The intriguing idea that strongly interacting electrons can generate spatially inhomogeneous electronic liquid-crystalline phases is over a decade old1,2,3,4,5, but these systems still represent an unexplored frontier of condensed-matter physics. One reason is that visualization of the many-body quantum states generated by the strong interactions, and of the resulting electronic phases, has not been achieved. Soft condensed-matter physics was transformed by microscopies that enabled imaging of real-space structures and patterns. A candidate technique for obtaining equivalent data in the purely electronic systems is spectroscopic imaging scanning tunnelling microscopy (SI-STM). The core challenge is to detect the tenuous but ‘heavy’ momentum (k)-space components of the many-body electronic state simultaneously with its real-space constituents. Sr3Ru2O7 provides a particularly exciting opportunity to address these issues. It possesses a very strongly renormalized ‘heavy’ d-electron Fermi liquid6,7 and exhibits a field-induced transition to an electronic liquid-crystalline phase8,9. Finally, as a layered compound, it can be cleaved to present an excellent surface for SI-STM.
The electronic structure of Sr3Ru2O7 is complicated owing to both its bilayer nature and the in-plane crystalline reconstruction caused by rotations of the RuO6 octahedra (Fig. 1, inset). Nevertheless, photoemission studies reveal at least five Fermi-surface pockets6, in agreement with de Haas–van Alphen data7. The most striking result of these k-space spectroscopic measurements is that the intense electron–electron interactions cause bands that are 10–30 times flatter than they would be if quasi-free electrons existed in the system. Thus, although its metallic state is a Fermi liquid, Sr3Ru2O7 is one of the most strongly renormalized heavy d-electron compounds known, and consequently, the spectral weight of k-space excitations should be markedly reduced. This system undergoes a series of metamagnetic transitions in fields between 7.5 and 8.1 T applied parallel to the crystallographic c axis. These transitions enclose an H–T plane region within which a large resistivity enhancement occurs8. Application of further in-plane fields produces an ‘easy’ transport direction for which this enhanced resistivity disappears10. Thus, electronic transport with 180∘ rotational (C2) symmetry exists within this narrow H–T region, whereas the surrounding regions exhibit transport with the expected 90∘ rotational (C4) symmetry. These are the transport characteristics expected of a field-induced electronic nematic10—in this case, one generated from a many-body state of Ru d-electrons comprising both r-space and k-space spectral contributions.
SI-STM is potentially an ideal technique for studying such systems because simultaneous studies of r-space and k-space electronic structure can be carried out. It was pioneered in studies of simple quasi-free-electron systems11,12 but, at least conceptually, might provide a route for imaging the far more complex many-body states of strongly correlated heavy-fermion systems. In SI-STM, measurement of the STM tip–sample differential tunnelling conductance dI/dV (r,V)≡g(r,E=eV) at locations r and sample-bias voltage V, yields an image proportional to the local density of electronic states LDOS(r,E). Moreover, k-space electronic structure elements can be determined simultaneously by using Fourier-transform scanning tunnelling spectroscopy (FT-STS). This is because the spatial modulations in g(r,V) due to interference of quasiparticles scattered by atomic-scale impurities are detectable in g(q,V), the Fourier transform of g(r,V). The k-space states undergoing the scattering interference have been deduced for both quasi-free electrons of noble metals11,12 and the weakly renormalized electronic states of superconducting cuprates13,14,15,16,17,18 in a process dubbed quasiparticle interference (QPI) imaging.
Multiple challenges exist in extending the basic techniques used for simple quasi-free electrons11,12 to many-body states in a strongly correlated heavy-fermion system such as Sr3Ru2O7. The larger the mass renormalization due to electron–electron interactions becomes, the less the one-electron spectral weight in the k-space quasiparticles. As SI-STM is a single-electron spectroscopy, this loss of spectral weight is potentially a severe problem for detectability of k-space states by scattering interference. Another challenge is the complexity of the band structure6,19. In the case of simple metals and the cuprates, the reasons for the success of QPI is the combination of simple, un-renormalized band structures and a large joint density of states (JDOS) to enhance the intensity of the experimental signal11,12,13,14,15. However, for multiple, topographically complex bands, one cannot assume that the equivalent process is even possible. Thus, detection of the heavy quasiparticles of a correlated many-body state using QPI seems extremely difficult, and, perhaps for this reason, it had not been attempted until now. Here, however, we are motivated by the possibility that if both the r-space and k-space electronic structure of Sr3Ru2O7 can be imaged using techniques that may be applied in high magnetic fields, this would open the way for this material to become a ‘hydrogen atom’ for the study of the many-body states forming electronic liquid crystals.
