A Diagrammatic rules

Transport through quantum-dot spin valves containing magnetic impurities

Abstract

We investigate transport through a single-level quantum dot coupled to noncollinearly magnetized ferromagnets in the presence of localized spins in either the tunnel barrier or on the quantum dot. For a large, anisotropic spin embedded in the tunnel barrier, our main focus is on the impurity excitations and the current-induced switching of the impurity that lead to characteristic features in the current. In particular, we show how the Coulomb interaction on the quantum dot can provide more information from tunnel spectroscopy of the impurity spin. In the case of a small spin on the quantum dot, we find that the frequency-dependent Fano factor can be used to study the nontrivial, coherent dynamics of the spins on the dot due to the interplay between exchange interaction and coupling to external and exchange magnetic fields.

pacs:
73.23.Hk,85.75.-d,72.70.+m,71.70.Gm

I Introduction

Recently, there has been growing interest in spin-dependent transport through nanostructures due to possible applications for spintronics devices. Transport through quantum dots coupled to ferromagnetic electrodes is particularly interesting due to the interplay of strong Coulomb interaction on the quantum dot and nonequilibrium physics. Such systems have been realized experimentally, e.g., by coupling self-assembled semiconducting quantum dots, (1); (2); (3); (4); (5) small metallic grains, (6); (7); (8); (9); (10) quantum dots defined in InAs nanowires, (11) carbon nanotubes, (12); (13); (14); (15); (16) or even single C molecules (17) to ferromagnetic electrodes. Quantum dots coupled to ferromagnets have been also studied extensively from a theoretical point of view. (18); (19); (20); (21); (22); (23); (24); (25); (26); (27); (28); (29); (30); (31) Especially interesting are quantum-dot spin valves, i.e., a single-level quantum dot coupled to noncollinearly magnetized electrodes. (32); (33); (34); (35); (36); (37); (38); (39) Here, spin-dependent tunneling leads to a spin accumulation on the dot that in turn influences the transport through the system by blocking the current. In addition, there is an exchange field acting on the dot spin. It is due to quantum charge fluctuations on the quantum dot and relies on the strong Coulomb interaction on the dot. It gives rise to a precession of the accumulated spin, thereby lifting the spin blockade of the dot. The resulting interplay of spin accumulation and spin precession leads to a number of interesting features in the transport characteristics, e.g., a shift of the peaks in the linear conductance with the angle enclosed by the magnetizations, (32); (33) a broad region of negative differential conductance in nonlinear transport (33) as well as to characteristic features in the finite-frequency Fano factor (35) and a splitting of the Kondo resonance. (27); (28); (29); (30); (31)

While in the above studies all the spin dynamics takes place in the singly occupied orbital of the quantum dot, more complex spin dynamics appears when additional spin excitations are possible. These may be spin waves in the ferromagnetic leads, as discussed in Ref. (40). In the present paper, we investigate a different situation, namely, the coupling to a magnetic impurity either with a large, anisotropic spin or with a spin .

We consider two different scenarios. In the first scenario, a magnetic impurity with a large, anisotropic spin is embedded in one of the tunnel barriers of a quantum-dot spin valve. Here, our main focus is on the spectroscopy of the impurity spin as well as on its switching by the spin-polarized current. We consider the impurity spin in one of the barriers as this leads to a simpler conductance pattern (only processes involving this particular barrier can excite the spin), thereby simplifying the analysis of the impurity spin behavior. Our system is somewhat related to the case of transport through single tunnel barriers containing a magnetic atom or a single molecular magnet, that has been investigated extensively in the recent past, both from a theoretical (41); (42); (43); (44); (45); (46); (47); (48); (49) as well as from an experimental (50); (51); (52); (53); (54); (55); (56) point of view. It was shown that the steps observed in the differential conductance can be used to extract magnetic properties such as anisotropies of the atomic spin. (53); (44); (45); (46); (47) Furthermore, the influence of nonequilibrium spin occupations was discussed, (49) explaining the overshooting observed at the conductance steps in experiment and predicting a super-Poissonian current noise. The absence of certain nonequilibrium features in turn was interpreted in terms of an anisotropic relaxation channel. For systems with magnetic electrodes, the possibility to switch the embedded spin by the spin-polarized current through the barrier was predicted theoretically (48); (42); (43) and observed in experiment. (56)

