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Observation of a nematic quantum Hall liquid on the surface of bismuth

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Science  21 Oct 2016:
Vol. 354, Issue 6310, pp. 316-321
DOI: 10.1126/science.aag1715

Relating interactions and nematicity

The electronic system in a strongly correlated material can sometimes be less symmetrical than the underlying crystal lattice. This loss of symmetry, caused by interactions and dubbed electronic nematicity, has been observed in a number of exotic materials. However, establishing a direct connection between the interactions and nematicity is tricky. Feldman et al. used scanning tunneling microscopy to image the wave functions of electrons on the surface of bismuth placed in an external magnetic field. The exchange interactions in the material caused a loss of symmetry, which was reflected in the orientations of the electrons' elliptical orbits.

Science, this issue p. 316

Abstract

Nematic quantum fluids with wave functions that break the underlying crystalline symmetry can form in interacting electronic systems. We examined the quantum Hall states that arise in high magnetic fields from anisotropic hole pockets on the Bi(111) surface. Spectroscopy performed with a scanning tunneling microscope showed that a combination of single-particle effects and many-body Coulomb interactions lift the six-fold Landau level (LL) degeneracy to form three valley-polarized quantum Hall states. We imaged the resulting anisotropic LL wave functions and found that they have a different orientation for each broken-symmetry state. The wave functions correspond to those expected from pairs of hole valleys and provide a direct spatial signature of a nematic electronic phase.

Nematic electronic states represent an intriguing class of broken-symmetry phases that can spontaneously form as a result of electronic correlations (1, 2). They are characterized by reduced rotational symmetry relative to the underlying crystal lattice and have attracted considerable interest in systems such as two-dimensional electron gases (2DEGs) (35), strontium ruthenate (6), and high-temperature superconductors (712). The sensitivity of electronic nematic phases to disorder results in short-range ordering and the formation of domains, making them difficult to study using global measurements that average over microscopic configurations. The effect of perturbations, such as crystalline strain, may be used to show a propensity for nematic order—that is, to provide evidence that vestiges of nematic behavior survive even in the presence of material imperfections (1). However, it is difficult to quantitatively correlate the experimental evidence of ordering with a microscopic description of the electronic states and the interactions responsible for nematic behavior. To put the study of nematic electronic phases on more quantitative ground, it is therefore important not only to perform local measurements, but also to find a material system for which theory can fully characterize the underlying broken-symmetry states and the electronic interactions.

Multivalley 2DEGs with anisotropic band structure have been anticipated as a model platform to explore nematic order in the quantum Hall regime (1317). The key idea is that Coulomb interactions can spontaneously lift the valley degeneracy in materials with low disorder and thereby break rotational symmetry. In contrast to previously studied metallic nematic phases, this leads to a gapped nematic state with quantized Hall conductance. We examined such a 2DEG on the surface of single crystals of bismuth (Bi), which is one of the cleanest electronic systems, with a bulk mean free path reaching 1 mm at low temperatures (18). Interest in Bi has recently been rekindled by bulk measurements showing phase transitions and anisotropic behavior, possibly related to nematic electronic phenomena, in the presence of large magnetic fields (1922). We focus here on the (111) surface of Bi, for which strong Rashba spin-orbit coupling results in a rich 2DEG consisting of spin-split surface states that produce multiple electron and hole pockets (23, 24). Scanning tunneling microscope (STM) images (Fig. 1A) show that the in situ cleaved Bi(111) surface has large (>200 nm × 200 nm) atomically ordered terraces that are separated by steps oriented along high-symmetry crystallographic directions (25). Angle-resolved photoemission spectroscopy (ARPES) measurements (23, 24, 26, 27) of this surface show that its Fermi surface consists of a hexagonal electron pocket at the Γ point, three additional elongated electron pockets around the M points, and six anisotropic hole pockets along the Γ-M directions (Fig. 1B, inset). The multiply degenerate anisotropic valleys and the low disorder of the Bi(111) surface make it an ideal system to search for nematic electronic behavior using the STM.

Fig. 1 Landau levels (LLs) of the Bi(111) surface states.

