Quantum behaviors in high-TC systems: Macroscopic and vortex quantum tunneling

Physica C 437–438 (2006) 303–308 www.elsevier.com/locate/physc

Quantum behaviors in high-TC systems: Macroscopic and vortex quantum tunneling F. Tafuri a,*, F. Lombardi b, T. Bauch b, D. Stornaiuolo c, D. Born c, D. Dalena a, A. Barone c, G. Rotoli d, P.G. Medaglia e, P. Orgiani e, G. Balestrino e, V. Kogan f, J.R. Kirtley g b

a Coherentia-Dip. Ingegneria Informazione, Seconda Universita` di Napoli, Via Roma 29, Aversa (CE) 81031, Italy Department of Microelectronics and Nanoscience, MINA, Chalmers University of Technology and Goteborg University, S-41296 Goteborg, Sweden c Coherentia, Dipartimento di Scienze Fisiche, Universita` di Napoli ‘‘Federico II’’, P.le Tecchio 80, Napoli 80125, Italy d Dipartimento di Energetica, Universita` of L’Aquila, Localita` Monteluco, L’Aquila, Italy e Coherentia, Dip. Ingegneria Meccanica, Universita` di Roma Tor Vergata, Roma, Italy f Ames Laboratory—DOE and Department of Physics and Astronomy, Iowa State University, Ames IA 50011-3160, United States g IBM Watson Research Center, Route 134 Yorktown Heights, NY, USA

Available online 14 February 2006

Abstract We describe the main ideas and some experimental outcomes of two experiments, both aimed to study quantum tunneling effects in high-TC structures. In the first experiment, macroscopic quantum tunneling is demonstrated in grain boundary YBaCuO Josephson junctions revealing dissipation levels lower than expected and opening novel perspectives for quantum circuitry. In the second, the low temperature dissipation is dominated by quantum tunneling of individual Pearl vortices in ultra-thin CaBaCuO systems characterized by extremely long Pearl lengths.  2006 Elsevier B.V. All rights reserved. Keywords: Josephson effect; Macroscopic quantum tunneling; Dissipation; Pearl vortices

1. Introduction The nature of superconductivity in oxide compounds is very intriguing and still an unsolved problem. The phenomenology of high critical temperature superconductors (HTS) encompasses a wide range of interesting issues at the border of our understanding of solid-state systems and at the limit of material science and nano-technology current capabilities. The complexity of these materials, the extreme values of their characteristic lengths and energies, the possibility to tune their order and their properties through the oxygen content are crucial ingredients for a composite and complicated phenomenology [1].

*

Corresponding author. Tel.: +39 081 7682584; fax: +39 081 2391821. E-mail address: [email protected] (F. Tafuri).

0921-4534/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2005.12.072

Relevant insights can be obtained by investigating quantum and macroscopic quantum effects in appropriate systems [2]. These frontiers have been recently opened by a series of improvements in the preparation of junctions and thin films, and an achieved maturity in the understanding of several properties of HTS systems [3–9]. Josephson junctions represent privileged systems to study quantum processes. In HTS a d-wave order parameter symmetry (OPS) [3] gives additional elements of interest related, for instance, to dissipation mechanisms due to mid-gap states, nodal quasi-particles, thus touching key problems related to coherence. Vortex matter also inspires another experiment aimed to investigate quantum tunneling of vortices (VQT), that we will describe below. Macroscopic quantum tunneling (MQT) and energy level quantization (ELQ) represent the first steps towards a quantum-state engineering, which has potential application as

