Exchange interactions from a nonorthogonal basis set

Eötvös Loránd University

Centre For Energy Research

Zoltán Tajkov

The Team

  • László Oroszlány, Zoltán Tajkov, János Koltai, Dániel Pozsár, Andor Kormányos, András Balogh, Tamás Véber, Marcell Sipos
     
  • Jaime Ferrer, Amador Garcia Fuente, Gabriel Martinez-Carracedo, Aurelio Hierro Rodriguez, Balázs Nagyfalusi, Rosa Eulalia González Ferreras
     
  • Felix Büttner, Kai Litzius, Steffen Wittrock
     
  • Efren Navarro-Moratalla, Marta Galbiati, Jose Joaquin Perez Grau
     
  • László Szunyogh, László Udvardi, Bendegúz Nyári, Anjali Jyothi Bhasu 
\mathcal{H}=-\frac{1}{2}\displaystyle\sum_{i\ne j}J_{ij}\,\vec{e}_{i}\vec{e}_{j}
E_{ij}^{\mathrm{int}}=\delta E(\vec{e}_{i},\vec{e}_{j})-\delta E(\vec{e}_{i})-\delta E(\vec{e}_{j})=-J_{ij}\,\delta\vec{e}_{i}\delta\vec{e}_{j}
\delta E_{\text{KS},ij}=\frac{1}{\pi}\displaystyle\int\limits _{-\infty}^{\varepsilon_\text{F}}\mathrm{d}\varepsilon\,\text{ImTr}\left(\delta\hat{V_i}\hat{G}(\varepsilon)\delta\hat{V_j}\hat{G}(\varepsilon)\right)

Heisenberg model and DFT perturbation theory

DFT through

RKKR

&

Liechtenstein, Katsnelson , Antropov,  Gubanov

J. Magn. Magn. Mater. 67 65 (1987)

 

Oroszlány, Ferrer, Deák, Udvardi, Szunyogh
Phys. Rev. B 99, 224412  (2019)

What is \(\delta \hat{V}_i\) ?

\left(\begin{array}{cc} V_{AA} & 0\\ 0 & 0 \end{array}\right),\ \text{vs.}\ \left(\begin{array}{cc} V_{AA} & V_{AR}/2\\ V_{RA}/2 & 0 \end{array}\right)

3) The definition of local operator in a non-orthogonal basis needs a pragmatic choice!

1) We need to rotate the magnetic moment!

2) We need to identify the magnetic entity!

Could be:

  • Single atom
  • Cluster of atoms
  • Certain orbitals inside an atom

Relativistic magnetic model parameters

\mathcal{H}=\frac{1}{2}\sum_{i\neq j}J_{ij}^{H}\boldsymbol{e}_{i}\cdot\boldsymbol{e}_{j}+\frac{1}{2}\sum_{i\neq j}\boldsymbol{e}_{i}\hat{J}_{ij}^{S}\boldsymbol{e}_{j}+\frac{1}{2}\sum_{i\neq j}\boldsymbol{D}_{ij}\cdot\left(\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}\right)+\sum_{i}\boldsymbol{e}_{i}\hat{K}_{i}\boldsymbol{e}_{i}

Udvardi, Szunyogh, Palotás, Weinberger

Phys. Rev. B 68, 104436 (2003)

 

Martínez-Carracedo, Oroszlány, García-Fuente, Nyári, Udvardi, Szunyogh, Ferrer
Phys. Rev. B 108, 214418  (2023)

Istropic

exchange

Symmetric  traceless exchange

Dzyaloshinskii - Moriya vector

On-site

anisotropy

Grogu

Fe, Co, Ni, KKR vs SIESTA

Oroszlány, Ferrer, Deák, Udvardi, Szunyogh;

Phys. Rev. B 99, 224412 ( 2019)

SKKR SIESTA
Fe 2.365 2.356
Co 1.542 1.580
Ni 0.675 0.626

\(\mu / \mu_B\)

\mathcal{H}=-\frac{1}{2}\displaystyle\sum_{i\ne j}J_{ij}\,\vec{e}_{i}\vec{e}_{j}
SKKR TB-LMTO SIESTA Experiment
bcc Fe 1478 1414 1330 1044-1045
hcp Co 1504  1645 1490  1388-1398
fcc Ni 348  397  389 624-631
T_C [\text{C}^\circ]

C\(_2\)F with SIESTA

A. N. Rudenko et al. 

Phys. Rev. B 88, 081405 (2013)

Oroszlány, Ferrer, Deák, Udvardi, Szunyogh; Phys. Rev. B 99, 224412 ( 2019)

