Magnetoresistance of Pr$_{1-x}$La$_x$Os$_4$Sb$_{12}$: Disentangling local crystalline-electric-field physics and lattice effects

0.6 MB·English

by  C. R. Rotundu

Magnetoresistance of Pr La Os Sb : Disentangling local crystalline-electric-field
1−x x 4 12
physics and lattice effects
∗
C. R. Rotundu, K. Ingersent, and B. Andraka
Department of Physics, University of Florida P.O. Box 118440, Gainesville, Florida 32611–8440, USA
(Dated: February 6, 2008)
7
0 Resistivity measurements were performed on Pr1−xLaxOs4Sb12 single crystals at temperatures
0 down to 20mK and in fields up to 18T. The results for dilute-Pr samples (x = 0.3 and 0.67)
2
areconsistent with modelcalculations performed assumingasinglet crystalline-electric-field (CEF)

n groundstate. Theresidualresistivityofthesecrystalsfeaturesasmearedstepcenteredaround9T,
a the predicted crossing field for the lowest CEF levels. The CEF contribution to the magnetoresis-
J tance has a weaker-than-calculated dependence on the field direction, suggesting that interactions
1 omitted from the CEF model lead to avoided crossing in the effective levels of the Pr3+ ion. The
1 dome-shaped magnetoresistance observed for x = 0 and 0.05 cannot be reproduced by the CEF
model, and likely results from fluctuations in thefield-induced antiferroquadrupolar phase.

]
n
PACSnumbers: 74.25.Ha,74.70.Tx
o
c
-
r I. INTRODUCTION rial down to 20mK.
p WefindthatthemagnetoresistanceofpurePrOs Sb
4 12
u
at 20mK is inconsistent with model calculations for ei-
s PrOs4Sb12,thefirstdiscoveredPr-basedheavyfermion
. and superconductor,1 remains a focus of extensive theo- ther the Γ3 or the Γ1 CEF ground state, and conclude
t
a retical and experimental investigation. Its significance thatthedomefeaturemostprobablyresultsfromfluctu-
m ations in the field-induced antiferroquadrupolar (AFQ)
lies in the fact that the origin of the heavy-fermion be-
- havior is associated with non-Kramers f-electron ions, phase. OndilutionofthePrlatticewithLa,thedomein
d themagnetoresistanceisreplacedbyasmearedstepthat
for which the conventional Kondo effect seems unlikely.
n
o Our previous specific-heat results in magnetic fields2 es- isconsistentwiththe pictureofaΓ1 singletCEFground
state but not with a Γ doublet. The dependence of the
c tablishedthatthecrystalline-electric-field(CEF)ground 3
[ state is a nonmagnetic Γ singlet. The field dependence f-electron contribution to the magnetoresistance on the
oftheCEFSchottkyanom1 alyforfieldsgreaterthan14T directionofthemagneticfieldissmallerthanispredicted
2
theoretically based on a CEF model. This discrepancy
is clearly inconsistent with the alternative scenario of a
v
suggests that interactions omitted from the CEF model
0 nonmagnetic Γ3 doublet ground state. This conclusion
leadtoavoidedcrossingintheeffectivelevelsofthePr3+
2 was independent of whether the exact Th point-group
5 symmetry or the higher (approximate) Oh symmetry ion.
0 was assumed for the Pr sites.3 The singlet nature of the
1
CEFgroundstatewassubsequentlyconfirmedbyinelas-
6
tic neutron scattering measurements and their analysis II. METHODS
0
/ within the Th symmetry scheme.4
t
a Despite the overwhelming evidence in favor of a sin- ResultsarepresentedbelowforPr1−xLaxOs4Sb12with
m glet CEF ground state, there are experimental results four different La concentrations: x = 0, 0.05, 0.3,
for PrOs Sb that seem to be better understood in and 0.67. For x = 0, 0.3, and 0.67, we grew large
- 4 12
d terms of a doublet ground state. For example, the single crystals (cubes with masses as large as 50mg)
n magnetoresistance1,5,6,7 at 1.4K exhibits a dome-like onwhichaccuratemagnetic susceptibility measurements
o
shape that is consistent with model calculations of the wereperformedupto300Kinordertoextracttheroom-
c
CEF resistivity for a Γ ground state and inconsistent temperature paramagnetic effective moment. In each
: 3
v withsimilarcalculationsforaΓ groundstate.5However, case, this moment was within 10% of that expected for
1
Xi theCEFresistivityisasingle-ionpropertythatmightbe Pr3+. (For the undoped compound, this finding con-
strongly affected in PrOs Sb by lattice coherence and tradicts a wide range of values reported in literature.)
