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


Limits of the upper critical ?eld in dirty two-gap superconductors
A. Gurevich
National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32310, USA (Dated: February 6, 2008)

arXiv:cond-mat/0701281v1 [cond-mat.supr-con] 12 Jan 2007

An overview of the theory of the upper critical ?eld in dirty two-gap superconductors, with a particular emphasis on MgB2 is given. We focus here on the maximum Hc2 which may be achieved by increasing intraband scattering, and on the limitations imposed by weak interband scattering and paramagnetic e?ects. In particular, we discuss recent experiments which have recently demonstrated ten-fold increase of Hc2 in dirty carbon-doped ?lms as compared to single crystals, so that the Hc2 (0) parallel to the ab planes may approach the BCS paramagnetic limit, Hp [T ] = 1.84Tc [K] ? 60 ? 70T . New e?ects produced by weak interband scattering in the two-gap Ginzburg-Landau equations and Hc2 (T ) in ultrathin MgB2 ?lms are addressed.
PACS numbers: PACS numbers: 74.20.De, 74.20.Hi, 74.60.-w

INTRODUCTION

It is now well established that superconductivity in MgB2 with the unexpectedly high critical temperature Tc ≈ 40K [1], is due to strong electron-phonon interaction with in-plane boron vibration modes. Extensive ab-initio calculations [2, 3, 4], along with many experimental evidences from STM, point contact, and Raman spectroscopy, heat capacity, magnetization and rf measurements [5, 6] unambiguously indicate that MgB2 exhibits two-gap s-wave superconductivity [7, 8]. MgB2 has two distinct superconducting gaps: the main gap ?σ (0) ≈ 7.2mV, which resides on the 2D cylindrical parts of the Fermi surface formed by in-plane σ antibonding pxy orbitals of B, and the smaller gap ?π (0) ≈ 2.3mV on the 3D tubular part of the Fermi surface formed by out-of-plane π bonding and antibonding pz orbitals of B. The discovery of MgB2 has renewed interest in new e?ects of two-gap superconductivity, motivating di?erent groups to take closer looks at other known materials, such as YNi2 B2 C and LuNi2 B2 C borocarbides [9] Nb3 Sn [10], or NbSe2 [11], heavy-fermion [12] and organic [13] superconductors, for which evidences of the two gap behavior have been reported. However, several features of MgB2 set it apart from other two-gap superconductors. Not only does MgB2 have the highest Tc among all noncuprate superconductors, it also has two coexisting order parameters Ψσ = ?σ exp(iθ1 ) and Ψπ = ?π exp(iθ2 ), which are weakly coupled. The latter is due to the fact that the σ and π bands are formed by two orthogonal sets of in-plane and out-of-plane atomic orbitals of boron, so all overlap integrals, which determine matrix elements of interband coupling and interband impurity scattering are strongly reduced [14]. This feature can result in new effects, which are very important both for the physics and applications of MgB2 . Indeed, two weakly coupled gaps result in intrinsic Josephson e?ect, which can manifest itself in low-energy interband Josephson plasmons (the Legget mode) [15] with frequencies smaller than ?π / . Moreover, strong static electric ?elds and currents can

decouple the bands due to formation of interband textures of 2π planar phase slips in the phase di?erence θ(x) = θ1 ? θ2 [16, 17] well below the global depairing current. In turn, the weakness of interband impurity scattering makes it possible to radically increase the upper critical ?eld Hc2 by selective alloying of Mg and B sites with nonmagnetic impurities. Despite the comparatively high Tc , the upper critical ?eld of MgB2 single crystals is rather low and anisotropic || ⊥ with Hc2 (0) ? 3 ? 5T and Hc2 (0) ? 15 ? 19T of [5, 6], where the indices ⊥ and || correspond to the magnetic ?eld H perpendicular and parallel to the ab plane, respectively. Since these Hc2 values are signi?cantly lower than Hc2 (0) ? 30T for Nb3 Sn [18, 19], there had been initial scepticism about using MgB2 as a high-?eld superconductor, until several groups undertook the wellestablished procedure of Hc2 enhancement by alloying MgB2 with nonmagnetic impurities. The results of high?eld measurements on dirty MgB2 ?lms and bulk sam⊥ ples has shown up to ten-fold increase of Hc2 as compared to single crystals [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], particularly in carbon-doped thin ?lms [28] made by hybrid physico-chemical vapor deposition [32]. This unexpectedly strong enhancement of Hc2 (T ) results from its anomalous upward curvature, rather di?erent from that of Hc2 (T ) for one-gap dirty superconductors [33, 34, 35, 36]. As shown in Fig. 1, Hc2 of MgB2 Cdoped ?lms has already surpassed Hc2 of Nb3 Sn, which could make cheap and ductile MgB2 an attractive material for high ?eld applications [37]. This radical enhancement of Hc2 shown in Fig. 1 is indeed assisted by the features of two-gap superconductivity in MgB2 . Fig. 2 gives another example of Hc2 (T ) for a ?ber-textured ?lm [25], which exhibits an upward curvature of Hc2 (T ) for H||c. This behavior of Hc2 (T ) and the anomalous temperature-dependent anisotropy ratio || ⊥ Γ(T ) = Hc2 (T )/Hc2 (T ) are di?erent from that of the one-gap theory in which the Hc2 (T ) has a downward ′ curvature, while the slope Hc2 = dHc2 /dT at Tc is proportional to the normal state residual resistivity ρn , and

