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Influence of different band masses on ballistic charge transport in ferromagnet–superconductor–ferromagnet trilayers
Physica C 469 (2009) 1915–1920
Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Influence of different band masses on ballistic charge transport in ferromagnet–superconductor–ferromagnet trilayers Z. Popovic´, M. Bozˇovic´ *, Z. Radovic´ Department of Physics, University of Belgrade, P.O. Box 368, 11001 Belgrade, Serbia
a r t i c l e
i n f o
Article history: Received 25 April 2009 Received in revised form 30 May 2009 Accepted 17 June 2009 Available online 23 June 2009 PACS: 74.45.+c 72.25.Ba 72.25.Dc
a b s t r a c t We study transport properties of clean FISIF double-barrier junctions consisting of metallic or semiconducting ferromagnets (F), a superconductor (S), and insulating interfaces (I). We solve the scattering problem based on the Bogoliubov–de Gennes equation and calculate differential conductance for arbitrary interface transparency, different effective band masses and Fermi wave vectors in the conductors. We analyze size and coherence effects that characterize ballistic transport: subgap transmission and geometrical oscillations of the conductance. We find that different band masses, as well as different Fermi wave vectors, affect the transport properties in a way similar to interfaces of a finite transparency. In all these cases, charge transport is reduced to resonant tunneling through the quasi-bound states in the superconducting film. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Spin-polarized ballistic transport Superconductor–semiconducting ferromagnet tunnel junctions
1. Introduction The interplay between ferromagnetism and superconductivity in thin-film heterostructures is a phenomenon that attracts considerable interest of researchers for some time already [1,2]. This increasing interest coincides with the progress in nanofabrication technology and a growing potential for application of such structures in quantum electronics and spintronics [3]. Small-scale systems exhibit ballistic charge transport, which is qualitatively different from diffusive behavior of charge carriers. Blonder et al. [4] were among the first to provide a detailed theoretical picture of phase-coherent charge transport through a junction between a normal metal (N) and a superconductor (S), with an insulating barrier (I) of arbitrary strength at the interface. The Andreev reflection [5] was recognized as the mechanism of normal-to-supercurrent conversion. It was shown later that the Andreev reflection can be modified by the injection of spin-polarized electrons and holes from a ferromagnetic metal (F) into a superconductor in FIS junctions [6,7]. Recently, ballistic transport through double junctions
of NISIN and FISIF type has been intensively studied both theoretically [8–14] and experimentally [15–17]. The purpose of this paper is to clarify the influence of different band masses [13,18,21] and/or Fermi wave vectors [9] on the properties of coherent electronic transport. The model we study is an FISIF heterostructure consisting of a superconducting film in contact with ferromagnetic metals or semiconductors, separated by the interface potential barriers of arbitrary transparency. We limit ourselves to clean, conventional (that is, isotropic and s-wave) superconductors and neglect non-equilibrium effects of charge and spin accumulation at the interfaces. To describe transport properties of such a system, we calculate the scattering probabilities from solution of the Bogoliubov–de Gennes equation. We find that difference in band masses between the layers, as well as the difference in Fermi wave vectors, affect the transport properties in a way similar to interfaces of a finite transparency. Namely, all these effects yield similar conductance spectra. In all cases such behavior can be attributed to a reduction of charge transport through quasi-bound states in the superconducting film.
2. Scattering probabilities * Corresponding author. Address: Department of Economics and Business, Universitat Pompeu Fabra, Rámon Trias Fargas 25–27, 08005 Barcelona, Spain. Tel.: +34 675049238. E-mail addresses: [email protected] (Z. Popovic´), [email protected], milos@ff. bg.ac.yu (M. Bozˇovic´), [email protected] (Z. Radovic´). 0921-4534/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.06.008
We consider a simple model of an FISIF double junction consisting of a clean superconducting layer of thickness d, connected to ferromagnetic metals or ferromagnetic semiconductors by thin,
Z. Popovic´ et al. / Physica C 469 (2009) 1915–1920
1916
Translational invariance of the junction in the x—y planes implies conservation of the parallel component of the wave vector kk;r . Consequently, the wave function
Wr ðrÞ ¼ eikk;r r wr ðzÞ;
ð2Þ
satisfies the following boundary conditions Fig. 1. A schematic of an FISIF double junction with parallel alignment of magnetizations.
