Giant orbital diamagnetism of three-dimensional Dirac electrons in Sr$ _3$PbO antiperovskite

In Dirac semimetals, inter-band mixing has been known theoretically to give rise to a giant orbital diamagnetism when the Fermi level is close to the Dirac point. In Bi$ _{1-x}$Sb$ _x$ and other Dirac semimetals, an enhanced diamagnetism in the magnetic susceptibility $\chi$ has been observed and interpreted as a manifestation of such giant orbital diamagnetism. Experimentally proving their orbital origin, however, has remained challenging. Cubic antiperovskite Sr$ _3$PbO is a three-dimensional Dirac electron system and shows the giant diamagnetism in $\chi$ as in the other Dirac semimetals. $ ^{207}$Pb NMR measurements are conducted in this study to explore the microscopic origin of diamagnetism. From the analysis of the Knight shift $K$ as a function of $\chi$ and the relaxation rate $T_1^{-1}$ for samples with different hole densities, the spin and the orbital components in $K$ are successfully separated. The results establish that the enhanced diamagnetism in Sr$ _3$PbO originates from the orbital contribution of Dirac electrons, which is fully consistent with the theory of giant orbital diamagnetism.

enhancement of diamagnetism towards the Dirac point reasonably and has been accepted as the theoretical picture behind the giant diamagnetism. One of the intuitive pictures for the theoretical understanding of giant orbital diamagnetism is based on the E-linear density of states D(E^), where E^ is a two-dimensional kinetic energy for the momentum perpendicular to the applied field. When Dirac electrons are confined in Landau levels under a magnetic field, the average energy gain and loss are not balanced, in contrast to the case for ordinary parabolic bands with constant D(E^) (Fig. 1B) (10), which increases the total free energy of Dirac bands under a magnetic field and gives rise to a diamagnetism. Note that the orbital diamagnetism comprises the contributions from all the electrons occupying the Dirac bands, not only from the electrons around the EF, as in the Pauli spin paramagnetism and the Landau (orbital) diamagnetism. This picture naturally explains the maximally enhanced orbital susceptibility when the EF lies in the Dirac mass gap.
Despite the progress in the theoretical understanding of the origin of giant diamagnetism, its experimental verification has remained challenging, as it requires a separation of the orbital component from the spin component. The expected orbital diamagnetism from Dirac electrons of the order of 10 -4 emu/mol is large but still could be comparable to the spin Pauli paramagnetism, for example, when bands other than the Dirac bands contribute and/or the g-factor is enhanced from 2. Magnetic resonance techniques in principle could analyze the contributions from different origins. The microscopic magnetism of Bi1-xSbx has been studied by nuclear magnetic resonance (NMR) (19), muon spin rotation (µSR) (20,21), perturbed angular distribution (22,23) and b-NMR (24). The verification of the orbital character of diamagnetism using these techniques, however, has been far from complete. In the case of NMR, the large electric quadrupole interaction of a nuclear spin I ³ 1 in 209 Bi NMR has imposed critical constraints on the detailed analysis of the electronic contribution and the separation of spin and orbital contributions. NMR study on an I = 1/2 nuclear spin system, without electric quadrupole and phonon interactions, should be a promising approach to verify the orbital origin of the giant diamagnetism in Dirac semimetals. Dirac semimetals containing appropriate nuclear species, however, have been limited.
Sr3PbO, the material we study here, is a member of the antiperovskite family A3TtO (A=Ca, Sr, Ba, Eu; Tt=Si, Ge, Sn, Pb) (25) and is theoretically proposed to be a three-dimensional massive Dirac electron system (26,27) with topological surface states (28,29). The cubic antiperovskite structure of Sr3PbO is shown in Fig. 2A, where the Pb atoms are on the corners of cubic unit cell and the Sr atoms form an octahedron surrounding the O atom at the center. In the ionic limit, the valence states of constituent ions can be expressed as Sr 2+ 3Pb 4-O 2-. In the reported band structures (26), the valence and the conduction bands indeed consist of the fully occupied 6p orbitals of Pb 4and the empty 4d orbitals of Sr 2+ , respectively. The 6p and the 4d bands overlap marginally and a gap opens almost everywhere on the band crossing plane. The C4 rotational symmetry, however, protects the band crossing at six equivalent points on G-X lines (Fig. 2B), which leads to six moderately anisotropic 3D Dirac bands free from the other parabolic bands. The 3D Dirac band has a very small mass gap of ~10 meV, which is created by the admixture of higher energy orbital states via spin-orbit coupling. The six Dirac bands merge at -125 meV below the Dirac points, giving rise to a saddle point. Below the saddle point, multi-band Fermi surfaces are expected to emerge when the Fermi level lies in this region. This region is essentially away from the Dirac physics. The presence of 3D Dirac electrons in Sr3PbO is supported by recent experiments (18,30) which show the presence of extremely light mass (~ 0.01me) holes and B-linear magnetoresistance. Angle-resolved photoemission spectroscopy on a sister compound Ca3PbO confirms the Dirac dispersion of the valence band predicted by band calculations (31).
