Electrical Transport During Phase Transformation in Metallic Glasses Citation Schulz, Robert (1984) Electrical Transport During Phase Transformation in Metallic Glasses. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/yxsm-vx37. https://resolver.caltech.edu/CaltechETD:etd-01252007-153245 Abstract Electrical resistivity measurements have been used as a primary tool to study relaxation phenomena and high temperature structural transformations in metallic glasses. We will try to correlate, throughout this work, the resistivity behavior with structural measurements from high and low angle X-ray scattering, transmission electron microscopy, Mossbauer experiments and low temperature superconductivity measurements to get a better understanding of the scattering mechanism during the phase transformation. We will distinguish between topological relaxation and chemical relaxation and in this latter case, emphasis will be given to phase separation. Three systems will be discussed extensively. Amorphous Cu 100-x Zr x around the equiatomic composition, (Mo 0.6 Ru 0.4 ) 100-x B x for boron concentrations ranging from x = 14 to x = 22 and (Zr 1-x Hf x ) 62 Ni 38 across the concentration range. We will show that upon annealing at temperature above 300°C, Cu 50 Zr 50 phase separates into two amorphous phases of concentration close to CuZr 2 and Cu 10 Zr 7 . During the phase separation an anomalous resistivity behavior, similar to the one observed during the early stage of Guinier-Preston zones formation in crystalline alloys is seen. In Mo-Ru-B alloys, the nonlinear behavior in resistivity is correlated with the presence of a peak in the small angle X-ray intensity and is explained in term of spinodal decomposition caused by the diffusion of boron in the amorphous matrix. In (Zr 1-x Hf x ) 62 Ni 38 , an exothermic transformation which gives rise to an increase in resistivity but no resolvable Bragg peaks in the high angle X-ray pattern is investigated in detail. The applicability of the Ziman formalism to successfully explain the behavior of the resistivity in all these cases is questioned and an alternative approach, based on d-band conduction, is proposed. New ways of understanding transport phenomena during structural relaxation in metallic glasses follow from these discussions. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Applied Physics) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Applied Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Johnson, William Lewis Thesis Committee: Duwez, Pol E. (chair) Goodstein, David L. Tombrello, Thomas A. Vreeland, Thad Johnson, William Lewis Defense Date: 23 May 1984 Funders: Funding Agency Grant Number Natural Sciences and Engineering Research Council of Canada (NSERC) UNSPECIFIED Department of Energy (DOE) UNSPECIFIED Record Number: CaltechETD:etd-01252007-153245 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-01252007-153245 DOI: 10.7907/yxsm-vx37 Related URLs: URL URL Type Description https://doi.org/10.1016/0022-3093(84)90673-2 DOI Article adapted for Section IV.1. Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 340 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 09 Feb 2007 Last Modified: 29 Oct 2025 21:47 Thesis Files Preview PDF (Schulz_r_1984.pdf) - Final Version See Usage Policy. 4MB Repository Staff Only: item control page

ELECTRICAL TRANSPORT DURING PHASE TRANSFORMATION IN METALLIC GLASSES Thesis by Robert Schulz in Partial Fulfiliment of the Requirements for the Degree of Doctor of Phi fosophy Callfornia Institute of Technology Pasadena, Callifornla 1984 (Submitted May 23, 1984) TO MY MOTHER -Ill- ACKNOWLEDGEMENTS 1 am very grateful to Professor W.L. Johnson and Professor Pol Duwez for [tntroductng me to the field of metallic glasses. 1 would especially Ifke to thank BIII Johnson for his support and for glving me the opportunity to pursue my own Interest in the fleld, His great knowledge In materlals, constant enthuslasm and fascination In science, together with his friendship makes him a great person to work with. ! would IJItke to thank Konrad Samwer for hfs help and encouragement with the work on Cu-Zr, Madhav Mehra and Bruce Clemens with Mo-Ru-B, ViadimiIr MatijJasevic wlth Zr-NI and the Mossbauer experiment, and finally Michael Atzmon and Pat Koen with the TEM results. For many useful discussions and experimental advice | would like to thank Art Willlams and Mike Tenhover who were very helpful! when | first Jolned the group. On the technical side | would like to thank Concetto Geremla and Angela Bressan, Thelr help and compantonshIp during the past years will always be cher!shed. 1 would IIke to acknowledge Johanne Lagantere for her great support during the writing of this thesis and also Dave Baxter for many useful comments. To conclude, J wish to thank all the members of BIII's group who hel ped In makIng my years at Caltech frultful and enjoyable. Support for thls work came from the Natlonal Research Councl! of Canada and from the UnIted States Department of Energy. -[v- ABSTRACT Electrica! resistivity measurements have been used as a primary tool to study relaxation phenomena and high temperature = structural transformations In metallic glasses. We wlll try to correlate, throughout thls work, the resistivity behavior with = structural measurements from high and low angle X-ray scattering, transmission electron microscopy, Mossbauer experfments and low temperature superconductivity measurements to get a better understanding of the scatter!ng mechanism during the phase transformation. We will distingulsh between topological relaxation and chemical relaxation and In this latter case, emphasis wlll be glven to phase separation. Three systems wlll be discussed extensively. Amorphous Cuj00-xZrx around the equlatomic composition, (Mop 6Rup.4)100-xBx for boron concentrations ranging from x=14 to x=22 and (Zry4_,Hfy)egNizg across the concentration range. We will show that upon annealing at temperature above 300°C, CusoZrsq phase separates Into two amorphous phases of concentration close to CuZro and Cu;oZr7. During the phase separation an anomalous resIstivity behavior, similar to the one observed durlng the early stage of Gulnler=-Preston zones formation [n crystallIne alloys Is seen. In Mo-Ru-B alloys, the nonlinear behavior In resistivity Is correlated with the presence of a peak In the small angle X-ray Intensity and [s explafned In term of spinodal decomposition caused by the diffusfon of boron In the amorphous matrix. In (Zry-yHfyx)62NI3g» en exothermic transformation which gives rise to an Increase In resistivity but no resolvable Bragg peaks In the high angle X-ray pattern Is_ Investigated In _ detall. ww Yo The applicabliity of the Ziman formalism to successfully explain the behav lor of the resistivity In all these cases Is questioned and an alternative approach, based on d=band conduction, Is proposed. New ways of understanding transport phenomena durtng structural relaxation In metallfc glasses follow from these discussions. V. Vi. ~«vl=- TABLE OF CONTENTS INTRODUCT I ON THEORETICAL ASPECTS 11.1 The Ziman Formalism 11.2 Topological Relaxation A. The Kelton=-Spaepen Approach B. Egami's Model for Structural Relaxation C. The Allla = Turtelll Theory D. Phenomenological Approach Based on the Cohen and Grest Theory of the Glass Transition 11.3 Chemical Relaxation A. The Rossiter and Wells Theory B. The Hillel, Edwards and Wilkes Theory C. Application to Nucleation In Topological ly Random Structures D. The Case of Spinodal Decomposition EXPERIMENTAL CONSIDERATIONS A. The Resistivity Measurements B. The Structural Measurements RESULTS AND DISCUSSIONS IV.1 Curso 9-xZl x Glasses We5 (Zr Hf JegNizg Glasses EVIDENCE FOR d-BAND CONDUCTION IN METALLIC GLASSES CONCLUSION REFERENCES -e 18 19 23 25 26 32 36 43 49 52 53 57 62 84 110 136 158 162 Table 11.3.1 Table 11.3.2 Table IV.1.1 Table IV.2.1 Table 1V.2.2 Table {V.3.1 evile- LIST OF TABLES Parameters used for calculating the excess resistivity during the growth of fcc microcrystals In a topologically random structure. Size and number of atoms for several fcc spherlcal microcrystals. G(r) parameters for ZrsoCusg metallic glasses (a) as quenched (b) annealed 11h, at 210°C and (c) subsequently annealed 9h. at 300°C, Crystallization temperatures and the % drop of the resistivity during the first two steps of the crystallization of (Mog 6Rug,4)100-»8x for x=14, 18, 22. G(r) parameters for (Mon 6RU9, 4) 100->8x after several heat treatments. Temperatures of the exotherms and the relative change In electrical resistance for each transition occurring In (Zry_ Hf, eoNIzg for x=0, 50, 100. Page 48 48 82 90 105 116 Table 1V.3.2 Table 1V.3.3 Table V.1 evill- Superconductivity transitions, critical field gradients and average electronic diffusivities for as quenched and heat treated Zrg2Nl38° Quadrupole splitting for different sites In (Zr9,62N10,38)99.8Fe0,2 alloys. Electronic density of states at the Ferm! level together with other parameters for Cujo9-»Zrx metallic glasses. 126 132 154 1.1a 1.2 1.3 2.2.1 2.2.2 Ze2e2d - x= LIST OF FIGURES Change of electrical resistance with temperature, normaj ized to the value at 300°K for CuzzTiz7 metallic glasses (Heating rate = 10°K/min.). Enlargement of (a) showlng the excess resistivity above the I[fnear contribution observed at low temperature. The structure factor S$(Q) obtained from neutron diffraction for CuggTi34 In the liquid state and In the glassy state. Excess res!stivity measured for Cuz3Tio7 glasses during the first and the second heating cycle (Heating rate = 10°*/miIn.). After the second cycle a reversible behavior In the resistivity can be achieved. Page Schematic Illustration of the first peak of the structure 2] factor S(Q) In the as quenched state, relaxed state and at high temperature. Relative change In resistivity as a function 9 of anneal Ing time at various temperatures for (Mog 6Rup.4)7gB22 glasses. The solld line Is a fit to the Kelton-Spaepen theory of structural relaxation, Relative change In resistivity for Aug. 776e9 .136SI9.094 alloys calculated for different heating rates together with the exper!mental data of Chen and Turnbull (ref. 31). 22 29 2.3.2 2.3.3 2.564 2.3.5 3el -x- Phase diagram for a binary alloy (or the Infinite three dimenstonal Istng model, ref. 36) and the evolution of a high temperature configuration follow!ng a quench to @ temperature T = 0.59 TL, Development with time of the structure factor S(K,+) (full ITne) corresponding to a quench from a temperature above the coexistence curve to a temperature T = 0.59 T, Inside the spinodal region (see fig. 2.3.1) (the data are from ref. 36). Also represented In the figure ts the function K3 Ju(k)] 2 = K3/(14(K/A)2)2 (dashed Ines) for different velues of the parameter . Structure factor S(K) for spherical fcc clusters. The hard sphere atomic radius Is taken to be 1.3 A. The first ten shells are represented, Normal lzed resistance trace taken at a heating rate of 10°C/min. for PdgsYz¢ metallic glasses. Calculated excess resistivity during the growth of fcc microcrystals In a topologically random structure. The excess resistivity Is plotted vs the size of the zones. All the parameters relevant to this calculation can be found In Table 11.3.1 and 11.3.2 Picture of the experimental setup. Page 33 35 42 44 47 55 3.2 3.3 4.1.1 4.1.2 4.1.5 4.1.6 4.1.7 -xl- Picture and schematic design of the tip of the resistivity probe. Mossbauer exper Imental setup and the drive electronics. The maln and the second band of the radial distribution function for the Cu-Zr system across the concentration range. Average nearest nelghbor distances vs Zr concentration for Cujoo-x2r~ metallic glasses. Relative change with time of the electrical resistance of CusoZrsp In Isothermal annealing Ro = R(t=0). HIgh angle X-ray scans with CuK radtation for CuZr (a) as quenched (b) annealed 10h. at 300°C (c) annealed 10h. at 350°C, Change In the electrical res!Istance with temperature of CusgZrs0 for different pre-annealted conditions. Ro = R(T=300K). X-ray diffraction pattern of (a) CuZro crystalline powder (b) CuZr annealed 12h. at 350°C then quenched after the first resistivity drop (c) same sample quenched after the second resistivity drop (d) Cu192°7 crystalline powder. Page 56 59 65 66 68 69 71 72 74 4.1.8 4.1.9 4.1.10 4.1.11 4.1.12 4.1.13 4.2.1 4.2.2 -xI fe X-ray diffraction scans with Mok ratiation for CuZr, CujoZrz and CuZro (a) as quenched (b) annealed at 210°C for 10h. (c) sample b annealed at 300°C for 9h. Palr distribution functions for Cusq7rsq (a) as quenched (b) annealed at 210°C for 9h. (c) sample b annealed at 300°C for Qh. Pair distribution function for (a) CuyoZr7 (b) CuZro- Patr distribution function for as quenched CusoZrs5q and the RDF between the annealed CusgZrs5q at 300°C for 9h. and the @s quenched sample. The full curve (a) which represents the sum of three G(r) [(b) (67.8% Zrsqcuso), (c) (26.9% Zr7Cuyg) and (d) (5.2% Zr2Cu)] corresponds to a least square fit to the data for the annealed CusgZrsq (cross polnts). (a) TEM of as quenched ZrCu (b) annealed 11h. at 210°C and 9h. at 300°C (c) diffraction pattern of the annealed sample. Reststance traces obtained at a heating rate of 15°C/min. for x=14 , 18 and 22 (Ro= room temperature resistance). The heat treatment, monitored by resistance changes, applied to (Mog ¢Rug,4)g6814 for the crystallization study. 75 76 78 79 80 8] 89 9] 4.2.3 4.2.4 4.2.5 4.2.6 4.2.7 4.2.8 =x Tle X-ray scan (CuK, radiation) of the crystalline phases obtained after each quench showed In flg. 4.2.2. Enlargement of the resistance traces around the crystallization temperature. Enlargement of the resistance traces below the erystalllzation temperature for x=14, 18 and 22. (Region "a" and "b" correspond to region [I and [Il In flg. 10 of ref. 4.2.16). The lower plot represents the excess resistivity above the extrapolated IInear temperature dependence character!Istic of the low temperature region. Percent change In resistance as a function of annealing time at various temperatures. The solld lInes are fit to the Kelton-Spaepen theory. Enfargement of the resistance traces below’ the crystallization temperature for x=14 and 22 annealed previously 12h. at varfous temperatures. Correlation between the excess resistivIlty and the change In the density of states (from 4.2.16) vs the reduced annealing temperature. Tor =i the crystallization temperature (°C) corresponding to the onset of the drastic drop In the measured property. 92 94 95 97 99 10] 4.2.9 4.2.10 4.3.1 4.3.2a 4.3.2b 4.3.3 4.5.4 4.3.5 =-xIve Normallzed specific heat coefficient vs conductlvity for (MOn 6RUQ 4) 1008 x (ref. 4.2.22 and 4.2.23) and (Zr2N1) 4 00-yBy glasses (ref. 4.3.31). The normal Ization constants correspond to x=18 and y=0 respectively. Reduced radial distribution function for x=14 and 22 annealed for different time at 525°C, Resistance and DSC traces on ZrgoQNl3zge (Hfo 52Zrg 5) 6oNI 3g and Hf oNiag. Enlargement of the resistance traces below’ the crystallIlzation temperature. The dashed line represents the extrapolated {Inear dependence at flow temperature. Excess resistivity above the I near dependence obtained from a feast square fit of the data at low temperature. Radial distribution function before (full IInes) and after (dashed IInes) the heat treatment represented fn the onset of each figure for (a) ZrgoNizg (b) (Hf 5Zrg segNizg and (c) HfgoNize- Superconducting transition measured resIistively for ZrggNl3g as quenched and heat treated beyond the first exotherm. Upper critical ffeld vs temperature for Zr62NI 3g as quenched and heat treated beyond the first exotherm at a heating rate of 15°C/mIn. 104 106 115 118 118 121 124 125 4.3.6 4.3.7 4.3.8 Del 5.2 563 5.4 5.9 -xv~- a) and b) TEM micrographs of heat treated ZrgoNI3g- Diffraction pattern of heat treated ZreoNizg (a) selected area In flg. 4.3.6 containing no pits and holes (b) area Including holes and sample's edges. Mossbauer spectra of (ZrgoNl3g)99,gFeo,2 8S quenched, heat treated beyond the first exotherm and fully crystallized after the second exotherm. Upper critical field gradient vs concentration for Mo~Ru-B and Zr=NI-B systems. NormalIzed density of states vs conductivity (or resistivity for x>55) for the NieZr system. The concentrations are Indicated In parenthesis, The data are normal lzed to the respective values for x=55, Upper critical fleld gradient vs concentration for NIi-Zr and Cu-Zr systems, Log-Log plot of the upper critical fleld gradient vs the density of states for the Cu-Zr and Ni-Zr systems. Positions of the first peak of the structure factor vs composition for the Zr=-NI system. 128 129 13] 14) 145 146 148 150 56 5.7a 5e7b 6.1 -xy fe Normalized density of states vs conductivity for the Cu-Zr system. The concentrations are Indicated In parentheses and the density of states are from efther calorimetric measurements or superconductivity measurements. The data are normallzed to the respective values for x=55, Calculated electronic diffuslvity for the s and d states vs the Cu concentration for different parameters (or s-d scattering matrix elements). Calculated conductivities for a parameter a=0.3 together with the experimental data points from Altountan et al. (ref. 70 and 75). Calculated local density of states: for nickel In a solld-like cell (hIlgh density region), for Ilquid nickel and for crystalline nickel. (The data are from ref. 83) Page 153 156 156 160 -1- 1. UNTRODUCT ION In September of 1984, It will be the twenty-fifth anniversary of the discovery of the first metallic glasses. It was, Indeed, In September of 1959 at Caltech, that Pol Duwez and his co-workers‘! found a broad diffuse band In the X-ray diffraction pattern of a gold- sillcon alloy rapidly quenched from the melt. Prior to this date, work on amorphous metals was [ImIited to very thin flims(2) obtained by techniques such as physical vapor deposition, sputtering and electrodeposition. Because of thelr geometry, these samples were restricted to a certain limited type of experiments. The Importance of the discovery of Duwez resides In the bulk nature of the samples which can be obtalned by quenching from the melt. The Interest of physicists and materfal sclentists tn the field has been growing almost exponentially and since the discovery of the "meltespinner" with which comes the possibility of massive production, the engineers as well contributed to Improve our knowledge on nonerystalline metals. Among all of the Interesting properties of amorphous metals, probably one of the most Important Is the electrical conductivity. However, the baslc theoretical difficulty which arlses from the lack of fong range order and the fact that the Bloch theorem Is not valid makes transport phenomena In these systems, a very controversial fleld. These difficulties have IImIited the use of electrical resistivity measurements as a structural probe for short range order In amorphous metals. Some experfmentalists even question usefulness of these -2- measurements In metallic glasses because of the difficulty’ In assigning an unequivocal, specific scattering mechanism. On the other hand, the measurements are relatively easy to perform and, despite the above theoretical considerations, large amounts of data and extensive work can be found In the I!terature. Most of the studies of the electrical transport In metallic glasses concern the low temperature behavior. Many models and scattering mechanisms have been proposed, with varying degrees of success, to explain the very low temperature dependence of the resistivity: Kondo effect(3), scattering from two-level tunneling states(4), coherent magnetic scattering(>), localization'§) etc. In the low and Intermediate temperature range, theories [Ike the localized spin-fluctuation scattering model?) and especially the diffraction model8) based on the extended Ziman metal theory have been used. Both of these theorles predict a T2 dependence of the resistivity at low temperature (T<®) and a linear dependence at high temperature (T>€6). © fs the Debye temperature. The diffraction model has been quite successful In explaining the temperature coefficient of the resistivity at hlgh temperature In many amorphous metallic systems and In particular, the negative temperature coefficient observed for the Cu-Zr and the Ni-Zr systems which we will discuss extensively In the followIng chapters. The purpose of this thesis Is not to study or criticlze any of these scatterIng mechanisms In particular, but Instead to use the most widely applied mode! (J.e, the extended Ziman theory), together with the electrical resistivity, as a too! to detect and study phase transformations whichoccur at high temperature In metallic glasses. -3- Because of thelr metastable nature, phase transformations, [n which | Include relaxation phenomena (topological as wel! as chemical) are Inherent to metallic glasses. Phase separation of a homogenous phase Into two distinct amorphous phases represents by Itself an Important problem In the fleld of glassy metals. Wewlll show that resistivity measurements can be very useful to locate, in the temperature-time diagram where such separation occurs, and to study Its kinetics. The behavior of the resistivity around the glass transition and Its dependence on the free volume content can also be used to test the various theorles proposed for the glass transition. Several authors realized how helpful high temperature resistivity measurements can be In followIng the crystallization processes of metallic glasses and numerous studies can be found In the literature(9-11), For this reason we will restrict our Interest In this thesis to amorphous phase transformations or at most, to the very early stage of crystallization. Fig. 1.1a shows a typical resistivity scan on a CuyzzTIo7 amorphous alloy. At first sight, the resistivity varles linearly with temperature, from room temperature to around 250°C, In agreement with the prediction of the diffraction model at high temperature, Around the crystallization temperature, the resistivity decreases by more than 40%. This Is because of the much greater static order of the polycrystalline phase. The typical! temperatures of Interest In the thesis wlll Include a range starting from a few hundred degrees below the crystallization temperature. Fig.1.1b shows that, on a_ small scale, the resIstivity Is actually highly nonlinear In this range of ary cow ] ¥ A b—— TEMPERATURE RANGE —— OF INTEREST Qo co vw R/R, lon] => T oy wn — Cu731197 -1,0 A. i 4 i 0 100 200 300 400 500 TEMPERATURE (°C) Fig. 1.1a & b Change of electrical resistance with temperature, normalized to the value at 300°K for CuzzTIo7 metallic glasses (Heating rate 10°K/min.). Enlargement of (a) show!Ing the excess resistivity above the linear contribution observed at low temperature. -5- temperature. The plot shows the excess resistivity above the l[Inear contribution due to phonon scattering. The Increase In the excess resistivity reflects an evolution towards a state of higher chemical short range order (CSRO). It has been demonstrated‘'2), Indeed, that CSRO exists In CugeTiz, alloys In the {Iquid as well as In the glassy phase. Fig.1.2 shows the structure factor S(Q) obtalned by neutron diffraction from & CugeTizg alloy In the IIquid and glassy state. The prepeak at Q=1.9A7! provides dlrect experlmental evI dence of CSRO or a tendency for unlike atoms to be nelghbors. This prepeak fs the diffraction evidence of a non-random, Iike-atom correlation at the second=-nelghbor distance (Ro = (5/4)(27/1.9) = 4.1K) and represents a pseudo-perlodicity A-B-A-B In the random structure, The size of the prepeak [n fIg. 1.2 shows that CSRO !s enhanced on vitrification. Therefore, by annealing below the crystallization temperature, the CSRO should grow and the Increase In resIstIvIity In flg. 1.1b Just reflects this tendency. Fig. 1.3 shows that after a non-reversible ordering process observed during the first heating cycle, we can achleve areversible behavior In the varlation of the resIstivity vs. temperature which can be thought to be a reflection of the " metastable equilibrium state" of chemical short-range order. The higher the temperature, the lower the degree of order and the smaller Is the resistivity. Balanzat et al.