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PHYSICAL REVIEW B

VOLUME 57, NUMBER 20

15 MAY 1998-II

Geometry-specific conductance fluctuations in semiconductor-superconductor microjunctions Y. Takagaki and K. H. Ploog Paul-Drude-Institut fu¨r Festko¨rperelektronik, Hausvogteiplatz 5-7, D-10117 Berlin, Germany ~Received 19 February 1998! The conductance of quantum dots contacted to a superconductor is found to exhibit geometry-specific fluctuations in low magnetic fields. In particular, the conductance fluctuations in a rectangular-shaped quantum dot develop hierarchical structures around zero magnetic field. The fluctuations are drastically suppressed when the lead is attached to the corner of the dot. These observations demonstrate significant modifications of the quantum interference effects by Andreev reflections. In addition, the self-similarity of the conductance fluctuations does not necessarily guarantee the underlying classical dynamics to be chaotic. @S0163-1829~98!51320-3#

The classical trajectories in stadium-shaped cavities are chaotic. There has been a considerable number of investigations on the quantum ballistic transport in semiconductor microcavities having such geometry.1–4 The exponential decay of the probability distribution of the enclosed area in chaotic cavities was predicted to lead to a Lorentzian shape of the magnetoresistance due to the weak-localization correction.2 This was compared with a linear shape in regular cavities.3 To study the weak-localization effect, the sample specific conductance fluctuations were eliminated by taking an average. The correlation function or the spectrum of the conductance fluctuations, alternatively, reflects the difference in the probability distribution,1 and so information on the fractal dimension may be extracted.5 The chaotic dynamics often involves self-similar structures in phase space. Consequently, a trajectory is altered enormously when one of the parameters is varied. However, the contribution of such hierarchical structures in the phase space is averaged out in the classical conductance. The variation when magnetic field B is changed, for example, is dominated by stable periodic orbits or a stable ergodic sea. A recent experiment of Sinai billiard4 reported the observation of a self-similar behavior of the conductance fluctuations around B50. However, the interpretation remains controversial as no theory has derived the self-similarity so far. In this paper, the transport properties are investigated in two-dimensional electron gas ~2DEG!—superconductor junction microstrutures6,7 rather than in the conventional system. Instead of the mirrorlike boundary reflection of electrons in the normal conductors, quasiparticles in normalconductor–superconductor ~NS! systems experience Andreev reflections from the NS interface. An electronlike excitation with the energy « above the Fermi level is reflected as a holelike excitation with the energy « below the Fermi level. In addition, the Andreev reflection is different from the normal reflection in two respects: ~1! the reflected quasiparticle exactly follows the incident trajectory ~retroproperty!; ~2! the phase shifts due to the electron and the hole cancel out in the Andreev-reflected trajectory when « 50. The retroproperty implies that the dynamics at B50 is perfectly regular when the Andreev reflection probability is unity, i.e., no quasiparticle tunneling or specular reflection occurs. Multiple reflections from the boundary take place 0163-1829/98/57~20!/12689~4!/$15.00

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when the retroproperty is broken by a magnetic field or a finite bias.8–10 We demonstrate that the conductance fluctuations are strongly influenced by the geometry of the cavity and the position of the lead. Surprisingly, hierarchical structures of the conductance fluctuations are found in a certain type of rectangular cavities, in which the classical dynamics in the normal counterpart is regular. We compare the transport properties in three NS structures depicted on the right-hand side of Fig. 1. A quantum dot patterned from a 2DEG is weakly coupled to a lead on one end and is terminated by a superconductor ~shaded area! on the other end. We calculate the conductance when a current is injected from the lead and extracted from the superconductor. In the quasiparticle language, the incident quasi-

FIG. 1. Magnetoconductance in three geometries depicted on the right-hand side. The shaded areas represent superconductors. The lead is attached to the 25th slice from the NS interface. R12 689

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particles are totally reflected from the NS interface, to the other type of quasiparticles for the Andreev reflection or as the same type of quasiparticles for the normal reflection, when « is less than the superconducting gap D 0 . The quasiparticle escapes from the cavity through the lead either as an electron or a hole. Therefore, the two-terminal conductance is given for «,D 0 by11 G5 ~ 4e 2 /h ! Tr@ s †he s he # ,

~1!

