condition (isentropic process), i.e., the thermodynamic
rela-tion (dS/dH)T = —(CH/T)(dT/dH)s holds approximately.
d T /d H < 0 in the helical phase and d T /d H > 0 in the ferromagnetically polarized phase. In 0.09 < ,u0H < 0.13 T, where the metamagnetic anomaly appears, d T/d H has larger negative values. With decreasing temperature, the irreversible heating contribution becomes pronounced, especially in the H regions where the metamagnetic anomaly appears. For T 1.5 K, the heating contribution masks the intrinsic cooling behavior in the H increasing process, while it cooperatively enhances the intrinsic heating behavior in the H decreasing process. For T 0.2 K, the irreversible heating at the metamagnetic transition (probably due to magnetic domain wall motion) becomes more dominant; the heating behavior appears in both sweeping processes and the intrinsic behavior of the isentropic process is no longer visible. From the H shift of the heating region between H increasing and decreasing processes, we infer that the hysteretic behavior of the order of
—0.02 T appears in the metamagnetic transition at 0.2 K.
From these results, the helical phase boundary in the H-T phase diagram has been determined as shown in Fig. 4(a). The critical field HM at T = 0 K is estimated to be 0.14 T.
0.2
E ,t ;I
t
I ~
.%
0.15
0.1
0.05
0
Helical phase
~LuN13A19
H YbNi3A19 H/I a
0 0.1
u0H(T)
10
FIG. 5. (Color online) H dependence of y. The dashed line is a guide to the eye.'
0.15
0.1
0.05
0~
2.5 r
(a)
^,.
Anomaly in MCE
Herical phase
YbNi3A19 H/I a
1 2 3 4
2
p 1.5
0.5
(b)fr
YbN i3A I9IE
H11 a! 4
I MF calculation f~
A
Schottky peak in C iAHerical phase
En _a_ . _• / transition in C
0 1 2 3
T (K)
4 5 6
FIG. 4. (Color online) H-T phase diagram for low fields (a) and high fields (b) determined by specific-heat measurements (A: Schottky peak in the FM phase, •: helical phase transition) and MCE (^). Dashed line shows a MF model calculation for the specific-heat peak position Tp(H) (see text).
IV. DISCUSSION
In the measured temperature range, C of YbNi3A19 has con-tributions from electrons (Cei = y T + C4 f), phonons (Cph = fT3), and nuclear spins (Cnuc = A/T2). Measured specific heat of LuNi3A19 can be well expressed as C = YL„T + I8L„T 3, with yLu = 6.4 mJ/K2 mol and /3Lu = 0.35 mJ/K4 mol at low temperatures." We tentatively use F'Lu for /3 of YbNi3A19. Cph has only a minor contribution to the total C in T < 10 K, as shown in Fig. 2 by a thin solid line. Cnuc appears as an upturn below —0.4 K for all H data. C4 f, representing contributions from 4f electrons of the Yb ions, can be phenomenologically expressed as BT" exp(— A / T) at low T below —2 K.16 The C(T) data in T < 2 K have been fitted by a sum of these terms and the H dependence of y is obtained as displayed in Fig. 5.
In zero field, y = 110 mJ/K2 mol 17yLu. This fact indicates that the quasiparticle mass is significantly enhanced even in the helical magnetic state. Note that y decreases gradually with increasing H without showing any noticeable anomalies across the phase boundary of the helical magnetic state, indicating that the mass enhancement mechanism is not directly associated with the HM ordering or HM fluctuations.
Using the determined values of A(H), Cei(T, H) data have been obtained. The electronic entropy Sei calculated using the
Cei(T,H) data is shown in Fig. 6. Sei is 4.2 J/K2 mol at T THM and reaches R ln 2 at —8.5 K. This is consistent with the fact that the valence of Yb ions is almost 3+,17'18 and the J = 7/2 multiplet of the Yb ions splits into four doublets due to the CEF effect (the site symmetry of the 6c site is C3) and the mag-netic behaviors in T < 10 K are dominated by the CEF ground-state doublet. The entropy released above THM, i.e., R ln 2 — Sel(THM), is attributable to the Kondo effect (TK 3 K)11 and/or the magnetic short-range ordering (ferromagnetic in the layers and helical between the layers).
