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PHYSICAL REVIEW B 1 APRIL 1998-IIVOLUME 57, NUMBER 14

Magnetoresistance and magnetization study of thulium

Mark Ellerby* and Keith A. McEwen*

Department of Physics, Birkbeck College, University of London, London WC1E 7HX, England

Jens JensenO” rsted Laboratory, Niels Bohr Institute, Universitetsparken 5, 2100 Copenhagen, Denmark

~Received 18 September 1997!

The results of a detailed resistance and magnetization study of thulium are presented. From these results wederive a magnetic phase diagram for thulium. Thec-axis component of the resistivity is strongly affected bythe superzone energy gaps induced by the modulated ordering of the magnetic moments in thulium in agree-ment with earlier experiments. According to the theory of Elliott and Wedgwood the superzones introduce anincrease of the resistivity proportional to the magnetization just belowTN , whereas experimentally the changeis found to be quadratic in the magnetization. Except for this principal discrepancy, the experimental results arewell accounted for when combining a variational calculation of the resistivity with the magnetic model derivedfor thulium from neutron scattering experiments.@S0163-1829~98!05009-7#

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I. INTRODUCTION

A detailed neutron-diffraction study of thulium was caried out by Koehleret al.,1 who found that belowTN556 Kthe magnetic structure is sinusoidally modulated along thcaxis, with a wave vectorQ.(2/7)c* and with the momentsconstrained to thec axis. As the temperature is reducehigher-order odd harmonics appear, and in the zetemperature limit the modulation of the magnetic momeapproaches a nearly perfect square wave which is commsurable with the lattice. Hence at zero temperature the stture is ferrimagnetic with the moments being of maximumagnitude and parallel to thec axis in four hexagonal layerfollowed by three layers with the moments antiparallel toc axis. These results were later confirmed by Brunet al.,2

who found that the structure only becomes commensurbelow ;32 K, and that the magnitude ofQ decreases lin-early by about 5% between 32 K andTN .

The most substantial study of the magnetization of tlium was performed by Richards and Legvold.3 They deter-mined the broad outline of the magnetic phase diagramconfirmed the ferrimagnetic structure determined byneutron-diffraction experiments. Thulium exhibits a firsorder transition from the ferrimagnetic structure to thec-axisferromagnet at a critical field applied along thec axis. In thecommensurable phase the critical field is found to be neconstant and about 2.8 T. The saturation moment measby Richards and Legvold was 7.14mB per ion where the extra0.14mB may be associated with the polarization of the coduction electrons. Zochowski and McEwen4 have recentlypresented results for the variation of the lattice parameterthulium obtained using capacitance dilatometry. They sumarized their results in a magnetic phase diagram whshows some complexity, and they found evidence of a 0increase in the length of the Tm crystal along thec axis onmaking the transition from the ferrimagnetic to the ferromanetic state.

In the low-temperature limit the magnetic excitations a

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spin waves. In the ferrimagnetic phase the spin wavestected by inelastic neutron scattering are found at enerbetween 8 and 10 meV.5–7 Since the magnetic periodicity iseven times that of the lattice along thec axis, the spinwaves are split into seven closely spaced energy bandslow temperatures, a relatively strong coupling betweenspin waves and the transverse phonons is observed. Incing the corresponding magnetoelastic coupling, the randphase-approximation~RPA! model8 developed by McEwenet al.7 explains most of the observations made both inlow-temperature spin-wave regime and at elevated temptures. Although thulium belongs to the heavy end of the raearth series the Ruderman-Kittel-Kasuya-Yosida~RKKY !-exchange interaction, proportional to (g21)251/36, is weakcompared to the crystal-field anisotropy energies. Thisplies that crystal-field excitations are important in thuliumelevated temperatures, both below and aboveTN .

