Competition between different long-distance superexchange couplings in a quadripartite spin-crossover molecular device

Long-distance superexchange coupling provides a powerful tool for scalable quantum processors required in quantum information and quantum computation. Quadripartite spin-crossover molecules are prototypical systems for studying novel quantum phenomena assisted by superexchange interactions. In this paper, we consider a quadripartite molecule where two central monomers are connected to two electrodes, whereas another two are side coupled to their neighboring central monomers. Numerical renormalization group results demonstrate that, when the energy level spacing between two central monomers $\mathrm{\Delta }$ turns on, spins on the two side monomers are organized ferromagnetically. The effective antiferromagnetic superexchange interaction between the side units and the electrodes then induces a long-distance ferromagnetic Ruderman-Kittel-Kasuya-Yosida interaction between two side monomers. For intermediate $\mathrm{\Delta }$, the linear conductance tends to reach its unitary limit in a moderate low-temperature regime due to a superexchange two-stage Kondo effect. When the hopping integral between two central monomers sweeps upwards, where $\mathrm{\Delta }$ is fixed at a small value, a quantum phase transition occurs. A long-distance antiferromagnetic exchange coupling between two side units is clarified, isolating the molecule from the electrodes with zero conductance. If $\mathrm{\Delta }$ is set to be an intermediate value, applying the hopping integral then triggers a singular quantum phase, where two central monomers are antiparallelly aligned, whereas two side ones are organized ferromagnetically, revealing well the competition between the above two kinds of long-distance superexchange couplings. Temperature-dependent images and numerical simulations confirm the above conclusions.

Figure 1

Schematic illustration of the quadripartite spin-crossover molecule (QSCM) connected to the source (S) and drain (D) electrodes. Molecular monomers (MMs) 1 and 2 connect directly to the electrodes, while each side monomer ($\mathrm{MM}3$ or $\mathrm{MM}4$) only connects with its neighboring central monomer. ${\epsilon }_i$ is the energy level of the $i\mathrm{th}$ monomer, $U$ is the on-site electron-electron repulsion, and $\mathrm{\Gamma }$ is the hybridization function between the central monomers and the conduction bands. $t_1$ illustrates the interorbital hopping integral (tunneling coupling) between the two central monomers, and $t_2$ is that between the central monomer and its corresponding side unit.

Figure 2

(a) The electron occupation number on each monomer $\langle n_i\rangle$, (b) spin-spin correlation between different monomers $i$ and $j\langle {\mathbf{S}}_i{\mathbf{S}}_j\rangle$, and (c) linear conductance $G(T\approx 0)$ at nearly zero temperature as a function of $\mathrm{\Delta }$. Here, $\langle n_4\rangle$ and $\langle {\mathbf{S}}_2{\mathbf{S}}_4\rangle$ are not shown due to $\langle n_3\rangle =\langle n_4\rangle$ and $\langle {\mathbf{S}}_2{\mathbf{S}}_4\rangle =\langle {\mathbf{S}}_1{\mathbf{S}}_3\rangle$. The other parameters are given by $\mathrm{\Gamma }=0.02$, $U=0.5$, $V_g=-U/2$, ${\epsilon }_1=V_g+\mathrm{\Delta }/2$, ${\epsilon }_2=V_g-\mathrm{\Delta }/2$, ${\epsilon }_3={\epsilon }_4=V_g$, $t_1=0,$ and $t_2=0.001$.

Figure 3

(a)–(c) The transmission coefficient through the QSCM $C\left(\omega \right)$ at nearly zero temperature for various of $\mathrm{\Delta }$. (d) The local density of states (LDOS) of $\mathrm{MM}1$ ($\mathrm{MM}2$) $A_1\left(\omega \right)$ at nearly zero temperature for $\mathrm{\Delta }$ in different regimes. Inset in (a) and (d): $C\left(\omega \right)$ and $A_1\left(\omega \right)$ for $\mathrm{\Delta }=0.128$ in an enlarged scale, which shows zero spectral weight at the Fermi level. The remaining parameters are given the same as in Fig. 2.

