Photon-assisted electronic and spin transport in a junction containing precessing molecular spin
We study the ac charge and -spin transport through an orbital of a magnetic molecule with spin precessing in a constant magnetic field. We assume that the source and drain contacts have time-dependent chemical potentials. We employ the Keldysh nonequilibrium Green’s functions method to calculate the spin and charge currents to linear order in the time-dependent potentials. The molecular and electronic spins are coupled via exchange interaction. The time-dependent molecular spin drives inelastic transitions between the molecular quasienergy levels, resulting in a rich structure in the transport characteristics. The time-dependent voltages allow us to reveal the internal precession time scale (the Larmor frequency) by a dc conductance measurement if the ac frequency matches the Larmor frequency. In the low-ac-frequency limit the junction resembles a classical electric circuit. Furthermore, we show that the setup can be used to generate dc-spin currents, which are controlled by the molecular magnetization direction and the relative phases between the Larmor precession and the ac voltage.
pacs:73.23.-b, 75.76.+j, 85.65.+h, 85.75.-d
Since the early 1970s, the potential use of molecules as components of electronic circuitry was proposed,(1) thereby introducing the field of molecular electronics. Since then, the goal of the field has been to create high-speed processing molecular devices with miniature size.(2); (3) In that respect, it is important to investigate the properties of transport through single molecules in the presence of external fields.(4); (5); (6); (7); (8) Single-molecule magnets are a class of molecular magnets with a large spin, strong magnetic anisotropy, and slow magnetization relaxation at low temperatures.(9) Due to both classical(10) and quantum(10); (11); (12); (13) characteristics of single-molecule magnets, their application in molecular electronics became a topic of intense research, considering their potential usage in creation of memory devices.(14) Several experiments have already achieved transport through single-molecule magnets.(15); (16); (17)
Time-dependent transport through molecular junctions has been theoretically studied using different techniques, such as nonequilibrium Green’s functions technique,(18); (19); (20); (21); (22) time-dependent density functional theory,(23); (24); (25); (26); (27) reduced density matrix approach,(28) etc. Time-dependent periodic fields in electrical contacts cause photon-assisted tunneling,(29); (30); (31); (4) a phenomenon based on the fact that by applying an external harmonic field with frequency to the contact, the conduction electrons interact with the ac field and, consequently, participate in the inelastic tunneling processes by absorbing or emitting an amount of energy , where Theoretically, photon-assisted tunneling through atoms and molecules was investigated in numerous works.(4); (32); (33); (34); (35); (36); (37) Some experimental studies addressed photon-assisted tunneling through atomic-sized(38); (39); (40) and molecular(41); (42) junctions in the presence of laser fields. Time-dependent electric control of the state of quantum spins of atoms has also been investigated.(43) In junctions with time-dependent ac bias, the presence of displacement currents is inevitable due to the charge accumulation in the scattering region.(44); (45) This problem can be solved either implicitly by including the Coulomb interaction in the Hamiltonian of the system(46); (47) or explicitly by adding the displacement current to the conduction current,(45); (48) thus providing the conservation of the total ac current.
Spin transport through magnetic nanostructures can be used to manipulate the state of the magnetization via spin-transfer torques (STTs).(49); (50) The concept of STT is based on the transfer of spin angular momenta from the conduction electrons to a local magnetization in the scattering region, generating a torque as a back-action of the spin transport, and thus changing the state of the magnetic nanostructure.(49); (50); (51); (52) Hence, current-induced magnetization reversal became an active topic in recent years.(53); (54); (55); (56); (57); (58); (59) The measurement and control of the magnetization of single-molecule magnets employing spin transport may bring important applications in spintronics.
In this work we theoretically study the charge and spin transport through a single electronic energy level in the presence of a molecular spin in a constant magnetic field. The electronic level may be an orbital of the molecule or it may belong to a nearby quantum dot. The molecular spin, treated as a classical magnetic moment, exhibits Larmor precession around the magnetic field axis. The Zeeman field and interaction of the orbital with the precessing molecular spin result in four quasienergy levels in the quantum dot, obtained using the Floquet theorem.(60); (61); (62); (63) The system is then connected to electric contacts subject to oscillating electric potentials, considered as a perturbation. The oscillating chemical potentials induce photon-assisted charge and spin tunneling. A photon-assisted STT is exerted on the molecular spin by the photon-assisted spin-currents. This torque is not included in the dynamics of the molecular spin, since the molecular spin precession is assumed to be kept steady by external means, thus compensating the STT. The precessing molecular spin in turn pumps spin-currents into the leads, acting as an external rotating exchange field. Some of our main results are as follows:
In the limit of low ac frequency, the junction can be mapped onto a classical electric circuit modeling the inductive-like or capacitive-like response.