r 4 12
a by strong quadrupolar and exchange interactions. To The superconducting transition temperatures Tc of the
probe this possibility, we have performed magnetoresis- large single crystals and of smaller resistivity bars, also
tance measurements on single-crystalPr1−xLaxOs4Sb12, obtained in the same growths, were checked via ac sus-
in which lattice translational symmetry is broken and ceptibility measurements. A good agreement between
intersite effects should be weaker than in the pure com- Tc values of large and small crystals confirmed the sto-
pound. Based on previously published magnetic suscep- ichiometry assigned to samples used in this study. The
tibility and specific heat results,8 we do not expect sig- residual resistivity ratio (the ratio of the resistance at
nificant changes in CEF energies (and eigenstates) of Pr room temperature to that extrapolated to T = 0) was
upon doping with La. In addition, we have extended RRR = 100, 50, 180, and 170 for x = 0, 0.05, 0.3, and
magnetoresistance measurements of the undoped mate- 0.7,respectively. ThevalueRRR=100exceedsthosere-
2

ported previously1,7,9 for pure PrOs Sb , indicative of
4 12
thehighqualityofoursamples. Thex=0.05crystalwas H = 10T
from the batch for which results were reportedin Ref. 8. 4.0
The resistivity was measured by a conventional four-
probe technique. The estimated uncertainty in the de-
termination of the absolute value of the resistivity was
30% due to the unfavorable geometry of the crystals. m) 3.5
c

Within this uncertainty, the resistivity at room temper-
(
ature was the same in all cases. In the plots below, we H = 18T
have scaled the resistivity of each sample to a zero-field
room-temperature value of 300µΩcm, in the range re- 3.0 H = 3T
portedpreviously. Itis importanttoemphasizethatthis
scaling procedure is in no way essential for the conclu-
sions of the paper, which are based on the temperature
and field dependence of the resistivity ofa givensample. 0.0 0.2 0.4 0.6 0.8 1.0
We calculated the CEF contribution ρ to the elec- T (K)
CEF
trical resistivity via the method applied by Fisk and FIG. 1: Resistivity vs temperature for PrOs4Sb12 in three
Johnston10 to the resistivity of PrB and by Frederick different magnetic fields, with both current and field along
6
andMaple5tothemagnetoresistanceofPrOs Sb . This the(001) direction.
4 12
method focuses on a single Pr ion, neglects intersite ef-

fects, and takes no account of the direction of the cur-
rent relative to the crystal axes or to the magnetic field.
H II (001)
Our calculations started with one or other of two forms
for Hˆ , the CEF Hamiltonian for Pr3+ in zero magnetic 3.5
0
field: that (corresponding in the Oh-symmetry notation
of Ref. 11 to W = −2.97K and x = −0.7225) deduced5 m)
by fitting the temperature dependence of the zero-field c
resistivity of PrOs4Sb12; or the Hamiltonian (described (0
inthe Th-symmetrynotationofRef.3by W =3.0877K,
3.0
x=0.45991,andy =0.10503)determinedfromelastic12
andinelastic4neutronscattering. Henceforth,wereferto
these cases as the “doublet” and “singlet” CEF scheme,
respectively,accordingtotheground-statedegeneracyof
0 5 10 15
Hˆ .
0 H (T)
The CEF states in a magnetic field H were obtained
by diagonalizing the Hamiltonian Hˆ0+gµBH·Jˆ, where FIG. 2: Residual resistivity vs magnetic field for PrOs4Sb12,
g = 4/5 is the Land´e g factor for Pr3+ and Jˆ is the withbothcurrentandfieldalongthe(001)direction. Arrows
indicate boundaries between paramagnetic and field-induced
f-electron angular momentum operator. The CEF re-
ordered phases.
sistivity is completely determined by these CEF states,
the temperatureT,andtwoconstantsρex andρA,which
parametrize, respectively, the overall strengths of mag-
between ρ obtained in this manner and ρ(T =20mK).
0
netic exchange and the aspherical Coulomb interaction
Thisresidualresistivity(orresistivityat20mK),when
betweenthe4f andconductionelectrons. FollowingRef.
plottedagainstmagneticfield(Fig.2),hasadomeshape
5, we took ρex =ρA =0.25µΩcm. However, our conclu- centeredaround9–10T.Suchadome-shapedmagnetore-
sionsareinsensitivetotheparticularchoiceofconstants.
sistance has been reported previously at the somewhat
highertemperaturesof1.4K(Ref.5)and0.36K(Ref.7).
Two explanations for this dome have been considered:
III. RESULTS AND DISCUSSION field-induced long-rangeantiferroquadrupolar(AFQ) or-
der, and crossing of the lowest CEF levels. It is striking
Figure 1 shows the resistivity of undoped PrOs Sb that ρ (H) rises sharply at the AFQ boundaries, indi-
4 12 0
for three representative fields, with both current and cated by arrowsin Fig. 2, and peaks around 10T, where
magnetic field oriented along the (001) direction. The the AFQ transition temperature is highest. However,
results are similar to those reported by other groups.1,13 Frederick and Maple have shown (see Fig. 2 of Ref. 5)
Below 200–300mK the resistivity saturates but has a that the width, peak position, and heightof the dome in
strong field variation. The residual resistivity ρ can be the magnetoresistance of PrOs Sb at 1.4K are repro-
0 4 12
obtainedusingthepreviouslynoted1,13temperaturevari- duced quite well by the single-ion CEF resistivity com-
n
ationatfixedfield: ρ(T)=ρ +BT withn>2. Within putedwithinthedoubletscheme. (Bycontrast,ρ for
0 CEF
the precision of our measurement, there is no difference thesingletschemeshowsastep-likeincreaseinthevicin-
3
T = 20 mK
T = 20 mK
4 Singlet CEF 6
H II (001) H II (010)
H II (011)
(011)
2 Pr0.7La0.3Os4Sb12
m) m) 4 (001)
c c
(CEF 40 Doublet CEF (
2
2
H II (001)
H II (011)
0
0
0 5 10 15 20 0 5 10 15 20
H (T) H (T)
FIG. 3: Theoretical CEF resistivity at 20mK vs magnetic FIG.5: Magnetoresistance ofPr0.7La0.3Os4Sb12 at20mKfor
field calculated within the singlet (upper panel) and doublet threedifferentorientationsofthemagneticfield. Thecurrent
(lower panel) CEF schemes, for fields along (001) ((cid:4)) and direction was (001).
(011) (▽).
a negligible temperature variation of the resistivity be-
15
Pr0.95La0.05Os4Sb12 H II (001) low 300mK in fields above-criticalfor superconductivity
(as evidenced by the overlap of the 20-mK and 310-mK
isotherms). However,the shapeofthedome forx=0.05
660 mK
ismuchlesssymmetricaboutthepeakfieldthanitsx=0
counterpart. Between2Tand10T,ρ(T=20mK)forthe
cm) 10 310 mK dopedsampleincreasesbyover80%,comparedto a25%
increasefortheundopedmaterial,whereastheresistivity
(
20 mK dropabove10Tis greaterinpercentagetermsforx=0.
