Shape of Cooper pairs in a normal-metal/superconductor junction
Yukio Tanaka,1,2Yasuhiro Asano,3and Alexander A. Golubov41Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan 2CREST, Japan Science and Technology Cooperation (JST), Nagoya 464-8603 Japan
3Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan
4Faculty of Science and Technology and Mesa⫹Institute for Nanotechnology, University of Twente, Enschede 7500AE, The Netherlands 共Received 10 April 2008; published 12 June 2008兲
In s-wave superconductors the Cooper pair wave function is isotropic in momentum space. This property may also be expected for Cooper pairs entering a normal metal from a superconductor due to the proximity effect. We show, however, that such a deduction is incorrect and the pairing function in a normal metal is surprisingly anisotropic because of quasiparticle interference. We calculate angle-resolved quasiparticle density of states in NS bilayers which reflects such anisotropic shape of the pairing function. We also propose a magnetotunneling spectroscopy experiment which could confirm our predictions.
DOI:10.1103/PhysRevB.77.220504 PACS number共s兲: 74.45.⫹c, 74.50.⫹r, 74.20.Rp
It is well known that Cooper pairs consisting of two elec-trons are characterized by electric charge 2e, macroscopic phase, internal spin, and by time and orbital structures.1The charge 2e manifests itself in various experiments, like Sha-piro steps, flux quantization and excess current due to the Andreev reflection. The macroscopic phase generates the Jo-sephson current.1The internal spin structure is classified into spin-triplet and spin-singlet states. Further, based on a sym-metry with respect to the internal time, superconducting state can belong to the even-frequency or the odd-frequency sym-metry class.2 The orbital degree of freedom is described by an angular momentum quantum number l.3,4The well estab-lished properties listed above hold in bulk superconductors. The presence of perturbations like spin-flip or interface scat-tering may change the symmetry of Cooper pairs. For in-stance, an unusual odd-frequency property of Cooper pairs in proximity structures was predicted in recent studies.5,6 The shape of Cooper pair wave function in nonuniform systems like superconducting junctions is not necessarily the same as that in the bulk state. Despite the extensive study of the proximity effect in several past decades, rather little attention has been paid to the problem of Cooper pair shape in non-uniform superconducting systems.7,8This issue is quite im-portant in view of current interest in the physics of supercon-ducting nanostructures.
The aim of the present Rapid Communication is to clarify the consequences of breakdown of translational symmetry in superconductors on the Cooper pair shape. For this purpose, we study the proximity effect in quasi-two-dimensional nor-mal metal / superconductor 共N/S兲 junctions by solving the Eilenberger equation. We analyze the pairing function and the local density of states 共LDOS兲 in N/S junctions with spin-singlet s-wave and dxy-wave superconductors. The shape of the Cooper pair deviates seriously from that of the bulk with the generation of the odd-frequency component of the pairing function due to the formation of the Andreev– Saint-James bound states.9 To detect the complex Cooper pair shape, we propose to use scanning tunneling spectros-copy in rotating magnetic field. We show that the calculated tunneling conductance exhibits complex patterns even in the s-wave case.
Let us consider a quasi-two-dimensional N/S junction as
shown in Fig. 1which is the simplest example of a nonuni-form superconducting system, where the S region is semi-infinite and the normal metal has finite length L. We consider a perfect N/S interface with perfect transmissivity, while it can be shown that characteristic behavior of Cooper pairs remains qualitatively unchanged even in the presence of a potential barrier at the N/S interface.
