Interrelations between atomic and electronic ...

PHYSICS REPORTS (Review Section of Physics Letters) 222, No. 2 (1992) 65—143. North-Holland

PH Y S IC S R E PORTS

Interrelations between atomic and electronic structures—Liquid and amorphous metals as model systems P. Häussler* Physikalisches Institut, Universitãt Karisruhe, P.O. Box 6380, 75CM) Karlsruhe 1, Germany Received April 1992; editor: B. Muhlschlegel

Contents: 1. Introduction 2. Theoretical remarks 2.1. Electronic influence on structure and phase stability 2.2. Structural influences on the DOS 2.3. Electronic transport 2.4. Collective density excitations 3. Structural properties of the amorphous and the liquid state 3.1. Atomic structure in real space 3.2. Atomic structure in reciprocal space 3.3. Why 1.8e/a instead of 2 c/a? 4. Experimental procedures 4.1. Preparation 4.2. Processing of the UPS data 5. Photoelectron spectroscopy 5.1. Au—(Al, Sn, Sb) alloys and (Ag~ 5Cu50)7~Ge25

68 71 71 76 77 78 79 79 82 84 84 84 85 87

5.2. Pure elements, Pb—Bi alloys, and TI—Bi alloys 5.3. Mg—Zn, Zn—Ca, and Ca—Mg alloys 5.4. (Au, Ag, Cu)—Sn alloys 6. Correlations between the DOS at EF and other properties 6.1. Electronic density of states at EF 6.2. Stability of liquid and amorphous alloys 6.3. Electronic transpo~of (Au, Ag, Cu)—Sn alloys 6.4. Scaling behaviour~’ersusZ 7. Magnetic amorphous alloys 7.1. Structure 7.2. Thermal stability and magnetism 7.3. UPS measurements of glassy magnetic Sn alloys 7.4. Transport properties 8. Concluding remarks

92 95 98 102 103 107 110 124 128 129 131 133 135 136

87

References

137

Abstract: In this paper we review an intimate interrelationship between atomic and electronic structures. Charge- and spin-density oscillations in the electron system, giving rise to so-called Friedel and RKKY oscillations in the effective pair potential or the effective magnetic interaction, cause indirect ion—ion and spin—spin interactions mediated by the conduction electrons. These interactions prefer positions of neighbouring ions and local moments which are commensurate with their minima and/or maxima. Structure itself has a strong influence on the electronic states. Structureinduced gaps in the electronic density of states occur at Brillouin-zone boundaries. Whereas for crystalline systems these interrelations are well established, they are still controversial for disordered systems. In this paper we discuss non-magnetic as well as magnetic amorphous glasses and liquid alloys. We will show that due to their isotropy the electron—structure interrelationships are observed even more easily than in crystalline systems. The systems under consideration may serve as models for electron—structure interrelationships and show many similarities to an isotropic three-dimensional Peierls system. After reviewing the electronic influence on structure, we mainly focus our attention on photoelectron spectroscopy of the UPS region. These measurements clearly show structure-induced pseudogaps in the electronic density of states. Concentration and temperature dependences are in good agreement with theory. One pseudogap is far below EF and hence unimportant for most properties. Another one is at EF, with strong influences on electronic transport and phase stability. The relation of the pseudogap at EF in non-magnetic alloys to the glass-forming ability, the thermal stability, and deviations of electronic transport properties from the free-electron behaviour are reported. The resistivity, the Hall coefficient and a new interpretation of the thermopower are discussed in some detail. Influences of the indirect magnetic interaction on the thermal stability, the atomic structure, and the magnetic behaviour are also briefly discussed.

*

Present address: Hoechst AG, Central Research, Applied Physics, 6230 Frankfurt, Germany.

0370-1573/92/$15.00

© 1992

Elsevier Science Publishers B.V. All rights reserved

INTERRELATIONS BETWEEN ATOMIC AND ELECTRONIC STRUCTURES Liquid and Amorphous Metals as Model Systems

P. HAUSSLER Physikalisches Institut, Universitàt Karisruhe, P.O. Box 6380, 7500 Karlsruhe 1, Germany

NORTH-HOLLAND

P. Hdussler, Interrelations between atomic and electronic structures

67

Abbreviations: B

02(T) upper critical field of the superconducting state DOS electronic density -of states EB binding energy f(K) local field correction g(r) pair distribution function g = N(EF)/No(EF) Mott’s g-value Mott’s g-value deduced from UPS data g,h Mott’s g-value deduced from specific heat data Mott’s g-value deduced from susceptibility data Iocrfr) effective indirect exchange interaction K absolute value of the scattering vector wave number of the first peak in S(K) K~0 wave number of an electron-induced peak in S(K), or diameter of the pseudo Brillouin or Jones zone N(EF) electronic density of states at EF No(EF) free-electron value of N(EF) n electron density n* renormalized electron density Q~ wave number of characteristic excitations (phonon rotons) q contribution of elastic umklapp scattering to ~ RKKY Ruderman—Kittel—Kasuya—Yosida RH Hall coefficient R~ free-electron value of RH r contribution to ~ caused by energy dependent scattering processes

r0 r~

a y(r)

p(T) p0 p,,(T) ~b0ff(r) ~(K)

M(T) e(K) K A AF

temperature coefficient of the resistivity reduced pair distribution function experimentally determined linear specific heat coefficient electronic contribution to ~ non-electronic contribution to y,,,,~ deviation of RH from R°H deviation of S(T) from S°(T) heat of mixing screening factor (Lindhard’s dielectric function) Debye temperature Knight shift electron—phonon coupling constant Friedel wavelength wavelength of density oscillations effective Coulomb potential Bohr magneton thermopower parameter

S(K) S(K, cv) S(T) S0(T) S°(T) S’(T) Sh(T) S,1 T0 T0 T~ TK U,0 U0 ~ U,, UE u(r) v(K) Z

x00,

~

XL

12

nearest-neighbour positions as measured nearest-neighbour positions caused by Friedel oscillations static structure factor dynamic structure factor thermopower diffusion thermopower free-electron value of S(T) low-temperature value of S(T) high-temperature value of S(T) local moments characteristic temperature of phonon rotons Curie temperature transition temperature to the superconducting state crystallization temperature total energy volume dependent part of the total energy band-structure energy magnetic energy Ewald energy bare pseudopotential Fourier transform of the bare pseudopotential mean number of conduction electrons per atom

resistivity mean particle density Ziman’s resistivity effective pair potential perturbation characteristic experimentally determined magnetic susceptibility electronic contribution to x~,,, ionic contribution to Pauli paramagnetic susceptibility Landau diamagnetic susceptibility susceptibility caused by electron—electron interaction band-structure (energy—wavenumber) characteristic atomic volume characteristic phonon roton frequency

1. Introduction From the theoretical as well as from the experimental point of view electron—structure interrelations are much in the focus of modern solid state physics. The electric and magnetic polarizations of the electron cloud around localized charges or local moments both affect atomic structure. The electronic structure itself, on the other hand, depends on atomic structure. Gaps in the electronic density of states (DOS) arise at Brillouin-zone boundaries. Accordingly, considerable band-structure energy contributions UbS (to be taken into account negatively) to the total energy U~0~ occur with phase stabilizing effects whenever the Fermi energy is located within a gap. In magnetic systems, a magnetic contribution Urn may further suppress ~ For a reasonable description of a system the interrelationships between atomic and electronic structures have to be considered self-consistently. In non-magnetic systems the underlying description [1] remains valid not only for metals but also for poor conductors or even semiconductors, i.e. covalently bonded systems. Thus it is more correct, independently of conductivity, to talk about systems with s- and p-states at EF, states with strong dispersion. Low-dimensional systems such as Peierls systems are best known with respect to electron—structure interrelations [2, 3]. Somehow they are easily understood due to the low dimensionality. Electron— structure interrelations and hence a decrease of the total energy due to band-structure effects cause superlattice ordering. In a purely one-dimensional system the opening of a gap at the Fermi energy causes a (Peierls) transition from a metallic to an insulating behaviour. Charge- as well as spin-density waves, accompanied by soft phonon modes as well as structural instabilities or lattice distortions, are consequences which are in the focus of much theoretical and experimental work [3,4]. Structural instabilities as well as electronic features of many of the new, exciting materials such as organic superconductors [5] or the layered high-To materials [6—8]are discussed along this line. The properties of the latter are extremely anisotropic and the electronic system can be considered as quasi two-dimensional along the a, b-axis. Structural transitions, e.g. from the tetragonal to the orthorhombic state, seem to be driven by charge-density or spin-density wave instabilities in a half-filled band [8]. Charge- and spin-density wave effects are even discussed for their superconductivity [6, 7, 9]. The quasicrystalline phase [10—14],a novel metastable state with five-fold rotational symmetry but no long-range order, as well as the so-called Frank—Kasper phases [11, 15] are also considered as being stabilized by the conduction electrons. A dip-like anomaly was reported recently in the DOS at EF of the former [12, 14]. The DOS seems to be only (30 ± 10)% of the normal DOS without the dip and has been claimed to be responsible for the phase stability of the quasicrystalline system [12, 13]. Multilayers, new artificial materials which have come much into the focus of recent solid state physics, show, in systems with alternating magnetic and non-magnetic stackings, clear indications of long-range magnetic ordering effects, attributed to the so-called Ruderman—Kittel—Kasuya—Yosida (RKKY) interaction [16], the indirect magnetic interaction mediated by the conduction electrons via spin-density oscillations. Whereas the Peierls systems as well as the high- T~materials have a conduction band that is close to half-filled, this is not the case for crystalline Hume-Rothery (HR) alloys. Their filling is between half-filled and nearly full. HR phases are model systems with respect to electronic influences on 68


69

structure and phase stability and hence are often called electron phases. Their electron system is clearly three-dimensional and the Fermi surface touches sharp zone boundaries in several directions. HumeRothery [17] was the first to point out that certain crystalline phases occur in binary alloys at well-defined mean numbers Z of conduction electrons per atom (e / a). A sequence of such phases exists for Z-values between 1 e / a (the pure noble metals) and approximately 1.8—2.0 e/a especially in binary alloys of the noble metals Au, Ag, Cu with polyvalent elements such as Zn, Al, Sn, Sb [17, 18]. One of the characteristic features of HR phases is the scaling behaviour of many physical properties with Z. In earlier publications, the present author showed that the amorphous phase of noble-metal— polyvalent-element alloys can be considered as the limiting case of the crystalline HR phases for Z 1.8 e/a [19,20]. Its physical properties can be scaled with Z quite similarly to those of the crystalline HR phases. The Fermi surface of this new Hume-Rothery phase with amorphous structure is sphere-like, touching a sphere-like pseudo Brillouin-zone boundary in all directions [19]. The amorphous phase, in some sense, behaves like an isotropic three-dimensional Peierls system after the transition to the superlattice (without a real metal—insulator transition). It can exist in a wide concentration range and the electronic influence can be strongly varied, going for example from Z = 1.8 e/a (Au80Sb20) to Z = 5 c/a (pure Sb). Amorphous as well as liquid systems are advantageous compared to the crystalline state. First of all, they are isotropic and electron—structure effects occur in all directions. Secondly, their atomic structure is not as rigid as in the crystalline state and hence can more readily adapt to electronic constraints. Electronic influences on structure as well as structural influences on the DOS, with consequences on electronic transport, may hence be observed over a wide range of composition [19—22]. The literature of liquid and amorphous metals is large. Even in the field of simple alloys many reviews exist [15, 23—28]. Here we will mainly concentrate on new results. The emphasis will be on general ideas and basic results described in terms of pseudopotential theory. More subtle descriptions such as multiple-scattering, mode-coupling, higher correlation functions than pair correlations are disregarded. The structure of disordered systems is of interest in its own right. The dominant influence of conduction electrons on the structure factor S(K) in k-space [19, 20, 29, 30] or their influence on the pair distribution function in r-space [15,20, 25—27, 31] at short- and medium-range distances have both successfully been used to understand important features of the atomic structure as well as the stability of the liquid and amorphous state against crystallization [19, 25, 26, 32—36]. K is the absolute value of the scattering vector defined by K k0~with k0 the initial and k the final wave vector. The influence of disorder on the DOS is of the equal basic theoretical interest. Some aspects have not yet been understood. Short- and medium-range order, still existing in the disordered phase and structure-induced pseudogaps in the DOS far below EF [37,38], as well as close to EF [21,22], have been reported for non-magnetic systems. Pseudogaps were first proposed by Mott to explain electronic properties of liquid mercury [39] and were later used by Nagel and Tauc for the discussion of the phase stability of disordered systems [32]. Disorder or, subsequently, the pseudogap at EF has strong influence on electronic transport. Under conditions where the Fermi sphere is touching the Brillouin zone so-called diffuse umklapp-scattering effects occur [40]. The resistivity p gets quite high [41] and the Hall coefficient RH [19] as well as the thermopower S(T) [42,43] deviate strongly from the corresponding free-electron value. *) —

*) S(K), the structure factor, and S(T), the thermopower, should not be mixed up. To avoid confusion, we will always display explicitly the appropriate variable.

70


RH of the liquid state, on the other hand, is close to the free-electron value R°H[23]. Small deviations of approximately 3—5% have been explained by mass-density anomalies [23] and larger ones for the heavy elements like Tl, Pb, and Bi by spin—orbit effects [44]. Unfortunately, the good agreement of RH with the free-electron model (FEM) was misleading for years and automatically led to the conclusion that the complete conduction band is free-electron-like. Conclusions like this are weakly founded since it is still controversial whether all electronic states or only those at the Fermi energy contribute to the Hall coefficient [45,46]. Large departures from free-electron behaviour of the DOS far below EF have been reported for liquid elements, although RH/R°~ is close to 1 [47].In amorphous alloys deviations of RH from R°Hare so large that mass-density anomalies have to be disregarded. Subsequently, it has been suggested that pseudogaps at EF, more pronounced in the amorphous than the liquid state, may be responsible [43]. The influence of pseudogaps on transport properties has been discussed many times and a lot of work is still done on this subject. Anomalous behaviour of various electronic properties such as p [39, 48, 49], S( T) [39, 49, 50], the electronic part of the magnetic susceptibility Xe [51], or RH [52] may occur. With the strong influence of conduction electrons on the interatomic potential and therefore on the structure of disordered metallic systems, it is quite obvious that their vibrational dynamics will also be affected [25, 26, 53—55]. Inelastic neutron-scattering experiments [56,57] have revealed roton-like short-wavelength collective density excitations as a characteristic features of disordered systems. Above a certain temperature T0 these excitations influence electronic transport [42, 43, 58]. Dynamic properties of disordered systems are of particular interest in their own right and have come much into the focus of recent theoretical [25, 48, 53, 54, 59—62] and experimental work [56,57]. Effects of phonon rotons on the temperature dependence of both p( T) as well as S( T) are discussed below. (Au, Ag, Cu)—Sn alloys serve as model substances for both the liquid [23] and amorphous state [19]. The advantage of using these alloys for UPS measurements is that Sn is a simple metal and noble-metal d-states do not contribute significantly to the DOS at EF. Over reasonable temperature and concentration ranges their properties are found to be in agreement with the FEM. As mentioned above, systematic deviations from the FEM also exist, which are more pronounced in the amorphous than in the liquid state. Replacing Sn (Z~~ = 4 c/a) by Al (ZAj = 3 c/a) or Sb (Zsb = 5 cIa) allows one to study the influence of Z. Whereas alloys containing noble metals can be prepared over large concentration ranges and clearly show the structure-induced features near EF, well below EF structural influences on the DOS are strongly superimposed by large d-state intensities. For this reason, glassy Ti—Bi, Pb—Bi, Mg—Zn, Zn—Ca, and Ca—Mg, systems without d-states in the valence-band region, are also reported [37, 38, 63]. The (11, Pb)—Bi alloys, in addition, allow UPS measurements in the liquid state up to high temperatures. A clear temperature dependence of their DOS at EF was reported quite recently [37, 38]. This dependence itself seems to be influenced by structural features and may have a strong impact on the theory of electronic transport. The inclusion of liquid systems and alloys without any transition metal or noble metal shows that the effects discussed in this review are quite general. After some theoretical remarks in chapter 2, in chapter 3 we review those structural aspects of disordered systems which are important for the present review. Experimental details of the UPS measurements are described in chapter 4. The UPS data themselves are presented in chapter 5. In chapter 6 the electronic DOS taken from other methods is related to the results of chapter 5. Additionally, consequences of a pseudogap at EF for various physical properties are described in the same chapter with a special emphasis on p, RH, and S(T). Replacing Au, Ag, Cu by Fe, Co, Ni, or Pd brings magnetic interactions into play, mediated by the conduction electrons as well. Whereas in

P. Hamster, Interrelations between atomic and electronic structures

71

chapters 3—6 the focus is on non-magnetic systems, in chapter 7 we briefly present magnetic systems, with the main emphasis on atomic structure, thermal stability, magnetism, and UPS. 2. Theoretical remarks In crystalline materials long-range order causes sharp Brillouin-zone boundaries in k-space. In a single-particle picture the electrons can be described as eigenstates and Bloch’s theorem allows the calculation, e.g., of the band structure by limiting the calculations to a unit cell. In glassy systems, Bloch’s theorem is inapplicable because long-range order is absent and a unit cell no longer exists. But, short- and medium-range order, still present, cause diffraction halos (fig. la) corresponding to pseudo Brillouin zones with blurred boundaries (fig. ib) [54, 55]. Anisotropies disappear and the zones are sphere-like. Electronic states at zone boundaries are heavily damped and a definite dispersion no longer exists (fig. ib). A serious problem of any theoretical description, e.g. the calculation of the DOS or phase stability, is caused by the structural input. The adjustment of the atomic structure to electronic constraints needs to be included. The electronic influence on structure and phase stability can qualitatively be seen in fig. ic. Electronic states get raised as well as depressed at any zone boundary, creating Van Hove singularities above and below the corresponding energy. If all the states up to infinite energy would be occupied, pseudogaps would have no effect on total energy. This is different if the states are occupied up to k = kF KpeI2 (Kpe 2kF), since their decrease is not counterbalanced by states above EF. Thus, peak positions in S(K) close to 2kF have the largest influence on stability [34] and subsequently are most favoured. As reported by the author the zone boundary at K~~/2 even adapts to varying kF in order to remain optimally situated [19, 20]. Ground-state properties as well as electronic transport properties are strongly affected under conditions where kF is close to the zone boundary [64]. This is valid for the disordered systems under consideration and will be part of the discussion below. 2.1. Electronic influence on structure and phase stability For a correct description of electron—structure interrelations, in principle, complex theoretical procedures need to be applied. Here we stick to the pseudopotential theory [1, 65]. For a discussion of structure and stability in the frame of pseudopotential theory the total energy is given by

SIK)

E (b)

(a)

J”~ 0

K

kF

~ 0

(c)

E

-~

!~a!5as.

k ME) 22 Fig. 1. Schematic electron—structure interrelationships of non-magnetic disordered systems. (a) Typical structure factors S(K) at two different temperatures with a split first peak. 2kF indicates the Fermi-sphere diameter. (b) Dispersion relation of electronic states. (c) Electronic density of states of a non-magnetic system with structure-induced pseudogaps indicated.

