Anomalous quantum criticality in an itinerant ferromagnet

Abstract

The dynamics of continuous phase transitions is governed by the dynamic scaling exponent relating the correlation length and correlation time. For transitions at finite temperature, thermodynamic critical properties are independent of the dynamic scaling exponent. In contrast, at quantum phase transitions where the transition temperature becomes zero, static and dynamic properties are inherently entangled by virtue of the uncertainty principle. Consequently, thermodynamic scaling equations explicitly contain the dynamic exponent. Here we report on thermodynamic measurements (as a function of temperature and magnetic field) for the itinerant ferromagnet Sr_1-x Ca_x RuO₃ where the transition temperature becomes zero for x=0.7. We find dynamic scaling of the magnetization and specific heat with highly unusual quantum critical dynamics. We observe a small dynamic scaling exponent of 1.76 strongly deviating from current models of ferromagnetic quantum criticality and likely being governed by strong disorder in conjunction with strong electron-electron coupling.

ARTICLE

Received 11 Sep 2014 | Accepted 27 Jul 2015 | Published 8 Sep 2015

C.L. Huang1,2,w, D. Fuchs1, M. Wissinger1, R. Schneider1, M.C. Ling3, M.S. Scheurer3, J. Schmalian1,3

& H.v. Lhneysen1,2

The dynamics of continuous phase transitions is governed by the dynamic scaling exponent relating the correlation length and correlation time. For transitions at nite temperature, thermodynamic critical properties are independent of the dynamic scaling exponent. In contrast, at quantum phase transitions where the transition temperature becomes zero, static and dynamic properties are inherently entangled by virtue of the uncertainty principle. Consequently, thermodynamic scaling equations explicitly contain the dynamic exponent. Here we report on thermodynamic measurements (as a function of temperature and magnetic eld) for the itinerant ferromagnet Sr1 xCaxRuO3 where the transition temperature becomes zero for x 0.7. We nd dynamic scaling of the magnetization and specic heat

with highly unusual quantum critical dynamics. We observe a small dynamic scaling exponent of 1.76 strongly deviating from current models of ferromagnetic quantum criticality and likely being governed by strong disorder in conjunction with strong electronelectron coupling.

DOI: 10.1038/ncomms9188

Anomalous quantum criticality in an itinerant ferromagnet

1 Institut fr Festkrperphysik, Karlsruher Institut fr Technologie, 76021 Karlsruhe, Germany. 2 Physikalisches Institut, Karlsruher Institut fr Technologie, 76128 Karlsruhe, Germany. 3 Institut fr Theorie der Kondensierten Materie, Karlsruher Institut fr Technologie, 76128 Karlsruhe, Germany. w Present address:

Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany. Correspondence and requests for materials should be addressed to C.L.H. (email: mailto:[email protected]

Web End [email protected] ) or to J.S. (email: mailto:[email protected]

Web End [email protected] ) or to H.L. (email: mailto:[email protected]

Web End [email protected] ).

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SrRuO3 is a three-dimensional itinerant ferromagnet with the Curie temperature TCE160165 K (refs 14). It crystallizes in an orthorhombic perovskite structure with

tilted RuO6 octahedra. In the 4d4 conguration of Ru4 , the t2g levels are occupied in a low-spin S 1 state5. Ferromagnetism of

SrRuO3 can be suppressed by applying hydrostatic pressure, down to TC E50 K at a pressure of 17 GPa6. Alternatively, TC is suppressed all the way to absolute zero on substitution of Sr2 by the smaller Ca2 ions at a critical concentration xcE0.7, where

Sr1 xCaxRuO3 (SCRO) turns from a ferromagnetic (FM) to a

paramagnetic ground state. Whether a quantum critical point (QCP) may be present at xc is a matter of current debate2,4,7,8.