The spin-wave excitations in the ferromagnetic phaseduced by an applied field parallel to thec axis, have beencompared with the excitations seen in the zero-field phaslow temperatures.9,10 The analysis shows that both thcrystal-field anisotropy and the exchange coupling is ratstrongly modified from one phase to the other, and thchanges must be related to the large shift in thec-axis latticeparameter~0.7%! observed at the transition.4

The earliest resistance study of a single crystal of thuliwas performed by Edwards and Legvold.11 They found thatthe resistivity parallel to thec axis showed a sharp upturn aTN . This increase in thec-axis resistivity belowTN is ex-plained by the reduction of the Fermi-surface area perpdicular to thec axis due to the intersection of new zonboundaries produced by the magnetic periodicity. Thisperzone effect was proposed by Mackintosh.12 The theorywas subsequently worked out in detail by Elliott anWedgwood,13,14 who applied the relaxation time approximation and assumed the conduction electrons to be frelectron-like.

Here we report a systematic study of the resistance

8416 © 1998 The American Physical Society

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57 8417MAGNETORESISTANCE AND MAGNETIZATION STUDY . . .

magnetoresistance in thulium, examining two particularometries both with the applied fieldB parallel with thecaxis. In the longitudinal geometry the currenti is parallel tothec axis and in the transverse casei is parallel to thea axis.In addition magnetization measurements were made in oto verify some of the finer aspects of the magnetic phdiagram and to allow consideration of the effects of demnetization fields. The theory of Elliott and Wedgwood is geeralized by using a variational calculation15 rather than therelaxation time approximation. Although the account of tsuperzone effect is essentially unchanged, the comparbetween theory and experiments is much improved whenresult of the variational calculation is combined with tRPA model for Tm derived from the neutron-scatteriexperiments.7,9,10

II. EXPERIMENTAL DETAILS

The crystals of thulium used in this study were cut frothe same ingot as that used in Refs. 4,7,9,10 and grown aAmes Laboratories. The dimensions of the sample usedthe magnetization measurements were 6.531.030.2 mm~mass 0.01212 g! with the c axis parallel to the longest dimension. The longitudinal resistance sample had a crsectional area of 0.20 mm2 and a length~separation betweenvoltage probes! of 4.5 mm. For the transverse resistansample the corresponding dimensions were 0.35 mm2 times5.5 mm.

The magnetization was studied using the 12 T vibratsample magnetometer~VSM! at Birkbeck College. The operation of the VSM is described in Ref. 16. The resistanand magnetoresistance measurements were made usinfour-probe dc method in conjunction with a Keithley 18nanovoltmeter and Keithley 220 constant current supply:propriate current reversals were made to eliminate therelectric effects. The cryostat was built by Oxford Instrumeconsisting of a variable temperature insert mounted in a 7 Tvertical-field magnet. The temperature was controlled ameasured using a calibrated carbon-glass thermometeconjunction with a Lakeshore DRC-93C Controller.

To ensure the same conditions for each isothermal mnetoresistance and magnetization measurements the sawere heated aboveTN ~annealed! and then cooled to therequired temperature. Measurements as a function of tperature were made while cooling in a constant field.

III. EXPERIMENTAL RESULTS AND THE PHASEDIAGRAM

In Fig. 1 we present the temperature dependence ofresistivity measured in the longitudinal configuration, wfield and current along thec axis. Figure 1~a! shows theeffect of an applied field. At zero field there is an upturnthe resistivity atTN;57.0 K corresponding to the behavioobserved by Edwards and Legvold.11 In contrast, the appli-cation of a fieldB54.0 T along thec axis, which drives thesystem ferromagnetic, quenches the upturn and the reszero-temperature resistivityr(T→0) becomes significantlylower. Figure 1~b! presents a detail of the first derivativethe resistivity with respect to temperature in an applied fiof 1.0 T. The figure shows the presence of two anomalie

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the ordered phase at 37.3 and 30.0 K. Evidence of simtransitions was also found in measurements made at ofields between 0 and 2.7 T.17 The effect of the applied fieldon the low-temperature behavior of the resistivity is psented in Fig. 1~c!: r(T→0) has been subtracted to eacomparison of data. Between 0 and 2.7 T there is a stechange in the curvature followed by a much more radichange between 2.7 and 4 T, in which interval the systmakes the transition from the ferrimagnetic to the ferromnetic phase.