72

P. Hdussler, interrelations between atomic and electronic structures

U~0~ = U,,

+ (UE + UbS + Urn)struc

(1)

U~is a volume-dependent term accounting for approximately 95% of the total energy. UE, UbS and Urn are structure dependent and responsible for eventual phase stability. UE, the so-called Ewald energy, favours simple crystalline phases and therefore requires diffusion of the constituents. UbS, the band-structure energy, arises from deviations of the electron gas from free-electron behaviour (fig. 1). It is negative and favours a phase in which it has a larger value. We assume UbS to be dominant if diffusion is suppressed as during the preparation of metallic glasses. A magnetic contribution Urn also has to be taken into account as soon as localized moments are involved. Below we discuss both effects but with more emphasis on UbS. There is a description in real as well as in reciprocal space. Here we start with the former.

2.1.1. Real space Due to both the charge and spin of electrons, the effect of conduction electrons on atomic structure and phase stability can be twofold. Around the ions or the local moments, the electron cloud gets electrically or magnetically polarized. Due to the limitation of electronic moments by kF and, subsequently, the incomplete screening of local distortions, the polarizations themselves are oscillating with distance r. The electron distribution, for example, around each ion screening its local charge shows density oscillations4~eft(1~ (fig. 2a). Poisson’s yieldsfree so-called oscillations inis the effective In systems with equation a long mean path ofFriedel the electrons ~eff(~) expressed as pair [66] potential 4~eff(~) occos(2kFr

+ ø)1r3

(2)

,

with AF=27r/2kF

(3)

the Friedel wavelength. At large distances the phase shift, 0, approaches zero but may differ significantly from that for short and medium distances [59]. Experimental indications of 0 = ~r/2 at (a)

(b) ‘~

-~

~~\S~’~’

“: ~

~-

~

~

.‘,

~

~)

‘5

V

..

+

.,

Fig. 2. (a) Charge-density oscillations, which cause an indirect electronic interaction between localized charges (ion i and ion j). The shaded halos around ion i correspond to concentric spheres with enhanced electron density. (b) Spin-density oscillations, which cause an indirect magnetic interaction between localized moments i, j, and k. The dots indicate electron spins “parallel” and crosses electron spins “antiparallel” with respect to the localized moment i. Charge or spin halos exist around any ion or local moment but are indicated in each figure around one only.

73

P. Häussler, Interrelations between atomic and electronic structures

short- and medium-range distances were first reported by the present author for liquid and amorphous metals [20]. Under these conditions the cosine function becomes a negative sine function, 2kFr) Ir3 , 4~eff(7) —sin(

(4)

qualitatively depicted in fig. 3b. Recently theoretical work has shown that eq. (4) is generally valid for systems with a short mean free path of the electrons [67]. For non-magnetic alloys the electronic influence on structure is seen in the commensurate position of (positive) ions at the minima of ~‘eff(~~The ions are forced into these minima, subsequently causing oscillations (wavelength Ape = 2 lTIKpe) of the particle density (fig. 3a). The index e to Ape is attached in order to denote the electronic influence. The particle density versus r is denoted by g(r), the pair distribution function, or y(r) = 4lTrp 0[g(r) 1], the reduced pair distribution function. p0 denotes the mean particle density. Without any additional phase shift the Friedel oscillations cause nearestneighbour positions at distances [20] —

r~(n+~)AF, n1,2,3,...,

(5)

which can easily be checked by diffraction experiments. The commensurate matching band-structure of the ions at short mediumtorange with the minima of equivalent to a large (negative) energyand according UbS

~

J

[g(r)

—

1]~eff(r)r2dr

~5

4~eft(7~)

(6)

(a)

—•1•

~

T

T

—

Kpe

I

I

-r

~

I~

-

5(r) (b)

—~

~

—r

-r

(c) Ieff(r)

~ -T

-r

-

/t 2kF I I

o

I ~ I ~

/\.~

V

—.

-

-

I

0 1pair potential 2 3 ~ Fig. 3. Idealized (a) pair distribution function, (b) effective

4

-

I

r 2kFJ

s 6 indirect 7 Lexchange interaction I, (c) effective 11(r) [71].Oscillations

in (b) and (c) are strongly enhanced at medium-range distances.

74

P. H/fussIer, Interrelations between atomic and electronic structures

and hence a low total energy with phase stabilizing effects [26, 34, 43]. Ionic arrangements with the same periodicity as 4)eff(r) are most favourable and Nagel—Tauc’s stability criterion in r-space (Ape = 27T/Kpe = 2ITI2kF = AF) is fulfilled [19,25,29,31,33]. Molecular-dynamic simulations by different authors confirm the influence of the Friedel minima on structure [27,68]. A strong correlation even between the next-nearest-neighbour distance and the Friedel oscillations has been theoretically predicted [69] as well as experimentally verified [20]. Whereas in non-magnetic alloys only the screening of localized charges needs to be considered, in magnetic systems in addition the spin polarization of the electron cloud around local moments has to be considered. Spin-density oscillations (fig. 2b) give rise to an oscillatory effective indirect exchange interaction (RKKY) between local moments [701,also mediated by the conduction electrons. The RKKY interaction is claimed to have additional effects on phase stability [71]. As shown by Kaneyoshi [72,73], the common RKKY interaction again changes drastically in systems with a short mean free path of the electrons. The effective exchange coupling coefficient jeff(~~)is no longer cosine-like and has to be replaced by evT,

Ieff(r)

(7)

a positive sine function with both a and f3 dependent on kF. With the approximation that f3 equals 2kF, the wavelength of the corresponding oscillation is again AF, the Friedel wavelength mentioned above. The corresponding ICff(r) is depicted in fig. 3c. In contrast to the indirect interaction between (positive) ions, there are two alternatives for the local moments. They can occupy positions either ferromagnetically or antiferromagnetically with respect to the local moment at the centre. Without any additional phase shift, ferromagnetic order will occur if the neighbouring local moments are at r = r~or antiferromagnetic order if the moments are at r = r~+ AFI2 (fig. 3c). The commensurate matching of local moments with the maxima and minima of Ietf(r) is equivalent to a large (negative) magnetic contribution, Urnre_~IeffSi~Sj,

(8)

i,~i

to the total energy. S denote the local moments. Whenever maxima of g(r) coincide with maxima of jeff(r), in addition to the decrease of the total energy due to the increase of UbS, the magnetic influence due to ferromagnetic interaction will decrease the total energy and enhance the stability even more. This is different for the antiferromagnetic position, which is unfavourable for (positive) ions. The variation of AF with composition will affect the positions of local moments and the increasing number of local moments, the value of Urn and hence the magnetic behaviour. Because of the complexity, molecular-dynamic simulations, taking self-consistently both indirect interactions into account, have not yet been done for disordered systems and are still, to our knowledge, outside the scope of present computing capabilities. 2.1.2. Reciprocal space The well-defined distance AF between nearest-neighbour shells induces in k-space a pronounced peak in S(K). Accordingly, in polyvalent liquid elements and alloys as well as many metallic glasses, the first peak in S(K) seems to be split into two (figs. 4a, 10, and 11) [19,20, 27, 29, 74, 75]. There is a peak at


S(K) (a)

K~

0(b)

v(K)

75

Koll

(c)

(d)

2k~2k~

Xl~~

Fig. 4. (a) Static structure factor of an amorphous or liquid metal, (b) band-structure characteristic, (c) pseudopotential, (d) perturbation characteristic. (a), (b) and (c) are qualitatively drawn in arbitrary units. Variations of v(K)5b,(K) with due composition to the variation are disregarded. of ~(K) with In (b), Z.(c) and (d) an alternative Fermi sphere diameter 2k~is indicated, showing a shift of the minimum in q

and another one, electron-induced, at Kpe close to 2kF. Electronic influences on atomic structure can directly be discussed in k-space with additional insights on electronic structure and transport. Only non-magnetic systems are discussed here. Pseudopotential theory relates physical properties rather directly to (fig. 4c) v(K)

~j

J

v(r) e~ d3r,

(9)

the Fourier transform of the bare pseudopotential v(r) [1]. 11 denotes the atomic volume. In k-space Ub, depends on S(K), v(K) and two functions e(K) and ~(K), which take into account the electronic response to the localized charge of the ions. s(K) is the screening factor or Lindhard’s dielectric function with modifications due to exchange and correlation [1], e(K)

= 1

—

8ii•e [1 —f(K)]x(K),

(10)

with /1

4k~—K2

K+2kF \

)~

8KkF ln K—2kF (11) ~(K), the perturbation characteristic, is shown in fig. 4d. It is negative and approaches zero for +

P. Haussler, Interrelations

76

between atomic and electronic

structures

K > 2kF. At 2kF there is a logarithmic singularity, which is responsible for the Friedel oscillations in r-space. f(K), the local-field correction, takes care of exchange and correlation. In k-space, UbS of disordered systems is written as

UbS

J

S(K)~b,(K)K2dK,

(12)

jv(K)12e(K)x(K)

(13)

with ~b,(1~)

=

.

43bS(K), the band-structure characteristic or energy—wavenumber characteristic shown in fig. 4b, describes the indirect ion—ion interaction mediated by the conduction electrons in k-space [26]. For crystalline systems K has to be replaced by g, the reciprocal lattice vector, and K2 dK by ~g’ a summation over all g [1]. The zero at K 2kF by 0 is given by the zero of v(K) and the increase for K > ~(K), both causing a minimum for K 2kF. The position of the minimum varies with Z. Before continuing with the disordered 0 < state, K < we shortly discuss crystalline HR alloys. In these alloys, the most important zone is not in all cases a true Brillouin zone [17, 18]. In general it is a zone built from additional parts of other zones. This so-called Jones zone [76] has the boundary close to kF and, from the point of view of a reasonable band-structure contribution, a large contribution to the total energy.*) To keep UbS as large as possible, variations of kF by alloying generally causes an adjustment of the Jones zone. Defect formation and lattice distortions have successfully been explained in these terms [18]. Because of the limited variability of crystalline structures these adjustments may occur only in limited concentration ranges along particular axes. Although for disordered systems the concept of Brillouin zones is not strictly valid, the Jones zone is still a valuable quantity. The peak in S(K) closest to 2kF (fig. 4a) indicates the diameter of the Jones zone of the disordered phase (henceforth called a pseudo Brillouin zone). Due to the minimum of

J

between I(~and 2kF and the weighting of eq. (12) with K2, both the proximity of S(2kF) to 2kF and its intensity have strong influence on UbS. Contributions of UbS to the total energy occur in all directions due to isotropic structural features. The disordered phase is quite open for adjustments of the pseudo Brillouin zone to the electronic constraints because of its poorly defined structure. Changing 2kF and hence the position of the minimum in ~bS(K) may induce changes of Kpe over wide concentration ranges by a continuous rearrangement of the atoms. For alloys, in principle, partial structure factors as well as partial pseudopotentials should be taken into account. Unfortunately, in most cases they are unknown or uncertain, which hampers detailed discussions. In this review we discuss the total functions only. 4bS(K)

2.2. Structural influences on the DOS Whereas in crystalline materials structural influences on the DOS are beyond any doubt, similar effects in disordered metals have, surprisingly, been denied for years, although it was shown *)

The Jones zone is empirically defined by Brillouin-zone boundaries which correspond to the most intense diffraction peaks.

77


theoretically that, under conditions where S(K) is large in regions where v(K) is large, deviations of the DOS from the free-electron behaviour should occur [45,77,78]. Calculations of both the atomic and the electronic structure have recently been published by Hafner and co-workers [79—82]yielding similar results. Although in crystalline systems real gaps in the DOS may occur in particular directions, they may hardly be resolved due to anisotropy. An increasing structural weight, i.e. many Brillouin-zone faces at the same ki-value, enhances the depth of the integrated total DOS. Due to isotropy this is automatically fulfilled in the disordered state. The depression of the DOS, hence, is expected to be well pronounced and directly related to S(K) [83]. In first approximation, the binding energy of electronic states which are influenced by the atomic structure can be estimated by the FEM according to

EB(K/2) = [fi2k~. h2(K/2)2] /2m,

(14)

—

with m the free-electron mass and K = K~or Kpe the position of peaks in S(K) [43]. Under conditions as shown in fig. 4, where two peaks in S(K) below or at 2kF exist, two pseudogaps are expected below or at EF. Such pseudogaps have clearly been observed for both the amorphous [21, 37, 38, 43] and the liquid state [47, 84, 85] whenever d-state contributions are far below EF or absent within the valence band. Structural changes with temperature, especially of the peak at 2kF, which disappears with rising T [75,86, 87], have been clearly shown to affect the corresponding pseudogap [37,38]. The pseudogap at EF is quite important for many properties. Its depth (1 g), indicated in fig. ic, is written in terms of Mott’s g~value*),defined as g = N(EF)INO(EF). Both, the depression of the DOS at EF as well as the intensity of S(K) at K = 2kF, are measures of UbS. —

2.3. Electronic transport The most common description of electronic transport of disordered metals is given by Ziman’s scattering or nearly free-electron (NFE) model, which follows from Boltzmann’s equation [41, 64]. Ziman’s model is based on weak (quasi) elastic scattering of the electrons. Accordingly, electronic transport is strongly influenced by the proximity of the Fermi sphere to the pseudo Brillouin-zone boundary (Kpe 2kF). The resistivity p can be interpreted as caused by a diffuse elastic umklapp scattering (Bragg, 2kv-scattering) well described by the static structure factor S(K). Additional effects due to phonons were introduced with Baym’s modification of Ziman’s theory by considering the dynamic structure factor S(K, w) instead of the static one [88]. Excellent agreement with the experimental results was found over large concentration and temperature ranges [89—92]. The Boltzmann equation as well as the NFE model both demand the Hall coefficient RH to be in agreement with R~,its free-electron value. And, in fact, in liquid simple elements or their alloys (e.g. Au, Si, Sn, Au—Sn) this was found to be the case [23]. But serious difficulties have been reported for amorphous alloys [19, 93]. Besides concentration regions where RH is close to R~,regions with large deviations exist. The thermopower S(T), another property deduced from Ziman’s scattering model, was found to be in complete disagreement with experimental data in the amorphous state at room temperature and even in the liquid state, although RH was often close to R°Hfor the latter, as just mentioned [94]. Both the *)

Mott’s g-value should not be mixed up throughout this paper with g(r), the reduced pair distribution function.

78


value and the sign disagree with theory. Quite recently the present author proposed a new interpretation of S(T), which yields excellent agreement with Ziman’s description at low temperatures [42, 43, 58]. At low temperatures, below a characteristic energy hw,~,only long-wavelength phonons exist. These phonons are ineffective with the result that the low-temperature part of the thermopower S 1(T) agrees with Ziman’s model of elastic scattering. The disagreement at elevated temperatures is, subsequently, thought to be caused by inelastic umklapp-scattenng effects of the electrons with short-wavelength collective density excitations. Below, we also briefly discuss the influence of shortwavelength excitations on the temperature dependence of p( T).

2.4. Collective density excitations Figure 5 shows a schematic drawing of collective density excitations of the crystalline (al) and the disordered state (bi). Here, S(K) is assumed with only one peak (at 2kF). Inelastic neutron scattering experiments on glassy Mg 70Zn30, performed by Suck et al. [56], have revealed energetically low-lying short-wavelength collective density excitations at wavenumbers Qpe where the main peak in the structure function is located (fig. 6). Simplifying the nomenclature according to Handrich and Resch [61], we refer to these states as phonon rotons, reminiscent of a similar dispersion relation of liquid helium. Their energy 11w0 (characteristic temperature T0) is at a few meV (some 10 K). With increasing intensity of S(Kpe)’ the minimum should shift to even lower energies (temperatures) [61]. In crystals these states go down to zero and are identical with those at long wavelengths after a reduction to the first Brillouin zone. Since in disordered systems similar reductions are invalid, a conceptional difficulty arises because the wavelength of these excitations would be smaller than the mean interatomic distance. In analogy with the crystalline case, we therefore remind the reader of the (elastic) diffuse umklapp scattering mentioned above. Large regions of the amorphous sample can participate in electronic scattering and the short-wavelength excitations get transferred into inelastic long-wavelength excitations around K = o~I’) There might even be a hierarchy of inelastic phonon rotons with different energies 11w1 when multiples of Kpe are transferred. Indeed, higher phonon rotons are predicted by the theory [60]. In the last few years phonon rotons have been the subject of many theoretical publications [48, 59—62] and have been revealed experimentally in amorphous [56] as well as liquid systems [57]. They are now believed to be a general feature of disordered systems. The corresponding vibrational density of states (fig. 5b2) is quite different at low w Below as compared to the crystalline state (fig. 5a2). 2kF is fulfilled. T In the present alloys Qpe = Kpe 0 both elastic umklapp scattering and

al)

a2)

bi)

b2)

-~

ii _

\// ~

_

0(E)

Ic

g

g/2

0

Qpe

KPQ/2 Kpe r2kt-~

iTh

DIE)

Fig. 5. Schematic dispersion relation of collective density excitations in (a!) the crystalline case and (bi) the disordered case. Phonon roton states are shown in (bi) at K,,,,. The corresponding dynamic densities of states are shown in (a2) and (b2). *)

To call those states short-wavelength states as above and in many papers is therefore somewhat misleading.

P. Haussler, Interrelations between atomic and electronic structures 8

79

I

I~~I’

T r 6

- (K)

~ 10~

~

:

~°

‘~2kF

Qpe

-

~°

-

0 —W’

I

2.3

2.7

0 (A~)

60

40 20

0

3.1

Fig. 6. Collective density excitations (phonon roton type) of Mg

2kF.

70Zn3,, metallic glasses [56]. K,,,, indicates the position of the experimentally

determined peak in S(K) close to

long-wavelength inelastic scattering with normal phonons occur. Above T 0, additionally, phonon rotons can be excited thermally as well as by electron scattering. Due to the matching of the wavelength of electronic, structural, and dynamic excitations resonance-like effects occur. Electronic transport properties versus temperature may therefore be strongly affected. Concluding our theoretical remarks we would like to emphasize that three characteristic wavenumbers agreethein Fermi-sphere the metallic glasses under consideration: 2kF, diameter, Kpe, the diameter of the pseudo Brillouin zone, Qpe’ the wavenumber of characteristic excitations (phonon rotons). This intimate interrelationship between electronic, structural and dynamical properties and their self-adjustment makes these alloys quite unique, justifying that we call them model systems. The amorphous and the liquid state behave like an isotropic three-dimensional Peierls system.

3. Structural properties of the amorphous and the liquid state As shown above, the dominant influence of conduction electrons on structure contains the key for the understanding of the disordered state. In the present chapter, we review for non-magnetic systems those structural properties which are relevant for the discussion below. Magnetic systems are separately discussed in chapter 7. Structural relations to the crystalline Hume-Rothery phases are discussed elsewhere [19, 43]. The disordered state can be seen as the limiting case of these phases for Z 1.8 e/ a, with the Fermi sphere touching the pseudo Brillouin-zone boundary in all directions. 3.1. Atomic structure in real space In real space the electronic influence on structure is expected to be seen in the commensurate position of ions at the Friedel minima of ~eff(~~ In fig. 7, representative for many others, the

80


•2 .(a)’

gtr:

(b~11Au

2

Sn

~

:E~I6~:E

c~eoIsiIo;i,

75(4,0)

trIm)

~

os—

~

~

~r

~

(nm]

Fig.

