Electronic properties from analysis of the harmonic content of the de Haas-van Alphen effect:...

Journal o[ Low Temperature Physics. VoL 32, Nos. 3/4, 1978

Electronic Properties from Analysis of the Harmonic Content of the de Haas-van Alphen Effect:

Application to Dilute Magnetic Alloys*

Yun Chung, D. H. Lowndes,t and Catherine Lin Hendel

Department of Physics, University of Oregon, Eugene, Oregon

(Received September 29, 1977; revised January 11, 1978)

A new technique, third harmonic de Haas-van Alphen (dHvA) wave shape analysis, is described for measurement of the spin-dependent interactions between conduction electrons and local moments in dilute alloys. We derive expressions for the harmonic content of the dHvA effect in a general case, including simultaneous contributions due to (1) magnetic interaction (MI or Shoenberg effect), (2) the spin-dependent scattering (SDS) of conduction electrons, and (3) exchange energy shifts in their Landau levels due to local moments, in addition to the usual Lifshitz-Kosevich harmonic content. The effects of M I and SDS mix nonlinearly in determining the observable amplitude and phase of each resultant dHvA harmonic. One important consequence of this mixing is that the observation of a spin splitting zero of the dHvA amplitude is not indicative of equal scattering rates for spin-up and spin-down electrons, in the presence of ML These techniques are then illustrated by applying them to studies of the dilute alloy systems Au(Fe ) and Au (Co), both of which are found to exhibit local magnetic moments, though apparently of quite different origin. For Au(Fe) the exchange energy shift (exchange field) and the spin-dependent scattering rates were determined as functions of (H,T). A 3:1 anisotropy of spin-up and spin-down electron scattering rates was observed for the (111) neck orbit. For A u ( Co ) we report the first dHvA observations of interaction effects between impurities, via measurements of the spin-dependent scattering of conduction electrons by magnetic pairs of Co impurities. We conclude that the dHvA effect appears to be a sensitive probe for determining impurity spin behavior in a magnetic field, and for measuring cyclotron orbitally averaged values of the exchange constant Jo,bit in very dilute local moment systems. Similarly, the ability to resolve spin-dependent information allows the onset of solute interactions leading to magnetism to be observed at very low solute concentrations. *Research supported by NSF research grant no. DMR-07652 AO2. ?Guest Professor of Physics, Physics Laboratory and Research Institute for Materials,

University of Nijmegen, Nijmegen, The Netherlands.

599

0022-2291/78/0800-0599505.00/0 �9 1978 Plenum Publishing Corporation

600 Yun Chung, D. H. Lowndes, and Catherine Lin Hendel

Finally, the possibility of applying these same third harmonic wave shape analysis procedures for the measurement of conduction electron orbital g- factors in metals is briefly discussed, and one example of such a measurement for a A u ( A g ) dilute alloy, is given

1. INTRODUCTION In 1956 Lifshitz and Kosevich 1 derived an expression for the de

Haas -van Alphen (dHvA) oscillations in the magnetizat ion of a pure metal single crystal which is essentially that in use today:

~ l = - r f i ~ C, Drcos (TrrS ) s in[2rrr (F /H-y )+pTr /4] (1) r = l

in which r is the d H v A harmonic index, and

vTF 1 Cr - (rgrH)l/2 2 sinh (rAt ,T /H)

D r = e -Xru~X/ I - l

S = lzgc/2 (2a)

u = 4kB(2~r)-l/2(e/hc)3/2= 1.304 • 10 -5 Oel /2 /K

A = 2rrakBmoc/eh = 146.9 k O e / K

where F is the d H v A frequency (=hcAext/2rre; Aext is the extremal cross-sectional area of the Fermi surface normal to the magnetic field), y is the Onsager phase factor (= �89 for free electrons), p = - 1 (+1) if the extremal Fermi surface cross section is a max imum (minimum) with respect to

2A 0 : variation of ka, ~ is the Fermi surface curvature factor, 10 ~xt/ knl, ~ is the cyclotron effective mass (in units of the free electron mass), kB is Bol tzmann 's constant,

rfi = I 2 1 - 1 0 F A 1 OF ~ (2b) F ~ O FsinO 0~b~b

gc is the electronic cyclotron orbitally averaged g-factor, and X is the Dingle temperature . 2 X is the cyclotron orbital average of the electronic scattering rate ~'~ at the point k on the Fermi surface, 3-5

X = (h/2rrkB)(1/~k) = F/rrka (3)

where (F) is the broadening (assumed to be Lorentzian) of the Landau levels. 2,5

Equat ions (1)-(3) demonst ra te that a number of different parameters describing the propert ies of electrons in metals can influence the amplitude

Analysis o | the Harmonic Content of the de Haas-van Alphen Effect 601

and phase of the dHvA oscillations. Since only those electrons lying on narrow orbits, at the extremal cross sections of the Fermi surface, produce observable oscillations, then Eq. (1), in principle, provides a way of comparing experimental and theoretical values for a variety of electronic properties, for electrons in well-defined states on the Fermi surface. This is an important advantage of dHvA measurements in comparison with bulk transport coefficient measurements, which provide only a weighted average of electronic properties over the entire Fermi surface.

From an experimental point of view, the greatest significance of the LK expression for M is that it is a harmonic series: Each term in the series depends in a complicated way on a wealth of electronic parameters. It is not possible to determine all of these parameters from measurements of the amplitude and phase of a single harmonic alone.

An additional complication is that Landau level broadening (primarily due to electron scattering by impurity atoms and by crystalline defects such as vacancies and dislocations) and the thermal broadening of the Fermi surface contribute to an approximately exponential attenuation of the higher harmonics:

D" rAtz(T + X ) 157"/'~1 oc ~ e x p (4)

2 sinh (rAt~T/H) H

Thus, to obtain information from the dHvA harmonics one is required to carry out accurate measurements of the (weak) harmonic amplitudes and phases in the presence of the strong fundamental oscillations. Such precise measurements of the higher harmonics may not even be possible, using presently available magnetic fields, in the case of orbits with large electronic mass or large Dingle temperature.

A final complication is that the LK expression (1) correctly describes the observed dHvA harmonic content only in the case of dHvA frequencies less than about 100T. For dHvA frequencies greater than 100T, magnetic interaction (MI) of the conduction electrons with their own oscillatory magnetization produces waveform distortion and appreciable changes in the higher harmonic content (r->2), though not in the fundamental (r = 1) oscillation itself. For dHvA frequencies in the range 102-104 T the changes in harmonic content produced by this MI (or Sho- enberg) effect 6 are comparable in magnitude to the harmonic amplitudes given by the LK expression, and for frequencies ~ 1 0 4 T the MI harmonic content is completely dominant.

For all of these reasons the principal applications of the dHvA effect for the measurement of electronic properties have until recently b een almost exclusively those using information obtainable from the fundamental oscillations alone. Indeed, because of the difficulty of measuring the


absolute amplitude of dHvA oscillations, such measurements have until very recently been limited to the information obtainable by measuring changes in the dHvA amplitude as a function of H and T. Such measurements, together with measurements of the fundamental dHvA period, do allow one to determine electronic effective masses, Dingle temperatures, and Fermi surface extremal areas, from which maps of the variation over the Fermi surface of point properties (Fermi velocity, Fermi surface radius vector, and point electronic lifetimes) may be constructed. For each of these three types of measurement, sophisticated experimental and analy- tical techniques have been developed, with some of the differentiation among techniques depending upon whether one was working with pure metals, metals containing defects (e.g., lifetime measurements in dilute alloys containing nonmagnetic impurities), or intermetallic compounds.

However, it is clear that much of the most detailed and fundamentally interesting information present, in principle, in the dHvA waveform is almost completely inaccessible if one is restricted to measurements of the fundamental (r = 1) oscillations alone. For example:

(1) Several years ago Shiba 4 showed how the theoretical expression for the dHvA effect was modified by the presence of spin-dependent conduction electron self-energy terms, due to interactions with local magnetic moments. The real parts of the ('I', ~, )-spin electron self-ener- gies cause shifts in the dHvA frequency and of the effective g-value (the latter is also describable as an effective exchange energy shift). The imaginary parts of the self-energy appear as spin-dependent parts of the total electronic scattering rate (up- and down-spin Dingle temperatures). Because the effective g-values and electronic scattering rates are generally expected to be both field and temperature dependent , and up- and down- spin scattering rates are usually unequal, "Dingle plots" (using the fundamental dHvA frequency) cannot be used to determine electronic lifetimes.

Shiba's theoretical work was actually preceded by pioneering experimental work in which anomalies in the dHvA effect in dilute alloys containing transition metal solute atoms were first detected. In a series of papers, Paton, Hedgecock, and Muir showed how the energy-dependent relaxation time associated with the Kondo effect could be included in the expression for the dHvA effect. They later reported a magnetic field and temperature dependence of the Dingle temperature for the electrons on the "needle" orbit in Zn(Mn) dilute alloys, 52 and pointed out the connection between their measurements and the field dependence of the electronic scattering rate resulting from the freezeout of spin-flip scattering in the magnetoresistance. This work has recently been extended in the measurements of Li 53 and the theoretical calculation by Harris et al. 54 for

Analysis of the Harmonic Content o! the de Haas-van Alphen Effect 603

Zn(Mn) and Zn(Cr). Paton also showed how localized spin fluctuation effects could be detected using the dHvA effect, via measurements of changes in the cyclotron effective mass and the Dingle temperature in dilute AI(Mn) alloys. 55

Studies of similar effects in noble metal-based dilute alloys began with Coleridge and Templeton's observation of a beat minimum in the field dependence of the dHvA amplitude in Cu(Cr). s6 This work was followed by Coleridge, Scott, and Templeton's (CST) systematic study of the dHvA effect in dilute alloys of Cr, Mr, Fe, and Co in Cu. These authors demonstrated the existence in general of spin-dependent scattering (SDS) and exchange energy shifts, using an experimental technique based on observations of shifts in the positions of spin splitting zeros (or minima) on the Fermi surface, as a function of solute concentration, and measurements of the depth of the spin splitting minima. Very shortly after CST's work, AUes eta/ . 8"9 showed that it was possible to obtain spin-dependent information (regarding both scattering rates and exchange energy shifts) from measurements of the harmonic content of the dHvA effect, via measurements of relative amplitudes and phases of the first and second dHvA harmonics.

Unfortunately, neither CST's nor Alles' technique has widespread application. The spin splitting technique is limited to very dilute alloys, and to those locations on the Fermi surface where spin splitting minima happen to occur (essentially accidentally), The second harmonic analysis used by Alles neglects the harmonic content due to magnetic interaction, and also provides only two measured quantities from each data block at (H, T). Since in the presence of local moments there are normally at least three parameters (up- and down-spin scattering rates and the exchange energy shift) whose combined (H, T) dependence is sought, Alles had to resort to a linearized model and a fitting procedure to determine average (H, T) dependences.