2
50 40
Hc2(T)/Hc2(0)
1

H
0.8

π σ

H , Tesla

30 20

H⊥ab

H||ab

0.6

0.4

c2

0.2
Nb Sn

H(π) c2

H(σ)
c2

10
NbTi

3

0 0

0.2

0.4

0.6

0.8

1

0 0

10

20

30

40

T/Tc

T, K
FIG. 3: The mechanism of the upward curvature of Hc2 (T ) illustrated by the bilayer toy model shown in the inset. The dashed curves show Hc2 (T ) calculated for σ and π ?lms in the one-gap dirty limit with the BCS coupling constants λσ = 0.81, λπ = 0.285, and Dπ = 0.1Dσ . The solid curve shows Hc2 (T ) calculated from Eq. (26) of the two-gap dirty limit theory for the BCS matrix constants from Ref. [77]

FIG. 1: Hc2 (T ) for carbon-doped MgB2 ?lms [28] in comparison with NbTi and Nb3 Sn. The red and blue lines show ?ts from Eq. (19) with g = 0.045 .
50 40

30 20 10 0 0

10

20

30

40

T, K

FIG. 2: Hc2 (T ) of a ?ber-textured MgB2 ?lm [25] both parallel (triangles) and parallel (squares) to the ab planes. The solid lines show calculations from Eq. (19) with g = 0.065, (c) (ab) (ab) Dπ ? Dσ for H||c and Dπ = 0.19(Dσ Dσ )1/2 for H ⊥ c.

′ Hc2 (0) = 0.69TcHc2 [33, 34, 35, 36]. However, the behavior of Hc2 (T ) in MgB2 can be explained by the two-gap theory in the dirty limit based on either Usadel equations [38, 39] or Eliashberg equations [9, 40]. The behavior of Hc2 (T ) can be qualitatively understood using a simple bilayer model shown in Fig. 3, which captures the physics of two-gap superconductivity in MgB2 , and suggests ways by which Hc2 can be further increased. Indeed, MgB2 can be mapped onto a bilayer in which two thin ?lms corresponding to σ and π bands are separated by a Josephson contact, which models the interband coupling. The global Hc2 (T ) of the such weakly-coupled bilayer is mostly determined by the (σ) (π) ?lm with the highest Hc2 , even if Tc and Tc are very di?erent. For example, if the π ?lm is much dirtier than (σ) the σ ?lm then Hc2 dominates at higher T, but at lower temperatures the π ?lm takes over, resulting in the up-

ward curvature of Hc2 (T ). If the σ ?lm is dirtier, the π ?lm only results in a slight shift of the Hc2 curve and a ′ reduction of the slope Hc2 near Tc . The bilayer model also clari?es the anomalous angular dependence of Hc2 (α, T ) for H inclined by the angle α with respect to the c-axis (parallel to the ?lm normal in (σ) (π) Fig. 3) [41]. In this case both Hc2 (α, T ) and Hc2 (α, T ) depend on α according to the temperature-independent one-gap scaling Hc2 (α) = Hc2 (0, T )/ cos2 α + ? sin2 α [42, 43], but with very di?erent e?ective mass ratios ? = mab /mc for each ?lm. Because the σ band is much more anisotropic than the π band, ?σ ? 1, and ?π ? 1 [44, 45], the one-gap angular scaling for the global Hc2 (α, T ) breaks down. For example, in the case shown in Fig. 3, Hc2 (T ) is anisotropic at higher T, but at lower T, the nearly isotropic π band reduces the overall anisotropy of || ⊥ Hc2 , so the ratio Γ(T ) = Hc2 (T )/Hc2 (T ) decreases as T decreases. This is characteristic of many dirty MgB2 ?lms like the one shown in Fig. 2, for which the π band is typically much dirtier than the σ band. By contrast, in clean MgB2 single crystals Γ(T ) increases from ? 2 ? 3 near Tc to ? 5 ? 6 at T ? Tc [46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56]. This behavior was explained by two-gap e?ects in the clean limit [57, 58]. Fig. 3 suggests that Hc2 (T ) of MgB2 can be signi?cantly increased at low T by making the π band much dirtier than the main σ band. This could be done by disordering the Mg sublattice, thus disrupting the pz boron out-of-plane orbitals, which form the π band. Achieving high Hc2 requires that both σ and π bands are in the dirty limit. Yet, making the π band much dirtier than the σ band provides a ”free boost” in Hc2 without too much penalty in Tc suppression due to pairbreaking interband scattering or band depletion due to doping

