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Positron annihilation experiments in metals; Electronic structure and fermi surface studies
Scripta
METALLURGICA
Vol. 14, pp. 23-29, 1980 P r i n t e d in the U.S.A.
P e r g a m o n Press Ltd. All r i g h t s r e s e r v e d .
VIEWPOINT
SET No.
2
POSITRON ANNIHILATION EXPERIMENTS IN METALS; ELECTRONIC STRUCTURE AND FERMI SURFACE STUDIES*
Stephan Berko Department of Physics, Brandeis University Waltham, Massachusetts 02254 (Received
November
I.
26,
1979)
Introduction
In this paper we discuss the use of the positron (e+) annihilation technique in the study of pure metals and of metallic alloys. The behavior of slow e+-s in condensed matter has been the subject of intense experimental and theoretical investigation during the last two decades and the field has been reviewed thoroughly (I). By studying the various properties of the annihilation quanta one obtains information about the electrons sampled by positrons injected from a radioactive source into the material under investigation. In metal physics the e+ technique has been successfully applied to problems in two distinct fields: electronic structure studies (momentum densities, Fermi surfaces, etc.) and defect studies (vacancy formation energies, dislocations, radiation damage, etc.). More recently, the development of monochromatic low energy positron beams has opened up the possibility of metallic surface studies by positrons. In the first part of our paper we sketch briefly the main experimental techniques used in the e+ experiments and outline the underlying physical concepts. The second part deals with applications, in particular to momentum density and Fermi surface studies. The reader is referred to the extensive review (I) and to the "mini-reviews" in previous issues of this Journal dealing with other applications of the e+ annihilation technique in more detail. 2.
Fundamental Concepts
In the usual experiments fast e+-s from a long-llved radioactive source such as gZNa or bSCo are injected into the metal sample. The positrons lose their energy by ionizing collisions and phonon scattering and reach thermal equilibrium with the metal in a few picoseconds (10-1Zs), penetrating the sample well into the bulk to mean depths of i0-I00 pm. In defect-free single crystals the e+ is described by a delocallzed Bloch wavefunctlon ~+(~), repelled by the positive ions and having maxima in the interstitial regions between the atoms. Being the antiparticle of the electron, each positron annihilates with an electron of the metal, emitting two or three annihilation quanta, with a characteristic annihilation rate (inverse lifetime) that depends on the electron density it overlaps with prior to annihilation. The annihilation process, well understood theoretically, results in two quanta (2Y) from spln-singlet overlaps, and in three quanta (3Y) from spln-trlplet overlaps. In a metal 2y decays predominate since the 3y decay is ~I0 ~ times less probable. Typical annihilation lifetimes in metals are a few hundred picoseconds. Prior to annihilation the positrons, being repelled by the positive ion cores can be trapped in low density regions of the crystal, such as a vacancy, the neighborhood of a dislocation, mlcrovolds, etc. The same repulsion leads to the possibility of slow positrons being ejected from a metal surface corresponding to a "negative work function" (2). 3.
Experimental Techniques
Four typical experiments can be performed with the annihilation y-s in order to gain information about the electrons the e+ annihilated with: I) 2y angular correlation experiments; 2) the measurement of the energy spectrum of one of the y-s from 2X annihilation; 3) e+ lifetime experiments; and 4) 3y/2y yield measurements. *Work supported by the National Science Foundation and the Army Research Office.
23 0036-9748/80/010023-07502.00/0 C o p y r i g h t (c) 1980 P e r g a m o n Press
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The 2y angular correlation of annihilation radiation (ACAR) depends, by momentum conservation, on the momentum distribution p(~) of the annihilating e+-e - pair, and is most often used in the study of electronic momentum densities. For zero momentum the two ~-s are emitted opposite to each other, each having energy ~mc z (0.51Mev) and thus momentum mc. Their deviation from collinearity for finite ~ can be measured by counting coincidences between two detectors C 1 and C 2 as in Fig. I. (The sample emitting the two y-s is at the origin of the coordinate system.) Because of the large y momenta (~ m c), the angles involved are only a few
FIG. 1 Geometry of 2y angular correlation experiments. Hatched squares indicate location of detectors C 1 and C2; double arrow represents the momentum ~. Typically distance L between the sample and the detectors is 3-I0 meters and angles 8 and # are a few milliradians.
milliradians; thus counters have to he positioned at a large distance from the samples (3-i0 meters). The small angles 8 and ~ are directly related to the Pz (Pz = mc~ and Pv (Pv = ~mc) components of ~. One milliradian corresponds to a momentum component of 8 ~ - mc x I 0 - ~ Most experiments to date have been performed with a "long-slit" geometry, using two long detectors with no energy resolution, thus measuring one component of ~; given the momentum distribution p(~) such a setup measures N(pz ) = IYp(~)dPxdpy. More recently new multiple detector systems (3) permit "polnt-sllt" geometry and measure two components, thus yielding N(py,pz) = Ip(~)dPx , two-dimensional ACAR surfaces. The Px component of ~ (Fig. I) produces a Doppler shift (typically a few keV) from the mc z (511 keV) energy of each y. N(Px) can be measured by observing the energy Doppler broadening of one of the Y-s with a high efficiency Ge spectrometer. Given the inherent limitations of the energy resolution of such spectrometers (~ I.I keV at 511 keV), the effective momentum resolution is an order of magnitude poorer than for an ACAR setup. Because of simplicity and fast data acquisition, Doppler broadening is often used in defect studies where details in p(~) are not of paramount importance. The lifetime of the e+ can be studied by using as a timing signal a nuclear y(l.2@ Mev) that follows promptly the emission of a e+ from ZZNa. One can obtain accurately the lifetime by measuring with fast electronic techniques the time interval distribution between the nuclear and one of the annihilation ~-s. The measurement of the 3y/2~ yield is only important if the e+ and e-spins are relevant, as in studies of magnetically ordered samples using sPin-polarized positrons (4). Before discussing momentum density measurements, we briefly outline the status of e+ experiments in defect studies. 4.
