Observation of Macroscopic Quantum Tunneling in a Single Bi

PRL 99, 037002 (2007)

PHYSICAL REVIEW LETTERS

week ending 20 JULY 2007

Observation of Macroscopic Quantum Tunneling in a Single Bi2 Sr2 CaCu2 O8 Surface Intrinsic Josephson Junction Shao-Xiong Li,1 Wei Qiu,1 Siyuan Han,1 Y. F. Wei,2 X. B. Zhu,2 C. Z. Gu,2 S. P. Zhao,2 and H. B. Wang3 1

2

Department of Physics and Astronomy, University of Kansas, Lawrence, Kansas 66045, USA Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China 3 National Institute for Materials Science (NIMS), Tsukuba 305-0047, Japan (Received 28 November 2006; published 16 July 2007) We report on the first unambiguous observation of macroscopic quantum tunneling (MQT) in a single submicron Bi2 Sr2 CaCu2 O8 surface intrinsic Josephson junction (IJJ) by measuring its temperaturedependent switching current distribution. All relevant junction parameters were determined in situ in the classical regime and were used to predict the behavior of the IJJ in the quantum regime via MQT theory. Experimental results agree quantitatively with the theoretical predictions, thus confirming the MQT picture. Furthermore, the data also indicate that the surface IJJ, where the current flows along the c axis of the crystal, has the conventional sin’ current-phase relationship. DOI: 10.1103/PhysRevLett.99.037002

PACS numbers: 74.72.Hs, 73.23.b, 73.40.Gk, 85.25.Cp

Coherent quantum dynamics have been demonstrated in a variety of Josephson qubits [1], thus making them strong candidates for implementing scalable quantum computing architectures. However, decoherence resulting from coupling macroscopic quantum variables to localized material defects is perhaps the greatest obstacle for Josephson qubits. Recently, it has been shown that defects in amorphous AlOx tunnel barriers in Al and Nb junctions act as microscopic two-level fluctuators (TLFs) which cause significant decoherence [2] while epitaxially grown Al2 O3 tunnel barriers display significantly longer coherence times on account of the lower density of TLFs [3]. Although certain types of superconducting qubits are less susceptible to decoherence caused by TLFs than others [4], eliminating defects in tunnel barriers will inevitably increase quantum coherence in all forms of superconducting qubits. For this reason we are motivated to study materials that inherently form clean Josephson junctions. An intrinsic Josephson junction (IJJ) is formed between two vicinal superconducting CuO2 double layers of a single crystal high-Tc superconductor such as Bi2 Sr2 CaCu2 O8 (Bi-2212). The tunnel barrier between layers is an integral part of the single crystal and thus is smooth on an atomic scale with extremely low defect density. This unique property makes IJJs promising for realizing superconducting qubits that possess much longer coherence times. However, before using IJJs as elements in qubits their quantum nature must be demonstrated. Macroscopic quantum tunneling (MQT) [5] is a robust quantum phenomenon and its observation in IJJs would provide solid evidence of their quantum nature. Previous experiments have revealed a number of practical issues that have complicated attempts to observe MQT in IJJs. Since these junctions inherently have high critical current density (Jc 4000 A=cm2 ) and short Josephson penetration depth (J 0:2 m) [6,7] in gen0031-9007=07=99(3)=037002(4)

eral IJJs with dimensions greater than 1 m can have highly nonuniform phase distributions across the area of the junction. Furthermore, it is difficult to electrically isolate a single junction from a stack of IJJs in a single crystal with similar critical current Ic . MQT measurements are also complicated by the fact that the escape rate can be significantly enhanced in a stack of IJJs [8]. In addition, high Jc and switching of many IJJs in a stack can result in significant self-heating [9]. Therefore, great care must be taken when making a quantitative comparison of experimental results to standard MQT theory. In this Letter, we report measurements of escape rates from the zero-voltage (V 0) state using a single submicron surface IJJ to circumvent the problems discussed above. The surface IJJ had Jc 92:3 A=cm2 and J 1:3 m which improved phase uniformity. Ic of the surface IJJ was less than 0:8 A while that of the subsurface inner IJJs in the stack was about 20 A. By keeping bias current below 1 A in our experiments all of the inner IJJs remained in the V 0 state when the surface IJJ switched to the V 0 state, thus enabling us to isolate the behavior of the surface IJJ. Furthermore, the top electrode of the surface IJJ was in direct contact with a thin film Au pad which provided more efficient cooling via electronic heat transfer. Finally, the low Ic of the surface IJJ allowed us to access the classical phase retrapping-diffusion regime at T * 1 K so that the effective damping resistance, R! !p ; could be determined in situ [10,11], facilitating a parameter-free comparison of the experimental data to MQT theory. Our results agree quantitatively with MQT and thermal activation (TA) theory in the quantum and classical regime, respectively. To the best of our knowledge, this is the first unambiguous experimental evidence of MQT in Bi-2212 IJJ. Our results also support the hypothesis that the low Jc surface IJJ, where the current flows along the c axis of the crystal, has the conventional sin’

