Low Frequency Spin Dynamics in the CeMIn_5 Materials

4 Pages·0.16 MB·

by  N. J. Curro

Low Frequency Spin Dynamics in the CeMIn Materials
5
N. J. Curro, J. L. Sarrao, and J. D. Thompson
Condensed Matter and Thermal Physics, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
P. G. Pagliuso
Instituto de F´ısica “Gleb Wataghin”, UNICAMP, 13083-970, Campinas-SP, Brazil
3 Sˇ. Kos, Ar. Abanov, and D. Pines
0
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
0
(Dated: February 1, 2008)
2
WemeasurethespinlatticerelaxationoftheIn(1)nucleiintheCeMIn materials,extractquanti-
n 5
a tativeinformationaboutthelowenergyspindynamicsofthelatticeofCemomentsinbothCeRhIn5
∗
J and CeCoIn5, and identify a crossover in thenormal state. Abovea temperature T theCe lattice
exhibits ”Kondo gas” behavior characterized by local fluctuations of independently screened mo-
8
ments; below T∗ both systems exhibit a ”Kondo liquid” regime in which interactions between the

] local moments contribute to the spin dynamics. Both the antiferromagnetic and superconducting
l groundstatesinthesesystemsemergefromthe”Kondoliquid”regime. Ouranalysisprovidesstrong
e
- evidencefor quantumcriticality in CeCoIn5.
r
t
s PACSnumbers: 76.60.-k,75.20.-g,75.25.+z,71.27.+a
.
t
a
m The CeMIn5 heavy fermion materials possess a rich cal fluctuations of the Ce moments. Below T∗ the q-
- phase diagram revealing a fascinating interplay between independent local moment contribution to T1−1 becomes
d the antiferromagnetic behavior of incompletely screened temperatureindependent, andandasecondqandT de-
n
Ce local moments, an unconventional normal state, and pendent component emerges. We identify this emergent
o
superconducting behavior of the heavy electrons that is component with the heavy electrons, that is the itiner-
c
[ reminiscentofthatfoundinthecuprates[1,2]. Thenon- ant component of the Ce 4f electrons arising from their
Fermiliquidbehavioroftheitinerantheavyelectronsand couplingto one anotherandto the conductionelectrons.
2
the possible d-wave symmetry of their superconducting This extra contribution to the relaxation agrees quanti-
v
state has led to the proposal that the superconducting tatively with inelastsic neutron scattering (INS) results
4
5 pairingmechanismarisesfromtheircouplingtospinfluc- inCeRhIn5,andsuggeststhattheantiferromagneticcor-
3 tuations [1, 3, 4, 5, 6]. In this Letter, we consider the relations in CeCoIn5 are primarily 2D, and that in the
5 information about the coupled system of Ce local mo- absence of superconductivity one would have a quantum
0 ments and itinerant heavy electrons that can be derived critical point corresponding to a transition from Fermi
2 from Nuclear Magnetic Resonance (NMR) experiments liquid to antiferromagnetic behavior at T =0.
0 on the spin lattice relaxationrate (T−1) of the In(1) nu-
/ 1 In orderto developa model for the hyperfine coupling
t clei located in the plane of the Ce moments. We show
a intheCeMIn5 materials,itisnecessarytodeterminethe
that the In(1) nuclei are strongly coupled to their four
m number of spin degrees of freedom, and the hyperfine
nearest neighbor Ce spins by an anisotropic hyperfine
- couplings to each degree of freedom. For all three mate-
d interaction, and that the resulting anomalous behavior rials (M=Rh, Ir, Co), the bulk susceptibility χ is dom-
n of T1T provides important information on the low fre-
inated at high temperatures by localized Ce moments,
o quencydynamicsofthelatticeofCespins,andthequasi-
and can be adequately fit by an expression for these
c particles to which they couple. Because the coupling is
: moments in a tetragonal crystalline electric field (CEF)
v anisotropic, it does not vanish for antiferromagnetically
i correlated Ce moments, so that T−1 provides valuable [11, 12, 13, 14, 15]. Neutron diffraction (ND) in the or-
X 1 dered state of CeRhIn5 indicates that the magnetism is
information on the dynamics of their magnetic ordering,
r localized on the Ce sites, albeit with reduced moments
as well as on the influence of Kondo screening of the
a [16]. Thereducedmomentplausiblyreflectssomedegree
moments on their relaxation rate, and departures from
ofhybridizationwithincompletescreeningoftheCemo-
Kondo behavior at low temperatures [7, 8]. We present
ment in the CEF ground state doublet. (For the CEF
our results for both the antiferromagnet CeRhIn5 and
scheme measured in [15] the ground state doublet has
the superconductor CeCoIn5.