Defect Studies
Lifetime as well as angular correlation data were known to depend on the state of the metal under investigation, but prior to 1967 were analyzed in terms of bulk properties. In 1967 MacKenzie et al (5) attributed the temperature dependence of e+ lifetimes in metals to lattice vacancies, and Berko and Erskine (6) proposed that positrons localize in dislocations as well as in vacancies prior to annihilation. Since then e+ annihilation has become a much used tool in metallurgical defect studies. The many applications, ranging from vacancy formation energy measurements to radiation produced void studies have been thoroughly reviewed (7). All these studies rely on the fact that when positrons annihilate in a metal containing defects, they exhibit a complex lifetime spectrum; the ACAR curves also depend on the nature and the density of the defects. Specifically, if a positron is trapped at a defect, its lifetime ~D is longer than the lifetime ~B characteristic of annihilation of delocalized "Bloch state" positrons, since the trapped positron "sees" fewer electrons than the delocalized one. Similarly, the ACAR curves and Doppler profiles from trapped states exhibit more annihilation
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with conduction electrons (low momenta) and less with core electrons (high momenta), since the trapped positron overlaps fewer ionic cores. A simple parametrization of the ACAR or Doppler profile curves leads to "line-shape parameters", such as the "S" parameter (7), a measure of conduction vs core annihilation intensity. It is important to realize that the absolute value of ~D and T B as well as the exact shape of the ACAR curves, or even the value of the S parameter is not easy to predict theoretically - these values depend on the electronic band structure of the metal, on the electronic structure around the defect, as well as on the details of the e+-e interaction prior to annihilation. It is the chanse in these values caused by changes in defect concentration and defect characteristics due to temperature, impurities, annealing cycles, radiation damage, etc. that is interpretable in terms of metallurgically significant parameters. Defect studies in metals are usually analyzed by the "trapping model", originally developed by Brandt (8) for studying e + annihilation in alkali halides. The trapping model as used for metals (9) describes by coupled rate equations the time dependence of an ensemble of positrons. The solution of these rate equations correlates the observed complex lifetime with the physically important parameters T D and TB, the trapping rate per defect ~D and the defect concentration C D. It is usually assumed that the positrons slow down rapidly to their delocalized ground state, i.e. that at time t ~ 0 all e+-s are in the Bloch state. It has been found that positrons are indeed very sensitive to vacancies (i.e. that PD is high): vacancy concentrations as low as i0 -! can be detected, and at a vacancy concentration of 10-4 the probability for e + trapping approaches unity. In vacancy studies one can obtain the formation energy of mono-vacancies directly from an Arrhenius plot of the proper combination of the measured parameters. Mono-vacancy activation energies have been measured by the e + technique in many metals, with resulting values ranging from 0.44 eV in Cd to 3.75 eV in W -see complete list in a recent review by Doyama (I0). In some metals, in particular in the alkali metals, positrons do not seem to trap in vacancies (7). The e+ annihilation technique has been applied to many other metallurgical problems, such as divacancy formation, vacancy-impurity interaction, plastic deformation, fatigue and creep, various quenching and annealing processes~ phase transitions, precipitation, etc. (7,10). The first observation by positrons of radiation produced microvoids in Mo (ii) led to several electron and neutron radiation damage and annealing kinetics studies, as discussed in a review by Gauster (12). More recently it was also demonstrated that positrons can be used to detect microporosity in Pb (Cd) alloys (13). Many of the above studies are, however, of an exploratory nature; much more detailed work with carefully characterized specimens is required in order to further develop the e + technique into a high precision tool in defect studies. 5.
Momentum Density and Fermi Surface Studies
As we have indicated in Section 3, the angular correlation of 2y annihilation radiation (ACAR) depends on the momentum distribution p (~) of the annihilating electrons as sampled by the positron. As discussed in reviews of ACAR experiments (14), the measurement of p(~) contains two sets of information: a) sharp breaks in p(~) reflect the topology and size of the Fermi surface (FS); b) the actual shape of p(~) depends on the wavefunctions of the electrons and of the positron. Many ACAR experiments have been performed on various metals using the "long-slit" geometry, and have been analyzed using various electronic band theory techniques. Recently we have completed at Brandeis University a Nal multidetector two-dimensional ACAR apparatus that has led to new, high precision 2D ACAR measurements of N(py,pz) = fp(~)dp x (3). These new experiments, as well as several other 2D ACAR machines that have been completed last year in other laboratories have been discussed in a recent review (15). In order to indicate the data obtainable with the 2D multidetector machine, we show in Figure 2 the 2D ACAR result for an oriented single crystal of AI. The distribution N(py,pz) is nearly spherical in the extended zone in momentum space, reflecting a nearly spherical FS due to the "free-electron-like" bands of AI. The detailed shape of N(py,pz) agrees quite well with the results of an OPW band computation that includes the positron wavefunction (16).
26
ELECTRONIC
STRUCTURE
Vol.
~=~)
14, No.