Figure 2 presents isothermal magnetoresistance and mnetization results obtained at 5 K@Figs. 2~a!–2~c!# and at 35K @Figs. 2~d!–2~f!#. The samples used for the magnetizatiand the longitudinal magnetoresistance measurements hdemagnetization factorD!1 and the consequent demagntizing fields were negligible. In the transverse configurati@Figs. 2~c! and 2~f!# demagnetization corrections have bemade using18 D50.28. The magnetization data at 5 K showvery clearly the transition from the ferrimagnetic phase, wa moment of approximately 1mB per ion, to the ferromag-netic phase where the magnetization saturates at a valu7.14mB ~in a field of 7 T!. The transition occurs at 2.8 T foan increasing field and at about 2.0 T in a decreasing fiAt the transition thec-axis resistivity at 5 K@Fig. 2~b!# de-

FIG. 1. The temperature dependence of thec-axis resistivity inthulium ~a! showing the effect of an applied field;~b! the first de-rivative for B51.0 T; ~c! in various magnetic fields.

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8418 57MARK ELLERBY, KEITH A. MCEWEN, AND JENS JENSEN

creases from approximately 4.6 to 1.1mV cm, whereas thechange of thea-axis resistivity@Fig. 2~c!# is one to two or-ders of magnitude smaller. At low temperatures the transiappears to occur in a single stage for both the magnetortance and the magnetization. Figures 2~d!–2~f! show themeasurements made at 35 K for an increasing field.results indicate a low-field transition as marked by arrowsB50.4 T. As the field is increased there is a second tration at 2.8 T after which the magnetization slowly beginssaturate, reaching 6.6mB/ion at a field of 7 T. The hysteresiin this transition to the ferromagnetic phase was about 0.at 5 K, and decreased to about 0.1 T at 35 K. Note thattransverse magnetoresistance in Fig. 2~f! shows more or lessthe opposite behavior of that observed at 5 K though on adifferent scale.

Figure 3 presents the detail and temperature dependof the low-field transition observed in the temperature intval 33–40 K. The magnetization measurements at 35.037.0 K clearly shows that there is hysteresis of about 0.associated with this transition, thus indicating that it isfirst order. The transition corresponds to the anomaly at 3K in the derivative of the resistivity at 1 T@see Fig. 1~b!#.

The relative variations of the resistivities obtained in tpresent experiments at zero field compare well with the pvious results obtained by Edwards and Legvold,11 but theabsolute magnitudes are smaller, by nearly a factor of 2the c-axis case and about a factor of 1.3 for thea-axis com-ponent. Although we cannot be sure of the origin of thedifferences, we suspect that they arise from possible errovariations in the cross-sectional area of the samples inexperiment of Edwards and Legvold. We note that opresent results are consistent with the averaged result19 ob-tained on a polycrystalline sample.

FIG. 2. Isothermal magnetization and longitudinal and traverse magnetoresistance at 5 and 35 K.

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Figure 4 presents the magnetic phase diagram derfrom the resistivity and magnetization measurementsscribed above. The positions of the phase boundaries wdefined by taking the midpoint of a transition for increasifield. The general form of the phase diagram is consistwith that derived from the magnetostriction and thermexpansion measurements.4 There are however differences ithe details. The first of these differences is the boundbetween the ferrimagnetic and ferromagnetic phase. In Rethere are two additional phases between the two structwithin an interval of the field which corresponds to the dmagnetization field of 0.8 T not corrected for in that studThe present experiments show that the extra phases dooccur for samples where the demagnetization factorD!1,and the transition between the ferrimagnetic and the femagnetic phase is accomplished in a single step. Anodifference in the two phase diagrams relates to the regmarked A in Fig. 4. In the study by Zochowski anMcEwen4 there was an additional phase betweenA and thec-axis modulated~CAM! phase. The hysteresis and the stewise change shown by the magnetization, Figs. 3~a! and 3~c!,suggests that the demagnetization field, which was signcant in the magnetostriction measurements, may also because for this subdivision of theA phase.

