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Submitted on 1 Jan 1989
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Normal and superconducting properties of UBe13
J.P. Brison, O. Laborde, D. Jaccard, J. Flouquet, P Morin, Z. Fisk, J.L. Smith
To cite this version:
Normal
and
superconducting properties
of
UBe13
J. P. Brison
(1,
*),
O. Laborde(1),
D. Jaccard(1,
**),
J.Flouquet
(1),
P. Morin(2),
Z. Fisk(2)
and J. L. Smith(3)
(1)
Centre de Recherches sur les Très BassesTempératures,
C.N.R.S., BP 166 X, 38042 GrenobleCedex,
France(2)
Laboratoire Louis-Néel, C.N.R.S., BP 166 X, 38042 Grenoble Cedex, France(3)
Los Alamos NationalLaboratory,
Los Alamos NM 87545, U.S.A.(Reçu
le 25 avril 1989,accepté
sousforme définitive
le 30 mai1989)
Résumé. 2014
Nous décrivons des
expériences
demagnétorésistivité
dans unelarge
gamme detempérature
(30 mK T 80 K)
et dechamp magnétique (H 20 T).
A hautetempérature
(T > 4 K),
lamagnétorésistance
suit une loi d’échelle enH/H*
à un seulparamètre
H*
(T). Cependant,
il n’existe pas de liensimple
entremagnétorésistance
et aimantation. A bassetempérature,
les termesA (H)
T2 de
la résistivité varient fortement souschamp
tandis que lachaleur
spécifique
de laphase
normale estindépendante
duchamp.
Cette forte variation de A(H)
est due à la fortedépendance
enchamp
de l’interaction directe entrequasi particules.
Cespropriétés
de laphase
normale sont reliées à celles de laphase supraconductrice (notamment,
la chaleurspécifique
et la conductionthermique).
Abstract. 2014 Extensive
magnetoresistivity experiments
inlarge temperature (30
mK T 80K)
and field(H
20T)
ranges arereported.
Athigh
temperatures(T >
4K),
themagnetoresistivity
obeys
ascaling
law inH/H*
with one temperature adjustable parameter H*(T). However nosimple
relation can be found betweenmagnetoresistivity
andmagnetization.
At low temperatures the A(H)
T2 terms of theresistivity
arestrongly
fielddependent although
thespecific
heat of the normalphase
is fieldindependent.
It isproposed
that the field variation ofA (H)
reflects the strong fielddependence
of the direct interactions betweenquasi particles.
The low temperaturemagnetoresistivity
data are connected with the unusualsuperconducting properties
ofUBe13
(specific
heat and thermalconductivity).
ClassificationPhysics
Abstracts72. 15Gd - 72.150m - 74.70Tx - 65.40f
Introduction.
The
heavy
fermionUBe13
isthought
to be one of thegood examples
of a « Kondo lattice »system.
This means that the occurrence of theheavy quasi-particles
at lowtemperatures
would be due to thecoupling
between localized f electrons and conduction electrons. One is led todistinguish
between a« high
temperature
regime »
where the uranium ions areconsidered like
magnetic impurities scattering
the conductionelectrons,
and a « lowtemperature
coherent state », withheavy
quasiparticles
in a« perfect »
lattice,
describedwithin the framework of
strongly interacting
Fermiliquids
and in anon-magnetic ground
state.
In this article we
report
mostly magnetoresistivity
andmagnetization
(M)
experiments
performed
in alarge
range ofmagnetic
fields(H)
andtemperatures
(T ).
These measurementsare
appropriate
to check thepreceding
ideas andexplore
theproperties
ofUBe13
in bothregimes.
Theiranalysis
suggests
links between normal andsuperconducting properties
like the upper critical field(Hc2)’
the lowtemperature
specific
heat(C )
and the thermalconductivity
K.1.
Expérimental
set up.The
magnetoresistivity
measurements wereperformed
on the samepolycrystalline sample
using
an acbridge.
The contacts were realized withindium ;
the current wasparallel
toH.
The different
temperature
ranges were obtained in aHe4
bulb(for
T . 4.2K)
with thesample
immersed inpumped He4
(1.5
K -T - 4.2 K)
or in dilutionrefrigerators
(30
mk : T 4.2K).
