Microwave surface resistance in superconductors with grain boundaries

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by  Yasunori Mawatari

Microwave surface resistance in superconductors with grain boundaries
Yasunori Mawatari
Energy Technology Research Institute,
National Institute of Advanced Industrial Science and Technology,
Tsukuba, Ibaraki 305–8568, Japan
5 (Dated: Dec. 28, 2004)
0
0 Microwave-fielddistribution,dissipation,andsurfaceimpedancearetheoreticallyinvestigatedfor
2 superconductorswithlaminargrainboundaries(GBs). Inthepresenttheoryweadoptthetwo-fluid
modelforintragraintransportcurrentinthegrains,andtheJosephson-junctionmodelforintergrain
n
a tunnelingcurrentacrossGBs. ResultsshowthatthesurfaceresistanceRsnonmonotonicallydepends
J on the critical current density Jcj at GB junctions, and Rs for superconductors with GBs can be
1 smaller than thesurface resistance Rs0 for ideal homogeneous superconductors without GBs.
1
PACSnumbers: 74.25.Nf,74.20.De,74.50.+r,74.81.-g

]
n
o I. INTRODUCTION the superconductors. Magnetic induction B is assumed
c to be less than the lower critical field, such that no vor-
r- High-temperaturesuperconductorscontainmanygrain ticesarepresentinthe superconductors. (See Ref.20for
p boundaries (GBs), where the order parameter is locally microwave response of vortices.)
u suppressed due to the short coherence length.1 GBs The GBs are modelled to have laminar structures as
s
. have attracted much interest for their basic physics as in Ref. 21; the laminar GBs that are parallel to the
at well as for their applications in superconductors,2,3,4 xz plane are situated at y = ma, where a is the spac-
m and play a crucial role in microwave response and sur- ing between grains (i.e., effective grain size) and m =
- face resistance Rs of high-temperature superconducting 0,±1,±2,···,±∞. The thickness of the barrier of GB
d films.5,6,7,8,9,10,11,12,13 junctions, dj, is much smaller than both a and the Lon-
n Electrodynamicsof GB junctions canbe described us- don penetration depth λ, and therefore, we investigate
o
ing the Josephson-junction model, and one of the most the thin-barrier limit of dj → 0, namely, GB barriers
c
[ important parameters that characterize GB junctions is situated at ma−0<y <ma+0.

1 the critical current density Jcj for Josephson tunneling
current across GBs.14,15,16 The J strongly depends on
5v the misorientation angle of GBs.c1j7,18 In YBa2Cu3O7−δ B. Two-fluid model for intragrain current
2 films, Jcj can be enhanced19 and Rs reduced13 by Ca
2 doping. TheinvestigationoftherelationshipbetweenRs Weadoptthestandardtwo-fluidmodel15,16 forcurrent
1 and J is needed to understand the behavior of R and transport in the grain at ma+0 < y < (m+1)a−0.