In the field-induced ferromagnetic (FM) phase (H > HM), the peak in C/ T becomes broader and shifts to higher temperatures with increasing H, as shown in Fig. 2. This peak results from the Schottky-type thermal excitations between the two Zeeman-split energy levels of the CEF ground-state doublet; excitations to the first excited CEF level can only be seen above 7 K as a slight increase in C in zero field.
The peak height Cpeak shown in Fig. 7 is much higher than 3.65 J/K mol, which is expected for a doublet with a fixed energy separation, and depends on H significantly. This
155106-3
RYOICHI MIYAZAKI et al. PHYSICAL REVIEW B 86, 155106 (2012)
0 E
ti
6
5 4
3 0.5 T 1T 2
1
0
0 2 4
T (K) Sph
6 /2 T
4 T/
YbNi3A1 H 1I a
8 10
FIG. 6. (Color online) The temperature dependence of the elec-tronic entropy Sei. For comparison, the lattice contribution Sph is drawn by a thin line.
behavior indicates pronounced ferromagnetic interactions among Yb ions.
In the paramagnetic (PM) or field-induced FM phase, we analyze the C(T, H) data taking into account the ferromagnetic interactions in a mean-field (MF) approximation. In our model, the Hamiltonian can be expressed as
= —gliBs(H + Ag,uBg) + 2.(giuBg)2, (1)
where a Yb magnetic moment is represented by gµBs using an in-plane pseudospin s (g: thermal average), effective g factor, and the Bohr magneton µB. The MF coefficient X is described as X = 2JFM/(gp,B)2 using the exchange integral JFM. When T is decreased, g(T,H) shows a significant development at a certain T range depending on the applied H. Because the energy separation of the CEF ground-state doublet (a H + ,lg aBg) develops accordingly, this behavior
0 ti
12
10
8
6
4
2
0 0
Helical phase
0.1
,u,H (T)
10
FIG. 7. (Color online) H dependence of the Schottky peak height Cpeak. A mean-field model calculation is drawn with a dashed line.
Note that the enhancement in Cpeak (>3.65 J/Kmol) is due to pronounced FM interactions.
results in the enhancement in Cpeak. The parameter set A and g have been determined so that the MF calculations reproduce reasonably well the Schottky peak position Tp(H) in the H-T phase diagram and the value of magnetization. The best fit has been obtained with A = 1.466 T/AB and g = 3.6, and the MF calculation of Tp(H) is shown in Figs. 4(a) and 4(b).
In the H-T phase diagram of Fig. 4, the Schottky peak position is nicely reproduced by the calculated Tp(H) in the PM phase. In the H —± 0 limit, the MF-calculated Tp(H) provides a fictitious FM transition temperature TFM = 3.18 K in zero field. The H dependence of Cpeak shown in Fig. 7 is also qualitatively reproduced. In YbNi3A19, however, the g value should depend on H to some extent, since the CEF excited levels mix gradually into the ground-state doublet due to the Zeeman effect. Deviations from the model calculation visible in Figs. 4 and 7 may be partly due to the simplification of the constant g. Another factor neglected in this model is the Kondo effect. According to the exact solution for the s-d impurity model,19 Cpeak is suppressed below 3.65 J/Kmol in zero field and it increases gradually with increasing H, approaching 3.65 J/K mol for g p,B H/ kB TK >> 1. The slow decrease in Cpeak in high fields shown in Fig. 7 might be due to combination of the Kondo effect and the FM interactions.
The helical magnetic ordering indicates the existence of mutually competing interlayer magnetic couplings. We use a simple model-° which includes Jo (> 0), J1 and J2 representing exchange constants between magnetic moments in a FM Yb plane, with the adjacent Yb planes and with the next-nearest Yb planes, respectively. This model has an energy minimum solution with an HM structure with an interlayer magnetic moment turn angle 0 (cos 4) = —J1/ 4.I2) given by
Eex = —s2(Jo + J1 cos 4) + J2 cos 20). (2) The propagation vector q = 0.8 x c* determined by the neutron-scattering study 1 3 corresponds to 0 = 96°. Inserting this value into Eq. (2), the effective exchange integral for the HM phase is given by JHM - Jo — 2.43 Jl . In the field-induced FM phase (4) = 0°), the effective exchange integral is given by JFM - Jo + 3.38J1. From THM : TFM = 3.41 K:
3.18 K =J11M:JFM, the ratio Jo:J1:J2=1:-0.01:
—0.03 is obtained. Significantly weak interlayer magnetic couplings, reflected in the 2 orders of magnitude smaller values of J1 and J2 than that of J0, is consistent with the fact that Yb layers are largely separated; note that the strength of Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction decays with the distance as 1/ r3.