7.

Pair distribution functions g(r) of amorphous Au—Sb and Au—Sn alloys [31,86]. Au,,,Sb1,, is partly crystalline. The numbers in parentheses

denote the Z-values.

experimentally determined pair distribution functions g(r) of amorphous Au—Sb and Au—Sn are shown. The short vertical lines indicate the positions r~of the Friedel minima estimated according to eq. (5). The atomic positions r~as well as their shifts versus composition are obviously connected with r~.r~is calculated within the FEM using atomic densities of the liquid state. 11) In fig. 8, r~and r~are shown for different systems versus composition. The systems are chosen such that the valency of the polyvalent component varies from 2 e / a (Mg) to 5 e Ia (Sb), including the alloys shown in fig. 7.If known, liquid-structure data are included. The vertical thin solid lines indicate compositions with Z = 1.8 eI a. Whereas the crystalline Hume-Rothery phases exist between the pure noble metals (Z = 1 e I a) and this special Z-value [17], the amorphous Hume-Rothery phase exists from here up to the vertical broken lines [19]. Three important features characteristically of all alloys are worth noting. First of all the shift of the atomic positions with r~is most pronounced at medium-range distances, more obvious for Au (e.g. Au—Sn and Au—Sb) than Cu-alloys. This is also true for the liquid state as seen, e.g., for Au—Sn. At medium-range distances, the electronic constraints on atomic positions are obviously more easily fulfilled since free space between the neighbours can, to some extent, be occupied. The amorphous range in Cu—Mg is too small to draw similar conclusions. Within the amorphous state (see, for example, *)

Liquid-state densities are found to give best agreement with other experimental data [19,20].


07~°

0.6

I

r~

:

,

81

0.7

0.7

0.7

0.6

0.6

0.6

~, I -~

0.5 ~

0.5

04

04

I •

0.5 -•~~•

‘ ::~H0~~t ::~L~~1 ~ ::~T:: 04

Cu

at%Mg

Cu

amorphous state;positions open symbols: liquid state. Fig. 8. Atomic r,, compared with the positions r

at%At

04

Au

ot%Sn

Au

at6Sb

4)AF of the minima of ~,,ff(r),eq. (5) (thin curves) [20]. Full symbols:

=

(n + lI

Cu—Al) the second or third nearest-neighbour distance is sometimes split into two, although their centre-of-mass seems to stay as close to r~as possible. The second important feature is the disagreement of the nearest-neighbour distance r 1 with r~= ~ AF, the first Friedel minimum, over most of the composition range although it seems to approach it as much as possible. This is understood as an effect of the hard-core diameter of the ions. From this point of view, it is even more surprising that special compositions exist, namely those with Z = 1.8 eIa, where r1 equals r~.We call amorphous alloys under these conditions ideally amorphous. The scaling behaviour of structural properties with Z is the third important characteristic feature claimed above. It is easily seen in fig. 8 that different amounts of polyvalent elements are needed to obtain the ideal amorphous state. Similar to crystalline HR phases, structural properties can obviously be scaled versus Z. For all systems known to us, r1 can be described in units of AF and hence indicates scaling (fig. 9). Additional data ofternary alloys such as (Ag50Cu50)1_~M1(M: Mg, Al, Ge) obtained by Mizutani and Yoshida [95]were analysed and reported elsewhere [19]. It is remarkable that all systems exhibit the first nearest-neighbour distance r1 in exact agreement with ~ at Z = 1.8 ela. The uniqueness of Z = 1.8 e/a is seen in both the amorphous At thisofZ-value 21T/Kpe) and andthe the liquid meanstate. wavelength 4)etf(T),both are internal length scales, the mean ionic distance ( identical. Whereas the liquid state is stable over the whole composition range this no longer holds for the amorphous state. The latter, necessarily, requires r 1 ~AF or Z 1.8 e I a.

82

P. Haussler, Interrelations between atomic and electronic structures SI

Au Ag Cu

At Ga I~

Mg

7/4—~~I

Sn

Au Ag Cu

Sb

I

At Ga In

Mg

Ge Sn Pb

Sb Bi

~

4 tX

(Xl

) F

1

6/4

7

F

•

6/4

••/

/f .7 /j~ /

~

a)

•‘

b)

Fig. 9. r~ in units of A,, versus Z for (a) the amorphous, and (b) the liquid state. The vertical lines indicate Z = 1.8 e/a. References to the individual systems are given in refs. [19,20].

3.2. Atomic structure in reciprocal space The structure factors of Au—Sb and Au—Sn, corresponding to the pair distributions of fig. 7, are shown in fig. 10. At Z 1.8 eI a a single first peak exists, with its position in exact agreement with 2kF. Under these conditions, all ions are located at their corresponding Friedel minimum as shown above. Convincing evidence of the electronic influence on structure is the shift of Kpe nearly parallel to the change of 2kF over the whole amorphous region. The pseudo Brillouin zone adjusts to the Fermi sphere as discussed above in order to reduce the total energy as much as possible. With increasing Z an additional peak at I(~appears far below 2kF. This peak has no relation to 2kF and indicates the importance of other effects such as directed bonds or atomic size differences whenever Z is far from 1.8 ela. Although, within experimental resolution, the intersection of Kpe with 2kF occurs in both the liquid and the amorphous state at compositions with Z = 1.8 eIa, there are characteristic differences between both phases. Far from the composition characterized by Z = 1.8 eIa, the peak at Kpe is less pronounced (fig. lOb) and in the liquid state at greater distances to 2k~than in the amorphous state (see also fig. 11). We conclude from this that the electronic influence on structure is T-dependent and less pronounced in the liquid state. In general, the peak far below EF gets closer to 2kF as one goes from the amorphous to the liquid state. The shift of Kpe parallel to 2kF, the characteristic scaling with Z, and differences between the liquid and amorphous phases are better seen in fig. 11. The thin vertical solid and broken lines again enclose the range of the homogeneous amorphous phase. In fig. 12 the differences Kpe 2kF of all liquid and amorphous alloys known to us are plotted versus Z. The uniqueness of the alloys with Z = 1.8 e/ a is obvious. At this special Z-value the Fermi sphere is in excellent agreement with the pseudo Brillouin-zone boundary, indicating the optimal condition for phase stability. -

—


(a)

2kF

I

I

11

I

I

I

amorphous

—

~:

S(K)

at i4Sb(2)

10

(b)

~

Au-Sb

83

10)1,4)

5

20)18)

4

u,d

at % S

9

678

i~ii:

__

32

30(2~ 40)26)

,JKp 20 I:o~3~ Pe 40 liquid-Sn 60

~25~35(

80 I

)
1.8 e/a the pseudogap at EF should disappear with increasing T and the pseudogap far below EF should shift closer to EF. Noble-metal—polyvalent-element alloys were systematically studied during the last decade [21,22,42,43, 98]. With regard to UPS measurements, these alloys show a big disadvantage, namely noble-metal d-state contributions within the valence band. These strongly overlap the pseudogap far below EF and may even disturb a detailed analysis close to EF. 5.1. Au—(Al, Sn, Sb) alloys and (Ag 50 Cu50)75Ge25 5.1.1. Total energy distribution curves

In fig. 15 the total EDCs of Au—(Al,Sn,Sb) are shown in the as-quenched and the well-annealed states. *) The spectra are normalized at binding energies between 1 eV and 2.5 eV (indicated in fig. 18 by *1 Pure Al is extremely sensitive to oxygen

and therefore shows a large peak of oxygen at E0 7 eV, whereas Au—Al is quite insensitive.

88

P. H/fussIer, Interrelations between atomic and electronic structures I

I

I

~ (a)

I

I

I

I

I

I

I

I

Au-AL

I

I

I

I

I

I

I

I

I

Au—Sn

I

I

I

I

Au-Sb

C

aJ

~IIIIIl

10

8

4

6

_________ 2

60 90 01~lO8

I

6

4

EB(eV)

I

I

I

I

I

~2 (b)

I

I

0

2

_

~50 70 60 80 100 10 8

6

2

4

E~(eV)

I

I

I I

I

I

I

Au-At

I

I

I

0

60 40 90 100

E

8(eV)

I

I

I

I

I

I

I

Au-Sn

I

I

I

I

Au—Sb

VI

111

40~40 ~30 60 50

60 50

60 50 70

80 90 I I 108

I

I 6

I

I 4

I

I 2

I

I 0

E5(eVl

108

I

6

4

2 E9(eVl

10

_______________________________ I I

8

6

4

2

90

E5(eV)

Fig. 15. Normalized UPS spectra versus binding energy [42].Satellite and secondary-electron corrections have been applied as described in section 4.1. The numbers denote the content of the polyvalent elements in at%. (a) As-quenched; (b) crystalline state.


89

thin broken lines) to intensities which themselves vary with composition as the DOS varies with the FEM. This normalization procedure depends primarily on the s- and p-state intensities. The different d-state intensities are therefore caused by an artificial enhancement, compared to the s- and p-states, due to different photoionization cross-sections in Al, Sn and Sb [99]. The structural state strongly affects the DOS. Comparing the as-quenched (fig. 15a) with the well-annealed (crystalline) samples (fig. 15b), the differences are obvious and both states are easily distinguished. Below we only focus on the as-quenched samples. 5.1.2. Noble-metal d-states The Au 5d-states, split by spin—orbit coupling into Sd3,2 and 5d512 states, are not at the focus of the present review but are discussed for two reasons. First of all, it is shown that these states are situated at reasonable distances from EF. Only minor contributions, if any, occur at EF. Secondly, their shape

indicates that the EDS of the amorphous state can be interpreted in terms of the band DOS masked by photoionization cross-sections. In fig. 16 the Au d-states, taken roughly from the total spectra by subtraction of a square-root-like DOS for the s- and p-states, are plotted versus EB. In this figure the intensities are normalized with the

Au content at the Au 5d312-peak in order to avoid the artificial enhancement mentioned above. Within the amorphous phase, the d-states are symmetric, indicating that UPS data can be interpreted in terms of the band DOS [100]. In the partly crystalline samples, at the Au-rich side, the most significant difference to the amorphous samples is seen in the peak closest to EF.It becomes flat on top (fig. 15) and approaches the shape of polycrystalline Au, indicating phase segregation or clustering effects even

during deposition at 77 K. *) Another difference is the decrease of the half-width of the Au 5d312-peak and its rapid shift closer to EF as compared to the amorphous region (fig. 16). (a) at%Ai

1

Au—At

/I

Au—Sn

I~IIIl

‘‘‘k’

‘

Au-Sb

i)i~t ot% so

0 E~(eV)

(nI¼4~

164

0 EB(eV)

at%Sb

10

~~‘I~I

E~(eV)

Fig. 16. Au d-states, split by spin—orbit effects into Au Sd,, 2 at large and Au 5d,,2 at smaller binding energies [42]. Solid lines indicate the amorphous samples and broken lines the partly crystallized samples.

In fig. 16 some of these peaks are hidden by spectra of the amorphous state.


90

The noble-metal d-states are located in regions where one expects the pseudogap which is caused by the structural peak at K~.For Au—Sb alloys close to pure Sb the low intensity of the Au 5d512 peak (figs. 15 and 16) is thought to be caused by its position right in the middle of such a deep pseudogap (fig. 23). In the following subsection the discussion is focussed only on details near EF. 5.1.3. Structure-induced pseudogaps at EF The regions near EF are drawn enlarged in figs. 17 and 18. In the latter, the spectra are numerically smoothed. Pure Al as well as pure Sn are metallic with a DOS which increases towards EF. *) In contrast, pure antimony grows in an amorphous semiconducting state with a strong decrease. * *) The UPS spectrum of pure Au goes down continuously with no indications of a pseudogap. Within the amorphous range, on the other hand, all three systems show a significant decrease towards EF, interpreted as part of a structure-induced minimum in the DOS [21]. Inverse photoelectron spec2) estimated troscopy may reveal the part above EF. In figs. 17 and 18 the binding energies EB(Kpe/ according to eq. (14), are indicated as full dots, and clearly show the close relationship of the pseudogap to the corresponding structural data. Whereas the free-electron model used in these calculations is quite a good approximation for the metallic state, this is no longer true for the semiconducting state. EB(KPeI2) of pure Sb, for example, is expected to be exactly at EF. Accordingly, the use of the FEM for Sb gives a value for 2kF which is approximately 5% too large (see also figs. 10 and 11). I

~

.!

\

[)~II

I

~I

~\ Au-At

\~I\\

I~ \tII1

I

\I

at/

I

~I ~

\\~\\‘

Au-Sn

At

,,

I~\ I I~l

\

)\ ‘

at / sn

t~\\~~\ ~ I

~

\

\

~I

\~\,

I

I

Au—Sb

at /

Sb

\

I~

~—~----.~~

--

~

~

\

\

-

1

.,

L

/

\\

I

—

~

I

‘0~°’

~

-

\

1

‘/

L

~ ~-< (~)

L

__________

E 8(eV)

E6(eVl

E0(eV)

Fig. 17. Enlarged regions of some UPS spectra near EF [42].The numbers denote the content of the polyvalent elements in at%. Satellite and secondary-electron contributions are both subtracted. The energies E0(K,,,,12) are indicated as full dots (•). *) Some **)

structures in these curves may be caused by direct transitions in the crystalline state. After crystallization, pure Sb is a semimetal and shows a much higher DOS at EF (fig. 15b).

P. Häussler. Interrelations between atomic and electronic structures

at%AL

0

at/

91

~

0Sn

20

20

40

at%Sb

~

20

40

Au-Al

40

Au-Sn

60

,

60

AuSb

80

80

80 100

/

~—-—

0

-.

/

0

,.

0

40 20 E

(~f)

60

II)’

80

4

3 2

3m

~

60

100

100

E

(~)

4

3

60 80 at%Sb

80 af% Sn

at/0A1

1 ou00 E~(eV)

20

4

3

2

1 0100 E5leV)

2

1

0100 E6CeV)

Fig. 18. Three-dimensional plots of some UPS data near EF. The thin horizontal solid lines, as a guide to the eyes, indicate approximate levels of equal densities of states. The thin broken curves indicate where the normalization to the FEM has been performed. The shaded areas indicate regions of the pseudogap at E,,. Only satellite intensities were subtracted.

The existence of a pseudogap close to EF is obviously typical for the amorphous HR phase under consideration.2kF) Differences between with structural differences. In isfig.obvious. 19 the are depicted versustheZ.systems At Z = may 1.8 ecorrelate I a the universal behaviour, once more, intensities S( Above this value the three systems become different. Over the whole amorphous region S(2kF) of Au—Al is h~hand nearly constant. *) Au—Sb as well as Au—Sn both show a decrease of S(2kF) with increasing Z, and the former system shows, in addition, an upturn of S(2kF) at Z 3 e/a, possibly caused by the increasing influence of covalent bonds. Accordingly, in Au—Al the pseudogap at EF is quite deep over the whole amorphous region (figs. 17

and 18) and in Au—Sn becomes less pronounced close to pure Sn. Whereas at the Au-rich side the pseudogap of Au—Sb is similar to those of Au—Al and Au—Sn, close to pure Sb it becomes more pronounced, in agreement with the approach to the semiconducting state. In this region S(2kF) is no longer directly related to the depth of the pseudogap. Similar to the scaling behaviour of structural properties, figs. 17 and 18 give the impression that when

pure Au is approached the composition range where the pseudogap exists depends on the valency of the polyvalent element. In Au—Sb it occurs up to a smaller content of the polyvalent element than in Au—Sn or Au—Al, due to the higher valency of Sb. 5.1.4. Scaling with Z This scaling behaviour may be seen more clearly in fig. 20. Three Au alloys with Z -

1.8 e/a and

various compositions are depicted. Irrespective of the strong differences at other Z-values (figs. 17 and 18), these alloys seem to show a nearly identical pseudogap at EF. The small differences, disregarded in *1

Accordingly, Au—Al has a high resistivity and a large negative temperature coefficient of the resistivity over the whole amorphous region

[101,102] in agreement with Ziman’s theory [64].

92

P. H/fussIer, Interrelations between atomic and electronic structures Au 3

St2k~l

aAL

.Sn

I

I

•Sb

I

I

a •

I

I

I

Al40 2-si

.

1—_.

.

~

Ag 35Cu 37.s6e2s

~

.5

Ii

Z=1.8e/a

I

0— 1

I

2 3 2kF), the intensity of the structure factor at 2k,,, versus Z. Fig. 19. The vertical S( line indicates Z 1.8 eta. The data are from refs.

[31, 86, 101].

=

I 432100

L~

00

Ee~(eV)

Fig. 20. Different amorphous Hume-Rothery alloys with Z [98].The spectra are normalized to the same intensity at EB

1.8 eta

= 1.5 eV.

the past because of limited experimental resolution [22,98], might be real as other measurements below indicate (section 6.4). (Ag 50Cu50)75 Ge25 is included for comparison and shows similar depressions at EF [98]. The latter system is not directly comparable with the Au alloys since, generally, quantitative differences occur between alloys with different noble metals (see also sections 5.4 and 6). 5.2. Pure elements, Pb—Bi alloys, and TI—Bi alloys

Elements like P, Si, Ge, or Sb, so-called glass formers, are easy to prepare in the glassy state but are insulating or semiconducting due to covalent bonds. With the pioneering work of Buckel and Hilsch, Ga, Bi and Pb—Bi became the first amorphous metals ever known [103].Whereas Ga and Bi, unfortunately, crystallize below 20 K, Pb—Bi shows reasonably stability. Its structure shows similar 2kF and alarge smaller one close to it [104].For Pb—Bi features as mentioned above, a large well below and Tl-Bi, UPS measurements werepeak recently reported of both the liquid and the amorphous state [37,38, 84]. In principle, liquid metals can give similar information on electronic structure but are difficult to handle concerning photoemission experiments due to surface contaminations and a high vapour pressure of the constituents. The former problem has been surmounted by Indlekofer [47J, and extensive UPS measurements have since been reported by this author for those liquid elements with low vapour pressure. 5.2.1. Structure-induced pseudogaps far below EF As shown in fig. 21, a deep pseudogap far below EF occurs in the valence band of some liquid elements*), this being attributed to a dehybndization effect of the s- and p-states. The interpretation as a dehybndization effect was experimentally supported by UPS measurements at different excitation *1

Similar data exist for liquid Ge, Sn, and In [47].


M~4~EB~4~/~

________

Di 3O0C,hv~5O.I.V

~

93

j

10

I

I

gap

B

(eV)

I

/

Bi PbsOBisoIJ

—

-

(Au,Ag,Cu)

20Sn~

TI

3Io~(.ss’,v

212W

~

~

\\.I,_,.__,_._I__.:L_,%:sI~P~

______________ 12

10

8

6 1 2 BINDING ENERGY leVi

EF=O

Fig. 21. Valence-band spectra of different liquid elements [47]. UV excitation was performed by He I (hv = 21.2 eV), Hell (hv = 40.8 eV), or He 11* (ho’ = 48.4 eV), respectively. The binding energies EB(KP/2) are indicated by vertical arrows. The spectra are corrected for satellites only. Structure data are from ref. [105].

