Electron transport through nanoscopic spin valves

Journal of Alloys and Compounds 423 (2006) 244–247

Electron transport through nanoscopic spin valves a , M. Wilczy´ ´ R. Swirkowicz nski a , J. Barna´s b,c,∗ , W. Rudzi´nski b a

Faculty of Physics, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland Department of Physics, Adam Mickiewicz University, ul. Umultowska 85, 61-614 Pozna´n, Poland c Institute of Molecular Physics, Polish Academy of Sciences, ul. Smoluchowskiego 17, 60-179 Pozna´ n, Poland b

Received 2 August 2005; received in revised form 13 December 2005; accepted 21 December 2005 Available online 22 March 2006

Abstract Spin-polarized transport through a quantum dot strongly coupled to ferromagnetic electrodes with non-collinear magnetic moments is analyzed theoretically in terms of the non-equilibrium Green’s function formalism. The influence of an effective exchange field (due to coupling with ferromagnetic electrodes) on tunneling current, linear and non-linear conductance, and tunnel magnetoresistance is studied in detail. In noncollinear configurations we find negative differential conductance for sufficiently large bias voltage. Negative differential conductance can also occur in parallel configurations, when the bare dot level is located well above the Fermi level. Apart from this, a non-monotonic behavior of electric current with increasing angle between magnetic moments of the electrodes is found in systems with the bare dot level located close to the Fermi level. © 2006 Elsevier B.V. All rights reserved. Keywords: Nanostructures; Electronic transport

1. Introduction Transport characteristics of a nanoscopic tunnel junction consisting of a quantum dot (QD) or a molecule attached to ferromagnetic electrodes display features typical of spin valves [1]. Tunneling current flowing through such a system strongly depends on relative orientation of the magnetic moments of external leads. The current is usually maximal for parallel alignment of the magnetic moments, and is partly suppressed in other configurations [2]. In some systems, however, the situation may be reversed, i.e., tunneling current can be maximal for antiparallel configuration. Coupling between a QD and ferromagnetic electrodes gives rise to an effective exchange field exerted on the dot [3–5], which can lead to spin-splitting of the dot level. In systems with noncollinear magnetic moments, the exchange field may also lead to spin precession. Up to now, theoretical analysis of the spin precession was restricted mainly to the weak coupling regime [2,3,6,7]. When the dot is strongly coupled to ferromagnetic electrodes, the corresponding exchange field can lead to a significant spin-

∗

Corresponding author. Tel.: +48 61 8612300; fax: +48 61 8684524. E-mail address: [email protected] (J. Barna´s).

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splitting of the dot level, and thus can considerably influence the zero bias conductance [4,5]. Magnitude of the splitting, however, depends on the relative orientation of magnetic moments, which in turn can lead to a non-monotonic variation of the linear conductance with the angle between magnetic moments [4]. In this paper we study the interplay between spin-dependent tunneling processes and exchange field due to ferromagnetic electrodes. Both linear and non-linear response regimes are analyzed in the case of strong dot-lead coupling. The corresponding calculations are performed in the framework of non-equilibrium Green’s function formalism, based on the equation of motion method. Since the relevant Green functions are calculated in the Hartree–Fock approximation, the description is applicable well above the corresponding Kondo temperature. 2. Model and method The system under consideration consists of a QD attached to ferromagnetic electrodes and is described by the Hamiltonian: H = H L + HR + H D + H T .