(A) A typical cleaved Bi(111) surface, with crystallographic axes labeled. The data in (E) and (G) are an average of spectra measured along the blue line, and the conductance maps in Fig. 3 were performed in the area denoted by the black box. Surface defects are circled in purple; the inset shows a zoom-in on one defect (inset z height scale, 1.3 Å). (B) Conductance G as a function of energy E at magnetic field B = 0 (blue) and at 14 T (red). The curves are offset by 0.5 for clarity. At B = 0, the data are taken at temperature T ≈ 4 K. All other data throughout the manuscript are measured at 250 mK. The inset is a diagram of the Bi(111) first Brillouin zone, showing the electron (purple) and hole (blue) Fermi pockets of the surface states. (C) Landau fan diagram of G(E, B) that shows crossing electron- and hole-like LLs. The data are averaged over a 20-nm line, with individual spectra showing almost no spatial variation on this energy scale. Select orbital indices Ne and Nh of the respective electron and hole LLs are labeled. (D) Higher-energy resolution measurement of G(E, B) that clearly shows Fermi-level pinning of each hole LL. (E) High-resolution measurement of G(E, B) in a region where the LLs corresponding to Nh = 3, 4, and 5 each show splitting into a two-fold degenerate and a four-fold degenerate LL peak. Data are averaged over the blue line in (A). (F) Line cut of spectra showing strain-induced splitting of the six-fold degenerate Nh = 3 LL into two or three peaks, depending on position. Numbers in parentheses denote the degeneracy of each broken-symmetry state. (G) Zoom-in on G(E, B) in the same location as in (E). The four-fold degenerate peak further splits into two distinct LLs as it crosses the Fermi level, indicating broken symmetry states arising from exchange interactions. Arrows mark Δstrain and Δexch.

In the absence of magnetic field, spectroscopic measurements of the Bi(111) surface with the STM (Fig. 1B) show features in the tunneling conductance G that are related to van Hove singularities of the density of states (DOS), such as the sharp peak at energy E = 220 meV and the abrupt drop at 33 meV. These features correspond to the upper band edges of the surface states along the Γ-M direction (25, 28, 29). In the presence of a large magnetic field B, the electron and hole states of the Bi(111) surface are quantized into Landau levels (LLs), each with degeneracy geB/h, where e is the electron charge, h is Planck’s constant, and g accounts for the degeneracy arising from the valley degree of freedom (g = 6 for holes). At high magnetic field, the STM spectra show a series of sharp peaks (Fig. 1B) whose evolution with magnetic field can be used to distinguish between electron- and hole-like LLs, which disperse in energy with positive or negative slopes, respectively, as a function of magnetic field (Fig. 1, C and D). They do not exhibit avoided crossings, and the total conductance is additive when they cross, which suggests independent tunneling into each LL. LL spectroscopy on thin Bi(111) films was recently reported (28) but did not show evidence of symmetry breaking, which is the focus of our work.

Our first key observation is that the surface-state LLs do not disperse linearly with magnetic field. Instead, they are pinned to the Fermi level until they are fully occupied, as is clearly shown for the hole states in Fig. 1D. Such behavior is rarely observed in LL spectroscopy of ungated samples performed using a STM (30), and it indicates that the surface charge density is held constant in our system. Electron LLs exhibit pinning only when there are no proximal hole states, whereas they otherwise cross straight through the hole LLs at the Fermi level. This difference in behavior signals an intriguing competition between electron- and hole-like states in a magnetic field, and suggests charge rearrangement between pockets (29). We focus below on the hole states, for which the orbital index Nh is straightforward to assign, with the highest-energy peak corresponding to Nh = 0 closely matched to the zero-field drop in conductance at 33 meV. Using the values of the field and filling factor at which LLs cross the Fermi level, we determine the hole surface density to be p ≈ 7.1 × 1012 cm−2 (29).

High-resolution spectroscopic measurements provide an indication that both single-particle effects and electron-electron interactions break the six-fold symmetry of the hole LLs. Evidence of symmetry breaking can be seen in Fig. 1E, which shows the field evolution of the conductance spectra in one region of the sample where the LLs corresponding to Nh = 3, 4, and 5 are each split into two peaks with different amplitudes, indicating a lifting of the six-fold valley degeneracy of each level to form two- and four-fold degenerate LLs. The fact that the splitting (characterized by a gap Δstrain) occurs away from the Fermi level indicates that it is a single-particle effect. The very weak dependence of Δstrain on magnetic field and orbital index and the fact that we observe different magnitude gaps in different regions of the sample suggest that local strain underlies this partial symmetry breaking (29). As an illustration of the spatial dependence of this behavior, we show in Fig. 1F a spectroscopic line cut from a region of the sample in which the six-fold degeneracy of the Nh = 3 LL is lifted to produce either two or three broken-symmetry states, depending on location within the sample.

