Limits of the upper critical field in dirty two-gap superconductors

0.27 MB·English

by  A. Gurevich

Limits of the upper critical field in dirty two-gap superconductors
A. Gurevich
National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32310, USA
(Dated: February 6, 2008)
7 An overview of the theory of the upper critical field in dirty two-gap superconductors, with a
0 particularemphasisonMgB isgiven. Wefocus hereonthemaximum H whichmaybeachieved
2 c2
0 byincreasingintrabandscattering,andonthelimitationsimposedbyweakinterbandscatteringand
2 paramagneticeffects. Inparticular,wediscussrecentexperimentswhichhaverecentlydemonstrated
ten-foldincreaseofH indirtycarbon-dopedfilmsascomparedtosinglecrystals,sothattheH (0)
n c2 c2
a paralleltotheabplanesmayapproachtheBCSparamagneticlimit,Hp[T]=1.84Tc[K]≃60−70T.
J Neweffectsproducedbyweak interbandscatteringinthetwo-gap Ginzburg-Landauequationsand
H (T)in ultrathin MgB films are addressed.
2 c2 2
1
PACSnumbers: PACSnumbers: 74.20.De, 74.20.Hi, 74.60.-w

]
n
o INTRODUCTION decouple the bands due to formation of interband tex-
c tures of 2π planar phase slips in the phase difference
-
pr It is now well established that superconductivity in θ(x) = θ1 − θ2 [16, 17] well below the global depair-
ing current. In turn, the weaknessof interbandimpurity
u MgB2 with the unexpectedly high critical temperature
scattering makes it possible to radically increase the up-
s T 40K [1], is due to strong electron-phonon inter-
. c ≈ per critical field H by selective alloying of Mg and B
t action with in-plane boron vibration modes. Extensive c2
a sites with nonmagnetic impurities.
m ab-initio calculations [2, 3, 4], along with many experi-
mental evidences from STM, point contact, and Raman Despite the comparatively high Tc, the upper critical
-
d spectroscopy, heat capacity, magnetization and rf mea- fieldofMgB2 singlecrystalsisratherlowandanisotropic
n surements [5, 6] unambiguously indicate that MgB ex- with H⊥(0) 3 5T and H||(0) 15 19T of [5, 6],
o hibits two-gap s-wave superconductivity [7, 8]. M2gB where tch2e in≃dices− and coc2rresp≃ond t−o the magnetic
c 2 ⊥ ||
has two distinct superconducting gaps: the main gap field H perpendicular and parallel to the ab plane, re-
[
∆σ(0) 7.2mV,whichresidesonthe2Dcylindricalparts spectively. Since these Hc2 values are significantly lower
1 of the ≈Fermi surface formed by in-plane σ antibonding than Hc2(0) 30T for Nb3Sn [18, 19], there had been
v ≃
1 pxy orbitals of B, and the smaller gap ∆π(0) ≈ 2.3mV initial scepticism about using MgB2 as a high-field su-
8 on the 3D tubular part of the Fermi surface formed by perconductor, until several groups undertook the well-
2 out-of-planeπ bondingandantibondingpz orbitalsofB. established procedure of Hc2 enhancement by alloying
1 MgB withnonmagnetic impurities. The results ofhigh-
The discovery of MgB has renewed interest in new 2
0 2 field measurements on dirty MgB films and bulk sam-
effects of two-gap superconductivity, motivating differ- 2
7
ples has shown up to ten-fold increase of H⊥ as com-
0 entgroupsto takecloserlooksatotherknownmaterials, c2
paredto singlecrystals[20, 21, 22, 23, 24, 25, 26, 27, 28,
/ suchasYNi B CandLuNi B Cborocarbides[9]Nb Sn
at [10], or NbS2e 2[11], heavy-f2erm2 ion [12] and organic 3[13] 29, 30, 31], particularly in carbon-doped thin films [28]
2
m made by hybrid physico-chemical vapor deposition [32].
superconductors, for which evidences of the two gap be-
ThisunexpectedlystrongenhancementofH (T)results
- havior have been reported. However, several features of c2
d from its anomalous upward curvature, rather different
MgB set it apart from other two-gap superconductors.
n 2
from that of H (T) for one-gap dirty superconductors
o Not only does MgB2 have the highest Tc among all non- c2
[33, 34, 35, 36]. As shown in Fig. 1, H of MgB C-
c cupratesuperconductors,italsohastwocoexistingorder c2 2
v: parameters Ψσ = ∆σexp(iθ1) and Ψπ = ∆πexp(iθ2), doped films has already surpassed Hc2 of Nb3Sn, which
could make cheap and ductile MgB an attractive mate-
i which are weakly coupled. The latter is due to the fact 2
X rial for high field applications [37].
thattheσandπbandsareformedbytwoorthogonalsets
ar of in-plane and out-of-plane atomic orbitals of boron, so ThisradicalenhancementofHc2 showninFig. 1isin-
alloverlapintegrals,whichdeterminematrixelementsof deed assistedby the features of two-gapsuperconductiv-
interbandcouplingandinterbandimpurityscatteringare ity in MgB2. Fig. 2 gives anotherexample ofHc2(T) for
strongly reduced [14]. This feature can result in new ef- a fiber-textured film [25], which exhibits an upward cur-
fects, which are very important both for the physics and vature of Hc2(T) for H c. This behavior of Hc2(T) and
||
applications of MgB . Indeed, two weakly coupled gaps the anomalous temperature-dependent anisotropy ratio
2
result in intrinsic Josephson effect, which can manifest Γ(T) = H||(T)/H⊥(T) are different from that of the
c2 c2
itself in low-energy interband Josephson plasmons (the one-gap theory in which the H (T) has a downward
c2
Legget mode) [15] with frequencies smaller than ∆ /~. curvature, while the slope H′ = dH /dT at T is pro-
π c2 c2 c
Moreover, strong static electric fields and currents can portionalto the normalstate residual resistivityρ , and
n
2
50 1
H
π
0.8
40 σ
0)
a (20.6
sl 30 H⊥ab H||ab Hc
, Tec220 H(T)/c20.4
H NbSn 0.2 Hc(π2) Hc(σ2)
10 3
0
NbTi 0 0.2 0.4 0.6 0.8 1
T/T
0 c
0 10 20 30 40
T, K
FIG. 3: The mechanism of the upward curvature of H (T)
c2
illustrated by the bilayer toy model shown in the inset. The
FIG.1: H (T)forcarbon-dopedMgB films[28]incompar-
c2 2 dashed curves show H (T) calculated for σ and π films in
ison with NbTi and Nb Sn. The red and blue lines show fits c2
3 theone-gapdirtylimitwiththeBCScouplingconstantsλ =
from Eq. (19) with g=0.045 σ
0.81, λ = 0.285, and D = 0.1D . The solid curve shows
π π σ
. H (T) calculated from Eq. (26) of the two-gap dirty limit
c2
theory for theBCS matrix constants from Ref. [77]
50
40 ward curvature of Hc2(T). If the σ film is dirtier, the π
film only results in a slight shift of the H curve and a
c2
esla 30 reduction of the slope Hc′2 near Tc.
