Theory of Macroscopic Quantum Tunneling and Dissipation in High-Tc Josephson Junctions

0.21 MB·English

by  Shiro Kawabata

Theory of Macroscopic Quantum Tunneling and Dissipation in High-T Josephson
c
Junctions
Shiro Kawabata,1,2 Satoshi Kashiwaya,3 Yasuhiro Asano,4 Yukio Tanaka,5 Takeo Kato,6and Alexander A. Golubov1
1Faculty of Science and Technology, University of Twente,
P.O. Box 217, 7500 AE Enschede, The Netherlands
7
2Nanotechnology Research Institute (NRI), National Institute of Advanced
0
0 Industrial Science and Technology (AIST), Tsukuba, Ibaraki, 305-8568, Japan
2 3Nanoelectronics Research Institute (NeRI), National Institute of Advanced
Industrial Science and Technology (AIST), Tsukuba, Ibaraki, 305-8568, Japan
n 4Department of Applied Physics, Hokkaido University, Sapporo, 060-8628, Japan
a 5Department of Applied Physics, Nagoya University, Nagoya, 464-8603, Japan
J 6The Institute for Solid State Physics (ISSP), University of Tokyo, Kashiwa, 277-8581, Japan
0
(Dated: February 6, 2008)
3
Wehaveinvestigatedmacroscopicquantumtunneling(MQT)inin-planehigh-T superconductor
c
] Josephson junctions and the influence of the nodal-quasiparticle and the zero energy bound states
n
(ZES)onMQT.WehaveshownthatthepresenceoftheZESattheinterfacebetweentheinsulator
o
andthesuperconductorleadstostrongOhmicquasiparticledissipation. Therefore,theMQTrateis
c
- noticeably suppressed incomparison with the c-axisjunctionsin which ZES arecompletely absent.
r
p PACSnumbers:
u
s
.
t INTRODUCTION collective motion of the phase differences in the intrinsic
a
m junctions [15, 16, 17].
A mesoscopic single Josephson junction is an inter- In this paper, we will theoretically investigate the
-
d esting physical object for testing quantum mechanics at MQT inthe d-wavein-planejunctions paralleltothe ab-
n a macroscopic level. In current-biased Josephson junc- plane(seeFig.1)[18]. Insuchjunctions,the zeroenergy
o
tions, the measurements of macroscopic quantum tun- bound states (ZES) [19] are formed near the interface
c
[ neling (MQT) are performed by switching the junction between superconductorand the insulating barrier. ZES

fromitsmetastablezero-voltagestatetoanon-zerovolt- aregeneratedbythecombinedeffectofmultipleAndreev
1
agestate(see Fig.1(d)). Untilnow,experimentalinves- reflections and the sign change of the d-wave order pa-
v
4 tigations of MQT have been focused on s-wave (low-Tc) rameter symmetry, and are bound states for the quasi-
3 junctions only. This fact is due to the naive preconcep- particle at the Fermi energy. Below, we will show that
7 tionthattheexistenceofthelowenergyquasiparticlesin ZES give rise to Ohmic type strong dissipation so MQT
1
the d-wave order parameter of a high-T cuprate super- is considerably suppressed in compared with the c-axis
0 c
conductor [1] may preclude the possibility of observing and the d /d junction cases.
7 0 0
0 the MQT.
/ Recently we have theoretically investigated the effect
t
a of the nodal-quasiparticle on MQT in the d-wave c-axis
m
junctions (e.g., Bi2212 intrinsic Josephsonjunctions [12,
- 13] and cross whisker junctions [14]) [2, 3]. We have
d
shownthattheeffectofthenodal-quasiparticlegivesrise
n
o to a super-Ohmic dissipation [4, 5] and the suppression
c of the MQT due to the nodal-quasiparticle is very weak.
v: The first experimental observation of the MQT in the
Xi high-TcJosephsonjunctionwasmadebyBauchetal.,us-
ing a YBCO grain boundary bi-epitaxial junction [6, 7].