In the model studied in this paper, the tunnel barrier containing the magnetic impurity connects a lead with a quantum dot. The transport behavior is, in this case, more complex since the spin dynamics of the embedded impurity is coupled to the charge and spin degrees of freedom of the quantum dot. This has several consequences. First, interference between direct and exchange tunneling through the barrier plays already a role for nonmagnetic electrodes in transport to lowest order in the tunnel coupling. This contrasts with the simpler case of a single tunnel barrier with a magnetic impurity, for which this interference only contributes for ferromagnetic electrodes or in higher order transport. (41); (57); (48); (56) Second, we demonstrate that the Coulomb charging energy of the quantum dot can help to perform tunnel spectroscopy on the embedded spin. Even if the excitation energy between the ground state and the first excited state of the impurity spin is larger than any other spin excitation energy, all spin excitation energies are accessible when additional charge states of the dot contribute to transport through the system, which is not possible for the single-barrier case. Third, we discuss a current-induced switching of the impurity spin. The impurity state influence the spin accumulation on the quantum dot, which in turn acts back on the current through the system, leading to current oscillations as a function of the applied bias. Interestingly, these phenomena occur even for small polarizations of the leads. While the current is only sensitive to the average value of the spin, we find that the zero-frequency Fano factor also contains information about the spin dynamics. Finally, fourth, we point out how monitoring the exchange field peak in the frequency-dependent Fano factor can detect the switching of the impurity spin for noncollinear magnetizations.

In the second scenario, we consider a impurity side coupled to the spin of the electron on the quantum dot. Here, our main aim is to describe the coherent dynamics of the two spins on the quantum dot. We consider the case that the impurity spin is located on the quantum dot, as this allows us to take the exchange coupling between electron and impurity spin into account exactly. This model can serve to describe different situations. First, it can describe transport through a quantum dot that is doped with a magnetic atom. Transport through a quantum dot doped with a single manganese atom has already been studied theoretically. It was shown how the frequency-dependent shot noise can reveal the spin relaxation times. (58) Furthermore, the electrical control of the manganese spin state as well as the back action of the spin state on transport have been investigated in the absence of Coulomb interaction in the quantum dot. (59)

Second, our model can be used to describe the coupling of the electron spin on the dot to a nuclear spin via the hyperfine interaction. In general, such a coupling is disadvantageous as it leads to decoherence of the electron spin and therefore can lift, e.g., the Pauli spin blockade in a double quantum dot. (60); (61); (62); (63) However, it can also be used to dynamically polarize the nuclear spins in the quantum dot which in turn may be used to control and manipulate the electron spin. (64); (65); (66); (67); (68)

Transport through a quantum dot with a side-coupled spin was discussed in Ref. (69) for the case of nonmagnetic electrodes. It was shown how to extract the system parameters such as the exchange couplings, the factors and spin relaxation times from measurements of the current and Fano factor.

The case of noncollinearly magnetized ferromagnetic electrodes was recently investigated by Baumgärtel et al. (70) for a large ferromagnetic exchange interaction. It was shown that in addition to a spin dipole, a spin quadrupole moment accumulates on the quantum dot, driven by a quadrupole current.

In this work, we focus on the opposite regime of small exchange interaction between the side-coupled impurity and electron spin. This situation is particularly suited for the description of the weak hyperfine interaction. We discuss how the frequency-dependent Fano factor can be used to experimentally access the strength of the exchange coupling for large and small external magnetic fields. Furthermore, for the case of a weak external magnetic field, we show how the exchange field acting on the electron spin (but not on the impurity spin) gives rise to a highly nontrivial spin dynamics that manifests itself in the frequency-dependent Fano factor.