While specic heat3 and NMR4 measurements both concluded that the self-consistent renormalization theory9, equivalent to the HertzMillis model10,11, could describe the underlying physics of an FM QCP in SCRO, recent Kerr effect measurements on a composition-spread epitaxial lm showed that a possible quantum phase transition (QPT) around a (reduced) critical concentration xc were smeared by disorder originating from the difference of ionic radii between Sr and Ca ions8. In addition, on the basis of muon-spin-rotation (mSR) experiments, it was argued that a spontaneous phase separation may be a common feature in FM systems near their QPTs, leading to a suppression of dynamic critical behaviour7,12. From the theoretical point of view, the conventional HertzMillisMoriya model911 predicts, for three-dimensional itinerant magnets, dynamical critical exponents z 2

for antiferromagnets, as well as z 3 for clean and z 4 for

disordered ferromagnets10. While z 2 has been observed in

some antiferromagnets with spin-density-wave order13, there are important examples for deviations from the expectation for ferromagnets. Indeed, the model breaks down in the FM case,

because extra singular terms arising from fermionic modes, in addition to the order-parameter uctuations, lead to multiple time scales1416. Moreover, in some composition-driven QPTs, close to the critical concentration xc, disorder effects introduce additional uctuations, giving rise to a nonanalytic contribution to the free energy between the paramagnetic and locally FM-ordered regions, known as the Grifths rare regions1719.

In the following, we report on a consistent dynamical scaling analysis of the magnetization and specic heat for an x 0.7

sample, bearing all features of a QCP with, however, very unusual critical exponents. In particular, we observe a small dynamic critical exponent z 1.76, which strongly deviates from current

models, and may arise from an inhomogeneous electron liquid due to the strong disorder induced by Ca substitution.

ResultsMagnetic susceptibility and specic heat of Sr0.3Ca0.7RuO3. As

an overview, we show in Fig. 1a the magnetic d.c. susceptibility w M/B, where M is the magnetization and B is the magnetic

eld, and its reciprocal value w 1 versus temperature T for1.8 KrTr50 K. While w(T) increases with decreasing T down to the lowest temperature of 1.8 K, w 1(T) shows initially a downward curvature and levels off toward lower T. Notice the gradual change of w 1(T) from negative to positive curvature with decreasing temperature. The specic heat C(T) of Sr0.3Ca0.7RuO3

is shown in Fig. 1b in a plot of C/T versus T2. Between 20 and30 K, the specic heat can be described by C/T gh bphT2,

where the subscript h denotes the extrapolation of C/T from high temperatures to T 0 and bphT3 is the usual low-T Debye

phonon contribution (see solid line in Fig. 1b). Below E15 K, a positive deviation of C/T from the C/T versus T2 line is observed with a maximum of C/T for T-0, that is, a zero-temperature cusp. This deviation has been found before and was attributed to spin uctuations of Ru4 moments3. However, FM uctuations should lead to a divergence of C/T at xc for T-0, for example, C/TBlog(T0/T). This is not observed in Sr0.3Ca0.7RuO3 as will be discussed in more detail below.

Figure 2a shows that the magnetic susceptibility w(T) increases more slowly from T 30 to 1.8 K in weak elds. The divergence

of w(T) continuously weakens towards lower T, and is increasingly suppressed already in weak magnetic elds. Figure 2b shows that, correspondingly, the magnetic isotherms M(B) increase less rapidly with increasing T. The smooth curvatures of w(T, B)

and M(T,B) suggest the applicability of a scaling relation. We therefore assume that the x 0.7 sample is close to a QCP, see

inset of Fig. 1b, as will be substantiated below.

Quantum critical scaling of thermodynamic properties. At a QCP with hyperscaling, that is, below the upper critical dimension, the scaling relation for the critical part of the free energy

F T; B

reads

F T; B

b d zF bzT; bbd=nB

a

Sr0.3Ca0.7RuO3

0.4

0.2

0.0 0 10 20 30 40 500

100

3 )

3 mol1 )

(cm

1 (mol cm

50

T (K)

T 2 (K2)

b

C/T(J mol1 K2 )

0.20

0.15

0.10

0.05

0.00 0 200 400 600 800

200

0 0.0 0.5 1.0

x

; 1 where b is an arbitrary scale factor and z and bd/n are scaling exponents associated with the tuning parameters T and B, respectively. As usual, n is the correlation length exponent that can be obtained from an analogous scaling relation involving the distance r xc x from the critical concentration (x / r

j j n). d

and b describe the eld and concentration dependence of the order-parameter M(r 0, T 0, B)pB1/d and M(r, T 0, B 0)

prb, respectively. For the eld and temperature dependence of the magnetization M follows:

M T; B

@F

150

T c(K)

100

50

Figure 1 | Low-temperature magnetic susceptibility v and specic heat C for Sr0.3Ca0.7RuO3. (a) Left axis shows w and right axis shows w 1.

The data were measured in the eld-cooled mode at B 10 mT.

