Quantum critical fluctuations in layered YFe2Al10 L. S. Wua, M. S. Kima, K. Parka, A. M. Tsvelikb, and M. C. Aronsona,b,1 a Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794; and bCondensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, NY 11973
Edited by Zachary Fisk, University of California, Irvine, CA, and approved August 8, 2014 (received for review July 14, 2014)
The absence of thermal fluctuations at T = 0 makes it possible to observe the inherently quantum mechanical nature of systems where the competition among correlations leads to different types of collective ground states. Our high precision measurements of the magnetic susceptibility, specific heat, and electrical resistivity in the layered compound YFe2Al10 demonstrate robust field-temperature scaling, evidence that this system is naturally poised without tuning on the verge of ferromagnetic order that occurs exactly at T = 0, where magnetic fields drive the system away from this quantum critical point and restore normal metallic behavior. quantum criticality
| ferromagnet | dynamical scaling
T
he interplay of competing interactions is responsible for the array of ground states that are possible in correlated electron systems. It is of particular importance to understand how one such ground state gives way to another when the system is tuned at temperature T = 0, without the complications of thermal fluctuations. Consequently, much interest has focused on quantum critical points (QCPs), where an ordered phase can be created by an infinitesimal modifications of pressure, composition, or field. The onset of ferromagnetic order is perhaps the simplest T = 0 phase transition, and indeed much experimental and theoretical effort has been directed toward understanding its essential features (1–6). It is generally believed that ferromagnetic order occurs at T = 0 via a discontinuous or first order transition, as is observed in the clean ferromagnets ZrZn2 (7) and MnSi (8) under pressure. Disorder is known to render the ferromagnetic transition continuous, leading to the mean field behavior that is found when doping drives TC = 0 in Ni1−xPdx (9), Zr1−xNbxZn2 (10), and Nb1−yFe2+y (11). More controversial is the possibility that strong quantum fluctuations, such as those that destabilize order in low-dimensional systems, may be significant near the TC = 0 ferromagnetic transition and perhaps may even destroy its first-order character (5). A complete experimental investigation of the critical phenomena and their scaling behaviors in a carefully selected system where disorder is minimal is needed to establish that quantum critical fluctuations are both present and relevant to the destabilization of ferromagnetic order. Progress toward obtaining this information has been painstaking, although a number of systems have been identified where the Curie temperature TC has been driven to zero. High pressure experiments evade the disorder that necessarily accompanies doping, but thus far only resistivity and susceptibility measurements have been reported. Complete experimental access is possible when doping is used to suppress TC → 0, but even small amounts of compositional inhomogeneity can obscure any intrinsic critical fluctuations of TC = 0 ferromagnetic transitions, resulting in interesting complications such as the Griffiths phase (12, 13, 14), as well as short ranged order, including spin glasses (15). Alternative ordered states that range from unconventional superconductivity in UGe2 (16) and UCoGe (17), antiferromagnetic order in CeRu2Ge2 (18), hidden order in URu1−xRexSi2 (19), and spiral order in MnSi (20) may also emerge when ferromagnetism becomes sufficiently weak, potentially masking quantum critical fluctuations associated with the underlying TC = 0 ferromagnetic transition. Finally, the electronic 14088–14093 | PNAS | September 30, 2014 | vol. 111 | no. 39
delocalization via the Kondo effect in the f-electron based heavy fermions (21) or by proximity to a Mott transition in d-electronbased systems may also contribute to the destabilization of ferromagnetic order. The discovery of a new ferromagnetic system that is naturally tuned to the onset of order at TC = 0 would be an enabling development, because a wide range of experimental tools could then be used to assemble a holistic picture of the quantum critical fluctuations and their connection to the underlying criticality, providing important feedstock for future theoretical developments. We will present here experimental data and a detailed scaling analysis of the magnetization, specific heat, and electrical resistivity measured on YFe 2Al10 single crystals that establish its properties are dominated at the lowest temperatures and fields by the quantum critical fluctuations of a TC = 0 ferromagnetic transition. Although YFe2Al10 is not ordered above 0.1 K (22), the correlations that drive this quasi2D metal to the brink of ferromagnetic order are derived from the hybridization of Fe-based d-electrons with conduction electrons. Disorder effects are expected to be minimal in this system, as single crystal X-ray diffraction measurements find no evidence of departures from stoichiometry or site disorder (22, 23). The layered nature of YFe2Al10 is evident from its crystal structure (Fig. 1A), which features nearly square nets of Fe atoms that form the ac planes (23, 24). Magnetically, YFe2Al10 can be considered quasi-2D. Measurements of the alternating current (AC) magnetic susceptibility χ ac(T) (Fig. 1B) find a strong temperature divergence with χ ac ∼ T−γ (γ = 1.4 ± 0.05), but only when the ac field Bac = 4.17 Oe lies in the ac plane. When Bac is parallel to the b axis, χ ac does not diverge, and its value at 1.8 K is almost 30 times smaller than when the field is in the ac plane, where there is no measurable anisotropy. This divergence in χ ac does not culminate in magnetic order, at least for temperatures above 0.1 K (22). Instead it indicates that the Significance Temperature-driven phase transitions, such as the melting of ice or the boiling of water, are a familiar part of daily life. Much less is known about the most extreme phase transitions, which happen only at zero temperature, where quantum fluctuations limit the stabilities of different collective ground states such as magnetic order and superconductivity. We report an experimental investigation of YFe2Al10, where planes of Fe atoms are naturally poised on the verge of ferromagnetic order, exactly at T = 0. The thermal, magnetic, and electrical transport properties in YFe2Al10 all diverge as T → 0, a process that can be reversed by magnetic fields in a way that is dictated by the underlying system energy. Author contributions: L.S.W., K.P., and M.C.A. designed research; L.S.W., M.S.K., K.P., and M.C.A. performed research; L.S.W., A.M.T., and M.C.A. analyzed data; and L.S.W., A.M.T., and M.C.A. wrote the paper. The authors declare no conflict of interest. This article is a PNAS Direct Submission. 1
To whom correspondence should be addressed. Email: [email protected].
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 1073/pnas.1413112111/-/DCSupplemental.