The magnetoresistancebecomes qualitativelydifferent
at higher La doping. Figure 5 shows the 20-mK magne-
toresistance of Pr0.7La0.3Os4Sb12 for fields along (001),
5 (011), and (010); in each case, the current passed along
the (001) direction. All three isotherms exhibit a pro-
0 5 10 15 20 nounced but wide step, centered near 9–10T, superim-
H (T) posed on a linear background. In the investigated field
FIG. 4: (color online) Longitudinal magnetoresistance of range this x = 0.3 material does not exhibit the dome
Pr0.95La0.05Os4Sb12 at 20, 310, and 660mK, for current and structure characteristic of x = 0 and 0.05. For each
field along the(001) direction. curve, ρ versus H is approximately linear above 13T.
The resistivity of the non-f-electron analog LaOs Sb ,
4 12
measuredat 0.36K, has a quite large andapproximately
ity of the crossing field at which the lowest Th Γ(42) level linear field dependence.7 Furthermore, the directional
falls in energy below the Γ singlet.) dependence of the magnetoresistance of LaOs Sb —
1 4 12
We find that neither CEF scheme accounts satis- dρ/dH being larger along (011) than along (001)—is in
factorily for the magnetoresistance measured at 20mK agreementwiththetrendofthelinearbackgroundinFig.
(Fig. 2), which shows a dome of similar width to that at 5. It thus seems that the differences between the high-
1.4K.IrrespectiveoftheCEFscheme,ρ forfieldsori- field slopes of ρ(H) in Fig. 5 can be attributed primar-
CEF
entedalongthe(001)direction(squaresymbolsinFig.3) ily to non-f-electron contributions to the magnetoresis-
isdiscontinuousatthecrossingfieldandessentiallyflatat tance. Subtracting such linear parts results in very sim-
higherfields. Itthereforeseemsthatthelow-temperature ilar curves (not shown) for all three field directions. We
magnetoresistance of PrOs Sb is dominated by effects conclude that the f-electron magnetoresistance in this
4 12
beyond those considered in the single-ion CEF model. moderately doped material is nearly isotropic.
We now turn to the effects of La doping. Figure 4 Compared to the cases x = 0 and x = 0.05, the low-
shows the magnetoresistance of Pr0.95La0.05Os4Sb12 at temperature magnetoresistance of Pr0.7La0.3Os4Sb12 for
20, 310, and 660mK. Similarly to PrOs Sb , there is fieldsalong(001)ismuchclosertothatgivenbytheCEF
4 12
4
4 4
Pr0.7La0.3Os4Sb12
3 Pr0.33La0.67Os4Sb12
3
m) m)
c 2 c
( (
CEF
2
1
T = 0.35 K
T = 0.31 K
0
1
0 5 10 15 20 0 5 10 15
H (T) H (T)
FIG.6: MeasuredmagnetoresistanceofPr0.7La0.3Os4Sb12 at FIG.7: Magnetoresistance ofPr0.33La0.67Os4Sb12 at350mK
310mK for current and field along the (001) direction, and for current and field along the(001) direction.
theoretical CEF resistivity ρCEF for the same temperature
and field direction.
lowesttemperature (Fig. 7) exhibits essentiallyidentical
magnetic field dependence to that for x=0.3.
