Nonequilibrium states in quantum materials under time-period driving

Wang En; Dong Wen-Han; Zhou Hui; Liu Meng; Ji Hong-Yan; Meng Sheng; Sun Jia-Tao

doi:10.7498/aps.70.20201808

Abstract

The topology of quantum materials is the frontier research in condensed matter physics. In contrast with the conventional classification of materials by using the local symmetry breaking criterion, the states of quantum systems are classified according to the topology of wave functions. The potential applications of topological states may lead the traditional microelectronics to break through and accelerate the significant improvement in topological electronics. Most of the recent studies focus on the topological states of quantum systems under equilibrium conditions without external perturbations. The topological states of quantum systems far from the equilibrium under time-periodic driving have attracted wide attention. Here we first introduce the framework of Floquet engineering under the frame of the Floquet theorem. The nonequilibrium topological states of massless and massive Dirac fermions are discussed including the mechanism of phase transition. Light field driven electronic transition term in the quantum material gains extra time-dependent phase. Thereby the manipulation of effective transition term of the electron is realized to regulate the non-equilibrium topological states. We also mention how the photoinduced coherent phonon affects the nonequilibrium topological states of quantum systems from the perspective of atom manufacturing. Furthermore, research outlook on the nonequilibrium topological states is given. This review provides some clues to the design of physical properties and transport behaviors of quantum materials out of equilibrium.

Keywords:

Authors and contacts

1.
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
2.
MIIT Key Laboratory for Low-Dimensional Quantum Structure and Devices, School of Information and Electronics, Beijing Institute of Technology, Beijing 100081, China
3.
School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Corresponding author: Sun Jia-Tao, jtsun@bit.edu.cn

Funds: Project supported by the National Key R&D Program of China (Grant Nos. 2020YFA0308800, 2016YFA0301204), the National Natural Science Foundation of China (Grant No. 11974045), the Strategic Priority Research Program (B) of Chinese Academy of Sciences (Grant No. XDB30000000), and the Research Fund Program for Young Scholars of Beijing Institute of Technology, China (Grant No. 3050012222009)

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Cited By

图 1 弗洛凯工程研究框架示意图. 深蓝区域代表被驱动体系, 如玻色子系统、费米子系统等; 红色区域代表狄拉克以及外尔半金属、马约拉纳费米子、拓扑界面态等非平衡拓扑物态; 绿色区域代表弗洛凯工程在超快自旋电子学、谷电子学、瞬态超导特性、斯格明子等方面存在潜在应用. 图片素材来自文献[31, 39, 62, 68, 70, 71]

Figure 1. The schematic of the research framework of Floquet Engineering. The deep blue region denotes driven system including bosonic and fermionic system. The red region denotes the nonequilibrium states such as Dirac and Weyl semimetals, Majorana Fermions, topologically nontrivial interface states etc. And the green regions denotes the potential applications of Floquet engineering in ultrafast spintronics, valleytronics, transient superconductivity, Skyrmions etc. The pictures are adapted from Refs. [31, 39, 62, 68, 70, 71].

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图 2 (a) 在具有线性狄拉克锥的体系中施加圆偏光, 实现周期性外场驱动下的弗洛凯系统^[71]. (b) 弗洛凯系统中的能带结构和边带^[71]. (c) 弗洛凯系统能带结构起源示意图. 每一个方框分别代表静态系统哈密顿量子空间, 由于光场的引入, 使得相邻子空间的本征态之间通过吸收和发射虚光子过程发生耦合, 对应矩阵表达式中的非对角元项$ {H}_{\pm 1} $, 更高阶非对角元则对应多光子过程. (d) 利用Tr-ARPES技术在Bi₂Se₃表面观察到的弗洛凯-布洛赫能带结构^[84]. 其中颜色深浅代表对应光电子信号强度, 图中红色箭头表示能隙所在位置

Figure 2. (a) Floquet system driven by a periodic external field can be created by imposing circularly polarized laser on a linear Dirac cone^[71]. (b) Energy band structure and energy band replica in Floquet systems^[71]. (c) Schematic diagram of the origin of the energy side band of the Floquet system. Each framework represents one subspace from the Hamiltonian of the static system. The light field make the original eigenstates of different subspace coupled together through the process of absorbing and emitting virtual photons, corresponding to the off-diagonal element like $ {H}_{\pm n} $, n = 1. Other higher-order off-diagonal elements (n > 1) correspond to multiphoton processes. (d) Floquet-Bloch band structures of topologically nontrivial surface states on Bi₂Se₃ measured by Tr-ARPES^[84]. The magnitude of color bar denotes the intensity of photoemission signals. The red arrows denote the gap-opening regions.