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© 2007 The American Physical Society


PRL 99, 037002 (2007)

current-phase relationship (CPR), where ’ is the phase difference across the junction. The mesa-structured surface IJJ studied in this work is shown schematically in Fig. 1(a). The IJJ was fabricated from a close-to-optimally-doped Bi-2212 single crystal with bulk Tc 88 K. The crystal was cleaved in situ at 77 K to obtain stable and reproducible surface layers and surface IJJs with controllable Jc . Details of the surface IJJ fabrication process were described in [12]. In the present work, mesas of 0:9 0:9 m2 were fabricated on the crystal, with 5 IJJs in each stack. In order to separate the surface IJJs from the inner ones and to obtain lower Jc , the surface layer was intentionally underdoped [12], resulting in a reduced transition temperature Tc0 25 K. Our study has shown that the Au atoms on the top of the stack do not penetrate through the uppermost CuO2 double layer and properties of all layers beneath it are well preserved. The measured I-V curve for the device studied herein is shown in Fig. 1(b). The dynamics of a small Josephson junction, assuming a sin’ CPR, can be viewed as the motion of a fictitious particle with mass m C in a washboard potential U’ EJ s’ cos’. Where, C is the junction capacitance, EJ 0 Ic =2 is the Josephson coupling energy, s I=Ic is the normalized bias current, and 0 h=2e is the flux quantum. For s < 1 the potential has pmetastable wells with a barrier height U 2EJ 1 s2 scos1 s. Increasing the bias current tilts the potential and lowers the barrier which eventually vanishes at s 1. The plasma frequency is given by !p LJ C1=2 1 s2 1=4 , where LJ 0 =2Ic is the Josephson inductance of the junction. Because of thermal and quantum fluctuations the particle will have a finite a)

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FIG. 1 (color online). (a) Structure of the surface IJJ formed at the top of a Bi-2212 single crystal mesa. For bias current I < 20 A the inner IJJs function as a superconducting wire to connect the surface IJJ to the measurement circuit. Only one inner IJJ is shown for clarity; (b) I-V curve for the 0:9 m 0:9 m surface IJJ reported in this work; (c) Two complete cycles of a bias current waveform with 4.5 ms resting time.


probability of escaping from the well for s < 1. At temperatures higher than the quantum-classical crossover temperature Tcr @!p 1 2 1=2 =2kB , where 1=2Q and Q !p RC is the junction’s quality factor, escape from the potential well is dominated by thermal activation over the barrier [13]. The escape rate is given by [14] t !p =2at expU=kB T, where T is temperature and at (0 < at < 1) is a damping dependent factor. At T Tcr , MQT becomes the dominant escape mechanism. In the limit of weak and moderate damping and T 0 the tunneling rate is given by [5,15] q !p =2aq exp7:2bq U=@!p , where aq ’