(2) dHvA measurements of conduction-electron orbitally averaged g-values have recently proven to be a sensitive probe of departures of the electronic structure of metals from purely "band-structure" behavior. 1~ In the alkali metals K, Rb, and Cs, large departures of the g-value (from 2.00) have been interpreted as many-body effects, and values for some of the expansion coefficients in the Landau theory of isotropic Fermi liquids have been inferred. 1~ In Au, g-values have recently been measured for the first time over a complete nonspherical Fermi surface14; the large anisotropy (a factor of 2) in the observed g-values seems likely to find its explanation in a combination of one-electron relativistic effects (spin-orbit coupling) and (perhaps) many-electron exchange enhancement. All such effects are currently of great interest. It now appears likely that the relative


importance of one-electron and many-body effects can be assessed through a combination of band structure calculations, dHvA measurements, and bulk measurements such as conduction electron spin resonance. dHvA measurements provide the most detailed information at the present time, since they refer to only an orbital average, rather than an average over the entire Fermi surface. In experiments to date, the most precise and reliable dHvA g-value measurements have come directly from measurements of the dHvA (first and second) harmonic content, either alone or in conjunction with the initial calibration step in using dHvA "absolute ampli tude" measurement techniques. 13'14

Because of the possibility of obtaining detailed, fundamental information about electronic interactions with local moments from the harmonic content of the dHvA effect, on the one hand, and in view of the limitations of existing techniques, on the other hand, we have developed new techniques for "unfolding" this information from the first three harmonics of the dHvA effect. In comparison with the two dHvA techniques used in earlier studies of local moment-conduct ion electron interactions in dilute alloys, the third harmonic technique has the advantage that numerical values for four independent "observables" (the harmonic amplitude ratios M2/M1 and M3/Mb and the relative harmonic phase angles 2 0 1 - 02 and 3 0 1 - 03) may be determined from analysis of a single, short block at field H and temperature T. Thus, unique values for spin- dependent scattering rates and exchange energy shifts may be determined, and their changes with (H, T) mapped out. Because the theoretical parameters are usually overdetermined, accurate estimates of the effect of experimental errors on each of their values can also be obtained.

In this paper we present two principal results. First, the derivation of expressions for the harmonic content of the dHvA effect in the general case, including contributions due to magnetic interaction, spin-dependent scattering, and exchange energy shift, in addition to the usual Lifshitz- Kosevich harmonic content [Eq. (1)]. In particular, we show that the effects of MI and SDS mix nonlinearly in determining the observable harmonic content. Second, we derive equations which express the "observables" of wave shape analysis in terms of the microscopic electronic properties of interest (spin-dependent scattering rates and exchange energy shifts) and we show how these equations may quite conveniently be solved by graphic techniques, using (for example) a laboratory computer interfaced to a plotter. These results are then illustrated by applications to the dilute alloy systems Au(Fe) and Au(Co), both of which were found to exhibit local moments, though apparently of quite different origin. Spin- dependent scattering rates of conduction electrons and the exchange energy shift in their Landau energy levels are determined.

Analysis o! the Harmonic Content o! the de Haas-van Alphen Effect 605

Finally, we also point out the applicability of these same equations and harmonic analysis techniques (suitably simplified) to pure metals and intermetallic compounds, for example, for orbital g-factor measurements. I 'he simplified equations then provide a connection with, and generalization (to include third harmonic information) of the second harmonic techniques which have been used for g-factor measurements by Randles, 1~ Alles et al., ~ Crabtree, 13'~4 and Knecht. 12 We give as a simple illustration of our technique a g-value measurement for the (111) neck orbit in the nonmagnetic dilute alloy Au(Ag). We defer the discussion of a more extensive application of these technques, for measurements of orbital g-factor anisotropy, to a later paper.

2. E F F E C T OF L O C A L MOMENTS ON THE d H v A W A V E S H A P E

2.1. Exchange Energy Shitt and Spin-Dependent Scattering

It is well known that the observed dHvA oscillations are the coherent superposition of two sets of osciUations, 15 arising from the Zeeman-spli t spin-up and spin-down electronic Landau levels. This Zeeman splitting corresponds to phase shifts +(~rS) of the two sets of dHvA oscillations, leading to the appearance of the "spin splitting factor" cos (~rS) in the resultant rth harmonic dHvA amplitude. Thus, spin-dependent information is contained within the dHvA wave shape, but for ordinary metals is present in an almost trivial way (Fig. 1).

However , in dilute alloys containing local magnetic moments (for example, certain d-shell transition elements dissolved in the noble metals) much additional information about the spin-dependent interactions between conduction electrons and local moments is contained, in principle, in the dHvA wave shape. The local moment is associated with d states within the impurity's Winger-Seitz cell (assumed to be substitutional). 16 However , these localized states are embedded in the energy continuum of conduction electron states, and so are not bound states, but are "mixed" with the conduction states of the host metal and have a finite lifetime. This mixing interaction, occurring in the neighborhood of a local moment cell, can affect the energy levels of conduction electrons in a magnetic field in two ways: by producing an exchange energy shift Ae =/zBHex in the relative separation of the spin-up and spin-down Landau energy levels (an effect on the real part of the electronic self-energy) and by producing unequal scattering rates for the spin-up and spin-down electrons (an effect on the imaginary part of the electronic self-energy). 17-19


n

_ _ 1 _ _ _

l ( a ) ( b ) ,

Fig. 1. (a) Zeeman splitting into two sets of (1", $)-spin Landau levels; (b) modified Landau level splitting due to opposite exchange energy shifts of (#, $)-spin electrons interacting with dilute magnetic impurities. The presence of spin-dependent scattering is also indicated schematically via the unequal level broadenings (X~>Xt). The case of antiferromagnetic coupling (Jr < 0) is shown.

The spin-dependent relative shift of the electronic energy levels can be represented by an effective exchange Hamiltonian. 2~

~ x = - c * J S . s (5)

Here c* is the fractional local momen t concentrat ion (not necessarily the same as the impurity concentrat ion c if impuri ty- impuri ty interactions are important) , and J < 0 (>0) for anti(ferromagnetic) exchange. In an applied magnetic field the impurity m om en t becomes polarized and the change in the splitting of up- and down-spin electronic energy levels is then

a e = t - < eox> *} = -c*J<sz> = + c * l / l < s z > (6)

where the last expression assumes antiferromagnetic exchange. The opposi te (1', $)-spin exchange energy shifts in Eq. (6) correspond to opposi te shifts in the absolute phases of the (1', ,~)-spin contributions to the d H v A oscillations, and a modification of their Zeeman splitting (Fig. lb). For this reason, it is convenient to introduce an "effect ive" g-factor and an "effective exchange field" Hex defined so that the combinat ion of Zeeman and exchange energy shifts is

m E o . = 1 _ _ 1 ! ~o'(gdZBH + Ae ) = ~o'gdZBH (7)

where we define

g" = (go + H~x / H) , Hex = ( A e / . B ) (8)

Analysis o | the Harmonic Content oi the de Haas-van Alphen Effect 60"/

Note that g'c and gc are cyclotron orbitally averaged g-values and that g" (<go)> gc for (anti)ferromagnetic exchange.

The scattering rate of conduction electrons by impurities that carry a local moment also becomes spin dependent in an applied magnetic field, essentially because the impurity possesses an internal degree of freedom, its spin projection quantum number, and spin-flip scattering processes can occur. However, an applied magnetic field polarizes the impurity spin, leading to unequal spin-flip scattering probabilities for spin-up and spin- down conduction electrons. 21 These spin-dependent scattering channels can account for a large fraction of the total scattering rate, as is demonstrated by the large, negative "anomalous" magnetoresistance of dilute alloys such as Au(Fe), which results from the "freezing out" of spin-flip processes at low temperatures and in strong magnetic fields. 22 The presence of spin-dependent scattering (SDS) corresponds to unequal broadening of the up-spin and down-spin Landau levels (Fig. 1) and to different dHvA Dingle temperatures for spin-up and spin-down electrons on the same extremal Fermi surface orbit:

X" = ( h / 2 ~ r k ~ ) ( z - ~ ) ~ (9)

It is most convenient to instead define sum and difference Dingle temperatures X" and 8X,

2 = (X* +XS)/2, 8 X = (X t - X * ) / 2 (10)

since we shall see that 8X is then a measure of the spin-dependent part of the electronic scattering rate. (i.e., the difference of up- and down-spin scattering rates is just 28X).

In the presence of SDS the amplitudes M~ and M, ~ of the up-spin and down-spin components of the rth harmonic dHvA oscillations become unequal, so that a shift in the absolute phase of the resultant rth harmonic oscillation occurs, together with a change in its amplitude (Fig. 2). From

Fig. 2. Phasor diagram illustrating formation of the rth harmonic Lifshitz-Kosevich (LK) contribution to the magnetization, as the resultant of unequal spin-up and spin-down contributions in the presence of spin- dependent scattering (SDS), and in the absence of magnetic interaction (MI). d~, contains the dHvA phase shifts due to both Zeeman and exchange energy shifts of the Landau levels, with increasing dHvA phase plotted counterclockwise. Note that q', ~ q, if the exchange shift Ae is large enough to shift S through an rth harmonic spin-splitting minimum. AS, is given by Eq. (21a).

M ~ 1 i-" D r= - r / \ l x r x = )" ~'r t--

/ \ \ kAOB(LK)

Ae -/q' -qrl

/ Cr = Tl:r


measurements of the relative harmonic phase angles and relative harmonic amplitudes, values for P(, 8X, and Ae may be obtained. However, in order to obtain this information, it is first necessary to have an explicit and rigorous expression for the separate (and unequal) contributions of up-spin and down-spin electrons to the resultant, observable dHvA wave shape.

2.2. Theory: d H v A Effect in Dilute Alloys Containing Magnetic Impurities

Rigorous equations describing the dHvA effect have been derived within the Green 's function formalism by Engelsberg and Simpson 23 for the case of pure metals and nonmagnetic dilute alloys, and by Shiba 4 for the case of magnetic dilute alloys. According to the latter, the theoretical parameters that completely describe the interaction between conduction electrons and magnetic impurities are the spin-dependent changes (due to this interaction) in the real and imaginary parts of the electronic self- energy,

I ~ = A~ - i sgn F~ (11)

In the context of the dHvA effect, the quantities in Eq. (11) are orbital averages of the corresponding quantities at the point k. According to Shiba, the contributions to the real part of the electronic self-energy show up as changes in the dHvA frequency and as an effective g-shift,

AF/H = -(A~ + A~)/2hw~ (12a)

6g = g" - g~ = (A t - A~)/IxaH (12b)

while spin-dependent contributions to the imaginary part of the electronic self-energy show up as spin-dependent scattering,

J? = -(r~, + r~)/2,rkB (12c)

8X = ( r t - Vs)/27rkB (12d)

Thus, measurements of the four parameters AF, gg, X, and 8X are capable of completely determining the real and imaginary parts of ~,,, for both spin directions. Note, however, that z ~ and Js contain only spin- independent (spin-averaged) information. Only 8g and 8X contain spin- dependent information; i.e., 8g = 0 = 8X if E~ is independent of spin. Thus, these two parameters are of primary interest in connection with the magnetic properties of impurity atoms.

