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Formation of a Nematic Fluid at High Fields in Sr3Ru2O7

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Science  12 Jan 2007:
Vol. 315, Issue 5809, pp. 214-217
DOI: 10.1126/science.1134796

Abstract

In principle, a complex assembly of strongly interacting electrons can self-organize into a wide variety of collective states, but relatively few such states have been identified in practice. We report that, in the close vicinity of a metamagnetic quantum critical point, high-purity strontium ruthenate Sr3Ru2O7 possesses a large magnetoresistive anisotropy, consistent with the existence of an electronic nematic fluid. We discuss a striking phenomenological similarity between our observations and those made in high-purity two-dimensional electron fluids in gallium arsenide devices.

In the standard materials that form the basis of most of today's electronic technology, the Hamiltonian for the outer electrons is dominated by the attraction to the ions of the crystalline lattice. In “strongly correlated” materials, this is no longer true. The Coulomb interaction between electrons is large, so it might be expected to add a large term to the Hamiltonian that is not necessarily strongly related to the periodic potential. However, in most correlated electron systems studied to date, the many-electron collective states still retain strong links with the lattice, and the range of “correlated electron matter” identified so far is considerably less diverse than should, in principle, be possible. In itinerant systems, it almost always consists of electron liquids such as Fermi liquids or superfluids that respect the lattice symmetry; the identification of super-conductors in which the condensate breaks some lattice symmetries has been one of the triumphs of the field. [For a recent review, see (1).]

In recent years, there have been proposals that even more exotic electronic liquids might be observable. In an analogy with the nematic state of liquid crystals, which is characterized by orientational but not positional order, it might be possible to form nematic liquids in electronic systems with strong correlations (2). In the broadest sense, a correlated electron nematic is characterized by a lowering of rotational symmetry in its itinerant properties that is not simply a consequence of a symmetry lowering of the lattice.

We report electrical transport phenomena that show that the correlated electron system possesses this key property of a nematic fluid. In previous work, we have argued that a novel quantum phase forms in the vicinity of a metamagnetic quantum critical point in the correlated electron oxide Sr3Ru2O7 (35). Here, we show that this state is accompanied by pronounced magnetoresistive anisotropies that have two-fold symmetry and can be aligned using modest in-plane magnetic fields. Even in the presence of these two-fold anisotropies, neutron single-crystal diffraction resolves no change from the initial square symmetry of the lattice. The overall phenomenology of our observations bears a striking resemblance to that observed in gallium arsenide (GaAs) devices near the high-field limit (69), which suggests that the nematic phenomena previously thought to be specific to close proximity to a fractional quantum Hall state may be more general.

The single crystals used in the present work were grown in an image furnace with techniques described fully in (10). All transport data shown are measurements of the in-plane magnetoresistivity, denoted ρ (11). Crystal purity is crucial in Sr3Ru2O7. For a residual resistivity ρo ≈ 3 μΩ·cm (corresponding to a mean free path of ∼300 Å or less), the phase diagram contains a quantum critical point (QCP) that can be accessed by the application of a magnetic field of ∼7.8 T parallel to the crystalline c axis and the effects of which have now been studied with a variety of experimental probes (1217). As the purity is increased, first-order phase transitions appear as the QCP is approached, and measurements of at least five thermodynamic and transport properties contain features whose loci enclose a well-defined region of the phase diagram in the vicinity of the QCP. Previously, we have argued that these observations are indicative of the formation of a new quantum phase (35). For fields applied parallel to c, ρ has a pronounced anomaly when this phase is entered (Fig. 1A). The two steep “sidewalls” coincide with first-order phase transitions that can be observed using ac susceptibility or magnetization. The angle, θ, of the applied magnetic field to the ab plane of the crystal is a known tuning parameter in Sr3Ru2O7 (14). Previous work has shown that the large resistive anomaly of Fig. 1A disappears rapidly with tilt angle (18), leaving behind much weaker signals in ρ that trace the origin of the two first-order phase transitions as a bifurcation from a single first-order transition at θ ≈ 60° (5), marked by the white arrow in Fig. 2A. At first sight, this seems to contradict the identification of the bounded region between the two first-order lines as a single distinctive phase; there was no evidence that the sudden drop in resistivity with the angle at θ ≈ 80° coincided with any phase boundary. The apparent contradiction can be resolved by postulating the existence of domains of some kind. In such a picture, the behavior shown in Figs. 1A and 2A would be due to these domains producing the extra scattering. The fact that the anomalous scattering disappears so rapidly as θ increases would then be most straightforwardly interpreted as being due to the in-plane component of the tilted magnetic field (Hin-plane) destroying the domains.