Hc2, Tesla

3 [59, 60]. In fact, the interband scattering is weak for the same reason that Ψσ and Ψπ are weakly coupled, which may enable alloying MgB2 with more impurities to achieve higher Hc2 . Systematic incorporation of impurities in MgB2 has not been yet achieved because the complex substitutional chemistry of MgB2 is still poorly understood [61, 62, 63, 64]. Several groups have reported a signi?cant increase in Hc2 by irradiation with protons [65], neutrons [66, 67] or heavy ions [68], but so far the carbon impurities have been the most e?ective to provide the huge Hc2 enhancement shown in Figs. 1 and 2. The e?ect of carbon on di?erent superconducting properties can be rather complex [69, 70, 71] and still far from being fully understood. Yet given the indisputable bene?ts of carbon alloying, one can pose the basic question: how far can Hc2 be further increased? The bilayer model suggests that Hc2 increases if intraband scattering is enhanced. However, because intraband impurity scattering causes an admixture of pairbreaking interband scattering, the ?rst question is to what extent weak interband scattering in MgB2 can limit Hc2 . Another important question is how far is the observed Hc2 from the paramagnetic limit Hp . In the BCS √ theory BCS = ?/ 2, or Hp is de?ned by the condition: ?B Hp BCS Hp [T ] = 1.86Tc[K] [72], where ?B is the Bohr magneton. For Tc = 35K, this yields Hp = 65T, not that || far from the zero-?eld Hc2 (0) in Figs. 1 and 2. However, the BCS model underestimates Hp , which is signi?cantly enhanced by strong electron-phonon coupling [73]: Hp ? (1 +
BCS λep )Hp ,

Although the σ band is anisotropic, MgB2 is not a layered material [75, 76], so the continuum BCS theory is applicable because the c-axis coherence length ξc is much longer than the spacing between the boron planes ? 3.5?. InA || ⊥ deed, even for Hc2 (0) = 40T and Hc2 (0) = 60T in Fig. 1, the anisotropic Ginzburg-Landau (GL) theory [42] gives || ⊥ A ξc = (φ0 Hc2 /2π)1/2 /Hc2 ≈ 19?. Strong coupling in MgB2 should be described by the Eliashberg equations [40], but we consider here manifestations of intra and interband scattering and paramagnetic e?ects in Hc2 using the more transparent two-gap Usadel equations [38] ωf1 ?
αβ D1 [g1 Πα Πβ f1 ? f1 ?α ?β g1 ] 2 = Ψ1 g1 + γ12 (g1 f2 ? g2 f1 ) αβ D2

(2)

ωf2 ?

[g2 Πα Πβ f2 ? f2 ?α ?β g2 ] 2 = Ψ2 g2 + γ21 (g2 f1 ? g1 f2 ),

(3)

Here the Usadel Green’s functions fm (r, ω) and gm (r, ω) in the m-th band depend on r and the Matsubara freαβ quency ω = πT (2n + 1), Dm are the intraband di?usivities due to nonmagnetic impurity scattering, 2γmm′ are the interband scattering rates, Π = ? + 2πiA/φ0 , A is the vector potential, and φ0 is the ?ux quantum. Eqs. (2) and (3) are supplemented by the equations for the order parameters Ψm = ?m exp(i?m ),
ωD

(1)

Ψm = 2πT
ω>0 m

λmm′ fm′ (r, ω),

(4)

where λep is the electron-phonon constant. Taking λep ≈ 1 for the σ band [2, 3], we obtain Hp ? 130T, so there still a large room for increasing Hc2 by optimizing the intra and interband impurity scattering. For instance, ′ increasing Hc2 to a rather common for many high ?eld ′ superconductors value of 2T/K (much lower than Hc2 ? 5 ? 14T/K for PbMo6 S8 [74]) could drive Hc2 of MgB2 with Tc ? 35K above 70T. In the following we give a brief overview of recent results in the theory of dirty two-gap superconductors focusing on new e?ects brought by weak interband scattering and paramagnetic e?ects. The main conclusion is that, although interband scattering in MgB2 is indeed weak, it cannot be neglected in calculations of Hc2 (T ). We will also address the crossover from the orbitally-limited to the paramagnetically limited Hc2 in a two-gap superconductor.
THO-GAP SUPERCONDUCTORS IN THE DIRTY LIMIT

2 normalization condition |fm |2 + gm = 1, and the supercurrent density

J α = ?2πeT Im

αβ ? Nm Dm fm Πβ fm . ω m

(5)