The correspondence between the spin-dependent real and imaginary parts of the electronic self-energy and the parameters AF, dg, X, and 8X, which is given in Eqs. (12a)-(12d), is actually the result of comparing an

Analysis of the Harmonic Content of the de Haas-van Alphen Effect 609

experimentally convenient expression for the dHvA effect (Eq. (19) below] with Shiba's rigorous theoretical expression. Shiba's expression is a rather complex function of the self-energy E, and includes a summation over thermal frequencies ton = (2n + 1)1rkBT, with the self-energy being evaluated at each of the frequencies ton (n = 0, 1, 2 . . . . ). However, as we have already pointed out in the Appendix to Ref. 30, Shiba's expression reduces exactly to Eq. (19) in either of two cases: (1) If experimental conditions are such that 47r2kBT/htoc >> 1, then terms with n ~ 0 may simply be neglected. (2) If the n dependence of the self-energy is sufficiently weak that it can be evaluated (or approximated) by a single, constant value Z(to0) for low n, then Eqs. (19) and (12) also result. In practice, this requires that Z be independent of ton over a frequency range equivalent to a few Landau level separations, e.g., over an energy interval of 50-100 K for electrons with mass 0.3m0 in a 6-T magnetic field. Values of E obtained from the dHvA effect are subject to this interpretation.

Fenton has recently pointed out what appears to have been an erroneous assumption in several earlier calculations of Re E~, using the t-matrix formalism. 24 The most important conclusion of Fenton's calculation is that for both the large-U (Kondo) and small-U limits of the Anderson model,

Re Z,,(0) = A~ = -�89 )K,A + C V (13)

follows as an exact result for all temperatures and magnetic fields and in the presence of normal potential scattering (represented by V) from the magnetic impurity. In this equation c is the impurity (and local moment) concentration and the notation (S~)K,A refers to the impurity spin expectation value in the appropriate Kondo or Anderson limit. The notable feature of this result is that J is the bare exchange interaction coupling the impurity spin to the conduction electrons, while all "Kondo" effects are contained in the expectation value (Sz)K.A, to the extent that these effects cause a reduction from the free space value (S~)o. Using Eq. (13) and (12b), we see that

8g = --cJ(S~)K,A//tBH (14)

SO that Fenton's theory predicts that the field and temperature dependence of an impurity spin, including "Kondo" compensation effects, can be determined directly from dHvA measurements. In addition, Fenton makes clear that complications, due to the introduction of additional " K o n d o " terms in the coupling constant J in approximate theories, do not occur, and that these theories are incorrect. Thus, the dHvA effect appears to provide a sensitive direct probe for determining both the impurity spin behavior in Kondo systems (at least in a magnetic field) and the magnitude of the


exchange constant 3". According to Fenton, the field and temperature dependence of (S~}K.A should follow a (classical) Langevin function, rather than a Brillouin function.

Combining Eqs. (13) and (12a), we obtain

A F / H = -c(V/ho~c) 05)

so that the dHvA frequency shift, to lowest order, is simply a potential scattering (valence difference) effect, not carrying information about the specifically spin-dependent properties of the impurity. However, if a resonant exchange interaction is also present (as at a magnetic impurity site), the formalism developed by Mulimani 33 may be slightly extended to show that there is a small additional contribution to the dHvA frequency shift, proportional to the product of the T-matrix for potential scattering and the average over the two spin directions of the imaginary part of the T-matr ix operator for exchange interaction. So AF (and .17") remains a spin-independent quantity even in this case. These small changes in dHvA frequency with alloying have been precisely measured for various magnetic and nonmagnetic dilute alloys, using simultaneous NMR and dHvA measurements. 25

Finally, it is important to consider two questions which arise in dealing with experiments on real magnetic impurities and host metals, as opposed to somewhat idealized theoretical models for magnetic impurities: How does band structure information enter into the measured values of AF, 8g, .~, and 8X? And how are the above expressions for these parameters modified by the simultaneous presence of both potential scattering and exchange terms in the interaction Hamiltonian?

Partial wave analysis of the anisotropy of dHvA Dingle temperature measurements (and, to a lesser extent, of Fermi surface cross-sectional area changes) in dilute nonmagnetic alloys has shown that the quite different lifetime anisotropies over the Fermi surface that are produced by different substitutional solute atoms can be accurately described in terms of different strengths of scattering for the different angular momentum components in the incoming electron waves (l = 0, 1, 2). 36 B a n d states (Bloch waves) that are properly symmetrized with respect to the point group of the crystal at the impurity site must be used, and the effects of backscattering of outgoing waves, by the surrounding atoms, must be included in an accurate treatment (see Ref. 36 and references therein). For a given strength of scattering of any one part ia l / -wave (a property of the difference between impurity and host potentials) the different Dingle temperatures for different Fermi surface orbits then find a natural explanation in the differences of /-wave charge density on the different

Analysis of the Harmonic Content oi the de Haas-van Aiphen Effect 611

orbits (a host metal property). In general, several phase shifts will contribute appreciably to impurity scattering, the only exceptions observed being for scattering by a very short-range impurity potential (producing only an s-wave phase shift, with respect to the impurity site) or the resonant scattering by a transition metal impurity, in which case a single d-wave phase shift may dominate (e.g., Co impurities in Au34). Strong resonant scattering of d waves at the Fermi energy is expected for either transition metal or rare earth impurities in the noble metals. This interaction consists, in general, of both potential scattering and an exchange (spin-dependent) part. Schrieffer first pointed out 16 that the exchange interaction in real metals is primarily a resonant interaction between localized d states and the d-wave part of the conduction states, i.e., it is a d-d interaction, rather

"than the s-d interaction assumed in "s-d exchange" model calculations. Harris, Mulimani, and Zuckermann 26'33 (HMZ) have found theoreti-

cal expressions for the electronic self-energy due to the interaction between band electrons and magnetic impurities in cubic metals, calculated within the context of a d-d exchange interaction. They assume that only the d-wave part of the conduction electron wave function interacts with the impurity (d-wave resonance), but they include both potential and exchange scattering terms. Their calculations can be used to show that all four of the quantities AF, 8g, ,Y, and 8X then scale proportionally to the variation of d-wave charge density on different extremal orbits on the Fermi surface. The dependence of measured J values on d-wave charge density of the host metal is borne out by the few experimental determinations of J to date for different orbits, using dHvA spin-splitting zeros. 7

However, because of the simultaneous presence of both potential and exchange interactions, the expressions derived for these four parameters also involve additional interference terms between the potential and exchange interactions. X and AF have their leading contributions from the potential scattering, while 8g and 8X have their leading contributions from the exchange terms. [Equations (14) and (15) represent just these leading terms.] Thus, it is clear that although dHvA experiments do provide a direct separation of spin-independent and spin-dependent contributions to the electronic self-energy (via measurements of ,Y and AF vs. 8X and 8g, respectively), this is not the same as separating the potential interaction terms from exchange interaction terms, due to interference effects between these two when one calculates completely the real and imaginary parts of the self-energy.

HMZ also point out that their formalism can be extended to include s- and p-wave phase shifts for potential scattering, when necessary. 26"3~ They agree that finite potential scattering is necessary in order to have spin- dependent scattering (SX ~ 0), a result that was also obtained earlier by


Shiba within the context of the s-d exchange model. Mulimani 33 also stresses the point that if theory is to be used to obtain consistent values for exchange parameters (and for SDS!) from different experiments, then it is essential that band structure information be taken properly into account by including the momentum and angular dependence of both potential and exchange scattering. This point is also made by Caroli 35 in connection with the angular dependence of J(k, k') and the determination of consistent J values from different experiments. Most of the necessary orbital averages of partial wave charge densities, etc., have now been tabulated for noble metal hosts. 36 Shiba's earlier theoretical calculation was carried out using the s-d exchange model in which the momentum dependence of V(k, k') and J(k, k') is formally eliminated by setting them equal to constants for states lying within a finite energy range of the Fermi energy. However, his comparison with experiments on copper (transition metal) dilute alloys was then made by using the Fr iedel -Anderson scheme, in which the four independent conduction electron self-energy contributions simply become functions of only two parameters, the number of d electrons on the impurity site and the impurity spin. The semiquantitative agreement which he found between this model and the experiments must be considered rather good, in view of the limitations of the model summarized iaere.

3. d H v A W A V E S H A P E A N A L Y S I S ( W A )

3 .1 . "Observab le s" in d H v A W A

The large-amplitude field modulation technique (FMT) 27'2s is particularly useful for wave shape measurements because of its "Bessel function spectrometer" action, which makes it possible to enhance the higher dHvA harmonics at the expense of the fundamental. In using the FMT the directly measured quantity is an oscillatory voltage ~" whose relation to the oscillatory magnetization, M is

= ~ ~ r (16a) r = l

~'= ~ r~r=G ~ J.(rx)JVL, (16b) r = l r = l

where r is the dHvA harmonic number (r = 1, 2, 3), n is the detection harmonic number, J , is the ordinary Bessel function of order n, and x = 2zrFh/H 2, where h is the zero-to-peak modulation field amplitude.*

*we assume that the detection frequency is sufficiently low that the skin depth is much greater than sample dimensions. This condition is usually not difficult to fulfill in dilute alloys. Typically we modulate at 17 Hz and detect at either the second or fourth harmonic.

Analysis of the Harmonic Content of the de Haas--van Alphen Effect 613

The factor G, the absolute gain of the electronic system, is assumed to be unknown.* The dHvA wave shape is analyzed by taking a short block ( - 1 0 - 2 0 fundamental oscillations) of high-resolution digitized data at temperature T and mean field value H. This is Fourier-decomposed into sine and cosine components S, and C, relative to the left edge of the data block, for which the (arbitrary) initial field value was Ho. The directly measured quantities (either by Fourier transform or using a fitting routine) are the resultant harmonic voltage amplitudes V ~ = (S 2 + C 2 )1/2 and the harmonic phase angles 0, = tan-X(sr/c,).

However, because O, depends on the arbitrary (and unknown)'initial phase of the left edge of the data block, via the arbitrary initial value of Ho, it is convenient instead to define the relative harmonic phase angles 201- 02 and 301-03, which are independent of H0. 8

Similarly, because the gain factor G is unknown, it is convenient to define the relative harmonic amplitude ratios V~ ~ and V~ ~ in which the factor G cancels. These voltage ratios are directly proportional to the corresponding magnetization harmonic ratios, with a proportionality constant equal to the corresponding Bessel function ratios, as shown by Eq. (16). However, these proportionality constants can be directly determined by repeating the data block with twice and three times the original modulation current, and measuring the ratios of the fundamental voltage oscillations in all three data blocks. 8 A detailed discussion of a variety of purely experimental aspects of using the FMT in connection with wave shape analysis has been given elsewhere. 9A1"29

Thus, the four "observables" (using the first three dHvA harmonics) are two harmonic amplitude ratios and two relative harmonic phase angles. (A fifth observable, not independent of the first four, can also be defined; see Section 4.)

3.2. The Pure Lifshitz-Kosevich (LK) Limit

The goal of wave shape analysis in dilute magnetic alloys is to determine how 8X, X, and Ae vary as functions of (H, T) in different dilute alloy systems. In general, the observable effect of variations in these theoretical parameters is to alter both the relative amplitudes of different harmonics in the magnetization oscillations and to alter their relative phases. We therefore require a set of equations connecting the four "observables" of a single data block with these theoretical parameters.

