Thermodynamic evidence of magnetic-field-induced complete valley polarization in bismuth

We investigated the fundamental physical properties in the ultra-quantum limit state of bismuth through measurements of magnetoresistance, magnetization, magnetostriction, and ultrasound attenuation in magnetic fields up to 60T. For magnetic fields applied along the bisectrix direction of a single crystal, a drastic sign reversal in magnetostriction was observed at approximately 39T, which could be ascribed to the complete valley polarization in the electron Fermi pockets. The application of magnetic fields along the binary direction presented an anomalous feature at approximately 50T only in the magnetoresistance. The emergence of a field-induced splitting of a valley was proposed as a possible origin of this anomaly.

Scientific RepoRts | (2019) 9:1672 | https://doi.org/10.1038/s41598-018-38206-x (linear model) 16 . The difference between these two models becomes clear in the field range beyond 50T, and therefore, the accurate determination of the LL spectrum in this field range is crucially important to identify the actual style of valley polarization caused by magnetic fields.
In this report, we address the complete valley polarization in single crystals of bismuth through magnetotransport, magnetization, magnetostriction, and ultrasonic attenuation measurements in pulsed high magnetic fields up to 60T. Among these physical quantities, magnetostriction is especially sensitive to the filling of the electronic states, and hence, can provide us with the direct thermodynamic evidence of valley polarization in the ultra-quantum limit state as shown later. Figure 2 shows the transverse magnetoresistance (MR) measured at temperatures of 1.4, 2.1, and 4.2 K (for sample '#1-1') in magnetic fields applied along the binary axis (Fig. 2a) and bisectrix axis (Fig. 2b). In both cases, electric currents were applied along the trigonal axis. We measured two different sample pieces '#1-1' and '#2-1' for B || binary and '#1-2' and '#2-2' for B || bisectrix. All these samples were taken from the same crystalline bismuth ingot. In agreement with the previous measurements 10, 16,21 , all the samples exhibit clear Shubnikov-de Haas (SdH) oscillations. In the case of compensated semimetals such as bismuth, local minima in the transverse MR correspond to the field in which the LL crosses the Fermi energy (E F ). The local minima in the MR appear as local maxima in the second derivative of the MR, as indicated by solid black arrows in Fig. 2c,d. The indices shown in Fig. 2c,d were assigned based on the quadratic model. Here, subscripts ± represent spin splitting in each subband. The Zeeman splitting in the hole LLs may be caused by the finite misalignment of the field, which has also been observed in earlier studies 10,16,21 . The linear model results in almost the same assignment for B || binary up to 60T. Here, the emergence of the peak structure at approximately 50T for B || binary (B * ) cannot be explained in the both models (a thick red arrow in Fig. 2c). As seen in Fig. 2c, this peak structure moves to higher fields and is quickly damped with increasing temperature, contrary to the other peak structures originating from SdH oscillations.