The real and imaginary components of the dynamic conductance, associated with the resonant position of the chemical potentials with molecular quasienergy levels, are both enhanced around the ac frequency matching the Larmor frequency, allowing the detection of the internal precession time scale (see Fig. 4).
The setup can be employed to generate and control dc spin currents by tuning the molecular precession angle and the relative phases between the ac voltage and Larmor precession if ac frequency matches the Larmor frequency.
A part of this article is a complement to Ref.(64), representing the solution for the Gilbert damping coefficient,(65) nonperturbative in the coupling to the molecular magnet, in the absence of time-varying voltage. The other corresponding STT coefficients and an arising nonzero component of the STT are obtained as well.
The article is organized in the following way: We describe the model setup of the system in Sec. II. The theoretical formalism based on the Keldysh nonequilibrium Green’s functions technique(18); (19); (20) is introduced in Sec. III. Here we derive expressions for spin and charge currents in linear order with respect to ac harmonic potentials in the leads. In Sec. IV we obtain and analyze the dynamic conductance of the charge current using the current partitioning scheme developed by Wang et al.(48) This section is followed by Sec. V in which we analyze spin transport and STT under dc-bias voltage and in the presence of oscillating chemical potentials. We finally conclude in Sec. VI.
Ii Model setup
We consider a junction consisting of a single spin-degenerate molecular orbital of a molecular magnet with a precessing spin in a constant magnetic field along -axis, , coupled to two normal metallic leads. We assume the spin of the molecular magnet is large and neglecting the quantum fluctuations treat it as a classical vector , with constant length . The magnetic field does not affect the electric contacts, which are assumed to be noninteracting. An external ac harmonic potential is applied to each lead , modulating the single electron energy as , with being the single-particle energy of an electron with the wave number , in the absence of the time-varying voltage (see Fig. 1). Since we want to unravel the quantum effects induced by the tunneling electrons and the ac harmonic potentials, we consider a well coupled molecular orbital and treat it as noninteracting by disregarding the intraorbital Coulomb interactions between the electrons.
The junction is described by the Hamiltonian . Here is the Hamiltonian of lead . The subscript denotes the spin-up or spin-down state of the electrons. The tunneling Hamiltonian introduces the spin-independent tunnel coupling between the molecular orbital and the leads, with matrix element . The operators and represent the creation (annihilation) operators of the electrons in the leads and the molecular orbital. The next term in the Hamiltonian of the system is given by . Here, the first term describes the noninteracting molecular orbital with energy . The second term represents the electronic spin in the molecular orbital, , in the presence of the external constant magnetic field , and the third term expresses the exchange interaction between the electronic spin and the molecular spin . Here, represents the vector of the Pauli matrices. The proportionality factors and are the gyromagnetic ratio of the electron and the Bohr magneton, respectively, while is the exchange coupling constant between the molecular and electronic spins.
Presuming for simplicity, that the molecular spin factor equals that of a free electron, the term represents the energy of the classical molecular spin in the magnetic field . Accordingly, the field exerts a torque on the spin leading to its precession around the field axis with Larmor frequency . To compensate for the dissipation of magnetic energy due to the interaction with conduction electrons, we assume that the molecular spin is kept precessing by external means (e.g., rf fields).(66) Hence, we keep the tilt angle between and fixed and determined by the initial conditions. The dynamics of the molecular spin is then given by , where is the magnitude of the instantaneous projection of onto the - plane, given by , while the projection of the molecular spin on the -axis equals . The precessing spin pumps spin-currents into the system, but the effects of spin currents onto the molecular spin dynamics are compensated by the above-mentioned external sources.
Iii Theoretical Formalism
The ensemble and quantum average charge and spin currents from the lead to the molecular orbital are given by
with representing the charge and spin occupation number operator of the contact . The index takes values for the charge and for the components of the spin-polarized current. The prefactors correspond to the electronic charge and spin . Employing the Keldysh nonequilibrium Green’s functions technique, the currents can be calculated in units in which as (19); (20)
where is the identity operator, while are the Pauli matrices. In Eq. (III), are the retarded, advanced, and lesser self-energies from the tunnel coupling between the molecular orbital and the lead , while are the corresponding Green’s functions of the electrons in the molecular orbital. The matrices of the self-energies are diagonal in the electronic spin space with respect to the basis of eigenstates of , and their nonzero entries are given by , where are the retarded, advanced and lesser Green’s functions of the electrons in contact . The matrix elements of the Green’s functions are given by and , where denotes the anticommutator. The self-energies of lead can be expressed as(18); (19); (20)
Here we introduced the Faraday phases . From its definition, it follows that . Furthermore, is the Fermi-Dirac distribution of the electrons in the lead , with the Boltzmann constant and the temperature, while is the tunnel coupling to the lead . Using the self-energies defined above, and applying the double Fourier transformations in Eq. (III), in the wide-band limit, in which is energy independent, one obtains
with the abbreviations and . The generating function was used in Eq. (III), where is the Bessel function of the first kind of order .