*Crabtree 13 and Knecht 12 have recently described two independent techniques by which the gain of the detection system may be calibrated, in favorable circumstances, using the dHvA oscillations themselves and the FMT. It appears that such calibration can be done with sufficient accuracy to be genuinely useful only in the case when a single dHvA frequency (and its harmonics) is present.


In the presence of spin-dependent scattering the amplitudes M , t and M , ~ become unequal, and Eq. (1) takes the more general form 3~

i~l=-rh)~ ~ C,D'E~ 27rr F _ +p-~-trzrrS' (17) O" r = l

where we now sum explicitly over the two spins [the spin index tr is +1 ( -1) for spin up (down)] and where

D=exp(-Atz.,Y/H), E=exp(-AIzSX/H) (18)

using the definitions of .(" and 8X given by Eqs. (10) and (12) except that S'= (lz/2)g' now includes the exchange energy shift described by Eqs. (12b) and (14). All other quantities remain as defined in Eq. (2). Using Fig. 2, we can write the resultant LK harmonic content as

hTI = - r ~ ~. A ; C ~ ' sin 2zrr - +p-~+AO,+(1-q')-~] (19) r = l g J

with

A', = [E 2' + E - : " + 2 cos (2~rrS')]1/:/2 (20)

A0, = tan -1 [tan (IrrS') tanh (rA~ 8X/H)], -~r/2 -< A0, -< +r (2 la)

, [+1 for cos(~rrS')>0

q ' = ] - i for cos(TrrS')<0 (21b)

We take care to distinguish q', and S' (in the presence of SDS and exchange) from the corresponding unprimed quantities, in order to explicitly take into account the possibility that a shift through an rth harmonic spin-splitting zero could result from SDS and exchange effects. The factor (1-q',)Tr/2 simply adds 7r to the value of A0, given by Eq. (21), when necessary, depending on the quadrant in which (IrrS') lies, so that A', is always given by the positive square root [Eq. (29)] and A0, is restricted to the range given in Eq. (21a). We have already discussed the precise connection between Eq. (19), which is a convenient form for experimental analysis, and Shiba's expression for M in the presence of SDS and exchange energy shifts, in Section 2.2 and in the Appendix to Ref. 30.

A0, is the rth harmonic phase shift which is induced by SDS; in the absence of SDS (SX = 0) no phase shift occurs. Thus, were it not for the presence, in general, of additional phase shifts due to magnetic interaction (see below) the presence of the phase shifts A0, would be a clear indication of SDS. Note that although the spin-splitting angles 4~r = r~b do scale with r (Fig. 2), the phase shifts in the resultant dHvA harmonics do not, i.e., A02 r 2A01. This means that the relative phase shift "observables" defined above do contain information about SDS, which can be unfolded. Similar


arguments apply to the ha rmonic ampl i tude ratios, which are easily calculated f rom Eqs. (18)-(20). Howeve r , those equat ions can also be ob ta ined as a special case of the m o r e general results found in the next section, so we omit t hem here. (For a detai led discussion of this special case see Refs. 9 and 29.)

3.3. The General Case: Magnetic Interaction (MI) Effect Harmonic Content

Shoenbe rg 6 was the first to not ice and discuss the fact that the ha rmonic con ten t of the d H v A effect is usually s t ronger than that predic ted by the L K formula [Eq. (1) or (19)]. T he reason for this is that the driving field for the q u a n t u m oscillations is the magnet ic induct ion,

B = H + 4 ~ ( 1 - N . ) b ~ r (22)

ra ther than the applied field H. H e r e N , is the "effec t ive" demagnet iz ing factor for the d H v A sample.* W h e n Eq. (22) for B is subst i tuted for H in Eq. (19), the latter be c om e s only an implicit equa t ion for JlT/, because of the presence of )~r on the r igh t -hand side also. Because 14zr)~7/I << H, replacing H by B has a negligible effect on the ha rmonic ampl i tude factors C,, D ' , and E ' . Howeve r , if 14zr~7/I is an appreciable f ract ion of the per iod (HE~F) of the d H v A oscillation, then the effect of 4It.bY/', a l ternately adding to or subtract ing f rom a monoton ica l ly changing applied field H, is to p roduce large dis tor t ions in the shape of the d H v A wave fo rm when it is observed as a funct ion of H. This " s e l f -MI" effect p roduces a significant par t of the r = 2 (and higher) ha rmonic content , t hough with little change in the fundamen ta l d H v A ampl i tude .?

Howeve r , if the condi t ion

14~r(1 -Ne),t~rl << 1 (23)

is satisfied, then we can adop t the iterative expansion technique of Phillips and Gold , 31 subst i tut ing B [Eq. (22)] for H in the sine funct ion in Eq. (19),

*The factor (1 -N , ) was not present in the earliest analyses of MI effects, or in the treatment by Phillips and Gold, ax but is essential for the precise analysis of experiments using harmonic amplitudes and phases. In general the demagnetizing factor N, depends both on sample orientation and on skin depth at the detection frequency, because the local demagnetizing field is usually spatially nonuniform within the sample. Only for samples with surfaces described by a second-order equation (e.g., ellipsoids) is the demagnetizing coefficient independent of position. Crabtree has also shown that for a cubical sample N, is independent of orientation. For a discussion of these effects see Refs. 11 and 13.

tMI is also possible between dHvA oscillations arising at two different Fermi surface extremal cross sections. We designate these intermodulation effects as "other-MI," to distinguish them from the "self-induced MI" of interest here. See Ref. 31 for a complete discussion.


and expanding the argument, obtaining to lowest order*

~I= - ~ ~ A',C,D" sin [r(x -k '~,I)+p~r/4+AS,+(1 -q' ,)~-/2] (24) r= l

with k ' = (1 -Ne)8rr2F/H 2 and x = 2 r r ( F / H - y ) .

We define the "weak MI" limit by the condition 31

]k'A'~CID[ < 1 (25)

which corresponds physically to the requirement that the dHvA fundamental oscillations never become multivalued as a function of the applied field H. 32 If Eq. (25) is satisfied, then a general solution for ~t, a,s an explicit function of x, can be obtained to any desired degree of accuracy. Following Phillips and Gold, we define the nth approximation ~ r(n) to 57I as

h71~n) = -rh ~ A ',C,D" sin [ r(x - k')lTl~n-')) + p~r/ 4 + AO, + ( 1 - q ',)~'/ 2 ] r= l

(26) with/~r (~ 0. Carrying out this expansion procedure, we can collect terms that are of second and third order in the exponential scattering and thermal damping factors. The result is that the second- and third-order approxima- tions to M are

)VI ~2) = - rh{a'xC~D sin [x +p~r/4 + A01 + (1 -- q~ )zr/2]

+ A~CED 2 sin [2x +pcr/4 + A02 + (1 - q~)cr/2]

+�89 +pTr/4+AO~)+(l-q'~)cr]} (27)

)lTf 3) = -rh{A'ICID sin [x +pTr/4 + A01 + (1 - q~ )7r/2]

- ~(k ')2(A ~ CID )3 sin [x + pTr/4 + A01 + (1 - q ~ 7r/2]

I t t ,, 2 �9 -~k (AaCID)(A2C2D) sm [x + AOE-A01+(q~ -q~)Tr/2]

+ A~CED 2 sin [2x + pTr/4 + A02 + (1 -- q~)rr/2]

+ �89 C1D)2 sin [2(x + p~r/4 + A01) + (1 - q ~)Tr]

+ A~CaD 3 sin [3x + per~4 + AO3 + (1 - q~ )zr/2]

+pa(A'~C1D)a(k')2 cos [3x +pr + 3A01 + 3(1 -- q~ )~r/2]

+~k'(A'IC1D)(A~C2D 2) sin [3x + 2pr + A01 + A02

+ ( 2 - q ~ - q~)Tr/2]} (28) �9 Crabtree has pointed out (ReL 13) that the original Phillips and Gold expansion is not

completely general, in that the M and B vectors are assumed to be parallel, which is only true at orientations for which OF//O0 = 0. In our treatment [from Eq. (24) onward] we also make this assumption. The generalization, as outlined in the appendix to Ref. 13 for the second harmonic, is straightforward, though sufficiently cumbersome that we deliberately avoid its inclusion here.


The amplitude coefficients of the various terms in Eq. (28) are in error by no more than 2% if k'A'IC1D <- 0.2.

Therefore , the effect of SDS and he is to alter the resultant wave shape in both amplitude and phase from either of the usual LK or MI results. In particular, MI and SDS effects mix nonlinearly. If SDS (alone) produces phase shifts A01, A02, and A03 in the pure LK components, then the second harmonic MI component is shifted additionally in phase by 2AO~, and the two third harmonic MI components have additional phase shifts 3A0~, and (A0~ + A02), respectively, in the presence of SDS. Thus, the MI and LK components at each harmonic no longer differ in phase by a simple, fixed amount (such as p~r/4), as in a pure metal. Instead their phase difference, and the phase of each resultant observable harmonic amplitude, depends on both the extent of SDS and the magnitude of the Zeeman and exchange energy shifts, via the quantities A0,.

In order to calculate the combined effects of MI and SDS on the resultant second and third harmonic dHvA amplitudes, it is convenient to define all phase angles relative to the left edge of a data window (Xo = 27r(F/Ho- y)]. The phase angles can then be written as

Oti x = (Xo + p'rr/4 + A01)- (1 + q ~ )zr/2

01 = (xo + p /4 + a 0 1 ) + (1 -

0Mx~ _ (x0 + A02-- A0~) + (q ~ -- q ~)zr/2 1

0 Lx = (2Xo + pTr/4 + A02)-- (1 + q~)Tr/2

02 MI = (2Xo + 2pTr/4 + 2AO1)-- q ~ ~r

0 Lx = (3x0 + p~'/4 + A03)-- (1 + q~ )~r/2

0~ xq = [3Xo+ (p -- 2q~)3~-/4 + 3A01] + 7r/2

03 MI~ = (3x0 + p'rr/2 + A01 + h02)-- (q ~ + q~)Tr/2 (29)

using Eq. (28). Here we have absorbed all possible minus signs into the phase factors so that all coefficients are now treated as positive.

From Eq. (28) it is clear that MI corrections to the d H v A fundamental will generally be small, since they are third-order corrections. 31 However , we find that these corrections are significant, and cannot be ignored in doing wave shape analysis in crystals with low Dingle temperature.