Results
To obtain thermodynamic information, we performed magnetization (M) measurements up to 60T. The samples for the magnetization measurements were taken from the same ingot used for the transport measurements. Figure 3 presents the observed magnetization curves at 1.3 K for B || binary (Fig. 3a) and B || bisectrix (Fig. 3b). In addition to the distinct de Haas-van Alphen (dHvA) oscillations in the low-field range shown in the insets, they show considerable diamagnetism exhibiting slight sub-linear behaviour as in the case of graphite 22 . The magnitude of diamagnetism at 60T is comparable with that in the Meissner state of a superconductor at a field of 14 mT. We also show the differential magnetization for B || binary (Fig. 3c) and B || bisectrix (Fig. 3d). The dHvA oscillations are clearly resolved in the low-field range, as shown in the insets of Fig. 3c,d. The observed dHvA oscillations in the low-field range are consistent with those obtained by the torque and Nernst measurements 23,24 . For B || binary, local maxima in the dM/dB curve below 1.4T can mainly be attributed to the LLs of the light electrons (e2 and e3) 23,24 . It can be expected that for both the previous torque and Nernst measurements, for B || binary, two light electron pockets (e2 and e3) and one heavy electron pocket (e1) will reach the quantum limit at about 1.4 and 10T, respectively 11 . For B || bisectrix, one light electron pocket (e1) and two heavy electron pockets (e2 and e3) will reach the quantum limit at about 1.2 and 2.3 T 11 . Our magnetization results clearly resolve the dHvA signals from the electron pockets at low fields except for those from the heavy electron pocket (e1) at approximately 10T for B || binary. In contrast, we cannot resolve the dHvA signals from the hole pockets above 10T. Earlier magnetic measurements indicated that the dHvA signals from the hole pockets were smaller than those from the electron pockets 23,25 . To resolve the quantum oscillations in the high-field range, we should measure the other physical quantities. Thus, we measured the magnetostriction of bismuth. The blue solid lines in Fig. 4 represent the experimental results of the longitudinal magnetostriction at 1.4 K in pulsed magnetic fields up to 60T for B || binary (Fig. 4a) and B || bisectrix (Fig. 4b). The vertical axis represents the field-induced relative changes in the sample length (ΔL/L) along the binary (ε 11 ) and bisectrix (ε 22 ) axes, respectively. To eliminate any sample dependence, we used the same sample pieces as those used in the magnetotransport measurements (sample '#1-1' and '#1-2'). For both field directions, the samples monotonically shrink up to 20T, consistent with the earlier reports 26,27 . The experimental curve for B || binary exhibits kinks at 26 and 42T, which correspond to the level crossings of 2 h± and 1 h± Landau subbands, respectively. Therefore, magnetostriction resolves the quantum oscillations from the hole pockets more sensitively compared to the magnetization. The result for B || bisectrix also presents small kinks at 16 and 27T, which correspond to the level crossings of 3 h± and 2 h± subbands, respectively, whereas there is a kink immediately above the significant sign reversal at approximately 39T, which corresponds to the level crossings of both 1 h± and 0 e1− subbands in the quadratic model.
To more clearly resolve the quantum oscillations, we performed ultrasound measurements on bismuth. As shown in the earlier experiments, ultrasound attenuation can sensitively resolve the quantum oscillations in bismuth [28][29][30] . Figure 5 shows the ultrasound attenuation measured at 1.3, 2.5, and 3.0 K in pulsed magnetic fields up to 60T applied along the binary axis. As shown in Fig. 5, the ultrasound attenuation presents distinct peak structures at the hole LL crossing fields for small Landau indices. The sample used for the ultrasound measurements   (sample '#4') was taken from a different ingot than that used for the other measurements, but the peak fields reasonably coincide with those observed in the other experiments. Even in this highly sensitive experiment, there exists no distinct anomaly at approximately 50T, as observed in the second derivative of the MR shown in Fig. 2c. As indicated in Fig. 5, the ultrasound attenuation exhibits a steep increase in the field region above 50T, which implies the existence of an additional peak structure at fields higher than 60T.

