Montgomery [19] suggest that this is due to the occurrence of indirect transitions across the necks at L. These are made possible by the increased scatter- ing provided by the quadri and pentavalent solutes.

They suggest that the indirect transitions arise when the scattering is greater than a critical value (as mea- sured by a critical resistivity). The idea of a critical scattering for indirect transitions has also been invoked recently by DeReggi and Rea [29]. However there is no direct evidence for specific indirect transitions in the optical data so we shall not pursue this aspect further.

We conclude however that in these alloys the condition

Ak =

0 is relaxed.

Experiments on alloys based on Ag or Au are less extensive although the effect of temperature on the optical spectrum has been determined for both metals

;

Pells and Shiga [I81 Au

;

Liljenvall and Mathewson [30] Ag

;

Winsemius [31] Au and Ag.

Our comments will be brief. Just as for Cu the work on Ag shows the sensitivity of the L,,-Ly transition to temperature, the transition moving to lower energies as the lattice expands. Alloy data is limited but the addition of In to Ag, Morris and Lynch [32], Nilsson

[33]

has an analogous effect and although data for Ag based alloys are few there appears to be a close similarity with the Cu based alloys. As yet there has been no detailed study of the intra band properties of these alloys.

So far little has been said about the results of photo- emission experiments on these systems. Although the pure noble metals have attracted much attention only two alloys namely Ag,,Inl, and Cu9,Ge, have been studied by this technique Nilsson [33], whose results are in general agreement with conclusions drawn on the basis of direct optical studies.

On the other hand photoemission studies on the kind of alloy discussed in this section would be particu- larly rewarding if use were made of the higher photon

energies ho - 20-40 eV, since this, in say AgIn, would allow one to expose both the solvent and solute d states. To demonstrate what we mean by this let us return to the question raised at the beginning of this section namely, if the solute atom contains d states what happens to them in the alloy

?

Let us consider AgIn. In pure Ag the 4d states form a band some 3.5 eV wide and overlapping the lower half of the conduction band which has a breadth - ^{8 eV.}

In pure In the 4d states lie some 16 eV below the Fermi level. Thus when we add In to Ag the d levels of the solute lie far below those of the solvent (even after allowing for any renormalisation) and together they must form two independent systems of d states. As we have earlier argued, for a small addition of solute, the solute valency electrons must be crammed into the matrix band breadth and on this basis it is unlikely that there is any conduction electron level near those of the indium d electrons and since there are few or no indium nearest neighbours this implies that these is no major source of broadening for the indium d states which must therefore arise as atomic or quasi atomic levels and should therefore be observable as such in a photoemission experiment. Further addition of solute should cause a broadening, first through intersolute d-d overlap and later, as a local conduction band develops around the indium atoms, additional broadening through s-d resonance. Photoemission experiments should allow one to follow this process of solute

d

band formation and at the same time enable a study of the solvent d band.

That the d electrons do form independent or what are sometimes called split bands may be seen directly from the photoemission data for P brass Lindau and

I

-12 -10 -8 -6 -4 -2 0

Electron energy ( e V

FIG. 11. - Photoemission spectra for Cu, Zn and ,9 brass (after Lindau and Nilsson [34]). The energy of the initial energy state is shown in the abscissa the Fermi energy being taken as zero. This procedure is followed in all subsequent illustrations of

photoelectron spectra.

Nilsson [34]. Their data depicted in figure 11 clearly show the presence of two distinct and separate bands to be associated with the Cu and Zn atoms respectively.

The deeper lying Zn band in fi brass has both breadth and position little different from the d band of pure Zn whereas the d band associated with the Cu atoms has, in the alloy, a somewhat lower position in energy than in pure Cu. Such data is important in assessing the worth of band structure calculations for ordered intermetallic compounds particularly in connexion with establishing the position of the d states relative to the conduction states. A detailed discussion of theoretical and earlier optical work on fi brass can be found in the article by Lindau and Nilsson. Other intermetallic compounds that have been studied opti- cally are NiAl [35] CsAu [36, 371 RbAu [37] LiAg [37].

The data are important in establishing the positions of the d bands (which are often found to be in poor agreement with theoretical predictions) and the signifi- cance of spin orbit coupling in Au.

3.2 THE

NOBLE METALS AS SOLVENTS FOR THE TRAN- SITION METALS

e. g. CuNi., CuMn, AgPd.

