Local Density of States in a Dirty Normal Metal Connected to a Superconductor W. Belzig, C. Bruder, and Gerd Schíon ; Institut fíur Theoretische Festkíorperphysik, Universitíat Karlsruhe, D-768 Karlsruhe, Germany Department of Technical Physics, Helsinki University of Technology, FIN-050 Espoo, Finland A superconductor in contact with a normal metal not only induces superconducting correlations, known as proximity eæect, but also modiæes the density of states at some distance from the interface. These modiæcations can be resolved experimentally in microstructured systems. We, therefore, study the local density of states NèE;xè of a superconductor - normal metal heterostructure. We ænd a suppression of NèE;xè at small energies, which persists to large distances. If the normal metal forms a thin layer of thickness L n, a minigap in the density of states appears which is of the order of the Thouless energy ç çhd=l n. A magnetic æeld suppresses the features. We ænd good agreement with recent experiments of Guçeron et al. I. INTRODUCTION A normal metal in contact with a superconductor acquires partially superconducting properties. Superconducting correlations, described by a ænite value of the pair amplitude hè è èxèè " èxèi, penetrate some distance into the normal metal. This proximity eæect has been studied since the advent of BCS theory èsee Ref. and references thereinè. Recently, progress in lowtemperature and microfabrication technology has rekindled the interest in these properties í6. Interference eæects in a dirty normal metal increase the Andreev conductance 7;8. The eæect of the superconductor on the level statistics of a small normal grain has been investigated 9. Whereas the order parameter penetrates into the normal metal, the pair potential æèxè vanishes in the ideal metal without attractive interaction. Since æ yields the gap in the single-particle spectrum of a bulk superconductor, the question arises how the spectrum of the normal metal is modiæed by the proximity to the superconductor. Recently, this question has been investigated experimentally by Guçeron et al. 6. In their experiment, the local density of states of a dirty normal metal in contact with a superconductor was measured at diæerent positions and as a function of an applied magnetic æeld. In this paper, we evaluate the local density of states N èe; xè of a superconductor - normal metal heterostructure with impurity scattering in a variety of situations. We generalize earlier theoretical work 0í by applying the quasiclassical Green's function formalism and by including the eæect of a magnetic æeld. We compare with the experiment ofguçeron et al. 6 and ænd good qualitative agreement with the experimental data both in the cases with and without a magnetic æeld. ç =èd=æè = is the superconducting coherence length at T = 0 and l el is the elastic mean free path. The latter is related to the diæusion constant via D = v F l el. The density of states èdosè of this inhomogeneous system can be derived systematically within the quasiclassical real-time Green's functions formalism. In the dirty limit the equation of motion for the retarded Green's functions G E, F E reads 5 D ç çg E è r,ie ~ Aè ~ F E, F E ~r G E = èè è,ie +, in è F E, æ G E +, sf G E F E : The diagonal and oæ-diagonal parts of the matrix Green's function, G E and F E, obey the normalization condition G E + F E =; èè which suggests to parameterize them by a function çèe; xè via F E = sinèçè and G E = cosèçè. Inelastic scattering processes are accounted for by the rate, in = =ç in, while scattering processes from paramagnetic impurities are described by the spin-æip rate, sf ==ç sf. At low temperatures the former is very small è, in ç 0, æè, and will be neglected in the following. (a) (b) S -L 0 L s n A(y) z S x N W y N H x II. THE MODEL In the following we will consider geometries as shown in Fig.. The superconductor is characterized by a ænite pairing interaction ç and transition temperature T c é 0. In the normal metal we take ç = T c = 0. Here we restrict ourselves to the dirty èdiæusiveè limit, ç ç l el, where 0 FIG.. Geometries considered in this article. èaè A strictly one-dimensional geometry. èbè A more realistic geometry similar to experimental setup. For the geometry shown in Fig. the order parameter can be taken real. On the other hand, in the vicinity of an N-S boundary the absolute value of the order parameter is space dependent, and has to be determined