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FIG. 3. Details of the low-field transition indicated by an arroin Figs. 2~d!– 2~f!.

FIG. 4. The magnetic phase diagram constructed from thethermal magnetization (h), magnetoresistance (L), and resistancein a constant field (d). The division of the ferrimagnetic phase inttwo phases at nonzero field, which is indicated by the dashed lintentative.

,


The magnetization measurements at 35 K in Fig. 3~a! @orat 37 K in Fig. 3~c!# indicate that the phase at fields above0.4 T ~or 0.8 T! would have an extrapolated moment of about0.12mB/atom ~or 0.09mB/atom! in the limi t of zero field. Thiscorresponds very well to the moment expected in the ferri-magnetic phase if it was the stable zero-field phase at thesetemperatures. Hence, the so-called A phase in Fig. 4 must beclosely related to the commensurable ferrimagnetic phasefound at temperatures below about 30 K, whereas the CAMphase is the incommensurable phase which was observed2 byBrun et al. in the zero-field case. This raises the questionconcerning the precise nature of the A phase, and the signifi-cance of the nearly vertical dashed line in Fig. 4 between theferrimagnetic phase and the A phase. One possibility is thatthe commensurable ferrimagnetic structure has not devel-oped completely in the A phase; that this is a mixed phasemaintaining some incommensurable features. The other pos-sibility is that the anomalies defining this extra line do notarise from a phase transition but from some other drasticmodification of the system. For instance, the superzone gapsinduced by the periodic modulated moment may lead to asudden change in the properties of the conduction electrons,which at zero field may just happen to occur close to thecommensurable-incommensurable phase transition. Thisphase transition is influenced by the applied field, because ofthe net moment in the commensurable phase, whereas theposition and the magnitude of the superzone gaps is onlyweakly dependent on the field. A neutron-diffraction study ofthe magnetic structure of thulium in a field parallel to the caxis revealed no evidence of any anomalies,20 in either thewave vector or the intensities of the magnetic satellites ~fun-damental and third harmonics!, at the boundary between theA and the ferrimagnetic phases. In contrast, clear anomalieswere detected at the boundary between the A and CAMphases. Hence, the neutron experiments do not show anyindications of the appearance of an extra A phase when ap-plying a c-axis field.

IV . ANALYSI S OF THE RESISTANCE

Figure 1~a! shows the dramatic effect that magnetic orderhas on the c-axis resistivity. In the zero-temperature limi t theresidual resistivity is changed by a factor of nearly 4 at thetransition from the ferromagnetic phase to the phase with anoscillating c-axis moment. The explanation for this behavioris based on the superzone energy gaps created in the conduc-tion electron bands by the oscillating moment, which wasconsidered by Mackintosh.12 The sinusoidal polarization ofthe conduction electron spins induced by the RKKY interac-tion between the spins of the 4 f electrons and the conductionelectrons leads to energy gaps at the wave vectors(t6nQ!/2, wheret is a reciprocal lattice vector andQ is themagnetic ordering vector. The leading-order term corre-sponds to n51, but the squaring up of the magnetic orderingintroduces other odd integer values of n, and the higher-order coupling processes introduce both even and odd valuesof n. In the commensurable case, Q5(2/7)c*, the Brillouinzone is reduced by a factor 7 compared to the nonmagneticone and energy gaps may occur at the center and on theboundaries of the new Brillouin zone. Watson et al.21 haveestimated the effects of the magnetic ordering on the band

electrons in thulium within a nonrelativistic augmentedplane-wave approximation. Their calculations show the ap-pearance of many energy gaps of up to 0.085 eV, at or closeto the Fermi energy. These energy gaps result in a largereduction of the Fermi surface area perpendicular to the caxis. In order to introduce the essential effects due to themagnetic superzones Elliott and Wedgwood made the sim-plest possible assumptions in their calculations.13 They foundthat the conductivity component ~specified by the unit vectoru) perpendicular to the new zone boundary, decreases lin-early with the size of the energy gap D« in the case wherethe zone boundary touches the Fermi surface or cuts it intotwo parts:

suu.e2t

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~vk•u!2

u¹k«kudS.suu

0 ~12du! ~4.1!