The Pb antiperovskite should provide a promising arena for NMR studies of Dirac semimetals to verify the orbital character of the giant diamagnetism from Dirac electrons, as 207 Pb hosts I = 1/2 nuclear moment in contrast to 209 Bi. Here, we report 207 Pb NMR and magnetic susceptibility c studies of the 3D Dirac system Sr3PbO. An enhanced diamagnetism is observed in the magnetic susceptibility c as in Bi and other 3D Dirac systems. Using the Korringa relation with the spin-lattice relaxation rate T1 -1 , we show that the spin contribution Kspin in K cannot account for the enhanced diamagnetism. The K-c plot can be analyzed as the superposition of the spin and the orbital contributions with distinct hyperfine coupling constants, consistently with the analysis of the Korringa relation. The estimated orbital hyperfine constant indicates the delocalized nature of electrons in charge of the large orbital susceptibility. These results strongly affirm that the enhanced diamagnetism originates from the giant orbital susceptibility of Dirac electrons.
Five polycrystalline samples of Sr3PbO from different batches A-E with different hole densities (p) from ~ 10 18 to ~ 10 20 cm -3 were investigated. The Hall resistivity rxy in the zero-field limit gives positive Hall coefficients RH = +3.8, 0.13, 0.032 and 0.029 and cm -3 /C (Fig. 2C), yielding hole densities p = 1.6×10 18 , 5.0×10 19 , 2.0×10 20 and 2.2×10 20 cm -3 for samples A, C, D and E, respectively. Sample B should have a comparable but only slightly smaller p than sample C, judging from the NMR data. The donors very likely correspond to 0.01-1% level of cation defects and/or excess oxygens, which are introduced partially to relax the extremely reduced anionic state of  The magnetic susceptibilities c(T) for three samples A, C and E, shown in Fig. 2F, are found to be all diamagnetic. The magnitude of diamagnetism increases with decreasing the hole concentration p and hence increasing EF from E to A. The increase from samples C and E to sample A with EF in the Dirac bands is particularly significant and as large as of the order of 10 -4 emu/mol, which is comparable to the large diamagnetism observed in other Dirac semimetals (15,16). An appreciable temperature dependence is observed particularly for sample A. The c(T) of sample A shows a clear decrease with lowering temperature to ~30 K, which should be an intrinsic behavior of magnetic susceptibility. This is followed by a Curie-like increase very likely associated with magnetic impurities (0.01% level of s = 1/2 impurities) at low temperatures. The other samples with a high hole density (EF) show a monotonic increase of c(T) from room temperature down to 2 K, with a clear Curie-like behavior of similar magnitude as sample A at low temperatures.
It is not possible, however, to fit the c(T) behavior for samples C and E over entire temperature range only with a Curie-Weiss contribution and a constant offset, particularly at high temperatures above 100 K. This indicates the presence of very weak but appreciable increase of the intrinsic c(T) (broken lines in Fig. 2F) with lowering T in samples C and E at least at high temperatures above 100 K where the Curie contribution is negligibly small. Note that the weak temperature dependence of the intrinsic c(T) for samples C and E is negative, opposite to that of sample A.