Figure 4

Temperature-dependent (a) $G\left(T\right)$, (b) local magnetic moment ${\mu }^2\left(T\right)$, and (c) entropy $S_{mol}\left(T\right)$ for the QSCM as functions of temperature $T$ in terms of different $\mathrm{\Delta }$. Curves in (a) along the black arrow are for $\mathrm{\Delta }$ = 0, 0.304, 0.4, and 0.464, and curves along the olive arrow are for $\mathrm{\Delta }$ = 0.48, 0.528, and 0.8, respectively. (d) $\langle {\mathbf{S}}_i{\mathbf{S}}_j\rangle$ between MMs $i$ and $j$ versus ${\log }_{10}\left(T\right)$ for $\mathrm{\Delta }=0$, 0.464, and 0.8, respectively. Curves $1–3$ in panel (d) are for $\langle {\mathbf{S}}_1{\mathbf{S}}_2\rangle$, $1^{\prime }$–$3^{\prime }$ are for $\langle {\mathbf{S}}_1{\mathbf{S}}_3\rangle$, and $1^{″}$–$3^{″}$ are for $\langle {\mathbf{S}}_3{\mathbf{S}}_4\rangle$. Curves for each $\mathrm{\Delta }$ in the four panels are unified with the same color. The remaining parameters are the same as in Fig. 2 unless otherwise specified.

Figure 5

(a) $T_{K2}$ and (b) $T_{SERKKY}$ as functions of $\mathrm{\Delta }$. Scatter plots are captured from our NRG data, while the solid curves are the fitting functions. The other parameters are the same as in Fig. 2.

Figure 6

(a) $\langle {\mathbf{S}}_i{\mathbf{S}}_j\rangle$ at nearly zero temperature as functions of $t_1$. (b), (c) $C\left(\omega \right)$ at nearly zero temperature for various $t_1$. Inset in (b): $C\left(\omega \right)$ for $t_1$ = 0.0096 in an enlarged scale. The other parameters are given by $\mathrm{\Gamma }=0.02$, $U=0.5$, $\mathrm{\Delta }$ = 0, $V_g=-U/2$, ${\epsilon }_1=V_g+\mathrm{\Delta }/2$, ${\epsilon }_2=V_g-\mathrm{\Delta }/2$, ${\epsilon }_3={\epsilon }_4=V_g$, and $t_2=0.001$.

Figure 7

Temperature-dependent (a) $G\left(T\right)$, (b) ${\mu }^2\left(T\right)$, and (c) $S_{mol}\left(T\right)$ for the QSCM device in terms of different $t_1$. Curves in panels (a)–(c) are for $t_1=0$, 0.0168, 0.0216, 0.0336, and 0.0792, respectively. The remaining parameters are the same as in Fig. 6.

Figure 8

The temperature scale $T_{SSE}$, which refers to the spin singlet mediated long-distance antiferromagnetic exchange coupling between two side monomers $J_{SSE}$, as a function of $t_1$. Scatter plots are captured from our NRG data, and the solid curve is the fitting function. The other parameters are the same as in Fig. 6.

Figure 9

(a) $\langle n_i\rangle$, (b) $\langle {\mathbf{S}}_i{\mathbf{S}}_j\rangle$, and (c) $G(T\approx 0)$ at nearly zero temperature as functions of $t_1$ with fixed $\mathrm{\Delta }=0.428$. The other parameters are given the same as Fig. 6 unless otherwise specified.

Figure 10

Temperature-dependent (a) $G\left(T\right)$, (b) ${\mu }^2\left(T\right)$, and (c) $S_{mol}\left(T\right)$ of the QSCM device as functions of ${\log }_{10}\left(T\right)$ in terms of different $t_1$. Curves in these panels are for $t_1=0$, 0.0096, 0.0145, 0.0147, 0.0151, 0.0288, and 0.0408, respectively. (d) $\langle {\mathbf{S}}_i{\mathbf{S}}_j\rangle$ between $\mathrm{MM}i$ and $\mathrm{MM}j$ versus ${\log }_{10}\left(T\right)$ for $t_1=0.0096$, 0.0145, and 0.0288. Curves $1–3$ are for $\langle {\mathbf{S}}_1{\mathbf{S}}_2\rangle$, $1^{\prime }\text{–}{3}^{\prime }$ are for $\langle {\mathbf{S}}_1{\mathbf{S}}_3\rangle$, and $1^{″}\text{–}{3}^{″}$ are for $\langle {\mathbf{S}}_3{\mathbf{S}}_4\rangle$. Curves for each $t_1$ in these four panels are unified with the same color. The remaining parameters are the same as in Fig. 9.