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Standing wave patterns of microwaves propagating in Josephson tunnel junctions with integrable and chaotic billiard geometries
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Physica D 78 (1994) 214-221
Standing wave patterns of microwaves propagating in Josephson tunnel junctions with integrable and chaotic billiard geometries C . A . K r u e l l e a, T. D o d e r e r a, D. Q u e n t e r " , R . R H u e b e n e r a, R. P 6 p e l b j. N i e m e y e r b a Physikalisches Institut, Lehrstuhl Experimentalphysik 1I, Universittlt Tabingen, D-72076 T~bingen, Federal Republic of Germany b Physikalisch-TechnischeBundesanstalt, D-38116 Braunschweig, FederalRepublic of Germany Received 28 April 1994; revised 24 May 1994; accepted 17 June 1994 Communicatedby EH. Busse
Abstract
A new experimental system based on electron-beam scanning of Josephson tunnel junctions is introduced for studying standing wave patterns inside resonators with integrable and chaotic billiard geometry. The tunnel junctions are coupled to an external microwave source for controlled ff injection. Utilizing the process of photon-assisted tunneling, we obtain two-dimensional images of the local photon absorption density proportional to the square of the rf electric field across the junction. We observe forced standing wave patterns of the microwave field which depend sensitively on the boundary conditions and the external driving frequency. For rectangular resonators a theoretical model is developed for describing the multiple reflections of the microwave at the tunnel junction boundaries.
1. Introduction
In 1917 Albert Einstein pointed out that for nonintegrable mechanical systems like Poincar6's three-body problem the old quantization rules of Bohr and Sommerfeld do not hold [ 1 ]. In the language of modern quantum mechanics one has to ask how the classical chaotic motion of a particle in a nonintegrable system is related to the well defined eigenstates and eigenvalues describing its quantum mechanical behavior. With the growing interest in classical chaotic dynamical systems during the last two decades this old question of quantum mechanics attracted the attention of theorists as well as experimental physicists. The system most commonly studied is a free particle moving in two dimensions inside a billiard with stadium boundary. The reflections at the semicircles cause exponentially di-
verging trajectories and thus a chaotic motion of the particle inside the stadium. The wave mechanical description of this problem is governed by the Helmholtz equation ( V 2 + k 2) ~b( x ) = 0 for the two-dimensional wave propagation inside the resonator with suitable boundary conditions at the billiard "walls". Since the Helmholtz equation is mathematically identical with the time-independent Schr&linger equation for a particle with momentum p = hk moving in such a billiard (potential U ( x ) = 0 inside, U ( x ) = cc outside), the wave propagation in the stadium serves as an ideal model system for studying the properties of "quantum" or wave chaos. Because of the analogy to quantum mechanics so far all of the theoretical and experimental work has been dealing with free vibrations of the various oscillatory systems. These free vibrations are solutions of
0167-2789/94/$'/.00 t~ 1994 Elsevier Science B.V. All fights reserved SSDI 0167-2789(94)00156-K
C.A. Kruelle et al. / Physica D 78 (1994) 214-221
the homogeneous Helmholtz equation, i.e. they have to satisfy Dirichlet or Neumann conditions everywhere on the boundary. Solutions of this type, the eigenfunctions of the considered systems, can only be found for a discrete set of eigenvalues. In experiments, vibrating plates or microwave cavities were driven by a harmonic source oscillating with the various resonance frequencies of the system. However, this restriction in the choice of driving frequency is only due to the desired analogy with quantum states. Besides the investigation of quantum mechanical eigenfunctions we think that it is worthwhile studying wave phenomena inside resonators of integrable and chaotic geometry in a more general context. For example, if some parts of the boundary are forced to oscillate by an external driving force, vibrations.of a "wavy" medium can arise for any frequency (though when the forcing frequency equals one of the eigenfrequencies, the response may be enhanced). The driven system develops forced standing wave patterns which are solutions of the inhomogeneous Helmholtz equation (V 2 + k 2 ) ~ ( x ) = --4zrp(xb). Here p(Xb) stands for the distribution of "sources", i.e., parts of the boundary where energy interchange between the system and some external driving force takes place. Experimental imaging of solutions of the homogeneous Helmholtz equation has been performed in various systems. Chladni published his nodal patterns of vibrating plates already in 1787 [2]. Vibrational modes of several musical instruments like violins or especially drums could be made visible with the help of holographic interferograms. In a recent experiment Bltimel et al. demonstrated standing wave patterns on water surfaces of round and stadium-shaped ripple tanks [ 3]. However, for quantitative measurements of the spatial distribution of wave fields inside resonators of chaotic billiard geometry up to now the most promising systems seem to be microwave resonance cavities [4-11]. The dimensions of the top and ground plates of the cavities are usually in the cm-regime with a spacing d in z direction of several mm. For frequencies below ~',,~x = c/2d, c = speed of light, these resonators support two-dimensional transverse magnetic modes which obey Dirichlet boundary conditions (Ez = 0). Using a bead perturbation
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technique [4,5], Sridhar and Heller were able to observe the eigenfunctions of the Sinai billiard and found excellent agreement with their numerical calculations [6,7]. Stein and Sttckmann used a slightly different technique for the eigenfunctions of the stadium [9]. In this paper we show for the first time images of forced standing wave patterns, i.e. solutions of the inhomogeneous Helmholtz equation, in resonators of integrable and chaotic billiard shape. Our vibrating system is a Josephson tunnel junction coupled to an external microwave source acting as the driving force.
2. Experimental technique Irradiation of superconducting tunnel junctions with microwaves results in photon-assisted tunneling of quasiparticles [ 12,13] which can be observed as typical changes in the current-voltage characteristics (IVC) of the junction. Due to the strong impedance mismatch between the junction and the surrounding structures the incident microwave field is multiply reflected at the boundaries [ 14,15]. If the Josephson current (tunneling of Cooper pairs) is suppressed by a magnetic field, the tunnel junctions can be treated as parallel plate resonators and the propagation of electromagnetic waves is governed by the Helmholtz equation. However, compared with ordinary resonant cavities three basic differences must be taken into account: (1) The electromagnetic fields propagate inside the tunnel junction with the Swihart velocity ~ which is usually one or two orders of magnitude less than the vacuum speed of light [ 16]. This means a reduction in the length scale of the standing wave patterns. (2) Since reflection losses inside superconducting tunnel junctions are negligible the microwave is reflected at the open boundary with a reflectivity close to one and without a phase shift [ 17]. This implies that, in contrast to resonant cavities, standing wave patterns inside a Josephson tunnel junction have to satisfy Neumann conditions (VEz = 0 in the direction perpendicular to the boundary) at all parts of the boundary which do not act as sources. (3) The Q value of Josephson tunnel junctions
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C.A. Kruelle et al. / Physica D 78 (1994) 214-221 w = 125 ~ m
T>Tc !