Themagnetic
field wasproduced
eitherby superconducting
magnets
(for
fields up to 12T)
or in a resistive Bittermagnet,
at the S.N.C.I.(Grenoble-France)
for H up to 20 teslas. Particular attention wasgiven
to thethermometry
undermagnetic
field : the isothermal p(H)
curves were obtainedby
aregulation
of thetemperature
with the heliumbulb,
by
the vapor pressure of the bath ofby
the use of a referencecapacitor
which isonly
weakly
fielddependent.
The p(T)
curves at fixed field were made after calibration of thecarbon or
platinum
thermometers under H.Isothermal
magnetization
measurements were collected on asingle crystal coming
from the same batch as thepolycrystalline sample.
H(..--
8T)
wasapplied along
a (0, 0, 1 )
axis ;
thetemperature
varied from 1.5 to 300 K. 2.High température magnetoresistivity.
2.1 RESISTIVITY FOR H = 0. -
Figure
1 shows theresistivity
ofUBe13
in zeromagnetic
field. We see an increase oncooling
from T = 300 K down to 2.5 K wherep passes
through
aFig.
1. - Present determination of theresistivity
ofUBe13
in zero fieldshowing
an increase oncooling
down to 2.5 K, followed
by
a decrease attributed to the occurrence of coherence before themaximum. This
temperature
of 2.5 K is often called « coherencetemperature »
(Tc.h),
because it is assumed toseparate
thehigh
temperature
regime
from the lowtemperature
coherent one. Theupturn
of p below 10 K is reminiscent of that of thesingle impurity
Kondobehavior,
and the decrease of p below 2.5 K is ascribed to the formation of theheavy
quasiparticles
(due
to the Kondocoupling
between localized f electrons and conductionelectrons)
which restores the translationalsymmetry
of the lattice. In thisscheme,
most of theresistivity
aboveTc.h originates
from the diffusion of the conduction electrons on uraniumions considered as
magnetic impurities.
Thus,
magnetoresistivity
shouldprovide
useful information on thishigh
temperature
incoherentregime.
2.2 MAGNETORESISTIVITY FOR r>
Tc,h-
-Figure
2 shows thecorresponding
magnetoresis-tivity
p(H)
at a fixedtemperature
(T
= 1.6 K to 80K)
and from 0 to 20 teslas. It isnegative
for all
temperatures
and themagnitude
of6.p
increases oncooling.
The interest of thehigh
field measurements appears
clearly.
For T = 4.2K,
a value ofAp /p - p (0) p
(H)
of thep (0)
order of 0.45 is achieved for H = 20 T : this will allow us to check the models.
Fig.
2. -Magnetoresistivity
curves ofUBe13 (0 --
H , 20T),
for temperatures 1.6 K : F 80 K.When the
magnetoresistivity
is due to thepolarization
ofmagnetic impurities
and isgoverned by
one energyscale,
the isothermal p(H)
curves followscaling
laws onH with a
temperature
dependent
effective field H*(T).
In the reduced variableH/H*,
themagnetoresistivity
isgiven by
p (H, T) = p (0, T) 0 (HIH* (T»
where 0
is a universal function. For Kondoimpurities
[1],
the condition is satisfied at least for T ->TK
(where
TK
is the Kondotemperature).
Renormalizing
p(H, T)
to p(0, T )
takes into account thetemperature
dependence
of the effective interaction between theimpurities
and the conduction electrons.H* (T)
should vary like(T + TK) [1]
to be consistent with the behaviour of thesusceptibility
(X )
in thistemperature
range(the
magnetization
scales like M =Msat
g(H/H*(T))
and thus1/X
ocH* (T)).
To check thescaling
laws inUBe 13
in theThe best
superposition
of the p(H)/ p
(0)
curves(taken
fromFig.
2)
on p(H, 4.2)/p (0, 4.2)
through
ascaling :
H ---+ Hl a (T)
is shown infigure
3. It isperfect
withinexperimental
accuracy, in the whole range of fields andtemperatures
(4.2
K T -- 35K)
explored.
Thetemperature
dependence
ofH* (T)
is shown infigure
4 ;
the value at 4.2 K was fixed so thatp
(H)/p
(0)
= 0.5 at H = 0.64H*,
as forspin
1/2 Kondoimpurities
[2].
The results of[1]
arerecovered : above 15
K,
H*(T)
follows a linear Tdependence :
H * (T) - (T + TK)
withTK -
13 K while below 15 Kstrong
deviations fromlinearity
appear.Fig.