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logic devices and qubits, more easily embedded in electronic circuits and scaled up to large registers. ‘‘Qubit’’ proposals involving high-TC superconductors [10–14] mostly exploit Josephson junction circuits with an energy-phase relation with two minima in the absence of an external magnetic field. This means that there is no need to apply a constant magnetic bias, unlike in systems based on low temperature superconductors. Naturally degenerate states and violation of the time reversal symmetry become the keywords and concepts of all qubit proposals. On the other hand, it is commonly believed that the HTS quiet qubit designs belong to a higher complexity class than LTS charge and flux qubits, and their experimental realization may remain a challenge for some time [12]. This is also partly due to the fact that in a system with d-wave like order parameter symmetry and, therefore, in the presence of low energy quasi-particles, dissipation could prevent the occurrence of macroscopic quantum phenomena, the key element for qubits. The recent observation of MQT and ELQ may represent the start to study novel sophisticated quantum issues through HTS junctions and to really exploit the intrinsic bistable properties of HTS for quantum circuitry. Vortex quantum tunneling is a complementary example of how the ‘‘almost unique’’ properties of HTS allow the investigation of novel physical problems. 2. Towards ‘‘quantum’’ junctions The most recent results have to be considered as the product of several efforts of the scientific community and of an achieved ‘‘quantum awareness’’, which had two important developments, the demonstration of d-wave OPS, and the proposals for qubits which took advantage of this symmetry. The 45 grain boundary (GB) and s-wave superconductor–normal metal–d-wave superconductor (S–N–D) junctions offer the most favorable configurations for a naturally degenerate ground state [10–14]. In the S– N–D structure the d-wave electrode is misoriented by 45 to exploit d-wave OPS features. The whole body of work on these junctions gave remarkable encouraging results [3,4,6]. For instance, current vs. the superconducting phase (IC–u) measurements on high angle bicrystal grain boundary junctions, especially in 45 asymmetric and 45 symmetric configurations, have reported the prevalence of the 2u component under opportune conditions [15–17]. In one of these experiments [16], the 2u component results in a clear deviation of the magnetic pattern, which presents regular deviations from the expected cosine dependence. In this experiment, the possibility of using size effects in sub-micron junctions has been exploited to freeze out low energy quasi-particles, and to induce a prevailing 2u component in the IC–u dependence [16]. A SQUID with two grain boundary junctions, chosen to have a doubly degenerate state, forming a mesoscopic island may potentially represent a silent phase qubit whose operating point is stable and protected from external field fluctuations by its symmetry [18,19]. The read-out SQUID is pro-

posed to be in the same high-TC film as the qubit itself. The estimates of the decoherence due to fluctuations of the external flux show that an experimental observation of coherent quantum tunnelling and Rabi oscillations in the system is feasible [19]. Contributions to dissipation due to different transport processes, such as channels due to nodal quasi-particles, mid-gap states, or their combination, have been identified and distinguished [20]. Decoherence mechanisms can be reduced by selecting appropriate tunneling directions because of the strong phase dependence of the quasi-particle conductance in a d-wave GB junction [21]. However, other possible subtle mechanisms could contribute to freeze dissipation [8] and maintain coherence. These concepts will probably inspire possible lines of developments, which will follow the most recent observations of MQT and ELQ. These will be mostly directed to measure Rabi oscillations. These encouraging results have to be always counterbalanced by caution, which is not only due to the wellknown general questions concerning any possible use of any solid-state device for quantum computation, their protocols, and the nature of superconductivity of HTS, but also due to several technical problems. 3. YBCO biepitaxial junctions and macroscopic quantum tunneling In this section, we briefly describe biepitaxial Josephson junctions and the main ideas and outcomes of the MQT experiment. The biepitaxial Josephson junctions chosen for the MQT experiment were the result of efforts directed to produce tunnel-like junctions. In the first version, YBCO would grow along the c-axis (1 0 0) on the seed of MgO (1 1 0) oriented and along the (1 0 3) direction on the bare (1 1 0) SrTiO3 substrate. In the second version CeO2 replaces MgO as a seed layer. The presence of the CeO2 produces an additional 45 in-plane rotation of YBCO axes with respect to the in-plane directions of the substrate. The inset of Fig. 1 shows the OPS configuration for this type of configuration in the tilt limit (see below).

Fig. 1. Curves refer to CeO2-based biepitaxial junctions in the tilt (open gray circles) and twist (full black squares) limits, respectively. The arrows indicate the hysteretic behavior. In the inset the OPS configuration is shown for the tilt limit used for the MQT experiment.