where s he is the scattering coefficient for an injected electron to return as a hole. The phase coherent transmission of the quasiparticles is described by the Bogoliubov–de Gennes equation. We restrict our discussion to the zero-bias conductance, thus «50. We neglect the self-consistency of the pair potential amplitude D(x,y). A constant value D 0 is assumed in the superconductor, unless stated otherwise, and D50 in the 2DEG region. The conductance is determined by the scattering properties in the normal region, and so we do not take into account the magnetic field in the superconductor. These assumptions can be justified since the cavity size is typically smaller than the coherence length j 5\ v F /(2D 0 ).9 We simulate the NS junctions by a square lattice with the nearest-neighbor hopping amplitude t. We consider 50 lattice sites in the transverse direction. For simplicity, the normal lead is modeled by a 1D wire. In model A ~top in Fig. 1!, the cavity region consists of 50325 lattice sites and the 1D lead is attached to the last ~25th! slice from the NS interface. In model C ~bottom!, the lead is attached to the center ~the 25th slice from the NS interface among 50 slices! of the upper boundary of the square-shaped cavity. The classical dynamics in models A and C is expected to be regular. For model B ~middle!, a half of a stadium-shaped cavity is coupled to a superconductor. We expect the dynamics in this geometry to be chaotic ~when the retroproperty is broken by a magnetic field!, since a half-round ‘‘deflector’’ is attached to the lefthand side of model A. We calculate s he using the lattice Green’s-function technique.10 In Fig. 1, the conductance when the chemical potential m 52t is compared for the three geometries. We have calculated the conductance at various energies between 22t and 0, i.e., between the threshold energy in the 1D lead and the midband energy. Although we present results only for m 52t, the statements in this paper are unchanged qualitatively even when different energies are chosen. As D 0 50.01t is significantly smaller compared to the Fermi energy, the Andreev reflection probability is almost unity. Because of the retroproperty and the phase conjugation, the conductance at B50 is hence expected to be 4e 2 /h. However, the conductance is suppressed considerably because of the quantum-mechanical reflection from the junction of the 1D lead and the cavity. We have confirmed that the zerofield conductance increases with increasing the Fermi energy as the scattering from the junction becomes weak. The conductance fluctuations due to the quantum interference effect show up when B is larger than a certain value. A marked sensitivity of the fluctuations on the cavity geometry is apparent. Notice that the characteristics of the fluctuations change around BW 2 / f 0 54, where W is the width of the NS wire and f 0 5h/e the flux quantum. For BW 2 / f 0 .4, the fluctuations are similar in the three geometries. As we de-

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scribe below, the properties of these high-field fluctuations are in good agreement with existing theories. We thus conclude that the high-field fluctuations reflect the chaotic dynamics induced by the magnetic field. Before examining the low-field fluctuations in detail, let us summarize the characteristics of the high-field fluctuations. The amplitude of the fluctuations calculated for 5,BW 2 / f 0 ,10 is almost identical, i.e., 1.1e 2 /h. This value is in agreement with the predicted enhancement of the fluctuation amplitude in the NS systems.12,13 The correlation magnetic field B c ~Ref. 14! differs among the three geometries. We obtain B c S/ f 0 50.026, 0.067, and 0.060 from the top to the bottom. Here, S is the area of the 2DEG cavities. These values are an order of magnitude smaller than expected in a normal system, where the fluctuation pattern is altered by the addition of one flux quantum to the cavity. Brouwer and Beenakker15 calculated the magnetic-field correlation in NS systems using the random matrix theory. It was shown that B c becomes smaller when increasing the ratio of the numbers of the modes in the cavity and in the lead. Their estimate of B c is in good agreement with our numerical results, considering the fact that the theory is valid when the number of the modes is large. We have also confirmed that the correlation function is well described by the Lorentzian with a power 3/2, which has been predicted for the weak-coupling case.15 The low-field fluctuations, on the contrary, demonstrate drastic dependencies on the geometry and the lead position. The fluctuation pattern in Fig. 1~b! is the easiest to understand, where the characteristics of the fluctuations in both low and high magnetic field regimes are almost identical. The dynamics in model B is expected to be always chaotic ~except for the immediate vicinity of B50). The origin of the chaotic dynamics is due to the stadiumlike geometry in the low-B regime, whereas due to the magnetic field in the large-B regime. The microscopic details that determine the scattering matrix s he , however, are irrelevant as long as s he belongs to the same universality class.9 An unexpected feature of the fluctuations is found in rectangular-shaped quantum dots, where the classical dynamics is quasiregular, or at least not fully chaotic as we will show later. In Fig. 1~c!, the conductance fluctuates with finer magnetic-field scales. In fact, the low-field fluctuations appear to contain hierarchical structures around B50 as shown in Fig. 2.16 The B scale of the fluctuations becomes smaller for smaller B, in contrast to the conventional conductance fluctuations. On the contrary, the fluctuations are suppressed when the lead is attached to the corner of the rectangular dot, Fig. 1~a!. Examining various sizes and lead positions, we find that the variation of the fluctuation pattern is strongly related to the position of the lead. When the lead is moved away from the corner, rapid fluctuations, which can also be seen in Fig. 1~a! at BW 2 / f 0 ;1.2, emerge in the low-B regime and evolve to the hierarchical fluctuations. Lower levels of the hierarchy are generated by expanding the cavity on the left-hand side of the lead. We emphasize that the fluctuation pattern in Fig. 1~b! cannot be obtained in the rectangular cavities by placing the lead at an intermediate position. When the lead is located at the corner, the incident electron beam is forward-collimated as the geometry acts as a horn. The suppression of the fluctuations in Fig. 1~a! may suggest that the collimated beam can couple with only a small num-


