Heavy fermion fluid in high magnetic fields: an infrared study of CeRu$_4$Sb$_{12}$

0.18 MB·English

by  S. V. Dordevic

1
Heavy fermion fluid in high magnetic fields: an infrared study of CeRu Sb
4 12
S. V. Dordevic1,†,∗ K.S.D. Beach2, N. Takeda3, Y.J. Wang4, M.B. Maple1, and D.N. Basov1
1 Department of Physics, University of California, San Diego, La Jolla, CA 92093
2 Department of Physics, Boston University, Boston, MA 02215
3 Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwa, Chiba 277-8581, Japan and
4 National High Magnetic Field Laboratory, Tallahassee, FL 32310
(Dated: February 4, 2008)
6
0 Wereportacomprehensiveinfraredmagneto-spectroscopystudyofCeRu4Sb12compoundreveal-
0 ingquasiparticleswithheavyeffectivemassm∗,withadetailedanalysisofopticalconstantsinfields
2 upto17T.Wefindthattheappliedmagneticfieldstronglyaffectsthelowenergyexcitationsinthe
n system. In particular, the magnitude of m∗≃70mb (mb is the quasiparticle band mass) at 10K is
a suppressedbyasmuchas25%at17T.Thiseffectisinquantitativeagreementwiththemean-field
J solution of theperiodic Anderson model augmented with a Zeeman term.

1
2 PACSnumbers: 72.15.Jf,75.20.Hr,78.30.Hr

]
l Systematic investigations of physical properties of amongHFcompounds, atask ofmonitoringsubtle field-
e
- HeavyFermion(HF)materialshaveprovidedvaluablein- induced modifications of electrodynamics is becoming a
r
t sightsintotheroleplayedbymanybodyeffectsintrans- viable endeavor.
s
portand magnetic phenomena [1]. The HF groundstate
. Samples of CeRu4Sb12 were synthesizedusing Sb–flux
at is believed to be a manifestation of a delicate balance method with an excess of Sb (for more details on sam-
m betweencompetinginteractions[2]. Externalstimulican
ple growth see Ref. 8). Large polycrystalline specimens
- readily tip the balance yielding a variety of novel forms withatypicalsurfaceareaof4 4mm2 wereessentialfor
d of matter. Motivated by the exceptional richness and ×
magneto–optical experiments. The samples have been
n
complexity of effects occurring in the HF systems under
characterized by resistivity and magnetization measure-
o
applied external stimuli, many research teams have un-
c ments and found to be of similar quality as previously
[ dertakenthemethodicalinvestigationofpropertiesunder reported [8].

