Spin-bias driven magnetization reversal and nondestructive detection in a single molecular magnet

The magnetization reversal in a single molecular magnet (SMM) weakly coupled to an electrode with spin-dependent splitting of chemical potentials (spin bias) is theoretically investigated by means of the rate equation. A microscopic mechanism for the reversal is demonstrated by the avalanche dynamics at the reversal point. The magnetization as a function of the spin bias shows hysteresis loops tunable by the gate voltage and varying with temperature. The nondestructive measurement to the orientation of giant spin in SMM is presented by measuring the fully polarized electric current in the response to a small spin bias. For ${\text{Mn}}_{12}\text{ac}$ molecule, its small transverse anisotropy only slightly violates the results above. The situation when there is an angle between the easy axis of the SMM and the spin-quantization direction of the electrode is also studied.

Figure 1

(Color online) Schematic of energy configurations of our setup to manipulate the magnetization of an SMM. The splitting between $\uparrow$ and $\downarrow$ Fermi levels of the only source lead is phenomenologically denoted as $V$. Throughout the work the middle point of ${\mu }_S^{\uparrow /\downarrow }$ is set at 0 so that ${\mu }_S^{\uparrow /\downarrow }=\pm \frac{V}{2}$. The horizontal lines in the SMM region correspond to resonant energies to add an extra electron into the SMM via transitions from state $|0,m⟩$ to ${|1,m\pm \frac{1}{2}⟩}^{-}$, which is tunable with respect to ${\mu }_S^{\uparrow /\downarrow }$ by the gate voltage $V_g$.

Figure 2

(Color online) Left: The two branches of molecular many-body states considered in the simulations of this work. The basic parameters are given in Sec. 3B. $V_g=-20\text{mV}$. Arrows indicate all the steps required to reverse the SMM magnetization from $-21/2$ to 21/2. Right (four panels): The schematic of a single step that increases the SMM magnetization by 1. This mechanism is supported by the simulation shown in Figs. 5, 6.

Figure 3

(Color online) (a) Resonant energies to add an extra electron into SMM via transitions from state $|0,m⟩$ to ${|1,m\pm \frac{1}{2}⟩}^{-}$ for three different $V_g$. The middle point of ${\mu }_S^{\uparrow /\downarrow }$ is set as energy zero point so that ${\mu }_S^{\uparrow /\downarrow }=\pm V/2$. (b) Zoom-in when $V_g=-18.8\text{mV}$, where the middle point of the entire spectrum is aligned with 0, i.e., the middle point of the Fermi levels ${\mu }_S^{\uparrow /\downarrow }$. (c) Zoom-in of the lowest transitions in the middle panel. The notation ${|1,m\pm \frac{1}{2}⟩}^{-}-|0,m⟩$ is short for $E_{{|1,m\pm 1/2⟩}^{-}}-E_{|0,m⟩}$. Note that $E_{{|1,-\left(m\pm 1/2\right)⟩}^{-}}-E_{|0,-m⟩}=E_{{|1,m\pm 1/2⟩}^{-}}-E_{|0,m⟩}$ are degenerate in the absence of magnetic field. Basic parameters are given in Sec. 3B.

Figure 4

(Color online) (a) Magnetic hysteresis loops for different gate voltage $V_g$, when scanning the spin bias $V$ back and forth. $T=0.01\text{K}$. Arrows indicate the scanning direction of $V$. The scanning is assumed to be slow enough to allow the system relax to steady state. The triangle corresponds to the magnetic reversal point for the simulation in Figs. 5, 6. (b) The activation energy vs $V_g$. It consists of two slopes connecting at $V_g=-18.8\text{mV}$. The left slope is determined by the transition energy $E_{{|1,\pm 19/2⟩}^{-}}-E_{|0,\pm 10⟩}$; the right slope is determined by the transition energy $E_{{|1,\pm 5/2⟩}^{-}}-E_{|0,\pm 2⟩}$. Circles correspond to $V_g=-17.2$, $-18.8$, and $-19.6\text{mV}$ in the left panel. (c) Temperature-dependent magnetic hysteresis loops as a function of $V$ for $V_g=-18.8\text{mV}$. Results for experimental temperatures 3 K (Ref. 20) and 0.3 K (Ref. 21) are presented. Other parameters are given in Sec. 3B.

