/' JOURNAL OF RESEARCH of the National Bureau of Standards - A. Physics and Chemistry Val. 74A, No. 2, M a rch ·April 1970

Summary of the Conference on Electronic Density of States*l > H. Ehrenreich

Division of Engineering and Applied Physics, Harvard University, Cambridge, Massachusetts

(October 10, 1969)

It is very difficult to summarize a conference such as At the outset, let me thank the organizers on behalf this, involving as it did a many sple ndored array of top­ of everyone attending it for providing us with an out­ ics, both experimental and theoreti cal, expounded in no standing scien tific program that clearly focussed on the less than ninety papers. The organization of a con· principal questions which those of us who are grappling ference of such size is ordinarily impossible without with the electroni c structure of conde nsed matter are resort to simultaneous sessions. Whe n there are simul­ facing. During the time when we were not riding buses taneous sessions, of course, it is easier for a si ngle sum ­ or liste ning to papers, there was also a most pleasant marizer because he can only be at one place at one time social program, and mercifully, a few hours for sleep. and therefore can be excused for failing to do justi ce to I would regard the title of the present symposium, half of the pape rs. It is also easier for the audience " Electronic Density of States" a leitmotif rather than because if you permit people to resonate between ses­ an idee fixe because the subject matter presented at sions, you also allow them to become trapped in the the conference in fact was far more general tha n might halls, advertently or inadvertently as the case may be. be implied by the title. Si nce the de nsity of states is However, through the devilish cleverness of the or­ very influential in the determination of many basic ganizers of the present conference, a goodly fraction of physical properties, it provides an excelle nt focal point the papers were delivered by a rapporte ur. Ac­ for presenting some of the more recent developments cordingly, the entire audience, including the sum­ in the electronic properties of condensed matter. marizer were exposed to everything during a period of More importantly (and to my view this is one of the three and a half hard-working and elaborately or­ chief motivations of this conference) the density of ganized days. Furthermore, I lose my excuse for having states is a conveni e nt central qua ntity for confronting overlooked, as I undoubtedly did, some of the impor­ theory and experiment even though , unfortunately, it tant new developments presented or presaged here. never seems to be measured directly by any As it is , in the time of thirty-two minutes that have experiment. In the jargon of the modern theorist one been allotted to me it is, of course, impossible to me n- might phrase this diffic ulty in the following way. The > tion e ven a representative fraction of the contributions. density of states is proportional to the imaginary part of Indeed, even the various areas discussed he re can only the single particle Green's function, whereas many ex­ be sketched in broad outlines. Fortunately my task is periments determine a response function, which in­ considerably eased by the various excellent review lec­ vol ves Green's functions of two or more particles. tures and rapporteur summaries that punctuated the When suitable approximations are made, however, the conference. In order to avoid the risk of offending a state density enters in a fairly direct way into the ;;- few, I have decided instead to offend everybody by not theoretical interpretation of all of the various kinds of mentioning names in this talk, except when referring to measurement described at this conference. work which was not explicitly reported at this con­ The list of techniques available to the solid state ference which is appropriately referenced. physicist, which was extensively sampled here, is truly