, j 1~

:

~e

2

~////~“Tt In

0

2

Pb30Bi70

T150B150

4

6

—

E(~)

10

B2

(eV) Fig. 22. Comparison of ~ with E5(K,,/2) [38]. Open symbols indicate liquid-state data [47],closed symbols data of the amorphous state. The horizontal width of the symbol of liquid Ge indicates structure uncertainties when data from several sources are considered. Structural data are from refs. [31, 86, 87, 104]. The vertical length of the liquid-state symbols indicates the width of the pseudogaps as reported by Indlekofer [47].

energies, and was theoretically deduced from strong non-localities of the pseudopotential, resulting from relativistic effects which tend to bind the s-states more strongly than the p-states [83].Relativistic effects also influence the atomic structure, and are assumed to be responsible for the transition from an open structure of liquid Si and Ge to a more close-packed fcc-like structure of Sn and Pb in the IVth row of the periodic table, accompanied by the enhanced dehybridization of the s- and p-states. This interpretation, on the other hand, is inconsistent with the UPS data of amorphous Sb and liquid Bi. These both show even more pronounced pseudogaps far below EF than liquid Sn and Pb but a quite open structure with 3.8 and 8.8 nearest neighbours [31,105], certainly not close to fcc. 2) to the gap Indications of a21) direct structural influence are interpretation, seen in the close of EB(KpI position E~’~ (fig. [37,38]. According to this the relationship peaks on both sides of a pseudogap are interpreted as pseudo Van Hove singularities (see especially liquid Bi). The shift of K~closer to 2kF from Bi to Hg obviously forces the pseudogap closer to EF. Mercury exhibits only one peak in S(K), which is close to 2kF. Subsequently, the pseudogap exists at EF only. Figure 22 compares ~ (the positions of pseudogaps deduced from UPS measurements) with the corresponding binding energies EB(KPI2). The linear relationship clearly shows the structural origin of the pseudogap far below EF.


94

Figure 23 depicts the valence-band spectrum of pure Sb, one of the aforementioned semiconducting amorphous systems. It clearly shows both the structure-induced pseudogap far below EF and the deep pseudogap at EF. The structural relationship of the former is shown in fig. 22 and, obviously,

corresponds well to the other liquid and amorphous systems. Liquid Sb is known to be a good metal [23].The pseudogap at EF, hence, is expected to disappear from the amorphous to the liquid state. Whereas UPS data are yet unknown for liquid Sb due to its high vapour pressure, for Pb—Bi and Tl—Bi a direct comparison of the amorphous with the liquid state was published recently [37, 38]. 5.2.2. Comparison of the amorphous with the liquid state The valence-band regions of two different amorphous alloys are compared with those of the corresponding liquid state close to the melting point in fig. 24. In order to show general trends without further data treatments the spectra are shown as measured with only satellite intensities subtracted.*) The overall increase towards larger binding energies is caused by secondary electrons (fig. 14) and,

additionally, for T150Bi50 by a contribution of Ti 04,5 VV Auger transitions [47]. For a detailed discussion, secondary-electron contributions were subtracted as described above (see also the inset of fig. 24). The final corrected spectrum of Pb30Bi70 is shown in fig. 25. Similarly to pure Sb, there is a high intensity at small binding energies and, separated by a pseudogap, a minor intensity at larger energies. * The relations of ~ with EB(KPI2) of Pb50Bi50, Pb30Bi70 and Tl50Bi50 correspond well to the other data (fig. 22). A shift ~ of the pseudogap far below EF towards EF from amorphous the 2kF,the seems to occurto(fig. liquid state, as claimed above due to the corresponding shift of K~ towards 24), although a direct check is impossible due to the lack of structure data in the case of Tl—Bi alloys.” I

I

I

I

I

I

I

I

I

I

I

I

IA

Ea(eV)

77 K

Fig. 23. Valence-band spectrum of amorphous Sb at T = (Hel, ho’ = 21.2 eV) [221.Satellite intensities and secondary-electron contributions were subtracted as discussed in section 4. E, indicates the lower band edge according to the FEM, and E 0(K9/2) the approximate range of structure-induced electronic states deduced from eq. (14). The thin solid curve may schematically indicate the valence band, masked by partial photoionization cross-sections, without the pseudogap far below EF. *

1

* *1 “

Satellite intensities would be nearly invisible on this scale. The peak at E0 = eV is again interpreted as pseudo Van Hove singularity.

3

E8(K~’/2)of Tl,0Bi,0 in fig. 24 is deduced from an interpolation between structure data of pure liquid TI and Bi.


Pb Bi

‘

Ii uid

‘

‘

~

10

10

8

5 E,teV)

6

TL Bi

‘

~

~m~[Iq

95

h uid

0

~ E

0(eV)

010

8

6

4 E0(eV)

0

Fig. 24. A comparison of the amorphous with the liquid state of Pb,0Bi7, and Tl,0Bi,0 [38].The amorphous samples were measured at T = 77 K and the liquid samples just above their respective melting points (Hel, ho’ = 21.2 eV). The individual curves are vertically shifted with regard to each other and are normalized at E5 = eV. Liquid Pb,0Bi7, was measured with a different retardation energy and therefore shows different secondary-electron contributions. Liquid Tl,0Bi,0 is from ref. [84]. The inset shows the secondary-electron contributions of amorphous Pb~Bi70 fitted to the experimental data. at EB> E, 10.5 eV. E, indicates the lower band edge.

3

5.2.3. T-dependences of the pseudogap at EF As seen for both Pb30Bi70 and T150Bi50 (fig. 24), the pseudogap at EF partly disappears from the amorphous to the liquid state. Figure 26 shows the valence-band region of Pb30Bi70 near EF in more detail for different temperatures [38].A linear extrapolation towards EF serves as a reference. The inset shows the normalized deviation of the DOS at EF with temperature as a continuous variation from the amorphous to the liquid state. Well above the melting point the pseudogap obviously disappears corresponding to the disappearance of the peak in S(K) at K 2kF [74,86, 106]. T-dependences of the density of states at EF are of great importance for electronic transport properties versus T (ch. 6). Very

recently UPS measurements, together with resistivity measurements of Si-rich amorphous Au—Si alloys close to the metal—insulator transition, revealed an apparent close relationship between the pseudogap at EF, its variations during annealing and ion bombardment, and the corresponding resistivity [107]. 5.3. Mg—Zn, Zn—Ca and Ca—Mg alloys Mg, Zn and Ca are simple metals with no or minor d-state contributions in the valence-band region. Binary mixtures of these elements can be prepared in the amorphous state by several techniques such as vapour quenching, melt spinning, or splat cooling. Over the years, therefore, Mg70Zn30 has played an outstanding role in the physics of glassy metals. With the help of these alloys, a deep understanding was brought into the field of glassy metals by, among others, Hafner and coworkers [36, 55, 79, 108] on the theoretical side, and Mizutani with coworkers [109,110] on the experimental side. Pseudopotential theory is applicable and the atomic as well as the electronic structures were both deduced by means of ab-initio calculations. The peak in S(K) at K = K~,called 2kF dominates [110,111].a prepeak in these alloys, has a minor intensity, whereas the peak at Kpe In fig. 27 the valence-band spectra of differently prepared alloys of Mg 70Zn30 are compared with each other. Whereas for 3 eV> EB EF the spectra are in excellent agreement, at larger binding energies differences occur due to weak oxygen contaminations in two of the samples and a peak caused

96

P. Häussler, Interrelations between atomic and electronic structures I

I

I

Ii

I

I

I

Pb3~Bi7~

-~

E

~2LL

‘

~

I

10

8

6

I~

,II.III.111:SI

I

::~ Pb

30 Bi70

I

12

•‘

~

E~(~)

I

~:

4

EB(eV)

I

06

0

Fig. 25. Valence-band spectrum of amorphous Pb,0Bi7, with all corrections performed. E, and EB(KP/2) were estimated as in fig. 23.

I

20

I I

I I

1.5

I

I I 1.0

I

I

E8 1eV)

I

III

II

I

0.0

Fig. 26. Temperature dependence of the pseudogap of Pb,0Bi7, at E,, [38]. Secondary-electron contributions are subtracted and the intensities are normalized at E~= 1.6 eV. The inset shows ~N(EF)/NQ(EF) versus temperature. T, and Tm represent the crystallization temperature of the amorphous state and the melting point of the crystalline state.

by Cu impurities on the surface of one. *) After extensive sputter cleaning, the splat-cooled sample (+) seems to be free of oxygen and hence might come closest to the true valence-band spectrum. The width of the band is in excellent agreement with the free-electron value as indicated by E~. When EF is approached there is a weak but clear decrease of the EDC, indicating a structure-induced

pseudogap similar to that in the noble-metal—polyvalent-element alloys reviewed above. In contrast with these alloys, but in excellent agreement with the weak prepeak in S(K), there is no clear pseudogap well below EF. A small dip at E8 5 eV close to E~(K~I2) might indicate remaining structural influences of the weak prepeak [1111(see the crosses only). Due to the contamination with oxygen, only the region 3 eV> EB EF is considered and shown together with other compositions and the Ca alloys in fig. 28. Theoretical results are included for comparison. With increasing Zn content, the decrease towards EF gets even more pronounced. Unfortunately, structural data of the amorphous state of other2kF, Mg—Zn far state, from Mg70Zn30, are as in alloys, the liquid the pseudogap as yet unknown [1111. B ut, since Kpe is expected to be close to should stay close to EF [63]. The theoretical curves of Mg—Zn, apart from the region close to EF, where a weak but distinct structure-induced depression of the s- and p-states has been deduced, closely follow a free-electron square-root behaviour. Accounting for the partial photoionization cross sections, these calculations, including the pseudogap at EF, were found to be in excellent agreement with the experimental data [79]. Zn—Ca behaves differently from Mg—Zn. Whereas theoretically the s-states of Ca and Zn should still show a depression towards EF, the Zn p-states and in particular the Ca p- and d-states increase strongly, overlapping any pseudogap at EF for alloys with high Ca content [811.Similar features are seen in the *

The Cu peak occurs after sputter cleaning because the Ar beam strikes the Cu clamps holding the thin-film substrate, and redeposits

spurious amounts of Cu onto the film surface [63].

P. Hdussler, interrelations between atomic and electronic structures I

I

I

I

I

I

I

70Zn30

-02

Cu

6

4

II

/ ~

Mg

IS

97

4 0 Fig. 27.

10 A comparison of glassy Mg

8

E~

2

E8(eV)

70Zn30 prepared by different techniques [63].S: vapour quenched, sputtered for 2 mm (Art, 3 kV) [63],LI: melt spun, sputtered for 30 mm (Art, 5kV) [112J,+: splat cooled, sputtered for 235 mm (Ar*, 3kV) [113].Satellite and secondary-electron contributions were subtracted as described above. The spectra are normalized at E5 = eV to the same intensity. The strong increase at EB = 9 eV to larger binding energies, clearly separated from the valence band, is caused by Zn 3d-states.

1

~

Mg,,Zn100,,

‘

zn,,~a100,,

~

I

~

ca,~g100.~~

I

I ~I3

a

I

~

~

~

3

~

2

EB(eV) 0

I

Zn~Ca,00..~

3

2

E6(eV) 0

I

I

Ca,Mg100,

‘~

3

2 EB(eV)

[63]. All

~

0

2

Fig. 28. Valence-band spectra of different Mg—Zn, Zn—Ca, and Ca—Mg alloys near EF spectra are normalized at EB = eV to the same intensity. (a) Experimental data without accounting for partial photoionization cross sections. Satellite and secondary-electron contributions were subtracted. (b) Theoretical data 179—82, 1141. The dashed lines indicate the free-electron DOS.

98


experimental data although weak depressions towards EF still occur. Similar conclusions can be drawn for Ca13Mg87. These trends in the DOS at EF are supported by other measurements [63](see also ch. 6). After correction for electron—phonon mass-enhancement effects, the electronic specific heat of Mg70Zn30, for example, was found to be below the free-electron value whereas the Ca-rich Zn—Ca and Ca—Mg alloys exhibit a higher value due to the Ca 3d and 4p-states [63,115—117]. In agreement with the suggestions of Nagel and Tauc [32],the thermal stability increases with increasing depth of the pseudogap at EF [63].

Whereas Mg70Zn30 plays the role of a model system in the field of glassy metals, due to the high vapour pressure less is known about its liquid state. Accordingly, in the field of liquid metals the role of a model system is partly occupied by the noble-metal—Sn alloys [23].We finish the present chapter by giving details of the DOS of amorphous and liquid (Au, Ag, Cu)—Sn alloys. For the discussion of correlations of the electron—structure interrelation with other properties in ch. 6, the DOS at EF is described more quantitatively in terms of Mott’s g-value. A special normalization procedure for the UPS data is proposed and will find its justification when the g-data are compared with those from other methods. 5.4. (Au, Ag, Cu)—Sn alloys Disregarding the unfortunate position of the noble-metal d-states, which are right in the middle of the valence band, (Au, Ag, Cu)—Sn alloys have many advantages. After quenching they are amorphous between 27—80 at% Sn and metallic for all compositions. Besides these similarities, quantitative differences exist also2kF [42,93]. structure liquid[118].In Ag—Sn the on the Sn-rich side,amorphous for example, is than inThe liquid Au—Snfactor and of Cu—Sn corresponding state

less pronounced similar differencesat are expected [93]. Vapour quenching of pure Sn gives strongly distorted polycrystalline samples [119].Their electronic transport is quite different from those of the amorphous alloys but more free-electron-like than those of the well-annealed crystalline state [93].Moreover, a large peak in S(K) at K~and a small one at Kpe exist (fig. lob), the latter probably being caused by the high volume fraction of disordered regions. The valence-band spectrum (fig. 29b) is, at first glance, similar to that of amorphous Sb (fig. 23) with most significant differences at EF, where Sb shows a deep pseudogap whereas Sn is increasing. The separation of the s- and p-states is well pronounced and seems to be structure induced. The humps on both sides might again be caused by pseudo Van Hove singularities. Disregarding the d-states, the different alloys with 80 at% Sn, superimposed as in fig. 29a, seem to

form a common s, p-band. Fortunately, the position of the d-states is such that the superposition reveals almost completely the pseudogap at EB 4.5 eV, which is induced by the peak in S(K) at K = K~. EB(KPI2) is estimated as above with K~as the mean value from Au 20Sn80 and Cu20Sn80. Van Hove-like structures exist again on both sides. For a comparison with the other systems the relation between EOBaP 2) is included in fig. 22. andSimilar EB(KPIprocedures as just described fail for other compositions due to their less favourable d-state positions and due to the vanishing structural weight S(K~)with composition (fig. 10) and, hence, a

diminishing depth of the pseudogap at EB(KP/2). Indications of hybridization effects between the d- and s, p-bands as claimed recently [120]were not observed. The largest difference between pure Sn (fig. 29b) and the amorphous alloys (fig. 29a) exists at EF.

P. Häussler, Interrelations between atomic and electronic structures -;;

I

99

1’’

I

Au~sn~

A Ec~!

C ~

.~

0

b) EB(eV) Fig. 29. Valence-band spectra of pure Sn (at the bottom) and (Au, Ag, Cu) see the text. The alloys are normalized to the same intensity at E5

=

20Sn1, showing the basics of the normalization procedure [42].For details 1.5 eV.

Whereas pure Sn shows an increasing intensity towards EF, for the amorphous alloys the DOS gets

depressed. To reveal trends versus composition henceforth the spectra are normalized. 5.4.1. Normalization, Mott’s g-value

The unique s, p-band, shown in fig. 29a, is the basis for the procedure of normalization. Individual differences of the systems close to EF are obviously limited to a region approximately 1 eV below EF. At EB = 1.5 eV, in a region where differences no longer exist, a normalization of all spectra to a curve which varies versus composition as the free-electron value, has been used as a first step. As in the Bi alloys, an extrapolation of the EDCs along a straight line up to EF was further applied in order to construct a reference point at EF if no deviations from the free-electron value should exist. This procedure can be performed only within the amorphous phase close to pure Sn. For higher noble-metal contents, d-states overlap the slope around 1.5 eV due to their shift towards EF. The reference points at EF, constructed as just described for (Au, Ag, Cu)20Sn80 and (Au, Ag, Cu)30Sn70, are hence used to fix at EF a second FEM curve versus composition. Subsequently, the intensities at EF are compared with the FEM values thus constructed, yielding g~~5 as a crude estimate of Mott’s g-value. 5.4.2. Structure-induced pseudogaps at EF The spectra near EF are depicted enlarged in fig. 30. The intensities are normalized relative to each other as just described. As already discussed, Au—Sn shows a well-pronounced decrease near EF for all amorphous samples (figs. 17b and 30a). In Ag—Sn a real decrease near EF is absent. However, when EF

is approached the increase of the DOS seems to be flattened. The flattening is most pronounced at 35 at% Sn, indicating a depressed DOS at EF which is especially pronounced for this composition (see

II

P. H/fussier, Interrelations between atomic and electronic structures

100

I

I

Ii~\I

Au Sn

I

II

I

I

Ag Sn

1

-

I

j

-

—

~

-

:

I

~

~

~

,‘,.

~ -

at%Sn

1 ~

~

~\\

~-.

Cu - Sn

~ll

~‘

I

/

I

.-~

I

-

.

-

~-~Z~f

I

—

I 90 m/~Sn

—

—

a~

(a)

(b)

at%~

-

(~)

E 5leV)

EB(eV)

E9leV)

[42,

Fig. 30. Enlarged regions of the (Au, Ag, Cu)—Sn valence-band spectra near EF 43]. The numbers denote the Sn content in at%. Satellite intensities and secondary-electron contributions were both subtracted. Between 80 and 27 at% Sn the alloys are completely amorphous. The energies E9(K~,/2)of electronic states, corresponding to S(K~,),are indicated (S).

also fig. 13). The EDCs of Cu—Sn also show a flattened increase towards EF at small Cu concentrations and a decrease at high Cu concentrations, almost completely overlapping with the Cu d-states. In fig. 31 gups is shown versus composition. Obviously, in all three systems it is smallest near 27—30 at% Sn (Z 1.8 e/ a) and increases with increasing Sn content. Large deviations from the FEM (g = 1) exist in the whole amorphous region of Au—Sn and Cu—Sn, whereas in Ag—Sn gups tends to agree with the FEM at 80 at% Sn (see also fig. 29). Differences correspond to quantitative structural differences. In Au—Sn, for amorphous example, theregion, peak in in 2kF throughout the whole 5(K) at K =with Kpe the is relatively large and isvery to agreement pseudogap, which wellclose pronounced over the same region (figs. 30a, 31a). For Cu—Sn at the Sn-rich side, the peak at Kpe is weaker and not as close to 2kF as in Au—Sn, so the result is a slightly more free-electron-like DOS. Quite similar to liquid Ag—Sn, for amorphous Ag—Sn the peak close to 2kF is expected to be much less pronounced than in amorphous Au—Sn and Cu—Sn. Accordingly, the DOS of amorphous Ag—Sn at EF is less depressed compared with amorphous Au—Sn and Cu—Sn.

