⇦Chinese Physics Letters, 2020, Vol. 37, No. 8, Article code 087201 Magnetization Reversal of Single-Molecular Magnets by a Spin-Polarized Current Chao Yang (杨超)1, Zheng-Chuan Wang (王正川)1*, and Gang Su (苏刚)1,2* Affiliations 1School of Physical Sciences, University of Chinese Academy of Sciences, Beijng 100049, China 2Kavli Institute for Theoretical Physics, CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China Received 13 May 2020; accepted 1 June 2020; published online 28 July 2020 Supported by the National Key R&D Program of China (Grant No. 2018YFA0305804), the National Natural Science Foundation of China (Grant No. 11834014), and the Strategetic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB28000000).
*Corresponding authors. Email: wangzc@ucas.ac.cn; gsu@ucas.ac.cn Citation Text: Yang C, Wang Z C and Su G 2020 Chin. Phys. Lett. 37 087201    Abstract We study the magnetization reversal of single-molecular magnets by a spin-polarized current in the framework of the spinor Boltzmann equation. Because of the spin–orbit coupling, the spin-polarized current will impose a non-zero spin transfer torque on the single-molecular magnets, which will induce the magnetization switching of the latter. Via the $s$–$d$ exchange interaction between the conducting electrons and single-molecular magnets, we can investigate the magnetization dynamics of single-molecular magnets. We demonstrate the dynamics of the magnetization based on the spin diffusion equation and the Heisenberg-like equation. The results show that when the current is large enough, the magnetization of the single-molecular magnets can be reversed. We also calculate the critical current density required for the magnetization reversal under different anisotropy and external magnetic fields, which is helpful for the corresponding experimental design. DOI:10.1088/0256-307X/37/8/087201 PACS:72.25.-b, 75.75.-c, 75.78.-n © 2020 Chinese Physics Society Article Text A single-molecular magnet (SMM) is a big cluster containing magnetic ions and molecular ligands, which has a large spin due to its strong intramolecular exchange interactions and weak intermolecular exchange interactions.[1] With the double well of energy potential, the SMM possesses two metastable spin states at low temperature.[2,3] Thus, it exhibits the property similar to a nanoscale magnetic particle and these two spin states can be used to store information.[4] In recent years, much research indicate that SMMs can be a good candidate for future high density memory cells and new types of memory devices.[5–9] The magnetization states of SMMs can store information, accordingly, reading and writing information are operations to recognize and reverse the magnetization states. Because the SMM is small, it is impossible to switch the magnetization of SMMs by an external magnetic field, which is always performed by the spin-polarized current.[10] The spin of conducting electrons can interact with the local magnetic moment of SMMs by $s$–$d$ interaction, which will produce the spin transfer torque and cause the flip of magnetization. In 2007, Misiorny et al. investigated the spin-polarized current-induced magnetization reversal of SMM by the master equation.[11] The non-equilibrium Green's function method can also be used to study the spin-polarized transport in SMM-based spin valves,[12] which indicate that the spin-polarized current can switch the magnetization of SMM. Therefore, these SMM-based spin valves can be used as memory devices.[13–16] Because of the quantum properties, the motion of magnetization of SMM will not obey the classical Landau–Lifshitz–Gilbert (LLG) equation, which describes the magnetization dynamics of classical magnetic moments. In 2015, Zhang et al. derived a Heisenberg-like equation,[17] which contains the quantum correction terms, to replace the LLG equation. They combined the Heisenberg-like equation and the spin diffusion equation together to study the magnetization dynamics of SMM. The Heisenberg-like equation describes the magnetization dynamics of SMM, and the spin diffusion equation which is derived from the spinor Boltzmann equation is used to describe the motion of conducting electrons. The spinor Boltzmann equation is a powerful tool to investigate the spin-polarized transport.[18] Based on Kadanoff–Baym's treatment, one can obtain the spinor Boltzmann equation from non-equilibrium Green function (NEGF) formalism, and Sheng et al. derived the spinor Boltzmann equation by NEGF at a steady state.[19,20] We also derived the two-momentum spinor Boltzmann equation by the NEGF method.[21] Furthermore, the spin-polarized transport with spin-orbit interactions was also studied by the spinor Boltzmann equation.[22,23] In this letter, we study the magnetization reversal of SMMs by spin-polarized current. The system is schematically depicted in Fig. 1. A heavy metal thin layer is covered with a thin film of SMMs. The heavy metal thin layer will produce a Rashba spin-orbit coupling (SOC) owing to its structural inversion asymmetry.[24] When the current flows through the metal thin layer, the Rashba SOC will generate a non-equilibrium spin accumulation of conducting electrons.[24–26] This spin accumulation always concentrates at the interface, which will interact with the local magnetic moment of SMMs. Via the $s$–$d$ exchange interaction between the spin accumulation of conducting electrons and the SMMs,[11] we can manipulate the direction of magnetization of SMMs, and further reverse the magnetization under a certain critical current. Compared with the SMM-based spin valves, our design is more simple in structure and may have a higher spin injection efficiency.
cpl-37-8-087201-fig1.png
Fig. 1. The structure for the magnetization reversal of SMM by the spin-polarized current, in which a heavy metal thin layer is covered with a thin film of SMMs.