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T/B0.59 scaling, and indeed a remarkable scaling collapse of the magnetic susceptibility M/B (Fig. 2B), is found for 1.8 K < T < 30 K and B < 6 T, extending over more than three orders of magnitude in the scaling variable x = T/B0.59, where dχ 1:4 T − B = ψ 0:59 : dT B
magnetic properties of YFe2Al10 are dominated below ∼20 K by proximity to a T = 0 phase transition. What type of T = 0 phase transition is responsible for the quantum criticality in YFe2Al10? The magnitude of the uniform susceptibility and the strength of its divergence χ(T) ∼ T−1.4, as well as the divergence of the spin lattice relaxation time 1/T1T from NMR measurements (25), together suggest that it involves a uniform zero wave vector q = 0 or ferromagnetic instability. YFe2Al10 is a rare example of a system that forms naturally very close to a ferromagnetic QCP, without the need for tuning by pressure or composition. Our investigation of the field and temperature dependencies of the magnetization M(T, B) reveals that the field B suppresses the quantum critical fluctuations. Fig. 2A shows that the divergence of the direct current (DC) magnetic susceptibility χ = M/B is suppressed in field and that χ(T) becomes increasingly temperature independent and smaller in magnitude with increasing field, approaching the Fermi liquid (FL) behavior expected for a conventional metal, where χ is temperature independent. The broad maximum in χ(T), even more evident in the ac susceptibility χ ac (Fig. S1), defines a new scale T ⋆(B) that marks the cross-over between the quantum critical (QC) (T B) and FL (T B) regimes, and the inset of Fig. 2A shows that T⋆(B) ∼ B0.59. This duality between T and B suggests the possibility of
A
This scaling reveals an important property of the underlying free energy, which controls both the magnetic and thermodynamic properties. Namely, the quantum critical T/B0.59 scaling can be understood on the basis of a generic free energy F that assumes the validity of hyperscaling, where the spatial dimensionality d is augmented by a dynamic exponent z, i.e., deff = d + z near this T = 0 phase transition (2–4, 26): FðB; TÞ = T
d+z z
~f F
B
T yb =z
d+z
= B yb fF
T B z=yb
:
[2]
Here yb is the scaling exponent that relates to the tuning parameter B, which can alternatively be written in terms of the correlation length exponent ν, where z/yb = νz. Although it is believed that there are different critical time scales for the order parameter and for the fermionic degrees of freedom near a ferromagnetic QCP (5, 6), our analysis assumes that only one of these time scales dominates near the QCP in YFe2Al10, and consequently, only a single z is required. The field-temperature scaling of the magnetic susceptibility can be determined from that of the z=y free energy F, where x = T=B b and ~x = B=T yb =z : dχ d = dT dT
d2 F dB2
= Bd=yb −2 f χ′
T
B z=yb
:
[3]
The critical exponents can be read directly from the susceptibility scaling plot (Fig. 2B), giving d = z and νz = z/yb = 0.59. The precision of the experimental data confers a corresponding precision to our determination of these critical exponents, whose range of values is severely curtailed by the sharp minimum in the scaling residuals, shown in the inset of Fig. 2B and in more detail in Fig. S2. The values of d, z, and yb found in YFe2Al10 imply a particularly simple scaling form for the specific heat, which can be obtained from the same free energy F
B
Fig. 2. Field−temperature scaling and Fermi liquid breakdown: Magnetic susceptibility. (A) Temperature dependencies of the dc susceptibility χ = M/B measured in different fixed fields, as indicated. Solid red line has slope γ = 1.4. (Inset) Contour plot of the same data with T⋆(B) taken from maxima in dM/dT (red rimmed circles; Fig. S1A) and dM/dB (blue rimmed circles; Fig. 4B). Red and blue solid lines have T⋆(B) ∼ B0.59, indicating the cross-over from QC to FL regimes. (B) Scaling collapse of the temperature derivative of χ as a function of the variable T/B0.59. Red line indicates fit from the scaling function fM(x), as described in the text. An alternative plot of the data in B on linear scales is given in Fig. S2. (Inset) Net deviations from scaling determined for different values of the critical exponents 2 − d/yb and z/yb (Fig. S2E).
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Fig. 1. Quasi-2D quantum criticality. (A) Crystal structure of YFe 2Al 10, formed by stacking nearly square nets of Fe along the b axis. Gray solid line indicates unit cell. Each Fe atom is surrounded by a distorted octahedron of Al atoms, capped with Y atoms. (B) The ac susceptibility χac(T) diverges when the ac field Bac = 4.17Oe is in the ac plane, but not when Bac is parallel to the b axis. Red line is fit χac ∼ T−γ with γ = 1.4.
[1]
d−z CM ðB; TÞ ∂2 F d−z = − 2 = T z ~f C ð~xÞ = B yb fC ðxÞ: T ∂T
d+z
[4]
Using d = z, we find that ΔCM ðB; TÞ CM ðB; TÞ CM ð0; TÞ = − = T 0~f C ð~xÞ = B0 fC ðxÞ: T T T
C
−1
fM
T B z=yb
;
[6]
with −γ=2 fM ðxÞ= c x2 + a2 :
[5]
Assuming that ~f C ð~xÞ can be separated into field-dependent and field-independent parts ~f C ð~xÞ = ~f C ð0Þ + ~f C ðB=T yb =z Þ, we see from Eq. 5 that no power law divergence is possible for the B = 0 specific heat CM/T, which is instead predicted to be constant, implying that the observed temperature divergence of CM/T is weaker than any power law, consistent with its observed logarithmic temperature divergence (Fig. 3A and Fig. S3). Magnetic fields suppress the divergence in ΔCM/T, much as they do for the susceptibility χ(T), resulting in a FL-like temperature independence of CM/T ∼ γ(B) (Fig. 3B). Eq. 5 predicts that T/B0.59 also controls the scaling of ΔCM/T. Fig. 3C confirms this expectation for T < 10 K and fields as large as 7 T, over more than three orders of magnitude in the scaling variable x = T=Bz=yb . In addition to the free energy (Eq. 2) that underlies the observed scaling, the internal consistency of the magnetization and specific heat has been verified in SI Text, where we demonstrate that their field and temperature dependencies obey a Maxwell relation, as is required by thermodynamics. An intriguingly simple expression for the scaling function fF(x) can be found that not only describes the T/B0.59 scaling of M/B, but which also reproduces the observed B = 0 and T = 0 limits of both M/B and ΔCM/T. Inspired by similar expressions that were found for heavy fermions CeCu6−xAux (27) and β-YbAlB4 (28), we propose
A
M = B yb
Here c and a are the fitting parameters. The parameter a is depffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi termined by fitting the maxima T ⋆ =Bz=yb = x⋆ = a2 =ðγ + 1Þ ≈ 3 in dM/dT, where we have used the experimentally determined exponents d = z, z/yb = 0.59 and γ ∼ 1.4. The resulting parameter-free expression is Eq. 6 compared with the experimental data in Fig. 2B, and the agreement is excellent over the full range of x. This scaling function fM(x) implies a certain scaling function for the underlying free energy fF(x), which can in turn be related to the scaling function fC(x) for ΔCM/T. Although the details of this procedure are given in SI Text, its accuracy is demonstrated by the excellent agreement between the deduced expression for the scaled specific heat ΔCM/T = fC(x) and the scaled data (Fig. 3C). The internal consistency of the scaling functions fF(x), fM(x), and fC(x) that is implied by this comparison suggests that specific expressions for the temperature and field dependencies of ΔCM/T and M/B can be deduced in their respective QC (T B) and FL (T B) limits. The experimental scaling relations that have been established here provide a precise formulation of how the quantum critical fluctuations lead to the breakdown of the high field FL phase. In the QC limit, the expected field-independent divergence in χ ac(T, B → 0) ∼ T−γ is recovered (Fig. 1B), as well as the B = 0 logarithmic divergence CM/T ∼ –log(T) (Fig. 3A), albeit with a correction B2T−γ−2 in small magnetic fields. Deep in the FL (T B), the field dependencies of the Pauli susceptibility are
B
D
Fig. 3. Field−temperature scaling and Fermi liquid breakdown: Specific heat. (A) Temperature dependence of the B = 0 specific heat CM/T. An estimate of the phonon contribution has been subtracted to isolate the purely magnetic and electronic specific heat CM/T (22). (B) CM/T measured in different fixed fields, as indicated. (C) Scaling collapse of ΔCM/T = CM(B, T)/T − CM(0, T)/T, as a function of the variable T/B0.59. Red line indicates fit with scaling function fC(x), described in the text. An alternative plot of these data on linear axes appears in Fig. S2. (D) Temperature dependence of the magnetic Grüneisen parameter Γ/B, determined in different fixed fields (SI Text).
