我写好了一篇关于RuO2电子结构计算的文章，目前想投稿prb，初稿目前在附件中，我想让你逐段翻译里面...

我通读了你这份初稿，核心主线我理解为：用第一性原理把“RuO₂ 薄膜磁性为何实验上争议很大”归结到“外延整体应变 + 薄膜中不可避免的局部畸变”这两个因素的共同作用，并给出一个可以统一解释 AFM/SDW/低能激发的微观图像。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

下面是按文章结构整理的内容总结（相当于“我读懂了你在讲什么、结论是什么、证据链怎么搭的”）：

1) 研究背景与问题意识：RuO₂磁性争议来自“脆弱磁结构 + 薄膜非本征因素”

你在引言里先把 RuO₂ 放到 altermagnetism / 自旋电子学的大背景下：RuO₂ 被预测可能有较大的自旋劈裂和较高 Néel 温度，因此受到关注。(基于第一性原理计算的RuO2应变诱发磁有序.docx)
但紧接着强调：RuO₂ 的磁有序长期存在实验矛盾——有的实验看到反铁磁信号与 ~nm 级相关长度，有的（μSR/ARPES/QO 等）又更支持接近 Pauli 顺磁金属；同时理论也指出 RuO₂ 的磁序非常“脆弱”，对外界扰动敏感。(基于第一性原理计算的RuO2应变诱发磁有序.docx)
由此你把重点落到薄膜里常见的非本征因素（外延应变、缺陷/界面/对称性破缺等），并指出多数“看到磁性/自旋输运效应”的工作主要发生在薄膜体系。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

2) 方法：VASP/PBE，不把结论建立在 DFT+U 上（SOC 与 +U 放补充）

计算上你采用 VASP + PBE，给出赝势、截断能 520 eV、展宽 0.15 eV、k 点网格（单胞与沿 a 方向扩胞超胞的设置）、结构弛豫收敛策略，并说明正文主要不给 SOC/不给 +U，SOC 与 DFT+U 放补充材料对比。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

3) 关键模型：用“整体畸变参数 ε”把 RuO₂ 晶格从本征连续插值到 TiO₂

你把 TiO₂/RuO₂ 外延体系的“基底作用”第一步近似为对 RuO₂ 晶格的可控畸变，并定义整体畸变参量 ε：ε=0 回到本征 RuO₂，ε=1 趋近 TiO₂ 晶格，用于量化整体外延应变强度。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

4) 结果 A：只靠整体应变就能触发“顺磁 → AFM”的相变，并给出临界 εc

你计算 ε 从 0→1 的演化，得到：不考虑 SOC 时，随 ε 增大，RuO₂ 可从顺磁态转为 AFM，临界点 εc≈0.7404；(基于第一性原理计算的RuO2应变诱发磁有序.docx)
考虑 SOC 后，临界点几乎不变（εc≈0.7340），主要影响是磁矩幅值降低 ~10%，因此正文后续主要讨论无 SOC 情形。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

你还特别强调一个对“争议”很关键的点：在这些应变下，能带/费米面并没有发生大幅重整化，只有费米面局部区域变化更明显；因此即使 ARPES/QO 没看到“DFT+U 那种大改造的费米面”，体系仍可能在“整体应变 + 局部畸变”条件下产生磁有序。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

5) 结果 A’：在固定整体 ε 的前提下，加入“受控的周期性局部畸变”——它主要调幅，不太移动相变边界

考虑真实薄膜不可能只用一个 ε 描述，你进一步在固定平均晶格（保持 ⟨ai⟩=aε）的前提下，引入两类局部畸变并用超胞实现：

• 简谐畸变（连续起伏，幅值 Δh）
• δ 畸变（更像缺陷/局域强畸变，幅值 Δδ）(基于第一性原理计算的RuO2应变诱发磁有序.docx)

结论是：局部畸变确实会改变局域磁矩分布与幅值，但εc 只缓慢漂移，说明“是否出现宏观磁有序”主要仍由整体 ε 决定，而局部畸变更多影响磁性细节（调幅）。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

6) 结果 B：在临界应变附近，足够强的局部畸变可让 SDW 比 AFM 更稳定，并给出能量/波长尺度

你把后半部分的重点放到：实验上在 RuO₂/Py 等体系观察到与“传统 AFM 自旋波 THz 尺度”不一致的 GHz 低能激发，并提出 SDW 可能是关键。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

计算策略上，你指出直接 DFT 自洽搜索 SDW 很难，于是借鉴 Cr 的做法，通过“先验的周期性局部畸变”稳定 SDW，然后比较总能：定义 ΔE = E_SDW − E_AFM。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

主要发现包括：

• SDW 是否能成为更低能态取决于 ε 与局部畸变幅值 Δ；当 ε≈εc 且局部畸变达到一定强度时，会出现 ΔE<0，即 SDW 优于 AFM。(基于第一性原理计算的RuO2应变诱发磁有序.docx)
• 你给出的微观机制是：能量差主要来自 SDW 波节（node）附近。局部压缩会推动某些区域走向弱磁/近无磁，而 SDW 在 node 处本征上磁矩接近 0，因此局部压缩会让 SDW 在 node 处相对 AFM 更占优。(基于第一性原理计算的RuO2应变诱发磁有序.docx)
• 随周期 n（波长）变化，ΔE per cell 先降后升，并据此得到更利于 SDW 稳定的特征波长区间 约 7–10 nm。(基于第一性原理计算的RuO2应变诱发磁有序.docx)
• 在你强调的“临界应变 + 局部畸变”条件下，AFM 与 SDW 的能量差只有 10–100 μeV（~10 GHz），从而自然解释为何会出现异常低能激发。(基于第一性原理计算的RuO2应变诱发磁有序.docx)
• 你还做了更贴近真实薄膜的讨论：SDW 的波节会倾向钉扎在最大压缩区域，因此实际薄膜里 SDW 未必严格周期，其“波长”与局部畸变的空间分布/特征长度有关。(基于第一性原理计算的RuO2应变诱发磁有序.docx)
• δ 型局部畸变模型进一步表明：需要一定“宽度”的近无磁压缩区域，SDW 才更容易相对 AFM 稳定。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

7) 总结与文章主张

你的结论部分把文章主张收束成三句话：

1. 整体外延应变足够强：RuO₂ 顺磁 → AFM；
2. 局部畸变主要改变磁矩空间分布与幅值，但不显著改变整体临界应变；
3. 在顺磁–AFM 临界附近，如果存在足够强的局部畸变，SDW 可成为潜在基态；其能量尺度 10–100 μeV、特征波长 7–10 nm，可解释薄膜中低能激发与实验分歧。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

如果你看这份总结觉得“主线/关键数字/物理图像”都对，那我下一步就可以按你说的逐段翻译（我会尽量按 PRB 的写作习惯处理语气、逻辑连接与术语一致性，而不是逐字硬翻）。

Abstract (English translation)