The quasiclassical Green’s functions10 in a normal metal 共N兲 and a superconductor 共S兲 are parameterized as
gˆ⫾共i兲= f1共i兲⫾ˆ1+ f2共i兲⫾ˆ2+ g⫾共i兲ˆ3, 共gˆ⫾共i兲兲2= 1ˆ, 共1兲
where a superscript i共=N,S兲 refers to N and S, ˆj 共j=1–3兲 are the Pauli matrices, and 1ˆ is a unit matrix. The subscript + 共−兲 denotes a moving direction of a quasiparticle in the x direction,10 and⌬¯
+共x兲 关⌬¯−共x兲兴 is the pair potential for a left
共right兲 going quasiparticle. In a normal metal, ⌬¯⫾共x兲 is set to zero because the pairing interaction is absent there. The Green’s functions can be expressed in terms of the Ricatti parameters,11
f1⫾共i兲 = ⫿i关⌫⫾共i兲共x兲 +⫾共i兲共x兲兴/关1 + ⌫⫾共i兲共x兲⫾共i兲共x兲兴, 共2兲
f2共i兲⫾= i关⌫⫾共i兲共x兲 −共i兲⫾共x兲兴/关1 + ⌫⫾共i兲共x兲⫾共i兲共x兲兴, 共3兲
H φ Superconductor Normal metal x y z x=0 x =− L STM tip
FIG. 1. A schematic of a N/S junction. A normal metal 共−L ⬍x⬍0兲 is attached to a superconductor 共0⬍x⬍⬁兲. To detect the deformation of a Cooper pair, we propose a STS experiment in the presence of magnetic field共H兲 whose direction is perpendicular to the z axis and oriented by from the x axis.
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g⫾共i兲=关1 − ⌫⫾共i兲共x兲共i兲⫾共x兲兴/关1 + ⌫⫾共i兲共x兲⫾共i兲共x兲兴, 共4兲 with i= 1 for i = S and i= −1 for i = N. The parameters ⌫⫾共i兲共x兲 and
⫾
共i兲共x兲 obey the Eilenberger equation of the Ricatti
type,11
ivFxx⌫⫾共i兲共x兲 = − ⌬¯⫾共x兲兵1 + 关⌫⫾共i兲共x兲兴2其 + 2¯i⌫⫾共i兲共x兲,
ivFxx⫾共i兲共x兲 = − ⌬¯⫾共x兲兵1 + 关⫾共i兲共x兲兴2其 − 2¯i⫾共i兲共x兲, with¯=+i␦0, where is the energy of a quasiparticle
mea-sured from the Fermi level and␦0is the level broadening due to impurity scattering. vFx is the x-component of Fermi ve-locity. In the clean limit, we consider␦0Ⰶ⌬0. Boundary
con-dition at x = −L is given by ⫾共N兲共−L兲=−⌫⫿共N兲共−L兲. Boundary condition at the N/S interface becomes⫾共S兲共0兲=−⌫⫾共N兲共0兲 and ⫾共N兲共0兲=−⌫⫾共S兲共0兲. The pair potential ⌬¯⫾共x兲 is expressed by
⌬¯⫾共x兲=⌬共x兲⌽⫾共兲⌰共x兲, where a form factor ⌽⫾共兲 is given
by⌽⫾共兲=1 for s-wave symmetry and ⫾sin 2for dxy-wave one with being an incident angle of a quasiparticle mea-sured from the x direction. Bulk pair potential is⌬共⬁兲=⌬0, and we determine the spatial dependence ⌬共x兲 in a self-consistent way.
For xⰇL0, the angular structure of f2共S兲⫾follows that of the pair potential, whereas f1⫾共S兲 is zero with L0=vF/TC being a coherence length in N and TC being the transition tempera-ture. The pairing function f1⫾共i兲 is generated by inhomogeneity in a system and thus has a finite value only near the interface and in a normal metal. Recent studies6have shown that f
1⫾ 共i兲
has an odd-frequency symmetry since functions f1⫾共i兲 and f2⫾共i兲 have opposite parities. The induced odd-frequency compo-nent has the odd共even兲 parity, respectively. Pairing function f1 is defined in the angular domain of −/2ⱕ⬍3/2. We denote f1共兲 by f1+共兲 in the angle range −/2ⱕ⬍/2
and f1共兲= f1−共−兲 for/2ⱕ⬍3/2. The angular
struc-ture of functions f2 and g is defined in the same manner.
LDOS is given by the relation L共兲=Real关g共兲兴. In what follows, we fix temperature T = 0.05TC, the length of the nor-mal region L = 5L0, and␦0= 0.01⌬0.