To explore these issues, we have carried out a SI-STM study on single crystals of Sr3Ru2O7. The necessary scattering centres are introduced by replacing Ru atoms with Ti. To maximize the signal-to-background ratio of QPI, we needed to introduce a sufficient number of scattering centres to generate the interference patterns but sufficiently few not to significantly alter the electronic structure. Accordingly, we grew single crystals of Sr3(Ru0.99Ti0.01)2O7, that is, with 1% of the Ru atoms replaced by Ti.
Our samples are cooled in a cryogenic ultrahigh vacuum, cleaved between adjoining SrO planes, and inserted in the STM head at or below 4.2 K. Figure 1 shows a 28-nm-square topographic image of Sr3(Ru0.99Ti0.01)2O7. The quality of this image, and those previously obtained from Sr3Ru2O7 (ref. 20), is related to a key advantage of Ruddlesden–Popper ruthenates for SI-STM studies of complex electronic matter. Each contains square planar Ru4+ coordinated with four oxygen atoms in a basic RuO2 formula unit. This means that, unlike in the cuprates, all planes in the material are charge-neutral, maximizing the possibility of obtaining cleaved surfaces that are bulk-representative. A top-view schematic diagram of the crystal structure is shown in the inset; it illustrates how Ti atoms substitute Ru in the RuO2 plane and how the reconstruction consists of sequential rotations of the RuO6 octahedra about the c axis. In fact, the topographic data in Fig. 1 reveal all of these effects directly. Ti impurities from both upper and lower RuO layers are visible as dark and bright features respectively. The density of these features is in good agreement with the doped Ti density and they are absent in pure crystals. Moreover, the two different rotation angles corresponding to scattering from impurities in the two sublattices of each layer are also observable (labelled α and β for example in Fig. 1a).
Figure 1b shows the sub-unit-cell (see Supplementary Information) differential conductance mapping g(r,E) measurements of a typical 5×5 Å2 field of view far from Ti impurities. Clear sub-atomic features are seen in the spatial arrangements of electronic structure, and there are several energies revealing r-space patterns expected of dxz,dyz orbitals. The type of atomic-resolution orbital occupancy imaging recently demonstrated in cuprates21 may now become applicable to the Ru 4d states of Sr3Ru2O7. The rate at which the r-space patterns evolve with energy is remarkable but could perhaps have been anticipated because, as we have emphasized, the electronic bands in Sr3Ru2O7 are strongly renormalized by electron correlations6,7. In bands of mixed orbital character, the orbitals making the dominant contribution change as a function of k and energy. The images presented in Fig. 1b therefore probably represent the first real-space sub-atomic imaging of the energy evolution of strongly renormalized Wannier states; as such they invite more detailed theoretical investigation.
Fig. 2a–f shows a sequence of 28-nm-square g(r,E) images taken simultaneously and focused on the filled states. The most striking features are the strong g(r,E) oscillations near Ti atoms. Their dispersion is immediately obvious in the changes of g(r,E) in Fig. 2a–f. These changes occur because the LDOS(r,E) modulations exhibit characteristic wave vectors q(E) that disperse rapidly in energy. Figure 2g–l shows g(q,E), the Fourier-transformed images of Fig. 2a–f, respectively. The reciprocal-unit-cell locations of the Ru atoms are labelled as (0,2π/a0); (2π/a0,0) where a0 is the shortest inter Ru distance. Raw g(q,E) are presented in Supplementary Fig. S1. The dispersions of the QPI patterns are manifest by the rapidly changing g(q,E), as is the complexity of scattering processes. These data reveal, for the first time, scattering interference of very heavy d-electron quasiparticles at individual impurity atoms.