Our paper is organized as follows. In Sec. II, we present the models that describe a magnetic impurity with a large, anisotropic spin localized in one tunnel barrier and a small spin localized on the dot, respectively. We introduce the real-time diagrammatic technique (71); (72); (73); (74) that we use to calculate the transport properties in Sec. III. We discuss the form of the reduced density matrix of the quantum dot system and the generalized master equation it obeys for the two systems under investigation in Sec. IV. Our results for the transport properties are presented in Sec. V for an impurity in the barrier and in Sec. VI for an impurity on the dot. Finally, we conclude by giving a summary and comparing our results for the two models in Sec. VII.

Ii Model

In this paper, we consider transport through a quantum-dot spin valve, i.e., a single-level quantum dot tunnel coupled to noncollinearly magnetized ferromagnetic electrodes. We consider additional magnetic impurities either with a large, anisotropic spin localized in the tunnel barrier or with a spin on the quantum dot itself. In the following, we define the Hamiltonians for these two cases, respectively.

ii.1 Model A: Large spin in the barrier

Figure 1: (Color online) Schematic of model A, a quantum-dot spin valve with a magnetic impurity localized in the right tunnel barrier.

Our model A, which is schematically shown in Fig. 1, consists of a quantum-dot spin valve with an impurity embedded in the right tunnel barrier. As already mentioned in Sec.I, we choose the impurity to be in the tunnel barrier, as this will lead to a simpler conductance pattern, see Sec. V below, which allows to study the spin excitations more easily. In this case, the Hamiltonian can be written as the sum of four terms describing the two electrodes, the quantum dot, the spin and the tunneling between dot and lead,

(1)

The first term,

(2)

describes the ferromagnetic electrodes in terms of noninteracting electrons at chemical potential . We quantize the electron spin in the direction of the magnetization of the respective lead. The spin polarization is defined as , where is the constant density of states for majority () and minority () spin electrons.

The quantum dot is described by

(3)

Here, the first term characterizes the single, spin-degenerate quantum dot level with energy measured with respect to the Fermi energy of the leads in equilibrium. The charging energy is needed to occupy the quantum dot with two electrons. As indicated in Fig. 1, we choose the quantization axis of the dot parallel to the magnetization of the right electrode as this simplifies the expressions for the tunnel Hamiltonian, see below.

The third term in Eq. (1) describes the magnetic impurity embedded in the right tunnel barrier as a localized spin with Hamiltonian

(4)

We model the spin of magnitude as having a uniaxial anisotropy which favors the spin to be in the eigenstates pointing along the axis. We, furthermore, assume the presence of a magnetic field acting on the impurity spin. For concreteness, we assume this field to be pointing along the direction. This choice is motivated by the presence of stray fields from the ferromagnetic electrode which have the tendency to align the impurity along the magnetization of the electrode. As for the quantum dot, we quantize the impurity spin along the direction of the magnetization of the right electrode.

The last part of the Hamiltonian (1) describes the tunneling between the dot and the electrodes. For the coupling to the left lead, it takes the form

(5)

i.e., majority and minority spin electrons of the leads couple to spin up and spin down states of the quantum dot due to our choice of quantization axes. The coupling to the right lead consists of two terms,

(6)

Here, denotes the vector of Pauli matrices. The first part describes direct tunneling between the dot and the leads. The second term describes exchange scattering from the impurity spin.

The tunnel matrix elements and (which can be chosen real) are related to the spin-dependent tunneling rates via . The total tunnel coupling is then given by . Similarly, for the exchange tunneling, we relate the corresponding tunneling rate to its (real) matrix element by . Furthermore, there will be a contribution due to the interference between direct and exchange tunneling through the right barrier. It is characterized by . Here, determines the sign of the interference contribution which is governed by the relative sign of and . The upper sign, applies for equal signs of and while the lower sign, applies for different signs of and .

ii.2 Model B: Small spin on the dot

Figure 2: (Color online) Schematic of model B which consists of a quantum-dot spin valve with a magnetic impurity on the quantum dot.

