本論文探討二維電子氣體在空間變化磁場與賽曼場作用下的自旋極化行為，並考慮兩種不同的磁場配置。第一種配置中，面外磁場為均勻磁場，而面內賽曼場隨空間位置變化；第二種配置中，面外磁場在磁性介面兩側反轉方向，而面內磁場則保持均勻。我們利用微擾理論分析電子波函數的空間分布如何影響其自旋態。

在均勻面外磁場的配置下，電子波函數的導心位置取決於守恆動量 ky。因此，波函數與空間變化面內磁場之間的重疊積分也具有動量相依性，進而產生與動量相關的自旋混合，使自旋極化方向逐漸偏離面外方向。

在非均勻磁場的配置下，位於磁場變號介面附近的 snake states 具有相對於介面的對稱機率密度。由於面外賽曼場對介面呈反對稱分布，相應的微擾積分受到強烈抑制，因此 snake states 可維持接近完全沿面內磁場方向的自旋極化。相較之下，遠離介面的 bulk-like states 則會再次出現自旋混合。

最後，我們進一步利用以緊束縛模型為基礎的數值計算檢驗解析結果。所得到的能帶結構與自旋期望值與微擾理論分析一致。整體結果顯示，電子波函數的實空間分布與空間變化的賽曼場之間的交互作用，可以產生具有動量相依性的自旋極化，並形成一種不依賴傳統自旋-軌道耦合的等效自旋-動量耦合關係。

This thesis investigates the spin polarization of a two-dimensional electron gas under spatially varying magnetic fields and Zeeman fields by setting different magnetic-field configurations . In the first configuration, the out-of-plane magnetic field is homogeneous, while the in-plane Zeeman field varies spatially. In the second configuration, the out-of-plane magnetic field reverses direction across a magnetic interface, while the in-plane field is homogeneous. Perturbation theory is used to analyze how the spatial distribution of the wavefunctions affects the spin states.

In the homogeneous out-of-plane magnetic-field configuration, the guiding-center position depends on the conserved momentum ky . Consequently, the overlap between the wavefunction and the spatially varying in-plane field also depends on momentum. This produces momentum-dependent spin mixing and causes the spin polarization to tilt from the out-of-plane direction.

In the inhomogeneous magnetic-field configuration, the probability densities of the snake states are spatially symmetric near the interface of the magnetic-field dipole. Since the out-of-plane Zeeman field is antisymmetric, the corresponding perturbation integral is strongly suppressed. The snake states therefore remain nearly fully polarized along the in-plane field direction. In contrast, bulk-like states exhibit spin mixing.

The analytical results are further examined using numerical calculations based on a tight-binding model. The calculated energy bands and spin expectation values are consistent with the perturbative analysis. The results demonstrate a momentum-dependent spin polarization can emerge from the interplay between the real-space distribution of wavefunctions and spatially varying Zeeman fields. This mechanism provides an effective spin–momentum-coupled relation without relying on conventional spin–orbit coupling.

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