The enhanced diamagnetism in sample A with the EF in the Dirac bands should represent the same large diamagnetism observed in Bi1-xSbx and other Dirac semimetals, which cannot be described naively by the conventional kinds of magnetisms. The core diamagnetism is estimated to be -8.5×10 -5 emu/mol for Sr3PbO (32), which should not depend appreciably on the 1% level of cation defects or excess oxygens. The Pauli paramagnetism calculated from the density of states in the band calculation is only of the order of 10 -5 emu/mol assuming p ~ 2×10 20 cm -3 and g = 2 for sample E, not as large as the difference of susceptibilities between samples A and E. At this point, however, the possibility of an enhanced g-factor, which could account for the difference, cannot be excluded completely. 207 Pb NMR measurements for samples A-E were conducted to verify the orbital origin of giant diamagnetism experimentally. The NMR spectra at 150 K are shown in Fig. 3A. A systematic shift of the peak position as a function of p (and hence EF) is observed from sample A to E, implying that the NMR peaks originate from the bulk Sr3PbO. We note that the observed shifts are different from those of possible impurity phases such as 1.081%, -0.034% and 0.444% for metallic Pb, PbO and PbO2 (33) The Korringa behavior of the spin-lattice relaxation rate T1 -1 , (T1T) -1 = constant, is observed for all samples A-E over a wide temperature range as shown in Fig. 3C, which provides us with important hints to estimate Kspin and hence Korb. The magnitude of (T1T) -1 , which is proportional to the square of the electron density of states D(EF) at EF, increases systematically with increasing p (decreasing EF) from samples A to E, which can be reasonably understood as the increase of D(EF). Indeed, as seen in Fig. 3D, (T1T) -1/2 at 100 K for the samples with different EF are scaled well with the calculated D(EF) for the 6p orbitals of Pb, implying that (T1T) -1/2 captures the D(EF) that determines the spin contribution in c(T) and K(T). For sample A with the lowest hole density, a clear upward deviation from the Korringa behavior can be seen at high temperatures, which can be reasonably ascribed to the T-dependence of the thermally averaged density of states 2 around the Dirac points at E = EDP, the T-dependence of T1 -1 for sample A can be reproduced well (solid line in Fig. 3C), yielding EF-EDP ~ -60 meV (open square in Fig. 3D), roughly consistent with that estimated from the hole concentration and the band calculation (for details, see (37)). We also note that the orbital contribution to T1 -1 was theoretically estimated to be at least an order of magnitude smaller than the observed spin contribution (38), which can be neglected here.
Confirming that (T1T) -1/2 is a good measure of D(EF), we can estimate roughly the spin contribution Kspin in the observed K. The Korringa relation, T1TKspin 2 = S, yields the linear dependence of Kspin on (T1T) -1/2 with the slope S 1/2 . For a simple metal with an isotropic Fermi surface, the Korringa value S is determined by the gyromagnetic-ratio of the nuclei under observation, gn, and the gyromagnetic-ratio of an electron, ge, as S = ħ/4pkB × (ge/gn) 2 .
The p-dependence of the g-factor could modify the Korringa relationship as both Kspin and T1 -1 are in proportion to the square of g (37). The excellent scaling of T1 -1 with D(EF) over a wide variety of p values, however, indicates that the p-dependence of the g-factor is not appreciable within the range of p investigated here and that g is unlikely to be strongly modified from 2. The plot of K as a function of (T1T) -1/2 is shown in Fig. 4B. K decreases with decreasing (T1T) -1/2 , a measure of the density of states, from sample E to sample A, much more rapidly than the expected linear relationship Kspin = S 1/2 (T1T) -1/2 (gray dashed line). The strongly non-linear decrease of K from sample E can be naturally explained by the superposition of an additional orbital contribution Korb, which increases rapidly with increasing EF (decreasing p) towards the Dirac mass gap.
It is known that the effective Korringa value S * , inferred from an experimentally observed slope of K-(T1T) -1/2 , is often larger than S calculated from the gyromagnetic-ratio by more than a factor of 2 even in a simple metal. The black dashed line with an enhanced S * ~ 6.3S in Fig. 4B connects the data for all the heavily doped samples from E to B. This assumes that K is fully dominated by Kspin (Korb ~ 0) for these samples and therefore gives the upper-bound estimate for Kspin. Considering that the EFs of samples E-B are outside the Dirac band regime, the assumption of Korb ~ 0 for them highly likely captures the reality better than the S * = S limit. Even with the maximum estimate of Kspin, Fig. 4B clearly indicates that a large orbital contribution must be incorporated to account for the enhanced diamagnetic K for sample A.
Taking a closer look at Fig. 4B, we recognize the two important features to identify the orbital and the spin contributions in the K-c plot. First, within the Korringa relationship, the spin contribution in low-p sample A is negligibly small as compared with those of the other samples. Second, the K-(T1T) -1/2 line for each sample, representing the correlation between the T-dependences of K(T) and (T1T) -1/2 of a given sample, has a negative slope (dotted lines) for sample C and E and an almost infinite slope for sample A. They do not follow at all the linear behavior with a positive slope expected from the Korringa relationship. This very likely implies the presence of an additional T-dependent contribution to K(T) other than Kspin, which is small but dominates the slope of K-(T1T) -1/2 line for each sample and can be ascribed naturally to Korb(T).
