Stoner gap in the superconducting ferromagnet UGe2

0.28 MB·English

by  N. Aso

APS/123-QED
Stoner Gap in the Superconducting Ferromagnet UGe
2
N. Aso,1 G. Motoyama,2 Y. Uwatoko,3 S. Ban,4 S. Nakamura,4
T. Nishioka,5 Y. Homma,6 Y. Shiokawa,6 K. Hirota,1 and N.K. Sato4
1Neutron Sci. Lab., ISSP, University of Tokyo, Tokai, Ibaraki 319-1106, Japan
2Department of Material Science, Graduate School of Material Science, University of Hyogo, Hyogo 678-1297 Japan
3ISSP, University of Tokyo, Kashiwa 277-8581, Japan
4Department of Physics, Graduate School of Science, Nagoya University, Nagoya 464-8602, Japan
6 5Department of Material Science, Faculty of Science, Kochi University, Kochi 780-8520, Japan
0
6Oarai Branch, Inst. for Mater. Research, Tohoku University, Oarai, Ibaraki 311-1313, Japan
0
(Dated: February 2, 2008)
2

n Wereport thetemperature(T) dependenceof ferromagnetic Bragg peak intensities and dc mag-
a netization of the superconducting ferromagnet UGe2 under pressure (P). We have found that the
J low-T behavior of the uniform magnetization can be explained by a conventional Stoner model. A

1 functionalanalysisofthedataproducesthefollowingresults: Theferromagneticstatebelowacriti-
3 calpressurecanbeunderstoodastheperfectlypolarizedstate,inwhichheavyquasiparticlesoccupy
onlymajorityspinbands. AStonergap∆(P)decreasesmonotonicallywithincreasingpressureand

] increaseslinearlywithmagneticfield. WeshowthatthepresentanalysisbasedontheStonermodel
l
e isjustifiedbyaconsistencycheck,i.e., comparison ofdensityofstatesat theFermienergydeduced
- from the analysis with observed electronic specific heat coeffieients. We also argue the influence of
r
theferromagnetism on the superconductivity.
t
s
.
t PACSnumbers: 65.40.-b,71.28.+d,71.30.+h,71.27.+a
a
m
- I. INTRODUCTION magneticsuperconductor. Amicroscopiccoexistencebe-
d
tween weak ferromagnetism and superconductivity was
n
reported, but detailed neutron diffraction investigations
o Since a pioneer paper by Ginzburg on the coexistence
indicated that the magnetism coexisting with the super-
c
of ferromagnetism and superconductivity [1], the inter-
[ conductivityisnotpurelyferromagnetic[3],again. These
playbetweenthesetwolong-rangeorderingshasbeenan
examplesseemtoindicatethatsuperconductivityhardly
2 interesting topic in solid-state physics. Superconductiv-
coexistswithferromagnetism,eventhoughsuperconduc-
v ity and magnetism would be antagonistic because of the
tivity and ferromagnetism are carried by different types
6
competitive nature between the superconducting screen-
6 of electrons. Recently, Saxena et al. discovered a new
ing (Meissner effect) and the internalfields generatedby
2 type offerromagneticsuperconductorUGe2 in whichsu-
magnetic orderings. During the last three decades, how-
5 perconductivity occurs at high pressures [4]. It is par-
0 ever,thediscoveryofanumberofmagneticsuperconduc- ticularly interesting to note that both of ferromagnetism
5 tors has allowed for a better understanding of how mag-
and superconductivity may be carried by itinerant 5f
0 netic order and superconductivity can coexist. It seems
electrons,whichcanbehomogeneouslyspreadinthereal
/ to be generally accepted that antiferromagnetism with
t space,althoughit is stilla matter ofdebate andremains
a local moments coming from rare-earth elements readily
m toberesolved. Thisobservationhasrenewedourinterest
coexists with type II superconductivity. This is because
on the interplay of ferromagnetism and superconductiv-
- superconductivityandmagnetismarecarriedbydifferent
d ity.
