A Effective action and stochastic fields

Time-dependent spin and transport properties of a single-molecule magnet in a tunnel junction


In single-molecule magnets, the exchange between a localized spin moment and the electronic background provides a suitable laboratory for studies of dynamical aspects of both local spin and transport properties. Here we address the time-evolution of a localized spin moment coupled to an electronic level in a molecular quantum dot embedded in a tunnel junction between metallic leads. The interactions between the localized spin moment and the electronic level generate an effective interaction between the spin moment at different instances in time. Therefore, we show that, despite being a single spin system, there are effective contributions of isotropic Heisenberg, and anisotropic Ising and Dzyaloshinski-Moriya character acting on the spin moment. The interactions can be controlled by gate voltage, voltage bias, the spin polarization in the leads, in addition to external magnetic fields. Signatures of the spin dynamics are found in the transport properties of the tunneling system, and we demonstrate that measurements of the spin current may be used for read-out of the local spin moment orientation.

73.63.Rt, 75.30.Et, 72.25.Hg, 75.78.-n

I Introduction

Single-molecule magnets provide interesting workbench opportunities to study quantum phenomena related to their individual properties as well as promising potential for quantum information technology and quantum computation based on spintronics devices. Easy control of single magnetic moments paved the way for a deeper exploration of, e.g., magnetic anisotropies and exchange interaction, as well as new routes for significantly less energy consuming active electronics devices and information storage.

Molecular magnets offer a platform for studies of magnetic properties on a fundamental level due to their intrinsic discreteness. Experimentally, this has paved the way for electronic control and detection of the magnetization of individual molecules Hauptmann et al. (2008); Loth et al. (2010); Wagner et al. (2013), magnetic anisotropy, and exchange interaction of single atoms such as, e.g., Co and Mn on a surface Hirjibehedin et al. (2006); Wahl et al. (2007); Meier et al. (2008); Balashov et al. (2009); Voss et al. (2008), and tuning of the magnetic anisotropy in molecular magnets Bairagi et al. (2015). Furthermore, spatial anisotropies have been observed for the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction Zhou et al. (2010) as well as signatures of superexchange interaction and the long-range Kondo effect between single magnetic molecules Chen et al. (2008); Otte et al. (2009); Pruser et al. (2011). These advances in experimental techniques have led to realizations of magnetically stable atomic scale configurations Khajetoorians et al. (2011); Loth et al. (2012); Khajetoorians et al. (2013) that are important steps toward the creation of stable magnetic memory devices at the atomic scale. Magnetic molecules containing transition metal atoms, e.g., M-phthalocyanine and M-porphyrins where M denotes a transition metal element (Cr, Mn, Fe, Co, Ni, Cu) Wende et al. (2007); Fernández-Torrente et al. (2008); Chiesa et al. (2013); Raman et al. (2013); Fahrendorf et al. (2013), as well as single molecules comprising complexes of transition metal elements Carretta et al. (2004); Cornia et al. (2004) and antiferromagnetic rings van Slageren et al. (2002); Carretta et al. (2003); Troiani et al. (2005); Carretta et al. (2007); Wedge et al. (2012); Candini et al. (2010) have been explored in many different contexts.

For technological applications, on the other hand, the potential of molecular magnets and magnetic materials is unlimited. A range of different spintronics devices have been proposed, both using spin currents Chappert et al. (2007) or spin torque Locatelli et al. (2014). Such devices include molecular spin-transistors, molecular spin-valves, molecular multidot devices Bogani and Wernsdorfer (2008), etc. These can potentially be used both as building blocks of quantum computers Leuenberger and Loss (2001) and as quantum simulators Antropov et al. (1995). There are already several experimental realizations of these kinds of devices, including magnetic memories and spin qubits Mannini et al. (2009); Timco et al. (2009); Mannini et al. (2010); Carretta et al. (2006).

