皮米测量磁路 pm3Circuit: picoMeter Measurement Magnetic Circuit

Since 2025

1. Tool Overview（工具简介）

Over more than a century of modern magnetism, our understanding of magnetic matter has advanced from the micro/nanoscale to the sub-Ångström scale, and the magnetic-field distribution within the unit cell has accordingly become an unavoidable challenge in precision atomic manufacturing. The traditional scalar molecular-field theory, inferred phenomenologically along the energy scale, can no longer meet current urgent demands and must be reformed through first-principles and other foundational theories. Within the unified source–field framework of magnetism, microscopic magnetic moments are decomposed as a whole into spin-density and current-density distributions; combining the two yields a complete picture of the magnetic field and establishes a new molecular-field theory. pm3Circuit has been developed within this framework.

pm3Circuit (picoMeter Measurement Magnetic Circuit) is an open computational workflow that couples first-principles calculations with picoscopic magnetostatics. As a pioneering step, it starts from the spin density obtained from density functional theory (DFT): the spin-density distribution on the crystal lattice is first converted into distributed spin magnetic moments counted in fractional Bohr magnetons; the complete 3D vector magnetic field within the unit cell of a magnetic crystal (with B, H, and M mutually consistent) is then evaluated point by point according to the magnetic dipole model. No vector potential is introduced at any stage; the uniqueness of the magnetic field strength is guaranteed by gauge invariance and the variational principle. The workflow extends conventional micromagnetics from the nanometer scale down to the atomic (picometer) scale and interfaces seamlessly with first-principles methods, providing a computational tool for atomic-scale magnetic description, measurement, and control.

现代磁学走过百余年，我们对磁性物质的认知已从微纳尺度深入到亚埃尺度，单胞内的磁场分布也随之成为精细原子制造过程中不可回避的难题。沿能量标度唯象推算的传统标量分子场理论，已无法胜任当下的迫切需求，势必要通过第一性原理等底层理论加以改造。在磁学“源–场统一”框架下，微观磁矩可整体拆分为自旋密度与电流密度分布；将二者综合，即可得到完整的磁场图景，并由此构建起全新的分子场理论。pm3Circuit 正是在这一框架下发展起来的。

pm3Circuit（皮米测量磁路）迈出的第一步是把第一性原理计算与微观静磁学相结合的开放计算流程：从密度泛函理论（DFT）输出的自旋密度出发，先把晶格中的自旋密度分布转化为以分数玻尔磁子计的分布自旋磁矩，再按磁偶极子模型逐点计算磁性晶体单胞内完整的三维矢量磁场（B、H、M 自洽），全程不引入矢量势，由规范不变性与变分原理保证磁场强度取值的唯一性。它把传统微磁学从纳米尺度推进到原子（皮米）尺度并与第一性原理无缝对接，为原子级磁性描述、测量与调控提供计算工具。

2. Workflow (五步流程)

(1) Structure and DFT input preparation: crystallographic cif file (experimental lattice parameters) and CP2K all-electron GAPW input files (TPSS meta-GGA functional, pcseg-2 segmented contracted basis sets, DFT+U for the strong Fe 3d correlation, plane-wave cutoff of 350 Ry), assisted by Multiwfn for input preparation and analysis;
(2) Spin density to distributed moments: parse the spin-density cube grid output by CP2K (the hexagonal primitive cell is orthogonalized into a 60×120×160 grid) and synthesize distributed spin magnetic moments according to the grid-cell volumes;
(3) Magnetic-field calculation: a 3×3×3 orthorhombic supercell serves as the field source, with the magnetic dipole formula integrated point by point to obtain B and H (a 15 pm cutoff avoids source-point singularities; multithreaded slice-based computation); a 3×3×1 supercell is used for surface stray fields;
(4) Magnetostatic energy and configuration feedback: the magnetostatic energy is integrated over candidate spin configurations, and the energy minimum is adopted as the final result (the out-of-plane and in-plane antiferromagnetic configurations on either side of the Morin transition in hematite are selected in this way);
(5) Visualization and comparison with experiment: magnetic-field statistical distributions and isosurfaces (bin files converted to cube format and rendered with VMD), quantitatively compared with muon spin relaxation/rotation (μSR), neutron scattering, and diamond nitrogen-vacancy (NV) center magnetometry; the non-collinear vector spin density from Quantum ESPRESSO can also serve as a direct input.