Discussion
First, we consider valley polarization in the high-field range based on the magnetostriction results. Magnetostriction is determined by the valance between the elastic energy and total energy of charge carriers in the occupied states, and thus provides direct information of the valley polarization. According to Michenaud et al., the magnetostriction of bismuth is associated with the densities of electrons by the following equations 26 : e  22  1  2  3 here, ΔN e1 , ΔN e2 , and ΔN e3 denote field-induced changes of carrier densities in the three electron valleys (e1, e2, and e3, respectively). The coefficients α and β can be determined from the elastic compliance and deformation potential. As claimed by Michenaud et al., the estimation of α and β based on the early experimental data of compliance results in a slightly larger magnetostriction than the experimental results 26 . Therefore, we re-evaluated these coefficients with using the experimental results of ε 11 Fig. 4a,b. The dash-dotted line in Fig. 4a represents the calculated data with using the carrier densities in the linear model 27 . As shown in these figures, the overall trend of experimental data is reproduced better by the quadratic model than by the linear model. In particular, the abrupt sign reversal in ε 22 at approximately 39T for B || bisectrix is reasonably reproduced by the quadratic model. This sign reversal originates from the sign change in the first term (ΔN e1 < 0) at the right-hand side of equation (2), which reflects complete valley polarization. On the other hand, we do not observe sign reversal for B || binary up to 60T. This slight discrepancy may be solved by further fine tuning of the parameters used in the quadratic model. Secondly, we discuss the origin of the anomaly at approximately 50T for B || binary observed only in the MR (Fig. 2a,c). According to the quadratic model, the LL crossings of both the 0 e2− and 0 e3− levels take place at approximately 57T for B || binary 10,11 . Since the onset fields of this LL crossing move to higher fields by rotating the field direction in the binary-bisectrix plane 10,11 , the observed anomaly cannot be simply ascribed to this LL crossing caused by a misalignment of the field. The results of the magnetostriction and ultrasound measurements imply that the lowest LLs of the light electrons (0 e2− and 0 e3− ) will be away from the Fermi energy; in other words, the complete polarization of the electron valleys will be realized above 60T. As expected by equation (1), a drastic sign reversal will be observed at the field where the light electrons depopulate. The anomaly observed in MR at approximately 50T is strongly damped and moves towards a slightly higher field with increasing temperature, unlike the other peaks. Therefore, we ascribe this anomaly to a different origin from the others caused by SdH oscillations. In addition, our results of SdH oscillations and magnetostriction almost coincide with those observed in refs 10,26 . Therefore, we do not consider this anomaly as that from the secondary crystallographic domains caused by twinning 31 .
We infer the origin of this anomaly as a field-induced splitting of the electron valleys. As mentioned before, the application of a magnetic field for B || binary is regarded to reduce the gap between the conduction and valence bands at the L-points illustrated in Fig. 6a. The finite interaction between these two bands realizes avoided level crossing at a certain field B c , as schematically shown in Fig. 6b. Here, the horizontal axis represents the wavenumber along the field direction (k H ). In this case, the dispersion relation will deform to a 'camel's back' structure along the k H direction. As the field increases, Fermi energy (E F ) will pass through the local maxima in the 0 e2− and 0 e3− subbands at a field of B * prior to the complete depopulation of these subbands (Fig. 6c). During this process, each electron valley splits into to two valleys. This change in the FS topology is a kind of Lifshitz transition 32 , but is different from the magnetic breakdown that takes place in the reciprocal plane normal to the magnetic field [33][34][35] . Actually, the possible realization of the 'camel's back' structures of the lowest LLs of the light electron valleys (0 e2− and 0 e3− ) for B || binary has been proposed in ref. 36 . In addition, the field-induced change in the SdH frequency suggests that the topological change in the FSs is observed in antimony-doped bismuth under uniaxial compression 34,35,37 . As the two models used in our analysis do not consider the effect of k H -dispersion as k H = 0, the emergence of this additional anomaly cannot be anticipated in these models.
At the moment, it is not clear why we do not observe this anomaly in the physical quantities other than the MR. Contrary to thermodynamic measurements, anomalies in MR are comparatively magnified through the large factor of (ω c τ) 2 , where ω c and τ denote the cyclotron frequency and the relaxation time 38 . In recent experiments of heavy fermion systems, similar kinds of changes in topology of the FSs manifest as distinct anomalies in the transport properties, whereas there are no distinct anomalies in the thermodynamic quantities [39][40][41][42] . In addition, it is also not clear why we do not observe similar anomaly for B || bisectrix. The depopulation field for the 0 e1− valley (~40T) might be too small to realize the pronounced camel's back structure detectable in experiments. Moreover, MR for B || binary is more sensitive to the valley polarization than that for B || bisectrix because of the anisotropy in the mobility, as pointed out in ref. 10 . This may result in emergence of this anomaly only for B || binary. Although we propose topological change in FS as a possible origin for this anomaly, we do not rule out the other scenario. Further experimental studies are needed to clarify what is going on at around 50T for B || binary.
In summary, we studied various physical properties of bismuth single crystals in magnetic fields up to 60T. The observed quantum oscillations are reasonably explained by a theoretical model that includes a quadratic B dependence term in the Zeeman energy, except for an anomaly observed in the MR at approximately 50T for B || binary. We proposed field-induced splitting of the electron valleys as a possible origin for this anomaly. In contrast, our magnetostriction results for B || bisectrix present a sign reversal at approximately 39T, suggesting the complete depopulation of one of the three electron valleys at this field, which is also reasonably explained by the theoretical model.

Methods
Sample preparation. Sample specimens were spark-cut from the ingot of bismuth single crystals grown by the Czochralski method, and these surfaces were then etched in dilute nitric acid (HNO 3 :H 2 O = 3:7) for about 3-5 min. The principal axes of the crystals were determined within an accuracy of ±0.5° by the back-reflection Laue method. The ratio of room-temperature to residual resistivity was in the range of 80-200. Typical dimensions of the samples were 2 × 2 × 3 mm 3 for ultrasonic attenuation, 3 × 3 × 4-9 mm 3 for MR and magnetostriction, and 2 × 2 × 9 mm 3 for magnetization measurements. Samples '#1-1' , '#1-2' , '#2-1' , and '#2-2' were used for the MR measurements, '#1-1' and '#1-2' were also used for the magnetostriction measurements, '#3-1' and '#3-2' were used for the magnetization measurements, and '#4' was used for the ultrasonic attenuation measurement. The samples used for the MR, magnetization, and magnetostriction measurements were taken from the same ingot.
Measurement apparatus. MR measurements were performed in the transverse MR configuration with a standard four-probe technique. The resistance was measured using 100-kHz AC current and analysed using a custom low-noise digital lock-in technique. Electrical current was applied along the trigonal direction. Copper wires of 60 μm in diameter were attached to four electrodes on the sample using Wood's metal. Magnetization measurements were performed using standard induction methods. Magnetostriction measurements were performed in the longitudinal magnetostriction configuration by the capacitance method using a capacitance bridge (1615-A, IET LABS, Inc.). Ultrasound measurements were performed with the pulse-echo method at a fixed frequency of 33 MHz. A pair of LiNiO 3 single crystals with a thickness of about 0.1 mm was used as ultrasound transducers. All the high-fields experiments were performed in pulsed magnetic fields using a solenoid-type magnet that can produce 60T with a pulse duration of about 4 ms (for magnetization measurements) or 36 ms (for the other measurements) at the International MegaGauss Science Laboratory at ISSP of the University of Tokyo.