cj s
0 Jcj in Ca doped YBa2Cu3O7−δ films. The Jcj depen- The intragrain current J =Js+Jn is given by the sum
/05 disenncoet torfivRiasl, whohwetehveerr,GhBass nenohtaynectebteheenmcliacrroifiweadv,eadnidssiit- oJfnt=heσsnuEpe,rwcuhrerreentσsJs==1/iωσµsE0λ2anadndthσenniosrtmhaelncourrmreanlt-
t pation that is proportionalto R . fluid conductivity in the grains. The displacement cur-
a s
m In this paper, we present theoretical investigation on rentJd =−iωǫEwiththedielectricconstantǫcanbene-
the microwave field and dissipation in superconductors glectedforamicrowaverangeofω/2π∼GHz. Amp`ere’s
d- withlaminarGBs. Theoreticalexpressionsofthesurface law µ−01∇×B =(σn+iσs)E is thus reduced to
n impedance Z = R −iX of superconductors with GBs
s s s
o are derived as functions of Jcj at GB junctions. E =−iωΛ2g∇×B, (1)
c
: where Λ is the intragrainac field penetration depth de-
v g
i II. BASIC EQUATIONS fined by
X
r Λ−2 =ωµ (σ −iσ )=λ−2−iωµ σ . (2)
a A. Superconductors with grain boundaries g 0 s n 0 n
Combining Eq. (1) with Faraday’s law, ∇×E = iωB,
Weconsiderpenetrationofamicrowavefield(i.e.,mag- we obtain the London equation for magnetic induction
nreentticdeinndsiutcytiJo)ninBto=supµe0rcHon,deulcetcotrriscthfiaetldocEcu,paynadsecmuri-- B =Bz(x,y)zˆ for y 6=ma as
infinite area of x>0. We investigate linear response for B −Λ2∇2B =0. (3)
z g z
small microwave power limit, such that the time depen-
denceofthemicrowavefieldisexpressedbytheharmonic For ideal homogeneous superconductors without GBs,
factor, e−iωt, where ω/2π is the microwave frequency Eq. (3) is valid for −∞ < y < +∞ and the so-
that is much smaller than the energy-gap frequency of lution is simply given by Bz(x) = µ0H0e−x/Λg, and
2
the electric field is obtained from Eq. (1) as E (x) = where γ = 2πJ /ωφ = 1/2ωµ λλ2 . The total tun-
y sj cj 0 0 J
−iωµ0ΛgH0e−x/Λg. The surfaceimpedance Zs0 =Rs0− neling current across GB is thus given by
iX forhomogeneoussuperconductorsisgivenbyZ =
s0 s0
1 ∂B
Ey(x = 0)/H0 = −iωµ0Λg. The surface resistance − z =Jsj +Jnj =(iγsj +γnj)Vj. (10)
Rs0 = Re(Zs0) and reactance Xs0 = −Im(Zs0) of ideal µ0 ∂x (cid:12)y=ma
homogeneous superconductors without GBs are given (cid:12)
(cid:12)
by16 Integrationof(cid:12)Faraday’slaw,∂Ey/∂x−∂Ex/∂y =iωBz,
yields
R = µ2ω2λ3σ /2, (4)
s0 0 n E (x,y =ma+0)−E (x,y =ma−0)
x x
X = µ ωλ (5)
s0 0 ma+0 ∂E (x,y) ∂V (x)
y j
= dy −iωB (x,y) = ,(11)
z
forσn/σs ≪1wellbelowthe superconductingtransition Zma−0 (cid:20) ∂x (cid:21) ∂x
temperature T .
c whereweusedEq.(6). Thestaticversion(i.e.,ω →0)of
Eq. (11) corresponds to Eq. (4) in Ref. 22. Substitution
of Eqs. (1) and (10) into Eq. (11) yields the boundary
C. Josephson-junction model for intergrain current
condition for B at y =ma,
z
WeadopttheJosephson-junctionmodel14,15,16fortun- ∂B ∂B aΛ2∂2B
z z j z
− + = ,
nGeBlinjugnccutirornenstisadcreotesrsmGinBesdabtyyth=egmaua.ge-Binevhaarviiaonrtopfhtahsee ∂y (cid:12)(cid:12)y=ma+0 ∂y (cid:12)(cid:12)y=ma−0 Λ2g ∂x2 (cid:12)(cid:12)y=ma
difference across GBs, ϕj(x), and the voltage induced whereΛ (cid:12)(cid:12)is the character(cid:12)(cid:12)isticlengthforac field(cid:12)(cid:12)(cid:12)pene(t1r2a)-
across GB, V (x), is given by the Josephson’s relation, j
j
tion into GBs defined by
ma+0 φ0 Λ−2 = ωµ a(γ −iγ )
Eydy =Vj = (−iωϕj), (6) j 0 sj nj
Zma−0 2π = µ0a(2πJcj/φ0−iωγnj). (13)
