Physica status solidi / A.: Volume 76, Number 1 March 16 [Reprint 2021 ed.] 9783112501207, 9783112501191

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plrysica status solidi (a)

ISSN 0031-8965 * VOL. 76

NO. 1

MARCH 1983

Classification Scheme 1. Structure of Crystalline Solids 1.1 Perfectly Periodic Structure 1.2 Solid-State Phase Transformations 1.3 Alloys. Metallurgy 1.4 Microstructure (Magnetic Domains See 18; Ferroelectric Domains See 14.4.1) 1.5 Films 1.6 Surfaces 2. Non-Crystalline State 3. Crystal Growth 4. Bonding Properties 5. Mossbauer Spectroscopy 6. Lattice Dynamics. Phonons 7. Acoustic Properties 8. Thermal Properties 9. Diffusion 10. Defect Properties (Irradiation Defects See 11) 10.1 Metals 10.2 Non-Metals 11. Irradiation Effects (X-Ray Diffraction Investigations See 1 and 10) 12. Mechanical Properties (Plastic Deformations See 10) 12.1 Metals 12.2 Non-Metals 13. Electron States 13.1 Band Structure 13.2 Fermi Surfaces 13.3 Surface and Interface States 13.4 Impurity and Defect States 13.5 Elementary Excitations (Phonons See 6) 13.5.1 Excitons 13.5.2 Plasmons 13.5.3 Polarons 13.5.4 Magnons 14. Electrical Properties. Transport Phenomena 14.1 Metals. Semi-Metals 14.2 Superconductivity. Superconducting Materials and Devices 14.3 Semiconductors 14.3.1 Films 14.3.2 Surfaces and Interfaces 14.3.3, Devices. Junctions (Contact Problems See 14.3.4) 14.3.4 High-Field Phenomena, Space-Charge Effects, Inhomogeneities, Injected Carriers (Electroluminescence See 20.3; Junctions See 14.3.3) 14.4 Dielectrics 14.4.1 Ferroelectrics 15. Thermoelectric and Thermomagnetic Properties 16. Photoconductivity. Photovoltaic Effects 17. Emission of Electrons and Ions 17.1 Field Emission Microscope Investigations 18. Magnetic Properties 18.1 Paramagnetic Properties 18.2 Ferromagnetic Properties 18.2.1 Ferromagnetic Films 18.3 Ferrimagnetic Properties 18.4 Antiferromagnetic Properties (Continued on cover three)

physica status solidi (a) applied research

Board of Editors S. A M E L I N C K X , Mol-Donk, J. A U T H , Berlin, H. BETHGE, Halle, K. W. B Ö E R , Newark, P. GÖRLICH, Jena, G. M. H A T O Y A M A , Tokyo, C. H I L S U M , Malvern, B. T. KOLOMIETS, Leningrad, W. J . MERZ, Zürich. A. S E E G E R , Stuttgart, C. M. VAN V L I E T , Montréal Editor-in-Chief P. GÖRLICH Advisory Board L. N. A L E K S A N D R O V , Novosibirsk, W. A N D R Ä , Jena, E. B A U E R , Clausthal-Zellerfeld, G. C H I A R O T T I , Rom, H. C U R I E N , Paris, R. G R I G O R O V I C I , Bucharest, F. B. H U M P H R E Y , Pasadena, E. K L I E R , Praha, Z. M A L E K , Praha, G. O. M Ü L L E R , Berlin, Y. N A K A M U R A , Kyoto, T. N. RHODIN, Ithaca, New York, R. SIZMANN, München, J . S T U K E , Marburg, J . T. W A L L M A R K , Göteborg, E. P. W O H L F A R T H , London

Volume 76 • Number 1 • Pages 1 to 398, K 1 to K104, and A l to A8 March 16,1983 PSSA 76(1) 1 - 3 9 8 , K 1 - K 1 0 4 , A 1 - A 8 (1983) ISSN 0031-8965

AKADEMIE-VERLAG

BERLIN

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Contents Review Article R . E.

HUMMEL

Differential Reflectometry and I t s Application to the Study of Alloys, Ordering, Corrosion, and Surface Properties

11

Modification of t h e Pair Approximation of t h e Quasi-Chemical Ordering Theory of Binary Systems

45

Original Papers V . A . KLIMENKO

V . P . BARKHATOV, V . F . B A L A K I R E V , Y U . V . GOLIKOV, a n d E . G . K O S T I T S I N

X - R a y Diffraction Investigation of Cation Distribution in High-Temperature Cubic Spinels of M g - M n - 0 System

57

V . V . K A L I N I N a n d N . N . GERASIMENKO

The Interaction between the Point Defects Introduced by I m p l a n t a t i o n and Dislocations in Silicon

65

V . I . I V A N O V - O M S K I I , V . K . OGOKODNIKOV, a n d V . D . R O Z U M N Y I

Surface Defects of Closed-System-Grown C d x H g i _ z T e Epitaxial Layers

71

Y . IWAMA a n d Y . TAKENO

Formation Process of MnBi Thin Films b y Williams' Method

75

Y u . B . BOLKHOVITYANOV

The Lattice-Pulling Effect Induced by t h e Film Stresses. Conditions of Existence and Discovery

85

G . P . K R A M A R , Y A . I . PANOVA, a n d Y . V . P A S S Y N K O V

The Influence of Microstructure on t h e Dielectric Spectra of Ferrites . . .