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Wu et al.
B
A
Fig. 4. Field-driven Fermi liquid collapse. Field dependencies of χ = dM/dB (A) and CM/T (B) at different temperatures as indicated. Inset in B shows CM/T ∼ −log(B) (red line), although the cross-over to quantum critical behavior at the lowest fields is an inevitable result of the nonzero measuring temperature T = 0.55 K.
given by χðBÞ = dM=dB ∼ Bðd+zÞ=yb −2 (Fig. 4A) and the Sommerfeld coefficient is given by γ(B) = CM(B)/T ∼ −log(B) (Fig. 4B, Inset). By comparing the QC and FL expressions, it is clear that the field dependencies of CM(B)/T at fixed temperatures are nonmonotonic, explaining the maxima that separate the high field FL and low field QC phases in Fig. 4B. The logarithmic field divergence of the T = 0 Sommerfeld coefficient γ(B, T) = CM(B, T)/T (Fig. 4B, Inset) implies a weaker divergence of the quasi-particle mass at the QCP than the power law divergencies that are generally found in other QC systems (29, 30). However, the field divergence of the Pauli susceptibility (Fig. 4A) is too strong to be attributed entirely to that of the quasi-particle mass, reflecting instead a ferromagnetic enhancement of the susceptibility that is echoed in the unusually strong divergence of χ ac(T) ∼ T−1.4 found for B = 0. By the same token, the distinctly different temperature dependencies of C/T and χ(T) also rules out the possibility of a Griffiths phase, where both would have the same critical exponents (14, 31–33). It has been argued that the universal behaviors near different types of QCPs can be established through the divergence of the Grüneisen ratio (34). For field-tuned magnetic systems, the magnetic Grüneisen ratio Γ is given by (SI Text)
B
C
D
[7]
Because the power law divergence of the numerator is much stronger than the logarithmic divergence of the denominator, the magnetic Grüneisen parameter Γ/B is expected to be strongly divergent at low temperatures, at least in low fields. This divergence is confirmed by the experimental data (Fig. 3D), where Γ is calculated indirectly from the measured ac magnetic susceptibility χ ac with Bac = 4.17 Oe and the measured zero field specific heat. The strong temperature divergence for B = 0 establishes the quantum criticality of YFe2Al10, as well as the role of magnetic field as a scaling variable that suppresses the QC fluctuations. The electrical resistivity ρ(T) is a measure of how strongly coupled the quasi-particles are to the QC fluctuations that dominate the magnetic susceptibility and specific heat at low temperatures and fields. We carried out measurements of the anisotropy of ρ(T), using a modified Montgomery technique (35). Fig. 5A shows that there is virtually no resistive anisotropy in the ac plane, where the resistivities ρIka ∼ ρIkc are approximately a factor of 2 larger than the interplanar resistivity ρIkb, even at temperatures where the QC fluctuations no longer contribute to the magnetization or specific heat. This anisotropy is much weaker than that found Wu et al.
A
Fig. 5. Quantum critical electrical resistivity ρ(T). (A) The in-plane resistivities ρIka and ρIkc are larger than the interplanar resistivity ρIkb. Data obtained using the modified Montgomery method (35). (Inset) ρ(T) measured for four different YFe 2Al 10 samples, in each case with Ika. (B) The effects of a 9-T magnetic field on ρ(T ) where Ika. (C ) ρ(T ) measured with Ika in different fields Bka. (D) Scaling collapse of Δρ(B, T ) = ρ(B, T ) − ρ(0, T ) for temperatures 0.2–50 K and fields as large as 14 T. Different samples were used for A–C. We estimate a systematic error of ∼10–20% arises from uncertainties in sample dimensions for each voltage measurement, where the Montgomery method requires at each temperature six voltage measurements with different current directions to arrive at ρIka , ρIka, and ρIka. The T → 0 resistivities for all measured samples are consistent with the systematic errors.
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Γ ½dðM=BÞ=dT=T T −γ−2 T −3:4 =− ∝ ∼ : B C=T log T log T
in the magnetic susceptibility (Fig. 1B), suggesting that the quasi-particles are only weakly coupled to the critical fluctuations, reflecting instead the part of the quasi-2D character of YFe2Al10 that can be traced to its layered crystal structure and its influence on the underlying electronic structure. The large magnitudes of ρIka,c imply T = 0 sheet resistances for per square Fe atoms in the ac plane Rsquare ≈ 2 kΩ, and considering as well their relatively weak temperature dependencies, we infer that there is quite strong quasi-particle scattering in YFe2Al10, although little variation is found among crystals from different batches (Fig. 5A, Inset), making the role of residual disorder unclear. Both ρIkb and ρIka,c have metallic temperature
dependencies, indicating that the quasi-particles in YFe2Al10 are definitively not localized. The QC fluctuations provide an additional scattering mechanism at low temperatures and fields that is strongest in the ac planes. The T → 0 resistive upturn in YFe2Al10 is suppressed by fields in the ac plane, but it is much less sensitive to fields applied parallel to the b axis (Fig. 5B), the same anisotropy found in the ac susceptibility χ ac (Fig. 1B). The part of the resistive upturn that is diminished with increasing field (Fig. 5C) Δρ(B, T) = ρ(B, T) − ρ(0, T) undergoes a scaling collapse for 0.2 K < T < 50 K and B < 14 T, with the same scaling variable x = T/B0.59 that was found for M/B and CM/T (Fig. 5D), where ΔρðB; TÞ T = fR 0:59 : T B
[8]
This scaling confirms that the resistive upturn in YFe2Al10 results from increasingly strong scattering as the QCP is approached by decreasing either T or B. Theories of quasi-particle scattering from critical fluctuations in ferromagnets find that the specific heat CM(B, T) is proportional to ρ(B, T) (∂ρ/∂T) when long (short) wavelength critical fluctuations dominate the quasi-particle scattering (36, 37). However, the scaling functions fC(x) and fR(x) in Eqs. 4 and 8 are not equal and cannot be simply related, ruling out these scenarios. Spin disorder scattering of quasiparticles from fluctuations of the moments enforces a proportionality between Δρ(B, T) and the magnetization M. This explanation is impossible in YFe2Al10, because the scaling form (Eq. 8) cannot be reduced to a function of the magnetization (Eq. 6), due to the temperature prefactor of the latter. The traditional pictures fail badly in YFe2Al10, suggesting that the fluctuating moments and the conduction electrons may not be entirely separable. Indeed, the low temperature resistive upturn is reminiscent of the Kondo effect in normal metals, and the B = 0 ρ(T) can be satisfactorily fitted (22) by the Kondo expression (38). However, the resistivity for a fixed number of Kondo compensated moments in a conventional metallic host scales with T/B (39, 40), and so a different explanation is needed for the T/B0.59 scaling found in YFe2Al10 (Fig. 5D). The observation of T/B0.59 scaling in the electrical resistivity is evidence that the order parameter fluctuations are coupled to soft quasi-particle modes in YFe2Al10, although the influence of the QCP on the resistivity is much weaker than for the uniform susceptibility or specific heat, confirming that the quasi-particles are not coupled strongly enough to the QC fluctuations to themselves become fully critical. Our primary experimental results in YFe2Al10 are the divergencies in the uniform susceptibility χ ac ∼ T −γ (γ = 1.4) and in the specific heat CM/T ∝ −log(T), whereas field-temperature scaling is found in the magnetization, specific heat, and electrical resistivity, in each case giving νz ≡ z/yb ∼ 0.59. Are these critical exponents consistent with theoretical predictions? We begin with the standard Hertz-Moriya-Millis (HMM) theory of metals near QCPs (2–4), which is a mean-field theory that ignores the spatial and temporal fluctuations of the incipient ordered phase. HMM theory for a d = z = 3 ferromagnet predicts both a divergence of the magnetic susceptibility χ ac ∼ T −γ with γ = 4/3, close to the observed value γ = 1.4 ± 0.05 (Fig. 1B), and CM/T ∼ −log(T) (Fig. 3A). However, the observation of T/B0.59 scaling is itself inconsistent with HMM, where the truncated equation of state B = M/χ 0 + μBM3 with constant μ and χ 0 ∼ T −4/3 implies a scaling variable x = T/B0.5. As is evident from the inset of Fig. 2B, this choice of z/yb = 0.5 leads to a scaling of the magnetic susceptibility data that is markedly inferior to that with z/yb = 0.59 (Fig. S2 E and F). This failure of HMM implies that the critical fluctuations of a T = 0 phase transition are responsible 14092 | www.pnas.org/cgi/doi/10.1073/pnas.1413112111
