Pis'ma v ZhETF, vol.88, iss. 12, pp.927-933
© 2008 December 25
Proximity-driven source of highly spin-polarized ac current on the basis of superconductor/weak ferromagnet/superconductor voltage-biased Josephson junction
A. M. Bobkov, I. V. Bobkova1*!
Institute of Solid State Physics RAS, 142432 Chernogolovka, Moscow reg., Russia
Submitted 31 October 2008 Resubmitted 11 November 2008
We theoretically investigate an opportunity to implement a source of highly spin-polarized ac current on the basis of superconductor/weak ferromagnet/superconductor (SFS) voltage-biased junction in the regime of essential proximity effect and calculate the current flowing through the probe electrode tunnel coupled to the ferromagnetic interlayer region. It is shown that while the polarization of the dc current component is generally small in case of weak exchange field of the ferromagnet, there is an ac component of the current in the system. This ac current is highly spin-polarized and entirely originated from the non-equilibrium proximity effect in the interlayer. The frequency of the current is controlled by the voltage applied to SFS junction. We discuss a possibility to obtain a source of coherent ac currents with a certain phase shift between them by tunnel coupling two probe electrodes at different locations of the interlayer region.
PACS: 74.45,+c, 74.50.+r
Various proximity and transport phenomena in hybrid structures containing superconducting and ferromagnetic elements are currently in the spotlight. The equilibrium transport and proximity effect in such structures have been theoretically and experimentally investigated recently in details as for the case of weak ferromagnetic alloys (see Ref.[l] and references therein) so as for half-metals like Cr02 [2, 3].
Spin-dependent properties of nonequilibrium systems are also actively investigated. In particular, the spin imbalance induced in a ferromagnet (F) in nonequilibrium conditions was studied in ferromagnet-superconductor-ferromagnet (FSF) junctions [4-9]. It was found that antiferromagnetic alignment of the exchange fields of the ferromagnets strongly suppresses superconductivity, which leads to large magnetoresistive effect. The influence of the interplay between Andreev reflection at the interface and spin accumulation close to the interface was investigated [10]. The effect of spin injection in Josephson junctions was considered [11]. The papers [9, 12] have theoretically studied the spin relaxation due to spin-flip scattering.
The authors of [13, 14] propose the possibility of manipulating magnetization of a mesoscopic normal (N) region through the Zeeman splitting of superconducting density of states and applied voltage in voltage biased SNS or FSNSF tunnel junctions. Also, the spin-polarized transport through superconductor-
e-mail: bobkova0issp.ac.ru
normal metal hybrid structures, where the density of states in a superconductor is Zeeman-splitted, was investigated [13-15]. It was shown that, in alternative to half-metallic ferromagnets, such junction can be used to generate highly spin-polarized currents, which are tunable in magnitude and sign by the bias voltage and exchange field. In [13, 14] the normal metal region between the superconducting leads has been considered to be long enough in order to suppress the proximity effect and, consequently, ac Josephson effect in it. However, if the junction length is of the order of superconducting coherence length £ = s/HD/A, where D is the diffusion constant and A is the superconducting order parameter in the leads, the interplay between the proximity effect and the Zeeman exchange field in the interlayer can lead to a number of novel qualitative effects. In particular, it was shown that by tunnel coupling of the normal regions of two superconductor-normal metal-ferromagnet trilay-ers the absolute spin-valve effect can be realized for a certain interval of voltages applied between the normal regions [16]. For voltage-biased superconductor-weak ferromagnet-superconductor Josephson junction the interplay between the proximity effect and the Zeeman exchange field results in additional peak-like features in the I-V characteristics [17].
In the present paper we study voltage-biased superconductor-weak ferromagnet-superconductor (SFS) Josephson junction focusing on the short enough interlayer (which is considered to be of the order of superconducting coherence length). In this regime the
IlHCbMa b ?K3T<J> tom 88 Bbin.11-12 2008
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proximity effect between ferromagnet and superconductors is essential. One of the well-known manifestations of the proximity effect, which already takes place in equilibrium, is the so-called minigap in the local density of states (LDOS) of the interlayer [18, 19]. Another important manifestation of the proximity effect is an ac current, which appears in case of voltage-biased junction. We show that the interplay of two above mentioned phenomena in non-equilibrium SFS Joseph-son junction gives an opportunity to implement a source of highly spin-polarized ac current by tunnel coupling the interlayer region to an additional electrode. The frequency of this ac current is controlled by the voltage V, applied to the SFS junction. In addition the proximity effect in the interlayer causes substantial non-linearities in the I — characteristics of the ac current flowing through the additional electrode. Here Vp is the potential of the probe electrode with respect to the left superconducting lead. The dc current flowing through the additional electrode is also strongly non-linear function of Vp, although its spin polarization is obtained to be rather weak except for some narrow ranges of Vp. We also discuss the phase difference between the ac current flowing through the probe electrode and the ac Josephson current flowing across the SFS junction and show that it depends on the position in the interlayer. Therefore the system under consideration can be used as a source of phase-shifted coherent ac currents by tunnel coupling of two probe electrodes at different locations of the interlayer region.