(1) 结构与 DFT 输入准备：晶体 cif（实验晶格参数）与 CP2K 全电子 GAPW 输入文件（TPSS meta-GGA 泛函、pcseg-2 分段收缩基组、DFT+U 处理 Fe 3d 强关联，平面波截断 350 Ry），配合 Multiwfn 完成输入与分析； (2) 自旋密度到分布磁矩：解析 CP2K 输出的自旋密度立方格点（六方原胞正交化为 60×120×160 网格），按格点体积合成分布自旋磁矩； (3) 磁场计算：以 3×3×3 正交超胞为场源，磁偶极子公式逐点积分求 B 与 H（15 pm 截断避免源点奇异性，多线程分片计算），3×3×1 超胞用于表面杂散场； (4) 静磁能与构型反馈：对候选自旋构型积分静磁能，以能量最低者为最终结果（赤铁矿 Morin 转变两侧的面外/面内反铁磁构型即由此判定）； (5) 可视化与实验对照：磁场统计分布与等值面（bin 转 cube 后由 VMD 渲染），与缪子自旋弛豫/转动（μSR）、中子散射、金刚石 NV 色心磁测量等实验定量对照；Quantum ESPRESSO 的非共线矢量自旋密度亦可作为直接输入。

3. Main Results for the Hematite (α-Fe2O3) Case（赤铁矿示例）

Taking antiferromagnetic hematite as the model material, the work delivers the first sub-lattice-resolved map of the complete 3D vector magnetic field within the unit cell: no region of the cell has a vanishing total field, yet every component crosses zero; the most probable total-field strength is 1.5 T (low-temperature out-of-plane configuration) and 1.0 T (in-plane configuration above the Morin transition), deviating by no more than 4% from the zero-field μSR value (1.56 T) and the antiferromagnetic-resonance effective field (1.026 T), respectively; over 99% of the crystal volume has a magnetic flux density below 17 T, which clarifies the misconception of taking the Mössbauer “hyperfine field” (above 50 T at the iron nuclei) as the magnetic field inside the crystal; low-field isosurfaces form hexagonal mesh layers echoing the long-range lattice order, whereas high fields contract into orbital-like shapes around the iron nuclei; the energy analysis reproduces the physical picture of the Morin transition: the low-temperature out-of-plane configuration is dominated by single-ion anisotropy (about 1.5 meV/Fe lower than the in-plane configuration), the change in dipole-interaction energy amounts only to the μeV/Fe scale, and upon heating thermal fluctuations suppress the anisotropy and drive the spins toward the basal plane; the predicted surface stray fields agree quantitatively with NV center magnetometry.

以反铁磁赤铁矿为模式材料，首次给出子晶格分辨的单胞三维矢量磁场全图：单胞内不存在总磁场为零的区域，但每个分量都穿过零值；总磁场最可几强度为 1.5 T（低温面外构型）与 1.0 T（Morin 转变以上面内构型），分别与零场 μSR 实验值（1.56 T）和反铁磁共振有效场（1.026 T）的偏差不超过 4%；99% 以上晶体体积的磁通密度低于 17 T，澄清了把穆斯堡尔谱“超精细场”（铁核处大于 50 T）直接当作晶体内部磁场的误解；低场等值面呈现与长程晶格序相呼应的六方网格层，高场收缩为围绕铁核的类轨道形态；能量分析重现 Morin 转变的物理图像：低温面外构型由单离子非各向同性主导（相对面内构型低约 1.5 meV/Fe），偶极相互作用能变化仅为 μeV/Fe 量级，升温后热涨落压制非各向同性、驱动自旋转向面内；表面杂散场预测与 NV 色心磁测量定量一致。

Maintained by Sigma-Jun Lu (lujun@iphy.ac.cn)