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在超导和强关联电子材料的研究中, 引入压力和应变来改变晶格参数和对称性, 是调控体系电子性质的有效实验手段. 在静水压和外延薄膜面内应变的调控中, 晶格参数的变化可以引起电子结构的显著改变进而诱导出新奇的物理现象. 相比于这两种方法, 近年来开始被广泛采用的单轴应变调控方法, 除了可以改变晶格参数, 还可以直接破缺和调控体系的对称性, 影响体系的电子有序态乃至集体激发. 弹性单轴应变作为对称性破缺场, 可以作为电子向列相及其涨落的探针; 应变对超导和电子向列相的调控, 也可以为理解体系中电子态的微观机制提供实验依据. 本文将介绍单轴应变调控的基本概念、实验方法的发展, 以及采用这些方法调控铁基超导体中的超导和电子向列相等方面的一些研究进展, 并简单介绍单轴应变在其它量子材料中的应用.In the study of quantum materials, pressure and strain that can change lattice parameters and symmetry are effective experimental methods for manipulating the electronic properties of the systems. In the measurements under hydrostatic pressure or in-plane epitaxial strain, the change in lattice parameter could lead to significant alterations in the electronic structure, thereby inducing novel quantum phenomena and phase transitions. In comparison, the in-plane uniaxial strain, which has been widely employed in recent years, not only changes lattice parameters but also directly breaks and controls the symmetry of the system, thereby affecting the electronic ordered states and even collective excitations of the systems. This review article aims to provide a comprehensive overview of the basic concepts of uniaxial strain, the development of experimental methods, and some research progress in using these methods to tune superconductivity and electronic nematicity in iron-based superconductors. This review contains six sections. Section is a genetral introduction for the uniaxial strain techque and discuss the arrangement of this paper. Section (2) introduces basic concepts and formulas related to elastic moduli and the decomposition of uniaxial strain into irreducible symmetric channels under
$ D_{4h}$ point group. Section (3) gives an introduction to iron-based superconductors (FeSCs) and discuss the uniaxial-pressure detwinning method and related research progress. Section (4) introduce the establishment of the elastoresistance as a probe of the nematic susceptibility and discuss the key research works in this direction. Section (5) describes the research progress on the effects of uniaxial strain on superconductivity and nematicity. In sections (4) and (5), key experimental techniques, such as elastoresistance are discussed in detail. Section (6) expands the discussion to several class of quantum materials suitable for uniaxial-strain tuning method beyond the FeSCs. Finally, we give an brief summary and perspective to the uniaxial strain tuning technique. Overall, this review article aims to serve as a valuable resource for the beginners in the field of FeSC and those who are interested in using uniaxial strain to tune the electronic properties of quantum materials. By summarizing recent advancements and experimental techniques, this review hopes to inspire further research and innovation in studying correlated electron materials under uniaxial strain.-
Keywords:
- Strongly correlated electron /
- Iron-based superconductors /
- Spin excitations /
- Uniaxial strain
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图 2 应力和应变的定义, (a) 正应力和正应变; (b) 剪切应力和剪切应变, $ x_0 $和$ y_0 $是原始点的坐标, Δx和Δy是该点的位移, $ l_0 $是固体原本的长度, $ l_1 $是固体产生形变后的长度, F为对固体施加的应力, $ A_0 $为受力面的面积
Fig. 2. Definitions of stress and strain. (a) Normal stress and normal strain; (b) Shear stress and shear strain. $ x_0 $ and $ y_0 $ are the coordinates of the original point, and Δx and Δy are the displacements of the point. $ l_0 $ is the original length of the solid, and $ l_1 $ is the length of the strained solid. F is the stress applied to the solid, and $ A_0 $ is the area of the stressed surface.
图 3 $ D_{4 h} $和$ D_{6 h} $点群固体在不同对称性通道中的应变模式. (a) $ {\rm{BaFe}}_{2}{\rm{As}}_{2} $晶体的结构示意图; (b) $ {\rm{CsV}}_{3}{\rm{Sb}}_{5} $晶体的结构示意图; (c) 四方晶系材料中的应变在不可约表示$ A_{1 g, 1} $, $ A_{1 g, 2} $, $ B_{1 g} $和$ B_{2 g} $对称性通道中的应变分解. $ A_{1 g} $应变维持$ C_4 $旋转对称性, $ B_{1 g} $和$ B_{2 g} $应变将$ C_4 $对称性降低到$ C_2 $; (d) 六方晶系材料中的应变在不可约表示$ A_{1 g, 1} $, $ A_{1 g, 2} $和$ E_{2 g} $对称性通道中的应变分解. $ A_{1 g} $应变维持$ C_6 $旋转对称性, $ E_{2 g} $应变将对称性降低到$ C_2 $旋转对称性, 白色虚线表示对称轴
Fig. 3. Irriducible strains of $ D_{4 h} $ and $ D_{6 h} $ point group.(a), (b) Crystal structures of $ {\rm{BaFe}}_{2}{\rm{As}}_{2} $ (a) and $ {\rm{CsV}}_{3}{\rm{Sb}}_{5} $ (b). (c) In $ D_{4 h} $ materials, strain can be decomposed into the irreducible strains in the $ A_{1 g, 1} $, $ A_{1 g, 2} $, $ B_{1 g} $ and $ B_{2 g} $ symmetry channels. The $ A_{1 g} $ strain preserves the $ C_4 $ rotational symmetry, and the $ B_{1 g} $ and $ B_{2 g} $ strains lower the primary symmetry to the $ C_2 $ rotational symmetry. (d) In $ D_{6 h} $ materials, strain can be decomposed into irreducible strains in the $ A_{1 g, 1} $, $ A_{1 g, 2} $ and $ E_{2 g} $ symmetry channels. The $ A_{1 g} $ strain preserves the $ C_6 $ rotational symmetry, and the $ E_{2 g} $ strain lowers the symmetry to the $ C_2 $ rotational symmetry. The white dotted line indicates the axis of symmetry.