3. -Superposition
of the curves offigure
2 from T = 4.2 K to 40 K, through ascaling
on themagnetic
field asexplained
in the text.Fig.
4. - Behaviour of the effective characteristic field H*(T )
derived from thescaling
relations.2.3 MAGNETORESISTIVITY FOR T --
Tcoh.
- For rTcoh,
thedescription
of f electrons inone
impurity
schemes isexpected
to break down.Instead,
a Fermiliquid picture
ofstrongly
interacting quasiparticles might
berelevant,
as for otherheavy
fermionsystems
in their lowtemperature
regime.
Thus inUBe13,
achange
in the behaviour of the p(H)
curves at 1.6 and 2.1 K wasexpected.
Figure
5 showsthat,
indeed,
thescaling
laws on H do notwork
anymore on the last twocurves :
taking
the curve p(H, 1.6)lp
(0, 1.6)
as a referencespoils completely
thesuperposi-tion of the curves described
previously.
This alone does not mean that oneimpurity
schemesare obsolete in the coherent
regime :
even for the oneimpurity
Kondotheory
it is notexpected
thatscaling
laws work in the wholetemperature
range T>TK
down toFig.
5. - Bestsuperposition
of all the curves offigure
2, taking the curve at 1.6 K as a reference for thescaling
describedpreviously, showing
aspoiling
of the result when curves in the coherentregime
aretaken into account.
2.4 SUSCEPTIBILITY AND MAGNETIZATION. - Even at low
temperatures
themagnetization
isstrictly
linear in H. For H = 8T,
M reaches a value of 0.2p,B per uranium atom.
Using
Arrott’s method
(Fig. 6),
thelinearity
of M with H is obvious. However one notices a nonreversible behavior of M in
decreasing
field which exists in the wholetemperature
rangeinvestigated
and is notyet
explained.
It hasalready
beenreported
that M is linear in H up to 24 T down to 1.5 K[5].
Fig.
6. -Representation
of themagnetization
(M)
databy
Arrot’splot :
M2 versusHlM.
Arrott’s method allows an accurate determination of the
reciprocal
initialsusceptibility
x , x follows a Curie-Weis law above 150 K with 0 = - 85 K and J.Leff =3.36 9 B :
thus value isslighlty
different from the value J.Leff = 3.05J.L B of
[3]
but is consistent with the data of[4]
J = 4
(ILcff
=3.58 IL
s)
and J = 9/2(ILeff
= 3.62 gB).
Between 150 K and 20K,
1/X
seems tofollow a linear T
dependence
with 0 = - 130 K(Fig. 7).
Theapparent
increase of 0 may be due tocrystal
field effects. Below 20K,
our data agrees with those of[3].
Thestrong
downwards curvature of
11X
at lowtemperatures
is agood
indication of amagnetic ground
state for the
crystal
field levels of the U ions.Fig.
7. -Temperature dependence
of thesusceptibility X
in full line the data of reference[3].
Insert is the low temperatureregime.
2.5 DISCUSSION. - The observation of
scaling
laws on p(H)
and thetemperature
dependence
of thesusceptibility point
towards apicture
of a localizedmagnetism
for the Uions. This is consistent with
spectroscopic
measurements[6]
which favor a5f3
configuration.
A
crystal
field schemeF6-F8-F8
with a Kramers doubletT 6
wasproposed
forexplaining
specific
heat data[7].
However thesplitting r 6 ...
F8
of 180 K appears ratherlarge
in relationto the
strong
isotropy
of the U sites and incomparison
to theREBe13 isomorphous
compounds.
Indeed inNdBe13
and inPrBe13
[8],
the overallsplitting
reachesonly
70 and 125 Krespectively although they
are thelargest
across theREBe13
series. Thus thecomparison
ofUBe13
withREBe13
isdifficult ;
the nature of lowlying
levels inUBe13
remains an openquestion.
The energy scale
TK -
13 K deduced from thetemperature
dependence
of H*(i.e.
from thescaling
law on p(H,T»
is consistent with thelarge
increase of p oncooling
below 10 K androughly
with the estimation of the Kondotemperature
made fromspecific
heat data[7]
TK
= 7.7 K. Below 15K,
thedeparture
of H* (T )
from a linear Tdependence
isquite
naturalsince in the one
impurity
Kondotheory
a crossover occurs at T -TK
and there are no reasonsfor the same
scaling
lawbeing
valid for rTK.