(1)

ε c+ c represents Hamiltonian of the lead β (β = L, R) The term Hβ = ks kβs kβs kβs in the non-interacting quasi-particle approximation, with εkβs being the energy of an electron with the wave vector k and spin s (s = + for spin-majority electrons and s = − for spin-minority ones). The term
HD stands for the dot Hamiltonian and is usually written in the form HD = ε d + d + Ud↑+ d↑ d↓+ d↓ , where σ σ σ σ

´ R. Swirkowicz et al. / Journal of Alloys and Compounds 423 (2006) 244–247 εσ denotes the discrete energy level which can be spin dependent in a general case. Here, σ = ↑,↓ is the spin projection on the spin quantization axis of the dot. Finally, U stands for the electron correlation parameter in the dot. In turn, the tunneling part of the Hamiltonian (1) is assumed in the form

HT = W sσ c+ d + h.c., where the matrix Wkβ is given by: kβ sσ kβ kβs σ

 

⎛ Wkβ = ⎝

Tkβ+ cos Tkβ− sin

φβ 2

−Tkβ+ sin

φβ 2

Tkβ− cos

 ⎞ φ β

2

  ⎠.

 

(2)

φβ 2

Here, the local spin quantization axis in the lead β coincides with the unit vector nβ along the corresponding classical spin moment, and φβ denotes the angle between the vector nβ and the quantization axis for the dot (axis z). We consider here the case where the magnetic moments of electrodes and the quantization axis of the dot are in a common plane. The parameters Tkβs in Eq. (2) are the matrix elements which describe electron tunneling from the dot to the spin-majority (s = +) and spin-minority (s = −) electron
bands in the lead β. In the following, we introduce the parameters βs = 2π k |Tkβs |2 δ(E − εkβs ) and assume that βs are constant within the electron band of the leads and vanish outside the band. In this paper we investigate a non-magnetic dot, so the bare dot level ε0 (the level of the dot uncoupled to the electrodes) does not depend on the spin index. Coupling of the dot to ferromagnetic leads gives rise to an effective exchange field, Bex , which can be expressed in the form [3]: 1  nβ Re gµB

Bex =

×





dε (β+ − β− )fβ 2π

β



1 1 − , ε − ε0 − i¯h/τ0 ε − ε0 − U − i¯h/τ0

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3. Numerical results To illustrate the influence of exchange field Bex on electronic transport through QDs, we consider conductance in the linear and non-linear response regimes. Numerical calculations have been performed assuming: β± = 0 (1 ± p), where p denotes the spin polarization of the external electrodes and 0 is a constant. In the following the energy is measured in relative units, and numerical analysis is focused on highly polarized systems (half-metallic ferromagnets) corresponding to p = 0.95. Apart from this, the considerations are limited to the temperature kB T/0 = 0.5, which is higher than the relevant Kondo temperature. According to Eq. (3), the effective exchange field significantly depends on the correlation parameter U. To emphasize the influence of exchange field on electronic transport, we have performed numerical calculations for different values of U, and for two different values of the dot level energy ε0 measured from the Fermi level EF (taken here as zero in equilibrium). In the first situation we assume ε0 /0 = 0.1, which is very close to the Fermi level in the leads at equilibrium. In the second case, the dot level is well above the Fermi level, ε0 /0 = 2. The corresponding numerical results are presented in Fig. 1, where the linear conductance G is shown as a function of the

(3)

where fβ denotes the Fermi–Dirac distribution function in the electrode β, and τ 0 is the relaxation time (see Ref. [8]). The exchange field determines the quantization axis for the dot (which is opposite to the exchange field). For symmetrical and unbiased systems one can thus assume φL = −φR = −θ/2, where θ is the angle between classical spin moments of the electrodes. The exchange field leads to spin-dependent renormalization of the dot level: εσ = ε0 ∓ gµB Bex /2, with the upper (lower) sign corresponding to σ = ↑ (σ = ↓). The Zeeman-like splitting of the energy level is proportional to the coupling parameter  and gives rise to higher-order corrections in the coupling parameter. The exchange field is of crucial importance for the Kondo effect and leads to splitting and suppression of the zero bias Kondo anomaly [5]. In this paper the Kondo effect is neglected due to a relatively high temperature assumed in numerical calculations. Moreover, in systems with highly spin-polarized electrodes discussed in this paper, the Kondo anomaly is strongly suppressed, so the only relevant effect is the renormalization and spin-splitting of the dot energy level [4]. To describe electronic transport in a biased system, we use the nonequilibrium Green’s function formalism based on the equation of motion method [2]. Higher-order Green’s functions are decoupled according to the Hartree–Fock approximation. The electric current I and the occupation numbers nσ = dσ+ dσ  are then calculated from the following formulae (for details, see [2,7]): I=