Model B consists of a quantum-dot spin valve with an additional spin localized on the quantum dot as shown schematically in Fig. 2. Here, we restrict ourselves to the case of a impurity spin for two reasons. First of all, this keeps the size of the Hilbert space small while still giving rise to the nontrivial spin dynamics we are interested in. Second, for a larger spin that additionally has some anisotropy, spin states would not be degenerate any longer, thereby destroying the possibility to observe the coherent spin dynamics, see also the discussion in Sec. III. The total Hamiltonian now takes the form

(7)

describing the electrodes, the dot containing the impurity spin and the tunneling between dot and leads. The first part, , is identical to the one given in Eq. (2). For the second term, we have

(8)

The first two terms describe the single dot level with energy and Zeeman splitting due to an external magnetic field. For simplicity, we assume the magnetic field to point along the quantization axis of the dot which we choose perpendicular to the magnetizations of the leads. The third term describes the Coulomb interaction which is needed to doubly occupy the quantum dot. The fourth term describes the Zeeman energy of the additional spin on the quantum dot. Here, we assume the same factor for the electrons on the dot and the impurity spin, for a discussion of a system where electron and impurity spin have different factors, cf. Ref. (69). Finally, the last term describes an exchange interaction between the spin of the electron on the dot and the impurity spin.

The eigenstates of the dot Hamiltonian (8) and their corresponding energies are summarized in Table 1. The eight states can be classified according to the number of electrons on the dot. For the empty and doubly occupied dot, we have two states each that differ in energy by the Zeeman energy with the impurity spin pointing up or down. For the singly occupied dot, we have in total four states, three triplet () and one singlet state (). While the triplet states are energetically split by the Zeeman energy, singlet and triplet are split by the exchange coupling .

Eigenstate Energy
Table 1: Eigenstates of the dot Hamiltonian (8) and corresponding eigenenergies. The first entry in each ket denotes the state of the quantum dot while the second entry characterizes the impurity spin state.

The coupling between dot and leads is described by the tunneling Hamiltonian

(9)

where denotes half the angle enclosed by the magnetizations.(33) We relate the tunnel matrix elements to the tunnel couplings as for the model discussed in Sec. II.1. Instead of using the tunnel coupling strength to the left and right lead, we can characterize the dot-lead coupling alternatively by the total tunnel coupling and the asymmetry with .

Iii Technique

In order to evaluate the transport properties of the two systems under investigation, we make use of a real-time diagrammatic technique (71); (72); (73); (74) and its extension to systems with noncollinearly magnetized ferromagnetic electrodes. (32); (33) The basic idea of this approach is to integrate out the noninteracting degrees of freedom of the electrodes. We, then, arrive at an effective description of the remaining, strongly interacting subsystem in terms of its reduced density matrix .

We denote by the elements of the reduced density matrix, where and are eigenstates of the reduced system. The time evolution of the reduced density matrix elements is governed by a set of generalized master equations,

(10)

The first term on the right-hand side describes the coherent evolution of the reduced system. The second term describes the dissipative coupling to the electrodes. The generalized transition rates are defined as irreducible self-energies of the dot propagator on the Keldysh contour. They can be evaluated diagrammatically in a perturbation expansion in the tunnel coupling strength . The corresponding diagrammatic rules are summarized in Appendix A.

Expanding the density matrix elements as well as the generalized transition rates in a power series in the tunnel coupling , we find that in the stationary limit the master equation for the off-diagonal matrix elements to leading order in the tunnel couplings takes the form

(11)

if . As a consequence, coherent superpositions between states whose energy difference is large compared to the tunnel coupling have to be neglected in the sequential tunneling regime. On the other hand, for superpositions that satisfy , the master equation in the stationary limit takes the form

(12)

to lowest order in the tunnel coupling. Here, the generalized transition rates have to be evaluated at in order to consistently neglect all effects of order . Hence, in this case, the coherences will not vanish in general.

