Electrically driven singlet-triplet transition in triangulene spin-1 chains

Recently, graphene triangulene chains have been synthesized, and their magnetic response has been analyzed by scanning tunneling microscopy methods by Mishra et al. [Nature (London) 598, 287 (2021)]. Motivated by this study, we determine the exchange bilinear and biquadratic constants of the triangulene chains by calculating two-spin rotations in the spirit of the magnetic force theorem. We then analyze open-ended, odd-numbered chains, whose edge states pair up forming a triplet ground state. We propose three experimental approaches that enable us to trigger and control a singlet-triplet spin transition. Two of these methods are based on applying a mechanical distortion to the chain. We finally show that the transition can be controlled efficiently by the application of an electric field.

Figure 1

(a) Sketch of a horseshoelike $N=5$ GT chain, where the spins at each triangulene pair are coupled by the same exchange constants $J$ and $\beta$, and where the two end GT spins are coupled by a smaller exchange constant $J_{1N}$. (b) A singlet-triplet crossing occurs at a finite value of $J_{1N}/J$ smaller than 1.

Figure 2

(a) $J_{1N}^c/J$ as a function of $N$ for several values of $\beta$. (b) $J_{1N}^c/J$ as a function of $\beta$ for chains having different lengths.

Figure 3

(a) Lowest-lying energy states as a function of $d_{\text{H-H}}$ for a $N=11$ chain. The plot shows that the singlet and triplet edge states lie inside the Haldane gap. Inset: Blowup of the energy axis to highlight the singlet-triplet energy splitting. (b) $J_{1N}/J$ as a function of $d_{\text{H-H}}$. The horizontal line identifies the critical $J_{1N}$ separating the regions where the ground state is a triplet (pink) or a singlet (violet). Inset: Sketch of the horseshoe chain end where the distance $d_{\text{H-H}}$ is defined.

Figure 4

(a) and (b) Same as in Fig. 3, but where the horseshoe chain is linked by a sulfur atom.

Figure 5

(a) and (b) Same as in Fig. 3, but where in addition to a linking sulfur atom, an electric field $\mathcal{E}$ is being applied along the chain main axis.