a moment of 0.92µ , whereas the measured moment is
B
Previous reports of T1−1 in CeIrIn5 and CeCoIn5 by 0.74µB [16, 17, 18].) The magnetic shifts of the high
other authors have attributed an unusual temperature symmetry In(1) site in CeRhIn5, presented in Fig. (1),
dependence of T−1 in the normal state to antiferromag- are linear in χ in the paramagnetic state. If the main
1
netic fluctuations up to room temperature [9, 10]. We contribution to the magnetic shift arises from the local-
show in both CeRhIn5 and CeCoIn5 that above a tem- ized Ce moments, then Kα = K0,α+βαχα, where βα is
perature T∗ of the order 10K, T−1 is dominated by lo- an effective hyperfine coupling and α = c,ab. Note that
1
2
0.16 1000
8 1)
7 (a) -K 6 (b) 4ξ/a
H || c 20 0.14 -1c 4 2
6 6 0 0.12 -1T) (se 1002 410
K (%)c2 K (%)ab3 K (%) 864 1150 (x10 emu/mχ-3 TT ( sec K)1 000...100086 (T11.50.46001 T-T0N.1 (K) 1021
o
2 2 l) V)1.0
0 1 H0 || ab 00 χ5 (x1100-3 e1m5u/2m0ol2)5 5 0.04 Γ (me0.5
0.02
-2 0 (c)
0
0 10 20 30 40 50 0.00 0.0
0 20 40 60 80 0 20 40 60 80
T (K) T (K) T (K)
FIG. 1: Themagnetic shift of In(1) in CeRhIn (circles) and
5 FIG. 2: (a) T T versus T in CeRhIn and CeCoIn ; the
the bulk susceptibility (lines) versus temperature. The in- 1 5 5
CeRhIn data for T > 20K are taken from [20] (CeRhIn
set shows Kα versus χα. The solid (open) circles and solid NQR (•5), CeCoIn H ||c (solid squares), and H ||ab (open5
(dashed) lines are for H||ab (H||c). squares)). (b) (T T5)−10−111.43sec−1K−1 (•) and0ξ/a (open
1
squares, ref [7]) vs. T −TN in CeRhIn5, where the solid line
isafitasdescribedinthetext. (c)Γvs. T forCeRhIn . The
5
the hyperfine coupling is anisotropic (β = 26.4kOe/µ ,
c B dashedlineisalinearfittothedataabove20K,andthesolid
βab =19.6kOe/µB), and the size of these hyperfine cou- line is taken from [8] using T0 =5K.
plings aremorethananorderofmagnitude greaterthan
typicaldipolarfields(∼1kOe/µ ),sotheIn(1)mustbe
B
coupledtotheCeviaananisotropictransferredhyperfine field Hint at the In(1) site for T << TN is 1.7 kOe [19],
coupling of electronic origin. which corresponds to a value x=0.12.
We postulate thatthe In(1) nucleus is coupledto each The spin lattice relaxationof the In(1) nuclei is deter-
of the four nearest neighbor Ce spins through a tensor mined by the fluctuations of the Ce spins, and is given
A˜ = (cid:8)A||,A⊥,Ac(cid:9), where the principal axes lie parallel by the Fourier component of hH(t)H(0)i at the Larmor
and perpendicular to the Ce-In(1) bond axis in the ab frequency, where the time dependence arises from the
plane, and along the c axis: fluctuations of the Ce spins S . Moriya showed that the
i
spin lattice relaxation rate can be expressed in the gen-
H=γ¯h X I·A˜i·Si. (1) eral form [21]:
i∈n.n.