H, Tc220 depTehnedbeinlaceyeorfmHocd2e(lαa,lTso)cfloarriHfiesintchleinaendombyaltohuesaannggluelaαr
with respect to the c-axis (parallel to the film normal in
10
Fig. 3)[41]. InthiscasebothH(σ)(α,T)andH(π)(α,T)
c2 c2
depend on α according to the temperature-independent
0
0 10 T2,0 K 30 40 one-gap scaling Hc2(α) = Hc2(0,T)/ cos2α+ǫsin2α
[42, 43], but with very different effective mass ratios ǫ=
p
m /m for eachfilm. Because the σ bandis muchmore
ab c
FIG. 2: Hc2(T) of a fiber-textured MgB2 film [25] both par- anisotropic than the π band, ǫσ 1, and ǫπ 1 [44,
allel (triangles) and parallel (squares) to the ab planes. The 45], the one-gapangular scaling fo≪r the global H∼ (α,T)
c2
solid lines show calculations from Eq. (19) with g = 0.065,
breaks down. For example, in the case shown in Fig. 3,
D ≪D(ab) forH||candD =0.19(D(c)D(ab))1/2 forH ⊥c.
π σ π σ σ H (T) is anisotropic at higher T, but at lower T, the
c2
nearly isotropicπ band reduces the overallanisotropyof
H , so the ratio Γ(T) = H||(T)/H⊥(T) decreases as
Hc2(0)=0.69TcHc′2[33,34,35,36]. However,thebehav- Tcd2ecreases. This is charactecr2istic ofcm2any dirty MgB2
ior of Hc2(T) in MgB2 can be explained by the two-gap films like the one shown in Fig. 2, for which the π band
theoryinthedirtylimitbasedoneitherUsadelequations is typically much dirtier than the σ band. By contrast,
[38, 39] or Eliashberg equations [9, 40]. incleanMgB singlecrystalsΓ(T)increasesfrom 2 3
2
≃ −
The behavior of H (T) can be qualitatively under- near T to 5 6 at T T [46, 47, 48, 49, 50, 51, 52,
c2 c c
≃ − ≪
stood using a simple bilayer model shown in Fig. 3, 53, 54, 55, 56]. This behavior was explained by two-gap
which captures the physics of two-gap superconductiv- effects in the clean limit [57, 58].
ity in MgB2, and suggests ways by which Hc2 can be Fig. 3 suggests that Hc2(T) of MgB2 can be signifi-
further increased. Indeed, MgB2 can be mapped onto a cantly increased at low T by making the π band much
bilayer in which two thin films corresponding to σ and dirtierthanthemainσ band. Thiscouldbedonebydis-
π bands are separated by a Josephson contact, which orderingthe Mg sublattice, thus disrupting thep boron
z
modelstheinterbandcoupling. TheglobalHc2(T)ofthe out-of-plane orbitals, which form the π band. Achiev-
such weakly-coupledbilayeris mostly determined by the ing high H requires that both σ and π bands are in
c2
(σ) (π)
film with the highest Hc2, even if Tc and Tc are very the dirty limit. Yet, making the π band much dirtier
different. For example, if the π film is much dirtier than than the σ band provides a ”free boost” in H without
c2
theσ filmthenH(σ) dominatesathigherT,butatlower too much penalty in T suppression due to pairbreak-
c2 c
temperatures the π film takes over, resulting in the up- inginterbandscatteringorbanddepletiondue todoping
3
[59, 60]. In fact, the interband scattering is weak for Althoughtheσbandisanisotropic,MgB isnotalayered
2
the same reason that Ψ and Ψ are weakly coupled, material[75,76],sothecontinuumBCStheoryisapplica-
σ π
which may enable alloying MgB with more impurities blebecausethec-axiscoherencelengthξ ismuchlonger
2 c
to achieve higher H . Systematic incorporation of im- than the spacing between the boron planes 3.5˚A. In-
c2
∼
purities in MgB2 has not been yet achieved because the deed,evenforHc⊥2(0)=40TandHc||2(0)=60TinFig. 1,
complex substitutional chemistry of MgB2 is still poorly the anisotropic Ginzburg-Landau (GL) theory [42] gives
understood[61,62,63,64]. Severalgroupshavereported ξ = (φ H⊥/2π)1/2/H|| 19˚A. Strong coupling in
a significant increase in H by irradiation with protons c 0 c2 c2 ≈
c2 MgB should be described by the Eliashberg equations
2
[65], neutrons [66, 67] or heavy ions [68], but so far the
[40], butwe considerhere manifestations ofintraandin-
carbonimpuritieshavebeenthemosteffectivetoprovide
terbandscatteringandparamagneticeffectsinH using
c2
the huge H enhancement shown in Figs. 1 and 2. The
c2 the more transparent two-gapUsadel equations [38]
effect of carbon on different superconducting properties
canberathercomplex[69,70,71]andstillfarfrombeing Dαβ
fully understood. Yet given the indisputable benefits of ωf1 1 [g1ΠαΠβf1 f1 α βg1]
− 2 − ∇ ∇
carbonalloying,onecanposethebasicquestion: howfar
=Ψ g +γ (g f g f ) (2)
1 1 12 1 2 2 1
can H be further increased? −
Thecb2ilayermodelsuggeststhatHc2 increasesifintra- ωf D2αβ[g Π Π f f g ]
2 2 α β 2 2 α β 2
bandscatteringisenhanced. However,becauseintraband − 2 − ∇ ∇
impurity scattering causes an admixture of pairbreaking =Ψ2g2+γ21(g2f1 g1f2), (3)
−
interband scattering, the first question is to what ex-
tent weak interband scattering in MgB2 can limit Hc2. Here the Usadel Green’s functions fm(r,ω) and gm(r,ω)
Another important question is how far is the observed in the m-th band depend on r and the Matsubara fre-
Hc2 fromthe paramagneticlimit Hp. In the BCS theory quency ω =πT(2n+1), Dmαβ are the intraband diffusiv-
Hp is defined by the condition: µBHpBCS = ∆/√2, or ities due to nonmagnetic impurity scattering, 2γmm′ are
HBCS[T] = 1.86T [K] [72], where µ is the Bohr mag- the interband scattering rates, Π = +2πiA/φ0, A is
p c B ∇
the vector potential, and φ is the flux quantum. Eqs.
neton. For T = 35K, this yields H = 65T, not that 0
c p
far fromthe zero-fieldH||(0)in Figs. 1 and 2. However, (2) and (3) are supplemented by the equations for the
c2 order parameters Ψ =∆ exp(iϕ ),
theBCSmodelunderestimatesH ,whichissignificantly m m m
p
enhanced by strong electron-phonon coupling [73]:
ωD
H (1+λ )HBCS, (1) Ψm =2πT λmm′fm′(r,ω), (4)
p ≃ ep p ω>0 m
XX
whereλ istheelectron-phononconstant. Takingλ
ep ep
1 for the σ band [2, 3], we obtain Hp ∼ 130T, so ther≈e normalization condition |fm|2+gm2 = 1, and the super-
still a large room for increasing H by optimizing the current density
c2
intra and interband impurity scattering. For instance,
increasing Hc′2 to a rather common for many high field Jα =−2πeTIm NmDmαβfm∗Πβfm. (5)
superconductors value of 2T/K (much lower than H′ ω m
c2 ≃ XX
5 14T/K for PbMo S [74]) could drive H of MgB
6 8 c2 2
− HereN isthe partialelectrondensity ofstatesforboth
withT 35Kabove70T.Inthefollowingwegiveabrief m
c
≃ spins in the m-th band, and α and β label Cartesian
overview of recent results in the theory of dirty two-gap
indices. Eqs. (4) contains the matrix of the BCS cou-
superconductorsfocusingonneweffectsbroughtbyweak
interbandscatteringandparamagneticeffects. Themain pling constants λmm′ = λ(mepm)′ −µmm′, where λ(mepm)′ are
conclusionisthat,althoughinterbandscatteringinMgB2 electron-phonon constants, and µmm′ is the Coulomb
is indeed weak, it cannot be neglected in calculations pseudopotential. The diagonal terms λ11 and λ22 quan-
of Hc2(T). We will also address the crossover from the tify intraband pairing, and λ12 and λ21 describe inter-
orbitally-limited to the paramagnetically limited H in band coupling. Hereafter, the following ab initio values
c2
a two-gap superconductor. λσσ 0.81, λππ 0.285, λσπ 0.119, and λπσ 0.09
≈ ≈ ≈ ≈
[77] are used. There are also the symmetry relations:
THO-GAP SUPERCONDUCTORS IN THE N λ =N λ , N γ =N γ (6)
1 12 2 21 1 12 2 21
DIRTY LIMIT
where N 1.3N for MgB . Solutions of Eqs. (2)-(6)
π σ 2
≈
WeregardMgB asadirtyanisotropicsuperconductor minimize the following free energy Fd3r [17]:
2
with two sheets 1 and 2 of the Fermi surface on which
1 R
sthpeecstuivpeelryco(inndduiccetsin1gagnadps2tcaokreretshpeovnadluteosσ∆a1nadnπdb∆a2n,drse)-. F = 2 NmΨmΨ∗mλ−m1m′ +F1+F2+Fi (7)
mm′
X
4
Here F and F are intraband contributions,
1 2 40
F =2πT N [(ω(1 g ) (8)
m m m
− − 30
ω>0
X
Re(f∗∆ )+Dαβ[Π f Π∗f∗ + g g ]/4 K
m m m α m β m ∇α m∇β m T, 20
and F is due to interband scattering [78]:
i
10
F =2πqT [1 g g Re(f∗f )], (9)
i − 1 2− 1 2
ωX>0 00 0.5 1 1.5 2
g = γ /2πT
where 2q=N γ +N γ . The Usadel equations result + c0
1 12 2 21
from δF/δf∗ = 0, ∂F/∂Ψ∗ = 0, and J = cδF/δA.
