Effect of zero energy bound states on macroscopic quantum tunneling ...

Effect of zero energy bound states on macroscopic quantum tunneling in high-Tc superconductor junctions Shiro Kawabata1,2, Satoshi Kashiwaya3, Yasuhiro Asano4 , and Yukio Tanaka5 Nanotechnology Research Institute and Synthetic Nano-Function Materials Project (SYNAF), National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki, 305-8568, Japan 2 International Project Center for Integrated Research on Quantum Information and Life Science (MILq Project), Hiroshima University, Higashi-Hiroshima, 739-8521, Japan 3 Nanoelectronics Research Institute, AIST, Tsukuba, Ibaraki, 305-8568, Japan 4 Department of Applied Physics, Hokkaido University, Sapporo, 060-8628, Japan 5 Department of Applied Physics, Nagoya University, Nagoya, 464-8603, Japan (Dated: February 2, 2008) The macroscopic quantum tunneling (MQT) in the current biased high-Tc superconductor Josephson junctions and the effect of the zero energy bound states (ZES) on the MQT are theoretically investigated. We obtained the analytical formula of the MQT rate and showed that the presence of the ZES at the normal/superconductor interface leads to a strong Ohmic quasiparticle dissipation. Therefore, the MQT rate is noticeably inhibited in compared with the c-axis junctions in which the ZES are completely absent. PACS numbers: 74.50.+r, 03.65.Yz, 05.30.-d

Recently we have theoretically investigated the effect of the nodal-quasiparticle on the MQT in the d-wave c-axis junctions6 (e.g., the Bi2212 intrinsic junction7 and the cross-whisker junctions8 ). We have showed that the effect of the nodal-quasiparticle gives rise to a super-Ohmic dissipation9,10 and the suppression of the MQT due to the nodal-quasiparticle is very weak. In fact, recently, Inomata et al. have experimentally observed the MQT in the Bi2212 intrinsic junctions.11 They have reported that the effect of the nodal-quasiparticle on the MQT is negligibly small and the thermal-to-quantum crossover temperature is relatively high (∼ 1K) in compared with the case of s-wave junctions.2,3 In this paper, we will investigate the MQT in the d-wave junctions parallel to the ab-plane (see Fig. 1), e.g., the YBCO grain boundary junctions12,13 and the ramp-edge junctions.14 In such junctions, the zero energy bound states (ZES)15,16,17,18 are formed near the interface between superconductor and the insulating barrier. (Note that in the d0 /d0 junction (Fig. 1 (a)) no ZES are formed as will be mentioned later.) The ZES are generated by the combined effect of multiple Andreev reflec-

tions and the sign change of the d-wave order parameter symmetry, and are bound states for the quasiparticle at the Fermi energy. Therefore the ZES may give rise to a strong dissipation for the MQT. Note that recently Amin and Smirnov have theoretically calculated the decoherence time of a d-wave qubit and discussed the effect of the ZES on the qubit operation.19 They, however, phenomenologically assumed that the system coupled to an Ohmic heat bath. Instead, we will calculate the effective action starting from microscopic Hamiltonian without any phenomenological assumptions. Moreover by using this effective action we will derive the theoretical formula of the MQT rate and discuss the influence of the ZES on the MQT. The grandcanonical Hamiltonian of the d-wave junc-

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d0

d0 b

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Great attention has been attracted to theoretical and experimental studies of the effect of the dissipation on the macroscopic quantum tunneling (MQT) in superconductor Josephson junctions.1 In current-biased Josephson junctions, the macroscopic variable is the phase difference between two superconductors and measurements of the MQT are performed by switching the junction from its metastable zero-voltage state to a non-zero voltage state (see Fig. 1 (c)). Heretofore, experimental tests of the MQT have been focused on s-wave (low-Tc) junctions.2,3 This fact is due to the naive preconception that the existence of the low energy nodal-quasiparticle in the d-wave order parameter of a high-Tc cuprate superconductor4,5 may preclude the possibility of observing the MQT.

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arXiv:cond-mat/0501358v2 [cond-mat.supr-con] 1 Feb 2005

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I

ωp ZES

FIG. 1: Schematic drawing of the d-wave Josephson junction. (a) d0 /d0 junction and (b) d0 /dπ/4 junction. (c) Potential U (φ) v.s. the phase difference φ between two superconductors. U0 is the barrier height and ωp is the Josephson plasma frequency.