self-consistently. The self-consistency condition is conveniently expressed in the imaginary-time formulation, where T X æèxè lnè T c èxè è=çt F i!ç èxè, æèxè : èè! ç!çé0 Here,! ç = çtèç + è are Matsubara frequencies. The summation is cut oæ at energies of the order of the Debye energy. The coupling constant in S has been eliminated in favor of T c, while the coupling constant in N is taken to be zero. In the case where the interface between N and S has no additional potential, the boundary conditions are 6 F E è0, è=f E è0 + è ç s d G E è0, è dx F ç n d Eè0, è= G E è0 + è dx F Eè0 + è : èè Here, ç nèsè are the conductivities of the normal metal and the superconductor, respectively. The complete selfconsistent problem requires a numerical solution. Starting from a step-like model for the order parameter, selfconsistency was typically reached within 0 steps. Finally the DOS is obtained from NèEè = N 0 ReG E èxè, where N 0 is the Fermi level DOS in the normal state. We will present now results for three diæerent cases: A. The DOS near the boundary of a semi-inænite normal metal and superconductor. B. The DOS in a thin normal ælm in contact with a bulk superconductor. C. The eæect of a magnetic æeld on the DOS in an experimentally realized N-S heterostructure. In the following sections energies and scattering rates will be measured in units of the bulk energy gap æ and distances in units of the coherence length ç =èd=æè =. III. RESULTS AND DISCUSSION A. DOS in an Inænite System We assume that the normal metal and the superconductor are much thicker than the coherence length L s ;L n ç ç and investigate how the DOS changes continuously from the BCS form N BCS èeè=n 0 = jej =èe, æ è = deep inside the superconductor to the constant value N N èeè=n 0 = in the normal metal. In a ærst approximation, neglecting self-consistency and paramagnetic impurities, we can solve Eq. analytically, with the result çèe; xè = 8 é é: atanë tanèç 0 =è expè, p!=d n xèë ç s + atanë tanèèç 0, ç s è=èæ expè q xé0 p! +æ =D s xèë xé0: è5è Here sin ç 0, ç s! =,ie +, in ; æ ç s = atanè è;,ie +, in è,ie +, in è = = æ ç èè,ie +, in è +æ è =sin 0 : Several material parameters combine into the parameter æ =èç n ç s =ç s ç n è ; è6è measuring the mismatch in the conductivities and the coherence lengths of the two materials. Furthermore, ç sènè is deæned by èd sènè =æè =, where D sènè is the diæusion constant of the superconductor ènormal metalè. The resulting DOS, NèEè, in the normal metal at a distance x =:5ç n from the interface is shown in Fig. for diæerent values of the parameter æ. It shows a subgap structure with a peak below the superconducting gap energy Eéæ and a strong suppression at zero energy. The modiæcation of the DOS is most pronounced at small values of æ and at small distances. The smaller the energy, the larger is the distance where the modiæcations are still visible. In particular at E = 0 the DOS vanishes for all values of x. Pair-breaking eæects lead to a ænite zero-energy DOS, as will be shown later..5 γ=0 γ=0. γ= γ=5 FIG.. DOS in the normal metal at x =:5ç n. Next we solve the problem self-consistently and present some numerical results for the case æ =. We ærst concentrate on the superconducting side of the boundary. As shown in Fig. the peak in the DOS is strongly suppressed, changing from a singularity to a cusp, but it remains at the same position æ as one approaches the boundary. On the other hand, the density of states with energies below æ increases. The states with energies well below æ decay over a characteristic length scale q D s =è p æ, E è, see Eq. è5è.
F(x)/F bulk be most favorable. -5 0 x/ξ 5 bulk x=-.5ξ x=-0.75ξ x=0 0.5 FIG.. Density of states on the superconducting side of the N-S boundary. The inset shows the self-consistent pair amplitude. The DOS on the normal side at diæerent distances from the NS-boundary is shown in Fig.. The pronounced sub-gap structure found in the approximate solution is still present in the self-consistent treatment. The ægure shows how the peak height and position change with the distance. In the absence of pair-breaking eæects the DOS vanishes at the Fermi level for all distances èdotted curvesè. Inclusion of a pair-breaking mechanism èsolid curvesè regularizes the DOS at the Fermi level, and also the peak height is suppressed. The curves are in qualitative agreement with experimental data shown in Ref. 6. The self-consistent calculation presented here leads to a slightly better æt than the theoretical curves shown in Ref. 6 where a constant pair potential was used in the solution of the Usadel equation. In particular, the lowenergy behavior of the experimental curves is reproduced correctly. 0. 0. x=ξ x=.5ξ x=0.75ξ 0. 0. FIG.. Density of states on the normal side of an N-S boundary for two spin-æip scattering rates:, sf = 0 èdotted linesè and, sf =0:05æ èsolid linesè. At ænite temperatures èbut T ç T c èwe expect no qualitative changes in the behavior described above except that the structures in the DOS will be smeared out by inelastic scattering processes. Hence for an experimental veriæcation temperatures as low as possible would B. DOS in thin N-layers Next we consider a thin normal layer in contact with a bulk superconductor, L s ç L n ' ç. The boundary condition at x = L n is chosen to be dçèe; xè=dx =0, i.e., the normal metal is bounded by an insulator. In this case the DOS on the N side develops a minigap at the Fermi energy, which is smaller than the superconducting gap. If the thickness of the normal layer is increased, the size of this minigap decreases. Results obtained from the self-consistent treatment are shown in Fig. 5. Details of the shape of the DOS depend on the location in the N- layer 7.However, the magnitude of the minigap is spaceindependent, as shown in the inset of Fig. 5. The magnitude of the gap is expected to be related to the Thouless energy D=L n, which is the only relevant quantity which has the correct dimension. Of course the relation has to be modiæed in the limit L n! 0. Indeed as shown in Fig. 5 a relation of the form E g ç èconst ç + L n è, æts quite well. The sum of the lengths may beinterpreted as an eæective thickness of the N-layer since the quasiparticle states penetrate into the superconductor to distances of the order of ç. The eæect of spin-æip scattering in the normal metal on the minigap structure is also shown in the inset of Fig. 5. The minigap is suppressed as, sf is increased until a gapless situation is reached at, sf ç 0:æ. E g / 0. 