In our model, the magnetic dynamics of the SMM and the spin accumulation of conducting electrons should be considered together. We shall use the Heisenberg-like equation to describe the dynamics of SMM, and the spinor Boltzmann equation hence the spin diffusion equation to describe the motion of conducting electrons. Similarly to Levy's theory,[1] the $s$–$d$ interaction between conducting electrons and SMM can be written as $\hat{H}_{\rm int}=-J{\hat{\boldsymbol\sigma}}\cdot{\hat{\boldsymbol S}}$, where ${\hat{\boldsymbol\sigma}}$ is Pauli operator representing the spin of conducting electrons, ${\hat{\boldsymbol S}}$ is the spin operator of SMM, and $J$ is the coupling constant. The Hamiltonian of an SMM can be written as[1] $$ \hat{H}_{\rm SMM}=-D\hat{S}^2_z+E_{\rm S}(\hat{S}^2_x-\hat{S}^2_y)-J{\hat{\boldsymbol\sigma}}\cdot{\hat{\boldsymbol S}},~~ \tag {1} $$ where $\hat{S}_{x,y,z}$ are components of spin operator ${\hat{\boldsymbol S}}$, $D$ is the axial anisotropy constant, and $E_{\rm S}$ is the transverse anisotropy constant. As we know, the spin operator of SMM satisfies the following Heisenberg equation $$ i\hbar\frac{\partial{\hat{\boldsymbol S}}}{\partial t}=[{\hat{\boldsymbol S}},\hat{H}_{\rm SMM}].~~ \tag {2} $$ Substituting the Hamiltonian into the above equation, we can obtain $$\begin{alignat}{1} &i\hbar\frac{\partial\hat{S}_x}{\partial t}=(-E_{\rm S}-D)(\hat{S}_z\hat{S}_y+\hat{S}_y\hat{S}_z) -J(\hat{\sigma_y}\hat{S}_z-\hat{\sigma_z}\hat{S}_y),\\ &i\hbar\frac{\partial\hat{S}_y}{\partial t}=(D-E_{\rm S})(\hat{S}_z\hat{S}_x+\hat{S}_x\hat{S}_z)-J(\hat{\sigma_z}\hat{S}_x -\hat{\sigma_x}\hat{S}_z),\\ &i\hbar\frac{\partial\hat{S}_z}{\partial t}=2E_{\rm S}(\hat{S}_y\hat{S}_x+\hat{S}_x\hat{S}_y)-J(\hat{\sigma_x}\hat{S}_y -\hat{\sigma_y}\hat{S}_x).~~ \tag {3} \end{alignat} $$ Then taking average over the above equation and making a mean-field approximation,[17] we obtain $$\begin{alignat}{1} &i\hbar\frac{\partial S_x}{\partial t}=(-E_{\rm S}-D)(S_zS_y+i\hbar S_x)-J(m_yS_z-m_zS_y),\\ &i\hbar\frac{\partial S_y}{\partial t}=(D-E_{\rm S})(S_xS_z+i\hbar S_y)-J(m_zS_x-m_xS_z),\\ &i\hbar\frac{\partial S_z}{\partial t}=2E_{\rm S}(S_yS_x+i\hbar S_z)-J(m_xS_y-m_yS_x).~~ \tag {4} \end{alignat} $$ The above Heisenberg-like equation describes the motion of the average spin ${\boldsymbol S}=\langle{\hat{\boldsymbol S}}\rangle$, where $S_x=\langle\hat{S}_x\rangle$, $S_y=\langle\hat{S}_y\rangle$, $S_z=\langle\hat{S}_z\rangle$; and $m_x=\langle\hat{\sigma}_x\rangle$, $m_y=\langle\hat{\sigma}_y\rangle$, $m_z=\langle\hat{\sigma}_z\rangle$ are the components of spin accumulation ${\boldsymbol m}$. The spin accumulation ${\boldsymbol m}$ is the statistical average of spin operator ${\hat{\boldsymbol\sigma}}$, which can be calculated from the spinor Boltzmann equation that will be introduced in the following. For the conducting electrons in the system, the Hamiltonian is $$ \hat{H}_e=-\frac{\hbar^2}{2\mu}{\nabla}^2+J{\boldsymbol S}\cdot{\hat{\boldsymbol\sigma}}+\hat{H}_{\rm R},~~ \tag {5} $$ where $\mu$ is the effective mass of an electron, $\hat{H}_{\rm R}=\alpha({\nabla}\times{\boldsymbol z})\cdot{\hat{\boldsymbol\sigma}}$ describes the Rashba SOC, $\alpha$ represents the coupling constant, and ${\boldsymbol z}$ is a direction