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GEOMETRY-SPECIFIC CONDUCTANCE FLUCTUATIONS . . .

FIG. 2. Conductance of a rectangular cavity consisting of 50 360 lattice sites shown with expanded field scales. The lead is attached to the 25th slice from the NS interface.

ber of trajectories in the dot.17 It should be noted, however, that the suppression is not observed in normal cavities and is a characteristic of the NS system. In the upper inset of Fig. 3~a!, we plot the classical conductance corresponding to Fig. 1. Almost no difference between models B and C is found. The conductance for model B is slightly larger near zero magnetic field as the trajectories that do not encounter the NS interface are reduced by the deflector. For these two cases, the conductance becomes independent of B for BW 2 / f 0 .0.4 at '2e 2 /h. ~For the parameters used in this paper, BW 2 / f 0 514.1W/r c , where r c is the cyclotron radius.! This B independence is a consequence of the multiple Andreev reflections, which lead to the same probabilities of 0.5 for the quasiparticles to leave the system either as electrons or as holes. One can roughly estimate that the quasiparticles no longer return to the lead directly when W 2 /r c .d, where d is the width of the lead. The conductance in model A, however, shows a broad dip around BW 2 / f 0 51. When the quasiparticles are injected from the corner, symmetric trajectories illustrated in the inset of Fig. 3~b! become dominant at weak magnetic fields. The quasiparticle leaves the system after even number Andreev reflections, and so the conductance is reduced. Notice that the conductance in Fig. 1 initially decreases in model A, whereas it increases in models B and C. The initial decrease of the conductance in model A may be attributed to this classical trajectory. When the magnetic field is further increased (BW 2 / f 0 .3), the dynamics is completely determined by

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FIG. 3. Probability distributions of ~a! trajectory length L and ~b! enclosed area S 0 in a geometry corresponding to model C. Dashed, dotted, and solid lines in the inset of ~a! show the classical conductance in the three geometries of Fig. 1 from the top to the bottom, respectively. The inset in ~b! depicts a trajectory that causes the broad dip in the classical conductance near BW 2 / f 0 51.

the magnetic field, giving rise to the conductance value of 2e 2 /h irrespective of the geometry and the lead position. We show in Figs. 3~a! and 3~b! the probability distributions of, respectively, the trajectory length P(L) and the enclosed area of a trajectory P(S 0 ) in a geometry corresponding to model C at BW 2 / f 0 50.7. The phase modulation in the NS system for an Aharonov-Bohm–type interference effect of a closed loop is also given by the enclosed magnetic flux.18 Therefore, the trajectories which enclose more than one flux quanta do not contribute to the fluctuations. The probability distribution in regular cavities comprises d -function peaks, whereas that in chaotic cavities is described by a smooth exponentially decaying curve. One finds in Figs. 3~a! and 3~b! the long duration tail even when the magnetic field is weak because of the narrow lead. However, the large fluctuations and the nonexponential decays observed in both of the probability distributions indicate that the dynamics is not fully chaotic. The influence of the retroproperty, which gives rise to P(S 0 )} d (S 0 ) when B50, is manifested as the enhanced peak around S 0 50. For BW 2 / f 0 .3 ~not shown!, the distributions are described by smooth curves and clear exponential decays are found.19 We have also found that the enhancement around S 0 50 vanishes, indicating that the retroproperty plays no role when r c ,5W. It may be noteworthy that P(S 0 ) becomes fairly symmetric in the large-B regime.19 Brouwer and Beenakker9,15 demonstrated that the correlation magnetic


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FIG. 4. Modulation of the conductance by the phase difference 2 u between two superconductor segments.