Normalized magnetoresistance p(H)/p(0 T) for j II c and H II a (shown in Fig. 8) drops about 30% at HM, which is in marked contrast with the absence of any signature in y at the HM boundary. Based on this fact, we consider a simple two-carrier model, in which the Fermi sheets are separated into two parts FS1 and FS?. FS, disappears in the HM phase due to the nesting with the propagation vector q. In the nonordered phase, it is expected that FS, plays a considerable role in the electric transport along the c axis, since it should have a rather flat surface perpendicular to the c axis.21 FS2 carries heavy quasiparticles contributing dominantly to the y value.
In the Drude picture, the contribution from FS, to the electric conductivity can be expressed as cri = nie2ti/m*, where ni, ti, and m7 represent the carrier density, the relaxation time,
155106-4
HEAVY QUASIPARTICLES FORMED IN THE ... PHYSICAL REVIEW B 86, 155106 (2012)
F= o _ Q -Q
1.2 1.1 I 0.9 0.8 0.7 0.6 0
Helical phase
YbNi3A1 T=2 K H I/ a j//c
0.05 0.1 0.15 0.2 0.25 0.3 u (T)
FIG. 8. (Color online) H dependence of normalized magnetore-sistance p(H)/ p(0 T). The dashed line is a guide to the eye .
and the effective mass of FS,, respectively. From the 30%
drop in p(H)/p(0 T) at HM, (n2/m2)/(n n2/m2) = 0.7, tentatively assuming rl = r2. Because y does not show any noticeable increase at HM within the experimental ac-curacy, (n lm n2m2)/n2m2 < 1.02. These two equations yield m2/ml > 4.6 and n2/ni > 11, indicating that the heavy quasiparticles are formed mainly on FS2 not on FS1. Note that if r1 < r2 is assumed, taking into account that FS1 is a
"hot sheet" ass
ociated with HM fluctuations, the lower bound value of m2/ntt becomes even larger. This anisotropic mass enhancement is consistent with recent dHvA measurements, in which the cyclotron effective masses of mc*/mo = 3-12 and 1 -5 (mo: the free electron mass) have been observed for H c and H1c, respectively.14 The present model is also consistent with the fact that p(H) data for j II a with H I) b (not shown) does not show a noticeable jump at HM, since FS1 does not carry current along the a axis.
The realization of the helical magnetic ordering in YbNi3A19 is caused by the weak but competing interlayer
antiferromagnetic couplings (Ji , J2 < 0). Such magnetic frus-tration may be able to cause quasiparticle mass enhancement, as discussed for LiV2O422 and for geometrically frustrated systems.23 However, since the observed heavy quasiparticles do not reside in F51, which is responsible for the realization of the helical magnetic ordering, such a scenario is unlikely in YbNi3A19.
The present findings suggest that the heavy quasiparticles are bound in the 2D Yb ferromagnetic layers (on part of FS2). Heavy quasiparticles formed in ferromagnetic states have been reported so far in CeRu2Ge2 (y = 20 mJ/K2 mol, ferro-magnetic transition temperature TFM = 8 K),24 CeRuPO (y = 77 mJ/K2 mol, TFM = 15 K),25 CeAgSb2 (y = 65 mJ/K2 mol, TFM = 9.6 K),26 SmOs4Sb12(y = 820 mJ/K2 mol, TFM = 3 K),27 UIr2Zn20 (y = 450 mJ/K2 mol, TFM = 2.1 K),25 UGe2 (y = 35 mJ/K2 mol, TFM = 52 K, superconducting transi-tion temperature Tsc - 1 K at 1.3 GPa),29'30 URhGe (y = 164 mJ/K2 mol, TFM = 9.5 K, and Tsc = 0.27 K),31.32 and UCoGe (y = 57 mJ/K2 mol, TFM = 3 K, Tsc = 0.8 K).33 As the crystal structures of these compounds suggest, all of their electronic states have 3D characters. Even in such 3D systems, mechanisms of quasiparticle mass enhancement (or magnetic moment screening due to the Kondo effect) in FM states remain to be elucidated, not only experimentally but also theoretically.34 To our knowledge, YbNi3A19 is probably the first realization of the 2D version of such a system. We believe that YbNi3A19 will provide an unparallel opportunity to investigate 2D heavy quasiparticles in FM layers.