vector determined by the symmetry. This Rashba SOC is induced by the structural inversion asymmetry, e.g., a layer of heavy metal or an external electric field perpendicular to the surface will produce the Rashba SOC. From the above Hamiltonian, we can derive the spinor Boltzmann equation,[22,23] based on the NEGF formalism, as $$\begin{align} &\Big(\frac{\partial}{\partial t}+\frac{{\boldsymbol p}}{\mu}\cdot{\nabla}+e{\boldsymbol E}\cdot{\nabla}_p\Big)\hat{f}+\frac{i J}{\hbar}[{\boldsymbol S}\cdot{\hat{\boldsymbol\sigma}},\hat{f}]+\frac{\alpha}{2\hbar}\{({\nabla}\times{\boldsymbol z})\cdot{\hat{\boldsymbol\sigma}},\hat{f}\}\\ &+\frac{i\alpha}{2\hbar^2}[({\boldsymbol p}\times{\boldsymbol z})\cdot{\hat{\boldsymbol\sigma}},\hat{f}]=\Big(\frac{\partial \hat{f}}{\partial t}\Big)_{\rm collision},~~ \tag {6} \end{align} $$ where $e$ is the charge of an electron, ${\boldsymbol E}$ is the external electric field, $[\cdots,\cdots]$ represents the commutator bracket and $\{\cdots,\cdots\}$ represents the anticommutator bracket. The spinor distribution function $\hat{f}$ is a $2\times2$ matrix, which can be decomposed into the scalar distribution function $f$ and the vector distribution function ${\boldsymbol g}$ around the equilibrium state:[18] $$ \hat{f}=f^0+\left(-\frac{\partial f^{0}}{\partial\varepsilon}\right)[f\hat{\sigma}_0+{\boldsymbol g}\cdot{\hat{\boldsymbol\sigma}}] ,~~ \tag {7} $$ where $f^0$ is the equilibrium distribution function, $\hat{\sigma}_0$ is the identity matrix, and the collision term $(\frac{\partial\hat{f}}{\partial t})_{\rm collision}$ can be simplified with the relaxation approximation to $$\begin{alignat}{1} &-\Big(\frac{\partial\hat{f}}{\partial t}\Big)_{\rm collision}=\\ &\Big(-\frac{\partial f^0}{\partial\varepsilon}\Big)\Big[\frac{f\hat{\sigma}+{\boldsymbol g}\cdot{\hat{\boldsymbol\sigma}}-\langle f\hat{\sigma}+{\boldsymbol g}\cdot{\hat{\boldsymbol\sigma}}\rangle}{\tau}+\frac{2\langle{\boldsymbol g}\cdot{\hat{\boldsymbol\sigma}}\rangle}{\tau_{sf}}\Big],~~ \tag {8} \end{alignat} $$ where $\tau$ is the momentum relaxation time, $\tau_{\rm sf}$ is the spin-flip relaxation time. Separating the scalar distribution function and the vector distribution function, we have the equation for the scalar distribution as follows: $$ \Big(\frac{\partial}{\partial t}+\frac{{\boldsymbol p}}{\mu}\cdot{\nabla}+q{\boldsymbol E}\cdot{\nabla}_p\Big)f+\frac{\alpha}{\hbar}({\nabla}\times{\boldsymbol z})\cdot{\boldsymbol g}=-\frac{f-\langle f\rangle}{\tau} ,~~ \tag {9} $$ and the equation for the vector distribution function as $$\begin{split} &\Big(\frac{\partial}{\partial t}+\frac{{\boldsymbol p}}{\mu}\cdot{\nabla}+q{\boldsymbol E}\cdot{\nabla}_p\Big){\boldsymbol g}-\frac{J}{\hbar}{\boldsymbol S}\times{\boldsymbol g}+\frac{\alpha}{\hbar}({\nabla}\times{\boldsymbol z})f\\ &-\frac{\alpha}{\hbar^2}({\boldsymbol p}\times{\boldsymbol z})\times{\boldsymbol g}=-\frac{{\boldsymbol g}-\langle{\boldsymbol g}\rangle}{\tau}-\frac{2\langle{\boldsymbol g}\rangle}{\tau_{sf}}. \end{split}~~ \tag {10} $$ Integrating Eqs. (9) and (10) over ${\boldsymbol p}$, we can obtain the continuity equation for the charge and spin accumulation: $$ \frac{\partial n}{\partial t}+{\nabla}\cdot{\boldsymbol j}+\frac{\alpha}{\hbar}({\nabla}\times{\boldsymbol