95

S . V . SUDAREVA, V . A . RASSOKHIN, a n d A . F . P R E K U L

The Structure of Cr-Al Alloys Exhibiting Anomalous Physical Properties

101

E . Y . A N U F R I E V a n d V . A . GURTOV

Accumulation and Discharging of the Hole Charge in Oxide Layers of MIS Structures \

107

L . WOJTCZAK, A . URBANIAK-KUCHARCZYK, J . MIELNICKI, a n d B . MRYGON

The Magnetic Contribution to Specific H e a t in Thin Ferromagnetic Films

113

V . G . FEDOTOV, A . G . BYCHKOV, D . N . KARLIKOV, L . M . GORYNYA, A . U . SHELEG, E . M . SMOLYARENKO, a n d Z . K . ORLIK

Temperature Studies of Optical Properties of Cadmium Crystals

Diphosphide 121

A . RAJEWSKA, B . PURA, J . PRZEDMOJSKI, a n d R . D^BROWSKI

X - R a y Study of t h e Smectic A - N e m a t i c Phase Transition in 4-n-Pentylphenyl-4-(Trans-4'-n-Pentylcyclohexyl) Benzoate

127

R . S. BHATTACHARYA, A . K . R A I , S . C . LING, a n d P . P . PRONKO

Epitaxial Regrowth of Si Implanted (100) and (211) GaAs

131

Contents

4 L . KOMITOV a n d A . G . P E T R O V

Optical Method for Determination of Anchoring Energy of Tilted Nematic Layers

137

C . PASNICTT, D . C O N D U R A C H E , a n d E . LTJCA

M. E. FLEET

Influence of Substitution a n d Addition of Calcium on Magnetic Properties of Nio.245Zno.755Fe204 Ferrite

145

Preferred Crystallographic Orientation for Crystallisation under Nonhydrostatic Stress

151

M . H A L B W A C H S , P . MAZOT, a n d J . W O I R G A R D

Anelastic Relaxation Phenomena in Plasma-Sprayed (AI 2 0 3 ) 3 Mg0 Spinel with Reference to Diffusion Process

157

H . MIZUBAYASHI, T . AEAI, a n d S. OKUDA

Low-Temperature Dislocation Pinning and Breakaway Observed in Ta Containing H

165

L . N . ALEKSANDROV

Formation of Semiconductor Epitaxial Films by Pulse Heating Crystallization or Regrowth

179

S . B . N E W C O M B , W . M . STOBBS, a n d J . A . L I T T L E

The Transmission Electron Microscopy of 1 to 2 n m Rhodium Particles Dispersed on Y-A1203

191

V . M . P A N , I . Y A . D E K H T Y A R , M . E . OSTNOVSKII, S . E . L I T V I N , B . G . N I K I T I N , M . M . N I S H E N K O , a n d M . P . VORONKO

Thermodynamical and and Fe 80 P 13 C 7 Alloys A.

I . LANDAU

Structural Properties

of

Amorphous

Fe 8 0 B 2 0

Analytical Calculation of Parameter s of Thermally Activated Dislocation Motion through a R a n d o m Array of P o i n t Obstacles

197 207

V . B . D U D N I K O V A a n d V . S . LJRUSOV

H.

LEMKE

Solvus of KCl:Ca 2 + Solid Solution b y Flotation Measurements of Density of Single Crystals

217

Energieniveaus und Bindungsenergien von Ionenpaaren in Silizium . . .

223

T . NANBA a n d T . P . MARTIN

R a m a n Scattering from Metal Smokes

235

S . GROSSWIG, J . H A R T W I G , U . S C H E L L E N B E R G E R , a n d W . - D . Z I M M E R

Theoretical and Experimental S t u d y of Growth Sector Boundaries in Deuterated Potassium Dihydrogen Phosphate Single Crystals

241

H . FISCHER, G . GÖTZ, a n d H . K A R G E

Radiation Damage in Ion-Implanted Quartz Crystals (I)

249

N . E . BELOVA, F . E I C H H O R N , V . A . SOMENKOV, K . U T E M I S O V , a n d S . S H . S H I L S H T E I N

Analyse der Neigungsmethode zur Untersuchung von Pendellösungsinterferenzen von Neutronen und Röntgenstrahlen

257

F . W A L Z , H . J . B L Y T H E , a n d F . DWORSCHAK

Annealing Behaviour of Dilute FeTi, FeCu, and FeMn Alloys in the Temperature Range above Stage I I I Following Low-Temperature Electron Irradiation

267

U . MESSERSCHMIDT, Y . NISHINO, T . IMURA, a n d H . SAKA

X - R a y Topographic In-Situ Observation of Slip B a n d Propagation in MgO Single Crystals

277

5

Contents W . ZAG a n d K . U R B A N

Temperature Dependence of the Threshold Energy for Atom Displacement in Irradiated Molybdenum

285

S . N E O V , I . GERASSIMOVA, a n d B . SYDZHIMOV

H.

SODOLSKI

Neutron Diffraction Study of 2 Te0 2 • V 2 0 5 Glass

297

Depolarization Current in a Dielectric with Deep Trapped Charge Layers

303

A . N . G E O R G O B I A N I , M . V . G L U S H K O V , E . S . LOGOZINSKAYA, Z H . A . P U K H L I I , I . M . T I G I N Y A N U , Radiative a n d I . A . Recombination SHCHERBAKOV in y-La S Single Crystals 2 3

311

M . M . H A F I Z , A . A . AMMAR, A . I . A L - A D L , a n d ABOUTALEB MOHAMED

Effect of Heat Treatment on the Conduction and Structure of Ge-Se-Te Amorphous Alloys

319

A . I . P E K A R E V , G . F . KRASNOVA, a n d G . Z . N E M T S E V

Orientation Dependence of Oxidation Stacking Fault Density in Silicon . .

327

G . E F T E K H A R I , D . D E COG AN, a n d B . T U C K

Electrical Conduction through Anodic Oxides on I n P

331

A . G . K A Z A N S K I I , V . F . K I S E L E V , E . A . S I L A E V , a n d V . S . VAVILOV

On the Role of Surface Phenomena in the Staebler-Wronski Effect . . . .

