EUROPHYSICS LETTERS

15 February 1998

Europhys. Lett., 41 (4), pp. 425-429 (1998)

Possibility of macroscopic resonant
tunneling near the superconductor-insulatortransition in YBa2 Cu3O7−δ thin films
S. Tanda1 , K. Kagawa1 , T. Maeno1 , T. Nakayama1
K. Yamaya1 , A. Ohi2 and N. Hatakenaka3
1

Department of Applied Physics, Hokkaido University - Sapporo 060-8628, Japan
Fuji Electric Corp. Research and Development, Ltd.
Yokosuka, Kanagawa, 240-01, Japan
3
NTT Basic Research Laboratories - Atsugi, Kanagawa, 243-01, Japan2(received 9 September 1997; accepted 16 December 1997)
PACS. 74.76Bz – High-Tc films.
PACS. 71.30+h – Metal-insulator transitions and other electronic transitions.
PACS. 73.40Gk – Tunneling.

Abstract. – Experimental results of the I-V characteristics near the superconductor-insulatortransition observed for disorder-tuned YBa2 Cu3 O7−δ thin films are presented. The I-V characteristics exhibit new quasiperiodic structures as a function of the current. The current interval,
the number of the dI/dV peaks, and the magnetic-field dependence of the peaks are consistent
with the theoretical predictions of the resonant tunneling of a phase particle in a tilted-cosine
potential for a single Josephson junction with small capacitance.

Quantum phase transition in two-dimensional (2D) systems has been the subject of many
theoretical and experimental works for its abundant physical implications. In particular, thesuperconductor-insulator (S-I) transition related to the quantum unit of resistance h/4e2 ∼6.45 kΩ has been studied for various materials [1]-[8]. Among them, cuprate superconductors
with 2D CuO2 sheets are suitable to study this phase transition because disorder can easily becontrolled by oxygen deficiency. These systems also enable us to investigate the macroscopic
quantum tunneling (MQT) [9], in connection with the applicability of quantum mechanics
on a macroscopic scale, because of the spontaneous formations of Josephson networks withsmall capacitances due either to the presence of crystal grains or to oxygen contamination inYBa2 Cu3 O7−δ thin films, or both. In such a small Josephson junction, the phase differenceθ across the junction cannot be considered as a classical variable; it must be treated as
an operator that does not commute with the Cooper-pair-number operator n̂: [θ̂, n̂] = i.
Under current biases, the junction can be represented as a quantum particle moving in a
one-dimensional tilted cosine potential. The escape from the metastable state by quantum
tunneling is thus possible.
c EDP Sciences