'" An in vit ed paper presented at th e 3d Mate ria ls Research Symposium , Electronic Dcns£ty impressive and stands in contrast to the much more I o/S/fl l es. Nove mber 3- 6. 1969. Ga ithersburg, Md . limited variety available to our colleagues in the ele­ ( 1 Supported in part by G ranl No. GP-8019 of th e National Science Foundation and the Advanced Research Projects Agenc y. mentary particle field. We heard about optical absorp- 293 tion and reflectance, x-ray spectroscopy, photoemis­ Indeed, the simpler problem of calculating band sion, Fermi surface experiments, tunneling, measure­ structures for monatomic metals on the single particle ments of the electronic specific heat, magnetic suscep­ picture is still controversial. As we saw in connection tibility, superconducting critical fields, and transport with several of the contributions and much of the properties, positron annihilation, Compton scattering, spirited discussion that followed them, we still don't un­ ion neutralization spectroscopy among others, and how derstand clearly how, when, or why to localize the some of these are influenced by pressure, strain, and exchange interaction. To date no sufficient theoretical temperature. reason has been advanced for preferring either the The preceding list reflects the fact that photons con­ Slater or the Gaspar-Kohn-Sham versions of this poten­ tinue to be one of the favorite probes for studying the tial. Proponents of either point of view, or those favor­ microscopic properties of matter. It is therefore tempt­ ing intermediate values of the coefficients at present ing to use the 1965 Conference in Paris on the Optical usually support their position by comparison with ex­ Properties and Electronic Structures of Metals and Al­ periment rather than basic theoretical arguments. loys [1] which dealt with the same class of materials as In this same connection we might note the debate fol­ the present conference and similar ideas concerning lowing the introductory lecture concerning the relative the interpretations of experiments as a fiducial mark to merits of first principles and pseudo- or model-potential give us some indication of what we have learned about band calculations. The essential point made in that lec­ metals and alloys during the interim. ture, in my view, is that "pseudism" is important Pippard, in his summary of that conference, re­ because it provides eleme ntary insight into the mean­ marked on the extraordinary number of tImes the au­ ing of the results of the elaborate machine computa­ dience was shown the Cu band structure. Since then tions. The two approaches, in fact, are complementary the variety of band calculations, and, in particular the and the proponents of each have a genuine need for, kinds of materials considered has proliferated greatly. and indeed ideally should merge with the other. Pd has received a great deal of recent attention largely There is no doubt whatsoever that band theory has as a result of excellent high field susceptibility and done sufficiently well that it is worth using its results to photoemission measurements. Other examples calculate the density of states accurately. This is a dif­ discussed here involved more exotic materials such as ficult numerical problem, particularly if one resignedly AuAh, EuO and Ge02. Evidently the machinery for accepts spending ceilings that curtail the amount of doing such calculations on ordered alloys containing available computer time, for one needs to know the several atoms per unit cell, some of which are suffi­ energy at many millions of points in the Brillouin zone ciently heavy that relativistic effects become impor­ in order to construct adequate histograms that yield all tant, is now available at several laboratories in a readily the fine detail in the density of states that is often usable form. However, as in all band calculations, even necessary to interpret experimental information re­ if one accepts the Hartree-Fock approximation, the liably. We saw several examples of advances in per­ result obtained is only as good as the potential that is forming such calculations more economically at this used as input information. In metallic alloys one might conference. The QUAD scheme is one of these. expect some charge transfer among the atoms belong­ Another, which I will call the IBM scheme, not after the ing to a single unit cell. In my view this possibility has machine but the workers, is similar to QUAD in its not yet received adequate attention. ability to generate very detailed E(k) curves, but it For example, AuAh has the CaF2 structures and is avoids the use of histograms. All these result in very one of the few metals in which a Raman frequency has finely grained structure in the state density. been observed [2]. Remarkably, its magnitude is In this same connection, I think the importance of similar to that of CaF 2. One might ask whether the very learning how to sum functions of k over constant energy similar stiffness of the optical frequencies in these surfaces in the Brillouin zone efficiently and reliably materials is purely an accident or whether there could needs emphasis. This is important not only for calculat­ be enough charge transfer among the atoms in the cell ing the state density, but also for computing Green's to result in appreciable ionic character. I realize that I functions that are central to the solution of alloy band am undoubtedly not saying anything that is not already problems, frequency dependent dielectric functions familiar to band theorists. However, experimentalists that can be compared with optical data, susceptibili­ should be warned that the construction of alloy poten­ ties, and many other quantities. While many theorists tials, even for ordered systems, is still a problem that may think such problems as insufficiently dignified, I requires attention. would, nevertheless, stress that their solution is imp or- 294 tant if one ever expects to confront theory and experi­ tion supplied, even what would in a simple minded view me nt reali sti cally on a more complete bas is for more correspond to the same theore ti cal quantity, often dif­ complicated ystems_ fers from experiment to experime nt. The Fermi surface At the same time le t me temper this call to computer effective mass determined from electroni c specifi c heat and numeri cal analysis handbooks by reminding you of and Pauli susceptibility measureme nts is an example the obvious fac t stressed by many speakers that band of this_ calculations are single particle descriptions involving Even within a sin gle particle fram ework, the state electrons or holes as ideal quasi-particles which in­ density function is only characte ri sti c of a particular teract with a self-consistent field that is in practice type of experiment. For example, in x-ray e mi ssion ex­ determined more or less self-consistently_ However , periments, optical selection rules pick out only those due to electron interactions including those involving compone nts of the valence band state density having phonons, real quasi-particles acquire finite Lfetim es, appropriate symmetries with respect to the core hole_ except right at the Fermi s urface_ Some of the pa pers We saw that this fact has particular utility in providing presented here refle cted the fact that methods of taking in sight into the c haracter of the wave function overlap > quasi-particle effects into account more syste matically and hybridization among different components in both in band calculations are now bein g developed _ I would ordered and disordered alloys_ Since the core hole is lo­ look towards greater exploi tation of such techniques in calized in a given atomic site, the stud y of say the

