Superconductivity in topologically nontrivial material Au2Pb(3)

It is of particular interest to investigate the geometric phase of the bands near the Fermi surfaces.By projecting the Kohn –Sham wavefunctions into various local orbital basis (Supplementary Figure S8),we ?nd that the dominant band components near the Fermi surfaces are parity-even d -orbitals of Au and parity-odd p -orbitals of Pb.We choose two closed paths on the valence band around one of the T-sectors,and calculate topological invariants through the phase difference,φk ðÞ¼arg d 9k -arg p 9k

,of the projection coef ?cients of Kohn –Sham wavefunctions of the highest occupied onto the Au-d and Pb-p orbitals along these loops.The two loops are carefully chosen to avoid band degeneracy at the band crossing (see the dark blue curve in Figure 5a).In Figure 5b,a unit vector n =[cos φ,sin φ],is plotted for each k -point on the loop to visualise the orbital texture.

A few interesting features are observed for the projected orbital textures.In particular,the winding numbers are zero for the orbital textures on the inner loop (see e.g.,a typical orbital texture on the inner loop in Figure 5b).In contrast,for the outer loop,the orbital textures corresponding to various Au-d and Pb-p x orbitals show a nontrivial topology,with the winding number equals ?1.On the other hand,from Figure 6we ?nd that an ‘8’shape projected nodal line is numerically con ?rmed on a certain Brillouin zone slice,while on another Brillouin zone slice a

circle-shaped

Figure 4.The band structures of Au 2Pb along the Brillouin zone path R –Γ,which undergoes the most obvious change under 1%uniaxial compression.(a )Calculation was done with relaxed structure and without spin-orbital coupling.The loop A and loop B correspond to the outer loop and inner loop in Figure 5a,respectively.(b )Calculation was done with relaxed structure and with spin-orbit coupling.The result is quite similar to a .(c )Calculation was done with the structure under 1%uniaxial compression and with spin-orbit coupling.The point R is still de ?ned by R =(b 1+b 2+b 3)/2,as in the case of relaxed structure.Note that the conduction band and the valence band shift to each other as compared with (b ).This Brillouin zone region noticeably responsive to external pressure is referred to as the T-sector hereafter,and will be focused on in the investigations of orbital texture.(d )Comparison of the density of states of the relaxed structure (black curve)and the compressed structure (red curve).Spin-orbit coupling is included.The density of states near the Fermi level (shifted to 0eV)is enhanced under the applied external pressure.

Superconductivity in topologically nontrivial material Au 2Pb Y Xing et al

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Published in partnership with Nanjing University

npj Quantum Materials (2016)16005

projected nodal line is observed.These intriguing observations hint at nontrivial band structure and topological properties of the T-sectors.

To capture the main features of the band structure around the T-sectors,we introduce an ad hoc two-band effective Hamiltonian to describe the physics regarding the Au-d and Pb-p x orbitals:H eff ¼M ðk Þk U σ;

ð1Þ

where k denotes the momentum relative to the central point of the ‘8’shape nodal line shown in Figure 6a,and M (k )is a polynomial of k ,taken as M ðk Þ¼m 1-????????????????

m 22þk 2q with m 14m 2.The energy spectra read E 7¼7M ðk Þk j j ,exhibiting a nodal point at k =0and a nodal surface at k ¼????????????????

m 21-m 22p .One can verify that

the above Hamiltonian leads to the band structure and orbital texture consistent with the density functional theory (DFT)calculation results.Actually,the valence band wavefunction of the Hamiltonian is

u k j i ¼-sin y 2

e -i ?=2

cos y 2e

i ?=2

!;ð2Þwhere θand ?are the polar and azimuthal angles of k ,

respectively.From the wavefunction,we can see that the orbital texture of the Hamiltonian (1)is determined solely by the k U σfactor.The form of M (k )has been chosen to mimic the spectra from DFT calculation,reproducing the nodal surface near the Fermi level.It is easy to see that the projected orbital texture is dependent only on ?evolved along each path.The orbital texture on any loop that encloses the z axis (k x =k y =0)connecting the north and south poles of the Bloch sphere,has a winding number 1or ?1.In contrast,if the loop excludes the z axis,the orbital texture is trivial with the winding number being 0.Accordingly,a loop enclosing the nodal line on the k z =constant plane (i.e.,the crossing line between the plane with ?xed k z and the nodal surface)always encloses the z axis,giving a nonzero winding number of the orbital texture (the outer loop of Figure 5b).However,a loop inside the nodal line may or may not enclose the z axis.In the latter case,the winding number of the orbital texture is zero,which is understood to be the case of the inner loop of Figure 5b.

DISCUSSION

The nontrivial topology of the orbital texture suggests the possibility of topological superconductivity.Note that the whole-Brillouin zone includes eight different sectors around Fermi energy,with the other seven equivalent copies of the effective Hamiltonian equation (1)obtained by time-reversal and symmetry of the orthogonal space group.In particular,the time-reversal copy of the Hamiltonian (1)can be described by ΓH eff Γ-1¼M ðk Þ-k x σx þk y σy -k z σz ÂÃ,where the time-reversal operator Γ=iKs y where K is the complex conjugate operator,and s y is a Pauli matrix operating on real spin.We note that the Hamiltonian equation (1)and its time-reversed copy have opposite spin orientations,which are not explicitly described here.It is readily veri ?ed that the topology of the two time-reversed Fermi surfaces is the same.Note that the p …… 此处隐藏：6027字，全部文档内容请下载后查看。喜欢就下载吧 ……