On the theoretical side, we have witnessed great progress over the course of the past decade in developments of the theory for, e.g., single molecular magnets and magnetization dynamics. There have been several studies of magnetic exchange interaction and the possibilities for electrical control of the interaction and spin transport Bhattacharjee et al. (2012); Zhang and Zhang (2009); Szilva et al. (2013); Fransson (2008a); Filipović et al. (2013); Fransson et al. (2014). Under non-equilibrium conditions, magnetic molecules show signatures of intrinsic anisotropic exchange interactions that can be used to control molecular spin Misiorny et al. (2013); Szilva et al. (2013), something that may lead to read-and-write capabilities with currents in spintronics devices Timm and Elste (2006); Misiorny and Barnaś (2007); Moldoveanu et al. (2015). Non-equilibrium studies of transport properties have, moreover, suggested that vibrations coupled to the spin degrees of freedom may induce electrical currents that can provide interesting properties for, e.g., mechanical control of single magnetic molecules Härtle et al. (2011); Roura-Bas et al. (2013). Superconducting spintronics also paves the way toward enhancing central effects of spintronics devices Linder and Robinson (2015); Stadler et al. (2013); Holmqvist et al. (2011, 2014).

The majority of the reported theoretical progress is, however, has been limited to stationary, or Markovian, processes. Although this is an important regime, both for fundamental studies as well as for technological applications, it is nonetheless crucial to control also transient properties induced by sudden on-sets and variation of the external conditions applied to the system. Regarding spin dynamics, the Landau-Lifshitz-Gilbert (LLG) equation is often postulated as the platform for theoretical studies, despite the fact that the (exchange and damping) parameters for this equation are typically taken on phenomenological grounds or from experiments. These parameters are, in addition, assumed to have a negligible time-dependence, something that cannot be taken for granted in nanoscale systems. Previous derivations of the LLG equation Fransson and Zhu (2008); Bhattacharjee et al. (2012) clearly illustrate that the electronically mediated exchange interactions depend strongly on the magnetization dynamics and are, hence, intrinsically dynamical quantities as well. The non-linearity of the dynamical equations indicates, moreover, that it is non-trivial to decide whenever the time-dependence of the interaction parameters can be neglected.

To begin to depart from the ad hoc treatments of the dynamics of spins coupled to electron currents, in this paper we perform time-dependent studies and analyses beyond the Markovian and adiabatic approximations for both the spin-dynamics and the tunneling current. In addition, we include the interdependence between the current through the molecule and the localized magnetic moment by considering both action and back-action in the description. This can be regarded as the first loop in a self-consistent calculation, however, we do not perform our calculations to full self-consistency.

Figure 1: The system studied in this work consisting of a local magnetic moment coupled to a QD in a tunnel junction between ferro- and non-magnetic leads.

The model system, onto which we apply our developed method, is comprised of a magnetic molecule that is embedded in the tunnel junction between metallic leads. The leads themselves may support spin-polarized currents. Here, the magnetic molecule consists of two components, namely, a quantum dot (QD) level and a localized magnetic moment, that interact via exchange. The QD level is tunnel coupled to the leads. Hence, the current flowing through the metal-QD-metal complex is expected to probe the presence of the localized magnetic moment, and, vice versa, the localized magnetic moment is expected to depend on the current. Taking this observation as an initial condition for our studies, we construct a calculation scheme in which the dynamics of the localized magnetic moment is described by a generalized version of the Landau-Lifshitz-Gilbert equation Fransson (2008b, a); Fransson and Zhu (2008); Bhattacharjee et al. (2012). The effective spin-spin interactions are mediated by the tunneling current flowing across the junction. The current, on the other hand, depends directly on the presence and dynamics of the localized magnetic moment. We include this dependence by feeding the time-evolution of the spin dynamics into the current, which causes the current-dependent temporal spin fluctuations to generate signatures back into the current.

The effective spin model derived in Sec. II, depends only on the parameters included in our microscopical model – there are no ad hoc contributions in the description in addition to the basic model. However, within the realms of the model, there is a current mediated spin-spin interaction generated in the effective spin model, which describes interactions between the spin at time and time . Hence, although there is only one spin in the system, it is still justified to introduce the concept of spin-spin interaction since the spin at different times can be regarded as different spins.

Separation of the magnetic molecule into a QD level and a localized magnetic moment is justified for, e.g., M-phthalocyanines and M-porphyrins. In these compounds, the transition metal -levels, which are deeply localized, constitute the localized magnetic moment. The - and -orbitals in the ligands, on the other hand, generate the spectral intensity at the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) levels which may be considered as the QD level(s) in our model.