insulating interfaces, Fig. 1. To model the ferromagnetic layers we adopt the Stoner model, which describes the spin-polarization effect by a single-electron Hamiltonian with an exchange potential. The quasiparticle propagation is described by the usual Bogoliubov–de Gennes equation
DðrÞ Wr ðrÞ ¼ EWr ðrÞ; H0 ðrÞ þ qr hðrÞ
H0 ðrÞ qr hðrÞ
D ðrÞ
ð1Þ
2
with H0 ðrÞ ¼ rð h =2mðrÞÞr þ WðrÞ þ UðrÞ l, where UðrÞ and l are the Hartree and chemical potential, respectively. The interface c fdðzÞ þ dðz dÞg, where the z axis potential is modeled by WðrÞ ¼ W is perpendicular to layers and dðzÞ is the Dirac d function. The superconducting pair potential is taken in the form DðrÞ ¼ D0 HðzÞHðd zÞ, where HðzÞ is the Heaviside step function and D0 is the bulk superconducting gap. Note that self-consistency may be safely avoided when the proximity effect is weak between S and F layers. This includes considered junctions with tunnel barriers at interfaces, and/or narrow F constriction, and/or large mismatch in Fermi velocities. In Eq. (1), r is the quasiparticle spin (r ¼"; # ¼#; "), E is quasiparticle energy with respect to l, while and r qr ¼ 1ð1Þ for r ¼" ð#Þ. We assume that a uniform magnetization is parallel to the layers, such that the exchange potential hðrÞ is given by h0 fHðzÞ þ Hðz dÞg. To describe the difference in band masses we use the ratio mF =mS , where mF and mS are the electron band masses in the ferromagnets and in the superconductor, respectively. Similarly, ðSÞ l UðrÞ is the Fermi energy of the superconductor, E , or the " F ðFÞ # mean Fermi energy of a ferromagnet, EF ¼ EF þ EF =2. Next, to capture the Fermi wave vector mismatch (FWVM) we allow for differentq moduli of the Fermi wave vectors in the F and S layers, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðFÞ
kF ¼
ðFÞ
2
ðSÞ
2mF EF = h and kF ¼
ðSÞ
2
2mS EF = h , respectively. The misðFÞ
ðSÞ
match is measured by a dimensionless parameter kF =kF .
wr ðzÞjz¼0 ¼ wr ðzÞjz¼0þ ; ðSÞ 1 dwr ðzÞ 1 dwr ðzÞ ZkF ¼ wr ðzÞ mF dz z¼0 mS dz z¼0þ mS
ð3Þ ð4Þ
;
z¼0
wr ðzÞjz¼d ¼ wr ðzÞjz¼dþ ; 1 dwr ðzÞ 1 dwr ðzÞ ¼ mS dz mF dz z¼d
wr ðzÞ mS
ð5Þ
ðSÞ ZkF
z¼dþ
ð6Þ
; z¼d
2 ðSÞ c = where Z ¼ 2m W h kF is a dimensionless parameter measuring the strength of each interface barrier [9,18,21]. The four independent solutions of Eq. (1) correspond to four types of quasiparticle injection: an electron or a hole from either the left or from the right electrode. For the injection of an electron from the left, with energy E > 0, spin r, and angle of incidence h (measured from the z axis), solution from wr ðzÞ in various regions has the following form
þ þ 1 wr ðzÞ ¼ exp ikr z þ br ðE; hÞ exp ikr z 0 0 þ ar ðE; hÞ expðikr zÞ ; 1
ð7Þ
in the left ferromagnet ðz < 0Þ,
þ þ u wr ðzÞ ¼ C 1 ðE; hÞ exp iqr z þ C 2 ðE; hÞ exp iqr z v v þ C 3 ðE; hÞ exp iqr z þ C 4 ðE; hÞ exp iqr z ; u
ð8Þ
in the superconductor ð0 < z < dÞ, and
þ 1 0 wr ðzÞ ¼ cr ðE; hÞ exp ikr z þ dr ðE; hÞ exp ikr z ; 0 1
ð9Þ
in the right ferromagnet ðz > dÞ, for the parallel alignment of the magnetizations [9].
ðSÞ
ðFÞ
ðSÞ
Fig. 2. Scattering probabilities A through D for an NISIN double junction with thin superconducting film d=n0 ¼ 1; D0 =EF ¼ 103 ; mF =mS ¼ 1 and kF =kF ¼ 1. Solid curves: transparent interfaces, Z ¼ 0, dotted curves: low transparent interfaces, Z ¼ 10.
Z. Popovic´ et al. / Physica C 469 (2009) 1915–1920
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð1 þ X=EÞ=2 p Here, u and v ¼ ð1 X=EÞ=2 are the BCS ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi coherence factors, and X ¼ E2 D2 . The z components of the wave vectors are
kr ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðFÞ ð2mF =h ÞðEF þ qr h0 EÞ kk;r
ð10Þ
in ferromagnets, and
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðSÞ qr ¼ ð2mS = h ÞðEF XÞ kk;r
gating in the superconducting layer, are given by the coefficients C 1 through C 4 . Solution for the other three types of injection can be obtained by the same procedure. From the probability current conservation, the probabilities of outgoing particles satisfy the normalization condition
Ar ðE; hÞ þ Br ðE; hÞ þ C r ðE; hÞ þ Dr ðE; hÞ ¼ 1: ð11Þ
in superconductor, where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðFÞ jkk;r j ¼ ð2mF =h ÞðEF þ qr h0 þ EÞ sin h:
1917
Appropriate probabilities have the following form
Ar ðE; hÞ ¼ Re ð12Þ
The coefficients ar ; br ; cr and dr are, respectively, the probability amplitudes of Andreev reflection as a hole of opposite spin, normal reflection as an electron, transmission to the right electrode as an electron, and transmission to the right electrode as a hole of the opposite spin (the so-called crossed Andreev reflection). Amplitudes of the Bogoliubov electron-like and hole-like quasiparticles, propa-
ð13Þ
kr jar ðE; hÞj2 ; kr
ð14Þ
Br ðE; hÞ ¼ jbr ðE; hÞj2 ;
ð15Þ
2
C r ðE; hÞ ¼ jcr ðE; hÞj ; kr jdr ðE; hÞj2 ; Dr ðE; hÞ ¼ Re kr
ð16Þ ð17Þ
where kr is kr at the Fermi level, E = 0 (see, for example, Ref. [9]).
