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Asymmetry-induced effects in Kondo quantum dots coupled to ferromagnetic leads

We study the spin-resolved transport through single-level quantum dots strongly coupled to ferromagnetic leads in the Kondo regime, with a focus on contact and material asymmetry-related effects. By using the numerical renormalization group method, we analyze the dependence of relevant spectral functions, linear conductance and tunnel magnetoresistance on the system asymmetry parameters. In the parallel magnetic configuration of the device the Kondo effect is generally suppressed due to the presence of exchange field, irrespective of system's asymmetry. In the antiparallel configuration, on the other hand, the Kondo effect can develop if the system is symmetric. We show that even relatively weak asymmetry may lead to the suppression of the Kondo resonance in the antiparallel configuration and thus give rise to nontrivial behavior of the tunnel magnetoresistance. In addition, by using the second-order perturbation theory we derive general formulas for the exchange field in both magnetic configurations of the system.


Introduction
Transport properties of nanoscopic objects, such as quantum dots or molecules, tunnelcoupled to external leads have attracted a lot of attention from both theoretical as well as experimental sides [1,2,3,4,5,6]. This is mainly due to possible applications in nanoelectronics and spintronics, and due to a unique possibility to study various many-body correlation effects between single charges and spins. When the coupling between the quantum dot and external leads is relatively strong, the electronic correlations may lead to the Kondo effect if the dot's occupation number is odd [7,8,9,10]. For quantum dots coupled to ferromagnetic leads, on the other hand, it was shown that the Kondo resonance can be suppressed due to the presence of an effective exchange field, ∆ε exch , that leads to the spin splitting of the quantum dot level [11,12,13,14,15,16,17,18,19]. This suppression occurs if the magnetic moments of external leads form a parallel (P) magnetic configuration and when |∆ε P exch | T K , where T K is the Kondo temperature and ∆ε P exch denotes the level splitting due to exchange field in the parallel configuration [15]. For an antiparallel magnetic configuration of the device, the exchange field was found to vanish, ∆ε AP exch → 0, since the effective coupling to external leads becomes then spinindependent [12,13,15,20]. This is however true only for fully symmetric systems, while, as we show in this paper, for systems exhibiting some asymmetry, either the left-right contact asymmetry or the material's asymmetry, the exchange field may also develop in the antiparallel configuration. Since experimentally it is very difficult to build a truly symmetric device, it seems desirable to analyze the effects of contact and material's asymmetry on spin-resolved transport properties of quantum dots.
In the present paper we thus thoroughly study transport through quantum dots coupled to ferromagnetic leads in the Kondo regime, focusing especially on asymmetryinduced effects. To obtain the correct picture, we employ the Wilson's numerical renomalization group (NRG) method [21,22,23] with the idea of a full densitymatrix (fDM) [24]. By using NRG, we calculate the dependence of relevant spinresolved spectral functions, the linear conductance in the parallel and antiparallel configurations, and the tunnel magnetoresistance (TMR) on the asymmetry between the couplings to the left and right leads and for different spin polarizations of the electrodes. We show that even relatively small asymmetry may fully suppress the Kondo resonance in the antiparallel configuration, leading to nontrivial dependence of the TMR effect on the asymmetry parameters. We also show that although asymmetry generally destroys the Kondo effect in the antiparallel configuration, there is a range of asymmetry parameters, when the Kondo resonance can be restored. In addition, by using the second-order perturbation theory, we derive general formulas for the exchange field in both magnetic configurations depending on asymmetry parameters.

Theoretical description
The system consists of two ferromagnetic leads coupled to a single-level quantum dot, see Fig. 1. The magnetizations of the leads are assumed to be collinear and they can form two magnetic configurations: the parallel (P) and antiparallel (AP) ones. Switching between different magnetic configurations of the device can be obtained by sweeping the hysteresis loop, provided the left and right ferromagnets have different coercive fields. The system considered can be described by the single- impurity Anderson Hamiltonian where a † rkσ creates an electron with spin σ momentum k and energy ε rkσ in the left (r = L) or right (r = R) lead. The energy of the dot level is denoted by ε d , U describes the Coulomb interaction of two electrons residing on the dot, and n σ = d † σ d σ is the particle number operator, with d † σ creating a spin-σ electron on the dot. The last term of the Hamiltonian (1) describes tunneling processes between the dot and the ferromagnetic electrodes, with V rkσ being the relevant hopping matrix elements. Due to the coupling to external leads, the dot level acquires certain width, which is described by the spin-dependent hybridization function, Γ rσ = πρV 2 rσ , for lead r and spin σ. Here, ρ is the density of states at the Fermi level and we assumed that V rkσ is independent of momentum k [12,13]. The conduction bands of the leads are assumed to be energy and spin independent, ρ = 1/(2W ), where W ≡ 1 is used as energy unit. These assumptions are justifiable since we are interested in the Kondo regime which is mainly related with the conduction electron states around the Fermi energy. Next, by introducing the spin polarization of lead r, p r = (Γ r↑ − Γ r↓ )/(Γ r↑ + Γ r↓ ), the couplings can be expressed as, Γ rσ = (1 + σp r )Γ r , with Γ r = (Γ r↑ + Γ r↓ )/2. Here, Γ rσ denotes the coupling to the spin-majority (σ =↑) or spin-minority (σ =↓) conduction band of the ferromagnetic lead r. For convenience, we have incorporated the effect of leads' ferromagnetism into the spin-dependent tunnel matrix elements. This assumption is commonly used in NRG calculations [12,20].