z})\cdot{\boldsymbol m}=-\frac{n-\langle n\rangle}{\tau} ,~~ \tag {11} $$ and $$\begin{alignat}{1} &\frac{\partial {\boldsymbol m}}{\partial t}+{\nabla}\cdot\vec{\vec{j}}_s-\frac{J}{\hbar}{\boldsymbol S}\times{\boldsymbol m}+\frac{\alpha}{\hbar}({\nabla}\times{\boldsymbol z})n\\ &-\frac{\alpha}{\hbar^2}\int({\boldsymbol p}\times{\boldsymbol z})\times{\boldsymbol g}d{\boldsymbol p}=-\frac{{\boldsymbol m}-\langle{\boldsymbol m}\rangle}{\tau}-\frac{2\langle{\boldsymbol m}\rangle}{\tau_{sf}} ,~~ \tag {12} \end{alignat} $$ where $n$ and ${\boldsymbol m}$ stand for the charge density and spin accumulation, respectively, which are defined[18] as $$ n({\boldsymbol R})=\int f({\boldsymbol R},{\boldsymbol p})d{\boldsymbol p} ,~~ \tag {13} $$ and $$ {\boldsymbol m}({\boldsymbol R})=\int {\boldsymbol g}({\boldsymbol R},{\boldsymbol p})d{\boldsymbol p} .~~ \tag {14} $$ Similarly, the charge current density ${\boldsymbol j}$ and the spin current density $\vec{\vec{j}}_s$ can be defined as $$ {\boldsymbol j}({\boldsymbol R})=\int \frac{{\boldsymbol p}}{\mu}f({\boldsymbol R},{\boldsymbol p})d{\boldsymbol p} ,~~ \tag {15} $$ and $$ \vec{\vec{j}}_s({\boldsymbol R})=\int \frac{{\boldsymbol p}}{\mu}{\boldsymbol g}({\boldsymbol R},{\boldsymbol p})d{\boldsymbol p}.~~ \tag {16} $$ From the coupled Eqs. (9) and (10), we can obtain the scalar distribution function and the vector distribution function. According to the definitions (14) and (16), the spin accumulation and spin current density can then be obtained. Equation (12) is the so-called spin diffusion equation, and combining it with Heisenberg-like Eq. (4), we can study both the dynamics of SMM and conducting electrons. In order to investigate the dynamics of magnetization, we should solve the coupled spin diffusion equation and the Heisenberg-like equation, which are difficult to solve. Therefore, we must make an approximation to simplify the above equations. Since the electrons move much faster than the direction change of the magnetization of SMM, we can assume that the motion of electrons has reached a quasi-steady state before the direction of magnetization changes. Firstly, we solve the spinor Boltzmann equation at steady state for the given initial magnetization, and obtain the spin accumulation. Then we use this spin accumulation to solve the Heisenberg-like equation, and achieve the magnetization at next time. Repeatedly, we can obtain the whole dynamics of magnetization. In numerical calculations, we take the single-molecular magnet Fe$_8$ as an example. Its spin quantum number is $S=10$, the anisotropic parameters are $D=0.025$ meV and $E_{\rm S}=0.004$ meV,[27] and we assume the initial magnetization to be ${\boldsymbol S}=(0,0,10)$. For the conducting electrons in the system, we choose the Fermi energy $\varepsilon_{\rm F}=3$ eV, relaxation times $\tau=0.1$ ps and $\tau_{sf}=1$ ps, which are typical values of metals. The Rashba SOC coupling constant is adopted as $\alpha=10^{-11}$ eV$\cdot$m,[25] and the coupling constant of the interaction between electrons and the SMM as $J=0.015$ meV.[11]
cpl-37-8-087201-fig2.png
Fig. 2. The unit vector of magnetization of the SMM vs time $t$, (a) current density $j=0.51\,µ$A/m$^2$, (b) current density $j=0.53\,µ$A/m$^2$.