I
AI
i>
~
v
=
r
~ hr./°
-
[ool]
[om]~,~ FIG. 2 2D ACAR surface from AI. The orientation of the crystal is illustrated in terms of the Brillouin zone in the inset. Each crossing of lines is an independent measurement. Sample at I00 K. In metals with an anisotropic FS the ACAR results are correspondingly anlsotroplc; the well known FS of copper with its "necks" along the direction can be obtained directly from t~e ACAR results in copper (I). In most of the pure metals, however, the e+ annihilation technique cannot compete in precision with the standard methods of "Fermlology," such as the measurement of magneto-reslstance, of the deHaas-van Alphen effect (dHvA) (17), etc. During the last decade, however, the central interest in metal physics has shifted from the study of pure ordered metals to the investigation of disordered systems, such as random substitutional alloys and amorphous metals. The theory of the electronic structure of concentrated alloys has undergone a major development, with the coherent potential approximation 4CPA) playIng a central role. Realistic calculations of energy bands, density of states as well as of electronic lifetimes have been performed in transition and noble metal alloys, using the t matrix approximation 4ATA) (18), and, more recently, the full CPA treatment 419). One of the essential conclusions of these computations is that in disordered alloys the Fermi surface can still be well defined, since the smearing of the FS, although appreciably larger than the effect of the temperature, is still a small fraction (a few %) of the FS dimensions. It is well known that in disordered alloys the scattering of the conduction electrons drastically limits the usefulness of the standard FS techniques. The dRvA measurements can be only performed in very dilute alloys, with solute concentrations of less than a few percent. The positron technique, on the other hand, does not depend on large electron coherence lengths, and is thus ideally suited for FS studies of high concentration disordered alloys. Many ACAR measurements have been performed to date to study the change in shape and size of the FS upon alloying; these experiments have been reviewed in detail (20). Most experiments have been performed on copper based alloys. Usually the main features of the pure Cu FS, such as the neck radius k N and the Fermi surface radius vectors klO 0 and kil O are obtained as a function of increasing solute concentration, These FS measures are found to increase monotonically with electron per atom ratio e/a. As an example we plot in Fig, 3 the measured values of k N vs e/a for Cu-Zn. Data are compared with the rigid band model prediction 4RBI) , with two ATA computations (SMT) and (CR) 421) and with a "sinking band" model prediction (SB), originally used to predict Cu-AI optical data (22). In Fig. 4 we plot some available data for the kll 0 FS dimension in the ~-phases of Cu-En, Cu-AI, Cu-Ga and Cu-Ge. We note that high accuracy data will be required to clearly differentiate between the various theoretical models. The existing data for the klo 0 values vs e/a for these alloys vary more from experiment to experiment (20), but clearly indicate that no contact with the Brillouin zone is made up to e/a = 1.3, even for the Cu-Ge and Cu-Ga alloys. For copper based alloys with transition metal solutes such a s Cu-Ni and Cu-Pd, the data indicate large deviations from rigid band behavior, in qualitative agreement with CPA computations (23). More high precision data will be required to test the theoretical predictions quantitatively. Interesting recent work has been performed on the concentrated alloy phases having
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ELECTRONIC
STRUCTURE
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FIG. 3 Fermi surface neck radius vs. e/a in Cu-Zn alloys. Different data points as per Table II.I, Ref. 20.
HCP structure,
CuZn 2
1.3
ratio
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FIG. 4 The Fermi surface dimension kll 0 for various copped based alloys. Different data as per Table II.I, Ref. 20.
of Cu-Ge and Ag-AI, as well as on the Cd-Mg system (24).
[ T 2 ] / § ~ mcx,~3 FIG. 5 2D ACAR surface from Cu-30 at.% Zn. The orientation terms of the Brillouin zone inset. Sample at I00 K.
of the crystal is illustrated
in
Using the new 2D ACAR apparatus, we have recently performed detailed experiments on a-brass (25). In order to demonstrate the increased precision made available by the new 2D developments for future alloy studies, we show in Fig. 5 a 2D ACAR surface obtained from an oriented single crystal of Cu-30 at.% Zn. Analysis of these results indicate that a "Cu-like" FS is found to persist even for this high concentration Q-brass. Comparison with Cu is easiest when plotted as a contour map. In Fig. 6 we display such a contour map comparison. In order to demonstrate the anisotropies even at high momenta, we have subtracted a rotationally isotroplc component R(py,pz) from the N(py,pz) surfaces. It can be seen that the o~-brass ACAR data is very similar to hhat of pure Cu, wlth the main feature being an expansion of the FS. The high momentum anisotropies, due to the Fourier coefficients of the electronic wavefunctions, are clearly present, even for the e-brass sample. These 2D ACAR data were quantitatively analyzed to obtain the locus of the discontinuities due to the FS. These results are plotted in Fig. 7 in terms of the FS neck and FS projection onto the (III) plane. The projection of the Brillouin zone is also plotted. The solid line is the FS in the (III) plane as measured by the dHvA technique in pure Cu (17). The data indicate that the FS is growing essentially Isotroplcally with increasing e/a, at a rate somewhat lower than that predicted by the rigid band model. We also conclude
28
ELECTRONIC
Cu
STRUCTURE
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i
Cu- 30 ot % Zn
8T~'7 [/'~-' e~
~ ~
I
Py along [|21] in m=x~~
FIG. 6 The contour map of N(pypz) - R(pypz) from Cu and Cu-30 at.% Zn. The projectlon of the Brlllouin zones onto the (111) plane are also indicated.
2
"oCu Cu-11.6at %Zn o Cu-30at % Z,~
3 4 5 Pz in r n c x l O °3
6
7
8
FIG. 7 Fermi surface projection onto the (I11) plane and the FS neck radii on the hexagonal Brillouin face for pure Cu and ~-brass.
from the analysis that the FS smearing of kll 0 due to disorder is less than 3% of kll O. It is clear from thls example that future high precision 2D ACAR measurements will provide us with much valuable information regarding the electronic band structure of alloys and the stability of alloy phases. With improved resolution it will become feasible to study the smearing of the FS with alloying in non-dilute systems, equivalent to Dingle temperature studies by dHvA measurements at low concentrations. However, the theoretical interpretation of the e+ resuits in disordered alloys is more complicated than in pure metals, in view of the possible preference of e+-s for one of the alloy constituents (20), as well as of the random scattering of the e + in the alloy (26). These effects will have an influence on the shape of p(~), but will have little affect on the position of the FS discontinuities. Experiments on ordered alloy systems have also been performed, leading to interesting results. In particular the FS of the high temperature superconductor V3Si has been studied recently with 2D ACAR techniques (27). 6.