2 tions (Fig. 1 (a) and (b)) is given by 2 2 XZ ~ ∇ † H= dr ψσ (r) − − µ + W (r) ψσ (r) 2m σ Z 1X − drdr′ ψσ† (r) ψσ† ′ (r ′ ) g (r − r ′ ) ψσ′ (r ′ ) ψσ (r) , 2 ′ σ,σ

(1)

where µ is the chemical potential. This Hamiltonian describes conduction electrons in a potential W (r) which includes a boundary. The second term in H describes the anisotropic attractive interaction of strength g(r). This model also includes the insulating tunnel barrier I where g(r) = 0 by a suitable choice of W (r). Below,we will assume that the tunnel barrier is given by a delta function potential, i.e., W (r) = w0 δ(x) and consider the high barrier limit z0 ≡ mw0 /~2 kF ≫ 1 (m is the mass of the electron and kF is the Fermi wave number.) which corresponds to typical experimental situations. The partition function of the system can be written as a path integral over the Grasmmann fields,20 ! Z ~β Z 1 ¯ dτ L[τ ] , (2) Z = DψDψ exp − ~ 0 where βP=R 1/kB T and the Lagrangian is given by L[τ ] = dr ψ¯σ (r, τ )∂τ ψσ (r, τ ) + H[τ ]. In order to σ write the partition function as a functional integral over the macroscopic variable (the phase difference φ), we apply the Hubbard-Stratonovich transformation. This introduces a complex order parameter field ∆(r, τ ). Now the integrals over the Grassmann fields and |∆| can be performed by using the Gaussian integral and the saddle point approximation, respectively. Moreover, by following the method of Sch¨ onRand Zaikin,21 we obtain the partition function as Z = Dφ(τ ) exp (−Sef f [φ]/~), where the effective action Sef f in the high barrier limit is given by # 2 Z ~β " M ∂φ(τ ) Sef f [φ] = dτ + U (φ) 2 ∂τ 0 Z ~β Z ~β φ(τ ) − φ(τ ′ ) . (3) − dτ dτ ′ α(τ − τ ′ ) cos 2 0 0 In this equation, M = C (~/2e)2 is the mass (C is the junction capacitance) and the potential U (φ) is described by Z 1 ~ (4) dλ φIJ (λφ) − φ Iext , U (φ) = 2e 0 where IJ is the Josephson current and Iext is the external bias current, respectively. The dissipation kernel α(τ ) is related to the quasiparticle current Iqp under constant bias voltage V by Z ~ω ~ ∞ dω . (5) α(τ ) = Dω (τ )Iqp V = e 0 2π e

In this equation, Dω (τ ) is the Matsubara Green’s function of a free boson Dω (τ ) =

∞ 2~ω 1 X exp(iνn τ ), ~β n=−∞ νn2 + (~ω)2

(6)

where νn = 2nπ/~β is the bosonic Matsubara frequency (n is an integer). Below, we will derive the effective action for the two types of the d-wave junction (d0 /d0 and d0 /dπ/4 ) in order to investigate the effect of the ZES on the MQT (see Fig. 1). In the case of the d0 /d0 junction, the node-tonode quasiparticle tunneling can contribute to the dissipative quasiparticle current. However, the ZES are completely absent. These behaviors are qualitatively identical with that for the c-axis Josephson junctions.6 On the other hand, in the case of the d0 /dπ/4 junction, the ZES are formed around the surface of the right superconductor dπ/4 . Therefore the node to the ZES quasiparticle tunneling becomes possible. Firstly, we will calculate the potential energy U in the effective action (3). As mentioned above, U can be described by the Josephson current through the junction. In order to obtain the Josephson current we start from the Bogoliubov-de Gennes (B-dG) equation,17 u(r) δ(r − r ′ )h(r ′ ) ∆(r − r ′ )eiϕ dr′ v(r) ∆∗ (r − r ′ )e−iϕ −δ(r − r′ )h∗ (r ′ ) u(r) , (7) = E v(r) Z