Figures 2(a) and 2(b) show the dynamics of magnetization, i.e., the results corresponding to current density $j=0.51\,µ$A/m$^2$ and $j=0.53\,µ$A/m$^2$, respectively. By simultaneously solving Eq. (10) and the Heisenberg-like equation (4), we can obtain the magnetization $S_{ x,y,z}$. Firstly, we can see the time scale for the dynamics of magnetization is of nanoseconds. As we know, the electrons move in picoseconds scale, which implies that our quasi-steady approximation is reasonable. Secondly, the electric current drives the direction of magnetization to change periodically. Showing the results of Fig. 2 on a unit sphere, we can get the corresponding trajectory of the direction of magnetization in Figs. 3(a) and 3(b), respectively. When the current density $j=0.51\,µ$A/m$^2$, the trajectory is a closed curve on the upper hemisphere, and the magnetization remains $S_z>0$ with a periodic motion, which corresponds to the right well of double-well energy potential in Fig. 4(a). When the current density $j=0.53\,µ$A/m$^2$, the trajectory is a closed curve on the whole sphere, which means that the magnetization overcomes the energy barrier and jumps between the two potential wells, and will cause the magnetization reversal of SMM. The result indicates that we can use a spin-polarized current that is large enough to switch the magnetization between two stable states $S_z=10$ and $S_z=-10$ in a proper duration (Fig. 4). For the SMM-based memory devices, we can use a current which continues for an appropriate duration to reverse the local magnet and write information.
cpl-37-8-087201-fig3.png
Fig. 3. The trajectory of the direction of magnetization unit vector on a unit sphere, (a) current density $j=0.51\,µ$A/m$^2$, (b) current density $j=0.53\,µ$A/m$^2$.
cpl-37-8-087201-fig4.png
Fig. 4. The double well of energy potential of Fe$_8$ molecular magnet, (a) without external magnetic field, (b) under an external magnetic field along the $z$-axis.
cpl-37-8-087201-fig5.png
Fig. 5. The spin current density as a function of time.
Figure 5 shows the spin current density as a function of time. Although the current is continuous in our numerical calculations, the spin current changes periodically like a sequence of pulses. Because the spin current interacts with the SMM through spin transfer torque, the spin polarized direction of conducting electrons changes with the magnetization switching. Combining the spin-diffusion equation (12) satisfied by the spin current with the Heisenberg-like equation (4) satisfied by the SMM, we can study the variation of local moment of SMM and spin current simultaneously. We can see that the spin current changes periodically, the period of the spin current is conceded with the switching period of the SMM. In our system, $s$–$d$ interaction is much weaker than Rashba SOC, and Rashba SOC results in a spin-polarization along the $y$-axis.[23] Therefore, $j_{sx}$ and $j_{sz}$ are much smaller than $j_{sy}$. The above results show that we can achieve the magnetization reversal of SMM with a current beyond the critical current density $j_{\rm C}$. For the Fe$_8$ cluster, the critical current density is $j_{\rm C}=0.52\,µ$A/m$^2$. If we change the molecular ligands in the Fe$_8$ cluster, e.g., replace Br$^{-}$ with ClO$_4^{-}$,[28] the anisotropic parameters will change accordingly. Next, we will further investigate the critical current densities for different anisotropic parameters, which is important for relevant experiments and device designs.
cpl-37-8-087201-fig6.png
Fig. 6. (a) The critical current density $j_{\rm C}$ vs the axial anisotropy constant $D$ at $E_{\rm S}=0.004$ meV; (b) the critical current density $j_{\rm C}$ vs the transverse anisotropy constant $E_{\rm S}$ at $D=0.025$ meV.