337

D . V . KRISHNA SASTRY a n d P . JAYARAMA R E D D Y

G. L.

WHITTLE

Influence of Nitrogen Ion Implantation on the Electrical Transport Properties of InTe and InSe Thin Films

345

Magnetic Order in Isoelectronic Co(GaVFe) Alloys

351

N . STOJADINOVIO, S . D I M I T R I J E V , S . MIJALKOVI.fx R2I^)I2

__

R, -

n2

(R1 + R2)I2

_AJg R"

An oscilloscope, connected to the P M T output monitors the light scanning over the samples. I t allows a direct measurement of the normalized difference in reflectivities of the two specimens and an observation of whether or not the light beam rests an equal amount of time on each specimen. 2.2 Sample

positioning

The samples are clamped to a mounting stage which can be moved vertically and laterally. This makes it possible to select a specific area on the sample for a given investigation, and to center the two specimens about the scanning beam. The circuit of the differential reflectometer is arranged to provide a positive peak in A R / R if the sample on top of the mounting stage has a higher reflectivity at a given wavelength. (As a matter of consistency, an alloy with a higher solute concentration is always placed on top. I n corrosion experiments, the reference specimen is placed in the upper position.) 2.3 Other

designs

A differential reflectometer, using the configuration described above, has been demonstrated to be the most versatile and sensitive design, though other approaches have been proposed. I n one case, a stationary light beam and a rotating stage on which the two specimens were mounted, has been used [3, 4], The samples have been vapordeposited on round substrates of equal size and placed in two matching holes in the spinning disc. The light beam scans a relatively large arc-shaped segment of the sample as well as a substantial portion of the blackened spinning disc. I n another design, which utilizes a moving specimen stage, the samples were mounted on the vibrating ends of a tuning fork [5]. A stationary set of samples [1, 2] has, however, several advantages: Any sample size and sample configuration can be used, smaller areas can be investigated, the area to be studied can be specifically selected, measurements in vacuum or in an electrolyte (corrosion) are less of a problem and differences in temperature, stress, or electric field can be effortlessly accomplished in the two specimens. The main advantage of stationary samples, however, is that also bulk, in addition to vapor-deposited specimens may be effortlessly studied. Bulk specimens are essential for the investigation of alloys of exact composition, small grain size, smooth surface, good reproducibility of results, and ultimate homogeneity. Their solute content can be easily checked, utilizing chemical or microprobe analysis. Bulk specimens may be polished by various methods, thus eliminating a rough surface which is often caused by annealing. Furthermore, small changes in compositional difference can be easily obtained. (The latter item is of particular importance to reduce possible variations caused by oxidation, see below, and to obtain a more pronounced structure of the spectral dependence of AR/R.) Finally, bulk specimens can be heat treated, rolled, and quenched to obtain ordered, disordered, or any other configuration. 2.4 Methods

of sample

preparation

The preparation of the specimens, to be studied by means of differential reflectometry, varies somewhat with the type of experiments being carried out. For the investigations of alloys, for example a combination of heat treatments and cold rolling is necessary to assure absolute homogeneity of the alloy composition across the samples and to obtain a fine grain. Because of the high sensitivity of the differential reflectometer,

Differential Reflectometry and Its Application to the Study of Alloys

15

even the smallest inhomogeneity may lead to structure in a differential reflectogram when the light beam is scanned across one sample only [6], Thus, one commonly applies a solution heat treatment for about ten days slightly below the solidus temperature, followed by cold rolling and short time recrystallization at a lower temperature. Two specimens containing the same type solute but a slightly different composition (generally, not more than 1 or 2 % difference) are usually imbedded side by side in the same metallographic mount. The sample pair is then polished using standard metallographic procedures (ending, for example, with 1 [jim diamond polishing compound on felt cloth). Since both alloys undergo identical preparation procedures at the same time, and the difference in solute is small, any possible changes of the surfaces such as deformation, oxidation, surface roughening, etc. are nearly identical and subtract out due to the differential technique. The measurements can, therefore, be performed in air. Any oxidation or surface roughening of the alloys decreases the peak height only slightly, but does not alter the position of the peaks on the energy scale. Imbedding and polishing the two specimens in a common mount has another important advantage: The two alloys are situated in the same optical plane causing the reflected beams to reach virtually the same point on the PMT. F o r oxidation studies, a pure metal (or only one kind of alloy) is used. One-half of this specimen is covered (after polishing) with a protective lacquer whereas the other part is allowed to corrode. After that, the lacquer is peeled off, which results in corroded and uncorroded specimens next to each other. Since the metal substrate is common to both sides, the contribution of the substrate to A R j R cancels due to the differential nature of the technique, if the corrosion product film is thin enough. F o r electrochemical corrosion studies a more advanced method is used. R a t h e r than utilizing one specimen, the metal disc is in this case divided into two parts which are electrically insulated from each other by a thin Teflon foil. Different potentials can then be applied to each side. F o r example, one specimen half may be held at the protective potential (reference) and the other at the corrosion potential. The main advantage of this technique is that the corrosion cell window, as well as the electrolyte solution, does not affect the results because of the nature of the differential reflectometer (see equation (1)). In situ studies are therefore effortlessly achieved without the use of a protective layer on the reference half. F o r high temperature oxidation studies, the entire specimen is corroded. Subsequently, the corrosion product is removed from part of the specimen b y a "half polishing" technique. I t has been found that the method of sample preparation does not affect the peak position in differential reflectograms. 3. Study ol Alloys 3.1

Foundations

I t is well known that the electron structure of metals, alloys, and semiconductors in the vicinity of the Fermi surface can be studied by measuring the spectral dependence of the optical properties [7]. Photons of proper energy, impinging on a solid, exchange their energy with electrons of a filled state. The electrons are then excited across the Fermi level into an allowed, unfilled state. I n metals, these transitions are believed to occur predominantly by conserving momentum (no change in wave number k) and are, therefore, called "direct interband transitions" [7], I n semiconductors indirect interband transitions are possible, which implies that phonons are additionally involved [7], Conventional optical spectral reflectance measurements of metals and alloys lack

16

R . E. HUMMEL

%

r

yt Ef Xs

y

Z i

Fig. 2. Electron band structure for copper according to calculations by Segall [8]

sharp structure. This is in contrast to the narrow lines which are obtained when X-rays are generated. X-rays originate from electron transitions between the sharp energy levels of the core states. Optical transitions, instead, take place between valence states which are spread into energy bands (Fig. 2) causing broad structure in conventional spectral reflectivity curves. B y using differential reflectometry, however, sharp structure is observed. The differential reflectometer is capable of determining within a hundredth of an electron volt the energies for interband transitions of electrons. The technique involves the periodic variation of the solute content within the same binary alloy system ("compositional modulation"). The reflectivity is measured as this oscillating perturbation, which modulates the band structure of the alloy, is applied to it.

