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: