Analysis of Dirac and Weyl points in topological semimetals via oscillation effects
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Analysis of Dirac and Weyl points in topological semimetals via oscillation effects
G. P. Mikitik and Yu. V. Sharlai
B. Verkin Institute for Low Temperature Physics & Engineering,Ukrainian Academy of Sciences, Kharkov 61103, Ukraine
We calculate the extremal cross sectional areas and cyclotron masses for the Fermi-surface pocketsin Dirac and Weyl topological semimetals. The calculation is carried out for the most general form ofthe electron energy bands in the vicinity of the Weyl and Dirac points. Using the obtained formulas,one can find parameters characterizing the Dirac and Weyl electrons in the topological semimetalsfrom appropriate experimental data. As an example, we consider the W1 electrons in TaAs.
The topological Weyl and Dirac semimetals have at-tracted much attention in recent years; see, e.g., reviews[1–5] and references therein. In the Weyl semimetals,two electron bands contact at discrete (Weyl) points ofthe Brillouin zone and disperse linearly in all directionsaround these specific points. The same type of the bandcontact occurs in the Dirac semimetals, but the bands aredouble degenerate in spin. In other words, a Dirac pointcan be considered as a superposition of two Weyl pointsin the quasi-momentum space. The chemical potential ofelectrons, ζ , in the Weyl and Dirac semimetals is close tothe band-contact energy ε d . A number of the Dirac andWeyl semimetals were discovered in recent years [1, 2, 6].Various oscillation effects are widely used in experi-mental investigations of the topological semimetals. Inparticular, measurements of the quantum-oscillationsphase related to the so-called Berry phase [7] were car-ried out in a host of works in order to detect the Weyland Dirac electrons in semimetals, see, e.g., review [5]and references therein. Beside this, using the Shubnikov- de Haas and de Haas - van Alphen effects, the extremalcross sectional areas of the Fermi surfaces and the cy-clotron masses corresponding to these cross sections weremeasured for a number of the Weyl [8–15] and Dirac [16–25] semimetals. In this paper we present formulas forsuch areas and masses. These formulas will allow one toobtain the parameters characterizing the Dirac and Weylpoints in the topological semimetals from the experimen-tal data.The most general electron dispersion relations ε c,v ( p )of the two bands c, v in the vicinity of a Weyl (Dirac)point look as follows [5, 26, 27]: ε c,v ( p ) = ε d + a · p ± E ( p ) , (1)where the quasi-momentum p is measured from thispoint, and [ E ( p )] is a positively definite quadratic formin the components of the vector p . Below we shall choosethe coordinate axes along principal directions of thisform. In this case, one has[ E ( p )] = b p + b p + b p , (2)where b , b , b are the positive constants. The scalingof the coordinate axes, ˜ p i = p i √ b ii , transforms Eqs. (1), (2) into the form, ε c,v = ε d + ˜ a · ˜ p ± | ˜ p | , that depends only on the constant dimensionless vector˜ a ≡ (cid:18) a √ b , a √ b , a √ b (cid:19) . The vector ˜ a characterizes a tilt of the bands ε c,v ( p ), andits length is the most important parameter of dispersionrelation (1). When the length of ˜ a is less than unity,˜ a = a b + a b + a b < , the dispersion relations ε c,v ( p ) looks like in Fig. 1a. Inthis case, the Fermi surface is either a closed hole pocketif ζ < ε d or a closed electron pocket if ζ > ε d . When˜ a >