assuming a constant relaxation time t at the Fermi surfacewhere vk is the Fermi velocity and dS is a surface element ofthe Fermi surface. The relative reductiondu of the conduc-tivity is to a first approximation the sum of the contributionsfrom all the different energy gaps, among which the domi-nating terms are those linear in the energy gap. Further, theenergy gaps of major importance are those which are propor-tional to the main first harmonic of the magnetic momentsM1. Hence we may assume

du5Gu M1 /M10 , ~4.2!

where M10 is the zero-temperature saturation value of the first

harmonic. The superzone boundaries are perpendicular to thec axis and in the free-electron model the resistivity in thebasal plane parallel to the boundaries is only affected to sec-ond order in D«/«F . In principle, the coupling of the elec-trons to the lattice may introduce linear terms in the basal-plane resistivity, but, experimentally the superzones do notseem to have much effect on the basal-plane resistivity inany of the heavy rare-earth metals.13,14,22 Edwards andLegvold used the Eqs. ~4.1! and ~4.2! for analyzing theirexperimental results on thulium.11 They used the approxi-mate expressions for the different contributions to t whichwere considered by Elliott and Wedgwood.13 In the fit of thec-axis resistivity they used Gc50.86, and they assumed thebasal-plane resistivity to be unaffected by the superzonegaps, Gb50.

If the effects of the superzones are neglected the Boltz-mann equation determining the resistivity may be solved us-ing the variational principle.15,23,24 Assuming free-electron-like behavior and the RKKY coupling to be q independent,the magnetic part of the resistivity is8,24

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~4.3!

with the weighted q average of the susceptibility tensor com-ponents given by

^xaa9 ~q,v!&q512

~2kF!4E0

2kFqdqE dVq

4p~q•u!2xaa9 ~q,v!.

~4.4!

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In the high-temperature limi t these expressions predict thespin-disorder resistivity to saturate at

r→rspd5J~J11!rmag0 . ~4.5!

When the scattering of the electrons against impurities andphonons is included the total resistivity is25

r total5r res1rphon1rmag, ~4.6!

where the impurity contributionr res is considered to be in-dependent of temperature and applied field. The phonon con-tribution rphon is determined by the Bloch-Gruneisenformula15

rphon5rQS T

Q D 5E0

Q/T z5

sinh2~z/2!dz ~4.7!

with aDebye temperature26 of Q5167 K in thulium. If the qvariation of the magnetic contribution ~4.4! is neglected it isstraightforward to repeat the calculation of Elliott and Wedg-wood and include the effect of the superzone boundaries inthe variational calculation of the magnetic resistivity contri-bution. To leading order we may generalize their result andwrite the final u component of the resistivity as

r totaluu 5

r resuu1rphon

uu 1rmaguu

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with rmaguu determined by Eq. ~4.3! andrphon

uu by Eq. ~4.7!. Theq average ~4.4! wil l be affected by the superzones, but theextra effects may be unimportant compared with those al-ready neglected during the derivation of Eq. ~4.4!. In the caseof thulium most of the dispersive effects are weak because ofthe relatively small magnitude of the RKKY coupling. Forinstance, the width of the spin-wave energy band at low tem-peratures amounts to only about 20% of the energy gap inthe spin-wave spectrum. Hence, most of the contributions tormag

uu are reasonably well described in terms of the mean-fieldsusceptibility averaged over the different sites, i.e.,

^xaa9 ~q,v!&q.xaa9 ~v!uMF ~4.9!