where φ is the flux quantum. The tunneling current
0
parallel to the y axis is given by the sum of the su-
III. SURFACE IMPEDANCE
perconductingtunnelingcurrent(i.e.,Josephsoncurrent)
J = J sinϕ and the normal tunneling current (i.e.,
sj cj j
A. Microwave field and surface impedance
quasiparticle tunneling current) J = γ V . The crit-
nj nj j
ical current density J at GB junctions is one of the
cj
mostimportantparametersinthepresentpaper,andthe Equations(3)and(12)arecombinedintoasingleequa-
resistance-area product of GB junctions corresponds to tion for x>0 and −∞<y <+∞ as
1/γ . We neglect the displacement current across GBs,
Jdj n=j −iωCjVj where Cj is the capacitance of the GB B −Λ2∇2B =aΛ2 +∞ ∂2Bzδ(y−ma), (14)
junctions. z g z j ∂x2
m=−∞
Here we define the Josephson length λ and the char- X
J
acteristic current density J as whose solution is calculated as
0
∞
B (x,y) 2 cosh[K(y−a/2)]
λJ = (φ0/4πµ0Jcjλ)1/2, (7) z = e−x/Λg + dk
µ H π Λ2K2sinh(Ka/2)
J = φ /4πµ λ3. (8) 0 0 Z0 g
0 0 0
ksinkx
× (15)
The ratio Jcj/J0 = (λ/λJ)2 characterizes the coupling (2KΛ2g/aΛ2j)+k2coth(Ka/2)
strength of GB junctions.22 For weakly coupled GBs,
namely, Jcj/J0 = (λ/λJ)2 ≪ 1 (e.g., high-angle GBs), for0<y <a, whereK =(k2+Λ−g2)1/2. Theright-hand
electrodynamics of the GB junctions can be well de- side of Eq. (14) and the second term of the right-hand
scribed by the weak-link model.14,15,16 For strongly cou- side of Eq. (15) reflect the GB effects. See Appendix A
pled GBs, namely, Jcj/J0 =(λ/λJ)2 >∼1 (e.g., low-angle for the derivation of Eq. (15) from Eq. (14).
GBs), the Josephson-junction model is still valid but re- Electric field in the grains is obtained from Eq. (1) as
quires appropriate boundary condition at GBs, as given Ey = iωΛg2∂Bz/∂x, and voltage induced across GB is
inEq.(4)inRef.22,aspointedoutbyGurevich;seealso obtained from Eq. (10) as V = iωaΛ2∂B /∂x . The
j j z y=0
Refs. 21 and 23. mean electric field E¯ at the surface of the superconduc-
s (cid:12)
In the small-microwave-powerlimit suchthat sinϕj ≃ tor is thus calculated as (cid:12)
ϕ =2πV /(−iωφ ) for |ϕ |≪1, the J is reduced to
j j 0 j cj 1 a−0
E¯ ≡ dyE (x=0,y)
Jsj ≃Jcjϕj =iγsjVj, (9) s aZ−0 y
3
1 a−0 γ ∝ J . The dissipation in the grains, σ |E|2/2, and
= V (x=0)+ dyE (x=0,y) sj cj n
a(cid:20) j Z+0 y (cid:21) theintragraincontributiontothesurfaceresistance,Rsg,
therefore, tend to increase with increasing J . The dis-
= iω Λ2 ∂Bz + Λg2 a−0dy ∂Bz . sipationatGBs,γnj|Vj|2/2,andthe intergracijncontribu-
" j ∂x (cid:12)x=y=0 a Z+0 ∂x (cid:12)x=0# tion to the surface resistance, Rsj, on the other hand,
(cid:12) (cid:12) decrease with increasing J .
(cid:12) (cid:12) (16) cj
(cid:12) (cid:12) The surface reactance X = −Im(Z ) is also divided
s s
Substitution of Eq. (15) into Eq. (16) yields the surface into two contributions,
impedance Z =R −iX ≡E¯ /H as
s s s s 0
X =X +X , (23)
∞ s sg sj
Z 2 1
s
= 1+ dk
−iωµ Λ π Λ3K3 wheretheintragraincontributionX andthe intergrain
0 g Z0 g sg
1 contribution Xsj are given by
× . (17)
(KΛ2/Λ2)+(k2a/2)coth(Ka/2)
g j 1 ∞ a−0 ω
X = dx dy σ |E|2+ |B |2 ,
The surface resistance and reactance are given by Rs = sg a|H0|2 Z0 Z+0 (cid:18) s µ0 z (cid:19)
Re(Z ) and X =−Im(Z ), respectively.
s s s (24)
∞
1
X = dxγ |V |2. (25)
B. Microwave dissipation and surface resistance sj a|H0|2 Z0 sj j
Both X and X decrease with increasing J .