thevariationofrelevantcontrolparameters,suchaspres-
1 The electrodynamic response of CeRu4Sb12 was stud-
sure (P) or magnetic field (H) [3, 4, 5]. In this work we
v ied using infrared and optical reflectance spectroscopy.
1 focused on the analysis of the low energy excitations of
Zero field near–normal–incidence reflectance R(ω) was
9 the HF system CeRu4Sb12 in high magnetic field using measured in the frequency range 30–30,000cm−1 (ap-
4 infrared (IR) spectroscopy. This technique has emerged
1 proximately 4meV–4eV) from T=10K to room tem-
as a very powerful experimentalprobe of the HF ground
0 perature. These measurements were supplemented with
state [6, 7]. Essentially all hallmark features of the the
6 magneto–optic reflectance ratios R(ω,H)/R(ω,H=0T)
0 HFstate,suchasthequasiparticleeffectivemassm∗,the
obtained in zero–field cooling conditions at 10K [12].
/ optical gap ∆, and a narrow Drude–like peak (with the
t The absolute value of reflectance in magnetic field
a width 1/τ) can be simultaneously studied based on the
m R(ω,H)wasobtainedbymultiplyingtheseratioswiththe
analysis of the optical constants.
H=0 data R(ω,H)=[R(ω,H)/R(ω,H=0T)]R(ω). The
- ·
d Our IR magneto-optics results uncover robustness of complex optical conductivity σ(ω)=σ1(ω)+iσ2(ω) and
n the HF fluid in CeRu4Sb12 against magnetic field: a dielectric function ǫ(ω) = ǫ1(ω)+iǫ2(ω) were obtained
o
from reflectance using Kramers–Kronig analysis. For
fieldof17Tsuppressesthequasiparticleeffectivemassby
c
low-frequencyextrapolations,weusedtheHagen-Rubens
: about25%. ThereasonweselectedCeRu4Sb12 forhigh-
v
formula (thin line in Fig. 1). The overall uncertainty in
field investigations reported here is two-fold. First, the
i
X family of skutterudite compounds to which CeRu4Sb12 σ(ω)andǫ(ω)due tolow-andhigh-frequencyextrapola-
tionsrequiredforKramers–Kroniganalysisisabout8-10
r belongs reveals a broad spectrum of behavior, includ-
a %. This uncertainty does not affect any conclusions of
ing non-Fermiliquid powerlaws, superconductivity with
the paper in any significant way.
unconventionalorderparameter andantiferromagnetism
[8,9,10,11]. Thiscatalogueofenigmaticpropertiessup- PlottedinthetoppanelofFig.1isthetemperaturede-
portsthenotionofthecompetinggroundstates: aregime pendenceofreflectanceR(ω)overanextendedfrequency
where even modest external stimuli are likely to have a interval. The gross features of R(ω) are in good agree-
major impact on the response of a system. Second, our mentwiththespectraobtainedpreviouslyonsinglecrys-
earlierzero-fieldexperimentshaveuncoveredremarkably tals [7]. The general shape of the reflectance spectrum
large changes of reflectance R(ω) of CeRu4Sb12 at low at 300K is metallic, with high absolute values at low
temperatures: thereflectancedropsbymorethan20%in frequencies, and a well defined edge around 4,000cm−1
thefar-IR[7]. Withsuchastrongfeature,unprecedented (not shown). Below a coherence temperature T∗ which
2
is about 50 K for CeRu4Sb12 [7] there is a strong sup- ThebottompanelofFig.2displaystheopticalconduc-
pressionofreflectanceinthe rangebetween50and1,000 tivity σ1(ω) at 10 K in a magnetic field, along with 30K
cm−1. This result is indicative of the development of a data (in zero field). Only a limited frequency region is
gap in the density of states commonly observed in HF shownto emphasize the effects due to applied fields. We
systems. Simultaneously a new feature develops in the findthatthe onsetofthe gapstructureinσ1(ω)is virtu-
far-IR.This structurebearssimilaritieswith the conven- allyunaffectedevenbythe17Tfield. Ontheotherhand,
tional ”plasma edge” in metals and is established as a the coherent mode due to heavy quasiparticles broadens
spectroscopic signature of heavy quasiparticles [6]. The as the field increases. This is consistent with positive
reflectance edge softens with decrease of T down to 60 magneto-resistance in the DC data [9, 10, 13].
cm−1 in the 10 K spectra. Several narrow peaks in the Field induced effects in the optical constants can be
far–IRregionareduetoopticallyactivephononsandwill quantified using several complementary methods. The
