Relativistic Magnetic Interactions from a Nonorthogonal Basis Set

Eötvös Loránd University

Wigner Research Centre for Physics

László Oroszlány 

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{int}}_{\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)

Single collinear scf calculation needed!

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
E_{ij}^{\mathrm{int}}+\delta E(\vec{e}_{i})+\delta E(\vec{e}_{j})=\delta E(\vec{e}_{i},\vec{e}_{j})=0

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

Multiple collinear reference states needed!

Single collinear scf calculation needed!

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]

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

"simple" projection

"complicated"

projection

[meV]

I. V. Solovyev Phys. Rev. B 107, 054442 (2023)

meV J DM
1nn -0.34 0
2nn -1.14 0.32'
3nn 0.65 0

\( K^{xx} \)-\( K^{zz} \)=0.47 meV

CrI\(_3\) benchmarks

Experiment:  Huang et al. Nature  546,270 (2017): 45K

CrI\(_3\) Monte-Carlo simulation

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

Phys. Rev. B 110, 184406 (2024)

[meV]

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

Phys. Rev. B 110, 184406 (2024)

Beyond two spin interaction: 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

Bilinear-biquadratic exchange

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).

FM and AFM reference is needed!

joining through S \(\Rightarrow\) dipole  

Singlet-Triplet transition through electric field

Get Grogu(py)!

  • Very early release !!
    • https://github.com/danielpozsar/grogu
  • post. proc. for single DFT calculation
  • 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

Number of orbitals: 8276

UNDER 1 Hour on 8 GPUs

pip install grogupy

GROGU-showcase

By László Oroszlány

GROGU-showcase

  • 223

More from László Oroszlány