^-

Here we shall be occupied with the problem of what happens to the d states on alloying a noble metal with a tran- sition metal. It is preferable to begin with a discussion of a system such as AgPd or CuNi

;

we choose the former on grounds of~mplicity.~rior to any detailed optical study of this kind of alloy there was a good theoretical background covering the concept of the resonant bound d state, Friedel [38], Andersson [39]

in connexion with a transition metal dissolved in a noble metal. Furthermore there was much experimental data from transport and magnetic studies to support the concept. We imagine a discrete atomic degenerate d level overlapped by a continuum of itinerant levels.

There is overlap both in real space and energy space.

The d level interacts with the continuum via a mixing potential Vdk and broadens into a resonant bound state of Lorentzian form with a half width A pro- portional to both Vdk [2] and the local density of conti- nuum states. If the position of the centre of the state E, with the Fermi level is known together with A then one can calculate the occupancy of the resonant state. The concept was originally conceived for the single impurity atom but optical measurements show that it has signifiance at practical alloy concentrations in the range 0-20 %. The optical properties of the resonant bound state have been studied theoretically by Caroli [40] and Kjollerstrom [41] and optical data may be analysed to provide the parameters E,, A and Vdk, Kjollerstrom 1411. Optical absorption due to the presence of a resonant d level was first seen in AuNi Abelbs [42], and AuPd Abelbs and Thbye [43,-] figure 12. Although the absorption due to the Pd d levels overlaps the Au d band edge it is neverthe- less clearly observable, a similar situation arises in CuPd [45, 461 but in CuNi [47] and more particularly

-

AgPd

-

145, 481 the resonant bound state absorption

FIG.

12.

- The optical absorption of a AuPd alloy (after Abelks and Th6ye [43,

441).

The additionalabsorption due to transitions from the localised Pd d electrons is clearly seen.

stands proud of the matrix absorption. The optical properties of a wide range of AgPd alloys first demons- trated the persistence of the localised levels to high concentrations Myers, Wallden and Karlsson 1451 however we choose to describe here the results obtained via photoemission studies of AgPd alloys Norris and Myers [48]. Figure 13 shows a series of photoelectron energy distribution curves for AgPd alloys. The low lying d states of the Ag do not mask the presence of the resonantly broadened Pd d states. Recalling our earlier discussion of fi brass we see that for small amounts of Pd dissolved in Ag we have a similar situation of two almost independent systems of solvent (Ag) and solute (Pd) d levels. However in fi brass both Cu and Zn each form a continuous lattice and the d band breadths arise from both interatomic and intra-atomic interactions. In our discussion of what might be found in say AgIn alloys we suggested that the In d states might occur as atomically sharp levels.

This is not so for Pd in Ag since the Pd d levels lie in the midst of a local Pd conduction band.

The form of the resonance at infinite dilution may be inferred. Since Pd is a heavy metal Norris and Myers [48] assumed that spin orbit coupling contributed some 0.5 eV to the total breadth of the resonant level, they then found that

and 2 A

=

0.5 eV. In a similar study of CuNi alloys

Seib and Spicer [47] found the nickel regnant state

to lie a-t E,

= 1

eV and 2 A

=

0.8 eV spin orbit

splitting being negligible in this case. For concentra-

tions up to at least 30 % Pd the shape of the resonant

level remains essentially unchanged as does its position,

on the other hand the level broadens due to the increa-

sing number of nearest Pd neighbours. We can

consider this broadening as arising from the inter-

ference of the localised resonances at least in the

initial stages of the process. The photoemission data

for these CuNi and AgPd systems are significant in the

following-ways

: ^-

H. P. MYERS

I 11 - 1 .i .- _ I - -

I

-5 -4 -3 -2 -I

0

-6 -5

-4

-3 -2

-1 0

Energy of initial state (eV1 Energy of init~ll state CeV)

FIG. 13. - Photoemission spectra for AgPd alloys (after Norris and Myers

[48]).

For small Pd contents the solvent and solute d bands are clearly separate and we have a situation approaching the condition of split bands. For greater Pd contents the bands merge into a single combined band but with definite Ag and Pd components. The increase in the density of state at the

Fermi level for Pd contents >

0.4 is

clearly indicated.

1) From a calculation of the occupancy of the level they show that even the single impurity Ni or Pd atom possesses d state holes and a density of d state at the Fermi surface which can be correlated excellently with other physical properties such as electron heat and electrical resistance, Norris and Myers [48], Myers, Norris and WalldCn [49].

2) The existence and persistence of the resonant bound state to quite large solute concentrations precludes any application of the rigid band or virtual crystal approaches for these alloys. In fact both sys- tems, but especially AgPd, approach the split band

_-

model for d states.