e 2¯h



nσ = −i

dE Tr[
L Gr
R Ga +
R Gr
L Ga ](fL − fR ), 2π dE < G = 2π σσ



(4)

dE r [G (
L fL +
R fR )Ga ]σσ 2π

(5)

where Giσσ  = dσ : dσ+ E represents the retarded (i = r), advanced (i = a) and lesser (i = <) Green’s function, whereas elements of the matrix
β have the forms: β↑↑ = β+ cos2 (φβ /2) + β− sin2 (φβ /2), 2 2 and β↑↓ = β↓↑ = β↓↓ = β+ sin (φβ /2) + β− cos (φβ /2), 1  −  sin(φ ). The current and occupation numbers are calculated β+ β− β 2 i



in a self consistent way. The non-diagonal terms, nσ−σ = −i (dE/2π)G< −σσ , have also been taken into account [2], and the lesser Green’s functions G< σσ  have been calculated in terms of the Ng ansatz [9,10].

Fig. 1. Linear conductance as a function of the angle θ for ε0 /0 = 0.1 (a) and ε0 /0 = 2 (b). The different curves correspond to different values of the Hubbard correlation parameter U. The insets show the density of states (DOS) for eV = 0 and for two specified angles θ.

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angle θ between the classical spin moments of the external leads. When the dot level is close to the Fermi level (ε0 /0 = 0.1, Fig. 1(a)), the conductance G varies with the angle in a nonmonotonic way; G increases first slowly with θ and then rapidly decreases when the magnetic moments tend to the antiparallel configuration. Such a dependence is especially pronounced for strongly correlated systems corresponding to large values of the parameter U, and is also consistent with the behavior found by Fransson [4]. Qualitatively different behavior is found in the second case, ε0 /0 = 2 (Fig. 1(b)). Now, a monotonic decrease of G is obtained for all values of the parameter U considered. Such a dependence is typical of normal spin valves [2], where the conductance decreases monotonically when magnetic configuration varies from parallel to antiparallel. The difference between Fig. 1(a and b) can be understood by taking into account the fact that the effective exchange field decreases when the magnetic configuration varies from parallel to antiparallel. (In symmetrical systems the exchange filed disappears in the antiparallel configuration.) Due to the exchangefield-induced spin-splitting of the dot level, a gap near the Fermi level opens for ε0 /0 = 0.1, which diminishes transport through the QD in the linear response regime. The width of the gap is determined by the exchange field and is of the order of 0 . However, the gap decreases with increasing θ, as can be concluded

Fig. 2. Tunneling current as a function of the bias voltage calculated for U/0 = 5000 and for ε0 /0 = 0.1 (a) and ε0 /0 = 2 (b). The different curves correspond to different values of the angle between classical spin moments of the ferromagnetic electrodes.

from the inset in Fig. 1(a), where the corresponding density of states (DOS) is shown. This has a significant influence on the conductance, leading to non-monotonic behavior presented in Fig. 1(a). The situation is different for ε0 /0 = 2. Now, the lowenergy component of the exchange-split level is still above the Fermi level and moves away from the Fermi level when the angle θ increases (see the inset in Fig. 1(b)). This gives an additional contribution to the normal spin valve behavior. Consider now electronic transport in the non-linear response regime. The I–V characteristics calculated for different values of θ are shown in Fig. 2. In the case of ε0 /0 = 0.1 and for collinear and nearly collinear configurations, we find a monotonic increase of electric current with increasing bias in the whole voltage range shown in Fig. 2(a). In the high voltage regime the current in non-collinear configurations reaches a maximum at a certain voltage and then slightly decreases with a further increase in the bias voltage. The situation is different in the case of ε0 /0 = 2 (Fig. 2(b), where the current in parallel configuration reaches first a local maximum at a certain bias voltage, which is followed by a local minimum at a higher voltage, and then again the current increases with increasing bias. In both situations shown in Fig. 2 one finds negative differential conductance in non-collinear configurations, which occurs when the voltage exceeds a certain value. In the case shown in Fig. 2(b), the negative differential conductance also occurs in parallel configuration, but only in a certain voltage range. This behavior can be understood by taking into account spin