types of electrons; magnetism is connected with deeply
n
o seated4f electrons,whilesuperconductivityisfundamen- Figure 1 shows a temperature (T) vs pressure (P)
c tally related to the outermost electrons such as s, p, and
: d electrons. phase diagram of UGe2. A Curie temperature (TFM) is
v about 52 K at ambient pressure, and monotonically de-
i Inthe caseofaferromagneticsuperconductor,atrick- creaseswithincreasingpressure. Thenitcollapsestozero
X
ier negotiation is needed for the coexistence, because in- temperatureataferromagneticcriticalpressureP (
FM
r ∼
a ternal fields are not canceled out in the range of a su- 1.5 GPa). In the ferromagnetic phase, another phase
perconductingcoherencelengthincontrastwithananti- transition or crossover seem to appear at T ( 32 K
X
≃
ferromagnetic superconductor. In the classical magnetic at ambient pressure). This characteristic temperature
superconductorErRh4B4 withasuperconductingtransi- TX also decreases with increasing pressure and becomes
tiontemperature8.7K,forexample,once the Ersublat- suppressedtozeroatacriticalpressureP ( 1.2GPa).
X
∼
tice starts to orderferromagneticallybelow about 0.8 K, The transitionsat P and P are likely of the firstor-
X FM
the superconductivity is immediately destroyed, except derinnature[5]. Superconductivityemergesinthepres-
a very narrow coexistence region near 0.8 K [2]. Here sure range between 1.0 and 1.5 GPa. Since a maxi-
∼ ∼
we note that the magnetic structure coexisting with the mumsuperconductingtransitiontemperature(T 0.7
SC
∼
superconductivity is not purely ferromagnetic but spa- K) is observed at around P [4], we speculate that the
X
cially modulated. ErNi2B2C is a modern example of critical point PX plays an important role in the onset of
2
the superconductivity (see, for example, Watanabe and II. EXPERIMENT
Miyake [6], Sandeman et al. [7], and references therein).
Very recently, Nakane et al. provided a supporting evi- Single crystals were grown by the Czochralsky pulling
dence for the speculation by means of ac magnetic sus- method using a tetra-arc furnace installed at Oarai
ceptibility measurements under external magnetic fields Branch of Institute for Material Research, Tohoku Uni-
H; in a plot of TSC as functions of P and H, the super- versity [9]. The pressure was generated using a copper-
conductivity always appears at around the critical point beryllium(CuBe)basedpiston-cylinderclampdevice[10]
PX, not around PFM [8]. However, there are still many with Fluorinert FC-75 (3M Co. Ltd., Tokyo) as a pres-
unsolved questions in this unique material to be further sure transmitting medium. The low temperature pres-
clarified. To shed more light on the nature of the ferro- sure was determined by measurements of the change in
magnetism as well as its pressure variation, we present a lattice parameter of NaCl put together with the sam-
in this paper the T-dependence of the uniform magne- ple. Elastic neutron scattering experiments were done
tization under pressure by the neutron diffraction tech- on the ISSP cold neutron triple-axis spectrometer HER
nique, togetherwith the dc magnetizationmethod. Sim- (C1-1) installed at JRR-3M, JAERI, Japan, with a typ-
ilar measurements were already reported, however the ical configuration of energy k = 1.11 ˚A−1 or 1.555 ˚A−1
i
present experiment is so precise that we can analyze the and collimations of Guide-Open-80’-80’. A cooledBe fil-
functional dependence of the magnetization. Actually, ter was placed before the sample to remove higher order
we have found that the low T-dependence of the uni- contaminations. The crystals were oriented with the a-
form magnetization can be described by a conventional axis perpendicular to the scattering plane. Temperature
Stoner model. This enables us to extract new informa- wascooleddownto1.4Kusinga4He-pumpingILL-type
tionabouttheferromagnetismasfollows: Thelow-T and orange cryostat. The dc magnetization measurements
low-P region of the ferromagnetic state, i.e., the FM1 were carried out using a conventional vibrating sample
region in Fig. 1, is understood as a perfectly polarized magnetometer (VSM) [9].