The time-averaged electromagnetic energy passing sg sj cj
through the surface of a superconductor at x = 0 and
−0<y <a−0 is given by the real part of
C. Simplified expressions for surface impedance
1 a−0 a
E = dy(E B∗) = E¯ H∗, (18)
2µ0 Z−0 y z x=0 2 s 0 The following Eqs. (26)–(34) show simplified expres-
sions of the surface impedance Z , the surface resistance
where E¯s =ZsH0 is defined by Eq. (16), and (Bz)x=0 = Rs = Re(Zs), and the surface resactance Xs = −Im(Zs)
µ0H0. Poynting’s theorem24 states that E is identical to for certain restricted cases, assuming σn/σs ≪ 1 and
the energy stored and dissipated in the superconductor, γ /γ ≪1 well below the transition temperature.
nj sj
1 ∞ a−0 For small grains of a ≪ λ such that coth(Ka/2) ≃
E = dx dy(σ −iσ )|E|2 2/Ka, Eq. (17) is reduced to
n s
2
Z0 (cid:20)Z+0
+(γ −iγ )|V |2− a−0dy iω|B |2 .(19) Zs ≃−iωµ0 Λ2g+Λ2j 1/2. (26)
nj sj j z
Z−0 µ0 (cid:21) (cid:0) (cid:1)
Theright-handsideofEq.(14)isreducedtoΛ2∂2B /∂x2
TherealpartsofEqs.(18)and(19)showthatthesurface j z
fora≪λ,andtheeffectiveacpenetrationdepthisgiven
resistanceRs =Re(E¯s/H0)=Re(Zs)iscomposedoftwo by Λ = (Λ2 + Λ2)1/2 as in Ref. 21, resulting in the
terms: eff g j
surfaceimpedance givenbyEq.(26). TheR andX for
s s
small grains is obtained as
R =R +R . (20)
s sg sj
The intragraincontributionRsg is fromthe energydissi- Rs ≃ 1+ 2λ J0 −1/2 1+ 4λ2γnj J0 2 , (27)
pation in the grains,and the intergraincontribution Rsj Rs0 (cid:18) a Jcj(cid:19) " aσn (cid:18)Jcj(cid:19) #
is from the dissipation at GBs:
X 2λ J +1/2
s 0
Rsg = a|H1 |2 ∞dx a−0dyσn|E|2, (21) Xs0 ≃ (cid:18)1+ a Jcj(cid:19) , (28)
0 Z0 Z+0
1 ∞ where Rs0, Xs0, and J0 are defined by Eqs. (4), (5)
Rsj = a|H |2 dxγnj|Vj|2. (22) and (8), respectively. Equation (27) is decomposed
0 Z0 into the intragrain R and intergrain R contribu-
sg sj
Both the intragrain current |Jg| around GBs and the tions, asRsg/Rs0 ≃(1+2λJ0/aJcj)−1/2 and Rsj/Rsg ≃
intergrain tunneling current |Jj| across GBs are sup- (4λ2γnj/aσn)(J0/Jcj)2, respectively.
