Soft modes are intimately linked to structural instabilities and are key for the understanding of phase transitions. The soft modes in ferroelectrics, for example, map directly the polar order parameter of a crystal lattice. Driving these modes into the nonlinear, frequency-changing regime with intense terahertz (THz) light fields is an efficient way to alter the lattice and, with it, the physical properties. However, recent studies show that the THz electric-field amplitudes triggering a nonlinear soft-mode response are surprisingly low, which raises the question on the microscopic picture behind the origin of this nonlinear response. Here, we use linear and two-dimensional terahertz (2D THz) spectroscopy to unravel the origin of the soft-mode nonlinearities in a strain-engineered epitaxial ferroelectric ${\mathrm{SrTiO}}_3$ thin film. We find that the linear dielectric function of this mode is quantitatively incompatible with pure ionic or pure electronic motions. Instead, 2D THz spectroscopy reveals a pronounced coupling of electronic and ionic-displacement dipoles. Hence, the soft mode is a hybrid mode of lattice (ionic) motions and electronic interband transitions. We confirm this conclusion with model calculations based on a simplified pseudopotential concept of the electronic band structure. It reveals that the entire THz nonlinearity is caused by the off-resonant nonlinear response of the electronic interband transitions of the lattice-electronic hybrid mode. With this work, we provide fundamental insights into the microscopic processes that govern the softness that any material assumes near a ferroic phase transition. This knowledge will allow us to gain an efficient all-optical control over the associated large nonlinear effects.

中文翻译：

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铁电钙钛矿中太赫兹软模非线性的起源

软模式与结构不稳定性密切相关，是理解相变的关键。例如，铁电体中的软模式直接映射晶格的极性顺序参数。用太赫兹（THz）光场将这些模式驱动到非线性，频率变化的状态中，是改变晶格及其物理性质的有效方法。但是，最近的研究表明，触发非线性软模式响应的太赫兹电场幅度令人惊讶地低，这在此非线性响应的起源背后的微观图像上提出了问题。在这里，我们使用线性和二维太赫兹（2THz）光谱来揭示应变工程外延铁电体中软模非线性的起源。${\mathrm{钛酸锶}}_3$薄膜。我们发现，该模式的线性介电函数在数量上与纯离子运动或纯电子运动不兼容。相反，二维THz光谱揭示了电子和离子位移偶极子的明显耦合。因此，软模式是晶格（离子）运动和电子带间跃迁的混合模式。我们通过基于电子能带结构的简化伪势概念的模型计算来证实这一结论。它揭示了整个THz非线性是由晶格-电子混合模式的电子带间跃迁的非共振非线性响应引起的。通过这项工作，我们对控制柔软性的微观过程提供了基本见解。任何材料都假定在铁素体相变附近。这些知识将使我们能够对相关的大型非线性效应进行有效的全光学控制。