{15) found similar behavior for a CusgTisg amorphous al loy. This brief Introduction shows how sensitive and powerful resistivity measurements can be In detecting ordering processes In metallic glasses. Before moving further [nto the exper!mental results, |! wlll describe [mn chapter Il some of the theoretical 2.0 CuceT1 1,5$ PREPEAK M6134 1.0 a = AY LIQUID STATE o~ 0.5 p> f ~ 1,54+ a 1,0 a a 0.5 GLASSY STATE J / AFTER M. SAKATA ET AL. Loo? 0 ft | a | ft | a | _ St r" _f 7a 0 2 4 6 8 10 AW, a (A-}) Flg. 1.2 The structure factor $(Q) obtained from neutron diffraction Cuge Tig, In the IIqutd state and In the glassy state. for 1.5 ‘ qv 7? 4 1.0 r a FIRST CYCLE S P= S 0.5 F 4 <1 REVERSIBILITY .—= — \ "QUENCH 1 SECOND CYCLE \ ' - 0.5 L 1 1, _t 0 100 200 300 490 500 TEMPERATURE (°C) Fig. 1.3 Excess resistivity measured for CuzzTio7 glasses during the first and second heating cycle (Heating rate = 10°K/mtn.) After the second cycle a reversible behavior In the resistivity can be achieved. -8- background necessary In order to fully appreclate the results In this thesIs. I! will describe In detall, topological as well as chemical relaxation end ! will present the standard approach used to Investigate problems such as phase separation and phase segregation In amorphous structures. Chapter V represents an attempt to explain the behavlor of the electrical resistivity for the varlous systems studied In this research, from a different point of view than the one usually adopted f.e. the extended Ziman approach based on Boltzmann's transport theory. This will give rise, In conclusion, to new ways of Interpreting relaxation phenomena In amorphous structures. it. THEORET CAL ASPECTS 11.1 The Ziman Formal ism Transport theory In a highly disordered structure Is one of the major unresolved problems In solid state physics. Several models have been proposed to explain the behavior of the electrical resistivity In metallic glasses and In particular, the negative temperature coefficient of resistivity found when the magnitude of p Is larger than 150u2cm I will choose, In thls work, to Interpret the results within only one model: The Ziman Formalism. This approach has been particularly successful In explaining, at least qualitatively most of the transport phenomena observed In metallic glasses. It Is a sultable cholce because It relates electrical resistivity to atomic structure In a natural manner. We should keep In mind, however, that its applicability to metallic glasses, where the electron mean free path Is very short, Is questionable. Indeed, the Ziman formalism Is derived from the Boltzmann transport theory In the weak scattering regime which falls when the mean free path becomes comparable to the lattice spacing or when the resistivity reaches values on the order of 200 US? cm. Nevertheless, the Ziman theory Is still widely used In the fleld and for thls reason, | wlll address the Important problem of Its applicabl lity In chapter V. In the ortgtnal formulation, Ztman('4) considered the elastic scattering of nearly free electrons In plane wave states by the pseudo-potential associated with the fons In a sImple IIquid metal. Using the Boltzmann equation In the relaxation time approximation we can write In the Born approximation Pp = m/ne2t » wherem Is the electron mass, n Is the free electron density and e Is the -10- electronic charge. The scattering rate 1/T Is given by the followIng 2T relation 1/T -{ (l-cos 8) Pype( 8) dO where Perf 6) Is the fo) probablilty of scattering by unit time and unit solld angle. Using the Fermi Golden Rule we may write this scattertIng probability as 2m 2 = Peet A Ye | e(E = Es) where <U,,r> Is the matrix element for scattering from state k to k'! by the total potential and p(E=Ey? Is the density of states at the ferm] level. Assuming a quadratic dispersion relation and a spherical Ferm! surface we obtain mVk ¢ 0 (E=Es) = or one where V Is the total volume and k¢ Is the Ferm! 1 wave vector which Js related to the electron density by ke = (3ren)'/? « Using the well-known relation v¢_ = hk¢/m for the Ferm! velocity the resistivity becomes: 1 _ 1202 | 2/«\3./k f (e) K = k-k!' Is the scattering vector and 2 Is the average atomic volume. The matrix element <Up yt? can be written as: Vv 1 . <vpr [Ul y,> = yf er U(r) dor (2.1.2) oO where the total potential Js considered as a sum of pseudopotentials assoclated with each of the fons U(r) = » vuptrerp)« If uy(K) Ts the i Fourler transform of the pseudopotential, the Interference function -l1- I(k), which Iwill take to be equal to NIU, _t|4 becomes: = iKe 2 Kk) = 5 pie elKery (2.1.3) i For a monoatomic IIquid the structure factor Is defined as: 1 iKe a(K) “wide Ker ? (2.1.4) i therefore, the Interference function becomes: 1(K) = a(K) Ju(k)]? | (2.1.5) and the expression 2.1.1 for the resistivity becomes: 3 1200 K p= 2a a(ck-) (kz ) a(k) {u(k)|? (2.1.6) he Ve > f f This ts the classical Ziman formula In Its usual form. it Ts easy to show that the structure factor In eq. 2.1.4 can be related to the radial palr distribution function g(r) defined as follows: y 9(r) = ‘ < 2 orn +r3)> j through the relation: i*j f (g(r) - 1) Sintkr) gnp2ar (2.1.7) 0 <2 elt -]2- Faber et al.16) have extended this theory to binary simple liquid metals by Introducing the partial structure factor Gye (K) which Is related to the partial palr-correlation function Gyp (r) by an expression analogue to eq. 2.7. 9 4g (r) Is the probabllity of finding an ton of type a In a unlit volume at a distance 8 from the center of the Blon, normal!zed In such a way that If tends to unity for larger r. In this case eq.2.1.3 or a(k)|u(K)|2 In the expression for the resistivity (2.1.6), has to be replaced by: = 2 2 It Is [Interesting to note that for a substitutional alloy In which the two constituents have the same atomic volume and the same structure factor 8;4*a99=a2;=a, eq. 2.1.8 can be reduced to: 1(K) = (c,ue + cyu2) a(K) + c,¢(u, - uo)® (1 - a(k)) 174 * £242 16264, ~ U2 a 7 2 = (u*) a(k) + (u, = uy)® e(1-c) (1 ~ a(k)) (2.1.9) therefore, for the resistIvity one obtains: o = Py + Po where P, = lero < (cus +(1- c)us) a a(K) > he Ve and ra pee oC) < (u, - u,)? (1 = alk) > V Po gives rise to the Nordhelm's rule and Py varles [n a more or less JInear fashlon between the resistivity of the two pure constituents. In the above expressions <f (K)> means -13- 2k 2k formalism for binary alloys and Introduced new correlation functions 1 3 freo( K ) a K ) . Bhatia and Thornton''7) proposed a different which, In terms of the partial structure factors defined previously, can be written as; ~ 1), 2. 2 2 Any K) = WiC) | = cya), + Coan. + 2068) 5 a--(K) = J icc) |? = ¢,¢,(1 + c,c,(a,, + 2.1 cc N 1°2 12(844 + 899 - ayp)) (2.1.10) ayc(K) = Re(N(K)"C(K)) = cyep(cq(aqy = 849) = C290 - 245) these were called the density-densIity, concentration=concentration and density-concentration correlation functions. N(K) and C(K) are respectively the Fourler transform of the local deviation In the total number density and the mean concentration: a N(K) = >, e Kerry T,a=1,2 . } ; 2 (2.1.11) C(K) = t (co » elKry . Cy Deir) b | Using eq. 2.1.11, eq. 2.1.3 for the Interference function becomes: 1(K) 2 " zl~ | uN(K) + N(u, = uy) C(K) | (2.1.12) “4 ayy(K) + (Uy = UQ)® acc(k) + 2u(u, = Uy) aye(k) where u® Cyuytegu2 ° When the partial atomic volumes of the two spectes Ina binary alloy are nearly Identical as Is for the case of the regular solution, then ayc(K)~ 0 for all K and only two -14- Independent structure factors are needed to describe the resistivity completely. Furthermore, If we have an fdeal solution (heat of mixing * 0) the concentration-concentration correlation function Is given by uccl(K) = clt-c) and expression 2.1.12 becomes Identical to equation 2.1.9. The resistivity can therefore be expressed In terms of a topological and chemical part. P = Prspro * PcspRo 1 1272 i 2 K \3 K p = uy a,,,(K) () a ( ) 2.1.1 TSRO heev.? NN 2k¢ 2k, ( 3a) = me fo “uy? acclX) (ae -)*¢ (xt) (2.1.13b) During structural relaxation, the changes In bond distances and angles ~~ PCSRO which leads to a higher topological short range order, affect the resistivity through eq. 2.1.13a whereas the exchanges of chemically different nelghbors which leads to a higher chemical short range order, affect the resistivity through eq. 2.1.13b. In order to extend the Ziman formalism to Iiquid transition metals where the scatter!ng Is strong and the electron mean free path 1s very short, Evans et aj, (18) replaced In the orlgtnal ZIman formula (eq. 2.1.6) the weak pseudopotential u(K) by the tematrix for scattering from state k to k' by the muffiIn-tin potential of the Ions. 3 | t(K) = - oh >, (22 + T)sin(n, (Ey ))e "e P,(cos8) (2.1.14) mayan -15- For transition meta! elements, the d-phase shift "> dominates and stnce most of the contribution to the Integral In eq. 2.1.6 comes from the region near 2k, we can write to a reasonable approximation: 300°h? & - 2 p =z Sin (no) a(2ke) (2.1.15) me Qk Es For binary transition metal alloys, an expression similar to eq. 2.1.8, In which us(K) and ug(K) are replaced by single sIte t+ matrices, has to be used In the expression for the resistivity. The temperature dependence of the resistivity can be Included by takIng Into account the change In the structure factor (eq. 2.1.4) with temperature. a(K) has to be replaced by the resistivity static structure factor S(K) which can be expanded for an amorphous Debye solid as(19), S(K) = S_(K) + $4 (K) + S4(K) +... a(K) ene .. elastic term where S,(K) $, (Kk) a(K) AP(K) (1/8)*1,(6/T)e 4 ...one phonon term So(K)+..% 1-(1 + au) en 2 The Hernandez-Cal derone approx!mation(20) to the multiphonon serles has been used for the higher order terms. In the above expression anoKe mk6 where M Is the atomic welght, 9 Is the Debye a(K) = temperature and kp 1s the Boltzmann constant, e72W Is the Debye Waller damping factor with 2W = o(K)(T/8 )21(9/T) yA (K) Is the -16- average of the structure factor a(K) over the Debye sphere center on K and finally: 8/T 1.64 (T 1(6) = xx coth(x/2) dx "<< 8) P 6/T (T >> 6) T 1,(8) -[ xedx 3.29 (T << 6) [e*-1][e7*-1] e/T (T >> 8) ie) From these expressfons we can write the static structure factor In the high and low temperature IImIt as follows: a(K) + o(K)(3.29A°(K)-1.64a(K))(T/6)¢ +... (2.1.16a) For T<<@ S(K) " For T>>6 $(K) = a(K) + a(K) (AP(K)=a(K)) (T/8)40° (k) (1-A°(K) ) (2)? (2 1.160) The factor (1-A°(K)) In eq. 2.1.16b Is usually small so we can neglect the quadratic temperature dependence term at high temperature In most cases. We therefore conclude, after substitution of these expressions Into eq. 2.1.6 or eq. 2.1.15 that the diffraction model predicts a resistivity vary!Ing IIke T2 at low temperatures and IInear In T at high temperatures. Qa ] do o dT The stgn of the temperature coefficient of resistivity oa at high temperature depends whether a(2k,) Is greater or less than AP (2k¢) or, In other words, It depends on the position of 2k, with respect to the position of the main peak In the structure factor kpe When 2k¢ Is close to kp» a(2k¢) Is large and @ [s negative whereas -17- when 2ky !S far from the maln peak In the structure factor, a Is positive. In this thesis we are primarily Interested In the hich temperature region (T>® ) where the resistivity varies IInearly with temperature due to thermal broadening. ~18- 11.2 Topological Relaxation Topological relaxation and the concept of free volume are closely related. We usually assoclate the Increase In the topological short range order with a consequent reduction In the free volume content. The theory of free volume was originally developed for liquids(21), = The free volume was defined as the "macroscopic" excess volume of the IIquid (In the sense discussed by Egam! et al. (22)) compared to the extrapolated volume of the solid. This free volume Is distributed over the whole system In the form of vacant spaces between the atoms. The repulsive part of the Interatomic potential was assumed to be almost hard sphere IIke and the free volume was written as: Ve = 2-2 (2.2.1) where © Is the average atomic volume and 2, » the atanic volume at Ideal close packing, The theory was then applied to amorphous solltds by Spaepen and hIs co-workers who described the structural relaxation In metallic glasses In the following way: In a relaxation event, the atoms at a relaxation site collapse at about a focal free volume fluctuation. The collapsed volume Is transferred permanently to the specimen surface through many Individual atomic rearrangements. Following thls event, the total free volume Is reduced and the viscosity Increases. The viscosity can be expressed In the theory by: * YV /Ve Ne no (Te (2.2.2) where v* Is the critical free volume fluctuation for an atomic Jump producing flow, y Is a geometrical factor close to 1 andn, Is a structure Independent factor.) -19- A. he Kel ton-Spaepen Approach Kelton et al. studied the kinetics of IJsothermal structural] relaxation through changes In the electrical resistivity'25), Thelr understanding of the behavior of the resistivity Is based on the experimental observation that the viscosIty for Pd=S!=-V amorphous alloys Increases IInearly as a function of annealing time ( n Is a constant depending only on temperature). Using eq. 2.2.2 we can write: W'/ve(t) y"/ve(0) Noe = nt + Noe which can be Inverted to give Ve(t) _ r"/v,(o) (2.2.3) Ve(0) n -yv"/v_(0) * an(2- t e W/V) 4 1) 4 vy /v,(0) 0 According to eq. 2.1.15, the change In the electrical resistivity can be directly related to the change In the structure factor evaluated at 2kg, Ashcroft and Langreth'24) found that the Percus-Yevick hard- sphere model is a sultable approximation to the main peak of the structure factor. In thls model the alloy Is characterlzed by a packIng fraction paremeter € = m /2 where 2, is the hard sphere volume. The peak helght reduction and broadening of the structure factor with temperature can be approximated In the Percus-Yevick mode! by allow!Ing for a change In packing fraction with temperature. Kelton et al.'25) used this model to describe the structural relaxation In metallic glasses. Let us expand the Percus-Yevick structure factor In a Taylor series: S(K,E) = S(KsE,) + ae oe +... 0 -20- From this, using the preceding discussion, we can write Ap = 2 Th ( aay aa) ) fe) and by using equation 2.2.1 2, ve(o) v-(t) dp = a (1 Tey) (vplo) <9) (2.204) 0 With eq. 2.2.3 and eq. 2.2.4, we can fit the electrical resistivity change with three parameters: a pretactor, 1 / No and Yv"/v4(0). The sign of the resistivity change depends on the sign of 3S(2k¢) ; | aE IE, When kg Is close to the main peak In the structure factor, a Is negative and the resistivity Increases upon relaxation. When 2k: fs far from the malin peak, a Is positive and the resistivity decreases with topological relaxation. Fig. 2.2.1 shows a schematic plot of the structure factor In the as-quenched state, relaxed, and at high temperature. Fig. 2.2.2 shows a fit to the electrical resistivity for an (Mop 6RUg4)7g822 amorphous alloy using eq. 2.2.4. Even If It hes not been proven that the viscosity Increases {Inearly with time during Isothermal annealing In this specific alloy, the fit'Is very good at low temperature (450°C and 500°C), but becomes worse and even ImpossIble as we ralse the temperature to 600°C. Wewlll show tater that chemical relaxation and phase segregation Is responsible for thls behavior at high temperature. -2]- ho wn 4 « 2.0F RELAXED STATE 1,5 AS QUENCHED (UNRELAXED) 1.OfF STRUCTURE FACTOR AT HIGH TEMPERATURE 0 5 3,0 4,5 6.0 7.5 K (ATL Flg. 2.2.1 Schematic {Illustration of the first peak of the structure factor $(Q) In the as quenched State, relaxed state and at high temperature. -22- (Mog gRuUg, 4) 7gBo9 0 4 8 12 TIME (HRS) Fig. 2.2.2 Relative change In resistivity as a function of annealing time at varlous temperatures for (Moy 6RUg 4) 78822 glasses. The solld line 1s a fit to the Kelton-Spaepen theory of structural relaxation. -23- B. Egami's Model for Structural Relaxation Egan! et al.'23.25) discussed the short-comings of the macroscopic free volume theory when applied to metallic glasses and proposed Instead a microscopic model for structural relaxation tn those amorphous alloys. They Introduced several local parameters to describe the local atomic structure. Among the most Important Is the focal! hydrostatic stress parameter defined as: Py “ex Li ij (2.2.5) j ij Where 2; Is the focal atomic volume and $(r) Is the Inter-atomic potential. Because of the quas! harmonic!ty of this potential In an amorphous metal both positive and negative density fluctuations are present. The reglons In tensile stress are called n-type and are free-volume-Iike, whereas the regions In compressIve stress are called p-type and can be viewed as anti-free volume-Iike. The n and p type local density fluctuation contribute to the total volume In an opposite way. In this pleture, structural relaxation results from the recombination of the n-p pairs. Even If the microscople free volume, defined as <({. -2 2, ts greatly affected by this process, the macroscopic free volume or the total volume Is only changing slightly as a result of the anharmonicity of the I[nteratomic potential. Using a computer model contalning 2067 atoms for the amorphous structure, Srolovitz, Egam! and Vitek'26) simulated the effect of structural relaxation on the radial distribution function (RDF) by calculating the ARDF when one fixes the origin on each atom In the model and then excludes all the sites having the largest -24- tensile and compressive stress. They found good agreement between thelr model and an experimental. A RDF measured on an amorphous Fesonl 4oP16B6 alloy. To a good approximation, the local RDF centered on atoms with hydrostatic stress p can be expressed as(26); G(r) = G(r + yp) (2.2.6) where y Is a@ constant and Go(r) Is the RDF for atoms with p=0. G(r) Is related to the g(r) defined previously In eq. 2.1.7 by: G(r) = 4nr B (g(r)-1) (2.2.7) The effect of the local density fluctuation Is then to shift the local RDF by an amount proportional to p. If we expand 2.2.6 In a Taylor serftes around p=0, 2 aG a G = ) ] 0.22 G(r + yp) = G(r) + 5 yp + 2 YP the total RDF <Bir)>=f G(r + yp)N(p)dp (where N(p) Is the normaltzed distribution of the hydrostatic stress parameter) becomes: 2 _ 1.2/2 %\_ 2 <Go(r)> = G(r) +5 y 52 fe? (2.2.8) stnce J Pxcp apo. Expertmentally, It ts often observed that the effect of structural relaxation Is to sharpen the peaks of the RDF without shifting them. For this reason Egami et al. choose to Interpret the changes In the RDF solely on the bas!Is of the relaxation -25- of the density fluctuation <p2> and write: 2 2 3°G (r) ARDF = J-——°,— acp*> or (2.2.9) 2.22 where A<p'> = <P >a inealed ~ <P as quenched The success of Egami's model Is that It predicts correctly the small change In the total density of the amorphous alloy during structural relaxation and [t predicts a ARDF In agreement with the exper!ment. C. che Allia-Turtelii Theory Based on the previous microscopic model for structural relaxation, Allla et al. (27) proposed a correlation between the resistivity and the density fluctuation <p2> using the Ziman formalism. From eq. 2.2.7 and eq. 2.1.7 we can write: <a,(K)> =]+ t ff <6, (r)>sin(kr) dr (2.2.10) and by substitutlon of 2.2.8 Into 2.2.10, the equation after Integrating twice by parts becomes: <a,(K)> = ao(k) + 3 yk? (1 - ay(k) )<p*> (2.2.11) " where a (kK) 1+ a 6 (r)sin(Kr) dr Ustng expression 2.1.16b and eq. 2.1.15 for the resistivity at hich temperature we find: p(T) = <a, (2ky)> + a(2k_)(AP(2kg) - <ay (2kp)>)g (2.2.12) and therefore the change Inresistivity and coefficient of the -26- resistivity at a fixed temperature due to relaxation, can be written as: Ap « 5 ¥ 2(2k,)? (<0 ae ~ a, (2k,)) Acp*> (2.2.13) v' (2k, é and fa = a(t £2) « aor (1 - ay (2ke)) A<p*> where 0) = a, (2k -) +a (A°(2ke) - a, (2ke)) z and A<p2> has the same meaning as In eq. 2.2.9 . We see In eq. 2.2.13 that 49 and Aa have opposite signs. When 2k¢ Is close to ky (which corresponds to the maximum of ‘the structure factor), ao(2k¢)>1 and the resistivity Increases upon structural relaxation since A<p2> <O. The temperature coefficient of the resistivity on the other hand decreases, {n this case, with structural relaxation. These observations seem to agree with experimental measurements (27,28), Among the several theorles which have been proposed to explatn the IIquid-glass transition, one of the most extensive Is the mode! of Cohen andGrest'29), The theory Is based on the exIstence In dense IIquids and glasses of awell defined cellular structure which persists on a time scale comparable to the typical time for atomic motion within the cell. These cells are equivalent to the Vorono! Polyhedra In e dense random packing mode! of hard spheres. They assume that the local free energy of a cell f(v) depends only on Its volume, -27- and write: f(v) = f+ aK (v= v,)° V< Ve = Fo + zk (v - vo)? +E (v- Vo) v>v When v<vV_ the cells are called solid-I{ke and when v>vc, they are called IIquid-IIke. The free volume of a cell Is defined as Vg = vevo. The linearlty of f(v) for v>ve Implles that free exchange of free volume can take place between IIquid-llke nearest neigh- bor cells. If P(v) fs the probabllity that a cell has a volume v, the probabllity that a given cell Is IIquid-iIke Iss f P(v) dv (2.2.14) Ye Pp Furthermore, they define ae percolation problem where p, ts the critical !fquid=-like cell concentration, above which there exists an Infinite Ifquid-llke cluster. They found that the “equllfbrtum" fraction of the Iliquid-lIlke cells p, as a function of temperature, undergoes a dlIscontinuous change at a temperature Tp. This discontinufty Is taken to be evidence of an underlying equilibrium glass transition. This transition {fs never seen experimental ly because of kinetic effects. At a temperature called the experimental glass transition, the relaxation time of the system becomes comparable to the time of measurement and the system falls out of complete thermodynamic equlllbrium. Tp Is usually some 20° below Tg. The glass Is therefore characterlzed by a frozen fraction of IIquid-like cell Ptroz which relaxation toward the equlllbrium fraction Peg {T) upon anneallng below Tg: -28- In this plcture, a glass can be viewed as a binary alloy In which the two species are respectively the solld=like and the JIculd-like cells. Using the Faber-Z!man formaltsm'16) for the resistivity of binary simple [Iquild metals, {+t !s stralghtforward to show that the Interference function defined tn eq. 2.1.3 can be written as: =z N 26s 22 v2 (K) + Uy i(k) = N[Uy|? = u We apy (K) + Tgg(K) (2.2.15) s ss where us (K) and uy (K) are respectively the Fourler transform of the pseudopotential associated with an atom In a solld-IIke cell and Ina IIquid-ltke cell. N./N = (I-p) and No/N = p are the fractions of solld-llke and Ilquid=-like cells, €@ .