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1/2

1/2

2 1/4

(2pIC0/U0C) (1c ) is the plasma where xP(a) = c(a) frequency, pffiffiffi at is the3=2 thermal prefactor and DU ðaÞ ¼ cðaÞðEJ 4 2=3Þð1  cÞ is the p barrier height for c close to ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1=2 one (with cðaÞ ¼ ½ð0:5 þ 3=2 1 þ 32a2 Þ=2 and EJ = IC0U0/2p) [24]. The escape rate will be dominated by MQT at low enough temperature [2]: for Q > 1 and c close to one it is approximated by the expression for a cubic potential [2]    xP ðaÞ DU ðaÞ 0:87 exp 7:2 Cq ðaÞ ¼ aq 1þ ; ð2Þ 2p hxP ðaÞ  Q

where aq = (864pDU/hxP)1/2 and Q = xP RC is the quality factor (R is the resistance of the junction). For a fixed critical current IC0, the second harmonic (compared to the case of a pure sin u) reduces the barrier height DU and increases the plasma frequency xP at bias currents c close to one. Consequently, the escape rates in the thermal and quantum regimes will be larger for a > 0 than for a = 0, as can be seen from Eqs. (1) and (2). The measurements were performed in a dilution refrigerator with base temperature of 20 mK. The electrical lines to the sample have two stages of filtering: an RCL filter with a cut-off frequency of 100 MHz at the 1 K-pot, and a combination of thermocoax plus copper powder filter with cut-off frequency of 1 GHz at the mixing chamber. We record the switching current probability distribution P(I), which can be directly related to the escape rate C [25], using a technique similar to that described in [22]. The switching current probability distributions have been measured as a function of temperature. The dependence of the distribution width r on temperature is reported in Fig. 2. The measured r saturates below 50 mK, indicating a crossover from the thermal to the MQT regime. To rule out the possibility that the saturation of r is due to any spurious noise or heating in the measurement setup, the switching current probability distributions were measured for a reduced critical current IC0 = 0.78 l A by applying an external magnetic field B = 2 mT. The data in the presence of a magnetic field clearly show a smaller width r, which does not saturate down to the base temperature [8]. The extracted C = 2 ± 1 pF value gives a plasma frequency xP/2p = 7.8 ± 0.5 GHz and a quality factor Q of about 5, in the quantum regime. The crossover temperature from the thermal to the MQT regime is consistent with this value of Q. The extracted value for the RC product of 80 ps is a lower bound, as the damping is mainly due to the high frequency impedance in parallel with the junction. The ratio between the second and first harmonics ranges from

IC = 1.40 μA IC = 0.78 μA

60 50 σ (nA)

The tilt of one of the electrodes and the in-plane 45 rotation on the CeO2 are the main ingredients to achieve tunnel-like barriers. They both contribute to decrease the barrier transmission and the in-plane rotation enhances the desired d-wave features. A remarkable achievement has been to realize junctions whose critical current IC depends on the interface orientation h in complete agreement with the predictions of a d-wave OPS [9]. This means that d-wave effects are dominant for some types of junctions, and robust in the sense that interface microstructure cannot mask them. Finally, this implies that we can select the junction for the MQT experiment knowing the OPS configuration exactly. Since we are interested in those features that are distinct from the case of low TC superconductor (LTS) junctions, namely effects due to OPS, second harmonic component, and dissipation due to low energy quasi-particles [8], we select the junction in the tilt configuration (angle h = 0). This configuration (lobe to node) maximizes d-wave induced effects [8,9]. The measurement strategy was the same used for LTS JJs [22]. Relatively low dissipation, as evidenced by the I– V curve, with 90% hysteretic behavior (see Fig. 1) is a necessary prerequisite. We define the strength of the second harmonic components in the current-phase relation (CPR) by the parameter a = I2/I1, where I2 and I1 are the second and first harmonic components in the CPR, respectively. This results in I = I1(sin u  a sin 2u). For zero bias and a > 0.5, the potential has the shape of a double well. For a < 0.5, the potential is single welled. The analysis is restricted to a < 2; in the case 0.5 < a < 2 the phase will always escape into the running state from the lower lying well of the tilted ‘‘double-welled’’ washboard potential [8]. When the bias current c = I/IC0 is ramped from 0 to c < 1, the junction is in the zero voltage state in absence of thermal and quantum fluctuations. At finite temperature, the junction may switch into a finite voltage state for a bias current c < 1. This corresponds to the particle escaping from the well either by a thermally activated process or by tunneling through the barrier potential (MQT). In the pure thermal regime, the escape rate for weak to moderate damping is determined by [23]   xP ðaÞ DU ðaÞ Ct ðaÞ ¼ at exp  ; ð1Þ 2p kBT