The structural influence is more obviously seen in fig. 32, where the structure factor at 2kF is drawn ,( 1 g), the normalized deviation of the density of states being taken from fig. 31 [42, 43]. A linear relationship versus

1

—

—

g

= 1

—

N(EF)INO(EF)

S(2kF)

(16)


I

1~~’T’

‘II’

1S~~

I’I’I’

11

I 9

k

\~ ~ I

~

I

n

I i.oo.\

I -

Oihi.i1~

Au

9

1.0

—~O.—7A--~0

-

I’I’I’

I

9

i.e

101

0.5

\sh~,~Q

0.5

I~,~I/9

-

I

~

at%Sn

A9

at%Sn

Cu

at%Sn

Fig. 31. Electronic density of states at EF normalized to the FEM [42,43].The solid vertical lines indicate Z = 1.8 c/a and the broken vertical lines the amorphous alloys with the highest Sn content. ge,,,: metallic glasses, deduced from UPS measurements [42, 43]; g~~: metallic glasses, deduced from the specific heat without subtracting additional contributions due to low-lying excitations (section 6.1) [121, 122]; g~:liquid state, deduced from susceptibility data (section 6.1) [123].

seems to exist. With increasing S(2kF) the pseudogap becomes more pronounced and therefore indicates directly the influence of structure as claimed in section 2.2. Based on the limited number of structural data, Au—Sn as well as Cu—Sn fall on the same straight line. According to the linear relationship of eq. (16), in the following chapters we may also use (1 g) for the (Au, Ag, Cu)—Sn —

alloys whenever S(2kF) is needed for discussion. Equation (16) is not generally valid. Alloys with strong covalent bonds show no strict proportionality (see Au—Sb above). From the structural data of fig. lOb, which show a decrease of S(2kF) from the amorphous to the liquid state, and from the results just discussed, we expect (1 g) to become smaller from the amorphous to the liquid state, in a manner quite similar to that shown above for Pb—Bi alloys. —

5.4.3.

Comparing the amorphous with the liquid state

UPS spectra of liquid Au—Sn at the Sn-rich side were measured by Indleko~eret al. [124](fig. 33). A fairly close general similarity to the amorphous alloy is observed. Comparing both in more detail, one notes that the pseudogap at EF has disappeared within the liquid state. Similar temperature dependences to that in Pb—Bi seem to exist. Consequently, deviations of electronic transport properties from the

free-electron behaviour may disappear from the amorphous to the liquid state (ch. 6). Deviations of electronic transport properties of liquid and amorphous alloys from the FEM exist in particular for alloys with Z 1.8 e /a [19]. This may indicate deviations of the DOS from the free-electron value at this particular composition even in the liquid state close to the melting point. Unfortunately, UPS measurements of the liquid state at this composition do not exist. Concluding this chapter, we summarize that atomic structure influences the electronic DOS in a manner quite similar to that in the crystalline state. Pseudogaps exist far below EF as well as close to EF, with the latter shifting parallel 2kF) to EF as far a function of composition. Cu)—Snwith the and, from compositions with Z For = 1.8(Au, e/ a, Ag, disappears pseudogap at EF is proportional to S(

102


I

I

I

I UPS, He I

(hv=21.2eV) Au

30Sn70

3

I

I

I

amorphous

I

S(2kF)

35,

T~-193~CJ

//‘

0

0.1

‘.,,j

/

________

02

Q3

0.4 1

-

0.5

8

4

2

EF=0

BINDING ENERGY (eV)

g

Fig. 32. Relation between S(2kF) and (1 — g) of amorphous Au—Sn and Cu—Sn [42,43]. The numbers denote the Sn content in at%. A: Au—Sn, I: Cu—Sn.

6

Fig. 33. Temperature dependence of the valence-band region of amorphous and liquid Au—Sn [124].

rising temperature well above the melting point. Pseudogaps also exist, independent of noble metals, in alloys like Pb—Bi or Mg—Zn. The lack of a pronounced pseudogap at EF in the liquid state (far from compositions with Z = 1.8 eta) or its disappearance with rising T can be understood as the disappearance of electron— -

structure influences with T.

6. Correlations between the DOS at EF and other

properties

The purpose of the present chapter is fourfold. Firstly, measurements of the specific heat, the upper critical field B~2of the superconducting state, the magnetic susceptibility, and the Knight shift are considered in order to find direct support for a pseudogap at EF. Secondly, since a pseudogap at EF influences the band-structure energy and hence the total energy, we discuss consequences to the heat of mixing, the thermal stability, and the glass-forming ability. Thirdly, since electronic transport is dependent N(EF), influences the pseudogap or, synonymously, of thearestructural peak at 2kF, on theonresistivity, the Hall ofcoefficient and particularly on the influences thermopower presented. The effect of phonon rotons on p(T) and S(T) and their influence on specific-heat anomalies are discussed in some detail. Finally, it is shown that the scaling behaviour of atomic and electronic structure versus Z is reflected in these properties too.


6.1.

103

Electronic density of states at EF

6.1.1. Specific heat and the upper critical field B~2

After subtracting the lattice contribution, the low-temperature specific heat takes the form ~ T. In simple metallic glasses the experimental y-value has several contributions [24,125], 2k~N(EF). Yexp = Ye(1

+ A) + Ydis’

Ye =

(17)

~1r

The electronic contribution y~is proportional to the DOS at EF. With N(EF) = No(EF), A =0, and Ydis = 0, the free-electron value y~would occur. The electron—phonon coupling constant A is independently estimated from McMillan’s formula [126]

—

( 18

p,*ln(øI145T*I~)+104 (1_O.62~*)ln(øDI1.45T~)

and accounts for electron—phonon interaction. T~Cdenotes the transition temperature to the superconducting state, 9D the Debye temperature, and i * an effective Coulomb potential. Ydis in eq. (17) is a non-electronic contribution caused by energetically low-lying excitations of the disordered state. In the present alloys, Ydi. can be interpreted as being caused by phonon rotons [43],which are separately discussed in section 6.3. This contribution can easily be distinguished from the electronic contribution if specific heat is measured in the superconducting state well below T~C.Reliable results of N(EF) deduced from Yexp need both a subtraction of Ydis as well as a correction of (1 + A). Mott’s g-value from specific-heat measurements g 5~,subsequently, is defined as )‘~ gSh7

19

YexpYciis

(1+A)70

Specific-heat measurements in systems which become superconducting, as is the case for most alloys under consideration, require, if they are performed without an applied magnetic field, a proper extrapolation from above T~C down to T = 0 K. In a magnetic field, T~C gets suppressed and specific-heat measurements are free of those difficulties. The application of a magnetic field has a further advantage. Measurements of the upper critical field B~2give independent information on N(EF) according to [127] dB2(T) dT

4k e (20)

T=T~~P(FX)

Until recently, Ydis of metallic glasses was assumed to be orders of magnitude smaller than Ye [125] and therefore has been neglected [24].In Mg70Zn30 it is indeed negligible [125]and, additionally, T~C~ low enough to perform a reliable extrapolation down to T =0 K. Using A = 0.3, N(EF) was found roughly 30% below No(EF) [24, 110, 128], in agreement with a depressed DOS at EF reported above (section 5.3). In agreement with UPS the specific heat of Zn—Ca is different. A large y~~~-value has been attributed to the increasing contribution of Ca 3d, 4p-states at EF

[24, 110].

Specific-heat data of systems containing a noble metal are known for a number of ternary (AgCu)—M alloys (M = Mg, Si, Ge) [24,129] as well as for Cu—Sn [121]and Au—Sn [122].(AgCu)—Mg shows

104


y0 close to 1.5. This large ratio can only partly be attributed to 1 + A since A is expected to be about 0.2—0.3 [129]. Further corrections due to Ydis have been neglected and so the question of whether this system shows N(EF) < No(EF) remains unanswered. Ratios Yexp’YO of (AgCu)—Si and (AgCu)—Ge are Yexp’

close to unity and the situation is thus different. Since

Yexp needs

to be corrected by

Ydis

and further by

1 + A, Mizutani and Yoshino [129] reached the conclusion that N(EF) in these alloys is likely to be depressed by about 20—30% below No(EF), which agrees with UPS data (fig. 20). Similar conclusions were drawn for amorphous Cu—Sn [121].These data (g5~),corrected only for 1 + A, are included in fig. 31c. Above 30 at% Sn, there is excellent agreement with g~. Whereas the latter measurements are hampered by extrapolations from above T~C,measurements of Au—Sn, performed down to T = 300 mK in an applied magnetic field [1221,allow a more reliable extrapolation. After correcting the measured y-value by 1 + A, surprisingly good agreement with g = 1 was obtained over a large concentration range (fig. 31a). The strong increase close to Z = 1.8 e/a was found to be attributable to Ydis by measurements in the superconducting state [122]. After its subtraction, surprisingly, the agreement of g5~ with g = 1 improves further. Similar effects are also attributed to the strong increase in Cu—Sn around 30 at% Sn [122], with the curious result that, unlike the situation in Au—Sn, there is now excellent agreement of g5~with g~ down to 25 at% Sn. As a possible reason for the disagreement of g5~with in Au—Sn an improper evaluation of A from eq. (18) has been put forward [122].The discrepancies of g~between Au—Sn and Cu—Sn, therefore,

remain unclear and hence need further consideration. Specific-heat measurements, obviously, give contradictory results for Mott’s g-value. The situation might get somewhat clarified if gB~.,-data from critical field measurements are considered. for amorphous Au—Sn falls below ~ [1~2]and hence below the free-electron value by about 10%, whereas for amorphous Cu—Sn ~ is larger than g5~but still less than the free-electron value by about 20% [121], giving indications f~a pseudogap at EF in both systems. Whereas specific-heat measurements give N(EF) at T = 0 K, other methods such as the magnetic susceptibility or the Knight shift are capable of revealing the DOS at elevated temperatures. Electronic magnetic susceptibility The magnetic susceptibility Xexp of simple non-magnetic metals consists basically of two contribu-

6.1.2.

tions, Xexp = Xe +

Xi’

(21)

an electronic contribution

Xe and an ionic one composed of three contributions,

x1.

The electronic magnetic susceptibility itself is

XeXp+XL+Xe~e,

(22)

with X~the Pauli paramagnetic term, XL the Landau diamagnetic term, and Xe-c a contribution due to electron—electron interaction. With X~= ,4N(EF) and XL = X~the electronic contribution becomes —

Xe = ~p~N(E~)

+ Xe-e ,

(23)

with the band term X~= ~~N(EF) N(EF) = No(EF)

and Xee

= 0,

on the right-hand side. ~ denotes Bohr’s magneton. With

the free-electron value x0 will result. Experimental data, properly


105

corrected for the other contributions, reflect the DOS at EF and hence g~1Mott’s g-value from susceptibility measurements, gns=XexP~e_e_Xi

(24)

.

A detailed analysis of structural influences on Xe in terms of pseudopotential theory was performed by Dupree 2kF andand Sholl [123]for liquid Cu)—Sn. positioninofdetermining the main peaks of S(K) of with their intensity were(Au, bothAg, found to beThe important the variation Xe respect to with concentration. Similar relationships result from eqs. (16) and (23). The g~-values of liquid Au—Sn, Ag—Sn and Cu—Sn, taken just above the melting point, are included in fig. 31. Again structure-induced deviations occur, in particular for alloys close to Z = 1.8 e Ia. But they are less pronounced than in the amorphous state and hence are in qualitative agreement with the T-dependence of the UPS data (fig.

33). Similar anomalous variations with composition were observed by Takeuchi and coworkers [130, 131] for many liquid systems including Cu—Sn, Cu—In, Ag—Sn, and Ag—In, and by Terzieff and coworkers [132] for Au—In and Au—Ge. The ternary system (Ag 50Cu50)—Ge has been measured for several compositions in the amorphous as well as the liquid state [133]. The variations of Xexp with composition and temperature (fig. 34a) are interpreted as caused by N(EF). In the amorphous state close to compositions with Z = 1.8 e/ a, there is a large depression of Xexp. In the liquid state, the depression gets weaker and tends to disappear as T increases. Similar T-dependences were reported for liquid Cu—In and Cu—Sn [130]. The T-dependence for the liquid state is shown in fig. 34b. At Z 1.8 e/ a the extrapolation to the

amorphous state roughly agrees with the measured values. This, on the other hand, can no longer be I

I

‘III’!’

Tri230K~*~T5.Vj

X0~X

—10

T~Tm

a)

~

-~

-20 ~

02040608010: Ag50Cu50

at %Ge

Fig. 34. (a) As-measured magnetic susceptibility ~ as a function of Ge content in (Ag50Cu,0)—Ge alloys [133]. The vertical line indicates Z = 1.8 eta. •: amorphous state; ~‘: liquid state above the melting point and A: liquid state at 1230 K; — x~the ionic contribution; ———: x0 + x. (b) Temperature derivative ax.~~/ ~T in the liquid state as a function of Ge content.


106

true close to Ge, since amorphous Ge is a semiconductor, which should show a deep pseudogap at EF, similar to that of Sb (fig. 23).

-

Although the depression in the range close to Z = 1.8 e / a seems to be overemphasized as compared to Xt’ these measurements led to the conclusion that the free-electron value of the DOS at EF is reached in the liquid state at high temperatures, well above the melting point, and that depressions become evident with decreasing T in a manner similar to that shown above for Pb—Bi by UPS (fig. 26). 6.1.3. Knight shift

Experiments on the Knight shift K yield independent information on the Pauli paramagnetic susceptibility Xp’ which is not masked by diamagnetic contributions. In the simplest approximation K ~5 given by K= ~1T(kp(O)I2)EXp,

(25)

(l~(O)I2)EF

with X~= p~N(E~) as above. denotes the density of conduction electrons at the nuclear site averaged over the Fermi surface. Assuming (I ~(0)~2 ) EF to be constant, variations versus composition and T are attributed to variations of X~and hence N(EF) alone. Experimental data of liquid Tl—Bi have been reported and in fact indicate a pseudogap at EF [134].This pseudogap, on the other hand, is most pronounced at the TI-rich side and gets even more pronounced with rising T and thus contrasts with the UPS data. Unfortunately, similar measurements have not been extended closer to Bi and hence it remains unclear whether the DOS gets suppressed again when pure Bi is approached, as deduced from the UPS data above. The increasing strength of the pseudogap with T is in clear contradiction with the general observation that a peak in S(K) at K 2kF disappears with rising T; this needs further

consideration. The binary Cu—Al system and the ternary system (Ag

50Cu50)—Ge are both shown in fig. 35. Most significant deviations from the free-electron behaviour arise at compositions with Z = 1.8 e/a. The depths themselves are in the range of 20—30% and hence in excellent agreement with the g~-data of

liquid alloys shown in fig. 31. This behaviour, once more, strongly supports the existence of a pseudogap at EF also in the liquid state for alloys with Z close to 1.8 eta. 10

‘I

•

Cu—At

t~ç50Cu50l-Ge

0

0 ~

\

-10 -20 30 —

/

‘7

\

-

0

Lu

-

•

20

1.1

40

60

I~

80

100

at%AL

.

0

20

Ag50Cu50

I

40

I

60

•

80

100

at%Ge

Fig. 35. Deviations ~K/K0 of the Knight shifts of liquid Cu—Al and (Ag,0Cu,0)—Ge from the corresponding free-electron values versus composition [135].The vertical lines indicate compositions with Z = 1.8 eta.

P. Häussler, interrelations between atomic and electronic structures

107

Summarizing the first section of the present chapter, where the specific heat, the upper critical field, the electronic magnetic susceptibility, and the Knight shift have been reviewed in order to give additional support to the existence of a pseudogap at EF, its variation with composition and T, we may state that many indications from other than UPS data indeed exist for a pseudogap in the amorphous as well as the liquid state. 6.2. Stability of liquid and amorphous alloys 6.2.1. Enthalpy of mixing

The heat of mixing, measured from the temperature variation in a liquid system on mixing the pure elements, gives macroscopic indications of the microscopic binding properties, e.g. random mixing, compound or cluster formation [136],as well as electronically driven short- and medium-range order. A negative heat of mixing (or heat formation) is often observed for binary alloys of Au, Ag, and Cu with polyvalent elements. Their asymmetric concentration dependence has received considerable attention [94,131, 135—139]. Takeuchi et al. [136]have attributed these data to the formation of so-called pseudo molecules. A fraction of the atoms of such a system is thought to be bound by some sort of covalent bonds. The pseudo molecules were not considered as stable compounds, but as having flexible structure, different from the rigid structure of real molecules. With time, some dissociate into free atoms and others are formed. Pseudo molecules and free atoms are thought to be in a state of dissociative equilibrium. The lifetime of pseudo molecules decreases with increasing temperature. T-dependences of the heat of mixing were explained in these terms. In contrast, Sommer [139]has interpreted similar data by a real chemical short-range order, where associates with well-defined stoichiometry get formed (association model). The thermodynamic data of many systems have been explained and the model has been extended to explain their glass-forming ability. The asymmetric heat formation on mixing may also be due to the formation of short- and medium-range order caused by the Friedel oscillations. Or, in other words, by the increase of UbS due to the band-gap formation at EF as described above. Largest effects are expected for alloys with Z = 1.8 eta (Kpe = 2kF), where the atoms are ideally located with respect to the Friedel minima (sections 2.1 and 3.1) and the pseudogaps are most pronounced (figs. 31 and 34). If parts of the occupied states at EF are shifted downwards by an appreciable amount, the corresponding reduction of the electronic energy causes heat formation. In fig. 36 data of several Cu alloys are shown for various compositions. The ternary system (Ag 50Cu50)—Ge is again included. A considerable amount of heat is indeed formed in these alloys at compositions with Z = 1.8 e / a. Similar variations versus composition were observed for Ag—Sn and Ag—Sb [131].The ideal commensurability of the atomic positions with the Friedel minima and, hence, the large band-structure contribution to the total energy readily explains these data. A relatively high stability of the liquid state, e.g. a low crystallization temperature, may occur. And in fact, the low eutectic minima, e.g., in Au—Si and Au—Ge have been explained along this line [32].In most systems of fig. 36, only less pronounced eutectic minima exist at corresponding compositions. In many other systems, like Ag—Sn or Cu—Sn, eutectic minima are even missing. Since during crystallization both the liquid and the crystalline state are in thermodynamic equilibrium and the mobilities of the individual atoms are fairly high, a considerable influence of crystalline phases or compounds on the stability of the liquid phase may have to be taken into account.

108

P. Hdussler, Interrelations between atomic and electronic structures •

I

~

0

/

I

..1r.

•

//

~

-

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0

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.~Q 100 0

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80 100 at%Sb

Ag50 Cu50 Fig. 36. Heat of mixing of several liquid alloys (Cu—In [140],Cu—Al [135],Cu—Sn, Cu—Ge, (Ag,0Cu,0)—Ge [135],Cu—Sb [136]). The vertical thin lines indicate Z = 1.8 eta.