the band calculations of the near future_ It is important AI-L 2 ,:1 emission spectrum in systems such as AuAI 2 to reme mber that when we speak of dressin g effects, and others discussed he re, provide an indication of the say, due to electron-phonon interac tions, we ought to be amount of d wave function in these systems located on dressing the ri ght bare object, namely the correctly cal­ the Al sites_ c ulated quasi-particle appropriate to the stati onary As anothe r example, we might mention the k-conv er­ lattice_ sati on rule entering interba nd optical processes which One should again be reminded of the fact that very implies that the state density appearing in theoretical fe w if any of th e experime nts di scussed at this meeting expressions for the opti cal consta nts is the so-called correspond to creation of just one quasi-particle near "joint density of states_" This seemingly innocuous fact the Fermi surface_ This fact was also stressed already has led to a spirited controversy in connection with the ~ at the Paris Conference_ Optical experiments, for ex­ interpretation of photoemission experiments which was ample, correspond to the c reation of two quasi-particles already in full bloom at the Paris Conference_ As you all and ion ne utralization measure me nts to three_ These know by now, there are two schools of thought whose may interact with each other as well as with the other proponents we might call the k-conservationists and the particles in th e system_ This was illustrated in the k-nonconservationists_ The latter group has main­ di scussion of the various types of phenomena that can tained, on the basis of a considerable body of experi­ occur in x-ray emission, whic h lead to the conclus ion mental evidence, that particularly in materials having that the observed spectrum of the valence band may narrow bands s uch as the noble and transi tion metals, bear less resemblance than one would hope to what is the energy distribution of the photoemitted electrons calculated from band theory_ In addition to the long should directly reflect the structure in the density of famiuar Landsberg or Auger tails that s mear out the states pertaining to these bands_ The k-conservationists lower valence band edge, there are recently predicted on the other hand have asked, "Why should this con­ > elementary excitations suc h as the plasmaron and other servation law be violated?" Indeed, one of the papers, broad structures that also result from interactions with which represents the first altern pt at the form ulation of plasmons_ Another effect that was reported on here a systematic theory of the photoelectric effect in souds, results from exciton type interac tions between elec­ points to ways in which this might come about. trons near the F ermi s urface a nd the core holes with Several of the other contributions point to progress which they co mbine in an x-ray emission process_ towards a reconciliation of these viewpoints_ For exam­ ;, These may strongly affect the transition rate and lead ple, we have heard in connection with Cu that a direct to substantial enhanceme nt or dimunition of the ob­ transitions analysis using constant matrix elements ac­ served inte nsity near the Fermi surface_ As has been counts quite well for the observed energy distribution_ pointed out to us, these effects must be quantitatively Similar conclusions hav~ been reached on the basis of understood before information concerning band struc­ very detailed calculations for Pd_ The essential point, ture can be reliably extracted from such experiments_ which was emphasized by both camps, is that the stron­ > As a result of particle interaction effects, the inform a- gest peak in the joint density of states coincide with 295 peaks in the calculated state density, particularly in the parison between the results obtained by different types case of narrow valence bands. of experiments. There was only one noteworthy excep­ While in many cases the photoemission technique is tion, which was concerned with efforts to confront a very useful tool, this may not be the case universally. Knight shift data with those of soft x-ray emission ex­ It was suggested, for example, that it is less successful periments. There is a real need for more such detailed in providing information concerning f states in the Eu comparisons, even on the basis of band theory alone. chalcogenides and rare earth metals since these states According to a paper count, superconductors and are seen to give rise to abnormally low quantum yield semiconductors received less attention than the simple, relative to, for example, p states. It is also clear that the transition, and rare earth metals. However, there are variation of optical matrix elements with energy and good reasons for mentioning them even in this broad selection rules, which can also lead to structure in the summary. As appropriate, superconductivity was not observed spectra, needs further attention because in discussed as a phenomenon, but rather as a tool to ex­ many calculations this matrix element is still regarded tract information relevant to the state density. On the to be a constant. positive side, we heard how strong coupling theory can Before leaving the subject of optical properties of be used together with other measurements to obtain the crystal, two other points are worth making. Despite the electron-phonon coupling