~ffllllllllllllllllllilllllilA4
,~II I
high V~I ("belly") . . . . low V~I / ......... no V~I ( n o d a l ~
[
~. ~ ~ / ~
Fig. 1. Cross-sectionof a Nb/AlOx/Nb-tunneljunction with quarter stadium geometryconnected to a superconductingstripline for microwavecoupling (1: SiO2, 275 nm, 2: bib top electrode, 150 nm, 3: Nb base electrode, 150 nm, 4: SiO2, ~ 1.7 prn, 5: Nb2Os, 45 nm, 6: Nb microslripgronndplate, 150 nm, 7: silicon substrate). (typical value Q ~ 100) is much less than for resonant cavities. Therefore Josephson tunnel junctions are not well suited for the investigation of eigenmode distributions in these systems. The advantage of low Q resonators lies in the fact that the pattern formation of spatially varying electromagnetic fields is much less dominated by resonant eigenmodes. For this reason forced standing wave patterns oscillating at frequencies other than eigenfrequencies can be studied more easily than with high Q cavities. For our experiments we use a standard reflex klystron with fixed frequency of 70 GHz as microwave source. The Josephson tunnel junction is connected to a superconducting stripline for controlled microwave injection. Fig. 1 shows a schematic cross-section of a typical junction geometry. Details of the junction preparation technique can be found in Ref. [ 18]. The microwave field is coupled into the superconducting stripline via a micro-fabricated dipole antenna on the same chip. A small amount of the rf power is propagated through the small slit between the base electrode and the Pbln wiring layer to the right edge of the junction. We assume that the injected wavefront is approximately homogeneous over the whole width w = 125/zm of the coupling slit. Depending on the preparation parameters the microwave propagates inside the N b / A l O x / N b - j u n c t i o n with a normalized
~_..~.~. v -i'.V(:r, ~) Fig. 2. Local current-voltagecharacteristics for three regions of a microwaveirradiated superconductingtunnel junction with different local if-voltagelevels. The arrows indicate the Vrs-dependent electron-beaminduced transitions to the normal state inside areas locally heated by the beam. In our experimentsthe transition arrows are almost vertical. The -AV(x,y) signal level is in the /~V-regime, while the gap voltage 2~/e is typically a few mV. The slope of the arrows (here 1000× exaggeratedfor clarity) is determined by the ratio of the heated area ~-02 to the area of the whole junction. Swihart velocity -~/c between 0.029 and 0.033, the corresponding Swihart wavelength ~ is 125/~m and 140/tin, respectively. Our imaging technique for the microwave standing wave patterns uses the concept of low-temperature scanning electron microscopy (LTSEM) [ 19]. The Josephson tunnel junction is scanned with the electron beam of a scanning electron microscope, while the bottom of the substrate carrying the junction is in direct thermal contact with liquid helium. For a description of the experimental setup we refer to Ref. [20]. All experiments were performed at a He bath temperature of 1.9 K, sample temperature T ~ 6 K. The imaging method is based on detecting the local photon-assisted quasiparticle tunneling current [21,22]. For obtaining a two-dimensional voltage image, the top side of the current-biased junction is scanned with the electron beam, and the beam-induced change - A V ( x , y) of the junction voltage is recorded simultaneously as a function of the coordinate point (x, y) of the beam focus on the sample surface. Fig. 2 gives a scheme of the signal generation principle. Due to the local heating effect in an area ~rr/2 07 denotes the thermal healing length [23] ) around the beam
C.A. KrueUeet al. / Physica D 78 (1994) 214-221 focus the quasiparticle tunneling current for this perturbed area is increased by an amount 8leb(X,y). If the heated area is switched to the normal conducting state, the local current increment 81eb(X, y) is j USt the difference Inornm(X, y) --lqp(X,y), where lqp( X, y) denotes the quasiparticle tunneling current through the area ~r/2 without electron-beam irradiation. Assuming ,r~/2 small compared with the total tunneling area, the corresponding voltage change across the junction is given by [21]
- A V ( x , y ) = aV
.81eV(X,y)
(1)
vB and the structural information is provided by the spatial distribution of the local current change 81eb(x, y) in the perturbed area. In the presence of an inhomogeneous if-field Vrf(x,y) across the junction the photon-assisted quasiparticle tunneling current density jqp(X,y) is spatially modulated in the same way. Fig. 2 shows schematically the local current-voltage characteristics of three regions with different if-field amplitudes ("belly", intermediate field, and nodal line). For small incoming microwave fields Ez (i.e., the corresponding rf voltage across the junction V,f << 2Ale, A = energy gap of the superconducting layers) the if-induced local increase 31,f(x, y) of the tunneling current is proportional to the square of the local field Ez (x,y) across the junction [24]. From Fig. 2 we deduce that the electron-beam induced local current change 8Ieb ( X, y) is equal to 81max- Crf "E2z( X, y) ), where Crf denotes the conversion coefficient if field to photon-assisted quasiparticle tunneling current. Using Eq. ( 1) we finally obtain for the voltage signal
AV(x,y): aV(x, y) = C rf" a~l vs.E2(x,y) Y gain aV
•
~Imax
(2)
Hence, for obtaining a voltage image of the if field distribution Ez2 (x, y) the response signal AV(x,y) of
217
the whole junction is recorded as a function of the coordinate point (x, y) of the beam focus on the sample surface.