m m −
Taking f =sinα and g =cosα , we obtain
m m m m FIG. 4: Dependence of the critical temperature T on the
c
interband scattering parameter g calculated from Eq. (13)
ωsinα +γ sin(α α )=∆ cosα , (10)
1 12 1− 2 1 1 with theBCS matrix constants λmn from Ref. [77]
ωsinα +γ sin(α α )=∆ cosα . (11)
2 21 2 1 2 2
−
These coupled equations along with Eq. (4) define the UPPER CRITICAL FIELD FOR Hkc
two-gap uniform states for J =0.
H alongthe c-axisis the maximumeigenvalueofthe
c2
linearized Eqs. (2) and (3):
CRITICAL TEMPERATURE
D
(ω iµ H)f 1Π2f =∆ +(f f )γ , (17)
B 1 1 1 2 1 12
Eqs. (2) and (3) give the well-known results for T ± − 2 −
c
in two-gap superconductors [7, 8, 79, 80]. For negligible (ω iµ H)f D2Π2f =∆ +(f f )γ , (18)
B 2 2 2 1 2 21
interbandscattering,substitutionoff =∆ /ωandf = ± − 2 −
1 1 2
∆ /ω into Eq. (4) yields:
2 Here the Zeeman paramagnetic term µ H, which re-
B
±
quires summation over both spin orientations in Eq.
T =1.14~ω exp[ (λ λ )/2w], (12)
c0 D − +− 0 (4), is included. In the gauge Ay = Hx, the solu-
w(λh2−er+e4λλ±12=λ21λ)111/2±. λT2h2e, wint=erbλa1n1λd2c2o−upλli1n2gλ2i1n,craenadseλs0Tc=0 ∆t˜iomnsexapr(e−fπmH(xx2)/φ=0)f,˜mwhexerpe(−f˜mπHisxe2x/pφr0e)s,seadnvdia∆∆˜mm(xfr)om=
as compared to noninteracting bands (λ = λ = 0), Eqs. (17) and (18). The solvability condition (4) of two
12 21
while intraband impurity scattering does not affect Tc0, linearequationsfor∆˜1and∆˜2 givestheequationforHc2
in accordance with the Anderson theorem. Solving the [38],whichaccountsforinterbandandintrabandscatter-
linearized Eqs. (2) and (3) with γmm′ =0, gives Tc with ing and paramagnetic effects:
6
the account of pairbreaking interband scattering:
(λ +λ )(lnt+U )+(λ λ )(lnt+U )
0 i + 0 i −
−
U g = (λ0+wlntc)lntc, (13) +2w(lnt+U+)(lnt+U−)=0, (19)
t − p+wlnt
(cid:18) c(cid:19) c where t=T/T , and
2p=λ +[γ λ 2λ γ 2λ γ ]/γ , (14) c0
0 − − 21 12 12 21 +
− −
U(x)=ψ(1/2+x) ψ(1/2), (15) λ =[(ω +γ )λ 2λ γ 2λ γ ]/Ω , (20)
i − − − 12 21 21 12 0
− − −
2Ω =ω +γ Ω , (21)
wheretc =Tc/Tc0andγ± =γ12±γ21,g =γ+/2πTc0,and Ω =[(ω± +γ+)2++4γ± γ0 ]1/2, (22)
ψ(x) is a digamma function. The dependence of T on 0 − − 12 21
c
the interband scattering parameter g is shown in Fig. 4. ω =(D D )πH/φ , (23)
± 1 2 0
±
Asg →∞,Eqs. (13)and(14)giveTc →Tc0exp(−p/w), U =Reψ 1 + Ω±+iµBH ψ 1 . (24)
and for g 1, we have ±
2 2πT − 2
≪ (cid:18) (cid:19) (cid:18) (cid:19)
π
T =T [λ γ +λ γ 2λ γ 2λ γ ] (16) Ifinterbandscatteringandparamagneticeffectsareneg-
c c0 0 + − − 21 12 12 21
−8λ − −
0 ligible, Eqs. (19)-(24) reduce to a simpler equation
This formula can be used to extract the interband scat- [38, 39], which can be presented in the parametric form:
tering rates from the small shift of T [81]. However, as
c lnt= [U(h)+U(ηh)+λ /w]/2+ (25)
shown below, even weak interband scattering can signif- − 0
icantly change the behavior of Hc2(T), so it cannot be [(U(h)−U(ηh)−λ−/w)2/4+λ12λ21/w2]1/2,
neglected even though g 1. H =2φ T th/D , (26)
c2 0 c 1
≪
5
whereη =D /D ,andtheparameterhrunsfrom0to
2 1
∞
asTvariesfromTcto0. Forequaldiffusivities,η =1,Eq. 1
g = 0.03
(26) simplifies to the one-gap de-Gennes-Maki equation
lnt+U(h)=0 [34, 35, 36].