From Eq. (29) it follows that the observable relative phase angles take on the limiting values

20LK_o2={~+(2AO1--AO2)+�89 pure LK limit (30a)

pure MI limit (30b)


and

30 LK - 03 = f �89 + (3AOl - AO3) + �89 - q~ )~r

�88 + (2AOl -- aO2)+�89 -- 1)rr

0

pu re L K l imit (3 l a )

pu re M l p q l imit (31 b)

pu re M I q l imi t (31c)

where we have neglected the r = 1 M I terms in comparison with the r = 1 L K contribution in Eqs. (31a) - - (31c) and w h e r e all phase angles have b e e n wr i t t en m o d u l o (2zr). T h e " p u r e M I " and " p u r e L K " l imits in Eqs. (30) and (31) re fe r to the l imi t ing cases in which the h a r m o n i c c on t e n t is d o m i n a t e d by e i the r the L K or M I con t r i bu t i on (with SDS effects also p resen t ) . In genera l , b o t h M I and L K h a r m o n i c con ten t will be p r e s e n t and the o b s e r v a b l e r e l a t ive phase angles (201 - 02) ~ and (301 - 03) ~ will have va lues lying b e t w e e n these two l imits. I t is then c onve n i e n t to use p h a s o r d i a g r a m s to desc r ibe the r e su l t an t h a r m o n i c a m p l i t u d e s and phases in this gene ra l case, as shown in Fig. 3. W e def ine err as the phase angle by which the r e su l t an t r th h a r m o n i c a m p l i t u d e dev ia t e s f rom the phase of the pu re L K c o m p o n e n t . T h e n

Or = [ r x o + p r : / 4 - (1 + q ' r )~ /2] +A0r + a , (32)

so tha t

( 2 0 1 -- 0 2 ) ~ = ( 2 0 1 -- 02) MI'sDs

= (201 -- 02) LK + (2AOl - AO2) sos

+ (2~xl - a2) MI + (q~ - q2)'n'/2

(33)

/ / / / ~ M ~

A// M~( I

Fig. 3. Phasor diagram showing formation of the observable rth harmonic dHvA signal as the resultant of the LK and MI components, in the presence of SDS. All phases are shown with reference to the phase that the pure LK component would have in the absence o[ SDS (horizontal axis), and are given by Eq. (29) of the text. a, is the angle by which the rth harmonic resultant deviates from the phase of the LK component, due to MI, while ~--% is the phase angle between the MI and LK components (see text and the Appendix).

Analysis of the Harmonic Content of the de Haas--van Alphen Effect 619

Thus,

( 2 0 1 -- 0 2 ) O B - - ( 2 0 1 - - 02) LK -m- ( 2 A 0 1 - - A 0 2 ) SDS + (20~1 -- a 2 ) MI + ( q 2 -- q 2 ) T r / 2

(34a) �9 Similarly,

(301 - 03) ~ - (3 01 - 03) LK = (3A01 - A03) sDs + (3Otl -- a3) Mx (34b).

+ (q~ - q3)zr/2 - 3(q~ - qx)~-/2

where the MI phase shifts.or, are calculated in the presence of SDS and the exchange energy shift. Note that q', may be #q,, since Ae may cause a shift through an rth harmonic spin-splitting minimum. In Eqs. (33)-(34), we have used the definitions

(201 - 0 2 ) L K -~- p~r/4 + (1 + q2)~r/2 (35a)

(301 - 8 3 ) L K ~ p~/2 + (q3 -- ql)~r/2 (35b)

following f rom Eqs. (30) and (31), in the absence of SDS and exchange. We close this section with the reminder that the equations we have

obtained, and the wave shape analysis technique itself, are only valid in the "weak M I " limit defined by Eq. (25). We also note that an extension of the original Phillips and Gold expansion, to include terms up to order eight for the first four d H v A harmonics, has recently been published, s~

4. FUNCTIONAL DEPENDENCE OF OBSERVABLES ON MICROSCOPIC PARAMETERS

In the preceding section we have obtained equations giving the amplitude and phase of each con'cribution to the first three d H v A harmonics in terms of the three microscopic theoretical parameters (MTP) that are of main interest in describing the electron-local momen t interaction: X', 8X, and S' (or Hex). I t is clear that equations that express the four experimental observables (two relative phase angles and two harmonic ampli tude ratios) in terms of these three MTP can also be written down; we do this in the Appendix.

However , the resulting equations are complicated transcendental functions of the MTP, and cannot be analytically inverted to give closed expressions for the MTP in terms of a set of four measured values for the observables, as one normally wishes to do in analyzing the results of experimental measurements . Fur thermore , the complexity of the equations obscures the degree of functional dependence of each observable on each of the MTP: It is not clear a priori whether any given observable depends strongly or weakly on each of the MTP, nor is it obvious how to solve the equations most efficiently. In particular, it is not clear in the presence of MI

620 Yun Chung, D. H. Lowndes, and Catherine [,in Hendel

whether a subset of two (or more) of the observables can be used to directly determine values for one (or more) of the MTP.

We have overcome both of the above difficulties by developing a graphical technique for displaying the functional relationship between observables and MTP, and for inverting a set of measured values for the observables at magnetic field H and tempera ture T to obtain a set of corresponding values for the MTP. Thus, measurements on a single, short d H v A data block (typically - 1 0 - 2 0 oscillations) at field H and temperature T may be used to explore the dependence of the MTP on (H, T). In this section we describe these techniques.

4.1. Solutions Without MI

It is helpful to first display techniques for obtaining a solution for the MTP in the limit of no MI, i.e., the pure LK limit. We denote "observables" with the superscript OB. From Eqs. (17)-(21), (34), and (35) and Fig. 2 it follows that

T12 --- (201 - 82) ~ = (201 - 02) LK + (2AO1 - A02)+ (q~ - q2)Tr/2 (36)

T13 ------ (301 - 03) ~ = (301 - 03) LK + (3AOl - AO3) + (q ~ - q3)'cr/2

-- 3(q~ - ql)~r/2 (37)

[ M3/M1 ]~ M3122 = [(Mz----~-~)2 J \ - - - ~ - ]

[E 6 + E -6 + 2 cos (6rrS')][E 2 + E -2 + 2 cos (2zrS')]] 1/2 = { l (381 [ E 4 + E -4 + 2 cos (4zrS')] 2

( M z / M 1 ) OB D{ E4 + E-4 + 2 COS (4WS')~ 1/2

M 2 1 = C 2 / C ~ \E2+E-2+2 cos (2IrS'} (39)

(Ma/M1)On 6 --6 .q_ 2/E + E 2 cos (67rS')~ 1/2 M 3 1 = C3/C1 D k E 2 + E _ 2 + 2 cos (2rrS')] (40)

The three observables given by Eqs. (36)-(38) are independent of 3E, and depend only on 8X and S', since all of the quantities in the ratios of factors C, are either known from the conditions of the exper iment (T, H , / z ) or else cancel out in taking a ratio. The last two observables, Eqs. (39) and (40), depend in addition on ,~. [Equation (38) is of course not independent of Eqs. (39) and (40).] Note that the Fermi surface curvature factor, cr which is usualty not accurately known, cancels out in the right- hand sides of all of Eqs. (38)-(40).

A natural procedure in this case is to seek to determine 8X and S' f rom the measured values of (201-0~) ~ (301-03) ~ and


1.0

Od

e~ t D

I

O.5

0.1 0 0.5 1.0

5'

Fig. 4. The observable (281 - 82) as a function of S' for several fixed values of 8X. Broken lines are for the case of no MI harmonic content (Section 4.1), while solid lines give the solutions including the effects of MI (Section 4.2) with the sample's effective demagnetizing factor Are taken to be 0.2. Figures 4-9 are plotted for H = 40 kOe, T = 1.1 K, and other parameter values (where necessary) for the (111) neck orbit in Au: F = 1.532x 104T,/z = 0.28, ~-1/2=0.26, S=0.16, and p = + l . Figure 4 uses Y(=0.5, and Figs. 5-7 use Y(= 1.0, though they depend only very weakly on the value of Y(, as is shown in the Appendix.

[(Ma/M1)/(ME/M1)2] ~ and then to d e t e r m i n e Y~ f rom the two a m p l i t u d e ra t ios (Ma/M1) ~ and (ME~M1) 01~, using the now fixed va lues of 8 X and S ' . W e do this by p lo t t i ng Eqs . (36) - (40) , using p a r a m e t e r va lues a p p r o - p r i a t e to the cond i t ions of the e x p e r i m e n t a l d a t a b lock . T h e p lo ts a re m a d e by a l a b o r a t o r y m i n i c o m p u t e r e q u i p p e d with a d isk o p e r a t i n g sys tem and i n t e r f aced to an X - Y p lo t t e r . E x a m p l e s of such p lo t s a re shown in Figs. 4 - 9 .

F igu re s 4 and 5 show tha t the two re la t ive p h a s e angle o b s e r v a b l e s T 1 2 and T 1 3 a re p e r i o d i c func t ions of S ' wi th p e r i o d equa l to 1, an t i sym- me t r i c a b o u t the va lue S ' = 0.5, and a n t i s y m m e t r i c wi th r e spec t to change of

6 2 2 Y u n C h u g , D . H. L o w n d e s , and Cather ine Lin H e n d e l

d - ~ . ~

- - - / ~ I

I o o c~ o. o

-i ~ �9

~ ~ 0 ~ .

~'~ ~'~. ~

�9 ~ = . ' ,0 , ~ ".~ O"

~ . ~

I r r J F r t I ~ . ~ J I f J I r r J I r

~ - - I ~ . . . . . . .

"~ I

o d

o

0

0 I

Analysis o | the Harmonic Content oi the de Haas-van Alphen Effect 623

2.0

~. )

<S ,S

10 5 -

IE v

0 0.1 0.2

S' Fig. 7. As in Fig. 6, but illustrating the effect of small amounts of spin-dependent scattering on the quality of the third harmonic spin-splitting zero, both with (solid lines) and without (broken lines) magnetic interaction harmonic content. The lines are labeled by the values of 8X.

sign of 8X. Fur thermore , the sensitivity of these two phase angle observables to changes in 8X depends critically on the value of S'. For example, for 0 < S' < 1/6, (301 - 03) ~ is relatively insensitive to the extremes of zero spin-dependent scattering (SX = 0) and completely spin-dependent scattering (18x1=.r while for 1 / 6 < S ' < 1 / 2 , it is very sensitive to small changes in spin-dependent scattering.

The ampli tude ratio M 3 1 2 2 is mainly useful for determining S' when 8X is small, in which case it responds sensitively to proximity to S' values that produce spin splitting minima of any of the first three d H v A harmonics. However , when 8X is substantial, T12 and T13 by themselves are capable of determining a unique solution (8, X, S'), within error limits governed by the symmetr ies described above (Figs. 6 and 7). M 2 1 and M 3 1 (upon which M 3 1 2 2 is nonlinearly dependent) may then be used to determine )~'.


I I I I I I I I I r

ff

-2

I I I I I I I I I 0.5 1.0 1.5

Fig. 8. The observable M 2 1 [see Eq. (39)] as a function of ,~ for several values of 8 X and for $' = 0.16 (the latter value for the (111) neck orbi t in Au). The use of such a figure to determine the value of -,~ is described in Section 4. See also the legend for Fig. 4.

0 1,,,~ I I I I I t I ~ I

S'=0.16 -,

O

~'~'- 3

- 6 I I I 1 I I I I I O.5 10 1.5

Fig. 9. The observable M 3 1 [see Eq. (40)] as a function of Y~ for several values of 6 X and for S ' = 0.16. See the legend to Fig. 8.

Analysis o! the Harmonic Content o| the de Haas-van Alphen Effect 625

Figures 8 and 9 demonstrate that ,~" may be determined by plotting the logarithm of either M21 or M31 vs..,~ for a fixed value of S' and a family of values of 8X. The known values (S', 8X) can then be used to locate the solution for ,~.