in YFe2Al10 for the T = 0 divergencies of χ and CM/T and that they can be suppressed by magnetic fields to restore a featureless and conventional metallic state. A field theory approach has been taken to derive values of these QC exponents in clean and disordered ferromagnets in different dimensions d (5, 41). The theory reproduces the logarithmic divergence in C/T that we observe in YFe2Al10, as well as the observed d = z, for disordered ferromagnets with 2 < d < 6. The situation for disordered ferromagnets is complicated because the paramagnons and the diffusive quasi-particle modes have different dynamical exponents zc = d and zq = 2, respectively (5), and thus two different critical time scales may in principle be present. The experimental observation d = z suggests that the order parameter fluctuations may prevail in YFe2Al10, an observation that is supported by the noncriticality of the overall resistivity. However, we stress that the experimental values of the susceptibility exponent γ = 1.4 and the scaling variable νz ≡ z/yb ∼ 0.59 are not reproduced in this theory. The need to treat order parameter fluctuations and soft quasiparticles on equal footing greatly increases the difficulty of theoretical analysis in QC metallic ferromagnets, where a local Ginzburg-Landau effective action for the order parameter field does not exist (5). As this controversy continues, we resorted instead to a phenomenological description of the T/B0.59 scaling that was observed in YFe2Al10, based on a generic free energy with hyperscaling, where a single time scale was assumed to become critical. Augmented by a specific expression for the T/B0.59 scaling function, we reproduced the observed field and temperature dependencies of the quantum critical magnetization and specific heat using a single set of critical exponents (d = z, γ = 1.4, νz = 0.59). The general success of this heuristic analysis, based on the hypothesis of hyperscaling, suggests that YFe2Al10 is a system that is below its upper critical dimension, perhaps due to its 2D character. Hyperscaling generally implies universality, where the form of the free energy does not depend on the details of the system, implying that critical points can be classified into universality classes based on the values of the critical exponents themselves. We propose that YFe2Al10 is, to our knowledge, the first confirmed member of such a universality class. Mean field theories predict that the collapse of ferromagnetic order is via a first-order transition (5, 6) in the absence of disorder or by a continuous mean field transition in the presence of disorder (7–10). According to ref. 5, it remains the case when strong quantum fluctuations are present, as may be realized in low-dimensional systems. The data presented here establish that, although YFe2Al10 does not order, strong divergencies in the uniform susceptibility and specific heat indicate QC fluctuations are indeed dominant as this system approaches a T = 0 ferromagnetic transition. Of course, it would be of great interest to use compositional modifications to drive YFe2Al10 to the onset of ferromagnetic order, provided this could be done without introducing significant disorder. Whether or not this putative TC = 0 transition is ultimately found to be first order, it now seems likely that ferromagnetic quantum criticality is protected in low-dimensional systems like quasi-2D YFe2Al10, as it is in quasi-one dimensional YbNi4(P1−xAsx)2 (42). The evidence presented here shows that YFe2Al10 is a rare example of a quasi-2D metallic system where critical fluctuations associated with a TC = 0 ferromagnetic transition dominate the specific heat and uniform susceptibility at the lowest temperatures and fields, without the need for compositional or pressure tuning. Our magnetization, specific heat, and electrical resistivity measurements have allowed us to propose and experimentally establish an expression for an underlying QC free energy that unifies the scaling found in the susceptibility, specific heat, and the QC part of the resistivity while assuring a strong divergence of the magnetic Grüneisen parameter. These experimental results extend a challenge to theory to develop new frameworks in Wu et al.
ACKNOWLEDGMENTS. We thank M. Garst, M. Brando, and F. Steglich for useful discussions. Work at Brookhaven National Laboratory was carried out under the auspices of US Department of Energy, Office of Basic Energy Sciences, Contract DE-AC02-98CH1886.
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23. Kerkau A, et al. (2012) Crystal structure of yttrium iron aluminium (1/2/10), YFe2Al10. Z Kristallogr, New Cryst Struct 227:289–290. 24. Thiede VMT, Ebel T, Jeitschko W (1998) Ternary aluminides LnT2Al10 (Ln=Y, La-Nd, Sm, Gd-Lu and T=Fe, Ru, Os) with YbFe2Al10 type structure and magnetic properties of the iron-containing series. J Mater Chem 8(1):125–130. 25. Khuntia P, et al. (2012) Field-tuned critical fluctuations in YFe2Al10: Evidence from magnetization, 27Al NMR, and NQR investigations. Phys Rev B 86(22):220401-1– 220401-5. 26. Sachdev S (1999) Quantum Phase Transitions (Cambridge Univ Press, Cambridge, UK). 27. Schroder A, et al.; v. Lohneysen H (2000) Onset of antiferromagnetism in heavy-fermion metals. Nature 407(6802):351–355. 28. Matsumoto Y, et al. (2011) Quantum criticality without tuning in the mixed valence compound β-YbAlB4. Science 331(6015):316–319. 29. Shishido H, Settai R, Harima H, Onuki Y (2005) A drastic change of the Fermi surface at a critical pressure in CeRhIn5: dHvA study under pressure. J Phys Soc Jpn 74(4): 1103–1106. 30. Gegenwart P, Custers J, Tokiwa Y, Geibel C, Steglich F (2005) Ferromagnetic quantum critical fluctuations in YbRh2(Si0.95Ge0.05)2. Phys Rev Lett 94(7):076402–076404. 31. Castro Neto AH, Castilla G, Jones BA (1998) Non-Fermi liquid behavior and Griffiths phase in f-electron compounds. Phys Rev Lett 81(16):3531–3534. 32. Castro Neto AH, Jones BA (2000) Non-Fermi-liquid behavior in U and Ce alloys: Criticality, disorder, dissipation, and Griffiths-McCoy singularities. Phys Rev B 62(22):14975–15011. 33. Millis AJ, Morr DK, Schmalian J (2002) Quantum Griffiths effects in metallic systems. Phys Rev B 66(17):174433-1–174433-10. 34. Zhu L, Garst M, Rosch A, Si Q (2003) Universally diverging Grüneisen parameter and the magnetocaloric effect close to quantum critical points. Phys Rev Lett 91(6): 066404-1–066404-4. 35. Dos Santos CAM, et al. (2011) Procedure for measuring electrical resistivity of anisotropic materials: A revision of the Montgomery method. J Appl Phys 110(8):083703–083707. 36. de Gennes PG, Friedel J (1958) Anomalies de résistivité dans certains métaux magníques. J Phys Chem Solids 4(1):71–77. 37. Fisher ME, Langer JS (1968) Resistive anomalies at magnetic critical points. Phys Rev Lett 20(13):665–668. 38. Hamann DR (1967) New solution for exchange scattering in dilute alloys. Phys Rev 158(3):570–580. 39. Schlottmann P (1983) Bethe-Ansatz solution of the ground-state of the SU (2j+1) Kondo (Coqblin-Schrieffer) model: Magnetization, magnetoresistance, and universality. Z Phys B 51(3):223–235. 40. Batlogg B, et al. (1987) Superconductivity and heavy fermions. J Magn Magn Mater 63-64:441–446. 41. Kirkpatrick TR, Belitz D (1996) Quantum critical behavior of disordered itinerant ferromagnets. Phys Rev B Condens Matter 53(21):14364–14376. 42. Steppke A, et al. (2013) Ferromagnetic quantum critical point in the heavy-fermion metal YbNi4(P(1-x)As(x))2. Science 339(6122):933–936.