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图 3 (a) 蜂窝状晶格中的次近邻跃迁在施加圆偏光后对应虚光子的吸收和发射过程, 产生与Haldane模型类似的次近邻跃迁矩阵元^[71]; (b) 施加圆偏光后石墨烯纳米带能带中的手性边缘态(红线)^[93], 其在实空间对应两个边界上反向运动的手性边缘态(左下角插图); (c) 光子晶体中实现周期场驱动的弗洛凯系统^[62]; (d) 利用泵浦-探测技术探测出现在石墨烯中的反常霍尔电导^[94]; (e) 在不同光场强度下弗洛凯-布洛赫能带图及能隙内的手性边缘态电导平台^[94]. 图中红色、蓝色、黑色能带分别对应光场强度逐渐加大, 弗洛凯能隙亦逐渐增大. 最后一张图对应光场大小为0.23 mJ/cm²时不同费米面位置对应接近量子化的电导平台, 反应了费米面附近的系统能隙为拓扑能隙

Figure 3. (a) Laser irradiated honeycomb lattice have virtual photon absorption and emission processes. These effects lead to next-nearest-neighbor hoppings similar as that in Haldane model^[71]. (b) Band structures of laser irradiated graphene nanoribbon^[93]. The red lines in this panel denote the chiral edge states, which move along opposite directions of graphene nanoribbon edges (inset). (c) Photonic analog of laser irradiated honeycomb lattice^[62]. (d) The anomalous Hall conductance in laser irradiated graphene measured by pump-probe method^[94]. The sub-linear relationship between nearly quantized conductance and laser fluence has been observed. (e) Floquet band structure with different laser fluence^[94]. If increasing the laser fluence, the anticrossing gap in Floquet system is also enlarged. The last panel shows the calculated conductance at different Fermi levels with laser fluence 0.23 mJ/cm². These results indicate that the gap in Floquet system is topological nontrivial.

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图 4 光场驱动下三维狄拉克半金属Na₃Bi的原子结构和电子结构　(a) Na₃Bi原子结构示意图^[13]; (b) 利用角分辨光电子能谱测量得到Na₃Bi能带, 展现了其狄拉克点位置和线性能带色散关系^[13]; (c) 沿着Na₃Bi的x轴方向施加圆偏光, 三维狄拉克锥沿k_x方向劈裂为两个手性相反的外尔锥^[97]; (d) 沿着Na₃Bi的x轴方向施加泵浦圆偏振光, 沿着y轴方向施加探测圆偏振光^[97]; (e) TDDFT得到的弗洛凯-外尔点在倒空间中移动轨迹^[97]

Figure 4. Light field driven electronic phase transition of three-dimensional Dirac semimetal Na₃Bi: (a) atomic structures of Na₃Bi^[13]; (b) band structures of Na₃Bi near Dirac points measured by ARPES, The location of Dirac points and its linear dispersion are presented as well^[13]. (c) when the circularly polarized light is applied along to the x axis of Na₃Bi, the Dirac cone will inherently split into two Weyl cones with opposite chiralities along k_x direction^[97]; (d) schematics of the circularly polarized pump and probe laser irradiated along the x axis and y axis of Na₃Bi^[97]; (e) varying delay time during the laser pulses leads to dancing Floquet-Weyl points calculated by TDDFT^[97].

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图 5 有质量二维体系的光场修饰态　(a)同时考虑电场和光场硅烯的有效哈密顿量对应的相图^[29]. 横纵坐标分别对应电场强度$ {E}_{z} $和${{\cal{A}}}^{2}/\varOmega ~({{\cal{A}}}^{2}$为光强, $ \varOmega $为光场频率). 拓扑荷由$ ({\cal{C}}, {{\cal{C}}}_{\mathrm{s}}) $描述, $ {\cal{C}} $为陈数, $ {{\cal{C}}}_{\mathrm{s}} $为自旋陈数. (b) 硅烯中的单自旋狄拉克锥能带示意图^[29]. K' 能谷为能隙关闭的单自旋线性色散, K能谷则保持抛物线型能带色散. (c) WS₂谷选择的光学斯塔克效应能级示意图^[43]. (d) WS₂谷选择的布洛赫-西格特位移能级示意图. (e), (f) 不同能谷的手性弗洛凯拓扑态^[107]. 控制失谐量Δ诱导能级反转并改变导带与弗洛凯边带的杂化, 由于排斥作用K能谷的能带交叉被禁止而K' 能谷发生能带交叉, 形成手性边界态