RuO2 has been proposed as a candidate material for next-generation spintronics, yet its magnetic order remains under broad debate. Here, based on first-principles calculations, we systematically investigate the magnetic ordering of RuO2 under epitaxial strain. Using TiO2/RuO2 thin films as a representative system in which TiO2 imposes epitaxial strain on RuO2, we find that sufficiently strong global substrate-induced strain can drive RuO2 from a paramagnetic state into an antiferromagnetic (AFM) state. Near the critical strain for the magnetic transition, local lattice distortions of sufficient strength can stabilize a spin-density-wave (SDW) state that is energetically more favorable than the AFM state, with an energy difference on the order of 10–100 μeV and a characteristic wavelength of approximately 7–10 nm. This epitaxial-strain-induced magnetic-ordering mechanism provides a unified microscopic picture for the experimental controversies regarding magnetism in RuO2 thin films, and highlights the key roles of global strain and local distortions in controlling RuO2 magnetism and spin transport. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

我只做了PRB摘要常见的英文表达与句式调整（比如把“广泛质疑”译成“under broad debate”这类更学术的说法），没有改变任何物理含义或结论。

In recent years, research on altermagnetism has surged. Altermagnets exhibit no net magnetization in real space, yet their reduced crystal symmetries can induce a pronounced spin splitting in momentum space, endowing them with key characteristics of both ferromagnets and antiferromagnets; as a result, they have attracted strong interest in spintronics, where microscopic control of magnetic order is central. Meanwhile, the high-frequency response, stray-field-free advantage, and lattice-anisotropy-driven transport properties of altermagnets open new possibilities for extending the performance limits of magnetic materials. Related studies have reported or proposed phenomena such as the anomalous Hall effect [41–44], terahertz (THz) emission, and spin–charge interconversion [45–47], offering new ideas for next-generation spintronic devices and for improving technologies including MRAM, GMR, and TMR [48–50]. In this context, predictions of a large spin-splitting energy and a high Néel temperature in RuO2 [1], together with theoretical groundwork on RuO2 antiferromagnetism [2–4] and rapid experimental progress on the altermagnetic spin-splitting effect (ASSE) in RuO2 [5], have jointly propelled RuO2 to become a research hotspot in this field. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

我没有改变原意；唯一“额外做的一点”是把文中的 ASSE 顺手按你参考文献[5]的语境补全为 altermagnetic spin-splitting effect，方便英文读者理解，其它都是PRB常用的英文表述顺滑化处理。

In recent years, research on altermagnetism has rapidly gained momentum. Altermagnets exhibit no net magnetization in real space, yet their reduced crystal symmetries can induce a pronounced spin splitting in momentum space, thereby combining selected characteristics of both ferromagnetic and antiferromagnetic systems. Consequently, they have attracted considerable attention in spintronics and related fields that focus on microscopically controlling magnetic order in materials. Meanwhile, the high-frequency response, stray-field-free advantage, and lattice-anisotropy-driven transport properties of altermagnets open new opportunities for extending the performance limits of magnetic materials. Related studies have reported phenomena in altermagnets such as the anomalous Hall effect [41–44], terahertz (THz) emission, and spin–charge interconversion [45–47], offering new ideas for the development of next-generation spintronic devices and for improving technologies including MRAM, GMR, and TMR [48–50]. In this context, predictions of a large spin-splitting energy and a high Néel temperature in RuO2 [1], together with theoretical groundwork on RuO2 antiferromagnetism [2–4] and rapid experimental progress on the altermagnetic spin-splitting effect (ASSE) in RuO2 [5], have jointly propelled RuO2 to become a research hotspot in this field.

我这版只做了英文写作上的顺句与术语统一（例如 “altermagnetic spin-splitting effect (ASSE)” 这种括号处理），没有改变原意。

Building on these considerations, an increasing number of studies have begun to focus on how strain influences the magnetism of RuO2, and how strain-induced lattice distortions modify electrical transport and magnetic order to enable device-level tuning [21–23]. The necessity of this research direction is particularly evident: the fragile Fermi-surface structure of RuO2 renders its magnetic state unusually sensitive to the lattice constants. Previous work has shown that doping can markedly affect its magnetism by altering the lattice [53]; even in the context of chemical catalysis, the pronounced impact of RuO2 lattice parameters on the activation energy has been repeatedly emphasized [29]. Motivated by these insights, experiments have also started to explore fine control of substrate–film coupling via the film thickness [51,58]. Application-oriented studies have further proposed using epitaxial strain to control the details of spin transport in RuO2, thereby expanding the tuning dimensions of spintronic devices [52,59]. Overall, the influence of strain on RuO2 magnetism has received systematic attention at the theoretical, experimental, and application levels.

改动说明：我只做了英文论文常见的表达顺序调整（例如把“器件层面的调制”译为 device-level tuning，把“研究路线的必要性尤为突出”译为更学术的 particularly evident/compelling），不改变原意与逻辑关系。

Despite the great promise of RuO$_2$ for advancing spintronic platforms, its magnetic ordering has long remained highly controversial. Early studies suggested that RuO$_2$ hosts relatively weak antiferromagnetic order, and resonant x-ray scattering measurements were used to discuss antiferromagnetic signatures in both bulk and thin-film samples, yielding a magnetic correlation length of about 5 nm in films [2,6]. On the other hand, a substantial body of experiments supports RuO$_2$ being closer to a Pauli-paramagnetic metal: $\mu$SR results indicate the absence of observable internal dynamic magnetic fields and long-range magnetic order [7,8]; some ARPES measurements also show a band structure more consistent with a nonmagnetic state [9,10]; and quantum-oscillation (QO) experiments pose an even stronger challenge to the existence of antiferromagnetic order [11]. Meanwhile, several works have emphasized that the magnetic state of RuO$_2$ itself is extremely fragile—for instance, the relatively small Fermi velocity near the Fermi surface makes its magnetism highly susceptible to external perturbations [13,14]. Given that RuO$_2$ features a finite Coulomb repulsion together with a relatively wide bandwidth, placing it in a strongly covalent, intermediate-coupling regime, such complex magnetic behavior is not unexpected. Accordingly, it has been proposed that non-intrinsic factors—such as oxygen vacancies, epitaxial strain, and symmetry breaking at surfaces or interfaces—may play an important role in generating magnetism in thin films [13,15,16], where these effects are often pronounced. Indeed, most experiments reporting antiferromagnetic order are based on RuO$_2$ thin films, and spin-transport experiments that support altermagnetism in RuO$_2$ have also predominantly been carried out in film geometries [17,18]. Moreover, it has been reported that tuning the electron count via electron- or hole-type impurities can modify the Néel-vector orientation and enhance the experimental signatures of antiferromagnetism [19,20], further highlighting the critical impact of non-intrinsic effects on RuO$_2$.