Nevertheless,
it appears to be verified inUBe13
down toTK/3.
Below 15K,
the curvature ofH* (T)
is consistent with that of1/X.
But a further
comparison
of themagnetoresistivity
andmagnetization
isdisappointing.
Forexample,
at T = 4.2K,
M is linear in H almost up to 24 T[5]
whereas at the sametemperature, Ap
deviatessignificantly
from anH2
(i.e.
M)
behaviour for fields above 5 T.Kondo
impurities
at T = 0[2].
Thiscomparison
is somewhatquestionable
asp (H/H* )
iscertainly
different at 4.2 K and 0 K(scaling
laws do not work below 4.2K)
but shouldgive
good
« orders ofmagnitude
».(As
regards
tomagnetization,
it does not vary much oncooling
below 4.2
K).
Figure
8 shows in full lines the exact theoretical solutions for p(HIH*)Ip
(0)
andM(HIH*)IMsat’
with theexperimental
scaling
curvep (HIH*(T))lp (0)
and themagnetization
at 4.2 K as a function ofHIH* (4.2 K).
Fig.
8. - Full lines : theoreticalpredictions
for themagnetoresistivity
and themagnetization
ofspin
1/2 Kondoimpurities
at T = 0. Dots :experimental points
for themagnetoresistivity
ofUBe13
in the temperatureregion
wherescaling
laws work. Dotted line :magnetization
ofUBe13
at 4.2 K(from [5])
as a function ofH/H* (4.2 K)
and normalized so that itsslope
in H = 0 coincides with the theoretical one.This
strongly
suggests
that themagnetoresistivity
and themagnetization
ofUBe13
are notcontrolled
by
the same processes. This idea is alsosupported by
the different energy scalesrelevant for the
magnetoresistivity
(TK -
13K)
and themagnetization
(0 -
85 K to 130K) ;
the latterbeing
consistent with thelinearity
of M up tohigh
fields even at 4.2 K. Recent thermalexpansion
andmagnetostriction experiments
[9]
have also ruled out adescription
of thehigh
temperature
properties
ofUBe13
with aunique
parameter.
Furthermore,
thisstudy
ofUBe13
athigh
temperatures
shows no direct evidence of a Kondoeffect in this
compound.
The classificationof UBe13
as a « Kondo lattice » reliesessentially
onthe appearance of a coherent
regime
withoutlong
rangemagnetic ordering,
whereas it is described in terms of a localizedmagnetism
athigh
temperatures
(up
to now, neutrondiffraction measurements have failed to detect any static sublattice
magnetization
down to50 mK
[12]).
But theshape
ofp (H/H*(T))
cannot beprecisely compared
to theoreticalpredictions
for Kondoimpurities,
availableonly
at T =0,
and the curvature ofH* ( T)
below15 K is
opposite
to thatexpected
for anantiferromagnetic coupling
between conduction andlocalized electrons.
Moreover,
the existence of two energy scalespoints
out that the effects of Kondocouplings
(if any)
inUBe13
are at leaststrongly
alteredby competition
with sometype
of R.K.K.Y. interactions
(and
notonly by
the veryhigh
concentration of« magnetic
impurities
»).
Indeed,
the occurrence ofmagnetic
correlations inUBe13
wasperhaps already
directly
observed inpreliminary
neutron measurements[10]
andindirectly
in Hall effectexperiments
[11].
Still,
tokeep
asimple picture,
one could guess thatmagnetoresistivity
isTK -
13K)
whilemagnetization
ismostly governed by
the f electron-f electroncoupling
(energy
scale o - 85 to 130K).
Infact,
the need for two different energy scales is alsorecovered in the low
temperature
heavy
fermionregime.
3. Low
température magnetoresistivity.
3.1 MEASUREMENTS. - Below 1
K,
two differenttypes
of measurements have beenperformed
- for H ,12
T,
measurements at fixed fields to determine thetemperature
dependence
of theresistivity.
It was found to bequadratic
for T --900 mK,
and Thigher
than the criticaltemperature
of thesuperconducting
transition in thegiven
field,
restricting
theanalysis
toFig.