1, there is a direction in the p -space along whichthe dispersion relations ε c,v ( p ) look like in Fig. 1b, and“open” electron and hole pockets of the Fermi surfaceexist both at ζ < ε d and ζ > ε d . It is necessary toemphasize that the parameter ˜ a , which specifies the tiltof the bands, differs from zero for all the Weyl pointsand for the Dirac points induced by the band inversion [1]since all these points do not belong to the class of highly-symmetric points in the Brillouin zone of the topologicalsemimetals. If ˜ a <
1, a Weyl (Dirac) semimetal falls intothe type I, whereas the case ˜ a > | ζ − ε d | , the Fermi surfaces near the Weyl(Dirac) points are ellipsoids, with the center of the ellip-soids being displaced from these points (i.e., from p = 0)by the vector that is proportional to ( ζ − ε d ). Beside this,if at least two components of ˜ a differ from zero, the axesof the ellipsoid deviate from the axes of the coordinatesystem. The displacement of the Fermi surface leads tothe fact that its maximal cross section perpendicular toa unit vector n generally does not pass through the Weyl(Dirac) point p = 0, Fig. 1. Using the dispersion relation(1), (2), one can calculate both the maximal cross sec-tional area S max of the Fermi surface at an arbitrary di-rection n of the magnetic field H and the cyclotron mass a) pε − ε d ε c ε c ε v ε v b) pε − ε d ε c ε v ε v ε c FIG. 1: Dispersion relations ε c ( p ) and ε v ( p ) of the two con-tacting bands in the vicinity of a Weyl (Dirac) point in thecases of ˜ a < a > ζ − ε d < ζ − ε d > m ∗ = (1 / π )( ∂S max /∂ζ ) corresponding to this cross sec-tion, S max = π ( ζ − ε d ) R / n (1 − ˜ a ) , (3) m ∗ = ( ζ − ε d ) R / n (1 − ˜ a ) . (4)The angular-dependent factor R n in these expressionshas the form, R n = b b b [(1 − ˜ a )˜ n + (˜ a · ˜ n ) ] = X i,j =1 κ ij n i n j , (5) where ˜ n ≡ (cid:18) n √ b , n √ b , n √ b (cid:19) ,κ ij = b b b ( b ii b jj ) / (cid:2) (1 − ˜ a ) δ ij + ˜ a i ˜ a j (cid:3) , and δ ik is the Kronecker symbol.It follows from Eqs. (3) and (4) that | ζ − ε d | = S max π | m ∗ | = 2 e ~ Fc | m ∗ | , (6)where F is the frequency of the quantum oscillationsproduced by the cross-sectional area S max in a physicalquantity Q [i.e., the first harmonic of Q is proportionalto cos(2 πF/H + φ ) where φ is some phase]. There-fore, if the frequency F and the cyclotron mass m ∗ havebeen measured at least for one direction of the magneticfield, formula (6) enables one to find the position of thechemical potential ζ relative to the energy ε d of the Weyl(Dirac) point.The dispersion relation (1), (2) is determined by thesix parameters: b , b , b , ˜ a , ˜ a , ˜ a . Beside this,the orientation of the principal axes of the quadraticform [ E ( p )] relative to the crystallographic axes of thesemimetal can be described by three angles, and hencethe nine parameters define a Weyl (Dirac) point in thegeneral case. The angular dependences of the frequency F are specified by the factor 1 / √ R n in Eq. (3), and thisfactor is defined by the six constants κ ij . Hence, an ap-proximation of experimental angular dependences of thisfrequency with formulas (3), (5) together with Eq. (6)provides possibility to determine the six combinations ofthe parameters characterizing the dispersion relation.The densities n W and n D of the Weyl and Dirac chargecarriers