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

\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!