Previously, Bode et al. Bode et al. (2012) performed a similar theoretical treatment of this problem using a non-equilibrium Born-Oppenheimer approximation. Here, however, we go beyond the adiabatic limit and extend the model to the non-Markovian regime in order to treat memory effects and its impact on the exchange interaction. Hence, the interaction fields in the spin equation of motion are not only time-dependent but also dependent on their time-evolutions. A major difference with this formulation is that all retardation effects are included in the time integration of the interaction fields, and it is, therefore, not meaningful to discuss quantities such as Gilbert damping since such parameters are defined in the adiabatic limit.

In general, there also exist stochastic field acting on the localized spin as a result of its interaction with the surrounding electrons. Here, we have chosen to omit the action of these fields, despite their importance for a full description of the physics Arrachea and von Oppen (2015). However, since we consider the physics in the wide band limit, these electronically induced stochastic fields are of Gaussian white noise character with no voltage bias dependence; see Appendix A. The stochastic field in this limit will, therefore, merely play the role of a structureless thermal noise field. As the main focus of our work is on the dynamics of the localized magnetic moment and the exchange interactions, we notice that our results are valid whenever the energies of the interactions are larger than the corresponding energies of these thermal noise fields. Adding a Langevin term, which arises from the quantum fluctuation in the spin action Fransson and Zhu (2008), into the spin equation of motion could be an interesting extension of the model used in this work, which would be the objective for a separate study.

The paper is organized as follows. In Sec. II we discuss the basic set-up of the formalism we employ in this study. After defining the model for the magnetic molecular QD, we derive the equations for the spin moment and the tunneling current. Numerical results from these equations are presented in Sec. III and we summarize and conclude the paper in Sec. IV.

Ii Method

To be specific, we consider a magnetic molecule embedded in a tunnel junction between metallic leads that may support spin-polarized currents, see Fig. 1 for reference. The magnetic molecule comprises a localized magnetic moment coupled via exchange to the highest occupied molecular orbital (HOMO) or lowest unoccupied molecular orbital (LUMO) level, henceforth referred to as the QD level. We define our system Hamiltonian as


Here, is the Hamiltonian for the lead , where () creates (annihilates) an electron in the lead with energy , momentum k and spin . We have introduced the chemical potential for the leads and the voltage bias across the junction defined as . Tunneling between the leads and the QD level is described by , where . The single-level QD is represented by , where () creates (annihilates) an electron in the QD with energy and spin . We include the Zeeman split due to the external magnetic field where g is the gyromagnetic ratio and the Bohr magneton. The local spin is described by , where is the interacting rate between the local spin and the electron spin , whereas is the vector of Pauli matrices.

ii.1 Equation of motion of the local magnetic moment

Using the methods in, e.g., Refs. Fransson (2008b, a); Fransson and Zhu (2008); Bhattacharjee et al. (2012); Fransson et al. (2014) and Appendix A, we derive an effective spin model for the localized magnetic moment from which we obtain the equation of motion


Here, in order to arrive at this result we have neglected longitudinal spin fluctuations () and rapid quantum fluctuations. The effective magnetic field is defined as


where is the external magnetic field while the second term provides the internal magnetic field due to the electron flow, where


Here, and the charge , where is the identity matrix. This two-electron Green function (GF) is approximated by a decoupling into single electron GFs according to


where is the lesser/greater matrix GF of the QD defined by and . In Eq. (5) denotes the trace over spin 1/2 space.

The current is the electron spin-spin correlation function which mediates the interactions between the localized magnetic moment at times and . As with the internal magnetic field, we decouple this two-electron GF according to


This current mediated interaction can be decomposed into an isotropic Heisenberg, , interaction and the anisotropic Dzyaloshinski-Moriya (DM), , and Ising, , interactions. This can be seen from the product , which is the corresponding contribution in the effective spin model Fransson et al. (2014) to in the spin equation of motion. Using the general partitioning , where and describes the electronic charge and spin, it is straight forward to see that


where we have used the identity . As the Pauli matrices are traceless, the above expression reduces to


After a little more algebra we obtain the Heisenberg (), anisotropic Ising () and anisotropic Dzyaloshinsky-Moriya (D) interactions


This leads to that we can partition the current mediated spin-spin interaction in the spin equation of motion into


In absence of spin-dependence in the QD GF, that is, for , it is clear that only the Heisenberg interaction remains, since both and explicitly depend on . There are different sources that generates a finite , e.g., spin injection from the leads, Zeeman split QD level, but also the interaction with the localized magnetic moment gives an essential contribution. In this paper, we include effects from all three sources.