Figure 6(a) shows the critical current density as a function of axial anisotropy constant. We can see that the critical current density $j_{\rm C}$ increases with the axial anisotropy constant. Because the axial anisotropy constant describes the depth of potential well, a larger axial anisotropy constant means a higher axial energy barrier that needs to overcome for the reversal of the magnetization. Figure 6(b) shows the critical current density as a function of transverse anisotropy constant. The critical current density firstly increases to the maximum and then decreases. Because a larger $E_{\rm S}$ increases the transverse barrier, which makes it difficult to overcome the barrier via a transverse motion, $j_{\rm C}$ increases. On the other hand, as $E_{\rm S}$ is proportional to the nondiagonal element of the Hamiltonian, a larger $E_{\rm S}$ will increase the quantum tunneling probability. When $E_{\rm S}$ is large enough, the magnetization reversal mainly depends on quantum tunneling, thus $j_{\rm C}$ will decrease. Let us now consider an external magnetic field applied to the system. The Hamiltonian of an SMM can be written[1] as $$ \hat{H}_{\rm SMM}=-D\hat{S}^2_z+E_{\rm S}(\hat{S}^2_x-\hat{S}^2_y)-J{\hat{\boldsymbol\sigma}}\cdot{\hat{\boldsymbol S}}-\mu_{\rm B} g{\boldsymbol H}\cdot{\hat{\boldsymbol S}},~~ \tag {17} $$ where $\mu_{\rm B}$ is Bohr magneton, $g$ is the Lande factor, and ${\boldsymbol H}$ is the external magnetic field. From this Hamiltonian, we can obtain the Heisenberg equation for the spin operator of the SMM: $$\begin{align} i\hbar\frac{\partial\hat{S}_x}{\partial t}={}&(-E_{\rm S}-D)(\hat{S}_z\hat{S}_y+\hat{S}_y\hat{S}_z)-[(J\hat{\sigma_y}+\mu_{\rm B}gH_y)\hat{S}_z\\ &-(J\hat{\sigma_z}+\mu_{\rm B}gH_z)\hat{S}_y],\\ i\hbar\frac{\partial\hat{S}_y}{\partial t}={}&(D-E_{\rm S})(\hat{S}_z\hat{S}_x+\hat{S}_x\hat{S}_z)-[(J\hat{\sigma_z}+\mu_{\rm B}gH_z)\hat{S}_x\\ &-(J\hat{\sigma_x}+\mu_{\rm B}gH_x)\hat{S}_z],\\ i\hbar\frac{\partial\hat{S}_z}{\partial t}={}&2E_{\rm S}(\hat{S}_y\hat{S}_x+\hat{S}_x\hat{S}_y)-[(J\hat{\sigma_x}+\mu_{\rm B}gH_x)\hat{S}_y\\ &-(J\hat{\sigma_y}+\mu_{\rm B}gH_y)\hat{S}_x].~~ \tag {18} \end{align} $$ Similarly, taking average over the above equation and making a mean-field approximation, we obtain $$\begin{align} i\hbar\frac{\partial S_x}{\partial t}={}&(-E_{\rm S}-D)(S_zS_y+i\hbar S_x)-[(Jm_y+\mu_{\rm B}gH_y)S_z\\ &-(Jm_z+\mu_{\rm B}gH_z)S_y],\\ i\hbar\frac{\partial S_y}{\partial t}={}&(D-E_{\rm S})(S_xS_z+i\hbar S_y)-J[(Jm_z+\mu_{\rm B}gH_z)S_x\\ &-(Jm_x+\mu_{\rm B}gH_x)S_z],\\ i\hbar\frac{\partial S_z}{\partial t}={}&2E_{\rm S}(S_yS_x+i\hbar S_z)-[(Jm_x+\mu_{\rm B}gH_x)S_y\\ &-(Jm_y+\mu_{\rm B}gH_y)S_x].~~ \tag {19} \end{align} $$ This is the Heisenberg-like equation with an external magnetic field. For conducting electrons, we should also consider the influence of external magnetic field, thus the Hamiltonian of electrons is $$ \hat{H}_e=-\frac{\hbar^2}{2\mu}{\nabla}^2+(J{\boldsymbol S}+\mu_{\rm B}g{\boldsymbol H})\cdot{\hat{\boldsymbol\sigma}}+\hat{H}_{\rm R}.~~ \tag {20} $$ Similarly, we can reach the spinor Boltzmann equation $$\begin{align} &\Big(\frac{\partial}{\partial t}+\frac{{\boldsymbol p}}{\mu}\cdot{\nabla}+e{\boldsymbol E}\cdot{\nabla}_p\Big)\hat{f}+\frac{i }{\hbar}[(J{\boldsymbol S}+\mu_{\rm B}g{\boldsymbol H})\cdot{\hat{\boldsymbol\sigma}},\hat{f}]\\ &+\frac{\alpha}{2\hbar}\{({\nabla}\times{\boldsymbol z})\cdot{\hat{\boldsymbol\sigma}},\hat{f}\}+\frac{i\alpha}{2\hbar^2}[({\boldsymbol p}\times{\boldsymbol z})\cdot{\hat{\boldsymbol\sigma}},\hat{f}]=-\Big(\frac{\partial \hat{f}}{\partial t}\Big)_{\rm collision}.\\~~ \tag {21} \end{align} $$ Then, the spin diffusion equation is $$\begin{alignat}{1} &\frac{\partial {\boldsymbol m}}{\partial t}+{\nabla}\cdot\vec{\vec{j}}_s-\frac{1}{\hbar}(J{\boldsymbol S}+\mu_{\rm B}g{\boldsymbol H})\times{\boldsymbol m}+\frac{\alpha}{\hbar}({\nabla}\times{\boldsymbol z})n\\ &-\frac{\alpha}{\hbar^2}\int({\boldsymbol p}\times{\boldsymbol z})\times{\boldsymbol g}d{\boldsymbol p}=-\frac{{\boldsymbol m}-\langle{\boldsymbol m}\rangle}{\tau}-\frac{2\langle{\boldsymbol m}\rangle}{\tau_{sf}}.~~ \tag {22} \end{alignat} $$ Associating the spinor Boltzmann equation hence the spin diffusion equation with the Heisenberg-like equation, we can study the magnetization dynamics of SMM under an external magnetic field. A similar magnetization reversal is shown in Fig. 7, while an external magnetic field is applied to the system. Because the external magnetic field is along the $x$-axis, the trajectory is skewing. Figures 7(a) and 7(b) are similar to the case without an external magnetic field. The current we choose here is $j=0.53\,µ$A/m$^2$, we can see that the magnetization has reversed under this current.