(K) and 8,, (K) are the corresponding partial structure factors, and Iso (K) Is the Interference function between a solid-!ike and a IIquid=-IIke cell. For a segregated system (f.e. a system In which the solld-like cells and the [fquid-IIke cells are clustered), Is, is non negligible only at small K values I.e. for K << 27 /R where R_ I[s the average cluster size. Since the resistivity Is proportional to the structure factor evaluated at 2k,, Igy (2ke) will make a negligible contribution to the resistivity. For a segregated system, therefore, eq. 2.1.15 and 2.2.15 give: 2 2 2 paur acc(2k—) + Cupar e(2k¢) - ucas.(2k¢)) p 2 2 om « 5 (ubace (2ke) - usa. (2k~)) Ap (2.2.16) b> or Fig. -29- 11.2 * * hd = 8.4F Aug, 776Eg, 1365'0,094 , 5.6 ~ « 2.8f HEATING RATE Q (°C/mIN) ® 2 of Q@=5,5 | 2 Q@= 4.5 4 -2,8 r 4 Q= 3.3 x -5.6 P 7 ~8.47 x EXPERIMENTAL POINTS FROM CHEN AND TURNBULL 1 -11.2 ah PZ 4 r- 1 278 284 290 296 302 308 314 TEMPERATURE (°K) 2.2.35 Relative change In resistivity for Aug .776& 136°!o 094 alloys calculated for different heating rates together with the experimental data of Chen and Turnbull (ref. 31). -30- The sign of Ap during Isothermal annealing depends on the sign of the expression In parentheses [n €q. 2.2.16. When 2k¢ Is close to Kp» Bye (2ke) << agg(2kg) and the resistivity [Increases with structural relaxation since Ap<O. When a, (2k¢) > asg(Zk¢) equation 2.2.16 agrees with our Intultive feeling that an atom near an _ Interface between a solld-Iike anda IIquid=-like cluster wlll scatter the electrons more effectively In the [fquld-!Ike phase than In the solld- llke phase. In that case this approach [s simllar to Tsuel'ts mode!(30) which considered a disordered solld as a structure with a highly developed short range order contalning, however, some disordered sites, (the IlIiquid-like cells) which are considered to be scattering centers for the electrons, Fig. 2.2.3 shows a calculated resistivity curve using eq. 2.2.16 for a Aug .776€0,1365!10.094 amorphous alloy together with the experimental! data of Chen and Turnbull (31), In calculating p/p, we assume that the fraction of liquid-like cells relaxes towards Its equl i {fbrium value Peg?) given by the theory of Cohen and Grest with a stngle time constant 1(T) which scales with the viscosity: dp. (p - Peg(T)) dt 1(T) where t(T) = To exp(v_/ve) = 10°! exp(1360/(7 - 241.3) In order to take [nto account the contribution of crystal |Ization we assume that the equilibrium fraction of Jiquid-like cells not crystallized at a time t follows a Johnson=-Mehl-Avram! equation(32) | -3]- PeglT) exp (- (kt)") (1-X)PQ9(T) where k(T) kK, exp (- AE/kT) For Au-Ge-S! we have Tp = 280K and Tg ~ 295K. All the other parameters used In the calculation can be found In references 29, 31, and 33. The resistivity values of Chen and Turnbull are the average values taken from several measurements. ThIs approach Is particularly sultable for kInetic studies since It relates the resistivity to only one simple parameter: the number of IIquid-llke cells which can be considered as defects In the amorphous structure. -32- {1.3 Chemical Relaxation The broad fleld of chemical relaxation In alloys, which Includes phenomena IIke ordering, coarsening or phase segregation, sp!noda} decomposition and nucleation, Is of great practical as well as theoretical Interest. In Its slmplest form these phenomena occur when an AB alloy (such as AlZn or CuAu) IS quenched fron a high temperature, where the system Is chemically homogenous to a temperature T<Tg (where Te !s the critical temperature for phase separation or ordering). | Immediately after the quench, which Is Ideally Instantaneous, the system Is thermodynamically unstable and, therefore, willl undergo a _ process of phase transformation If the atomic mobility Is large enough. The same phenomenon also occurs obviously, for an homogenous system which has been quenched orlg!nal ly to a temperature where diffusion Is prohibited and subsequently annealed at high temperature. In this case the term chemical relaxation Is more appropriate. [t fs obvlously out of the scope of this chapter to discuss In detail! all of these phenomena, therefore, we wlll concentrate mainly on the problem of phase segregation. A good review of some recent results on the kinetics of phase segregation can be found In reference 34. The term "phase segregation" embraces phase segregation of an amorphous phase Inside another amorphous phase, formation of a crystalline phase Inside an amorphous phase or a crystalline phase [nsfde another crystalline phase. In thls research we are primarily Interested by the fIrst case: Phase segregation of an amorphous phase Inside another amorphous phase. However, most studies, especially those on spinodal decomposition, were performed using crystalline phases, Flg. 2.3.1 and -33- COEXISTENCE CURVE lof \ f A ‘ 7 0.8 i SPINODAL H ie i CURVE ‘ - 0.6 ® i (0,597) 0.4 . _ -1,.0 -0,5 0,5 1,0 Phase dlagram for a binary alloy (or the Infinite three dimensional Ising model, ref. 36) and the evolution of a high temperature configuration following a quench to a temperature T = 0.59 Ty -34- 2.3.2 show a computer simulated quench of an Ising model(36) from a temperature above the coexistence curve to a temperature (T = 0.597.) Inside the spinodal region. The alloy Is orlginally at the equlatonic composition. The spinodal curve Is defined by the locus of points where the second derlvative of the free energy with respect to concentration vanishes. Fig. 2.3.2 shows the evolution with time of the concentration=-concentration structure factor during the phase segregation. In the early time, when deviations from uniformity are small, the Cahn-Hillfard(55) Jinearlzed theory of spinodal decomposition gives: S(K,t) = S(K,0) exp [2R(K)t] ; 245) ; (2.3.1) (K) = = MK ( = + sx) c where M and S are positive constants respectively related to the diffusion coefficient and the surface tension and cfs the average composItion of the system. In the spfnodal region where 9?(t) ace < 0 (region Als fig. 2.3.1) the system Is unstable with respect to fong-wavelength fluctuations (R(K) > 0). When 92 (S)/ace > 0, the metastable region (region B In flg. 2.3.1) the system Is stable with respect to small fluctuations but unstable with respect to strong locallzed fluctuations such as a nucleus, provided an extra activation energy Is supplled. ThIs theory predicts the existence of a certaln Kmax ‘when 32F(S)/ae* <0) at which S(K,t) will have {ts most rap!d growth. This describes the Initial stages of coarsening but falls to describe the experimental data at all) times and, In particular, the shift of Kis.(+) toward small K values (flg. 2.3.2). Also represented [n this figure fs K3|v(k) (2 -35- mz J 4a mi U mJ no TEMPERATURE=.59T, | 40,75 ft , q STRUCTURE FUNCTION] 32.60 ¢ FROM J. MARROET ALL 2 = — > : > ex - = = = 24,45 A= 0.25 1 & x Pa * Es NS 16,30 S Seal 2 WN af 8,15 0 | K Fig. 2.3.2 Development with time of the structure factor S(K,t) (full Iltne) corresponding to a quench from a temperature above the coex! stent curve to a temperature T = 0.59 T, Inside the spinodal region (see fig. 2.3.1) (the dataare from ref. 36). Also represented In the figure ts the function K7|/v(k)|2 © K>/(14(K/4)%)2 (dashed ltInes) for different values of the parameter. - 36- for different screening parameters A (dashed curve). v(K) Is the Fourler transform of a screened Coulomb potential which Is glven by: 2 . 2 vir) = Ze grrr VK) Pee) (2.3.2) The factor K>|v(K)|2 was seen In the Integrand of equation 2.1.6 for the resistivity. Durfng the phase separation the resistivity goes first to @ maximum when the overlap between K3](v(K)12 and S(K,t) Is maximum, and then decreases as the peak of the structure factor moves further to the left In flg. 2.3.2. Such nonlinear behavior In the resIstivity Is often seen during the age-hardening of supersaturated solid solutions. To my knowledge, one of the first attempts to understand this behavior was due to N.F,. Mott (37), He evoked the wave nature of a beam of electrons and argued that the scattering power of a group of clustered atoms wif! be greatest when the size of the cluster {fs comparable to the wave-length associated with the electrons. Typlcally thls latter distance fs about four or five Inter-atomic distances. Several! authors (38-40) afterwards, developed vartious formalisms to explain this optimum In the resistivity and tried to correlate the reststivity to the size of the scattering entity. 1 will discuss briefly some of these approaches In the follow!ng sections. A. Ihe Rossiter and Wells Theory Rossiter and Wells studied the dependence of the electrical resistivity on the chemical short range order In crystalline binary alloy. They [ntroduced a formalism which relates directly the -37- resistivity to the Cowley short range order parameter. An average potential throughout the lattice Is defined as u(r) = yu, (r)tCouo(r) and the deviation from this average potential can be written as: on a site 1 V) = u;(r) - u(r) = Cou, - Up) (2.3.3) " on a site 2 Vo = u(r) - u(r) = - Cy (uy - up) The residual resistivity arises from the scattering of nearly free electrons by these fluctuations In the potential and therefore the Interference function (eq. 2.1.3) may be written: iKe(r.-r.) 10) = Daw (K)v, (Ke Vo3 oJ (2.3.4) iKeR. = C1Co(u - u,)? + DL <, v5 >e J j#0 where Ry = oly and <VoV > Is an average over all pairs of sites separated by R,. If we define py, as the probabil lity of finding an J 12 atom 1 at a distance Rj from an atom 2 we can write: KVQV5> = CqVq(PS1Vp + (1 = P2q)¥q) + CoVa(Pa¥y + (1 ~ Pia)v) 07 2 j j CyCo (uy = Up)” (1 - Pap ~ Poy) : J. but since CoPy 0 = Py <V V.> = C,Co(u, - U ye (1 - pJ /c,) = ¢y4C5 (uy - u yé a oj P2*") 2 12° "1% 7 “1r2 ] 2 j where a5 Is the Cowley short range order parameter for the atom at -38- distance Rj from an arbitrary origin, The average of eq. 2.3.4 over all possible directions between K and Ry gives, finally: in(kR.) 1(K) = ey¢p (u, = up)? ( 1+ >"; cr alain ) (2.3.5) j#0 , where the summation Is now over the ny atoms Is the jth shell and as Is the average value of %;j for this shell. By substitution of 2.3.5 In 2.1.1, the resistivity can be written In terms of the degree of short range order by: 0 = 12nn c1%2 ( > ny a; Y(kg oR) ) (2.3.6) he“v,? j=0 whe Y(k,,R.) = 4 K K 2 Sin{kR;) re feny? © | (xi) (x) (uy - up) KR, 8) and Cm = 1 by definition The function Yoke, Ry) Is a decaying, osclltatory function of R and therefore the predominant contribution to the residual resistivity comes usually from the first term. 1270 + PED, heeve Te 2 nya, Y(R, ) (2.3.7) Here, ny Is the coordination number of the first shel] and a Is the negative for an ordering process and positive for a clustering process. The value of Y(R,) can be elther positive or negative depending on the value of k¢- In the case of CuTl, which was -39- discussed briefly In the Introduction, the reversible behavior reflects reversible changes In the ordering parameter (1). Both On and Y(R;) are negative since the resistivity decreases with Increasing temperature. For the CuTIl system the term (uj-u5)? should be replaced by the difference between the t-matrices of the two elements. Rossiter and Wells(42) applted thelr theory to the problem of pre-preciIpItation during the agelng process and argued that the maximum In the resistivity occurs when the dimensions of the clusters become comparable wlth the conduction electron mean free path. For larger clusters, the zones could no fonger be treated as Isolated scatterers and the resistivity decreases because of the reduction In the overal! surface scattering at the boundarfes of each zone. Rossiter et al. also discussed the case of spinodal decompos!tion by replacing the order parameter In eq. 2.3.6 by: a5 = 2 aq cos(Q-R,) where the wavelength of each concentration fluctuation Is given by A = 217/Q and the amp! Itude Ta) Is Increasing with agelng time. They argued that whether there Is Gulnler-Preston zone formation or spInodal decomposition the resistivity still exhibits the same genera! behavior. B. The Hillel, Edwards and Wilkes Theory Hiltel et aj.(45? disagree with Rossiter on the reason for the decrease follow!Ing the maximum In the resistivity. They considered the theory discussed above, as applled to a crystalline lattice, only -40- valld In the smal] zone !ImIt, were the clusters can be assumed to be randomly orlented. Hittet(44) prefers to attack the problem using a different formalism and proposed that Bragg scattering, due to the micro- structure of the zones, leads to the observed Increase In resistivity while the decrease, as the Gulnier-Preston zones grow [n slfze, Is caused by the Increasing anlsotropy of the relaxation time. Since the microcrystals are coherent with the matrix structure, the Bragg scattering ultimately becomes [Imited to a vanishingly small fraction of the Ferm! surface. In other words, when the number of atoms [In the cluster Is small, the Bragg peaks are wide and cover the whole Ferm! surface, when this number Is large, the peaks are extremely narrow and the Bragg scattering decreases. Hitlel considers a very dilute substitutional alloy and uses the elementary transport theory for free electrons to calculate the residual resistivity. As was done In eq. 2.1.5, the square of the scattering matrix element Is factorlzed Into structure factor and form factor NI< k'fUfk >|? = Ju(k)|? s(x) where u(K) Is the Fourler transform of the difference potentia! for a single atomic site u(r) = UcoryteYsolvent 2nd the structure factor Is given by: ikKer. ,2 S(K) = e 4 (2.3.8) | solute atoms Hillel assumes a random arrangement of N. single solute atoms together -4]- with z clusters, each containing Nc atoms. He neglects al! Inter- particle Interference effects, except when they are within the same zone. All Gulnler-Preston zones have [nternal solute concentration of 100%. Under these conditions, the structure factor becomes: S(K) = No/N + Z/N Sy (K) (2.3.9) c N C iKer_ ,2 _ a where Ne = | >. e a=] Sy Is the structure factor of a single zone. At this poltnt c Hittel(44) uses a simple approxImation, which Is valld In the early stages of agelng, and consists of takIng the spherical average for the cluster's structure factor over. ali possible orientations =f do <Sy (K)> = T Sy (K) - Hillel emphasizes that we cannot expect a decrease In resistivity for large zones In thls approximation § since the decrease Is precisely related to the breakdown of this approximation. lf No No+zN. Is the total number of solute atoms, equation 2.3.9 can be rewrltten as: S(K) = No/N + 2/N (<Sy CK)? - NO) (2.3.10) The Debye formula‘45) can be used to calculate this average cluster structure factor, sin(Kd g) <Sy (K)> = No + 22. (2.3.11) c g where the summation Is over all InteratomIc dIstances, Fig. 2.3.3 -42- 3F STRUCTURE FACTOR FOR SPHERICAL FCC CLUSTERS (THE FIRST TEN SHELLS ARE SHOWN) S(K) x 1000 Flg. 2.3.3 Structure factor S(K) for spherical fcc clusters. The hard fe] sphere atomic radius Is taken to be 1.3 A. The first ten shells are represented. ~43- shows the results for an fcc spherical microcrystal. The first 10 shells are represented [n the figure and this corresponds to a maximum of about 200 atoms In the cluster. Note the formation of a Bragg peak around 3A7), C. Application to Nucleation In Topotogically Random Structures. When the host lattice Is topologically disordered, the previous assumption of randomly orlented Gulnler=Preston zones fs valid even for very large clusters. This makes the Hillel formallsm particularly sultable to treat the problem of nucleation In very dilute topologically random structures. An {deal system would be for Instance a blnary metallic giass contalning a few percent of a ternary element which would have the tendency to segregate at low temperatures. The electrical resistivity of metallic glasses shows, quite often, anomalous behavior around the crystallization temperature. For example, fig. 2.3.4 represents the normalized resistance trace taken at a heating rate of 10°C/min. for a PdgaY36 amorphous alloy. Up to 370°C, the resistivity Is almost constant, It decreases slightly by about 2% near 385°C and then Increases by about 10%, reaching a maximum around 403°C, To understand such behavior let us apply the previous mode! to the problem of nucleation of fec microcrystals In a random structure. From eq. 2.1.6, eq. 2.3.10 and 2.3.11 the resistivity during -44- 1,2 : - 0,9} a F . 4 = | 0,7} PDg 4" 36 : 0 ' 5 - _ ee 1 50. 470 290. 410 530 650 TEMPERATURE. (°C) Fig. 2.3.4 Normal lzed res!stance trace taken at a heating rate of 10°C/min. for Pde,Yz_ metallic glasses. -45- clustering can be written as: 1 _ 12792 No K_\/k af age Sin( Kay) herve N ik (zk-)(xk- ) | (u,-u5)| (1 ‘ele, ) (2.3.12) where No/N = c¢ Is the concentration of solute atoms and Z/No=r ts the ratio between the number of clusters and the number of solute atoms. Most of the metallic glasses Involve transition metal elements and therefore (uy-uz)2 should strictly be the difference between the solute and the solvent tematrices. However, since we are looking only for qualitative results, we will use a screened Coulomb potential with @ same screening parameter, » , for both constituents, From eq. 2.3.2: 2 2 2 Aze ( - = 3. 1 7 Ye) ea | (2.3.19) where AZ {fs the difference In the atomic number. This expression Is a monotonically decreasing function of K which fakes relatively smal] values near K = 2kz. Since the major contribution to the Integral 2.3.12 comes from the region near 2ks, where the function <Spy(K)> - No has @ peak, the use of a screened Coulomb potential can severely underestimate Ap , as It was pointed out In'45). Moreover, In topologically random systems, the electron mean free path Is_ usually very small. tn order to take Into account the finite electronic mean free path, we can apply the PIppard-Ziman approximation which consists In usIng a lower cut-off In the Integral 2.3.12 . Assuming a Gausstan probability distribution for the electron mean free path, we can -46- replace eq. 2.3.12 by: fy (K- K Ka) K_=2n/% where & is the average electronic mean free path and F(k) is the Integrand of eq. 2.3.12. Through Integrating by parts, eq. 2.3.14 can be rewrltten as: 2k e pe J ‘A (K)F(K)dK where H(K) = 1/2 + 1/2 erf ((K - K.)/W20) kK > K. V2 - 1/2 erf ((K-K)/Y20) ok < K. The excess resistivity can therefore be written as: 1 3 j ap = 2era(azyer | a( ) ee (2.3.15) hve 5 eke eke (x24 )2 ? Kd) a during the Fig. 2.3.5 shows the calculated excess resistivity Ap 12, growth of fcc microcrystals In a topologically random structure using eq. 2.3.15 and the parameters given In Table 11.3.1 -47- EXCESS RESISTIVITY DURING THE GROWTH OF FCC MICROCRYSTALS O57 J RADIUS OF THE ATOMS e IN A CLUSTER = 1,3 A (a) Excess RESISTIVITY AR/R, (2) 25 CLUSTER SIZE (A) Fig. 2.3.5 Calculated excess resistivity during the growth of fcc microcrystals In a topolofcally random structure. The excess resistivity fs plotted vs the size of the zones. A!I the parameters relevant to this calculation can be found In Table 11.3.1 and 11.3.2. -48- TABLE J1.5.1 Parameters used for calculating the excess resistivIlty during the growth of fcc microcrystals In a tepologtcally random structure. PackIng density of the matrix (atoms / AS) = ~ 06 Number of free electrons per atom = 1.9 Screening length (A) = 0.5 Radius of the atoms In a cluster (A) = 1.3 Electronic mean free path (A) = 3.0 Root mean square deviation of the distribution (AY = 1.5 Number of clusters / Number of solute atoms = 1074 Table 11.3.2 glves, for a given cluster dlameter, the number of atoms In the zone, TABLE 11.5.2 Stze and number of atoms for several fcc spherical microcrystals. fof Shells # of Atoms Cluster Diameter (A) 1 13 Ded 5 79 11.63 10 201 16.44 15 381 20.80 20 495 23.83 Because of the screened Coulomb potential this excess resistivity Is very small. For transition metal elements, however, eq. 2.1.15 ~49- Indicates that we should expect large excess resistivity whenever the difference In the d=-phase shift No of the two constituents is large. In the Cu-Zr system, for Instance, the phase shifts at the Ferm! energy for CugoZr4g are’46)no(cu)=2.996 and N7(Zr)=0.488 and we will show that, Indeed, a relatively large excess resistivity ts observed during phase separation In this system. |! would Ike to emphasize at this polInt that the previous formalism Is also valid for the growth of a topologically random cluster, I.e. In the case of nucleation of an amorphous phase Inside another amorphous phase. The only difference Is that the width of the Bragg peak near 2ky ts larger. The formation and the growth of amorphous zones with boron concentration close to 25% was found, for Instance, In FegoNt4oBoo metallic glasses during agetng'47), An anomalous excess resistivity Is assoclated with the growth of these clusters (48), The previous formalisms described to this point, are matnly concerned with the description of the resistivity during the Initial growth of localized fluctuations. They are, therefore, more appropriate for the treatment of nucleation. When an alloy ts quenched below the critical temperature for demIxIng Into the spInodal region, however, the early stages of phase separation occurs via long range compositional fluctuations. Binder et af, (49) discussed In detall the behavior of the resistivity In this case using the time dependent concentratlon=-concentration correlation function given fram both IInear'50) ang non! tnear>1) +heortes of splnodal decomposition -50- and from computer simulations (56), Binder uses eq. 2.1.6 for the resistivity together with a screened Coulomb potential (eq. 2.3.13) for the ulfference In potential between the two constituents. The structure factor S(K,t) given by eq. 2.3.8 Is then time dependent. As glven before, the {tnear!zed theory of spinodal decomposition predicts that(50), S(K,t) = S(K,0) exp(- Da@K?