state inwhichonly amajorityspinbandis occupied. As
the pressure increases toward P , a Stoner gap ∆(P) in
X
the heavy quasiparticle bands decreases monotonically, III. RESULTS AND DISCUSSION
similarly to T (P). When the pressure exceeds P , the
X X
gap seems to jump up, although the applicability of the
InFig.2weshowtheT-dependenceofmagneticBragg
Stoner model to this high pressure ferromagnetic state
peak intensities I (T) at Q = (0,0,1) for several pres-
B
(FM2) is less convincing compared to the region P < sures. All data were accumulated at k = 1.555 ˚A−1 in
i
P . From these results, we argue the influence of an
X the process of increasing temperature. In contrast with
effective internal field produced by the ferromagnetism,
a conventional dc magnetization measurement, neutron
which is found to be remarkably large below P , on the
X scattering experiments do not suffer from complications
superconductivity.
arising from a pressure cell contribution to the magne-
tization as well as a magnetic domain effect in a ferro-
magnetic sample, and hence the present results are not
60
obscuredatallby these effects. While there is no appar-
50 Paramagnetic T ent anomaly in the curve of P = 0.28 GPa, we clearly
FM
observe a steep increase below T 10 K at 1.1 GPa.
T X
40 x (In the present study, we define T ∼as a maximum tem-
X
) FM2 peratureappearingin the secondderivativeoftheI (T)
(K 30 PFM curve with respect to T; Note that this definition yBields

T
a T -value close to previously reported ones.) We note
20 X
P thatthe overallfeatureofthe presentresultis consistent
x
FM1 with the Bragg peak intensity and static magnetization
10
Superconductivity data previously reported in Refs. [9, 11, 12]. At 1.23
0 GPa, such an anomalous behavior was not observed in
0.0 0.5 1.0 1.5 2.0 accordance with P 1.2 GPa.
X
∼
P (GPa) Remembering that the neutron intensities are propor-
tionaltothesquareofmagnetizationM,wecalculatethe
magnetic Bragg peak intensities in terms of the Stoner
FIG. 1: (Color online) Phase diagram for UGe2 determined
model, which is expressed as follows (see, for example,
byourneutron diffraction measurements. Theshaded region
[13]);
between about 1.0 and 1.5 GPa shows a superconductivity
regiontakenfromtheliterature[8]. Thesolidlinesareguides 3
M =M0 1 α T2 exp( ∆/T) , (1)
totheeye. “FM1”denotesaperfectlypolarizedferromagnetic { − · · − }
state in which only majority spin bands are occupied. For
“FM2” state above PX, see thediscussion in thetext. α= 3√π 1 32,∆=2EF Θ′ 2−13 , (2)
4 {E } {E − }
F F
3
JRR-3M/ISSP-HER, k=1.555A-1, Open-BeF-80’-80’
i 1.2
Q x
u.) 20000 UGe2 000, ...2588 =GG (G0PP,Paa0a,1) ∆θ, ', T 10..08 Px
(a. 15000 01..917 G GPPaa zed 0.6
G 11..243 G GPPaa ali 0.4 ∆
B m θ'
sity- 10000 Nor 0.2 TTxx ((TTahtise iwwoar ek)t al.) (a)
n 5000 0.0
e 1.5
t
n Perfectly polarized
I
0 P
1.0 x 2-1/3
0 10 20 30 40 50 60 EF ~ 0.793
T (K) θ' / Imperfectly polarized
0.5 2/3
FIG. 2: (Color online) Temperature dependence of the fer- Unpolarized
romagnetic Bragg peak intensities at Q = (0,0,1) against the (b)
temperatureT measured at variouspressures. “BG” denotes 0.0
0.06
background intensities in the paramagnetic phase of about
120
1250, which arise from the incoherent scattering of both the 0.05 1/EF (This work)
cTrhyestsaolliidtsleilnfeasnadrethcaelcpurleastseudrerecseulll.tsNoontethtehbatasPisXof∼th1e.2StGoPnear. K) 0.04 γ (Tateiwa et al.) 100 (mγ
model described in thetext. (1/F 0.03 8600 J/K2
E /m
where M0 indicates the magnetization at zero temper- 1/ 00..0021 Px 4200 ole)
ature, ∆ a so-called Stoner gap, EF a Fermi energy, (c)
′
and Θ is a molecular field coefficient. The results are 0.00 0
0.0 0.5 1.0 1.5
shown in Fig. 2 by solid lines. Interestingly, we find
P (GPa)
good agreement between the low-T magnetization data