9. -T2 analysis
of themagnetoresistivity
ofUBel3,
for H between 12 and 4 T, and valid below 900 mK.-
Up
to 20T,
measurements at fixedtemperatures
wereperformed
(Fig. 10),
from which the values ofA(H)
andpo(H)
could be determined up to 20 teslas. Below 150mK,
theresistivity
does not show anytemperature
dependence, yielding directly
po (H)
between 15 and 20 teslas.Knowing
p o (H)
from 4 to 20T,
A(H)
was theneasily
extracted from the curvesat
620,
480 and 320 mK offigure
10.The
superconducting
transitionoccurring
at 950 mKprevents
a direct measurement ofA (H)
andp o (H) in
low fields.Nevertheless,
as shown infigure
2 with the curves for 2.1 KFig.
10. - Lowtemperature measurements in
high
fields up to 20 T of themagnetoresistivity
ofUBel3.
assuming
a weakertemperature
dependence.
This allows toextrapolate
A (H)
andp o (H)
down to H = 0 with an almostquadratic H
law,
consistent with the p(H )
curves at1.6 K. The field
dependence
of A (H)
andpo(H)
determined that way isreported
infigure
11.Fig.
11. - Determination ofA(H)
andpo(H)
through
the measurements at fixed field(some
of thempresented
inFig. 9)
and at fixed temperature(from
Fig. 10).
Below 4 T,extrapolated
values asexplained
in the text.3.2 COMPARISON WITH PREVIOUS RESULTS. - A
T2
behaviour of theresistivity
at lowtemperatures
is common inheavy
fermionsystems.
It is often associated with the Fermiliquid
behaviour of the
heavy quasiparticles.
Asimple
idea is then to considerA (H)
as an intrinsicproperty
of thesystem,
linked to the interactions betweenquasiparticles,
andpo(H)
as aresidual
resistivity,
determinedby
theimpurities
within thesample.
The
analysis
(1)
had beenperformed previously
on anothersample
ofUBe13 [5]
and it isthus
possible
to check thesample dependence
ofA (H )
andp()(H).
The « old » and « new »Fig.
12. -Comparison
of theprevious
results of[5]
and the present one from the fixed fieldmeasurements : while A
(H)
changed by
a factor 1.4,po (H)
decreasedby
a factor 3 on the newsample.
and a
stronger
decrease ofpo(H)
for this newsample
(factor 3) ;
this is ingood
agreement
with theorigin proposed
for bothquantities.
It had also been
reported
in[5]
that the limit ofvalidity
of theT2
law(almost
thesuperconducting
transitionTc
in zerofield)
was fieldindependent.
Thisimportant
feature isalso verified here at least up to H = 12 T.
3.3 DISCUSSION. - The
high
temperature
impurity
scheme is now left for a Fermiliquid
picture, although
other authors stillapply
thehigh
temperature
analysis
to this lowtemperature
regime
[1, 13].
Oneimportant
feature is that themagnetoresistivity
ofUBe13
isalways negative
in the whole range of fields andtemperatures yet
explored.
This ismore
easily
accounted for in the oneimpurity
picture
which neverthelessimplies
the occurrence ofsuperconductivity
at 950 mK in the presence of alarge
number of efficientparamagnetic
centers.In cubic
heavy
fermioncompounds
such asCeAl2,
Celn3, CePb3,
above their Néeltemperature
TN,
anegative
magnetoresistivity
is observed whilepositive
contributions of themagnetoresistivity
appear belowTN
[14].
InUBe13
at zero pressure(P),
the fact thatonly
negative magnetoresistivity
is detectedsuggests
thatUBe13
isparamagnetic
above the upper critical fieldHc2
(T - 0 ).
Recentspecific
heat measurements undermagnetic
field seem toindicate a
magnetic ordering
of the normalphase
ofUBe13
below 150 mK[12, 15].
Ouremergence of a
positive magnetoresistivity
below T « 4 K[14].
At P = 67kbar,
UBe13
may beantiferromagnetic ;
the appearance of thislong
rangemagnetic ordering
also coincides with the Pdisappearance
ofsuperconductivity.
InUBel3, superconductivity
andantifer-romagnetism
seem to beantagonistic,
in contrast to the situation observed when U atoms aresubstituted
by
Th atoms(Ul -,Th,,Bel3
with x :::.0.015) [16].
3.4 ANALYSIS OF THE
T2
TERM. - The discussion of the A(H)
T2
term hasalready
beenreported
in connection with the fielddependence
of the upper critical field[25].