The asymmetry induced effects to be studied in the present paper will include the left-right contact asymmetry, Γ L = Γ R , and the material asymmetry, i.e. different spin polarizations of the leads, p L = p R . In particular, we will study how the linear response transport properties depend on the parameters of the system, especially on the couplings' ratio Γ R /Γ L and spin polarizations' ratio p R /p L . The linear-response spin-dependent conductance of the system can be found from the Meir-Wingreen formula [25], where f (ω) denotes the Fermi function and A σ (ω) is the spectral function of dot level. The spectral function is given by Another quantity of interest is the tunnel magnetoresistance (TMR), which describes the change in spin-dependent transport properties when the magnetic configuration of the device is varied. The TMR is defined as [26], where G P/AP is the total linear conductance in the P/AP magnetic configuration. In addition, we will also study the spin filtering properties of the system, which can be related to the spin polarization of the linear conductance, where P P/AP denotes the spin polarization in the parallel or antiparallel magnetic configuration of the device. For the single-impurity Anderson model, it is very convenient to perform an orthogonal transformation from the left-right basis into an even-odd basis, where only the even linear combination of the lead operators couples to the dot, while the odd combination is completely decoupled. The new effective spin-resolved couplings for the parallel and antiparallel configuration can be then expressed as where p denotes the average spin polarization p = (p L + p R )/2 and Γ = Γ L + Γ R , while the parameter β P/AP is given by From the above formula follows that the couplings are generally different for each spin direction, Γ which results in different level renormalization for spin-up and spin-down. This difference leads to an effective spin-splitting of the dot level, known as contact-induced exchange field, ∆ε P/AP exch [12,13,14]. In the parallel configuration, Γ P ↑ = Γ P ↓ , and consequently, ∆ε P exch = 0, once the spin polarization is finite, p r > 0, for any value of left-right contact asymmetry. In the antiparallel configuration, on the other hand, the exchange field can occur, ∆ε AP exch = 0, provided there is an asymmetry in the system, so that Γ AP ↑ = Γ AP ↓ . The asymmetry can be either related to the contacts, Γ L = Γ R , or to the material, p L = p R . Moreover, it turns out that even in the presence of asymmetry, in the antiparallel configuration there is a parameter range where the exchange field can still vanish, which happens for β AP = 0. This occurs precisely when the following condition is met The spin-dependence of the effective couplings gives rise to the exchange field ∆ε exch , which can be found analytically by calculating second-order corrections δε dσ to the energy of quantum dot levels, ∆ε exch ≡ δε d↑ − δε d↓ . At low temperature, one then gets for the parallel and antiparallel magnetic configurations There are actually two factors that determine the strength of the exchange field: the first one, ∼ ln |ε d /(ε d + U )|, is related to the gate voltage, which can be used to change position of the dot level. This factor leads to the cancelation of ∆ε P/AP exch at the particle-hole symmetry point of the model, i.e. ε d = −U/2, irrespective of magnetic configuration and asymmetry. The second factor, ∼ β P/AP , on the other hand, is associated with asymmetry in the system and depends on magnetic configuration, see Eqs.(6)- (7). It may either increase or suppress the exchange field, depending on the magnetic configuration and parameters of the model.