6.2.2. Thermal stability of amorphous systems The crystallization temperature TK of an amorphous phase is similarly influenced by the crystalline

state. But due to the metastable nature of the amorphous phase, the transition to equilibrium may go through other metastable phases (Ostwald’s step rule). It is known from the literature [18,141, 142] that quenching techniques yield more uniform (metastable) phase diagrams. Therefore, due to the enhanced uniformity of the following crystalline state, different amorphous alloys are more comparable and the dominant electronic influence on the structural stability may, hence, become evident. In fig. 37 the crystallization temperatures of amorphous (Au, Ag, Cu)—Sn alloys are plotted versus 60G—’--—r-

ii.

I (K)

~

r

~i.

T~

TO

IX)

1K)

300

200

40G.

200-

~

•~I

-

300

-

200

-

-

1’ 100

100

100

I

0

____I_

0 Au

20

I

40

•

i’.

I

1.1.

60

80 100 at%Sn

0.._.___t_

0 Ag

20

I,I•I~

60

60

80 100 ot°A,Sn

....i...j.

0 Cu

20

I

40

•

I

6.0

•

80

100 at’!0 Sn

Fig. 37. Crystallization temperature TK of amorphous (Au, Ag, Cu)—Sn alloys versus composition [42,43].


109

composition. In all tl~reesystems the most stable state is given at 25—30 at% Sn, in the region where the pseudogap at EF is most pronounced (fig. 31). Whereas Au—Sn shows a stable crystalline compound at 50 at% Sn, in Ag—Sn and Cu—Sn similar compounds are absent. Subsequently, amorphous Au—Sn with 50 at% Sn has a crystallization temperature which is distinctly below the others. In Nagel and Tauc’s well-known publication on the stability of amorphous metals, glass-transition temperatures Tg are directly compared with S(2kF) [32].In the present review, for the (Au, Ag, Cu)—Sn alloys S(2kF) itself has been shown to be proportional to 1 g. Instead ofplotting Tg versus S(2kF), in fig. 38, therefore, TK versus 1 g is shown; this may be equivalent. The regions of the amorphous state are indicated by horizontal bars. For the reason just mentioned, Au 50Sn50 falls below the common curve. But for all the others a close relationship between TK and 1 g is obtained. It is remarkable that this relationship exists over the whole amorphous range. Different alloys are located on different parts of the same curve. Getting closer to 1 g = 0 apparently results in less stable systems, indicating that the stability of the amorphous state indeed requires a pseudogap at EF. —

—

—

—

6.2.3. Glass-forming ability

According to many authors, the occurrence of a deep eutectic minimum in the corresponding binary phase diagram is necessary to obtain a metallic glass [143].Of the present alloys, Au—Sn shows a eutectic minimum whereas Ag—Sn and Cu—Sn have none, although their amorphous phases can also be prepared over a wide range of composition. Accordingly, the existence of a eutectic minimum is more important for liquid quenching techniques than for the vapour quenching used for the amorphous alloys discussed here. Pseudogaps at EF or, synonymously, the existence of pronounced S(2kF)-values are thought to be the necessary conditions for phase stability of the present alloys. Amorphous alloys can be prepared for Z 1.8 e / a. Of the pentavalent elements a minimum of 20 at% has to be alloyed with 80 at% Au, Ag, Cu, and of the tetravalent, trivalent, and divalent alloys 26.7 at%, 40 at%, and 80 at%, respectively. The maximum content of the polyvalent partner, on the other hand, depends on the occurrence of covalent bonds. Elements like P, As, Sb, Si, and Ge, typical glass-formers with strong covalent bonds, can be prepared in a semiconducting amorphous state without any noble-metal contribution, whereas, e.g., Al, In, Sn, and Pb are metallic and need a content of at

400 TK

I

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I

I

I 28.1

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(K)

3ØØ

/ 28 S £

200

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i~I.k~32

100

~

Au/Sn

F

-i

I

Cu/Sn

0

0

I

0.1

I

02

I

0.3

I

0.4

I

05

0.6

1—g Fig. 38. Crystallization temperature TK of (Au, Ag, Cu)—Sn alloys versus 1 — g [42]. The numbers at some points indicate their Sn content in at%. A: Au—Sn, I: Ag—Sn, •: Cu—Sn. The horizontal bars indicate the regions of the amorphous state.

110


least 20 at% Au, Ag, Cu [19,93]. Bi and Ga are exceptions from this rule. Both are metallic after evaporation but exhibit extremely low_thermal stability [103]. For divalent elements both limits (Z = 1.8 e I a as the one and at least 20 at% noble-metal content as the other) tend towards one composition (Au, Ag, Cu)20M80 (M: Mg, Zn, Cd, Hg). Because of these limitations noble-metal—divalent-element alloys hardly become amorphous. And indeed, only Cu—Mg is capable of becoming clearly amorphous in a range close to Cu20Mg80 [144]. Firstly proposed by Nagel and Tauc [32]and in most cases never really confirmed due to strong influences of d-states at EF (a review is given by Oelhafen [145]),we conclude that the noble-metal containing alloys as well as the simple-metal alloys such as Mg—Zn and Pb—Bi show the electronic influence on thermal stability with high evidence. Obviously these alloys are model systems regarding electron—structure interrelations and their influence on phase stability. 6.3. Electronic transport of (Au, Ag, Cu)—Sn alloys

Electronic transport properties depend on electron—structure interrelations. Extensive work on the (Au, Ag, Cu)—Sn alloys has been reported elsewhere [19,23, 42, 43, 93]. Figure 39 shows the resistivity p( T), the thermopower S(T), and the Hall coefficient RH( T) of Sn-rich amorphous metals. Annealing hardly changes p( T) and RH( T) up to the crystallization temperature TK where pronounced variations occur (but see in figs. 41, 48 such data in enlarged versions). The resistivity is high and the samples show superconductivity at low temperatures. The crystallization temperature TK is defined as indicated. RH is in excellent agreement with— the free-electron value ~.

-

Cu25Sn75 TC~~

20

-0.2

Cu25Sn~

\

SIT) (IL)

\S0

0 ~

...

c)

~

R~

-10•

R8(T) (i&~~) 0

Au20Sn80 I

100

•

I

200

1(K)

300

Fig. 39. Electronic transport properties of Sn-rich metallic glasses versus temperature [43].(a) Resistivity, (b) thermopower, (c) Hall coefficient. In all figures only the temperature regions below the vertical bars are in an annealed state and hence reversible. The distinct changes at 180 K and 220 K indicate crystallization.


R~=—1_~j---withn=Zti0,

111

(26)

~ denoting the mean particle density. S(T) is proportional to T for T~C< T < T0. At T0 there is a knee, which results in a nearly linear dependence of S(T) on T for another 120 K. Henceforth we distinguish between S’( T), the lowtemperature thermopower below T0, and S’~(T), the high-temperature value above. S~(T), apparently, is close to the free-electron value [146] S°(T)=

T.

-

(27)

t(T) agree well with the corresponding free-electron value, Whereas at the Sn-rich both RH and S characteristic deviations arise with increasing content of noble metals. Figure 40 shows p, RH, and S’( T) IT of disordered (Au, Ag, Cu)—Sn alloys versus composition. We first discuss the concentration dependences and then characteristic T-dependences, which can only be partially seen in fig. 39. 6.3.1. Resistivity

The resistivity of liquid and amorphous alloys formed the focus of many studies and was reviewed in great detail over the years [64,89,90, 94, 151]. Disregarding multiple scattering effects, higher correlation functions, and non-local effects, Ziman’s weak-scattering description, in terms of nearly-free electrons, forms the basis for the following qualitative discussion. Strictly speaking it is valid for systems with a mean free path of the electrons much larger than the average interatomic distance. A more complete description is obtained from more subtle theories which often contain Ziman’s description as the limiting case [151].In general, the applicability of Ziman’s approach is assumed whenever RH equals R°H. For qualitative trends we simplify the discussion by disregarding partial structure functions or partial pair potentials. In reality these data are difficult to obtain and mostly unknown. Hard-sphere models, often used, are inapplicable since any electronic influence on structure is neglected. Ziman’s model gives 21

3irQ

Pz

=

e

S(K)lv(K)I .4. (K/2kF) d(K/2kF).

0m 2/13k2

F

(28)

0

Henceforth we shall use the following abbreviation for the integral: 2) = (S(K)Iv(K)1

f S(K)Iv(K)~.4. (KI2kF)3 d(K/2kF).

(29)

Ziman’s formula is generally assumed to be valid at high temperatures where electronic excitations are much larger than phonon energies (k 9D). Quasielastic scattering will result. This can be 8T> kBt described by the static structure factor S(K) and, due to the weighting with K3, strongly depends on S(2kF), the structural weight at the upper limit of the integral. At low temperature the static structure factor S(K) has to be replaced by the dynamic structure factor S(K, w), which includes inelastic scattering effects with rising T [88].


112 140’YI’I

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Fig. 40. Electronic transport properties of (Au, Ag, Cu)—Sn versus composition in the glassy and the corresponding liquid state [43]. (a) Resistivity of metallic glasses (full symbols) [147—149] and the liquid state (thin solid curves) [23, 150]. (b) Hall coefficient of metallic glasses (full symbols) [93] and the liquid state (open symbols) [23]. (c) Low-temperature thermopower coefficient of the amorphous state (full symbols) [147—149]. The vertical solid lines at 27 at% Sn indicate alloys with ~ = 1.8 eta and hence the minimal Sn content for homogeneous amorphous alloys. The vertical broken lines indicate the maximal Sn content up to which the amorphous phase exists. In (b) and (c) thin curves represent the FEM [eqs. (26) and (27)].


113

For non-simple elements such as transition or noble metals, the pseudopotential u(K) has to be replaced by the true scattering function and thus results in a more complex description than eq. (28) [152].With some simplifications, p(T) can be written in a similar form, the only difference being that v(K) is replaced by the t-matrix t(K).

In fig. 40a resistivity data of the amorphous state (full symbols)*) are compared with those of the liquid state (thin solid curves), measured just above the melting point. There is an overall similarity between the liquid and amorphous states which becomes even more pronounced if their temperature coefficients a are taken into account [23, 150]. As expected, the highest p for each system occurs when Z~u1.8e/ais fulfilled.

For liquid Cu—Sn p has been explained in the more subtle (-matrix formulation [152] although the Hall coefficient agrees with the FEM within 5% (fig. 40). For amorphous Au—Sn and Cu—Sn Ziman’s formulation has been used and, surprisingly, p was also found in agreement with experimental data [92], although RH deviates up to 40% (fig. 40a). Since S(2kF) itself is related to 1 g, p should also depend on 1 g. As the pseudogap at EF becomes more pronounced p does indeed increase. A correlation between the structural properties and the pseudogap has been proposed by_ Nicholson and Schwartz [78], introducing a novel theoretical description of p. The resistivity versus Z and T has directly been related to Z- and T-dependences of the pseudogap alone. For small g-values a high resistivity should result and vice versa. The greatest success of Ziman’s description is seen in the explanation of one of the most curious properties of metallic glasses, the well-known negative temperature coefficient (NTC) of p(T) in the amorphous state at high temperatures (T ~ ~9D)~ and very often in the liquid state close to the melting point [23,92, 152]. Whenever Kpe 2kF is fulfilled, due to the weighting of eq. (28) with K3, a decrease of S(2kF) with rising T causes a decrease of p. Equivalently, this behaviour can also be explained by a decrease of 1 g with T, as just mentioned. The T-dependence of the resistivity of amorphous Cu—Sn is shown in fig. 41 for different compositions. p(T) varies linearly with T at elevated temperatures. At low T the behaviour is more complex (see also fig. 48) and will be discussed below. Figure 42 shows a = (lip) dpidT at high T versus composition. Au—Sn shows an NTC over the whole amorphous region, both other systems change sign with composition. The NTC of Au—Sn over a wide range of composition arises from the close proximity of Kpe to 2kF over the whole amorphous range (fig. 11). —

—

—

6.3.2. Hall coefficient In contrast to the crystalline state, RH of the liquid state of simple elements or alloys is known to be close to R°H[23].This agreement can readily be explained: Upon melting, the complex anisotropy of the

Fermi surface is destroyed since sharp Brillouin-zone boundaries no longer exist. Fermi-surface distortions are smeared out and preferred directions are absent [156]. Therefore, it was surprising to find that the liquid heavy metals Tl, Pb and Bi and the non-simple metals (e.g. transition metals) are in disagreement with the FEM [23].Deviations ~sRHof the former into the direction of smaller values are approximately 10%. RH of the latter can even change sign and become positive. Whereas deviations of

the heavy metals have been explained by additional effects to the free-electron behaviour [44,157, 158], the positive values are still not convincingly explained. *

I Although more data than shown here are known from the literature [93, 153], only those which we now believe are the most reliable are

depicted.

114


Cu-Sn

102

~JiL

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~

~

-

•.

55 1~00 ~

75 099

-

ut%Sn

0 Fig. 41. Normalized T-dependence of p(T) of amorphous Cu—Sn [149]. At very low T the alloys become superconducting.

In fig. 40b the experimental data of the liquid and amorphous alloys of (Au, Ag, Cu)—Sn are compared with R~.RH is negative, indicating electrons as the current carriers. In the liquid state RH is close to R°H. Small deviations of the order of 5% exist. These deviations were disregarded in the past, and the liquid state was widely accepted to be in good agreement with the FEM [23]. Amorphous Ag—Sn on the Sn-rich side agrees also with R~,whereas for Au—Sn this happens only close to the composition with 80% Sn. Cu—Sn shows a behaviour somewhere between these cases. Close to Z = 1.8 e i a, the region where the Fermi sphere touches the pseudo Brillouin-zone boundary, deviations up to 40% arise. •

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115

The small deviations of the liquid state go in the same direction as in the amorphous case and seem to be most pronounced_at similar compositions. Accordingly, questions of T-dependences arise. For amorphous Au70Sn30 (Z 1.8 ela) a small temperature coefficient indeed has been reported, this, however, being approximately a factor of four too small [93].Within experimental resolution, RH of Au20Sn80m=(ZR~’ = 3.4 ela)is has, in contrast, to be and thus of agrees with the fact = R°H already fulfilled.been We found conclude thatT-independent systematic deviations RH exist in both that R~ states, this being more pronounced in the amorphous than in the liquid state. Similar deviations occur in all amorphous and liquid alloys containing noble metals [19,23, 93] (see section 6.4). Within the liquid state the small deviations have been explained by mass-density anomalies versus composition. This trivial explanation casts doubt on the calculation of R°H to which the experimental data are compared. Within the amorphous state, deviations are so large that this explanation fails. A two-band mechanism is also unlikely since Fermi-surface distortions are smeared out. Hole-like electronic states require a well-defined dispersion relation for the electronic states at EF [156,157].*) Since at present there is no real theoretical understanding, it is of particular importance to reveal empirical trends and correlations of ~RH to other properties. The present author recently reported relations between ~RH and 1 g [42, 43, 58]. For the noble-metal—Sn alloys ~RHIR°Hobeys a parabolic relation with 1 g or S(2kF) (fig. 43), —

—

(30)

RHRH

The latter proportionality follows from fig. 19, but is not generally fulfilled as shown in section 6.4. ARH seems to depend on the DOS at EF and not on all electrons of the conduction band. This has been the subject of discussions for quite some time [46,44, 157, 160, 161]. Additional support of this observation comes from liquid elements like In, Sn, and Ge, which show RH close to R~,although deep pseudogaps far below EF exist [47].Deviations of the heavy elements Tl, Pb, and Bi from the free-electron value are not contradictory, since these elements show also pseudogaps at EF as shown above (figs. 21 and 26). 0.5

I

I

I

£

~/

0.3

j

Au-Sn

~/

Ag-Sn

/

:~__

-

/,,~~h..(AuA9.Cu)SnIIq

—01

0

I

I

I

0.1

0.2

0.3

(1 —g)2

04

Fig. 43. Deviations of the Hall coefficient from R° 2 S2(2kF) [43,42]. Deviations of the corresponding liquid samples (fig. 44(b) are confined within the box close to zero. g 0versus (1 — g) 0-data (fig. 31) have been used for the latter. *

F The influence of hole-like states can disappear even in polycrystalline materials. Noble metals, evaporated onto liquid-helium-cooled

substrates, show excellent agreement with R~,although RH of bulk samples disagrees significantly [19,159](fig. 40b).

116


After reviewing the experimental facts, we will now briefly mention some more subtle explanations. The emphasis is mainly on theories which relate /~RHto deviations of N(EF) from No(EF). As 1 g is proportional to S(2kF), our interest is also concentrated on theories relating ~RH to S(2kF). Before doing so, another model which is often used to explain deviations of R~from R~is briefly discussed. Morgan and coworkers [162]gave numerical arguments for the occurrence of an anomalous —

but well-defined dispersion relation of the electrons in disordered systems, yielding a generalized concept of effective mass. According to these authors a negative value of dEidk for s, p-states (S-shaped dispersion relation) and a pseudogap underneath the d-band is expected due to s, p—dhybridization effects. If EF is located within this region, it was claimed that positive Hall coefficients may occur. We have shown above that the d-states are well apart from EF. Their positions relative to EF are very different for the (Au, Ag, Cu)—Sn sytems. The d-states shift closer to EF with increasing

temperature [124],although, e.g., for Au205n80 the deviations stay constant, or for Au70Sn30 get even smaller. Both observations contradict the large t~RHiR~-valuesfor all three systems in the amorphous state. Thus, we claim s, p—d-hybridization effects to be unimportant in the present alloys. Theoretical work by Bose et al. [163]casts further doubt on this approach. Hybridization effects between s- and p-states, on the other hand, which were proposed by Beck et al. [164], have not yet been taken into account in detail and so are still open for discussion. 2], Jan [165]has shown the Fermi surface as sphere-like [isotropic, not necessarily E(k) cx k thatAssuming Boltzmann’s equation predicts the classical value R° 11 even for the NFE model, with no modi-

fications from effective mass or relaxation time. After the failure of the Boltzmann type of approach, two explanations remain. One depends on additional effects, which might still be treated with the Boltzmann equation, the other one on a description which goes further. Ballentine [157,158] focussed

on the former and summarized them as skew-scattering effects. The only important source of skew scattering in non-transition, non-magnetic metals is spin—orbit (so) interaction [44,157, 158]. Some features, on the other hand, cast serious doubt on its influence in the present alloys. As so interaction decreases with decreasing atomic number, the skew-scattering contribution to RH should become less pronounced for lighter elements. As shown above for the amorphous state, no distinct differences are found if Au is replaced by Ag or Cu. For alloys with approximately 30 at% Sn, the deviations are nearly equal whereas the atomic weights of the noble metals differ by a factor of more than two. With these results, one might assume Sn itself to be responsible for so contributions. In an earlier paper we showed that replacing Sn by lighter as well as heavier elements also seems to have no effect on ~RH [19].