(K2 - kK? )t) (2.3.16) 1 2 — - {|_1 af [2 where, In terms of the parameters In eq. 2.3.1, K. = is sce Is the critical wave number for growth, D = 2MS/a2 ts ai diffuston constant and, a Is the fattice spacing. Let € be the correlation length for the fluctuation of concentration above the critical -r/E temperature and before the quench (T.e.9 0 (r) = F € , ). Therefore, If we assume an Infinite cooling rate: B §(K,0) = —7 3 (2.3.17) [K°+e “J Using eq. 2.3.17 and 2.3.16 to calculate the excess resistivity during the early stages of phase separation we obtaln when Kc<<A , Ky, 1/6 the followIng results: 2.2 Ap(t) - Ap(0) = taht 1 5e?K2 exp(yt) erf(yt) (2.3.18) VE ens 2 _ 2,4 _M of \2 where y = 1/4Da Ke = 55 (=) This equation predicts that the excess resistivity Increases | Inearly with time for small times (yt << 1). Wewlll see [n section [V that -51- this Is actually the case for the Cu-Zr system during the early stage of phase segregation. For larger times the ‘Inearlzed theory of sp!nodal decomposition falls to describe the exper!Imental S(K,t) and the exponential growth In elther S(K,+) or Ap has to be discarded. In this regime fluctuations Interact and macroscopic grains form. The shift In the peak of the structure function with time (fig. 2.3.2) Is associated with an Increase In the dimensions of the phase separated regions. Binder'49) showed, from a scaling assumption, ‘that the excess resistivity, Inthis regime, decreases IIke the Inverse mean radius of the precIpitated zones Ap(t) - dp(~) = q(t) « t7)/3 We have seen that, whether we decide to describe the early stage of decomposition via nucleation or splInodal decomposition both can give to an [Increase [n the resistivity. The physical reason for the decrease Is more controversial but It Is, In any case, associated with the shift of the structure functlon toward smal! K values and the Increase In the sfze of the precipitated zones. Ustng the experimental structure factor obtained from X-ray scattering analysis to calculate the excess resistivity, J. Mimault et al.(92) found a maximum In resistivity only In the case of spinodal decomposition. The reason for thls discrepancy can be related to the finite quenching rate In real experlments. These authors actually found [n thelr samples quenched [nto the nucleation region, smal! preclpitates which were formed almost Instantaneously during the quench, -52- $it. EXPERIMENTAL CONSIDERATIONS The alloys used In thls study were prepared by melting the appropriate amounts of the constituent elements on a water cooled silver boat under an argon atmosphere. The purity of the starting materlals was better than 99.9%. Ingots, of typically 2 or 3 grams were melted several times to Insure homogenel ty. Two quenching techniques were used In this study to prepare amorphous _— alloys. Cu73T!27, Pd64Y36, CujogexZrx «= ane (Zrs_ Hf, )ggNizg were obtafned In forms of ribbons by the melt- spinning technique!5>), In this technique a jet of molten metal Is projected onto a rotating copper whee! turning at approximately 10,000 RPM. The sample In contact with the Cu surface solldiffies, then comes off as a ribbon of about 2mm In width, 30um In thickness and several feet In length. The cooling rate achieved by this technique Is estimated to be on the order of 106K/min. The (Mop Rug 4) 199-.8x and (Zr62Nl3g)o9.8Fe0.2 amorphous samples were produced by the Piston and Anvil method'94), The melted alloy Is ejected from a quartz tube In the form of a droplet which Is caught between a rapidly moving piston and an anvil whose surfaces are both made of a CuBe alloy. The samples prepared thls way were In the form of folls approximately 1cm In diameter and 40um thick. After preparation, all samples were checked for crystalline Inclusion using X-ray diffraction and CuK, radiations. Samples whose diffraction pattern consisted of a smooth featureless band and no sharp Bragg reflections were considered as amorphous and kept for further studies. -53- A. Ihe resistivity measurements The resistivity measurements were done using a standard four point dc method. The samples were elther, ribbons of typically 2cm jong or, strips cut with awlre saw from an as quenched foll. Hewlett-Packard 6177¢c De current source was used together with a 3490A Hewlett-Packard multimeter to measure the voltage drop across the sample. Typical current and measured voltage were 10mA and 5mV respectively. An error of less than .04% on the normal Ized resIstance can be achieved this way. The digital output of the voltmeter was fed Into a PDP 11/23 "Minc™ computer which was used to control the exper!ment. With the help of a D/A output module on the MiInc, for each deta polnt, the current was reversed and the voltage averaged In order to eliminate any thermal Induced voltage. Areal time display of the resistance on the screen of a terminal was provided, and found very useful In determining an appropriate time to quench the sample from a specific state. The temperature of the furnace could also be controlled by the computer. Two different heat!ng modes were often used: constant heating rate (typically 100C/mtn.) or constant annealing temperature. With careful adjustment of the feedback parameters In the program, temperature fluctuations could easlly be limited to the order of 1%. For the Mo-Ru-B system, Pt wires were spot-welded on the sample to make the electrical contact. The measurements were performed In a vertical furnace In a slight over-pressure of hellum. The chamber was evacuated and back-fliled with He several times before the -54- measurements. For the other systems a special arrangement has been bullt In order to facliitate sequential annealing and quenching experlIments. A quartz tube golng through a vertical furnace Is attached to the top of a dewar. The probe can slide freely on a vertical axis Inside the quartz tube. A high vacuum valve and an O-ring seal at each end of the tube permits measurements In any type of atmosphere. As before a slight over-pressure of hellum Is usually employed In thls case. Flg. 3.1 shows a picture of the experimental arrangement. The system Is particularly sulted to measurement of the residual resistivity at low temperature after varlous annealing performed at high temperature. The tip of the probe Is made out of a single plece of nickel to provide rapld heat transfer. Usually the temperature of the sample reaches Its equllfbrium value In less than 10 min. The sample sits on quartz sifde and pressure contacts are made via a set of four Cu clamps. A Pt=PtRh thermocouple Is used to monitor the temperature and resides In a Cu block located at one end of the sample. Mullite rods are used to Isolate the wires Inside the probe. Because of thelr zero thermal coefficient of expansion mullite rods are very resistant to thermal shock. Fig. 3.2 shows a picture and a schematic design of the tip of the resistivity probe. A reel on which a long plece of ribbon can be mounted Is attached to the end of the probe for annealing purposes. This way, the resistivity can be accurately correlated with other types of measurements [Ike X-ray diffraction. -55- Fig. 3.1 Picture of the exper!Imental setup. -56- CU BLOCK NICKEL { eT efi fee fe ft EO ___] L ELECT, WIRES | P THERMOCOUPLE GLASS PLATE CU REEL » NS i \ i. SEE Gy IF) & (ES)— chan Fig. 3.2 Picture and schematic design of the tip of the resistivity probe. -57- B. Ihe Structural Measurements The alm of this research Is to understand the behavior of the resistivity In terms of structural changes. Several type of structural Investigations were performed. They Include: X-ray diffraction, Mossbauer experiments, transmissfon electron microscopy | Ow temperature superconductivity measurements and finally Differential Scanning Calorimetry (DSC). | will describe briefly tn the following each of these experiments and | refer the reader to the experimental section corresponding to each alloy studied In chapter IV for more detalls. X-ray diffraction: In order to calculate the Radial Distribution Function (RDF), detalled X-ray studies were performed on several samples [n the reflection geometry using a GE XRD-5 scanning diffractometer or a PhiltI ps Norelco diffractometer. The samples were prepared by stacking several! fayers of ribbons together with thinned Duco cement on a glass plate. The sample were thick enough to eliminate any scattering from the substrate and to sImp!lfy corrections for sample absorption. Molybdenum Ky radiation ( 4 = 0.7107 A) was used with a LIF focussing crystal! monochromator placed In the diffracted beam to eliminate scattering contributions from other energles. A typicel run was done by scanning In angles from 10° to 80° In steps of 0.25° and from 80.5° to 150° In steps of 0.5°. The Interference function can be measured, this way, for K values up to 17 Ant, A time Interval of 7 min. was chosen for each step In order to reduce the statistical error to Jess than one percent on the average. A micro-processor controlled -58- the experiment and stored the recorded Intensity digitally. The data obtafned were then transfered to a PDP 11/23 computer for analysis. After correction for background, polarization, absorption and Compton scattering, the data are normalized usIng the "high angle method"(55) _ Convergence factors ranging from 0.005 to 0.015 were used to calculate the Fourler transform In specific cases. The detalls of the RDF calculation and the data handling are given In (56), Mossbauer Experlments Mossbauer exper!tments were performed at room temperature on (Zr6gNi3g)99,.eFeo.2 alloys using a Co?7(Rh) source In constant acceleration mode. An old Mossbauer drive was first rebullt In order to do so. The new electronic feedback system can now achleve an error signal of jess than 1% of the reference sIgnal. ! would [Ike to emphasize that without the help of Viadimir Matijasevic during the reconstruction of the drive this experiment would probably have not been possible. A Canberra multichannel analyzer was used to collect the data. Fig. 3.3 shows a sketch of the experimental setup and the drive electronics. A complete explanation of the detalls of the drive can be found tn (97), Amorphous folls quenched by the Piston and Anvil technique were used directly as absorbers. Because of the very small concentrations of Iron In the sample, Fe enriched In the Mossbauer Isotope (Fe?7) was used to make the Ingot but even so, collection of a typical spectrum required roughly two weeks. The photons are detected by a Kr(C0x) proportional counter. Before each measurement, the drive was callbrated using the splitting of the Inner two |Ines of the magnetic hyperfine spectrum of Fe?’ In metallic Fe. The analysis was performed -59- DETECTOR COLLIMATOR POWER TRANSDUCER AMPLIFIER OURS weed J BOUREE +0 aa DRivE ABSORBER COIL gene DIFFERENTIAL FROM VELOCITY SENSING COIL} AMPLIFIER CHENKEL AMPLIFIER ANALYZER MULTICNENNEL INTEGRATOR ANALYZER nnn AANA ~ REFERENCE SIGNAL SIGNAL TO POWER FROM CA sen AMPLIFIER peFERENCE (YOO ERROR SIGNAL SIGNAL Fy 7 7 . Fig. 3.3 Mossbauer experlmental setup and the drive electronics. -60- by fitting n Independent Lorentzian lines with varlable width and helght, where n Is the lowest even number of Iines needed to obtain a visually good fit. 1 am grateful to Carl Unruh for providing me with his program and helping me to perform this analysts. Transmisston Electron Microscopy: A series of Cuyoo.,Zr, and (Hfy-yZr,JggNizg alloys were studied under a transmissfon electron microscope. For Cu-Zr we used a Slemens ELMISKOP 1!, whereas for Hf=-Zr-NI a Phillips 300 electron microscope was used. Typical magnification tn this last case Is 210,000. Prior to the experfment the samples were electro-pollshed using a 550B Jet Thinning Instrument (South Bay Tech. Inc.) and kept In methanol until the time of the experiment to prevent oxidation. For Cu-Zr a solution of 20% nitric acid, 20% glycerin, 60% methanol was used at ~30°C. For Hf-Zr-NI we used a solution 80% methanol, 20% perchloric acid at -159C, | am very grateful to Michael Atzmon and Pat Koen for thelr help with these measurements. Low temperature superconductivity measurements were performed on Zr62Nl38 alloys. In order to measure the upper critical fleld, a magnetic fleld produced by a superconducting NbTI solenold was appl ied normal to the direction of the current [n the sample. The critical fleld was determined by plotting the resistance of the sample as a function of the applied field. Hoo Is defined as the mid-point of the resistive transition. The temperature was maintained constant using a pressure contro! valve over the He bath. The temperature was calculated using the vapor pressure over the IIqutd. The superconductivity transition temperature was obtained by extrapolating -61- dH ec the graph of H.o versus T to vanishing fleld. aT 1 Is just the slope of the curve at T=Te- DSC measurements were .btalned on a commercially avallable unit (PerkIns-Elmer DSC-!) at the Jet Propulsion Laboratory. In these measurements the temperature of the sample Is Increased at a uniform rate and the heat evolved during varfous transformations Is recorded, A scanning rate of 20K/mIn. was used. -62- !1V¥. RESULTS AND DISCUSSIONS WV.1 0 0 Cuso0-»Zrx [This section is essentially an article with the same title by Robert Schulz, Konrad Samwer and W. L. Johnson published in Journal of Non-Crystalline Solids 61 & 62 (1984) 997.] -63- KINETICS OF PHASE SEPARATION IN CusoZrso METALLIC GLASSES ABSTRACT An excess resistivity above the usual topological relaxation contribution Is seen In CusoZrsq glassy alloy upon annealing at temperatures far below the glass transition. This excess resistivity Is correlated with the appearance of a well defined two step crystal|ization when the annealed sample Is heated through To at a rate of 15°C/min. X-ray diffraction suggests that phase separation Is taking place and Is responsible for this anomalous resIstlvIty behavior. Results from transmisston electron microscopy are consistent with this Interpretation. 1. INTRODUCTION Evidence for phase separation In vapor quenched Cu-Zr amorphous alloy has been reported'!), The phase separation occurred during synthes!s and the composition of the two amorphous phases are Identiflable with those of ordered crystalline phases CuZro and CusZr. It Is belleved that the phase separated structure represents a lower free energy state for thls nonerystallIne alloy system and Is a result of the enhanced kinetics found In "vapor quench film The authors also belleve that phase separation could be achleved In I!quid quenched materlal by extended annealing at temperatures considerably below Ta and Te of the two resulting amorphous phases. Recently Altounlan et al.'2) suggest that the equlatonic CuZr does not exist as a single phase In the crystalline state and show In ~64- fact that the crystal! ization products of the amorphous Cus59Zr50 Is @ mixture of CuioZrz and CuZr2, and not CuZr as It was previously reported(>), The crystallization temperature of CuyoZr7 and CuZr2 glassy alloys are respectively "360°C and "455°C as measured by the onset of the resistivity drop at Tc for a heating rate of about 10°C/mIn. Fig. 4.1.1 shows the main and the second band of the radial distribution function for Cu-Zr glasses across the concentration range. Fig. 4.1.2 shows that the average nearest nelghbor distance undergoes an abrupt change around the equlatomic composition. If CusoZrsq phase segregates Into Cu rich and Cu poor zones, we should expect the formation of a split first peak In the X-ray pattern during anneal Ing. In this section we report measurements of electrical resistivity, X-ray diffraction and TEM In order to find phase separation In CusqZrsq at temperature below 360°C. 2. EXPERIMENTAL The Ingots were prepared by RF Induction meiting the appropriate amounts of Cu and Zr ona silver boat. Amorphous ribbons were ob- tained by the melt-spInning technique under reduced hellum pressure. The samples were stored under vacuum and used within a few days after preparation. All measurements of resistivity and X-ray diffraction with annealing temperature have been done on the same ribbon to avold fluctuation In properties from sample to sample. The kinetics of phase separation appears to be very sensitive to the state of relaxation of the amorphous alloy. The resistivity was measured using -65- G(R) : cree Snen ate Se ee ee eee eee meek eee R (Ancstroms) Fig. 4.161 The matin and the second band of the radial distribution function for the Cu-Zr system across the concentration range. -66- 3.21 e 12H - 200 C ZrCuy_, x AFTER WASEDA-CHEN sputtered ———— ® 5 3.0} —_« 7% & re x [ am) 2,.8h ._ ae 4 xX ° 0,30 0.40 0.50 0,69 0.70 0,80 X (CONCENTRATION) Flg. 4.1.2 Average nearest nel ghbor Cus o9~>ZF x metallic glasses. distances vs Zr concentration for -67- @ four points computerlzed DC method described 1nf4), Speclal care has been taken agalnst oxidation of the samples during the exper!ment, Measurements of the scattered X-ray Intensity were performed In the reflection geometry usIng Mok radiation on a Norelco diffractometer. The analytical procedure used to compute the palr correlation function was described elsewhere (5). Samples for electron microscopy were electropollshed In a 20% nitric acid, 20% glycerin, 60% methanol solution at -30°¢, 3. RESULTS AND DISCUSSION Using the ZIman theory to describe the electrical resistivity of an amorphous alloy and the free volume model, Kelton et al.(6) relate the kinetics of the resIstivity changes to the kinetics of the viscosity changes upon annealing [In PdSI. The resistivity vs time, In terms of viscosity, can be written as: sel — (4.1.1) fe) c + an(A/n, t exp(-c) + 1) where c {fs a constant depending on the quenched-in free volume. Flg. 4.1.3a and b are typical representation of such topological relaxation for CusoZrepj. As the temperature Is Increased further, chemical relaxation occurs and departure from this model can be seen In fig. 4.1.3c and d. At the annealing temperature of 300°C, the excess resistivity above the usual free volume relaxation Increases Initially linearly with time, then bends over. This Is In agreement with Binder et al.(7) who describe the time-dependent resistivity In phase separation kinetics In stmple binary systems !Ike Al-Zn. Using a -68- ,97 F10C G 0 — 250C b 84 7 be of 280 c - a 45 een oe — oc J ol 300C d ; 0 4 8 12 TIME (HouRS) Fig. 4.163 Relative change with time of the electrical resistance of CusoZr5q In tsothermal annealing Ry = R(t=0). 