Small errors in the prior determination of 8X will of course "feed through" to produce corresponding errors in PC. The error in X can be minimized by using that one of M21 or M31 that is [urthest from a second/ third harmonic spin splitting minimum condition: This of course depends on the value of S'. E.g., if S' is believed to be near 0.15, the occurrence of a third harmonic minimum at S'= 0.167 means that M2 1 should be used for the determination of X', not M31.

A similar statement applies to describe the way in which errors in the value of S' affect the determination of .,~: errors in S' have the largest effect when the amplitude ratios M21 and M31 are the most rapidly varying functions of S', i.e., near any of the values of S' that would, in the absence of spin-dependent scattering, produce first, second, or third harmonic spin splitting zeros.

4.2 . So lut ions Inc luding MI

The question which immediately arises in the context of the graphical method of solution just outlined is whether the presence of additional MI harmonic content makes the graphical method workable. Equations (32)- (34) show that the effect of MI is to shift the phase of the resultant rth harmonic dHvA signal by an amount or, relative to the phase angle of the pure LK component , and to modify the resultant dHvA amplitude accord- ingly (Fig. 3). In the Appendix we show that the phase angles t~2 and a3 depend only on 8X and S', while or1 depends (weakly) in addition on X. Thus, the three observables (201-02) ~ (301--03) ~ and [(M3/M1)/(M2/M1)2] ~ remain essentially only functions of 8X and S', and these three observables may still be used to determine 8X and S' by the graphical method described above. Similarly, P( may then be determined from the measured values of (M2/M1) ~ or (M3/M1) ~ also essentially as described above.

Figures 4-7 illustrate that, although the harmonic content due to MI is far from dominant for the (111) neck orbit in Au, its effect on the observables of wave shape analysis is certainly nonnegligible, and MI must be included in any analysis in which it is hoped to determine accurate values for ,~, 8X, and Ae as functions of H and T.

Figures 4 and 5 illustrate that the effect of MI on the two phase-angle observables is particularly important over those ranges of values of S' for which it is difficult to detect small changes in 8X [e.g., for 0 < S ' < 1/6 in the case of (301- o3)OB].


In all previous analyses of the spin-dependent scattering of conduction electrons using the dHvA effect, it has been assumed that the condition for the occurrence of a spin-splitting zero was that the scattering rates for spin-up and spin-down electrons should be equal, i.e., that the dHvA amplitudes contributed by the spin-up and spin-down electrons should be equal. Figure 7 shows that this is not true in general: The nonlinear mixing produced by the magnetic interaction effect is capable of producing a clean spin-splitting zero (not just an amplitude minimum) for a nonzero value of 8X. Thus, observation of a clean spin-splitting zero in a dilute magnetic alloy does not constitute evidence that X t = X "~, i.e., that spin-dependent scattering is absent. Instead, it seems that a full wave shape analysis, of the type described here, is necessary to reach such a definite conclusion.

Figure 7 also shows that the position of the apparent third harmonic spin-splitting zero is shifted (by about +0.011) away from the value S ' = 0.167 for a pure metal.

In a dilute magnetic alloy the observation of such a shift in position would ordinarily be interpreted as an exchange energy shift; yet it arises here in part as a consequence of (rather weak) MI. Again it is clear that a full wave shape analysis, using several observables, is necessary in general.

Finally, Figs. 8 and 9 are another way of showing that MI effects are most pronounced near values of S' that would give rise to harmonic spin-splitting zeros in the absence of MI. The shifts between the broken (no MI) and solid (MI) lines are larger in Fig. 9 (using the third harmonic) than in Fig. 8 (using the second harmonic), because the value S ' = 0.16 used in these figures is close to the value S ' = 0.167 at which the third harmonic zero would occur for no MI and 8 X = O.

5. A P P L I C A T I O N S

As was described in the last section, there exists a natural procedure to follow in determining 8X, ,,~, and S' from Fourier analysis of a single, short block of data at temperature T and average field H : The measured values of any two of the three observables (201-02) OB, (301--03) OB, and [(M3/M1)/(M2/M1)2] ~ may be used first to determine values for 8 X and S', and then the final unknown .~ may be determined from the measured values of the amplitude ratios (Ma/M1) ~ and (ME~M1) ~

A laboratory computer interfaced to an X - Y plotter can be used to carry out the first step of this solution in a particularly appealing way, one which displays not only the unique solution, but which also allows a direct, visual estimate of the way in which errors in the observables feed through to .p roduce errors in 8 X and S'. Our procedure is simply to define a two-dimensional space (SX, S') and then to calculate the values of the two


selected observables as a function of (SX, S'). Whenever a calculated value is obtained which is equal to the experimental ly measured value (within the est imated experimental errors), a point is plotted at the location (SX, S'). For each observable calculated in this way, there results a locus of points, whose width is a measure of the est imated errors. The intersection of two such loci gives the solution (SX, S'). Figures 10 and 12 are examples of solutions generated in this way.

5.1. Cyclotron Orbit g-factor Measurement: Au(Ag)

In a case when no local m om en t is present, it is expected that 8 X = 0 (X t = X s) and Ae = 0, so that the measured value of S' = S = (/x/2)gc and the cyclotron orbit g-factor can be determined by wave shape analysis.

Primarily as a check on the above procedures, we have measured the orbital g-factor for the (111) neck orbit in a dilute alloy of Au(Ag). The Ag solute was introduced in order to make the sample 's skin depth much greater than its dimensions, and thus to minimize problems associated with incomplete penetra t ion of the sample at the modulat ion or detection frequencies (17 and 68 Hz, respectively), using the field modulat ion technique, ll Figure 10 shows the result of one such g-value determination: 8 X = - 0 . 0 0 7 + 0 . 0 0 8 K (zero, within experimental error) and S = (0.158+0.0015, corresponding to a cyclotron orbit g-value 1.13 for the (111) neck orbit (using /z =0.28) , in good agreement with the results of Crabt ree et al. for the neck orbit in m u . 13

+ 0.050

+ 0.025

0.0

- 0.025

":: : i ! ! i i i i i lz: . . . " ' ' i i l i i i i i : . ~ ,

' ' ' ! l i i : i i i i ~ . . o : : . i : . : : i : O : o ~

I I I I I I I I 0.12 0.14 0.16 0.18 O. 20

S'

Fig. 10. Simultaneous graphical solution, using an X - Y plotter interfaced to a laboratory computer, of two of the equations analogous to (36)-(38) (but including MI effects), which relate pairs of values (SX, S') to the measured values of two observables. Th~ data are for the (111) neck orbit in a nonmagnetic Au(Ag) dilute alloy.


The Au(Ag) sample used in these measurements had been spark-cut in the same size and shape as the Au(Fe) and Au(Co) samples, so that the Au(Ag) sample could be used as a reference sample for an experimental est imate of the samples ' effective demagnetizat ion factor Are. As has been discussed elsewhere, 11 Ne may also be calculated as the volume average of the spatially nonuniform demagnetizing factor for a sample in the shape of a rectangular prism. The use of such a calculation necessarily assumes that the detection frequency is low enough to give full, uniform penetrat ion of the sample, so that the experimental demagnetizat ion factor also corresponds to a (uniform) volume average. In fact, Ne was also exper iment- ally est imated for Au(Ag) by comparing measured values of (201-02) ~ with calculated values over a wide field range and adjusting the value of Ne for best agreement . This resulted in a value Are = 0.2 • 0.1, in comparison with the calculated volume average value Ne =0.15 . The value N~'=0.2 was used in subsequent analysis of our results for Au(Co) and Au(Fe) dilute alloys.

5.2. C o n d u c t i o n E l e c t r o n - L o c a l M o m e n t Interact ions

5.2.1.. Au(Fe)

Dilute alloys of Au(Fe) exhibit the Kondo effect, with TK well below 0.5 K. Spin-glass freezing also occurs at very low tempera tures for Fe concentrations of order 500 at ppm or greater. Since our experiments were carried out on samples with approximately 100 at p p m Fe, at tempera tures between 1 and 2 K, and in magnetic fields in the range 30-75 kOe (/.*BH >> kBTx), we expect to see behavior characteristic of the interaction of conduction electrons with a strong, isolated local magnetic moment .

Measurements of 2(, 8X, and S' were carried out for the (111) neck orbit in several single-crystal samples of Au(100 at ppm Fe). P( was found to be essentially independent of H or T, with the value X = 0 .58+0 .1 K. The results for 8X and for H,x [Eq. (8)] are shown in Fig. 11. There is no experimental ly significant H or T dependence for 8)(, but H,x clearly exhibits both H and T dependences.

According to Eqs. (6) and (8), or Eq. (14), the H and T dependences of the effective g-shift (Hex) arise from polarization of the Fe local momen t in the applied magnetic field. Loram et al. 37 have used magnetization measurements to determine that S = 1 .29• 0.03 for Fe in Au; their results show a Brillouin function behavior for the impurity magnetization, which is not yet saturated at 2 K, but is approaching saturation at 1 K. Our own measurements (Fig. 11) show very similar behavior, although the necessity to have d H v A third harmonic content restricts us to H ~> 40 kOe, so that we cannot really distinguish between a Brillouin function and the Langevin


30 .4

25

2O

E ~15

x

2= 1C

! o

o o o o o , , o

~ o o

T

40 ~'o ' do H { K G )

o .3

.1

7,0 � 8 4 1 8 4

Fig. 11. Left-hand ordinate: effective "exchange field" [Eq. (8)] describing the antiferromagnetic shift in the relative spacing of the spin-up and spin-down electronic Landau levels, due to exchange interaction with approximately 100 atomic ppm Fe local moments in Au. II, 1.11 K; Q, 1.98 K. Right-hand ordinate: difference 8X in spin-up and spin-down conduction electron scattering rates (Eq. (10)]..Fq, 1.11 K; O, 1.98 K.

function behavior for (Sz) which is predicted in Fenton's theory. 24 The change in the Zeeman splitting of the electronic Landau levels due to exchange interaction with the Fe local moment is given by Ag/g = H~x/H, and is about 20% at H = 60 kOe for our 100 at ppm Fe concentration.

Since [(gSlzBH)2+(lrkr~T)2] ~ >>kaTK under the conditions of our measurements, we may use the results of perturbation theory calculations for 8X and H~x, in order to obtain a direct estimate of the conduction electron-local moment exchange coupling strength. From Refs. 38--40 we find

kB 8 X = 2cj2Cap(Sz) (41)

I~Hex = cJCa( Sz ) (42)

s o t h a t

kB 8X/tzBHex = 2Jp (43)

where c is the fractional impurity concentration, J is the full exchange integral, including the effects of orbital degeneracy, p is the conduction electron density of states for a single spin direction, and (Sz> is the impurity spin. In these equations we have kept only the leading terms, and have also


expressed them in a form appropr ia te for use with the orbitally averaged propert ies measured by the d H v A effect, by introducing the factor Ca, which is the fractional d-wave charge density. Following Schrieffer 16 and Harris et al. 26 (see the discussion in Section 2.2), we assume the exchange interaction proceeds through only the d-wave part of the conduction electron wave functions, so that d H v A measurements are weighted by the variation of the fractional d-wave charge density on different orbits. From Ref. 41 we calculate Cd~0.46 for the neck orbit, relative to the belly orbits.