instead of Eq. ~4.4!. In thulium the mean-field approximationprovides agood estimate of the q averaging except for twofeatures. Near TN , the critical fluctuations at small v for qclose to Q may be so large that they may dominate the be-havior of the q-averaged susceptibility. Secondly, the disper-sive effects cannot be neglected when considering the cou-pling between the magnetic excitations and the phonons. Thecoupling between the transverse phonons and the magneticexcitations propagating along the c axis in thulium isimportant,7 and there are indications that the same couplingis just as large for the excitations propagating in the basalplane.9,10 For the purpose of making an order of magnitudeestimateof this contribution,rm-p

uu , wehaveassumed thecou-pling to be isotropic in q space and 2kF in Eq. ~4.4! to be ofthe order of 2p/c. These approximations are crude but pserve the essential feature that there is magnetic scatteringintensity occurring at all the energies of the phonons, wherethe low-energy part has an exponentially strong weight inEq. ~4.3! at low temperatures. The mixing of the scattering atdifferent places in q space which occurs in the ferrimagnetic

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phase has two consequences. It makes the calculated resultsless sensitive to the precise way the weighting in q space isperformed. Secondly, it increases the volume in the Brillouinzone where low-energy scattering may occur ~this being ofthe order of the factor of 7 by which the Brillouin zone isreduced in the ordered phase!. The calculations show that themodification of the magnetic part of the resistivity due to themagnon-phonon interaction is of no importance in the ferro-magnetic ~or paramagnetic! phase, but that the q mixing inthe ferrimagnetic phase increases the effect of this couplingso much that its contribution to rmag

uu is estimated to be thedominating one below ;12 K.

The resistivity in the different phases of thulium has beencalculated as described above. The mean-field value of thesusceptibility tensor used in Eq. ~4.9! was obtained in theferrimagnetic phase from the model established by McEwenet al.7 This model predicts the critical field for the ferri- toferromagnetic transition to be 4.2 T, which is 50% largerthan the experimental value. This discrepancy may be re-moved by including the large magnetoelastic contributionassociated with the 0.7% change of the c-axis lattice param-eter which occurs at the transition.4,9 The neutron-scatteringexperiments show that the spin-wave energy gap at low tem-peratures changes from about 8.5 meV in the ferrimagneticphase to about 6.7 meV in the ferromagnetic phase at a fieldof 4 T. We have included this shift in the anisotropy energyby introducing, somewhat arbitrarily, the appropriate modi-fication of the crystal-field parameter B6

0 proportional to^O6

0&. This model is a preliminary one which indicates thatthe change of the spin-wave energy gap is not necessarily ofmuch importance for the resistivity, since in practice we ob-tained almost the same result whether the energy shift wasincluded or not.

Figure 5 shows the comparison between the calculatedresults and the c-axis resistivity measurements in a c-axisfield of 0 and 4 T. The resistivity parameters used in the fitare given in Table I, and the calculated contributions to theresistivity in zero field are presented separately in Fig. 6. Inthis figure the magnetic part is divided into three compo-nents, arising from the coupling between the magnetic exci-tations and the phononsrm-p the longitudinal partrmag

L due tothe cc susceptibility component and the transverse part rmag

T

deriving from the sum of the two basal-plane components.Surprisingly, the longitudinal fluctuations dominate, not onlyclose to TN but also at low temperatures ~above 12 K!. Thevariation of the mean field from site to site leads to strongerlongitudinal fluctuations in the ferrimagnetic phase at lowtemperatures than in the ferromagnetic one. Figure 6 indi-cates that there is still an appreciable variation left in rmag

T

and rmagL at 80 K ~notice that in the high-temperature limit

rmagT is going to be twice as large as rmag

L ). However, con-sidering the total contribution r

mag5rmag

T 1rmagL then it is

already about 95% of the saturation value rspd at TN andhence the phonon contribution completely dominates thetemperature derivative of the resistivity above TN . In thezero-field case, the resistivity measurements have been ex-tended up to room temperature and the Bloch-Gruneisen for-mula Eq. ~4.7! produces a nearly perfect fit of the data be-tween TN and room temperature.