305

40 14 13 12 11 10 9

30 20 10 0

200

T

*

20 40 60

400 600 Tbath (mK)

800

Fig. 2. Temperature dependence of width r of the switching current probability distribution for IC0 = 1.40 lA and B = 0 T (open points) and for IC0 = 0.78 lA and B = 2 mT (full points). The inset shows r in the temperature range below 75 mK.

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a  0 at T = 900 mK to a saturated value a = 0.77 ± 0.06 below T = 100 mK [8]. Therefore, at low temperatures the fundamental state is naturally double degenerate. The low average barrier transparency (D  104) strongly reduces dissipation from nodal quasi-particles. This is one of the main arguments to understand why dissipation mechanisms related to a d-wave order parameter in the junction do not prevent the observation of MQT [8]. The low dissipation argues against the common belief that the presence of low energy quasi-particles in HTC systems can prevent the macroscopic quantum behavior required for solid-state quantum computers. Macroscopic energy quantization in the presence of radio-frequency radiation has been recently demonstrated [34]. We also expect novel insights from the study on Nb– Au–YBCO (S–N–D) ramp-junctions, which have already given remarkable results to generate and manipulate halfflux quanta [7], but are suitable for further progress in the study of quantum effects [26]. The ideas coming from the experiments on the biepitaxial junctions could also stimulate other experiments on bicrystal junctions. The latter, even if not suitable for complicated circuit designs because of the constraints imposed by the bicrystal line, still offers junctions of great quality. 4. CaBaCuO ultra-thin films and junctions Apart from the progress on bicrystal junctions mentioned above, bicrystal junctions have been recently realized also by using ultra-thin films, i.e. artificially layered HTS [27], such as [Ba0.9Nd0.1CuO2+x]m/[CaCuO2]n(CBCO-m · n). These films (similarly to all existing HTS cuprates) are composed by a stacking sequence of two structural subunits having different functions, namely the charge reservoir (CR) block and the infinite layer (IL) superconducting block (see Fig. 3a). The IL block always consists of CuO2 planes separated by an alkaline earth (mostly Ca) plane, while the structure and the chemical composition of the CR block vary from compound to compound. The structural and transport properties of these compounds have been discussed elsewhere [27]. We have for the first time realized Josephson junctions composed of only a few superconducting CuO2 planes (6 layers, since we have focused in particular on ultrathin [Ba0.9Nd0.1CuO2+x]5/[CaCuO2]2/[Ba0.9Nd0.1CuO2+x]5/ [CaCuO2]2/[Ba0.9Nd0.1CuO2+x]5 (5/2/5/2/5) structures). The junction schematic is shown in Fig. 3b, where the six superconducting CuO2 planes are shown. We have measured both the asymmetric and symmetric configurations (total misorientation angle 24). The CBCO film is only 8 nm thick. The simple structure of the GB composed of 6 superconducting CuO2 layers (due to the highly controlled structure of the 5/2/5/2/5 artificial structure) also allows a reliable estimation of the coupling along the a–b planes of two CuO2 layers separated by a 24 asymmetric or symmetric GB: in particular we calculate a critical current density per plane of about 0.20.3 · 102 A/cm2.