3. Results and discussion
3.1. lntegrable billiards The eigenfunctions of a rectangular resonator (dimensions Ix, ly) are well known. Depending on whether Dirichlet or Neumann boundary conditions must be satisfied, Ez cx sin(m,rx/lx) sin(n~'y/ly) or Ez cx cos(m*rx/lx) cos(n~ry/ly), m,n = O, 1,2 . . . . is expected, respectively. With the help of the bead perturbation technique Sridhar mapped the (9,2)Dirichlet mode in a resonant cavity [6]. Using LTSEM, Mayer et al. [25] and Lacbenmann et al. [26] were able to image the magnetic field distribution H~(x,y) for several (m,n)-Neumann modes (Fiske modes [27] ) in Josephson tunnel junctions. Considering forced standing wave patterns in ifdriven Josephson tunnel junctions the situation is more complex. There exists no system of separate "modes". Rather, the standing wave field is not only governed by the shape of the resonator but also depends in a sensitive, "continous" way on the distribution of sources and the external driving frequency w. Before solving this boundary problem in general, we consider the elementary problem of one single point source oscillating with frequency w at the center (0,/y/2) of one side of the rectangle. The outgoing cylindrical wave, the solution of the Helmholtz equation for a point source in two dimensions, is the Hankel function HCol)(kr), where k = to/~ + lot, a = damping constant, r = distance to the point source. For describing the multiple reflections of this outgoing wave at the boundaries the method of images [28], which is known for solving the problem of induced charges in electrostatics, can be applied. Multiple imaging of the original point source at the four sides of the rectangle generates an infinite rectangular lattice of point sources at positions (2mlx, (n + 1 ) ly), m , n = 0, 4-1,4-2 . . . . The superposition
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C.A. Kruelle et al. / Physica D 78 (1994) 214-221 oo
Ez(X,y)=Ez,o" rrnn =
[ (X
--
Y~
Y ~ H(ol)(krmn),
2mlx) 2 + (y - (n
+
t~
l)/y)2] 1/2 (3)
of all cylindrical waves starting at the lattice points results in a standing wave pattern inside the rectangle, which satisfies Neumann conditions at all parts of the boundary except the position of the point source. Each image source in the virtual lattice corresponds to a ray which starts at the original source and reaches the considered point (x, y) inside the resonator after multiple reflections at the boundary. For solving the more general case when the incident microwave field enters the rectangular resonator through one or more slits of finite extension, we have to integrate over the distribution of sources p ( x b ) at the boundary:
E z ( x ) = Ez,o" ~ p(xb) m=- ~ n=- C~oundary × [H
(4)
where x,,m = (2mlx, 2nly), m, n = 0, +1, -t-2. . . . This expansion is useful when only the direct source and perhaps the first few reflections are important. This is the case for a damped wave with an attenuation length 1 / a comparable with the resonator dimensions. As an example, Fig. 3 shows the forced standing wave pattern generated by a microwave field entering the Josephson tunnel junction through a slit from the left. For the numerical solution using Eq. (4) and shown in Fig. 3a three approximations had to be taken into account: (1) We assume a homogeneous distribution of sources inside the coupling slit. The integral can be converted into a sum of point sources if the separation between them is chosen much less than the wavelength. (2) Instead of evaluating the Hankel function H
(a)
Fig. 3. Forced standing wave field inside a Josephsontunnel junction with rectangular geometry.The microwaveenters through a slit at the left, junction dimensions= 500 x 425/zm ~. (a) Numerical calculation of E2z(x, y) according to Eq. (2). The Swihart wavelength ~ = 140 ~ra and the damping constant a = 1/500 /~rn-l are taken from the experiment. (b) Experimental voltage image AV(x, y). T ~ 6 K. V~rrCOS(krq-4)
e -at
for distances r larger than the resonator dimensions. (3) Due to the damping the higher terms ( Iml, Int > 3) of the doubly infinite series do not influence the appearance of the standing wave pattern substantially but form a small background correction and are therefore neglected. With these three approximations the numerical effort of evaluating Eq. (4) for achieving sufficiently satisfying results is rather small. Two parameters must be taken from the experiment, the Swihart wavelength and the damping constant a. The shape of the standing wave pattern is more sensitive to the microwave wavelength ~, which governs the lateral distribution of maxima and nodal lines, while the damping constant a basically determines the relative height of the maxima. For comparison with this theoretical calculation Fig.
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C.A. Kruelle et al. / Physica D 78 (1994) 214-221
130 ~m
rf---Y,~ rf---~
l 500 ~m
Fig. 4. Voltage image AV(x, y) of the forced standing wave pattern inside a rectangular Josephson tunnel junction coupled to two slits for controlled microwave injection, Bright areas indicate regions of high ff field ampH_tude. Junction dimensions = 500 x 400/zm 2, Swihart wavelength A = 140/~m. T ~-, 6 K.
3b shows the corresponding experimental image. The spatial resolution of the LTSEM technique is typically a few /zm. This is at least one order of magnitude less than the intrinsic length scale A/2 of the standing wave patterns. For best results the scanning grid (here 640 × 512) is chosen in such a way that the grid step size is less than the physical resolution. The data acquisition for such an LTSEM image takes about 8 minutes. The good agreement between the numerical simulation and the experimental image can be taken as proof of validity of the theoretical approach as well as the experimental technique. A second example for an rf driven standing wave pattern in a resonator of rectangular shape is shown in Fig. 4. Here the microwave field enters the tunnel junction through two equal slits at the left side. The modulation of the microwave field in one direction at the top and bottom edges of the rectangle results from the interference of wave trains originating from each slit and bouncing back and forth along the edges of the junction. This feature has the character of the so-called "bouncing-ball" state first described by McDonald and Kaufman [29] for the stadium billiard. Fig. 5 shows one more example for a forced microwave standing wave pattern in a Josephson tunnel junction of integrable billiard shape. The quarter circle is just a symmetry-reduced version of a circular membrane whose vibrational modes are well known
b:
125 #m - - - ' ~
Fig. 5. Voltage image AV(x, y) of the forced standing wave pattern inside a Josephson tunnel junction of quarter circle shape, the radius r is equal to the Swihart wavelength A = 125/zm. T ~ 6 K. Bright areas indicate regions of high ff field amplitude.
from drum acoustics [ 30]. For homogeneous Dirichlet boundary conditions the system of eigenfunctions is a subset of the eigenfunctions for the full circle with nodal lines at the two straight edges of the quarter circle. However, this symmetry of the boundary conditions is broken, if the resonator is coupled to an external driving force acting upon the boundary. In Fig. 5 the microwave field enters the tunnel junction as a plane wave from the left side. At the bottom and the round edge Neumann boundary conditions must be satisfied. The superposition of all multiply reflected wave trains results in a standing wave pattern which cannot be associated with one of the eigenmodes of a circular membrane. Since here the radius of the quarter circle is chosen equal to the Swihart wavelength --- 125 g m we observe a rather simple standing wave pattern. From the relative height of the two maxima we can conclude that the microwave field inside the tunnel junction is damped with an attenuation length 1/a of about several hundred/zm.