Here Nm is the partial electron density of states for both spins in the m-th band, and α and β label Cartesian indices. Eqs. (4) contains the matrix of the BCS cou(ep) (ep) pling constants λmm′ = λmm′ ? ?mm′ , where λmm′ are electron-phonon constants, and ?mm′ is the Coulomb pseudopotential. The diagonal terms λ11 and λ22 quantify intraband pairing, and λ12 and λ21 describe interband coupling. Hereafter, the following ab initio values λσσ ≈ 0.81, λππ ≈ 0.285, λσπ ≈ 0.119, and λπσ ≈ 0.09 [77] are used. There are also the symmetry relations: N1 λ12 = N2 λ21 , N1 γ12 = N2 γ21 (6)

We regard MgB2 as a dirty anisotropic superconductor with two sheets 1 and 2 of the Fermi surface on which the superconducting gaps take the values ?1 and ?2 , respectively (indices 1 and 2 correspond to σ and π bands).

where Nπ ≈ 1.3Nσ for MgB2 . Solutions of Eqs. (2)-(6) minimize the following free energy F d3 r [17]: F = 1 2 Nm Ψm Ψ? λ?1 ′ + F1 + F2 + Fi m mm
mm′

(7)

4 Here F1 and F2 are intraband contributions, Fm = 2πT
? Re(fm ?m ) +
40

+ ?α gm ?β gm ]/4

T, K

ω>0 αβ ? Dm [Πα fm Π? fm β

Nm [(ω(1 ? gm ) ?

(8)

30

20

and Fi is due to interband scattering [78]: Fi = 2πqT [1 ? g1 g2 ?
? Re(f1 f2 )],

10

(9)
0 0 0.5

ω>0

where 2q = N1 γ12 + N2 γ21 . The Usadel equations result ? from δF/δfm = 0, ?F/?Ψ? = 0, and J = ?cδF/δA. m Taking fm = sin αm and gm = cos αm , we obtain ω sin α1 + γ12 sin(α1 ? α2 ) = ?1 cos α1 , (10) (11)

g = γ+/2πTc0

1

1.5

2

FIG. 4: Dependence of the critical temperature Tc on the interband scattering parameter g calculated from Eq. (13) with the BCS matrix constants λmn from Ref. [77]

ω sin α2 + γ21 sin(α2 ? α1 ) = ?2 cos α2 .

These coupled equations along with Eq. (4) de?ne the two-gap uniform states for J = 0.
CRITICAL TEMPERATURE

UPPER CRITICAL FIELD FOR H c

Hc2 along the c-axis is the maximum eigenvalue of the linearized Eqs. (2) and (3): (ω ± i?B H)f1 ? D1 2 Π f1 = ?1 + (f2 ? f1 )γ12 , (17) 2 D2 2 Π f2 = ?2 + (f1 ? f2 )γ21 , (18) (ω ± i?B H)f2 ? 2

Eqs. (2) and (3) give the well-known results for Tc in two-gap superconductors [7, 8, 79, 80]. For negligible interband scattering, substitution of f1 = ?1 /ω and f2 = ?2 /ω into Eq. (4) yields: Tc0 = 1.14 ωD exp[?(λ+ ? λ0 )/2w], (12)

where λ± = λ11 ± λ22 , w = λ11 λ22 ? λ12 λ21 , and λ0 = (λ2 + 4λ12 λ21 )1/2 . The interband coupling increases Tc0 ? as compared to noninteracting bands (λ12 = λ21 = 0), while intraband impurity scattering does not a?ect Tc0 , in accordance with the Anderson theorem. Solving the linearized Eqs. (2) and (3) with γmm′ = 0, gives Tc with the account of pairbreaking interband scattering: (λ0 + w ln tc ) ln tc g =? , tc p + w ln tc 2p = λ0 + [γ? λ? ? 2λ21 γ12 ? 2λ12 γ21 ]/γ+ , U U (x) = ψ(1/2 + x) ? ψ(1/2), (13) (14) (15)

Here the Zeeman paramagnetic term ±?B H, which requires summation over both spin orientations in Eq. (4), is included. In the gauge Ay = Hx, the solu? tions are fm (x) = fm exp(?πHx2 /φ0 ), and ?m (x) = ? ? m exp(?πHx2 /φ0 ), where fm is expressed via ?m from ? ? Eqs. (17) and (18). The solvability condition (4) of two ? ? linear equations for ?1 and ?2 gives the equation for Hc2 [38], which accounts for interband and intraband scattering and paramagnetic e?ects: (λ0 + λi )(ln t + U+ ) + (λ0 ? λi )(ln t + U? ) +2w(ln t + U+ )(ln t + U? ) = 0, where t = T /Tc0, and λi = [(ω? + γ? )λ? ? 2λ12 γ21 ? 2λ21 γ12 ]/?0 , 2?± = ω+ + γ+ ± ?0 , ?0 = [(ω? + γ? )2 + 4γ12 γ21 ]1/2 , ω± = (D1 ± D2 )πH/φ0 , 1 1 ?± + i?B H + ?ψ U± = Reψ 2 2πT 2 (20) (21) (22) (23) (24)