New Developments:
Slow Positron Beams
Before concluding this "minl-review" we would like to mention briefly the most recent development in e+ physics, that of slow, mono-energetlc e+ beams. Using a radioactive e+ source followed by a "fast-slow converter," it has now become possible to produce a beam of variable energy (from zero to several keV) positrons of sufficiently high intensity to be used for experiments (28). The converters are based on the observation that when fast positrons are introduced into a solid, a small fraction is re-ejected from the surface due to an effective negative positron work function (2). The first experiment designed to study the interaction of these slow positron beams with solid surfaces indicated that a high percentage of positrons form posltronlum in colllson with the surface (28); positronium is the hydrogen-like bound state between an e- and a e+. Recently experimental programs have been started in various laboratories designed to develop slow positron interactions with surfaces into a new tool of surface physics (29). The recent experlments, performed in ultra-high vacuum with well characterized surfaces indicate a positron surface state in pure metals, thermally activated positronium emission from this state, and a considerable sensitivity to surface impurities. Low energy positron diffraction experiments from surfaces are also in progress. It is hoped that in the near future it wlll also be possible to use these variable energy beams to study surface defects in metals. 7.
Conclusion and Acknowledgements
We hope that is was possible in this short review to indicate some of the important appllcations of positron physics to various aspects of metal physics and metallurgy. We have mainly
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highlighted the experimental results. Much theoretical work has been performed to understand the quantum behavior of low energy positrons in metals (I). The defect trapping mechanism as well as the surface behavior is intimately tied to problems of the diffusion mechanism of light particles in solids (30). In the near future we can expect to see many new applications to defect studies, high precision studies of alloys by the 2D ACAR technique as well as surface investigations with slow positron beams. The author is indebted to F. Sinclair for a critical reading of this manuscript. References I. For up-to-date reviews see Positrons in Solids, Topics in Current Physics 12, P. Hautojarvi, ed. (Springer-Verlag, Berlln-Heidelberg, 1979); also Positron Annihilation,---Proc. V th Int. Conf. on e+ Ann., R.R. Hasigutl and K. Fujiwara, eds. (The Japan Inst. of Metals, 1979). 2. A.P. Mills, Jr., P.M, Platzman and B.L. Brown, Phys. Rev. Lett. 41, 1076 (1978) and references therein. 3. S. Berko, M. Haghgooie and J.J. Mader, Phys. Lett. 63A, 335 (1977); M, Haghgooie, J.J. Mader and S. Berko, Phys. Lett. 69A, 293 (1978). 4. S. Berko and A.P. Mills, Jr., J. Physique 32, CI-287 (1971) and references therein. 5. I.K. MacKenzie, T.L. Khoo, A.B. McDonald and B.T.A. McKee, Phys. Rev. Lett. 19, 946 (1967). 6. S. Berko and J.C. Erskine, Phys. Rev. Lett. 19, 307 (1967). 7. See for example A. Seeger, Appl. Phys. ~, 183 (1974); M. Doyama and R.R. Hasiguti, Cryst. Lett. Defects ~, 139 (1973); also relevant papers in Ref. I. 8. W. Brandt in Positron Annihilation, A.T. Stewart and L.O. Roellig, eds. (Ac. Press, N.Y. 1967) p. 155. 9. D.C. Connors and R.N. West, Phys. Lett. 30___A, 24 (1969); B. Bergersen and M.J. Stott, Solid State Comm. 7, 1203 (1969). I0. M. Doyama in--Positron Annihilation, Proc. V th Int. Conf. on e+ Ann., R.R. Hasiguti and K. Fujiwara, eds. The Japan Inst. of Metals, 1979), p. 13. ii. O. Mogensen, K. Petersen, R.M.J. Cotterill, B. Hudson, Nature 239, 98 (1972); R.M.J. Cotterill, I.K. MacKenzie, L. Smedskjaer, G. Trumpy, J.H,O.L. Traff, Nature 239, 99 (1972). 12. W.B. Gauster, J. of Vac. Sci. and Techn. 15, 688 (1978). 13. C.K. Hu, S. Berko, G.R. Gruzalski and D. Turnhull, Solid State Comm. 31, 65 (1979). 14. S. Berko in Compton Scattering, B. Williams, ed. (McGraw Hill, London, 1977), p. 273; S. Berko, J. Physique, 39, C6-1568 (1978); see also relevant reviews in Ref. I. 15. S. Berko in Positron~nihilation, Proc. V th Int. Conf. on e+ Ann., R.R. Hasiguti and K. Fujiwara, eds. (The Japan Inst. of Metals, 1979), p. 65. 16. J. Mader, S. Berko, H. Krakauer and A. Bansil, Phys. Rev. Lett. 37, 1232 (1976). 17. A.P. Cracknell and K.C. Wang, The Fermi Surface (Clarendon Press, Oxford, 1973). 18. H. Ehrenreich and L. Schwartz, Solid State Phys., H. Ehrenreich, F. Seitz and D. Turnbull, eds. (Ac. Press, N.Y. 1976), Vol. 31. 19. B.L. Gyorffy and G.M. Stocks in Electrons in Disordered Metals and at Metallic Surfaces, NATO Adv. Study Institutes Series B42 (Plenum Press, New York, 1979), p. 89. 20. S. Berko in Electrons in Disordered Metals and at Metallic Surfaces, NATO Adv. Study Institutes Series B42 (Plenum Press, New York, 1979), p. 239. 21. A. Bansil, H. Ehrenreich, L. Schwartz and R.E. Watson, Phys. Rev. Bg, 445 (1974). 22. R.S. Rea and A.S. DeReggi, Phys. Rev. Bg, 3285 (1974). 23. B. Gordon, W.M. Temmerman, B.L. Gyorffy and G.M. Stocks, Inst. Phys. Conf. Ser. 39, 402 (1978); see also Ref. 19. 24. M. Hasegawa, M. Hirabayashi and S. Kolke, in Positron Annihilation, Proc. V th Int. Conf. on e+ Ann., R.R. Hasigutl and K. Fujlwara, eds. (The Japan Inst. of Metals, 1979), 673. 25. M. Haghgooie and S. Berko in Positron Annihilation, Proc. V th Int. Conf. on e+ Ann., R.R. Haslguti and K. Fujiwara, eds. (The Japan Inst. of Metals, 1979), p. 291. 26. K.M. Hong and J.P. Carbotte, Can. J. Phys. 55, 1335 (1977); C. Koenig, Phys. Star. Sol. B88, 569 (1978). 27. W.S. Farmer, F. Sinclair, S. Berko and G.M. Beardsley, Solid State Comm. 31, 481 (1979). 28. K.F. Canter, A.P. Mills, Jr. and S. Berko, Phys. Rev. Lett. 33, 7 (1974); see refs. therein. 29. A.P. Mills, Jr., Phys. Rev. Lett. 41, 1828 (1978); l.J. Rosen-berg, A.H. Weiss, K.F. Canter in Positron Annihilation, Proc. vt1~--Int. Conf. on e+ Ann., R.R. Haslgutl and K. Fujiwara, eds. (The Japan Inst. of Metals, 1979), p. 883; K.G. Lynn, Phys. Rev. Lett. 43, 391 (1979). 30. T. McMullen, J. Phys. F7, 2041 (1977); W. Brandt and N.R. Arista, Phys. Rev.--Al9, 2317 (1979); R. Paulln in Positron Annihilation, Proc. V th Int. Conf. on e+ Ann., R-$R. Haslguti and K. Fujiwara, eds. (The Japan Inst. of Metals, 1979), p. 601.