图 4 (a), (b) Ba(${\rm{Fe}} _{1-x} $${\rm{Co}} _{x})_2 $As2和FeSe1–xSx的相图[41,42]; (c)—(e) 正交相孪晶的形成和单轴压力退孪晶示意图[43]; (f) 单晶样品的单轴加压装置示意图[30], 适用于电阻、中子和X射线衍射测量; (g) 适用于多片BaFe2As2及类似单晶样品的退孪晶装置[44]; (h) 表面粘满FeSe单晶样品的BaFe2As2单晶[45]; (i) RIXS实验中, 采用BaFe2As2作衬底退孪晶FeSe的装置示意图[31]; (j) 采用因瓦合金固定铝带来施加各向异性应变进而退孪晶大量FeSe单晶的装置示意图[33]
Fig. 4. (a), (b) Phase diagram of Ba(${\rm{Fe}} _{1-x} $${\rm{Co}} _{x})_2 $As2 and FeSe1–xSx[41,42]. (c)–(e) Schematic diagrams of the formation of orthogonal twins and detwinning under uniaxial pressure[43]. (f) Schematic diagram of a uniaxial-pressure device[30], suitable for resistance, neutron, and X-ray diffraction measurements. (g) A uniaxial-pressure device suitable for detwinning multiple pieces of BaFe2As2[44]. (h) A BaFe2As2 single crystal with many pieces of FeSe single crystals glued on the surface[45]. (i) A device used to detwin FeSe with a BaFe2As2 substrate in RIXS experiments[31]. (j) Schematic diagram of a device using Invar alloy to fix an aluminum strip to apply anisotropic strain for detwinning large amounts of FeSe single crystals[33].
图 5 (a) $ {\rm{BaFe}}_{2}{\rm{As}}_{2} $中向列响应$ d \eta / d \varepsilon_P $的温度依赖性关系, 竖线表示结构相变温度$ T_s = 138 K $, 红线表示平均场模型拟合, 拟合公式为$ \frac{d \eta}{d \varepsilon} = \frac{\lambda}{a_0\left(T-T^*\right)+3 b \eta_0^2}+\chi_0 $[26]; (b)—(e) 弹性电阻测量示意图[69]. (b) 纵向弹性电阻测量($ I\parallel\varepsilon_{xx} $); (c) 横向弹性电阻测量($ I\perp\varepsilon_{xx} $); (d) 四方相$ [100] $与$ \varepsilon_{xx} $成θ角的纵向弹性电阻测量示意图; (e) 两片单晶安装在压电陶瓷堆表面的照片, 用于同时测量纵向(左)和横向(右)弹性电阻; (f) BaFe2As2中$ B_{2 g} $弹性电阻$ -2 m_{66} $(正比于向列极化率$ \chi_{N\left(B_{2 g}\right)} $)的温度依赖性关系, 黑线表示居里-外斯拟合, 拟合公式为$ 2 m_{66} = \frac{\lambda}{a_0\left(T-T^*\right)}+2 m_{66}^0 $; (g) 实验装置示意图; (h) 考虑逆极化率$ -\left(2 m_{66}-2 m_{66}^0\right)^{-1} $(左轴)和居里常数$ -\left(2 m_{66}-2 m_{66}^0\right)^{-1}\left(T-T^*\right) $可以更好地了解拟合质量[27]
Fig. 5. (a) Temperature dependence of the nematic response $ d \eta / d \varepsilon_P $ of BaFe2As2. Vertical line marks the structural transition temperature $ T_s = 138 $ K. Red line shows fit to mean field model. The fitting formula is $ \frac{d \eta}{d \varepsilon} = \frac{\lambda}{a_0\left(T-T^*\right)+3 b \eta_0^2}+\chi_0 $[26]. (b)–(e) Schematic diagrams of elastoresistance measurement[69]. (b) Longitudinal elastoresistance measurement ($ I\parallel\varepsilon_{xx} $); (c) transverse elastoresistance measurement ($ I\perp\varepsilon_{xx} $); (d) longitudinal elastoresistance for $ \varepsilon_{xx} $ aligned along an arbitrary in-plane direction with an angle θ with respect to $ [100] $; (e) a photograph of two crystals mounted on the surface of a PZT piezo stack for simultaneous measurement of the longitudinal (left) and transverse (right) elastoresistance. (f) Temperature dependence of the $ B_{2 g} $ elastoresistance $ -2 m_{66} $ which is proportional to the nematic susceptibility $ \chi_{N\left(B_{2 g}\right)} $ of BaFe2As2. The black line shows the Curie-Weiss fit, the fitting formula is $ 2 m_{66} = \frac{\lambda}{a_0\left(T-T^*\right)}+2 m_{66}^0 $. (g) Schematic diagram of experimental setup. (h) The quality of fit can be better appreciated by considering the inverse susceptibility $ -\left(2 m_{66}-2 m_{66}^0\right)^{-1} $(left axis) and the Curie constant $ -\left(2 m_{66}-2 m_{66}^0\right)^{-1}\left(T-T^*\right) $(right axis)[27].