Figure 5

(Color online) The time-dependent magnetization and the spin-resolved currents during the magnetic reversal. Solid and dashed lines represent the cases in the absence and the presence of a weak transverse anisotropy, respectively. The initial state is $P_{|0,-10⟩}=1$ and $P_{i\ne |0,-10⟩}=0$. $T=0.01\text{K}$, $V_g=-18.8\text{mV}$, $V=1.8\text{meV}$ [the triangle in Fig. 4a]. Other parameters are given in Sec. 3B. The positive sign of current stands for flowing from source to SMM and negative for from SMM to source.

Figure 6

During the magnetic reversal, the time-dependent probabilities of the branches $|0,m⟩$ and ${|1,m⟩}^{-}$. Thick and thin lines represent the cases when $B_2=B_4=0$ and ${10}^2{B}_2={10}^4{B}_4=D$, respectively. Note that the states from $|0,-2⟩$ through $|0,2⟩$ are explicitly reshaped by the $B_2$ and $B_4$. All the parameters are the same as Fig. 5.

Figure 7

(Color online) Nondestructive detection to the giant spin orientation. (a) Transition energies $E_{{|1,m\pm 1/2⟩}^{-}}-E_{|0,m⟩}$, when $V_g=-18.8\text{mV}$. (b) Zoom-in of (a) near $E_{|1,\pm 21/2⟩}-E_{|0,\pm 10⟩}$. By tuning the gate voltage $V_g$, the Fermi levels of leads ${\mu }_{S/D}^{\uparrow /\downarrow }$ can be located aligned with the two degenerate transition energies $E_{{|1,\pm 21/2⟩}^{-}}-E_{|0,\pm 10⟩}$. [(c1) and (d1)] Schematics of how fully polarized electric current flows when $S_z=\pm 10$. [(c2)–(c4) and (d2)–(d4)] For $S_z=\pm 10$, numerical results of source-to-drain current, SMM magnetization, and LUMO occupation as functions of $V_g$ in the presence of a small $V=0.05\text{meV}$. $T=0.1\text{K}$. Other parameters are given in Sec. 3B.

Figure 8

The energies of the perturbed branches ${|0,m⟩}_p$ and ${|1,m⟩}_p^{-}$ as a function of perturbed projection $m_p$. Circle and triangle: $B_2={10}^{-2}D$, $B_4={10}^{-4}D$. Solid and dashed lines: $B_2=B_4=0$. Other parameters are the same as Fig. 2.

Figure 9

(Color online) The same as Fig. 4 except $B_2={10}^{-2}D$ and $B_4={10}^{-4}D$.

Figure 10

(Color online) (a) In the presence of an angle $\theta$ between the easy axis of SMM and the spin-quantization direction of the lead, there are four possible injection and leakage processes marked by the arrows. ${\text{cos}}^2\left(\theta /2\right)$ or ${\text{sin}}^2\left(\theta /2\right)$ indicates the relative probability of each process. (b) The reversal time will be prolonged by $1/\text{cos}\left(\theta \right)$. [(c) and (d)] Magnetization and spin-resolved currents for the same situation shown in Fig. 5 for different $\theta$.

Figure 11

(Color online) (a) Current tends to flow from drain to source when $E_{{|1,\pm 21/2⟩}^{-}}-E_{|0,\pm 10⟩}$ is between ${\mu }_S^{-}$ and ${\mu }_D^{\uparrow /\downarrow }$. (b) Current tends to flow from source to drain when $E_{{|1,\pm 21/2⟩}^{-}}-E_{|0,\pm 10⟩}$ is between ${\mu }_S^{+}$ and ${\mu }_D^{\uparrow /\downarrow }$. [(c1)–(c3)] The same measurement scheme as Fig. 7(c2) except for $\theta \ne 0$. [(d1)–(d3)] The same measurement scheme as Fig. 7(d2) except for $\theta \ne 0$.

Figure 12

(Color online) (a)–(c) Energy schemes employed by the previous authors and (d) in this work. (a) Nonmagnetic leads (Ref. 37), (b) ferromagnetic leads with antiparallel polarizations, and (c) ferromagnetic leads with parallel polarizations (Ref. 41); (d) nonmagnetic lead with spin-dependent splitting of chemical potentials.