where ϕ is the phase of order parameter, u(v) is the amplitude of the wave function for the electron (hole)like quasiparticle,P h(r) = −~2 ∇2 /2m − µ + w0 δ(x), and ′ ′ −1 ∆(r − r ) = Ω k ∆k exp [ik · (r − r )] (Ω is the volume of the superconductor). In the superconductor regions (d0 and dπ/4 ), the B-dG equation (7) can be transformed into the eigenequation uk uk ξk ∆k eiϕ , (8) =E vk vk ∆k e−iϕ −ξk where, ξk = ~2 k 2 /2m + ~2 p2 /2m − µ (p = 2πn/D and D is the width of the junction). The amplitude of the order parameter ∆k is given by ∆0 cos 2θ ≡ ∆d0 (θ) for d0 and ∆0 sin 2θ ≡ ∆dπ/4 (θ) for dπ/4 , where cos θ = k/kF . The Andreev reflection coefficient for the electron (hole)like quasiparticle rhe (reh ) is calculated by solving the eigenequation (8) together with the appropriate boundary conditions. By substituting rhe (reh ) into the formula of the Josephson current for unconventional superconductors (the Tanaka-Kashiwaya formula),17,22,23,24 e X 1 X ∆+ ∆− IJ = (9) rhe − reh , ~ p β ω Ω+ Ω− n

we can obtain φ dependence ofpthe Josephson current. Here, ∆± = ∆(±k,p) , Ω± = (~ωn )2 − |∆± |2 , ωn =

3 (2n + 1)π/β~ is the fermionic Matsubara frequency. In the case of low temperatures (β −1 ≪ ∆0 ), we get I1 sin φ for d0 /d0 IJ (φ) ≈ , (10) −I2 sin 2φ for d0 /dπ/4 2 where I1 ≡ 3π∆0 /10eRN , I2 ≡ π 2 ~β∆20 /35e3 Nc RN 2 2 (RN = 3π~z0 /2e Nc is the normal state resistance of the junction and Nc is the number of channel at the Fermi energy). Note that Eq. (10) is only valid in the case of the high barrier limit (z0 ≫ 1). By using Eq. (4), we obtain the analytical expression of the potential U, i.e.,  Iext ~I1   cos φ + φ for d0 /d0  − 2e I1 (11) . U (φ) ≈ Iext ~I2   − cos 2φ + 2 φ for d0 /dπ/4 − 4e I2

As in the case of the s-wave25,26 and the c-axis junctions,6 U can be expressed as the tilted washboard potential (see Fig. 1(c)). Next we will calculate the dissipation kernel α(τ ) in the effective action (3). In the high barrier limit (z0 ≫ 1), the quasiparticle current is given by17,18 Z ∞ 2e X 2 Iqp (V ) = |tN | dENL (E, θ)NR (E + eV, θ) h p −∞ × [f (E) − f (E + eV )] ,

(12)

where tN ≈ cos θ/z0 is the transmission coefficient of the barrier I, NL(R) (E, θ) is the quasiparticle surface density of states (L = d0 and R = d0 or dπ/4 ), and f (E) is the Fermi-Dirac distribution function. The explicit expression of the surface density of states is obtained by Matsumoto and Shiba.27 In the case of d0 , there are no ZES. Therefore the angle θ dependence of Nd0 (E, θ) is the same as the bulk and is given by |E| Nd0 (E, θ) = Re p . E 2 − ∆d0 (θ)2

(13)

On the other hand, Ndπ/4 (E, θ) is given by q E 2 −∆dπ/4 (θ)2 Ndπ/4 (E, θ) = Re +π|∆dπ/4 (θ)|δ(E).(14) |E| The delta function peak at E = 0 corresponds to the ZES. Because of the bound state at E = 0, the quasiparticle current for the d0 /dπ/4 junctions is drastically different from that for the d0 /d0 junctions in which no ZES are formed. By substituting Eqs. (13) and (14) into Eq. (12), we can obtain the analytical expression of the quasiparticle current in the limit of low bias voltages (eV ≪ ∆0 ) and low temperatures (β −1 ≪ ∆0 ) as  9π 2 eV 2   √ for d0 /d0  256 2 ∆0 RN . (15) Iqp (V ) ≈ 2 3π V   √ for d0 /dπ/4  16 2 RN