Cr trimer on top of Au [111]

hℎhitalic_h (Å) JHsuperscript𝐽𝐻J^{H}italic_J start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT D12zsubscriptsuperscript𝐷𝑧12D^{z}_{12}italic_D start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT D12ysubscriptsuperscript𝐷𝑦12D^{y}_{12}italic_D start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT
2.83 177 / 157 -3 / -2.5 0.8 / 0.6
2.36 159 / 143 0.1 / 0.3 -1.9 / -2
2.06 145 / 131 1.2 / 0.8 -7.2 / -7

[meV]


Martínez-Carracedo, Oroszlány, García-Fuente, Nyári, Udvardi, Szunyogh, Ferrer

Phys. Rev. B 108, 214418  (2023)

 CrGeX\(_3\) (X = Se,Te) and Janus Cr\(_2\)Ge\(_2\)(Se,Te)\(_3\) monolayers

Phys. Rev. B 110, 184406 (2024)

CGS CGT CGST
Lattice constant (Å) 6.44 6.98 6.71
Spin moment (μBsubscript𝜇𝐵\mu_{B}italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT) 3.531 3.749 3.643
Orbital moment (μBsubscript𝜇𝐵\mu_{B}italic_μ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT) 0.017 0.035 0.025
MAE (meV/Cr) -0.05 0.80 0.15
Band gap (eV) 0.7 0.2 0.3
CGS CGT CGST
Kcx⁢x−Kcz⁢zsuperscriptsubscript𝐾𝑐𝑥𝑥superscriptsubscript𝐾𝑐𝑧𝑧K_{c}^{xx}-K_{c}^{zz}italic_K start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x italic_x end_POSTSUPERSCRIPT - italic_K start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z italic_z end_POSTSUPERSCRIPT -0.03 -0.47 -0.15
Jc⁢1Hsubscriptsuperscript𝐽𝐻𝑐1J^{H}_{c1}italic_J start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c 1 end_POSTSUBSCRIPT -0.86 -7.04 -4.38
Jc⁢1x⁢x−Jc⁢1z⁢zsuperscriptsubscript𝐽𝑐1𝑥𝑥superscriptsubscript𝐽𝑐1𝑧𝑧J_{c1}^{xx}-J_{c1}^{zz}italic_J start_POSTSUBSCRIPT italic_c 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x italic_x end_POSTSUPERSCRIPT - italic_J start_POSTSUBSCRIPT italic_c 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z italic_z end_POSTSUPERSCRIPT -0.01 -0.06 0.03
Jc⁢1y⁢y−Jc⁢1z⁢zsuperscriptsubscript𝐽𝑐1𝑦𝑦superscriptsubscript𝐽𝑐1𝑧𝑧J_{c1}^{yy}-J_{c1}^{zz}italic_J start_POSTSUBSCRIPT italic_c 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_y italic_y end_POSTSUPERSCRIPT - italic_J start_POSTSUBSCRIPT italic_c 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z italic_z end_POSTSUPERSCRIPT 0.09 0.72 0.58
Sc⁢1xsubscriptsuperscript𝑆𝑥𝑐1S^{x}_{c1}italic_S start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c 1 end_POSTSUBSCRIPT -0.01 -0.02 -0.04
Dc⁢1ysuperscriptsubscript𝐷𝑐1𝑦D_{c1}^{y}italic_D start_POSTSUBSCRIPT italic_c 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT - - 0.70
Dc⁢1zsuperscriptsubscript𝐷𝑐1𝑧D_{c1}^{z}italic_D start_POSTSUBSCRIPT italic_c 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT - - -0.19
Jc⁢2Hsubscriptsuperscript𝐽𝐻𝑐2J^{H}_{c2}italic_J start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c 2 end_POSTSUBSCRIPT 0.28 0.19 0.72
Jc⁢2x⁢x−Jc⁢2z⁢zsuperscriptsubscript𝐽𝑐2𝑥𝑥superscriptsubscript𝐽𝑐2𝑧𝑧J_{c2}^{xx}-J_{c2}^{zz}italic_J start_POSTSUBSCRIPT italic_c 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x italic_x end_POSTSUPERSCRIPT - italic_J start_POSTSUBSCRIPT italic_c 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z italic_z end_POSTSUPERSCRIPT <<<0.01 -0.02 -0.04
Jc⁢2y⁢y−Jc⁢2z⁢zsuperscriptsubscript𝐽𝑐2𝑦𝑦superscriptsubscript𝐽𝑐2𝑧𝑧J_{c2}^{yy}-J_{c2}^{zz}italic_J start_POSTSUBSCRIPT italic_c 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_y italic_y end_POSTSUPERSCRIPT - italic_J start_POSTSUBSCRIPT italic_c 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z italic_z end_POSTSUPERSCRIPT -0.01 -0.05 -0.04
Dc⁢2xsuperscriptsubscript𝐷𝑐2𝑥D_{c2}^{x}italic_D start_POSTSUBSCRIPT italic_c 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT -0.16 -0.19 -0.16
Dc⁢2ysuperscriptsubscript𝐷𝑐2𝑦D_{c2}^{y}italic_D start_POSTSUBSCRIPT italic_c 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT - - 0.07
Dc⁢2zsuperscriptsubscript𝐷𝑐2𝑧D_{c2}^{z}italic_D start_POSTSUBSCRIPT italic_c 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT -0.19 -0.42 -0.25
Jc⁢3Hsubscriptsuperscript𝐽𝐻𝑐3J^{H}_{c3}italic_J start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c 3 end_POSTSUBSCRIPT 0.30 -0.25 0.24
Jc⁢3x⁢x−Jc⁢3z⁢zsuperscriptsubscript𝐽𝑐3𝑥𝑥superscriptsubscript𝐽𝑐3𝑧𝑧J_{c3}^{xx}-J_{c3}^{zz}italic_J start_POSTSUBSCRIPT italic_c 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x italic_x end_POSTSUPERSCRIPT - italic_J start_POSTSUBSCRIPT italic_c 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z italic_z end_POSTSUPERSCRIPT <<<0.01 -0.11 -0.12
Jc⁢3y⁢y−Jc⁢3z⁢zsuperscriptsubscript𝐽𝑐3𝑦𝑦superscriptsubscript𝐽𝑐3𝑧𝑧J_{c3}^{yy}-J_{c3}^{zz}italic_J start_POSTSUBSCRIPT italic_c 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_y italic_y end_POSTSUPERSCRIPT - italic_J start_POSTSUBSCRIPT italic_c 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z italic_z end_POSTSUPERSCRIPT -0.02 0.04 -0.05
Sc⁢3xsubscriptsuperscript𝑆𝑥𝑐3S^{x}_{c3}italic_S start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c 3 end_POSTSUBSCRIPT -0.02 -0.14 -0.10
Dc⁢3ysuperscriptsubscript𝐷𝑐3𝑦D_{c3}^{y}italic_D start_POSTSUBSCRIPT italic_c 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT - - -0.02
Dc⁢3zsuperscriptsubscript𝐷𝑐3𝑧D_{c3}^{z}italic_D start_POSTSUBSCRIPT italic_c 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT - - -0.28
Jc⁢4Hsubscriptsuperscript𝐽𝐻𝑐4J^{H}_{c4}italic_J start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c 4 end_POSTSUBSCRIPT 0.21 0.19 0.24