can be expressed in terms of directly-measurablefrequencies of the quantum oscillations, n W = N W V (2 π ~ ) , n D = 2 N D V (2 π ~ ) , (7)where V is the volume of a Weyl or Dirac pocket in theBrillouin zone, V =4[ S (1)max S (2)max S (3)max ] / π / = 8 √ π ( e ~ ) / ( F F F ) / c / , (8) N W and N D are the numbers of the equivalent pockets, F and F are the maximal and minimal frequencies pro-duced by the pocket when the magnetic field rotates invarious planes, and F corresponds to the direction of H perpendicular to the directions at which F and F occur. The cross-sectional areas S ( i )max correspond to thefrequencies F i , and these cross sections are mutually or-thogonal.Consider now two special cases in more detail. Dirac point
In the Dirac semimetals induced by the band inversion,the Dirac points can lie only in symmetry axes of thethird, fourth or sixth order [1]. The well-known Diracsemimetals Na Bi and Cd As just fall into this class. Inthis case, one of the principal axes of [ E ( p )] coincideswith the symmetry axis, which we designate as the axis3. The symmetry also imposes the restrictions: b = b = b ⊥ (generally b ⊥ = b ), a = (0 , , a ). Withthese restrictions, formulas (3) and (5) give the followingexpressions for S max ( θ ), the maximal area of the crosssection that is perpendicular to the magnetic field tiltedat the angle θ to the symmetry axis, S max (0) = π ( ζ − ε d ) b ⊥ (1 − ˜ a ) , (9) S max ( θ ) S max (0) = 1 p cos θ + ǫ sin θ , (10)where ǫ = (1 − ˜ a ) b /b ⊥ . Thus, if S max (0), m ∗ (0), S max ( π/
2) are measured, one can find | ζ − ε d | , b ⊥ (1 − ˜ a ),and ǫ with formulas (6), (9), (10). Note that dependence(10) has the standard form typical of an ellipsoidal Fermisurface. However, the anisotropy of the Fermi surface ǫ = S max (0) /S max ( π/
2) contains the factor p − ˜ a which iscaused by the tilt of the bands ε c,v ( p ). The density n D of the Dirac charge carriers can be found with Eqs. (7),(8) where S (1)max S (2)max S (3)max = S max (0)[ S max ( π/ now. Weyl point near a reflection plane
Consider a Weyl point for which the parameters meetthe following restrictions: b , b ≫ b and a ≪ a ,but ˜ a ≡ a / √ b can have any value satisfying the con-dition (˜ a ) < − (˜ a ) − (˜ a ) . Such a point may appearif it results from a nodal line that lies in the reflectionplane 1 − a lies inthe reflection plane (i.e., a = 0), whereas one of the lo-cal values of b , b is equal to zero (for definiteness, let b = 0) [5, 26, 27]. A nonzero strength of spin-orbit in-teraction lifts the degeneracy of the electron bands alongthe nodal line and can lead to the appearance of two Weylpoints disposed near the reflection plane (symmetricallyrelative to it) [11]. If the spin-orbit interaction does giverise to the Weyl point slightly displaced from the plane,one may expect that a , a , b , b will experience smallchanges, and the condition b , b ≫ b will hold truefor the point. The fact of the appearance of the closedFermi pocket surrounding the Weyl point provides thefulfilment of the condition (˜ a ) < − (˜ a ) − (˜ a ) whichmeans that such a pocket can occur near the point of theline where a is relatively small, a . √ b ≪ √ b ∼ a . θ / π F ( θ ) / F ( ) FIG. 2: The frequency of quantum oscillations, F = cS max / (2 πe ~ ), versus the angle θ between the direction of themagnetic field and the axis 3, Eqs. (3) and (5). Here ˜ a = 0 . a = 0 .
47, ˜ a = 0, b /b = 4, b /b = 0 .