and the calculation. (The observation of the exponen-
tial like T-dependence of the magnetization, instead of FIG. 3: (Color online) Pressure dependence of the obtained
a conventional T-power law behavior due to spin wave parametersfromtheStonermodel. Thesolidlinesareguides
excitaions, is probably related to a huge uniaxial mag- to the eye. (a) ∆, Θ′, and TX plotted are normalized with
respect to a respective ambient pressure value; ∆ = 39.5 K,
netic anisotropyofUGe2.) Thisagreementsuggeststhat Θ′ = 83.4 K, and T = 30.2 K. We also plot T taken from
the decreaseinthe magnetizationatlowtemperatures is Ref.[14]. (b)RatioXof Θ′/E is plottedagainstXP below P .
mainly causedby electron-holeexcitations in quasiparti- F X
Note that the pressure region of P < P corresponds to the
X
cle bands. perfectly polarized state in the Stoner model, i.e., Θ′/E >
F
From the least square fitting of the data, we estimate 2−1/3. For the region above P , see the discussion in the
X
a set of parameters α and ∆ in eq. (1), which further text. (c) Inverse Fermi energy 1/E is plotted against P
F
enables us to evaluate E and Θ′ using eq. (2). First we belowP ,togetherwithanelectronicspecificheatcoefficient
F X
concentrate on the pressure region below P . In Fig. 3 γ taken from Ref. [14].
X
′
(a) we show ∆ and Θ together with T , each of which
X
is normalized to unity at ambient pressure. It is inter-
esting to note that these quantities seem to lie on a sin- ratio is smaller than 2/3, the system is paramagnetic.
gle line, suggesting that the characteristic energy scale Asseeninthe figure,ouranalysisindicates thatthe per-
of unknown origin, TX, is related to the Stoner gap ∆ fectly polarizedstate is realizedbelow PX. This resultis
′
(equivalently Θ). supportedbybandstructurecalculationsindicatingthat
In Fig. 3 (b) we plot a ratio of Θ′/E against P. Fermisurfaceshavea predominantlymajorityspinchar-
F
According to the Stoner model, the ratio greater than acter [15, 16].
−1
2 3( 0.793)meansthatthesystemisinaperfectlypo- In Fig. 3 (c) we plot an inverse of Fermi energy 1/E
∼ F
larized ferromagnetic state, where only a majority spin deduced from the above analysis as a function of P. To
band is occupied. When the ratio lies between 2/3 and estimate a density of states atthe Fermi energy, D(E ),
F
−1
2 3,animperfectlypolarizedferromagneticstateoccurs, we assume that EFD(EF) is a constant value indepen-
where a minority spin band becomes to be partially oc- dentofpressure. Then,1/E correspondstoD(E ). On
F F
cupied by quasiparticles. Further in the case that the the other hand, D(E )canbe directly obtainedfroman
F
4
electronic specific heat coefficient γ, which is also shown 1.2 30
(a)
in Fig. 3 (c) for camparison [14]. As is clearly seen,
)
i1inn/tdEeerFpperinesdtaeptnriotopncoobrntaissotenadnatlontco.thγeT,hSi.iteso.n,ceo1ri/nmEcioFddee=nlc.ecγjuwstiitfihesaoPur- U) 10..08 75 k5O0e kOe ∆ (K 2100 (b)(cid:13)
/B 0
Figure4(a)showsthe dcmagnetizationM(T)at1.18 µ 1 kOe
mGPagan(e<tizPatXio)nunofdetrheexptreersnsaulrme aceglnlewtiacsfiseuldbstrHacetxetd. (fTrohme M ( 00..64 103 k0O keOe 1/K)00..0064 (c) 19200 (mγ
the total measured magnetization.) The magnetic field 0.2 E (F0.02 60J/K
was applied along the magnetization easy a-axis. We 1.18 GPa 1/ 30m 2
observedthat the M(T) curve shows a step-like increase 0.0 0.00 0 o
at lower fields similarly to the IB(T) curve, and that 0 10 20T 3(0K)40 50 60 0 2 H0 4 (0kOe6)0 80 le)
T exhibits an increase with H in accordance with ext
X ext
the previous results [5, 8]. We find that the static low-T
FIG. 4: (Color online) (a) Temperature dependence of the
magnetization can also be well described by the Stoner
static magnetization at 1.18 GPa under various magnetic
model (see dotted lines).