In Figs. 2 and 3 we show polar plots of f1, f2, and L in s-wave and dxy-wave junctions for various choices of and x. Dashed, solid and dotted lines represent, respectively, the results for x =⬁ 共superconductor兲, x=0 共interface兲, and x = −L/2 共normal metal兲. The odd-frequency component is always absent for x =⬁. Since L is independent of x in N, the resulting value ofLat x = 0 is equal to that at x = −L/2. First, we focus on the s-wave case 共Fig.2兲. As shown in
Fig.2共a兲, at=0 and x=⬁,0 the even-frequency component f2has a circular shape reflecting the s-wave symmetry.
How-ever, at x = −L/2, the shape is no longer a simple circular one but has a form of a doubly distorted circle. The shape of f2in
a superconductor always has the circular shape indepen-dently of as shown with dashed lines in Figs. 2共a兲, 2共d兲, and2共g兲. At =0.1⌬0 in Fig.2共d兲, f2 at the interface共solid
line兲 slightly deviates from the circular shape, while the shape in N drastically changes. The tendency is more re-markable at=0.5⌬0as shown in Fig.2共g兲. The butterflylike pattern of f2 at the interface关solid line in Fig.2共g兲兴 is
com-pletely different from the original circular shape in a super-conductor. At =0, function f1 at the interface becomes el-lipsoidal as shown by the solid line in Fig.2共b兲. The shape of f1at x = 0 and x = −L/2 exhibits the butterflylike pattern 关Fig.
2共e兲兴. For =0.5⌬0, the line shape of f1has many spikes as
shown in Fig. 2共h兲. Such anisotropic property of f1 and f2 affects the LDOS as shown in Figs. 2共c兲,2共f兲, and 2共i兲. In particular, LDOS at the interface for =0.5⌬0 关solid line in
共i兲兴 strongly deviates from the circular shape. At the inter-face, the shape ofLin Fig.2共c兲is quite similar to that of f1
shown in Fig. 2共b兲.
These profiles can be qualitatively understood as follows. At x = 0, the relations f1共N兲⫾=⫾⌫共1−␣2兲/⌶, and g
⫾ 共N兲=共1
+␣2⌫2兲/⌶ are satisfied with ⌶=1−⌫2␣2, ⌫=⌫⫾共S兲共0兲 and ␣ = exp关2iL/共vFcos兲兴. For Ⰶ⌬0, the relations⌫⬃1/i and
f1⫾⬃ig⫾are satisfied. Thus shape of function f1is similar to
that ofL. This argument is valid even for=0.1⌬0 in Figs.
2共e兲 and2共f兲, and for =0.5⌬0 in Figs. 2共h兲 and 2共i兲. The oscillating behavior in f1, f2, andL is more remarkable at =0.5⌬0. Although we do not present calculated results of f1
and f2at x = −L/2 for =0.5⌬0, the butterflylike pattern with
many spikes in the pairing functions can be seen also in a normal metal. The directions of the spin projections in LDOS are characterized by small value of⌶, which has close rela-tion to the formarela-tion of the Andreev–Saint-James bound states.9,12 For ⬃ ⫾/2, ␣ oscillates rapidly with small
ε=0.0∆0 Even−frequency 1 0 | |fsin2 θ | |f2cosθ −1 1 0 −1 (a) 1 | | sin θ ρ L | |ρLcosθ Odd−frequency 1 | |fsin1 θ | |f1cosθ 0 −1 1 0 −1 (b) LDOS 0 −1 1 0 −1 (c) Even−frequency ε=0.1∆0 2 0 −2 2 0 −2
Even−frequency Odd−frequencyε=0.5∆0 LDOS
2 0 −2 2 0 −2 (h) 2 0 −2 2 0 −2 (i) | |fsin2 θ 2 0 −2 (d) 2 0 −2 | |fsin2 θ | |f2cosθ | |f2cosθ | |fsin1 θ | |f1cosθ Odd−frequency 2 0 −2 2 0 −2 (e) | |fsin1 θ | |f1cosθ | | sin θ ρL | | sin θ ρL LDOS 2 0 −2 2 −2 (f) | |ρLcosθ | |ρLcosθ 0 (g)
FIG. 2. 共Color online兲 The results for s-wave symmetry. The shape of the even-frequency pair amplitude 关共a兲,共d兲,共g兲兴, the odd-frequency one关共b兲,共e兲,共h兲兴, and the angle-resolved local density of states关共c兲, 共f兲, 共i兲兴. Solid lines: x=0 共at N/S interface兲, dotted lines: x = −L/2 共in a normal metal兲 and dashed lines: x=⬁ 共in a supercon-ductor兲. =0 for 共a兲, 共b兲 and 共c兲, =0.1⌬0for共d兲, 共e兲 and 共f兲, and =0.5⌬0for共g兲, 共h兲 and 共i兲. The angle is measured from the x axis.