The spin equation of motion derived here goes far beyond the LLG equation as it includes all retardation effects under the time-integration, something which is essentially missing in the LLG equation except for the static exchange interaction and Gilbert damping. Starting from Eq. 2 and restricting to the adiabatic limit it is possible to derive the conventional LLG equation, see Ref. Bhattacharjee et al. (2012). For clarity this is also done in the Appendix B. This also implies that Eq. 2 includes the important Gilbert damping and spin-transfer torque, as discussed in Ref. Bode et al. (2012) and Ralph and Stiles (2008). Higher order retardation effects (dissipation, moment of inertia, etc) are included in the time-integral .

ii.2 Quantum dot GF

Bare quantum dot Green function

Next, we derive the GF for the QD, which is defined as where T is the contour-ordering operator. We introduce a bare GF as the solution to the equation


The bare GF then describes the electronic structure of the QD when coupled to the leads through the self-energy , however, without any coupling to the local spin moment, as illustrated in Fig. 2 (a). Here,


is the GF for the lead , including the time-dependence imposed by the voltage bias.

The self-energy is treated in the wide-band limit (WBL), which for the retarded/advanced and lesser/greater forms are given by


where and , whereas


Here, is the Fermi function. The WBL allows to write the retarded/advanced zero GF as


By defining the coupling parameters and and introducing the spin-polarization in the leads , such that , we can write . With this notation we can introduce the coupling matrix , where and . Analogously, we write the retarded/advanced and lesser/greater self-energies as and , where

Figure 2: Sketch of the system without (a) and with (b) a local magnetic moment and coupled to the leads. In the latter case the interactions with the spin moment induce an effective Zeeman split.

Using this notation we partion the bare GF in terms of its charge and magnetic components according to . The retarded/advanced form of can then be written


Analogously, the lesser/greater forms of are given by


where (time-dependence of the propagators in the integrands is suppressed)


Dressed quantum dot Green function

The next step is to include the interactions with the local magnetic moment into the description. We achieve this goal by defining the dressed QD GF as the first order expansion in terms of the local moment, that is,


where is the bare GF and is the correction from the interactions with the local magnetic moment. As above, we write , where and , whereas the corrections are given by


We refer to Appendix C for details about the lesser/greater forms of the charge and magnetic components of .

It should be noticed that the presence of the local spin moment gives rise to a spin-polarization of the QD level due to the local exchange interaction, see Fig. 2 (b) for an illustration. The effect is particularly strong whenever there is an intrinsic spin-polarization in either the leads and/or the QD, in which case . Then, the local spin moment affects the properties of both the charge and magnetic structure of the QD. Nevertheless, even for spin-degenerate leads and QD, that is, for , the QD level acquires a finite spin-dependence. This is legible in the expression for , where the first term only depends on the magnetic properties of the local spin moment and the charge density in the QD. Thus, by calculating the electronic structure in the QD as function of the local spin moment opens for tracing signatures of the local spin dynamics in the properties of the QD.

ii.3 Current

The properties of the QD are probed by means of the electron currents flowing through the system. In this way, the goal is to pick up signatures of the spin dynamics in the transport properties as these should influence the electronic structure of the QD. The electron currents can be decomposed into charge and spin currents, and , respectively. Here, we calculate the currents flowing through the left interface between the leads and the QD. Accordingly, we define


Using standard methods we can write the charge current as


Following the same route as initiated above, we partition the current into a spin-independent and spin-dependent part according to , where


Analogously to the charge current, we write the spin current as


where and


These expressions for the charge and spin currents suggest that any local dynamics that is picked up by the electronic structure of the QD should provide signatures in its transport properties. Next, we analyze the impact of the local dynamics on the transport properties.