fields along the magnetization easy a-axis. (b) Stoner gap
In Fig. 4 (b) we plot ∆ as a function of H , which
ext ∆isplottedasafunctionoftheexternalmagneticfieldH .
ext
was obtained in the same manner as above. It is found
(c) Inverse Fermi energy 1/E is plotted together with the
F
that ∆ increases almost linearly with Hext, as shown by electronic specific heat coefficient γ measured at 1.15 GPa
a broken line. This is highly expected from the Stoner taken from Ref. [17]. Note that 1/E , which is proportional
F
model;the gapinthequasiparticlebandsshouldlinearly to the density of states at E , agrees well with the observed
F
increasewiththemagneticfieldduetotheZeemaneffect γ value.
as follows,
comparedwiththat forP < P , for whichthere aretwo
∆(H)=∆+2gµ SH. (3) X
B possibleexplanations: First,thelowT-dependenceofthe
Here, ∆ is a value at zero magnetic field, i.e., ∆(H =0), uniformmagnetization(forP >PX)cannolongerbede-
gandS denoteag-factorandthemagnitudeofthequasi- scribed by the Stoner model. Second, the Stoner model
particlespin,respectively,andµ istheBohrmagneton. is still applicable to the FM2 region, but a pressure dis-
B
Indeed,thevalueof∆ 12Kestimatedfromtheextrap- tributionwithinthesamplewillcausetheM(T)curveto
olation to zero field is≃consistent with a value obtained deviate from the Stoner model. (The Curie temperature
from the Bragg peak intensity (at H = 0) mentioned decreases steeply above PX (see Fig. 1). In such a case,
above. A set of parameters, g = 6/7 and S = 5/2 cor- theexperimetalresultscouldbeobscuredbyevenasmall
responding to an f electron, produces better agreement pressure distribution within the sample [10, 18].) Since
between the observation and the calculation than a dif- it is unclear which is dominant, we tentatively tried to
ferent set of parameters, g = 2 and S = 1/2 for a free applytheStonermodeltotheFM2region. Theobtained
electron. This may reflect that the heavy quasiparticle ∆-values are as follows; ∆ = 40 ( 6), 25 ( 5), and 7
± ±
arises from f electrons. ( 7) K at P = 1.23, 1.28, and 1.40 GPa, respectively.
±
The slope of the brokenline in Fig. 4 (b) is calculated We find that ∆ shows a jump near at PX on going from
tobeabout0.3K/kOe. Itisveryinterestingtonotethat the FM1 to FM2 region,and that ∆decreasesmonoton-
this value is almost the same as the slope of curves in a ically with further increasing pressure and finally tends
plot of T vs H (see, for example, Ref.[6] and references toward zero in the vicinity of the ferromagnetic critical
X
therein). This clearly supports that TX is related to the pressurePFM. Note thatthe jump of∆reflectsthe sud-
Stoner gap ∆, as mentioned above. den change in the solpe of the IB(T) curves below and
Figure 4 (c) shows the Hext-dependence of 1/EF (at above PX. A further investigation is needed to clarify
1.18 GPa) obtained from the least square fitting of the whether the magnetizationinthe FM2 regioncanbe de-
M(T) data to the Stoner model. We also plot reported scribed by the Stoner model or not.