We define the current through the system as the average of the currents through the left and right tunnel barrier, . It is given by

(13)

Here, we introduced the vector which contains all density matrix elements written as a vector to allow for a compact notation. The vector is a vector containing a if the corresponding entry in is a diagonal density matrix element and a otherwise. Finally, the quantity contains the current rates which can be obtained diagrammatically similarly to the generalized transition rates by replacing one tunneling vertex by a current vertex. The corresponding diagrammatic rules are given in Appendix A.

The frequency-dependent current noise is defined as the Fourier transform of the symmetrized current-current correlation function . In the sequential tunneling regime and for low frequencies, , we can write it as

(14)

where is obtained from by replacing two tunnel vertices by current vertices. The frequency-dependent free dot propagator on the Keldysh contour is given by

(15)

We stress that the frequency-dependent current noise for is only sensitive to coherences between states with , i.e., we can savely neglect all other coherences in the calculation of as in the evaluation of the master equation and the stationary current. While our discussion of the real-time diagrammatic technique so far was rather general, in the next section, we turn to the explicit form of the density matrix as well as the generalized master equations for the two systems under investigation.

Iv Reduced density matrix and master equation

iv.1 Model A: Large spin in the barrier

For a quantum-dot spin valve with a large, anisotropic impurity spin embedded in the tunnel barrier, the eigenstates of the reduced system consisting of quantum dot and impurity spin are products of dot eigenstates and impurity spin eigenstates , . Assuming the energies of states with different impurity states to differ more than the tunnel coupling, , we have to neglect coherent superpositions between states with different impurity states.

The reduced density matrix therefore takes a block diagonal form given by

(16)

In order to give a physically intuitive interpretation of the generalized master equations, we introduce the probabilities to find the dot empty, , singly occupied, , and doubly occupied, , with the impurity in state . We collect these quantities in the vector . We furthermore introduce the average spin on the quantum dot , , and . The set of master equations can then be split into one determining the occupation probabilities and one set governing the average dot spin. In the following, we will keep the time derivative on the left-hand side of the master equations explicitly to allow for a physically intuitive interpretation of the master equations. For the numerical discussion below, these derivatives are equal to zero, however. The master equations for the occupations are given by

(17)

Here, is a matrix which describes processes that transfer an electron between the dot and lead and change the state of the impurity spin from to . Since the impurity is located in the right tunnel barrier, tunneling through the left lead cannot change the impurity state and therefore . Changing the impurity state is possible, however, for tunneling through the right barrier which provides a coupling between and . Similarly, is a vector which describes the coupling of the occupation probabilities to the spin on the dot, a feature characteristic of a quantum-dot spin valve. Again, we have . The precise form of and is given in Appendix B.1.

The time evolution of the dot spin obeys a Bloch-type equation,

(18)

The first four terms on the right-hand side describe the non-equilibrium spin accumulation on the quantum dot due to the tunneling from and to the spin-polarized leads. Similarly to the master equation for the occupations, we get accumulation terms that change the state of the impurity when tunneling takes place through the right barrier.

The next two terms account for the relaxation of the spin on the dot due to the tunneling out of an electron or the tunneling in of an electron forming a spin singlet with the electron already present on the dot. As these terms arise from generalized transition rates which start and end in a singly occupied state, in the sequential tunneling approximation the state of the impurity spin cannot be changed in these processes. We give the explicit forms of the accumulation and relaxation terms in Appendix B.1.

The last term describes the precession of the dot spin in an exchange field due to virtual tunneling to the leads. For the coupling to the left lead, we find the usual exchange field (33) which is independent of the state of the impurity spin. It is given by

(19)

where , and denotes the digamma function. While the first term arises from the spin-dependent level renormalization of an electron virtually tunneling to the lead and back, the second term stems from processes where an electron first tunnels onto the dot and then back into the lead.