To identify the bands responsible for these interferences, we consider the Sr3Ru2O7 band structure6,19 as outlined in Fig. 3a, where it would seem that numerous scattering q-vectors are possible. However, careful examination of the lengths, direction and dispersions of all of the observed q(E) in Fig. 2 directs our focus to the α2 band, which has primarily 4dxz,dyz orbital character. Its k(E∼Ef) contour is shown in comparison with the local-density-approximation band structure6 in Fig. 3a, and throughout the larger unreconstructed Brillouin zone in Fig. 3b. The α2-band spectral function A(k,E) is calculated from the E(k) estimated from angle-resolved photoemission spectroscopy (ARPES; see Supplementary Information) and the resulting regions of high JDOS, which dominate the QPI processes14,16,17 as estimated by taking the autocorrelation22,23 of A(k,E) are shown in Fig. 3c. Overlaid are a sequence of inequivalent scattering vectors qi:i=1,…,10 between the regions of high JDOS; these same vectors are shown in Fig. 3b to indicate how they interconnect the α2 band. Remarkably, all of the inequivalent maxima in the complex g(q,E) between 5 meV and 13 meV below EF can be accounted for using this model. This is shown, for example in Fig. 3d, by overlaying the positions of these same qi on g(q,E=−9 meV) from the model as open circles. The validity of the α2 QPI model is then borne out by the good agreement and highly overdetermined internal consistency between the model in Fig. 3b and data in Fig. 3d. Moreover, when the dispersions of the two basic vectors q1 and q2 (Fig. 3b–d) are measured from Fig. 2g–l and plotted in Fig. 3e, they agree well with the directly measured dispersion of the α2 band from ARPES (see Supplementary Information). In fact, dispersions of the qi(E) (which will be described elsewhere) are also in good agreement with these conclusions. Thus, we find that the α2 band is detectable directly by FT-STS in Sr3Ru2O7 and that it dominates QPI at Ti atoms. The velocities identified from the dispersion in Fig. 3e are approximately 1×104 m s−1, in good agreement with estimates based on ARPES and de Haas–van Alphen data6,7. These are ten times smaller than those seen in the α or β bands of Sr2RuO4, confirming the high mass and strong correlations of the quasiparticles in Sr3Ru2O7. As to why the α2 band should be predominant in QPI, one possibility is that strong momentum selective scattering effects due primarily to the correlations have a role24.
These observations open the way for powerful modern SI-STM techniques13,14,15,16,17,18 to be applied to many-body states not just in Sr3Ru2O7, but also to a much wider class of strongly correlated heavy-fermion systems, including those that are spatially inhomogeneous. With regard to the problem of electronic nematicity in Sr3Ru2O7, studies using the techniques introduced here but at high magnetic fields will be crucial. The current findings give at least two routes to microscopic investigation of a nematic state. First, the ability to resolve sub-atomic structure representative of the underlying orbitals offers the possibility of direct observation of lowered symmetry in their arrangement within the nematic region. Second, it is conceivable that the heavy d-electron QPI response will also show lowered symmetry. The density of Ti atoms chosen for the work reported here increases the residual resistivity of Sr3Ru2O7 to ∼10 μΩ cm. This does not cause either an electronic or structural phase transition or a change to the Fermi energy, but would wash out the bulk signals of nematicity. However, the Ti level in the crystals can be controlled with high precision, and now that the basic characteristics of heavy d-electron QPI have been established, it will be possible to study samples with an order of magnitude fewer Ti inclusions.
The observations that the strongest QPI signals come from the α2 band, and that r-space structures associated with the same dxy,dyz orbitals are detectable, are both potentially important. Early theoretical studies of the metamagnetism in Sr3Ru2O7 focused on a proposed van Hove singularity and explained metamagnetism as being due to spin-splitting of its high density of states25. After the discovery of coexisting nematicity10, a heuristic quadrupole electron–electron interaction connecting metamagnetism and nematicity was proposed26. More recent theoretical advances propose spatially heterogeneous states caused by these singularities27, or model the fundamental microscopic cause of metamagnetic nematicity in Sr3Ru2O7 as being due to the natural uni-directionality of underlying dxz dyz orbitals and the related quasi-one-dimensional van Hove singularity28,29. Thus, in addition to our results having broad significance to the study of strongly renormalized metals, both the r-space and k-space electronic structure we report here may have relevance to the formation of an intriguing electronic liquid-crystal phase.