3.2. Chaotic billiards For the theoretical calculation of eigenfunctions in billiards with chaotic geometry several numerical techniques like the Green's function method of McDonald and Kaufman [29,31 ] or Heller's superposition of waves with random direction [32] have been developed with great success. Up to now, these methods have not been applied to the case of forced standing wave patterns in externally driven systems. This seems
220
C.A. Kruelle et al. / Physica D 78 (1994) 214-221 l =
~<~"-" rf
~---r = 125/~rn--~ Fig. 6. Voltage image AV(x,y) of the forced standing wave patterninsidea Josephson tunneljunctionof quarterstadium shape (l = r = 125 /an). The microwave (A ---125/zm) enters from the right.T ~ 6 K. Bright areas indicateregions of high rf field amplitude. The lower contrast in the square part of the tunnel junction is caused by the Pbln wiring layer partiallycovering the Nb top electrodeas can be seen in Fig. I.
to be a promising task for the future. The method of images described above does not work well in this case since it is restricted to boundaries which are composed of straight lines. Of course this does not mean that the method of images cannot be applied to resonators of chaotic billiard shape at all, but that it can be applied only as an approximation. Due to these theoretical difficulties the experimental approach becomes important. Preparation techniques for Josephson tunnel junctions are highly developed and any shape of the tunneling area and rf injection geometry can be fabricated with great accuracy. Here we report on experiments for two basic resonator geometries: the quarter stadium and the"elbow" billiard. (1) Quarter stadium: The system most commonly studied in this context is the Bunimovich stadium billiard consisting of two semicircles separated by a square. Since most authors are interested in eigenfunctions which underly certain symmetry conditions, they usually consider one quarter of the full stadium. These eigenfunctions obey Dirichlet conditions at all parts of the boundary. The corresponding eigenfunctions of the full stadium show nodal lines at the two symmetry axes. For our experiment we have chosen a quarter stadium (Fig. 6) based on the same dimensions and a similar coupling geometry for the rf injection as in the quarter circle case. The microwave field enters from the right side. At this edge periodic boundary conditions with an oscillating frequency f = 70 GHz must be satisfied whereas Neumann conditions hold for all
~[oIOlr.mn Fig. 7, Voltage image AV ( x, y ) of the forced standing wave pattern inside a Josephson tunnel junction of elbow shape (1 = 400/,~m). The width of the coupling slit located on the left side is w = 40 /tin, Swihart wavelength ~ = 125 /zm. T m 6 K. Bright areas indicate regions of high ff field amplitude. The lower contrast in
the L-shaped area on the right and bottom edges of the tunnel junction is caused by the Pbln wiring layer partially coveringthe Nb top electrode. other parts of the boundary. Fig. 6 shows the resulting standing wave pattern. In the square part of the stadium the incident plane wave is only slightly disturbed while the standing wave exhibits an irregular pattern inside the quarter circle part. Due to the small junction dimensions compared with the microwave wavelength we cannot expect to see any "marring" of the wave pattern. (2) Elbow billiard: As a final example Fig. 7 shows the rf induced standing wave pattern inside a resonator of "elbow" shape first studied by Doron, Smilansky, and Frenkel [ 11 ]. The elbow is again a symmetryreduced version of the well known geometry of the Sinai billiard which in its original form consists of a circular disk in the center of a square. The Sinai billiard can be considered as the elementary cell for the scattering problem of a particle in a square lattice of circular targets. In Figure 7 the microwave field enters through the small straight edge on the left side. Since the tunnel junction dimensions are large compared to the Swihart wavelength A we observe a whole chain of maxima all evenly spaced with a separation of A/2. In summary, these experiments show that depending on the boundary conditions the propagation of waves inside Josephson tunnel junctions of various
C A . Kruelle et al. / Physica D 78 (1994) 214-221
geometries can develop a much richer variety of standing wave patterns than just a system of eigenmodes which are usually associated with quantum mechanical states. For inhomogeneous boundary conditions with periodically oscillating parts of the boundary the Helmholtz equation has solutions for any external driving frequency regardless of the specific resonator shape. Since theoretical solutions of these boundary problems for non-integrable geometries are not yet well established, our experiments can serve as test objects for numerical algorithms to be developed in the future.
Acknowledgements We thank A. Kittel, S. Lachenmann, and R. Gross for stimulating discussions and critical comments. Financial support of this work by the Deutsche Forschungsgemeinschaft and the Studienstiftung des deutschen Volkes is gratefully acknowledged.
References [1] A. Einstein, Verhand. Dtsch. Phys. Ges. 19, 82 (1917). [ 2 ] E. E E Chladni, Entdeckungen fiber die Theorie des Klanges, Leipzig 1787, reprint Leipzig 1980. [3] R. Bltlmel, I. H. Davidson, W. E Reinhardt, H. Lin, and M. Sharnoff, Phys. Rev. A 45, 2641 (1992). [4] L. C. Maier and J. C. Slater, J. Appl. Phys. 23, 68 (1952). [5] J. C. Amato and H. Herrmann, Rev. Sci. lnstrum. 56, 696 (1985). [6] S. Sridhar, Phys. Rev. Lett. 67, 785 (1991). [7] S. Sridhar and E. J. Heller, Phys. Rev. A 46, R1728 (1992). [8] H.-J. StOckmann and J. Stein, Phys. Rev. Lett. 64, 2215 (1990).