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variation of , which explains the fine structures in LDOS around =⫾/2. The quasiparticle interference effect is a source of Andreev–Saint-James bound state formation in a normal metal. As a result, the circular shape of Cooper pairs in s-wave superconductor is modified into the butterflylike pattern in a normal metal.
Next, we discuss the results for dxy-wave junctions shown in Fig.3. In a superconductor共x=⬁兲, functions f2and g are given by ⌬0sin 2/
冑
2−⌬02sin22 and /冑
2−⌬02sin22,respectively. As shown by dashed lines in Figs. 3共d兲, 3共f兲,
3共g兲, and3共i兲, the amplitudes of f2and g become large along the directions =关sin−1共/⌬
0兲兴/2. At =0 and x=0,
forma-tion of a mid-gap Andreev resonant state13 significantly en-hances the amplitudes of f1andLcompared to that of f2as
shown by solid lines in Figs. 3共a兲–3共c兲. For =0 and = 0.1⌬0, the shapes of f1 and L at the N/S interface are similar to those in s-wave superconductor junctions 关solid lines in Figs.3共b兲,3共c兲,3共e兲, and3共f兲兴. At =0.5⌬0, similar
to the s-wave case, functions f1 and f2 in N have complex
line shapes with many spikes.
Here, we propose an experimental setup to measure the complex Cooper pair shape, based on magnetotunneling spectroscopy, i.e., scanning tunneling spectroscopy共STS兲 in the presence of magnetic field. As shown in Fig.1, magnetic field is applied parallel to the N/S plane. Tunneling current at a fixed bias voltage is measured as a function of the angle between the x axis and the direction of magnetic field. The vector potential in this configuration is given by 共Ax, Ay兲= −H exp共−z/兲共sin, cos兲.14We assume that thickness of a quasi-two-dimensional superconductor is sufficiently small compared to a magnetic field penetration depth . Magnetic field shifts the quasiparticle energy to −H⌬0sin共−兲/
B0, where B0= h/共2e2兲 and=បvF/⌬0. Here, to
evalu-ate the order of magnitude of B0, we explicitly write the
Plank constant. For typical values of ⬃⬃100 nm, the magnitude of B0 is of the order of 0.02 Tesla. Local density of states measured in the considered magnetotunneling STS configuration is given by15
S共兲=兰−3/2/2L共,兲d. At suffi-ciently low temperatures the applied bias voltage V satisfies the relation eV =.S共兲 is a periodic function of and we denote its maximum value asM. In the following, we focus on the normalized value共兲 defined by共兲=S共兲/M.
In Fig. 4, =共兲 is plotted as a function of . In the s-wave case,共兲 in bulk is always unity due to the isotropic nature of the s-wave pairing as shown by curve B. On the other hand, at the N/S interface has an oscillatory depen-dence due to the deviation of the Cooper pair shape from the circular one. It is remarkable that the line shape of curve A changes drastically with the increase in as seen from Figs.
4共a兲–4共c兲. This sensitivity to variation reflects the complex shape ofLshown in Fig.2.
For dxy-wave case, the line shape of in the bulk has periodic oscillations with the period 0.5. The amplitude of the oscillations is reduced with the increase in as shown by the curves B in Figs. 4共d兲–4共f兲. On the other hand, the line shapes of the curves A change drastically with the increase in . This sensitivity originates from the complex patterns ofL shown in Fig. 3. Although we do not discuss in detail, the line shape of at the N/S interface is also sensitive to a magnetic field H for a fixed.