As can be seen from Eq. (2), the main quantity to be calculated is the spin-resolved spectral function of the dot-level. This is performed with the aid of the numerical renormalization group method with full density matrix (fDM-NRG). This method, known as the most powerful and versatile to study various quantum impurity problems, allows us to determine the dependence of the spectral function on parameters of the system in most accurate and reliable way. The starting point for the NRG is logarithmic discretization of the conduction band of the leads and mapping of the initial Hamiltonian to the Hamiltonian of a tight-binding chain with exponentially decaying hoppings, the so-call Wilson chain [21]. The chain Hamiltonian is then solved in an iterative manner and its discarded eigenstates are used for the construction of a full density matrix [24], which enables the calculation of relevant static and dynamic quantities at arbitrary temperature. In our calculations we have in particular employed the flexible density-matrix numerical renormalization group code [27]. In calculations we kept 1024 states at each iteration and used the Abelian symmetries for the total spin zth component and the total charge.

Results and discussion
In the following we present and discuss the numerical results on the linear conductance and spin polarization of the current in both magnetic configurations as well as the resulting TMR effect. First, we analyze the effects related with the contact asymmetry, Γ L = Γ R , and then the effects due to different spin polarizations of the left and right lead, p L = p R , are discussed. Finally, we present the general case when both asymmetries are present.
3.1. Effects of left-right contact asymmetry, Γ L = Γ R By changing the ratio Γ R /Γ L one changes both the magnitude of exchange field as well as the Kondo temperature. However, the dependence of both quantities on the strength of coupling Γ = Γ L + Γ R is different: while |∆ε exch | ∼ Γ, the Kondo temperature T K depends on Γ in an exponential way [28,29], . Tuning the contact asymmetry ratio will thus change the ratio |∆ε exch |/T K , which conditions the occurrence of the Kondo effect, as discussed and presented in the following.  Figure 2 shows the linear conductance in the parallel and antiparallel configuration, as well as the resulting TMR effect as a function of the couplings' ratio Γ R /Γ L and the level position ε d /U . Experimentally, the position of the dot level can be changed by tuning the gate voltage. When Γ R /Γ L = 0, the dot is coupled only to a single (left) lead and the conductance through the system is obviously equal to zero. With increasing the coupling to the right lead, the linear conductance becomes finite and exhibits a strong dependence on the level position. In the elastic cotunneling regime, i.e. for ε d /Γ ≫ 0 or (ε d + U )/Γ ≪ 0, the conductance in the parallel configuration is G P ∼ (1 + p L p R )Γ L Γ R , while for the antiparallel configuration one gets, [20,30], which for  parameters assumed in Fig. 2 gives, TMR ≈ 0.38. Note that this value is independent of the couplings asymmetry, provided the current is mediated by elastic cotunneling events.
In the Coulomb blockade regime, on the other hand, the situation is more complex, since the dot is singly occupied and the electronic correlations may lead to the Kondo effect. Moreover, the occurrence of the Kondo effect is conditioned by the ratio of the Kondo temperature and the exchange field induced splitting. In the parallel configuration, the exchange field is always present, irrespective of Γ R /Γ L , and the Kondo resonance is suppressed, except for the particle-hole symmetry point, ε d = −U/2. By moving away from this point, the conductance drops once |∆ε P exch | T K . Since T K depends on the coupling strength Γ, increasing Γ R /Γ L raises the Kondo temperature, which leads to larger width of the Kondo peak in the middle of the Coulomb blockade regime, see Fig. 2(a). In the antiparallel configuration, the exchange field vanishes if the system is symmetric, and there is a broad Kondo resonance in the whole local moment regime, see Fig. 2(b) for Γ R /Γ L ≈ 1. Nevertheless, if Γ R /Γ L = 1, the Kondo effect becomes suppressed when |∆ε AP exch | T K . This happens faster for Γ R /Γ L < 1 than for Γ R /Γ L > 1, since for smaller coupling Γ the Kondo temperature is lower and the above condition can be fulfilled for relatively small asymmetries of the couplings. As already mentioned, for the particle-hole symmetry point the exchange field vanishes in both magnetic configurations, see Eq. (9), the zero-temperature conductance is then just given by, G P/AP = e 2 /h σ 4Γ Lσ Γ Rσ /(Γ Lσ + Γ Rσ ) 2 , with the couplings correspondingly dependent on magnetic configuration of the device. For symmetric couplings, Γ R /Γ L = 1, both G P and G AP reach the maximum, with G P = 2e 2 /h and G AP = (1 − p L p R )2e 2 /h. The respective behavior of conductance in both magnetic configurations leads to the corresponding dependence of the TMR, which is shown in Fig. 2(c). Generally, the TMR is negative in the whole blockade regime, except for the particle-hole symmetry point, which is associated with the fact that |∆ε P exch | > |∆ε AP exch |, and consequently G P < G AP . Only for ε d = −U/2, when the exchange field is suppressed, one finds a typical spin-valve effect with positive tunnel magnetoresistance. Thus, tuning the position of the dot level and the asymmetry factors, one can obtain a device with desired magnetoresistive properties.