Fig. 3. Bias dependence of the TMR for two indicated values of the dot level ε0 . The calculations are performed with and without taking into account the exchange field Bex .

´ R. Swirkowicz et al. / Journal of Alloys and Compounds 423 (2006) 244–247

moment accumulated in the dot, m = n↑ − n↓ , and the exchange field. Detailed calculations show that the spin moment accumulated in the dot increases in the small bias regime, then it strongly decreases changing its sign. Similarly, Bex at first increases and then starts to decrease. The charge accumulated on the dot is practically constant at small voltages (Coulomb blockade regime) and considerably increases at higher bias. The I–V characteristics shown in Fig. 2 reveal a strong dependence of electric current on the angle θ between magnetic moments of the two electrodes. Such a dependence leads to a tunnel magnetoresistance, which can be defined as: TMR = [I(θ = 0) − I(θ)]/I(θ). Here, I(θ) is the current flowing through the system when spin moments form the angle θ. The TMR effect significantly depends on the position of the dot level ε0 , and is also strongly influenced by the exchange field Bex . To display this dependence we present in Fig. 3 the TMR ratio versus bias voltage, calculated for several specific situations. To be more transparent, only TMR associated with the transition from parallel (θ = 0) to antiparallel (θ = π) configurations is shown there. One can see that TMR is relatively small for ε0 /0 = 0.1 and is additionally diminished near the zero bias regime. This diminished value of TMR is a result of the exchange field, which opens a gap due to spin-splitting of the bare dot level and suppresses transport in the parallel configuration at low bias voltages. When Bex could be neglected, the

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TMR curve would be more flat, though it would have a minimum for zero bias (Fig. 3). On the other hand, for ε0 /0 = 2 the TMR effect is substantially enhanced near the zero bias regime. This strong enhancement is also due to the presence of exchange field. Considerable splitting of the bare dot level in the parallel configuration partially removes the Coulomb blockade for electrons with majority spins, leading to some increase of the current. No splitting appears in the antiparallel configuration. As a result one finds a considerable enhancement of TMR due to the exchange field. References [1] W. Rudzi´nski, J. Barna´s, Phys. Rev. B 64 (2001) 085318. ´ [2] W. Rudzi´nski, J. Barna´s, R. Swirkowicz, M. Wilczy´nski, Phys. Rev. B 71 (2005) 205307. [3] M. Braun, J. K¨onig, J. Martinek, Phys. Rev. B 70 (2004) 195345. [4] J. Fransson, cond-matt/0502288 (2005). [5] J. Martinek, Y. Utsumi, H. Imamura, J. Barna´s, S. Maekawa, J. K¨onig, G. Sch¨on, Phys. Rev. Lett. 91 (2003) 127203. [6] J. K¨onig, J. Martinek, Phys. Rev. Lett. 90 (2003) 166602. ´ [7] W. Rudzi´nski, R. Swirkowicz, J. Barna´s, M. Wilczy´nski, J. Magn. Magn. Mater. 294 (2005) 1. [8] Y. Meir, N.S. Wingreen, P.A. Lee, Phys. Rev. Lett. 70 (1993) 2601. [9] T.K. Ng, Phys. Rev. Lett. 70 (1993) 3635. [10] N. Sergueev, Q.-F. Sun, H. Guo, B.G. Wang, J. Wang, Phys. Rev. B 65 (2002) 165303.