Spin electric effects in molecular antiferromagnets

Molecular nanomagnets show clear signatures of coherent behavior and have a wide variety of effective low-energy spin Hamiltonians suitable for encoding qubits and implementing spin-based quantum information processing. At the nanoscale, the preferred mechanism for the control of a quantum systems is the application of electric fields, which are strong, can be locally applied, and rapidly switched. In this work, we provide the theoretical tools for identifying molecular nanomagnets suitable for electric control. By group-theoretical symmetry analysis we find that the spin-electric coupling in triangular molecules is governed by the modification of the exchange interaction and is possible even in the absence of spin-orbit coupling. In pentagonal molecules the spin-electric coupling can exist only in the presence of spin-orbit interaction. This kind of coupling is allowed for both $s=1∕2$ and $s=3∕2$ spins at the magnetic centers. Within the Hubbard model, we find a relation between the spin-electric coupling and the properties of the chemical bonds in a molecule, suggesting that the best candidates for strong spin-electric coupling are molecules with nearly degenerate bond orbitals. We also investigate the possible experimental signatures of spin-electric coupling in nuclear magnetic resonance and electron spin resonance spectroscopy, as well as in the thermodynamic measurements of magnetization, electric polarization, and specific heat of the molecules.

Figure 1

(Color online) Schematics of the $s_i=1∕2$ triangular molecule in electric field. The antiferromagnetic exchange couplings, represented by the bonds with thickness proportional to $J_{ii+1}$, are modified in electric field. In the absence of electric field, exchange couplings are equal $J_{ii+1}=J_{jj+1}$, here represented in gray (fade blue online). The black (blue online) triangle represents the exchange interaction strengths in electric field.

Figure 2

(Color online) The spin transitions in the $s_i=1∕2$ triangle induced by electric and magnetic fields. The electric field causes transitions between the states of opposite chiralities $C_z$ and equal spin projections $S_z$ (horizontal arrows), while the magnetic field instead causes transitions between the states of opposite spin projections $S_z$ and equal chiralities $C_z$ (vertical arrows).

Figure 3

(Color online) Schematics of a pentagonal spin ring molecule in electric field $\mathbf{E}$, light (green) arrow. The molecule in the absence of electric field is depicted in fade colors, while the full colors represent the molecules in electric field. Thickness of the bonds represents the strength of antiferromagnetic exchange interaction between the spins. An electric field modifies the strengths of spin exchange couplings $J_{ii+1}$.

Figure 4

(Color online) Dipole moment of an unperturbed bridge. In the first order, external electric fields couple to the electric dipole moments of the bridges that connect the spins at magnetic centers. The different orientations of the bridge dipole moments ${\mathbf{d}}_{ij}\ne {\mathbf{d}}_{kl}$ lead to inhomogeneous variations of the resulting exchange interactions between the spins $\delta J_{ij}\ne \delta J_{kl}$.

Figure 5

(Color online) Spin Hamiltonian limit. Expectation values of chirality $⟨C_z⟩$ (full lines) and the their bounds of uncertainty $⟨C_z⟩\pm \Delta C_z$ (dotted lines), see text, in the low-energy states of the Hubbard model as a function of the on-site repulsion $U$ at the fixed hopping matrix element $t=1$ (left scale). The dashed line shows dependence of the double occupancy probability $P_{\mathrm{D}}$ in the ground state on the right scale. The spin Hamiltonian description becomes accurate in the $U\to \infty$ limit. The approach to this limit is slow, and the double occupancy probability is proportional to $t∕U$.

Figure 6

(Color online) Geometry of the bond and reduction of symmetry. (a) Electric field $\mathbf{E}$ in the $y$ direction, leaves the $C_{2v}$ symmetry unbroken. (b) An electric field $\mathbf{E}$ in the $z$ direction, normal to the bond plane, reduces the symmetry to $\{E,{\sigma }_v\}$. (c) An electric field $\mathbf{E}$ in the $x$ direction, along the line connecting the magnetic centers, reduces the symmetry to $\{E,{\sigma }_h\}$. (d) In an inhomogeneous staggered electric field $\mathbf{E}$, the reduced symmetry group is $\{E,R_{y,\pi }\}$.

