| Topological insulator is a new quantum state with internal insulation and surface conduction.With its novel physical properties,topological insulator has become a very popular research field in recent years.Moreover,the surface states are protected by time-reversion symmetry and robust against impurities or disorders.Therefore,topological insulators have huge application prospects in the fields of spintronics,new two-dimensional electronic devices,quantum computers and so on.Rare earth hexaborides are strongly correlated systems containing f electrons,and therefore exhibit various exotic electronic properties:superconductivity,excellent electron emission properties,Kondo effect,mixed valence and topological properties,etc.,making rare earth hexaborides widely attention and research.In 2010,the concept of topological Kondo insulator was first proposed and theoretically predicted that samarium hexaboride(SmB6)may be a candidate material.After more than ten years,a large number of theoretical and experimental reports have successively confirmed that SmB6 became the first topological Kondo insulator.Later,ytterbium hexaboride(YbB6)was also predicted to be a topological insulator.However,the debate on whether SmB6 and YbB6 are topological insulators from both theoretical calculations and experiments has never stopped in the past decade.Moreover,there are relatively few studies on three-dimensional topological insulators,and the research of topological properties of strongly correlated systems is even more difficult.In this paper,the electronic structure and surface states for different surfaces of SmB6 and YbB6 are systematically studied by density functional theory.Firstly,the analyzing the electronic structure of SmB6 shows that due to the spin-orbit coupling(SOC)effect,orbital hybridization occurs between the 4f and 5d electrons of Sm,resulting in band inversion and opening a small bulk gap of 19 meV(experimental values range from several meV to 20 meV)at the high symmetry point X.Through the surface energy calculation,it is found that the slab structures of the SmB6(001)surface may appear with Sm and B6 as the terminated surface,but the Sm terminated surface is more stable.And these two terminated surfaces have metallic surface states in the bulk energy gap region.The Z2 topology number of SmB6 is(1;111).There are surface states on polar(001),non-polar(110)and(111)surfaces;and each surface has three Dirac cones.On the(110)surface,the surface state of Y point has the property of "topological crystal insulator",and the absolute values of the three mirror Chern numbers of SmB6 are 2,1 and 1.All these indicate that SmB6 is a topological insulator,providing some strong evidences for proving the topological insulator properties of SmB6.Secondly,different from SmB6,in YbB6,band inversion occurs between the Yb 5d and B 2p under the action of SOC,which also opens a bulk gap at the X point,resulting in the insulating property of bulk YbB6.The Z2 topology number of YbB6 is(1;111).There are also surface states on three different surfaces,and each surface has three Dirac cones,indicating the topological properties of YbB6.Under the action of SOC,the 4f band is split into 4f j=5/2 and j=7/2.The bandwidth of the splitting is consistent with the experiment,but there is still a large gap in the energy position relative to the Fermi level;When GGA+SOC+U(U=7eV)is used,4fj=7/2 band moves down to the vicinity of-1.0eV below Fermi level,which is in good agreement with the experiment.This indicates that the 4f electrons of YbB6 have strong correlation,and since the 4f band of Yb is located at-1.0eV of the valence band,YbB6 has no Kondo effect.Then,through the analysis of the influence of U value on the band structures of YbB6,it is found that with the increase of U value,YbB6 changes from semiconductor to semi-metal.When the U value is 14 eV,the band inversion disappears,which means that the topological properties disappear at the same time,and YbB6 changes from a semi-metal to an ordinary insulator.These have important theoretical significance for further understanding the topological properties of YbB6.Finally,we explored the influence of strain on SmB6 and YbB6.The structure of SmB6 is stable in the strain range of-16%to 12%,and Mulliken charge analysis shows that the charge of Smf orbital decreases with increasing compressive strain,which is consistent with the experiment.Under tensile strain,the energy gap decreases monotonically approximately,but does not close,and remains topologically insulating.However,under compressive strain,the energy gap first increases and then decreases.When the compressive strain is-12%,the energy gap disappears,and SmB6 transforms from an insulator to a metal.In the strain range of-10%to 10%,YbB6 also has good structural stability.when using the GGA+SOC method,it is found that with the increase of compressive strain,the charge of the Yb f orbital decreases from 13.73 to 13.49.In the whole strain range,the energy gap decreases monotonically without closing,indicating that the bulk gap is not sensitive to the strain,and the topological insulating properties of YbB6 are hardly affected.However,when the GGA+SOC+U(U=7 eV)method is used,in the compressive strain range of 0 to-5%,the band inversion energy range between Yb 5d and B 2p increases,indicating that the topological properties of YbB6 still exist.However,with the increase of tensile strain,the energy band inversion range is reduced,and when it reaches 3%,the energy band inversion disappears,which indicates that the topological properties of YbB6 disappear and become an ordinary insulator.The research results reveal the possibility of topological regulation of rare earth hexaborides by strain,and provide a research direction for the realization of topological insulator materials. |