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Figure 7 shows the low-temperature behavior ofc-axis resistivity at various fields. The most striking effehere as in Fig. 5 is the strong difference between thec-axisresistivity in the two phases of thulium produced by theperzone energy gaps in the ferrimagnetic phase. Thereadditional effects due to the reduction of the longitudinfluctuations and of the contributions due to the magnphonon interaction at the transition to the ferromagnephase, which lead to a visible change in the low-temperavariation of the resistivity. The good agreement betwetheory and experiments shown in Fig. 7 indicates thatchanges predicted by the theory are indeed occurring.

Figure 8 shows the behavior of the resistivity in the baplane, and the calculated results at zero field and atillustrate those differences between the two phases whichstill left when the superzone effects are neglected. The cparison with the experimental results at zero field showbetter overall agreement but a less satisfactory account olow-temperature behavior than obtained in the case ofc-axis resistivity. The low-temperature discrepancy maydue to the fact that the superzone energy gaps in the seorder may increase the conductivity in the basal plaWhether or not this is the right explanation is difficult to sa

FIG. 5. Thec-axis resistivity at zero field and in a field of 4applied along thec axis. The experimental results are the same aFig. 1~a!, which are compared with the calculated behavior shoby the solid lines. The dashed line shows an alternative accouthe superzone effects which assumesr total

cc to be proportional to (12aM1

2) instead of using Eq.~4.8!.

TABLE I. Resistivity parameters in units ofmV cm andGu

used in the calculations.

Comp. r res rQ rspd Gu

cc 1.22 9.45 7.40 0.73aa 0.92 31.0 21.2 0

et

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lTre-

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because the applied field removing the superzone gapsgives rise to an additional effect not included in the analyabove. Due to the Lorentz force there is an increase ofresistivity in the transverse geometry,23 which according toKohler’s rule is proportional to the square of the field at lofields. This effect is responsible for most of the field depedence observed at 5 K and shown in Fig. 2~c! and is impor-tant in most of the low-temperature regime. The largchanges observed at elevated temperatures and illustratFig. 2~f! are reproduced by the present theory. For instathe slope of the resistivity as a function of field in the fermagnetic phase changes sign from being negative betw;15–40 K~opposite sign of the contribution due to the Loentz force! to being positive above 40 K and belowTN .

The value ofrQ has been adjusted so that the calculaa-axis resistivity agrees with the experimental one at rotemperature. This leads to a less satisfactory fit just aboveTNin Fig. 8 than in thec-axis case. One way of removing thdiscrepancy would be to assume an anisotropic RKKY cpling, so that the contributionrmag

T to thea-axis resistivity isenhanced by a factor of about two in comparison withrmag

L .However, this seems to be a much too drastic modificationthe magnetic properties in comparison with the rather michange of thea-axis resistivity produced by the modification

V. DISCUSSION AND CONCLUSION

From the magnetoresistance and magnetization measments the phase diagram of thulium in the presence of a fiin thec direction has been derived. At low temperatures afields the system is ferrimagnetic, corresponding to a comensurable square-wave modulated ordering of thec com-ponent. At high temperatures but belowTN the modulationof the c-axis moment becomes incommensurable with

nnof

FIG. 6. The different contributions to the calculated variationr total

cc at zero field. The impurity termr reswhich is assumed constanwhen the superzone gaps are neglected, shows the variation ofactor (12GcM1 /M1

0)21 in Eq. ~4.8!.

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lattice. The experiments indicate an intermediateA phasebetween these two phases in the presence of ac-axis field.However, this interpretation is uncertain, and it requiresditional experimental investigations to decide whetheranomalies observed in the derivative of the resistivity, orthe thermal expansion,4 at the boundary between the ferrmagnetic phase and theA phase, are reflecting a true phatransition or are due to other drastic modifications ofsystem. We note that the neutron-diffraction experimeshow no anomalies at this boundary.20

The most striking effect observed in the magnetoretance of thulium is the large change in thec-axis resistivityat the transition between the ferri- and ferromagnetic phaThe superzone gaps reduce thec-axis conductivity by thefactor (12Gc)