Fig. 1.

The two diagonal components ρaa and ρbb of the in-plane magnetoresistivity tensor of a high-purity single crystal of Sr3Ru2O7. (A) For an applied field parallel to the crystalline c axis (with an alignment accuracy of better than 2°), ρaa (black) and ρbb (red) are almost identical. (B) With the crystal tilted such that the field is 13° from c, giving an in-plane component along a, a pronounced anisotropy is seen, with the easy direction for current flow being along b, perpendicular to the in-plane field component (18). If the in-plane field component is aligned along b instead, the easy direction switches to current flow along a.

Fig. 2.

Three-dimensional plots of the magnetoresistivity components ρbb (A) and ρaa (B) of a single crystal of Sr3Ru2O7 as the external magnetic field is rotated from alignment along the crystalline a axis (0°) to alignment along the crystalline c axis (90°), at a constant temperature of 100 mK. The quantity Hc(θ) that normalizes h is the main metamagnetic transition (i.e., the one that dominates the change in the magnetic moment). It varies smoothly from 5.1 T at 0° to 7.87 T at 90°. The same data plotted without this normalization are shown in SOM Text 3.

However, instead of simply removing the anomalous peak, Hin-plane exposed an intrinsic asymmetry of the underlying phase, defining “easy” and “hard” directions for magnetotransport. These easy and hard directions are shown in Fig. 1B. In previous experiments (35), we had worked with the current I // bHin-plane, so that the standard metallic transverse magnetoresistance ρbb could be studied across the whole phase diagram. In that configuration (red traces in Fig. 1), the anomalous scattering disappears rapidly as Hin-plane increases. However, the behavior of ρaa (measured with I // a // Hin-plane) is completely different. As shown by the black traces in Fig. 1, the scattering rate remains high even for an angle at which the anomalous scattering is absent for IHin-plane.

Pronounced in-plane resistive anisotropy can have a number of origins. There is known to be a strong magnetostructural coupling in Sr3Ru2O7, so one possibility is a symmetry-lowering structural phase transition giving the resistive anisotropy due to a corresponding anisotropy in the hopping integrals. Another is the formation of field-alignable magnetic domains such as those examined in the context of itinerant metamagnetism in recent theoretical work (19, 20).

To investigate these possibilities, we first carried out measurements of ρ(H) and magnetic susceptibility χ(H) at temperatures between 20 mK and 4 K on 20 samples from three different batches with a wide variety of shapes, 6 of which were cut from the same piece of crystal (SOM Text 3). Shapes vary from square plates with sides ABC to rectangular plates with side ABC to long cylinders with ABC. In each class of sample, dimension C is aligned with the crystalline c axis, but we deliberately tested combinations of A and B that were aligned or misaligned with the crystalline a and b axes (SOM Text 1). An important result of all these experiments is that, apart from a small region of hysteresis (maximum width ∼80 mT), all the first-order phase boundaries observed in Sr3Ru2O7 were invariant under changes to the sample shape. This firmly rules out demagnetization effects as playing a major role in determining the physics and, hence, directly contradicts predictions concerning magnetic domains (20). To check for a large spontaneous lattice parameter anisotropy, we performed elastic neutron scattering measurements for H // c (SOM Text 2). Within our experimental resolution of 4 × 10–5 Å, we see no evidence of any difference in lattice parameters a and b in the anomalous region.