The step beyond Boltzmann’s approach uses Kubo’s formalism. Since even the trend of theoretically predicted deviations is in disagreement with the measured data, some of the many proposals can immediately be disregarded. Mainly for heavily doped semiconductors, Matsubara and [160] [166]. 2 Kaneyoshi With Mott’s as well as Friedmann [161] proposed RH = R°~Ig, and Fukuyama et al. RH = R°~ig g-value shown above, both expressions should yield R 11~> IR~I,contrary to the experimental results shown above, where IRH~was found smaller than R~J~ A recalculation 2-term, giving of Fukuyama’s results, performed by Itoh [167],found a cancellation of the 1/g RH = ROHnin* with n~ k~i3ir2, (31) =

the effective carrier density, modified by a renormalization effect of the electronic states. kE takes care of the shift of 2kF when structures (e.g. minima) occur in the DOS. But the large z~RH-valuesof the present amorphous alloys are outside the scope of this model [168].

P. Hdussler, lnterrelations between atomic and electronic structures

117

First attempts of a microscopic treatment of RH in terms of Edwards’ theory*) have recently been reported by Itoh et al. [45].Besides the deformation of the electronic dispersion from the free-electron parabola, the full broadening of the spectral density has additionally been accounted for. A clear correlation was observed between ~RH and the pseudogap at EF in these calculations, which were based on hypothetical single-component systems and hard-sphere structure factors. Whenever EF is situated close to a pseudogap, RH deviates from R°H. This happens in particular if Z is fairly close to 1.8 eI a. The direction of these deviations, however, was found to be opposite to the experiments reported above. On the other hand, Itoh’s results indicate that the structure in the DOS at EF and the transport anomalies likely come from the same physical origin: the broadening of the spectral distribution. In order to obtain deviations of RH in the other direction, it was speculated that calculations based on real multi-component systems are needed. Another approach beyond Boltzmann’s equation has its origin in the fluctuation of local forces seen by the electrons [169]. These fluctuations represent deviations from perfect structure. Both a force— force correlation approach as well as an alternative derivation [170]could deduce ~RH in terms of phase shifts and the structure factor S(2kF), 2

z~RH =

—

RH

3

11

irn

1~/ ~ ~ (~E~ 2(1 + /=1

1)

sin2(~~+—1 ô~)S(2kF))]

.

(32)

2kF have a drastic influence on ARH [169]. The positions peaks theformulated structure factor relative Equation (32) hasofnot yet of been for alloys but to may also be a promising theoretical approach. Cluster calculations by means of Kubo’s formalism have recently found RH of other alloys to be in agreement with experimental data [164,171]. These calculations, however, are unsuitable for giving simple criteria even for the sign of the Hall coefficient and hence need further consideration.

6.3.3. Thermopower In contrast to the resistivity and even the Hall coefficient, the thermopower S(T) of amorphous systems has scarcely been studied. Most measurements had been performed on complex alloys containing elements with d-states at EF. Only few were known of simple alloys such as Ca—Al [172], Mg—Zn [173], or Cu—Sn [174]. Very recently the thermopowers of Au—Sn [147], Ag—Sn [148], Cu—Sn [149], Ag—In [175], Cu—Al [149], and Mg—Zn [149,176] were systematically studied. The present review will highlight the results for (Au, Ag, Cu)—Sn and Cu—Al. The other systems show similar features but are reported elsewhere [177]. S( T) is a measure of the electric field set up when electrons diffuse down a thermal gradient with the magnitude of diffusion ultimately limited by elastic and inelastic scattering of the electrons. The thermopower, in addition to the resistivity, thus contains further information on electron-scattering mechanisms. Different mechanisms are additive and umklapp-scattering effects again play an important

role [146,178]. From liquid systems it is known that even for simple elements the signs of RH and S( T) are very often opposite to each other. This is contrary to the theory and initiated many discussions [94, 179]. The basic expression of the thermopower is given by Mott’s formula [180] (33)

*) A combination of Green function and diagram techniques.

118


where the scattering behaviour of the electrons is described by =

EF a ln p(E)/aE~E

•

(34)

If the scattering mechanism of the electrons is independent of T, ~ is constant and S( T) should be proportional to T yielding SD(T), the diffusion thermopower. In crystalline systems at low temperatures, SD(T) is often accompanied by phonon-drag contributions, which cause deviations to the proportionality to T [146]. In disordered systems, phonon-drag effects disappear because the scattering of the phonons by the high degree of disorder keeps them essentially in local thermal equilibrium [181]. S( T) of disordered systems, therefore, should basically be described by SD ( T). In fact, Ca—Al shows a thermopower which is nearly proportional to T [172]. Small positive deviations at low T are explained by many-body effects [182]. Many—body effects should affect S( T) but not the resistivity [183]. In this description S(T) is formulated as [184] S(T) =

SD(T)[1

+ A(T)] + 2aTA(T) + yTA(T).

(35)

1 + A(T) describes mass-enhancement effects due to electron—phonon interaction. The second term corresponds to a renormalization of velocity and relaxation time, and the third term takes care of higher-order corrections. All these effects depend on the electron—phonon coupling constant A( T), and therefore on the vibrational density of states. Since A( T) is significant only at low temperatures, corrections should vanish at higher temperatures (T> 19D) and it is assumed that SD( T) is approached [184]. The crucial point for the use of eq. (35) is, in fact, the assumption of an underlying proportionality to T. This assumption fails if above a characteristic temperature, T 0, a new scattering process arises [146]. SD( T) as well as A( T = 0), both obtained by a fit to the measured thermopower, can be examined by a comparison with S°(T),the simplest expression for the diffusion thermopower, and by measurements of the superconducting properties as mentioned above (eq. 18). Although their deviations from the proportionality to T are negative, the thermopowers of Ag—Sn and Au—Sn, which look similar to those of Cu—Sn (fig. 39), were described in terms of many-body effects [147,148]. Although reasonable fits to the measured data were obtained, serious difficulties appeared. First of all, SD(T) was found to be positive over the whole composition range of the amorphous phase, although 1 — g (fig. 31) as well as zIRH (fig. 40b) are both small at the Sn-rich side. The vanishing 1 — g value as well as the vanishing ~RH-value both indicate agreement with the FEM, in which SD(T) should be negative, as is the Hall coefficient, and close to S°(T). Secondly, -y, which is theoretically only allowed to be positive, was found to be negative. Thirdly, at high temperature, S(T) is by no means proportional to T and gives rise to reasonable A( T) -values at high temperatures [149]. By including phonon rotons, the present author recently proposed a novel interpretation of S(T) on the basis of Ziman’s description [42,43]. This new interpretation takes into account the characteristic electronic, structural, and vibrational properties of the disordered state. Moreover, a solution of the long-standing problem of the different signs of RH and S(T) is proposed. Ziman’s nearly-free electron mode], written in terms of non-local pseudopotentials, yields [146] ~=3—2q—~r

(36)


119

with 2

(37)

S(2kF)li.’(2kF)1 q

r=

J~S(K12kF)kF alv(K)l

I8kIk •4. (K12kF)3 d(K/2kF)

(38)

.

(S(K)lv(K)12)

The q-term describes elastic umklapp scattering and the r-term energy dependences. S(T) is therefore sensitive to details of S(K) as well as v(K) at K = 2kF. The q-term, for example, is proportional to S(2kF) v(2kF) 2 and therefore gives positive values only, causing ~to become smaller than 3. As S(2kF) disappears, this term vanishes. The r-term may be positive or negative, again giving main contributions from K = 2kF. If u(K) is nearly independent of energy, or its derivative at 2kF is close to zero, this term effectively vanishes too. With both q and r being zero, ~equals 3 and S°(T) (eq. 27), quoted as the thermopower of free electrons [94, 146] will result. Both q and r describe deviations from the FEM. As seen from the UPS measurements (fig. 31) as well as from the Hall coefficient (fig. 40b), the noble-metal-containing alloys under consideration can be prepared in concentration ranges where the free-electron model provides a good description. Continuous transitions can be made to ranges where deviations exist. In some concentration ranges S( T) should therefore agree with S°(T), in others deviations may arise. Measurements of several amorphous Cu—Sn alloys and of pure polycrystalline Sn are shown in fig. 44. Whereas pure Sn was measured during first heating, the amorphous samples were annealed prior to the measurements. Only minor differences between the annealed and the as-quenched state were observed for the latter [147,149, 175]. Immediately after condensation, S(T) of pure Sn is negative and proportional to T over a large temperature range. The measured thermopower is in reasonably good agreement with S°(T). In the amorphous state, the T-dependence changes distinctly. Whereas for the Cu-rich alloys the thermopower SIT) (iiV/K)

at%Sn

0.2 0

~

-

Cu-Sn

.-

_______________

32

__

-0.4

•

0

S000

0

0

I

100

• I

I

200 IlK)

Fig. 44. Thermopower of amorphous Cu—Sn and polycrystalline Sn versus T [147,149).S°(T)is included for comparison.

120


at T < T0 remains in good agreement with S°(T), deviations occur at higher temperatures. With increasing Cu content, additional deviations from S °(T)arise not only at high temperatures but also at low temperatures. The thermopower of Cu68Sn32 is positive over the whole temperature range. Ono and Taylor [185]have shown that the inelastic scattering with long-wavelength phonons (Debye phonons) is ineffective for the thermopower and hence plays a minor role. The different effectiveness of the phonons together with the elastic and the inelastic umklapp scattering of electrons opens up the new description of 5(T). We will show that below T0 elastic umklapp deviations S( T) 2kF,scattering or 1 — g dominates as a substitute, will of help to from the behaviour.above The Tstructure factor at reveal thisfree-electron proposal. Deviations 0 are caused by inelastic umklapp scattering of the electrons with phonon rotons [42,43]. First we focus our attention on the low-temperature behaviour. The influence of phonon rotons is separately discussed in the next subsection. In fig. 40c the slope of S~(T)versus T is compared with S°(T)I T. S~(T)IT behaves in some respects similarly to the g-value (fig. 31) or the Hall coefficient (fig. 40b): For large Sn contents the FEM is closely approached. With decreasing Sn content, large deviations are observed and S~(T)IT ultimately becomes positive. Whereas Ag—Sn and Cu—Sn show comparable features, Au—Sn is different in the same way as its g-value and RH. S’( T) / T deviates more and the change of sign occurs at larger Sn content. As a special feature the thermopower stays nearly constant at a positive value over S°(T)can a broad range of composition. 1(T) from be described by the q-term of eq. (36). If S(2kF) or 1 g The deviations of S are finite, Se(T)~should become smaller than IS°(T)l.Assuming r = 0, the deviations, normalized in the same way as other properties above, should obey —

1— S~(T)iS°(T) =

cc S(2kF) ccl

— g.

(39)

The other factors in eq. (37) have been neglected here due to the lack of any knowledge of the corresponding mean pseudopotential. A close relationship between 1 — S’( T) IS°(T) and 1 g becomes evident from fig. 45. Although the proportionality is not exactly fulfilled, we take fig. 45 as a strong indication of the applicability of eqs. (36)—(38) for T < T 0. *) —

~M~l

05

~

/U

O~5

06

(1-g)

Fig. 45. Deviations of the low-temperature thermopower S’(T) from the FEM value plotted versus 1 — g [43,42]. A: Au—Sn [147],U: Ag—Sn [148], •: Cu—Sn [149). The dashed horizontal line separates the region of positive from the region of negative S’(T)tT. *1 Due to the small range of linearity between T~and T0, the extremely small values of S(T) at low temperatures, and in particular the fact that the thermopower of the reference material (Pb in this case) is an order of magnitude larger than of the amorphous alloys and hence needs to be known with high accuracy, the exact values are masked by relatively large uncertainties.


121

We would like to emphasize that these results indicate that the thermopower below T0 is well described by the free-electron value S°(T)and elastic umklapp scattering effects of the electrons on the pseudo Brillouin-zone boundary. The zone boundary itself corresponds to static density oscillations caused by the Friedel minima at short- and medium-range 2.1). Alloys with a weak 2kF or, synonymously, a small pseudogap distances at EF, are(section well described by S°(T)alone. peak in S(K) at 6.3.4. Collective density excitations and electronic transport versus T The Ziman description of thermopower fails at high temperatures as Laakkonen and Nieminen [186] showed for Mg 70Zn30 (Z = 2 eia). This glassy metal shows anomalous features such as a knee in S(T)2kF at T0, comparable to the noble-metal—polyvalent-element alloys [149, 173, 176]. Kpe is close to (Kpe = 0.94 x 2kF) [187] and the existence of a structure-induced pseudogap at EF has been reported (section 5.3). Phonon-roton states were observed as mentioned above (section 2.4). Due to these and many other similarities to the noble-metal-containing alloys, we expect phonon-roton states also in the latter. Below we give further indications from specific heat measurements and electronic transport. Due to phonon-roton states above T 0, excited both thermally and by inelastic electron scattering, we assume temperature effects on many physical properties [48, 50]. The inelastic umklapp scattering of conduction electrons, e.g., is claimed to be responsible for the knee in S(T) [42, 43]. The electrons at EF, the phonon2kF, rotons, structure may, indeed, interfere strongly since their characteristic Qpe’ and andthe Kpestatic are close to unity. wavenumbers T 0 depends strongly on structural features. For alloys with a high ~ieakin S(K) the phonon-roton minimum in the dispersion relation of collective density excitations will be deep in energy but it will stay at energies(at(temperatures) alloysrotons with acan small peak. Inwith the the former case if, electrons in addition, 2kFrather = Kpehigh is fulfilled 1.8 eia), the for phonon interfere conduction at rather low T. For (Au, Ag, Cu)—Sn, as well as other alloys of similar type, structural features change with composition (ch. 3) and so should the phonon-roton effects.

A direct proof of the existence of phonon-roton states can be given by both inelastic-scattering experiments, measuring the dynamic structure factor S(K, w), and by the specific-heat method. Unfortunately, large quantities of (Au, Ag, Cu)—Sn, necessary for neutron scattering, cannot be prepared by vapour quenching. As is well known from the literature, disordered systems show non-electronic contributions Ydis in excess of the linear specific-heat coefficient Ye of the electrons [125] (eq. 17). The physical picture is primarily based on a description in terms of two-level states [188]. Besides this explanation, which in many cases leaves the type of the tunnelling Species unresolved, descriptions in terms of fractals [189], and phonon rotons [48, 190] have also been proposed. In a paper published quite recently [191], the relationship between two-level states, fractals, and phonon-roton states to the spectrum of collective density excitations (figs. 5 and 6) were theoretically discussed. The roton part of the dispersion relation was shown in this paper to play a major role with regard to excess heat capacity. It is claimed that a combined description in terms of two-level states, fractals, and phonon rotons gives the most comprehensive explanation. As already mentioned above, amorphous Au—Sn as well as Cu—Sn indeed show large non-electronic, non-Debye-like excess contributions Ydis to the specific heat [122]. The experimental data are shown in fig. 46. Whereas close to Z = 1.8 e I a (ns30 at% Sn) the excess contribution is extremely large, below T = 5 K (where the specific heat has been measured), due to the shift of T 0 to higher temperatures at the Sn-rich side, excess contributions get smaller because phonon-roton states longerlow-lying become 2kFcan andnohence thermally excited. The cross-over from a system with a large peak at

122


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~dis[MoI

1\

K2

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liii TSITI •‘I Io/•

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/~~TR(T)

020

•

0 20 Au,Cu

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I

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•

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80 100 at%Sn

Fig. 46. Energetically low-lying non-electronic excess contributions Ydis to the specific heat of Au—Sn (•) and Cu—Sn (A, V) [122].

0

0 Cu

I

I

20

•

.1 60

•

I

60

•

80 100 at%A[

Fig• 47. Characteristic temperature T0 of Cu—Al alloys as determined from the thermopower data (T~T)) and from resistivity data (T~T)) versus the Al content [1921.The vertical solid line indicates Z = 1.8 eta and the vertical broken line the maximum content of Al for a homogeneous amorphous phase.

2kF and thus phonon-roton states phonon-roton states at rather low T,reflected to a system withexcess a small peak at at higher temperatures is therefore in the specific heat. A more indirect proof of the existence of phonon rotons is given by their effects on electronic transport. Whereas below T 0 the main contribution to S’( T) is given by the free-electron value and elastic umklapp scattering according to the description above, the onset of deviations from this behaviour with rising T at T0 is caused by the onset of inelastic umklapp scattering of the electrons with phonon rotons. The variation of T~T) (the characteristic temperature obtained from thermopower data) with composition is shown in fig. 47 for glassy Cu—Al [149].At Z = 1.8 ela, T0 seems to be close to T = 0 K. It rises strongly with increasing Al content [decreasing S(2kF)] as would be expected from the above consideration. The sudden occurrence of phonon rotons versus composition at Z = 1.8 e / a seems to be related to the transition from the crystalline to the amorphous state. Phonon rotons, in fact, have been said to be essential for order—disorder transitions [50]. As mentioned above, it is thought that many-body effects, the possible alternative source of the knee in S(T) at low T, should have no effect on p( T) [183]. Relations of phonon-roton effects to anomalous effects of the resistivity may therefore additionally support the exclusion of many-body effects as the major cause of the knee. In fig. 48 the normalized resistivity, its second derivative, the thermopower, and the Hall coefficient of glassy Cu20A180 are plotted against T in a strongly enlarged version. The resistivity was measured with high resolution (only about 0.3% of the signal is shown) in order to perform its second derivative with high accuracy. Below T= 15K, a magnetic field was applied to suppress superconductivity. p( T) can be interpreted as follows. Below T0 the main contribution (suppressed in fig. 48a), is given by elastic umklapp scattering. The increase with decreasing T [p(T) cx —ln T] is ascribed to electron— electron interaction in two dimensions [192, 193] and is not further discussed here*) The increase with rising T is caused by the inelastic scattering of the electrons with normal (Debye) phonons. Debye phonons, however, leave the thermopower unaffected [185]. *1 Weak localization, hypothesized in Mg70Zn30 for similar dependences, has to be disregarded due to the presence of the magnetic field.

P. Hdussler, Interrelations between atomic and electronic structures 9(fl-QlO) I

~I

123

-,

/ 0.001

-

0

Cu,

0

0 AIR, ~1

dl’

0

o

b)

.-

-02

.-

~‘~\

-

Sm

~

0

~ ‘~

-0.6

S(T)

N..

(1)

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•

08

R~IT)-RM(OI R,1(0)

‘-

-

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..•

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d)

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S IT)

~

N

0

-.- •

_____

.~..

.

.~

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.

.

0

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IlK)

100

Fig. 48. Enlarged plots of transport properties of glassy Cu,0Al80. (a) Resistivity [192],(b) the second derivative of the resistivity, (c) thermopower [149],(d) Hall coefficient [192].All measurements were performed with samples of the well-annealed amorphous state.

The inflection point of p( T) at T0 is obviously closely related to the knee in S( T) and therefore to the onset of inelastic scattering effects of the electrons with phonon rotons. *) Accordingly, the NTC may basically be seen as resulting from the phonon rotons, which cause a decrease of the static structure factor and hence a decrease of the resistivity in Ziman’s description. A decrease of the resistivity above can be understood if elastic umklapp scattering, getting weaker on account of inelastic umklapp scattering, contributes more to the resistivity than the latter. Close to Z = 1.8 eI a (Cu60Al40), the phonon-roton states get excited at the lowest temperatures and, subsequently, the negative temperature coefficient of p(T) exists down to the lowest temperatures. Some characteristic temperatures T~T), taken as the inflection point of the resistivity, are included in fig. 47 and, within experimental resolution, agree well with T~T). Summarizing the results of t(x, this T) section, we may a complete (qualitative) of was given. Duestate to thethat reinterpretation of S(T), the description long-standing p(x, T), RH(x, T), as well as S problem of different signs of RH and S( T) seems to be resolved if RH and S( T) are compared below T 0. The inclusion of phonon rotons explains anomalous features of p( T) and S( T). The existence of non-electronic excess contributions to the specific heat at low T also seems to be induced by phonon rotons. Deeper theoretical descriptions of RH, and specific influences of phonon rotons on thermopower *

I The

Hall coefficient, shown for completeness but known with much less accuracy, seems to be unaffected.