PHYSICS
which to understand the emergence of ferromagnetic order and, in particular, its impact on the underlying quasi-particles in metallic systems.
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PNAS | September 30, 2014 | vol. 111 | no. 39 | 14093
Edited by Zachary Fisk, University of California, Irvine, CA, and approved August 8, 2014 (received for review July 14, 2014)
The absence of thermal fluctuations at T = 0 makes it possible to observe the inherently quantum mechanical nature of systems where the competition among correlations leads to different types of collective ground states. Our high precision measurements of the magnetic susceptibility, specific heat, and electrical resistivity in the layered compound YFe2Al10 demonstrate robust field-temperature scaling, evidence that this system is naturally poised without tuning on the verge of ferromagnetic order that occurs exactly at T = 0, where magnetic fields drive the system away from this quantum critical point and restore normal metallic behavior. quantum criticality
| ferromagnet | dynamical scaling
T
he interplay of competing interactions is responsible for the array of ground states that are possible in correlated electron systems. It is of particular importance to understand how one such ground state gives way to another when the system is tuned at temperature T = 0, without the complications of thermal fluctuations. Consequently, much interest has focused on quantum critical points (QCPs), where an ordered phase can be created by an infinitesimal modifications of pressure, composition, or field. The onset of ferromagnetic order is perhaps the simplest T = 0 phase transition, and indeed much experimental and theoretical effort has been directed toward understanding its essential features (1–6). It is generally believed that ferromagnetic order occurs at T = 0 via a discontinuous or first order transition, as is observed in the clean ferromagnets ZrZn2 (7) and MnSi (8) under pressure. Disorder is known to render the ferromagnetic transition continuous, leading to the mean field behavior that is found when doping drives TC = 0 in Ni1−xPdx (9), Zr1−xNbxZn2 (10), and Nb1−yFe2+y (11). More controversial is the possibility that strong quantum fluctuations, such as those that destabilize order in low-dimensional systems, may be significant near the TC = 0 ferromagnetic transition and perhaps may even destroy its first-order character (5). A complete experimental investigation of the critical phenomena and their scaling behaviors in a carefully selected system where disorder is minimal is needed to establish that quantum critical fluctuations are both present and relevant to the destabilization of ferromagnetic order. Progress toward obtaining this information has been painstaking, although a number of systems have been identified where the Curie temperature TC has been driven to zero. High pressure experiments evade the disorder that necessarily accompanies doping, but thus far only resistivity and susceptibility measurements have been reported. Complete experimental access is possible when doping is used to suppress TC → 0, but even small amounts of compositional inhomogeneity can obscure any intrinsic critical fluctuations of TC = 0 ferromagnetic transitions, resulting in interesting complications such as the Griffiths phase (12, 13, 14), as well as short ranged order, including spin glasses (15). Alternative ordered states that range from unconventional superconductivity in UGe2 (16) and UCoGe (17), antiferromagnetic order in CeRu2Ge2 (18), hidden order in URu1−xRexSi2 (19), and spiral order in MnSi (20) may also emerge when ferromagnetism becomes sufficiently weak, potentially masking quantum critical fluctuations associated with the underlying TC = 0 ferromagnetic transition. Finally, the electronic 14088–14093 | PNAS | September 30, 2014 | vol. 111 | no. 39
delocalization via the Kondo effect in the f-electron based heavy fermions (21) or by proximity to a Mott transition in d-electronbased systems may also contribute to the destabilization of ferromagnetic order. The discovery of a new ferromagnetic system that is naturally tuned to the onset of order at TC = 0 would be an enabling development, because a wide range of experimental tools could then be used to assemble a holistic picture of the quantum critical fluctuations and their connection to the underlying criticality, providing important feedstock for future theoretical developments. We will present here experimental data and a detailed scaling analysis of the magnetization, specific heat, and electrical resistivity measured on YFe 2Al10 single crystals that establish its properties are dominated at the lowest temperatures and fields by the quantum critical fluctuations of a TC = 0 ferromagnetic transition. Although YFe2Al10 is not ordered above 0.1 K (22), the correlations that drive this quasi2D metal to the brink of ferromagnetic order are derived from the hybridization of Fe-based d-electrons with conduction electrons. Disorder effects are expected to be minimal in this system, as single crystal X-ray diffraction measurements find no evidence of departures from stoichiometry or site disorder (22, 23). The layered nature of YFe2Al10 is evident from its crystal structure (Fig. 1A), which features nearly square nets of Fe atoms that form the ac planes (23, 24). Magnetically, YFe2Al10 can be considered quasi-2D. Measurements of the alternating current (AC) magnetic susceptibility χ ac(T) (Fig. 1B) find a strong temperature divergence with χ ac ∼ T−γ (γ = 1.4 ± 0.05), but only when the ac field Bac = 4.17 Oe lies in the ac plane. When Bac is parallel to the b axis, χ ac does not diverge, and its value at 1.8 K is almost 30 times smaller than when the field is in the ac plane, where there is no measurable anisotropy. This divergence in χ ac does not culminate in magnetic order, at least for temperatures above 0.1 K (22). Instead it indicates that the Significance Temperature-driven phase transitions, such as the melting of ice or the boiling of water, are a familiar part of daily life. Much less is known about the most extreme phase transitions, which happen only at zero temperature, where quantum fluctuations limit the stabilities of different collective ground states such as magnetic order and superconductivity. We report an experimental investigation of YFe2Al10, where planes of Fe atoms are naturally poised on the verge of ferromagnetic order, exactly at T = 0. The thermal, magnetic, and electrical transport properties in YFe2Al10 all diverge as T → 0, a process that can be reversed by magnetic fields in a way that is dictated by the underlying system energy. Author contributions: L.S.W., K.P., and M.C.A. designed research; L.S.W., M.S.K., K.P., and M.C.A. performed research; L.S.W., A.M.T., and M.C.A. analyzed data; and L.S.W., A.M.T., and M.C.A. wrote the paper. The authors declare no conflict of interest. This article is a PNAS Direct Submission. 1
To whom correspondence should be addressed. Email: [email protected].