The experiments described above therefore rule out two of the more standard explanations (magnetic domains and structural change) for the anisotropy shown in Fig. 1B. In Fig. 2, we show the complete magnetoresistive field-angle “phase diagrams” for fields between 4.2 T and 9 T rotating the entire 90° from parallel to a to parallel to c (θ = 90°). The temperature is 100 mK. Figure 2A shows the magnetoresistivity for the easy direction, ρbb, and Fig. 2B that for the hard direction, ρaa.

Figure 3 shows typical data for the difference in ρ between the hard and easy directions (main figure) and the temperature dependence of ρ in a field of 7.4 T along both directions (upper inset). Along the easy direction, standard metallic behavior is seen, whereas a nonmetallic temperature dependence is seen for the hard direction [in agreement with results reported previously (3, 4)]. In the presence of such temperature-dependent anisotropy, it is reasonable to plot the difference between ρaa and ρbb, normalized by their sum, as a phenomenological order parameter. This is done in the lower inset to Fig. 3 at a field applied at θ =72°.

Fig. 3.

The temperature dependence of the difference between ρaa and ρbb for fields applied at θ = 72° such that the in-plane field component lies along a. (Upper inset) The temperature dependence of ρaa (black) and ρbb (red) for μoH = 7.4 T applied in the direction specified above. For this field orientation, ρbb has a clearly metallic temperature dependence, whereas ρaa has the mildly nonmetallic temperature dependence previously reported for H // c in (3, 4). If the in-plane field component is instead oriented along b, ρbb and ρbb switch in both magnitude and temperature dependence. (Lower inset) The temperature dependence of the difference between the two magnetoresistivities shown in the upper inset, normalized by their sum, which is similar to that expected for the order parameter associated with a continuous thermal phase transition.

Several striking features are evident from our data: (i) A key finding is that strong scattering can be observed in the whole region of the phase diagram enclosed by the two first-order phase boundaries that bifurcate from a single first-order transition line at θ ≈ 60° (5). If there is an in-plane field component, this scattering becomes strongly anisotropic. The present observations are important because if one interprets the lower inset of Fig. 3 as evidence for an order parameter, one sees that the phase that it describes is bounded (at low temperatures and constant θ) by well-defined first-order phase transitions that are independent of extrinsic parameters such as sample shape. (ii) Although the easy direction in Fig. 2 is for currents passed along the b axis, this is determined by the in-plane field component having been directed along a for the data shown. If the direction of the field is rotated by 90°, the anisotropy is reversed and a becomes the easy direction. Checks show that the rotation is not smooth; the easy direction is either along a or b but cannot be made to lie, for example, at 45°. (iii) As can be seen in the lower inset to Fig. 3, the presence of a (small) symmetry-breaking Hin-plane slightly rounds off the transition in temperature, giving a “tail” above 800 mK. This effect, which highlights the fact that an in-plane field (that breaks rotational symmetry) is conjugate to the order parameter, becomes more pronounced for lower angles, that is, for higher Hin-plane. (iv) Another feature, weaker, broader, but still very noticeable, is seen in the hard direction for θ < 40° and h ≈ 1.2 (Fig. 2B). (v) The anisotropy described in point (iv) is, like the one described in point (i), extremely sensitive to sample purity [(21) and SOM Text 4], strongly suggesting a common origin for the two. Its breadth in field and in temperature (not shown) is also consistent with its proximity to the ab plane, that is, in the presence of a large in-plane magnetic field. Specific heat data taken cooling down at its central field (h ≈ 1.2) show a logarithmic divergence of C/T down to 1 K (21), giving good evidence that this feature, like that for fields parallel to c, is related to incipient quantum criticality.

The combination of susceptibility, neutron scattering, and transport data gives strong evidence for the spontaneous formation of a structured, anisotropic state in the correlated electron fluid as a quantum critical point is approached in Sr3Ru2O7. Although the anisotropy is seen explicitly in the presence of a weak symmetry-breaking in-plane field component, there is compelling evidence that the symmetry breaking of the correlated electron state is spontaneous and exists even for H // c. The data shown in Fig. 1A can be reconciled with those at other parts of the phase diagram (Fig. 1B and Fig. 2) if the hard axis is randomly oriented along the a and b directions in different regions of the sample, leading to overall isotropy despite strong local anisotropy. Such behavior is commonplace in symmetry-broken states, for example, in simple ferromagnets in zero applied magnetic field (22).