(b) Zero-eld C/T versus T2. The solid line represents the t ofC/T gh bphT2 between T 20 and 30 K with gh 0.064 J mol 1 K 2

and bph 1.3 10 4 J mol 1 K 4. The inset shows TC as a function of

Ca concentration x (ref. 21).

@B bbd=n d zM bzT; bbd=nB

: 2

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a

B (T ) =

Sr0.3Ca0.7RuO3

8 )

9.69.28.88.4

0.65 0.70 0.75

d/z 1

r.m.s. (10

3 mol1 )

C/T(J mol1 K2 )

0.1

(cm

0.011 3 10 30

0 1 2 3 4 5 6 7 B (T )

T (K)

0 5 10 15 20

T (K)

b

Figure 3 | Zero-eld-specic heat DC of Sr0.3Ca0.7RuO3. The phonon

contribution has been subtracted. The solid line represents a t ofDC/T g0 aT0.7 between 1.8 and 13 K. The inset shows how the quality of

the t varies with (d z)/z in equation (5) by checking the smallest mean

square deviation wr.m.s..

0.10

0.08

0.061 3 10 30 T (K)

0.2

0.1

0.0

10 K

T = 1.8 K 30 K50 K

Sr0.3Ca0.7RuO3

0.01

0.1

0.5 12 47

20 K

M ( Bf.u.1 )

a

Sr0.3Ca0.7RuO30

C/T(J mol1 K2 )

(C e+ C m)/Td/z (mJ mol1 Kd/z1 )

8 T

Figure 2 | Magnetic susceptibility v and magnetization M of Sr0.3Ca0.7RuO3. (a) Field-cooled w versus temperature T measured at several different magnetic elds. (b) Magnetic isotherms measured at several different temperatures. Data are shown in part only for clarity.

0.10

0.08

0.06

With the choice of bzT T0 with the cutoff energy kBT0, we getM T; B

Tb=nzF B=Tbd=nz

: 3 Hence, M/Tb/nz should be a function of B/Tbd/nz. Our data obey the scaling behaviour (Equation (3)). As discussed in the Supplementary Note 1, the smallest error bars occur when the ratio of the two exponents b/nz and bd/nz is close to 1.6. This determines the exponent dE1.6 in good agreement with a previous estimate by Itoh et al.20 To determine the other exponents, we rst resort to quantum critical scaling properties of the specic heat that allows an unequivocal determination of d/z and bd/nz.

Then we will return to a discussion of the magnetization data.

The scaling behaviour of the critical part of the specic heat is given by

Ccr T; B T

@2F

b

0

d/z = 1.7, / z = 1.95

B/T / z(T/K / z)

2

/ z

4

1.8 2.0

1.5

1.0

r.m.s.(10

6

8 )

8

10 1E3 0.01 0.1 1

@T2 b dCcr bzT; bbd=nB

: 4

Putting again bzT T0 yields

Ccr T; B Td=zC B=Tbd=nz

Figure 4 | Critical contribution to the specic heat C. (a) DC (phonon contribution subtracted) of Sr0.3Ca0.7RuO3 plotted as DC/T versus logT at elds B 0, 0.5, 1, 2, 3, 4, 6 and 8 T from top to the bottom. (b) Scaling of

the eld-dependent specic heat data from (a). The inset shows how the quality of the scaling collapse varies with bd/nz (with xed value d/z 1.7)

by checking the smallest mean square deviation wr.m.s..

: 5 As long as d4z, this critical contribution at zero eld is subleading to the quasiparticle contribution to the specic heat. Nevertheless, equation (5) allows an unequivocal determination of d/z from the zero-eld-specic heat, that is, Ccr(T, 0) Td/zC(0).

Figure 3 shows that DC/T C/T bphT2 (with the phonon

contribution subtracted) is well described by DC/T g0 aT0.7

with g0 106 mJ mol 1 K 2 and a 6.0 mJ mol 1 K2.7 between

1.8 K up to 13 K. Hence, the subleading contribution Ccr/TpT0.7

to the specic heat is clearly visible in the zero-eld data. The inset of Fig. 3 displays the root-mean square (r.m.s.) deviation wr.m.s. versus d/z 1 yielding d/z 1 0.700.04 (see

Supplementary Fig. 1). The striking result following immediately is that the dynamic critical exponent for d 3 is z 1.760.04.

