Graphene triangulenes embedded in hexagonal

boron nitride 

Eötvös Loránd University

Dániel Pozsár

László Oroszlány, Viktor Ivády

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 

Antiferromagnetic spin-1 chains

  • Haldane gap

  • BLBQ nearest neighbor model

  • Measurement based quantum computing
\hat{H}_{BLBQ} = J \sum_{i=1}^{N-1} \left[ \hat{\mathbf{S}}_i \cdot \hat{\mathbf{S}}_{i+1} + \beta \left( \hat{\mathbf{S}}_i \cdot \hat{\mathbf{S}}_{i+1} \right)^2 \right]

Martínez-Carracedo, Gabriel, et al. "Electrically driven singlet-triplet transition in triangulene spin-1 chains." Physical Review B 107.3 (2023): 035432.

Mishra, Shantanu, et al. "Observation of fractional edge excitations in nanographene spin chains." Nature 598.7880 (2021): 287-292.

Determination of ground state spin in triangulenes

Angew. Chem. Int. Ed. 10.1002/anie.23783​

  • Lieb's theorem
  • Ovchinnikov's rule for bipartite lattices

​

S = \frac{1}{2} \vert A - B \rvert

Embedding in hexagonal boron nitride

Park, Hyoju, et al. "Atomically precise control of carbon insertion into hBN monolayer point vacancies using a focused electron beam guide." Small 17.23 (2021): 2100693.

Dai, Chunhui, et al. "Evolution of nanopores in hexagonal boron nitride." Communications chemistry 6.1 (2023): 108.

Result of the embedding process

What is a localised magnetic entity?

DFT perturbation theory

H_{ii}^{\sigma} = \frac{H_{ii}^{\uparrow} - H_{ii}^{\downarrow}}{2}
D_{nm} = \frac{2}{\pi} \int_{-\infty}^{\varepsilon_F} d\varepsilon \text{Im} \, \text{Tr}_{L} \left[ H_{nn}^{\sigma} G_{nm}^{\dagger}(\varepsilon) H_{mm}^{\sigma} G_{mn}^{\dagger}(\varepsilon) \right]
\delta E_{nm}^{(2)} = J_{nm} \left[ 1 + 2\beta_{nm} \left( \boldsymbol{S}_n \cdot \boldsymbol{S}_m \right) \right] \delta \boldsymbol{S}_n \cdot \delta \boldsymbol{S}_m
\hat{H}_{BLBQ} = J \sum_{i=1}^{N-1} \left[ \hat{\mathbf{S}}_i \cdot \hat{\mathbf{S}}_{i+1} + \beta \left( \hat{\mathbf{S}}_i \cdot \hat{\mathbf{S}}_{i+1} \right)^2 \right]

Oroszlány, László, et al. "Exchange interactions from a nonorthogonal basis set: From bulk ferromagnets to the magnetism in low-dimensional graphene systems." Physical Review B 99.22 (2019): 224412.

Perturbation of classical Hamiltonian

Energy of infinitesimal rotations from Kohn-Sham Hamiltonian

\delta E_{nm}^{(2)} = D_{nm}^{(2)} \delta\mathbf{S}_n \cdot \delta\mathbf{S}_m

Collinear DFT:

Magnetic force theorem:

Modelling infinite chains

Experiment Triangulene in hBN Triangulene in vacuum Tetramer in hBN
18 meV 20.22 meV 19.75 meV 12.54 meV
0.09 0.01 0.05 0.21
J
\beta

Mishra, Shantanu, et al. "Observation of fractional edge excitations in nanographene spin chains." Nature 598.7880 (2021): 287-292.

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

\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

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!

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

\( K^{xx} \)-\( K^{zz} \)=0.31 meV

CrI\(_3\) benchmarks