Research On The Magnetic Exchange Interaction Of Local Spins In Several Non-equilibrium Systems

With the development of micro electronic technique,controlling magnetic exchange interactions is an important field on the micro scale.One of the most effective way to control the magnetic exchange interactions is through the indirect exchange interaction of Ruderman-Kittel-Kasuya-Yoshida(RKKY),an indirect exchange interaction between two magnetic impurities mediated by conduction electrons.In the early years,there have been several studies of the RKKY interaction in the equilibrium electron gas,graphene,and quantum dots,etc..Recently,people have been studies the RKKY interaction controlled at nonequilibrium open systems,they found that the RKKY interaction presents the anisotropic Heisenberg interaction and the indirect DM interactions.In this thesis,we will study the indirect magnetic exchange interactions of local spins in the nonequilibrium open quantum systems.Firstly,we have studied the nonequilibrium indirect exchange interactions of local spins that embedded in the nonequilibrium open quantum systems with the Rashba spinorbit coupling.We use the nonequilibrium Green's functions method to derive the analytic expression of the magnetic interaction coefficient under the second order perturbation approximation.And by symmetry analysis of Green's functions in non-equilibrium state,we gave the conditions that appear these interactions.When the system has neither a current nor a spin-polarized current(i.e.,an equilibrium state)or the system has only a current and without a spin current(i.e.,a bias voltage independent of the spin),it can appear the anisotropic Heisenberg interactions and the antisymmetric DM exchange interactions.Moreover,the DM vector has only one component,which direction is related to the form of spin-orbit coupling.This is different from an electronic system without spin-orbit coupling,which only present isotropic Heisenberg interactions but no DM interactions.When the system has no current but has a spin-polarized current(i.e.,a spin voltage bias related to the spin),it has not only anisotropic Heisenberg interactions and antisymmetric DM interactions,but also symmetric KSEA interaction(its coefficient is a second-order symmetric tensor with zero diagonal elements).Moreover,the DM vector has two components,and the second order tensor of KSEA interaction coefficient has a nonzero non-diagonal element,which is also related to the form of spin-orbit coupling.This is different from an electronic system without spin-orbit coupling,which only present DM interactions and no KSEA interaction.When the system has both current and spinpolarized current,anisotropic Heisenberg interaction,antisymmetric DM interaction and symmetric KSEA interaction will also be appeared.However,the DM vector has three components,and two nondiagonal elements are nonzero in the second order tensor of KSEA interaction coefficient.Finally,we use the two-lattice system as an example to numerically calculate the magnetic interaction under various conditions,we found that the coefficients of these magnetic interactions can be changed from ferromagnetic(antiferromagnetic)to antiferromagnetic(ferromagnetism)or positive(negative)to negative(positive)as the parameters increase.And these phenomena are explained by analyzing the filling of electrons in the energy levels.So,we can control the anisotropic Heisenberg interaction,DM interactions and KSEA interactions by bias voltage and spin voltage bias.Then,we have studied the nonequilibrium indirect exchange interactions of local spins that embedded in the nonequilibrium open superconducting electronic systems.We also use the nonequilibrium Green's functions method to derive the analytic expression of the magnetic interaction coefficient under the second order perturbation approximation.Because of the system has the superconducting term,anomalous Green's functions will appear in the coefficient expression.The Green's function of our system has the same symmetry as the Green's function of an electron system without superconductivity.So,when the system has no spin-polarized current(i.e.,an equilibrium state or only a bias voltage independent of the spin),it only appear isotropic Heisenberg interactions,no antisymmetric DM exchange interactions.The DM interactions present only the system has the spin-polarized current(i.e.,a bias voltage independent of the spin).This is the same as the system without superconducting electronic.Similarly,we use the small system as an example to numerically calculate the magnetic interaction under various conditions.We found that the coefficient of the antiferromagnetic exchange interaction of the Heisenberg interactions decreases,the coefficient of ferromagnetic exchange interaction increases,and the range of parameters of appear ferromagnetism