Turning to the eld dependence of the specic heat, Fig. 4a shows C(T,B) plotted as DC/T versus logT. With increasing eld, the cusp of C/T for T-0 (see Fig. 3) becomes rounded and, for BZ1 T, DC/T even falls off slightly towards the lowest measuring temperature. The gradual decrease of DC/T from B 0 to 8 T

amounts to only 15% in this T range. To check for QCP

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temperature-eld scaling of the specic heat, we plot (C(T, B)

C(T,0))/Td/z versus B/Tbd/nz in Fig. 4b. The zero-eld contribution is subtracted to eliminate the non-critical quasiparticle contribution to C. With d/z 1.70 as determined above, we

nd very good scaling over more than three orders of magnitude in B/Tbd/nz with bd/nz 1.950.1 (Supplementary Fig. 1). The

inset of Fig. 4b shows a plot of wr.m.s. versus the tting exponent

bd/nz. Due to the small critical contribution to C, the scatter of the scaling plot is somewhat larger.

Critical scaling of the susceptibility and magnetization. We are now in the position to obtain the quantum critical scaling plots for the susceptibility and magnetization data. Figure 5a,b shows the corresponding plots obtained, respectively, from the w(T, B)

data of Fig. 2a measured as a function of T for different elds and from the M(T, B) data of Fig. 2b measured as a function of B for different temperatures. Here we have omitted the data of very low elds (Br0.1 T of Fig. 2a) where scaling is not expected because of domain effects. Very good scaling over more than three orders of magnitude of B/Tbd/nz is found for both data sets with b/nz 1.20.05 using bd/nz 1.95 determined from the specic

heat (Supplementary Fig. 2 and Fig. 3). Again, the determination of b/nz is corroborated by wr.m.s. minima shown in the plots of

Fig. 5a,b. Our result for b/nz slightly larger than unity is consistent with the experimental observation that TC(r)prnz is

roughly proportional to the low-T magnetizationprb (ref. 21). The comparatively small value d 1.630.15 that emerges

embodies the soft divergence of the susceptibility mentioned earlier. Remember that d 3 is the mean-eld value for the

exponent, a value that is usually enhanced via critical uctuations. Hence, the anomalously small dynamic exponent z obtained from the specic heat results is accompanied by an equally unusual

small exponent d. Using the usual scaling laws, our results further imply a rather large value of the exponent Z E0.73 that determines the spatial/momentum variation of the susceptibility w(q)pq (2 Z) and gives evidence for highly non-local quantum uctuations.

DiscussionThe consistent dynamic scaling analysis of magnetization and specic heat of SCRO at the QCP xc 0.7 raises questions to the

origin of this highly unusual behaviour. The very observation of scaling of the susceptibility suggests that we are below the upper critical dimension. Together with the numerical value for z, this fact implies that the universality class of the transition cannot be of the conventional j4 type, where d z44 would place the

system in the mean-eld regime. In addition, the negative sign of

the critical contribution suggests that one cannot interpret Ccr(T)

as specic heat of isolated collective degrees of freedom but, rather, that the critical dynamics is intertwined with the dynamics of quasiparticles. One possible interpretation is clearly related to the emergence of inhomogeneities. Scale-dependent strain elds, caused by the different values of the ionic radii of Sr and Ca can lead to modied effective exponents22. The mSR relaxation rate

T1 1 of SCRO for x 0.7 was found to exhibit at low T a

behaviour T1B constant12. A scaling analysis leads to T1pTy with yz (d 1)b/n d, which for our data yields yD 1. Given

this nding it is important to further experimentally investigate T1 in the low-temperature regime.

The key observation of this paper are the small values of the exponents d and z, if compared with the mean-eld

HertzMoriyaMillis theory10 for clean (d 3, z 3) or weakly

disordered (d 3, z 4) systems. Important deviations from

mean-eld behaviour, based on nonanalytic corrections to Fermi-liquid theory23, have been investigated in the vicinity of a ferromagnetic critical point and yield weak rst-order transitions2325, instead of fundamentally new critical exponents that are found experimentally in SCRO. Different values for critical exponents have in fact been discussed as a result of the vicinity to a quantum tricritical point26 or due to preasymtotic critical behaviour in disordered systems27. Both approaches yield values for d rather close to our ndings.

However, the substantially different value of the dynamical critical exponent found here cannot be explained in refs 26,27.