0..0.5 x=0 =0 =0 =0. = Numerical Data (+L N /ξ N ) 0 6 L n /ξ FIG. 5. Minigap E g as a function of the normal-layer thickness. Inset: local DOS of an N-layer of thickness L n =:ç in proximity with an bulk superconductor. We would like tomention that a similar feature had been found before by McMillan 0 within a tunneling model ignoring the spatial dependence of the pair amplitude. We have considered here the opposite limit, assuming perfect transparency of the interface but accounting for the spatial dependence of the Green's functions. For, sf = 0 our results for the structure of the DOS agree further with previous ændings of Golubov and Kupriyanov and Golubov et al.. Recently, a minigap
in a two-dimensional electron gas in contact to a superconductor has also been studied 8. C. Density of states in a magnetic æeld An applied magnetic æeld suppresses the superconductivity in both superconductor and normal metal. To study the eæect of the magnetic æeld on our system we consider the geometry shown in Fig. b. Because in the experimental setup the thickness of the ælms is much smaller than the London penetration depth, we can neglect the magnetic æeld produced by screening currents. Therefore it is reasonable to assume a constant magnetic æeld, which is present in both S and N. The vector potential is then chosen to be ~A = Aèyè~e x ; Aèyè =Hy : è7è Eq. èè can be considerably simpliæed in the case that the size of the system in y-direction is smaller or of the order of ç. The system is limited to,w= éyéw=, where W ' ç. Therefore the Green's functions do not depend on y and the equation can be averaged over the width W. The equation reduces to the eæective onedimensional equation D, æ GE @ x F E, F E @ x G E = è8è è,ie +, in è F E, æ G E +, eæ G E F E : Here,, eæ =, sf + De H W = acts as an eæective pairbreaking rate, which depends on the transverse dimension and the applied magnetic æeld. If we approximate the Green's functions in the superconductor by their bulk values, the DOS in the normal metal at zero energy can be calculated analytically: where Nè0è N 0 = è tanhè p,eæ =D xè, eæ é æ è, æ è=è + æ è, eæ é æ ; è9è æ = æ expè,p p, eæ =D xè : è0è, eæ +,, æ eæ In Fig. 6 the dependence of the DOS on, eæ at x = :5ç is shown for two diæerent spin-æip scattering rates èequal rates for normal metal and superconductorè. At, eæ =0:5æ the æeld dependence of the DOS shows a kink. This kink arises because above this value of, eæ the zero-energy DOS in the superconductor is nonzero ègapless behaviorè, which leads to an even stronger suppression of the proximity eæect. N(0)/N 0 Γ eff / FIG. 6. Zero-energy DOS in the normal metal at x =:5ç as a function of, eæ. Figure 7 shows a quantitative comparison of these results with experimental data taken by the Saclay group 9. In this experiment 6, the diæerential conductance of three tunnel junctions attached to the normal metal part of the system was used to probe the DOS at diæerent distances from the superconductor. Accordingly, wehave calculated the self-consistent DOS in the presence of a magnetic æeld throughout the system for all energies and determined the diæerential conductance 0.We used x = :8ç, consistent with an estimate from a SEMphotograph, and used a spin-æip scattering rate of, sf = 0:05æ in the normal metal as a æt parameter. This is necessary in the framework of our approach to explain the ænite zero-bias conductance at zero æeld. We, furthermore, assumed ideal boundary conditions at the NS interface, i.e., æ =, the motivation being that great care was used in the experiment to produce a good metallic junction, and signiæcant Fermi velocity mismatches are not to be expected. At low and high voltages the agreement with the experimental data is good for all three æeld values. On the other hand, the maximum in the DOS is not reproduced well by our calculation. Including the eæect of a nonideal boundary, i.e., æ é leads to an increase of the peak in the DOS but to a less satisfactory æt at low voltages. We cannot resolve this discrepancy, but we would like to point out that our theory is comparatively simple and does not include all the geometric details of the experiment èe.g., the geometry of the overlap junction is not really one-dimensional and would be diæcult to treat realisticallyè. Our intention is to show that theoretical treatment described here contains the physical ingredients to explain the basic features of the experimental data. The overall agreement between theory and experiment demonstrated in Fig. 7 shows this to be the case.
