Local Observations of Orbital Diamagnetism and Excitation in Three-Dimensional Dirac Fermion Systems Bi_{1-x}Sb_x
Yukihiro Watanabe, Masashi Kumazaki, Hiroki Ezure, Takao Sasagawa, Robert Cava, Masayuki Itoh, Yasuhiro Shimizu
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b Journal of the Physical Society of Japan
Local Observations of Orbital Diamagnetism and Excitation inThree-Dimensional Dirac Fermion Systems Bi − x Sb x Yukihiro Watanabe , Masashi Kumazaki , Hiroki Ezure , Takao Sasagawa , Robert Cava ,Masayuki Itoh and Yasuhiro Shimizu Department of Physics, Nagoya University, Nagoya, 464-8602 Japan Department of Physics, Tokyo Institute of Technology, Yokohama, 226-8503 Japan Department of Chemistry, Princeton University, Princeton, NJ, 08544, USA
Dirac fermions display a singular response against magnetic and electric fields. A distinctmanifestation is large diamagnetism originating in the interband e ff ect of Bloch bands, asobserved in bismuth alloys. Through Bi NMR spectroscopy, we extract diamagnetic or-bital susceptibility inherent to Dirac fermions in the semiconducting bismuth alloys Bi − x Sb x ( x = . − . Bi hyperfine coupling constant provides an estimate of the e ff ectiveorbital radius. In addition to the interband diamagnetism, Knight shift includes an anoma-lous temperature-independent term originating in the enhanced intraband diamagnetism un-der strong spin-orbit coupling. The nuclear spin-lattice relaxation rate 1 / T is dominatedby orbital excitation and follows cubic temperature dependence in the extensive tempera-ture range. The result demonstrates the robust diamagnetism and low-lying orbital excitationagainst the small gap opening, whereas x -dependent spin excitation appears at low tempera-tures.Relativistic Dirac fermions exhibit large diamagnetism at room temperature, as observedin bismuth alloys Bi − x Sb x
1, 2) and graphite.
3, 4)
Similar to supercurrent of the Meissner e ff ect,the diamagnetism of Dirac semimetals with linearly crossing bands comes from dissipation-less orbital current in thermodynamic equilibrium under magnetic field. In contrast to theLandau-Peierls diamagnetism of conducting electrons, the diamagnetism in Bi − x Sb x is en-hanced as the chemical potential µ is located close to the Dirac point
1, 5) or inside the bandgap. The interband e ff ect of Bloch bands has solved the mystery based on the exact formulaof orbital susceptibility for three-dimensional (3D) Dirac fermions
6, 7) and relates to the giantspin Hall e ff ect observed in Bi − x Sb x . Since the transport properties include a significant * [email protected] 1 /
10. Phys. Soc. Jpn. contribution of topological surface, the enhanced diamagnetism can be a complementary bulksensitive probe of Dirac fermions.Bi − x Sb x is semimetallic for x = T and L points of the Brillouin zone.
10, 11)
The Sb substitution induces the band inversionacross x ∼ .
05 where the system becomes Weyl semimetal showing negative magnetore-sistance.
The band gap opens in the bulk for x > .
05, leading to a three-dimensional(3D) topological insulator involving a gapless surface state with extremely high mobilityand quantum oscillations.
The diamagnetism becomes largest around x ∼ . ∆ ∼
10 meV, consistent with the interband orbital susceptibility.In general, the orbital susceptibility χ orb consists of four components: the Landau-Peierlsdiamagnetism χ LP of conduction electrons, the interband orbital susceptibility χ inter equiva-lent to the Van-Vleck susceptibility, the atomic core diamagnetism χ core , and the geometricsusceptibility χ geo due to the Berry phase.
7, 19)
In massive Dirac fermion systems such assemiconducting Bi − x Sb x ( x > .
05) with small band gap, the first term χ LP and the spin sus-ceptibility χ spin are negligible. Thus χ inter is expected to dominate the large diamagnetism ofBi − x Sb x , whereas it cannot be distinguished from the other two diamagnetic factors by bulkmagnetization measurements.Nuclear magnetic resonance (NMR) spectroscopy in principle can extract χ orb compo-nents independent of temperature and detect low-lying orbital excitation. Despite thelong history of the material, the comprehensive NMR study is still lacked. The β -NMR mea-surement shows a small negative Li + Knight shift with the unknown hyperfine interactionnear the surface of Bi . Sb . . Furthermore, orbital fluctuations can be probed by the nu-clear spin-lattice relaxation rate 1 / T in Dirac and Weyl semimetals,
20, 23) whereas the mate-rials so far reported involve significant spin excitation in 1 / T due to the existence of Fermisurface. In this Letter, we demonstrate the diamagnetic hyperfine fields and the orbital excitationby
Bi NMR experiments in Bi − x Sb x for x = . − .