The small value of z seems, however, consistent with a strong-coupling perspective of the coupling between collective magnetic degrees of freedom and incoherent quasiparticle excitations. Away from the critical point, it was shown28,29 that the appropriate description of metallic quantum ferromagnets is in terms of quasiparticles that interact with the coherent magnetization motion via an abelian gauge eld. For such a gauge coupling between critical degrees of freedom and the quasiparticle excitations, one would naturally expect a reduced value of the quasiparticle damping and, thus, of the dynamic critical exponent z. Naively, this is due to the fact that in the strong-coupling regime the magnitude of the magnetization is expected to be large even at the QCP, while long-range order is destroyed via directional uctuations. Hence, the coupling to the Stoner continuum is suppressed by the large magnitude of magnetization, similar to what happens in the ordered state. Our measurement strongly suggests to extend this strong-coupling description of quantum ferromagnetism all the way to the QCP. Regardless of the details, the very observation of dynamic scaling is strong evidence for a second-order transition settling an ongoing debate of this issue7. Further work should lead to a detailed understanding of the anomalously low dynamic critical exponent.

a

Sr0.3Ca0.7RuO3 From (T ) data50 K T 1.8 K

7 T B 0.5 T

0.1

0.01

0.01

MIT / z ( B f.u. 1 K / z )

MIT / z ( B f.u. 1 K / z )

/ z = 1.2 / z =1.95

B (T) / z

From M (B) data

0.5 47 1.0 1.2 1.4

2

1

0

B/T /z (T /K /z)

1

2

r.m.s.(106 )

1E3

1E3

1E4

1E4

b

/ z = 1.2 / z =1.95

1.82.5

3

4

5

67.5 10

15

20

30

40

0.1

T (K)

/ z

1.0 1.2 1.4

2

1

0 r.m.s.(106)

1E3

0.01 0.1 1

Figure 5 | Scaling of the magnetic susceptibility v and magnetization M as a function of temperature T and magnetic eld B. (a) Data fromFig. 2a). (b) Data from Fig. 2b. Insets show how the quality of the scaling collapse varies with b/nz (with xed value bd/nz 1.95) by checking the

smallest mean square deviation wr.m.s..

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Methods

Sample synthesis and experiments. The SCRO polycrystals were prepared by solid-state sintering using SrCO3, CaCo3 and RuO2 powders as described in ref. 21.

From the structural renement of powder X-ray diffraction spectra, electron dispersive X-ray spectroscopy and wavelength dispersive X-ray spectroscopy, a stoichiometric single phase (space group Pbnm) was conrmed. The magnetization M and the specic heat C were measured in the temperature range T 1.8300 K

and in magnetic elds up to B 8 T. We used the same pieces of plate-like samples,

with a typical size of 2 2 1 mm3, oriented parallel to the applied eld in both

measurements. The use of polycrystalline samples does not affect the possibility to reliably determine scaling exponents of phase transitions21, yet the concentration dependence of the critical exponents near the classical, nite-T transition is consistent with strain-induced anomalous scaling discussed in ref. 22, hinting at the role of inhomogeneities.

Determination of the critical concentration. Since one might wonder whether different values of critical concentration, that is, xca0.7, might lead to a different conclusion on anomalously small dynamic scaling exponent, we further discuss this issue in detail in Supplementary Note 2. From Supplementary Fig. 4, xc may be accurate within 3% only. To check the ramications of this uncertainty, we have additionally carried out a scaling analysis of a sample with x 0.75 (Supplementary

Figs 5 and 6). This analysis suggests that while the critical exponents vary slightly when assuming xc 0.75, they are still highly anomalous compared with those of

the conventional HertzMillisMoriya model. Using this variation, we show that even if the critical concentration was x 0.65, this would still give anomalous

exponents.

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Acknowledgements

This work was supported by the Deutsche Forschungsgemeinschaft through the Research Unit FOR 960. We thank J. L. Her, K. Yoshimura, C. Meingast and R. Heid for helpful discussions.

Author contributions

C.L.H. set up and carried out the magnetization and specic heat measurements. C.L.H., D.F. and H.v.L. discussed the experimental data. D.F., M.W. and R.S. synthesized and characterized the samples. M.C.L., M.S.S. and J.S. performed the dynamic scaling analysis. H.v.L. planned and headed the project. C.L.H., J.S. and H.v.L. wrote the paper.

Additional information

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