Figure 5. Photon dressed states in two-dimensional systems with massive Dirac fermions: (a) Phase diagram of silicene by using effective Hamiltonian considering both electrical field and light field^[29]. (b) Sketch of silicene in the single Dirac cone state^[29]. The K valley exhibits a parabolic dispersion while the K' valley remains the linear dispersion. (c) Schematics of the valley-selective OSE in WS₂^[43]. (d) Schematics of the Bloch-Siegert shift in WS₂. (e), (f) Valley-specific Floquet topological phase in WS₂^[107]. The band inversion and hybridization of Floquet sidebands is tuned by Δ and the chiral edge state is formed due to OSE.

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图 6 压缩应变黑磷中的弗洛凯狄拉克费米子和拓扑相变　(a) 沿x方向压缩3.72%的黑磷原子结构. (b) 平衡态压缩黑磷的体相布里渊区以及(100)投影面^[69]. 拓扑节点环出现在Γ-Z-W平面. (c) 光子能量0.5 eV下的黑磷的弗洛凯态的相图^[69]. 变量为激光振幅$ {A}_{0} $在y-z平面内的入射角$ \theta $, 如图(a)所示. (d)—(f) 光子能量0.5 eV下由激光驱动的弗洛凯狄拉克费米子发生拓扑相变^[69] (d) ${A}_{0}=50~\mathrm{V}/c$; (e) ${A}_{0}=263~\mathrm{V}/c$; (f) ${A}_{0}=300~\mathrm{V}/c. \, c$为光速, 施加的圆偏光为$ {{A}}\left(t\right)={A}_{0}(\cos\left(\omega t\right), \sin\left(\omega t\right), 0) $. (g) 通过在节点环能带结构中施加激光照射, 可以实现高陈数外尔点^[125]

Figure 6. Floquet-Dirac fermions and topological phase transition in compressed black phosphorus. (a) Atomic structure with 3.72% compressive strain along x direction. (b) Bulk first Brillouin zone and projected (100) surface. Here, the topological nodal ring appears in the Γ-Z-W plane. (c) Topological phase transition driven by laser with varying laser amplitude and incident angle of y-z plane $ \theta $ under a fixed photon energy 0.5 eV^[69]. (d)–(f) Floquet-Dirac band structure under different laser parameter^[69]: (d) ${A}_{0}=50~\mathrm{V}/c$; (e) ${A}_{0}=263~\mathrm{V}/c$; (f) ${A}_{0}=300~\mathrm{V}/c.\, c$ is the speed of light. (g) Construction of high Chern number Weyl points in nodal ring under incident light^[125].

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图 7 光场驱动下相干声子驱动的弗洛凯态　(a) 由石墨烯E_2g振动模中简并的横向和纵向声子模(相位差$ \mathrm{\pi }/2 $)产生的手性圆偏声子^[48]; (b) 手性相干声子驱动下计算得到的石墨烯Tr-ARPES以及TDDFT结果与弗洛凯能带相符合^[48]; (c) ZrTe₅的原子结构和层间振动模式A_1g示意图^[134]; (d) 平衡态下单层MoS₂的能带结构, 蓝框标注的是K能谷附近的导带底^[40]; (e) MoS₂自旋倾向角与上下自旋能隙$ \Delta \varepsilon $关于沿声子振动模E'' 方向位移Δd_s的关系图, 插图为E'' 的振动模式^[40]; (f) MoS₂在K和K' 能谷由右旋声子定义的两个自旋-弗洛凯本征态^[40]

Figure 7. Floquet states induced by coherent phonons driven by periodic light field: (a) Circularly chiral phonons generated from the degenerate LO and TO phonons (with phase difference π/2) of E_2g vibration mode of graphene^[48]; (b) calculated Tr-ARPES of phonon-driven graphene fits well with TDDFT simulations^[48]; (c) atomic structures and interlayer vibration mode A_1g of ZrTe₅^[134]; (d) band structure of monolayer MoS₂ under equilibrium^[40], the blue box marks the lowest conduction band near K valley; (e) relationship between the spin inclination angle and the up/down spin splitting Δε with respect to the displacement Δd_s along the phonon mode E'', the inset shows the E'' vibrational mode^[40]; (f) spin-Floquet eigenstates of MoS₂ at K and K' valleys induced by the right circularly polarized phonon^[40].

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Vol. 70, Issue 13 - 2021-07-05

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