2.1 Ab initio linear response
Featured work
![Stoner excitations and spinwave excitations calculated from $Im[\chi_{0}( {\bf q}, \omega )]$ and $Im[\chi( {\bf q}, \omega )]$ in Rhobohedral CrI$_3$. Electron correlations beyond DFT are needed to better describe the magnetic interactions and excitations.](images/research/ke2020arxiv.png)
Figure 2.1: Stoner excitations and spinwave excitations calculated from \(Im[\chi_{0}( {\bf q}, \omega )]\) and \(Im[\chi( {\bf q}, \omega )]\) in Rhobohedral CrI\(_3\). Electron correlations beyond DFT are needed to better describe the magnetic interactions and excitations.
Electron correlation effects on exchange interactions and spin excitations in 2D van der Waals materials
Liqin Ke and Mikhail I. Katsnelson
npj Comput Mater 7, 4 (2021)
Features and Capabilities
- Bare and full dynamic transverse spin susceptibilities: \(\chi_0^\pm({\bf q},\omega)\) and \(\chi^\pm({\bf q},\omega)\)
- Exchange coupling \(J_{ij}({\bf R})\)
Brief descriptions
- Dynamic transverse spin susceptibility \({ \chi^{+-} \left( {\bf q}, \omega \right) }\)
Starting from an \(\textit{ab initio}\) single-particle Hamiltonian \(H\), such as the LDA Hamiltonian, the non-interacting transverse spin susceptibility \(\chi_{0}^{+-} \left( {\bf q}, \omega \right)\) can be calculated using the eigenvalues and eigenfunctions of \(H\) within linear response theory~. \[\begin{eqnarray} \chi_0^{+-}({\bf r},{\bf r}',{\bf q}, \omega) &=& \sum^{\rm occ}_{{\bf k}n \downarrow} \sum^{\rm unocc}_{{\bf k}' n'\uparrow} \frac{ \Psi_{{\bf k}n\downarrow}^*({\bf r}) \Psi_{{\bf k}' n'\uparrow}({\bf r}) \Psi_{{\bf k}' n'\uparrow}^*({\bf r}') \Psi_{{\bf k}n\downarrow}({\bf r}') }{\omega-(\epsilon_{{\bf k}' n'\uparrow}-\epsilon_{{\bf k}n\downarrow})+i \delta} \nonumber\\ &+& \sum^{\rm unocc}_{{\bf k}n \downarrow} \sum^{\rm occ}_{{\bf k}' n'\uparrow} \frac{ \Psi_{{\bf k}n\downarrow}^*({\bf r}) \Psi_{{\bf k}' n'\uparrow}({\bf r}) \Psi_{{\bf k}' n'\uparrow}^*({\bf r}') \Psi_{{\bf k}n\downarrow}({\bf r}') }{-\omega-(\epsilon_{{\bf k}n\downarrow}-\epsilon_{{\bf k}' n'\uparrow})+i \delta}, \label{generalchi01q} \end{eqnarray}\] where \({\bf k}' = {\bf q}+ {\bf k}\). Here we assume the response is stationary and system posseses translational symmetry. Within RPA, the full susceptibilities are calculated using \(\chi=\chi_0 + \chi_0 I \chi\). Within time-dependent LDA, the exchange correlation kernel can be explicitly calculated using \(I_{\rm xc}^{+-}={B_{\rm xc}({\bf r})}/{m}({\bf r})\) for a collinear spin structure. In general, we represent two-particle quantities \(\chi^{0}\), \(\chi\) and \(I\) using a mixed basis~ \(\{B_{i}({\bf r})\}\), which consists of the product basis~ within the augmentation spheres and interstitial plane waves. The product basis itself is constructed by the products of wavefunctions and their energy derivatives within the augmentation spheres. Since magnetic moment and response are nearly completely confined within the magnetic Cr sites, we further project \(\chi( \mathbf{q},\omega )\) onto the functions \(\{m_i({\bf r})/||m_i||\}\) representing normalized local spin densities on each magnetic site, which gives a matrix \(\chi _{ij}( \mathbf{q},\omega )\) in magnetic basis site indices~. This projection corresponds to the rigid spin approximation (RSA), which is a modest approximation for magnetic semiconductors such as \(\text{CrI}_3\).
The sum rule~ used can be written as \[\begin{equation} \frac{M({\bf r}_1)}{\omega} = \int_\Omega \,\mathrm{d}{\bf r}_2\ \chi^{+-} ({\bf r}_1,{\bf r}_2,{\bf q}=0,\omega), \label{eqsumm} \end{equation}\] where \(\Omega\) denotes the unit-cell volume. Within RSA, reduce to two equations for each \(\omega\). We further assume that kernel \(I(\omega)\) is site-diagonal and \({\bf q}\)-independent, allowing it to be determined solely by . Note that ensures the Goldstone magnon mode at \({\bf q}=0\) and \(\omega=0\).
- Pair exchange parameters for Heisenberg model
In adiabatic approximation we map spin susceptibility into a classical Heisenberg model \[\begin{equation} H=-\sum_{i\ne j} J_{ij}\,\hat{\bf e}_i \cdot \hat{\bf e}_j, \label{eq:Hamiltonian} \end{equation}\] where \(\hat{\bf e}_i\) is the unit vector of magnetic moment on site \(i\). Effective pair exchange parameters are calculated as \[\begin{equation} J_{ij} = \frac{1}{4}\int_{\Omega_i} \,\mathrm{d}{{\bf r}_1} \int_{\Omega_j} \,\mathrm{d}{{\bf r}_2}\, m_i({{\bf r}_1}) J({{\bf r}_1},{{\bf r}_2}) m_j({\bf r}_2), \end{equation}\] where \(m_i({{\bf r}})\) is the density of magnetic moment on Cr site \(i\) and \(J({\bf r}_1,{\bf r}_2)\) is related to \(\chi^{+-}({\bf r}_1,{\bf r}_2)\) and satisfies \[\begin{equation} \int_{\Omega} \,\mathrm{d}{{\bf r}_2}\, J({\bf r}_1,{\bf r}_2) \chi^{+-}({\bf r}_2,{\bf r}_3,\omega=0) = \delta({\bf r}_1-{\bf r}_3). \end{equation}\] Thus, the effective pair exchange parameters can be obtained from the matrix elements of the inverse of spin susceptibilities matrix~ with a subsequent Fourier transform, \[\begin{eqnarray} &&J_{ij}({\bf R})= \frac{1}{\Omega _{\rm BZ}} \int \,\mathrm{d}\mathbf{q}\, e^{i2\pi\mathbf{ q\cdot R}} J_{ij}(\mathbf{q}) \\ &=&\frac{1}{\Omega _{\rm BZ}} \int \,\mathrm{d}\mathbf{q}\, e^{i2\pi\mathbf{ q\cdot R}} ||m_i|| \left[ \left( \chi^{+-}(\mathbf{q},\omega=0)\right)^{-1}\right]_{ij} ||m_j||. \nonumber \end{eqnarray}\] Here, the \(\chi_{ij}^{+-}\) matrices are obtained by projecting onto the functions \(\{m_i({\bf r})/||m_i||\}\) representing normalized local spin densities on each magnetic Cr site, which gives a matrix \(\chi _{ij}( \mathbf{q},\omega )\) in magnetic basis site indices~. This projection corresponds to the rigid spin approximation (RSA).
- Others
Relevant publications:
(Ke and Katsnelson 2021; Li et al. 2021) (Ke and Schilfgaarde 2012) (Ke et al. 2011)