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Another quantum-mechanical process, resonant tunneling, is possible in Josephson systems.
When the charging energy becomes much larger than that of MQT, the quantized energy
levels formed in the Josephson potential become important for tunneling. Under the current
bias, the alignment of the levels between the adjacent wells rules the tunnel phenomena.
This is known as macroscopic resonant tunneling (MRT) [10]. Giordano and Schuler haveobserved the quasiperiodic structures of current-voltage characteristics due to the resonant
features of the tunneling processes, i.e. the enhancement and suppression of 2π quantum
phase slippages in one-dimensinal superconducting Pb wires [11]. Although the system issimilar to Josephson systems, theory remains ambiguous, especially regarding the periodicityof the potential. This letter shows MRT phenomena near the S-I transition in YBa2 Cu3 O7−δthin films under magnetic fields. First, we will show that the YBa2 Cu3 O7−δ thin films can be
regarded as a series of Josephson junctions. Next, we present the I-V characteristics of thesystem. We found that the I-V curves near the S-I transition exhibit quasiperiodic structures
as a function of the current similar to the Giordano and Schuler observation. The structuresstrongly depend on the magnetic field. Last, we compare our results to theoretical predictionsof the resonant tunneling of a phase particle in a small Josephson junction.
The YBa2 Cu3 O7−δ thin films were deposited on MgO (100) substrates by rf-magnetron
sputtering using a YBa2 Cu3 O7−δ disk target. The sputtering gas was a mixture of Ar(50%) and O2 (50%) with total pressure of 0.5 Pa. The substrate temperature was kept
at 650 C during the deposition process. The temperature dependence of resistivities ρ(T )was measured with evaporated gold electrodes by the standard four-terminal method, andthe applied current was typically 1 to 10 mA. The differential resistances were measured bya lock-in amplifier with a frequency of 364 Hz. Figure 1 shows the sheet resistances as afunction of temperature for six samples (A-F). The average mean of the sheet resistance perCuO2 layer in YBa2 Cu3 O7−δ thin films and the parameter representing the intensity wereobtained from the relation R2 = ρ/d, where d (= 5.9 Å) is the lattice spacing between CuO2layers. Although the disordered parameters in these films were not systematically tuned,the observed normal-state sheet resistances above Tc are different enough that the degree ofdisorder of these samples is widely distributed. The critical sheet resistance has a value ofabout 10 kΩ close to the value h/4e2 ∼ 6.45 kΩ. The curves B-E show local coherence. Each
resistance exhibits a pronounced drop within the 60 to 70 K range, while these samples did
not show global coherence and had finite resistance. In addition, a decrease in temperature
led to an increase in resistance.
In fig. 2 we plotted the normalized conductivity σ/σ0 for sample F as a function of (T −Tc )/Tc on a logarithmic scale, where σ0 is the normal-state conductivity. The data clearlyshow the power law in the regime (T − Tc )/Tc ≤ 1, and we have the relation σ ∝ (T − Tc )−swith s = 1.36±0.03. This estimated value is in good agreement with the theoretical predictions = 1.30 [12]. The agreement indicates that the superconductor in sample F can be regarded
as a 2D superconducting percolation system, which is the fundamental model for strongly
disordered 2D systems.
As the temperature is reduced, the superconducting network gradually extends. At thecritical temperature Tc , the superconducting path extends from top to bottom and from left
to right of the sample, and the whole sample becomes globally coherent. The superconducting
percolation network is comprised of links —one-dimentional chains— and blobs —dense regionswith more than one connection between two points. If one link is cut at Tc , the global coherenceis broken and this link constitutes a backborn structure. Compared with globally coherent
sample F, the superconducting path is reduced considerably in the large-disordered samples
B-E. Therefore, links are regarded as Josephson junctions in series and blobs are regarded
as superconductors with zero resistivity. Locally coherent samples B-E with large disorders

s. tanda et al.: possibility of macroscopic resonant tunneling etc.
04–2

C
10D
10–4EF
10102104/ 0cm)
Resistivity (106)10B(CuO 2 layer unit :AYBa2Cu3O7–percolation model(T–Tc)–s
Tc=28.5K
102
s=1.36 0.03

R10427–6100
050

10–2100

100

101(T–Tc)/TcT(K)

Fig. 1

10–1

Fig. 2

Fig. 1. – The temperature dependence of the resistivity of disorder-tuned YBa2 Cu3 O7−δ thin films.
Film thickness were 300 Å (A, B), 1000 Å(C, D, E, F).
Fig. 2. – The temperature dependence of the conductivity of sample F. The solid lines show the powerlaw with s = 1.36 ± 0.03.

are considered ultrasmall Josephson junctions in series, I-V characteristics are determinedby the smallest Josephson junction. The multi-junction system is nonetheless preferable forobserving the behavior of a single Josephson junction because the junctions in series provide
high resistance and suppress the effect of the electromagnetic environments [13].
We measured the I-V characteristics near the S-I transition under various magnetic fields(fig. 3). We found that the dV /dI-I curves exhibit quasiperiodic structures as a function of thecurrent. These structures are similar to the Giordano and Schuler observation, which stronglydepends on the magnetic field. Let us explain these new structures from the standpoint ofMRT. First we calculated EJ using the Ambegaokar-Baratoff equation [14], EJ = πh̄∆/4e2 RN ,at zero temperature, where ∆ is the gap energy and RT is the tunnel resistance. The RT isroughly given by the normal resistance of the system, RT ' RN , which is different from R2 . In
sample E, the resistance is RN = 120 Ω, and the gap energy ∆ = 20 meV. Thus, the Josephson
coupling energy EJ is about 8.6 × 10−20 J. In contrast to the Josephson coupling energy, it isdifficut to estimate the charging energy Ec of the junction because no techniques have been
established for estimating the capacitance of the junctions in random media. The junction
capacitance is usually evaluated from the junction geometry or the Coulomb gap in I-V
characteristics. We applied the Coulomb gap technique to our system, but we could not obtainthe junction capacitance because sample E does not satisfy the Coulomb blockade conditions,
RN > h/e2 ∼ 25 kΩ. Thus we were obliged to infer the capacitance from the value of the othersamples. Fortunately, the capacitance of sample B, with large resistance RN = 40 kΩ, can beobtained. Figure 4 shows the I-V characteristics of sample B. The curve clearly shows theCoulomb gap due to single-electron tunneling, indicating the formation of small junctions inthe sample and strongly supporting our Josephson model for YBa2 Cu3 O7−δ thin films. The
capacitance of the junction in sample B, estimated from the Coulomb gap (e/2C = 71 mV), isC = 1.1 × 10−18 F. The charging energy Ec is about 1.1 × 10−20 J. The capacitance of sample

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50
B=0 Tesla

I()

dV/dI (arb.)