1 γ2k χ ”(q,ω)
Here γ is the gyromagnetic ratio, I is the nuclear spin = B lim F2(q) ab (2)
of the In(1), Si is the spin on the ith Ce, and the sum T1T 2 ω→0Xq,α α ω
is over the four nearest neighbor Ce sites. Note that
whereαissummedoverthetwodirectionsperpendicular
there should also exist a direct hyperfine coupling be-
to the applied static field. Here F2(q) are form factors,
tween the In nuclei and the conduction electrons. How- α
ever, the fact that T−1 drops by almost two orders of χα(q,ω) is the dynamical susceptibility, and the sum is
1
over the Brillouin zone. The form factors are the spatial
magnitude belowT suggeststhat this coupling is small
N
Fourier transforms of the coupling in Eq. (1), and are
compared to the coupling between the nuclei and the
given by:
Ce moments. For notational simplicity, we define A
ab
caannd sxhoaws tAh|a|,t⊥K=c =Aa4bA(1cχ±c xa)n.d KUasbing=E4qA.abχ(a1b); ounse- Fa2b(q) = 16A2abcos2(cid:16)qx2a(cid:17)cos2(cid:16)qy2a(cid:17) (3)
ing the slopes of Kα versus χα given above, we have 2 2 2 qxa 2 qya
Aab = 4.9kOe/µB and Ac = 6.6kOe/µB. The pa- +16Aabx sin (cid:16) 2 (cid:17)sin (cid:16) 2 (cid:17)
rameter x can be determined from the internal field q a q a
F2(q) = 16A2cos2 x cos2 y . (4)
in the antiferromagnetic state. The moments are lo- c c (cid:16) 2 (cid:17) (cid:16) 2 (cid:17)
calized on the Ce sites, with the structure given by:
S(ri) = S0cos(πx/a)cos(πy/a){cos(q0z),sin(q0z),0}, Since Fa2b(q) >0 for all q, T1−1 will pick up fluctuations
where S0 = 0.74µB, and q0 = 0.2972cπ [16, 17, 18, 19]. atallwavevectors. In particular,the In(1) is sensitive to
For this structure, the internal field is given by: H = the critical slowing down of the spin fluctuations above
int
2AabxS0{sin(q0z),cos(q0z),0}. The measured internal TN.
3
We now can determine quantitatively the low fre-
5
quency spin spectrum of CeRhIn5. As may be seen in 4
Fig. (2), T1T varies quadratically with T for T > 20K, 3
whichiswhatmighthavebeenanticipatedforacollection
2
of independent Ce moments that are weakly coupled to
the conduction electrons via an interaction H = Jσ·S.
1
To see this we note that the susceptibility of the local
moments is given by: TB0 56 100(T
χ (ω)= χ0(T) , (5) Γ/k 4 80 1AT)
L 3 60 F
1−iω/Γ(T) -1
2 40 (se
where χ0(T) is the bulk susceptibility (∼ 1/T for T > 20 c-1
20K), Γ(T)=πJ2N2(0)k T, and N(0) is the electronic K
B 0.1 0 -1
density ofstates atE . FromEqs. (2,3,5)wethen have: 0 10 20 30 )
F
6 T (K)
(T1T)−1 =4γ2kBA2ab(1+x2)χ0(T)/Γ(T)∼T−2. (6) 5 2 3 4 56 2 3 4 56 2 3
0.1 1 10
Fig. (2c)showsΓ(T)asa function oftemperature. Note T/T
0
that Γ ∼ T for T > 20K, with a slope determined by
JN(0)∼0.24. AlthoughINSdataofΓ(T)isunavailable FIG.3: Γ(T)/kBT0 versusT/T0. (◦)CeRhIn5 withT0 =5K,
in this material, Bao and coworkershave estimated that and (solid squares) CeCoIn with T = 17K. The solid line
5 0
Γ<<3meV for T <7K [7]. is from Cox [8]. The inset shows the build-up of antiferro-
We interpret the crossover at 20K to an almost con- magnetic correlations below ∼30K follows the 1/T behavior
stantvalueofΓasreflectingtheonsetofKondoscreening (solid line) described in the text, and extend to Tc = 1.8K,
the transition temperature for the 5T magnetic field used in
of the local moments. The relaxation of independently
this experiment.
screened local moments, a ”Kondo gas”, has been cal-
culated by Cox et al. [22], who find Γ(T) = T0f(T/T0)
where f is a universal function reflecting a temperature
dependent effective coupling, and T0 is proportional to behavior, then (T1T)−AF1 ∼ ξ(T). Fig. (2b) shows
the Kondo temperature. For T > T0, Qachaou et al. (T1T)−1 −(T1T)−01 versus T, where (T1T)0 = 0.00897
have shown that this scaling behavior holds for several sec K. The solid line is a fit to ξ ∼ (T −TN)−ν, where
mixed-valent and heavy electron materials [23]. As may ν = 0.30 ± 0.05. As seen in Fig. (2b), the tempera-
beseeninFigs. (2c)and(3),agoodfittotheexperimen- ture dependence of the NMR measurements agrees well
tal data may be obtained with T0 = 5K over the range with INS measurements of ξ. Note that for a single crit-
8K<T <40K. ical point, a mean field analysis gives ν = β = 1/2
Below T∗ ∼ 8K one begins to see the ”Kondo liquid” (Msublattice ∼ (TN − T)β). The fact that we observe
expected when the interaction between the Ce local mo- ν ∼ 0.3 suggests that either the point TN(H = 0) is a
ments induced by their coupling to the conduction elec- multicritical point, or that fluctuations change the ex-
trons plays a dominant role in the relaxation of the mo- ponents. The former is a plausible explanation for the
ments. Indeed, for T < T < T∗, T−1 shows a strong critical exponent β = 0.25 observed in NMR and ND
N 1
divergence associated with the critical slowing down of measurements [16, 19], and is consistent with the phase
the spin fluctuations of this Kondo liquid. In this region diagram in [24].