METALLURGICA
Vol. 14, pp. 23-29, 1980 P r i n t e d in the U.S.A.
P e r g a m o n Press Ltd. All r i g h t s r e s e r v e d .
VIEWPOINT
SET No.
2
POSITRON ANNIHILATION EXPERIMENTS IN METALS; ELECTRONIC STRUCTURE AND FERMI SURFACE STUDIES*
Stephan Berko Department of Physics, Brandeis University Waltham, Massachusetts 02254 (Received
November
I.
26,
1979)
Introduction
In this paper we discuss the use of the positron (e+) annihilation technique in the study of pure metals and of metallic alloys. The behavior of slow e+-s in condensed matter has been the subject of intense experimental and theoretical investigation during the last two decades and the field has been reviewed thoroughly (I). By studying the various properties of the annihilation quanta one obtains information about the electrons sampled by positrons injected from a radioactive source into the material under investigation. In metal physics the e+ technique has been successfully applied to problems in two distinct fields: electronic structure studies (momentum densities, Fermi surfaces, etc.) and defect studies (vacancy formation energies, dislocations, radiation damage, etc.). More recently, the development of monochromatic low energy positron beams has opened up the possibility of metallic surface studies by positrons. In the first part of our paper we sketch briefly the main experimental techniques used in the e+ experiments and outline the underlying physical concepts. The second part deals with applications, in particular to momentum density and Fermi surface studies. The reader is referred to the extensive review (I) and to the "mini-reviews" in previous issues of this Journal dealing with other applications of the e+ annihilation technique in more detail. 2.
Fundamental Concepts
In the usual experiments fast e+-s from a long-llved radioactive source such as gZNa or bSCo are injected into the metal sample. The positrons lose their energy by ionizing collisions and phonon scattering and reach thermal equilibrium with the metal in a few picoseconds (10-1Zs), penetrating the sample well into the bulk to mean depths of i0-I00 pm. In defect-free single crystals the e+ is described by a delocallzed Bloch wavefunctlon ~+(~), repelled by the positive ions and having maxima in the interstitial regions between the atoms. Being the antiparticle of the electron, each positron annihilates with an electron of the metal, emitting two or three annihilation quanta, with a characteristic annihilation rate (inverse lifetime) that depends on the electron density it overlaps with prior to annihilation. The annihilation process, well understood theoretically, results in two quanta (2Y) from spln-singlet overlaps, and in three quanta (3Y) from spln-trlplet overlaps. In a metal 2y decays predominate since the 3y decay is ~I0 ~ times less probable. Typical annihilation lifetimes in metals are a few hundred picoseconds. Prior to annihilation the positrons, being repelled by the positive ion cores can be trapped in low density regions of the crystal, such as a vacancy, the neighborhood of a dislocation, mlcrovolds, etc. The same repulsion leads to the possibility of slow positrons being ejected from a metal surface corresponding to a "negative work function" (2). 3.
Experimental Techniques
Four typical experiments can be performed with the annihilation y-s in order to gain information about the electrons the e+ annihilated with: I) 2y angular correlation experiments; 2) the measurement of the energy spectrum of one of the y-s from 2X annihilation; 3) e+ lifetime experiments; and 4) 3y/2y yield measurements. *Work supported by the National Science Foundation and the Army Research Office.
23 0036-9748/80/010023-07502.00/0 C o p y r i g h t (c) 1980 P e r g a m o n Press
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I
The 2y angular correlation of annihilation radiation (ACAR) depends, by momentum conservation, on the momentum distribution p(~) of the annihilating e+-e - pair, and is most often used in the study of electronic momentum densities. For zero momentum the two ~-s are emitted opposite to each other, each having energy ~mc z (0.51Mev) and thus momentum mc. Their deviation from collinearity for finite ~ can be measured by counting coincidences between two detectors C 1 and C 2 as in Fig. I. (The sample emitting the two y-s is at the origin of the coordinate system.) Because of the large y momenta (~ m c), the angles involved are only a few
FIG. 1 Geometry of 2y angular correlation experiments. Hatched squares indicate location of detectors C 1 and C2; double arrow represents the momentum ~. Typically distance L between the sample and the detectors is 3-I0 meters and angles 8 and # are a few milliradians.