Let us now return to the K-c plot in Fig. 4A with the information from Fig. 4B. As Kspin ~ 0, the K-c relationship for sample A should represent that for the non-spin contributions, K = Kchem+Korb = Kchem+Aorbcorb, which is indicated by the red solid line in Fig. 4A and gives an estimate of Aorb = 88±14 kOe/µB. We can ascribe the slopes of the K-c lines for samples C and E (green and blue dashed lines) close to Aorb = 88±14 kOe/µB to the The estimated Aorb ~ 88 kOe/µB from the K-c plot implies the unconventional character of giant orbital diamagnetism. Korb is normally driven by a van Vleck paramagnetic susceptibility with Aorb = 2ár -3 ñ (41) determined by the distance r between the nuclei and the orbiting electrons. Aorb ~ 2000 kOe/µB is estimated for 6p orbitals of Pb (42), which is one order of magnitude larger than the experimentally observed Aorb ~ 88 kOe/µB for the Dirac semimetal Sr3PbO. The hybridization of Pb 6p Dirac holes with Sr 4d states and other orbital states could reduce the calculated Aorb but not an order of magnitude. The small Aorb ~ 88 kOe/µB therefore implies that the orbiting of spatially spread itinerant electrons, not of those completely confined within the atomic orbitals, is in charge of the observed large orbital diamagnetism. If Dirac electrons were uniform free electron gas and not confined in the atomic orbital at all, on the other hand, we would have an estimate of Aorb < 1 kOe/µB (43), orders of magnitude smaller than the experimentally observed Aorb ~ 88 kOe/µB. The drastic enhancement from the free electron estimate is reasonable as the Dirac electrons in Sr3PbO are not completely free from the atomic orbital and hopping from one atomic orbital to the others. These comparisons are fully consistent with the theoretical picture of giant orbital diamagnetism based on the inter-band mixing of itinerant Dirac electrons on the crystal lattice, which is distinct from the conventional orbital magnetism of Van Vleck type.
The hole-concentration p (hence Fermi level EF) dependence of the magnetic susceptibility c(T) in Fig. 2F and the predominance of the orbital contribution in the enhanced diamagnetism are reproduced by a theoretical calculation of the magnetic susceptibility c cal based on the expression in Ref. (9), which explicitly includes the interband effects. Figure 5 indicates c cal (solid lines) and its deconvoluted orbital component c cal orb (broken lines) as a function of EF at T = 232 K and 348 K, calculated for the tight binding bands of Sr3PbO (27). Note that the calculated c cal does not include the contribution from the core electrons c0, a p-and T-independent constant, and therefore represents c-c0. An enhanced diamagnetism in c cal shows up when EF is in the Dirac band. Apparently, the orbital component c cal orb dominates the enhanced diamagnetism. When EF lies below the Dirac band regime, the calculated c cal orb is much smaller than that in the Dirac band regime, which justifies the assumption of Korb ~ 0 for samples C and E.
A small but appreciable T-dependence of c cal is seen particularly in and near the Dirac band regime in Fig. 5, which originates from the orbital contribution c cal orb(T) and changes sign from positive to negative upon going away from the mass gap. This is consistent with the increase and the decrease of experimental corb(T) and c(T) with increasing T for sample A (EF = -45 meV) and sample C (EF = -125 meV), respectively. These qualitative agreements between the theory and the experiment provide a further support for the validity of the above analysis of NMR results.
Quantitatively, however, the calculation based on the tight binding model does not allow us to capture the details of experimental results. c cal for EF = -45 meV (corresponding to sample A) shows an additional diamagnetic contribution of ~ -2×10 -4 emu/mol (essentially orbital in origin) as compared to those for EF < -100 meV (samples B-E). This is almost a factor of four larger than the experimental orbital contribution corb ~ 0.5 ×10 -4 emu/mol for sample A, which is difficult to account for only by the strong EFdependence of c cal and the ambiguity in the estimate of EF. c cal orb and c cal appear to be overestimated within the framework of the present calculation.
In conclusion, our 207 Pb NMR study of the 3D Dirac electron system Sr3PbO antiperovskite clearly revealed the orbital origin of large diamagnetism observed in the bulk magnetic susceptibility when its EF lies in the Dirac bands. This orbital diamagnetism is distinct from the ordinary orbital magnetism in that the orbiting electrons are not confined within the atomic orbitals but hop between the atomic orbitals. These observations are fully consistent with the microscopic picture of giant orbital diamagnetism of Dirac electrons established theoretically after the debates over decades and provide the first firm experimental evidence of such. The calculated orbital susceptibilities as a function of EF and T, based on the theories, indeed reproduce qualitatively the experimentally isolated orbital contribution to the magnetic susceptibility. Our results open up a fascinating possibility to further explore not only the intra-band effects but also the inter-band effects in topological semimetals. represent the kinetic energy originating from the two-dimensional momentum perpendicular to the applied field and the density of states as a function of E^, respectively. The Landau levels in a magnetic field with spin up and down are indicated by the blue and the red lines respectively. The average E^ of the Landau levels in a magnetic field, to which the gray shaded area of the zero field D(E^) condenses, is indicated by the gray broken lines. They are larger than the average E^ of the corresponding gray shaded area (black broken line) for the Dirac band but identical to that for the normal parabolic band.