12 7 340 C a 0 | 1 6.2 b ~ 350 aes BS fo) <7 4 365C C 0 J 0 | d 385C -39 ; / 0 4 8 12 TIME (HOURS) Fig. 4.1.4 Relative change with time of the electrical resistance of CusoZrs 9 In Isothermal annealing R, = R(t=0). -70- general expression for the excess resistivity and the | Inear theory of sptnodal decomposition they predict an excess resistivity which, at smal | times Increases | Inearly In time with a slope related to the diffuston coefficlent D: 2 So(t) - Ap(0) «= Ke exp (Da? Ke t/4) erf(Da°K? t/4) (4.1.2) als the lattice spacing and K, Is the critical wave number. If we assume this expression valid for the Cu-Zr system, D would be the diffuston coefficient of Cu. Fig. 4.1.3e shows the relative changes In resistivity for a sample Initially annealed 10h, at 210°C. The change In the slope of the excess resistivity could reflect the change In the diffusion coefficient of Cu upon free volume relaxation, which In our case, seems to Increase contrary to the usual observation seen in other systems (8), While the early stage of phase separation occurs via nucleation or sp!nodal decomposition for later times macroscopic grains form (Guinfter Preston zones) and due to the Interaction between fluctuations, the resistivity reaches a max!mum and then decreases (Fig. 4.1.4b). High angle X-ray diffraction shows no trace of crystals whlie this process Is happening (Fig. 4.1.5). Fig. 4.1.6 shows the crystall zation behavior for samples Initially annealed "Ith at different temperatures, For T, < 280°C no excess resistivity vs time fs seen during annealing and all the crystal] lzation curves fall on the same trace, characteristic of topologically relaxed structures. Crystallization Is taking place In almost a single step. The smal! shoulder corresponds to the small subpeak seen In a DSC scan of an as-splat CuspZrsp alloy’, For T, > 280°C we observe the onset of the excess resistivity and a well defined two step crystal |Izatlon. -71- Lt ts wee INTENSITY (ARBITRARY UNIT) iene ee = or nae pire pene etna Fig. 4.1.5 High angle X-ray 60 90 120 Two THETA scans with CuK, radiation for CuZr (a) és quenched (b) annealed 10h. at 300°C (c) annealed 10h. at 350°C. -72- 1.1 Lo AS QUENCHED ; f -B0 C 300 C ao ae) - 7 > AS QUENCHED 1 BL A (210C, 250 C, 280 C,)—— i/ i al 1 280 380 480 580 680 780 880 989 TEMPERATURE (K) Fig. 4.1.6 Change In the electrical resistance with temperature of CusoZrsy for different pre-annealed conditions. R, = R(T=300K). -73- After the first step X-ray diffraction shows the presence of CuZrg2 plus a few faint lines cue to Cujg2r7, The second step Is assoclated with the crystallization of Cuj,gZr7 %In agreement with'2) (fig. 4.1.7). Note that for annealing temperature > 365°C, which correspondsto the crystallization temperature of glassy CuZr2 alloys, the resistivity maximum Is not seen and the amorphous phase separation ts by-passed by precipitation of CuZrg crystallites. These are responsible In fig. 4.1.4c for the increase In resistivity as It was In the case of Fe-P-C°9). At 385°C crystallization takes place In less than 3h. Fig. 4.1.8 represents the maln peak of the Interference function of CusoZr5q after different annealing treatments. The peak shifts to higher q values, gets wider (In contrast to the usual relaxation behavior described by Egami(10)), and becomes more asymmetric. Also represented are the main band of as quenched CuZr5 and Cuj;oZr7 glassy alloys. The figure suggests that the annealed CusoZrs might be composed of a mixture of amorphous CuZrg and Cu;oZr7- Fig. 4.1.9 shows the radial distribution function of CusoZrsq at different annealing stages. The positions of the maxima are tabulated In Table 1V.1.1. The second band consists of two subpeaks, nearly resolved, anda shoul der at high r values. We note, In the second band of the as quenched CusgZrsqs that the first subpeak Is less Intense than the second one. This Is typical of an unrelaxed Bernal structure and ts In agreement with the results obtained by Chen (11), Upon relaxation, the first subpeak Increases In Intensity and the second decreases. This Is also consistent with computer mode! of relaxation of Bernal structures(!2), The Inabliity to resolve the first split-peak of the second band after annealing might be partly -74- 4 J 7 A I(6) | cen ©] (181) (pe) INTENSITY (ARBITRARY UNIT) Two THETA Fig. 4.1.7 X-ray diffraction pattern of (a) CuZry crystalline powder (b) CuZr annealed 12h. at 350°C then quenched after the first resistivity drop (c) same sample quenched after the second resistivity drop (d) Cu, 9Zr7 crystalline powder. -75- INTENSITY (ARBITRARY UNIT) 15 17 19 21 25 4 W Two THETA Fig. 4.1.8 X-ray diffraction scans with Mok, ratiation for CuZr, Cuyo2Zrz and CuZra (a) as quenched (b) annealed at 210°C for 10h. (c) sample b annealed at 300°C for gh, -76- 3,/ ee a 0 {\ L\ Lo YU v= 3.54 ] \ b e 9 IX AOA 4,0} ; Cc 0 IN ALA = v= 0 Ly 8 12 16 R (ANGSTROMS) Fig. 4.1.9 Palr distribution functions for CusoZrsp. (a) as quenched (b) annealed at 210°C for 9h. (c) sample b annealed at 300°C for 9h. -77- due to the relatively high convergence factor (.017) used to calculate G(r) due to the poor statistics at high angle. The second band shows @ more pronounced shoulder at large r than reported [n(11) for CusoZrs5o- This shoulder Is due to the second band of the Zr-Zr partial palr distribution function, Fig. 4.1.10 shows G(r) of glassy CuyoZr7 and CuZrg- The first maximum occurs respectively at R= 2.76A and R = 3.02A. Fig. 4.1.11 represents AG(r) = G(ranneated "G(r das quenched’ The first peak Is mainly due to the formation of CuyoZr7, the little bump at "2.9A Is due to unchanged ZrsqCusg remaining In the matrix and the shoulder at "3.3A Is caused by the formation of CuzZro, This suggests that one fit the annealed G(r) of CuspZrsp to a IInear combination of CusoZrso, CuZro and CuygZr7. Fig. 4.1.12 shows a least square fit with 5.2% of CuZr2, 26.9% of CuygZr7 and 67.8% of CuZr. The crosses correspond to experimental data points. This good fit suggeststhat the following reaction Is taking place In the amorphous structure amorphous ZrCu + amorphous Cuyg2ry + amorphous CuZr, (4.1.3) Fig. 4.1.13 shows (a) TEM of as=-quenched ZrCu and (b) annealed 11h. at 210°C and 9h. at 300°C. The diffraction pattern of the annealed sample shows only a2 broad diffuse band (c). Those results suggest phase separation on a scale of a few 100A. -78- 3.5 a ——_—— — 0 4 8 12 R (ANGSTROMS) Fig. 4.1.10 Patr distribution function for (a) Cuyo2rz (b) CuZro. 16 -79- 3,5 , — = aN 0.8 | So [ (\ R (ANGSTRomsS) Fig. 4.1.11 Palr distribution for as quenched CusgZrsy and the ARDF between the annealed CusqZrs, at 300°C for 9h. and the as quenched sample, -80- 2.50 : = “1,75 . | | | 2 38 4 5 6 7 R (ANGSTROMS) Fig. 4.1.12 The ful! curve (a) which represents the sum of three G(r) [(b) (67.8% ZrsqCusg), (c) (26.9% ZrzCu,9) and (d) (5.2% ZroCu)] corresponds to a least square fIlt to the data for the annealed ZrspCusg (cross points). as quenched Fig. 4.1.13 (a) TEM of as quenched Z2rCu (b) annealed 11h. at 210°C and 9h. at 300°C (c) diffraction pattern of the annealed sample, ~82- Table 1V.1.1 G(r) parameters for CusgZrsy metallic glasses (a) as-quenched (b) annealed 11h, at 210°C and (c) subsequently annealed 9h. at 300°C, Alloy Ry(A) R2/Ry R3/Ry CAN. | Alloy Ry(A) Ro/RyCANe a 2.83 1.73 1.79 13.02 Zr, Cu 3.02 1.73 13.00 b 2.81 1.74 - 12.43 - - - Cc 2.83 1.68 - 12.85 ZrzCujg 2-76 1.73 12.01 ACKNOWLEDGEMENT: The authors would I Ike to thank Michael Atzmon for his help with the TEM results, Angela Bressan for typing this manuscript and the National Research Counctl of Canada for @ fellowshIp during which this work was carrled out, *Permanent Address: |. PhysIkalfsches Institut, Bunsen str. 9, D-3400 Gottingen, West Germany. Work supported by the Department of Energy, Proj. Agree. No. DE-ATO3- B1ER10870 under Contract DE-AMO3-76SF00767. 11. 12. ~83- REFERENCES TW. Barbee, Jr., R.G. Walmsley, A.F. Marshall, D.L. Kelth, and D.A. Stevenson, App!. Phys. Lett. 38 (1981) 132. © Z. Altountan, Tu Guomhua, and J.0. StromOlsen, J. Appl. Phys. 53 (1982) 4755. R.L. Freed and J.B. Vander Sander, J. Non-Cryst. Solids 27 (1978) 9. M. Mehra, R. Schulz, and W.L. Johnson, J. Non Cryst. Solids 61 & 62 (1984) 859-864, G.S. Cargill, Sol. State Phys. 30 (1975) 227. K.F. Kelton and F. Spaepen, Proc. 4th Int. Conf. on Rapidly Quenched Metals (Sendal 1981) p. 527. G. K. Binder and D. Stauffer, Z. Physik B 24 (1976) 407. H.S. Chen, L.C. Kimmerling, JM. Poate, and W.L. Brown, Appl. Phys. Lett. 32 (1978) 461. P.K. Rastog! and Pol Duwez, J. Non-Cryst. Solids 5 (1970) 1. D. Srolovitz, T. Egaml, and V¥. Vitek, Phys. Rev. B 24 (1981) 6936. H.S. Chen and Y. Waseda, Phys. Stat. Sol. 51 (1979) 593. L.v. Helmendahl, J. Phys. E.5 (1975) L141. ~84- 1¥. RESULTS AND DISCUSSIONS IV.2 (Mop. 6Ru9, 4) 100-x8x -85- ELECTRICAL RESISTIVITY AND STRUCTURAL CHANGES UPON RELAXATION AND CRYSTALLIZATION OF (Mop 6RuUp. 4)109—,Bx METALLIC GLASSES ABSTRACT Crystallization of (Mop 6Rup 4) 19908. glasses takes place In three steps. The first step, corresponds to the precipitation of the sigma phase MosRuz which decomposes at higher temperatures. The second step Is associated with the formation of an hcp solld solution of Mo In Ru, and In the last step the remaining amorphous matrix crystallize In an fcc boride. Detalled electrical resistivity measurements taken below the crystallization temperature reveal an excess resistivity above the usual IInear temperature dependence predicted by the Ziman theory. The Increase {tn resistivity Is assoclated with the onset of long range compositional Inhomogene!ty (spInodal decomposition) and the decrease with the beginning of crystallization. The eJectrical behavior for samples pre-annealed for 12 hours at varfous temperaturessuggest that boron migration Is partly responsible for this excess resistivity. The linear relationship between the conductIvity and the density of states across the concentration range, and the fact that upon annealing (Moy 6RuUp 4) g2B1g the maximum In the excess resistivity Is assoclated with amlnimum In the density of states, Indicates that Mott's model Is not applicable and suggests that the d electrons are Important In the conduction. The changes In the radial distribution funct lon during anneal Ing at 525°C are also reported in this section. -86- 1. INTRODUCTION It ts well known that electrical resistivity measurements form a sensitive probe for the experimental Investigations of relaxation and crystal |Ization processes In metallic glasses. Both topological (free volume annihilation) and chemical relaxation (ordering, phase separation...) Influence the electrical resistivity. The problem arfses In the Interpretation of the data, In other words when we try to assign a_ specific mechanism or structural transformation to a particular feature In the electrical resIstivlity. Reversible changes are usually considered as chemical ordering!!) and Irreversible changes as topological relaxation(2). The latter has been studied extensively. P. Allfa et al.(3) use a theory based on the ZIman approach to the resistivity and the mlcroscopic model for structural relaxation proposed by Srolovitz, Egaml and Vitek(4) +o explain the free volume dependence of the electrical resistivity. They usually attributed the Increase of resistIvIity In thelr low temperature Isothermal measurements to changes’ In compositional short range order (clustering of Mo atoms In the case of Fe-Nl-Mo-B for Instance) and the decrease at high temperature to topological relaxation'9), Thefr results however, are In dl sagreement with those of A.K. Bhatnagar et al.5), Kelton and Spaepen assign the Increase as wej! as the decrease of the resistivity In Pd-Si-V to free volume relaxation and not the clustering of vanadium atoms. The reason for many of these dIsagreements Is the lack of direct structural Informations correlated with the resistivity measurements. The effect of chemical relaxation on the electrical transport In metallic glasses, on the other hand, [fs poorly understood. Even when -87- atomic clustering?) and phase separation(8-10) had been observed tn metall!c glasses, their effects on the electrical resistivity were never carefully Investigated In the amorphous’ state untt! recentiy(11). In the crystalline state however, the behavior of the resistivity during Gulnier-Preston zone formation, spl nodal decomposition etc. Its well known{!2-15), Considerable work has already been done on the effect of annealing on the structure of (Moo,6RuQ,4)100-»8x ‘metallic glasses(16-18), It has been suggested that upon annealing at temperature suffictently high to permit the diffuston of boron over atomic distances, but low enough to prevent crystallization, this glass phase separates {nto boron-rich and boron=-poor zones. Thus Mo~Ru-B represents an [deal candidate to study the effect of chemical segregation on the electrical properties of this glass. This section reports electrical resistivity and high angle X-ray diffraction measurements upon annealing (Mop.6Ru9.4)100-x8x for x=14,18 and 22 and fnterprets the results In terms of phase separation caused by diffuston of boron In the amorphous matrix, 11. EXPERIMENTAL The Ingots were prepared by Rf Induction melting the different constituents on a silver boat In an argon atmosphere. The Ingots were remelted several times to Insure homogenelty. The rapIdly quenched amorphous alloys were obtained by the "piston and = anvii" technique!!9), The folls were typically 40 In thickness and 1.5 to 2 cm In diameter. They were examined by a high angle X-ray diffractometer us!ng Cuky radiation and only the folls showIng a broad -88- band with no resolvable Bragg peaks were kept for further study. The detalled diffraction studies of the crystalline phases obtalned from the amorphous samples by heating through the’ erystallization temperature were performed on the same X-ray diffractometer by scanning at 0.0590 Intervals. The X-ray data used In computing the RDF were obtained tn the transmisston geometry on a Norelco scanning gonlometer using Mok radiation and a focussing LIF monocromator. The detalls of the data handifng and RDF calculation were described elsewhere(20,21), The resistivity was measured using a four point de method with Pt leads spot welded on to the samples, also described e! sewhere(11), ltl. RESULTS and DISCUSSION There fs now much evidence to belfeve that alloys with low boron content (x<18) have predominantly one type of local structure while those with hIgh boron content (x>18) have a distinct, second type of local structure(17,22,25), We then expect different crystallization behavior as we go through the concentration range. Fig 4.2.1 shows the normalized electrical resistivity for x=14, 18 and 22 as we heat the samples through the crystallization temperature at a heating rate of about 15°C/min. For the low boron concentration alloys we find a three step process, the first two becoming almost a single step for x=22. A shoulder {fs nevertheless dIistingulshable. Table 1V.2.1 g!lves the crystalllzatlon temperatures and the relative percentage drop of the first two. steps. -89- 1.0f — : oh og, 6p, weeBry NG <> (Mog GRU, 4) goByg \ | (Mlog Rup, 478822 0.8 5 NN d 0.6 + 200 600 1090 1400 TEMPERATURE (K) Fig. 4.2.1 Resistance traces obtalned at a heating rate of 15°C/mIn. for x=14, 18 and 22. (Ry = room temperature resistance). -90- Crystal!Ization temperatures and the percentage drop of the resistivity during the first two steps of the crystallIlzation. Alloys TI(°K) ~ drop T2(°K) ~ drop x=14 1020 60 1135 40 x=18 1045 45 1135 55 x=22 1085 27 1120 73 It Is clear from the data In the table that the first transition around 1020K Is characteristic of primary crystallization from glasses with low boron concentration and the second around 1130K, from glasses with a high boron content. In order to analyze the crystalllne phase corresponding to each step we applied to a foll of (Moo .6Ru 4) 36814 the heat treatment Indicated by the arrows shown [n flg. 4.2.2 and took an X-ray scan after each quench. Fig. 4.2.3 represents the results for quenches from 1075K, 1170K and 1300K. The crystalline phase Identified after the first step Is the boron free sigma phase MosRuz. During the second step an hcp solid solution of Mo In Ru appears In the structure and from the peaks position we estimate the Mo concentration at about 36%. In the final step the Ru hcp phase grows, the sigma phase disappears almost completely and an fcc boride of lattice parameter a=7.81A can be Identified. Thus In glasses of low boron concentration, there !s_ first primary crystallization of the sigma phase, MosRuz» which transforms at higher temperature [nto an hcp Ru. The remalning amorphous state crystallizes Into a boride phase. In a high boron concentration glass, crystallization takes place by first a formation of an hcp solld solution of Mo In Ru and then the crystallization of an fcc boride. -9]- 1.1 . | ; (Mog Rug, 4) g6Byy a = nied ws ; \ - | \ O.8t > a —— 0,7- 0.6 | | | | 200 600 1000 ” TEMPERATURE (K) Fig. 4.2.2 The heat treatment, monitored by resistance changes, applled to (Moo 6RUp 4) g6814 for the crystallization study. -92- (Mog gRug 4) ggBiy AS QUENCHED ac a UW Q ™ ™~ im) rt re = = = o ro) a oc ua Li © (05h) (005) (ZT) 0(0£) (005) (ZT) (222) 2(22) o(Tgs) oOcIge) O(TTh) AACTOT) o(TTh) (02h) (2TZ) 0(029) (272) 0 (202) ¢ 98200) (202 0 (0££) o (0£5) (211) | (ZIT) (00T) (201) 2 (200) (200) o(TTg) © (02¢) ~ ny(210) (201) < (on) OQ 4 i my ed 6 a(0Z4) f& a(TE€) 0(0£n) (005) (ZTE) | a(00%) 0( 222) 0(02h) (212) Y¥(O00T) o(20T) a(02Z) ALISNSLNE (321 TWWYON 60 50 40 30 TWO THETA Fig. 4.2.3 obtal ned radiation) of the crystal!Ine phases a scan (CuK X-ray In flg. 4.2.2. after each quench Shown -93- As mentioned eariler In the case of (MOQ 6RUp, 4) 78822 the first two transitions occur [In a broad single step. However, If we anneal the sample for 12h. at 500°C we recover awel!l defined two step process typical of crystalllzation of low boron phases, fig. 4.2.4. This ts what we would expect If annealing Induced phase separation by dif fuston of boron In the amorphous matrix occurred. Fig. 4.2.5 shows an enlargement of the resistivity scans Just before crystallization. Up to around 600K the resistivity varles almost IInearly with T as It fs found to In many metallic glasses at high temperatures (T > 8q) and as [s predicted by the ZIman-Faber theory In the Debye approxtmation(24,25) , Around 800K, region "a" on the plot, a plateau or a departure from the IInear temperature dependence Is observed. Then the resistivity falls again first slowly In region “b" then rapidly as crystallization proceeds. C.C. Koch et al. (16) studied the smal! angle scattering on an (Moo Rup 4)g7B;g— and found three regions of Interest (flg. 10 In ref. 16). For low temperature annealing (<450°C) IIttle difference Is observed In the scattering Intensity from that of the as-cast condition. For higher annealing temperatures or longer times (reglon 11) peaks In the [(K) curves, which can be assumed to represent a "pseudo-Bragg peak" attributed to a perfodicity In the scatterIng entity are observed. For the duration of a resistivity scan, typically two hours, this region corresponds to the plateau "a" In fig. 4.2.5. After still higher annealing temperatures or longer times (region [If In ref. 16) the peaks In I(K) disappear and the curves can be fitted to the Gulnier and Porod law of discrete scatterers. Some crystalline precipitates can be observed by TEM In this region of the temperature-time diagram -94- 1,000 7 7 7 (Mog 6RUp, 4) 86814 0,925 } f 1.00 | AS QUENCHED {=} = —— AS QUENCHED 4 / 0,85 r = ANNEALED 12H J AT 500C (Mog gRUg, 47989 0.70 1 , 4 1000 1050 1100 1150 1200 TEMPERATURE (K) Fig. 4.2.4 Enlargement of the resistance traces around the crystal lszation temperature. -95- 1.02 7 ; (Mog 6RUg, 4 1-yBy “~~! x=18 0,97 + cline, x=22 | Sf | ol \| | 3.0 fo x=14 os “ se I | <J mae 1 | | 200 400 600 800 1000 1200 TEMPERATURE (K) Fig. 4.2.5 Enlargement of the resistance traces below the crystallization temperature fo x=14, 18, 22. (Region "a" and "b" correspond to the region I! and IIIf In flg. 10 of ref. 4.2.16 ). The lower plot represents the excess resistivity above the IInear temperature dependence characteristic of the flow temperature region. ~96- which correspond to region "b" on our resistivity scan. The lower portion of fig. 4.2.5 shows the excess resistivity above the I[Inear temperature dependence due to thermal scattering. The Increase In the excess resistivity seems then related to spinodal decomposition In the amorphous matrix caused by boron diffusion and the decrease, to the onset of crystallization. Such maximum In the excess resistivity Is well known In the crystalline state. Mimault et al.(15) explain the connection between the anomalous Inittal Increase of the electrical resistivity and the presence of a peak In the X-ray small angle scatter!ng for an Al-Zn system when quenched [nto the spinodal region. Recently Schulz et al.(11) found stmilar behavior In the amorphous state during phase separation of a CusoZr50 metallic glass. For Cu-Zr the Increase of resIstivity due to topological relaxation and the Increase due to chemical segregation are well separated In time during [sothermal measurements. ThIs [ts not the case for (Mop 6Rup.4)g2B1g as we can see In flg. 4.2.6. At low temperature (<450°C) only topological relaxation (free volume annihifation) ts observed. The full lines represents the best fit of the data using the Kelton-Spaepen model for topological relaxatton'26), At higher temperature both topological and chemical relaxation contribute to Increase the resIstIlvIty over about the same +ime Interval. The decrease of the resistivity at 600°C after two hours ts due to the onset of crystallization and colncides with the boundary between region |! and region III In flg. 10 of ref. 16. Note that an Increase of the electrical resistivity has been also reported during tsochronal annealing of amorphous (Moo 6Rup, 4)g2B18 flims( 18), From fig. 4.2.5 It seems that there Is no correlation between the -97- 0.4 (Mog, 6RUg, 4) g2Big 0 4 g 12 TIME (HRS) Fig. 4.2.6 Percent change In resistance as a function of annealing time at varfous temperatures, The solld JInes are fit to the Kel ton-Spaepen theory. -98- excess resistivity observed during a scan and the boron concentration of the glass. Fig. 4.2.7, however, shows that this anomalous behavior of the resistivity Is very sensitive to the state of relaxation and that annealing has a slightly different effect on boron 14 and boron 22 glasses. If spinodal decomposition Is responsible for this resistivity anomaly, topological relaxation or free volume annthilation should slow down the diffuston(27) of boron and thus affect the behavior. Johnson and Wilitams(23) have calculated the effective boron volume Vb across the concentration range. They found that It decreased linearly with Increasing boron concentration up to a value close to the effective volume of boron In a close packed metallic arrangement Vp) near the upper limit of compositions for which a glass can be made (I.€. x,=24). If we consider Vp-Vp as a structural excess volume we canwrite an effective free volume associated with the boron atom [n terms of a thermal free volume and this structural term'17), Ve = a(T, - T)) Q + B(x. - x) 2 (x < x) (4.2.1) Ty Is the fictive temperature associated to the quench and T, an appropriate reference temperature at which the thermal free volume Is effectively zero. % Is the average atomic volume and a,8 are constants. The diffuslon of boron In the matrix depends on this free volume. In the as quenched condition the thermal term Is farge and masks, any effect of the structure! component on the excess resistivity (ffg. 4.2.5). If we anneal outthe free volume at low temperature however, we can reduce the thermal term to alTa-To)2» Ta -99.