## Abstract

Because of the long Fermi wavelength of itinerant electrons, the quantum limit of elemental bismuth (unlike most metals) can be attained with a moderate magnetic field. The quantized orbits of electrons shrink with increasing magnetic field. Beyond the quantum limit, the circumference of these orbits becomes shorter than the Fermi wavelength. We studied transport coefficients of a single crystal of bismuth up to 33 tesla, which is deep in this ultraquantum limit. The Nernst coefficient presents three unexpected maxima that are concomitant with quasi-plateaus in the Hall coefficient. The results suggest that this bulk element may host an exotic quantum fluid reminiscent of the one associated with the fractional quantum Hall effect and raise the issue of electron fractionalization in a three-dimensional metal.

Electronic properties of bismuth have been extensively studied during the 20th century. As early as 1928, Kapitza discovered that bismuth's resistivity increases by several orders of magnitude in the presence of a large magnetic fieldand that it shows no sign of saturation (*1*). Two years later, studies of bismuth led to the discovery of quantum oscillations in both magnetization (*2*) and resistivity (*3*). Bismuth was the first metal whose Fermi surface was experimentally identified (*4*). Commenting on the exceptional role played by bismuth in the history of metal physics (*5*, *6*), Falkovskii wrote in 1968, “It is easiest to observe in bismuth the phenomena that are inherent in all metals” (*7*).

An extremely small Fermi surface and a very long mean-free path distinguish bismuth from other metals. The Fermi surface occupies 10^{–5} of the Brillouin zone (*5*), an order of magnitude lower than graphite, the closest rival and another celebrated semi-metal. The mean-free path at room temperature exceeds 2 μm (*8*), which is almost two orders of magnitude longer than in copper. Because of the low carrier density, the quantum limit in bismuth can be reached by the application of a magnetic field as small as 9 T along the trigonal axis. In this limit, electrons are all pushed to the lowest Landau level, and the magnetic length (the radius of the lowest-energy quantized isolated electron orbit in a magnetic field) becomes shorter than the Fermi wavelength. As recently noted (*9*), the quasi-linear magnetoresistance of bismuth (*1*) in this limit does not fit into the quasi-classical theory of electronic transport. The last experimental investigation of high-field magnetoresistance in bismuth, in the 1980s, found no evidence of saturation up to 45 T (*10*). Notably, this was contemporaneous with the discovery of fractional quantum Hall effect (FQHE) (*11*). Soon, the many-particle quantum theory succeeded in providing an elegant solution to this unexpected experimental finding (*12*). Today, the FQHE ground state is an established case of a quantum fluid whose elementary excitations are fractionally charged. Such a fluid emerges in high-mobility two-dimensional (2D) electron systems formed in semiconductor heterostructures in the presence of a magnetic field exceeding the quantum limit. In contrast with the integer quantum Hall effect (IQHE), which can be explained in a one-particle picture, the occurrence of FQHE implies strong interaction among electrons and their condensation to a many-body quantum state (*13*).

Recently, we reported on the giant quantum oscillations of the Nernst coefficient in bismuth in the vicinity of the quantum limit (*14*). The Nernst signal (the transverse voltage produced by a longitudinal thermal gradient) was found to peak markedly whenever a Landau level meets the Fermi level. Otherwise, it is severely damped. This observation was in qualitative agreement with a theoretical prediction (*15*) invoking a “quantum Nernst effect” associated with the IQHE. Here, we present measurements resolving distinct peaks in the Nernst signal deep in the ultra-quantum limit. Measurements of the Hall coefficient in the same field range reveal a series of quasi-plateaus extending over a window marked by fields at which the Nernst peaks occur. These findings raise the issue of the relevance of the FQHE physics in a clean 3D compensated semi-metal. They suggest that electron correlations in bismuth are stronger than what has been commonly assumed and that this elemental metal may host an exotic quantum fluid.

The lower panel of Fig. 1 contains our primary experimental observation: the detection of three peaks occurring at 13.3, 22.3, and 30.8 T in the Nernst response (*16*). These three peaks follow the rich structure found in the field dependence of the Nernst signal in the previous study limited to 12T (*14*) and emerge well beyond the quantum limit.