改动说明：我仅做了英文论文写作中常见的语序与衔接优化（例如把“更接近”处理为 closer to，把“提出更强挑战”写成更学术的表达），没有改变原意、结论或因果逻辑。(基于第一性原理计算的RuO2应变诱发磁有序.docx)

Taking the commonly studied TiO$_2$/RuO$_2$ epitaxial thin-film system as an example, we first approximate the substrate effect on RuO$_2$ as a controllable distortion of the RuO$_2$ lattice, and on this basis systematically examine how lattice distortions influence the magnetism of RuO$_2$. We show that, even without invoking the still-debated DFT+$U$ treatment, RuO$_2$ can undergo a transition from a paramagnetic state to an antiferromagnetic (AFM) state once the lattice distortion exceeds a certain level. Furthermore, when the local distortion is sufficiently strong, a spin-density-wave (SDW) state may become energetically more stable than the AFM state, with an energy difference on the order of 10–100 $\mu$eV. These results provide a consistent microscopic picture for the rich non-intrinsic magnetic ordering observed in RuO$_2$ thin films, and help clarify the mechanisms by which stress/strain affects the magnetism and spin-transport behaviors of RuO$_2$ films. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

改动较大的地方：我没有改动大意；仅做了英文论文常见的措辞规范化，比如把“非本征”译为 non-intrinsic（也可换成 extrinsic，若你更偏好 PRB 常用说法我可以统一替换），以及把“第一步近似”译为 first approximate … as 来更符合学术语气。

Our first-principles calculations were performed using the VASP package. Within the framework of density functional theory (DFT), we employed the PBE exchange–correlation functional and the corresponding projector-augmented-wave pseudopotentials to describe the interaction between valence electrons and ionic cores [28–33]. The pseudopotentials used were Ru_pv (4p$^6$4d$^6$5s$^1$), O (2s$^2$2p$^4$), and Ti (3p$^6$3d$^3$4s$^1$). The plane-wave energy cutoff was set to 520 eV. A smearing width of 0.15 eV was adopted with the ISMEAR = 1 scheme. The lattice constants of RuO$_2$ and TiO$_2$ involved in this work are listed in Graph 1.

For calculations on the rutile primitive cell, the Brillouin zone was sampled using a Γ-centered 9×9×13 k-point mesh; for density-of-states (DOS) calculations, a denser 18×18×26 mesh was used. For supercells constructed by expanding along the a axis, the corresponding k-point mesh was chosen as Γ-centered 1×9×13, and the DOS was computed using a 2×18×26 mesh.

Regarding structural convergence, for primitive-cell calculations we set the convergence criterion for ionic relaxation to a Hellmann–Feynman force below 0.001 eV/Å. For scanning calculations involving a large number of supercells, in order to control the computational cost while maintaining accuracy, we adopted a compromise force-convergence strategy: the forces on O atoms were constrained to be within 0.04 eV/Å, and the validity of this threshold—as well as comparison results obtained using stricter relaxation settings—are discussed in the Supplementary Material. To further reduce the overhead associated with performing full ionic relaxations for every supercell, we employed an initialization strategy of “treating the primitive cell first and then constructing the supercell”, such that in most cases the supercells could meet the ~0.04 eV/Å force level without additional relaxation. Without this strategy, the forces on O atoms in some supercells could exceed 0.30 eV/Å, leading to unnecessary numerical errors.

Data post-processing and visualization were mainly carried out using VESTA and VASPKIT [34,60]. The main text focuses on results without spin–orbit coupling (SOC) and without introducing DFT+$U$; comparisons and supplementary analyses involving SOC and DFT+$U$ are consistently presented in the Supplementary Material.

改动说明：我只做了英文论文里常见的措辞顺滑与格式规范化（例如 “k 点网格”写成 k-point mesh、把参数用更标准的写法如 “Γ-centered 9×9×13”），没有改变任何数值、计算设置或逻辑含义。

RuO$_2$ is highly strain sensitive. The fragility of its Fermi-surface structure makes its magnetic state strongly tunable by lattice strain; experiments have even reported superconductivity in RuO$_2$ thin films under specific strain conditions. (基于第一性原理计算的RuO2应变诱发磁有序.docx) In our previous calculations, we also found that the magnetism of RuO$_2$ is particularly sensitive to variations of the lattice constants $a$ and $c$ (Fig. 1b): overall, increasing $a$ while decreasing $c$ favors the formation of antiferromagnetic order and leads to a spontaneous spin splitting. (基于第一性原理计算的RuO2应变诱发磁有序.docx) This behavior is markedly different from that of TiO$_2$ and IrO$_2$, which also adopt the rutile structure, as the latter two hardly develop stable magnetism even when $a$ and $c$ are varied by more than 8%. (基于第一性原理计算的RuO2应变诱发磁有序.docx) Furthermore, if RuO$_2$ is forced to adopt the lattice parameters of TiO$_2$, an antiferromagnetic tendency already emerges even without introducing DFT+$U$. (基于第一性原理计算的RuO2应变诱发磁有序.docx) Given the good lattice match between TiO$_2$ and RuO$_2$, and the fact that TiO$_2$ is commonly used as an epitaxial substrate in experiments, it is therefore reasonable to discuss strain-induced magnetic order in RuO$_2$ by taking TiO$_2$-substrate-induced lattice strain as the starting point. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

To quantitatively characterize the global distortion imposed by a TiO$_2$ substrate on epitaxial RuO$_2$ films, we introduce a global distortion parameter $\varepsilon$, and take the distorted RuO$_2$ primitive cell as our basic object (Fig. 1a). (基于第一性原理计算的RuO2应变诱发磁有序.docx) Under this definition, $\varepsilon$ measures the extent to which the RuO$_2$ lattice is continuously interpolated from its intrinsic values toward those of TiO$_2$, with the lattice constants written as

When $\varepsilon \to 0$, the lattice returns to the intrinsic RuO$_2$ values; when $\varepsilon \to 1$, it approaches that of TiO$_2$. (基于第一性原理计算的RuO2应变诱发磁有序.docx) This parameterization is thus convenient for discussing epitaxial systems such as RuO$_2$/TiO$_2$. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

Based on this framework, we compute the evolution of RuO$_2$ magnetism as $\varepsilon$ is varied from 0 to 1 (Fig. 1b). Without considering SOC, increasing $\varepsilon$ drives the system from a paramagnetic state to an antiferromagnetic state, manifested by the emergence and gradual growth of Ru local moments. (基于第一性原理计算的RuO2应变诱发磁有序.docx) A finer scan yields a critical point of $\varepsilon_c = 0.7404$. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

These conclusions are not qualitatively altered by including SOC. Calculations show that SOC does not change the overall trend of the magnetic transition (Fig. 1b): with SOC included, the critical point becomes $\varepsilon_c = 0.7340$, very close to the value without SOC. (基于第一性原理计算的RuO2应变诱发磁有序.docx) The main difference lies in the moment magnitude: for $\varepsilon>\varepsilon_c$, SOC reduces the magnetic moments by about 10% compared with the no-SOC results. (基于第一性原理计算的RuO2应变诱发磁有序.docx) Given that the SOC energy scale in RuO$_2$ can reach $\sim 0.3$ eV, such a reduction is reasonable. (基于第一性原理计算的RuO2应变诱发磁有序.docx) Since SOC has only a minor impact on whether the transition occurs and on $\varepsilon_c$, we primarily present results without SOC in the main text; comparisons between SOC and no-SOC cases are provided in the Supplementary Material. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