ðSÞ
Fig. 3. Scattering probabilities Ar through Dr ; r ¼", for an FISIF double junction with transparent interfaces, Z ¼ 0, thin superconducting film, d=n0 ¼ 1; D0 =EF ¼ 103 , strong ðFÞ ðSÞ ðFÞ ferromagnet h0 =EF ¼ 0:5, and equal Fermi wave vectors, kF =kF ¼ 1. Curves correspond to three different mass ratio: mF =mS ¼ 0:1 (dashed), mF =mS ¼ 1 (solid), and mF =mS ¼ 10 (dotted).
Fig. 4. Same as in Fig. 3 for
r ¼#.
Z. Popovic´ et al. / Physica C 469 (2009) 1915–1920
1918
It can be shown that Ar ¼ Dr ¼ 0 whenever
d qþr qr ¼ 2np
ð18Þ
for n ¼ 0; 1; 2; . . .. Therefore, both direct and crossed Andreev reflection vanish at the energies of geometrical resonances in quasiparticle spectrum of the S layer [9]. The absence of direct and crossed Andreev processes implies that all quasiparticles with energies satisfying Eq. (18) will pass unaffected from one electrode to another, without creation or annihilation of Cooper pairs. Both the presence of insulating barriers and exchange interaction reduce Ar and C r and enhance Br and Dr . Approaching the tunnel limit ðZ ! 1Þ, the spikes in Ar ; C r , and Dr , as well as the dips in Br , occur at the energies given by the quantization conditions þ
dqr ¼ n1 p; dqr ¼ n2 p;
ð19Þ
which correspond to the bound-state energies of an insulated superconducting film. In this case, these bound states are the only conducting channels, both for supercurrent and quasiparticle current [9].
The influence of different band masses and/or Fermi wave vectors on the scattering probabilities is illustrated for double junctions with fully transparent ðZ ¼ 0Þ and low-transparent ðZ ¼ 10Þ interfaces. This is shown in Figs. 2–7 for h ¼ 0. In all illustration 3 D0 =EðSÞ and the thickness of the superconducting layer is F ¼ 10 1 v F =pD0 is the superconducting d ¼ n0 ¼ 636kF , where n0 ¼ h coherence length and v F is the Fermi velocity. The ferromagnet is ðFÞ assumed to be strong, h0 =EF ¼ 0:5. In our example of a relatively thin superconducting film, subgap electronic tunneling is significant and Andreev reflection is reduced. The influence of interface transparency on scattering probabilities for equal band masses and Fermi wave vectors is shown in Fig. 2 for h0 ¼ 0 (NISIN heterostructure). For transparent interfaces ðZ ¼ 0Þ Ar and C r exhibit oscillatory behavior due to the coherent tunneling through the heterostructure. The probability of transmission C r is highest close to the resonant states above the gap. The Andreev reflection, measured by the probability Ar , is thus entirely suppressed near these energies. The probabilities of normal reflection and crossed Andreev reflection, Br and Dr , respectively,
Fig. 5. Same as in Fig. 3 for interfaces of low transparency, Z ¼ 10.
ðSÞ
Fig. 6. Scattering probabilities Ar through Dr ; r ¼", for an FISIF double junction with transparent interfaces, Z ¼ 0, thin superconducting film d=n0 ¼ 1; D0 =EF ¼ 103 , strong ðFÞ ðSÞ ðFÞ ferromagnet h0 =EF ¼ 0:5, and mF =mS ¼ 1. Curves correspond to three different Fermi wave vector ratio: kF =kF ¼ 0:5 (dotted), j ¼ 1 (solid), and j ¼ 2 (dashed).
Z. Popovic´ et al. / Physica C 469 (2009) 1915–1920
1919
Fig. 7. Same as in Fig. 6 for interfaces of low transparency, Z ¼ 10.
are different than zero only when the interfaces are non-transparent or when h0 – 0 (FISIF heterostructure). On the other hand, in the case of interfaces of low transparency ðZ ¼ 10Þ the probability of tunneling is highest close to the quasi-bound states of an isolated superconducting film. As the transparency of the interfaces increases (equivalently, as Z decreases), the probabilities around quasi-bound states shift and widen. The influence of different band masses on probabilities Ar through Dr in FISIF heterostructure is shown for r ¼"; # (Figs. 3 and 4) for transparent interfaces. For mismatched effective masses, mF =mS ¼ 0:1 and mF =mS ¼ 10, the probabilities exhibit a pronounced oscillatory behavior close to the resonant states above the gap. In the case of interfaces of low transparency ðZ ¼ 10Þ differences in band masses have a qualitatively negligible influence on the probabilities. Also, we find that the probabilities are practically independent on the spin orientation. Hence, in Fig. 5 we show only the case r ¼". For equal band masses, the influence of different Fermi wave vectors on scattering probabilities is shown for r ¼" and for interfaces that have either full or low transparency, Figs. 6 and 7, respectively. It can be seen that influence of different band masses and Fermi wave vectors leads to a qualitatively similar behavior of scattering probabilities as the influence of lower transparency. In both cases resonant tunneling through quasi-bound states above the superconducting gap induces oscillations in energy dependence of scattering probabilities.