Figure 3: Exchange field due to virtual tunneling through the right barrier as a function of bias voltage for (upper panel) and (lower panel). Other parameters are , , , , , , , .

The exchange field due to the coupling to the right lead which points in the direction of is given by

(20)

where , and .

Here, the first term on the right-hand side describes exchange field contributions due to virtual tunneling between dot and lead that does not change the state of the impurity spin. The other two terms are due to virtual tunneling where the intermediate state has an increased/decreased impurity spin state. These processes give rise to a logarithmic divergency of the exchange field cut off by the bandwith of the lead electrons. To understand this, we consider the impurity in the state . In this case, only the sequences and of (virtual) transitions are possible while there are no such processes starting from . Hence, these processes only renormalize the energy of the spin down state, giving rise to the logarithmic divergency. Similarly, when the impurity is in any other state, the logarithmic contributions to the exchange field do not cancel between processes that increase and decrease the intermediate impurity spin state. As we only consider sequential tunneling, it is clear that our results are only valid if and , otherwise higher order logarithmic corrections become important.

We, therefore, find that the presence of the impurity spin in the tunnel barrier has two basic effects on the exchange field. First of all, it alters its strength. Second, due to the presence of the spin-flip processes, it also alters its energy dependence (cf. Fig. 3), giving rise to additional peaks and dips whose separation is governed by the anisotropy , the Zeeman energy , and the size of the impurity spin . Since the transition energies between the various impurity states depend on the states itself, the position of the additional peaks and dips depends on the value of .

iv.2 Model B: Small spin on the dot

Case Parameters Superpositions of
(i)
(ii) , ,
(iii) , ,
(iv) , ,
, ,
,
(v) ,
, , ,
,
Table 2: Coherent superpositions that have to be taken into account in the sequential tunneling regime for different values of the external magnetic field and the exchange coupling between the two spins.

For the case of a impurity localized on the quantum dot, the reduced density matrix in the most general case takes the form

(21)

i.e., apart from the eight diagonal matrix elements that describe the probability to find the system in one of its eigenstates there can be up to 16 coherences. Coherences between states with different electron numbers vanish due to the conservation of total particle number. As discussed above, depending on the energy differences of the states forming the coherent superposition, we either have to take them into account or neglect them in the sequential-tunneling regime. In Table 2, we summarize the different transport regimes that arise consequently.

In the following, we will only consider the cases (ii) and (v), i.e., we only consider the case of weak exchange couplings, . When a large magnetic field is applied, , only - coherences have to be taken into account. When the externally applied magnetic field is weak, , we have to take into account all coherences. There are two reasons focusing on the two cases. On the one hand, they are particularly suited to demonstrate the information about the transport processes contained in the finite-frequency noise. On the other hand, the cases of small exchange couplings are suited to describe the influence of nuclear spins, that couple to the electron spin via hyperfine interaction, on transport through the quantum dot.

Case (ii): Large magnetic field

We first turn to the discussion of the master equation in case (ii) where , . In this case, there are only superpositions of and present. It is therefore natural to introduce the isospin via

to bring the master equation into a physically intuitive form. Similar to the case of an ordinary quantum-dot spin valve, (33) we can now split the master equation into one set governing the occupation probabilities that we summarize in the vector and one set governing the time evolution of the isospin . However, there is an important difference. While in the ordinary quantum-dot spin valve, there is a real spin accumulating on the quantum dot, here we have an isospin accumulation as a real spin accumulation is suppressed by the large external magnetic field. The master equation of the occupation probabilities reads

(22)

where denotes a matrix that contains the golden rule transition rates between the various dot states and is a vector that characterizes the influence of the isospin on the dot occupation whose precise form is given in Appendix B.2.