Figure 7

(Color online) Energy $\left(\omega \right)$ of the ESR transitions in a triangle of $s=1∕2$ spins as a function of the applied electric field $\mathbf{E}$ that lies in the molecule’s plane so that $d\left|\mathbf{E}\right|=dE=\mathcal{E}$. The magnetic field is $\mathbf{B}\parallel \widehat{\mathbf{z}}$ and ${\omega }_0=g{\mu }_{B}B$, see Eqs. (120, 121). The diameter of the circles is proportional to the transition amplitudes $\left|⟨\alpha |S_x|{\alpha }^{\prime }⟩\right|$, Eqs. (118, 119). Here, $|\alpha ⟩$ are the eigenstates of $H$ in the lowest energy $S=1∕2$ multiplet. Inset: eigenvalues (in units of $D_z$) as a function of $\mathcal{E}=d\left|{\mathbf{E}}_{\parallel }\right|$, in units $3D_z∕4$.

Figure 8

(Color online) Energy $\left(\omega \right)$ of the ESR transitions in a triangle of $s=1∕2$ spins as a function of the applied in-plane electric field $\mathbf{E}$ so that $d\left|\mathbf{E}\right|=dE=\mathcal{E}$, and in the presence of the in-plane magnetic field $\mathbf{B}\parallel \widehat{\mathbf{x}}$. The diameter of the circles is proportional to $\left|⟨\alpha \left|S_z\right|{\alpha }^{\prime }⟩\right|$, Eqs. (120, 121). The states $|\alpha ⟩$ are the eigenstates of $H$ in the lowest $S=1∕2$ multiplet. Inset: eigenvalues (in units of $D_z$) as a function of $d\left|{\mathbf{E}}_{\parallel }\right|=\mathcal{E}$, in units $3D_z∕4$.

Figure 9

(Color online) Energy $\left(\omega \right)$ of the ESR transitions in a pentagon of $s=1∕2$ spins as a function of the electric field applied in the molecule’s plane $d\left|\mathbf{E}\right|=dE=\mathcal{E}$. The Zeeman splitting, ${\omega }_0=g{\mu }_{B}B$ is set by the magnetic field $\mathbf{B}\parallel \widehat{\mathbf{z}}$, orthogonal to the molecule’s plane. The considered transitions are those between eigenstates $\left(|\alpha ⟩\right)$ belonging to the $S=1∕2$ multiplet of the spin Hamiltonian (figure inset). Unlike the case of the spin triangle, these are coupled to each other by the electric field via eigenstates belonging to other multiplets, and therefore depends also on the exchange constant $J$ (here $J∕{\Delta }_{\mathrm{SO}}=100$). The diameter of the circles is proportional to $\left|⟨\alpha |S_x|{\alpha }^{\prime }⟩\right|$, and therefore to the transition amplitude.

Figure 10

(Color online) Energy $\left(\omega \right)$ of the ESR transitions in a pentagon of $s=1∕2$ spins as a function of the applied in-plane electric field $\mathbf{E}$ so that $d\left|\mathbf{E}\right|=dE=\mathcal{E}$. The Zeeman splitting is set by an in-plane magnetic field $\mathbf{B}\parallel \widehat{\mathbf{x}}$, and ${\omega }_0=g{\mu }_{B}B$. The considered transitions are those between eigenstates $\left(|\alpha ⟩\right)$ belonging to the $S=1∕2$ multiplet of the spin Hamiltonian (figure inset). Unlike the case of the spin triangle, these are coupled to each other by the electric field via eigenstates belonging to other multiplets and therefore depends also on the exchange constant $J$ (here $J∕{\Delta }_{\mathrm{SO}}=100$). The diameter of the circles is proportional to $\left|⟨\alpha |S_z|{\alpha }^{\prime }⟩\right|$, and therefore to the transition amplitude.

Figure 11

(Color online) Expectation values of the $z$ component of $s=1∕2$ spins in a triangular molecule as a function of applied electric field. The magnetic field is perpendicular to the ring plane $(\mathbf{B}\parallel \widehat{\mathbf{z}})$; the electric field is parallel and perpendicular to ${\mathbf{r}}_{12}$ in the upper and lower panel, respectively. In the electric field along one of the bonds (lower panel), the spins that lie on that bond have the same out-of-plane projections. The shadings (colors online) denote the different spins.

Figure 12

(Color online) Expectation values of the $z$ component of $s=3∕2$ spins in a triangular molecule as a function of applied electric field. The magnetic field is perpendicular to the ring plane $(\mathbf{B}\parallel \widehat{\mathbf{z}})$; the electric field is parallel and perpendicular to ${\mathbf{r}}_{12}$ in the upper and lower panels, respectively. The shadings denote the different spins.