04 (these valuescorrespond to the first set of the parameters in Table I). Theangle θ changes either in the 1 − − Although one may also expect that ˜ a ≈ b and ˜ a , and sowe do not impose any restriction on these parameters.For the Weyl point that results from the nodal line,the perpendicular to this line in the reflection plane, thenormal to the plane, and the direction along the lineare close to the directions of the axes 1 , F ( θ ) = cS max ( θ ) / (2 π ~ e ) where θ is the anglebetween the magnetic field and the axis 3. This anglechanges either in the 1 − − H rotating in each of the planes. In particu-lar, the dependences shown in Fig. 2 are similar to thosefound for the so-called W1 electrons in TaAs; see Fig. 3ain Ref. [11]. In Fig. 2 we take ˜ a = 0. If ˜ a = 0, theminimum value of F ( θ ) is reached at the nonzero angle θ m in the plane 1 − θ m ≈ ˜ a ˜ a − ˜ a − ˜ a r b b . (11)A similar formula describes the position of the minimumof F ( θ ) in the plane 2 − a , ˜ a and b are replaced by ˜ a , ˜ a and b , respectively). Thesenonzero θ m are due to the above-mentioned deviationof the Fermi-surface axes from the coordinate axes. Example: W1 electrons in TaAs
As an example, let us analyze the known experimentaldata for the W1 electrons in the Weyl semimetal TaAs[11]. Near the W1 points the appropriate nodal lines areparallel to the c axis, and therefore it is reasonable tosuppose that the directions of the axes 1 , , a, b, c .Arnold et al. [11] found that the frequency of quantumoscillations F changes like in Fig. 2 when the directionof the magnetic field varies in the reflection plane c − a from the c axis ( θ = 0) to the a axis ( θ = π/ F (0) ≈ m ∗ (0) /m ≈ . θ = 0,the frequency F ( θ ) splits into the two branches F a ( θ ), F b ( θ ) associated with the ellipsoids lying near the axes a and b (Fig. 3), and the frequencies F a ( π/
2) and F b ( π/ F (0) and m ∗ (0), we arrive at ζ − ε d ≈ . m ∗ ( π/ /m ≈ .
24 and 0 .
48 which correspond to thefrequencies 29 T and 59 T, respectively.With ζ − ε d ≈ . R / n (1 − ˜ a ) at θ = 0, R / c (1 − ˜ a ) = c ( ζ − ε d ) e ~ F (0) ≈ . · m s , where R c ≡ R n | θ =0 . On the other hand, Eq. (5) gives R / c (1 − ˜ a ) ≈ q b b (1 − ˜ a − ˜ a )(1 − ˜ a ) , (12)and hence we have found the value of the right hand sideof this expression.According to Eq. (5), the factor R / n at θ = π/ p R a ≈ q b b (1 − ˜ a − ˜ a ) , p R b ≈ q b b (1 − ˜ a − ˜ a ) , for the W1 ellipsoids lying near the a and b axes, re-spectively. Since there are no visible displacements ofthe minima of F a ( θ ) and F b ( θ ) from the point θ = 0in Fig 3a of Ref. [11], we conclude that the parameter˜ a is small for the W1 electrons, i.e., ˜ a ≈
0, and so˜ a ≈ (˜ a ) + (˜ a ) . Therefore, the ratios R / c /R / a and R / c /R / b reduce to s b (1 − ˜ a − ˜ a ) b (1 − ˜ a ) , s b (1 − ˜ a − ˜ a ) b (1 − ˜ a ) . (13)However, these ratios determine F a ( π/ /F (0), F b ( π/ /F (0), and so they are equal to 29 / ab FIG. 3: The outline of the cross sections of the W1 ellipsoidsby the a − b plane in TaAs. / / / a − b plane fromthe a axis ( φ = 0) to the direction [110] ( φ = π/ F a ( φ ) and F b ( φ ) splits into the twobranches F a ( φ ), F a ( φ ) and F b ( φ ), F b ( ϕ ) if ˜ a = 0,i.e., if the principal axes of the ellipsoids deviate from the a and b axes. The φ -dependences of these four branchesare determined by the factors, R a ,a ( φ ) = b b (1 − ˜ a ) sin φ + b b (1 − ˜ a ) cos φ ± p b b b ˜ a ˜ a sin φ cos φ, (14) R b ,b ( φ ) = b b (1 − ˜ a ) cos φ + b b (1 − ˜ a ) sin φ ± p b b b ˜ a ˜ a sin φ cos φ. (15)At φ = 0, the two factors R a ( φ ) and R a ( φ ) reduce to R a , whereas R b (0) and R b (0) coincide with R b . When φ = π/
4, the four frequencies partly merge again since R a ( π/
4) = R b ( π/
4) and R a ( π/
4) = R b ( π/ φ , R a ( φ ) + R a ( φ ) + R b ( φ ) + R b ( φ ) = 2 R a + 2 R b . This equality leads to the relation between the appropri-ate four branches of the frequency,1[ F a ( φ )] + 1[ F a ( φ )] + 1[ F b ( φ )] + 1[ F b ( φ )] = 2[ F a ] + 2[ F b ] , (16) TABLE I: The two possible sets of the parameters specifyingthe W1 electrons in TaAs.Set √ b √ b √ b ˜ a ˜ a ˜ a m/s 10 m/s 10 m/s1 3 .