3) The data clearly show the separate effects of intra and interatomic interactions on the breadth of d states. At small concentrations of solute the solute d states are broadened only by the intraatomic s-d resonance. As more and more solute is added the resonant level broadens on account of solute-solute interactions

-

this is the interatomic contribution.

The photoemission data allow us to separate these processes. However the interatomic contribution can be thought of as arising from two different me- chanisms

- a

direct d-d overlap between d states on adjacent atoms in a conventional tight binding manner and an indirect interaction via the conduction electrons d, si-sj dj. There is still considerable discussion as to which is the major mechanism for the origin of d band

widths in the pure metals Friedel [I], Ziman [50].

Photoemission tells us nothing about how the inter- atomic broadening arises but does tell us that it gives the larger contribution to d band widths.

It is known from magnetic, electrical resistance and electron heat data that a significant change in the density of d state at the Fermi level arises when the solute (Ni or Pd) becomes greater than 40 %, this is also seen in the photoemission data as a large increase in the number of electrons emitted from the Fermi level. Unfortunately the photon energies used in this study were insufficient to expose the whole of the d bands in the concentrated Pd alloys but they do show the sharp peak in the density of states just below the Fermi level in Pd. It would be worth while repeating this work using a higher photon energy.

In one sense this has been done for CuNi using X-ray photoemission (Hiifner

et

al.

[51])

but the resolution is much poorer than would be obtained with UV light.

We shall return to these alloys again later.

The above CuNi and AgPd alloys do not exhibit

local magneticmoments (we neglect the anomalous

magnetic behaviour of CuNi) as do CuMn or AgMn

for example. Optical aniphotoemission studieshave

been of use in deciding the degree of spin splitting

which arises on the transition metal atom

;

in parti-

cular the photoemission experiments on CuMn and

AgMn, Norris and WalldCn [52] show that the occu-

pied spin states on the Mn atom lie - 2.8 eV below the Fermi level

;

it is difficult to obtain an accurate estimate of the width of the filled spin state but it appears that A - 0.5 eV and with the help of magnetic data we deduce the energy difference between the centres of the filled and empty spin states to be

- ^{5.5 eV.}

How about transitions between the conduction band states in these alloys

?

They are in fact observed in the photoemission data for a 5 % Pd in Ag alloy but in much weakened form compared with pure silver [48].

The disorder scattering in this type of alloy is parti- cularly strong especially for CuPd, AuPd, CuNi where there is significant d state density at the Fermi level even in the noble metal rich alloys. This is seen in the optical data too. Figure 14 shows the optical

FIG. 14. - Optical absorption in CuPd alloys (after Myers, Walldin and Karlsson [45]). Compare with the data for CuGa and CuAs figs. 8 and 9. Note the absence of any splitting of the 5 eV peak. We conclude that direct transitions between conduc- tion band states (LpL?) are now absent. The observed features are associated wholly with transitions from the Cu and Pd d

levels.

absorption of a series of CuPd alloys. Note that in comparison with say CuGa the addition of Pd does not cause a splitting of the prominent peak at 5 eV.

We conclude the s-d scattering is sufficient to remove all direct conduction band transitions. There is a strong similarity to the spectra of CuAs alloys. In fact the only prominent features of the alloy spectra are those attributable to transitions from the d levels to the Fermi level and these show very little change in position with composition.

3.3 Cu, Ag, Au

MUTUAL ALLOYS.

-These alloys have received much attention [53, 54, 55, 561 but they are not so simple to understand in detail as those of pre- vious sections. The main reason for this is that the d bands of the pure metals fall in roughly the same energy interval. Thus the principal optical edges arise at. 2.0 eV, 3.9 eV and 2.4 eV whereas the

d

band breadths are 3.0 eV, 3.5 eV and 5.7 eV for Cu, Ag and Au respectively. The conduction band breadths vary from - 7.5 eV in Cu and Ag to approximately 9 eV in Au. So far we have avoided any discussion of what happens when the d levels of alloy components overlap. It might be thought that the situation is suited to the virtual crystal approach but even in CuAu alloys this is not the case. Consider AgAu alloys for which extensive optical and photoemission data are available [53,

561.

Although the absorption edge of Ag moves continuously to lower photon energies as Au is added it is found that the alloys in the mid-composition range do not form one com- mon absorption edge but show a duplex structure the upper portion of the edge arising primarily from the gold atoms.