- 1,02 0.97 fT 0,96 } v + qT q (Mog GRU 4) ggBry AS QUENCHED (Mog, gRUg 4) 79822 “> AS QUENCHED 200 Fig. 4.2.7 Enlargement of 400 600 800 1000 TEMPERATURE (K) 1200 the resistance traces below the erystal|llzation temperature for x=14 and 22 annealed previously 12h. temperatures, at various -100- ts the anneal!lng temperature, and thus observe the effect of the concentration on the resistivity bump. Fig 4.2.7 shows that In the case of (Mon eRup,4)7g822 for which the structural component van! shes (x® Xm), the resistivity anomaly completely disappears after annealing for 12h. at 450°C. The effective boron free volume vanishes preventing Its diffusion, and the resistivity varies IInearly all the way up to the crystallization temperature. For higher annealing temperatures, followed by a quench to room temperature, we observe the gradual effect of the Increase [In the quenched-In thermal free volume O(Tg-To) » the resistivity anomaly becoming more significant at higher annealing temperature. For (Moo ¢Rup.4)86814 the structural component Is not negligible (x<<xm) and low temperature annealing Is not sufficient to stop the diffusion of boron as we can see in flg. 4.2.7. Unilke tn the crystalline state, the very short electronic mean free path (3-4A) In these glasses cast doubt on an approach using the Ziman scattering theory for resistivity and small angle X-ray scattering data from spinodal decomposition theory to theoretical ly explain the resistivity anomaly observed here. Fig. 4.2.8 compares the excess resistivity wlth the change In the density of states Interms of the reduced annealing temperature. Tor is the crystallization temperature corresponding to the onset of the drastic drop In _ the measured property. We see that the maximum In the resistivity corresponds to a minimum In the density of states. Based on a correlation between the fractional change In magnetic susceptibility and resistivity upon crystallization, Z. Altountan et aJ,(28) suggest that s-d scattering which Is proportional to the d density of states -101- (Mon ¢RuUq 4) eB 1.1t 0,60 0.A'B2r18 2 RESISTIVITY J 4.99 > 1,0} 41.00 2 > Pa 0.9 4 0,98 i Y From SPECIFIC-HEAT Y From SUPERCONDUCTING PROPERTIES 0.0 0.5 1.0 1.5 REDUCED TEMPERATURE (T/Tcr) Fig. 4.2.8 Correlation between the excess resistivity and the change In the density of states (from 4.2.16) vs the reduced annealing temperature. T Is the crystalllzation temperature (°C) cr corresponding to the onset of the drastic drop In the measured property. -102- makes an Important contribution to the resistivity for Cu=Zr and Ni-Zr glasses. Our results on Mo-Ru-B show that even If resistivity and density of states are correlated during crystallization they behave [n opposite directions In the amorphous state. Recently L.E. Ballentine et al.(29) have calculated the conductivities of several Iiquid transition metals characterized by strong scattering and sd hybridization using the Kubo formula and clusters of 365 atoms whose palr distribution functions agree with the measured values. They treated the S and d states on an equal footing In order to evaluate the relative contributions of the s and d states to the conductivity. They found that although the diffusivity of d states Is less than that of s states, the d states dominate conductivity because of thelr much greater density. They wrote(29); 2 o= = (DN (Ep) + DyNG(Ee)] (4.2.2.) Where Ne and Ng are the s and d partial densities of states and Do» Dg the respective diffusivity. The sd contribution to the conductivity was found to be neg! Igible For }Ilquid La, Cr, Mn and Fe the d states contribute about 85% of the conductivity. | If this Is also the case In (Mop 6Rup 4) yo9-,8, glasses we can write [n a first approximation: ee o = o DNG(E,) (4.2.3) The average electronic diffusivity D = 1/3vg2 (where v¢ Is the average group velocity at the Fermi surface and &, the average electron mean free path) Is related to the critical fleld gradient -103- by(30): 4kec ~ “Feb Tr “aT (4.2.4) if the electrons Involved In the superconductive palring are also the ones responsible for the conductivity In the normal state, It follows from 4.2.3 and 4.3.4 that: bi9) dHe2 NylE¢) = Gek,c aT 0 (4.2.5) T It as been shown(25) that for (Mop 6Rup.4)109-,8x the critical field gradient does not vary with the concentration to within experimental uncertainty. A plot of the density of states vs the conductivity for different concentration glves, therefore, a stralght line as seen In Flg 4.2.9. An analog relationship Is also observed for @ (ZrgN!)100~yBy metal lic glasses(31), The Increas!ng slope at high dHo2 boron concentration Is assoclated with a general Increase of —q77]_ . 1, These results Indicate that the Mott mode! for s-d scattering Is not appl Icable In these glasses and suggest Instead a d-band conduction. More evidence for d band conduction In metalilc glasses wlll be presented in chapter V. The right explanation of this resistivity anomaly must then be the following: Annealing Induces phase separation Into boron-rich and boron-poor zones. This’ spatial varlation [In the metallold concentration affects the density of states at the Ferm! level Ng(E¢) and thus the conductivity In a d-band conduction picture, -104- 1,3} lif Y/Y 0.97 x=20 0.7} yer? “FO, 66710, 333? 1-vBy 0,5 Ll i i al 0,5 0./ 0,9 1,1 1,3 1,5 o/o. Normal{zed specific heat coefficient vs conductivity — for glasses (ref. 4.2.31). The normallzation constants correspond to x=18 and y=0 respectively. ~105- Fig. 4.2.10 shows the changes In the radial distribution function after 2h. and 11h. annealing at 525°C ang Table 1V.2.2 {ists the parameters obtained from the G(r). Table [V¥s2.2 G(r) parameters after several heat treatments. Alloy Heat Treatment = Ry Ro Ro/Ry C.N. x=14 As quenched 2.79 4.62 1.66 15.3 2h 2.78 4.64 1.67 15.7 1th 2.79 4.59 1.65 15.1 x=22 As quenched 2.76 4.61 1.67 14.1 2h 2.76 4.57 1.66 13.8 1th 2.79 4.63 1.66 14.0 The first band remains almost unchanged. The first peak [In the second band tends to split Into two subpeaks located for x=14 around 4.30A and 4.82A (l.e. centered around the max! mum corresponding to the as quenched condition) and for x=22, around 4.12A and 4.57A (I.e. centered at lower distance than the maximum corresponding to the as quenched condition). The result Is then a broad first peak In the second band for x=14 and a sharp peak wlth a shoulder at smal! R value for x=22. The second peak of the second band Increases slightly In Intensity In both cases, malnly on the large distance side. Such behavior of the second band upon annealing Is unusual for topological relaxation alone(2), ~106- 4 q ma T “T T T 3 (Mog Rug, 4) g6B1y 0 3+ AS QUENCHED 0 3 2u at 525 C 0 ce +t d]H at 525 C ] < (Mog 6RUg, 4) 78899 J 3 5 0 3 f AS QUENCHED J 0 zt 2H AT 525 C 0 aL. i 1 1. dd AT 525 C 0 y 8 12 16 R(A) Fig. 4.2.10 Reduced radtal distribution functilonsfor x=14 and 22 annealed for different times at 525°C, -107- CONCLUSION 1. Primary crystallization of a boron free phase, hcp solid solution of Mo In Ru for high boron concentration glasses and sigma phase MosRusz, which then decomposes at higher temperature Into the hcp Ru for low boron concentration Is responsible for the two first steps In the resistivity drop around the crystalllzatlon temperature, [n the final step, the remaining boronerich amorphous matrix crystallizes Into an fcc boride. 2. The anomalous excess resistivity above the usual I[Inear temperature dependence predicted by the Ziman theory at hlIgh temperature Is explained In terms of phase separation caused by long range diffusion of boron which affects the density of states at the Ferm!I leve! and thus the number of carrfers [nad band conduction model. 3. The changes In the radial distribution function upon anneal Ing at 525°C cannot be explatned by topological relaxation alone, Some types of chemical relaxation probably caused by phase separation must occur as well. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19, -108- REFERENCES M. Balanzat, Scripta Metal 74 (1980) 173-178 and E. Balanzat, C. Malry and J. Hillafret, J. de Phys. 41, C8 (1980) 871- 874, M. Marcus, Acta Metall. 22 (1979) 879. P. Allla, R. Sato Turtelll and F. Vinal, Solid State Comm. 43, 11 (1982) 821 ~824. D. Srolovitz, T. Egam! and V. Vitek, Phys. Rev. 8, 24, 12 (1981) 6936 - 6944, P. Allta, O. Andreone, R. Sato Turtelll and F. Vinal, J. Appl. Phys. 53 (1982) 12. S. Venkataraman, K.V. Reddy, U.N. Virataswaroop, G.V. Rao and A.K. Bhatnagar, J. Non Cryst. Solids Proc. of the 5th Intl. Conf. on Liquid and Amorphous Metals (Aug. 1983) H.S. Chen and S.Y. Chuang, App!. Phys. Lett. 31 (1977) 4. C.P. Chen and D. Turnbull, J. Non Cryst. Sollds 17 (1975) 169. JL. Walter, D.G. Legrand and F.E. Luborsky, Mater. Scl. Eng. 29 (1977) 161. J .Piiler and P. Haasen, Acta Met. 30 (1982) 1. R. Schulz, K. Samwer and W.L. Johnson, J. Non Cryst. Solids, 61 & 62 (1984) 997-1002. P.L. Ross!ter and P. Wells, Phil. Mag. 24 (1971) 425; J. Phys. C: Solld State Phys. 4 (1971). A.J. Hillel, J.T. Edwards and P. Wilkes, Phil. Mag. 32 (1975) 189; Phil. Mag. 35, 5 (1977) 1221-1229. K. Binder and D. Stauffer, Z. Phystk B 24 (1976) 407 -415. Je Mimault, J. Delafond, A. Junqua, A. Naudon and J. Grille, C.C. Kock, D.M. Kroeger, J.S. Lin and J.0. Scarbrough, Phys. Rev. B 27 (1983) 3. M.Mehra, R. Schulz and W.L. Johnson, J. Non Cryst. Solids, 61 & £2 (1984) 859-864, L. Soldner, H. Adrian and G. Saemann-Ischenko, App!. Phys. Lett. (In press). P. Pletrokowsky, Rev. Scl. Instrum. 34 (1963) 445, 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. -109- A. Willlams, PhD Thes!s, California Inst. of Tech., Pasadena Ca. (1981) G.S. CargilI III, Sol. St. Phys. 30 (1375) 227. S. Hopkins and W.L. Johnson, Solld State Comm. 43, 2 (1982) 537. W.L. Johnson and A.R. Wilifams, Phys. Rev. B 20 (1979) 1640. P.J. Cote and L.V. Melsel, Phys. Rev. Lett. 39, 2 (1977) 102; Phys. Rev. B 16, & (1977) 2978. S.R. Nagel, Phys. Rev. B 16, 4 (1977) 1694. K.F. Kelton and F. Spaepen, Proc. 4th Intl. Conf.on Rapidly Quenched Metals (Sendal 1981), eds. T. Masumoto and K. Suzuki, pp. 527-530 H.S. Chen, L.C. KimerlIng, J.M. Poate and W.L. Brown, App!. Phys. Lett. 32(8) (1978) 15. Z. Altountan, Tu Guo-hua and J.0. Strom-O!lsen, J. Appl. Phys., 53(7) (1982) 4755 and Z. Altounlan, Tu Guo-hua and J.0. Strom Olsen (preprint) L.E. Ballentine, S.K. Bose and J.E. Hammerberg, J. Non Cryst. Solftds, 61 & 62 (1984) 1195-1200. E. Helfand and N.R. Werthamer, Phys. Rev. Lett. 13 (1974) 686; Phys. Rev. 147 (1966) 288. A. Mak and W.L. Johnson, Phys. Rev. Lett. 89A (1983) 353, -110- 1¥. RESULTS AND DISCUSSIONS V3 (ZrqoyHf x) 62N138 -111- THE EARLY STAGE OF CRYSTALLIZATION OF (ZryoyHf,ggNisg METALLIC GLASSES ABSTRACT This section reports a directly observable correlation between the chemical short range order and the electrical resistivity In metallic glasses . The phase transition corresponding to the first exotherm observed In a DSC scan on (ZryoyHfyegNi3g IS pecullar In a sense that, contrary to usual metallic glasses, this transition ts associated with an Increase In electrical resistivity and X-ray diffraction taken Just after the DSC peak shows only the broad diffuse band characteristic of the glassy phase. Electrica} resistivity, differential scanning calortmetry, Jow ‘temperature superconducting measurements, htgh angle X-ray’ diffraction, transmisston electron microscopy and Mossbauer spectroscopy are used to study this transition In detall,. 1. INTRODUCTION (Zry_ Hf )eoNigg represents an attractive system In the fleld of metalllc glasses for many reasons. First of all, It can be made amorphous over a very wide range of composition’ !) and contrary to the better known Cu-Zr system It [s more stable and not as sensitive to oxygen contamination. Second, thls glass fs probably the Ideal candidate for the Isomorphic substitution technique(2Z) which consists of gradually replacing Zr by Hf In the amorphous matrix andusing the different X-ray patterns to deconvolute the data Into partial palr -112- distribution functions of the different constituent elements, Zr-Zr, Zr-NI and NieNI. This technique Is based on the assumption that we can replace Zr by Hf without InfluencIng the structure of the glass. In fact Zr and Hf are similar In many aspects, they form a continuous serles of solid solution, they have similar Goldshmitt radif, electronegativity etc.. A study of the stability and the crystallization process of this system for different Zr to Hf ratlo constitutes then a good test for the validity of this technique. Buschow et al.(5) were the first to observe the pecullar behavior assoclated with the first DSC peak In Zr-NI and Hf-NI systems near 38% NI. SiImilar observations were reported recently by Z. Altounlan et ale(1) on the same system. The former authors suggest that the first transition in those amorphous alloys ts a transition within the amorphous state and comprises malnly short range atomic rearrangement. Y.D. Dong et al.‘4) on the other hand claim from X-ray diffraction analysis, that for this composition the first exotherm arftses from the crystallization of NiZra and the second, from crystallization of NiZr from the nickel enriched glass. Summarizing the results of several! authors''r4s9) on ZryNiq-x we conclude that for x>66 (up to 78) a sIngle sharp exotherm arlses from the s!multaneous formation of the equilibrium phases %Zr and NiZrg- For 60<x<66 we observed two distinct exotherms which for x>62=-63 can be Identified as crystallization of first NiZrg then NiZr. For x<60 (down to 50) Initial crystallization of ZrNI followed almost Instantaneously by the crystallization of ZrgNl from the Zr enriched glass explains the asymmetric single DSC peak, with a shoulder at high temperature observed In this range. Between 60 and 62-635 the nature of the first -113- exotherm changes from crystallization of NiZr to crystallization of NIiZrg and It Is In this range that the high angle X-ray pattern remains broad after the exotherm without any resolvable Bragg peaks. It is the purpose of this section to study this pecullar transition [hn thls range of concentration. EXPER | MENTAL The Ingots were prepared by rf [Induction melting the appropriate amount of each constituents on a silver boat [n an argon atmosphere. They were melted several times to ensure homogenelty. Glassy ribbons of typically 30:m In thickness, 2mm In width and several feet long were prepared by melt-spinning. They were examined by high angle X-ray diffraction using Cuk, radiation. Only the samples showlng a broad band character!stic of amorphous samples were kept for further study. For each composition al! measurements were done on the same ribbon to eliminate fluctuations In the free volume content. The electrical resistivity was measured using a four point de method with pressure contact. The probe Is free to move on a vertical axIs Inside of aquartz tube, which goes through a furnace before belng flxed at the top of a dewar. The system can achleve very fast heating or coollng rates. The measurements were done Ina slight over-pressure of hellum. Other detalls are described tn'®). The DSC traces were obtalned using a Perkin -Elmer DSC-! under argon atmosphere at a heating rate of 20°C/min. Superconductivity down to around 1.6K was studied through changes In the electrical resistance. The eritical fleld was determined by plotting the resistance of the sample as a function of the magnetic fleld at a constant temperature malntalned using a pressure control valve over the He bath. The temperature -114- was calculated using the vapor pressure of the helium. Samples for X-ray diffraction were prepared by stacking three layers of ribbons together with Duco cement on a glass slide. The data were obtalned In the reflection geometry on a Norelco Scanning Gonlometer using Mok, radiation and focuss{ng LIF monocromator. The detalls for data correction and RDF calculations were described elsewhere{7). For transmission electron microscopy the specimens were thinned by Jet pollshing using 80% methanol and 20% perchloric acid at -15°C. Samples for Mossbauer spectroscopy were prepared by Introducing 0.2% of enriched Fe’ In Zr-NI as a probe element. The experiments were performed at roon temperature on folls obtaln by the "piston and anvil" technique. The source used was Co?7(Rh) and the experiments were done [In the constant acceleration mode. RESULTS AND DISCUSSION Fig.4.3.1 shows the DSC traces and the changes In resistance as we heat the samples through the crystallization temperature at 15K/min. The DSC data were corrected for the difference In heating rate. Table IV.3.! gives the temperatures corresponding to the onset of the two exotherms and the corresponding relative changes In electrical resistance for each transition, -115- 1,15 q mi T “Tt T ZReoNiz9 0.901 J | nN s< is 1.20 — 4 (HesgZr5g) 62N138 2 = o > <= 9,95} > faa < e a = 1,30 + S LL HF goNi3zg i 1,05 } ve us 500 700 900 1100 TEMPERATURE (K) Fig. 4.3.1 Res!tstance and DSC traces on ZreoNlzg> (Hf 52Zro 5) 6oNl zg and Hf poNizge -116- TABLE JV.3.1 Temperatures of the exotherms and the relative change In electrica! resistance for each transItion. ALLOYS THK) $ T2(K) $ ZrEQNI 3g 675 +6.2 700 ~16.7 (Hfg 5270 ,.5)62N138 730 +7.8 750 + 6.5 Hf oNI zg 775 +845 825 +16.8 We see In this table that Hf Increases the stabllIty of the glass by moving the peaks position to higher temperatures. Generally, the resistance decreases on crystallization sInce the crystalline state has much greater atomic order. In our case however, the first transition 1s accompanied by an Increase of resIstivity by roughly the same amount In all three alloys. An Increase of resistivity upon crystallization has been seen also In few other systems: CuZr In the vicinity of Cuyozr7(8), ZrsRh and ZrzFe'9). Altountan et al. (8) explained this behavior fram the fact that the density of states at the Ferm! level Increases upon crystal{ization which [mpltes an Increase of resIstivity within the framework of the s-d scattering model. Large changes In the resistivity can be observed after the second exotherm, the resIstance decreases by 17 In the case of ZreoNizg and Increases by 17% In the case of HfgoNIzg.- Bragg peaks corresponding to the equilibrium phases ZrNf and Zr2Ni or HfNi_ and HfONi can now be observed from high angle X-ray scattering contrary +o samples quenched Just after the first exotherm. This farge difference In behavior between the Zr and the Hf based alloys brings some doubts on the validity of the IsomorphIc substitution technique In this system. -117- Flg. 4.3.2a shows an enlargement of the resistivity scan below the crystallization temperature. Below 400-450 K the resistivity varles, roughly, [Inearly with the temperature, In agreement with the ZItman-Faber theory which predicts Iinear dependence of § the resistivity (10,11) above the Debye temperature, due to phonon scattering. The |inear dependence Is represented on the figure by the dashed IIne for the case of ZreoNizg. Above 400-450K a departure from the I[Inear behavior can be observed In all three cases. Fig. 4.3.26 represents the excess resistivity above the phonon contribution. Such behavior has been seen In a number of metallic glasses(!2,13,14), Je Hillatret et al.{15) betteve that the Increase In resistivity reflects an evolution towards an "equilibrium state" of chemical short range order. They propose the existence of ordered domalns but they do not specify the exact nature of this equilibrium state. Based on an original Idea from Mott(16), pol Duwez et al.(17) suggest that mlcrocrystallIine precipitates are responsible for the additional Increase In resistivity above the I fnear Increase [n amorphous Fe-P=C. Those precipitates would act as additional scattering centers for the conduction electrons In a manner similar to that found In the early stage of Guilnier-Preston zones formation In supersaturated crystalline solld solutions. In many cases however, microcrystalline precipitates can not be found and al! those Interpretations then become speculation due to the lack of structural measurements coupled to the resistivity behavior. Recently, two of us (18) showed, that the excess resIstivity observed In (Mog.6Rud.4)100-»8x Is correlated with the formation of domains of high and tow boron concentration within the amorphous -118- 1 ' 02 4 me i i t qt i fo ~ a She te 7 o he ~. a : sen, Pe ; “te, ed : 3 ee . ed (1) ZReNIz9 2.07 (2) (HF EQZR5Q) GON I 39 aaa en ae | * Lh b 250 450 650 850 TEMPERATURE (K) Fig. 4.3.2a & b Enlargement of the res|stance traces below the crystallization temperature. The dashed |Ine represents the extrapolated | Inear dependence at low temperature. Excess resistivity above the linear dependence obtalned from a least square fit of the data at low temperature. -119- matrix. {nm CusgZr5p we Showed that the anomalous resistivity behavior Is due to phase separation [nto Cu rich and Cu poor zones of concentration close to CujoZr7 and Cuzr2 respectively(!9), From low temperature specific heat Kroeger et al.