The master equation that governs the time evolution of the isospin is given by

(23)

Here, the first term,

(24)

describes the accumulation of the isospin along the axis due to electrons tunneling onto and off the dot. Similarly, the second term describes a relaxation of the isospin

(25)

Finally, the last term describes the precession of the isospin in an exchange field that is given by

(26)

The exchange field describes the level splitting between and which is due to the finite exchange coupling as well as due to virtual tunneling processes that renormalize the energies of the two states in a different way. As can be inferred from Eq. (23), it gives rise to a precession of the accumulated isospin around the exchange field.

The current through the quantum dot is given by

(27)

It depends on the occupation probabilities as well as on the component of the accumulated isospin. This resembles the normal quantum-dot spin valve where the current also depends on both, the dot occupations as well as on the dot spin. (33)

Case (v): Small magnetic field

We now turn to the discussion of the master equation in the case . In this case, we have to include all coherences of the reduced density matrix, Eq. (21).

To allow for a physical interpretation of the different matrix elements, we introduce the probabilities to find the quantum dot empty, , singly occupied, , and doubly occupied, . Furthermore, we introduce the expectation values of the electron spin on the dot, , as well as the expectation values for the impurity spin when the dot is empty, , singly occupied, , and doubly occupied, . While for a single spin the description of its density matrix in terms of spin expectation values is sufficient, this is in general no longer true for a system of two spin particles. (70) For the case of small magnetic fields that we consider here, we therefore have to introduce in addition the expectation values of the scalar product between electron and impurity spin, , and their vector product, . Finally, we also need to introduce the quadrupole moment (75); (70)

(28)

The quadrupole moment is a symmetric tensor, . Its diagonal elements are not independent of each other as they satisfy the sum rule , i.e., the trace of vanishes.

In Appendix C we give the explicit expressions that relate the above quantities to the density matrix elements in Eq. (21). We note that in the case , where we only have taken into account the - superpositions, we could have expressed the reduced density matrix in terms of the quantities just introduced as well. However, by choosing a description in terms of the isospin, we obtain a much simpler master equation.

Using the physical quantities we just discussed, we can split the master equation into several sets. The first set

(29)

describes the evolution of the occupation probabilities. It takes a form identical to the case of a normal quantum-dot spin valve, i.e., it exhibits a coupling of the occupations to the spin accumulated on the quantum dot. Interestingly, the occupations do not couple neither to the impurity spin, the scalar or vector product of and nor to the quadrupole moments directly. They are only influenced by these quantities due to their influence on the accumulated electron spin .

The equation governing the time evolution of the electron spin in the dot is given by

(30)

where and is the usual exchange field acting on the electron spin accumulated on the dot. Again, we find a strong similarity to the case of the normal quantum-dot spin valve. While the first term in brackets describes the accumulation of spin on the dot along due to spin-dependent tunneling of electrons between dot and leads, the second term describes a relaxation of the dot spin due to tunneling. The third term in brackets describes the precession of the dot spin in the exchange field generated by virtual tunneling between dot and leads. The last two terms finally describe the influence of an external magnetic field and the exchange coupling to the impurity spin.

The master equations for the impurity spin in the presence of zero, one and two electrons on the dot can be written as

(31)

Here, the first term on the right-hand side describes transitions between the three quantities by tunneling of electrons in analogy to the first term in the equation for the occupations, Eq. (29). The second term characterizes the coupling to the quadrupole moments as well as the scalar and vector product of and . This resembles the coupling of the dot occupations to the electron spin on the dot in Eq. (29). Finally, the terms in the third line describe the precession of the impurity spin in an externally applied magnetic field as well as the influence of the exchange interaction between electron and impurity spin.

The master equations governing the time evolution of the scalar and vector product between the electron and impurity spin are given by

(32)
(33)

Their structure closely resembles Eq. (30) in that there are terms which describe the accumulation, relaxation, and the influence of the spin precession due to the exchange field. Furthermore, the vector product turns out to be sensitive to an external magnetic field as well as to the exchange coupling between the spins.