图 6 (a) Sr2RuO4中, 理论上破缺对称性的应变 $ ( \varepsilon_{xx} -\varepsilon_{yy} ) $ 对$ T_c $的调控效应及对应的配对对称性. (b)—(d) 单轴应变装置的工作原理. (b) 样品处于零应变的状态, 此时压电陶瓷上的电压为0. (c) 电压驱动中间的压电陶瓷堆伸长, 样品受压应变. (d) 电压驱动两端的压电陶瓷堆伸长, 使得左侧的钛桥向外, 样品受到拉应变. 图摘自文献[34]
Fig. 6. (a) General phase diagram expected for $ p_x\pm i p_y $ pairing symmetry in a tetragonal crystal subject to a small symmetry-breaking strain $ (\varepsilon_{xx}-\varepsilon_{yy}) $ in Sr2RuO4. (b)–(d), Working principle of the strain cell. (b) Sample at zero strain. (c) The sample is compressed by extending the middle piezoelectric actuator. (d) The sample is tensioned by extending both outer actuators and pushing the bridge piece out. The figure is from ref.[34]
图 7 (a) 在单轴应变$ \varepsilon_{[110]} $调控下, Ba(${\rm{Fe}} _{1-x} $${\rm{Co}} _{x})_2 $As2最佳掺杂(x = 0.071)中电阻率作为温度的函数[101]; (b) Ba(${\rm{Fe}} _{1-x} $${\rm{Co}} _{x})_2 $As2(x = 0.071)超导相变温度$ T_c $作为$ \varepsilon_{B_{2 g}} $(下轴)和$ \varepsilon_{A_{1 g, 1}} $(上轴)的函数[101]; (c) 粘贴在钛合金衬底表面的FeSe单晶, 上面做了电极以测量沿长度方向的电阻率[101]; (d) FeSe样品中$ \varepsilon_{B_{1 g}} $对$ T_c $的调控, $ T_c $定义为电阻与具体电阻值相交的温度(右上角电阻曲线中的横线所示)[29]; (e) $ {\rm{CsV}}_3{\rm{Sb}}_5 $单晶粘贴在钛衬底上的照片, 用于测量单轴应变下的电阻率 (左图) 和互感信号(交流磁化率实部$ \chi' $)(右图); (f) $ {\rm{CsV}}_3{\rm{Sb}}_5 $沿[110]方向单轴应变下的电阻率测量结果; (g) $ {\rm{CsV}}_3{\rm{Sb}}_5 $沿$ [110] $方向单轴应变下的互感测量, (f)和(g)图中水平虚线表示用于跟踪$ T_c $的相对变化的值[105]
Fig. 7. (a) Resistivity as a function of temperature under uniaxial stress for optimally Ba(${\rm{Fe}} _{1-x} $${\rm{Co}} _{x})_2 $As2 (x = 0.071)[101]. (b) Superconducting transition temperature $ T_c $ as a function of the two irreducible strain components $ \varepsilon_{B_{2 g}} $ (bottom axis) and $ \varepsilon_{A_{1 g, 1}} $ (top axis)[101]. (c) Photograph of FeSe sample glued on a tatanium bridge, with contacts attached for measuring resistivity[29]. (d) $ T_c(\varepsilon_{B_{1 g}}) $, determined as the temperature where the resistivity crosses specific values, as shown in the inset[29]. (e) Photos of the $ {\rm{CsV}}_3{\rm{Sb}}_5 $ single crystals attached on titanium platforms for the measurements of resistivity (left panel) and mutual inductance (ac $ \chi' $) (right panel) under uniaxial strains[105]. (f) Resistivity measurements under the uniaxial strain along the [110] direction. (g) Mutual inductance (ac $ \chi' $) under uniaxial strains along the [110] direction, the horizontal dashed lines mark the values used to track the relative change of $ T_c $[105].
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