124


and excess specific heat need further consideration. The Hall coefficient, in particular, remains a challenging theoretical problem. 6.4. Scaling behaviour versus Z Basic features such as the influence of conduction electrons on structure, the existence of structureinduced pseudogaps, and phonon rotons are independent of the involvement of noble metals. Similar effects have been observed for disordered Mg—Zn and Pb—Bi. Other properties such as the scaling behaviour with Z are sensitive to the type and content of noble metals. The scaling behaviour of both the structural properties (figs. 9 and 12) and the glass-forming limit at the noble-metal-rich side (where Z equals 1.8 e/a) were reviewed above. As seen in fig. 49, similar features appear to exist, e.g., for p, TK, RH, and Se(T)/T. Recognizing how different the individual systems are at values far from Z 1.8 e /a, it is remarkable how similar they become at this special Z-value. The resistivities of pure Si and Ge, for example, differ by many orders of magnitude from that of pure Sn, whereas at Z = 1.8 e /a the differences are less than 10%. At this Z-value, shown for the Au alloys, the RH have been reported as identical within experimental resolution [19] and the S’( T) / T as showing singularities, both with strong deviations from the FEM [43]. Since for the Au-containing amorphous alloys the thermopower is only known for Au—Sn, low-temperature data of other alloys containing Ag or Cu are included in fig. 49d. Even Mg70Zn30 shows a good fit into this general scheme. -

A scaling behaviour of electronic transport properties with Z may easily be understood because of the uniqueness of structural and electronic properties. But scaling occurs even for the thermal stability (fig. 49b), clearly indicating, once more, the dominant electronic influence on the stability of the amorphous Hume-Rothery phase. Similar scaling is known for the melting temperature of crystalline Hume-Rothery phases [17, 18]. As reported elsewhere, the values of p, TK, and RH at Z = 1.8 e/a are slightly different if Au is replaced by Ag or Cu [19]. This already indicates specific influences of the noble metals. It is of particular interest how scaling is fulfilled if the polyvalent component varies from the pentavalent to the divalent elements and, subsequently, the noble-metal content from 80 at% to 20 at% only. These studies, on the other hand, are strongly hampered by the vanishing stability range versus composition for the divalent elements in alloys with the noble metals as reported above. In the past, therefore, noble-metal-containing amorphous Hume-Rothery alloys with Z = 1.8 e / a have mainly been prepared with penta-, tetra-, and trivalent elements. The noble-metal content of these alloys is limited to a range of 80—60 at%, not very different for the different systems, whereas the inclusion of the divalent elements would allow a large extension of the noble-metal content down to just 20 at%. Several questions immediately arise because, due to scaling, we expect well-defined values of the different properties at Z = 1.8 e/a. However, as shown in figs. 40b and c, alloys with only 20 at% Au, Ag, Cu, such as (Au, Ag, Cu)20Sn80 and many others like (Au, Cu)20Al80 [19, 194] and Au20In80 [195], show transport properties in good agreement with the FEM. This good agreement at small noble-metal content was found as a general feature whenever metallic polyvalent elements are involved [19]. Thus, the inclusion of noble-metal—divalent-element alloys with only 20 at% Au, Ag, Cu provokes the question of which feature finally dominates—the good agreement with the FEM or the scaling behaviour with deviations from the FEM. In section 6.3, the good 2kF agreement with the has of beentheascribed disappears on FEM account peak to at both the fact that S(K)from at K2kF. = KpeThis cannot be valid for (Au, Ag, Cu) K = K~ ~ 2kF, and the its peak shift in away 20M80


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0 I

RH

[io.i~.~.l [ As] -5

-_0J~\

Cu

10

‘Sn

•Sn

AAI I

‘Sn

I

I

I

I-l

a

a

Ag

oBj

•

am

Au

I / I• LI-0~°~, /

-o-”~

I

-

(~~) T

-

1

-

a’~’\ •

010

•

/~0

~

\

15.

\

•

I

2

I

3

I

-10

-

•“.

I

4

-

0 ‘a

-

•

____ ~2O1 I

-5

a

~

—is

I

1

2

I

3

I

4

2

5

Fig. 49. Electronic transport properties and the thermal stability of amorphous alloys versus Z [43, 93]. (a) Resistivity of Au alloys, (b) crystallization temperature of Au alloys, (c) Hall coefficient of Au alloys, (d) thermopower of Au, Ag, Cu alloys, and Mg70Zn30.

126


(M: Mg, Zn, Cd). In these alloys the peak at K~has nearly disappeared and the peak at Kpe 21~F shows a high intensity [196,197]. Whereas Cu—Mg forms an amorphous phase in a narrow range around Cu 20Mg80 [144,198], other systems such as Au—Zn are unstable and therefore inhomogeneous close to the same composition [199].*) The crystallization temperature of Cu20Mg80 is in excellent agreement with all the other Cu alloys at Z = 1.8 e/a, as expected if scaling occurs [19, 194]. RH deviates 15% from R~[24].**) Hall effect measurements, unfortunately, are rarely known for other Cu alloys. So, we are forced to study the Au alloys, where homogeneous amorphous alloys with 20 at% Au and Z = 1.8 e / a are not known to us. The use of the ternary system Au—(ZnGa), with ZnGa as a kind of artificial polyvalent element with a varying mean valency 3 eta ZZnGa 2 e/a, surmounts these difficulties. By adjusting Zn and Ga, a continuous variation of the Au content from 60 to 20 at% can be obtained [202] along a line in the ternary system which obeys Z 1.8 e / a. The binary phase diagrams of the ternary system are shown in fig. 50.~The triangle between these diagrams shows both the stability limit along Z = 1.8 e/a from Au60Ga40 towards Au20Zn80 and the stability limit along 80 at% Zn, Ga with the amorphous phase enclosed (shaded area). Au68 (Ga50Sn50)32 is another ternary alloy with a mean valency of Ga50Sn50 of ZGaSn = 3.5 eta and an overall mean valency of Z= 1.8e/a. The study of the ternary alloys and their comparison with other alloys at Z = 1.8 e / a has some further advantages. Whereas in the binary alloys above, a variation of the noble-metal content,

_______________

0

Go Fig. 50. Binary phase diagrams of the ternary Au—(ZnGa) system. The triangle includes schematically the stability range of the crystalline and the amorphous Hume—Rothery phases, with the boundary along Z = 1.8 eta, where scaling occurs.

*1 Au—Zn as well as Ag—Mg seem to show further amorphous phases, which are unrelated to the amorphous Hume-Rothery phase 3’—-y Hume-Rothery phases (Au—Zn) [199] or discussed of the a—)3’in Hume-Rothery this review. These phases phases (Ag—Mg) exist between [196,200] Z =exist. 1.6—1.4 eta, where miscibility gaps of the ) **) In contrast to Mizutani [24], the free-electron value R°H is calculated with the mass densities of the liquid state. ‘>The complexity of the Au—Zn as well as the Au—Ga phase diagram is caused by the electronic influence on the phase stability of crystalline Hume-Rothery alloys.


127

unavoidably, varies the mean Z-value as well as the height of S(2kF), in the alloys along Z = 1.8 e / a both stay constant. Moreover, the condition Kpe = 2kF remains fulfilled. In figs. 51 and 52, the scaling behaviour of the crystallization temperature and the Hall coefficient of different Au alloys with Z = 1.8 e /a are shown versus the content of the polyvalent elements. In these figures the range from 40 to 80 at% M (3 eta ZM 2 eta, see the upper scale) is covered by the ternary system Au—(ZnGa). It is surprising, how well TK of all the different alloys is located within a band between 280—320 K over a large composition range. The ternary system Au 68(Ga50Sn50)32 fits well into this band and Au—(ZnGa) seems to form the natural extension of the binary alloys. Au73Si27 is the only system which shows a distinctly Thefor nearly a similar 2kF) as higher shown TK. above the constant (Au, Ag,thermal cu)—Snstability alloys, indicates since S(2kF) was relationtobetween TK and S( shown be identical for different alloys (fig. 19) at this particular Z-value. The crystallization temperature, defined as shown in fig. 39a by a resistance drop versus T, is quite insensitive to crystalline inclusions. Homogeneous amorphous Au alloys, in fact, only exist up to 67 at% M. Above this limit the samples are inhomogeneous and partly crystalline. The Hall coefficient, shown in fig. 52, is such more sensitive and therefore varies strongly in this range. Due to limited experimental resolution (arising from the limited accuracy of film-thickness measurements), RH has been assumed in the past to be constant from 20—40 at% M, in agreement with scaling, although Au 60A140 and Au60In40 both seem to deviate from this common behaviour. Including the new results from the ternary alloys, there now appears to be a downward curvature. The deviation of the systems from the corresponding FEM value, which is indicated for each by a horizontal bar marked by the corresponding symbol, is different for the individual systems. ~RH gets smaller with increasing noble-metal content and approaches the free-electron value. This behaviour is more clearly shown in fig. 53, where ~RH is drawn versus at% M. In spite of the above question, RH seems roughly to stay constant (fig. 52), and at the same time becomes closer to the OSI

~AI

•Sb DOe OBI ‘Sn

aIn 3

54

400

(K)

•Ga

I•I

-

200

-

100

-

xZn 2

•

LMIQJ

5

—

F•j~~.~_+~j. -

SnGa

GuZn

[1o41A~o~J

-

-

a

I

3

4

•

I

2

g I

•

— ~- -•~~‘~%~ ~ a

-

~

e

0

I

I

-

0 20

10

I

•

0 Au

e

~

60

60

80 100 at%M

Fig 51. Crystallization temperature of Au alloys at Z = 1.8 eta versus at% M (M: Sb, Bi, Si, Ge, Sn, GaSn, Al, In, Ga, GaZn, Zn) [202]. Au—(ZnGa) is inhomogeneous above 67 at% ZnGa [202].Above the figure, the symbols are explained by denoting the corresponding polyvalent element. The upper scale indicates the valency of the polyvalent constituents.

—

•

0

I

I

20

40

•

I

60

80 100 at%M

Au Fig. 52. Hall coefficient of Au alloys at Z [202].Same symbols as fig. 51.

=

1.8 eta versus at% M

128

P. Hdussler, interrelations between atomic and electronic structures ~

AD

54

3

II,,

I

2

j6

H

to,,

I

-

-

1+)

I /01

0

-40

LM~~

-60

~ -

°

-

•

I

6’

Au Cu Fig. 53. Deviations ARH from the FEM of different amorphous Au alloys at Z = used. V denotes Ag, 0Mg80 and ~ Cu,0Mg80.

I

80

1100

ut%M at%Mg

1.8 eta versus at% M. The same symbols as in figs. 51

and 52 were

FEM. Obviously, over a large composition range it is proportional to the noble-metal content. The vertical line at 80 at% M indicates the composition where both stability limits merge, as discussed above. As this line is approached, it appears unclear whether z~RHbecomes zero, as may be indicated by Ag20Mg80, or continues to vary in proportion to the noble-metal content, as is indicated by Cu20Mg80. 2kF) itself, as far as is known, is equal for all the Au alloys Since Z staysthis constant 1.8 eta, and S( that, apart from the findings in section 6.3, z~tRHis not only at Z = 1.8e/a, gives aatstrong indication dependent on 52(2kF) but on other properties too. The pseudopotentials of the noble metals at K = 2kF, for example, are larger than those of the polyvalent elements. This is due to the proximity of their noble-metal d-shells to the Fermi level, which induces a resonance scattering of the conduction electrons at EF [201]. For alloys close to 1.8 eta, in the light of the new experiments, eq. (30) may only be valid in its first part, z~R~IR°~cx(1_g)2.

(40)

The disappearance of the pseudogap at EF, indicated by the decrease of ~XRHtR°H if eq. (40) holds valid, may be supported by the UPS spectra of fig. 20, where small differences of the pseudogap at EF exist for the three Au alloys. In the past, these differences have been disregarded in spite of the fact that at other concentrations the differences are very much larger (fig. 17 and 18). Summarizing this section, we may state that the scaling behaviour of Hume-Rothery phases, including the amorphous phase, is still a fascinating subject. Further diffraction experiments as well as UPS measurements, particularly of the ternary system Au—(ZnGa), are needed to clarify some of the remaining questions. 7. Magnetic amorphous alloys After the discussion of the non-magnetic alloys, we finally turn to the more complex systems containing Fe, Co, Ni, or Pd, instead of the noble metals. Extensive research on these metallic glasses


129

has been performed for many years by Janot, Piecuch, Mangin and coworkers [203,204] and many others. The theoretical background of their magnetic properties is given by Kaneyoshi [73]. Ni—P, Pd—Si and many other alloys have successfully been prepared by different techniques. Alloys of more technical importance such as (AXBI_X)7SPI6B6AIO, where (A, B) denote (Fe, Ni), (Co, Ni), or (Fe, Mn) [205, 206], are of the same type and may be described as pseudo binary with, for example, A~B1_5as some sort of artificial transition element and P16B6A13 as some sort of artificial polyvalent element. There have been many reviews about these alloys [73,205]. For the samples reported here, codeposition with two magnetron sputter sources [71]as well as sequential flash-evaporation have been applied with no major differences in the sample properties [207—211].For the UPS measurements, the latter technique has again been applied inside the preparation chamber of the spectrometer [211]. During deposition the substrate temperature was held at T = 77 K or below. The main concern of the present chapter is threefold. First of all, it is shown that electronic influences on structure are comparable to those of non-magnetic alloys with indications of additional effects of the RKKY interaction on the position of local moments. Secondly, influences of these additional effects on thermal stability and on magnetic properties are shown. In the ferromagnetic case, the influence on magnetic properties is observed through T~,the Curie temperature [72, 73],

T~cx

f

dr. 2 g(r)I~~0(r)r

(41)

T~indicates commensurability between g(r) and Ieff(r). The ideal matching between maxima of g(r) and the maxima of j’eto(r) (fig. 3) enhances T~[71] and is also expected to cause a high crystallization temperature due to an enhancement of Urn (eq. 8). Thirdly, we will show that minima, well in the middle of the valence band, can be attributed to the peak in S(K) at K~~ 2kF similar to the noble-metal-containing alloys discussed above. 7.1. Structure The main structural features of Fe—(Sn, Ge, Sb) are shown in figs. 54 to 57. Most data were taken at room temperature by electron diffraction. In fig. 54 the total structure factors S(K) are shown at different compositions. Fe—Ge is in excellent agreement with literature data [214]. Thin vertical lines, calculated with the assumption that ZFe = 1.0 e/a, indicate 2kF. The finding of an effective valence for the transition elements is one of the main problems. In the following, we assume ZFe = 1.1 eta, which has been proposed for the liquid state of Fe [215]. For (Co, Ni, Pd)—Sn, discussed below, Z~ 0,ZN,, and 2kF are fulfilled for Zpd have been adjusted fitting procedure, the conditions Kpe = composition hold [as compositions where both in theahigh thermal stabilitysuch andthat the stability limit versus for the (Au, Ag, Cu)—Sn and Fe—(Sn, Ge, Sb) alloys]. But, since only few Co, Ni, Pd systems have been measured systematically over large composition ranges, their Z-values are still under doubt. Again the first peak of S(K) seems to be split into two, with the electron-induced peak close to 2kF (fig. 54). Figure 55 shows its shift parallel to 2kF over the whole region of the amorphous state. Nagel—Tauc’s criterion is best fulfilled at compositions between the vertical solid lines. At these compositions S(Kpe) itself is large (fig. 54), indicating the optical matching of the position of ions (and/or local moments) with ~ett(r) [and/or IDff(r)]. The corresponding effective Z-value, dependent on the choice of ZFe, is between 1.9 and 2.1 e/a and hence larger than that in the noble-metal alloys.


130

5)6) 11

I I

26,

-

SIK

I

Fe Sn

-

I

11

I

I

SIkI

I

~

I

I

26,

11 -

I

Fe-Sb

~.

it

10

25

-

:

_~,_._.,_,_.it*/*.~

2:

---::

6

-

-

_20

40

—~ 56

60

80

36

at’/Sb 45

10~

-

63 56

100 6 120

0

nm’I

20

40

60

80

100 6 120

(nrrrI

~O

4,, 20

4t

50

100 6 120

80

nw’)

Fig. 54. Total structure factors S(K) of quench-condensed Fe—(Sn, Ge, Sb) measured at room temperature. Fe—Sn and Fe—Ge are from ref. [212], Fe—Sb from ref. [71], pure Ge from ref. [213] and pure Sb from ref. [31]. The latter both were measured at T = 4K. Fe 80Sn20, Fe4oSnw, and 2kF, calculated with the assumption of ~ = lets. FeHISbI, are partly crystalline. The thin vertical lines indicate

‘r

35

I

~•

Fe:Sn

I

•

Fe-Ge

•

~

‘I

I~

I

•

IFe—S

2kF Liq.

Inmul ~

30

2kF

/ -

Kpe 1.1 1.0

1.1

1.0

I

•~S

(

~~_a•

—0

25

~

‘~‘~

Kpe

-

1.0 1.1

‘N

.

pe

Liq ~

)iq. • Liq. 0

liq. D

20

0

___20 _________ 60 80 100

Fe

.~

40

I

I

I

at / Sn

0 Fe

20

1.0

I

1.1,

60

80

100

at%Ge

0

.~....J-

Fe

20

•

I

1.0

•

I

60

•

I 60

100

at%Sb

Fig. 55. Positions of the first two peaks in S(K) versus composition [71,203, 212]. Open rectangular boxes denote partly crystalline films, open circles the liquid state. The thin curves indicating 2k,, were calculated with the assumption of ZF~= 1.0 or 1.1 eta. The ranges between the vertical lines approximate those compositions where K~, 2k. is approximately fulfilled and TK has a maximum.


131

Figure 56 shows the corresponding reduced pair distribution functions y(r). With changing r~due to variations of 2kF the peak positions also change. Between r~and r~,calculated according to eq. (5), Fe—Sb shows a clearly separated additional peak at the antiferromagnetic position, not observed for Au—Sb or pure Sb (fig. 7a). In Fe—Ge and Fe—Sn, instead of a clearly separated antiferromagnetic peak, the nearest-neighbour peaks show asymmetric contributions at the large-r side not seen in the Au—Sn system at comparable compositions (fig. 7b). With decreasing content of the polyvalent elements, the peaks at the antiferromagnetic position finally disappear, leaving exclusively the ferromagnetic positions occupied (Fe 77Sn23). This correlation is best observed in fig. 57, together with the fact that the matching of r1 with r~.occurs especially in the range between the solid vertical lines (1.9 eta < Z