16 under magnetic field parallel andperpendicular to the c axis. By comparing the Knight shift with the magnetic susceptibility,we analyze the temperature ( T ) dependent and the anomalous T -invariant components. The T and x dependences of χ orb and 1 / T are compared with the theoretical calculation based onthe 3D Dirac fermions.Single crystals of Bi − x Sb x ( x = ∆ =
16 meV for x = x = /
10. Phys. Soc. Jpn. concentration was estimated as 10 − cm − for x = ≃ cm − for x = .
16 fromthe Hall resistance (Fig. S2). Magnetization was measured with a superconducting quantuminterference device under the magnetic field for 0.1 – 7 T. The
Bi NMR Knight shift K andnuclear spin-lattice relaxation rate 1 / T were obtained in a static magnetic field H = t π/ − τ − t π/ − τ with the pulse duration t π/ = µ s and the interval time τ = − µ s.We checked the negligible rf heating e ff ect on spin-echo signals by reducing the pulse power.The origin of K = ( ν − ν ) /ν was calibrated from the resonance frequency of the referenceBi(NO ) aqueous solution, ν = s electron contribution wasimplicitly subtracted. The nuclear magnetization relaxation recovery for the central line wasfitted with a multi-exponential function for the nuclear spin I = / Magnetic susceptibility χ was measured as a function of T and H normal ( χ ⊥ ) and par-allel ( χ k ) to the ab plane of the hexagonal R ¯3 m lattice in Bi − x Sb x , as shown in Fig. 1. Herethe c axis is taken along the diagonal direction of the rhombohedral lattice. The calculatedcore diamagnetic susceptibility − × − emu mol − for Bi + was already subtracted. Weobserved anisotropic diamagnetism for three samples with the di ff erent x : the amplitude of χ k is greater than χ ⊥ at low temperatures. The result is consistent with the theoretical cal-culation, where the anisotropy is explained by the band structure. Namely, electron (hole)dominates the orbital motion around the L ( T ) valley having nearly gapless (gapped) excita-tion under the magnetic field along the a ( c ) axis.
6, 36)
The T dependence of the susceptibilityis attributed to the thermal excitation comparable to a band gap energy. For x = . χ k reaches − . × − emu mol − below 10 K. The amplitude is 2–3 times greater than theprevious report. The diamagnetism is weakened as x increases, consistent with the chemicalpotential µ dependence.
6, 21)
The experimental result of χ orb is compared with the theoretical calculation for χ inter in3D Dirac fermion systems. χ inter at finite temperatures is expressed as χ inter = = − α π c ∗ c " ln E Λ ∆ − Z ∞ ∆ d ǫ − f ( − ǫ ) + f ( ǫ ) √ ǫ − ∆ , (1)using the fine structure constant α , c ∗ ≡ √ ∆ / m ∗ with the e ff ective electron mass m ∗ , the bandwidth E Λ , and the Fermi distribution function f ( ǫ ) for an energy ǫ . As shown in Fig. 1(a), thenumerical calculation (a solid curve) using Eq.(1) qualitatively reproduces the T dependenceof χ orb for 2 ∆ =
16 meV and the energy cut o ff E Λ /∆ = M divided by H weakly depends on H [Fig. 1(b)]. For x = M / H along the ab plane exhibits a minimum ( − . × − emu mol − ) around 1 T and gradually /
10. Phys. Soc. Jpn. -8-6-4-20 c ( - e m u m o l - ) T (K) Bi x Sb x x c || c ┴ M / H ( - e m u m o l - ) H (T)(b) L T m Fig. 1. (a) Magnetic susceptibility χ plotted against temperature T in Bi − x Sb x ( x = . , . , .