37 6 .
74 6 . . .
47 02 6 .
88 3 .
30 6 . . .
47 0 where F a and F b are equal to 29 and 59 T. According toFig. 3a in Ref. [11], F a ( π/
4) = F b ( π/ ≈
33 T. Thiscondition gives the fourth relation on the five parameters˜ a , ˜ a , b , b , and b ,[ R c ] / [ R a ( π/ / = F a ( π/ F (0) ≈ . (17)The parameter ˜ a can be obtained from the band struc-ture calculation along the a axis. In particular, figure 1in Ref. [11] permits one to obtain the following crude es-timate: ˜ a ≈ .
5. Taking into account the above fourrelations between b , b , b , ˜ a , ˜ a , we find two possi-ble sets of the parameters characterizing the W1 pointsin TaAs, Table I. The dependences of the frequencies F a , F a , F b , F b on φ for the first set of the parameters arepresented in Fig. 4.At nonzero values of ˜ a and ˜ a , the principal axes ofthe ellipsoid in the a − b plane deviate from the a and b axes (Fig. 3). A simple analysis leads to the followingformula for the deviation angle ψ :tan(2 | ψ | ) = [ F a ( π/ − − [ F a ( π/ − [ F a ] − − [ F b ] − , (18)where the value of F a ( π/
4) = F b ( π/ ≈ . | ψ | ≈ ◦ or | ψ | ≈ ◦ . These two values of | ψ | correspond tothe two sets of the parameters in Table I. For the firstset, the orientation of the ellipsoids in the a − b plane isschematically shown in Fig. 3. In this case, the maximalaxis of their cross sections by the a − b plane is inclinedat the angle of 11 ◦ to the a and b axes. For the secondset, this angle is equal to 79 ◦ .Knowing the angle ψ , the frequencies F and F informula (8) can be calculated, and we arrive at F F = F a F b (cid:20) − ( F a − F b ) F a F b tan (2 | ψ | ) (cid:21) − / . (19)Eventually, expressions (7), (8), (19) with F = F ( θ =0) = 7 T give the density n W ≈ . · cm − pro-duced by the eight equivalent pockets of the W1 electronsin TaAs. [1] N.P. Armitage, E.J. Mele, A. Vishwanath, Rev. Mod.Phys. , 015001 (2018). φ / π F ( φ ) / F ( θ = ) FIG. 4: The frequencies F a ,a (the solid lines) and F b ,b (thedashed lines) versus the angle φ between the a axis and themagnetic field lying in the a − b plane. The parameters arethe same as in Fig 2 (i.e., they coincide with the first set inTable I).[2] A. Bernevig, H. Weng, Z. Fang, X. Dai, J. Phys. Soc.Jpn. , 041001 (2018)[3] H. Gao, J.W.F. Venderbos, Y. Kim, A.M. Rappe, AnnualReview of Materials Research , 153 (2019).[4] S. Wang, B.-C. Lin, A.-Q. Wang, D.-P. Yu, Z.-M. Liao,Advances in Physics: X , 518 (2017).[5] G.P. Mikitik, Yu.V. Sharlai, J. Low Temp. Phys. ,272 (2019)[6] C.-L. Zhang, C.M. Wang, Z. Yuan, X. Xu, G. Wang, C.