values of the γ-coefficient observed at 1.15 GPa under Finally we discuss the correlation between the ferro-
external magnetic fields [17]. Again, we find the same magnetismintheFM1regionandthesuperconductivity.
relation 1/E = cγ with the same scale factor c as the ItisevidentthattheStonergapbehaviorintheFM1re-
F
above. Note thatthereis noadjustableparameteratall, gion is firmly established by the consistency check, i.e.,
neverthelesswefindthe goodagreementbetweenD(E ) thecomparisonofourresultswiththeγ-coefficients. Us-
F
′
estimated from the Stoner model and deduced from the ingtheparameterΘ ineq.(2)obtainedfromthefitting,
heatcapacityexperiments. Thisclearlyprovesthe valid- we can estimate an effective internal field H seen by
eff
ity of our model analysis. the itinerant electrons due to the ferromagnetismby the
′
Let us return to the pressure region of P > P . As definitionofµ H =k Θ. Thenwefinditto bevery
X B eff B
maybeseenfromFig.2,theaccordancebetweenthecal- large;forexample,H 100Tatambientpressureand
eff
≥
culated and the experimental results is less convincing H 40 T at 1.1 GPa (see Ref. [19] for detail). This
eff
∼
5
may explain an asymmetric shape of the superconduct- the majority spin band. The Stoner gap ∆ in the heavy
ing dome with respect to P in the T-P phase diagram, quasiparticle bands was estimated to be about 40 K at
X
ifweassumethatthesuperconductivitydoesnotsurvive ambientpressure,and∆wasfoundtodecreasemonoton-
under such a strong internal field. In the literature, it ically withincreasingpressureP andto increaselinearly
has been speculated that the nonunitary superconduct- with magnetic field H. The similarity between the P-
ingstatewouldberealizedinUGe2;otherwisethesuper- andH-dependences of∆andTX suggeststhatthe char-
conductivity would not coexist with the feromagnetism. acteristicenergy T ofunknownorigincanbe relatedto
X
However, it seems very unlikely that the strong internal the Stoner gap. Assuming that the product E D(E )
F F
field mentioned above dose not kill the superconductiv- is constant, we evaluated the P- and H-dependence of a
ity,evenifthespin-tripletpairingstatewouldbeformed. density of states at the Fermi energy D(E ) using the
F
This leads us to suggestspatially inhomogeneouscoexis- Stoner model. Then we found that D(E ) = cγ with
F
tence of the ferromagnetism and the superconductivity, the same constant c for both the P- and H-dependence
provided that the superconductivity below P is intrin- of the electronic specific heat coefficient γ. This justifies
X
sic, but not due to the pressure inhomogeneity. We need our interpretation based on the Stoner model. Finally
a further experiment to confirm this possibility. we argued the relationship between ferromagnetism and
superconductivity;theeffectiveinternalfieldseenbyitin-
erantelectrons is estimatedto be sufficiently strongthat
IV. SUMMARY the superconductivity would hardly survive, which leads
us to suggest the spatially inhomogeneous coexistence
We investigated the uniform magnetization of the of ferromagnetism and superconductivity. We hope that
pressure-induced superconductor UGe2 by the neutron these results stimulate further theoretical investigations.
diffraction technique together with the dc magnetiza-
tion measurements under pressure. For this strongly
anisotropic ferromagnet, we found that the low-T be- Acknowledgments
havior in the magnetization of the FM1 region can be
explained by the conventional Stoner model. Our anal- We thank K. Miyake and S. Watanabe for useful dis-
ysis based on the Stoner model produces the following cussions. This work was supported by a Grant-in-Aid
results: The ferromagnetic state below the critical pres- fromtheMinistryofEducation,Culture,Sports,Science
sureP ( 1.2GPa)isunderstoodastheperfectlypolar- andTechnology,Japan. NKSalsothanksDaikoFounda-
X
∼
ized state in which the heavy quasiparticles occupy only tion for partial financial support.
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