16) at 7T. Magnetic field was applied parallel or normal to the ab plane of the hexagonal R ¯3 m lattice. A solid curveis the numerical calculation based on Eq.(1) for 3D Dirac fermions. (b) Magnetic field H dependence ofmagnetization M divided by H for 0–7 T at 2.0 K. Inset: the schematic band structure around the L and T points. increases at high fields. Similar behavior is seen for x = is absent in the measured field range.Although the theoretical calculation of χ orb under the intense field is absent, the interbande ff ect in Eq.(1) is governed by low-lying energies ( ǫ ∼ ∆ ) and hence the transition between n = The population of the n = χ inter is expected to increase with √ H , which may explain the field dependent behavior of M / H in a low field range.The Bi NMR spectrum (Fig. 2) consisting of a sharp central line and broad satellitelines represents the first-order quadrupolar splitting in the presence of electric field gradi-ent at the nuclear spin I = /
2. The maximum splitting along the c axis gives the nuclear /
10. Phys. Soc. Jpn. Sb H || c H || a
Fig. 2.
Bi NMR spectrum of the Bi . Sb . single crystal at 100 K under a magnetic field H = c axis. Triangles denote the central peak position that defines Knight shift K . A dotted line denotes the K = ν = .
214 MHz). quadrupole frequency ν Q = . H k c , the central resonance fre-quency is located close to ν , while it significantly shifts to a lower frequency for H k a .The Bi Knight shift K = ( ν − ν ) /ν is determined from the central peak position aftersubtracting the second-order quadrupole contribution ( < . K a and K c , measured for H k a and c also exhibit anisotropic behavior,as shown in Fig. 3(a). K a reaches − .
8% nearly independent of x , while K c remains slightlypositive for x = .
08 and 0.1. The diamagnetic K a is consistent with χ k . However, the depen-dence of K a against T is much weaker than that of χ k . It points to a sizable T -independentcomponent included in the Knight shift. The origin likely comes from the intraband orbitalsusceptibility, as discussed below.Here the uniform diamagnetic shielding due to the bulk susceptibility χ k = − . × − in the dimensionless unit ( x = . K dia = πχ = − . a axis, which is an order smaller than the observed T -dependent componentof the Knight shift. Therefore, K is mostly dominated by the hyperfine interaction with theorbital angular moment of Dirac fermions. As shown in Fig. 3(b) and Table S1, the K − χ linearity gives the hyperfine coupling constant A DF = ± . ± . / µ B for the a and c axes in Bi . Sb . , respectively. They are much smaller than the atomic orbital hyperfineconstant A orb = N u B h r − i =
170 T / µ B for the mean atomic radius of bismuth, h r − i ∼ (1 / cm ), where N is the Avogadro number and µ B is the Bohr magneton. Since the meanorbital radius scales to third root of A DF , the orbital radius of Dirac fermions reaches 3–3.5times larger than the atomic radius of bismuth for H k a . The x dependence of A DF along the c axis may come from the di ff erence in the trap potential of the orbital motion. /
10. Phys. Soc. Jpn. K ( % ) c (10 -4 emu mol -1 ) T (K) x a c K ( % ) -1.00.01.0-6 -5 -4 -3 -2 -1 0-1.00.01.0 300250200150100500-0.50.5-0.50.5(a)(b) K c c inter K inter Fig. 3. (a) Temperature dependence of
Bi Knight shift K in Bi − x Sb x . Magnetic field was applied alongthe c axis (open symbols) and the ab plane (filled symbols). (b) K plotted against χ as an implicit function oftemperature. Black and red dotted lines are the linear fitting result for x = . A DF =
170 T / µ B , respectively. The temperature dependent term scaling to χ isassigned to K inter . The crossing point gives the constant o ff sets of χ and K . The low-lying magnetic excitation is investigated by the nuclear spin-lattice relaxationrate 1 / T .
20, 21, 23)
As shown in Fig. 4, 1 / T behaves nearly isotropic and follows T depen-dence for x = .
08 and 0.10 in an extensive T range. 1 / T continues to decay more than fiveorders of magnitude for 5 – 290 K. The result is consistent with the behavior expected for theintraband orbital excitation in massless 3D Dirac and Weyl fermion systems, which isdistinct from that of the spin excitation (1 / T ∼ T ). For x = .
16, 1 / T deviates from the T dependence below 100 K and becomes close to the Korriaga’s law (1 / T ∼ T ), implyingthe residual density of states. /
10. Phys. Soc. Jpn. -5 -4 -3 -2 -1 / T ( s - )
10 100 T (K) 0.08 0.10 0.16 10 -5 -4 -3 -2 -1 / T ( s - )
10 100 T (K) 0.08 0.10 0.16(a) (b) (c) H ┴ c H || c~T ~T -1 / T ( a r b . un it s )
10 100 T (K) D = 0 m = 0 D = 16 m = 0 D = 5 m = 10 D m
Fig. 4.