The photoemission data for AgAu alloys figure 15 clearly show the composite structure of the combined

lnltlal State energy (eV1 Initial slate energy (eV)

FIG. 15. - Photoemission spectra for CuAu and AgAu alloys (after Nilsson [56]). See text for detailed discussion.

d band. In CuAu alloys the separate identity of each

d electron component is not obvious but is in fact

inferred from other arguments in particular the theore-

tical basis for an understanding of this kind of alloy

as provided by the coherent potential approximation

[57, 58, 591. This theoretical approach provides a

method for following the expected changes in d band

structure as one proceeds from the true split band

situation via overlapping bands to the virtual crystal

limit. In its simplest form the effects of s-d resonance

are neglected and it is assumed that the pure compo-

nents, A and B, have the same d band width, W, but

centred on different energies and &I. The latter may

C4-44 H. P. MYERS

be considered as the positions of the atomic d states for the atoms A and B prior to switching on interatomic broadening processes that give rise to the band width W. The

8,

are not those corresponding to the binding energies of the d states of the free atoms but the renormalised values after having taken account of the change in the volume density of the s electrons when the atoms form part of a solid [60]. The impor- tant parameter of the theory is 6IWwhere 6

= E A -

EB.

If 6/W < 0.25 then the virtual crystal approximation holds whereas for 6/ W > 0.5 we obtain the condition of split bands (in practice the presence of s-d resonance demands considerable strengthening of this condi- tion [59]). For 0.25 < 61 W < 0.5 there may be sepa- rate bands in dilute alloys but for more concentrated solutions there will be one combined band. In the latter instance the d states are to be found in the union of the two bands appropriate to the pure components but they are distributed in a non uniform manner both in real space and in energy space. Figure 16.

Crudely speaking we can say in the pure components the tight binding overlaps give rise to dA-d, and dB-dB resonances producing the band widths W.

FIG. 16. - Schematic presentation of the results of the CPA for an alloy the pure components of which have similar and overlapping d bands. The sketches show the density of d state per component atom and the density per atom of alloy is then a weighted average. For similar band width but greater separation of ^EA and EB the dilute alloys would show separate systems of d levels for the two components. s-d resonance however is

neglected.

In the alloy the overlapping d bands provide additional dA-dB resonances which couple all the electrons in both solvent and solute bands. (This cross coupling is further aided by s-d resonance). Thus both the solvent and solute d bands have a width equal to the union of the two bands and electrons from both components will contribute to the state density at all energies within the composite band

:

however the upper and lower portions of the composite d band will contain a greater proportion of the d electrons from the element which in the pure state has its d band in the higher and lower position respectively. The dis- tribution of the d states is not a simple smeared compo- sitional weighting of the pure element d bands (although this is not a bad first approximation particularly if some attempt to estimate d band narrowing is made).

Qualitatively the model gives a good idea of how the d bands are changing in the noble metal alloys although no accurate quantitative appreciation has yet been given

;

a major difficulty being uncertainties in the important quantities W,

8:

and

^8:.

It has had its grea- test success in the application to the CuNi [61] and AgPd [62] systems. The coherent potential approxi- mation is dependent upon the use of configurational averages both in connexion with the scattering of an electron from a particular lattice site and with regard to the medium in which the site is embedded. This means that what we have called the dA-dB coupling resonance is overemphasized at the expense of cluster resonances of dA-dA and dB-dB type. This is particular true for the dilute alloy. Thus the possibility that the atoms of an alloy may tend to form clusters of similar atoms is neglected and the homogeneous solid solution presupposed. So far it has not proved possible to remedy this disadvantage of the model but this is not particularly serious at our present level of under- standing.

Thus in spite of the detailed experimental data for the mutual noble metal alloys we cannot provide other than a very qualitative appreciation of the behaviour of the d bands. Note however that this is still within the bounds of charge neutrality for the atoms. Charge transfer undoubtedly exists in many alloys particularly those based on the noble metals such as CuZn, CuIn, etc. but the optical data give no direct measure of this effect. On the other hand the displacement of the principal absorption edge in such alloys cannot be explained in detail without consideration of both lattice expansion and changes in the potentials at the different atoms as the following account illustrates.

The most detailed attempt to interpret the optical spectrum of a noble metal alloy is that of Levin and Ehrenreich [59] who within the framework of a model Hamiltonian in combination with the CPA approxi- mation attempted to derive the variations in the posi- tions of the underlying atomic d levels (i. e. edAg and

&),

and the breadths of the component d bands

(derived from the atomic levels on the assumption