model. The 20-mKmeasurements(Fig.5)aremorecon-
sistent with the singlet CEF scheme than with the dou- Themaindifferencebetweenthef-electronmagnetore-
bletscheme,inthatthe latterpredictsasharppeakthat sistance inferred for dilute-Pr alloys and that calculated
is absent in the data. A second and stronger argument inthesingletCEFschemerelatestothelow-temperature
in favor of the singlet scheme is provided by the near- width of the step along the (001) field direction. The
isotropyofthef-electronmagnetoresistancenotedinthe CEF model predicts an almost discontinuous jump of
previous paragraph. Figure 3 plots the CEF resistivity the magnetoresistance at 20mK at the level-crossing
for fields along (001) and (011). The doublet scheme field, while the rise in the measured magnetoresistance
(lowerpanelinFig.3)predictsahighlyanisotropicρCEF of Pr0.7La0.3Os4Sb12 takes place over 3–4T. Since there
stemming from the fact that the lowest two CEF lev- is better agreement between the measured and theoret-
els cross at 8.5T along (001), but instead diverge along ical magnetoresistance along the (011) direction, where
(011). In the singlet CEF scheme (upper panel in Fig. level anticrossing is expected, we speculate that interac-
3), the anisotropy is much smaller because the lowest tionsomittedfromthe CEFmodelmix thelowestlevels,
CEF levels cross at 8.6T along (001), while along the preventing any crossing even along high-symmetry field
(011) direction they anticross at 8.3T with a minimum directionsandtherebyproducingisotropicmagnetoresis-
gap of only 0.7K. Unlike the doublet CEF scheme, the tance. These interactions are most likely nonlocal. We
singlet scheme does a reasonable job of reproducing the note, however, that a mean-field treatment of intersite
measured(011)magnetoresistance. However,itunderes- magnetic and quadrupolar interactions between Pr ions
timates the width of the step for fields along (001), and did not find avoided crossing.2,12
hence stilloverestimatesthe anisotropyin the f-electron Figure6showsthatthemeasuredmidpointfieldforthe
magnetoresistance extracted from Fig. 5. magnetoresistanceriseinPr0.7La0.3Os4Sb12 is about1T
Figure 6 shows that at the higher temperature of higher than is predicted based on the CEF level scheme
310mK, there is much better agreement between the determined for pure PrOs4Sb12. This perhaps points to
magnetoresistance of Pr0.7La0.3Os4Sb12 along (001) and a small shift in the CEF levels upon doping to 30% La.
and ρ calculated within the singlet CEF scheme. At The similarity between the magnetoresistance steps ob-
CEF
this temperature, the thermal smearing of the step in served for x = 0.3 and 0.67 suggests that there is little
ρ matchesquitewellthewidthoftheriseinthemea- further evolution of the CEF energies over this doping
CEF
sured magnetoresistance. The results of similar calcula- range. A weak dependence of CEF levels on La dop-
tionsforthedoubletscheme(notshown,butverysimilar ing is in agreement with our specific-heat measurements
tothe350-mKresultsinFig.2ofRef.5)areingrossdis- of the Schottky anomaly in lightly doped alloys,14 and
agreement with the measurement. is also supported by the nearly invariant temperature of
The character of the magnetoresistance seems to be the maximum in the magnetic susceptibility.8
little changed by further La dilution. The longitudinal Thecontrastbetweentheresistivityversusfieldcurves
magnetoresistanceforx=0.67wasinvestigateddownto for the x = 0 and x = 0.3 samples is striking. The ob-
0.35Kandinfieldsto14T.Themagnetoresistanceatthe viousdifferences betweenthese twocompositions arethe
5
presenceofafield-inducedorderedphaseforthepurema- plied fields. At the lowest temperatures, the magnetore-
terial and the absence of translational symmetry in the sistance for x = 0 and x = 0.05 is inconsistent with the
diluted case. Our previous specific heat measurements14 results of CEF model calculations. This discrepancy is
indicate that the field-induced ordered(AFQ) phase dis- attributed to the long-range order present in the pure
appears somewhere near x = 0.2. Since the model for and lightly doped materials.
the CEF resistivity are single-site in character, it seems
likely that the dome-shaped magnetoresistance observed
forx=0and0.05isassociatedwiththepresenceoflong-
rangeorderinthesematerials,perhapsthroughenhanced
Acknowledgments
scatteringofconductionelectronscausedby fluctuations
in the AFQ order parameter.
In summary, we have shown that a simple CEFmodel Stimulating discussions with P. Kumar are acknowl-
accountsquite wellforthef-electroncontributiontothe edged. This work has been supported by U.S. Depart-
magnetoresistanceofPr1−xLaxOs4Sb12 withx=0.3and mentofEnergy(DOE)GrantNo.DE-FG02-99ER45748,
0.67. The weak dependence of this contribution on field byNationalScienceFoundation(NSF) GrantNo.DMR-
directionisconsistentwiththeexistenceofasingletCEF 0312939,and by the National High Magnetic Field Lab-
in zero magnetic field, with avoided level crossing in ap- oratory.
∗ Electronic address: [email protected]fl.edu 73, 014515 (2006).
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