ACKNOWLEDGMENTS
This work was supported by Grants-in-Aid for Sci-entific Research on Innovative Areas "Heavy Electrons"
(20102007,A01-23102712) from MEXT and (C: 23540421) from JSPS.
*[email protected] taoki@tmu .ac.jp
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16Note that the obtained value of y depends on the functional form of C41. The data shown in Fig. 5 have been obtained with setting A = 0. If A is included in the fitting parameter set, y = 140 and 44 mJ/K2 mol for 11,0H = 0 and 5 T is obtained, respectively.
Nevertheless, the overall behavior of y vs H does not change
155106-5
RYOICHI MIYAZAKI et al. PHYSICAL REVIEW B 86, 155106 (2012) and no noticeable jump still appears at HM. In JL0 H 2r 4 T, C4f
is strongly suppressed and the yT term dominates around 1 K, where C/T appears to be almost T independent, as shown in Fig. 2. Thereby, the y value can be obtained more accurately as reflected in the smaller error bar. In p0H = 5 T, yy,/yL„ ti 17.
This value is reasonable since the ratio of the dHvA cyclotron effective mass mc*(Yb)/mc*(Lu) = 3-25 (depending on the branch) has been observed in this field region.14
17Y. Utsumi, H. Sato, S. Ohara, T. Yamashita, K. Kimura, S. Motonami, K. Shimada, S. Ueda, K. Kobayashi, H. Yamaoka et al., Phys. Rev. B (to be published).
18This feature is in line with the fact that YbNi
3A19 does not show any noticeable quantum critical behaviors. In contrast, YbA1B4 shows pronounced quantum critical behaviors and they are considered to be associated with the strong valence fluctuation of Yb ions.
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155106-6
International Conference on Strongly Correlated Electron Systems (SCES 2011) IOP Publishing Journal of Physics: Conference Series 391 (2012) 012046 doi: 10.1088/1742-6596/391 /1/012046
Ru su bstitution effect on the peak effect in superconducting PrOs4Sb12
Ryoichi Miyazaki', Ryuji Higashinakal, Yuji Aoki', Hitoshi Sugawara2 and Hideyuki Sato'
'Department of Physics
, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan 2Department of Physics
, Kobe University, Kobe 657-8501, Japan E-mail: ryoichi -miyazaki@ed . tmu . ac . j p , aoki@tmu . ac . j p
Abstract. Ac susceptibility (xac) measurements on Pr(Osi_xRux)4Sb12 single crystals for x = 0.05 and 0.1 were performed by mutual inductance method. A peak structure caused by anomalously enhanced flux pinning force, which is so-called the peak effect, appears in the H dependence of xac. The peak structure shifts to lower fields as T increases and disappears in T > 1 K in both samples, while in PrOs4Sb,2 it is observable up to near Tc. This fact indicates that the Ru substitution suppresses the peak effect. We demonstrate that the observed T dependent behavior of the peak structure can be explained roughly by the synchronization model although there remains deviation from the model curve, suggesting some modification may be needed in the model to be applied to Pr(Os,_xRux)4Sb12.
1. Introduction
Since the filled skutterudite compound PrOs4Sb12 was reported to be the first heavy fermion (HF) superconductor among Pr-based compounds, PrOs4Sb12 has been attracting much attention [1]. The superconducting transition temperature 71 = 1.85 K being higher than TT = 0.74 K of no-4f-electron reference LaOs4 Sb12 [ 1, 2] demonstrates clearly the involvement of Pr 4f no-4f-electrons in the superconductivity. Actually, several measurements performed so far suggest that it is of unconventional type. The Sb NQR spin-lattice relaxation rate indicates strong-coupling isotropic superconducting gap [3]. The existence of multiple superconducting phases with point nodes is indicated by angle-resolved thermal conductivity measurements [4], although angle-resolved specific heat suggests fully opened gap [5]. The appearance of spontaneous internal fields is indicated by zero field p,SR measurements [6].