field of the conductance fluctuations is B independent when the dynamics in the cavity is chaotic, as is the case for BW 2 / f 0 .4 in Fig. 1. The enhanced S 0 '0 peak due to the retroproperty is, thus, anticipated to be responsible for the hierarchical fluctuations. However, it seems unlikely that the hierarchical fluctuations can be attributed to certain kinds of trajectories as the classical conductance does not show such a behavior. In the experiments by Hartog et al.6 and by Morpurgo et al.,7 the conductance modulation was induced by the phase difference between the two superconductor segments

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C. M. Marcus, A. J. Rimberg, R. M. Westervelt, P. F. Hopkins, and A. C. Gossard, Phys. Rev. Lett. 69, 506 ~1992!. 2 H. U. Baranger, R. A. Jalabert, and A. D. Stone, Phys. Rev. Lett. 70, 3876 ~1993!. 3 A. M. Chang, H. U. Baranger, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 73, 2111 ~1994!. 4 R. P. Taylor, R. Newbury, A. S. Sachrajda, Y. Feng, P. T. Coleridge, C. Dettmann, N. Zhu, H. Guo, A. Delage, P. J. Kelly, and Z. Wasilewski, Phys. Rev. Lett. 78, 1952 ~1997!. 5 R. Ketzmerick, Phys. Rev. B 54, 10 841 ~1996!. 6 S. G. den Hartog, C. M. A. Kapteyn, B. J. van Wees, T. M. Klapwijk, W. van der Graaf, and G. Borghs, Phys. Rev. Lett. 76, 4592 ~1996!; 77, 4954 ~1996!. 7 A. F. Morpurgo, S. Holl, B. J. van Wees, T. M. Klapwijk, and G. Borghs, Phys. Rev. Lett. 78, 2636 ~1997!. 8 B. J. van Wees, P. de Vries, P. Magne´e, and T. M. Klapwijk, Phys. Rev. Lett. 69, 510 ~1992!. 9 C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 ~1997!. 10 Y. Takagaki and Y. Tokura, Phys. Rev. B 54, 6587 ~1996!.

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as well as by the magnetic field penetrating through the cavity region. We have so far considered the effect of the magnetic field through the cavity region. In the remainder of the paper, we examine the influence of the phase of the superconductor. We divide the superconductor strip into two segments, @ 0,y,W/2# and @ W/2,y,W # . We assign a phase u to the pair potential in each segment as D 0 e 2i u and D 0 e i u . Due to the symmetry G(B, u )5G(2B,2 u ), the conductance is identical when the lead is attached to the upper boundary or the lower boundary of the dot. Numerical results are presented for the stadium-like-shaped cavity as they turned out to be similar in all the geometries. The conductance is plotted in Fig. 4 as a function of 2 u for various values of the magnetic fields. The conductance modulation contains many harmonics because of the multiple Andreev reflection that stemmed from the narrow lead.15 Although the oscillation amplitude was maximum near B50 in the experiments, the amplitude is minimum in our calculation around B50. Morpurgo et al.7 reported that the experimental result is well explained by a calculation where the contribution due to the multiply reflected trajectories is omitted despite the presence of the constriction in the device. As this calculation predicts the maximum amplitude near B50,7 the discrepancy may be ascribed to the dominance of the multiple Andreev reflection in our models. The interference effect is significantly modified by the multiple Andreev reflection as evidenced in Fig. 4. To test our theory in experiments, the constriction width has to be reduced to contain only a few modes.

Y. Takane and H. Ebisawa, J. Phys. Soc. Jpn. 61, 1685 ~1992!. Y. Takane and H. Ebisawa, J. Phys. Soc. Jpn. 61, 2858 ~1992!. 13 P. W. Brouwer and C. W. J. Beenakker, Phys. Rev. B 52, 16 772 ~1995!. 14 P. A. Lee, A. D. Stone, and H. Fukuyama, Phys. Rev. B 35, 1039 ~1987!. 15 P. W. Brouwer and C. W. J. Beenakker, Phys. Rev. B 54, 12 705 ~1996!. 16 Part of the conductance fluctuations may be originating from the energy band structure inherent in the tight-binding model. See, for example, U. Sivan, Y. Imry, and C. Hartzstein, Phys. Rev. B 39, 1242 ~1989!. 17 R. Akis, D. K. Ferry, and J. P. Bird, Phys. Rev. B 54, 17 705 ~1996!. 18 Y. Takagaki, Phys. Rev. B 57, 4009 ~1998! 19 The characteristic length L s associated with the exponential decay ;exp(2L/Ls) is 150W at BW 2 / f 0 53. Similarly defined characteristic area is 1.24W 2 and 1.13W 2 for the positive and negative signs, respectively. 11 12