through the probabilities of local Andreev reflection and direct transmission of electrons injected from the left electrode. Without solving the suitable transport equation, we take df ðk; VÞ ¼ f0 ðE eV=2Þ f0 ðE þ eV=2Þ [4]. In this approach, the charge current per orbital transverse channel is given by
IðVÞ ¼
1 e
Z
1
dE½f0 ðE eV=2Þ f0 ðE þ eV=2ÞGðEÞ;
ð21Þ
1
where
GðEÞ ¼ G0
X r¼";#
Pr
Z
2
d kk;r 2
2pkF
ðAr þ C r Þ
ð22Þ
is differential charge conductance at zero temperature and is the conductance quantum. Eq. (22) is a simple generG0 ¼ e2 =ph
3. The differential conductance When voltage V is applied to the junction symmetrically, the charge current density can be written in the form [10,14,19]
JðVÞ ¼
Z
e 3
ð2pÞ h
1
1
dE
X
Pr
Z
2
d kk;r ð1 þ Ar Br þ C r
r¼";#
Dr Þdf ðk; VÞ Z Z 1 X e 2 dE P ¼ d kk;r ðAr þ C r Þdf ðk; VÞ; r 4p3 h 1 r¼";#
ð20Þ
ðFÞ
where Pr ¼ ð1 þ qr h0 =EF Þ=2, and df ðk; VÞ is the asymmetric part of the non-equilibrium distribution function of current carriers. In the last equality the normalization condition given by Eq. (13) was taken into account. Note that both electron and hole contribution to the current are included, although the final result is expressed
Fig. 8. Differential conductance spectra of an FISIF double junction with interfaces with (a) full, Z ¼ 0, and (b) low, Z ¼ 10, transparency. Other parameters have values ðFÞ ðSÞ ðSÞ ðFÞ d=n0 ¼ 1; D0 =EF ¼ 103 ; h0 =EF ¼ 0:5, and kF =kF ¼ 1. Curves correspond to three different mass ratio: mF =mS ¼ 0:1 (dashed), mF =mS ¼ 1 (solid), and mF =mS ¼ 10 (dotted).
Z. Popovic´ et al. / Physica C 469 (2009) 1915–1920
1920
dence of the conductance. This implies that in a double tunnel junction (Figs. 8b and 9b) conductance spectra are practically independent of band masses and Fermi wave vectors ratio. 4. Conclusions We have analyzed the influence of different band masses and/or Fermi wave vectors on the ballistic transport properties of FISIF heterostructures and found that the effect should be observable only for the interfaces of low transparencies. In a double tunnel junction conductance spectra are practically independent of the difference in band masses or Fermi wave vectors ratio. However, if a junction is transparent, these differences can significantly affect the properties of the charge transport. This is manifested in a way similar to the presence of non-transparent interfaces in a junction with equal band masses and Fermi wave vectors. The similarity occurs because in all the cases charge transport is limited to the quasi-bound states in the superconducting film. Acknowledgement
Fig. 9. Differential conductance spectra of an FISIF double junction with interfaces with (a) full, Z ¼ 0, and (b) low, Z ¼ 10, transparency. Other parameters have values ðSÞ ðFÞ d=n0 ¼ 1; D0 =EF ¼ 103 ; h0 =EF ¼ 0:5, and mF =mS ¼ 1. Curves correspond to three ðFÞ ðSÞ ðFÞ ðSÞ different Fermi wave vector ratio: kF =kF ¼ 0:5 (dotted), kF =kF ¼ 1 (solid), and ðFÞ ðSÞ kF =kF ¼ 10 (dashed).