3

E iel/J k 5 6

-Mnrn) Fig. 3

Fig. 4

Fig. 3. Schematic representation of a portion of a band diagram as it can be found for dilute copper-based alloys. Solid and dashed lines are assumed to be electron bands of alloys 1 and 2, respectively. Alloy 2 is thought to have a higher solute concentration. Note that equivalent interband transitions, as those shown here may also occur at other places in the band diagram, see Fig. 2 Fig. 4. Experimental differential reflectogram of a copper-0.25 at% silicon alloy. The light beam was scanned between pure copper and a Cu-0.5 at% Si alloy [9]

Differential Reflectometry and Its Application to the Study of Alloys

17

By observing the change in reflectivity (AR) the derivative with respect to the perturbation is essentially obtained. Fig. 3 illustrates this for a specific case. The solid lines in this schematic band diagram are thought to be the allowed electron states of alloy 1, whereas the dashed lines are assumed to be those of alloy 2. The Fermi level, E-g, is also indicated for both cases. For clarity the states are drawn farther apart than one would expect for a 1 or 2% difference in solute concentration. An allowed interband transition from the upper d-band to an s-state just above the Fermi level is shown for both alloys. These transitional energies, ET, are slightly different in magnitude. By varying periodically the solute concentration, X, the transition energy oscillates likewise around a medium value. When the derivative of the reflectivity is formed with respect to X and is plotted versus the photon energy, any interband transition shows up as a maximum (or minimum). The resulting differential reflectogram is equivalent to the response from one sample of composition (Xx + X2)/2 modulated periodically by a compositional increment ±AX/2 = (X2 — XJ/2 if AX is small. As an example, Fig. 4 shows a differential reflectogram as it is obtained by compositional modulation of many a-phase copper-based alloys. Four distinct peaks can usually be observed which will be designated as peaks A through D. It has been customary to assume that peaks of this type are caused by various interband transitions of the valence electrons [7], as explained above. However, any correlation between structure such as seen in Fig. 4 and optical transitions can only be decisively established by a thorough lineshape analysis. Such lineshape analysis has been performed for d-band to s-band transitions [10]. The main suppositions and results of this calculation are summarized next. The change in reflectivity caused by compositional modulation is related to the change of the dielectric function by AR AX, (2) dx dx R where e1 = n* - k2 (3) and e2 = 2nk (4) are the real and imaginary parts of the complex dielectric function e = e1—ie2 (5) and ix and /? are weighting factors, called the Seraphin coefficients [11]. The spectral dependence of the Seraphin coefficients for copper is shown in Fig. 5.

2

physica (a) 76/1

R. E. HtJMMBL

18

For the calculation in question, it is assumed that structure in A R / R is caused by transitions between d-bands and the s-band just above the Fermi level [EA(k, x) ——• Es(k, x)]. For simplicity only transitions between d-bands and s-derived states at the Fermi energy are considered, even though it is realized that other parts of the Fermi surface, particulary the ones with p-like symmetry are contributing to the structure. The results of this calculations can be applied without change to transitions into final states with p-like and other symmetries. I t is further assumed that any change in composition varies only the initial d-band, the final s-band, and the Fermi energy (see Fig. 3). Using the spheric parabolic band model one can write = E°a,.(z) + -5

. (6) a The Fermi energy is varied through solute additions according to Ej,(x) = E%(x) + E°(x) , (7) whereas E^(x) is the Fermi energy with respect to the bottom of the s-band. The interband part of the dielectric function (which is assumed to be the only part relevant in the spectral region under consideration) is expressed by the following equation: EA,s(k,x)

¿inter) =

_

hayhn?

f ^ /(fld(fc, x)) - f(E,{k, x)) J 4jr 3 hco — [ E s ( k , x) — Ed(k, x)] + ihf

'

'

K

Here, f(E) is the Fermi distribution, p^ is the interband momentum matrix element, and h r the lifetime broadening energy. The derivative de/dx of (8) yields in combination with (2) ^

= A A

[JB0 {x)

+ JS»^)]

whereas F(s, 8) = sin

z ' 270°\

V

r

\ 0

i i i i 2 4

1 i i ii -2

0

1111

Most experimental differential reflectograms have curve slopes which suggest 6 values other than 0° and 90°. These 6 values and the appropriate transition energies E T can be found by curve fitting in the following way: In Fig. 7, part of an experimental differential reflectogram is shown. The minimum of (ARjR) is used as the zero point of the vertical axis. Then three critical frequency values are extracted: ct>m is the frequency at which (AR/R) has its maximum value; a>' " is '). The ratio (ARIR)m»l(ARIR)m is a function of 6 only and is plotted in Fig. 8. Once 6 is known, coT (and thus ET), can be found as a value between ay' and a>„ where (14) ^ ( 1 + sinSK p / R l raT 2 \ R Ir The lifetime broadening can be calculated by r=(a>m-caT)

tenx(]-6

+ y).