(19)

where tc = Tc /Tc0 and γ± = γ12 ±γ21 , g = γ+ /2πTc0 , and ψ(x) is a digamma function. The dependence of Tc on the interband scattering parameter g is shown in Fig. 4. As g → ∞, Eqs. (13) and (14) give Tc → Tc0 exp(?p/w), and for g ? 1, we have Tc = Tc0 ? π [λ0 γ+ + λ? γ? ? 2λ21 γ12 ? 2λ12 γ21 ] (16) 8λ0

.

This formula can be used to extract the interband scattering rates from the small shift of Tc [81]. However, as shown below, even weak interband scattering can significantly change the behavior of Hc2 (T ), so it cannot be neglected even though g ? 1.

If interband scattering and paramagnetic e?ects are negligible, Eqs. (19)-(24) reduce to a simpler equation [38, 39], which can be presented in the parametric form: ln t = ?[U (h) + U (ηh) + λ0 /w]/2 +
2 2 1/2

(25) (26)

[(U (h) ? U (ηh) ? λ? /w) /4 + λ12 λ21 /w ] Hc2 = 2φ0 Tc th/D1 ,

,

5 where η = D2 /D1 , and the parameter h runs from 0 to ∞ as T varies from Tc to 0. For equal di?usivities, η = 1, Eq. (26) simpli?es to the one-gap de-Gennes-Maki equation ln t + U (h) = 0 [34, 35, 36]. Now we consider some limiting cases, which illustrate how Hc2 depends on di?erent parameters. Fig. 5 shows the evolution of Hc2 as g increases for ?xed D1 and D2 and negligible paramagnetic e?ects. Interband scattering reduces the upward curvature of Hc2 (T ), Hc2 (0), and ′ Tc , while increasing the slope Hc2 at Tc . Notice that the signi?cant changes in the shape of Hc2 (T ) in Fig. 5 occur for weak interband scattering (g ? 1), which also provides a ?nite Hc2 (0) even if D2 → 0. For example, the high-?eld ?lms in Fig. 1 and 2 have g ? 0.045 and 0.065, respectively. For g ? 1, Eq. (19) yields the GL linear temperature dependence near Tc : Hc2 = 8φ0 (Tc ? T ) π 2 (s1 D1 + s2 D2 ) (27)

1

g = 0.03

H /H (0,0)

c2 c2

0.5

0

0

10

20 T, (K)

30

40

1 D = 0.05D H (T,g)/H (0,0.01)
π σ

c2

0.5

where Tc is given by Eq. (16), s1 = 1 + λ? /λ0 and s2 = 1?λ? /λ0 . Eq. (27) is written in the linear accuracy in g ? 1. Higher order terms in g not only shift Tc but ′ also increase the slope Hc2 at Tc , as evident from Fig. 5. ′ For s1 ? s2 , the slope Hc2 is mostly determined by the cleanest band with the maximum di?usivity. However, because of weak interband coupling in MgB2 , the values of s1 and s2 are very di?erent. For λ11 = 0.81, λ22 = 0.285, λ12 = 0.119, λ21 = 0.09 [77], we get λ? = λ11 ? λ22 = 0.525, λ0 = (λ2 + 4λ12 λ21 )1/2 = 0.564, thus s1 = ? ′ 1 + λ? /λ0 = 1.93, s2 = 1 ? λ? /λ0 = 0.07. Thus, Hc2 is mostly determined by D1 of the σ band. Yet, if the σ band is so dirty that D1 /D2 < s2 /s1 ? 0.04, the slope ′ Hc2 is determined by the much cleaner π band. At low T both the Zeeman and interband scattering terms in Eq. (19) can be essential. Eq. (19) reduces to the following equation for Hc2 (0):
2 2 ?2 Hp ?2 Hp B B (λ0 + λi ) ln 2 2 + (λ0 ? λi ) ln 2 2 ?B H + ?2 ?B H + ?2 + ? 2 2 ?2 Hp ?2 Hp = w ln 2 B ln 2 B 2 + ?2 2 + ?2 ?B H ?B H + ?

c2

0

0

10

20 T, (K)

30

40

FIG. 5: E?ect of interband scattering and the di?usivity ratio on the evolution of Hc2 (T ). Upper panel shows Hc2 (T, η) for the ?xed g = 0.03 and di?erent η = D2 /D1 : 0; 0.05; 0.1; 0.5 (from top to bottom curves). Lower panel shows Hc2 (T, g) for the ?xed D2 /D1 = 0.05 and di?erent g = 0.01; 0.05; 0.1; 0.5 (from top to bottom curves).

and m is the bare electron mass. Eq. (30) follows from the basic di?usion relation l2 = D0 t, and the energy uncertainty principle 2 /2ml2 = /t for a particle con?ned in a region of length l. For g = 0, Eq. (28) yields Hc2 (0) = f φ0 Tc exp( ), 2 ? ? 1 D2 2γ D
1/2

(31) λ0 . w

(28)

f=

? ? D2 2λ? D2 λ2 0 + ln2 + ln 2 ? ? w w D1 D1

?