alization of the Landauer formula [20], with terms that take into account transmission of the current through Cooper pairs and quasiparticles. For E < D, the subgap transmission of quasiparticles (without conversion into Cooper pairs) suppresses the Andreev reflection, while for E > D all the probabilities oscillate with E and ds due to the interference of incoming and outgoing particles. In the following we analyze the conductance spectra for onedimensional double junction with a normal incidence of charge carriers, h ¼ 0, which corresponds to a single-channel flow in FISIF nano-structure. The conductance spectrum is given by
GðEÞ ¼ G0
X
Pr ðAr þ C r Þ:
ð23Þ
r¼";#
Influence of different band masses and Fermi wave vectors on zerotemperature conductance spectrum for transparent and low-transparent interfaces is illustrated in Figs. 8 and 9 for the thickness of ðFÞ the superconducting layer d ¼ n0 and for h0 =EF ¼ 0:5. It can be seen that influence of different band masses and Fermi wave vectors on differential conductance spectra is qualitatively the same as the influence of lower transparency. In both cases resonant tunneling through quasi-bound states induces oscillations in energy depen-
The work was supported by the Serbian Ministry of Science, Project No. 141014. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
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Contents lists available at ScienceDirect
Physica C journal homepage: www.elsevier.com/locate/physc
Influence of different band masses on ballistic charge transport in ferromagnet–superconductor–ferromagnet trilayers Z. Popovic´, M. Bozˇovic´ *, Z. Radovic´ Department of Physics, University of Belgrade, P.O. Box 368, 11001 Belgrade, Serbia
a r t i c l e
i n f o
Article history: Received 25 April 2009 Received in revised form 30 May 2009 Accepted 17 June 2009 Available online 23 June 2009 PACS: 74.45.+c 72.25.Ba 72.25.Dc
a b s t r a c t We study transport properties of clean FISIF double-barrier junctions consisting of metallic or semiconducting ferromagnets (F), a superconductor (S), and insulating interfaces (I). We solve the scattering problem based on the Bogoliubov–de Gennes equation and calculate differential conductance for arbitrary interface transparency, different effective band masses and Fermi wave vectors in the conductors. We analyze size and coherence effects that characterize ballistic transport: subgap transmission and geometrical oscillations of the conductance. We find that different band masses, as well as different Fermi wave vectors, affect the transport properties in a way similar to interfaces of a finite transparency. In all these cases, charge transport is reduced to resonant tunneling through the quasi-bound states in the superconducting film. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Spin-polarized ballistic transport Superconductor–semiconducting ferromagnet tunnel junctions
1. Introduction The interplay between ferromagnetism and superconductivity in thin-film heterostructures is a phenomenon that attracts considerable interest of researchers for some time already [1,2]. This increasing interest coincides with the progress in nanofabrication technology and a growing potential for application of such structures in quantum electronics and spintronics [3]. Small-scale systems exhibit ballistic charge transport, which is qualitatively different from diffusive behavior of charge carriers. Blonder et al. [4] were among the first to provide a detailed theoretical picture of phase-coherent charge transport through a junction between a normal metal (N) and a superconductor (S), with an insulating barrier (I) of arbitrary strength at the interface. The Andreev reflection [5] was recognized as the mechanism of normal-to-supercurrent conversion. It was shown later that the Andreev reflection can be modified by the injection of spin-polarized electrons and holes from a ferromagnetic metal (F) into a superconductor in FIS junctions [6,7]. Recently, ballistic transport through double junctions
of NISIN and FISIF type has been intensively studied both theoretically [8–14] and experimentally [15–17]. The purpose of this paper is to clarify the influence of different band masses [13,18,21] and/or Fermi wave vectors [9] on the properties of coherent electronic transport. The model we study is an FISIF heterostructure consisting of a superconducting film in contact with ferromagnetic metals or semiconductors, separated by the interface potential barriers of arbitrary transparency. We limit ourselves to clean, conventional (that is, isotropic and s-wave) superconductors and neglect non-equilibrium effects of charge and spin accumulation at the interfaces. To describe transport properties of such a system, we calculate the scattering probabilities from solution of the Bogoliubov–de Gennes equation. We find that difference in band masses between the layers, as well as the difference in Fermi wave vectors, affect the transport properties in a way similar to interfaces of a finite transparency. Namely, all these effects yield similar conductance spectra. In all cases such behavior can be attributed to a reduction of charge transport through quasi-bound states in the superconducting film.
2. Scattering probabilities * Corresponding author. Address: Department of Economics and Business, Universitat Pompeu Fabra, Rámon Trias Fargas 25–27, 08005 Barcelona, Spain. Tel.: +34 675049238. E-mail addresses: [email protected] (Z. Popovic´), [email protected], milos@ff. bg.ac.yu (M. Bozˇovic´), [email protected] (Z. Radovic´). 0921-4534/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2009.06.008
We consider a simple model of an FISIF double junction consisting of a clean superconducting layer of thickness d, connected to ferromagnetic metals or ferromagnetic semiconductors by thin,
Z. Popovic´ et al. / Physica C 469 (2009) 1915–1920
1916
Translational invariance of the junction in the x—y planes implies conservation of the parallel component of the wave vector kk;r . Consequently, the wave function
Wr ðrÞ ¼ eikk;r r wr ðzÞ;
ð2Þ
satisfies the following boundary conditions Fig. 1. A schematic of an FISIF double junction with parallel alignment of magnetizations.