constants, and how measure­ fact that one learns only about the joint density of states ments of the critical field at very low temperatures can in such experiments, it is, in fact, possible to derive the be made to yield the electronic specific heat as a func­ conventional state density from optical data by a more tion of pressure with high accuracy. On the negative circuitous route. The usefulness of differential side, it was pointed out in connection with a general reflectance techniques is now well establishpd and was review of the information provided by tunneling experi­ illustrated in several of the contributions presented ments, that such measurements for superconductors do here which even extended to the x-ray case. Informa­ not really tell us all that much about the normal state tion from such measurements can be used as input for properties of metals and semiconductors. pseudopotential band calculations or those based on It is clear from the exquisitely detailed interpretable the k . p approximation. The problem of constructing information being currently obtained from cyclotron potentials which plague first principle band calcula­ resonance, magneto-optical, and even nonlinear optical tions is thereby avoided. Since the secular equations data in simple semiconductors and semimetals, that for such problems are generally smaller, they can be our understanding of these materials is still in a solved at sufficiently large numbers of points in the somewhat more mature state than that of most metals. Brillouin zone to obtain the state density. Of course this applies only to the classical and long stud­ The other point concerns another recent develop­ ied materials like Ge and InSb and not to amorphous ment. A fact that has been distressing to many theorists semiconductors which were discussed in only a single is that, while in semiconductor calculations of optical paper. However, the very beautiful interplay and agree­ coefficients there was always good agreement between ment between theory and experiment must still be re­ theory and experiment in regard to both the position garded as serving as a standard of excellence which and magnitude of the observed structure, this has not solid state physics in general must continue to emulate. been the case in metals. For the case of Al we saw quite Surfaces also were not discussed extensively here, convincing evidence that such discrepancies are on the largely, I think, because the theoretical ideas and point of disappearing, largely as a result of better calcu­ techniques for dealing with such problems in realistic lations which deal more adequately with the k-depend­ systems are only beginning to be developed_ However, ence of the momentum matrix element. Indeed, other some very promising experimental techniques, notably recent investigations have shown that electron-electron ion neutralization spectroscopy and resonance tunnel­ scattering effects, which lead to vertex corrections that ing which shed light on the nature of surfaces and im­ might be expected to be stronger in metals than portant phenomena like chemisorption were described. semiconductors, are, in fact, very weak in this material This area will surely see a great deal of activity in the [3] _ Even though the so-called Mayer-El Naby near future_ resonance is probably no longer with us [4], our un­ Since the discussion of a large variety of alloy derstanding of the alkali metals is unfortunately still systems occupied so much of the conference, let me not in as good a shape. conclude this summary with some remarks concerning It is regrettable that relatively few papers presented this subject. We were exposed to a wide variety of data at the conference attempted to provide a detailed com- concerning many alloy systems, most of them involving 296 tranSItIOn metals. Certainly there are many more than volved the so-called average t-matrix approach which were consid ered at the Paris Conference. Howeve r, Soven [7] applied to brass. An extension of such calcu· this compariso n is somewhat unfair since even now, lations was di scussed here. Subsequently, Soven [8] with one or two notable exceptions which were formulated a more ge neraJ elf-consistent effective presented here, there is still a dearth of optical informa­ fi eld approach that he termed the coherent pote ntial tion concerning disordered alloys. This is to be con­ approximation which is more general than the other. trasted with the situation involving specific heat and While this was mentioned in severaJ of the papers, it is transport measurements about which we heard a great perhaps worthy of some additional commentary .,. deal. Indeed, most of the papers dealing with the elec­ because of its possible applicability to realistic alloy tronic specific heat were concerned with alloy systems. systems. A most interesting effect that was described dealt with In this approximation the alloy is replaced by an ef­ the recently di scovered magnetic clusters in NiCu al­ fective medium described by a single particle non-Her­ loys which can make their own appreciable contribu­ mitian and complex Hamiltonian which, however, is tion to the specific heat. This conjecture is quite new still periodic in the case of substitutional alloys. The ) and deserves detailed theoreti cal treatment. There was self·consistency condition determining this Hamiltoni­ also an intriguing