[meV]

 CrGeX\(_3\) (X = Se,Te) and Janus Cr\(_2\)Ge\(_2\)(Se,Te)\(_3\) monolayers

Phys. Rev. B 110, 184406 (2024)

Triangulene chains  as S=1 Haldane chain

Mishra et al. Nature 598, 287 (2021).

fit to BL-BQ model:

J=18 \mathrm{meV}\, ,\beta=0.09

Beyond two spin models

Dimer Infinite chain Experiment
17.7 meV 19.75 meV 18 meV
0.03 0.05 0.09

\( \beta \)

\( J \)

\displaystyle \hat H = \sum_{nm} {\color{red}J_{nm}} \left (\vec{S}_n \cdot \vec{S}_{m} + {\color{red}\beta_{nm}} (\vec{S}_n \cdot \vec{S}_{m})^2 \right )

Martínez-Carracedo, Oroszlány, García-Fuente, Szunyogh,  Ferrer

Phys. Rev. B 107, 035432 (2023)

Mishra et al.

Nature 598, 287 (2021).

Relativistic magnetic interactions

  • Very early release !!
    • https://github.com/danielpozsar/grogu
  • Single DFT calculation
  • Pair creation is extremely cheap
  • parallel BZ integral with MPI or CUDA
  • Generalised Heisenberg model
H(\{\mathbf{S}_i\}) = \frac{1}{2} \sum_{i \neq j} \mathbf{S}_i \mathcal{J}_{ij} \mathbf{S}_j + \sum_i \mathbf{S}_i K_i \mathbf{S}_i

UNDER 1 Hour on 8 GPUs