be discussed below. extendedDrudemodel(EDM)offersaninstructiveanal-
The bottom panel of Fig. 1 displays the reflectance ysisprotocolandisroutinelyusedtoquantifycorrelation
data in magnetic field, plotted on a linear scale over a effects in HF systems [6]. Within the EDM the opti-
narrower frequency interval in order to emphasize field- cal constants are expressed in terms of the quasiparticle
induced changes. Only a few in-field curves are shown scattering rate 1/τ(ω) and the effective mass spectrum
for clarity. For comparison, the plot also includes sev- m∗(ω) [6]. Specifically, m∗(ω) can be obtained from the
eral zero field spectra at higher temperatures. The thin following equation:
line is a Hagen–Rubens extrapolationused for the 300K
spectrum. Forfieldsbelow5Tnosignificantchangeinre- m∗(ω) ω2 1
p
flectancecanbedetected,withintheerrorbarsoftheex- = Im , (1)
periment. At higher fields, the frequency position of the mb −4πω hσ(ω)i
lowlying”plasmaedge”hardens,abehaviorwhichispar- where the plasma frequency
alleledbyzero-fieldspectratakenathigherT.Comparing ω = 4πe2n/m =10,744cm−1 is estimated from
p b
temperature- and field-induced modification of R(ω), we the ipntegration of σ1(ω) up to the frequency of the
notice thatthe neteffectofapplyinga 17Tfieldis func- onset of interband absorption. The spectra of m∗(ω) in
tionally equivalent to a temperature increase of 20K. HF compounds usually reveal a complicated frequency
At higher frequencies (ω >110cm−1) the reflect≈ance is dependence which is particularly true in the vicinity
field independent for H<∼17T, within the error bars of of the energy gap [6]. However, an extrapolation of
the present experiment. m∗(ω) spectra towards zero frequency yields a reliable
The top panel of Fig. 2 displays the temperature de- estimate of the quasiparticle effective mass m∗ that has
pendence of the dissipative part of the optical conduc- been shown to agree with the thermodynamic mass for
tivity σ1(ω). The dominant feature in the low-T data a variety of HF compounds [6, 7]. The estimated value
is a sharp increase in the conductivity above 80cm−1 of 70mb (at 10K) CeRu4Sb12 is somewhat smaller
with an overshoot of the 300 K spectrum at 550cm−1. the≃n previously reported 80m for single crystals [7].
b
This behavioris usually assignedto the formationof the This effect can be assigned to increased disorder and/or
(hybridization) gap in the excitation spectrum of a HF variations of the Kondo temperature in polycrystalline
system[6]. Becauseoftheimprovedsignal-to-noiseratio, specimens compared with previously studied single
it became possible to extend earlier zero field measure- crystals. In the middle panel of Fig. 3 we plot the
ments [7] to lower frequencies. These new results allow extracted values of the mass in zero frequency limit as
us to resolve the onset of a narrow Drude-like feature, a function of magnetic field. A 17T field is capable of
usually associated with the response of heavy quasipar- suppressing the mass by about 25% from its zero-field
ticles at low T [6]. The gradual suppression of the con- value.
ductivity at ω <60cm−1 at low T is consistent with the Alternatively, mass renormalization can be inferred
decrease of the quasiparticle scattering rate 1/τ(ω) be- from the analysis of the loss function spectra Im[1/ǫ(ω)]
low the low-ω cut-off of our data (30 cm−1). The latter (toppanelofFig.3). Thelossfunctionoffersaconvenient
effect will lead to an increase of the conductivity in the waytoexplorelongitudinalmodesintheopticalspectra,
ω 0 limit, in accord with the DC resistivity measure- including heavyelectronplasmons inHF compounds[6].
→
ments [9, 10, 13] displayed with full symbols. The nar- These longitudinal modes produce Lorentzian-like peaks
row resonances at 88, 116, 131, 205, 221 and 248 cm−1 in Im[1/ǫ(ω)] spectra centered at ω∗= 4πe2n/ǫ m∗,
p ∞
are phonon modes. Some of these modes, including the where ǫ∞ is the high-frequency contribuption to the di-
peaks at 88, 131 and 205cm−1, have not been observed electric function. In Fig. 3 the plasmon due to heavy