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[9] J. Stein and H.-J. St6ckmann, Phys. Rev. Lett. 68, 2867 (1992). [10] H.-D. Gr/if et al., Phys. Rev. Lett. 69, 1296 (1992). [ 11 ] E. Doron, U. Smilansky, and A. Frenkel, Phys. Rev. Lett. 65, 3072 (1990). [12] A. H. Dayem and R. J. Martin, Phys. Rev. Lett. 8, 246 (1962). [13] E K. "lien and J. E Gordon, Phys. Rev. 129, 647 (1963). [14] N. R. Werthamer, Phys. Rev. 147, 255 (1966). [ 15] C. A. Hamilton and S. Shapiro, Phys. Rev. B 2, 4494 (1970). [16] J. C. Swihart, J. Appl. Phys. 32, 461 (1961). [17] A. Barone, G. Patem6, Physics and Applications of the Josephson Effect, John W'fley, New York, 1982, p. 298 [ 18] R. P6pel, J. Niemeyer, R. Fromknecht, W. Meier, and L. Grimm, J. Appl. Phys. 68, 4294 (1990). [19] R. P Huebener, in Advances in Electronics and Electron Physics, Vol. 70, edited by P. W. Hawkes, Academic Press, New York, 1988, p. 1 [20] T. Doderer, H.-G. Wener, R. Moeck, C. Becker, and R. E Huebener, Cryogenics 30, 65 (1990). [21] H. Seifert, R. E Huebener, and P. W. Epperlein, Phys. Lett. A 95, 326 (1983). [22] C. A. Krulle, T. Doderer, D. Quenter, R. E Huebener, R. P6pel, and J. Niemeyer, Appl. Phys. Lett. 59, 3042 (1991). [23] J. R. Clem and R. E Huebener, J. Appl. Phys. 51, 2764
(1980). [24] J. R, Tucker and M. J. Feldman, Rev. MOd. Phys. 57, 1055 (1985). [25] B. Mayer, T. Doderer, R. R Huebener, and A. V. Ustinov, Phys. Rev. B 44, 12463 (1991). [26] S. G. Lachenmann, T. Doderer, R. P. Huebener, D. Quenter, J. Niemeyer, and R. POpel, Phys. Rev. B 48, 3295 (1993). [27] M. D. Rske, Rev. Mod. Phys. 36, 221 (1963). [28] E M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953, p. 812. [29] S. W. McDonald and A. N. Kaufman, Phys. Rev. Lett. 42, 1189 (1979). [30] T. D. Rossing, Physics Today, March 1992, p. 40. [31] S. W. McDonald and A. N. Kaufman, Phys. Rev. A 37, 3067 (1988). [32] E. J. Heller and P. W. O'Connor, Nuel. Phys. B 2, 201 (1987).
Physica D 78 (1994) 214-221
Standing wave patterns of microwaves propagating in Josephson tunnel junctions with integrable and chaotic billiard geometries C . A . K r u e l l e a, T. D o d e r e r a, D. Q u e n t e r " , R . R H u e b e n e r a, R. P 6 p e l b j. N i e m e y e r b a Physikalisches Institut, Lehrstuhl Experimentalphysik 1I, Universittlt Tabingen, D-72076 T~bingen, Federal Republic of Germany b Physikalisch-TechnischeBundesanstalt, D-38116 Braunschweig, FederalRepublic of Germany Received 28 April 1994; revised 24 May 1994; accepted 17 June 1994 Communicatedby EH. Busse
Abstract
A new experimental system based on electron-beam scanning of Josephson tunnel junctions is introduced for studying standing wave patterns inside resonators with integrable and chaotic billiard geometry. The tunnel junctions are coupled to an external microwave source for controlled ff injection. Utilizing the process of photon-assisted tunneling, we obtain two-dimensional images of the local photon absorption density proportional to the square of the rf electric field across the junction. We observe forced standing wave patterns of the microwave field which depend sensitively on the boundary conditions and the external driving frequency. For rectangular resonators a theoretical model is developed for describing the multiple reflections of the microwave at the tunnel junction boundaries.
1. Introduction
In 1917 Albert Einstein pointed out that for nonintegrable mechanical systems like Poincar6's three-body problem the old quantization rules of Bohr and Sommerfeld do not hold [ 1 ]. In the language of modern quantum mechanics one has to ask how the classical chaotic motion of a particle in a nonintegrable system is related to the well defined eigenstates and eigenvalues describing its quantum mechanical behavior. With the growing interest in classical chaotic dynamical systems during the last two decades this old question of quantum mechanics attracted the attention of theorists as well as experimental physicists. The system most commonly studied is a free particle moving in two dimensions inside a billiard with stadium boundary. The reflections at the semicircles cause exponentially di-
verging trajectories and thus a chaotic motion of the particle inside the stadium. The wave mechanical description of this problem is governed by the Helmholtz equation ( V 2 + k 2) ~b( x ) = 0 for the two-dimensional wave propagation inside the resonator with suitable boundary conditions at the billiard "walls". Since the Helmholtz equation is mathematically identical with the time-independent Schr&linger equation for a particle with momentum p = hk moving in such a billiard (potential U ( x ) = 0 inside, U ( x ) = cc outside), the wave propagation in the stadium serves as an ideal model system for studying the properties of "quantum" or wave chaos. Because of the analogy to quantum mechanics so far all of the theoretical and experimental work has been dealing with free vibrations of the various oscillatory systems. These free vibrations are solutions of
0167-2789/94/$'/.00 t~ 1994 Elsevier Science B.V. All fights reserved SSDI 0167-2789(94)00156-K
C.A. Kruelle et al. / Physica D 78 (1994) 214-221
the homogeneous Helmholtz equation, i.e. they have to satisfy Dirichlet or Neumann conditions everywhere on the boundary. Solutions of this type, the eigenfunctions of the considered systems, can only be found for a discrete set of eigenvalues. In experiments, vibrating plates or microwave cavities were driven by a harmonic source oscillating with the various resonance frequencies of the system. However, this restriction in the choice of driving frequency is only due to the desired analogy with quantum states. Besides the investigation of quantum mechanical eigenfunctions we think that it is worthwhile studying wave phenomena inside resonators of integrable and chaotic geometry in a more general context. For example, if some parts of the boundary are forced to oscillate by an external driving force, vibrations.of a "wavy" medium can arise for any frequency (though when the forcing frequency equals one of the eigenfrequencies, the response may be enhanced). The driven system develops forced standing wave patterns which are solutions of the inhomogeneous Helmholtz equation (V 2 + k 2 ) ~ ( x ) = --4zrp(xb). Here p(Xb) stands for the distribution of "sources", i.e., parts of the boundary where energy interchange between the system and some external driving force takes place. Experimental imaging of solutions of the homogeneous Helmholtz equation has been performed in various systems. Chladni published his nodal patterns of vibrating plates already in 1787 [2]. Vibrational modes of several musical instruments like violins or especially drums could be made visible with the help of holographic interferograms. In a recent experiment Bltimel et al. demonstrated standing wave patterns on water surfaces of round and stadium-shaped ripple tanks [ 3]. However, for quantitative measurements of the spatial distribution of wave fields inside resonators of chaotic billiard geometry up to now the most promising systems seem to be microwave resonance cavities [4-11]. The dimensions of the top and ground plates of the cavities are usually in the cm-regime with a spacing d in z direction of several mm. For frequencies below ~',,~x = c/2d, c = speed of light, these resonators support two-dimensional transverse magnetic modes which obey Dirichlet boundary conditions (Ez = 0). Using a bead perturbation
215
technique [4,5], Sridhar and Heller were able to observe the eigenfunctions of the Sinai billiard and found excellent agreement with their numerical calculations [6,7]. Stein and Sttckmann used a slightly different technique for the eigenfunctions of the stadium [9]. In this paper we show for the first time images of forced standing wave patterns, i.e. solutions of the inhomogeneous Helmholtz equation, in resonators of integrable and chaotic billiard shape. Our vibrating system is a Josephson tunnel junction coupled to an external microwave source acting as the driving force.