As seen from the above results, by changing the magni-tude of the applied magnetic field H, the bias voltage V, and the rotation angle , it is possible to clarify the remarkable deformation of Cooper pairs. Therefore, magnetotunneling
Even−frequency ε=0.0∆0 2 0 −2 2 0 −2 Even−frequency ε=0.1∆0
Even−frequency Odd−frequencyε=0.5∆0 LDOS
2 0 −2 2 0 −2 (h) 2 0 −2 2 0 −2 (i) 2 0 −2 (a) (d) 2 0 −2 Odd−frequency | |fsin2 θ 2 0 −2 2 0 −2 (g) | |fsin2 θ | |fsin2 θ | |f2cosθ | |f2cosθ | |f2cosθ | |fsin1 θ 4 0 −4 Odd−frequency 4 0 −4 (b) | |fsin1 θ | |f1cosθ 2 −2 2 −2 0 0 (e) | |fsin1 θ | |f1cosθ | |f1cosθ LDOS 2 0 −2 4 0 −4 (c) | | sin θ ρL | | sin θ ρ L | |ρLcosθ | | sin θ ρL LDOS 2 0 −2 2 0 −2 (f) | |ρLcosθ | |ρLcosθ
FIG. 3. 共Color online兲 The results for dxy-wave symmetry. The
notations are the same as in Fig.2.
0.8 1.0
ρ
s-wave (a) (b) 1.2 ε=0.0∆0 A B A B 0.6 s-waveε=0.1∆ 0 0.8 1.0ρ
0.6 0.8 1.0ρ
0.6 0 1 2 φ / π (c) s-waveε=0.5∆ 0 0.8 1.0ρ
0.6 (d) ε=0.0∆0 d -wavexy 0.8 1.0ρ
0.6 (e) ε=0.1∆0 d -wavexy 0.8 1.0ρ
0.6 0 1 2 φ / π (f) d -wavexy ε=0.5∆0 1.2 A A A A B B B BFIG. 4. 共Color online兲 Normalized local density of states ob-tained in magnetotunneling spectroscopy as a function of the orien-tation angle of magnetic field in a spin-singlet s-wave junction 关共a兲, 共b兲, and 共c兲兴 and in a spin-singlet dxy-wave one 关共d兲, 共e兲, and
共f兲兴. =0 for 共a兲 and 共d兲, =0.1⌬0for共b兲 and 共e兲, and =0.5⌬0for 共c兲 and 共f兲. A: x=0 共at N/S interface兲 for H=0.1B0, and B: x =⬁ 共in a superconductor兲 for H=0.1B0.
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spectroscopy provides the way to detect bulk symmetry of the pair potential.
We have not explicitly discussed the influence of impurity scattering in N, but one can show that the oscillatory behav-ior of can be detected if the following two conditions are fulfilled: l⬎L and l⬎L0, where l is the mean free path in N.
To satisfy these conditions, normal metals with high mobility are desirable. Superconducting junctions with 2D-electron gas realized in InAs 共Ref. 16兲 or graphene17 are possible candidates due to high electronic mobility in both types of materials. Furthermore, large magnitude of TCor⌬0helps to
satisfy the second condition. From this viewpoint, junctions with high TC cuprates extensively studied by now18 are promising candidates. Surface roughness could also lead to a broadening of the oscillatory behavior of . This effect is controlled by an effective scattering length within the surface
layer共see Refs. 8 and19兲. Both bulk and surface scattering
lead to mixing of quasiparticle trajectories at different angles, while the general angular shape of the Cooper pair does not change and can be determined at realistic experi-mental conditions.
In summary, we have studied the Cooper pair shape in normal-metal/superconductor 共N/S兲 junctions by using the quasiclassical Green’s function formalism. The quasiparticle interference leads to striking deformations in the shape of a Cooper pair wave function in a normal metal. We also show that the anisotropic shape of Cooper pairs could be resolved by scanning tunneling spectroscopy experiments in magnetic field. The Cooper pair deformation is a common feature of nonuniform superconducting systems in the clean limit. This provides a key concept to explore unknown quantum inter-ference phenomena in superconducting nanostructures.
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