Finally, the master equation for the quadrupole moment takes the form

(34)

The first three terms on the right-hand side describe the accumulation of quadrupole moment on the quantum dot. Similarly, the fourth term is related to the relaxation of the quadrupole moment. Finally, the other terms describe the precesional motion of the quadrupole moment in the exchange field as well as due to an external magnetic field.

The current through tunnel barrier is given by

(35)

Although this is precisely the same form as for the normal quantum-dot spin valve, the current nevertheless contains information about the nontrivial spin dynamics on the dot, as the master equation for the dot spin couples to the other density matrix elements.

V Results - Large spin in the barrier

In this section we discuss the transport properties of a quantum-dot spin valve with a large, anisotropic impurity spin located in the right tunnel barrier. We will focus our attention on systems that are symmetric in the sense that and . Furthermore we assume the bias voltage to be applied symmetrically, .

v.1 Collinear magnetizations

In the following, we are going to discuss transport for collinear magnetizations. Most of the time we will restrict ourselves to the case of parallel magnetizations as we focus on moderate polarizations where the effects due to the impurity spin are much more important than the effects due to the relative orientation of the leads.

Interference between direct and exchange tunneling

From the form of the tunneling Hamiltonian (6) it is obvious that interference can take place between electrons tunneling directly through the barrier and electrons experiencing an exchange interaction with the impurity spin. For transport through a single tunnel barrier containing a localized spin (76); (77); (41); (43); (49), the interference contributions cancel between the spin-up and spin-down channel. Only for ferromagnetic leads (57); (48); (56); (78) or in the presence of spin-orbit interactions, one is sensitive to the interference terms.

This is different for the system under investigation here. We find that the interference terms influence the current even for unpolarized leads. In contrast to the single-barrier case where we just have to sum up the contributions from spin-up and spin-down electrons to the current, in the quantum dot case we have to separately compare the rates for tuneling in and out of the dot for each spin direction. While in the nonmagnetic case equal amounts of spin-up and spin-down electrons enter the dot from the left lead, the rates for leaving the dot are different due to the interference terms. This in turn gives rise to a spin accumulation on the quantum dot which reduces the current through the quantum dot.

Unfortunately, in our system there is no way to tune the phase of the interference terms experimentally as is possible, e.g., in an Aharonov-Bohm interferometer and thereby check the influence of the interference terms on the current. Nevertheless, it should be possible to detect the presence of the interference term experimentally and to detect its sign. Approximating the Fermi functions as step functions which is reasonable away from the threshold voltages, we can calculate the current through the system analytically in the various transport regions.

We first consider the case of unpolarized leads, . In this case, the current in region I where transport takes place through the singly and doubly occupied dot (cf. Fig. 4) is given by

(36)

i.e. it is sensitive to the couplings and the size of the barrier spin but not to the interference term. Similarly, in region II where spin excitations become possible, the current turns out to be insensitive to the interference term,

(37)

This is different in region III where transport takes place through the empty and singly-occupied dot but the spin cannot be excited. Here, the current is given by

(38)

i.e., the current now also depends on the interference term. From Eq. (38), we infer that for the current vanishes exactly in region III. Equation (38) also shows that the current in region III is only sensitive to the absolute value of the interference term but not to its sign.

This is different in the regime where the empty and singly occupied dot contribute to transport and spin excitations are possible. As the analytic result for the current in this regime is rather lengthy, we do not give it here. Instead, we now focus on transport for parallely magnetized leads. In region I, the current is now given by

(39)

Here, the current is clearly sensitive to the sign of the interference term which provides a way to access it in experiments. Similar expressions for the current in regions II and III can be found for parallely magnetized leads. As these expressions are rather lengthy, we do not give them here.

Spin spectroscopy

Figure 4: (Color online) Schematic of the differential conductance as a function of level position and applied bias voltage. Thick black lines mark the onset of transport through the dot. Blue dashed lines indicate the onset of impurity excitations. Dashed lines mark the gaining of energy from the impurity to allow all three charge states of the dot.

Inelastic spin tunneling spectroscopy (50); (51); (53); (54); (55); (79); (80);