The matrix components of the retarded Green’s function of the electrons in the molecular orbital, in the absence of the ac harmonic potentials in the leads, can be obtained exactly by applying Dyson’s expansion and analytic continuation rules.(20) Their double Fourier transforms are written as(67)
with and . The matrix elements of the corresponding lesser Green’s function are obtained using the Fourier transfomed Keldysh equation .(20) Here and is the lesser self-energy originating from the orbital-lead coupling in the absence of harmonic potentials in the leads. The retarded Green’s functions of the electrons in the molecular orbital, in the presence of the static component of the molecular spin and the constant magnetic field , are found using the equation of motion technique(68) and, Fourier transformed, read ,(67); (59) where and .
For a weak ac field , the retarded and lesser Green’s functions of the electrons in the molecular orbital can be obtained by applying Dyson’s expansion, analytic continuation rules, and the Keldysh equation.(20) Keeping only terms linear in they read
In the rest of the paper we will stay in this limit.
The particle current contains the following contributions:
The first component represents the transport in the absence of ac voltages in the leads. It has a static and a time-dependent contribution, which are both created by the precession of the molecular spin. This precession-induced current reads
In the limit , Eq. (III) reduces to the result obtained previously.(64) The second term of Eq. (10) is induced when an ac voltage is applied to lead and can be expressed in linear order with respect to using Eqs. (III), (8), and (III) as
These expressions for the currents constitute the main results of the article. They allow us to calculate the dynamic charge conductance and spin transport properties of our molecular contact. Note that spin currents are more conveniently discussed in terms of the spin-transfer torque exerted by the inelastic spin currents onto the spin of the molecule, given by (49); (50); (51); (52)
Hence, in the remainder of the article we will concentrate on the ac charge conductance and the dc spin-transfer torque.
Iv Charge transport
iv.1 Dynamic charge conductance
The time-dependent particle charge current from the lead to the molecular orbital is induced by the ac harmonic potentials in the leads and can be written as
where is the conductance between leads and .
In order to determine the dynamic conductance under ac bias-voltage conditions, one also needs to take into account the contribution from the displacement current. Coulomb interaction leads to screening of the charge accumulation in the quantum dot given by . According to the Kirchhoff’s current law, . The following expression defines the total conductance of charge current, :
while the displacement conductance is given by
The conservation of the total charge current and gauge invariance with respect to the shift of the chemical potentials lead to and .(45) These equations are satisfied by partitioning the displacement current into each lead,(48) , or, equivalently, , in such a way that the sum of the partitioning factors obeys . Using the sum rules given above one obtains the expression for the dynamic conductance,(45); (48)
where , , and . The first term of Eq. (17) represents the dynamic response of the charge current, while the second term is the internal response to the applied external ac perturbation due to screening by Coulomb interaction. Note that the dynamic conductance consists of a real dissipative component , and an imaginary nondissipative component indicating the difference in phase between the current and the voltage. Due to the total current conservation, the two terms in Eq. (17) should behave in a way that a minimum (maximum) of corresponds to a maximum (minimum) of for both real and imaginary parts.
iv.2 Density of states in the quantum dot
Since the dynamic conductance is an experimentally directly accessible quantity, we hope that a measurement can help to reveal the internal time scales of the coupling between the molecular and electronic spins in the transport. We begin by analyzing the density of states available for electron transport in the quantum dot
There are four resonant transmission channels. They are positioned at quasienergy levels (spin down), (spin up), (spin down) and (spin up).