pressed by the GBs, and are increasing functions of Equation (26) is further simplified when a ≪ 2λ2/λ
J
J . With increasing J , the intragrain electric field for small grain and weakly coupled GBs as
cj cj
|E| = |J /(σ +iσ )| also increases, whereas the inter-
g n s
grain voltage |V | = |J /(γ +iγ )| decreases because Z ≃−iωµ Λ , (29)
j j nj sj s 0 j
4
and we have
103 (a)
R 2γ λ 2λ 1/2 J 3/2 R =R +R
s ≃ nj 0 , (30) s sj sg
Rs0 σn (cid:18) a (cid:19) (cid:18)Jcj(cid:19) 102 Rsj
Xs 2λ 1/2 J0 1/2 Rsg
≃ . (31) 0
Xs0 (cid:18) a (cid:19) (cid:18)Jcj(cid:19) RR/ss 101 Rs ~ Jcj–1.5 a /λ = 0.1
Thus, we obtain the dependence of R and X on the
s s
material parameters as R ∝ γ a−1/2J−3/2 and X ∝
a−1/2J−1/2, which are indsependnejnt of λc.jThe R gisven 100
cj s
by Eq. (30) for the small grain and weakly coupled GBs
is mostly caused by intergrain dissipation, Rs ≃ Rsj ≫ 10–1
R . For X given by Eq. (31), on the other hand, both
sg s
intragrainX andintergrainX contributetothe total 103 (b)
sg sj
Xs =Xsg+Xsj. a/λ = 0.1

For large Jcj (i.e., strong-coupling limit) such that a/λ = 1
KΛ2/Λ2 ≫(k2a/2)coth(Ka/2), Eq.(17) for the surface 102
g j a/λ = 5
impedance Z is simplified as
s 0
s
R
Zs ≃−iωµ0 Λg+Λ2j/2Λg , (32) R/s 101
and we have (cid:0) (cid:1)
100
R λ J 4λ2γ J 2
s 0 nj 0
≃ 1− + , (33)
R aJ aσ J
s0 cj n (cid:18) cj(cid:19)
X λ J 10–1
s 0
≃ 1+ . (34)
Xs0 aJcj 102 a/λ = 0.1 (c)
X ~ J –0.5
The first and second terms of the right-hand side of s0 s cj a /λ = 1
wEqh.er(e3a3s)tchoerrtehsiprdontedrmtoctohrereisnptorangdrsatinotchoentinritbeurgtiroanin, Rcosng-, XX/s 101 a /λ = 5
tribution, R .
sj
100
IV. DISCUSSION 10–2 10–1 100 101
J /J
cj 0
Figure 1(a) and (b) shows J dependence of R . As
cj s
shown in Fig. 1(a), the intergrain contribution Rsj is FIG. 1: Dependence of surface resistance Rs = Re(Zs) and
dominant for weakly coupled GBs (i.e., small Jcj/J0 surface reactance Xs =−Im(Zs) [i.e., Eq. (17) with Eqs. (2)
regime),whereastheintragraincontributionRsg isdom- and (13)] on critical current density Jcj at GB junctions. Rs
inant for strongly coupled GBs (i.e., large Jcj/J0). The is normalized to the surface resistance without GB, i.e., Rs0
R decreases with increasing J as R ∝ J−1.5 [see given by Eq. (4), Xs is normalized to Xs0 given by Eq. (5),
Eqsj. (30)], whereas Rsg increasescjwith Jscjj. Thcej result- aωn/d2πJc=j is10noGrHmza,liλzed=t0o.2J0µmde,fiσnned=as1E07q.Ω(−81).mP−a1r,aamnedteγrnsjar=e
ing surface resistance Rs =Rsj +Rsg nonmonotonically 1013Ω−1m−2, which yield Rs0 =0.25mΩ, Xs0 =16mΩ, and
depends on Jcj and has a minimum, because Rs is de- J0 = 1.6 ×1010A/m2. (a) Total surface resistance Rs =
termined by the competition between Rsj and Rsg. As Rsj+Rsg,intergraincontributionRsj givenbyEq.(22),and
shownin Fig.1(c), on the other hand, Xs monotonically intragrain contribution Rsg given by Eq. (21) for a/λ = 0.1.
decreaseswithincreasingJcj [i.e.,Xs ∝Jc−j0.5 forweakly (b) Rs and (c) Xs for a/λ=0.1, 1, and 5.
coupled GBs as in Eq. (31)].
The nonmonotonicdependence ofR on the grainsize
s
a is also seen in Fig. 1(b). For small Jcj/J0 the Rs de- a/λ=5, Rs/Rs0 ≈0.86for a/λ=1, and Rs/Rs0 ≈0.59
creases with increasing a as Rs ∝ a−0.5 [see Eq. (30)], for a/λ = 0.1. The minimum Rs/Rs0 is further reduced
whereas Rs increases with a for large Jcj/J0. when λγnj/σn is further reduced.