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 1073/pnas.1413112111/-/DCSupplemental.
www.pnas.org/cgi/doi/10.1073/pnas.1413112111
T/B0.59 scaling, and indeed a remarkable scaling collapse of the magnetic susceptibility M/B (Fig. 2B), is found for 1.8 K < T < 30 K and B < 6 T, extending over more than three orders of magnitude in the scaling variable x = T/B0.59, where dχ 1:4 T − B = ψ 0:59 : dT B
magnetic properties of YFe2Al10 are dominated below ∼20 K by proximity to a T = 0 phase transition. What type of T = 0 phase transition is responsible for the quantum criticality in YFe2Al10? The magnitude of the uniform susceptibility and the strength of its divergence χ(T) ∼ T−1.4, as well as the divergence of the spin lattice relaxation time 1/T1T from NMR measurements (25), together suggest that it involves a uniform zero wave vector q = 0 or ferromagnetic instability. YFe2Al10 is a rare example of a system that forms naturally very close to a ferromagnetic QCP, without the need for tuning by pressure or composition. Our investigation of the field and temperature dependencies of the magnetization M(T, B) reveals that the field B suppresses the quantum critical fluctuations. Fig. 2A shows that the divergence of the direct current (DC) magnetic susceptibility χ = M/B is suppressed in field and that χ(T) becomes increasingly temperature independent and smaller in magnitude with increasing field, approaching the Fermi liquid (FL) behavior expected for a conventional metal, where χ is temperature independent. The broad maximum in χ(T), even more evident in the ac susceptibility χ ac (Fig. S1), defines a new scale T ⋆(B) that marks the cross-over between the quantum critical (QC) (T B) and FL (T B) regimes, and the inset of Fig. 2A shows that T⋆(B) ∼ B0.59. This duality between T and B suggests the possibility of
A
This scaling reveals an important property of the underlying free energy, which controls both the magnetic and thermodynamic properties. Namely, the quantum critical T/B0.59 scaling can be understood on the basis of a generic free energy F that assumes the validity of hyperscaling, where the spatial dimensionality d is augmented by a dynamic exponent z, i.e., deff = d + z near this T = 0 phase transition (2–4, 26): FðB; TÞ = T
d+z z
~f F
B
T yb =z
d+z
= B yb fF
T B z=yb
:
[2]
Here yb is the scaling exponent that relates to the tuning parameter B, which can alternatively be written in terms of the correlation length exponent ν, where z/yb = νz. Although it is believed that there are different critical time scales for the order parameter and for the fermionic degrees of freedom near a ferromagnetic QCP (5, 6), our analysis assumes that only one of these time scales dominates near the QCP in YFe2Al10, and consequently, only a single z is required. The field-temperature scaling of the magnetic susceptibility can be determined from that of the z=y free energy F, where x = T=B b and ~x = B=T yb =z : dχ d = dT dT
d2 F dB2
= Bd=yb −2 f χ′
T
B z=yb
:
[3]
The critical exponents can be read directly from the susceptibility scaling plot (Fig. 2B), giving d = z and νz = z/yb = 0.59. The precision of the experimental data confers a corresponding precision to our determination of these critical exponents, whose range of values is severely curtailed by the sharp minimum in the scaling residuals, shown in the inset of Fig. 2B and in more detail in Fig. S2. The values of d, z, and yb found in YFe2Al10 imply a particularly simple scaling form for the specific heat, which can be obtained from the same free energy F
B
Fig. 2. Field−temperature scaling and Fermi liquid breakdown: Magnetic susceptibility. (A) Temperature dependencies of the dc susceptibility χ = M/B measured in different fixed fields, as indicated. Solid red line has slope γ = 1.4. (Inset) Contour plot of the same data with T⋆(B) taken from maxima in dM/dT (red rimmed circles; Fig. S1A) and dM/dB (blue rimmed circles; Fig. 4B). Red and blue solid lines have T⋆(B) ∼ B0.59, indicating the cross-over from QC to FL regimes. (B) Scaling collapse of the temperature derivative of χ as a function of the variable T/B0.59. Red line indicates fit from the scaling function fM(x), as described in the text. An alternative plot of the data in B on linear scales is given in Fig. S2. (Inset) Net deviations from scaling determined for different values of the critical exponents 2 − d/yb and z/yb (Fig. S2E).
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PNAS | September 30, 2014 | vol. 111 | no. 39 | 14089
PHYSICS
Fig. 1. Quasi-2D quantum criticality. (A) Crystal structure of YFe 2Al 10, formed by stacking nearly square nets of Fe along the b axis. Gray solid line indicates unit cell. Each Fe atom is surrounded by a distorted octahedron of Al atoms, capped with Y atoms. (B) The ac susceptibility χac(T) diverges when the ac field Bac = 4.17Oe is in the ac plane, but not when Bac is parallel to the b axis. Red line is fit χac ∼ T−γ with γ = 1.4.
[1]
d−z CM ðB; TÞ ∂2 F d−z = − 2 = T z ~f C ð~xÞ = B yb fC ðxÞ: T ∂T
d+z
[4]
Using d = z, we find that ΔCM ðB; TÞ CM ðB; TÞ CM ð0; TÞ = − = T 0~f C ð~xÞ = B0 fC ðxÞ: T T T
C
−1
fM
T B z=yb
;
[6]
with −γ=2 fM ðxÞ= c x2 + a2 :
[5]
Assuming that ~f C ð~xÞ can be separated into field-dependent and field-independent parts ~f C ð~xÞ = ~f C ð0Þ + ~f C ðB=T yb =z Þ, we see from Eq. 5 that no power law divergence is possible for the B = 0 specific heat CM/T, which is instead predicted to be constant, implying that the observed temperature divergence of CM/T is weaker than any power law, consistent with its observed logarithmic temperature divergence (Fig. 3A and Fig. S3). Magnetic fields suppress the divergence in ΔCM/T, much as they do for the susceptibility χ(T), resulting in a FL-like temperature independence of CM/T ∼ γ(B) (Fig. 3B). Eq. 5 predicts that T/B0.59 also controls the scaling of ΔCM/T. Fig. 3C confirms this expectation for T < 10 K and fields as large as 7 T, over more than three orders of magnitude in the scaling variable x = T=Bz=yb . In addition to the free energy (Eq. 2) that underlies the observed scaling, the internal consistency of the magnetization and specific heat has been verified in SI Text, where we demonstrate that their field and temperature dependencies obey a Maxwell relation, as is required by thermodynamics. An intriguingly simple expression for the scaling function fF(x) can be found that not only describes the T/B0.59 scaling of M/B, but which also reproduces the observed B = 0 and T = 0 limits of both M/B and ΔCM/T. Inspired by similar expressions that were found for heavy fermions CeCu6−xAux (27) and β-YbAlB4 (28), we propose
A
M = B yb
Here c and a are the fitting parameters. The parameter a is depffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi termined by fitting the maxima T ⋆ =Bz=yb = x⋆ = a2 =ðγ + 1Þ ≈ 3 in dM/dT, where we have used the experimentally determined exponents d = z, z/yb = 0.59 and γ ∼ 1.4. The resulting parameter-free expression is Eq. 6 compared with the experimental data in Fig. 2B, and the agreement is excellent over the full range of x. This scaling function fM(x) implies a certain scaling function for the underlying free energy fF(x), which can in turn be related to the scaling function fC(x) for ΔCM/T. Although the details of this procedure are given in SI Text, its accuracy is demonstrated by the excellent agreement between the deduced expression for the scaled specific heat ΔCM/T = fC(x) and the scaled data (Fig. 3C). The internal consistency of the scaling functions fF(x), fM(x), and fC(x) that is implied by this comparison suggests that specific expressions for the temperature and field dependencies of ΔCM/T and M/B can be deduced in their respective QC (T B) and FL (T B) limits. The experimental scaling relations that have been established here provide a precise formulation of how the quantum critical fluctuations lead to the breakdown of the high field FL phase. In the QC limit, the expected field-independent divergence in χ ac(T, B → 0) ∼ T−γ is recovered (Fig. 1B), as well as the B = 0 logarithmic divergence CM/T ∼ –log(T) (Fig. 3A), albeit with a correction B2T−γ−2 in small magnetic fields. Deep in the FL (T B), the field dependencies of the Pauli susceptibility are
B
D
Fig. 3. Field−temperature scaling and Fermi liquid breakdown: Specific heat. (A) Temperature dependence of the B = 0 specific heat CM/T. An estimate of the phonon contribution has been subtracted to isolate the purely magnetic and electronic specific heat CM/T (22). (B) CM/T measured in different fixed fields, as indicated. (C) Scaling collapse of ΔCM/T = CM(B, T)/T − CM(0, T)/T, as a function of the variable T/B0.59. Red line indicates fit with scaling function fC(x), described in the text. An alternative plot of these data on linear axes appears in Fig. S2. (D) Temperature dependence of the magnetic Grüneisen parameter Γ/B, determined in different fixed fields (SI Text).
