| Since1980, the Integer quantum Hall effect (IQHE) and the fractional quan-tum Hall effect(FQHE) had been discovered, there is tremendous interest currently in topological nontrivial quantum states and their novel properties in condensed matter physics. Recently, there are some generalizations of IQHE which are named as topo-logical insulator(TI) in three dimensions and quantum spin Hall effect(QSHE) in two dimensions if the time reversal symmetry is preserved. TI(QSHE) is fully gapped in the bulk, as the normal insulator, but gapless on the surface(edge), as a massless Dirac Fermion, these surface(edge) states are protected by time reversal symmetry. Soon af-ter the discovery of TI, the aspects of non-trivial topology have been investigated in superconductivity systems. Similar to TI, topological superconductor(TSC) has a non-zero superconductivity gap in the bulk but a gapless Andreev bound state(ABS) on the surface, and it is characterized by an integer in three dimensions and a Z2index in two dimensions. Recently, the copper doped Bi2Se3is reported to be superconductor with non-trivial topological properties. The doctoral dissertation is organized as follows.A brief review both on theoretical and experimental researches are given in chap-ter one. The theoretical aspects include:the quantum Hall effect, quantum anomalous Hall effect (Haldane Model), quantum spin Hall effect (Kane-Mele model), Fu-Kane-Mele’s topological band description and Qi-Hughes-Zhang’s topological field descrip-tion of three dimensional topological insulator. On the experimental aspects:some candidates of TI are introduced. The first example is HgTe/CdTe quantum well, the band structure and experimental evidence for topological non-trivial edge state will be shown. The second example is the second generation of TIs, including Bi2Se3, Bi2Te3and Sb2Te3. These materials are described by the low energy effective Hamiltonian proposed by H. Zhang et. al. The following evidences, which support Bi2Se3etc. to be TIs, are presented:(1) Surface band structure from angle-resolved photoemission spectroscopy (ARPES),(2) Spin-momentum locked surface state from spin-resolved-angle-resolved photoemission spectroscopy. The topological megneto-electric effect is introduced in the last section of this chapter.In chapter two, I report our researches on Casimir effect between TIs. A brief review on Casimir effect is given in the first section. Our researchs on Casimir inter-action between TIs with finite surface band gap at zero-temperature are given in the second section. In the third section, we generalize Casimir interaction between TIs to finite temperature case. We deduce an effective theory to describe the electromagnetic response on the TI-vacuum interface with a finite surface band gap. By using this effec-tive theory, we find Maxwell equations with non-trivial surface corrections to describe the reflection and refraction coefficients of electromagnetic waves on the TI-vacuum interface, the reflection coefficients are related to the Casimir energy between TIs by Lifshitz formula. Our calculations show that the following conditions are profitable for repulsove Casimir force:(1) large topological magneto-electric polarizations,(2) large surface band gap,(3) small oscillation strength in TI bulk,(4) low temperature. We find that if the topological magneto-electric polarizations of two TIs have opposite signs and the product of TI surface band gap m and distance between TIs a is greater than1/2, the Casimir force between TIs maybe repulsive, this universal relationship is independent on topological magneto-electric polarization, oscillation strength in TI bulk and temperature. One can use this relation to estimate the critical surface band gap for practical materials, we find mc～1eV for repulsive Casimir force between TlBiSe2dielectrics.In chapter three, I introduce systematically on our research about momentum de-pendent pairing potentials in CuxBi2Se3. In the first section, I give a brief review on the experimental research of superconductor CuxBi2Se3, including the following results,(1) the superconductivity transition temperature is about3.8K,(2) in the superconduc-tivity phase Cu atoms are intercalated between quintuple layers of Bi2Se3,(3) such an intercalation will make the chemical potential upper shift into the conduction band. In the second section, some theoretical investigations are introduced. Fu and Berg pro-posed a fully gapped odd-parity pairing potential to describe the topological non-trivial superconductivity in CuxBi2Se3, however, Fu and Berg’s theory is inconsistent with a recent experiment investigated by Sasaki et. al., there are some other explanations pro-posed by Sasaki et. al.. We find that the pairing potential proposed by Sasaki et. al. dose not satisfy the C3rotation symmetry of the rhombohedral lattice, and momentum dependent pairing potentials are indispensable for pairing potentials satisfy the follow-ing conditions:(1) it is topological non-trivial,(2) the band structure has some point nodes in the plane paralleling the (111)-surface,(3) C3rotation symmetry is preserved. Our analyses on momentum dependent pairing potentials in CuxBi2Se3are given in the third section. We find that the parity-even momentum dependent pairing potentials with non-vanishing gap are topological trivial. and the parity-odd momentum depen-dent pairing potentials with non-vanishing gap are topological non-trivial. We inves- tigate some typical momentum dependent pairing potentials in CuxBi2Se3, calculate the winding number, surface and bulk spectral function and corresponding density of states. We find there are some typical momentum dependent pairing potentials which exist similar spectral functions of momentum independent pairing potentials proposed by Fu and Berg, and Sasaki et. al.. In addition, we find a solusion for the three condi-tions proposed above.In the last chapter, I give a conclusion and some outlook for Casimir effect be-tween TIs, pairing symmetry and pairing mechanism of CuxBi2Se3. |
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