The Hamiltonian of the molecular orbital is a periodic function of time , with period . Its Fourier expansion is given by . Applying the Floquet theorem one can obtain the Floquet quasienergy corresponding to the Floquet state in the Schrödinger equation
where .(60); (61); (62); (63) The Floquet Hamiltonian matrix is block diagonal, with matrix elements given by ,(61) where describes the Floquet states, while denotes the electron spin states. For restricted Floquet quasienergies to the frequency interval a block is given by
with . The corresponding Floquet quasienergies are eigenenergies of the matrix (20), equal to and . The precessing component of the molecular spin couples states with quasienergies and to states with quasienerges and , which differ in energy by an energy quantum . Namely, due to the periodic motion of the molecular spin an electron can absorb or emit an energy , accompanied with a spin flip. Spin-flip processes due to rotating magnetic field were analyzed in some works.(67); (64) A similar mechanism was discussed in a recent work for a nanomechanical spin-valve, in which inelastic spin-flip processes are assisted by molecular vibrations.(69)
iv.3 Analysis of dynamic conductance
Now we analyze the charge conductance in response to the ac voltages. The suppression of dc conductance of charge current due to photon-assisted processes in the presence of an ac gate voltage, or a rotating magnetic field, was discussed in Ref.(63). Here we consider ac conductance in a double-driving experiment, where we first induce molecular spin precession at Larmor frequency , and then turn on the oscillating fields with frequency in the leads. Assuming equal chemical potentials of the leads , we analyze the dynamic conductance at zero temperature. Since we work in the wide-band limit, this symmetry simplifies the partitioning factors to . Hence, Eq. (17) can be transformed into
Here is the effective transmission function that can be expressed as , which reads
The real part and imaginary part of the dynamic conductance versus chemical potential are plotted in Figs. 2(a) and 2(b). Both and achieve their maximum at , where the resonance peaks are positioned. In accordance with Eq. (21) the electrons in lead , with energies , can participate in the transport processes by absorbing a photon of energy . For the dynamic conductance reduces to dc conductance , and reaches its maximum at resonances given by the Floquet quasienergies.(63) The imaginary part of the dynamic conductance approaches zero for [black line in Fig. 2(b)]. The considerable contribution of the displacement current to the total current is reflected in the decrease of , and the increase of near resonances with increasing , as the displacement current opposes the change of the particle charge current under ac bias [red and blue dot-dashed lines in Figs. 2(a) and 2(b)]. For a small value of both and , shows sharp resonant peaks. However, with the increase of , each of the peaks in broadens [green line in Fig. 2(a)]. It approaches a constant value around the corresponding resonant level, with the width equal to , since the inequality
is the condition for the inelastic photon-assisted tunneling to occur.
iv.4 Frequency dependence of the ac conductance and equivalent circuit
The behavior of the ac-conductance in the low-ac-frequency regime can be understood using a classical circuit theory.(70) Namely, at small ac frequencies , the molecular magnet junction behaves as a parallel combination of two serial connections: one of a resistor and an inductor and the other of a resistor and a capacitor, i.e., as a classical electric circuit (see Fig. 3). Depending on the phase difference between the voltage and the current, the circuit shows inductive-like (positive phase difference) or capacitive-like (negative phase difference) responses to the applied ac voltage. Thus, the dynamic conductance can be expanded up to the second order in in the small-ac-frequency limit as
where , , L, and C denote the resistances, inductance and capacitance of the circuit. In our further analysis we will assume that . The first term of Eq. (IV.4) represents the dc conductance . The second, imaginary term, linear in , is in the low-ac-frequency limit.
Depending on the sign of , the linear response is inductive-like () while decreases, or capacitive-like () while increases with the increase of . For the system behaves like a resistor with . The nondissipative component shows inductive-like behavior for
as we have observed in Fig. 2(b) (red line), and capacitive-like or resistive behavior otherwise.
The behavior of the dynamic conductance components and as functions of the ac frequency for and , with two values of at zero temperature is presented in Fig. 4. The real part is an even, while the imaginary part is an odd function of . In the low-ac-frequency regime , is a quadratic function, while is a linear function of ac frequency (solid and dashed black lines in Fig. 4). By fitting parameters of these functions and using Eq. (IV.4), one obtains circuit parameters , , and , confirming that in this limit the ac conductance of the system resembles the previously described classical circuit model. The circuit parameters can be calculated in terms of the dynamic conductance according to Eq. (IV.4). Note that they depend on the relative position of the Fermi energy of the leads with respect to the molecular quasienergy levels.
Near the four resonances we expect the system to be highly transmissive and therefore to conduct well. This is confirmed by Figs. 2 and 4. Namely, the imaginary conductance component around resonances and is a positive linear function of in the low-ac-frequency limit [see Fig. 4(b), black solid line]. This implies that the behavior of the system is inductive-like, since the displacement current tends to reduce the charge current, as electrons reside awhile in the quantum dot, causing the delay in phase between the voltage and the current. Accordingly, the real component decreases quadratically from initial value upon switching on the ac frequency [black solid line in Fig. 4(a)]. However, the off-resonance behavior is capacitive-like resulting from intraorbital Coulomb interactions, included via displacement current.(48) Hence, in the low-ac-frequency limit is negative and decreases linearly with the increase of for Fermi energies of the leads which are far from the resonant energies [black dashed line in Fig. 4(b)]. In this case increases quadratically with [black dashed line in Fig. 4(a)]. Obviously, the molecular magnet junction behaves as a classical circuit only in the low-ac-frequency regime.