increases as the superconducting strength increases.However,the magnitude of DM interaction is larger than that of non-superconducting electron systems,and increases with the increase of superconducting parameters.But the DM interaction is always positive as the average chemical increases.Similarly,these phenomena are explained by analyzing the filling of electrons in the energy levels.Thirdly,we have studied the nonequilibrium indirect exchange interactions of local spins that embedded in the nonequilibrium open periodically driven electronic systems.Because the system is periodically driven,we use the Floquet theory.We also use the nonequilibrium Green's functions method to derive the analytic expression of the magnetic interaction coefficient of Floquet representation under the second order perturbation approximation,then we take the periodic average and get the expression of the average magnetic interaction coefficient.Under Floquet representation,the periodically driven does not change the symmetry of Green's function.So,when the system has no spinpolarized current(i.e.,an equilibrium state or only a bias voltage independent of the spin),it only also appear isotropic Heisenberg interactions,no antisymmetric DM exchange interactions.The DM interactions present only the system has the spin-polarized current(i.e.,a bias voltage independent of the spin).This is the same as the case of non-periodically driven electronic systems.Then,we calculate an example of a small system driven by a single frequency,and discuss the system at high frequencies and low frequencies.We found that the expression of the magnetic interaction coefficient contains the higher harmonic of the periodic driving frequency.But in the calculation,we need to cut off the higher harmonics,and found that the Floquet Green's function would converge well as long as the number of the Floquet sidebands was large enough.At high frequencies,it only need a few number of the Floquet sidebands,then the Floquet Green's function will converge.And the behavior of the magnetic interaction as the parameters increases is similar to that of the system without periodically driven.However,at low frequencies,it needs more number of the Floquet sidebands,then the Floquet Green's function will converge.Moreover,the magnetic interaction will oscillate many times as the chemical formula and bias voltage increases,and the lower the frequency,the more intense the oscillation.And the system with periodically driven,the range of bias voltage and spin bias of present the magnetic interactions becomes wider.Finally,we have studied the nonequilibrium indirect exchange interactions of local spins that embedded in the nonequilibrium open periodically driven electronic systems with the Rashba spin-orbit coupling.Using the same method,we derive the analytical expression of the magnetic interaction coefficient under the second order perturbation approximation.Similarly,under Floquet representation,the periodically driven does not change the symmetry of Green's function.So,when the system has no spin-polarized current(i.e.,an equilibrium state or only a bias voltage independent of the spin),it only also appear anisotropic Heisenberg interactions and antisymmetric antisymmetric DM exchange interactions.The KSEA interactions present only the system has the spinpolarized current(i.e.,a bias voltage independent of the spin).This is the same as the case of non-periodically driven electronic systems with the spin-orbit coupling.Then,we calculate an example of a small system driven by a single frequency,and discuss the system at high frequencies and low frequencies.We found that the expression of the magnetic interaction coefficient also contains the higher harmonic of the periodic driving frequency.But in the calculation,we need to cut off the higher harmonics.Similarly,at high frequencies,it only need a few number of the Floquet sidebands,and the behavior of the magnetic interaction as the parameters increases is similar to that of the system without periodically driven.However,at low frequencies,it needs more number of the Floquet sidebands,and the magnetic interaction will oscillate many times as the chemical formula and bias voltage increases,and the lower the frequency,the more intense the oscillation.And the system with periodically driven,the range of bias voltage and spin bias of present the magnetic interactions also becomes wider.This is the same as the case of periodically driven electronic systems without the spin-orbit coupling.Therefore,in these four systems,Heisenberg interaction,DM interaction and KSEA interaction can be controlled by bias voltage and spin bias voltage.In the periodically driven electronic systems,we can also control the Heisenberg interaction,DM interaction and KSEA interaction through the frequency and amplitude of the periodically driven.