insulating interfaces, Fig. 1. To model the ferromagnetic layers we adopt the Stoner model, which describes the spin-polarization effect by a single-electron Hamiltonian with an exchange potential. The quasiparticle propagation is described by the usual Bogoliubov–de Gennes equation
DðrÞ Wr ðrÞ ¼ EWr ðrÞ; H0 ðrÞ þ qr hðrÞ
H0 ðrÞ qr hðrÞ
D ðrÞ
ð1Þ
2
with H0 ðrÞ ¼ rð h =2mðrÞÞr þ WðrÞ þ UðrÞ l, where UðrÞ and l are the Hartree and chemical potential, respectively. The interface c fdðzÞ þ dðz dÞg, where the z axis potential is modeled by WðrÞ ¼ W is perpendicular to layers and dðzÞ is the Dirac d function. The superconducting pair potential is taken in the form DðrÞ ¼ D0 HðzÞHðd zÞ, where HðzÞ is the Heaviside step function and D0 is the bulk superconducting gap. Note that self-consistency may be safely avoided when the proximity effect is weak between S and F layers. This includes considered junctions with tunnel barriers at interfaces, and/or narrow F constriction, and/or large mismatch in Fermi velocities. In Eq. (1), r is the quasiparticle spin (r ¼"; # ¼#; "), E is quasiparticle energy with respect to l, while and r qr ¼ 1ð1Þ for r ¼" ð#Þ. We assume that a uniform magnetization is parallel to the layers, such that the exchange potential hðrÞ is given by h0 fHðzÞ þ Hðz dÞg. To describe the difference in band masses we use the ratio mF =mS , where mF and mS are the electron band masses in the ferromagnets and in the superconductor, respectively. Similarly, ðSÞ l UðrÞ is the Fermi energy of the superconductor, E , or the " F ðFÞ # mean Fermi energy of a ferromagnet, EF ¼ EF þ EF =2. Next, to capture the Fermi wave vector mismatch (FWVM) we allow for differentq moduli of the Fermi wave vectors in the F and S layers, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðFÞ
kF ¼
ðFÞ
2
ðSÞ
2mF EF = h and kF ¼
ðSÞ
2
2mS EF = h , respectively. The misðFÞ
ðSÞ
match is measured by a dimensionless parameter kF =kF .
wr ðzÞjz¼0 ¼ wr ðzÞjz¼0þ ; ðSÞ 1 dwr ðzÞ 1 dwr ðzÞ ZkF ¼ wr ðzÞ mF dz z¼0 mS dz z¼0þ mS
ð3Þ ð4Þ
;
z¼0
wr ðzÞjz¼d ¼ wr ðzÞjz¼dþ ; 1 dwr ðzÞ 1 dwr ðzÞ ¼ mS dz mF dz z¼d
wr ðzÞ mS
ð5Þ
ðSÞ ZkF
z¼dþ
ð6Þ
; z¼d
2 ðSÞ c = where Z ¼ 2m W h kF is a dimensionless parameter measuring the strength of each interface barrier [9,18,21]. The four independent solutions of Eq. (1) correspond to four types of quasiparticle injection: an electron or a hole from either the left or from the right electrode. For the injection of an electron from the left, with energy E > 0, spin r, and angle of incidence h (measured from the z axis), solution from wr ðzÞ in various regions has the following form
þ þ 1 wr ðzÞ ¼ exp ikr z þ br ðE; hÞ exp ikr z 0 0 þ ar ðE; hÞ expðikr zÞ ; 1
ð7Þ
in the left ferromagnet ðz < 0Þ,
þ þ u wr ðzÞ ¼ C 1 ðE; hÞ exp iqr z þ C 2 ðE; hÞ exp iqr z v v þ C 3 ðE; hÞ exp iqr z þ C 4 ðE; hÞ exp iqr z ; u
ð8Þ
in the superconductor ð0 < z < dÞ, and
þ 1 0 wr ðzÞ ¼ cr ðE; hÞ exp ikr z þ dr ðE; hÞ exp ikr z ; 0 1
ð9Þ
in the right ferromagnet ðz > dÞ, for the parallel alignment of the magnetizations [9].
ðSÞ
ðFÞ
ðSÞ
Fig. 2. Scattering probabilities A through D for an NISIN double junction with thin superconducting film d=n0 ¼ 1; D0 =EF ¼ 103 ; mF =mS ¼ 1 and kF =kF ¼ 1. Solid curves: transparent interfaces, Z ¼ 0, dotted curves: low transparent interfaces, Z ¼ 10.
Z. Popovic´ et al. / Physica C 469 (2009) 1915–1920
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð1 þ X=EÞ=2 p Here, u and v ¼ ð1 X=EÞ=2 are the BCS ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi coherence factors, and X ¼ E2 D2 . The z components of the wave vectors are
kr ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðFÞ ð2mF =h ÞðEF þ qr h0 EÞ kk;r
ð10Þ
in ferromagnets, and
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðSÞ qr ¼ ð2mS = h ÞðEF XÞ kk;r
gating in the superconducting layer, are given by the coefficients C 1 through C 4 . Solution for the other three types of injection can be obtained by the same procedure. From the probability current conservation, the probabilities of outgoing particles satisfy the normalization condition
Ar ðE; hÞ þ Br ðE; hÞ þ C r ðE; hÞ þ Dr ðE; hÞ ¼ 1: ð11Þ
in superconductor, where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðFÞ jkk;r j ¼ ð2mF =h ÞðEF þ qr h0 þ EÞ sin h:
1917
Appropriate probabilities have the following form
Ar ðE; hÞ ¼ Re ð12Þ
The coefficients ar ; br ; cr and dr are, respectively, the probability amplitudes of Andreev reflection as a hole of opposite spin, normal reflection as an electron, transmission to the right electrode as an electron, and transmission to the right electrode as a hole of the opposite spin (the so-called crossed Andreev reflection). Amplitudes of the Bogoliubov electron-like and hole-like quasiparticles, propa-
ð13Þ
kr jar ðE; hÞj2 ; kr
ð14Þ
Br ðE; hÞ ¼ jbr ðE; hÞj2 ;
ð15Þ
2
C r ðE; hÞ ¼ jcr ðE; hÞj ; kr jdr ðE; hÞj2 ; Dr ðE; hÞ ¼ Re kr
ð16Þ ð17Þ
where kr is kr at the Fermi level, E = 0 (see, for example, Ref. [9]).