-C. Lee, L. Pi, C. Xi, H. Lin, N. Harrison, H.-Z. Lu, J.Zhang, S. Jia, Nat. Commun. , 1028 (2019).[7] G.P. Mikitik, Yu.V. Sharlai, Phys. Rev. Lett. , 2147(1999).[8] X. Huang, L. Zhao, Y. Long, P. Wang, D. Chen, Z. Yang,H. Liang, M. Xue, H. Weng, Z. Fang, X. Dai, G. Chen,Phys. Rev. X , 031023 (2015).[9] Y. Luo, N.J. Ghimire, M. Wartenbe, H. Choi, M. Neu-pane, R.D. McDonald, E.D. Bauer, J. Zhu, J.D. Thomp-son, F. Ronning, Phys. Rev. B , 205134 (2015).[10] C. Shekhar, A.K. Nayak, Y. Sun, M. Schmidt, M. Nick-las, I. Leermakers, U. Zeitler, Yu. Skourski, J. Wosnitza,Z. Liu, Y. Chen, W. Schneller, H. Borrmann, Yu. Grin,C. Felser, B. Yan, Nature Physics , 645 (2015).[11] F. Arnold, M. Naumann, S.-C. Wu, Y. Sun, M. Schmidt,H. Borrmann, C. Felser, B. Yan, E. Hassinger, Phys. Rev.Lett. , 146401 (2016).[12] J. Hu, J.Y. Liu, D. Graf, S.M.A. Radmanesh, D.J.Adams, A. Chuang, Y. Wang, I. Chiorescu, J. Wei, L.Spinu, Z.Q. Mao, Sci. Rep. , 18674 (2016).[13] J. Du, H. Wang, Q. Chen, Q. Mao, R. Khan, B. Xu, Y.Zhou, Y. Zhang, J. Yang, B. Chen, C. Feng, M. Fang,Sci. China-Phys. Mech. Astron. , 657406 (2016).[14] P. Sergelius et al., Sci. Rep. , 33859 (2016).[15] Z. Wang, Y. Zheng, Z. Shen, Y. Lu, H. Fang, F. Sheng,Y. Zhou, X. Yang, Y. Li, C. Feng, Z.-A. Xu, Phys. Rev.B , 121112 (2016). [16] L.P. He, X.C. Hong, J.K. Dong, J. Pan, Z. Zhang, J.Zhang, S.Y. Li, Phys. Rev. Lett. , 246402 (2014).[17] A. Pariari, P. Dutta, P. Mandal, Phys. Rev. B , 155139(2015).[18] T. Liang, Q. Gibson, M.N. Ali, M. Liu, R.J. Cava, N.P.Ong, Nature Materials , 280 (2015).[19] Y. Zhao, H. Liu, C. Zhang, H. Wang, J. Wang, Z. Lin, Y.Xing, H. Lu, J. Liu, Y. Wang, S.M. Brombosz, Z. Xiao,S. Jia, X.C. Xie, J. Wang, Phys. Rev. X , 031037 (2015).[20] A. Narayanan, M.D. Watson, S.F. Blake, N. Bruyant, L.Drigo, Y.L. Chen, D. Prabhakaran, B. Yan, C. Felser, T.Kong, P.C. Canfield, A.I. Coldea, Phys. Rev. lett. ,117201 (2015).[21] Z.J. Xiang, D. Zhao, Z. Jin, C. Shang, L.K. Ma, G.J.Ye, B. Lei, T. Wu, Z.C. Xia, X.H. Chen, Phys. Rev. lett. , 226401 (2015).[22] W. Desrat, C. Consejo, F. Teppe, S. Contreras, M.Marcinkiewicz, W. Knap, A. Nateprov, E. Arushanov, J. of Phys.:Conf. Ser. , 012064 (2015).[23] J. Cao, S. Liang, C. Zhang, Y. Liu, J. Huang, Z. Jin, Z.-G. Chen, Z. Wang, Q. Wang, J. Zhao, S. Li, X. Dai, J.Zou, Z. Xia, L. Li, F. Xiu, Nat. Commun. , 7779 (2015).[24] L.-P. He, S.-Y. Li, Chin. Phys. B , 120302 (2018).[26] G.P. Mikitik, I.V. Svechkarev, Fiz. Nizk. Temp. , 295(1989) [Sov. J. Low Temp. Phys. , 762(1996) [Low Temp. Phys.527