The crystals used for this work were grown in St Andrews in a NEC Machinery Corporation Image furnace (SCI-MDH-11020). Batches of Sr3(Ru1−xTix)2O7 were prepared by mixing RuO2, TiO2 and dried SrCO3 powders in the ratio 2.52(1−x):2x:3. The mixed powder was then reacted in air at 1200 ∘C for 12 h, reground, formed into a feed-rod and sintered on a layer of strontium ruthenate bedding powder for 2 h at 1420 ∘C, again in air. The rod was then melted in the image furnace, with crystalline growth seeded from the molten zone using a ruthenate seed crystal, and the polycrystalline rod fed into the molten zone at 25 mm h−1. Gross phase purity was established using X-ray diffraction of powdered segments of the resultant single crystals and electron probe microanalysis was used to determine the Ti concentration. The residual resistivity was measured using standard four-probe techniques and an a.c. method. Typical currents were 100 μA and frequencies were in the range 50–100 Hz. Data were taken in three cryostats: a continuous flow 4He cryotstat with a base temperature of 3.8 K, an adiabatic demagnetization refrigerator (Cambridge Magnetic Refrigeration) with a base temperature of 100 mK and a 3He–4He dilution refrigerator (Oxford Instruments) with a base temperature of approximately 30 mK.
Magnetic characterization was used to measure and then minimize the levels of SrRuO3, Sr2RuO4 and Sr4Ru3O10 impurity phases, which are always a challenge when growing this member of the Ruddlesden–Popper series of layered perovskite ruthenates. Sr2RuO4 inclusions can be identified by studying the diamagnetic a.c. response on entering the superconducting state. For quantitative work, care must be taken to operate in the low-demagnetizing-factor configuration in which the a.c. field is applied parallel to the ab planes. SrRuO3 and Sr4Ru3O10 are ferromagnetic with different Curie temperatures, so d.c. magnetization measurements in a commercial superconducting quantum interference device magnetometer (Quantum Design MPMS) allow a sensitive determination of small impurity contents. Full details of the characterization procedures can be found in ref. 30.
To obtain QPI in q-space, we calculate the absolute intensity of the two-dimensional fast Fourier transform of the conductivity map g(r,E). This shows an extremely bright spot in the centre due to long-wavelength fluctuations. To remove these very small q-vector elements in q-space, we first subtract the data value averaged with a Gaussian filter width of about 6 Å from the original conductance value at every pixel. The two-dimensional Fourier transform of this resulting value effectively gets rid of the low-q vector components, showing clearer individual QPI features.
Owing to drift during a scan, the raw g(r,E) does not show perfect square symmetry, but is slightly elongated in one direction. To correct this effect, we use a conformal mapping algorithm to make g(r,E) a square. Here, we use information about the atomic Bragg peak positions to calculate the necessary parameters of the conformal transformation. Although the resulting Fourier-transform g(q,E) shows clear dispersive QPI features with fourfold symmetry, we attempted to enhance the signal-to-noise ratio even further. We selectively averaged symmetrical data points over two octants of g(q,E) showing more discernible intensity peaks which, in this study, happened to be along the vertical direction. Mathematically this operation corresponds to symmetrically averaging (averaging only among symmetrically equivalent points under fourfold symmetry) over the q-space region, which satisfies |qy|>|qx|, and using only this part to construct complete g(q,E).
We acknowledge and thank E. Fradkin, T. Hanaguri, C. A. Hooley, E.-A. Kim, S. A. Kivelson, Y. Kohsaka, M. J. Lawler, A. J. Millis, S. Raghu, T. M. Rice, S. Sachdev, K. M. Shen, H. Takagi, A. Tamai and F.-C. Zhang for helpful discussions and communications. These studies are carried out with support from NSF DMR-0520404 to the Cornell Center for Materials Research, from Brookhaven National Laboratory and from the UK EPSRC, Royal Society and Leverhulme Trust.