Another quantity describing the spin-resolved transport properties of the system, interesting from an application point of view, is the spin polarization P P/AP of the current flowing through the device, which is shown in Figs. 3(c) and (f) for both magnetic configurations. The behavior of spin polarization can be understood from the spin-resolved conductance. In the parallel magnetic configuration the coupling of the spin-up level is much stronger than the coupling of the spin-down level, since spin-up electrons belong to the spin-majority band. As a consequence, the spin-up channel gives the main contribution to the conductance, see Figs. 3(a) and (b). The difference between G P ↑ and G P ↓ is most visible around the resonances, ε d ≈ 0 and ε d ≈ −U , where the spin polarization P P takes large positive values, approaching unity for Γ R /Γ L ≪ 1. This is contrary to the region around the particle-hole symmetry point where G P ↑ ≈ G P ↓ and the spin polarization is suppressed, P P → 0. In the antiparallel configuration the situation is slightly more complex, since the spinresolved couplings depend greatly on the system's asymmetry, see Eq. (5). For equal spin polarizations of the leads, p L = p R , as assumed in Fig. 3, one finds the effective couplings, Γ AP σ = Γ + σp(Γ L − Γ R ). In consequence, the spin-resolved couplings fulfill the following relations, Γ AP ↑ > Γ AP ↓ for Γ R /Γ L < 1, and Γ AP ↑ < Γ AP ↓ for Γ R /Γ L > 1, with Γ AP ↑ = Γ AP ↓ for the symmetric case, Γ L = Γ R . This gives rise to the corresponding behavior of the spin polarization: P AP ≶ 0 for Γ R /Γ L ≶ 1, and P AP ≈ 0 for Γ R /Γ L ≈ 1. In the region around ε d = −U/2, where the effective field vanishes, the behavior of P AP is however different. The spin-dependent conductances are then given by, , which yields negative spin polarization P AP < 0 for Γ R /Γ L < 1, and positive spin polarization P AP > 0 for Γ R /Γ L > 1. Note that this is just opposite to the case when the exchange field is present, i.e. ε d = −U/2. The above analysis clearly demonstrates that by properly engineering the couplings between the dot end electrodes, and by tuning the occupancy of the dot with the gate voltage, one can obtain desired spin polarizations of the flowing current, spanning almost the whole range from −1 to 1.
From an experimental point of view, it may be important to know how large the asymmetry should be to suppress the Kondo resonance in the antiparallel configuration. To address this question, in Fig. 4 we plot the linear conductance as a function of Γ R /Γ L for different values of the coupling Γ L . Since T K depends exponentially on Γ, changing Γ L corresponds to a huge change in the Kondo temperature (note that T K also depends on Γ R /Γ L ). On the other hand, the dependence of the exchange field on Γ L and the ratio Γ R /Γ L is only algebraic. It can be seen that with lowering Γ L , the suppression of the Kondo effect occurs for smaller asymmetries, e.g. for very weak coupling even relatively small asymmetry between the left-right contacts can fully suppress the linear conductance in the antiparallel configuration. Proper and very careful implementation of a quantum dot/molecular device is therefore necessery in order to observe desired effects, such as e.g. restoration of the Kondo effect when switching the magnetic configuration from parallel into antiparallel one [15].