2. Experimental technique Irradiation of superconducting tunnel junctions with microwaves results in photon-assisted tunneling of quasiparticles [ 12,13] which can be observed as typical changes in the current-voltage characteristics (IVC) of the junction. Due to the strong impedance mismatch between the junction and the surrounding structures the incident microwave field is multiply reflected at the boundaries [ 14,15]. If the Josephson current (tunneling of Cooper pairs) is suppressed by a magnetic field, the tunnel junctions can be treated as parallel plate resonators and the propagation of electromagnetic waves is governed by the Helmholtz equation. However, compared with ordinary resonant cavities three basic differences must be taken into account: (1) The electromagnetic fields propagate inside the tunnel junction with the Swihart velocity ~ which is usually one or two orders of magnitude less than the vacuum speed of light [ 16]. This means a reduction in the length scale of the standing wave patterns. (2) Since reflection losses inside superconducting tunnel junctions are negligible the microwave is reflected at the open boundary with a reflectivity close to one and without a phase shift [ 17]. This implies that, in contrast to resonant cavities, standing wave patterns inside a Josephson tunnel junction have to satisfy Neumann conditions (VEz = 0 in the direction perpendicular to the boundary) at all parts of the boundary which do not act as sources. (3) The Q value of Josephson tunnel junctions
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C.A. Kruelle et al. / Physica D 78 (1994) 214-221 w = 125 ~ m
T>Tc !
~ffllllllllllllllllllilllllilA4
,~II I
high V~I ("belly") . . . . low V~I / ......... no V~I ( n o d a l ~
[
~. ~ ~ / ~
Fig. 1. Cross-sectionof a Nb/AlOx/Nb-tunneljunction with quarter stadium geometryconnected to a superconductingstripline for microwavecoupling (1: SiO2, 275 nm, 2: bib top electrode, 150 nm, 3: Nb base electrode, 150 nm, 4: SiO2, ~ 1.7 prn, 5: Nb2Os, 45 nm, 6: Nb microslripgronndplate, 150 nm, 7: silicon substrate). (typical value Q ~ 100) is much less than for resonant cavities. Therefore Josephson tunnel junctions are not well suited for the investigation of eigenmode distributions in these systems. The advantage of low Q resonators lies in the fact that the pattern formation of spatially varying electromagnetic fields is much less dominated by resonant eigenmodes. For this reason forced standing wave patterns oscillating at frequencies other than eigenfrequencies can be studied more easily than with high Q cavities. For our experiments we use a standard reflex klystron with fixed frequency of 70 GHz as microwave source. The Josephson tunnel junction is connected to a superconducting stripline for controlled microwave injection. Fig. 1 shows a schematic cross-section of a typical junction geometry. Details of the junction preparation technique can be found in Ref. [ 18]. The microwave field is coupled into the superconducting stripline via a micro-fabricated dipole antenna on the same chip. A small amount of the rf power is propagated through the small slit between the base electrode and the Pbln wiring layer to the right edge of the junction. We assume that the injected wavefront is approximately homogeneous over the whole width w = 125/zm of the coupling slit. Depending on the preparation parameters the microwave propagates inside the N b / A l O x / N b - j u n c t i o n with a normalized
~_..~.~. v -i'.V(:r, ~) Fig. 2. Local current-voltagecharacteristics for three regions of a microwaveirradiated superconductingtunnel junction with different local if-voltagelevels. The arrows indicate the Vrs-dependent electron-beaminduced transitions to the normal state inside areas locally heated by the beam. In our experimentsthe transition arrows are almost vertical. The -AV(x,y) signal level is in the /~V-regime, while the gap voltage 2~/e is typically a few mV. The slope of the arrows (here 1000× exaggeratedfor clarity) is determined by the ratio of the heated area ~-02 to the area of the whole junction. Swihart velocity -~/c between 0.029 and 0.033, the corresponding Swihart wavelength ~ is 125/~m and 140/tin, respectively. Our imaging technique for the microwave standing wave patterns uses the concept of low-temperature scanning electron microscopy (LTSEM) [ 19]. The Josephson tunnel junction is scanned with the electron beam of a scanning electron microscope, while the bottom of the substrate carrying the junction is in direct thermal contact with liquid helium. For a description of the experimental setup we refer to Ref. [20]. All experiments were performed at a He bath temperature of 1.9 K, sample temperature T ~ 6 K. The imaging method is based on detecting the local photon-assisted quasiparticle tunneling current [21,22]. For obtaining a two-dimensional voltage image, the top side of the current-biased junction is scanned with the electron beam, and the beam-induced change - A V ( x , y) of the junction voltage is recorded simultaneously as a function of the coordinate point (x, y) of the beam focus on the sample surface. Fig. 2 gives a scheme of the signal generation principle. Due to the local heating effect in an area ~rr/2 07 denotes the thermal healing length [23] ) around the beam
C.A. KrueUeet al. / Physica D 78 (1994) 214-221 focus the quasiparticle tunneling current for this perturbed area is increased by an amount 8leb(X,y). If the heated area is switched to the normal conducting state, the local current increment 81eb(X, y) is j USt the difference Inornm(X, y) --lqp(X,y), where lqp( X, y) denotes the quasiparticle tunneling current through the area ~r/2 without electron-beam irradiation. Assuming ,r~/2 small compared with the total tunneling area, the corresponding voltage change across the junction is given by [21]
- A V ( x , y ) = aV
.81eV(X,y)
(1)
vB and the structural information is provided by the spatial distribution of the local current change 81eb(x, y) in the perturbed area. In the presence of an inhomogeneous if-field Vrf(x,y) across the junction the photon-assisted quasiparticle tunneling current density jqp(X,y) is spatially modulated in the same way. Fig. 2 shows schematically the local current-voltage characteristics of three regions with different if-field amplitudes ("belly", intermediate field, and nodal line). For small incoming microwave fields Ez (i.e., the corresponding rf voltage across the junction V,f << 2Ale, A = energy gap of the superconducting layers) the if-induced local increase 31,f(x, y) of the tunneling current is proportional to the square of the local field Ez (x,y) across the junction [24]. From Fig. 2 we deduce that the electron-beam induced local current change 8Ieb ( X, y) is equal to 81max- Crf "E2z( X, y) ), where Crf denotes the conversion coefficient if field to photon-assisted quasiparticle tunneling current. Using Eq. ( 1) we finally obtain for the voltage signal
AV(x,y): aV(x, y) = C rf" a~l vs.E2(x,y) Y gain aV
•
~Imax
(2)
Hence, for obtaining a voltage image of the if field distribution Ez2 (x, y) the response signal AV(x,y) of
217
the whole junction is recorded as a function of the coordinate point (x, y) of the beam focus on the sample surface.