Taking Hex(sat)-~Hex(70 kOe, 1.1 K ) = 13.5 kOe from Fig. 11, we obtain JNeck =JCa = - 0 . 7 eV, JB~lly = - 1 . 5 eV, and JFS = --1.2 to --1.3 eV for a Fermi surface average of the coupling strength. This latter value is in excellent agreement with the values J = -1 .34 and J = -1 .25 derived f rom fitting the bulk magnetizat ion 37 and resistivity, 42 respectively, of dilute Au(Fe) alloys by Loram et al. From our Eq. (43) we also obtain p = 0.14 s t a t e / eV-a tom (using JFs = --1.2), very close to the value p = 0.15 for Au.

From Fig. 11 we obtain 8 ) ( = 0 . 3 K, which together with P ( = 0.6 K gives X t = 0.9 K and X ~ = 0.3 K, a 3:1 spin dependence of the conduction electron scattering rates. This result clearly demonstrates the importance of spin-dependent scattering in these alloys, even in relatively strong magnetic fields. Conventional " s lope" Dingle tempera ture measurements are clearly inadequate as a measure of conduction electron scattering rates in a system such as Au(Fe).

5.2.2. A u ( C o )

Isolated Co impurities in a Au host display no local magnetic momen t at low temperatures , the characteristic Kondo or spin-fluctuation tempera ture for this system being greater than room temperature .

Thus, no exchange energy shift or spin-dependent scattering is expected to be found in d H v A wave shape measurements as a result of the presence of isolated Co solute atoms. However , the Fr iede l -Anderson condition for magnet ism at a solute a tom site depends directly upon the local density of d states at the Fermi level, which may be locally modified by interactions with neighboring atoms. Evidence that these interactions can and do occur comes f rom bulk measurements of the low-tempera ture magnetizations of alloys such as Au(Fe), 43 Au(Co), 44'45 Cu(Fe), 46"-48 and Cu(Co), 44"49 which have been shown to contain contributions proport ional to c 2 and c 3 (c is the solute concentration) at tempera tures below TK. These terms are interpreted as arising f rom interacting pairs or triplets of solute atoms, respectively.

It is of particular interest to try to distinguish between two types of interactions: nearest-neighbor interactions (perhaps of d - d overlap type)


0.075

0.060

0.045 >< ,.0

0.030

0 1 0 1 5

J I I I I I

o o

O o

~ o o o

o o o o o o o o o

o o o

o o o~ ~.

o o o o O o o o o

O o o o o o o o

o o e o e e o o o oo ~ " ~ .

o o ~ o o o c o o o o o o o o o o o ~ o o o ~ . . . .

o o o o o 0 o o 0 o o o o o o o o o o 0

o o o o o o o o o o

I i I I I 0.12 0.14 0.16

5'

I 0.18

Fig. 12. Simultaneous computer-generated solution of two of equations (36)-(38) to obtain unique values of 8 X and $' for the (111) neck orbit in a sample of Au(150 at ppm Co). (See also Fig. 10 and Section 5 of the text.) The unique solution occurs at 8 X = 0 . 0 3 0 + 0 . 0 0 2 K and $ ' = 0.163 + 0.002.

and longer range interactions (perhaps of RKKY type). Since the conduction electrons presumably directly mediate an interaction of the latter type, it is of considerable interest to try to directly observe the onset, at very low Co concentrations, of interactions between the conduction electrons and the magnetic pairs, triplets, etc., which result, using the wave shape analysis technique.

Third harmonic wave shape analysis measurements for six different single-crystal samples of Au(Co), with three different solute concentrations (0, 150, and 500 at ppm), were carried out for the (111) neck orbit, with results shown in Figs. 12-14. Significant amounts of spin-dependent scattering (SX # 0) were found even in the 150-ppm alloys (Figs. 12 and 13).

.25

.2(

Fig. 13. The difference 8 X in the spin-up and spin- down conduction electron Dingle temperatures for .1~ the (111) neck orbit in Au(Co) at T = 1.1 K. The .~ data were obtained from three different single- ~o.10 crystal samples for the 150 at ppm alloy and two different samples for the 500 at ppm alloy. The error bars represent the probable random .05 uncertainties in 8X resulting from the estimated experimental errors in the relative dHvA harmonic phase angle measurements. O, 150ppm; n , 500 ppm.

D �9

{ %

40 5'0

�9 I

6'0 7'o ' H( K(;)


3.0

2.0 , , r - v

1.0 6

1

F I I I I 0 100 200 300 400 500

0.3 I i i i

0.1

i < / x -.~'---~ - --x I I I 0 5 10 15 20 25

Fig. 14. Top: .~" vs. c (Co impurity concentrat ion, in units of at ppm). Bot tom: 8X vs. c 2. The data are the average of the data for the five different samples described in the legend for Fig. 13, plus measurements made on a single, pure Au "reference sample" at zero impurity concentrat ion. The arrow in the lower figure shows where the 8 X data for the 150-ppm samples would appear if the abscissa in the lower figure was linear in the concentrat ion c.

When the results for all the samples were plotted as a function of Co concentration, c, it was found that the spin-independent part of the scattering rate X scaled linearly with concentration, as would be expected for normal potential scattering by solute atoms. However, 8X scales very nearly as the solute atom concentration squared. We interpret this result as providing clear evidence for the spin-dependent scattering of the conduction electrons by local moments associated with interacting pairs of Co impurities (pair, or local moment, concentration c*oC c2).

These moments are induced moments, resulting from both the applied magnetic field and from stabilization of the condition for the occurrence of magnetism, by impurity-impurity interactions. This picture of induced moments associated with interacting Co pairs is consistent with the results of Boucai et al., at much higher Co concentrations, according to which isolated Co impurities carry no local moment, but groups of three or more Co atoms are spontaneously magnetic. 45


Thus, dHvA techniques are able to detect the very onset of magnetism arising from interactions between pairs of impurities, and so provide a link between observations of isolated local moments, associated with clusters larger than pairs, and the eventual occurrence of magnetic freezing of the spin-glass type. The sensitivity of the dHvA measurements of 8X is indicated by the fact that X t - X ~ = 28X is only 4% and 13% of 3~ in the 150- and 500-ppm Co alloys, respectively.

According to Eqs. (41) and (42) these results can be used to estimate the concentration c*, of local moments, and hence the effective range of the interaction producing magnetic pairs of Co impurities (with c* now replacing c in these equations). For random nearest neighbor pairs the pair concentration expected is c* = 6c 2, while for random pairs with an effective interaction range of 10 A, the pair concentration is c* = 125c 2. For single Fe moments in Au our measurements showed that IJ~l = 1.2 eV. If we assume a similar value for the exchange coupling between conduction electrons and the local moment associated with a Co pair, then Eq. (41) gives c*(Sz)=480c 2 (using p =0.15 state/eV-atom and Ca =0.5 for the neck orbit). Taking (Sz)=2S---3.0 (assuming saturated moments for 50 kOe) gives c* - 160c 2, seemingly in good agreement with the result for pairs formed via a long-range interaction. (Taking {Sz)<2S gives still larger values for c*.)

However, Eq. (42) then predicts an exchange energy shift of magnitude Hex --- 7.5 kOe; in fact, we observed no such shift, though we probably .would not have detected a shift of 2.5 kOe or less. Equations (42) and (43) can both be simultaneously satisfied, with Hex<2.5 kOe, if IJI--4 eV. In this case c * - 1 0 c 2 if (S~) has its saturation value, but c* is again larger than the near-neighbor pair limiting value if (Sz) is less than saturated. Thus, only if IJI has the unreasonably large value of - 4 eV can the local moment concentration approach the low value describing nearest neighbor pairs.

We believe, instead, that our measurements of 8X do support the idea of a much higher concentration of local moments (and a reasonable IJI value), and that Eqs. (41)-(43), derived from a perturbation approach for isolated local moments, are simply invalid in the case of a spatially extended pair moment formed via a long-range interaction. For example, the assumption that T >> TK is almost certainly not true for the TK associated with Co pairs in Au. 4s

6. S U M M A R Y

A new technique of third harmonic de Haas-van Alphen wave shape analysis (THWA) for measurement of the spin-dependent interactions


between conduction electrons and local moments in dilute alloys has been described.

We have derived expressions for the harmonic content of the dHvA effect in a general case, including the simultaneous contributions due to (1) magnetic interaction (MI or Shoenberg effect), (2) the spin-dependent scattering (SDS) of conduction electrons, and (3) exchange energy shifts in their Landau levels due to local moments, in addition to the usual Lifshitz- Kosevich (LK) harmonic content. The effects of MI and SDS mix nonlinearly in determining the observable amplitude and phase of each resultant dHvA harmonic. One important consequence of this mixing is that the observation of a spin splitting zero of the dHvA amplitude is not indicative of equal scattering rates for spin-up and spin-down electrons, in the presence of MI.

However, using information contained in the first three dHvA harmonics, one can define four "observables" (two relative harmonic phase angles and two harmonic amplitude ratios), in terms of which the three microscopic electronic properties of interest (the two spin-dependent scattering rates X t and X ~ and an exchange energy shift Ae) may be determined. The two relative phase angles depend only on the spin- dependent part 8 X = (Xt-X~)/2 of the scattering rate and upon the exchange energy shift Ae; the two harmonic amplitude ratios depend in addition upon the mean (spin-independent) scattering rate . ~= (Xt+XS)/2. Thus, unique values for the three microscopic parameters may be determined, using the information contained in harmonic analysis of a short block of dHvA data at (H, T), and so the (H, T) dependence of the microscopic parameters may be determined. Since the microscopic parameters are in fact overdetermined, an estimate of the effect on them of measurement errors in the "observables" is also obtained.

We have described a convenient sequence, and demonstrated some graphical methods, for solving for the microscopic parameters. These techniques were then illustrated by applying them to studies of the dilute alloy systems Au(Fe) and Au(Co), both of which were found to exhibit local magnetic moments, though apparently of quite different origin.

For Au(Fe) the exchange energy shift (exchange field) and the spin- dependent scattering rates were determined as functions of (H, T). A 3 : 1 anisotropy of spin-up and spin-down electron scattering rates was observed for the (111} neck orbit. The exchange energy was found to behave like an impurity spin near saturation (Brillouin- or Langevin-like behavior), and the strength of the conduction electron-local moment exchange coupling was found to be JN~ck = --0.7 eV. We also showed that the assumption that the exchange interaction proceeds through the d-wave part of the conduction electron wave functions allows one to calculate the Fermi surface- averaged exchange coupling strength JFS. The resulting value, JFS = --1.2 to

Analysis o| the Harmonic Content o| the de Haas-van Alphen Effect 635

-1.3 eV, is in excellent agreement with the values needed to explain both bulk magnetization and resistivity measurements in Au(Fe). Thus, the dHvA effect appears to be a sensitive probe for determining impurity spin behavior in a magnetic field, and for measuring cyclotron orbitally averaged values of the exchange constant Jorbit in very dilute local moment systems.