cpl-37-8-087201-fig7.png
Fig. 7. (a) The unit vector of magnetization vs time $t$; (b) the trajectory of the direction of magnetization unit vector on the unit sphere; where $H_y=0$, $H_z=0$, and $\mu_{\rm B}H_x=0.005\,µ$eV, $j=0.53\,µ$A/m$^2$.
cpl-37-8-087201-fig8.png
Fig. 8. The critical current density for magnetization reversal vs the external magnetic field. (a) $H_y=0$, $H_z=0$; (a) $H_x=0$, $H_z=0$; (c) $H_x=0$, $H_y=0$.
In Fig. 8, we plot the critical current density as a function of the external magnetic field, where we take $D=0.025$ meV and $E_{\rm S}=0.004$ meV. Figure 8(a) shows that when $|H_x|$ is larger, the critical current density $j_{\rm C}$ is smaller. Because $H_x$ makes the transverse motion easier, $j_{\rm C}$ will decrease. Figure 8(b) shows the critical current density as a function of $H_y$, where $j_{\rm C}$ decreases with $H_y$. The conducting electrons move along the $x$-axis, and will produce a spin accumulation $m_y$ along the $y$-axis because of the Rashba SOC.[23] Via the spin accumulation $m_y$, the current can switch the direction of magnetization. $H_y$ has the similar contribution with $m_y$ as shown in Eq. (19), so applying an external magnetic field $H_y$ can be regarded as increasing $m_y$, which has the same effect with increasing the current. Therefore, the current density required for the magnetization reversal will decrease. Figure 8(c) shows the critical current density as a function of $H_z$, where $j_{\rm C}$ decreases gradually to the minimum. As shown in Fig. 4(b), the double well of energy potential will change under an external magnetic field along the $z$-axis. When applying an external magnetic field along the $z$-axis, the energies of states $S_z>0$ will increase and the energies of states $S_z < 0$ will decrease. It breaks the symmetry of the double well. Thus the energy barrier from $S_z=10$ to $S_z=-10$ is lowered, which makes $j_{\rm C}$ decrease. When $H_z$ is large enough, the axial barrier is small. In this case, the magnetization reversal mainly depends on the transverse barrier, which is determined by transverse anisotropic parameter $E_{\rm S}$, instead of the axial barrier. Thus, $j_{\rm C}$ remains nearly unchanged, when $H_z$ is large enough. In summary, we have investigated the magnetization reversal of single-molecular magnets by spin-polarized current under the influence of Rashba spin-orbit coupling. We use the Heisenberg-like equation to describe the dynamics of SMM, and the spinor Boltzmann equation hence the spin diffusion equation to describe the motion of conducting electrons. Because of the spin-orbit coupling, the conducting electrons will produce a nonzero spin transfer torque on the SMM. Via the $s$–$d$ interaction between the conducting electrons and single-molecular magnets, we can manipulate the dynamics of magnetization of SMM. Our results indicate that we can use a spin-polarized current that is large enough to switch the magnetization between the two stable states in a proper duration. In contrast, a small current cannot reverse the magnetization. Furthermore, we calculate the critical current density required for the magnetization reversal under different anisotropic parameters and external magnetic fields. These results are valuable for relevant experiments and device design. We thank Professor Zhen-Gang Zhu and Professor Bo Gu for their helpful discussions. References Single-Molecule MagnetsEffects of intrinsic spin-relaxation in molecular magnets on current-induced magnetic switchingMolecular Electronics: From Single‐Molecule to Large‐Area DevicesNanoscience and TechnologyTheory for transport through a single magnetic molecule: Endohedral N @ C 60 Electron Transport through Single Mn 12 Molecular MagnetsCoupling Single Molecule Magnets to Ferromagnetic SubstratesThe Quantum Magnetism of Individual Manganese-12-Acetate Molecular Magnets Anchored at SurfacesSpin-Transfer