ðSÞ
Fig. 3. Scattering probabilities Ar through Dr ; r ¼", for an FISIF double junction with transparent interfaces, Z ¼ 0, thin superconducting film, d=n0 ¼ 1; D0 =EF ¼ 103 , strong ðFÞ ðSÞ ðFÞ ferromagnet h0 =EF ¼ 0:5, and equal Fermi wave vectors, kF =kF ¼ 1. Curves correspond to three different mass ratio: mF =mS ¼ 0:1 (dashed), mF =mS ¼ 1 (solid), and mF =mS ¼ 10 (dotted).
Fig. 4. Same as in Fig. 3 for
r ¼#.
Z. Popovic´ et al. / Physica C 469 (2009) 1915–1920
1918
It can be shown that Ar ¼ Dr ¼ 0 whenever
d qþr qr ¼ 2np
ð18Þ
for n ¼ 0; 1; 2; . . .. Therefore, both direct and crossed Andreev reflection vanish at the energies of geometrical resonances in quasiparticle spectrum of the S layer [9]. The absence of direct and crossed Andreev processes implies that all quasiparticles with energies satisfying Eq. (18) will pass unaffected from one electrode to another, without creation or annihilation of Cooper pairs. Both the presence of insulating barriers and exchange interaction reduce Ar and C r and enhance Br and Dr . Approaching the tunnel limit ðZ ! 1Þ, the spikes in Ar ; C r , and Dr , as well as the dips in Br , occur at the energies given by the quantization conditions þ
dqr ¼ n1 p; dqr ¼ n2 p;
ð19Þ
which correspond to the bound-state energies of an insulated superconducting film. In this case, these bound states are the only conducting channels, both for supercurrent and quasiparticle current [9].
The influence of different band masses and/or Fermi wave vectors on the scattering probabilities is illustrated for double junctions with fully transparent ðZ ¼ 0Þ and low-transparent ðZ ¼ 10Þ interfaces. This is shown in Figs. 2–7 for h ¼ 0. In all illustration 3 D0 =EðSÞ and the thickness of the superconducting layer is F ¼ 10 1 v F =pD0 is the superconducting d ¼ n0 ¼ 636kF , where n0 ¼ h coherence length and v F is the Fermi velocity. The ferromagnet is ðFÞ assumed to be strong, h0 =EF ¼ 0:5. In our example of a relatively thin superconducting film, subgap electronic tunneling is significant and Andreev reflection is reduced. The influence of interface transparency on scattering probabilities for equal band masses and Fermi wave vectors is shown in Fig. 2 for h0 ¼ 0 (NISIN heterostructure). For transparent interfaces ðZ ¼ 0Þ Ar and C r exhibit oscillatory behavior due to the coherent tunneling through the heterostructure. The probability of transmission C r is highest close to the resonant states above the gap. The Andreev reflection, measured by the probability Ar , is thus entirely suppressed near these energies. The probabilities of normal reflection and crossed Andreev reflection, Br and Dr , respectively,
Fig. 5. Same as in Fig. 3 for interfaces of low transparency, Z ¼ 10.
ðSÞ
Fig. 6. Scattering probabilities Ar through Dr ; r ¼", for an FISIF double junction with transparent interfaces, Z ¼ 0, thin superconducting film d=n0 ¼ 1; D0 =EF ¼ 103 , strong ðFÞ ðSÞ ðFÞ ferromagnet h0 =EF ¼ 0:5, and mF =mS ¼ 1. Curves correspond to three different Fermi wave vector ratio: kF =kF ¼ 0:5 (dotted), j ¼ 1 (solid), and j ¼ 2 (dashed).
Z. Popovic´ et al. / Physica C 469 (2009) 1915–1920
1919
Fig. 7. Same as in Fig. 6 for interfaces of low transparency, Z ¼ 10.
are different than zero only when the interfaces are non-transparent or when h0 – 0 (FISIF heterostructure). On the other hand, in the case of interfaces of low transparency ðZ ¼ 10Þ the probability of tunneling is highest close to the quasi-bound states of an isolated superconducting film. As the transparency of the interfaces increases (equivalently, as Z decreases), the probabilities around quasi-bound states shift and widen. The influence of different band masses on probabilities Ar through Dr in FISIF heterostructure is shown for r ¼"; # (Figs. 3 and 4) for transparent interfaces. For mismatched effective masses, mF =mS ¼ 0:1 and mF =mS ¼ 10, the probabilities exhibit a pronounced oscillatory behavior close to the resonant states above the gap. In the case of interfaces of low transparency ðZ ¼ 10Þ differences in band masses have a qualitatively negligible influence on the probabilities. Also, we find that the probabilities are practically independent on the spin orientation. Hence, in Fig. 5 we show only the case r ¼". For equal band masses, the influence of different Fermi wave vectors on scattering probabilities is shown for r ¼" and for interfaces that have either full or low transparency, Figs. 6 and 7, respectively. It can be seen that influence of different band masses and Fermi wave vectors leads to a qualitatively similar behavior of scattering probabilities as the influence of lower transparency. In both cases resonant tunneling through quasi-bound states above the superconducting gap induces oscillations in energy dependence of scattering probabilities.
