Research On Quantum Hall Effects In Topological Matter

The quantum Hall effect was first observed in the two-dimensional electron gas under a strong magnetic field.The quantum Hall conductance can be expressed as ne²/h,where n is an integer.This result creates a new method of directly measuring fundamental constants.With the deepening of the research,it was discovered that the integer n in quantum Hall conductance can be expressed as integrals of the Berry curvature of all occupied states in the Brillouin zone,which opened the prelude to the study of topological matter.Topological matter is a state of matter with exotic topological properties,and its topological properties can be described by topological invariants.Because the topological properties of topological matter are not affected by the geometry of the devices and have great application prospects,researchers are constantly searching for new topological matter and studying their unique topological properties.After the discovery of the quantum anomalous Hall effect,because its non-dissipative boundary state has great potential in manufacturing low-energy electronic devices,how to realize the quantum anomalous Hall effect in topological materials has become an important research direction.Recently,it has been reported that MnBi₂Te₄is an intrinsic antiferromagnetic topological insulator.Theoretical calculations show that its surface states have an energy gap of size 88 meV,and the quantum anomalous Hall effect has been observed in its thin film.However,the latest results of angle-resolved photoemission spectroscopy shows that the surface states may be gapless,which is far from the previous theory.Therefore,studying which factors will affect the topological properties of the surface states of MnBi₂Te₄and how to adjust the quantum anomalous Hall effect has not only important theoretical value,but also huge application value.In addition,the two-dimensional electron gas will form Landau levels in a strong magnetic field,and the Landau levels will deform and intersect the Fermi energy at the edge of the sample,forming the one-dimensional topological edge states.The topological edge states move in a dissipationless manner and can generate quantum Hall conductance of integer multiples of e²/h.However,in the three-dimensional electron gas,the dimension along the direction of the magnetic field prevents the quantization of Hall conductance.Therefore,in the study of three-dimensional topological materials,what conditions are needed to form a three-dimensional quantum Hall effect,not only have important value in understanding the unique topological state of matter,but also have application value in the aspect of control of quantum states.In summary,this dissertation will study how to realize the three-dimensional quantum Hall effect in three-dimensional topological semimetals,and the factors that affect the topological properties of MnBi₂Te₄and how to control quantum anomalous Hall effect in MnBi₂Te₄thin films.The main research results are as follows:We have studied the three-dimensional quantum Hall effect of topological semimetals in an external magnetic field.We show that the surface states of Fermi arcs in topological semimetals can produce a unique three-dimensional quantum Hall effect through the Weyl nodes.Due to the topological constraint,the Fermi arc on a single surface is open and cannot support the quantum Hall effect.The Fermi arcs on the upper and lower surfaces can complete the closed Fermi loop through the"wormhole"tunneling via Weyl nodes to produce the quantum Hall effect.The boundary states of the Fermi arc exhibit a unique three-dimensional distribution,which can be regarded as another example of the(d-2)-dimensional boundary states.In addition,when the Fermi surface gradually changes and cuts through the Weyl node,the Hall conductivity of the film evolves from the 1/B dependence to a quantized platform.This phenomenon can be achieved by adjusting the gate voltage of the topological semimetal flake.We have also studied the topological properties of the surface states of the newly discovered intrinsic antiferromagnetic topological insulator MnBi₂Te₄.Previously,it was predicted that the surface states of MnBi₂Te₄would open a large energy gap.However,the latest angle-resolved photoemission spectroscopy measurements show that the energy gap of the surface states is not always observable.In order to resolve this discrepancy between the theoretical prediction and experimental results,starting from the three-dimensional bulk Hamiltonian and considering the spatial distribution of bulk magnetization,we analytically deduce the two-dimensional effective model for the surface states of a single surface for a semi-infinite geometry.Our calculated results show that the small surface gap observed in the angle-resolved photoemission spectroscopy experiments may be caused by a much smaller and more localized intralayer ferromagnetic order.In addition,we also calculate the spatial distribution and penetration depth of the surface states,which shows that the surface states are mainly localized in the first two septuple layers of the sample surface.From our analytical results,the influence of bulk parameters on surface states can be clarified.At the same time,we also deduce the effective model of the surface state for the MnBi₂Te₄thin films,which shows that the finite size effect decreases as the thickness increases.When the sample is thicker,the more localized the bulk magnetization is,the smaller the effective magnetization will be.We also show that the Chern number oscillates between-1 and 0 as the number of odd and even layers changes.In addition,we have studied the influence of the external electric field on the quantum anomalous Hall effect in the intrinsic antiferromagnetic topological insulator MnBi₂Te₄thin films.For the odd-layer MnBi₂Te₄thin film,if the potential energy V induced by the electric field is large enough,the Hall conductance will change from the original e²/h to0,that is,the Chen number will change from 1 to 0.In other words,with the increase of V,a topological quantum phase transition from a topologically non-trivial state to a topologically trivial state occurs.In addition,because the even-layer MnBi₂Te₄thin films has the PT(spatial inversion and time-reversal combined)symmetry,the Berry curvature of the energy band is zero.However,when an electric field is applied in the direction perpendicular to the surface,the PT symmetry of the system will be destroyed.At this time,the energy band will lift the degeneracy and split,and the Berry curvature of the energy band will no longer be zero.Therefore,for even-layer thin films,when the Fermi energy cuts to the energy band,the applied electric field will cause the anomalous Hall conductance of the even-layer thin films to change from zero to a finite value.Unlike the odd-layer thin films,when the Fermi energy falls in the surface gap,its Hall conductance is still 0,which does not change with the change of V,that is,no topological phase transition occurs.