Interesting comparisons can be made between the present data and those in other correlated systems. In-plane transport anisotropies have been observed in both cuprates and manganites. In those systems, the crystals are always orthorhombic, but in some cases the anisotropy increases while the degree of orthorhombicity decreases, and strong anomalies have been seen in the Hall effect, both of which have been interpreted as evidence for spontaneous charge-stripe formation (2325). A much stronger similarity exists between our observations and those on two-dimensional (2D) GaAs devices. For example, it was shown (69) that if the devices can be prepared with ultrahigh mobility such that the fractional quantum Hall effect (FQHE) could be observed in the upper two Landau levels, the correlated electron system does not make a simple FQHE-Fermi liquid crossover as the field is reduced and the filling is increased. In the N =2 to N = 5 Landau levels, the FQHE is replaced by a strong spontaneous resistive anisotropy aligned with principal in-plane crystal axes, even though the crystal symmetry shows no evidence of orthorhombicity. The anisotropy exists even for fields perpendicular to the plane of the device (presumably because of some symmetry-breaking gradients introduced during device fabrication) but can be rotated by 90° by applying a modest inplane field. Just as in the present observations on Sr3Ru2O7, the GaAs data have strong temperature and purity dependencies, and the easy direction lies perpendicular to an in-plane field applied along one of the two relevant crystalline principal axes.

The phenomenological similarity between the GaAs and Sr3Ru2O7 results suggests a common origin for the observations. The disorder dependence gives an important clue, because strong sensitivity to elastic scattering is the signature of a state that is anisotropic in k-space, as is well known in unconventional superconductivity (26). The challenge is how to reconcile what are, apparently, large differences in the starting physical situations. To promote self-organization of a correlated electron system, one must tune the ratio of a potential energy term often summarized by the parameter U (the Coulomb repulsion, which tends to localize) to the kinetic energy, often denoted by W, which tends to delocalize. If this is done “chemically,” by forming new compounds, the change in the U/W ratioisstrongly linked to a change to the electron-lattice coupling, because increasing U and decreasing W also involves increasing the strength of the periodic potential. In the GaAs devices, it is possible to increase the U/W ratio by quenching the kinetic energy by going to very low Landau levels. This leads to a relatively high effective correlation strength without an increase in the effective strength of the periodic potential. In Sr3Ru2O7, similar basic physics is taking place but with a different kind of tuning. It is intrinsically a strongly correlated material, so the starting periodic potential is much larger than in GaAs. However, the existence of an underlying metamagnetic quantum critical point makes the quasiparticle mass m* diverge on the approach to criticality (15, 27). This mass divergence is another route to increasing U/W without increasing the strength of the periodic potential, hence freeing the correlated electron fluid from its rigid link to the underlying lattice.

We have shown that in highly restricted parts of its phase diagram, in proximity to metamagnetic quantum critical points, the electron fluid in Sr3Ru2O7 develops a strong resistive anisotropy, whose hard and easy axes can be interchanged by the application of modest in-plane magnetic fields. The data are consistent with the formation of a nematic state with broken rotational symmetry. Intriguingly, a correlated electron nematic arising from a Pomeranchuk-like Fermi surface distortion (4, 2836) possesses two of the key features that are present in our data and those from GaAs, namely the k-space anisotropy that would give a strong disorder dependence and the possibility of anisotropic transport, intrinsic or through domain formation (37). Whatever the detailed microscopic origin (38), our data suggest that nematic behavior is a feature of ultraclean low-dimensional correlated electron systems in which the bandwidth can be reduced independently of changes to the strength of the periodic potential.

Supporting Online Material

www.sciencemag.org/cgi/content/full/1134796/DC1

SOM Text

Figs. S1 to S4

References and Notes

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