For higher ac frequencies we use Eq. (21) to analyze the behavior of and , where the dynamic response of the system remains predominantly inductive-like for . With further increase of , the ac conductance vanishes asymptotically. Upon turning on the ac frequency, while the system is on resonance , the imaginary component increases quickly from 0 to a local maximum and then decreases to its minimum value around [green and blue lines in Fig. 4(b)]. The real part decreases to a local minimum and then has a steplike increase towards a local maximum around [green and blue lines in Fig. 4(a)]. This behavior of the dynamic conductance can be understood as follows. For , at , besides the resonant level with quasienergy , the upper level with quasienergy becomes available for photon-assisted electron transport. It is then distanced by the energy from the chemical potential . Consequently, an electron with Fermi energy equal to can absorb a photon of energy in the lead and tunnel into the level with quasienergy . This leads to an enhancement of the response functions and , after going to a local minimum, with features corresponding to photon-assisted tunneling processes. Each steplike increase of and the corresponding dip of in Fig. 4 are determined by the difference between the quasienergy levels and the chemical potential , viz. . Thus, for and the set of parameters given in Fig. 4, they are positioned around and . For the larger tunnel couplings each steplike increase in is broadened due to the level broadening . We notice that the enhancement of the dynamic conductance is higher around than around the subsequent frequency . This is due to the fact that the frequency has to traverse one resonant peak in , or dip in , to reach the second one. We need to mention that the off-diagonal conductances , where , and hence have a behavior that opposes that of the diagonal ones.
In the spirit of the scattering matrix formalism, the dynamic conductance of our molecular magnet junction, in the low-ac-frequency regime, can be expanded as(71)
where is the dc conductance. The quantity is called the emittance.(71) It contains the contribution from the displacement current and the partial density of states that characterize the scattering process.(46); (72); (73) The partial density of states can be calculated using the scattering matrix, and can be understood as density of states due to electrons injected from lead , and leaving through lead .(46); (72); (73) The emittance measures the dynamic response of the system to an external oscillating ac field and, depending on its sign, the response is capacitive-like or inductive-like.(71) The matrix element of the third term, , represents the correction to the real part of the dynamic conductance and describes the dynamic dissipation in the low-ac-frequency regime.(71) Both and obey the sum rules, since the total current conservation and gauge invariance conditions have to be satisfied.(45) According to Eq. (26), their diagonal elements and can be approximated as and in the low frequency limit.(71) Based on the analyzed and the behavior of and can be examined. Around all resonances the emittance (inductive-like response) and since , while off resonance (capacitive-like response) and (see Figs. 2 and 4).
iv.5 Effects of the molecular magnetization direction on the ac conductance
Now we analyze the ac conductance components and as functions of the tile angle of the molecular spin from the external field , plotted in Figs. 5(a) and 5(b). For , the peaks of both and in Figs. 2(a) and 2(b) at are much lower than those at , implying that the molecular magnet junction is less transmissive at the upper two mentioned resonances. This can be qualitatively understood by looking at Fig. 5. The behavior of the conductance components near the resonances for (solid lines in Fig. 5) and (dot-dashed lines in Fig. 5), depends on the direction of with respect to the external magnetic field . For the molecular spin is static and the only two levels available for electron transport are Zeeman levels and . In this case, when the system is at the resonance , the components and take their maximum values, and displaying an inductive-like behavior. For and , both and take their minimum values. There is no transmission channel at this energy for , but is relatively large, and displays a capacitive-like response. With the increase of , the additional two channels at energies and appear and become available for electron transport. This leads to the increase of conductance components and at , and their decrease at , as functions of (see Fig. 5). For , in the case of small the complex components of the effective transmission function approach the same height at resonant energies , so the probability of transmission reaches equal value at each level. Thus, both and show peaks of the same height at the resonances. The points of intersection of solid and dot-dashed lines of the same color in Fig. 5 correspond to this particular case. For larger frequencies , these points are shifted away from , since the peaks broaden and overlap and the suppression or increase of and is much faster. Finally, for the situation is reversed compared to the one with , as again the static spin is in the direction opposite that of the external field . The Zeeman splitting in this case is equal to , so the only two levels available for electron transport are and . Therefore, for