milliradians; thus counters have to he positioned at a large distance from the samples (3-i0 meters). The small angles 8 and ~ are directly related to the Pz (Pz = mc~ and Pv (Pv = ~mc) components of ~. One milliradian corresponds to a momentum component of 8 ~ - mc x I 0 - ~ Most experiments to date have been performed with a "long-slit" geometry, using two long detectors with no energy resolution, thus measuring one component of ~; given the momentum distribution p(~) such a setup measures N(pz ) = IYp(~)dPxdpy. More recently new multiple detector systems (3) permit "polnt-sllt" geometry and measure two components, thus yielding N(py,pz) = Ip(~)dPx , two-dimensional ACAR surfaces. The Px component of ~ (Fig. I) produces a Doppler shift (typically a few keV) from the mc z (511 keV) energy of each y. N(Px) can be measured by observing the energy Doppler broadening of one of the Y-s with a high efficiency Ge spectrometer. Given the inherent limitations of the energy resolution of such spectrometers (~ I.I keV at 511 keV), the effective momentum resolution is an order of magnitude poorer than for an ACAR setup. Because of simplicity and fast data acquisition, Doppler broadening is often used in defect studies where details in p(~) are not of paramount importance. The lifetime of the e+ can be studied by using as a timing signal a nuclear y(l.2@ Mev) that follows promptly the emission of a e+ from ZZNa. One can obtain accurately the lifetime by measuring with fast electronic techniques the time interval distribution between the nuclear and one of the annihilation ~-s. The measurement of the 3y/2~ yield is only important if the e+ and e-spins are relevant, as in studies of magnetically ordered samples using sPin-polarized positrons (4). Before discussing momentum density measurements, we briefly outline the status of e+ experiments in defect studies. 4.
Defect Studies
Lifetime as well as angular correlation data were known to depend on the state of the metal under investigation, but prior to 1967 were analyzed in terms of bulk properties. In 1967 MacKenzie et al (5) attributed the temperature dependence of e+ lifetimes in metals to lattice vacancies, and Berko and Erskine (6) proposed that positrons localize in dislocations as well as in vacancies prior to annihilation. Since then e+ annihilation has become a much used tool in metallurgical defect studies. The many applications, ranging from vacancy formation energy measurements to radiation produced void studies have been thoroughly reviewed (7). All these studies rely on the fact that when positrons annihilate in a metal containing defects, they exhibit a complex lifetime spectrum; the ACAR curves also depend on the nature and the density of the defects. Specifically, if a positron is trapped at a defect, its lifetime ~D is longer than the lifetime ~B characteristic of annihilation of delocalized "Bloch state" positrons, since the trapped positron "sees" fewer electrons than the delocalized one. Similarly, the ACAR curves and Doppler profiles from trapped states exhibit more annihilation
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with conduction electrons (low momenta) and less with core electrons (high momenta), since the trapped positron overlaps fewer ionic cores. A simple parametrization of the ACAR or Doppler profile curves leads to "line-shape parameters", such as the "S" parameter (7), a measure of conduction vs core annihilation intensity. It is important to realize that the absolute value of ~D and T B as well as the exact shape of the ACAR curves, or even the value of the S parameter is not easy to predict theoretically - these values depend on the electronic band structure of the metal, on the electronic structure around the defect, as well as on the details of the e+-e interaction prior to annihilation. It is the chanse in these values caused by changes in defect concentration and defect characteristics due to temperature, impurities, annealing cycles, radiation damage, etc. that is interpretable in terms of metallurgically significant parameters. Defect studies in metals are usually analyzed by the "trapping model", originally developed by Brandt (8) for studying e + annihilation in alkali halides. The trapping model as used for metals (9) describes by coupled rate equations the time dependence of an ensemble of positrons. The solution of these rate equations correlates the observed complex lifetime with the physically important parameters T D and TB, the trapping rate per defect ~D and the defect concentration C D. It is usually assumed that the positrons slow down rapidly to their delocalized ground state, i.e. that at time t ~ 0 all e+-s are in the Bloch state. It has been found that positrons are indeed very sensitive to vacancies (i.e. that PD is high): vacancy concentrations as low as i0 -! can be detected, and at a vacancy concentration of 10-4 the probability for e + trapping approaches unity. In vacancy studies one can obtain the formation energy of mono-vacancies directly from an Arrhenius plot of the proper combination of the measured parameters. Mono-vacancy activation energies have been measured by the e + technique in many metals, with resulting values ranging from 0.44 eV in Cd to 3.75 eV in W -see complete list in a recent review by Doyama (I0). In some metals, in particular in the alkali metals, positrons do not seem to trap in vacancies (7). The e+ annihilation technique has been applied to many other metallurgical problems, such as divacancy formation, vacancy-impurity interaction, plastic deformation, fatigue and creep, various quenching and annealing processes~ phase transitions, precipitation, etc. (7,10). The first observation by positrons of radiation produced microvoids in Mo (ii) led to several electron and neutron radiation damage and annealing kinetics studies, as discussed in a review by Gauster (12). More recently it was also demonstrated that positrons can be used to detect microporosity in Pb (Cd) alloys (13). Many of the above studies are, however, of an exploratory nature; much more detailed work with carefully characterized specimens is required in order to further develop the e + technique into a high precision tool in defect studies. 5.
Momentum Density and Fermi Surface Studies
As we have indicated in Section 3, the angular correlation of 2y annihilation radiation (ACAR) depends on the momentum distribution p (~) of the annihilating electrons as sampled by the positron. As discussed in reviews of ACAR experiments (14), the measurement of p(~) contains two sets of information: a) sharp breaks in p(~) reflect the topology and size of the Fermi surface (FS); b) the actual shape of p(~) depends on the wavefunctions of the electrons and of the positron. Many ACAR experiments have been performed on various metals using the "long-slit" geometry, and have been analyzed using various electronic band theory techniques. Recently we have completed at Brandeis University a Nal multidetector two-dimensional ACAR apparatus that has led to new, high precision 2D ACAR measurements of N(py,pz) = fp(~)dp x (3). These new experiments, as well as several other 2D ACAR machines that have been completed last year in other laboratories have been discussed in a recent review (15). In order to indicate the data obtainable with the 2D multidetector machine, we show in Figure 2 the 2D ACAR result for an oriented single crystal of AI. The distribution N(py,pz) is nearly spherical in the extended zone in momentum space, reflecting a nearly spherical FS due to the "free-electron-like" bands of AI. The detailed shape of N(py,pz) agrees quite well with the results of an OPW band computation that includes the positron wavefunction (16).