Iii Results

iii.1 Stationary limit

Before embarking into the full time-dependent properties of the system, we review some of the expected results for the stationary regime in order to provide a benchmark for our calculations. In the stationary limit all the time-dependences induced from the on-set of the applied voltage bias have decayed which leads to the bare QD GF becomes time local and we can, therefore, study the energetic properties of the QD. Then, the local magnetic moment, , can be regarded as a constant spin-polarization and a source for coupling between the spin states, in agreement with Ref. Filipović et al. (2013). The Fourier transform of the bare QD GF is, therefore, written on the form




and the self-energies become


since in the stationary limit. In the stationary limit the interaction parameters, moreover, simplify in the limit to

Figure 3: Charge and spin current for a static local magnetic moment in a tunnel junction. (a) Charge current as function of and (b) spin current as function of . Here, we used meV, = 1 K, = 1 T, and = 2 mV such that and .
Figure 4: Heisenberg interaction of a static local magnetic moment in a tunnel junction in the z-direction, . Panel (a) shows the Heisenberg interaction for non-magnetic leads as a function of bias voltage V. Here, the gate voltage is set to meV and meV and the plots shifted for clarity (scale is the same). Panel (b) shows the Heisenberg interaction for antiferromagnetic leads, , as a function of bias voltage V. Here, we used meV, = 1 K, = 0 T and = 2 mV.
Figure 5: Ising interaction of a static local magnetic moment in a tunnel junction in the z-direction, . Panel (a) shows the Ising interaction for non-magnetic leads as a function of bias voltage V. Here, the gate voltage is set to meV and meV and the plots shifted for clarity (scale is the same). Panel (b) shows the Ising interaction for antiferromagnetic leads, , as a function of bias voltage V. Other parameters as in Fig. 4.
Figure 6: DM interaction D of a static local magnetic moment in a tunnel junction in the z-direction, , for antiferromagnetic leads, . Panel (a) shows the DM interaction as a function bias voltage V where the gate voltage is set to meV and meV. Panel (b) shows the DM interaction different gate voltage . Other parameters as in Fig. 4.

Considering a symmetric and spin-independent background, i.e. non-magnetic contacts , and a constant local magnetic moment, , the local spin-polarization gives rise to finite spin currents in the system, see Eqs. (25) – (26) (note ). In Fig. 3 we plot the calculated (a) charge () and (b) spin current () as a function of the gate voltage, , for a QD with a bare level at . While the charge current behaves as expected for a single-level QD, given by


the features in the spin current for gate voltages near zero give a clear indication of the induced spin-polarization from the local spin moment. Due to the local spin moment induced effective Zeeman split in the QD, as shown in Fig. 2 (b), the spin current is strongly peaked at . As can be seen in Fig. 2, either one of the spin up- or down channels will be more favorable for the tunneling electrons, thus causing a net spin current in either direction depending on the configuration of the electron level of the leads. This is an important feature as it can be used in order to read out the state of the local spin moment from the spin current.

Figure 7: Time-dependent evolution of the exchange interaction parameters as a function of gate voltage after an onset of a step-like finite bias voltage of V = 2 mV. Panel (a) shows the strength of the Heisenberg interaction, panel (b) shows the part of the Ising interaction and panel (c) shows the z-component of the DM interaction. Here we used the parameters meV, = 1 K, = 1 T, .

Regarding the Heisenberg interaction, recalling that, e.g., for non-magnetic leads, it can be readily seen that the charge contribution to the Heisenberg exchange is given by


This suggests a spin-spin interaction which is strongly peaked around . Similarly, the contribution from the local spin-polarization, , acquires the form


which is also strongly peaked at . However, as the integrand of this component changes sign at , the contribution from the QD spin-polarization goes through local minima at and a local maxima at , as a function of the chemical potential . We therefore expect a competition between the charge and magnetic components which may lead to a change of sign in the Heisenberg interaction, depending both on the properties of the system as well as on the external conditions. This is illustrated by the computed Heisenberg exchange plotted in Fig. 4 (a) as a function of the voltage bias for different gate voltages, showing the changing character from negative to positive interaction as the chemical potential approaches the QD level. For ferromagnetic leads aligned anti-ferromagnetically in Fig. 4 (b), , we notice an anisotropic behavior as the sign of the interaction switches with respect to the polarity of the voltage bias. This is in agreement with previous studies of anti-ferromagnetically aligned leads coupled to molecular spins Misiorny et al. (2013).