the susceptibility takes the form: We next consider the superconductor CeCoIn5. In or-
der to deduce the hyperfine tensor, three independent
χ (q,ω)= αξ2(T)µ2B , (7) experimental quantities are required. From the Knight
AF 1+ξ2(T)(q−Q)2−iω/ω (T) shifts in this material measured in [12] we have A =
sf ab
3.0kOe/µB and Ac = 2.2kOe/µB. As discussed there,
which adequately describes the INS data [7]. Here α is the Knight shift measurements show that for T < 50K
a temperature independent constant, and ωsf ∼ 1/ξz, Ac vanishes whereas Aab does not change; this behavior
where z is the dynamical scaling constant. In the mean is reflected as the anomalous rise in the T1T data be-
field regime, z = 2 and we have (T1T)−1 = (T1T)−01 + low 40K in Fig. (2a) for H0||ab. A direct measurement
(T1T)−AF1, where (T1T)−01 is the q−independent, ”lo- of the parameter x is lacking since there is no magnetic
cal”, contribution assumed constant below T∗, and order [25]. For concreteness we assume x = 0.12, as in
(T1T)−AF1 is the contribution from the antiferromag- CeRhIn5. Fig. (3) shows Γ(T)/kBT0 versus T/T0 for
netic fluctuations. The latter is given by (T1T)−AF1 ≈ CeCoIn5 andCeRhIn5,whereT0 =5KforCeRhIn5,and
γ2A2abx2kBαµ2B/2π2ωsfξ, where we have assumed 3D T0 = 17K, for CeCoIn5. The CeCoIn5 data were deter-
fluctuations, and that the correlation length along the mined using the T1T data for H0||c. For different values
c direction, ξ ∼ ξ ∼ ξ. If we assume mean field of x, the overall magnitude of the Γ(T) data, and hence
c ab
4
T0,willchangebyatmostafactoroftwo,buttheoverall It is also consistent with experiments at fields H > Hc2
temperature dependence will be the same and the con- thatshowthespecificheatincreasesatlowtemperatures
clusions will not change. as though there were an ordering temperature at T ≈ 0
Note that in both materials the data scale with the [2]. The temperature dependence of (T1T)−AF1 is signifi-
Cox formula for T > T∗ ≈ 1.5T0, however below this cant. In the CeRhIn5, (T1T)−AF1,3D ∼ ξ, as expected for
temperature Γ(T) is less than the Cox prediction. As 3D fluctuations. On the other hand, for quasi-2D fluc-
was the case for CeRhIn5, Γ(T) drops markedly below tuations (T1T)−AF1,2D ∼ ωs−f1 ∼ ξ2. In fact, the data in
the Cox ”Kondo gas” prediction below T∗, a change we CeCoIn5 are best understood in a picture of 2D fluctu-
canidentifywiththeonsetofa”Kondoliquid”regimein ations, where ω ∼ T, which is exactly the result ob-
sf
which the interactions between the local moments gives servedin the cupratesabovethe pseudogaptemperature
rise to a new physical regime. We suggest that just as [28]. In light of the behavior observed in CeRhIn5 such
in the Rh material, T∗ in the Co material also marks correlationsprovideanaturalexplanationofthesuppres-
the onset of antiferromagnetic correlations, and proceed sionofΓ,as wellasa possible mechanismforthe d-wave
to analyze the data in CeCoIn5 below T∗ in the same superconductivity [29]. Indirect evidence for such corre-
fashion as in CeRhIn5. lations has been seen in [14]. Measurements of Knight
The inset of Fig. (3) shows (T1T)−AF1 = (T1T)−1 − shiftandT1−1 underpressureandasafunctionofdoping
(T1T)−01 versusT,where we have taken(T1T)0 =0.0793 shouldproveinvaluabletounderstandingtheevolutionof
sec K, the value at 30K. The solid line is (T −T )−1, these antiferromagnetic correlations as the ground state
N
with T = 0. Importantly, this result suggests that in changes from antiferromagnetic to superconducting.
N
the absence of superconductivity CeCoIn5 would order
magnetically atT ∼0, so that in this materialone has a We thank P. C. Hammel, C. P. Slichter, W. Bao, A.
quantum critical point that is obscured by superconduc- Balatsky, C. Varma, J. Lawrence, Z. Fisk, and Y. Bang
tivity[26]. Quantumcriticalityisalsoconsistentwiththe for useful discussions. This work was performed under
non-Fermi liquid behavior seen in the normal state [27]. the auspices of the U.S. Department of Energy.
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