Torque on a Single Magnetic AdatomSelf-consistent treatment of nonequilibrium spin torques in magnetic multilayersSpin polarized transport through a single-molecule magnet: Current-induced magnetic switchingFirst-Principles Study of Electron Transport through the Single-Molecule Magnet Mn 12 Single-Molecule NanomagnetsTunneling through magnetic molecules with arbitrary angle between easy axis and magnetic fieldTransformation of spin information into large electrical signals using carbon nanotubesMethods in Molecular BiologyCurrent-induced magnetic switching of a single molecule magnet on a spin valveIdentification of Transverse Spin Currents in Noncollinear Magnetic StructuresQuasiclassical approach to magnetotransport in magnetic inhomogeneous systemsBoltzmann equation for spin-dependent transport in magnetic inhomogeneous systemsThe Boltzmann equation for a spinor distribution function with spinor momentum at the Fermi levelThe spinor Boltzmann equation with Rashba spin–orbit couplingGeneralized spin-orbit torques in two-dimensional ferromagnets with spin-orbit couplingInterface-Generated Spin CurrentsTheory of nonequilibrium intrinsic spin torque in a single nanomagnetTheory of spin torque due to spin-orbit coupling Fe 57 NMR and relaxation by strong collision in the tunneling regime in the molecular nanomagnet Fe8Tuning the Magnetic Properties of the High-Spin Molecular Cluster Fe8
[1] Christou G, Gatteschi D, Hendrickson D N and Sessoli R 2000 MRS Bull. 25 66
[2] Misiorny M and Barnaś J 2008 Phys. Rev. B 77 172414
[3] Vuillaume D 2019 ChemPlusChem 84 1215
[4] Bogani L and Wernsdorfer W 2009 Molecular Spintronics Using Single-Molecule Magnets in Nanoscience and Technology (Singapore: World Scientific) p 194
[5] Elste F and Timm C 2005 Phys. Rev. B 71 155403
[6] Heersche H B, de Groot Z, Folk J A, van der Zant H S J, Romeike C, Wegewijs M R, Zobbi L, Barreca D, Tondello E and Cornia A 2006 Phys. Rev. Lett. 96 206801
[7] Rizzini A L, Krull C, Balashov T, Kavich J J, Mugarza A, Miedema P S, Thakur P K, Sessi V, Klyatskaya S, Ruben M, Stepanow S and Gambardella P 2011 Phys. Rev. Lett. 107 177205
[8] Kahle S, Deng Z, Malinowski N, Tonnoir C, Forment-Aliaga A, Thontasen N, Rinke G, Le D, Turkowski V, Rahman T S, Rauschenbach S, Ternes M and Kern K 2011 Nano Lett. 12 518
[9] Delgado F, Palacios J J and Fernández-Rossier J 2010 Phys. Rev. Lett. 104 026601
[10] Shpiro A, Levy P M and Zhang S 2003 Phys. Rev. B 67 104430
[11] Misiorny M and Barnaś J 2007 Phys. Rev. B 76 054448
[12] Barraza-Lopez S, Park K, García-Suárez V and Ferrer J 2009 Phys. Rev. Lett. 102 246801
[13] Friedman J R and Sarachik M P 2010 Annu. Rev. Condens. Matter Phys. 1 109
[14] Timm C 2007 Phys. Rev. B 76 014421
[15] Hueso L E, Pruneda J M, Ferrari V, Burnell G, Valdés-Herrera J P, Simons B D, Littlewood P B, Artacho E, Fert A and Mathur N D 2007 Nature 445 410
[16] Jonchhe S and Mao H 2019 Single-Molecule Mechanochemical Sensing Using DNA Origami Nanostructures in Biomimetic Sensing ed Fitzgerald J and Fenniri H (New York: Springer) p 171
[17] Zhang X, Wang Z C, Zheng Q R, Zhu Z G and Su G 2015 Phys. Lett. A 379 848
[18] Zhang J, Levy P M, Zhang S and Antropov V 2004 Phys. Rev. Lett. 93 256602
[19] Sheng L, Xing D Y, Wang Z D and Dong J 1997 Phys. Rev. B 55 5908
[20] Sheng L, Teng H Y and Xing D Y 1998 Phys. Rev. B 58 6428
[21] Wang Z C 2012 Eur. Phys. J. B 85 303
[22] Wang Z C 2018 Physica A 492 395
[23] Yang C, Wang Z C, Zheng Q R and Su G 2019 Eur. Phys. J. B 92 136
[24] Amin V P, Zemen J and Stiles M D 2018 Phys. Rev. Lett. 121 136805
[25] Manchon A and Zhang S 2008 Phys. Rev. B 78 212405
[26] Manchon A and Zhang S 2009 Phys. Rev. B 79 094422
[27] Baek S H, Borsa F, Furukawa Y, Hatanaka Y, Kawakami S, Kumagai K, Suh B J and Cornia A 2005 Phys. Rev. B 71 214436
[28] Barra A L, Bencini F, Caneschi A, Gatteschi D, Paulsen C, Sangregorio C, Sessoli R and Sorace L 2001 ChemPhysChem 2 523