The R for strongly coupled GBs can be smaller Theoretical results shown above may possibly be ob-
s
thanR foridealhomogeneoussuperconductorswithout served by measuring R , X , and J in Ca doped
s0 s s cj
GBs, namely, Rs/Rs0 <1 for Jcj/J0 >∼1. The minimum YBa2Cu3O7−δ films. The enhancement of Jcj (Ref. 19)
surfaceresistanceforλγ /σ =0.2isR /R ≈0.97for and reduction of R (Ref. 13) by Ca doping are individ-
nj n s s0 s
5
ually observedin YBa2Cu3O7−δ, but simultaneous mea- Bz(x,ma)=Bz(x,0) as
surements of J and R are needed to investigate the
cj s
relationshipbetween Rs andJcj. The nonmonotonicJcj ˜b(k,q) = ∞dx +∞dyB (x,y)e−iqysinkx, (A1)
dependence of R for strongly coupled GBs may be ob- z
s Z0 Z−∞
sweirtvhedlairngehiJgchjqounaltihtyefiolrmdserwoitfhJs0m∼all1g0r1a0iAns/ma2<aλt alonwd ˜b0(k) = ∞dxBz(x,0)sinkx= +∞ dq ˜b(k,q),
temperatures. Z0 Z−∞ 2π
(A2)
V. CONCLUSION respectively. The Fourier transform of Eq. (14) leads to
We havetheoreticallyinvestigatedthe microwave-field ˜b(k,q) k
= 2πδ(q)
distribution in superconductors with laminar GBs. The µ0H0 K2
field calculation is based on the two-fluid model for αk k˜b (k)
current transport in the grains and on the Josephson- + e−imqa 1− 0 (,A3)
K2+q2 µ H
junctionmodelfortunnelingcurrentacrossGBs. Results m " 0 0 #
whereK =(k2+Λ−2)1/2 anXdα=aΛ2/Λ2. Substituting
showthat the microwavedissipationatGBs is dominant g j g
Eq. (A3) into Eq. (A2), we have
for weakly coupled GBs of J ≪ J , whereas dissipa-
cj 0
tion in the grains is dominant for strongly coupled GBs
of Jcj ≫ J0. The surface resistance Rs nonmonotoni- ˜b0(k) = k +αk 1− k˜b0(k) +∞ dq e−imqa ,
JcallyasdeRpen∝dsJo−n1.5Jcfjo;rtJhe R≪s dJec,rweahseerseawsitRh inincrceraeasisnegs µ0H0 K2 " µ0H0 #Xm Z−∞ 2πK2+q2
cj s cj cj 0 s (A4)
with J for J ≫J . The intragraindissipation can be
cj cj 0
suppressed by GBs, and the surface resistance of super-
which is reduced to
conductors with GBs can be smaller than that of ideal
homogeneous superconductors without GBs. ˜b (k) 1 2 1
0
= − . (A5)
µ H k kKΛ22K+αk2coth(Ka/2)
0 0 g
Acknowledgments B (x,y) is calculated from˜b(k,q) given by Eq. (A3) as
z
I gratefully acknowledgeH. Obara,J.C. Nie, A. Sawa, B (x,y) 2 ∞ +∞ dq ˜b(k,q)
M. Murugesan, H. Yamasaki, and S. Kosaka for stimu- z = dk eiqysinkx
lating discussions. µ0H0 π Z0 Z−∞ 2π µ0H0
2α ∞ k˜b (k)
= e−x/Λg + dkksinkx 1− 0
π µ H
APPENDIX A Z0 " 0 0 #
+∞ dq eiq(y−ma)
× . (A6)
Equation (15) is derived by solving Eq. (14) with the −∞ 2π K2+q2
m Z
boundary condition of B =µ H at x=0, as follows. X
z 0 0
We introduce the Fourier transform of B (x,y) and Substitution of Eq. (A5) into Eq. (A6) yields Eq. (15).
z
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