1207:2U=@!p 1=2 and bq ’ 1 0:87=Q. For EJ kB T a single escape event, either due to thermal activation or MQT, will cause the junction to switch rapidly from the V 0 state to the running state (V 5 mV below 0.1 K for the junction tested). Performing a statistically large number of measurements yields the probability of escape as a function of current bias PI: It can be shown that for T Tcr the width () of PI, which is a measure of the strength of thermal fluctuations, is given by / T 2=3 while for T Tcr the width is independent of T because MQT is the dominant escape mechanism. On the other hand, for EJ & 10kB T the particle has a significant probability of being retrapped in a subsequent well after the initial escape. Thus, escaping from the initial well does not necessarily cause the junction to switch to the running state. The state in which repeated escape and retrapping occurs is called the phase diffusion state. Effects of phase diffusion on PI include an increase of mean switching current Is , a reduction of , and more dramatically, a negative @=@T [10]. In our experiments we measured PI using a time-offlight technique [16,17]. The bias current was ramped linearly from 0 to 0:8 A in 5 ms. The sudden voltage jump produced when the IJJ escaped from the V 0 state was used to register the value of switching current. To reduce statistical uncertainty, each measured PI contains 50 000 escape events. The escape rate is related to PI R via I PIdI=dt= 1 I0 PI 0 dI 0 . The sample was mounted in a dilution fridge. An rf shielded room, 2 mumetal cylinders, and a niobium enclosure shielded the sample from fluctuations of the ambient electromagnetic field. All signal lines were filtered by RC low-pass filters and copper powder microwave filters anchored to 1 K pot and mixing chamber, respectively. Battery-powered lownoise amplifiers were employed. The temperature of the mixing chamber was regulated to within 1% and 0:3% of the set point at T below and above 0.6 K, respectively. PI with widths as narrow as 0.1 nA have been obtained using our circuitry, thus verifying that extraneous noise has a negligible effect on our data. It is important to point out that, in addition to MQT, temperature-independent PI can be generated by experimental artifacts such as self-heating, extraneous noise, and

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PRL 99, 037002 (2007)

inadequate measurement resolution. Hence, a quantitative comparison of experimental results with the predictions of MQT theory with no free parameters is invaluable. This, of course, requires rather accurate knowledge of junction parameters Ic , C, and R. In our experiment Ic 0:748 0:003 A was determined from fitting the distribution at T 0:62 K (in the classical regime) to TA theory where PI depends very weakly on C and R. The junction capacitance C 88 5 fF was estimated from the junction’s geometry and from resonant activation [18,19]. The damping resistance around the plasma frequency, R 250 10 (corresponding to Q 2:5), was determined by comparing data obtained in the phase diffusion regime to results of numerical simulations [10]. We emphasize that the measured value of R does not represent intrinsic damping of the surface IJJ but the impedance of the bias circuit used in our experiment. In order to measure the level of intrinsic damping of a surface IJJ one must incorporate it into circuits with much higher impedance. Figure 2 shows the measured PI between 22 mK and 4.2 K as well as the predictions of MQT (T 0:1 K) and TA (0:1 K < T 0:1 K) theory with no adjustable parameter and the dashed lines are best fits to TA theory using Ic and T as free parameters. Inset: reduced 2 from the fits. A sudden increase of 2 at 0:9 K presumably correlates with the transition from TA to phase diffusion regime.

FIG. 3 (color online). The switching rate obtained from measurements (symbols) and calculated curves using MQT (solid line) and thermal activation (dashed lines) theory. Deviation between data and TA theory at 200 mK and 300 mK is due to quantum corrections to TA. The downward bending near the high barrier end of data measured at 0.80 K indicates the onset of retrapping/phase diffusion which became more prominent at 1 K. For clarity, data above 1 K are not shown.

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PRL 99, 037002 (2007) Is (µΑ)

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FIG. 4 (color online). Temperature dependence of the width (main panel) and mean (inset) of the switching current distribution (symbols) and the prediction of TA and T 0 MQT theory (lines). The same values of Ic , C and R were used for the entire temperature range. Negative @=@T above 0.8 K is a hallmark of phase diffusion.

see that T has three distinctive regimes: for T > 0:8 K, T is a decreasing function of temperature; for 0:3 K < T < 0:8 K, T / T 2=3 ; and, for T < 60 mK, T becomes T independent within the uncertainties of the results. As discussed previously, the different temperature dependences are ascribed to phase diffusion, thermal activation, and quantum tunneling, respectively. In the temperature range of 60 mK < T < 3Tcr 300 mK the data deviate noticeably from calculations based on the T 0 MQT and TA rate formulae because they do not account for the presence of both processes at finite T [13,20]. In conclusion, by using a low Jc (92 A=cm2 ) Bi-2212 surface intrinsic Josephson junction with both linear dimensions