Hereonly
complementary points
will begiven.
Usually,
it seemspossible
to describe the lowtemperature
coherentregime
ofheavy
fermionsystems
as a Fermiliquid
with one energy scalegoverning
both the bandproperties
(like
thespecific
heat)
and the interactions betweenquasiparticles (as
seen intransport
properties).
InUBe13,
asalready
found for thehigh
temperature
regime
(T > Tc.h),
there are no connections between the field variation of itsspecific
heat(at
least up to 8 teslas[15,
17,
18])
and thestrong
decrease of A(H)
in field. One of thesimplest
way to account for this is torely
upon the distinction between the different bare interactions made in thehigh
temperature
regime :
1)
The conduction electron-f electron interactions whichgive
rise to thestrong
resistivity
ofUBe13
above 2.5 K and are assumed to beresponsible
for the formation of theheavy
quasiparticles
and thus of the bandproperties
in the lowtemperature
regime.
The observation ofscaling
laws shows that these interactions areweakly
fielddependent.
(The
basicassumption justifying
thescaling
is that themagnetoresistivity
arisesonly
from thepolarization
of the «impurities
»).
2)
f electron-f electron interactions whichproduce magnetic
correlations. When theheavy
quasiparticles
areformed,
the conduction electrons and f states arehybridized.
Thus,
theseinteractions could
give
an effectivequasiparticle-quasiparticle scattering
in the coherentregime.
We propose that the latter interactions are fielddependent
and dominate themagnetoresistivity
at lowtemperatures,
the effect ofprevious
interactionsbeing mostly
suppressed by
the formation of thequasiparticles
(coherence).
Two different mechanisms may lead
yet
to aT2
term in theresistivity
[19].
The more common isthrough
the diffusion of thequasiparticles by thermally
excitedmagnetic
fluctuations. If the
high
value of the Curie-Weisstemperature
0 derived athigh
temperatures
(85 K)
is taken as a characteristic ofmagnetic
fluctuations,
few thermalmagnetic
fluctuationsshould remain at low
temperatures,
resulting only
in a small contribution of this mechanism tothe
large
A(H)
T2
term.Besides,
current theories on weak ornearly antiferromagnets
predict
noT2
term aboveTN
[20, 21].
The other mechanism is a direct
quasiparticle-quasiparticle
interaction mediatedby
virtualspin
fluctuations. It has beenargued
[25]
that inUBel3,
it couldproduce
alarge
T2 term. Notably,
thestrength
of this interactionmight
beemphasized by
theproximity
ofT,
andT,.h
(the
T2
term is not even observed atT,
in zerofield) :
the attractivepotential
leading
tosuperconductivity
could be due to such acoupling.
In thisscheme,
theT2 dependence
of p arises from the usualphase
spacearguments
of Fermiliquid theory
[19],
the limit ofvalidity
of theT2 law
could then be related to bandproperties
(thus
its fieldindependence
and also its pressuredependence).
On this lastpoint,
it is worthwhile to notice that the pressuredependence
of the A coefficient seems to follow that of the lineartemperature
term y of thespecific
heataccording
to thescaling
law A -y 2.
Ourextrapolation
of A for H = 0 is 70tflcmk -2
while A is foundequal
to 20tflcmk -2
andcompress-ibility
K =10-6
bar ,
that leads to a Grüneisenparameter
at P - 0 of ,fZ=
Log
V
8LogV
equal
to43,
in relativegood
agreement
with the values 60 and - 50 foundrespectively by
specific
heat[24]
and thermalexpansion
[9]
experiments.
A
previous
paper[25]
provided
support
to the idea that the field variation of Areproduces
the fielddependence
of the interactionpotential, notably
the attractive« B.C.S. »
potential
V (H).
The unusual curvature of the upper critical field ofUBe13
nearTc
could beexplained assuming
that A(H)
ocV 2(H)
and thenusing
a fielddependent
« bare » transition
temperature :
Tc(H)
ocexp(- ’IPd
V(H» = exp(-
a /F),
where P d is thedensity
of states at the Fermi level. Infact,
notonly
can we assumeA oc
V 2,
but also A ocp d
V 2 :
using
thesimple
kinetic formula : p= m * 2
(1/T )
one can seene
that the
scattering
rate contributes a factorP d 1 V 12
(cf.
the Fermi GoldenRule)
and the« effective mass » the other factor pd. The interest is that the pressure measurements confirm these ideas. Under pressure, both pd
(cf.