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Wu et al.
B
A
Fig. 4. Field-driven Fermi liquid collapse. Field dependencies of χ = dM/dB (A) and CM/T (B) at different temperatures as indicated. Inset in B shows CM/T ∼ −log(B) (red line), although the cross-over to quantum critical behavior at the lowest fields is an inevitable result of the nonzero measuring temperature T = 0.55 K.
given by χðBÞ = dM=dB ∼ Bðd+zÞ=yb −2 (Fig. 4A) and the Sommerfeld coefficient is given by γ(B) = CM(B)/T ∼ −log(B) (Fig. 4B, Inset). By comparing the QC and FL expressions, it is clear that the field dependencies of CM(B)/T at fixed temperatures are nonmonotonic, explaining the maxima that separate the high field FL and low field QC phases in Fig. 4B. The logarithmic field divergence of the T = 0 Sommerfeld coefficient γ(B, T) = CM(B, T)/T (Fig. 4B, Inset) implies a weaker divergence of the quasi-particle mass at the QCP than the power law divergencies that are generally found in other QC systems (29, 30). However, the field divergence of the Pauli susceptibility (Fig. 4A) is too strong to be attributed entirely to that of the quasi-particle mass, reflecting instead a ferromagnetic enhancement of the susceptibility that is echoed in the unusually strong divergence of χ ac(T) ∼ T−1.4 found for B = 0. By the same token, the distinctly different temperature dependencies of C/T and χ(T) also rules out the possibility of a Griffiths phase, where both would have the same critical exponents (14, 31–33). It has been argued that the universal behaviors near different types of QCPs can be established through the divergence of the Grüneisen ratio (34). For field-tuned magnetic systems, the magnetic Grüneisen ratio Γ is given by (SI Text)
B
C
D
[7]
Because the power law divergence of the numerator is much stronger than the logarithmic divergence of the denominator, the magnetic Grüneisen parameter Γ/B is expected to be strongly divergent at low temperatures, at least in low fields. This divergence is confirmed by the experimental data (Fig. 3D), where Γ is calculated indirectly from the measured ac magnetic susceptibility χ ac with Bac = 4.17 Oe and the measured zero field specific heat. The strong temperature divergence for B = 0 establishes the quantum criticality of YFe2Al10, as well as the role of magnetic field as a scaling variable that suppresses the QC fluctuations. The electrical resistivity ρ(T) is a measure of how strongly coupled the quasi-particles are to the QC fluctuations that dominate the magnetic susceptibility and specific heat at low temperatures and fields. We carried out measurements of the anisotropy of ρ(T), using a modified Montgomery technique (35). Fig. 5A shows that there is virtually no resistive anisotropy in the ac plane, where the resistivities ρIka ∼ ρIkc are approximately a factor of 2 larger than the interplanar resistivity ρIkb, even at temperatures where the QC fluctuations no longer contribute to the magnetization or specific heat. This anisotropy is much weaker than that found Wu et al.
A
Fig. 5. Quantum critical electrical resistivity ρ(T). (A) The in-plane resistivities ρIka and ρIkc are larger than the interplanar resistivity ρIkb. Data obtained using the modified Montgomery method (35). (Inset) ρ(T) measured for four different YFe 2Al 10 samples, in each case with Ika. (B) The effects of a 9-T magnetic field on ρ(T ) where Ika. (C ) ρ(T ) measured with Ika in different fields Bka. (D) Scaling collapse of Δρ(B, T ) = ρ(B, T ) − ρ(0, T ) for temperatures 0.2–50 K and fields as large as 14 T. Different samples were used for A–C. We estimate a systematic error of ∼10–20% arises from uncertainties in sample dimensions for each voltage measurement, where the Montgomery method requires at each temperature six voltage measurements with different current directions to arrive at ρIka , ρIka, and ρIka. The T → 0 resistivities for all measured samples are consistent with the systematic errors.