discussion concerning rare earth an is simply that an effective electron wave travelling metals which raised the question as to whether the f through the crystal which impinges on an atomic site electrons could possibly be at least partially itinerant in suffers no further scatteri ng due to the random some of these systems. character of the crystal potential. Put another way, the One fairly obvious thing that needs emphasis in con­ effective wave, just like a Bloch wave in a crystal, is not nection with these papers and that was stressed in a scattered by the atoms. However , unlike the Bloch number of them is the need for data of single crystal wave, the effective wave may be damped as it specimens having known phases, and how crucial it is propagates through the crystal. The present limitation to avoid samples involving mixture of phases. Without of this description is that it is only applicable to certain these precautions, the overanxious theorists will, as classes of Hamiltonians in which the random character they did in the case of the Mayer-EI Naby anomalies, is cell localized. The theory has the virtue of correctly find themselves in the awkward position of explaining reducing to the known results for s mall impurity con­ what Pippard already warned in 1965 might be non­ centrations and arbitrary scattering strengths on the facts. one hand, and for arbitrary concentrations but s mall As we saw, many of the experimental techniques ap­ scattering strengths on the other. It interpolates in a plicable to pure metals are relevant for di sordered al­ physically reasonable way between these limits, yield­ loys as well. We have already mentioned optical and x­ ing results that are valid for arbitrary alloy concentra­ ray data in this connection. The fruitful and relatively tions and reasonably strong scattering strengths. It fails easily interpretable Fermi surface experiments, alas, in predicting band tailing effects, experimental seem to be much more difficult for many alloy systems. evidence for which we heard described here in connec­ However, measurements such as those involving tion with semiconductor tunneling experiments. Also, positron annihilation which also probe the Fermi sur­ it does not yield strictly localized states except in the face geometry are not restricted by such criteria. They limit of very small impurity concentrations. have already been very successfully used to investigate We should note parenthetically that while the ex· > detailed Fermi surface changes in Cu-Al [5] and as we istence of such states near band edges is generally be­ heard here, to brass. lieved, some questions were raised here as to whether A great deal of progress in this area since the Paris or not such states can exist in the middle of a tight bind­ Conference has come along the theoretical front. Until ing band and whether or not the frequently made a few years ago the only theoretical models available for hypothesis that there exists a sharp demarcation describing alloy behavior involved perturbation theory, between localized and nonlocalized states is correct. As the virtual crystal, or the ri gid band models. However, we were reminded, localized states such as those due recently a number of rather effective techniques based to f-electrons, can exist even in periodic systems when on scattering theory have been adapted to this problem the Coulomb interaction is sufficiently strong. As was and implemente d by calculations for both model and shown, conventional band descriptions break down reali s ti c systems. These all transcend the earlier, more under these circumstances. limited approaches. The first incisive contributions to To obtain the effects omitted by the coherent poten­ electronic theory, made by Edwards and Beeby [6], in· tial description of di sordered alloys it is necessary to 297 allow for the possibility of statistical clustering effects. To give these remarks a sharper focus, I should like I This is a much more difficult proble m. But as we heard to show you by means of one example the results of an in two of the papers presented here, some very promis­ application of this theory to C uNi alloys. Figure 1 ex­ ing progress is beginning to be made in these hibits S. Kirkpatrick 's calculations [10] for the density "( directions. Indeed, the early work of L M. Lifshitz [9] of states of these alloys and also the results of has already given us an indication of the sorts of res ults photoemission experiments by Seib and Spicer [11] , to be expected. which, for reasons already mentioned, should only be Because of the previously stated limitations, the compared qualitatively with the theoretical results. It coherent pote ntial theory in its present form is strictly should be emphasized that these calculations do in­ speaking applicable only to isoelectronic alloys like volve approximations, most of the m probably not too GeSi, where the random part of the potential is substan­ serious. Since it would be inappropriate to discuss tially confined to the core region at each site, or to 3d these in the present context, I would like to confine my transition-noble metal alloys in which the d-states that remarks to a few brief comments. The first is that the are most affected by the disorder are substantially coherent potential approximation is e vide ntly applica­ localized. ble to quite complicated density of states functions