Now we consider some limiting cases, which illustrate 0)
0,
how Hc2 depends on different parameters. Fig. 5 shows H(c20.5
the evolution of H as g increases for fixed D and D /2
c2 1 2 Hc
andnegligibleparamagneticeffects. Interbandscattering
reduces the upward curvature of H (T), H (0), and
c2 c2
T , while increasing the slope H′ at T . Notice that
c c2 c
the significant changes in the shape of H (T) in Fig. 5 0
c2 0 10 20 30 40
occur for weak interband scattering (g 1), which also T, (K)
≪
provides a finite H (0) even if D 0. For example,
c2 2
the high-field films in Fig. 1 and 2 h→ave g 0.045 and 1
≃
0.065, respectively. For g 1, Eq. (19) yields the GL Dπ = 0.05Dσ
linear temperature depende≪nce near Tc: 0.01)
0,
Hc2 = π2(8sφ0D(Tc+−sTD) ) (27) g)/H(c20.5
1 1 2 2 T,
(2
Hc
where T is given by Eq. (16), s = 1 + λ /λ and
c 1 − 0
s =1 λ /λ . Eq. (27)iswritteninthelinearaccuracy
2 − 0
−
in g 1. Higher order terms in g not only shift T but 0
≪ c 0 10 20 30 40
alsoincreasethe slope H′ atT , asevident fromFig. 5. T, (K)
c2 c
For s s , the slope H′ is mostly determined by the
1 ∼ 2 c2
cleanest band with the maximum diffusivity. However,
FIG.5: Effectofinterbandscatteringandthediffusivityratio
because of weak interband coupling in MgB , the values
2 on theevolution of H (T). Upperpanel shows H (T,η) for
of s and s are very different. For λ = 0.81, λ = c2 c2
1 2 11 22 the fixed g = 0.03 and different η = D /D : 0;0.05;0.1;0.5
2 1
0.285, λ = 0.119, λ = 0.09 [77], we get λ = λ
12 21 − 11 − (fromtoptobottomcurves). LowerpanelshowsHc2(T,g)for
λ22 = 0.525, λ0 = (λ2−+4λ12λ21)1/2 = 0.564, thus s1 = the fixed D2/D1 = 0.05 and different g = 0.01;0.05;0.1;0.5
1+λ /λ = 1.93, s = 1 λ /λ = 0.07. Thus, H′ (from top to bottom curves).
− 0 2 − − 0 c2
is mostly determined by D of the σ band. Yet, if the σ
1
band is so dirty that D /D < s /s 0.04, the slope
1 2 2 1
≃ and m is the bare electron mass. Eq. (30) follows from
H′ is determined by the much cleaner π band.
c2 the basic diffusion relation l2 =D t, and the energy un-
At low T both the Zeeman and interband scattering 0
certainty principle ~2/2ml2=~/t for a particle confined
terms in Eq. (19) can be essential. Eq. (19) reduces to
in a region of length l. For g =0, Eq. (28) yields
the following equation for H (0):
c2
φ T f
0 c
µ2H2 µ2H2 Hc2(0)= exp( ), (31)
(λ +λ )ln B p +(λ λ )ln B p 2γ D˜ D˜ 2
0 i µ2H2+Ω2 0− i µ2H2+Ω2 1 2
B + B − 1/2
=wlnµ2Hµ2B2H+p2Ω2 lnµ2Hµ2B2H+p2Ω2 (28) f = wλ202 +ln2 DD˜˜21 +p2wλ− lnDD˜˜21! − λw0. (32)
B + B −
If D D , Eqs. (31)-(32) reduce to the result of Ref.
where µBHp = πTc0/2γ is the field of paramagnetic in- [38],0a≪nd fomr the symmetric case, D˜ = D˜ , Eqs. (31)-
stability of the superconducting state, and lnγ = 0.577. 1 2
(32) give the one-band result H (0) = φ T /2γD˜ [35].
Wefirstconsiderthelimitg 0,whichdefinesthemax- c2 0 c
→ However for D˜ = D˜ , H (0) can be much higher than
imumHc2(0) achievableina dirtytwo-gapsuperconduc- H (0)=0.69H1′6 T .2Indece2d, if the effective diffusivities,
tor with no T suppresion. In this case Ω = πD H/φ c2 c2 c
c + 1 0 D˜ and D˜ are very different, Eqs. (31)-(32) yield
andΩ =πD H/φ ,soforT T ,paramagneticeffects 1 2
− 2 0 c
≪
just renormalize intraband diffusivities in Eq. (28): Hc2(0)= 2φγ0DT˜c e−(λ−+λ0)/2w, D˜2 ≪D˜1e−λw0, (33)
2
Dm →D˜m =qDm2 +D02, (29) Hc2(0)= 2φγ0DT˜c e−(λ0−λ−)/2w, D˜1 ≪D˜2e−λw0.(34)
1
where D =µ φ /π is the quantum diffusivity
0 B 0
Thus, H (0) is determined by the minimum effective
c2
D =~/2m, (30) diffusivity, but unlike the limit D 0, H (0) remains
0 0 c2
→
6
canbe higher than the bulk H =φ /2πξ2 [76, 82]. Let
c2 0
1.2 us see how this result is generalized to two-gap super-
conductors. For a thin film of thickness d<max(ξ ,ξ ),
1 1 2
the functions f and f are nearly constant, so integrat-
1 2
0) 0.8 ing Eqs. (17) and (18) over x with ∂ f( d/2) = 0, re-
(p x ±
H sults in two linear equations for f and f with Π2 =
/20.6 1 2
Hc (πHd/φ0)2/3. Thus, we obtain the previous Eq. (19)-
0.4
(24) in which one should make the replacement
0.2
ω(f) (πHd/φ )2(D D )/6 (39)
0 ± → 0 1± 2
0.2 0.4 0.6 0.8 1
T/Tc Wefirstconsiderthecaseofnegligibleinterbandscatter-
ing and paramagnetic effects. Then Eq. (19) and (39)
give the square-root temperature dependence near T
c
FIG. 6: Crossover from the orbitally to paramagnetically
Dlimit=ed0H.05c2D(T)ancadlcDula/tDed f=rom0,E1,q5s,.20(2f6r)omandtop(37to)-(b3o8t)tofomr H(f) = 4φ0 3Tc(Tc−T) (40)
2 1 1 0 c2 π3/2d(s D +s D )1/2
curves, respectively. p1 1 2 2
characteristic of thin films [82] instead of the bulk GL
linear dependence (27). From Eqs. (31) and (39) we can
finite even for D 0 or D 0. In fact, if both
1 → 2 → also obtain H(f)(0) for D D :
D1 D0 and D2 D0, we return to the symmetric c2 0 ≪ m
case≪D˜ = D˜ , for w≪hich Eqs. (31)-(32) yield the result
1 2 φ 3T 1/2 exp(f/4)
of the one-gap dirty limit theory [36] H(f)(0)= 0 c (41)
c2 d πγ (D D )1/4
(cid:18) (cid:19) 1 2
H (0) H =φ T /2γD =πT /2γµ (35)
c2 → p 0 c 0 c B Nextweconsiderthecrossovertotheparamagneticlimit
in thin films at low temperatures. For neglect interband
For a one-band superconductor, Eq. (35) can also
scattering, the expressions µ2H2 +Ω2 under the loga-
be written as the paramagnetic pairbreaking condition, B
rithms in Eq. (28) become µ2H2 +(πHd/φ )4D2/36.
µBHp = ∆(0)/2, where ∆(0) = πTc/γ is the zero- B 0
temperature gap. For two-band superconductors, the Substituting here Hc(2f) ∼ φ0/ξd, we conclude that para-
meaning of H is less transparent, yet the maximum H magnetic effects become essential if
p p
expressed via T is given by the same Eq. (35) as for
c
min(D ,D )<D ξ/d. (42)
one-band superconductors. 1 2 0
Finallyweconsiderhowparamagneticeffectsaffectthe
Thus, reducing the film thickness extends the region of
shapeofH (T)inthelimitg 0. Thiscaseisdescribed
c2 → theparameterswhereHc2 islimitedbytheparamagnetic
by Eq. (26) modified as follows:
effects rather than by impurity scattering.