Fig. 3. (a) The typical stacking sequence of the charge reservoir (CR) block and the infinite layer (IL) superconducting block characteristic of HTS compounds and of the CaBaCuO compound; (b) the grain boundary schematic: in the 5/2/5/2/5 configuration only six superconducting CuO2 planes are present on each electrode (24 and asymmetric) and (c) scheme of the 5/2/5 CaBaCuO structure used in the vortex quantum tunneling experiment.

Josephson coupling has also been confirmed by the presence of Josephson vortices imaged through Scanning SQUID Microscopy (SSM). 5. Vortex quantum tunneling and CaBaCuO ultra-thin films Among other unusual properties, CaBaCuO thin films have extremely large Pearl lengths [28,29] (see Fig. 4). We have used this property to perform a novel experiment on quantum tunneling of vortices. The idea of vortex quantum tunneling in films was first considered by Glazman and Fogel (G–F) [30]. This idea was studied extensively in theory and experiment in relation to finite flux-creep rates in

Fig. 4. Pearl vortices (in the middle) imaged through SSM in CaBaCuO ultra-thin films, are compared with conventional Abrikosov vortices (on the left). Each image is 100 lm wide. The Abrikosov vortices couple a peak flux of 0.42U0 into the SQUID pickup loop. The left and middle images have the same vertical scale. The Pearl vortices in the right image are amplified by a factor of ten for better clarity.

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hard type-II superconductors for T ! 0. For thermally activated creep, the rates should go to zero in this limit [31,32]. Within creep models, the barriers through which the vortices tunnel are due to material disorder and are treated statistically. Narrow current-carrying thin-film bridges offer a unique opportunity to study both thermal activation and quantum tunneling through well-defined potential barriers that are tunable by current and temperature. We take advantage here of new advances in thin film technologies, which make possible ultra-thin and stable films, and of the lowering of the barrier due to the extremely large Pearl lengths K [33] (vortex energy e / U20 =K, where K = 2k2/d, k is the bulk penetration depth and d is the film thickness respectively). Within the London approach, the barrier shape is well known for bridges narrow relative to K. Details on e and its spatial dependence can be found in [33]. To realize these conditions, we use ultra-thin high-TC films with Pearl lengths K  100 lm. The critical temperature of these films TC  40 K provides a relatively easy T-domain to work in, and the data reported are quite robust. When current I flows through the bridge at high temperatures, vortices are thermally activated at the strip edges and pushed in, causing dissipation and a non-zero voltage V. These processes should lead to a power-law dependence V / In with the exponent n determined by the film parameter K and temperature [31,33]. n¼

U20 2 8p KðT Þk

BT

þ 1;

307

Fig. 5. (a) SSM image with a 104 Hz, 100 lA r.m.s. a.c. current passing through the bridge at T = 20 K; (b) cross-section through the data along the dashed line in (a); the solid line is calculated for uniform current through the bridge, with z = 5 lm.

ð3Þ

where U0 is the flux quantum and kB is the Boltzmann constant. This is confirmed by the high-temperature IVs and by the independently measured Pearl length. However for low temperatures, the IVs show a slower current dependence than required by thermal activation. Instead, a quantum tunneling model utilizing the ideas of [2,30], along with our knowledge of the barrier shape, provides a good representation of the data. Here, we present measurements from a ‘‘wide’’ bridge approximately 85 lm across. After measurement, this bridge was photolithographically narrowed by a factor of 2 and then remeasured for the reasons that will be clear below. Fits to scanning susceptibility measurements of a 5/2/5 CBCO microbridge (85 lm wide) gave K = 200 ± 20 lm (see [29] for details of these procedures). The length K is therefore longer than the bridge width W, and we expect the current through the bridge to be uniform, as confirmed by the SSM image and fit of Fig. 5. All I–V curves are nearly straight on a log–log scale, i.e., V / In at high currents. The exponent n extracted from power–law fits to the data is shown in Fig. 6, for both the wide and narrow bridges. The solid line is n(T) of Eq. (3). Fitting the high-T part of the data for the wide bridge we obtain TC  38 K and K(0)  320 lm in agreement with K(0)  200–400 lm independently measured by SSM. Thus, the thermal activation model works well above 15 K, where it gives correct