(15)

For illustration, a lineshape analysis, as described above, has been applied to an experimental differential reflectogram for a copper-3.5 at% aluminium alloy. Fig. 9 shows the experimental spectral dependence of A RjR in the vicinity of peak A (solid line) and some calculated AR/R values (dots) for 8 = 55°. The agreement is remarkably good. One can state therefore with reasonable confidence that peak A and the substructure around this peak are caused by electron transitions from a d-band to s-states

Fig. 7. Schematic representation of a differential reflectogram in the vicinity of peak A. Specific w and (\B/E) values are shown which are extracted from the differential reflectogram. This procedure allows to determine 6. The photon energy is E = ha> [10] 2*

20

R . E . HUMMEL

Fig. 8

Fig. 9

Fig. 8. Calculated dependence between 0 and (AR/it)m"/(AR/R)m

[10]

Fig. 9. Experimental differential reflectogram, A R/R versus E (solid line), and calculated versus s (dots) for a Cu-3.5 a t % A1 alloy [10]

F(s)

just above the Fermi energy. Comparison of the experimental transitional energy ET with that deduced from calculated band diagrams [8, 24] suggestes that peak A is caused by transitions from the upper d-bands to the Fermi energy. Detailed results are given in Section 3.2. Experience has shown that the difference between Em and ET is usually only between two and six hundredths of an eY [9] — a difference which is too small to be recognized in current band calculations. Thus, the peak energy, Em can be used in many cases as a good approximation for the true transition energy, ET. The lineshape in the vicinity of peaks B and C has been analyzed by considering the L^t-Ey —» Li as well as the L 2 —> L x transition [12] (Fig. 2). Using transition energies and other variables as adjustable parameters an experimental differential reflectogram could be essentially reproduced. An alternative path to obtain transition energies [3, 12 to 14] is to convert the measured static and modulated reflectivity data in AE2 spectra by means of a KramersKronig analysis [7, 15], Difficulties usually arise with this method since R is measured only in a limited energy range, whereas the Kramers-Kronig relation requires data for wavelengths from zero to infinity. In practice, it is attempted to overcome this limitation by extrapolating R and A RjR beyond the experimental range usingtheoretical or phenomenological considerations. Such an extrapolation would not cause substantial error if one can assume that no structure exists beyond the measured spectral range. This assumption is probably valid only in rare occasions. However, the energy position of singularities is always reproduced with good accuracy even if A RjR is known only in a limited energy range [13]. Prange et al. [14] avoided the extrapolation of A RjR and extrapolated instead Aex the spectral dependence of which is generally assumed to be simpler. Other investigators (e.g. [16]) estimated the ratio of alloy to pure reflectivity, RJRp, from their differential reflectivities, then calculated £ip and e 2p for the pure metal from an independent measurement of i?p by means of a Kramers-Kronig analysis, and finally calculated eia and s 2 a from _Ra employing another Kramers-Kronig procedure. This yielded Ae2 = e2p — £2a from which the free electron part was separated to yield Ae^t,. Rosei et al. [12] calculated separately the imaginary part of the dielectric constant for copper and for the alloys. Then they convoluted them with different "broadening parameters" and subsequently obtained Ae2 by subtraction. In concluding this section, it is noted that inaccuracies and possibly loss (or even gain) of structure may be introduced in any experimental data by digitizing or handl-

Differential Reflectometry and Its Application to the Study of Alloys

21

ing them. Needless to s a y t h a t it is always desirable to use t h a t method of d a t a reduction which involves the least a m o u n t of manipulations of the measured curves while obtaining the s a m e t y p e of results. Compositional modulation (small AX) a n d the analysis described a t the beginning of this section (Fig. 6 to 9) provides direct, reliable, and accurate d a t a on the shift rate of some interband transitions which are caused b y solute additions. 3.2

3.2.1

Results

Copper-zinc

and similar

alloys

I n this section, experimental results dealing with the optical a n d electronic properties of some « - p h a s e copper-based alloys are summarized. T h e solutes which were studied include elements having two, three, four, or five valence electrons. The differential reflectograms for copper alloys containing zinc, gallium, a l u m i n u m , tin, silicon, or germanium are qualitatively similar [9] thus reflectograms of only one of these binary s y s t e m s are presented a s an e x a m p l e . I n F i g . 10, a series of A R / E v e r s u s E curves taken on copper-zinc alloys are shown, in which the zinc content increases from top to b o t t o m . The characteristic p e a k s A to D , mentioned in Section 3.1, can be recognized. One observes t h a t the energies of these p e a k s shift with variation of t h e solute concentration. I n particular, peak A, a t a b o u t 2.2 eV, is seen to shift to higher electron energies. This p e a k has been identified in Section 3.1 to result f r o m electron transitions between the upper d-bands to the conduction b a n d s j u s t a b o v e the F e r m i surface. I t is believed [17] t h a t they occur near the L s y m m e t r y point, a n d also to a smaller extent, near X a n d other points (see F i g . 2). I n F i g . 11, this shift in transition energy, Z?T, is shown (along with ET's measured on other copper-based alloys) a s a function of solute content, X. Essentially a linear increase of E T with increasing X is observed. The slopes are f o u n d to be steeper for solutes having successively larger electron to a t o m ratios (with the exception of c o p p e r - a l u m i n u m alloys). The rise in energy difference between upper d - b a n d a n d F e r m i level c a u s e d b y solute additions can be explained in a first a p p r o x i m a t i o n b y suggesting a rise in the F e r m i energy, which results when e x t r a electrons are introduced into the copper m a t r i x from the higher v a l e n t solutes (rigid band model) [18], T h e slopes in the i? T = f ( X ) curves in F i g . 11 are, however, considerably smaller t h a n predicted b y the rigid band model. These results suggest, therefore, t h a t the d - b a n d s are likewise raised with

Fig. 10. Experimental differential reflectograms for various bulk copper-zinc alloys. The parameter on the curves is the average zinc concentration of the two alloys in a t % . (The curve marked 0.5% for example resulted by scanning the light beam between pure copper and a Cu-1% Zn alloy. The compositional difference between the two alloys was characteristically 1 at%.) The individual curves have been shifted for clarity [9]

-3lnm!

