Time-reversal symmetry breaking and gapped surface states due to spontaneous emergence of new order in d-wave nanoislands
Time-reversal symmetry breaking and gapped surface states due to spontaneous emergence of new order in d-wave nanoislands
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DOI：
10.1103/physrevb.96.060503
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发表时间：
2016-10
影响因子：
8.6
通讯作者：
Y. Nagai;Y. Ota;K. Tanaka
中科院分区：
文献类型：
--
作者：
Y. Nagai;Y. Ota;K. Tanaka
我们在具有[110]表面的纳米尺度$d$波系统中自一致地求解了$d$波阶参数的Bogoliubov-de Gennes方程，并证明了由于自发诱导的扩展$s$波对称的复阶参数，在低温下会发生自发的时间反转对称（TRS）破破。在TRS存在下，Andreev表面束缚态受到一维（1D）拓扑不变量的保护，由于这个新阶参量的出现而被打破。扩展的$s$-波阶参数局域化在表面附近的一个狭窄区域内，这与无间隙Andreev表面态的拓扑保护具有一维拓扑不变量的特性是一致的。在这个trs破缺阶段，不仅诱发了复杂阶数参数，而且d -波阶数参数本身也变得复杂。此外，拓扑保护的消失导致了新的表面附近涡旋现象。研究表明，当纳米岛的边长或无限长纳米带的宽度较大时，沿表面沿扩展的$s$波阶参数会形成涡-反涡对。
We solve the Bogoliubov-de Gennes equations self-consistently for the $d$-wave order parameter in nanoscale $d$-wave systems with [110] surfaces and show that spontaneous time-reversal symmetry (TRS) breaking occurs at low temperatures due to a spontaneously induced complex order parameter of extended $s$-wave symmetry. The Andreev surface bound states, which are protected by a one-dimensional (1D) topological invariant in the presence of TRS, are gapped by the emergence of this new order parameter. The extended $s$-wave order parameter is localized within a narrow region near the surfaces, which is consistent with the fact that topological protection of the gapless Andreev surface states is characterized by the 1D topological invariant. In this TRS-breaking phase, not only is the complex order parameter induced, but also the $d$-wave order parameter itself becomes complex. Furthermore, the disappearance of topological protection brings about novel vortex phenomena near the surfaces. We show that vortex-antivortex pairs are formed in the extended $s$-wave order parameter along the surfaces if the side length of a nanoisland or the width of an infinitely long nanoribbon is relatively large.