For Au(Co) we have reported the first dHvA observations of interaction effects between impurities, via measurements of the spin-dependent scattering of conduction electrons by magnetic pairs of Co impurities. The induced local moment associated with Co pairs produces the only spin- dependent scattering in these dilute alloys, because isolated Co impurities (which are also present) carry no local moment at low temperatures. Bulk measurements have already provided evidence of the spontaneous magnetism associated with larger clusters of Co impurities (triplets, etc.), at much higher Co concentrations, so that our measurements of induced pair magnetism provide a link between those observations and the nonmagnetic behavior of isolated Co atoms in Au. Measurements of the nonlinear dependence of spin-dependent scattering rates on Co concentration can be used to estimate either the range of the pairing interaction or (equivalently) the concentration of the magnetic Co pairs. Thus, dHvA techniques allow the onset of magnetism to be observed at very low solute concentrations.

Finally, the possibility of applying these same third harmonic wave shape analysis procedures for the measurement of conduction-electron orbital g-factors in metals has been briefly discussed, and one example of such a measurement, for a Au(Ag) dilute alloy, has been given.

A P P E N D I X

A 1 . Introduct ion

In this section we derive equations for the relative magnitudes of the rth harmonic LK and MI components, for r = 1, 2, and 3, in the presence of SDS and exchange energy shift. We also calculate the phase angles ar by which the resultant rth harmonic amplitude is shifted by MI, measured relative to the phase angle of the pure LK component (Fig. 3). We show that oL2 and a3 can always be calculated for given values of 8X and S', while a l depends very weakly in addition upon X. In practice this last dependence may be either ignored or approximated. Thus, Eqs. (34a) and (34b) continue to provide a direct connection between the two relative phase angle observables (201-02) and (301-03) and the two theoretical parameters 8X and S', assuming that the effective demagnetizing factor Ne of the sample is known. Finally, we sketch a quick sequence of steps for handling the equations, to generate the graphical solutions represented by Figs. 4-9.


A 2 . Second H a r m o n i c

the rat io of second ha rmonic M I and L K componen t s We define R (N. x _ M M I I M LI': 2~ , ) - 2 / 2 �9 Then, using Fig. 3 and Eqs. (29) and (32),

~2 = p~r/4 + 2A01 - (2q~ - 1 - q2)~r/2 (A1)

y2 = (Tr - p c r / 4 ) - (2A01 - A02)+ (1 - q~)~r/2 (A2)

tan dE = (sin y2)/[(1/RE)-COS y2] (A3)

sin t~2 = (sin y2)/[1 + ( l / R 2 ) 2 - (2 /R2) cos y2] 1/2 (A4)

Thus, the r igh t -hand side of Eq. (34a) m a y be calculated for given values of 8 X and S ' (assuming that Are is known).

Actual ly , f rom Fig. 3 it follows that

ME M' k ' ( A ~ C 1 ) 2 s in 0r 2 (A5) R2=~-~-2LK= 2 A~C2 = s i n ( a z + y 2 )

and combin ing Eqs. (34a) and (A2), we obta in

(a2 + y 2 ) - 2 a l = (201 - 02) Lt<- (201 - 02) ~ + (or - p c r / 4 ) + (1 - qE)lr/2 (A6)

so that a2 + "Y2 = "rr -- ]~2 is a directly measurab le quanti ty, if we neglect the small correc t ion 2a l . (a l is appreciable only if the th i rd-order M I correct ion to the d H v A fundamenta l cannot be neglected in compar i son with the fundamenta l ampli tude.)

A 3 . Third H a r m o n i c

There are two third ha rmonic M I contr ibut ions in Eq. (28), which we designate in Fig. 15 as M ~ I~ ocMLKM LK M ~ Iq ( , 2 ) and as [oc(MLK)3]. *

Using Fig. 15, their resul tant is found to be

M MI = D3F3 = Da[(3k' /2)2(A'lCIA~C2) 2

+ (]k'E)2(A ~ C1) 6 + 2(3k'/2)(]k'E)(A ~ C1A~C2)(A ~C1) 3 cos d)4] 1/2 (A7)

with phase angle

~3 = (r nl- ~5) = (~q - ( ~ 4 - r (AS)

*The terminology in which the MI components are referred to as p (in-phase) and q (out-of-phase) channels with respect to the LK component is carried over from Phillips and Gold for convenience. However, this classification is no longer exact, because of the different combinations of SDS-induced phase shifts which occur in the LK and the two MI components, so that they no longer differ in phase by precisely 0, ~r/4, or ~r/2 rad.

Analysis of the Harmonic Content of the de Haas-van Alphen Effect 63'7

i / i I

/ \ / \

i / I i l ~ \ i l l l I

Fig. 15. Phasor diagram showing formation of the resultant third harmonic MI from the two independent "q" and "pc/" channel contributions [Eq. (28)]. The phases are shown with reference to the phase that the pure LK third harmonic would have in the absence of SDS [horizontal axis = 3Xo + p~r/4- (1 + q3)~'/2]. Solutions for the phase angles are given in the Appendix.

w h e r e

r ~ = pit~4 + A01 + A02- - (q ~ + q~ -- 1 -- q3)1r/2

~bq = (1 + p)'rr/2 + 3A01 -- (3q~ -- 1 -- q3)zr/2 ( A 9 )

~b4 = (4Jq - 4 ~ ) = 7r/2 + p~r/ 4 + (2A01 - A 0 2 ) - (2q~ - q~ )~r/2

sin ~b5 = (sin r x = (sin rb4~k'2(A~C1)3/F3

tan 465 = (sin 464)/[(! / r3) + cos ~b4] (AIO) IIA-Mlql llzrMIpq 1 t t 2 t

1"3=1,13 /xvl 3 =~k (A1C1) /A2C2=(sinlbs) /s in(d~a-dJs)

~" r l = �89 T h e n , the angle ~3 can be d e t e r m i n e d us ing Fig. 3, as

sin a3 = (sin "y3)/[(1/R3)2+ 1 + ( 2 / R 3 ) cos T3] I/2 (All)

tan ~3 = (sin "ya)/[(1/R3)-cos 3t3]

wi th MI LK

R 3 = M3 / M 3 = (sin a 3 ) / s i n (a3 + ~/3)

3'3 = ~" - ( a 3 + / 3 3 )

( A 1 2 )

(M3)


with ~:3 given by Eqs. (A8)- (A10) . The observable third harmonic amplitude is then

M o B = [(M3LI~ )2 + (MMI)2 _ 21M~K i ~r~3, I cos y31 ~/2

3 ,' 2 2 "]/3] 112 (A14) = D [(A3C3) +F3-2A~C3Facos

A4. First Harmonic

It is usually not necessary to calculate the MI contribution to the d HvA fundamental, since it is smaller than the LK contribution by the square of the combined Dingle and temperature factors [Eq. (28)]. However, the angle 0~' 1 can be calculated, if necessary, first using Fig. 16 to combine the two MI contributions, then obtaining their resultant:

M~ I ------ D 3F1 = D 3[(~k 'E)2(A ~ C 1)6 .+ (�89 ~ C1)2 (A 2C2) 2

+~(k')3(A'~C1)4(A~Cz) cos &4] 1/2 (A15)

where we note that Fa = F 3 / 3 and the phase angle

~1 = t~/n/ + (~5 = t~p -- (t~4 -- t~5 ) ( A 1 6 )

where

q ~ = - p w / 4 + (A02 - m01) "[- (1 + ql + q i -- q~)~r/2

~bp = A0I + (2 -- q'x + qa ) z r /2

rk4 = (d~p - rb~) = p ~ ' / 4 + (2A0a -- A02) + (1 - 2q ~ + q-~ )7r/2

(A17)

~ / t ~ ~ M 1

Fig. 16. Phasor diagram showing formation of the resultant first harmonic (fundamental) MI from the two independent "p" and "pq" channel contributions [Eq. (28)]. The phases are shown with reference to the phase that the pure LK fundamental would have in the absence of SDS (horizontal axis = x o + p l r / 4 -

(1 +qa)cr/2]. Solutions for the phase angles are given in the Appendix.

Analysis of the Harmonic Content of the de Haas-van Aiphen Effect 639

and

sin ~b5 = (sin ~b4)M~I"/M~ I = (sin ~b4)(k') 2(A ~ Cl)3 / 8F1

tan ~b5 = (sin daa)/[(1/rl)+COS ~b4]

r l = M M I O l M M i I ~ 1 , , 2 , =xk (A1C1) /A2C2 t

= (sin ~bs)/sin (4,4- ~bs)= r3 = 1R2

(A18)

Then, a l is determined from Fig. 3, using equations identical to Eqs. (A l l ) - (A13) with subscript 3 everywhere replaced by 1, and ~ given by Eqs. (A16) and (A17).

Finally, the observable fundamental amplitude is

MoB = [(M].K )2 + (M]n~)z_ 21MLK I IM~II cos 3,d 1/2

= D[A'ICa) z + (FxD2) 2 - 2A]CIF1D2 cos 3/1] 1/2 (A19)

A 5 . C o n v e n i e n t Steps for H a n d l i n g MI Correct ions

For the second harmonic MI corrections, Eqs. (A1)-(A5) may be used just as they are written down. For the first and third harmonics, there are two MI components contributing, and the resulting complexity of the above equations makes it slightly unclear how to proceed. For the third harmonic correction the sequence we have used is to first obtain ~:3 using Eqs. (A8) and (A9), then obtain 3'3 from (A13), a3 from (Al l ) , and/33 from (A13). A similar procedure is used for the first harmonic. Note that in the case of the first harmonic it is in using the analog of Eq. ( A l l ) that the procedure becomes weakly dependent on 7(, since

R1 =- M ~ / M LK ~ C2D 2 (A20)

so that otl depends weakly on X. For a sample with reasonable Ix, T, and X, this dependence (and a 1 itself) may be ignored or approximated.

We have assumed throughout this Appendix that the sample's effective demagnetizing coefficient Ne is known. For a sample in the shape of a cube or a general second-order surface, or a rectangular prism with aQ along a principal axis, Ne can be calculated and this value used, provided that the modulation frequency is low enough to give complete penetration of the sample. N~ may also be measured directly, using a "reference sample" of similar size, shape, and orientation. For a more complete discussion of these problems see Refs. 11-14.


A C K N O W L E D G M E N T S

W e w o u l d p a r t i c u l a r l y l ike to t h a n k D r . P a u l R e i n d e r s fo r his c a r e f u l

r e a d i n g of t h e m a n u s c r i p t a n d fo r m a n y h e l p f u l c o m m e n t s . W e also wi sh to

t h a n k D r . I. M. T e m p l e t o n fo r t h e l o a n o f t h e A u ( A g ) s a m p l e , a n d o n e of

us ( D H L ) wi shes to t h a n k t h e U n i v e r s i t y of N i j m e g e n fo r its h o s p i t a l i t y a n d

t h e use o f its fac i l i t i es d u r i n g t h e t i m e t h a t this p a p e r was b e i n g w r i t t e n .

R E F E R E N C E S

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Analysis of the Harmonic Content of the de Haas-van Aiphen Effect 641

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Electronic properties from analysis of the harmonic content of the de Haas-van Alphen effect:...

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