22

R . E . HUMMEL

Fig. 11. Threshold energies, E j , for interband transitions for various copper-based alloys as a function of solute content, X (A Zn, • Ga, o Al, V Sn, x Si, + Ge). The E T values are extracted from differential refleotograms similar to those shown in Pig. 10. For this the structure around peak A is used in conjunction with a lineshape analysis, as shown in Fig. 7. The difference between Ey and the energy corresponding to the maximum of peak A (E m ) is about 0.03 eV. The possible error in marking the maxima Em is indicated by the error bar a. The possible error introduced b y the lineshape analysis is indicated by,b. The rigid band line (R.B.) for Cu-Zn is added for comparison (from [9], see in this context also older results in [19, 20] which differ in approach and result from those reported here)

increasing X and/or that the Fermi level is shifted up much less than anticipated. Band calculations [21] substantiate this suggestion. They reveal that upon solute additions to copper, the d-bands become narrower (which results from a reduction of Cu-Cu interactions) and that the d-bands are lifted up as a whole. Furthermore, the calculations show that solute additions to copper cause a rise in E¥ and a downward shift of the bottom of the s-band (^ri)- Fig. 3 reflects these results. Because of the lowering of E r i , the Fermi level rises much less than would be expected if E r i had remained constant. An important characteristic of all ET = f(X) curves is that the threshold energy for interband transitions, E r , does not vary appreciably for solute concentrations up to slightly above 1 at% (Fig. 11). Friedel [22] predicted just this type of behavior and related it to "screening" effects. He argued that for the first few atomic percent solute additions to copper, the additional charge from the higher valent solute is effectively screened and the copper matrix behaves as if the impurities were not present. The matrix remains essentially unperturbed as long as the impurities do not interact. Rosei et al. [12] who re-analyzed one differential reflectogram (copper versus Cu1 at% Zn) of Hummel and Andrews [23] arrived at a different conclusion on this point. Rosei deduced, even for this low concentration, a shift rate of 0.014 eV/at% Zn, a value which is characteristic of slightly higher zinc concentrations, see Fig. 11. It is believed that the data reduction [12, 13] (Kramers-Kronig analysis, etc., see Section 3.1) may have simulated this relatively large shift for small solute concentrations. We turn now to electron transitions which cause peak D in the differential reflectograms and which take place when electrons absorb approximately 5 eV from the impinging photons. This peak involves transitions from the lower d-bands to the Fermi energy [24 to 28] (Fig. 2). Thus, considerations can be employed that are similar to those used to explain the 2.2 eV peak. Two interesting variations should be pointed out, however. First, for photon energies around 5 eV, the /?/» ratio is negative (Fig. 5) which results in a lineshape that is inverted with respect to the one observed in the vicinity of peak A. Second, it has been found [9, 25] that in copper-based alloys, in which the solvents possess a large valence electron concentration, the transition -^d, lower E-g is much more pronounced than in alloys containing solutes with a small number of valence electrons. (This might be attributed to an increase in the density of states of the d-bands at this energy for alloys with large valence electron concentration.) For example, in Fig. 12 (copper-arsenic), feature D is a sharp minimum, whereas in Fig. 10 (copper-zinc) the same feature consists merely of a shoulder. Thus, copper alloys containing silicon, germanium, or arsenic are used preferentially to

Differential Reflectometry and Its Application to the Study of Alloys 1.6

1

EleW3 4 56

1

AC

A X'0.25% As

075

23

Pig. 12. Experimental diferential reflectograms for various bulk copper-arsenic alloys. The parameter is the average arsenic content of the two alloys in at%. The difference between the two alloy compositions is equal to or smaller than 1 at % rai

r/

1

- J J p - ^ A V '

I

\

study the -Ed, lower —*• E-g transition. I n Fig. 13, the energy of minimum D is depicted as i 600 WO 800 200 a function of solute concentration. This plot is similar to -J (nm) the one of Fig. 11 which suggests t h a t the lower d-band —• E-g transitions behave similarly to the upper d-band —* E-g transitions. Specifically, one observes only a very small shift in transition energy up to about' 1 a t % solute, followed by a steady rise in interband energy amounting 16 X 10" 2 eV/at% solute. The slope in Fig. 13 is about ten times steeper t h a n in Fig. 11 (E&t upper —• E v transition) in which Er increases by only 1.2 X 10 - 2 eV/at% solute. This variation can only be attributed to a difference in the behavior of the upper compared to the lower d-bands. Thus, the results seem to indicate t h a t the lower d-bands, at the points of origin for the electron transitions, move only small amounts upon alloying (see Fig. 3). One more structural feature is contained in the differential reflectograms shown in Fig. 10. I t is of the %-type (see Section 3.1) and involves peaks B and C. This structure around 4 eV is commonly ascribed to transitions near the L symmetry point [12, 24, 30 to 33]. Rosei et al. [12] identify the L'2(E¥) — L^ and the critical point L2 —» Li transitions (Fig. 14) as the cause for peaks B and C and the substructure around these peaks. Chen and Segall [27] found t h a t the h'^E^) L t transition energy for pure copper is 4.26 eV which is exactly halfway between peaks B and C (for low solute concentrations). When the energies of peaks B and C are plotted versus the solute concentration, both curves run essentially parallel to each other [29]. Therefore, the information about the energy shift of the transition around 4 eV may be taken to be contained in peak B or peak C. i

i

i

i

i

Fig. 13. Energy (E m ) of peak D and shift in energy (A£?m) compared to pure copper as a function of composition for various copper-based alloys ( • Si, • As, A Ge) [9]

24

Fig. 14

R . E . HUMMEL

Fig. 15

Fig. 14. Schematic band structure near L for copper (solid lines) and an assumed dilute copperbased alloy (dashed lines) Fig. 15. Energy of peak B and shift in energy compared to pure copper as a function of composition for various copper-based alloys (A Zn, O Al, • Ga, V Sn) [9], A similar shift is found for peak C [29]