Effects of different spin polarizations of the leads, p L = p R
Up to now we have focused on the asymmetry related to the left-right contacts, however, the asymmetry can be also present if the electrodes have different spin polarizations, p L = p R . The corresponding transport characteristics are shown in Fig. 5 for equal couplings Γ L = Γ R and different spin polarization ratio p R /p L , with p L = 0.4. The maximum value, p R /p L = 2.5, corresponds then to fully spinpolarized right lead, while p R /p L = 0 corresponds to nonmagnetic right electrode. Now, by changing p R /p L , one can tune the magnitude of the exchange field, while the relevant Kondo temperature is constant. Generally, with increasing the ratio p R /p L , the average spin polarization increases, and so does the exchange field. This intuitive behavior is however only valid for the parallel magnetic configuration and is nicely visible in Fig. 5(a). It can be seen that G P displays a Kondo resonance at the particle-hole symmetry point, whose width decreases with increasing strength of the exchange field, i.e. with increasing the ratio p R /p L . In the antiparallel configuration, on the other hand, the exchange field is a nonmonotonic function of p R /p L : it is maximum for p R = 0, vanishes for p R = p L , and again reaches local maximum for p R = 1. Consequently, G AP displays the Kondo effect in the whole Coulomb blockade regime when p L ≈ p R , which becomes then suppressed with changing the materials' asymmetry ratio from the point p R /p L = 1, see Fig. 5(b). Moreover, it can be seen that the Kondo resonance around ε d = −U/2 is now broader than in the case of parallel configuration since generally, |∆ε P exch | > |∆ε AP exch |. The different dependence of G P and G AP on the spin polarization ratio p R /p L reflects itself in a nontrivial behavior of the TMR effect, see in the local moment regime for p R /p L ≈ 1 when TMR → −1, and becomes again positive in the middle of the Coulomb blockade regime. The spin polarization of the conductance in both magnetic configurations as a function of level position and the leads' spin polarization ratio p R /p L is shown in Fig. 6. In the parallel configuration, the dependence of P P is quite intuitive: the spin polarization increases with raising the average spin polarization p, i.e. increasing p R /p L , and reaches unity for p R → 1, since then the right electrode supports only spinup electrons. Finite spin polarization is mainly due to the presence of exchange field. Consequently, in the middle of the Coulomb blockade valley the spin polarization is suppressed, because ∆ε P exch → 0. In the antiparallel configuration, the spin polarization changes sign with increasing the ratio p R /p L . When, p L > p R , there are more spin-up tunneling processes than spin-down ones and the spin polarization is positive, while for p L < p R , the sign of P AP exch is determined by the majority band of the right lead (spin-down electrons), therefore P AP exch < 0, reaching −1 for p R → 1. On the other hand, once p R ≈ p L , the spin polarization vanishes since the resultant couplings to spin subbands are comparable. 3.3. Effects of both contact and material asymmetry, Γ L = Γ R and p L = p R From our discussion follows that asymmetry can destroy the Kondo resonance in the antiparallel configuration if |∆ε AP exch | T K . However, it turns out that even if there is an asymmetry in the system, either related to the contacts or to the material, there is a parameter range when the exchange field can still vanish in the antiparallel configuration. This happens precisely when the condition (8) is met, so that Γ AP ↑ = Γ AP ↓ . Figure 7 shows the dependence of the linear conductance in the antiparallel magnetic configuration as a function of the relevant asymmetry parameters, i.e. p R /p L and Γ R /Γ L , for ε d = −U/3. Away from the particle-hole symmetry point, ε d = −U/2, the presence and the strength of the exchange field are conditioned by the system's asymmetry parameters, see Eq. (9). For p R /p L → 0, the conductance G AP behaves similarly as that for the parallel configuration, since then trivially there is only one magnetic electrode. The conductance is then generally suppressed and starts increasing only for Γ R /Γ L > 1, when the Kondo temperature increases, so that the condition |∆ε AP exch | T K is only weakly met or even not met. For p R /p L > 0, the case becomes much more interesting, since for certain asymmetry parameters one finds an increased conductance due to the Kondo effect. This happens when the condition (8) is satisfied, as presented by a dashed line in Fig. 7. The exchange field increases then as one moves away from this line. The width of the restored Kondo resonance increases with increasing Γ R /Γ L , since a larger asymmetry is then needed to suppress the Kondo effect, which happens for |∆ε AP exch | T K .

Conclusions
In this paper we have studied the Kondo effect in quantum dots asymmetrically coupled to ferromagnetic leads. The calculations were performed with the aid of the numerical renormalization group method. In particular, we have determined the dependence of the linear conductance in different magnetic configurations of the device, the TMR and current spin polarization on the contacts' and material's asymmetry parameters. For quantum dots symmetrically coupled to external leads, the conductance in the parallel configuration is suppressed due to the presence of exchange field, while in the antiparallel configuration the exchange field is absent. On the other hand, when the dot is coupled asymmetrically to the leads, we have shown that the exchange field can also develop in the antiparallel configuration. If the magnitude of exchange field induced level splitting is larger than the Kondo temperature, the Kondo effect becomes suppressed, which may happen even for relatively small asymmetry between the couplings to the left and right lead. This leads to a nontrivial dependence of the tunnel magnetoresistance on the asymmetry parameters. In addition, we have demonstrated that even if the system is asymmetric, there is a range of parameters in the antiparallel configuration where the exchange field can still vanish. We have also derived approximate analytical formulas for the exchange field in both magnetic configurations depending on the asymmetry parameters.
Concluding, the presented analysis shows that by properly tuning the parameters of the system, one can in principle construct a magnetoresistive device of desired properties, including a source of spin polarized electrons with basically any designed spin polarization. Moreover, a very careful fabrication of a sample seems inevitable, if one wants to observe the suppression and restoration of the Kondo effect by changing the magnetic configuration of the device.