r
Recently,Inomataetal.[8],Jinetal.[9],andKashiwaya
a
et al. [10, 11] have experimentally observedthe MQT in
the c-axis (Bi2212 intrinsic) junctions. They reported
that the effect of the nodal-quasiparticle on the MQT
FIG. 1: Schematics of the in-plane d-wave Josephson junc-
isnegligiblysmallandthethermal-to-quantumcrossover
tion. (a) d0/d0, (b) d0/dπ/4, and (c) dπ/4/dπ/4 junction. In
temperature is relatively high (0.5∼1K) compared with thecaseof thed0/dπ/4 anddπ/4/dπ/4 junctions,theZES are
the case of the low-Tc and the YBCO bi-epitaxial junc- formed near the boundary between superconductor dπ/4 and
tions. In Jin et al.’ s experiment, O(N2) (N is the num- insulating barrier I. (d) Potential U(φ) v.s. the phase differ-
ber of the stacked junctions) enhancement of the MQT enceφbetweentwosuperconductors. U0 isthebarrierheight
and ω is theJosephson plasma frequency.
rate was reported. This enhancement is attributed to p
2
EFFECTIVE ACTION
δ(r r )h(r ) ∆(r r )eiϕ u(r)
By using the method developed by Eckernet. al., [20] dr′ ∆ (r− r′ )e ′iϕ δ(r −r )′h (r ) v(r)
∗ ′ − ′ ∗ ′
Z (cid:18) − − − (cid:19)(cid:18) (cid:19)
the partition function of the system can be described by u(r)
a functional integral over the macroscopic variable (the = E v(r) , (5)
phase difference φ), (cid:18) (cid:19)
where ϕ is the phase of order parameter, u(v) is the
Seff[φ] amplitude of the wave function for the electron (hole)-
Z = φ(τ)exp . (1)
D − ~ like quasiparticle, h(r) = ~2 2/2m µ+w δ(x), and
Z (cid:18) (cid:19) ∆(r r′) = Ω−1 k∆k−exp∇[ik (r −r′)] is0the order
In the high barrier limit, i.e., z0 mw0/~2kF 1( m param−eter (Ω is the volume of th·e su−perconductor). In
is the mass of the electron, w0 is t≡he height of t≫he delta thesuperconductorPregions(d0anddπ/4),theB-dGequa-
function potential I, and kF is the Fermi wave length), tion (5) can be transformed into the eigenequation
the effective action S is given by
eff
ξk ∆keiϕ uk uk
=E , (6)
~β M ∂φ(τ) 2 (cid:18)∆ke−iϕ −ξk (cid:19)(cid:18) vk (cid:19) (cid:18) vk (cid:19)
S [φ] = dτ +U(φ) +S [Φ],
eff Z0 " 2 (cid:18) ∂τ (cid:19) # α where,ξk =~2k2/2m+~2p2/2m−µ(p=2πn/D andD
~β ~β φ(τ) φ(τ ) is the width of the junction). The amplitude of the or-
Sα[Φ] = − dτ dτ′α(τ −τ′)cos −2 ′ . der parameter ∆k is given by ∆0cos2θ ≡∆d0(θ) for d0
Z0 Z0 (2) aTnhdeA∆n0dsirnee2vθr≡efl∆ecdtπio/4n(θco)efffiorcideπn/t4f,owrthheereelceocstrθon=(kh/okleF)-.
like quasiparticle r (r ) is calculated by solving the
Inthis equation,β =1/k T,M =C(~/2e)2 isthe mass he eh
B eigenequation (6) together with the appropriate bound-
(C is the capacitance of the junction) and the potential
aryconditions. Bysubstitutingr (r )intotheformula
he eh
U(φ) is described by
oftheJosephsoncurrentforunconventionalsuperconduc-
tors (the Tanaka-Kashiwayaformula) [19],
~ 1
U(φ)= dλ φI (λφ) φ I , (3)
2e(cid:20)Z0 J − ext(cid:21) IJ = ~e β1 ∆Ω+rhe− ∆Ω−reh , (7)
whereI istheJosephsoncurrentandI istheexternal Xp Xωn (cid:18) + − (cid:19)
J ext
bias current, respectively. The dissipation kernel α(τ) is we can obtain φ dependence of the Josephson current.