Therefore from Eq. (5), the asymptotic form (τ ≫ ~/∆0 ) of the dissipation kernel for the d-wave junctions is given by  9 ~2 RQ 1    √ for d0 /d0 128 2 ∆0 RN |τ |3 , (16) α(τ ) ≈ 3 ~RQ 1   √ for d /d  0 π/4 16 2 RN τ 2

where RQ = h/4e2 is the resistance quantum. These results (Eqs. (15) and (16)) indicate that in the case of d0 /d0 junctions, the dissipation is the super-Ohmic type as in the case of the c-axis junctions.6 This can be attributed to the effect of the node-to-node quasiparticle tunneling. Thus the quasiparticle dissipation is very weak. On the other hand, in the case of the d0 /dπ/4 junctions, the node-to-ZES quasiparticle tunneling gives the strong Ohmic dissipation which is similar to that in normal junctions (α(τ ) ∼ 1/τ 2 ).25,26 Therefore the dissipation for the d0 /d0 junctions is enormously weaker than that for the d0 /dπ/4 junctions. Let us move to the calculation of the MQT rate Γ for the d-wave Josephson junctions. At the zero temperature Γ is given by Γ = limβ→∞ (2/β) ln Z.1 Within the WKB approximation the partition function Z is evaluated by the saddle-point approximation. Then Γ in the low viscosity limit is obtained by a perturbative treatment28 as Γ ≈ A exp (−SB /~), where SB ≡ Sef f [φB ] and φB is the bounce solution. Following the above procedures, we finally obtain the central results of this paper, i.e., the analytical formulae of the MQT rate for the d-wave junctions as29 RQ ∆ δM 2 5 ~ 3π 4 Γ √ + 12 0 − c 81π 5 ∆0 CRN (1−x ) ≈ e 64 2 RN 5 ~2 Γ0 for d0 /d0 , (17) Γ 81ζ(3) RQ ≈ exp − √ (1 − x2 ) for d0 /dπ/4 ,(18) Γ0 32 2π 2 RN R∞ where c = 0 dy y 4 ln(1 + 1/y 2 )/ sinh2 (πy) ≈ 0.0135, ζ(3) ≈ 1.20 is the Riemann zeta function, x = Iext /I1(2) , p and Γ0 = 12ωp 3U0 /2π~ωp exp (−36U0 /5~ωp ) is the MQT rate without the dissipation (U0 is the barrier height of the potential U and ωp is the Josephson plasma frequency. Note that U0 and ωp depend on the type of the junction). In Eq. (17) δM is the renormalized mass due to the high frequency components (ω ≥ ωp ) of the quasiparticle dissipation and is given by

iq

h

Z 1 3 ~2 RQ 1+y √ dy y 2 √ 1−y 16 2 ∆0 RN −1 ∆0 Z ~ω p 2 dz z 2 K1 (z|y|) , ×

δM =

(19)

0

where K1 is the modified Bessel function. In order to compare the influence of the ZES and the nodal-quasiparticle on the MQT more clearly, we will

4 estimate the MQT rates (17) and (18) numerically. For a mesoscopic bicrystal YBCO Josephson junction30 (∆0 = 17.8 meV, C = 20 × 10−15 F, RN = 100 ΩAx = 0.95), the MQT rate is estimated as Γ 83% for d0 /d0 ≈ . (20) 25% for d0 /dπ/4 Γ0 As expected, the node-to-ZES quasiparticle tunneling in the d0 /dπ/4 junctions gives strong suppression of the MQT rate in compared with the d0 /d0 junction cases. In conclusion, the MQT in the high-Tc superconductors has been theoretically investigated. The node-tonode quasiparticle tunneling in the d0 /d0 junctions gives rise to the weak super-Ohmic dissipation as in the case of the c-axis junctions.6 For the d0 /dπ/4 junctions, on the other hand, we have found that the node-to-ZES quasiparticle tunneling leads to the strong Ohmic dissipation. We have also analytically obtained the formulae of the MQT rate which can be used to analyze experiments. In the context of an application to the d-wave phase qubit,19,31,32,33 it is desirable to use the d0 /d0 or the c-

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axis Josephson junctions in order to avoid the strong Ohmic dissipation. However the effect of the ZES can be abated by several mechanisms (e.g., by applying magnetic field or by a disorder in the interface).17,18 Therefore it is interesting to investigate the MQT of the d0 /dπ/4 junction in such situations. Finally, we would like to comment about a recent experimental research. Bauch et al. have succeeded to observe the MQT in a YBCO grain boundary bi-epitaxial Josephson junction.34 In their junction, however, the dissipation is mainly due to not intrinsic mechanisms (the ZES and the nodal-quasiparticles) which discussed in this paper but an extrinsic impedance in parallel to the junction. Therefore the development of the junction fabrication and the measurement techniques will enable us to directly compare our theory with experimental results. We would like to thank S. Abe, P. Delsing, N. Hatakenaka, K. Inomata, T. Kato, and A. Tanaka for useful discussions. This work was partly supported by NEDO under the Nanotechnology Program.

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