The quantum limit in bismuth is set by the well-known topology of the Fermi surface in this compensated semi-metal (*5*, *17*): An ellipsoid associated with hole-like carriers around the T-point of the Brillouin zone and elongated along the trigonal axis and three cigar-like slightly tilted electron ellipsoids along the L points. The cross-section of the hole ellipsoid perpendicular to the trigonal axis is *A*^{h} = 0.0608 nm^{–2}. For each of the three electron ellipsoids, the corresponding area is *A*^{e} = 0.0836 nm^{–2} (*17*) (Fig. 1A). These numbers set the quantum limit, which is attained by a magnetic field equal to *B*_{QL} =(*A*/2π) (*ħ*/*e*), where *ħ* is the reduced Planck constant and *e* is the charge of electron, which is 6.4 T for holes and 8.6 T for electrons. When *B* = *B*_{QL}, the condition λ_{F} ^{⊥} = 2π*l*_{B} is realized: The circumference of the quantized electronic orbit becomes equal to the Fermi wave-length. Here, λ ^{⊥}_{F} is the Fermi wavelength of the electrons traveling perpendicular to the field, and is the magnetic length. In a 2D system, this corresponds to a Landau-level filling factor of unity. In a 3D system, there is an infinite degeneracy along the *z* axis. Nevertheless, analogous to the 2D case and in the absence of an established terminology, we use the expression “filling factor” for the ratio ν =(2π*l*_{B}/λ_{F}^{⊥})^{2}.

The Fermi surface cross-sections correspond to the low-field response of the system, however. Strong magnetic field is known to modify the two Fermi surfaces in order to maintain charge neutrality (*18*, *19*). This feature together with Zeeman splitting leads to a slight enhancement of the quantum limit. It occurs at 8.9 T, and is marked by the most notable peak in the Nernst signal (*S*_{xy}). All previous studies (*10*, *18*–*20*) converge in detecting a dip in resistivity at 8.85 ± 0.25 T (±SD) and identifying it as the one corresponding to the first Landau level [supporting online material (SOM) text].

Bismuth is host to surface states quite distinct from the bulk semi-metal and with a much higher carrier density (*21*). It is very unlikely that their existence is relevant to our observations. The metallic state resolved by photoemission on the 111 surface (normal to the trigonal) has a Fermi surface who seradius is on the order of 0.5 to 3 nm^{–1} (*22*) and the expected quantum oscillations would have a frequency range of 100 to 1000 T.

Therefore, we conclude that the three peaks we observed emerge in the ultraquantum limit. Given the distance between the four distinct orbits in the reciprocal space, magnetic breakdown is an unlikely explanation. Moreover, the peaks resolved here do not display a *B*^{–1} periodicity. Figure 2 presents the high-field data as a function of *B*^{–1}. The peaks are situated at rational fractions (2/3, 2/5, and 2/7) of the first integer peak. The low-field data are presented in Fig. 2B. As seen in Fig. 2C, their *B*^{–1} positions are close to those of the dips resolved in resistivity (at temperature *T* = 25 mK and for *B* <18T) (*19*). In addition to these peaks [already identified in the previous report (*14*)], there are two unidentified peaks between ν = 1 and ν = 2 anomalies and one between the ν = 2 and ν = 3. Assuming that λ_{F} is constant between two successive integer peaks, these peaks occur close to ν = 4/3, ν = 5/3,and ν = 5/2. Figure 2C summarizes the position of all Nernst peaks (both integer and fractional) and resistivity dips. The upward curvature was previously reported and attributed to the field-induced modification of the carrier density (*19*). This feature would imply a field-induced change in λ_{F}. Therefore, the values of ν for fractional peaks, which are extracted by linearly extrapolating the position of adjacent integer peaks are subject to caution. However, as there is no visible phase transition and the change in λ_{F} is continuous, the extracted values of ν are not expected to differ much from the effective ones.

We also performed low-resolution measurements of resistivity and the Hall coefficient on our sample at 0.44 K. In agreement with the previous high-field study (*10*), resistivity does not show any strong feature beyond the quantum limit (fig. S1). The Hall response, ρ_{xy} (Fig. 3) is strongly nonlinear for fields exceeding 3 T. Above this field, a rich structure including a sharp peak at 9.8 T is resolved. At still higher fields—i.e., in the ultraquantum limit—a succession of fast and slow regimes in the field-dependence of ρ_{xy} is visible. Comparing the relative position of these quasiplateaus and the Nernst peaks clearly shows that each Nernst peak occurs between two successive Hall plateaus in agreement with the theoretical prediction invoking IQHE (*15*).