rbeialastevdolttaogethVe qbuyasiparticle current Iqp under constant (H2enre+, 1∆)π±/β=~ ∆is(t±hke,p)fe,rmΩ±ion=ic Ma(~tsωunb)a2r−a f|r∆eq±u|2e,ncωyn. I=n
p
the case of low temperatures (β 1 ∆ ) and the high
− 0
α(τ)= ~ ∞ dωe−ωτIqp V = ~ω , (4) barrier limit (z0 ≫1), we get ≪
e 2π e
Z0 (cid:18) (cid:19) I1sinφ for d0/d0
at zero temperature. IJ(φ)≈ −I2sinφ2φ for d0/dπ/4 , (8)
thrBeeelotwyp,eswoef wthielldd-weraivvee juthnectieoffnec(tdiv/eda,cdtio/nd for, atnhde  I3sin 2 for dπ/4/dπ/4
0 0 0 π/4
dqπu/a4s/ipdaπr/t4i)cliensoarndderZtEoSinovneMstiQgaTt.eItnhteheeffceacsteoofftthheendo0d/adl0- Iw3h≡ere3Iπ1z0≡∆30π/∆4e0R/N10e(RRNN,=I23≡π~πz202~/β2∆e220N/3c5ies3NthceRnN2o,ramnadl
junction, the node-to-node quasiparticle tunneling can state resistance of the junction and N is the number of
c
contribute to the dissipative quasiparticle current. How- channel at the Fermi energy).
ever, ZES are completely absent. These behaviors are By substituting the Josephsoncurrent into eq. (3), we
qualitatively identical with that for the c-axis Joseph- can obtain the analytical expression of the potential U,
son junctions [2, 3]. On the other hand, in the case of i.e.,
the d0/dπ/4 and dπ/4/dπ/4 junction, the ZESare formed ~I I
1 ext
around the surface of the superconductor d . There- cosφ+ φ for d /d
π/4 − 2e I 0 0
fore the node to ZES (d0/dπ/4) and the ZES to ZES  ~I (cid:18) 1I (cid:19)
(dπF/i4r/stdlπy/,4w)eqwuailslipcaarlctuicllaetetutnhneepliontgenbteiacolmeneesrgpyosUsibinle.the U(φ)≈ − 4~e2I(cid:18)−cosφ2φ+12I Ie2xtφ(cid:19) for d0/dπ/4 .(9)
3 ext
effective action (2). As mentioned above, U can be de- cos + φ for d /d
sthcreibheidghbybatrhreieJrolsiempiht.soInncourrdreenrttothorboutagihntthheejJuonscetpiohnsoinn As in the ca−se oef t(cid:18)he s-w2ave2anId3 th(cid:19)e c-axis juπn/c4tioπn/s4[2],
current we start from the Bogoliubov-deGennes (B-dG) U canbeexpressedasthetiltedwashboardpotential(see
equation [19], Fig. 1(d)).
3
DISSIPATION DUE TO This can be attributed to the effect of the node-to-node
NODAL-QUASIPARTICLES AND ZES quasiparticle tunneling. Thus the quasiparticle dissipa-
tion is very weak. On the other hand, in the case of the
Next we will calculate the dissipation kernel α(τ) in d0/dπ/4 junctions,the node-to-ZESquasiparticletunnel-
the effective action (2). In the high barrier limit, the ing gives the Ohmic dissipation which is similar to that
quasiparticle current is given by [19] in normal junctions [20]. Therefore the dissipation for
the d /d junctions is strongerthanthat for the d /d
0 π/4 0 0
I (V) = 2e t 2 ∞ dEN (E,θ)N (E+eV,θ) junctions. Moreover, in the case of the dπ/4/dπ/4 junc-
qp N L R
h | | tions,theZES-to-ZESquasiparticletunneling dominates
Xp Z−∞ thequasiparticledissipation. ThebroadeningoftheZES
[f(E) f(E+eV)], (10)
× − peakǫistypicallyoneordermagnitudesmallerthan∆ .
0
where tN cosθ/z0 is the transmissioncoefficientofthe Therefore, due to the prefactor (∆0/ǫ)2 in Eq. (12), the
barrier I,≈NL(R)(E,θ) is the quasiparticle surface den- quasiparticle dissipation in the dπ/4/dπ/4 junctions be-
sity of states (L = d0 and R = d0 or dπ/4) and f(E) is comes enormously stronger than that for the d0/d0 and
the Fermi-Dirac distribution function. The explicit ex- d0/dπ/4 cases.
pression of the surface density of states was obtained by From Eq. (4), the asymptotic form of the dissipation
Matsumoto and Shiba [21]. In the case of d , there are kernel is given by
0
no ZES. Therefore the angle θ dependence of Nd0(E,θ) 32~2 RQ 1
is the same as the bulk and is given by  27√3~2∆R0RN1|τ|3 for d0/d0
Nd0(E,θ)=Re E2−|E∆|d0(θ)2. (11) α(τ)≈ 25~24√∆20RNQ2 R|τQ|2 1 for d0/dπ/4 .(15)
for d /d
On the other hand, Ndπ/4(pE,θ) is given by The resultf3o5rπd0(cid:18)/dǫ0 j(cid:19)uncRtiNon|τis|2inagreemeπn/t4wiπth/4previ-
E2−∆dπ/4(θ)2 ous works [4, 5, 22, 23].