3. Results and discussion
3.1. lntegrable billiards The eigenfunctions of a rectangular resonator (dimensions Ix, ly) are well known. Depending on whether Dirichlet or Neumann boundary conditions must be satisfied, Ez cx sin(m,rx/lx) sin(n~'y/ly) or Ez cx cos(m*rx/lx) cos(n~ry/ly), m,n = O, 1,2 . . . . is expected, respectively. With the help of the bead perturbation technique Sridhar mapped the (9,2)Dirichlet mode in a resonant cavity [6]. Using LTSEM, Mayer et al. [25] and Lacbenmann et al. [26] were able to image the magnetic field distribution H~(x,y) for several (m,n)-Neumann modes (Fiske modes [27] ) in Josephson tunnel junctions. Considering forced standing wave patterns in ifdriven Josephson tunnel junctions the situation is more complex. There exists no system of separate "modes". Rather, the standing wave field is not only governed by the shape of the resonator but also depends in a sensitive, "continous" way on the distribution of sources and the external driving frequency w. Before solving this boundary problem in general, we consider the elementary problem of one single point source oscillating with frequency w at the center (0,/y/2) of one side of the rectangle. The outgoing cylindrical wave, the solution of the Helmholtz equation for a point source in two dimensions, is the Hankel function HCol)(kr), where k = to/~ + lot, a = damping constant, r = distance to the point source. For describing the multiple reflections of this outgoing wave at the boundaries the method of images [28], which is known for solving the problem of induced charges in electrostatics, can be applied. Multiple imaging of the original point source at the four sides of the rectangle generates an infinite rectangular lattice of point sources at positions (2mlx, (n + 1 ) ly), m , n = 0, 4-1,4-2 . . . . The superposition
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C.A. Kruelle et al. / Physica D 78 (1994) 214-221 oo
Ez(X,y)=Ez,o" rrnn =
[ (X
--
Y~
Y ~ H(ol)(krmn),
2mlx) 2 + (y - (n
+
t~
l)/y)2] 1/2 (3)
of all cylindrical waves starting at the lattice points results in a standing wave pattern inside the rectangle, which satisfies Neumann conditions at all parts of the boundary except the position of the point source. Each image source in the virtual lattice corresponds to a ray which starts at the original source and reaches the considered point (x, y) inside the resonator after multiple reflections at the boundary. For solving the more general case when the incident microwave field enters the rectangular resonator through one or more slits of finite extension, we have to integrate over the distribution of sources p ( x b ) at the boundary:
E z ( x ) = Ez,o" ~ p(xb) m=- ~ n=- C~oundary × [H
(4)
where x,,m = (2mlx, 2nly), m, n = 0, +1, -t-2. . . . This expansion is useful when only the direct source and perhaps the first few reflections are important. This is the case for a damped wave with an attenuation length 1 / a comparable with the resonator dimensions. As an example, Fig. 3 shows the forced standing wave pattern generated by a microwave field entering the Josephson tunnel junction through a slit from the left. For the numerical solution using Eq. (4) and shown in Fig. 3a three approximations had to be taken into account: (1) We assume a homogeneous distribution of sources inside the coupling slit. The integral can be converted into a sum of point sources if the separation between them is chosen much less than the wavelength. (2) Instead of evaluating the Hankel function H
(a)
Fig. 3. Forced standing wave field inside a Josephsontunnel junction with rectangular geometry.The microwaveenters through a slit at the left, junction dimensions= 500 x 425/zm ~. (a) Numerical calculation of E2z(x, y) according to Eq. (2). The Swihart wavelength ~ = 140 ~ra and the damping constant a = 1/500 /~rn-l are taken from the experiment. (b) Experimental voltage image AV(x, y). T ~ 6 K. V~rrCOS(krq-4)
e -at
for distances r larger than the resonator dimensions. (3) Due to the damping the higher terms ( Iml, Int > 3) of the doubly infinite series do not influence the appearance of the standing wave pattern substantially but form a small background correction and are therefore neglected. With these three approximations the numerical effort of evaluating Eq. (4) for achieving sufficiently satisfying results is rather small. Two parameters must be taken from the experiment, the Swihart wavelength and the damping constant a. The shape of the standing wave pattern is more sensitive to the microwave wavelength ~, which governs the lateral distribution of maxima and nodal lines, while the damping constant a basically determines the relative height of the maxima. For comparison with this theoretical calculation Fig.
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C.A. Kruelle et al. / Physica D 78 (1994) 214-221
130 ~m
rf---Y,~ rf---~
l 500 ~m
Fig. 4. Voltage image AV(x, y) of the forced standing wave pattern inside a rectangular Josephson tunnel junction coupled to two slits for controlled microwave injection, Bright areas indicate regions of high ff field ampH_tude. Junction dimensions = 500 x 400/zm 2, Swihart wavelength A = 140/~m. T ~-, 6 K.