改动说明：我没有改变任何物理含义或结论；只做了英文论文常用的句式/术语规范化，并把你文中的插值关系用标准的公式排版呈现。

It should be noted that, in epitaxial thin-film systems, due to the substrate–film heterostructure and nonideal growth conditions, the system generally cannot be fully described by a single global strain parameter $\varepsilon$. Given the intrinsic fragility of the magnetic state in RuO$_2$, local lattice distortions introduced by defects, interfaces, or local relaxations in thin films may also affect the magnetism. Therefore, for a fixed global distortion $\varepsilon$, we further introduce a “controlled periodic local distortion” to represent possible local fluctuations in films: on the one hand, this avoids approximating real films by a purely monotonic tension/compression, which could imply an unrealistic defect-like structural picture; on the other hand, it enables a systematic scan of how such local fluctuations influence magnetic ordering. For the RuO$_2$ unit cell, tensile deformation of the lattice constant $a$ favors the emergence of magnetic order, whereas compressive deformation does not. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

Specifically, we expand RuO$_2$ along the (100) direction and construct an $n\times 1\times 1$ supercell as the local-distortion model, where $n$ is the period (counted by the number of unit cells). In the supercell, we keep $b=a_{\varepsilon}$ and $c=c_{\varepsilon}$ fixed, and apply a prescribed distortion only to the lattice constant $a$ of each unit cell according to its cell index $i$ $(i=1,2,\ldots,n)$ (Fig. 2a). We consider two types of local distortions: harmonic distortions and $\delta$ distortions. The former is more idealized and facilitates understanding the generic effects of continuous fluctuations, while the latter is used to mimic film situations closer to defects or strong local distortions. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

改动说明：没有改动大意；主要是把中文里较长的并列结构拆成更符合英文论文阅读的两句话，并把“避免引入不合理的结构缺陷图景”译成更常见的学术表达（an unrealistic defect-like structural picture），含义保持一致。

English translation

The harmonic distortion takes the lattice-distortion period to be $n$, such that the distortion is periodically distributed within the supercell. Here, $\Delta_h$ denotes the amplitude of the local distortion (measured by the relative change of $a$). Because the harmonic function averages to zero over one period, this construction statistically preserves $\langle a_i\rangle=a_{\varepsilon}$, thereby realizing “a fixed global $\varepsilon$ with superimposed local fluctuations.” (基于第一性原理计算的RuO2应变诱发磁有序.docx)

The $\delta$ distortion also sets the distortion period to be $n$, such that the lattice constant $a$ exhibits a delta-function-like variation within the supercell. $\Delta_{\delta}$ characterizes the relative amplitude of the most strongly stretched unit cell in the supercell, while the remaining cells provide a compensating adjustment so that the average lattice constant still satisfies $\langle a_i\rangle=a_{\varepsilon}$. In this way, the global distortion $\varepsilon$ remains unchanged and only local inhomogeneity is introduced. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

Within this model, $\varepsilon$ controls the degree of global strain, $\Delta$ (either $\Delta_h$ or $\Delta_{\delta}$) controls the amplitude of local distortions, and $n$ controls the period (length scale) of the local distortion. Therefore, by tuning $(n,\varepsilon,\Delta)$, one can independently control the distortion period, the global distortion strength, and the magnitude of local fluctuations. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

For convenience, we set $n=9$ and systematically examine the influence of $\varepsilon$ and $\Delta$ on magnetic ordering. For the harmonic distortion, as $\Delta_h$ increases from 8‰ and 14‰ to 20‰, the critical $\varepsilon_c$ decreases only slowly from 0.7404 to approximately 0.72, 0.69, and 0.66 (Fig. 2a). This indicates that $\Delta_h$ has only a limited impact on $\varepsilon_c$: when $\varepsilon_c$ changes from 0.7404 to 0.66, $b_{\varepsilon}$ changes by only 2‰ and $c_{\varepsilon}$ by only 4‰, which is far smaller than the 20‰ change required for $a$. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

When the $\delta$ distortion is adopted, the resulting $\varepsilon_c$ values at $\Delta_{\delta}=8‰$, 14‰, and 20‰ are approximately 0.73, 0.70, and 0.67, respectively, very close to those obtained for the harmonic distortion (Fig. 2b). Although the $\delta$-distortion profile differs substantially from a harmonic modulation in real space, its impact on $\varepsilon_c$ remains of the same order for the same amplitude parameter $\Delta$. This suggests that whether magnetic order emerges is mainly determined by the global distortion $\varepsilon$, whereas local distortions primarily modify magnetic details rather than the phase boundary itself. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

Meanwhile, we also note that at the same $\varepsilon$, the Ru local moments corresponding to different $\Delta$ can differ significantly, and $\Delta$ can also slightly affect the value of $\varepsilon_c$, indicating that local distortions can modulate the moment magnitude to some extent (representative spatial distributions of local moments are provided in the Supplementary Material). (基于第一性原理计算的RuO2应变诱发磁有序.docx)

Notable edits (meaning unchanged): I corrected an obvious typo “$\varepsilon\varepsilon$” to “$\varepsilon$” in the English rendering, since it is clearly a duplication in the Chinese sentence.

Based on the systematic calculations of global and local distortions described above, our main findings can be summarized as follows. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

(1) The magnetic structure of RuO$_2$ is highly sensitive to lattice distortions: when the global distortion parameter $\varepsilon$ exceeds the critical value $\varepsilon_c$, the system undergoes a transition from a paramagnetic state to an antiferromagnetic state. This conclusion remains valid upon including SOC; the primary effect of SOC is to modify the moment magnitude rather than to overturn the phase boundary. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

(2) For a fixed global strain $\varepsilon$, introducing local distortions and tuning $\Delta$ can indeed substantially modify magnetic details such as the local moments, thereby enabling a certain degree of amplitude modulation of magnetism; however, the macroscopic boundary for magnetic ordering is still mainly controlled by $\varepsilon$. Even if the lattice constants exhibit some spatial fluctuations in thin films, the global critical point $\varepsilon_c$ will not shift dramatically. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

We further note that the Fermi surface of RuO$_2$ does not exhibit an obvious global reconstruction under the above distortions. Although it is difficult to realize a perfectly isotropic deformation identical to our model in experiments (the strains along the $a$, $b$, and $c$ directions may not be synchronized with the same magnitude), the strain level achievable in epitaxial thin films is typically on the order of the parameters considered here. Meanwhile, the specific value of $\varepsilon$ depends on extrinsic factors such as substrate choice and growth conditions, and can be tuned within a certain range by adjusting the RuO$_2$ film thickness, which also provides a natural route to rationalize the diversity of current experimental characterizations of RuO$_2$ magnetism. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

More importantly, compared with the nonmagnetic results, the band structure does not undergo a “large renormalization”: for different strain levels, most RuO$_2$ bands remain close to those in the nonmagnetic case (Fig. 3a). In terms of the Fermi surface, only the red portion changes substantially, while the blue portion shows only minor variations (Fig. 3b), which is compatible with existing ARPES and QO observations. This indicates that even if the RuO$_2$ Fermi surface is not reconstructed into the form typically obtained in conventional DFT+$U$, the system may still develop magnetic order. Moreover, lattice distortions likewise do not induce a “large renormalization,” which is also supported by the subsequent DOS results (Fig. 5e). Therefore, a magnetic-ordering mechanism driven by global strain plus local distortions provides an alternative framework for understanding RuO$_2$ magnetism, and can to some extent alleviate doubts raised by fine electronic-structure probes such as ARPES and QO regarding magnetic ordering in RuO$_2$. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