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PHYSICS
Γ ½dðM=BÞ=dT=T T −γ−2 T −3:4 =− ∝ ∼ : B C=T log T log T
in the magnetic susceptibility (Fig. 1B), suggesting that the quasi-particles are only weakly coupled to the critical fluctuations, reflecting instead the part of the quasi-2D character of YFe2Al10 that can be traced to its layered crystal structure and its influence on the underlying electronic structure. The large magnitudes of ρIka,c imply T = 0 sheet resistances for per square Fe atoms in the ac plane Rsquare ≈ 2 kΩ, and considering as well their relatively weak temperature dependencies, we infer that there is quite strong quasi-particle scattering in YFe2Al10, although little variation is found among crystals from different batches (Fig. 5A, Inset), making the role of residual disorder unclear. Both ρIkb and ρIka,c have metallic temperature
dependencies, indicating that the quasi-particles in YFe2Al10 are definitively not localized. The QC fluctuations provide an additional scattering mechanism at low temperatures and fields that is strongest in the ac planes. The T → 0 resistive upturn in YFe2Al10 is suppressed by fields in the ac plane, but it is much less sensitive to fields applied parallel to the b axis (Fig. 5B), the same anisotropy found in the ac susceptibility χ ac (Fig. 1B). The part of the resistive upturn that is diminished with increasing field (Fig. 5C) Δρ(B, T) = ρ(B, T) − ρ(0, T) undergoes a scaling collapse for 0.2 K < T < 50 K and B < 14 T, with the same scaling variable x = T/B0.59 that was found for M/B and CM/T (Fig. 5D), where ΔρðB; TÞ T = fR 0:59 : T B
[8]
This scaling confirms that the resistive upturn in YFe2Al10 results from increasingly strong scattering as the QCP is approached by decreasing either T or B. Theories of quasi-particle scattering from critical fluctuations in ferromagnets find that the specific heat CM(B, T) is proportional to ρ(B, T) (∂ρ/∂T) when long (short) wavelength critical fluctuations dominate the quasi-particle scattering (36, 37). However, the scaling functions fC(x) and fR(x) in Eqs. 4 and 8 are not equal and cannot be simply related, ruling out these scenarios. Spin disorder scattering of quasiparticles from fluctuations of the moments enforces a proportionality between Δρ(B, T) and the magnetization M. This explanation is impossible in YFe2Al10, because the scaling form (Eq. 8) cannot be reduced to a function of the magnetization (Eq. 6), due to the temperature prefactor of the latter. The traditional pictures fail badly in YFe2Al10, suggesting that the fluctuating moments and the conduction electrons may not be entirely separable. Indeed, the low temperature resistive upturn is reminiscent of the Kondo effect in normal metals, and the B = 0 ρ(T) can be satisfactorily fitted (22) by the Kondo expression (38). However, the resistivity for a fixed number of Kondo compensated moments in a conventional metallic host scales with T/B (39, 40), and so a different explanation is needed for the T/B0.59 scaling found in YFe2Al10 (Fig. 5D). The observation of T/B0.59 scaling in the electrical resistivity is evidence that the order parameter fluctuations are coupled to soft quasi-particle modes in YFe2Al10, although the influence of the QCP on the resistivity is much weaker than for the uniform susceptibility or specific heat, confirming that the quasi-particles are not coupled strongly enough to the QC fluctuations to themselves become fully critical. Our primary experimental results in YFe2Al10 are the divergencies in the uniform susceptibility χ ac ∼ T −γ (γ = 1.4) and in the specific heat CM/T ∝ −log(T), whereas field-temperature scaling is found in the magnetization, specific heat, and electrical resistivity, in each case giving νz ≡ z/yb ∼ 0.59. Are these critical exponents consistent with theoretical predictions? We begin with the standard Hertz-Moriya-Millis (HMM) theory of metals near QCPs (2–4), which is a mean-field theory that ignores the spatial and temporal fluctuations of the incipient ordered phase. HMM theory for a d = z = 3 ferromagnet predicts both a divergence of the magnetic susceptibility χ ac ∼ T −γ with γ = 4/3, close to the observed value γ = 1.4 ± 0.05 (Fig. 1B), and CM/T ∼ −log(T) (Fig. 3A). However, the observation of T/B0.59 scaling is itself inconsistent with HMM, where the truncated equation of state B = M/χ 0 + μBM3 with constant μ and χ 0 ∼ T −4/3 implies a scaling variable x = T/B0.5. As is evident from the inset of Fig. 2B, this choice of z/yb = 0.5 leads to a scaling of the magnetic susceptibility data that is markedly inferior to that with z/yb = 0.59 (Fig. S2 E and F). This failure of HMM implies that the critical fluctuations of a T = 0 phase transition are responsible 14092 | www.pnas.org/cgi/doi/10.1073/pnas.1413112111
in YFe2Al10 for the T = 0 divergencies of χ and CM/T and that they can be suppressed by magnetic fields to restore a featureless and conventional metallic state. A field theory approach has been taken to derive values of these QC exponents in clean and disordered ferromagnets in different dimensions d (5, 41). The theory reproduces the logarithmic divergence in C/T that we observe in YFe2Al10, as well as the observed d = z, for disordered ferromagnets with 2 < d < 6. The situation for disordered ferromagnets is complicated because the paramagnons and the diffusive quasi-particle modes have different dynamical exponents zc = d and zq = 2, respectively (5), and thus two different critical time scales may in principle be present. The experimental observation d = z suggests that the order parameter fluctuations may prevail in YFe2Al10, an observation that is supported by the noncriticality of the overall resistivity. However, we stress that the experimental values of the susceptibility exponent γ = 1.4 and the scaling variable νz ≡ z/yb ∼ 0.59 are not reproduced in this theory. The need to treat order parameter fluctuations and soft quasiparticles on equal footing greatly increases the difficulty of theoretical analysis in QC metallic ferromagnets, where a local Ginzburg-Landau effective action for the order parameter field does not exist (5). As this controversy continues, we resorted instead to a phenomenological description of the T/B0.59 scaling that was observed in YFe2Al10, based on a generic free energy with hyperscaling, where a single time scale was assumed to become critical. Augmented by a specific expression for the T/B0.59 scaling function, we reproduced the observed field and temperature dependencies of the quantum critical magnetization and specific heat using a single set of critical exponents (d = z, γ = 1.4, νz = 0.59). The general success of this heuristic analysis, based on the hypothesis of hyperscaling, suggests that YFe2Al10 is a system that is below its upper critical dimension, perhaps due to its 2D character. Hyperscaling generally implies universality, where the form of the free energy does not depend on the details of the system, implying that critical points can be classified into universality classes based on the values of the critical exponents themselves. We propose that YFe2Al10 is, to our knowledge, the first confirmed member of such a universality class. Mean field theories predict that the collapse of ferromagnetic order is via a first-order transition (5, 6) in the absence of disorder or by a continuous mean field transition in the presence of disorder (7–10). According to ref. 5, it remains the case when strong quantum fluctuations are present, as may be realized in low-dimensional systems. The data presented here establish that, although YFe2Al10 does not order, strong divergencies in the uniform susceptibility and specific heat indicate QC fluctuations are indeed dominant as this system approaches a T = 0 ferromagnetic transition. Of course, it would be of great interest to use compositional modifications to drive YFe2Al10 to the onset of ferromagnetic order, provided this could be done without introducing significant disorder. Whether or not this putative TC = 0 transition is ultimately found to be first order, it now seems likely that ferromagnetic quantum criticality is protected in low-dimensional systems like quasi-2D YFe2Al10, as it is in quasi-one dimensional YbNi4(P1−xAsx)2 (42). The evidence presented here shows that YFe2Al10 is a rare example of a quasi-2D metallic system where critical fluctuations associated with a TC = 0 ferromagnetic transition dominate the specific heat and uniform susceptibility at the lowest temperatures and fields, without the need for compositional or pressure tuning. Our magnetization, specific heat, and electrical resistivity measurements have allowed us to propose and experimentally establish an expression for an underlying QC free energy that unifies the scaling found in the susceptibility, specific heat, and the QC part of the resistivity while assuring a strong divergence of the magnetic Grüneisen parameter. These experimental results extend a challenge to theory to develop new frameworks in Wu et al.
ACKNOWLEDGMENTS. We thank M. Garst, M. Brando, and F. Steglich for useful discussions. Work at Brookhaven National Laboratory was carried out under the auspices of US Department of Energy, Office of Basic Energy Sciences, Contract DE-AC02-98CH1886.
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PHYSICS
which to understand the emergence of ferromagnetic order and, in particular, its impact on the underlying quasi-particles in metallic systems.
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PNAS | September 30, 2014 | vol. 111 | no. 39 | 14093