2153.7 in the zero-temperature limit. The dscription of the magnetoresistance in thulium obtained bypresent calculations is satisfactory if the effects of the supzone gaps are small or vanish, as is the case for thea-axisresistivity or for thec-axis resistivity in the ferro- or paramagnetic phase. This circumstance allows an accuratesessment of the superzone effects, which according totheory of Elliott and Wedgwood13,14 should scale linearlywith the modulated magnetization, Eq.~4.8!. The compari-son at zero field in Fig. 5 shows that this is roughly true,there are systematic deviations between the effects predby the theory and the observed behavior. The higherharmonics developed due to the squaring up of thec-axismoment may give rise to a more complex variation of tsuperzone energy gaps, but just belowTN the higher harmon-ics may be neglected, and in this temperature regime

FIG. 7. The comparison between theory and the experimec-axis resistivity obtained at low temperatures at the different vues of thec-axis field. In comparison with Fig. 1~c! the experimen-tal results at 2.7 T~showing a mixed-phase behavior! have beenreplaced by the results obtained at 3 T, and the residual resistivhave not been subtracted in the present figure.

-en

ets

-

s.

er-

s-he

tedd

e

resistivity does not change proportionally toM1 but ratherproportionally toM1

2 as indicated by the dashed line in Fi5. The critical effects expected in this temperature regiseem to be unimportant: at least these effects should enhthe resistivity in comparison with the mean-field behavrather than the opposite. Thermal activation across the suzone energy gaps neglected in the theory should be unimtant as soon asM1 /M1

0 is larger than;0.06 ~0.1 K belowTN).

Apart from the different influence the superzone enegaps have on thec-axis resistivity, the present account of thmagnetoresistance of thulium is acceptable and it is mimproved in comparison with the previous one.11 The resis-tance in thulium behaves in a way which is consistent wthe RPA model established by McEwenet al.7 The compari-son between the theory and the experiments is improwhen including the isotropic extrapolation of the magneelastic contribution derived for the excitations propagatingthe c direction.7 However, the improvement is marginal because the effect is small and only of slight importance atvery lowest temperatures.

In an anisotropic ferromagnet the longitudinal part of tsusceptibility in Eq.~4.3! may be neglected to a first approxmation in comparison with the transverse spin-wave conbutions. In this case27,28 the magnetic resistivity becomeproportional toTexp(2D/kBT) whereD is the energy gap inthe spin-wave spectrum. At low temperatures thulium iswell-defined spin-wave system, but the approximation whapplies for instance in terbium, does not apply here for seral reasons. The modulation of the mean field from sitesite allows large longitudinal fluctuations in the ferrimanetic phase and it turns out that the contribution from thfluctuations dominates in the ferrimagnetic phase at temptures above;12 K. The other reason is the weakness of texchange coupling compared to the crystalline anisotr

all-

es

FIG. 8. The experimental and calculated magnetoresistancthulium when the current is in thea direction.

id

tsas

x-nd

forac-a-


terms, which implies that poles, other than the spin-wapole, in the transverse component of the susceptibility tenrapidly become important when the system is heated.

The present work on thulium has revealed that thereneed for a closer examination of the effects which the molated magnetic ordering have on the conduction electroWe have indicated that it should be possible to get a sizareduction of the Fermi-surface area which is quadratic inmagnetization rather than linear. The recent improvementthe description of the magnetic properties of the other herare-earth metals8 may be utilized for similar improvement

vesor

s au-ns.blehein

vy

in the analysis of the superzone effects occurring, for eample, in the magnetically modulated phases of erbium aholmium.

ACKNOWLEDGMENTS

We thank S. Zochowski, M. de Podesta, and E. Gratzuseful discussions and comments during this study. Weknowledge support from the ESPRC, the Wolfson Foundtion, and the Danish Natural Science Research Council.

E.

O.

*Present address: Department of Physics and Astronomy, Univsity College London, Gower Street, London WC1E 6BT, Englan1W. C. Koehler, J. W. Cable, E. O. Wollan, and M. K. Wilkinson

Phys. Rev.126, 1672~1962!.2T. O. Brun, S. K. Sinha, N. Wakabayashi, G. H. Lander, L. R

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