In Fig. 15 the energy of peak B is shown to decrease sharply with increasing solute concentration suggesting a lowering of particularly the Lx symmetry point due to alloying (Fig. 14). Band calculations confirm this suggestion [34], The transitions involving the conduction bands are affected by solute effects, for example lattice dilations. Different solute elements having different atomic radii must, therefore, influence the L2 —• Lx transitions in a different way. On the other hand, Fig. 15 suggests that these transitions are also strongly influenced by the electron concentration of the solute: the decrease of the energy of peak B is more substantial for alloys which contain solutes with larger valence electron concentration. Further, both Cu-Ga and Cu-Al alloys, which possess identical electron concentrations per atom, behave alike. It is important to note that in copper-based alloys, containing silicon, germanium, or arsenic, having solute concentrations of more than 1 at%, minimum B (and the 4.2 eV absorption edge [16, 25]) becomes less distinguishable, (Fig. 12) which may suggest that for higher electron concentrations the L^JEJJ-) —>- Lx transition loses some strength. One further piece of information can be taken from the differential reflectograms. It pertains to the "lifetime broadening" energy: The initial and the final states of any interband transition are generally not sharp, but are somewhat broadened because of the limited time an electron remains at an excited energy level [7]. This lifetime broadening can be calculated using equation (15) and has been found to increase from 1.2 X 1014 s - 1 for pure copper to 1.9 X 1014 s _ 1 for a-phase copper alloys containing about 9 at% of the solute [9]. The lifetime broadening energy h r increases likewise with solute concentration after an initial flat portion up to approximately 1 at% solute [10]. In summarizing this section, it can be stated that differential reflectometry is well capable of identifying the electron transitions for copper-zinc and similar alloys. The results presented so far confirm convincingly the findings which have been obtained by theoretical means. Furthermore, the information contained in Fig. 10 to 15 add

Differentia] Reflectometry and Its Application to the Study of Alloys EleV) 3 '4 55 1 r 1 11 e

1

0.5 X-0.25ZAU 0.5 0

b

a 0.75

[

^

/-\ ,

A

y \

V M

7 0 T 0\ T

25

Fig. 16. Experimental differential refleotograms for various bulk copper-gold alloys. The peaks are labelled with lower case letters to emphasize the difference to Fig. 10 [9]

quantitative, experimental data about finer details of the change in electron configuration which happens due to the alloying process. We have seen t h a t in the range between 1.5 and 6.2 eV electron transitions occur between the upper d-bands and the Fermi energy, between the lower d-bands and the Fermi energy, and between some conduction bands. 3.2.2

Copper-gold

alloys

The differential reflectograms of dilute alloys of copper with gold are somewhat different from those presented in the foregoing section. The reflectograms 0 are characterized by an %-type structure near 2 eV containing a minimum a and a maximum b, a broad maximum c around 3 eV, and gj-type structures d - e and 400 ZOO f - g at higher energies (Fig. 16). This behavior suggests - Mnml a slightly different electron structure of Cu-Au alloys. Copper and gold possess the same number of valence electrons. Therefore, one might expect that the Fermi level would not be altered substantially when small amounts of gold are added to copper. I n addition, one might assume, as the virtual bond state model does, t h a t the copper and gold d-states retain their identities for small solute concentrations (see Section 3.2.3). The differential reflectograms in Fig. 16 seem to suggest just this behavior: the threshold energy for interband transitions (peaks a and b) remains essentially constant in Cu-Au alloys; see also Fig. 17, b. (Conflicting results on this point have been reported in the literature. The findings have been extracted, however, mainly from conventional optical data. Both a decrease [17, 5] and an increase [35] in the threshold energy for interband transitions, as well as a "nearly zero slope" [36, 37] have been observed, when gold was added to copper.) The structure around 4 eV has been assigned in the previous section to conduction band to conduction band transitions between the L2 and L x symmetry points. Fig. 16 and 17, d, show that, as before, the energy difference between these symmetry points decreases with increasing gold additions. This shift is, however, one order of magnitude smaller than t h a t observed for copper-aluminium, suggesting only small changes in the conduction bands when gold is added to copper. Since the electron concentration per atom stays essentially the same in these alloys, only solute effects seem to be responsible for the changes in this transition.

10 X(at%)~

Fig. 17. Energy of peaks b, c, and d as a function of composition for copper-gold alloys. Note the scale difference for the graph b. The limit a shown next to this curve is the possible error in marking Eb [9]

26

R. E.

HUMMEL

The £i-type structure around 5 eV involving peaks f and g is interpreted, as before, to be caused b y transitions from t h e lower copper d-bands to t h e Fermi level. These transitions become weaker with increasing gold additions, as can be inferred from t h e disappearance of this structure a t higher gold concentrations. Finally, Fig. 16 shows broad structure around 3 eV which has not been observed for copper-zinc or similar alloys. This broad peak m a y be the result of several closely spaced i n t e r b a n d transitions which cannot be individually resolved a n d which m a y originate from t h e gold d-bands (d-band splitting). The m a x i m u m c increases with increasing gold concentration (Fig. 17, c) suggesting a lowering of t h e gold d-bands due t o alloying.

