Spin State Switching in Heptauthrene Nanostructure by Electric Field: Computational Study

Abstract

Recent experimental studies proved the presence of the triplet spin state in atomically precise heptauthrene nanostructure of nanographene type (composed of two interconnected triangles with zigzag edge). In the paper, we report the computational study predicting the possibility of controlling this spin state with an external in-plane electric field by causing the spin switching. We construct and discuss the ground state magnetic phase diagram involving S=1 (triplet) state, S=0 antiferromagnetic state and non-magnetic state and predict the switching possibility with the critical electric field of the order of 0.1 V/Å. We discuss the spin distribution across the nanostructure, finding its concentration along the longest zigzag edge. To model our system of interest, we use the mean-field Hubbard Hamiltonian, taking into account the in-plane external electric field as well as the in-plane magnetic field (in a form of the exchange field from the substrate). We also assess the effect of uniaxial strain on the magnetic phase diagram.

Keywords: Hubbard model; graphene magnetism; heptauthrene; magnetic phase diagram; nanographene; spin switching.

Figure 1

(a) View of the full heptauthrene nanostructure, consisting of carbon atoms (larger balls) and hydrogen atoms (smaller balls) [70]; (b) schematic view of the heptauthrene nanostructure with carbon atom positions marked with circles. Open and filled circles indicate the majority and minority carbon sublattice, respectively. Two triangles connected with a central dimer of carbon atoms are encircled with dashed lines. The labels of carbon sites at the longest zigzag edge are shown. The directions of in-plane electric fields and in-plane magnetic field are marked.

Figure 2

The ground-state magnetic phase diagram for the electric fields of magnitude not exceeding 1 V/Å and arbitrary in-plane orientation. Four values of exchange energy are selected: 0 meV (a), 100 meV (b), 150 meV (c), and 200 meV (d). The borders of phases with various values of total spin S are marked with solid lines. The dashed lines delimit the antiferromagnetic (AFM) and non-magnetic (NM) phase for $$S = 0$$.

Figure 3

The phase diagram showing the borders between the state with $$S = 1$$ and the state with $$S = 0$$ as a function of the electric field along the x-direction and along the y-direction, for three values of exchange energy.

Figure 4

The phase diagram showing the borders between the state with $$S = 1$$ and the state with $$S = 0$$ as a function of the exchange energy and electric field along the x-direction (a) and along the y-direction (b), for various values of electric field along the other direction.

Figure 5

The phase diagram showing the borders between the state with $$S = 1$$ and the AFM and NM states with $$S = 0$$ as a function of the electric field along the x-direction and along the y-direction, for four values of exchange energy: 0 meV (a); 20 meV (b); 50 meV (c); and 100 meV (d). The solid lines delimit $$S = 1$$ and $$S = 0$$ NM state; the dashed lines delimit $$S = 1$$ and $$S = 0$$ AFM state; the dotted lines delimit $$S = 0$$ AFM and NM state.

Figure 6

The dependence of the critical electric field along the given direction on the strain along the same direction: (a) x direction and (b) y direction, in the absence of other fields. The critical field corresponds to switching between $$S = 1$$ and $$S = 0$$ state.

Figure 7

Spin density distribution across the nanostructure for various values of the electric field, in the absence of the exchange energy: $$E_x = 0$$, $$E_y = 0$$ (a); $$E_x = 0.4$$ V/Å, $$E_y = 0$$ (b); $$E_x = 0.3$$ V/Å, $$E_y = 0$$ (c); $$E_x = 0.3$$ V/Å, $$E_y = 0.05$$ V/Å (d). Red and blue colours mark two antiparallel spin orientations. The radius of the circle is proportional to the spin density (the scale is shown at the bottom of the plot).

Figure 8

Spin distribution along the longest zigzag edge of the nanostructure in the absence of the exchange energy, for electric field $$E_y =$$ 0 and a few selected values of $$E_x$$ (a) and for electric field $$E_x =$$ 0.3 V/Å and a few selected values of $$E_y$$ (b). Lines are guides to eyes only.

Figure 9

Total spin of the nanostructure and spins of sublattices and triangles as a function of: (a) electric field $$E_x$$ for $$E_y = 0$$ and $$\Delta = 0$$; (b) electric field $$E_x$$ for $$E_y = 0$$ and $$\Delta = 100$$ meV; (c) electric field $$E_y$$ for $$E_x = 0.3$$ V/Å and $$\Delta = 0$$; (d) exchange energy $$\Delta$$ for $$E_x = 0.4$$ V/Å and $$E_y = 0$$. Vertical dotted lines mark the points at which the state changes.

Figure 10

Single-electron eigenenergies as a function of: (a) electric field $$E_x$$ for $$E_y = 0$$ and $$\Delta = 0$$; (b) electric field $$E_x$$ for $$E_y = 0$$ and $$\Delta = 100$$ meV; (c) electric field $$E_y$$ for $$E_x = 0.3$$ V/Å and $$\Delta = 0$$; (d) exchange energy $$\Delta$$ for $$E_x = 0.4$$ V/Å and $$E_y = 0$$. Vertical dotted lines mark the points at which the state changes. The solid lines denote the states occupied by electron, while the dashed lines mark empty states. The arrows indicate the spin direction of the state. The blue color marks the lower energy and the red colour the higher energy state for the given spin orientation. Only two occupied states highest in energy and two empty states lowest in energy are shown.