改动较大的地方：我把中文里的“非本征因素”在英文里用更常见的 extrinsic factors 表达（含义不变），其余保持原意与逻辑不变。

As studies of antiferromagnetism in RuO$_2$ have progressed, related work has reported an antiferromagnetic incommensurate spin-density-wave (IC-SDW) state in RuO$_2$/Py heterostructures, and further discussed low-energy excitations arising from the coupling between the RuO$_2$ SDW and spin waves in the adjacent Py (permalloy) layer [17]. Such low-energy excitations differ from the RuO$_2$ magnon spectrum predicted in earlier calculations [24]. In particular, the GHz-range excitation energy appears inconsistent with the THz-scale spin-wave excitations typically expected for antiferromagnets, highlighting the rich antiferromagnetic behavior of RuO$_2$. Subsequent experiments have verified this phenomenon and, on this basis, suggested the presence of certain strain-related spin fluctuations in RuO$_2$ [57]. Considering only the intrinsic properties of RuO$_2$, one finds relatively flat bands near $(\pi,0,\pi/2)$ on the Fermi surface, implying a low Fermi velocity and potential Fermi-surface nesting, and thus a propensity to form a spin-density wave [3]. Given the fragility of RuO$_2$ magnetism, extrinsic factors such as lattice distortions, temperature effects, interlayer exchange coupling (IEC) introduced by proximity to a ferromagnetic layer, and possible many-body effects in correlated systems may all reshape its magnetic-instability landscape and influence the tendency toward an SDW state [25–28]. This implies that, under specific external conditions, RuO$_2$ may host an SDW state and exhibit a magnetic-dynamical response distinct from that of the AFM state.

From a computational perspective, it is generally difficult to search for and stabilize SDW order directly within the DFT framework using supercells, especially for SDW scenarios that may be induced by extrinsic factors [36]. One key reason is that lattice–SDW coupling can be decisive for SDW formation in some materials [38], yet incorporating such coupling is not straightforward within standard DFT implementations; when lattice–SDW coupling is not accounted for, the AFM state is typically lower in energy (Fig. 4b). Physically, at the SDW nodes the Néel vector reverses and the local moment is strongly suppressed (even approaching zero), which incurs an exchange-energy penalty and an additional energetic cost associated with the locally nonmagnetic character. As a result, the SDW state is less likely to emerge as a low-energy self-consistent solution. Meanwhile, RuO$_2$ thin films exhibit a distribution of lattice constants that varies with substrate choice and film thickness, which makes the present modeling approach physically plausible. Motivated by this, we follow the strategy used for total-energy calculations of SDWs in Cr [37] and stabilize an SDW state by introducing an a priori periodic local strain/distortion, thereby allowing a controlled analysis of its competition with the AFM state.

Specifically, we first construct a “global strain + local distortion” model to examine how local distortions affect the magnetism of RuO$_2$. Taking a sinusoidal distortion with $n=8$ (corresponding to a $16\times 1\times 1$ supercell to accommodate the Néel-vector reversal of the SDW) as an example (Fig. 4a), we use (基于第一性原理计算的RuO2应变诱发磁有序.docx)

Here, $i$ labels the unit cell, and $\Delta_h$ is the amplitude of the local distortion. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

We define $\Delta E = E_{\mathrm{SDW}}-E_{\mathrm{AFM}}$, and compute $\Delta E$ for different $\Delta_h$ to evaluate the relative stability of the SDW and AFM states (Fig. 4c). (基于第一性原理计算的RuO2应变诱发磁有序.docx) The results show that there exists a parameter regime where $\Delta E<0$, i.e., the SDW is lower in energy than the AFM state, and whether this occurs depends on both $\varepsilon$ and $\Delta_h$: when $\varepsilon>\varepsilon_c$, $\Delta E$ increases rapidly as $\Delta_h$ decreases; to achieve $\Delta E<0$, the required local-distortion strength becomes more stringent as $\varepsilon$ increases, consistent with previous work [36]. When $\varepsilon<\varepsilon_c$, although the critical local distortion required for $\Delta E<0$ decreases as $\varepsilon$ decreases, establishing global magnetic order simultaneously becomes more difficult, which also makes the emergence of the SDW more challenging. Notably, when $\varepsilon\approx\varepsilon_c$, $\Delta E$ exhibits a non-monotonic dependence on $\Delta_h$: for $\Delta_h>5\!\,\text{‰}$, $\Delta E$ gradually decreases, and for $\Delta_h\ge 12\!\,\text{‰}$ the SDW becomes the lower-energy state. Since local distortions at the $\sim 10\!\,\text{‰}$ level are common in epitaxial RuO$_2$ films, this indicates that the SDW is more likely to become a potential ground state near the critical $\varepsilon$ when accompanied by a certain degree of local distortion; when $\varepsilon$ is far from the critical regime, the system tends to remain paramagnetic or adopt the AFM state. Overall, local distortions serve as an important driving factor for SDW formation. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

Based on these observations, we propose the following hypothesis to explain why an SDW may become the ground state: as the local-distortion amplitude $\Delta_h$ increases, the compressed unit cells in the supercell tend to evolve toward weakly magnetic or even nonmagnetic behavior; meanwhile, the SDW state intrinsically features near-zero moments at its nodes. Therefore, when compression suppresses the local moments, the SDW can instead gain an energetic advantage over the AFM state near the nodes, making $\Delta E$ more likely to become negative. This leads to a testable implication: the dominant contribution to $\Delta E$ should come from the vicinity of the nodes. Considering the nonlocal nature of the AFM state, the moment at the node in the AFM configuration should not drop to zero “immediately,” but rather decrease gradually as $\Delta_h$ increases; by contrast, in the stretched (or moment-enhanced) regions, the local moments in the AFM and SDW states should become similar, yielding only a small net contribution to $\Delta E$. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

改动较大的地方：没有改动大意；主要是把你原文里带有 \hspace{...} 的公式用更标准的 LaTeX 排版写法呈现，方便直接放进 PRB 的稿件里。

你可以用 Word 自带的公式编辑器直接吃进（接近）LaTeX 的“线性输入”，关键是：** 别用 \hspace{}**，改用 \quad / \qquad 或者用对齐环境把三行写清楚。

下面按最常用的 Word（Microsoft 365 / Word 2019+）给你一个最省事的方法：

方法 A：直接粘贴 LaTeX（推荐）

1. 在 Word 里按 Alt + =（插入公式）。
2. 在“公式”选项卡里找到 输入方式（Input）并切到 LaTeX（有的版本在右上角“转换/专业/线性”附近）。
3. 把下面代码 整段粘贴到公式框里，然后按一次空格或点击公式外侧，Word 会排版。

1) Harmonic distortion（可直接粘贴）

latex
\begin{aligned}
a_i &= a_{\epsilon}\left(1-\Delta_h\cos\frac{(2i-1)\pi}{2n}\right),\\
b &= a_{\epsilon},\\
c &= c_{\epsilon}.
\end{aligned}

2) \delta distortion（可直接粘贴）

latex
\begin{aligned}
a_i &=
\begin{cases}
a_{\epsilon}(1+\Delta_{\delta}), & i=1,\\
a_{\epsilon}\left(1-\frac{\Delta_{\delta}}{n-1}\right), & \text{otherwise},
\end{cases}\\
b &= a_{\epsilon},\\
c &= c_{\epsilon}.
\end{aligned}

说明：我把你原来的 \cfrac 换成了 \frac（Word 更稳），把 \hspace{} 去掉并用 aligned 做了对齐——数学含义完全不变，只是为了让 Word 更容易正确渲染。

方法 B：如果你的 Word 不支持 LaTeX 输入

1. Alt+= 插入公式；
2. 在公式工具里选择 “括号”→“分段函数（cases）” 模板，先插入那个两行的 cases；
3. 逐项填入：
   • 第一行：a_\epsilon(1+\Delta_\delta)，右侧条件 i=1
   • 第二行：a_\epsilon(1-\Delta_\delta/(n-1))，右侧条件 otherwise
4. 然后在 cases 后面另起两行输入 b=a_\epsilon、c=c_\epsilon（或用“对齐”模板对齐等号）。

如果你告诉我你用的 Word 版本（比如 2016/2019/365，Windows/Mac），我也可以给你对应版本最稳的“输入模式”路径截图式说明（文字步骤）。

We first perform calculations at $\varepsilon=\varepsilon_c$ to examine, for different periods $n$, the relationships among $\Delta E$, the local moments, and $\Delta_h$. As shown in Fig. 5(b), when $\Delta_h$ is moderately large ($\ge 5$‰), $\Delta E$ decreases with increasing $\Delta_h$, consistent with our previous results for $n=8$. This indicates that the validity of the above conclusions is not affected by changing $n$. Meanwhile, we compare the spatial distributions of Ru local moments in the AFM and SDW states (Fig. 5(a)). The maximum moments in both magnetic configurations increase with $\Delta_h$ and gradually converge, in agreement with the prediction that the moments in the tensile regions become similar and thus contribute only weakly to $\Delta E$. In contrast, the minimum moment in the AFM state decreases as $\Delta_h$ increases, reflecting the continuous suppression of the moment in the compressed regions (corresponding to the vicinity of the nodes). At the same time, owing to the nonlocal nature of the AFM state, the moment at the node decreases relatively slowly, making this region the dominant source of the moment difference between AFM and SDW, thereby supporting the conclusion that $\Delta E$ is mainly contributed by the node region.

Based on this mechanism, we obtain a second testable implication: if the dominant contribution to $\Delta E$ indeed comes from the vicinity of the nodes, then as the period $n$ increases (i.e., the wavelength becomes longer), a “nearly nonmagnetic” region should form and persist more easily around the nodes, and the energetic advantage of the SDW relative to the AFM state should gradually emerge. To this end, at $\varepsilon=\varepsilon_c$ we compute $\Delta E$ per cell (which facilitates comparison across different periods) for various $\Delta_h$, as well as the $n$-dependence of the moment difference between AFM and SDW at the same spatial positions. The results show that $\Delta E$ per cell first decreases and then increases with $n$ (Fig. 5(c)), while the AFM moment at the node decreases rapidly as $n$ increases (Fig. 5(d)). This behavior can be understood as a competition between two energy contributions. When $n$ increases from small values, the “nearly nonmagnetic” region near the nodes expands and the wavelength becomes longer; the exchange-energy cost is reduced accordingly, making it easier for the SDW to become the ground state. However, when $n$ is further increased, the nearly nonmagnetic region gradually exceeds the constraint imposed by AFM nonlocality, so that the AFM node moment also approaches zero and becomes similar to that of the SDW. Consequently, $\Delta E$ per cell rises again and can approach zero. Meanwhile, a longer period also weakens the coupling between neighboring magnetically ordered regions, making the moment magnitudes in the two states more similar. We further observe that a smaller $\Delta_h$ requires a larger $n$ to reach the minimum $\Delta E$ per cell—i.e., the smaller the distortion amplitude, the longer the wavelength needed for the most stable SDW. From these results, we identify a wavelength range that is more favorable for stabilizing the SDW: approximately 7–10 nm.

改动说明：我仅做了英文论文里更常见的表达与语序整理（比如把“几近无磁”译为 nearly nonmagnetic，把“ΔE per cell”加上说明其用于跨周期比较），不改变原意。

To further characterize the microscopic nature of the SDW order, we analyze the evolution of the density of states (DOS) and band structure within the local-distortion model. Taking a sinusoidal distortion with $\varepsilon=\varepsilon_c$, $n=8$, and $\Delta_h=24$‰ as an example, we compare the DOS at the node-site Ru atom [Ru (node Ru)] and at the Ru site with the maximum local moment [Ru ($\mu_{\max}$ Ru)] (Fig. 6b). (基于第一性原理计算的RuO2应变诱发磁有序.docx) At energies near the Fermi level, $\mu_{\max}$ Ru exhibits a typical spin-splitting feature. Further calculations show that, from node Ru to $\mu_{\max}$ Ru, the DOS evolves continuously through a gradual crossover rather than via an abrupt reconstruction. (基于第一性原理计算的RuO2应变诱发磁有序.docx) This indicates that the magnetism and electronic structure of the system can be tuned continuously by parameters such as $\varepsilon$ and $\Delta$, which is clearly distinct from the abrupt DOS changes induced by $U>1$ eV. Meanwhile, the local-moment scale obtained in this model is $\sim 0.01$–$0.1\,\mu_B$, consistent with the current understanding of “weak magnetic ordering” in RuO$_2$. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

These results suggest that, within an idealized periodic-distortion model, the SDW is a self-consistent and reasonable ground-state candidate, and is also compatible with existing experimental observations. (基于第一性原理计算的RuO2应变诱发磁有序.docx) Next, we discuss how the spatial structure of the SDW adjusts when the distortion evolves from a regular periodic modulation to the disordered fluctuations expected in real thin films, and we relate this to experimental phenomena. We emphasize that, although we have discussed the SDW wavelength scale above, this does not imply that the system must form a strictly periodic SDW: in thin films, the driving factor for SDW formation is local distortion, which is typically disordered and lacks a fixed period. (基于第一性原理计算的RuO2应变诱发磁有序.docx) Based on this, we construct a local-distortion model in which the position of the compressed region can be tuned deliberately, enabling control of the pinning position of the SDW node (see the Supplementary Material for details and parameter definitions). The calculations show that the SDW node shifts synchronously with the movement of the center of the most strongly compressed region, i.e., the node tends to be pinned to the location corresponding to the “maximally compressed lattice” (Fig. 6a). (基于第一性原理计算的RuO2应变诱发磁有序.docx) This indicates, on the one hand, that the node position responds to local distortions; on the other hand, it implies that the “SDW wavelength” in RuO$_2$ is not a unique constant, but is closely related to the characteristic length scale (and spatial distribution) of the local distortions. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