3.2.3 Copper-nickel

and similar

alloys

Nickel has one electron less t h a n copper. The rigid band model would suggest, therefore, t h a t b y adding nickel a t o m s to a copper matrix, some of t h e copper 4s-electrons would fill the e m p t y nickel 3d-states. Thus, the Fermi energy would be lowered a n d t h e transition energy between upper d-bands and Fermi energy would be reduced. This would cause a decrease of the threshold energy (around 2.2 eY) in a differential reflectogram. An alternate model, t h e virtual bound state model, has been suggested b y Friedel [38] and Anderson [39] who assume t h a t t h e 3d-orbitals of nickel form highly localized levels around the i m p u r i t y atoms. Copper and nickel d-electrons would t h u s form essentially independent d-bands. The energy separation between Cu d-states and E ¥ would be expected to remain unchanged and t h e peak energy around 2.2 eV would stay constant with increasing nickel additions to copper. Differential reflectometry is likely to decide between these two models. I n Fig. 18, a series of differential reflectograms, taken on various copper-nickel alloys, is shown [29]. These reflectograms resemble in i m p o r t a n t features those, obtained for c o p p e r gold alloys (Fig. 16). Of particular interest is the e ^ t y p e structure, involving peaks a and b around 2.2 eV. As already pointed out, the energy EleV)— for interband transitions, for this lineshape is located 1.6 2 3 k 5 6 near the center between minimum and m a x i m u m , see Fig. 6. For the dilute copper-nickel alloys shown in Fig. 18 with minima and m a x i m a a t 2.1 and 2.3 eV, respectively, a n Et of 2.2 eV can be deduced in agreement with t h e

800 - — 2

600

Inm)

hOO

200 ,

Fig. 18. Experimental differential reflectograms of various bulk copper-nickel alloys. The parameter on the curves is the average nickel content of the two alloys in at%. The difference between the two alloy compositions varies between 0.5 and 5% (from [29] and new results by Hummel et al., see also [42])

Differential Reflectometry and Its Application to the Study of Alloys

27

threshold energy found for other copper-based alloys. The structure around 2.2 eV is thus again ascribed to transitions from the top of the copper d-bands to the Fermi surface. I t is noted in Fig. 18 t h a t ET (2.2 eV) remains constant with increasing solute content, which suggests t h a t virtual bound states are indeed formed when nickel is alloyed to copper. Further, it is observed t h a t the structure around 2.2 eV becomes broader with increasing solute concentration, which is attributed to an increase in lifetime broadening. The virtual bound state parameters, t h a t is the energy E d below the Fermi energy of the broadened peak in the electronic density of states, and its halfwidth A have been found to be about 0.8 and 0.2 eV, respectively [40 to 42], Differential reflectograms show a shallow peak [40] in the vicinity of 0.8 eV which is interpreted to be caused by transitions from the nickel d-bands (virtual bound states) to the Fermi surface. Photo emission studies [43, 44] on Cu-Ni alloys have also suggested the validit y of the virtual bound state model. As for the previously discussed copper-based alloys, the structure slightly above 5 eV, involving peaks f and g, is ascribed to transitions from the lower copper d-bands to the Fermi level. Fig. 18 indicates t h a t the energy for this transition increases with higher nickel content. However, the rise in ET is about one order of magnitude smaller t h a n that observed for, say, copper-aluminum alloys. The conduction band to conduction band transitions around 4 eV involving the L symmetry points (peaks d and e) seem to be only weakly represented for small nickel concentrations, whereas maximum e is the dominant peak for nickel-rich alloys. Transitions which cause the broad structure around 3.5 eV (peak c) decrease in strength with increasing solute concentration. Peak c is attributed (as in copper-gold) to new transitions involving the nickel d-bands. A transition around 6 eV (minimum h) which has not been observed in other copper-based alloys seems to decrease in energy with increasing nickel concentrations. We turn now to alloys which are in many respects similar to copper-nickel. Alloys consisting of Au-Fe, Au-Ni, Cu-Pd or A u - P d seem to have one important feature in common: the d-bands of the solutes are situated above the host d-bands. They hybridize only weakly the s-derived bands of the solvent and form virtual bound states. Thus, the optical spectra for these alloys display structure at low energies (below the interband edge of the solvent), i.e. at energies for which the optical spectrum of the solvent is normally structureless [41, 42, 45 to 51]. The behavior of gold-iron alloys should be similar to copper-nickel alloys — both consist of a noble metal solvent and a transition metal solute. Nevertheless, a rise of the gold upper d-band —> 2?F absorption edge has been reported which amounted to about 0.011 eV/at% iron [52], This is 16

2

3

k 56

tt:

10 Fig. 19. Experimental differential reflectogram for bulk gold-0.5 at% iron. The specimens consisted of pure gold and an Au-1 at% Fe alloy, respectively (unpublished data by R. E. Hummel and M. Goho)

0 800 600 -—Mnm)

400

200

28

R . E . HUMMEL

approximately the same rise as found in copper-zinc (Fig. 11). A 3.5 eY edge, assigned to transitions between conduction bands, was found to decrease nonlinearly with solute concentration, averaging to 0.022 eV/at% iron. Additional absorption was detected between 3 and 6 eV which was attributed to transitions from the iron d-states to the Fermi surface [52], A differential reflectogram for a gold-0.5 a t % Fe alloy is shown in Fig. 19. 3.2.4 Alloys between noble metals The spectral differential reflectivity of some binary alloys of copper, silver, and gold has been investigated [16]. I n all six alloy systems a change in the d-band —• E F gap energy with increasing solute content has been found ranging from —0.0033 eV/at% for copper-gold to —0.025 eV/at% for silver-gold alloys. (The former result is in variance with that of other findings [9, 36, 37].) Of interest are the silver-copper alloys because of the extremely small mutual solubility of their components. The main edge was found not to shift with increasing solute content; instead the strength of the absorption peak decreased [16]. Extended investigations along the lines described in the previous sections (bulk alloys, small AX, less involved data reduction, larger solute concentration range) may shed more light on the behavior of these alloys. 3.2.5 Nickel-based

alloys

As already mentioned above, the d-bands of nickel are situated closer to the Fermi level, than in copper. They are split; some spin bands are filled, lying completely below the Fermi level; the others (the minority spin d-bands) extend slightly above the Fermi energy [53, 54], Thus, electron transitions having relatively small energies are possible. Fig. 20 shows the spectral dependence of the differential optical conductivities A a = (v/2) Ae2 of some nickel-based alloys derived from the differential reflectivity [55, 56]. Tokumoto [55] assigned the minima in A