~

I

~ --2= = ="-- f-= ='-'-' :=C> PURE Ni PURE Nt <./)-- = f-= ~ '-'-' 10 % Cu II %Cu :>-

<./) '-'-'= 20 % CU 19%Cu = '-'-' ::::f- <./) ~

..J 30 % CU «: <..> f- =a.. ~--~L-----~----~-----L----~-4 0%Cu 39%Cu --~----~------~----~----~--~ -.4 -.2 - .1 -4 ENE RGY (Ry)

CPA CALCUL AT ION PHOTO EMIS SION RE SULTS FOR NiCu FOR Ni Cu (S eib and Spice r )

FIGURE L The hybridized tl state densities , calcu.lated in the coherent potential approximotion, are compared with the optical state density obtained from. photoemission eiperiments [10 , /1]. 298 which include d egenerate d-bands and hybridization which will place the theory of alloys on a firmer footing. with conduction bands. Second, the only input informa­ After these somewhat discursive concluding re­

) tion needed for ni ckel-rich alloys is the hybridized marks, let me close the conference by once again ni ckel state density, the positions of the resondnt Ni th anking everyone; organizers, s peakers, and rappor­ and Cu d-l evels which give rise to the d-bands, and the teurs, questi oners, commenters, and li ste ners for hav­ concentration. Third, the distortion in the state density ing made it as stimulating as it turned out to be. curve with increased alloying shows the rigid band model, which has been particularly popular for this References alloy system, does not really apply. This is seen even more clearly in the results of calculations of the mag­ [1] The Proceedings of that Conference were edited by F. Abeles netic properties [12]. Finally, the prominent calculated a nd published under that title by North·Holland Publishing Company (A msterda m) in 1966. structures and their behavior is qualitatively in accord [2] Feldman, D. W. , Parker, J. H., and Ashkin, M., Phys. Rev. Let· with the experimental observations. The principal t ers 21, 607 (1968). peaks remain stationary, but they change in intensity [3] Beeferma n, 1., and Ehrenreich, H., (phys. Rev. , to be > and s hape in both the calculations as well as the experi­ published ). [4] Ma ye r, H. , and EI Naby, M. H., Z. Pl,ysik 174,289 (1963); ment. The contribution of copper to the state density S mith , N. V. , Ph ys. Rev. Lett e rs 21, 96 (1968). turns out to be broad and relatively structureless and [5] Fujiwa ra, K. , Sueo ka, 0 .. and Imu ra , T., J. Ph ys. Soc. Japan comes principally from the lower regions of the d·band. 24, 467 (1968). While this kind of theory predicts a wealth of detail, [6] Beeby , J. L. , Proc. Roy. Soc. (London) A279 , 82 (1964) and its quantitative validity remains an open question that cited papers. [7] Soven, P. , Phys. Rev. 151, 539 (1966). must be explored further. More important, some of its [8] Soven, P. , Phys. Re v. 156, 809 (1967). present limitations must be overcome to render it ap­ [9] Lifshitz, I. M., Soviet Physics- Uspekhi 7, 549 (1965). plicable to a wider class of alloys. However, given the [10] Kirkpatric k, S., Vel icky, B. , a nd Ehre nreich, H. (to be publi shed) .. improving theoretical and experimental situation that [11] Seib, D. H., and Spicer, W. E., Phys. Re v. Lelters 22, 711 is clearly evidenced from this confe rence, it seems (1969). clear that at the next meeting of this type we will surely [12 ] Kirkpatrick, S., Ve l ic ky, B. , La ng, N. D. , a nd Ehre nreich" hear about further, more extensive developments H., J. Appl. Phys . 40,1283 (1969).

(Paper 74A2 - 601)

299