T=42K.01 Tesla

071mV.05 Tesla–.05

0I (mA)–50–0.5.05

Fig. 30V (V)

0.5

Fig. 4

Fig. 3. – dV /dI-I curves of sample E. The temperature was 4.2 K.
Fig. 4. – I-V characteristics of sample B. The threshold voltage was 71 mV.

E can be evaluated through the ratio of resistance between the junctions on different samples,
since both capacitance and resistance are characterized by the junction’s cross-section [15].
The charging energy of sample E is of the order of 4.2 × 10−21 J.
Now let us explain our experimental results in terms of macroscopic resonant tunneling.
First, we check whether our experimental conditions were such that MRT occurred. MRT
requires quantized energy levels in the Josephson potential. The number of quantum levels(n) in the potential is obtained as the ratio of the height of the Josephson potential 2EJ tothe level spacing, roughly h̄ωJ , in the harmonic approximation of the potential:
r2EJ
EJ=.
(1)n∼h̄ωJ2Ec
Here ωJ is the Josephson plasma frequency. The number of levels n is at least more than 2for MRT to occur. Moreover, the temperature must be held low enough to prevent maskingby the thermal fluctuations, i.e. kB T  h̄ωJ . In sample E, all of the above conditions are
satisfied: n ' 3 and T = 4.2 K. In fact, we observed that periodic structures are broadened
as the temperature increases, and disappear at T = 55 K. This implies that the origin of thestructures stems from the quantum-mechanical effects.
Next, we have compared our results to the MRT predictions. MRT occurs when the energy
levels in adjacent wells are coincident. These coincidences occur at special values of bias
currents which reflect the quantized energy levels. First, we checked the periodicity of theoscillations in dV /dI-I curves. The current intervals can be determined by setting the levelspacing equal to the energy gain from the battery, h̄ωJ = h̄∆I/2e; thus,∆I =2e p
8Ec EJ ,h(2)√where we use the relation ωJ = 8Ec EJ /h̄. The current interval is ∆I ∼ 2.4 mA. This is close
to the experimental value ∆I ∼ 6 mA, in spite of the rough estimation of the capacitance.
Second, the number of the observed peaks is consistent with the number of quantum levelsin the potential calculated above. Third, the upper peak position is also consistent withs. tanda et al.: possibility of macroscopic resonant tunneling etc.

429

the MRT predictions; as the current bias increases, the tunneling mechanism changes from
MRT to usual MQT. We can determine the threshold current by considering the potential’sarrangement [10]. In our case, Ec /EJ = 0.07 and the threshold current is therefore about0.17Ic = 44 mA, where Ic is the Josephson critical current. The current of third peak, 25 mA,
is less than the threshold current. Furthermore, the magnetic-field dependence of the dV /dI-I
curves is also consistent with the theoretical predictions; as the magnetic field was increased,
the number of peaks reduced from three to two and the intensity of the peaks gradually
weakened. Hence, the strong magnetic field reduces the Josephson coupling energy EJ andprevents the occurrence of the MRT. Finally, the structure disappeared at 0.25 T. Just as withthe temperature, this reflects quantum-mechanical effects. Thus, we have concluded that thenew quasiperiodic structures in I-V characteristics are due to the MRT.
In summary, we have found that the I-V characteristics near the universal critical sheet
resistance exhibit new quasiperiodic structures as a function of the current in YBa2 Cu3 O7−δthin films. The periodic structures are in good agreement with theoretical predictions of theMRT in small Josephson junctions.
***We would like to thank Prof. N. Giordano of Purdue University for his invaluable discussion. We also thank the Kurata Foundation and the Iketani Science Foundation.
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[15] In general, both the capacitance and the resistance of the junction are the function of both tunnel
distance d and cross-section S, and can be described by C ∼ S/d and RN ∼ e−d /S, respectively.
Our 2D Josephson networks are made by introducing oxygen deficiencies in the CuO2 planesof YBa2 Cu3 O7−δ . Since the deficiency has the same effect as cutting the bonding, only thecross-section of the junction is changed while tunneling distance seems to be constant.