U(h) Reψ[1/2+h(i+p)] ψ(1/2) (36)
→ −
U(ηh) Reψ[1/2+h(pη+i)] ψ(1/2) (37) ANISOTROPY OF H AND H
→ − c1 c2
H =H =2φ T th/D , (38)
c2 c2 0 c 0
For anisotropic one-gap superconductors, the angular
where p = D1/D0, and η = D2/D1. Fig. 6 shows how dependence of the lower and the upper critical fields is
Hc2(T)evolvesfromtheorbitally-limitedHc2(T)withan given by [42, 43]
upward curvature at D D to the paramagnetically-
1 2
limitedHc2(T)ofaone-g≫apsuperconductorforD1 <D0 H (α,T)= Hc1(0,T), H (α,T)= Hc2(0,T) (43)
c1 c2
[72]. The nonmonotonic dependence of H (T) in Fig. 6 R(α) R(α)
c2
indicates the first order phase transition, similar to that
in one-gap superconductors. where R(α) = (cos2α+ǫsin2α)1/2, ǫ = mab/mc. Here
the anisotropy parameter Γ(T) = H||/H⊥ = ǫ−1/2 is
c2 c2
independent of T for both H and H . By contrast,
c1 c2
THIN FILMS IN A PARALLEL FIELD Γ (T)=H||/H⊥ forMgB singlecrystalsincreasesfrom
2 c2 c2 2
2 3 at T to 5 6 at T T , but Γ (T)=H||/H⊥
∼ − c − ≪ c 1 c1 c1
Hc2 canbesignificantlyenhancedinthinfilmsormul- decreases from 2 3 to 1 as T decreases [46, 47,
≃ − ≃
tilayers,inwhichMgB2layersareseparatedbynonsuper- 48, 49, 50, 51, 52, 53, 54, 55, 56]. This behavior was
conducting layers. It is well known that in a thin film of explained by the two-gap theory in the clean limit [57,
thickness d < ξ in a parallel field, H(f) = 2√3H ξ/d 58, 83, 84].
c2 c2
7
ThedirtylimitismoreintricateinthesensethatΓ(T) ǫ < 0.04 and α = π/2. For a stronger anisotropy, the
1
caneither increase ordecreasewith T, depending onthe condition ζ(α) 1 can still hold in a wide range of α,
≪
diffusivity ratio D /D . However,the physics ofthis de- except a vicinity of α π/2. In this case the calcu-
2 1
≈
pendence is rather transparent and can be understood lation of H (α,T) requires a numerical solution of the
c2
using the bilayer toy model as discussed in the Intro- matrix equation for H [39]. However, the Usadel the-
c2
duction. Indeed, for very different D and D , both the ory can only be applied to dirty MgB samples which,
1 2 2
angular and the temperature dependencies of H (α,T) contrary to the assumption of Ref. [39], usually ex-
c2
are controlled by cleaner band at high T and by dirtier hibit much weaker anisotropy (Γ 1 2) than single
2
≃ −
band at lower T. For instance, if D D , the high- crystals. Perhaps, strong impurity scattering and ad-
2 1
≪
T part of H (α,T) is determined by the anisotropic σ mixture of interband scattering reduce the anisotropy
c2
band,whilethelow-Tpartisdeterminedbytheisotropic of D(c)/D(ab) 0.2 0.3 as compared to that of the
1 1 ≃ −
π band. In this case Γ(T) decreases as T decreases, as Fermi velocities v2 / v2 0.02 predicted by ab-
h ciσ h abiσ ∼
characteristic of dirty MgB2 films represented in Figs. 1 initio calculations for single crystals [44]. The moderate
and 2. If the π band is cleaner than the σ band, Γ(T) anisotropy of D in dirty MgB makes the scaling rule
1 2
increases as T decreases, similar to single crystals. (44) a very good approximation, as was recently con-
For the field H inclined with respect to the c-axis,the firmed experimentally [30].
first Landau level eigenfunction no longer satisfies Eqs.
(17), (18) and (4). In this case f (ω,r) are to be ex-
m
pandedinfullsetsofeigenfunctionsforallLandaulevels, GINZBURG-LANDAU EQUATIONS
andH becomesarootofamatrixequationMˆ(H )=0
c2 c2
[38, 39]. As shown in Ref. [38], this matrix equation for The two-gapGL equations were obtained both for the
H greatly simplifies for the moderate anisotropy char- dirty limit without interband scattering [38, 85], and for
c2
acteristic of dirty MgB for which all formulas of the the cleanlimit [86]. Hereweconsiderthe GL dirtylimit,
2
previoussection canalsobe used for the inclined field as focusing on new effects brought by interband scattering.
wellbyreplacingD1 andD2 withtheangular-dependent For γmm′ =0, the Usadel equations near Tc yield
diffusivities D (α) and D (α) for both bands:
1 2 f =Ψ /ω+D Π2Ψ/2ω2 Ψ Ψ 2/2ω3, (47)
m m mα α − m| m|
Dm(α)=[Dm(a)2cos2α+Dm(a)Dm(c)sin2α]1/2 (44) wheretheprincipalaxisofDαβ aretakenalongthecrys-
talline axis. For weak interband scattering, the free en-
In terms of the bilayer model shown in Fig. 1, Eq. (44)
ergyF =F +F containsthefreeenergyF Ψ ,Ψ for
just means that Eq. (43) should be applied separately 0 i 0{ 1 2}
for each of the films. For g = 0, Eqs. (27) and (44) γmm′ = 0 and the correction Fi{Ψ1,Ψ2} linear in γmm′.
determinetheangulardependenceofH (α)nearT ,and Here F0 does not have first order corrections in γmm′ if
c2 c Ψ satisfies the GL equations, so F can be calculated
the London penetration depth Λ is given by [38] m i
αβ by substituting Eq. (47) into Eq. (9) and expanding
Λ−2 = 4π4 N Dαβ∆ tanh∆1 +N Dαβ∆ tanh∆2 gm ≈1−|fm|2/2−|fm|4/8:
αβ φ20 1 1 1 2T 2 2 2 2T Fi =πqT [f1 f2 2+(f1 2 f2 2)2/4], (48)
(cid:2) (4(cid:3)5) | − | | | −| |
ω>0
Eqs. (27), (44), and (60) show that the one-gap scaling X
where q =(N γ +N γ )/2. Combining F with F in
(43) breaks down because the behavior of H (α,T) is 1 12 2 21 i 0
c1
the dirty limit for g = 0 [38], we arrive at the GL free
mostlycontrolledbythecleanerbandforallT,whilethe
energy FdV for g 1:
behavior of H (α,T) is determined by the cleaner band
c2 ≪
at higher T, and by the dirtier band at lower T. Thus, R F =a1 Ψ1 2+c1α ΠαΨ1 2+b1 Ψ1 4/2
Γ (T) and Γ (T) for H and H in the two-gap dirty | | | | | |
1 2 c1 c2 +a Ψ 2+c Π Ψ 2+b Ψ 4/2
2 2 2α α 2 2 2
limit are different. Temperature dependencies of Γ(T) | | | | | |
a Re(Ψ Ψ∗)+c Re(Π Ψ Π∗Ψ∗)
were calculated in Refs. [38, 39]. − i 1 2 iα α 1 α 2
Eqs. (44) and(19)describe wellboth the temperature bi Ψ1 2 Ψ2 2+2bi(Ψ1 2+ Ψ2 2)Re(Ψ1Ψ2) (49)
− | | | | | | | |
andtheangulardependenciesofH (α,T)indirtyMgB
c2 2 Here the GL expansion coefficients are given by
films[25,28,29,30]. Eq. (44)isvalidiftheσbandisnot
N T πγ
tζomo+nanairsoetnroegpliicg,ibalnedprtohveidoeffd-dtihaagtoζnal e1le[m38e]n.tsHeMremn ∼ a1 = 21 lnT1 + 4T12 , (50)
≪ N (cid:2) T πγ (cid:3)
2 21
ζ = (ǫ1−ǫ2)2sin4α , (46) a2 = 2 lnT2 + 4T , (51)
[ cos2α+ǫ sin2α+ cos2α+ǫ sin2α]4 π(cid:2) 7ζ(3)γ (cid:3)
1 2 c =N D 12 , (52)
1α 1 1α 16T − 8π2T2
ǫ1 = Dp1(c)/D1(ab) and ǫ2 = Dp2(c)/D2(ab). For ǫ2 = 1, the (cid:2) π 7ζ(3)γ21(cid:3)
parameter ζ(α) < 0.45 for a rather strong anisotropy c2α =N2D2α 16T − 8π2T2 , (53)
(cid:2) (cid:3)
8
b =N 7ζ(3) 3πγ12 , (54) which reduces to Eq. (27) near Tc to the linear accuracy
1 1 16π2T2 − 384T3 in γmm′. However, GL calculations of Hc2(T) in MgB2
b2 =N2(cid:2)176ζπ(23T)2 − 338π4γT213(cid:3), (55) biteeydoanpdptlhiceabliinlietayr,Tsicn−ceTat(eTrm) a[8n7d,a88(]Th)avcehaanrgaethsiegrnlsimat-
1 2
a = N1 λ12 + πγ12 (cid:2)+ N2 λ21 + πγ21(cid:3), (56) very different temperatures T1 and T2. For λmn of Ref.