Electron-electron interactions further lift the LL degeneracy and are manifested in spectroscopic measurements by the appearance of energy gaps when the LLs cross the Fermi level. Figure 1G shows a high-resolution measurement of the Fermi-level crossing of the Nh = 4 LL (in the same area as in Fig. 1E), where over a range of 0.5 T, the four-fold degenerate peak develops an exchange energy gap (Δexch = 450 μeV) that is coincident with the Fermi level. Although there are spatial variations in the exact magnitude of the gaps between the broken-symmetry LLs, exchange interactions consistently enhance gaps between LLs that are already split by strain and induce a gap between previously degenerate levels when they cross the Fermi level. The magnitude of the exchange gap is consistent with that estimated theoretically for the hole pockets of Bi(111), and it is not related to an Efros-Shlovskii Coulomb gap (29). These observations demonstrate that a combination of a single-particle effect, likely strain, and many-body interactions lift the six-fold valley degeneracy of the hole LL to produce three broken-symmetry states.

We performed spectroscopic mapping with the STM to directly visualize the underlying quantum Hall wave functions and to demonstrate the breaking of crystalline symmetry in these phases. Conductance maps at energies corresponding to each of the three broken-symmetry hole LLs show anisotropic ellipse-like features that point along high-symmetry crystal axes, with relative angles rotated by 120° with respect to each other (Fig. 2, A to D). The elliptical features are centered on atomic-scale surface defects, and the same defects produce rings in all three directions. This suggests that ellipse orientation is not associated with symmetry breaking from the defect itself, which is further confirmed by atomic-resolution topographs (29). As we show below, the three different directionalities arise from cyclotron orbits in pairs of hole valleys that are elongated in the same direction. More important, such spatially resolved measurements enable us to directly visualize the spontaneous breaking of the LL degeneracy by electron-electron interactions. By tuning the magnetic field to adjust the occupancy of two of the three broken symmetry states, we can contrast spatial maps of the LLs with and without exchange splitting. The measurements in Fig. 2, E to G, obtained in the same region as those in Fig. 2, A to D, show that the elliptical features in the conductance maps can occur as a superposition of two different orientations, indicating that the symmetry between these two orientations is not broken in the absence of an exchange gap. A comparison of Fig. 2F with Fig. 2, B and C, clearly shows that unidirectional elliptical features emerge as the exchange gap opens, providing a direct manifestation of nematic valley-polarized states on the Bi(111) surface.

Fig. 2 Rotational symmetry breaking and local domains of a nematic electronic phase.

(A) Average conductance spectrum at 12.9 T, measured over a 100-nm line cut (which exhibits little spatial dependence) near the start of the line cut in Fig. 1F, showing three broken-symmetry hole LLs, two of which are split by exchange interactions at the Fermi level. (B to D) Spatial maps of conductance normalized by its average value Embedded Image at energies corresponding to the three split hole LL peaks. Ellipses of reduced conductance are centered on surface defects, with different orientations at each energy. (E) Average conductance spectrum at 14 T, measured in the same location as in (A). The spectrum shows restored symmetry of the exchange-split LLs in (A) to produce a four-fold degenerate LL. (F) Spatial map of Embedded Image at the energy of the four-fold degenerate LL peak, which shows ellipses with two orientations. (G) Spatial map of Embedded Image at the energy of the two-fold degenerate LL peak that is split from the four-fold degenerate peak by strain, showing the same unidirectional behavior as in (D). The spatial maps in (F) and (G) are measured in the same area as (B) to (D). (H) Average conductance spectrum (measured over a 100-nm line cut that exhibits little spatial dependence) at 12.9 T in a location about 1 μm away from the region shown in (A) to (G). (I to K) Spatial maps of Embedded Image in the new location at energies corresponding to the three split hole LL peaks. The energetic order of the three directions is different, with the first two orientations switched, demonstrating the presence of domains. For all conductance spectra, the electron LLs are labeled, and the hole LL degeneracy is denoted in parentheses near each peak.

Another key feature of a nematic electronic phase without long-range order is the presence of domains, which we observe in our system by performing spatially resolved spectroscopy with the STM. We find that the sequence in energy of the three broken-symmetry hole LL states can change depending on the location within the sample. An example of this behavior can be seen by contrasting the spectrum and corresponding conductance maps in Fig. 2, A to D, to those measured about 1 μm away, shown in Fig. 2, H to K. These data reveal that the orientations of the two broken-symmetry states corresponding to the first two peaks in the spectra have switched between the two locations on the Bi surface. Thus, our STM measurements not only show that electron-electron interactions drive nematic behavior, but also illustrate the formation of local nematic domains.