26
ELECTRONIC
STRUCTURE
Vol.
~=~)
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I
AI
i>
~
v
=
r
~ hr./°
-
[ool]
[om]~,~ FIG. 2 2D ACAR surface from AI. The orientation of the crystal is illustrated in terms of the Brillouin zone in the inset. Each crossing of lines is an independent measurement. Sample at I00 K. In metals with an anisotropic FS the ACAR results are correspondingly anlsotroplc; the well known FS of copper with its "necks" along the
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ELECTRONIC
STRUCTURE
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27
6.2
,-~,2.0
6.0
.. SB
¢ =
R S - -
..
1.8
• "" / • -- / ~
O
SMT "
--~ 5 . 8
SMT - -- -
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o
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~
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i
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A CuGe I m CuA~ 4
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u
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• CuZ. 2
~
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Electr0n/at0m
ratio
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such as the ~-phase
IJ
1.2
Electron/atom
e/a
FIG. 3 Fermi surface neck radius vs. e/a in Cu-Zn alloys. Different data points as per Table II.I, Ref. 20.
HCP structure,
CuZn 2
1.3
ratio
L4
e/a
FIG. 4 The Fermi surface dimension kll 0 for various copped based alloys. Different data as per Table II.I, Ref. 20.
of Cu-Ge and Ag-AI, as well as on the Cd-Mg system (24).
[ T 2 ] / § ~ mcx,~3 FIG. 5 2D ACAR surface from Cu-30 at.% Zn. The orientation terms of the Brillouin zone inset. Sample at I00 K.
of the crystal is illustrated
in
Using the new 2D ACAR apparatus, we have recently performed detailed experiments on a-brass (25). In order to demonstrate the increased precision made available by the new 2D developments for future alloy studies, we show in Fig. 5 a 2D ACAR surface obtained from an oriented single crystal of Cu-30 at.% Zn. Analysis of these results indicate that a "Cu-like" FS is found to persist even for this high concentration Q-brass. Comparison with Cu is easiest when plotted as a contour map. In Fig. 6 we display such a contour map comparison. In order to demonstrate the anisotropies even at high momenta, we have subtracted a rotationally isotroplc component R(py,pz) from the N(py,pz) surfaces. It can be seen that the o~-brass ACAR data is very similar to hhat of pure Cu, wlth the main feature being an expansion of the FS. The high momentum anisotropies, due to the Fourier coefficients of the electronic wavefunctions, are clearly present, even for the e-brass sample. These 2D ACAR data were quantitatively analyzed to obtain the locus of the discontinuities due to the FS. These results are plotted in Fig. 7 in terms of the FS neck and FS projection onto the (III) plane. The projection of the Brillouin zone is also plotted. The solid line is the FS in the (III) plane as measured by the dHvA technique in pure Cu (17). The data indicate that the FS is growing essentially Isotroplcally with increasing e/a, at a rate somewhat lower than that predicted by the rigid band model. We also conclude
28
ELECTRONIC
Cu
STRUCTURE
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Cu- 30 ot % Zn
8T~'7 [/'~-' e~
~ ~
I
Py along [|21] in m=x~~
FIG. 6 The contour map of N(pypz) - R(pypz) from Cu and Cu-30 at.% Zn. The projectlon of the Brlllouin zones onto the (111) plane are also indicated.
2
"oCu Cu-11.6at %Zn o Cu-30at % Z,~
3 4 5 Pz in r n c x l O °3
6
7
8
FIG. 7 Fermi surface projection onto the (I11) plane and the FS neck radii on the hexagonal Brillouin face for pure Cu and ~-brass.
from the analysis that the FS smearing of kll 0 due to disorder is less than 3% of kll O. It is clear from thls example that future high precision 2D ACAR measurements will provide us with much valuable information regarding the electronic band structure of alloys and the stability of alloy phases. With improved resolution it will become feasible to study the smearing of the FS with alloying in non-dilute systems, equivalent to Dingle temperature studies by dHvA measurements at low concentrations. However, the theoretical interpretation of the e+ resuits in disordered alloys is more complicated than in pure metals, in view of the possible preference of e+-s for one of the alloy constituents (20), as well as of the random scattering of the e + in the alloy (26). These effects will have an influence on the shape of p(~), but will have little affect on the position of the FS discontinuities. Experiments on ordered alloy systems have also been performed, leading to interesting results. In particular the FS of the high temperature superconductor V3Si has been studied recently with 2D ACAR techniques (27). 6.
New Developments:
Slow Positron Beams
Before concluding this "minl-review" we would like to mention briefly the most recent development in e+ physics, that of slow, mono-energetlc e+ beams. Using a radioactive e+ source followed by a "fast-slow converter," it has now become possible to produce a beam of variable energy (from zero to several keV) positrons of sufficiently high intensity to be used for experiments (28). The converters are based on the observation that when fast positrons are introduced into a solid, a small fraction is re-ejected from the surface due to an effective negative positron work function (2). The first experiment designed to study the interaction of these slow positron beams with solid surfaces indicated that a high percentage of positrons form posltronlum in colllson with the surface (28); positronium is the hydrogen-like bound state between an e- and a e+. Recently experimental programs have been started in various laboratories designed to develop slow positron interactions with surfaces into a new tool of surface physics (29). The recent experlments, performed in ultra-high vacuum with well characterized surfaces indicate a positron surface state in pure metals, thermally activated positronium emission from this state, and a considerable sensitivity to surface impurities. Low energy positron diffraction experiments from surfaces are also in progress. It is hoped that in the near future it wlll also be possible to use these variable energy beams to study surface defects in metals. 7.