Further the model and the method we use are described. We study a voltage-biased SFS junction, where F is a diffusive weak ferromagnet of length d coupled to two identical superconducting reservoirs. The superconductors are supposed to be diffusive and have s-wave pairing. We assume the SF interfaces to be not fully transparent and suppose that the resistance of the SF boundary Rg dominates the resistance of the ferromagnetic interlayer Rp. We assume the parameter (Rp/Rg)(<jF/a»), where <jp and crs stand for conductivities of ferromagnetic and superconducting materials respectively, to be also small, what allows us to neglect the suppression of the superconducting order parameter in the S leads near the interface. In addition, a normal voltage-biased "probe" terminal is tunnel coupled to the interlayer through a junction of a resistance Rp^> Rg.
We use the quasiclassical theory of superconductivity for diffusive systems in terms of time-dependent Usadel equations [20]. The fundamental quantity for diffusive transport is the momentum average of the quasiclassical Green's function g(x,e,t) = (g(pf,x,e,t))Pf. It is a 8 x 8 matrix form in the product space of Keldysh,
particle-hole and spin variables. In general the quasi-classical Green's functions depend on space R, time t variables and the excitation energy e. The considered problem is effectively one-dimensional and R = x, where x - is the coordinate measured along the normal to the junction.
In order to solve the Usadel equation it is convenient
to express quasiclassical Green's function g in terms of
R A.
Riccati coherence functions and 7 ' , which measure the relative amplitudes for normal-state quasiparti-cle and quasihole excitations and distribution functions xK and x . All these functions are 2x2 matrices in spin space and depend on (x,e,t). The corresponding expression for g is given in Ref. [21]. Riccati coherence and distribution functions obey Riccati-type transport equations [21, 22], which should be solved together with the boundary conditions at SF interfaces. As it was mentioned above we consider the case when the dimen-sionless conductance of the boundary G = Rp/Rg < 1, so the interface transparency Ts/■ ~ G(l/d) -C 1. Due to the smallness of the interface transparency Ts /■ we can use Kupriyanov-Lukichev boundary conditions at SF boundaries [23]. In terms of Riccati coherence and distribution functions they are presented in Ref. [17]. As it was already mentioned above, under the condition (Rp¡Rg)((iF/crs) -C 1 we can neglect the suppression of superconducting order parameter in the leads and, moreover, take the Green's functions at the superconducting side of the boundaries to be equal to their bulk values.
If one takes into account the explicit form of the Green's function for the uncoupled normal probe electrode g^R'A = =Fi7rr3<Jo and = ^2i7rtanh[(e — — eVp)/2T]f3<Jo, then in the first order on the transparency Tpp of the junction between the interlayer and the probe electrode we get the following expression for the current flowing through this electrode:
= (1)
• tanh
£ 27^' *)t)®tanh£ 2^Vp
gp'A'K(x,e,t) is the upper left part of the interlayer Green's function g^' 'K(x,e,t) in the particle-hole space. The product ® of two functions of energy and time is defined by the noncommutative convolution A®B = e«9?9?-^f)/2A(e,t)B(e,t). ff and a, are Pauli matrices in particle-hole and spin spaces respectively. The spin current jsp/se (se = 1/2 is the electron spin) can be calculated from Eq. (1) with the substitution 03 for (Jq.
1.2
1.1
1.0
-J5-. 0.9
0.8
0.7
ÉÔTU—Г^т-
h eV= 3.5 Г,
"J-ff-p
eV= 2.5 Г„
tl_i_I_i_i_i_I_i_i_i_I_i_i_i_I_i_i_l_
-2 0 2 4
0.02
0.01
0
-0.01
-0.02
-0.03
НИМ4-
. eV=2.5Tr
-Л-Л^-
-i_i_i_I_i_i_i_I_i_i_i_I_i_i_i_I_i_i_i_L
-2 0 2 4 6
eVp/Tc
1.0
-
0.8 - (C)
~ 0.
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