We show below that the elliptical features in our STM conductance maps arise from cyclotron orbit wave functions of the broken-symmetry quantum Hall phases that are pinned by surface defects. To characterize these features in detail, we studied them in an area with few surface defects (box in Fig. 1A) and examined their dependence on orbital index at a constant magnetic field (14 T) around the same defects (circled in Fig. 1A). The conductance maps shown in Fig. 3, A to E, were obtained at the energies of the strain-induced broken-symmetry LLs for Nh = 0 to 4, and they revealed concentric ellipses of suppressed conductance similar to those in Fig. 2, with a consistent orientation for all the orbital indices. The size of the outermost ring increased with increasing orbital index, as did the number of concentric rings of suppressed conductance. Around these same surface defects, we observed approximately circular rings in conductance maps measured at the nearby electron LL peak (Fig. 3F), which further confirms that the defects themselves do not break rotational symmetry.

Fig. 3 Isolated anisotropic cyclotron orbits and theoretical modeling.

(A to E) Spatial maps of Embedded Image at 14 T in the area denoted by the black box in Fig. 1A, at energies corresponding to the strain-induced broken-symmetry hole LL for orbital indices Nh = 0 to 4. Isolated anisotropic cyclotron orbits are present around surface defects. (F) Spatial map of Embedded Image in the same area at the energy of the Ne = 8 LL, showing circular rings of suppressed conductance (black arrows) that occur around the same surface defects. The weak elliptical feature around the lower defect is related to a missing cyclotron orbit from the Nh = 3 LL at a nearby energy. The trapezoidal feature in the background conductance results from the shape of the terrace because the LL visibility is suppressed near step edges. (G) Amplitude Embedded Image of the m = Nh = 4 cyclotron orbit wave function. (H to K) Simulated maps of the expected conductance, Embedded Image, with individual cyclotron orbits centered on the surface defects circled in Fig. 1A. The size and shape of the simulated conductance are a good match to the data in (B) to (E). (L) Semimajor axis size of the cyclotron orbits for Nh = 4 (blue) and ring size of those from electron LLs near the Fermi level (red) as a function of magnetic field. Dashed lines are fits to the field dependence of the extracted sizes.

The rings of suppressed conductance for both electron and hole LLs can be understood as a consequence of cyclotron orbits that are shifted in energy because of the sharp potential produced by the atomic surface defects. In the symmetric gauge, the cyclotron orbits of each LL can be labeled by a second orbital quantum number m (31, 32). Only the m = N cyclotron orbit has weight at the defect, so it is the only state whose energy is shifted by the defect potential, which we modeled as a delta function (29). Without the defect, conductance maps measured at the LL peak would include DOS contributions from all cyclotron orbits, and no spatial variation would be expected. However, because the m = N orbit is shifted to a different energy by the defect, it becomes visible as a decreased conductance in the shape of the wave function when measurements are performed at the unperturbed LL energy.

A theoretical model of cyclotron orbit wave functions for the surface states of Bi(111) can be used to capture the elliptical features in the STM conductance maps near individual defects with excellent accuracy. The anisotropy of the surface state hole pockets is reflected in their cyclotron orbit wave function, as exemplified by the m = Nh = 4 state, whose amplitude Embedded Image (where lB = Embedded Image is the magnetic length) is plotted in Fig. 3G. The number of elliptical features in these wave functions increases with orbital index and is a reflection of the spatial oscillations of the m = Nh wave function, which is proportional to a Laguerre polynomial with Nh + 1 peaks (29). Using the defects marked in Fig. 1A as the centers of such cyclotron orbits, we simulated the expected conductance pattern by subtracting Embedded Image from a uniform background (Fig. 3, H to K). The similarity to the experimental data in Fig. 3, B to E, for different Nh states is remarkable, especially given that the only adjustable fit parameter is the anisotropy of the hole pocket effective mass. We extract a ratio of 5 for the hole pocket anisotropy, in good agreement with previous ARPES measurements (23, 26, 27) and calculations (33). Our model also captures the field dependence of the cyclotron orbit size of the hole LL for Nh = 4, as well as that of the electron LLs near the Fermi level. Figure 3L shows the experimentally measured size of the outermost rings for both sets of orbits. They follow the expected Embedded Image or 1/B scaling for hole and electron LLs, respectively, which reflects the dependence of the cyclotron orbit wave functions on magnetic length and orbital index (29).