beforeinmeasurementsonsmallersinglecrystals[7]. We quasiparticles is clearly observed at 74cm−1 at 10K.
alsoobserveanadditionalmodeat40cm−1 whichcanbe Astemperatureincreasestheplasmon≃shiftstohigheren-
identified only in the low-T spectra once the electronic ergies, implying a decrease of the quasiparticle effective
background in the conductivity is diminished. mass (ω∗ 1/√m∗). Similarly, as the magnetic field
p ∼
3
increases, the mode hardens to 90 cm−1, which again quadratic term is due to higher-order changes in the hy-
indicates that the heavy quasiparticles reduce their ef- bridization energy.
fective mass. In the top panel of Fig. 3, we plot the field The quasiparticle contribution to the optical conduc-
dependence of the mass determined from the frequency tivity σ(ω) of the Zeeman-split heavy electron metal is:
of the peak in the loss function spectra. In full agree-
ment with the values obtained by extrapolating m∗(ω) e2 n
s
spectra to zero frequency, we observe a marked depres- σ(ω)= , (4)
m 1/τ iω(m∗/m )
sionoftheeffectivemassbynearly25%at17T,withthe e Xs − e s
characteristic downwardcurvature. where1/τ isthequasiparticlescatteringrate. Thequasi-
We have also employed two additional methods of particles have a total density n = n +n = n +n ;
↑ ↓ c f
quantifying the field-induced changes of m∗. The sim- in a Zeeman field, they are spin-polarized according to
plest estimate of renormalization is the (inverse of the) n↑ n↓ =H/H0. In the limit ωτ 1, Eqs. 3 and 4 can
− ≫
positionoftheminimuminreflectance. Theminimumin be combined to give:
reflectance,theso-calledplasmaminimum,isdetermined
bythescreenplasmafrequency,whichontheotherhand ω2 1 m∗ 5 2n H 2
is proportional to the ratio n/m∗, where n is the carrier p Im = 1 f ,
density and m∗ their effective mass. The plasma mini- − 4πω hσ(ω)i (cid:18)me(cid:19)0(cid:20) −(cid:18)2 − n (cid:19)(cid:18)H0(cid:19) (cid:21)
(5)
mum points (from Fig. 1) are also plotted in the middle
which for H=0T reduces to Eq. 1. According to Eq. 5,
panel of Fig. 3 and they agree well with previous two
the effective mass probed in an IR experiment is aver-
data sets. Finally the partial sum rule:
aged between the two spin channels and its magnitude
is reduced as (H/H0)2, in full agreement with the ex-
ωc (ωp∗)2 πne2 perimental observation (top panel of Fig. 3). The fit
σ1(ω)dω = = , (2)
Z0 8 2m∗ (grayline) correspondsto H0=35T. This estimate of H0
is lower then the Kondo temperature T 100K [10]
can also give an estimate of the effective mass [6]. The K ≃
forCeRu4Sb12 butis closeto the coherencetemperature
upper integration limit ω was set just below the onset
c T∗ 50 K. The Zeeman splitting also implies an overall
of hybridiztion gap. The values of the effective mass ex- ≃
reduction of the energy gap between the spin-down sub-
tracted this way are also in good agreement with the
band at E > E and the spin-up sub-band at E < E
other three methods (Fig. 3, middle panel). The scatter F F
[16]. As pointed out above, appreciable reduction of the
ofdata points inFig.3servesasanestimate ofthe error
gap is not observed in our data. This is consistent with
bars.
the notionthat in anopticalexperiment one probes sep-
The Periodic Anderson Model (PAM) is believed to
aratelygapsinthe spin-upandspin-downchannels,pro-
capture the essence of the physics in HF metals [1]. We
videdthatspinflipscatteringdoesnotsignificantlyalter
will show below that this model when augmented with
the selection rules. Neither of the two transitions is ex-
the Zeeman term elucidates the behavior of the effec-
pected to significantly change in fields smaller than the
tive mass in a magnetic field. At the mean-field level,
hybridizationgap. Furthertheoreticalstudiesareneeded
theheavyfermionstateischaracterizedbyhybridization
to address the issue of spin-flip scattering, as well as the
of the free (c) and localized (f) electrons [14, 15]. Hy-
problem of the field dependence of the hybridization en-
bridization breaks the conduction band into disjoint up-
ergy.
perandlowerbrancheswithveryshallowdispersionnear
In conclusion, we have presented a comprehensive set