i 2 w 4T 2 w 4T [77],T1 0.9Tc0andT2 0.1Tc0sohigherordergradient
∼ ∼
(cid:2)7ζ(3) (cid:3) (cid:2) (cid:3) terms (automatically taken into account in the Eliash-
c = (D +D )(γ N +γ N ), (57)
i (4πT)2 1 2 12 1 21 2 berg/Eilenberger/Usadelbased theories) become impor-
π tant. For example, at T T1 where a2(T1) a1(T),
b = (γ N +γ N ), (58) ≈ ≫
i 384T3 12 1 21 2 retainingthefirstgradientterm c2 requirestakinginto
∝
account a next order term H2 in the first brackets in
∼
where T1 = Tc0exp[ (λ0 λ−)/2w], and T2 = Eq. (61),whichisbeyondtheGLaccuracy. Thus,apply-
− −
Tc0exp[ (λ0+λ−)/2w]. The GLequationsareobtained ing the GL theory in a wider temperature range [87, 88]
−
byvarying FdV. Iwouldliketopointoutthemisprints makes it a procedure of unclear accuracy, which can re-
with wrongRsigns of ai, c1 and c2 in Eqs. (13), (14) and sultinaspuriousupwardcurvatureinHc2(T)notalways
(20) in Ref. [38] (see also Ref. [86]). present in a more consistent theory (for example, in the
The first two lines in Eq. (49) are the GL intra- dirty limit at D D ). In addition, the anisotropy of
1 2
band free energies and the term aiRe(Ψ1Ψ∗2) describes D1(α) may further≃limit the applicability of the GL the-
the Josephson coupling of Ψ1 and Ψ2. Interband scat- ory for H ab, as for c1 c2 higher order gradientterms
|| ≫
tering increases a1 and a2, and the interband coupling in the π band become important [85].
constant a . The net result is the reduction of T deter-
i c
mined by the equation 4a (T )a (T )=a2, which repro-
1 c 2 c i
duces Eq. (16). Besides the renormalization of am, bm DISCUSSION
and c , interbandscattering produces new terms,which
m
describe the mixed gradient coupling and the nonlinear
Theremarkableten-foldincreaseofH (T)inC-doped
c2
quatric interaction of Ψ and Ψ . Similar terms were in-
1 2 MgB films [25, 26, 27, 28, 29] has brought to focus new
2
troduced in the GL theories of heavy fermions [12] and
and largely unexplored physics and materials science of
borocarbides [89], and phenomenological models of H
c2 two-gap superconducting alloys. Moreover, the observa-
inMgB [87]. Thesetermsresultfrominterbandscatter-
2 tions of H close to the BCS paramagnetic limit poses
c2
ing, so both c and b vanish in the clean limit [86]. The
i i the important question of how far can Hc2 be further
mixed gradient terms in Eq. (49) produce interference
increased by alloying. This possibility may be naturally
terms in the current density J= cδF/δA:
builtinthebandstructureofMgB ,whichprovidesweak
− 2
interband coupling and weak interband scattering, thus
J= [(2c ∆2+c ∆ ∆ cosθ)Q +
− 1 1 i 1 2 1 allowing MgB2 to be alloyed without strong suppression
(2c2∆22+ci∆1∆2cosθ)Q2+ ofTc. Forexample,for the C-dopedMgB2 film shownin
ci(∆2∇∆1−∆1∇∆2)sinθ]2πc/φ0 (59) Fyeigt.T1,wρansownalys irnecdruecaesdeddofrwonmto≃305.K4µ[2Ω8c]m. Ittois5t6h0eµwΩecamk-,
c
where Q = θ +2πA/φ , and θ = θ θ . Here ness of interband scattering, which apparently makes it
m m 0 1 2
∇ −
J is no longer the sum of independent contributions of possibleto takeadvantageofverydirtyπ bandto signif-
two bands, because phase gradients in one band pro- icantly boost Hc2 in carbon-doped films which typically
duce currents in the other. Moreover, J acquires new have Dπ 0.1Dσ. The reasons why scattering in the π
∼
cosθ terms and the peculiar sinθ interband Josephson- band of C-doped MgB2 films is so much stronger than
like contribution for inhomogeneous gaps. For currents in the σ band has not been completely understood, but
well below the depairing limit, both bands are phase- anotherimmediate benefit for high-fieldmagnetapplica-
locked (θ = 0), and Eq. (59) defines the London pene- tions[37]isthatcarbonalloyingsignificantlyreducesthe
tration depth Λ2 =cφ0Q/8π2|J|: anisotropy of Hc2 down to Γ(T)≃1−2.
Despitemanyyetunresolvedissuesconcerningthetwo-
Λ=φ /4π[2π(c ∆2+c ∆ ∆ +c ∆2)]1/2 (60) gap superconductivity in MgB alloys, H of C-doped
0 1 1 i 1 2 2 2 2 c2
MgB has already surpassed H of Nb Sn (see Fig.
2 c2 3
where c1, c2 and ci depend on the field orientation ac- 1). Given the intrinsic weakness of interband scattering,
cording to Eq. (44). Eq. (47) can be used to calculate whichenablestuning MgB byselectiveatomicsubstitu-
2
Hc⊥2(T)fromthelinearizedGLequations,whichgiveHc2 tions on Mg and B sites, there appear to be no funda-
as a solution of the quadratic equation [89] mentalreasonswhyH ofMgB alloyscannotbepushed
c2 2
furtheruptowardthestrong-couplingparamagneticlimit
2πc H 2πc H 2πc H 2 (1). Thus,understandingthemechanismsofintraandin-
1 2 i
4 +a +a = a + (61)
(cid:20) φ0 1(cid:21)(cid:20) φ0 2(cid:21) (cid:20) i φ0 (cid:21) terband impurity scattering in carbon-dopedMgB2, and
9
the competition between scattering and doping effects [3] H.J.Choi,D.Roundy,H.Sun,M.L.Cohen,S.G.Loule.
becomes an important challenge for the computational Nature418(2002)758;Phys.Rev.B66(2002)R020513.
physics. For instance, it remains unclear why the multi- [4] A.Floris et al. Phys.Rev.Lett. 94 (2005) 037004.
[5] C. Buzea, T. Yamashita, Supercond. Sci. Technol. 14
phased C-doped HPCVD grownfilms [32] exhibit higher
(2001) R115.