through the probabilities of local Andreev reflection and direct transmission of electrons injected from the left electrode. Without solving the suitable transport equation, we take df ðk; VÞ ¼ f0 ðE eV=2Þ f0 ðE þ eV=2Þ [4]. In this approach, the charge current per orbital transverse channel is given by
IðVÞ ¼
1 e
Z
1
dE½f0 ðE eV=2Þ f0 ðE þ eV=2ÞGðEÞ;
ð21Þ
1
where
GðEÞ ¼ G0
X r¼";#
Pr
Z
2
d kk;r 2
2pkF
ðAr þ C r Þ
ð22Þ
is differential charge conductance at zero temperature and is the conductance quantum. Eq. (22) is a simple generG0 ¼ e2 =ph
3. The differential conductance When voltage V is applied to the junction symmetrically, the charge current density can be written in the form [10,14,19]
JðVÞ ¼
Z
e 3
ð2pÞ h
1
1
dE
X
Pr
Z
2
d kk;r ð1 þ Ar Br þ C r
r¼";#
Dr Þdf ðk; VÞ Z Z 1 X e 2 dE P ¼ d kk;r ðAr þ C r Þdf ðk; VÞ; r 4p3 h 1 r¼";#
ð20Þ
ðFÞ
where Pr ¼ ð1 þ qr h0 =EF Þ=2, and df ðk; VÞ is the asymmetric part of the non-equilibrium distribution function of current carriers. In the last equality the normalization condition given by Eq. (13) was taken into account. Note that both electron and hole contribution to the current are included, although the final result is expressed
Fig. 8. Differential conductance spectra of an FISIF double junction with interfaces with (a) full, Z ¼ 0, and (b) low, Z ¼ 10, transparency. Other parameters have values ðFÞ ðSÞ ðSÞ ðFÞ d=n0 ¼ 1; D0 =EF ¼ 103 ; h0 =EF ¼ 0:5, and kF =kF ¼ 1. Curves correspond to three different mass ratio: mF =mS ¼ 0:1 (dashed), mF =mS ¼ 1 (solid), and mF =mS ¼ 10 (dotted).
Z. Popovic´ et al. / Physica C 469 (2009) 1915–1920
1920
dence of the conductance. This implies that in a double tunnel junction (Figs. 8b and 9b) conductance spectra are practically independent of band masses and Fermi wave vectors ratio. 4. Conclusions We have analyzed the influence of different band masses and/or Fermi wave vectors on the ballistic transport properties of FISIF heterostructures and found that the effect should be observable only for the interfaces of low transparencies. In a double tunnel junction conductance spectra are practically independent of the difference in band masses or Fermi wave vectors ratio. However, if a junction is transparent, these differences can significantly affect the properties of the charge transport. This is manifested in a way similar to the presence of non-transparent interfaces in a junction with equal band masses and Fermi wave vectors. The similarity occurs because in all the cases charge transport is limited to the quasi-bound states in the superconducting film. Acknowledgement
Fig. 9. Differential conductance spectra of an FISIF double junction with interfaces with (a) full, Z ¼ 0, and (b) low, Z ¼ 10, transparency. Other parameters have values ðSÞ ðFÞ d=n0 ¼ 1; D0 =EF ¼ 103 ; h0 =EF ¼ 0:5, and mF =mS ¼ 1. Curves correspond to three ðFÞ ðSÞ ðFÞ ðSÞ different Fermi wave vector ratio: kF =kF ¼ 0:5 (dotted), kF =kF ¼ 1 (solid), and ðFÞ ðSÞ kF =kF ¼ 10 (dashed).
alization of the Landauer formula [20], with terms that take into account transmission of the current through Cooper pairs and quasiparticles. For E < D, the subgap transmission of quasiparticles (without conversion into Cooper pairs) suppresses the Andreev reflection, while for E > D all the probabilities oscillate with E and ds due to the interference of incoming and outgoing particles. In the following we analyze the conductance spectra for onedimensional double junction with a normal incidence of charge carriers, h ¼ 0, which corresponds to a single-channel flow in FISIF nano-structure. The conductance spectrum is given by
GðEÞ ¼ G0
X
Pr ðAr þ C r Þ:
ð23Þ
r¼";#
Influence of different band masses and Fermi wave vectors on zerotemperature conductance spectrum for transparent and low-transparent interfaces is illustrated in Figs. 8 and 9 for the thickness of ðFÞ the superconducting layer d ¼ n0 and for h0 =EF ¼ 0:5. It can be seen that influence of different band masses and Fermi wave vectors on differential conductance spectra is qualitatively the same as the influence of lower transparency. In both cases resonant tunneling through quasi-bound states induces oscillations in energy depen-
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