Cp
measurements[24])
and,
verylikely,
V will vary ; but still the link betweenTc
and A should remain(as
long
as the same « virtual »interactions control
A).
With the value a = 3.3 x10-2
used to fitHc2 in [25]
one can thenpredict
a relation betweenTc
and A in zero field under pressure :Careful measurements will be needed to check this
relation,
but at least the order ofmagnitude
estimates deduced from the data of[22]
are not indisagreement
with this relation. Let usfinally
stress that the pressure increase of thequasiparticle
Fermi energy(or
the pressure decrease ofy)
is associated with a P decrease of thesuperconducting
temperature
[24].
Similarphenomena
are found forUPt3
andliquid
3He.
Such effects areopposite
to thegeneral tendency recently reported
thatTc
increases when y decreases[26].
3.5 ANALYSIS OF
po(H).
- Inheavy
fermioncompounds,
the residualresistivity
is anon-trivial
quantity
since it is connected with the latticeproperties.
Forexample
inCeAl3,
the pressure and fielddependences
of A alsocorrespond
tolarge
variations of po[27].
Incompounds
wheretiny magnetic
moments areordered,
one can ask if this would not be drivenby impurities.
InUBel3,
the occurrence ofsuperconductivity
at 950 mKprevents
us fromobserving
the low field behavior ofpo(H)
directly
but itslarge
decrease above 4 T can be alsointerpreted
as aninterplay
between extrinsic and latticeproperties.
In thosesystems
considered as « Kondo lattices », it has beenargued
[28]
that in the lowtemperature
coherentregime,
asingle impurity
is amajor
defect in theregular
array of Ce or Uions ;
it could be viewedby
thequasiparticles
as a Kondo hole with aphase
shift near7T /2
at the Fermi level. The idea would bethat,
at H =0,
theimpurities
could have a relativephase
shift ofir/2
which moves away from this value as H is increased(giving
rise to anegative
magnetoresistivity).
The saturation ofp 0 (H) - 10
cm
above 10 teslas would coincide with the recovery of the true(bare)
residualresistivity. Again,
theproperties
of thesuperconduct-ing
stateprovide
a check for thishypothesis.
3.6 LOW TEMPERATURE SPECIFIC HEAT AND THERMAL CONDUCTIVITY IN THE SUPERCON-DUCTING STATE. - In
anisotropic superconductors,
evensimple impurities
havestrong
«pair
breaking »
effects. As aresult,
a kind ofgap-less
behaviour is observed inimpure samples
forconductivity
(instead
ofhigher
order power lawbehaviours) [28-31],
with ahigh
sensitivity
tothe
phase
shift of theimpurities
[29, 31].
Specific
heat measurements(on
asample
from the same batch as the one used for thepresent
magnetoresistivity
measurements)
show aminimum of
Cp/T
at 90 mK before it reaches a residual finite term yo - 70mJ. mole - 1 K .
This « resonance » structure at low values of
T /Tc
is inagreement
with theoreticalpredictions
on the effect of a small amount of
impurities
onCp/T
in theunitary
limit[29, 311 (Fig. 13).
Fig.
13. - At H = 0,temperature variation of
C /T.
Theextrapolation
ofC / T
to 0 K above 200 mKgives
avanishing
residual term[12].
The insert shows the unusualsharp anomaly
ofCPIT
atT,
due to the occurrence oflarge
critical fluctuations(see [25]).
Furthermore,
this can becompared
with measurements on the samepolycrystalline sample
of the thermal
conductivity
( ).
Theproblem
with the latterquantity
is that inUBe 13,
K is notsimply proportional
to T atTc.
Between 1 K and 4.2K,
K can beanalyzed
as :K = aT+ 13T 2 ; Cî =
0. 29mW/K2.cm ;
/3
= 0.26mW/K3.CM
(Fig. 14).
TheT 2
term islikely
to be attributed to
phonons
(diffused
by
electrons)
whereas the a T termrepresents
the electronic contribution.Thus,
atTc,
around 50 % of the thermalconductivity
is still mediatedby
thephonons
(the
resistivity
ofUBe13
at 1 K is several orders ofmagnitude higher
than that ofordinary
metals).