H and weaker T suppression [28] than uniform car-
c2 c [6] P.C. Canfield, S.L. Bud’ko, D.K. Finnemore. Physica
bon solid solutions [26, 69, 70]. This unexpected result
C385 (2003) 1.
mayindicateotherextrinsicmechanismsofHc2 enhance- [7] H. Suhl, B.T. Matthias, L.R. Walker, Phys. Rev. Lett.
ment, which are not accounted by the simple two-gap 3 (1959) 552.
theory presented here. Among those may be effects of [8] V.A. Moskalenko. Fiz. Met. Metalloved. 8 (1959)
electron localization or strong lattice distortions in mul- 503 [Sov. Phys. Met. Metallog. 8 (1959) 25]; V. A.
Moskalenko, M.E. Palistrat, V.M. Vakalyuk, Usp. Fiz.
tiphased C-doped films which can manifest themselves
Nauk,161 (1991) 155 [Sov. Phys. Usp.34 (1991) 717].
in the buckling of the Mg planes observed in the dirty
[9] S.V. Shulga, S.-L. Drechsler, G. Fuchs, K.-H. Mu¨ller,
fiber-textured MgB films shown in Fig. 2 [25]. Such
2 K.Winzer, M. Heinecke,K. Krug.Phys. Rev.Lett. 80,
buckling may enhance scattering in the π band formed (1998) 1730; S.V. Shulgaet al. cond-mat/0103154.
by out-of-plane pz boron orbitals. [10] V. Guritani, W. Goldacker, F. Bouquet, Y. Wang, R.
Lortz, G. Goll, A. Junod, Phys. Rev. B 70, (2004)
Recently significant enhancements of vortex pinning
184526.
and critical current densities J in MgB [90, 91, 92, 93,
c 2 [11] Y. Yokoya, T. Kiss, A. Chainani, S. Shin, M. Mikara,
94, 95, 96] has been achieved, particularly by introduc- H. Takagi. Science 294 (2001) 2518; E. Boaknin et al.
ing SiC [92] and ZrB2 [95] nanoparticles. Given these Phys.Rev. Lett.90 (2003) 117003.
promising results combined with weak current blocking [12] G. Seyfarth et al. Phys.Rev.Lett. 95 (2005) 107004.
by grain boundaries [97], the lack of electromagnetic [13] J.Singleton,C.Mielke.Contempor.Phys.43(2002)63.
granularity [98], and very slow thermally-activated flux [14] I.I. Mazin et al. Phys. Rev.Lett.89 (2002) 107002.
[15] A.J. Legget. Prog. Theor. Phys. 36 (1966) (901); Rev.
creep [99, 100], it is not surprising that MgB is being
2 Mod. Phys. 47 (1975) 331.
regarded as a strong contender of traditional high-field
[16] A.Gurevich,V.M.Vinokur.Phys.Rev.Lett.90(2003)
magnet materials like NbTi and Nb Sn. Despite these
3 047004.
achievements, a detailed theory of pinning in MgB2 is [17] A.Gurevich,V.M.Vinokur.Phys.Rev.Lett.97(2006)
trill lacking. Such theory should take into account a 137003.
composite structure of the vortex core, which consists of [18] T.P. Orlando, E.J. McNiff, S. Foner, M.R. Beasley.
concentric regions of radius ξ and ξ where ∆ (r) and Phys.Rev. B19 (1979) 4545.
σ π σ
[19] A.Godeke. Supercond. Sci. Technol. 19 (2006) R68.
∆ (r) are suppressed [101, 102, 103, 104, 105, 106]. For
π [20] S.Patnaiketal.Supercond.Sci.Technol.14(2001)315.
example, in MgB single crystals the larger vortex cores
2 [21] C.B. Eom, et al. Nature411 (2001) 558.
in the π band start overlapping above the ”virtual up- [22] V.Ferrandoet al.Phys. Rev.B 68 (2003) 0945171.
per critical field” Hv = φ0/2πξπ2 ∼ 0.5T, causing strong [23] F. Bouquet, et al. Physica C 385, (2003) 192.
overallsuppressionof∆ wellbelowH [102,103]. This [24] E. Ohmichi, T. Masui, S. Lee, S. Tajima, T. Osada. J.
π c2
effect can reduce J at H > H , however both H and Phys.Soc. Japan, 73 (2004) 2065.
c v v
H canbegreatlyincreasedbyappropriateenhancement [25] A. Gurevich et al. Supercond. Sci. Technol. 17 (2004)
c2
278.
of impurity scattering in MgB alloys.
2 [26] R.T.H. Wilke, S.L. Bud’ko, P.C. Canfield, D.K.
Recently there has been an emerging interest in mi- Finnemore, R.J. Suplinskas, S.T. Hannahs. Phys. Rev.
crowave response of MgB [107, 108, 109] and a possi- Lett. 92 (2004) 217003.
2
bility of using MgB in resonant cavities for particle ac- [27] A.V. Pogrebnyakov et al. Appl. Phys. Lett. 85 (2004)
2
2017.
celerators[110, 111]. These issues require understanding
[28] V.Braccini et al. Phys.Rev.B 71, (2005) 012504.
nonlinear electrodynamics and current pairbreaking in
[29] M. Angst, S.L. Budko, R.H.T. Wilke, P.C. Canfield.
two-gap superconductors [112, 113], in particular, band
Phys.Rev. B 71 (2005) 144512.
decouplingandtheformationofinterbandphasetextures [30] H.-J.Kim,H.-S.Lee,B.Kang,W.-H.Yim,Y.Jo,M.-H.
at strong rf currents [16, 17]. Jung, S.-I.Lee. Phys. Rev.B 73 (2006) 064520.
[31] P. Samuely, P. Szabo, Z. Holanova, S. Budko, P. Can-
Thisworkwaspartiallysupportedbyin-houseresearch
field. Physica C435 (2006) 71.
program at NHMFL. NHMFL is operated under NSF
[32] X.X.Xi et al. Supercond.Sci. Technol. 17 (2004) 196.
GrantDMR-0084173withsupportfromstateofFlorida.
[33] A.A.Abrikosov,L.P.Gor’kov.Zh.Exper.Teor.Fiz.,36
(1959) 319.
[34] P.G. De Gennes. Phys.Cond. Materie 3 (1964) 79.
[35] E.Helfand,N.R.Werthamer.Phys.Rev.147(1966)288;
N.R. Werthamer, E. Helfand, P.C. Hohenberg. Phys.
[1] J.Nagamatsu,N.Nakagawa,T.Muranaka,Y.Zenitani, Rev.147 (1966) 295.
J. Akimitsu. Nature410 (2001) 63. [36] K.Maki. Phys. Rev.148 (1966) 362.
[2] A.Liu,I.I.Mazin,J.Kortus.Phys.Rev.Lett.87(2001) [37] D. Larbalestier, A. Gurevich, D.M. Feldmann, A.
087005. Polyanskii. Nature414 (2001) 368.
10
[38] A. Gurevich.Phys. Rev.B 67 (2003) 184515. 027002.
[39] A.A. Golubov, A.E. Koshelev. Phys. Rev. B 68 (2003) [75] R.A.Klemm,A.Luther,M.R.Beasley.Phys.Rev.B12
1045031. (1975) 877.
[40] M. Mansor, J.P. Carbotte. Phys. Rev. B72 (2005) [76] S.Takachashi,M.Tachiki.Phys.Rev.B33(1968)4620.
024538. [77] A.A.Golubov et al.J.Phys: Condens.Matt.14 (2002)
[41] This toy model should not be taken too literally, since 1353.
H (α) in real multilayers in the inclined field can de- [78] F of Ref. [17] should be multiplied by −1/2.
c2 i
pend on the film thickness due to the effect of surfaces [79] N. Schophol, K. Scharnberg. Solid State Comm. 22
on thenucleation of vortices. (1977) 371.
[42] L.P.Gor’kov,T.K.Melik-Barkhudarov.JETP18(1964) [80] A.A.Golubov,I.I.Mazin.Phys.Rev.B55(1997)15146.