Temperature dependence of
Bi spin-lattice relaxation rate 1 / T under the magnetic field along (a) a and (b) c axes of Bi − x Sb x ( x = .
08, 0.10, 0.16). (c) Calculated 1 / T for 3D Dirac fermions with ∆ = µ = ∆ = µ =
10 meV.
Inset: schematic illustration of the band gap ∆ andthe chemical potential µ around the L point. To compare the result with the theoretical calculation for 3D Dirac fermions, we employan expression of orbital excitation,
20, 28) T T ! orb = π µ γ n e c ∗ Z ∞−∞ d ǫ " − ∂ f ( ǫ, µ ) ∂ǫ g ( ǫ ) ǫ ln 2( ǫ − ∆ ) ω | ǫ | , (2)where g ( ǫ ) is the density of states. It gives 1 / T ∼ T down to temperatures for ∆ ∼
0, asshown in Fig. 4(c), while 1 / T decays exponentially in the presence of ∆ ∼
100 K at low tem-peratures. However, we observed a deviation from the T law only below 20 K for x = . H k c . It suggests residual low-lying levels inside the band gap analogous togapless excitation in the topological surface state of Bi − x Sb x . If there is nonuniform relax-ation process around the local distortion, 1 / T would be spatially inhomogeneous. However,we observed only a single component down to low temperatures, indicating the uniform fluc-tuations distinct from the relaxation around the impurity.The T behavior of 1 / T has been also observed in Weyl semimetals at high tempera-tures,
25, 28) whereas the spin contribution becomes significant at low temperatures due to theexistence of the Fermi surface. Then 1 / T obeys the Korringa’s relation 1 / T T ∝ K invariantagainst T . In Bi − x Sb x , the Korringa law is grossly violated down to low temperatures owingto the predominant orbital fluctuations. The Korringa’s constant ∼ / ( T T K ) approaches tounity only near room temperature (Fig. S4).Finally, we discuss the origin for the anisotropic T -invariant Knight shift K deduced from /
10. Phys. Soc. Jpn. the K − χ analysis in Fig. 3(b) and Table S1. For explaining the large constant term, one hasto consider the other factors of intraband orbital susceptibility such as χ core and χ geo , whichcan be enhanced under strong spin-orbit coupling ( λ ∼ .
19, 41, 42) χ core is isotropic for s electrons but becomes anisotropic for partially filled p orbitals in bismuth. Furthermore,the contribution is enhanced by the relativistic nuclear shielding e ff ect. χ geo due to Berryphase in Dirac fermions may also play a significant role for the anisotropic K .
7, 19, 43)
Herewe consider that the Van-Vleck susceptibility identical to the interband e ff ect gives the T -dependent diamagnetism for ∆ comparable to the T scale.
7, 20)
In addition to these orbitalsusceptibilities, a spin-orbital cross term χ so may also relate to the constant K and χ in sec-ond order,
39, 43) where the positive and negative susceptibility is expected for electron and holebands, respectively. However, the positive (negative) K along the c ( a ) axis is inconsistentwith the behavior of χ so . Further numerical calculations are needed to explain quantitativelythe anisotropy and sign of K .In conclusion, the orbital diamagnetism and excitation were studied through the magneti-zation and Bi NMR measurements in the single crystals of bismuth alloys Bi − x Sb x . We ob-served the anisotropic and field-dependent diamagnetism in the bulk magnetic susceptibility.The temperature-dependent diamagnetic Knight shift proportional to the bulk susceptibilitygives the hyperfine coupling constant of Dirac fermions with the e ff ective orbital length scaleover the unit cell. The anisotropic constant shift suggests the enhanced nuclear shielding dueto strong spin-orbit coupling on bismuth. The nuclear spin-lattice relaxation rate governedby orbital current of Dirac fermions is distinct from the spin excitation following the Kor-ringa’s law. These results provide a new microscopic insight into the real-space picture of thediamagnetic orbital current in three-dimensional Dirac fermion systems.We thank S. Inoue and T. Jinno for technical support. We are also grateful to A.Kobayashi, T. Hirosawa, and M. Ogata for fruitful discussions. This work was supportedby JSPS KAKENHI (Grants No. JP19H01837, JP16H04012, and JP19H05824). /
10. Phys. Soc. Jpn.
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