3b shows the corresponding experimental image. The spatial resolution of the LTSEM technique is typically a few /zm. This is at least one order of magnitude less than the intrinsic length scale A/2 of the standing wave patterns. For best results the scanning grid (here 640 × 512) is chosen in such a way that the grid step size is less than the physical resolution. The data acquisition for such an LTSEM image takes about 8 minutes. The good agreement between the numerical simulation and the experimental image can be taken as proof of validity of the theoretical approach as well as the experimental technique. A second example for an rf driven standing wave pattern in a resonator of rectangular shape is shown in Fig. 4. Here the microwave field enters the tunnel junction through two equal slits at the left side. The modulation of the microwave field in one direction at the top and bottom edges of the rectangle results from the interference of wave trains originating from each slit and bouncing back and forth along the edges of the junction. This feature has the character of the so-called "bouncing-ball" state first described by McDonald and Kaufman [29] for the stadium billiard. Fig. 5 shows one more example for a forced microwave standing wave pattern in a Josephson tunnel junction of integrable billiard shape. The quarter circle is just a symmetry-reduced version of a circular membrane whose vibrational modes are well known
b:
125 #m - - - ' ~
Fig. 5. Voltage image AV(x, y) of the forced standing wave pattern inside a Josephson tunnel junction of quarter circle shape, the radius r is equal to the Swihart wavelength A = 125/zm. T ~ 6 K. Bright areas indicate regions of high ff field amplitude.
from drum acoustics [ 30]. For homogeneous Dirichlet boundary conditions the system of eigenfunctions is a subset of the eigenfunctions for the full circle with nodal lines at the two straight edges of the quarter circle. However, this symmetry of the boundary conditions is broken, if the resonator is coupled to an external driving force acting upon the boundary. In Fig. 5 the microwave field enters the tunnel junction as a plane wave from the left side. At the bottom and the round edge Neumann boundary conditions must be satisfied. The superposition of all multiply reflected wave trains results in a standing wave pattern which cannot be associated with one of the eigenmodes of a circular membrane. Since here the radius of the quarter circle is chosen equal to the Swihart wavelength --- 125 g m we observe a rather simple standing wave pattern. From the relative height of the two maxima we can conclude that the microwave field inside the tunnel junction is damped with an attenuation length 1/a of about several hundred/zm.
3.2. Chaotic billiards For the theoretical calculation of eigenfunctions in billiards with chaotic geometry several numerical techniques like the Green's function method of McDonald and Kaufman [29,31 ] or Heller's superposition of waves with random direction [32] have been developed with great success. Up to now, these methods have not been applied to the case of forced standing wave patterns in externally driven systems. This seems
220
C.A. Kruelle et al. / Physica D 78 (1994) 214-221 l =
~<~"-" rf
~---r = 125/~rn--~ Fig. 6. Voltage image AV(x,y) of the forced standing wave patterninsidea Josephson tunneljunctionof quarterstadium shape (l = r = 125 /an). The microwave (A ---125/zm) enters from the right.T ~ 6 K. Bright areas indicateregions of high rf field amplitude. The lower contrast in the square part of the tunnel junction is caused by the Pbln wiring layer partiallycovering the Nb top electrodeas can be seen in Fig. I.
to be a promising task for the future. The method of images described above does not work well in this case since it is restricted to boundaries which are composed of straight lines. Of course this does not mean that the method of images cannot be applied to resonators of chaotic billiard shape at all, but that it can be applied only as an approximation. Due to these theoretical difficulties the experimental approach becomes important. Preparation techniques for Josephson tunnel junctions are highly developed and any shape of the tunneling area and rf injection geometry can be fabricated with great accuracy. Here we report on experiments for two basic resonator geometries: the quarter stadium and the"elbow" billiard. (1) Quarter stadium: The system most commonly studied in this context is the Bunimovich stadium billiard consisting of two semicircles separated by a square. Since most authors are interested in eigenfunctions which underly certain symmetry conditions, they usually consider one quarter of the full stadium. These eigenfunctions obey Dirichlet conditions at all parts of the boundary. The corresponding eigenfunctions of the full stadium show nodal lines at the two symmetry axes. For our experiment we have chosen a quarter stadium (Fig. 6) based on the same dimensions and a similar coupling geometry for the rf injection as in the quarter circle case. The microwave field enters from the right side. At this edge periodic boundary conditions with an oscillating frequency f = 70 GHz must be satisfied whereas Neumann conditions hold for all
~[oIOlr.mn Fig. 7, Voltage image AV ( x, y ) of the forced standing wave pattern inside a Josephson tunnel junction of elbow shape (1 = 400/,~m). The width of the coupling slit located on the left side is w = 40 /tin, Swihart wavelength ~ = 125 /zm. T m 6 K. Bright areas indicate regions of high ff field amplitude. The lower contrast in
the L-shaped area on the right and bottom edges of the tunnel junction is caused by the Pbln wiring layer partially coveringthe Nb top electrode. other parts of the boundary. Fig. 6 shows the resulting standing wave pattern. In the square part of the stadium the incident plane wave is only slightly disturbed while the standing wave exhibits an irregular pattern inside the quarter circle part. Due to the small junction dimensions compared with the microwave wavelength we cannot expect to see any "marring" of the wave pattern. (2) Elbow billiard: As a final example Fig. 7 shows the rf induced standing wave pattern inside a resonator of "elbow" shape first studied by Doron, Smilansky, and Frenkel [ 11 ]. The elbow is again a symmetryreduced version of the well known geometry of the Sinai billiard which in its original form consists of a circular disk in the center of a square. The Sinai billiard can be considered as the elementary cell for the scattering problem of a particle in a square lattice of circular targets. In Figure 7 the microwave field enters through the small straight edge on the left side. Since the tunnel junction dimensions are large compared to the Swihart wavelength A we observe a whole chain of maxima all evenly spaced with a separation of A/2. In summary, these experiments show that depending on the boundary conditions the propagation of waves inside Josephson tunnel junctions of various
C A . Kruelle et al. / Physica D 78 (1994) 214-221
geometries can develop a much richer variety of standing wave patterns than just a system of eigenmodes which are usually associated with quantum mechanical states. For inhomogeneous boundary conditions with periodically oscillating parts of the boundary the Helmholtz equation has solutions for any external driving frequency regardless of the specific resonator shape. Since theoretical solutions of these boundary problems for non-integrable geometries are not yet well established, our experiments can serve as test objects for numerical algorithms to be developed in the future.
Acknowledgements We thank A. Kittel, S. Lachenmann, and R. Gross for stimulating discussions and critical comments. Financial support of this work by the Deutsche Forschungsgemeinschaft and the Studienstiftung des deutschen Volkes is gratefully acknowledged.
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