1031. [81] M. Iavarone et al. Phys.Rev.B 71 (2005) 214502.
[43] L.N. Bulaevskii. Zh.Exp.Teor. Fiz. 64 (1973) 2241; 65 [82] M. Tinkham. Phys. Rev.B 129 (1963) 2413.
(1973) 1278 [JETP 37 (1973) 1133; 38 (1974) 634]. [83] V.G. Kogan. Phys. Rev. B 66 (2002) R020509; V.G.
[44] D.K. Belashchenko, M. van Schilfgaarde, V.P. Kogan, N. Zhelezina. Phys.Rev.B 69 (2004) 1235061.
Antropov.Phys. Rev.B 64 (2001) 092503. [84] A.A. Golubov, A. Brinkman, O.V. Dolgov, J. Kortus,
[45] I.I. Mazin, V.P.Antropov.Physica C 385 (2003) 49. O. Jepsen. Phys.Rev. B 66 (2002) 054524.
[46] S.L. Bud’ko, V.G. Kogan, P.C. Canfield. Phys. Rev. B [85] A.A. Golubov, A.E. Koshelev. Phys. Rev. Lett. 68
64 (2001) 180506. (2003) 104503.
[47] S.Lee, H.Mori, T. Masui, Y.Eltsev, A.Yamamoto, S. [86] M.E. Zhitomirsky, V.H. Dao. Phys. Rev. B 69 (2004)
Tajima. J. Phys. Soc. Jap. 70 (2001) 2255. 054508.
[48] S.L. Bud’ko, P.C. Canfield. Phys. Rev. B 65 (2002) [87] I.N. Askerzade, A. Gencer, N. Gu¨clu´. Supercond. Sci.
212501. Technol. 15 (2002) L13; I.N. Askerzade. Physica C 390
[49] A.K. Pradhan et al.Phys. Rev.B 64 (2001) 212509. (2003) 281.
[50] M. Angst et al. Phys.Rev. B 88 (2002) 167004. [88] V.H. Dao, M.E. Zhitomirsky. Eur. J. Phys. 44 (2005)
[51] A.V.Sologubenko,J.Jun,S.M. Kazakov,J.Karpinski, 183.
H.R.Ott. Phys.Rev. B65 (2002) 180505. [89] H.Doh,M.Sigrist,B.K.Cho,S.-I.Lee.Phys.Rev.Lett.
[52] K. Takahashi, T. Atsumi, N. Yamamoto, M. Xu, H. 83 (1999) 5350.
Kitazawa, T. Ishida. Phys.Rev. B66 (2002) 012501. [90] A. Gu¨mbel, J. Eckert, G. Fuchs, K. Nenkov, K.H.
[53] M. Zehetmayer, M. Eisterer, J. Jun, S.M. Kazakov, J. Mu¨ller, L. Schultz,Appl.Phys.Lett. 80 (2002) 2725.
Karpinski, A. Wisniewski, H.W. Weber. Phys. Rev. B [91] K.Komori et al. Appl.Phys. Lett. 81 (2002) 1047.
66 (2002) 052505. [92] S.X.Dou et al. Appl.Phys. Lett.81 (2002) 3419.
[54] S.L.Bud’ko,P.C.Canfield,V.G.Kogan.PhysicaC387 [93] J. Wang et al. Appl.Phys.Lett. 81 (2002) 2026.
(2002) 85. [94] R. Flukiger, H.L. Suo, N. Musolino, C. Beneduce, P.
[55] A. Rydhet al. Phys.Rev.B 70 (2004) 132503. Toulemonde, P. Lezza. Physica C 385 (2003) 286.
[56] L. Lyard et al. Phys. Rev.Lett. 92 (2004) 057001. [95] M. Bhatia, M.D. Sumption, E.W. Collings, S. Dregia.
[57] V.G. Kogan. Phys.Rev.B 66 (2002) 020509. Appl.Phys. Lett.87 (2005) 042505.
[58] T. Dahm, N. Schopohl. Phys. Rev. Lett. 91 (2003) [96] B.J. Senkowicz, J.E. Giencke, S. Patnaik, S.B. Eom,
017001. E.E.Hellstrom, D.C.Larbalestier. Appl.Phys.Lett.86
[59] J. Kortus, O.V. Dolgov, R.K. Kremer, A.A. Golubov. (2005) 202502.
Phys. Rev.Lett. 94 (2005) 027002. [97] D.C. Larbalestier et al. Nature410 (2001) 186.
[60] M.Puttiet al.PhysRev.B70(2004)052509 (2004); B [98] A.A.Polyanskiietal.Supercond.Sci.Technol.14(2001)
71 (2005) 144505. 811.
[61] R.J. Cava, H.W. Zandbergen, K. Inumaru. Physica C [99] J.R.Thompson,M.Paranthaman,D.K.Christen,K.G.
385 (2003) 8. Sorge, J.G. Ossandon. Supercond. Sci. Technol. 14
[62] R.A. Ribeiro, S.L. Bud’ko, C. Petrovic, P.C. Canfield. (2001) L17.
Physica C 385 (2003) 16. [100] S. Patnaik, A. Gurevich, S.D. Bu, J. Choi, C.B. Eom,
[63] J. Karpinski et al. Supercond. Sci. Technol. 16 (2003) D.C. Larbalestier. Phys.Rev. B70 (2004) 064503.
221. [101] E. Babaev. Phys.Rev.Lett. 89 (2002) 0670011.
[64] J. Kim, R.K. Singh, N. Newman, J.M. Rowell. IEEE [102] M.R. Eskildsen et al. Phys. Rev. Lett. 89,187004-4
Trans. Appl. Supercond13 (2003) 3238. (2002).
[65] G.K. Perkins et al. Nature, 411 (2001) 561. [103] S.Serventiet al. Phys. Rev.Lett. 93 (2004) 217003.
[66] M.Eistereretal.Supercond.Sci.Technol.15(2002)L9. [104] A.E. Koshelev, A.A. Golubov. Phys. Rev. Lett. 90
[67] Y. Wanget al. J. Phys.: Cond. Matter. 15 (2003) 883. (2003) 177002.
[68] R.Gandikotaetal.Appl.Phys.Lett.86(2005)012508. [105] M. Ichioka, K. Machida, N. Nakai, P. Miranovic. Phys.
[69] S. Lee, T. Masui, A. Yamamoto, H. Uchiyama, S. Rev.B 70 (2004) 1445081.
Tajima. Physica C 397 (2003) 7. [106] A. Guman, S. Graser, T. Dahm, N. Schopohl. Phys.
[70] T. Masui, S. Lee, S. Tajima. Phys. Rev. B 70, 0245041 Rev.B 73 (2006) 104506.
(2004). [107] A.T. Findikoglu et al. Phys. Lett.83 (2003) 108.
[71] T. Kakeshita, S. Lee, S. Tajima. Phys. Rev. Lett. 97 [108] B.B. Jin et al. Appl.Phys. Lett. 87 (2005) 092503.
(2006) 037002. [109] G. Cifaiello et al. Appl.Phys. Lett. 88 (2006) 142510.
[72] G. Sarma. J. Phys. Chem. Solids 24 (1963) 1029. [110] E.W. Collings, M.D. Sumption, T. Tajima. Supercond.
[73] M. Schossmann, J.P. Carbotte. Phys. Rev.B 39 (1989) Sci. Technol. 17 (2004) 595.
4210. [111] A.Gurevich. Appl.Phys. Lett.88 (2006) 12511.
[74] H.J. Nui, D.P. Hampshire. Phys. Rev. Lett. 91 (2003) [112] M.N. Kunchur.J. Phys.Cond. Mat. 16 (2004) R1183.