{20) observed phase separation and chemical short range ordering In Zr-NI glasses between 60 and 66 AtZ Zr. They have found In a glass of 61.2 At% Zr, show!Ing no evidence of phase separation as quenched, that annealing one hour at 200°C Induced phase separation Into two phases of concentration near ZraNio and Zr2Nli. A heat treatment as low as 150°C for 30 min, produced great changes [In the volume fraction of these two phases In a glass of 62.9 At% Zr which was already phase separated as-cast. The lowest anneal tng temperature at which they report phase separation ("150°C or 423K) cotncides with the onset of the resistivity Increase In fig. 4.3.2. We then conclude that the excess resistivity observed In (Zr4~,Hf x) 62N138 Is caused by phase separation. We also see In flg. 4.3.2b that Hf based glasses are more stable against phase separation Into HfzNiz and HfoNI than Zr based glasses. In order to study the structural transformation which takes place during the first exotherm, we heat the samples to different locations within the crystallization range at a heating rate of 15°C/min. then quench raplidly to room temperature. The full lines In flg. 4.3.3 represent the radial distribution functions for as quenched a) ZrgoNlag b) (Hf 529 5) 62Ni zg and c) HfgoNizg. The results are very sImllar to the ones reported by Wagner and tee(21) on (Zr 4~yHf x 65N135¢ The reduced radtal distribution function exhibits a split first peak. The first maximum at r1=2,69h decreases In helght with Increasing Hf content and corresponds to Ni-(Zr,Hf) nelghbors. -120- The second at ry '=3.20A becomes larger In helght with Increasing Hf concentration and It corresponds to Zr-Zr nelghbor pairs. The dashed IInes show the reduced radial distribution function after the heat treatment represented In the onset of each figure. For ZreoNizg (fig. 4.3.3a) the sample has been quenched Just at the onset of the second exotherm (or at the onset of the resistivity drop). The helght of the first peak Is reduced by about 13% which Indicates a reduction In the number of unlike nelghbors (Zr-NI) and the second peak Increases by about 6% reflecting an Increase In the number of Zr-Zr nelghbors. (Zro,5Hfo.5)62NI3g (fig. 4.3.5b) has been quenched tn between the two exotherms or right at the maximum of the first resistivity Increase. The changes In the first band are sImilar to the previous case (reduction In the first peak by "15% and Increase of the second peak by "5%) which strongly suggests that Hf behaves like Zr during the first phase transition. Finally we quenched Hf gE oNI 38 at the onset of the first exotherm. The two sub-peaks of the maln_ band ere now better resolved compared to the as quenched sample and the respective helght of each peak remains unchanged contrary to the previous cases. The shoulder at large r of the second band Is also better defined and slightly more Intense. Those are typical of topological relaxation and do not reflect any change In the chemical short range order. We then conclude that the Increase of resIistIvity during the first exotherm Is due to a reduction In a number or Zr-Ni nelghbors and an Increase In the number or Zr-Zr nelghbors. In ZresNiz5, Wagner et at.'21) found a Warren chemical short-range order par ameter of -0.04 fndicating a slight preference for unlike nelghbors In the first coordination shell. The follow!ng values were -121- R (AncstRoms) Fig. 4.3.3 Radial distribution function before (full lines) and after (dashed tines) the heat treatment represented In the onset of each figure for (a) ZreoNizg (bd) (Hfg 5Zro s)eoNlzg and -122- obtained for the coordination numbers: N(NI-NI)=3.1, N(NI-Zr)=8.6 and N(Zr=-Zr)=9.2. Roughly speaking our results suggest that the number of Zr-NI nelghbors should decrease to around 7.4 and the number of Zr-Zr Increase to around 9.7. This study could actually provide an Interesting way to test the applicability of the Faber-Ziman theory In this amorphous alloy. The resistivity In Its most general form can be written as(23), _ 2 2 p= B{cy <|t,| a> t Cy <|to| aoo> + (4.3.1) C9 (<|t |2(1-a,5)>+<|t, |? (may )ote(ty tootyty) (95-1 )>)} ¢; and cg are the concentrations of the two constituents, +; and +2 are the single site scatter!ng matrix, the ats are the various partial structure factors, 8 depends on the average atomic volume, the electron density and the Ferm! velocity and the average <f(K)> means'29); 2k f sins f #(K)K7dK (4.3.2) f°) UstIng the Isomorphic substitution technique to measure the partial structure factor before and after the first exotherm and the tematrices for NI(24) and Zr(25) we can write In the lImlt of a constant prefactor: e ° fo cf <lty [70a 4> + 3<[tz| "290° + eye patty ty-tyty bay} (4.3.3) This approach Is presently under Investigation. -123- Fig. 4.3.4 shows the superconducting transition In ZrggNi3g 5 measured by electrical resIstance changes at a temperature of 1.6K for different applled magnetic fields. The as quenched sample shows a very broad transition which has been seen before In other Zr _ based metallic glasses'26), [+ was suggested that such behavior, coupled with other superconducting anomalles |Ilke the enhancement of Hea at low temperature, may be due to spatial Inhomogene!ties on a scale of the order of the superconducting coherence lenght{27), They proposed the existence of a distribution of electrontc diffusivity D=1/3v¢% due to varfations [In short-range order or to fluctuations In composition, This assumption would be In agreement with Kroeger et al.(9) who proposed the ex!stence of clusters of ZrsNl and ZroNI together wlth unassoclated atoms In the amorphous matrix. After the heat treatment beyond the first exotherm, we observe a= sharp transition characterIstic of homogenous phases, even though the heat treated sample shows clearly In flg. 4.3.5 two distinct phases. The upper critical flefd Indlcated by the arrows In flg. 4.3.4, Is shown vs the temperature In flg. 4.3.5. The anomalous [Inear HeoolT) behavior Is noticeable In the as quenched sample since curvature should be observable for the lowest measured temperatures. dH 2 Table IV.3.2 gives the T,, critica! field gradient aT and T, the average electronic diffusivity De1/3v,* ( 21s the average electronic mean free path, v¢ Is the average Ferm! velocity) for the different phases using the relation(28), dH» 4kpc dT (4.3.4) * ned 1, ~124- ZReoNi zg 1,0f _— J AS QUENCHED i ‘s 0 e 5 P “ bs = ae 2 a He, 8 —— co 1 s 0 r “ oe ‘ M™~™ = : = 0.5} : HEAT TREATED / S . Hc, TEMPERATURE 1,60 K 0 10 20 30 40 H (KOe) Fig. 4.3.4 Superconducting transition measured resistively for Z2rgoNlzg 4s quenched and heat treated beyond the first exotherm. -125- 25 qT ' T qv T 4 7 qT ZReoN 139 20 fF 4 ~ 15+ AS QUENCHED | S oY 10} ; HEAT Sf TREATED 1 1,5 1.8 2.1 2.4 TEMPERATURE (K) Fig. 4.3.5 Upper critical fleld vs temperature for ZreoNlag as quenched and heat treated beyond the first exotherm at a heating rate of 15°C/mIn. -126- Where c Is the speed of light, kp Is the Boltzmann constant and e Is the electronic charge. Table 1V.3.2 Superconductivity transitions, critical fleld gradients and average electronic diffusivities for as quenched and heat treated ZreoNizg- dH a2 2 PHASE TQ (K) “at {T, D(cm“/s) (KG/K) As quenched 2.36 28.6 0.384 Phase 1 1.92 25.5 0.430 Heat treated Phase 2 2.09 8.0 1.371 The Tc of the as quenched sample Is slIghtly lower than the one reported by Kroeger et al.") for the same composition (2.50K). Many possible reasons can account for this discrepancy: 1) The state of relaxation of the glass. We know that T, decreases upon free volume relaxation and we use melt-spun rlbbons whereas Kroeger's specimens were quenched by arc-hammer. 2) The difference In the measur!ng technique, Our result comes from the extrapolation of the Hog vsT curve whereas, Kroeger's result comes from specific heat. Finally, 3) we cannot omit the possIbIlIty of a slight offset In the concentration of the alloy. After heat treatment, phase 1, with the Te of 1.92K has almost the same critical fleld gradient or average electronic diffusivity as the amorphous as quenched sample whereas, phase 2, with the T, of 2.09K has an average electronic diffusivity more than three times larger. We thus consider phase 1 as the remalning amorphous state after the phase transition. From a plot of T. vs compos! tion’? -127- we evaluated hIs concentration at "56 At% Zr. It Is worth mentioning that this concentration corresponds to the maximum In the resistivity vs concentration curve for the Zr-NI system(29), The resistivity of ZrssNlq5 1s 185yQem whereas the Initial amorphous ZrgoNizg Is 174u00m''s29), This represents an Increase of 6.3% which 1s exactly the Jump helght observed In the resistivity dur{ng the first exotherm. This correlation suggests that phase 1 dominates the transport properties at high temperature. The resistivity would seem to be unaffected by the relatively high conducting phase 2 even though phase 2 Is observable at low temperature from superconducting measurements monitored by resistance changes. Fig. 4.3.6a and b show transmission electron micrographs for a heat treated Zrg2Ni3zg- The pictures show a distribution of dark speckles ranging from 10 to 30A tn diameter. Agglomeration of these speckles produces the dark patches (average slze 100-200A) tn the micrograph . Fig. 4.3.7a shows the electron diffraction pattern of a selected area of flg. 4.3.6a free of pits and holes. The pattern shows only a broad single halo and from It, there Is no evidence for phase separation or crystallization. When the area under Investigation Includes holes or large pIlts we can detect crystals In the diffraction pattern, fig. 4.3.7b. This reflects probably the presence of easy crystailfzation sites for heterogenous nucleation or oxide flims. J.L. Walter et al, (50) noticed oxide films at the bottom of deep pits In thelr thinned TEM samples of ZroNl. Fig. 4.3.6 (a) and (b) TEM micrographs -128- of heat treated -129- Flg. 4.3.7 Diffraction pattern of heat treated ZreoNizg (a) selected area In flg. 4.3.6 containing no pits and holes (b) area Including holes and sample's edges. -130- In order to Investigate the local short range order of this new phase we Introduce Fe as a probe element In the matrix and perform Mossbauer exper!tments. Fig. 4.3.8 shows spectra of (Zrg2Nl3g)99.8Fe0.2 In the as quenched condition, heat treated beyond the first exotherm and fully crystallized after the second. The spectra were analyzed usIng least square fit of multiple Lorentzian lines with varlable helght and width. Table 1V.3.3 gives the average quadrupole splitting corresponding to the different sites. Our ‘results are very similar to the ones reported by Wagner et al.(31) on Zrg5-yNI,Fes These authors discuss the similarities between the Mossbauer parameters of the stable crystalline phase (A Eq = 0.537) which, as In the present case, Is mafnly tetragonal ZroNI(Fe) (CuAl2 type structure) and the average parameters observed In amorphous Zr-Ni-Fe (A Eq =0.608). They suggest that the average Fe environment In the glass might be IIike that of Fe In crystalline ZroNi(Fe). This would be [n agreement with the presence of ZrgNi type associates In the amorphous state as suggested by Kroeger et al. (>), In this environment each Fe atom [s surrounded by Zr atoms only, with the average coordination of 8 Zr atoms arranged In an archimedian antiprism capped by 2 Fe atoms. The Zr sites have 11 Zr and 4 Fe atoms as first nelghbors. The heat treated sample shows two very distinct sItes. Site 1 has almost the same average quadrupole splitting as In the as quenched sample. This site corresponds to phase 1 In Table 1V¥.3.2 and represents the NI enriched amorphous matrix. Site 2, which must be representative of phase 2 In table 1V.3.2 has a relatively large quadrupole splitting (0.93!mm/s). This value Is almost Identical to the one reported by Vincze et al.{32) for TRANSMISSION CarBITRARY UNITS) -131- Premaonatnarnan | FOO a ones sen een mend ~\ Ps AS QUENCHED rn ar Phang, r , fee . | 3 HEAT TREATED . * ry 4 i. é wo ea PAPA ELLER LAP LN poet pogo meee ——_ - % f CRYSTALLINE - , em F ; of £7 ¥ «© (ZRE NI 3g)g9, gFEQ, 2 VELOCITY (mm/s) Fig. 4.3.8 Mossbauer spectra of (Zr Ni zg)o9,gF eg. 2 as quenched, heat treated beyond the first exotherm and fully crystallized after the second exotherm. ~132- Fe In orthorhombic ZrzFe (Re38 type). If Fe substitutes for NI In the amorphous matrix and behaves [n the same manner as NI during the phase transition associated with the first exotherm, the above results would suggest the existence of a new metastable phase Zr3ani orthorhombic. In this environment the Fe site has 6 Zr atoms first nelghbors and 3 Fe atoms somewhat further, while the two Zr sites have 2 Fe and 12 Zr In one case, 3 Fe and 11 Zr [n the other case. This structure has a lower number of Zr-Ni(Fe) nelghbors and a larger number of Zr-Zr nelghbors than the one found by Wagner and Lee'21) for amorphous ZresNizg5. Precipitation of an orthorhombIc ZraNi(Fe) would then be consistent with the reduction In the number of Zr-NI and the Increase In the number of Zr-Zr observed In the RDF after heating beyond the first exotherm. Table 1V¥.3.5 Quadrupole splitting for different sites In (2rp 6oNIo. 3899, 8Feq.2 alloys. ALLOY'S TREATMENT QUADRUPOLE SPLITTING (mm/s) REMARK As quenched 0.608 Fit to two Lorentzlans Heat 1 stte 0.615 Fit to four Treated 2 site 0.931 Lorentzlans Crystalline 0.537 Fit to two Lorentzlans -133- CONCLUSION The phase transition which occurs during the first exotherm In (Zry_ HfL) goNi 39 metallic glasses implies precipitation of very smal] "mi crocrystal lites" (10-20A slze) too smal! to give rise to sharp rings In the diffraction pattern. The overall number of Zr-Ni nelghbors decreases and the number of Zr-Zr nelghbors Increases. The resistivity [Increases because the NI enrfched amorphous matrix that remains after precipitation has a higher electrical resis+ivity than the orlg!inal amorphous phase, thus overcompensating the Influence of the highly conducting precipitates. Agglomeration of the "microcrystallites" Into zones larger than the superconducting coherence length glves rise to two distinct observable phases from the He2(T) curve, The Mossbauer results on samples with Fe Introduced as a probe element might suggest that precipitation of metastable very fine grained microcrystalline ZrzNi(Fe) is occurring during the first exotherm. 1. 2. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19, 20. -134- REFERENCES Z. Altountan, Tuo Guo-hua and J.0. Strom Olsen (preprint). O. Chipman, L. Jennings and B.C. Giessen, Bull. Am Phys. Soc. 23 (1978) 467. K.H.J. Buschow, B.H. Verbeek and A.G. Dirks, J. Phys. D j4 (1981) 1087 - 97. Y.D. Dong, G. Gregan and M,C. Scott, J. Non Cryst Solids, 43 (1981) 403 = 415. D.M. Kroeger, C.C. Koch, C.G. McKamey and J. 0. Scarbrough (preprint) M. Mehra, R. Schulz and W.L. Johnson, J. Non Cryst. Solids, 61 & 62 (1984) 859 = 864, G.S. Cargi!! Ill, Sol. St. Phys. 30 (1975) 227, and A.WIilIlams, PhD Thesis, Callfornila Inst. of Tech., Pasadena Ca. (1981). Z. Altountan, Tu Guomhua and J. 0. Strom Olsen J. Appl. Phys. 53 (1982) 7. R. Schulz (unpubl ished results). P.J. Cote and L.V. Melsel, Phys. Rev. Lett., 39 (1977) 102-105; Phys. Rev. B 15 (1977) 2970; 16 (1977) 2978, S.R. Nagel, Phys. Rev. B 16, 4 (1977) 1694. H.S. Chen and D. Turnbull, J. Chem, Phys. 48 (1968) 2560. P.L. Maltreplerre, J. Appl. Phys. 41 (1970) 267. E. Balanzat, J. Hillalret, J. Phys. F 12 (1978) 2907. J. Hillatret, Emmanuel Balanzat, N. Derradji and A. Chamberod, J. Non Cryst. Sollds 61 & 62 (1984) 781 - 786. N.F. Mott, J. Inst. Metals 60 (1937) 267. P.K. Rastogi! and Pol Duwez, J. Non Cryst. Solids 3 (1970) 1 = 16. R. Schulz, M. Mehra and W.L. Johnson (In press). R. Schulz, K. Samwer and W.L. Johnson, J. of Non Cryst. Sollds 61 & 62 (1983) 997 = 1002. D. M. Kroeger, C.C. Koch, J.0. Scarbrough and C.G. McKamey Phys, Rev. B (In press). 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. -135- C.N.J. Wagner and D. Lee, J. de Phys, C8 41 (1980) 242-245, D. Srolovitz, T. Egam!, and V. Vitek, Phys. Rev. B 24 (1981) 6936 - 6944, T.E. Faber and J.M. Ziman, Philos. Mag 11 (1965) 153. 0. Drefrach, R.Evans, H.J. Guntherodt and J-U Kunzl, J. Phys. F2 (1972) 709. Y. Waseda and H.S. Chen, Phys. Status. Solid! A 49 (1978) 387. M.G. Karkut R.R. Hake, Phys. Rev. (1983) (In press). W.L. Carter, S.J. Poon, G.W. Hull Jr., and T.H. Geballe Solid State Comm, 39 (1981) 41. E. Helfand and N.R. Werthamer, Phys. Rev. Lett. 13 (1974) 686; Phys. Rev. 147 (1966) 288. Z. Altountan, C.L. Folles, W.B. Mulr and J.0. Strom-Olsen, Phys. Rev. B 27 (1983) 1955 = 1958, JL. Walter, Z. Altountan and J.0O. Strom-Olsen, J. Non Cryst. Solids 61 & 62 (1984) 469 - 474, H.G. Wagner, M. Ghafarl, H.P. Klefn, U. Gonser, J. Non Cryst. Solids 61 & 62 (1984) 427 = 432, I. Vincze, F. Van der Woude and M.G. Scott, Solid State Comm., 37 (1981) 567-570. -136- V. EVIDENCE FOR d-BAND CONDUCTION IN METALLIC GLASSES {In this chapter a discussion will be given on the electrical resistivity of metallic glasses containing transition metal elements, The case of (M09 6RUg 4) 100-8 xe Zr 1 00-xN! y and Zr 199 -xCUy» which were discussed [n chapter !V, will be treated In detall. These systems are characterlzed by strong scattering and have a conventional mean free path on the order of the Interatomic spacing. In this limit, as discussed before, the applicability of the Boltzmann formalism Is doubtful and therefore the Ziman theory should be applied with caution. Most of the theoretical understanding of the electrical transport In metallic glasses, Involving transition metal elements, Is based on the extended ZIman theory {14+18,58) | In this theory, Evans et a}, (18) reinterpret Mott's classic explanation of the high resistivity of crystalline transition metals'®) and apply It to [Iquid transition metals using the ZIman formalism originally developed for simple liquid metals. First, we shall then briefly review Mott's theory. Transition metals are characterized by a high density of d states at the Ferm! level Ng(E,). Some parts of the Ferm! surface are s-IIke and some parts d-Ifke. The total potential (phonon, Impurities...) Induces transitions from the s-IIke part of the Ferm! surface to the d-like part with a high probability since the rate Is_ proportional to the density of the final states, N,(Ey). The relaxation time of an s electron In transition metals Is then much less than an s electron In s!tmple metals because of the absence of the empty d states at the Fermi level In thislattercase. The relaxation time Ty for the d-IIke part of the Ferm! surface Is comparable wlth Te because It -137- Implles malnly d=-d scattering for which the probabllity of transition ts also proportional to NgfEe)- Fluctuations In the distance between nelghboring atoms wlll Induce changes [n the width of the d=band by modifying the energy overlap Integrals, I, In the tight binding approximation. These perturbations will affect the "hopping frequency" of the electrons towards the nearby sites and represent the starting potnt for calculating the d-d scatter ing660.61), The conductivity of a crystalline transition metal, In thls two band model, can be separated Into an s part and ad part. c= Oo + 0, The velocity v, of an s electron at the Ferm! surface Is much greater than vy therefore Its mean free path Re = Ve Te IS also much larger than the mean free path of a d electron. It follows that: > Os 9g In the generallzed Ziman theory Evans et al.''8) considered the current to be carrled by free electrons In plane wave states. These waves are scattered resonantly on the Fermi sphere by a "“muffin-tin" potential. They replaced [In the original Ziman formula the weak pseudopotential by a tematrix. For transition metals, the d-phase shift "ys dominates and therefore, In this plcture, the resistivity comes from the d resonant scattering of the conduction electrons. These electrons are assumed to be s-like, the d electrons are considered to be Immoblle. -138- Let us write the s-d matrIx element as: * ve fly v tg where represents the conduction electron wave function made up Ys, p of s and p orbitals, and V Is the total potential. If, as before | represents the energy overlap Integrals between adjacent atoms, Mott '62) potnted out that the Evens theory Is valld If and only If the time taken for an electron to move from a d state In one atom toa d state In an adjacent atom, h/zl, (where z Is the coordination number ) Is larger than the Jump time from ans toad state, h/J. If this condition Is not fullfilled, the resonance required by the Evans theory cannot be bullt up. For this reason Mott belleves that thelr theory Is Incorrect for pure liquid transition metals because of the large coordination number and overlap Integral (62), In amorphous transition metals alloys, for which the average coordination number can be even farger, the same argument should apply and amode!l Involving s-d transitions and different mean free paths for s-p and d electrons should be more appropriate. Amorphous metals alloys are known to approach the extreme dirty IImit. The scattering Is so strong that the conventional mean free paths of both s and d electrons are on the same order of magnitude as the Interatomic spacing. In this Uimit of Inclplent Anderson localization, perturbation theory and the Bo! tzmann-equation formal ism break down, and the conductivity must be evaluated using the Kubo formula. In this extreme regime, the electrons should be thought of as diffusing around rather slowly or even as hopping between quas! localized states rather than belng occasionally scattered between -139- nearly coherent plane-wave states. GIrvin and Jonson‘®3) showed that the adlabatic-phonon assumption also falls In this [ImIlt and phonons actuslly assist the moblIlity, producing an anomalous negative temperature coefficient of resistivity. Such negative temperature coefficient [Is observed In all the samples discussed In thls thesls, This phenomenon Is known to occur for localized states In the form of phonon=assIsted hopping but It actually occurs even before localization Is reached. Recently R.W. Cochrane’ et al. (6) reinterpreted the low temperature resistivity anomaly found [In several amorphous metals as belng a precursor effect of a localization transition tn a three-dimensfonal Interacting electron system. They found that the conductivity scales wlth the ones observed In other systems having typical resistivity an order of magnitude larger, and In which, metal-Insulator transition Is found. Ustng the Kubo formaltsm, Ballentine et al. 664) have calculated, for clusters of 365 atoms, the conductivities of several Iiquid transition metals. They treated the s and d states on an equal footing and evaluated thelr relative contributfons to the conductivity. They found that the sd contribution to the conductiv! ty Is usually very small and thus wrote: 2 e oF (0,565) + DgNa(Es) | (5.1) N (Ez) and N{CE;) are the s and d partial densities of states and D, and Dy are the respective electronic diffusivity. They found that for liquid La the d states comprises 90% of the density of states at the Ferm! energy and contributes to 87% of the conductivity. Including hybridization, a diffusivity ratio D./Dg of only 1.1 was found -140- Indicating very little dynamical difference between the s-!Ike and the d-ltke states at the Ferm! energy. Even {tf the diffusivity ratio Is much larger for other IIquid transition metals |ike Cr, Mn and Fe the d states are still responsible for about 85% of the conductivity because of the very large d component of the density of states. Ina first approximation for those liquid transition metals we have: ef of DyNglEe) (5.2) Let us now return to metallic glasses and let us first cons!der the case of (Mop 6Rug 4) 199-.8, glasses which has been extensively studied In the literature 65-67) and also discussed In chapter IV. Fig. 4.2.9 In chapter IV showed a linear relationship between the conductivity and the density of states at the Ferm! level for (Mog 6RUg 4) 100-28 x across the whole concentration range for which the glass can be made amorphous. The values of Y were obtalned from calorImetric measurements’©5) and the data were normalized to the respective values for boron 18. We observe In thls figure a straight positive slope. The s-d scattering model would predict an opposite behavior (l.e. an Increase of resistivity with an Increase [n the density of d states at the Ferm! fevel) If the number of free carriers Is constant fn thls range of concentration. Fig. 5.1 shows the upper critical fleld gradient which fs related to the average electronic diffusivity py’68), dHe9 4kpc (5.3) -141- x Boron CONCENTRATION (AT,A) 13 15 17 193 21 23 25 30 q ' T 7 T (Mog | 6RUp , u?1-xBx 25 + ; 4 . 