Conclusion and Acknowledgements
We hope that is was possible in this short review to indicate some of the important appllcations of positron physics to various aspects of metal physics and metallurgy. We have mainly
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highlighted the experimental results. Much theoretical work has been performed to understand the quantum behavior of low energy positrons in metals (I). The defect trapping mechanism as well as the surface behavior is intimately tied to problems of the diffusion mechanism of light particles in solids (30). In the near future we can expect to see many new applications to defect studies, high precision studies of alloys by the 2D ACAR technique as well as surface investigations with slow positron beams. The author is indebted to F. Sinclair for a critical reading of this manuscript. References I. For up-to-date reviews see Positrons in Solids, Topics in Current Physics 12, P. Hautojarvi, ed. (Springer-Verlag, Berlln-Heidelberg, 1979); also Positron Annihilation,---Proc. V th Int. Conf. on e+ Ann., R.R. Hasigutl and K. Fujiwara, eds. (The Japan Inst. of Metals, 1979). 2. A.P. Mills, Jr., P.M, Platzman and B.L. Brown, Phys. Rev. Lett. 41, 1076 (1978) and references therein. 3. S. Berko, M. Haghgooie and J.J. Mader, Phys. Lett. 63A, 335 (1977); M, Haghgooie, J.J. Mader and S. Berko, Phys. Lett. 69A, 293 (1978). 4. S. Berko and A.P. Mills, Jr., J. Physique 32, CI-287 (1971) and references therein. 5. I.K. MacKenzie, T.L. Khoo, A.B. McDonald and B.T.A. McKee, Phys. Rev. Lett. 19, 946 (1967). 6. S. Berko and J.C. Erskine, Phys. Rev. Lett. 19, 307 (1967). 7. See for example A. Seeger, Appl. Phys. ~, 183 (1974); M. Doyama and R.R. Hasiguti, Cryst. Lett. Defects ~, 139 (1973); also relevant papers in Ref. I. 8. W. Brandt in Positron Annihilation, A.T. Stewart and L.O. Roellig, eds. (Ac. Press, N.Y. 1967) p. 155. 9. D.C. Connors and R.N. West, Phys. Lett. 30___A, 24 (1969); B. Bergersen and M.J. Stott, Solid State Comm. 7, 1203 (1969). I0. M. Doyama in--Positron Annihilation, Proc. V th Int. Conf. on e+ Ann., R.R. Hasiguti and K. Fujiwara, eds. The Japan Inst. of Metals, 1979), p. 13. ii. O. Mogensen, K. Petersen, R.M.J. Cotterill, B. Hudson, Nature 239, 98 (1972); R.M.J. Cotterill, I.K. MacKenzie, L. Smedskjaer, G. Trumpy, J.H,O.L. Traff, Nature 239, 99 (1972). 12. W.B. Gauster, J. of Vac. Sci. and Techn. 15, 688 (1978). 13. C.K. Hu, S. Berko, G.R. Gruzalski and D. Turnhull, Solid State Comm. 31, 65 (1979). 14. S. Berko in Compton Scattering, B. Williams, ed. (McGraw Hill, London, 1977), p. 273; S. Berko, J. Physique, 39, C6-1568 (1978); see also relevant reviews in Ref. I. 15. S. Berko in Positron~nihilation, Proc. V th Int. Conf. on e+ Ann., R.R. Hasiguti and K. Fujiwara, eds. (The Japan Inst. of Metals, 1979), p. 65. 16. J. Mader, S. Berko, H. Krakauer and A. Bansil, Phys. Rev. Lett. 37, 1232 (1976). 17. A.P. Cracknell and K.C. Wang, The Fermi Surface (Clarendon Press, Oxford, 1973). 18. H. Ehrenreich and L. Schwartz, Solid State Phys., H. Ehrenreich, F. Seitz and D. Turnbull, eds. (Ac. Press, N.Y. 1976), Vol. 31. 19. B.L. Gyorffy and G.M. Stocks in Electrons in Disordered Metals and at Metallic Surfaces, NATO Adv. Study Institutes Series B42 (Plenum Press, New York, 1979), p. 89. 20. S. Berko in Electrons in Disordered Metals and at Metallic Surfaces, NATO Adv. Study Institutes Series B42 (Plenum Press, New York, 1979), p. 239. 21. A. Bansil, H. Ehrenreich, L. Schwartz and R.E. Watson, Phys. Rev. Bg, 445 (1974). 22. R.S. Rea and A.S. DeReggi, Phys. Rev. Bg, 3285 (1974). 23. B. Gordon, W.M. Temmerman, B.L. Gyorffy and G.M. Stocks, Inst. Phys. Conf. Ser. 39, 402 (1978); see also Ref. 19. 24. M. Hasegawa, M. Hirabayashi and S. Kolke, in Positron Annihilation, Proc. V th Int. Conf. on e+ Ann., R.R. Hasigutl and K. Fujlwara, eds. (The Japan Inst. of Metals, 1979), 673. 25. M. Haghgooie and S. Berko in Positron Annihilation, Proc. V th Int. Conf. on e+ Ann., R.R. Haslguti and K. Fujiwara, eds. (The Japan Inst. of Metals, 1979), p. 291. 26. K.M. Hong and J.P. Carbotte, Can. J. Phys. 55, 1335 (1977); C. Koenig, Phys. Star. Sol. B88, 569 (1978). 27. W.S. Farmer, F. Sinclair, S. Berko and G.M. Beardsley, Solid State Comm. 31, 481 (1979). 28. K.F. Canter, A.P. Mills, Jr. and S. Berko, Phys. Rev. Lett. 33, 7 (1974); see refs. therein. 29. A.P. Mills, Jr., Phys. Rev. Lett. 41, 1828 (1978); l.J. Rosen-berg, A.H. Weiss, K.F. Canter in Positron Annihilation, Proc. vt1~--Int. Conf. on e+ Ann., R.R. Haslgutl and K. Fujiwara, eds. (The Japan Inst. of Metals, 1979), p. 883; K.G. Lynn, Phys. Rev. Lett. 43, 391 (1979). 30. T. McMullen, J. Phys. F7, 2041 (1977); W. Brandt and N.R. Arista, Phys. Rev.--Al9, 2317 (1979); R. Paulln in Positron Annihilation, Proc. V th Int. Conf. on e+ Ann., R-$R. Haslguti and K. Fujiwara, eds. (The Japan Inst. of Metals, 1979), p. 601.