The Ising interaction essentially behaves in a similar manner, however, this contribution requires a finite spin-polarization () to become non-vanishing. For non-magnetic leads, , this spin-polarization is provided by the local spin moment and we find that the Ising interaction acquires the form given in Eq. (33), up to multiplying constants. This is also verified by the numerically computed Ising interaction, shown in Fig. 5 (a) as function of the voltage bias for different gating conditions. Again, for ferromagnetic leads in anti-ferromagnetic alignment, , there is a switching behavior with respect to the polarity of the voltage bias.

A similar switching behavior appears in the DM interaction D, which is only considered for ferromagnetic leads aligned anti-ferromagnetically, see Fig. 6 (a), where the DM interaction is plotted as a function of the voltage bias and for different gating conditions. Varying the gate voltage, it can be seen that there is a finite DM interaction only whenever the QD electron level, , lies in the window between the chemical potentials in the leads spanned by the voltage bias. This is understood since the DM interaction results from net current flow interacting with the local spin moment, as it requires simultaneous breaking of time-reversal and inversion symmetries to be finite.

We comment finally on the relevance for calculating the interaction parameters in the stationary limit. This question is justified since the effective spin Hamiltonian in the stationary limit would assume the form


Here, one can notice that , which is a constant of motion, whereas as . Both these identities relies on the fact that the spin is time-independent in the stationary limit. Actually, only the Ising interaction is physically motivated, providing an anisotropy field on the spin. For collinear spin-polarization in the surrounding system, this contribution reduces to the form , which is the ordinary Ising Hamiltonian.

We justify the calculations and analysis of the stationary limit interaction parameters by that we can understand and interpret much of the time-dependent features, discussed in the remainder of this paper, from the results obtained in the stationary limit. In addition, our results also demonstrate that despite the dynamics may be trivial, the fields that mediate the interactions between the dynamical object need not be trivial.

Figure 8: Contribution to the local magnetic moment equation of motion for different exchange interaction parameters as a function of gate voltage . In (a) the change in z-direction depending on the induced internal magnetic field due to the charge flow. In (b)-(d) the change in the z-direction depending on the Heisenberg, Ising and DM interaction is shown respectively. Other parameters as in Fig. 7.
Figure 9: Time-dependent evolution of the field from the anisotropic DM interaction in x- and y-direction as a function of gate voltage after an onset of a step-like finite bias voltage of V = 2 mV. Other parameters as in Fig. 7.

iii.2 Time-dependent exchange interaction

As we are interested in the transient dynamics, we study the effect of an abrupt on-set of the voltage bias applied as a step-like function symmetrically over the junction such that . Before the on-set of the voltage bias, the local spin is subject to the static external magnetic field , giving , and where , whereas and , and we assume an initial polar angle of .

The time-dependence of the interaction parameters, cf. Eq. (9), has to be calculated as function of the gate voltage and voltage bias at each time-step. In Fig. 7 we plot the time-evolution of the Heisenberg, Ising, and DM interaction parameters as function of the gate voltage, where we integrated over all , hence, showing , and . Considered in this fashion, the plots illustrate the time-evolution of the exchange interaction that would be expected in the adiabatic approximation, that is, . In the transient regime, the interaction parameters changes continuously, both due to the changing characteristics of the system and the feed-back through the system from the changing local magnetic moment. In the long time limit, it may be noticed that the interaction strength peaks for all three types of interactions when the QD electron level is resonant with one of the chemical potentials of the leads . We, hence, retain the properties of the system in the stationary regime.

When going beyond the adiabatic approximation, one cannot strictly separate the interaction parameters from the time-evolution of the spin, see for instance Eq. (2). It is then more comprehensible to directly study and analyze each component of the equation of motion. Accordingly, in Fig. 8 we plot the rates of change in the respective panels

  • ,