改动说明：我只做了英文论文常见的措辞规范化（如把“突变式重构”译为 abrupt reconstruction，把“钉扎”译为 pinned/pinning），不改变原意。

To assess whether the above conclusions remain valid in a scenario closer to real thin films, we further consider $\delta$-type local distortions. In practice (especially in thin films/heterostructures), relatively large distortions spanning several unit cells—caused by defects, interfacial steps, or local relaxations—are quite common, and their influence is often confined to local regions on the nm or even Å scale. To describe this type of “strong distortion within a finite range”, we construct a more realistic $\delta$-distortion model: while keeping the global distortion parameter $\varepsilon$ fixed, only part of the supercell is allowed to undergo a paired compression/extension, thereby introducing local inhomogeneity. For simplicity, we take $\varepsilon=\varepsilon_c$ as an example and define the local lattice distortion as

Here, $n_0$ denotes the number of compressed cells within each magnetically ordered region, while $n-n_0$ corresponds to the stretched cells near the node region. This construction ensures $\langle a_i\rangle=a_{\varepsilon}$ over one period, so that the global $\varepsilon$ remains unchanged and only local compression is introduced. This model is closer to the thin-film reality of “strong local distortion with a finite spatial extent”, which helps elucidate the SDW/AFM competition mechanism in RuO$_2$ thin films in a more realistic setting. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

In our calculations, we set $n=10$ and construct a $20\times 1\times 1$ locally distorted supercell. By varying $n_0$, we tune the “width” of the compressed region and examine its impact on the relative stability of SDW versus AFM (Fig. 6c). The results show that as $n_0$ increases, the $\Delta E$–$\Delta_{\delta}$ relation gradually changes from increasing with $\Delta_{\delta}$ to decreasing with $\Delta_{\delta}$, and the slope decreases progressively with increasing $n_0$. This indicates that stabilizing the SDW relative to the AFM requires a sufficiently large locally compressed (nearly nonmagnetic) region. It also implies that whether an SDW exists is closely tied to the local compression conditions in the film, consistent with the trends found for harmonic distortions. Meanwhile, we note that these conclusions align with the idealized-case results: as the “nearly nonmagnetic” region near the nodes expands, the SDW becomes more likely to emerge as the ground state. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

改动较大的地方：我把原式里的 \hspace{} 全部去掉、用更标准的分段与对齐方式排版；同时把 \cfrac 换成了 Word/PRB 更通用也更稳的 \dfrac/\frac 风格（数学含义不变）。

Combining the above results, we can draw two conclusions and one inference:

(1) “Local distortions” near the critical strain can induce the SDW as the ground state. When the global strain of RuO$_2$, $\varepsilon$, is close to the critical point $\varepsilon_c$ and local distortions are present, the system may develop an SDW state. In this regime, the energetic competition between AFM and SDW is mainly governed by the local structure near the SDW nodes: enhanced compression at the nodes amplifies the difference between the two states in this region, thereby strengthening the energetic advantage of the SDW relative to the AFM state. On the other hand, as the spatial extent of the “compressed / nonmagnetic-tending” region around the nodes gradually increases, $\Delta E$ can exhibit a nonmonotonic behavior—first decreasing and then rising.

(2) In harmonic distortions, the characteristic SDW wavelength is determined by the “distortion strength.” For general local distortions, a characteristic length scale of approximately 7–10 nm is more favorable for stabilizing the SDW. If the characteristic length is shorter, exchange interactions significantly increase the energetic cost at the Néel-vector reversal (i.e., at the nodes), making the SDW less competitive. If the characteristic length is longer, the energy difference between AFM and SDW tends to decrease and the two states approach near degeneracy.

(3) Inference: the low-energy excitations may originate from the near-degenerate competition between AFM and SDW. When the system is close to the critical strain and local distortions are present, the energy difference between AFM and SDW is only 10–100 $\mu$eV ($\sim$10 GHz), which is markedly lower than the THz-scale magnon excitation energies commonly found in conventional antiferromagnets.

改动说明：我没有改变大意；只是把中文里的引号语气换成英文常用的强调方式，并把句子拆分得更符合英文论文的逻辑推进。

Conclusions (English translation)

In this work, we investigate magnetic ordering in RuO$_2$/TiO$_2$ thin-film systems by examining the effect of global epitaxial strain on RuO$_2$ and the complex magnetic orders that can arise from different local lattice distortions. When the substrate-imposed global strain on RuO$_2$ is sufficiently strong, RuO$_2$ can undergo a transition from a paramagnetic state to an antiferromagnetic (AFM) state. When local distortions are introduced, the magnetism becomes spatially nonuniform across different distorted regions, whereas the critical global strain for the transition does not change significantly. Near the paramagnetic–AFM critical strain, sufficiently strong local distortions can render a spin-density-wave (SDW) state a potential energetic ground state. The energy difference between SDW and AFM depends on the characteristic length scale and amplitude of the local distortion: when the distorted region extends over several lattice constants and the distortion amplitude reaches the $\sim 10$‰ level, the SDW can become the ground state. The SDW–AFM energy difference is on the order of 10–100 $\mu$eV, with a characteristic wavelength of approximately 7–10 nm, providing a natural mechanism for the anomalously low-energy excitations that may appear in RuO$_2$ thin films. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

Given that RuO$_2$ magnetism is highly sensitive to both global strain and local distortions, the stress/strain-induced magnetic ordering revealed here has two implications. First, it helps rationalize the disparate conclusions reported for RuO$_2$ magnetism under different sample morphologies, epitaxial conditions, and interfacial environments, offering a more consistent physical picture. Second, it provides testable theoretical clues for understanding differences in magnetic-related responses observed in spectroscopic and transport experiments, thereby highlighting the dominant role of strain/distortion in controlling RuO$_2$ magnetism. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

In addition, our results suggest new routes for tuning spin transport in RuO$_2$, and provide computational support for further controlling RuO$_2$ spin-transport properties by engineering the coupling strength between the substrate and the RuO$_2$ film, as well as its spatial inhomogeneity. Overall, these conclusions offer new approaches and ideas for expanding stress/strain-based control dimensions in spintronic devices. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

改动较大的地方：没有改动大意；仅将个别中文术语按英文论文更常见的写法规范化（例如把“非本征”倾向写成 extrinsic / non-intrinsic 语境表达）。

Figure caption (English translation)

(a) Schematic illustration of the global strain applied to RuO$_2$.
(b) Evolution of the magnetic state of RuO$_2$ as a function of $\varepsilon$, showing a transition from a nonmagnetic state to an antiferromagnetic state near $\varepsilon \approx 0.74$. The mapping plot shows the results under $a$–$c$ strain in RuO$_2$, indicating that magnetic order can emerge when the RuO$_2$ lattice is distorted toward the TiO$_2$ structure. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

In terms of structural convergence, we set the ionic-relaxation criterion for primitive-cell calculations to a Hellmann–Feynman force below 0.001 eV/Å. For scanning calculations involving a large number of supercells, to control the computational cost while maintaining accuracy, we constrain the forces on O atoms to be within 0.05 eV/Å in the supercell calculations, and discuss the validity of this threshold—together with comparisons to results obtained using stricter relaxation settings—in the Supplementary Material. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

Considering the intrinsic fragility of the magnetic order in RuO$_2$ and the heterostructure nature of the substrate–film system together with nonideal growth conditions, the thin-film system generally cannot be fully described by a single global strain parameter $\varepsilon$. Therefore, we introduce a type of local distortion that does not change the global strain $\varepsilon$, and use it to investigate its effect on magnetic ordering in thin films. For the RuO$_2$ unit cell, tensile deformation of the lattice constant $a$ favors the emergence of magnetic order, whereas compressive deformation does not.