5 a 2) Se; o = 30 : NS 25 7% (289 66710, 333)1-yBy 25 t+ J 20 i. wf i t t 0 4 8 12 16 20 24 Y Boron CONCENTRATION (AT, A) Flg. 5.1 Upper critical fleld gradient vs concentration for Mo-Ru-B and Zr-NI-B systems. ~142- This upper critical fleld gradient Is Independent of the boron concentration for the Mo-Ru-B system. Based on an abrupt discontinulty In the slope of the resistivity vs concentration at x=18, Johnson et al. 665) proposed the existence of two different types of defects with two different scattering mechanisms for x<18 and x>18, Figs. 4.2.9 and 5.1 suggest, that only one mechanism Is valid over the whole concentration range. Because the density of d states at the Fermi level Is fairly large In Mo-Ru-B glasses (Nj(E¢) = 1.57 states/ev-atom for x=14), the average electronic diffusivity represents basically the d state diffusivity (as long as the D./Dy ratio Is relatively small Ifke In the case of IIquid La). If eq. 5.2 ts valid for Mo-Ru-B, and If D=Dy (5.4) It follows from 5.2, 5.3 and 5.4 that dH - m "C2 NglEe) = Tekge OT |, ]° (5.5) Cc This equation Is basically analogue to the one predicted by the GInsburg-Landau-Abrikosov-Gor'kov theory for the density of states In the extreme dirty IIm!it. The slope of the density of states vs conductivity Is then proportional to the upper critical fleld gradient dH c2 at T.. For (Mon 6RUp 4) 100-2 x2 aT Is constant and we T Cc observe a linear relationshIp between N, and 9 across the whole concentration range. The least square flit between the densIty of -143- statesand the conductivity data corrected by the average atomic volume at each composition, glves an upper critical fleld gradient of 24.03 KG/°K, In good agreement with the experimental average data of 24.21 KG/°K. Fig. 5.1 shows that for (ZrgNI),9g-yB, glasses, the upper critical fleld gradient increases IInearly with concentration from 23 KG/°K for y=5 to about 30 KG/°K for y=25. This gives rise to the curvature observed In flg. 4.2.9 for the Zr-NI-B system. The data, In thts case, are from reference 69 and they are normalized to the density of states and conductivity of amorphous ZroNlI. Among all the metallic glasses of the type ET;g9_,LT, where ET refers to an early transition metal and LT a late transition metal the best known are Zr-Cu and Zr=NI. Based on the strong correlations which exist between the fractional change In magnetic susceptibilIty and resistivity upon crystallization Z. Altounian et a}. (70,71) suggest that s-d scattering effects make an Important contribution to the resistivity In both Zr-Cu and Zr-NI. This suggestion Is based on the assumptions that the Boltzmann equation and the weak scattering approximation are valid In the amorphous state as well as In _ the crystalline state. It assumes also that the s-d scattering matrix element Is the same In the two phases. There are no a prior! reasons to belleve that these assumptions are true, and In fact, we showed In chapter IV that even If the change In the resistivity Is correlated with the change In the density of d states during crystallization In an (Moo 6RUp 482818 glass, they behave In opposite direction upon relaxation In the amorphous state (i.e. an Increase In resistivity Is associated with a decrease In the density of states). Altounlan et al.(72) also showed recently that thelr resistivity data In -144- glassy Zr-NI are consistent with the Faber-Ziman model for liquid metals and attributed the discrepancles to errors Introduced by the simplifying assumptions made In applylIng the model. Kroeger et al. (76), on the other hand, explain some pecullarities [n the variation of Altountan's resistivity data for Zr concentration larger than 55 by the presence of clusters with definite stoichiometries. Flg. 5.2 shows a plot of the density of states vs the conductivity for the Zr-NI system. The data are normalized to the respective values for x=55, For Zr concentration larger than 55, fig. 5.3 shows that the upper critical fleld gradient and, therefore the average electronic diffusivity (from eq. 5.3) Is Independent of concentration to within experfmental uncertalnty. We thus observe, as for the Mo-Ru-B case, a linear relationship between the densIty of states and the conductivity for x>55 (flg. 5.2). The destates seem to donfnate the conductivity In this range of concentration. For x<55, the critical fleld gradient decreases rapidly with decreasing Zr concentration, Indicating a rapid Increase In the average electronic diffusivity (flg. 5.3). The conductivity Increases while the density of states at the Ferm! level decreases. In this regime (x<55) flag. 5.2 shows that, In fact, If we plot the density of states vs the resistivity Instead of the conductivity, we get a IInear relationshIp. The resistivity Is directly proportional to the d density of states suggesting that Mott s-d scattering of nearly free electrons Is predominant In this range of concentration. In this regime 5g << %, and equation 5.1 becomes: s 2 e -145- O/O(x = 55) 0.98 1,02 1,06 1,12 116 1,20 1.6 “T T T T T T T T T T NT100-x2Rx . J 1.4 (71) X (69) J VE (63.5) ™ CONDUCTIVITY 1.2 0. 7 ~ (60) tA ; tl a 1.0 B 7 Zz VERSUS ~ i RESISTIVITY ; Zz \ (45) 0.3 : 0.6 f+ (36.3) 1 (33) (33) X FROM D. M, KROEGER ET AL 7 O FRom Z. ALTOUNIAN ET AL 0.4 _t | l L _t 1 1 1 t at. 0,90 0,92 0,94 0,97 0,99 1,01 P/ P(x = 55) Fig. 5.2 Normalized density of states vs conductivity (or res!stivity x>55) for the Ni-Zr system. parenthes!s. x=55. for The concentrations are Indlcated In The data are normalized to the respective values for -146- 34 T T T T T is! 6B =O) ° 0 0 0 261 | N17_,ZRy ¥ a} S & ~ 325 1 a 5 |e 26+ | 20 i l J i t 30 40 50 60 70 80 90 ZR CONCENTRATION (AT. &) Fig. 5.3 Upper critical fleld gradient vs concentration for Ni-Zr and Cu- Zr systems. -147- No represents the density of s electrons or free carrlers and De Is the s state diffusivity which Is proportional to the mean free path ds Altounlan et al. (72), In Zr-Ni the ration Z/2 (where Z Is the valence or the lifetime of the carriers 1. As It was pointed out by and 2 Is the atomic volume) Is almost the same for both constituents so kg and the number of free carriers should be essentially constant over the concentration range. On the NI rich side, the density of the d states at the Ferm! level is relatively smal! and If the diffusivity of the s states Is much larger than that of the d states, the average diffusivity measured from the slope of the upper critical fleld will be essentially dominated by D, and therefore equation 5.3 can be written as dH.» ; aT fy med. 5 (5.7) Cc The upper critical fleld gradtent Is proportional to the s-d scattering rate which, In turn, using the golden rule Is proportiona! to the d density of states. Fig. 5.4 shows the log of the upper critical fleld gradient at Te vs the log of the density of states. For the Zr-NI system the least square fit of the data for x < 55 gives a slope of 1.015 In agreement with the previous discusslon, Thus for a fixed number of carriers Ne and a diffusivity D. varying, lIke TT, as 1/NG(E¢) we find from equation 5.6 a direct proportionality between the resistivity and the density of states, as Illustrated In flg. 5.2 Note that even If the d states diffustvity makes a non- negligible contribution to the average electronic diffustvity D, we would still! get a slope of one In the case where the IIfetime of the -148- 0 O01 O02 93 #04 O.5 0.6 0,7 3,45 T “T T “T “t ——T Cuj_,ZRy a - ~ 129 Ny(O) FROM SUPERCONDUCTING (70) PROPERTIES (65) 3.95 | SsLope : 0.504 ; A”(55) ¥(50) oO —_ N,(O) FROM SPECIFIC = (40) ™~ HEAT MEASUREMENTS OI? 3.05 I SLOPE : 0,756 : o 3 (55) 3.10 r xX 7 2.90 | scope : 1,015 (36.3) 2./0 _L _t Jt i l 1 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Loc (N,(0)) Fig. 5.4 Log-Log plot of the upper critical fleld gradient vs the density of states for Cu-Zr and Zr=-N! systems. -149- d-IIke states Is determined by d-d scattering for which, as discussed previously, the probability of transition Is also proportional to the density of d states at the Ferm! energy. We can wonder why the transition between the range where the d~states dominate the conductivity (x>55) and the one where the s~-states (x<55) dominate Is so sharp. A possible explanation comes from looking at fig. 5.5. Dong et al.'73) measured the positions of the peak of the first halo, Qp, In the x-ray diffraction pattern as a function of composition for the Zr-N{ system and found an apparent discontinulty around x=55. He suggests that the nature of the short range order might be different In the two ranges of concentration and this could explain why there Is predominantly d=-band conduction on one s!de and s~band conduction on the other. The fast system to be discussed Is the Cu-Zr glasses. Fig. 5.3 shows that the upper critical field gradient [ncreases continuously with Zr content over the whole range of concentration for which the alloy can be made amorphous. The rate of Increase with respect to Zr concentration Is, however, much lower than what Is found for the Zr-NI system. The slope of the fog-log plot between the upper critical fleld gradient and the density of states (measured from superconducting properties) Is 1/2 Instead of 1 IIke for the Zr-N!I case (fig. 5.4). By writing the average electronic diffusivity as D= 1/3¥_8 and the average electron mean free path as Qe V_t (where t Is the average lifetime of the electrons at the Ferm! surface) It follows from equation 5.3 and the last result that the average scattering rate 1/T ts proportional to the square root of the d density of states at the Ferm! level, assuming a constant average group velocity of the -150- 3.0 ¢ \ From Y, D, Done ET AL, e 2,.9h ¢ \ * @ 2.8- ot N1z-yZRy 27° SN, e 2.07 oe 2.5 l i t Se. 30 40 50 60 70 80 ZR CONCENTRATION (AT. &) Fig. 5.5 Positions of the first peak of the structure’ factor composition for the Zr-NI system, -151- electrons at the Ferm! surface. This suggests that the s-d scattering model of nearly free electrons Is not the only Important mechanism of conduction In Cu-Zr and that the d electrons must make a non- negligible contribution, as well, In the transport. it Is Interesting to note that If the density of states at the Ferml level Is obtalned from specific heat measurements, the slope of the log-log plot Is 3/4 Instead of 1/2 as we can see In flg. 5.4 For x < 50-55, the values of NY determined from calorimetric measurements are all targer than the ones obtalned from superconducting properties by using the Ginsburg-Landau~Abrikosov-Gortkov theory. For x > 50-55, It Is the Inverse, Discrepancles between the values obtalned by both (74,76) and attributed to methods have been observed before uncertainties In the resistivity measurement (74), differences between specimens (state of relaxation, phase separation...) {7), We belleve that the trend observed here Is too systematic to be explained by any of the above arguments. It Is possible, however, that the states giving rise to the heat capacity might not be all the same as the one Involved [nm superconducting palring due to IncIpfent focal ltzation. Ching, Song and Jaswal'77) have recently calculated the electronic states In Curpoo.,Zr, metallic glasses using the orthogonal Ized LCAO method on perlodic structural models contalning 90 atoms. There Is no assumption tn the theory which makes It applicable only to weak scattering cases. They analyze each electronic state In terms of a localization Index and show that the Cu d states are localized and the Zr d states are relatively delocalized. At the Ferm! level, the states are delocalized but there Is a tendency for a slight Increase In thelr locallzation Index when the Cu concentration -152- Is Increased. Since localization destroys superconductivity, this tendency of localization with Increasing Cu concentration Is probably related to the fact that the density of states seems to decrease faster with Cu concentration when measured from superconducting properties. Fig. 5.6 shows that the conductivity decreases {Inearly with the density of states over the whole range of concentration with a different slope depending, again, whether the denslty of states Is obtained from specific heat of from the slope of Hoos The behavior Is similar to the Mo~Ru-B system even though, the upper critical field gradient Is not aconstant. All these results on Cu-Zr can be understood If we assume that both s and d electrons are Important in the conduction. Let us write the average electronic diffusivity as arising from contribution of both s and d states N N De =: D. + goo D (5.8) ; Ns op + Na s Ns ip + Ny d where Ns, p and Ny represents the partial density of states at the Ferm! level and D, and Dy have thelr usual meaning. Table V.1 Iists the bare partial density of states obtained by Ching et al.!/”? together wlth some other relevant data for the Cu;o, Zr, system. The average electronic diffusivity has been calculated usIng eq. 5.3 and the upper critical fleld gradient from ref. 75. The conductivity, density and mass enhancement factor (1+) .¢+A ep? are from reference 7O and 75. The mean atomic volume 2 has been determined from the density and the average atomic welght. -153- 1.5 J 4 LJ q Ly mm | q 4 LAF = Cuy_yZR 1 1-x*"x (75) 1.3r A From Heo 1.2+ MEASUREMENTS 4 cA 1.1 4 il =< “ 1.Q0F 4 & a 7 7 11+ - 1.0F (41) : 0,9F From SPECIFIC HEAT O.8F © (41) MEASUREMENTS , 0.7 1 l t i l i _t 4 0.92 0.98 1,04 1.10 O/o (x = 55) Fig. 5.6 Normalized density of states vs conductivity for the Cu-Zr system. The concentrations are Indicated In parentheses and the denstty of states are from elther calorimetric measurements or superconductIvity measurements. The data are normalized to the respective values for x=55. -154- TABLE Y.1 Electronic density of states at the Ferm! ltevel together with other parameters for Cu;go.,Zr, metallic glasses. N(E,) states/ev atom _ x1000 Cu4s Zr5s ° a ° 3 23 fete st¥ep x Cu4p Zr5p Cu3d Zr4d (cm°/s) (Siem) (g/cm) (AX) 67 .115 .187 .026 .713 0.391 6.08 6.95 19.61 1.67 1.87 50 .126 .106 .050 .577 0.451 5.65 7.25 17.72 1.50 1.68 33.200 .057 .060 .289 0.533 5.30 7.57 15.94 1.33 1.61 The s states diffusivity Is glven by D, =1/3v,%, = V/3Vet « where v Is the s states veloclty at the Ferm] fevel, Re Is the mean free s path and t, Is the lifetime. If we assume that the Mott s-d scattering mode! fs the matn scattering mechanism for the s electrons, It follows using the golden rule that 2 1 _ 2n (5.9) then, 2 hv D. = $ : Js (5.10) en |<¥ylVI¥, >| Ni (Ee) Nj (Ep) In ref. 46, Waseda assumed that Zr has two valence electrons per atom and Cu one. The ratio between the effective valence electrons and the average atomic’ volume, In that case, Is almost Independent of the concentration, This means that we can consider the ~155- Ferm! wave vector kg = (37 22/2 )1/3 and the s states Ferm! velocity Ve = h2k¢/m as roughly constant across the concentration range. For a given s-d scatter!Ing matrix element, we can calculate the s and d states diffusIvity from eq. 5.8, eq. 5.10 and the values, In table V.1 for the partial density of states and average electronic diffusivity. Fig. 7a shows the results for three different parameters. The dropping off In the d states electronic diffusivity with Increasing Cu concentration Is associated with the Increase In the d_ states localization Index mentioned earller, the degree of locallzation belng related to the electron mean free path. It ts Interesting to note that when a=0.4, the d states diffusivity vanishes when the Cu concentration reaches 67 At.%. Beyond this polnt the d electrons are essentially Immoblle and the conductivity Is assumed entirely by the s and p electrons. if we were to Increase the Cu concentration beyond this point, the density of d states at the Ferm! level would still decrease and the conductivity would Increase according to the Mott s-d scattering model. We would then expect a minimum In the conductivity versus concentration around 67 At.% Cu and, tndeed, we find experimentally a maximum tn the resistivity curve around this concentration(70+78) , Fig. 7b shows the calculated conductivities usIng eq. 5.1 for a parameter a=0.3 together with the exper!mental potnts from Altountan et al.(7075), the bare density of states glven In Table V.1 has been corrected by the mass enhancement factor (1+. <4 6,) before being used In eq. 5.1. The conductivity Is found Pp to decrease over the whole range of concentration and the results are In fairly good agreement with the experimental data. ~156- 1.2 T T 7 7 "T a 1.07 xD. s STATES , ~ DIFFUSIVITY | KZ 0.8 F 4 “ (S) a 0.6 t l= 0.44 0.4 5 = L A 0.2 @D) OD STATES / 0 DIFFUSIVITY WN . a x ~ 9 7 6.0 fF ~ . = ~ 9 = + Gis.5 f ~ ww ‘ ™~ b ~ i 5.9 | 98 CALcuLATED FoR @&= 0.3 6 ; + EXPERIMENTAL 45 n 1 1 __t 1 20 30 40 50 60 70 80 Cu ConcENTRATION (aT, 2) Fig. 5.7a & b Calculated electronic diffusivity for the s and d states vs the Cu concentration for different parameters % (or s-d_ scattering matrix element). Calculated conductivities for a parameter & = 0.3 together with the experimental data polnts from Altountan et al (ref. 70 & 75). -157- Even If the electrical resistivity of Cu-zr 646) ang Zr-ni (7% has been found to be In reasonable agreement with the extended Faber-ZIman theory, the question of the applicability of this theory to systems wlth such a short mean free path still remains puzzling. This chapter represents an attempt to explain the behavior of the electrical resistivity In those systems from a different point of view and emphasizes the possIbllIty of d-band conduction In these glasses. -158- VI. CONCLUSION We have shown that the electrical resistivity can be used as a very convenient experimental tool to study phase separation, ordering and relaxation phenomena [In metallic glasses. Each of these problems Is, by ftself, very Important and because of the topological randomness of the structure, they are also very difficult to detect using conventional techniques IIke X-ray diffraction. The ability to locate them In the temperature-time dlagram [ts already a big step. Because of the great sensitivity of electrical resistivity measurements It Is posslble to observe s!Ight changes In the short range order which are Invisible to other techniques. The lack of a good and reliable theory relating atomic order to electrica! resistivity In metallic glasses, however, IImits the power of such measurements, In chapter Il, I described some of the theoretica! approaches which have been used to study these phenomena In the crystalline state. To what extent these formulas can be extended and applied to metallic glasses fs stlil a very controversial subject. Chapter V Is a warning for those who applied the Ziman formal!sm In the Born approximation and the free electron picture, too extensively. it ts In fact, not to hard to think of alternative ways to Interpret the behavior of the resistivity during = structural relaxation, without reference to the Ziman formalIsm. For Instance, It was shown In chapter V that, quIte often, In metallic glasses the conductivity Is directly proportional to the density of states at the Ferm! fevel with a constant slope glven by the upper critical field gradient at T,. -159- o = « N(Ee) (6.1) Severa! authors have calculated, using different techniques, the electronic density of states In disordered systems. In particular, the method of expansion [fn moments ‘>>? using a tight binding approximation has been quite successful for studying the local dens! ty of states In amorphous metals'94), {t+ was found that the shape of the density of states on a given site Is governed by the focal radial distribution of near nelghbors G,(r) and Is fess sensitive to the focal symmetry '?>), High-density regions glve rise to sharp peaks In the density of states while low density reglons produce a rather flat distri bution. Fig. 6.1, for Instance, shows a@ typical! example of the local density of states on a NI site In a solld-like cell (1I.@. high density region) compared with one for a IIquid-like cell. Golng back to the theory of Cohen and Grest (11.2.D), let N.(E;) be the average density of states for the solld-Ifke sites and No (E;), the average density of states for IIquid-lIke cells. By rewriting the total average density of states, eq. 6.1 becomes o = « ((I-p) No(Ee) + P Ne (Eg) (6.2) where p Is the number of IIquid-IIke cel! and the normalized change In resistance during structural relaxation can be written as (N(E,) ~ Np (Eg) Ap (6.3) Ne + p (Ny - Ne) 4p . Po -160- DENSE Ni SITE —————~] (Solid like cel]) CRYSTALLINE Ni LIQUID Ni (Liquid like cell) 02 Fig. 6.1 Calculated focal density of states: for nickel In a solld-!ike cell (high densIty region), for IIquid nickel and for crystalline nickel (the dataare from ref, 83). -161- When Er Is close to a band-edges, N.(E,) > No (Ee) (fig. 6.1) and the resistivity should decrease upon structural relaxation whereas when it 1s close to the middie of the band the resistivity should Increase during Isothermal! annealing. it Is not surprising that eq. 6.3 Is sImifar to eq. 2.2.16 since the local density of states Is directly related to the local radial distribution function. The connection with Egami's model for structural relaxation Is also evident since the local density of states ona site with hydrostatic stress p Is governed by Gor) given In eq. 2.2.6 and 2.2.8 . Because of the great potential application of metallic glasses and the Importance of the character of the homogenelty of a materia! In any Industrial application, wewlll probably see, in the future, a growling Interest in the use of electrical resistivity as a tool to detect phase transformation. Other techniques, more sophIsticated and probably more powerful, have been used to study Inhomogenelty In metallic glasses, however, electrical resistivity measurements are attractive since they can be performed so easlly. It Is not necessary to emphasize the great use of such measurements In studying age- hardening In conventional crystalline alloys. There remains much work to be done from both the experimental as well as theoretical point of view. 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