Thispart
should decrease at lowertemperatures
because the mean freepath
of thephonons
should then be limitedby
the size of thecrystallites
(with
f3 T2
= 1/3Cp
vsf,
one can estimatethat f
isalready
1 f.Lm at 1K).
Thus,
eventhough
thegeneral shape
of K( T)
between 0 K andTc might
be difficult tointerpret,
it appearsreasonable to assume that below 300
mK,
K isessentially
controlledby
the electroniccontribution.
Then,
following
[33],
one can compare the effect of theimpurities
onC p
and K at lowtemperatures
in thesuperconducting
state. For an axial state, wedeterminated,
from thespecific
heat measurements, apair
breaking
parameter
y =Fig. 14.
-Temperature dependence
of the thermalconductivity
K below 4.2 K at H = 0. Insert is theLorentz number after subtraction of the
phonon
contribution to the thermalconductivity.
a o is the coefficient of the residual linear T term of K at low
temperatures.
Thus,
a o should be verysmall,
of the order of 4f..LW/K2.cm.
Experimentally,
no linearT term has been observed on this
sample
down to 30 mK(Fig. 15),
allowing
us toput
anupper limit on a o of 6
f..LW/K2.cm.
This is another confirmation that thephase
shift of theimpurities
is very near-u/2,
since the calculationspredict
a drastic increase of a o when thisphase
shift deviates fromTT /2 [29, 31].
Let us
emphasize
that these low values of thepair breaking
parameter
yimply
veryhigh
values of the mean free
path
of thequasiparticles,
of the order of105
À
[34].
On the onehand,
itclearly
rules out the values off
of a fewangstrôems
deduced from free electron models. This is consistent with what had beenpreviously
discussed[25]
on estimates of the mean freepath
from theresistivity
aboveT,01,
and use of recent band calculations[35],
showing
thatUBe13
was in the clean rather thandirty
limit. On the otherhand,
105 A
would still be more than one order ofmagnitude higher
than the moreoptimistic
estimate of
f,
even if the value of the residualresistivity
were taken in these estimates(instead
of thehigh
temperature
value).
In
previous experiments
on a less puresample
(po (H --+ 00 ) ’"
20f..LOcm) [32],
residuallinear
temperature
terms in C and K were observed.Yo’" 110 mJ.mole-1 K-2
anda o = 30
f..LW/K2 cm.
The emergence of also a nonvanishing
a o coefficient isdirectly
connected with an increase of the
pair breaking
parameter
by
a factor 2.5by comparison
tothe
present
work. Evidence for resonantimpurity scattering
was also found in thespecific
heat of
UBe13
[36]
with a low value of yo - 20mJ.mole-1
K-2
up to a ratherhigh
temperature
(T
= 180mK).
Progress
must be made in thepurity
of the bare lattice forselecting
the natureand the content of the defects. There are clear
experimental
proofs
that theregime
of astrong
Fig ;
15. - At H = 0, the temperature variation ofK/ T
shows the weakness of any linear temperatureterm
by
contrast to the effectreported
onfigure
13 forC / T.
temperatures.
Thespecific
heat in thetemperature
range(150 mK-500 mK)
has aT3
variation with nosupplementary
lineartemperature
contribution. Conclusion.In this paper an
attempt
was made for a coherentinterpretation
of thehigh
and lowtemperature
data ofmagnetoresistivity
andmagnetization
inUBel3.
Theproperties
of the normal andsuperconducting phases
were correlated. Athigh
as well as at lowtemperatures
two different energy scales occur. Even if
UBe13
isthought
to be aKondo-lattice,
magnetic
correlations should exist and
play
animportant
role in its behaviour. A « classical » Fermiliquid picture
has beenproposed
for theA(H) T 2
term of p. The field variation of A could be related to the unusualshape
of the upper critical field while thelarge
fielddependence
of the residualresistivity
underlines thestrong
interplay
of defects with the latticeproperties. Microscopically,
there is a need for inelastic neutron measurements.Notably,
subtle effects between
antiferromagnetic
andferromagnetic couplings
(neglected here)
mayplay
animportant
role inUBe 13.
Acknowledgements.
One of us
(JPB)
thanks Profs. P. Wôlfle and P. Hirschfeld forstimulating
discussions. JF isdeeply
indebted to Prof. J. Friedel for his continuoussupport
in our« experimental
approach
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