N (E,θ)=Re +π ∆ (θ)δ(E).(12)
dπ/4 q E | dπ/4 |
| |
The delta function peak at E = 0 corresponds to the MQT IN IN-PLANE d-WAVE JUNCTIONS
ZES. Because of the bound state at E = 0, the quasi-
particle current for the d /d and d /d junctions Let us move to the calculation of the MQT rate Γ for
0 π/4 π/4 π/4
is drastically different from that for the d /d junctions the d-wave Josephson junctions based on the standard
0 0
in which no ZES are formed. By substituting Eqs. (11) CaldeiraandLeggetttheory[24]. AtzerotemperatureΓ
and (12) into Eq. (10), we can obtain the analytical ex- is given by
pressionof the quasiparticle currentI (V). In the limit
qp S
B
of low bias voltages (eV ∆ ) and low temperatures Γ Aexp , (16)
≪ 0 ≈ − ~
(β−1 ∆0), this can be approximated as (cid:18) (cid:19)
≪
where S S [φ ] and φ is the bounce solution.
B eff B B
32π2 eV2 ≡
Following the procedures as above, we obtain the ana-
for d /d
0 0
 28√3π22∆0VRN lyticalformulaeofthe MQTrate for the in-planed-wave
Iqp(V)≈ 25π24√∆20RN2 V ffoorr dd0/d/πd/4 . (13) junctioenxspas− c02375√π2∆~η0 + 158δM~ MUω0p for d0/d0
In the calculati3o5n o(cid:18)f Iǫqp(cid:19)forRtNhe dπ/4/dπ/π4/4juncπt/i4ons, we ΓΓ0≈ exphex−(cid:16)p2h83−353πζ233(543√ζ)2(3π)∆2ǫη0(12η−(1x2−(cid:17))ix2) i ffoorr dd0π//4d/πd/4π/4 ,
have replaced the ZES delta function δ(E) in Eq. (12)
with the Lorentz type function, i.e.,  h (cid:0) (cid:1) i (17)
1 ǫ where c0 = 0∞dy y4ln(1+1/y2)/sinh2(πy) ≈ 0.0135,
δ(E) , (14) ζ(3)istheRiemannzetafunction,η =R /R isthedis-
→ πǫ2+E2 R Q N
sipationparameter,U isthebarrierheightofthepoten-
0
in order to avoid a mathematical difficulty and model tialU,ωp istheJosephsonplasmafrequency,x=Iext/Ii
the real systems (which include e.g. disorder and many (i=1,2,3), and
body effects). It is apparent from Eq. (13) that, in the
case of d0/d0 junctions, the dissipation is of the super- Γ =12ω 3U0 exp 36U0 (18)
Ohmic type as in the case of the c-axis junctions [2]. 0 ps2π~ωp (cid:18)−5~ωp(cid:19)
4
is the MQT rate without the dissipation. In Eq. (17) [2] S. Kawabata, S. Kashiwaya, Y. Asano, and Y. Tanaka,
Phys. Rev.B 70, 132505 (2004).
δM = 3 ~2η 1 dy y2 1+y ~∆ω0p dz z2K1(z y )2. [3] SJ..PKhaywsa.bCahtaem, Y. S.oAl.sa6n7o,,1S2.0K(2a0s0h6iw).aya, and Y. Tanaka,
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p
In order t≥o compare the influence of the ZES and the P. Delsing, and T. Claeson, Phys.Rev.Lett. 94, 087003
(2005).
nodal-quasiparticle on the MQT more clearly, we will
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estimate the MQT rate (17) numerically. For a meso-
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×
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25% for d /d . (20) (2006).
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 π/4 π/4
H.Eisaki,Y.Yoshida,S.Kawabata,andY.Tanaka,sub-
mitted to Physica C (2006).
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particle tunneling in the d /d and d /d junc-
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014502 (2007).
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will be investigated in future articles.
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