On the basis of the model described above, we anticipate that the suppression we have detected in the conductance maps at the LL peaks should be accompanied by an enhanced conductance relative to the background at other energies. An example of such contrast reversal is shown in Fig. 4, A to I, which displays conductance maps near an isolated defect over a range of energies within one broken-symmetry LL peak with orbital index Nh = 4 (Fig. 4J). The maps measured at the LL peak and at higher energies show ellipses of suppressed conductance that correspond to a missing cyclotron orbit, whereas at lower energies, such maps show ellipses of higher conductance that indicate the lower energy to which this orbit has been shifted by the defect potential. This reversal of the contrast is clearly illustrated by the energy-averaged line cuts shown in Fig. 4K, which demonstrate that the cyclotron orbit energy has been lowered by about 300 μV by this particular defect. Examining different defects, we have found evidence for both attractive and repulsive potentials from the contrast reversal in the conductance maps (29).

Fig. 4 Energy shift of the cyclotron orbits.

(A to I) Spatial maps of Embedded Image around an isolated impurity at B = 10 T with energy spaced by 100 μeV throughout one broken-symmetry Nh = 4 LL peak. These maps show the shift to lower energy of the m = N cyclotron orbit. (J) Corresponding conductance spectrum (averaged over a 12 nm × 2.5 nm area centered about 5 nm underneath the defect) marked with colored circles for each mapped energy. (K) Oscillations of Embedded Image along the semiminor axis, averaged over 100 and 200 μeV (blue) and over 400 and 500 μeV (red), respectively, highlighting the contrast reversal in the maps.

Our measurements are in the clean regime where signatures of isolated cyclotron orbits are visible around individual defects, in contrast to previous studies of DOS modulations from drift states moving along equipotential lines in the disordered limit (3436). Cyclotron orbits that are shifted in energy by an isolated defect have been explored in graphene (32), and other measurements have indirectly probed the size and shape of cyclotron orbits (3638) by examining LL spatial dependence caused by potential modulations. We performed direct two-dimensional mapping of isolated cyclotron orbits, which enabled us to visualize nematic order on the Bi(111) surface, where the anisotropic hole mass leads to anisotropic cyclotron orbits.

The Bi(111) 2DEG represents an interesting venue to explore electron-electron interactions within anisotropic valleys. The ability to bring the lowest hole-like LL to the Fermi level, either by external gating or doping, may allow for direct visualization of fractional quantum Hall states and Wigner crystallization with a STM. In addition, the boundaries between different nematic domains are expected to harbor low-energy edge modes that are analogous to topologically protected states (13). The ability to generate a valley-polarized nematic phase that can be externally tuned with strain makes Bi(111) surface states ideally suited for controlled engineering of anisotropic physical properties. The predicted semimetal-to-semiconductor transition with decreasing thickness in bulk Bi (18) means that the transport properties of thin Bi(111) crystals will be dominated by the surface states, yielding further prospects for integration into devices that exploit the unique physical properties reported here.

Supplementary Materials

www.sciencemag.org/content/354/6310/316/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S8

Tables S1 and S2

References (3943)

References and Notes

  1. See supplementary materials on Science Online.
Acknowledgments: We thank D. A. Abanin, S. A. Kivelson, S. A. Parameswaran, S. L. Sondhi, and A. Yacoby for helpful discussions. Work at Princeton was supported by the Gordon and Betty Moore Foundation as part of the EPiQS initiative (GBMF4530) and by the U.S. Department of Energy (DOE) Office of Basic Energy Sciences. Facilities at the Princeton Nanoscale Microscopy Laboratory are supported by NSF grant DMR-1104612, NSF-MRSEC programs through the Princeton Center for Complex Materials (DMR-1420541, LPS and ARO-W911NF-1-0606), ARO-MURI program W911NF-12-1-0461, and the Eric and Wendy Schmidt Transformative Technology Fund at Princeton. Also supported by a Dicke fellowship (B.E.F.), a NSF Graduate Research Fellowship (M.T.R.), and DOE Division of Materials Sciences and Engineering grant DE-FG03-02ER45958 and Welch foundation grant F1473 (F.W. and A.H.M.).
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