Below 3 T, the slope of ρ_{xy} yields the Hall number *R*_{H} = 1.5 × 10^{–5} m^{3}/C corresponding to a carrier density of 3.9 × 10^{17} cm^{–3}, only 30% larger than the hole carrier density. In the high-field regime, ρ_{xy} is 0.129 ohm·cm at ν = 1/2 and 0.158 ohm·cm at ν = 1/3. Therefore, when the filling factor passes from 1/2 to 1/3, ρ_{xy} jumps by 0.029 ohm·cm. Let us recall that, in a 2D electron gas in the FQHE, the magnitude of ρ^{2D}_{xy} at filling factor ν would be *h*/(ν*e*^{2}) and the passage from the ν = 1/2 to ν = 1/3 plateau would lead to a jump of *h*/*e*^{2} = 25.8 kilo-ohms in ρ^{2D}_{xy}. Because in our 3D system, the jump in ρ_{xy} is 11.2 nm times *h*/*e*^{2}, it is tempting to consider the bulk crystal as an assembly of coherent 2D sheets each 11.2 nm thick. This length scale is very close to λ ^{∥}_{F} = 14 nm, the Fermi wavelength of holes along the trigonal axis and the magnetic field and much longer than the atomic distance between layers (∼0.2 nm).

These observations raise the issue of relevance of FQHE physics to bulk bismuth. The former is found in the context of high mobility, 2D and interacting electronic systems. To what extent do electrons in bismuth qualify for these attributes? The electronic mobility is undoubtedly large enough. In our crystal, it exceeds—by two orders of magnitude—the mobility of the GaAs/AlGaAs sample in which the FQHE was discovered in 1982 (*11*) (SOM text). It is true that bismuth is not commonly considered as a strongly interacting electron system. However, the very low level of carrier density undermines screening and favors Coulomb repulsion. Given that electrons in bismuth and in GaAs are comparable in their concentration and in their effective mass, it is reasonable to assume that Coulomb interaction is sufficiently strong to allow electron fractionalization.

The most serious obstacle for the realization of FQHE physics in bulk bismuth is dimensionality. IQHE (also a purely 2D effect) has already been observed in an anisotropic 3D electron gas (*23*) as well as a number of bulk systems (SOM text), and 3D FQHE has been a topic of theoretical investigation (*24*). Bismuth, with its weak anisotropy, was recently proposed as a candidate to exhibit the quantum spin Hall effect (*25*). However, according to our preliminary studies, in the vicinity of the quantum limit, the in-plane conductivity in bismuth is orders of magnitude lower than the perpendicular conductivity along the field axis. Instead of being an assembly of weakly coupled 2D sheets perpendicular to the field, the system is closer to a set of 1D wires oriented along the field (*26*). To the best of our knowledge, there is no appropriate theoretical frame for in-plane transport in such a context. Thus, bismuth in the ultraquantum limit emerges as an experimental playground for two distinct routes toward the electron fractionalization, the FQHE and the field-induced Luttinger liquid (*26*).

Several other questions remain. What happens to the electron-like carriers? For this field orientation, the three electron ellipsoids have been invisible in all studies of quantum oscillations. The complex structure of ρ_{xy}(*B*) for *B* < 14 T suggests competing responses of electrons and holes. However, in the ultraquantum limit, there is no definite signature of their presence in spite of the larger total area occupied by the three electron ellipsoids. This may be a consequence of their lower mobility. The absence of strong features in the raw ρ_{xx}(*B*) data are also noteworthy. In 2D systems, the quantum Hall plateaus are associated with absence of dissipation and vanishing longitudinal conductivity. In bulk systems showing IQHE, plateaus in ρ_{xy} are concomitant with minima in ρ_{xx}. Extensive high-resolution studies of resistivity at lower temperatures on cleaner samples may help to identify the source of resistive dissipation at high fields. The 80-year-old mystery of magnetoresistance in bismuth (*1*, *9*) needs fresh experimental and theoretical attention.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/1146509/DC1

Materials and Methods

SOM Text

Fig. S1

Table S1

References