R i di/dv. 0.9 0.7 h=0. (H exp =800G) h=0.5 (H exp =00G) h=0 (H exp =0) 0. 0. ev/ FIG. 7. Quantitative comparison of experiment 9 èdotted linesè and theory èsolid linesè. The experimental magnetic æelds are H = 0, 00, and 800G; h = HeWèD=æè =. The theoretical curves have been normalized by R i ç di=dv expèev=æ =:5è. IV. CONCLUSIONS AND OUTLOOK In conclusion, we have given a theoretical answer to the question asked in the introduction, viz., what is í beyond the proximity eæect í the eæect of a superconductor on the spectrum of a normal metal coupled to it. Using the èreal-timeè Usadel equations, we have calculated the local density of states in the vicinity of an N-S boundary in both ænite and inænite geometries. It shows an interesting sub-gap structure: if the normal metal is inænite, the density of states is suppressed close to the Fermi energy, but there is no gap in the spectrum. This is the behavior found in a recent experiment 6. In thin normal metals we ænd a mini-gap in the density of states which isofthe order of the Thouless energy. We have also investigated the suppression of these eæects by an applied magnetic æeld and ænd good agreement with experiment. We are grateful to D. Esteve and H. Pothier for raising the questions leading to this work and for many inspiring discussions. We would also like to acknowledge helpful discussions with N. O. Birge, M. Devoret, S. Guçeron, and A. D. Zaikin. The support of the Deutsche Forschungsgemeinschaft, through SFB 95, as well as the A. v. Humboldt award of the Academy of Finland ègsè is gratefully acknowledged. several articles in Mesoscopic Superconductivity, edited by F. W. J. Hekking, G. Schíon, and D. V. Averin, Physica B 0 p. 0-5 è99è. A. C. Mota, P. Visani, A. Pollini, and K. Aupke, Physica B 97, 95 è99è. V. T. Petrashov, V. N. Antonov, S. V. Maksimov, and R. Sh. Shaikhaidarov, Pis'ma Zh. Eksp. Teor. Fiz. 58, 8 è99è ëjetp Lett. 58, 9 è99èë. 5 H. Courtois, P. Gandit, and B. Pannetier, Phys. Rev. B 5, 6 è995è. 6 S. Guçeron, H. Pothier, N. O. Birge, D. Esteve, and M. Devoret, preprint. 7 F. W. J. Hekking and Yu. V. Nazarov, Phys. Rev. Lett. 7, 65 è99è. 8 H. Pothier, S. Guçeron, D. Esteve, and M. H. Devoret, Phys. Rev. Lett. 7, 88 è99è. 9 K. M. Frahm, P. W. Brouwer, J. A. Melsen, and C. W. J. Beenakker, Phys. Rev. Lett. 76, 98 è996è. 0 W. L. McMillan, Phys. Rev. 75, 57 è968è. A. A. Golubov and M. Yu. Kupriyanov, J. Low Temp. Phys. 70, 8 è988è; Zh. Eksp. Teor. Fiz. 96, 0 è989è ësov. Phys. JETP 69, 805 è989èë; Pis'ma Zh. Eksp. Teor. Fiz. 6, 80 è995è ëjetp Lett. 6, 85 è995èë. A. A. Golubov, E. P. Houwman, J. G. Gijsbertsen, V. M. Krasnov, J. Flokstra, H. Rogalla, and M. Yu. Kupriyanov, Phys. Rev. B 5, 07 è995è. A. I. D'yachenko and I. V. Kochergin, J. Low Temp. Phys. 8, 97 è99è. G. M. Eliashberg, Zh. Eksp. Teor. Fiz. 6, 5 è97è ësov. Phys.-JETP, 668 è97èë. 5 K. D. Usadel, Phys. Rev. Lett. 5, 507 è970è. 6 M. Yu. Kupriyanov and V. F. Lukichev, Zh. Eksp. Teor. Fiz. 9, 9 è988è ësov. Phys. JETP 67, 6 è988èë. 7 W. Belzig, Diploma thesis, Universitíat Karlsruhe, August 995, unpublished. 8 A. F. Volkov, P. H. C. Magnçee, B. J. van Wees, and T. M. Klapwijk, Physica C, 6 è995è. 9 Unpublished experimental data taken by the authors of Ref. 6. 0 The diæerential conductance also shows single-electron effects caused by the small capacity of the tunnel junctions. It is necessary to include them for a quantitative comparison, and we have done so using the prescription given in Ref. 6. G. Deutscher and P. G. de Gennes, in Superconductivity, edited by R. D. Parks èmarcel Dekker, New York, 969è. 5