(32)

where ?B Hp = πTc0 /2γ is the ?eld of paramagnetic instability of the superconducting state, and ln γ = 0.577. We ?rst consider the limit g → 0, which de?nes the maximum Hc2 (0) achievable in a dirty two-gap superconductor with no Tc suppresion. In this case ?+ = πD1 H/φ0 and ?? = πD2 H/φ0 , so for T ? Tc , paramagnetic e?ects just renormalize intraband di?usivities in Eq. (28): ? Dm → Dm =
2 2 Dm + D0 ,

If D0 ? Dm , Eqs. (31)-(32) reduce to the result of Ref. ? ? [38], and for the symmetric case, D1 = D2 , Eqs. (31)? (32) give the one-band result Hc2 (0) = φ0 Tc /2γ D [35]. ? ? However for D1 = D2 , Hc2 (0) can be much higher than ′ Hc2 (0) = 0.69Hc2 Tc . Indeed, if the e?ective di?usivities, ? ? D1 and D2 are very di?erent, Eqs. (31)-(32) yield Hc2 (0) = φ0 Tc ?(λ? +λ0 )/2w e , ? 2γ D2 φ0 Tc ?(λ0 ?λ? )/2w Hc2 (0) = e , ? 2γ D1
0 ? ? D2 ? D1 e? w , (33) 0 ? ? D1 ? D2 e? w . (34) λ λ

(29)

where D0 = ?B φ0 /π is the quantum di?usivity D0 = /2m, (30)

Thus, Hc2 (0) is determined by the minimum e?ective di?usivity, but unlike the limit D0 → 0, Hc2 (0) remains

6 can be higher than the bulk Hc2 = φ0 /2πξ 2 [76, 82]. Let us see how this result is generalized to two-gap superconductors. For a thin ?lm of thickness d < max(ξ1 , ξ2 ), the functions f1 and f2 are nearly constant, so integrating Eqs. (17) and (18) over x with ?x f (±d/2) = 0, results in two linear equations for f1 and f2 with Π2 = (πHd/φ0 )2 /3. Thus, we obtain the previous Eq. (19)(24) in which one should make the replacement ω± → (πHd/φ0 )2 (D1 ± D2 )/6
(f )

1.2 1

Hc2/Hp(0)

0.8 0.6 0.4 0.2 0

(39)

0.2

0.4

0.6

0.8

1

T/Tc

FIG. 6: Crossover from the orbitally to paramagnetically limited Hc2 (T ) calculated from Eqs. (26) and (37)-(38) for D2 = 0.05D1 and D1 /D0 = 0, 1, 5, 20 from top to bottom curves, respectively.

We ?rst consider the case of negligible interband scattering and paramagnetic e?ects. Then Eq. (19) and (39) give the square-root temperature dependence near Tc Hc2 =
(f )

4φ0 3/2 d(s π

3Tc (Tc ? T ) 1/2 1 D 1 + s2 D 2 )

(40)

?nite even for D1 → 0 or D2 → 0. In fact, if both D1 ? D0 and D2 ? D0 , we return to the symmetric ? ? case D1 = D2 , for which Eqs. (31)-(32) yield the result of the one-gap dirty limit theory [36] Hc2 (0) → Hp = φ0 Tc /2γD0 = πTc /2γ?B (35)

characteristic of thin ?lms [82] instead of the bulk GL linear dependence (27). From Eqs. (31) and (39) we can (f ) also obtain Hc2 (0) for D0 ? Dm : Hc2 (0) =
(f )

φ0 d

3Tc πγ

1/2

exp(f /4) (D1 D2 )1/4

(41)

For a one-band superconductor, Eq. (35) can also be written as the paramagnetic pairbreaking condition, ?B Hp = ?(0)/2, where ?(0) = πTc /γ is the zerotemperature gap. For two-band superconductors, the meaning of Hp is less transparent, yet the maximum Hp expressed via Tc is given by the same Eq. (35) as for one-band superconductors. Finally we consider how paramagnetic e?ects a?ect the shape of Hc2 (T ) in the limit g → 0. This case is described by Eq. (26) modi?ed as follows: U (h) → Reψ[1/2 + h(i + p)] ? ψ(1/2) Hc2 = Hc2 = 2φ0 Tc th/D0 , (36) (37) (38)