Considering singlet-triplet crystalline electric field (CEF) level scheme of Pr 4f electrons[7], pairing mediated by quadrupole excitons has been discussed [8].
Another feature that characterizes the unconventional superconducting state in PrOs4Sb12 is the so-called "peak effect", i.e., anomalously enhanced flux pinning force in the mixed state below the upper critical field Hc2. An anomalous irreversible behavior is observed in the magnetization curve in the mixed state [9], a considerable peak structure in the field dependence of the surface impedance [ 10], and minima in the H dependence of the flux-flow resistivity [11]. Similar anomalies have been observed in many of the HF superconductors [ 12-14] and therefore it is expected that the observed peak effect in PrOs4Sb12 reflects the involvement of the heavy quasiparticles in the HF superconductivity . In this paper, we report the Ru substitution effect on the peak effect of PrOs4Sb12 by ac susceptibility measurements.
Since the unconventional HF and conventional superconducting states appear at the opposite ends in the
Published under licence by IOP Publishing Ltd I
International Conference on Strongly Correlated Electron Systems (SCES 2011) IOP Publishing
Journal of Physics: Conference Series 391 (2012) 012046 doi:10.1088/ 1742-6596/391 / 1 /012046
Pr(Osi_xRux)4Sb12 series with continuous Tc variation in between [15-17], Pr(Osi_xRux)4Sb12 is an appropriate system to study how the anomalous flux pinning depends on the type of superconductivity.
2. Experimental
Ac susceptibility has been measured by mutual inductance method with ac field Hac 0.3 Oe down to 0.1 K using a dilution refrigerator equipped with a 8-T superconducting magnet. Single crystals of Pr(Osi_xRux)4Sb12 with x = 0.05 and 0.1 were grown by Sb self flux method. Obtained single crystals with the size of about 1 x 1 x 2 mm3 exhibit sharp superconducting transitions in T dependence measurements under zero field, indicating good sample quality.
3. Results and discussion 3.1. Temperature dependence
Figure 1 shows the T dependence of xac (= x' + ix") in zero field, where x' and x" correspond to the real and the imaginary parts of xac, respectively. x' represents the equilibrium differential susceptibility
x' =dM/dH while x" represents the energy dissipation W = irx"Hac. In both samples, x' decreases below the superconducting transition temperature TT — 1.81 K (x = 0.05) and 1.76 K (x = 0.1), respectively. Broad peaks also appear in the T dependence of x" just below T, in both samples.
•
crl
1 1.2 1.4 1.6
T (K)
Pr(Osi _zRuu)4Sb WI= OT
1.8 2
Figure 1. Temperature dependences of xac of the single crystals of Pr(Osi_xRux)4Sb12 with x = 0.05 and 0.1 measured in zero field. x' and x" represent the equilibrium differential susceptibility and the energy dissipation, respectively.
3.2. Field dependence
The H dependences of xac at several temperatures are shown in fig.2. Specific peak structures indicating the peak effect are observed in x' (H) just below Hc2 in both samples. With increasing T, the peak structures shift to the lower fields and becomes suppressed gradually. The H dependence of x" also shows a broad peak at the field where x' shows the peak structure. These behaviors of xac in the Ru doped samples are similar to those of PrOs4Sb12 [18]. However, while the peak structure remains up to T ti Tc in PrOs4Sb12, the peak structure is not visible in x'(H) and x" (H) for T > 1 K in x = 0.05 and 0.1. To analyze the feature of the peak structures quantitatively, Hc2, Hmin and Hpk representing the kink structure just below Hc2, the local minimum of x'(H) and the local peak in x'(H) are determined respectively by using field derivative dx'/dH.
2
International Conference on Strongly Correlated Electron Systems (SCES 2011) IOP Publishing
Journal of Physics: Conference Series 391 (2012) 012046 doi : 10.1088/1742-6596/391/1/012046
ct
0 0.5
1 1.5
/10H (T)
2
cA
4, czt
0 0.5
1
,uHH (T)
1.5 2