the gapedge(Fig.3bottompanel);this accountsforthe
ofinfrareddataobtainedonCeRu4Sb12 inhighmagnetic
very large effective mass of the quasiparticles. The state
field. The external field is found to affect the low-energy
ismetallicandpossessesawell-definedFermisurfacethat
excitations of the system, which leads to the renormal-
incorporates both the c and f electrons. In the presence
izationofthequasiparticleeffectivemass. Indeed,a17T
of a Zeemanterm – which consists of a magnetic field H~
fieldwasfoundtodiminishm∗byabout25%. Mean-field
coupled to the total magnetic moment at each site – the
solution of the PAM with a Zeeman term offers a quan-
mass enhancement factor behaves as [16]:
titative account for the field dependence of the effective
mass.
2
m∗ m∗ H 3 H The research supported by NSF.
= 1+2s + , (3)
(cid:18)me(cid:19)s (cid:18)me(cid:19)0(cid:20) (cid:18)H0(cid:19) 2(cid:18)H0(cid:19) (cid:21) †Present Address: Department of Physics, The Uni-
versity of Akron, Akron, OH 44325
where m is the free electron mass, s = ( , ) and
e
± ↑ ↓
H0 is on the order of the Kondo energy in magnetic
units. The linear term in Eq. 3 results from the split-
ting of the Fermi surface into separate spin-up and spin-
downsurfaces,eachwithadifferentbandcurvature. The ∗ Electronic address: [email protected]
4
[9] N. Takeda and M. Ishikawa, Physica B 281– 282, 388
FIG. 1: Temperature and field dependenceof the reflectance
(2000).
of CeRu4Sb12. Top panel: temperature dependence with [10] E.D. Bauer et al., J.Phys.:Condens.Matter 13, 5183
H=0. Bottom panel: high field reflectance at T=10K, along
(2001).
with several zero field curvesat higher temperatures. [11] E.D. Bauer et al. Phys.Rev.B65, 100506 (2002).
[12] W.J. Padilla, Z.Q. Li, K.S. Burch, Y.S. Lee, K.J. Miko-
laitisandD.N.Basov,Rev.Sci.Instrum.75,4710(2004).
FIG. 2: Temperature and field dependence of the real part [13] K.Abeet al.,J.Phys.:Condens.Matter 14,11757(2002).
of the optical conductivity σ1(ω). Top panel: temperature [14] P. Coleman, Phys.Rev.B 35, 5072 (1987).
dependencein zero field. Bottom panel: optical conductivity [15] A.J. Millis and P.A.Lee, Phys.Rev.B 35, 3394 (1987).
inhighmagneticfieldatT=10K,alongwithseveralzerofield
[16] K.S.D. Beach, cond-mat/0509778.
curvesat higher temperatures.
[1] A.C. Hewson, The Kondo Problem to Heavy Fermions,
(Cambridge University Press,1997).
[2] S.Doniach, Physica B 91, 231 (1977). FIG. 3: The top panel presents the loss function Im[1/ǫ(ω)],
[3] P.Gegenwart et al., J. Low Temp. Phys. 133, 3 (2003). which has pronounced peaks at the frequencies of the lon-
[4] K.H.Kim et al. Phys.Rev.Lett.91, 256401 (2003). gitudinal modes, such as the heavy plasmon ωp∗. All field
curves are at T=10K. The middle panel shows the field de-
[5] N.Harrison,M.JaimeandJ.A.Mydosh,Phys.Rev.Lett.
∗
90, 096402 (2003). pendence of quasiparticle effective mass m extracted from:
[6] L. Degiorgi, Rev.Mod.Phys. 71, 687 (1999). 1) the m∗(ω) spectra, 2) the maximum in the loss function
Im[1/ǫ(ω)], 3) minimum in reflectance R(ω) and 4) the inte-
[7] S.V.Dordevic, D.N.Basov, N.R.Dilley, E.D.Bauer and
M.B. Maple, Phys.Rev.Lett. 86, 684 (2001). gral of the narrow Drude mode (Eq. 2). The bottom panel
[8] N. Takeda and M. Ishikawa, Physica B, 259- 261, 92, schematically displays thePAMbandstructure, bothin zero
and high magnetic field.
(1999).
-1
Frequency [cm ]
100 1,000
1.0
300 K
80 K
0.9
50 K
e
c 30 K
n
a 10 K
t 0.8
c
e
l
f
e
R
0.7

1.0
300 K
e
0.9
c
n
a
t
c
e 17 T 50 K
l
f
e 0.8 10 T
R
0 T
30 K
0.7
0 30 60 90 120 150
-1
Frequency [cm ]
Figure 1
-1
Frequency [cm ]
100 1000
3
10
]
m
300 K
c

80 K
W[

1 50 K
s
30 K
10 K
2
10
3
10
30 K
]
1
-
m
c

W[

1
s
17 T
10 T
0 T
2
10
30 60 90 120 150
-1
Frequency [cm ]
Figure 2
3
-
0
17 T
1
6

x
10 T 30 K

)
4
e/ 0 T 50 K
1

(
2
m
I
-
0
30 60 90 120 150
-1
Frequency [cm ]
) 1.0
T

0
(
0.9 Fit from Eq. 5
*
m
m* (B) from m*(w )

/
) 0.8 m* (B) from Im[1/e (w )]
H
m* (B) from min in R(w )
(
*
m m* (B) from integral of s (w )
0.7
1
0 5 10 15
Magnetic Field [T]

„ ›
H 0, spin

y
H „ 0, spin fl
g
r
e
n
E
E EEE
=
f H 0
Wave vector
Figure 3