Next we consider the crossover to the paramagnetic limit in thin ?lms at low temperatures. For neglect interband scattering, the expressions ?2 H 2 + ?2 under the logaB rithms in Eq. (28) become ?2 H 2 + (πHd/φ0 )4 D2 /36. B (f ) Substituting here Hc2 ? φ0 /ξd, we conclude that paramagnetic e?ects become essential if min(D1 , D2 ) < D0 ξ/d. (42)

Thus, reducing the ?lm thickness extends the region of the parameters where Hc2 is limited by the paramagnetic e?ects rather than by impurity scattering.
ANISOTROPY OF Hc1 AND Hc2

U (ηh) → Reψ[1/2 + h(pη + i)] ? ψ(1/2)

where p = D1 /D0 , and η = D2 /D1 . Fig. 6 shows how Hc2 (T ) evolves from the orbitally-limited Hc2 (T ) with an upward curvature at D1 ? D2 to the paramagneticallylimited Hc2 (T ) of a one-gap superconductor for D1 < D0 [72]. The nonmonotonic dependence of Hc2 (T ) in Fig. 6 indicates the ?rst order phase transition, similar to that in one-gap superconductors.

For anisotropic one-gap superconductors, the angular dependence of the lower and the upper critical ?elds is given by [42, 43] Hc1 (α, T ) = Hc1 (0, T ) , R(α) Hc2 (α, T ) = Hc2 (0, T ) (43) R(α)

THIN FILMS IN A PARALLEL FIELD

Hc2 can be signi?cantly enhanced in thin ?lms or multilayers, in which MgB2 layers are separated by nonsuperconducting layers. It is well known that in a thin ?lm of √ (f ) thickness d < ξ in a parallel ?eld, Hc2 = 2 3Hc2 ξ/d

where R(α) = (cos2 α + ? sin2 α)1/2 , ? = mab /mc . Here || ⊥ the anisotropy parameter Γ(T ) = Hc2 /Hc2 = ??1/2 is independent of T for both Hc1 and Hc2 . By contrast, || ⊥ Γ2 (T ) = Hc2 /Hc2 for MgB2 single crystals increases from || ⊥ ? 2 ? 3 at Tc to 5 ? 6 at T ? Tc , but Γ1 (T ) = Hc1 /Hc1 decreases from ? 2 ? 3 to ? 1 as T decreases [46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56]. This behavior was explained by the two-gap theory in the clean limit [57, 58, 83, 84].

7 The dirty limit is more intricate in the sense that Γ(T ) can either increase or decrease with T, depending on the di?usivity ratio D2 /D1 . However, the physics of this dependence is rather transparent and can be understood using the bilayer toy model as discussed in the Introduction. Indeed, for very di?erent D1 and D2 , both the angular and the temperature dependencies of Hc2 (α, T ) are controlled by cleaner band at high T and by dirtier band at lower T. For instance, if D2 ? D1 , the highT part of Hc2 (α, T ) is determined by the anisotropic σ band, while the low-T part is determined by the isotropic π band. In this case Γ(T ) decreases as T decreases, as characteristic of dirty MgB2 ?lms represented in Figs. 1 and 2. If the π band is cleaner than the σ band, Γ(T ) increases as T decreases, similar to single crystals. For the ?eld H inclined with respect to the c-axis, the ?rst Landau level eigenfunction no longer satis?es Eqs. (17), (18) and (4). In this case fm (ω, r) are to be expanded in full sets of eigenfunctions for all Landau levels, ? and Hc2 becomes a root of a matrix equation M (Hc2 ) = 0 [38, 39]. As shown in Ref. [38], this matrix equation for Hc2 greatly simpli?es for the moderate anisotropy characteristic of dirty MgB2 for which all formulas of the previous section can also be used for the inclined ?eld as well by replacing D1 and D2 with the angular-dependent di?usivities D1 (α) and D2 (α) for both bands: Dm (α) =
(a)2 [Dm

?1 < 0.04 and α = π/2. For a stronger anisotropy, the condition ζ(α) ? 1 can still hold in a wide range of α, except a vicinity of α ≈ π/2. In this case the calculation of Hc2 (α, T ) requires a numerical solution of the matrix equation for Hc2 [39]. However, the Usadel theory can only be applied to dirty MgB2 samples which, contrary to the assumption of Ref. [39], usually exhibit much weaker anisotropy (Γ2 ? 1 ? 2) than single crystals. Perhaps, strong impurity scattering and admixture of interband scattering reduce the anisotropy (c) (ab) of D1 /D1 ? 0.2 ? 0.3 as compared to that of the 2 2 Fermi velocities vc σ / vab σ ? 0.02 predicted by abinitio calculations for single crystals [44]. The moderate anisotropy of D1 in dirty MgB2 makes the scaling rule (44) a very good approximation, as was recently con?rmed experimentally [30].
GINZBURG-LANDAU EQUATIONS