Fig. 6. The exponent n extracted from V / In for the wide (85 lm) bridge (circles) and the narrow (42 lm) bridge (triangles). The solid line is n(T) of Eq. (3) with the fit parameters TC = 37.6 K and K(0) = 316.8 lm. The two-fluid K = 2k2/d(1  (T/TC)4) has been used for simplicity.

values of the exponent n(T). At low temperatures, the thermal activation model fails. Similar results were found on samples of typically the same width and extremely large Pearl lengths, but of slightly different configurations (for instance CBCO 5/2/5/2/5). The fit values for n(T) in Fig. 6 do not diverge as thermal activation requires. Hence, we turn to the possibility of quantum tunneling. In the G–F theory [30], the signature of vortex quantum tunneling is a temperature independence of the shape of the I–V curves at low temperatures. This temperature independence has been found and is widely discussed in [33]. Further, vortex quantum tunneling should depend sensitively on the width of the tunneling barrier. We have also verified a particular dependence of the low temperature limit of the log V vs. I: lg

V gW 2 U2 /  2 20 V0 hI K

ð4Þ

on the width of the microbridge W and the Pearl length, and demonstrate that this dependence is roughly obeyed when the bridges are lithographically narrowed [33] (in Eq. (4), g is the drag coefficient).

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6. Conclusions The MQT experiment argues against the common belief, that in HTC systems the presence of low energy quasi-particles can prevent any true quantum behavior required for solid-state quantum computers. This finding may open up the way to experiments aimed at demonstrating coherence in systems taking advantage of the ‘‘quiet’’ configuration offered by the d-wave symmetry. In the vortex quantum tunneling experiment, the low temperature data are consistent with the overdamped quantum tunneling of individual Pearl vortices, the size of which in our films is macroscopic since K  0.1 lm. This conclusion is based on a special form of the low-T IVs, for which both current and width dependences are consistent with the data. Acknowledgments This work has been partially supported by the ESF projects ‘‘Pi-Shift’’, ‘‘QUACS’’, by ‘‘STINT IG 2004–2075’’ and by MIUR under the project ‘‘Quantum effects in Nano-structures and Superconducting devices’’. References [1] P.W. Anderson, Sources of Quantum Protection in High-Tc Superconductivity, Science 288 (2000) 480; D.J. Scalapino, Phys. Rep. 250 (1995) 329; J.F. Annett, N.D. Goldenfeld, A.J. Leggett, in: D.M. Ginsberg (Ed.), Physical Properties of High-Temperature Superconductors V, World Scientific, Singapore, 1996, p. 376. [2] A.J. Leggett, J. Phys. (Paris) Colloq. 39 (1980) C6-1264; A.O. Caldeira, A.J. Leggett, Ann. Phys. (NY) 149 (1983) 374. [3] C.C. Tsuei, J.R. Kirtley, Rev. Mod. Phys. 72 (2000) 969. [4] H. Hilgenkamp, J. Mannhart, Rev. Mod. Phys. 74 (2002) 485. [5] M. Sigrist, Progr. Theor. Phys. 99 (1998) 899; T. Lofwander, V. Shumeiko, G. Wendin, Supercond. Sci. Tech. 14 (2001) R53; A.A. Golubov, M.Yu. Kupryanov, E. Ilichev, Rev. Mod. Phys. 76 (2004) 411; S. Kashiwaya, Y. Tanaka, Rep. Prog. Phys. 63 (2000) 1641. [6] F. Tafuri, J.R. Kirtley, Rep. Prog. Phys. (2005). [7] H. Hilgenkamp, Ariando, H.J.H. Smilde, D.H.A. Blank, G. Rijnders, H. Rogalla, J.R. Kirtley, C.C. Tsuei, Nature 422 (2003) 50.

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