【作者】 陈亮；

【导师】 完绍龙；

【作者基本信息】 中国科学技术大学 ， 理论物理， 2012， 博士

【摘要】 自上世纪80年代发现整数量子霍尔效应和分数霍尔量子效应以来,寻找具有非平凡拓扑序的材料和研究这些材料的物理性质,一直是现代凝聚态物理学的一个重要课题。近年来,作为量子霍尔效应概念的一个延伸和发展,拓扑绝缘体在实验上和理论上都取得了重大的进展。拓扑绝缘体是这样一种材料,在这种材料的体内存在能隙,而在这种材料的表面(边界)存在无能隙的表面态(边界态),并且这些表面态(边界态)是由拓扑绝缘体(量子自旋霍尔体系)的能带结构具有非平凡的拓扑所决定的。在拓扑绝缘体被发现之后,特别是超导趋肤效应可能导致拓扑绝缘体表面出现Majorana束缚态之后,人们发现,超导体系中也可能存在拓扑非平凡的性质,这些材料被定义成拓扑超导体,即材料的体里面存在一个超导能隙,而表面上存在一个无能隙的Majorana表面态,近年来的实验研究表明,CuxBi2Se3(?)根可能是这样一种拓扑超导体。我攻读博士学位期间,主要研究了零温和有限温度情况下拓扑绝缘体之间的Casimir效应,以及CuxBi2Se3中的可能存在的拓扑非平凡超导配对势。本博士论文主要包含以下内容：第一章主要介绍拓扑绝缘体的一些背景知识：理论方面,主要介绍量子霍尔,反常量子霍尔效应中的Haldane模型,量子自旋霍尔效应中的Kane-Mele模型,Fu,Kane和Mele对三维拓扑能带绝缘体的定义以及Qi,Hughes和Zhang对三维拓扑绝缘体的拓扑场论描述；实验方面,主要介绍几种已经发现的拓扑绝缘体材料,包括两维情况下的HgTe/CdTe量子阱中观测到的量子化纵向电导,以及一系列三维拓扑绝缘体,特别是Bi2Se3,的能带结构和自旋分辨-角分辨光电子能谱；最后介绍与拓扑绝缘体相关的最重要的反常现象——拓扑磁电效应。第二章主要介绍我第一方面的研究工作,拓扑绝缘体之间的Casimir效应。本章第一节介绍Casimir效应相关的理论和实验方面的研究工作。第二节介绍我对零温情况下拓扑绝缘体之间Casimir效应的研究。我们从拓扑绝缘体表面存在有限质量的Dirac费米子出发,推导出一个具有特殊边界条件的电磁场理论。利用这个有效电磁场理论,我们计算了电磁波在拓扑绝缘体表面的反射系数和透射系数。利用得到的反射系数和Lifshitz公式,我们计算了Casimi r势能密度Ec作为拓扑绝缘体之间距离a,拓扑磁电极化θ,拓扑绝缘体表面Dirac费米子质量m以及拓扑绝缘体内光学振子的振动强度g的函数EC(a,θ,m,g。我们发现拓扑磁电极化θ越大,表面Dirac费米子质量越大,光学振子的振动强度g越小,则越有可能观测到排斥Casimir力。此外,我们还得到了一个判定拓扑绝缘体之间能否观测到排斥Casimir力的临界关系：(ma)c=1/2,当ma>(ma)。时,拓扑绝缘体之间的Casimir力可能是排斥力。利用这个临界关系,我们估算了实际材料TlBiSe2中出现排斥Casimir力所需要的表面Dirac费米子的质量(m>leV).本章第三节介绍有限温度时拓扑绝缘体之间的Casimir效应。因为实际测量Casimir力时温度是一个必须考虑的参量,所以研究温度对拓扑绝缘体之间Casimir力的影响是有意义的。我们将零温情况下拓扑绝缘体之间Casimir效应的研究工作推广到有限温度,发现如果零温情况下拓扑绝缘体之间存在排斥Casimir力,那么随着温度的升高,拓扑绝缘体之间的排斥Casimir力会减弱,但前面给出的临界关系(ma)。=1/2并不随温度变化。第三章介绍我第二方面的工作,超导材料CuxBi2Se3中动量相关的拓扑非平凡配对势。本章第一小节简要回顾近两年对CuxBi2Se3研究的一些实验结果,包括超导转变温度,铜的参杂方式和最佳参杂浓度等。第二小节介绍与这个材料相关的理论。目前人们对CuxBi2Se3的配对方式和导致超导的吸引相互作用还不清楚,Fu和Berg提出动量无关的奇宇称超导配对势来描述其超导对称性,但这一配对势与Sasaki等人实验的结果存在不符的地方,虽然Sasaki等人又提出了一些动量无关的奇宇称配对势来解释这个实验现象,但他们的配对势给出的谱函数并不保持晶格本身所具有的C3旋转对称性。我们研究发现,要得到具有C3旋转对称性的谱函数,则必须考虑动量相关的超导配对势。第三小节介绍我们对CuxBi2Se3中可能存在的动量相关的超导配对势所进行的系统的研究工作。对其中几种典型的配对势,我们计算了体系的拓扑不变量,表面态和体里面Bloch态的谱函数和态密度。我们发现,对于偶宇称配对,当体系在整个布里渊区存在不等于零的能隙时,其拓扑不变量——绕数总是等于零,说明这种配对势是拓扑平庸的,不存在拓扑非平凡的表面态。对于奇宇称配对,当体系在整个布里渊区存在不等于零的能隙时,相应的绕数总是不等于零,说明这种配对势是拓扑非平凡的。对于奇宇称配对势,我们发现当体系在整个布里渊区存在有限数目个能隙节点时,绕数仍然是可以定义的,并且不等于零,此时虽然能隙闭合但仍然存在拓扑非平凡的表面态。特别的,我们发现有几种特殊的动量相关的超导配对,它们能给出与Fu和Berg以及Sasaki等人提出的动量无关的超导配对相似的谱函数和态密度。这说明目前的实验结果无法甄别动量相关的配对势和动量无关的配对势,我们的工作为CuxBi2Se3中的超导配对势提供了新的研究方向。最后我们提出一种配对势,不仅具有拓扑非平凡的表面态,给出的谱函数与Sasaki等人实验结果相符,而且还不破坏晶格结构本身所具有的C3旋转对称性,我们认为CuxBi2Se3中实际的配对势可能具有这种形式。第四章对我博士期间所做的工作做一个总结,并指出一些可能的进一步研究方向。更多还原

【Abstract】 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.更多还原