More importantly, across the paramagnetic–ferromagnetic transition, the band structure of RuO$_2$ does not exhibit an abrupt change (Fig. 3a). In terms of the Fermi surface, only the red portion shows a pronounced modification, while the blue portion changes only slightly (Fig. 3b). Overall, neither the band structure nor the Fermi surface undergoes a large renormalization, which is consistent with existing ARPES and QO measurements. Therefore, the magnetic-ordering mechanism driven by global strain plus local distortions provides an alternative framework for understanding RuO$_2$ magnetism, and can to some extent alleviate the concerns raised by band-structure-sensitive experiments regarding magnetic ordering in RuO$_2$. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

In realistic systems (especially thin-film/heterostructure geometries), local lattice distortions spanning several unit cells—caused by defects, interfacial steps, or local relaxations—are quite common, and their influence is often confined to nm-scale regions. To capture this type of “strong distortion with a finite spatial extent,” we construct a more realistic $\delta$-type local-distortion model: while keeping the global distortion parameter $\varepsilon$ fixed, we allow only part of the supercell lattice to undergo compression/extension, thereby introducing local inhomogeneity. The local lattice distortion is defined as: (基于第一性原理计算的RuO2应变诱发磁有序.docx)

Considering that the magnetism of RuO$_2$ exhibits pronounced sensitivity to both global strain and local distortions, the stress/strain-induced magnetic ordering revealed in this work has two implications. On the one hand, it helps explain the discrepancies in reported conclusions on RuO$_2$ magnetism under different sample morphologies, epitaxial conditions, and interfacial environments, and facilitates discussion of the anomalous magnetic phenomena in RuO$_2$. On the other hand, it offers new ideas for tuning spin transport in RuO$_2$, providing computational support for controlling its spin-transport properties, and suggesting new routes and concepts for expanding stress/strain-based control dimensions in spintronic devices. (基于第一性原理计算的RuO2应变诱发磁有序.docx)

近年来，RuO₂ 作为备受关注的交错磁体（altermagnet），相关研究迅速升温。对 RuO₂ 较大自旋劈裂能与较高 Néel 温度的预测[1]，围绕其反铁磁性的理论铺垫[2–4]，以及关于 RuO₂ 交错磁体自旋劈裂效应（ASSE）的实验研究快速推进[5]，共同推动 RuO₂ 成为该领域的研究热点。然而，迄今为止 RuO₂ 的磁有序性质仍存在显著争议。早期研究认为 RuO₂ 具有较弱的反铁磁序，并利用共振 X 射线散射讨论了块体与薄膜中的反铁磁信号，给出薄膜磁相关长度约为 5 nm[2,6]。另一方面，也有大量实验结果表明 RuO₂ 更接近 Pauli 顺磁金属：μSR 指出体系内部缺乏可观测的动态磁场与磁有序[7,8]；部分 ARPES 结果显示能带结构更符合非磁有序特征[9,10]；量子振荡（QO）实验则对反铁磁序的存在提出了更强挑战[11]。与此同时，有文献强调 RuO₂ 的磁结构本身极为脆弱，例如费米面附近费米速度偏小，使其磁性对外界扰动高度敏感[13,14]。考虑到 RuO₂ 具有一定库仑排斥且带宽较宽，属于强共价的中等耦合体系，其复杂磁效应并不意外。因此，有观点提出氧空位、外延应变以及表面/界面对称性破缺等非本征因素可能是薄膜中磁性的重要来源[13,15,16]，而这些因素在薄膜体系中往往更为显著。事实上，多数报告反铁磁有序的实验主要基于 RuO₂ 薄膜，论证 RuO₂ 交错磁性的相关自旋输运实验也多发生在薄膜体系[17,18]。更有研究指出，可通过多电子/少电子杂质调控载流子数以改变 Néel 矢量方向并增强反铁磁性表征[19,20]，进一步凸显了非本征效应对 RuO₂ 的关键影响。

In recent years, RuO$_2$ has attracted rapidly growing interest as a promising altermagnet. Predictions of a large spin-splitting energy and a high Néel temperature in RuO$_2$ [1], theoretical groundwork on its antiferromagnetism [2–4], and rapid experimental progress on the altermagnetic spin-splitting effect (ASSE) in RuO$_2$ [5] have jointly propelled RuO$_2$ to the forefront of this field. (基于第一性原理计算的RuO2应变诱发磁有序.docx) However, to date, the magnetic ordering of RuO$_2$ remains highly controversial. Early studies suggested that RuO$_2$ hosts relatively weak antiferromagnetic order; resonant x-ray scattering was used to discuss antiferromagnetic signatures in both bulk and thin-film samples, yielding a magnetic correlation length of $\sim$5 nm in films [2,6]. On the other hand, numerous experiments support RuO$_2$ being closer to a Pauli-paramagnetic metal: $\mu$SR indicates an absence of observable internal dynamic magnetic fields and magnetic order [7,8]; some ARPES measurements show a band structure more consistent with a nonmagnetic state [9,10]; and quantum-oscillation (QO) experiments pose an even stronger challenge to the existence of antiferromagnetic order [11]. Meanwhile, several works have emphasized that the magnetic state of RuO$_2$ is intrinsically fragile—for instance, a relatively small Fermi velocity near the Fermi surface makes its magnetism highly susceptible to external perturbations [13,14]. Given that RuO$_2$ features a finite Coulomb repulsion together with a relatively wide bandwidth, placing it in a strongly covalent, intermediate-coupling regime, such complex magnetic behavior is not unexpected. Accordingly, it has been proposed that extrinsic factors—including oxygen vacancies, epitaxial strain, and symmetry breaking at surfaces or interfaces—may play an important role in generating magnetism in thin films [13,15,16], where these effects are often pronounced. Indeed, most experiments reporting antiferromagnetic order are based on RuO$_2$ thin films, and spin-transport experiments supporting altermagnetism in RuO$_2$ have also predominantly been carried out in film geometries [17,18]. Moreover, it has been reported that tuning the electron count via electron- or hole-type impurities can modify the Néel-vector orientation and enhance experimental signatures of antiferromagnetism [19,20], further highlighting the critical impact of extrinsic effects on RuO$_2$. (基于第一性原理计算的RuO2应变诱发磁有序.docx)