Figure 13

(Color online) Expectation values of the $z$ component of $s=1∕2$ spins in a pentagon as a function of applied electric field. The magnetic field is perpendicular to the ring plane $(\mathbf{B}\parallel \widehat{\mathbf{z}})$; the electric field is parallel $(\theta =0)$ and perpendicular $(\theta =\pi ∕2)$ to ${\mathbf{r}}_{12}$ in the upper and lower panel, respectively. The shadings (colors online) denote the different spins.

Figure 14

(Color online) Electric polarization $P_x$ ($x$ component) in Eq. (134) as a function of the magnetic field in the $x$ direction. The three lines correspond to various values of an additional external electric field in the $z$ direction. The plot is for the temperature $k_{\mathrm{B}}T=0.001{\Delta }_{\mathrm{SO}}$, and the electric field $d{E}_x=0.1{\Delta }_{\mathrm{SO}}$.

Figure 15

(Color online) Electric polarization $P_x$ ($x$ component) in Eq. (134) as a function of the magnetic field in the $z$ direction. The three lines correspond to various values of the external magnetic field in the $x$ direction. The plot is for the temperature $k_{\mathrm{B}}T=0.001{\Delta }_{\mathrm{SO}}$, and the electric field $d{E}_x=0.1{\Delta }_{\mathrm{SO}}$.

Figure 16

(Color online) In-plane magnetization $M_x$ in the $x$ direction in Eq. (135) as a function of the electric field $E_x$ in the $x$ direction. The three lines correspond to a fixed value of an additional magnetic field in the $z$ direction. The assumed temperature is $k_{\mathrm{B}}T=0.001{\Delta }_{\mathrm{SO}}$, while in the inset it is at higher temperature $k_{\mathrm{B}}T=0.1{\Delta }_{\mathrm{SO}}$.

Figure 17

(Color online) Electric susceptibility ($xx$ component), Eq. (136), as a function of the electric field in $x$ direction. The three lines correspond to various values of the external magnetic field in the $x$ direction. The plot is for the temperature $k_{\mathrm{B}}T=0.001{\Delta }_{\mathrm{SO}}$. In the inset, the same quantity is plotted at a higher temperature, $k_{\mathrm{B}}T=0.1{\Delta }_{\mathrm{SO}}$.

Figure 18

(Color online) Electric susceptibility ($xy$ component), Eq. (136), as a function of the electric field in $x$ direction. The three lines correspond to various values of the external magnetic field in the $x$ direction. The plot is for the temperature $k_{\mathrm{B}}T=0.001{\Delta }_{\mathrm{SO}}$. In the inset the same quantity is plotted at a higher temperature $k_{\mathrm{B}}T=0.1{\Delta }_{\mathrm{SO}}$.

Figure 19

(Color online) Linear magnetoelectric tensor ($xx$ component) in Eq. (138) as a function of the electric field in the $x$ direction. The three lines correspond to various values of the external magnetic field in the $x$ direction. The plot is for the temperature $k_{\mathrm{B}}T=0.001{\Delta }_{\mathrm{SO}}$.

Figure 20

(Color online) Linear magnetoelectric tensor ($xz$ component) in Eq. (138) as a function of the electric field in the $x$ direction. The three lines correspond to various values of the external electric field in the $x$ direction. The plot is for the temperature $k_{\mathrm{B}}T=0.001{\Delta }_{\mathrm{SO}}$.

Figure 21

(Color online) Linear magnetoelectric tensor ($xx$ component) in Eq. (138) as a function of the magnetic field in the $x$ direction. The three lines correspond to various values of the external electric field in the $x$ direction. The plot is for the temperature $k_{\mathrm{B}}T=0.001{\Delta }_{\mathrm{SO}}$.

Figure 22

(Color online) Magnetic susceptibility ($zz$ component) in Eq. (137) as a function of the magnetic field in the $x$ direction. The three lines correspond to various values of the external electric field in the $x$ direction. The plot is for the temperature $k_{\mathrm{B}}T=0.001{\Delta }_{\mathrm{SO}}$. The inset represents the same quantity at a higher temperature $k_{\mathrm{B}}T=0.1{\Delta }_{\mathrm{SO}}$.

Figure 23

(Color online) Magnetic susceptibility ($xx$ component), Eq. (137) as a function of the electric field in the $x$ direction. The three lines correspond to various values of the additional magnetic field in the $z$ direction. The plot is for the temperature $k_{\mathrm{B}}T=0.001{\Delta }_{\mathrm{SO}}$.

Figure 24

(Color online) Heat capacity, Eq. (156), as a function of temperature in various electric fields.

Figure 25

(Color online) Heat capacity, Eq. (156), at low temperature as a function of external electric field.