The two-gap GL equations were obtained both for the dirty limit without interband scattering [38, 85], and for the clean limit [86]. Here we consider the GL dirty limit, focusing on new e?ects brought by interband scattering. For γmm′ = 0, the Usadel equations near Tc yield fm = Ψm /ω + Dmα Π2 Ψ/2ω 2 ? Ψm |Ψm |2 /2ω 3 , α (47) where the principal axis of Dαβ are taken along the crystalline axis. For weak interband scattering, the free energy F = F0 + Fi contains the free energy F0 {Ψ1 , Ψ2 } for γmm′ = 0 and the correction Fi {Ψ1 , Ψ2 } linear in γmm′ . Here F0 does not have ?rst order corrections in γmm′ if Ψm satis?es the GL equations, so Fi can be calculated by substituting Eq. (47) into Eq. (9) and expanding gm ≈ 1 ? |fm |2 /2 ? |fm |4 /8: Fi = πqT
ω>0

cos α +

2

(a) (c) Dm Dm

sin α]

2

1/2

(44)

In terms of the bilayer model shown in Fig. 1, Eq. (44) just means that Eq. (43) should be applied separately for each of the ?lms. For g = 0, Eqs. (27) and (44) determine the angular dependence of Hc2 (α) near Tc , and the London penetration depth Λαβ is given by [38] ?2 ?1 4π 4 αβ αβ + N2 D2 ?2 tanh N1 D1 ?1 tanh φ2 2T 2T 0 (45) Eqs. (27), (44), and (60) show that the one-gap scaling (43) breaks down because the behavior of Hc1 (α, T ) is mostly controlled by the cleaner band for all T, while the behavior of Hc2 (α, T ) is determined by the cleaner band at higher T, and by the dirtier band at lower T. Thus, Γ1 (T ) and Γ2 (T ) for Hc1 and Hc2 in the two-gap dirty limit are di?erent. Temperature dependencies of Γ(T ) were calculated in Refs. [38, 39]. Eqs. (44) and (19) describe well both the temperature and the angular dependencies of Hc2 (α, T ) in dirty MgB2 ?lms [25, 28, 29, 30]. Eq. (44) is valid if the σ band is not too anisotropic, and the o?-diagonal elements Mmn ? ζ m+n are negligible provided that ζ ? 1 [38]. Here Λ?2 = αβ ζ= [ cos2 α + ?1 sin2 α +
(ab) (c) D1 /D1

[|f1 ? f2 |2 + (|f1 |2 ? |f2 |2 )2 /4],

(48)

where q = (N1 γ12 + N2 γ21 )/2. Combining Fi with F0 in the dirty limit for g = 0 [38], we arrive at the GL free energy F dV for g ? 1: F = a1 |Ψ1 |2 + c1α |Πα Ψ1 |2 + b1 |Ψ1 |4 /2 +a2 |Ψ2 |2 + c2α |Πα Ψ2 |2 + b2 |Ψ2 |4 /2

?ai Re(Ψ1 Ψ? ) + ciα Re(Πα Ψ1 Π? Ψ? ) 2 α 2 ?bi |Ψ1 |2 |Ψ2 |2 + 2bi (|Ψ1 |2 + |Ψ2 |2 )Re(Ψ1 Ψ2 ) Here the GL expansion coe?cients are given by T πγ12 N1 ln + 2 T1 4T N2 πγ21 T a2 = + ln 2 T2 4T 7ζ(3)γ12 π ? c1α = N1 D1α 16T 8π 2 T 2 7ζ(3)γ21 π ? c2α = N2 D2α 16T 8π 2 T 2 a1 = , , , ,

(49)

(50) (51) (52) (53)

(?1 ? ?2 )2 sin4 α

cos2 α + ?2 sin2 α]4

, (46)

For ?2 = 1, the and ?2 = ?1 = parameter ζ(α) < 0.45 for a rather strong anisotropy

(ab) (c) D2 /D2 .

8 3πγ12 7ζ(3) ? , 2T 2 16π 384T 3 3πγ21 7ζ(3) ? , b2 = N2 16π 2 T 2 384T 3 N1 λ12 N2 λ21 πγ12 πγ21 ai = + , + + 2 w 4T 2 w 4T 7ζ(3) (D1 + D2 )(γ12 N1 + γ21 N2 ), ci = (4πT )2 π (γ12 N1 + γ21 N2 ), bi = 384T 3 b1 = N1 (54) (55) (56) (57) (58) which reduces to Eq. (27) near Tc to the linear accuracy in γmm′ . However, GL calculations of Hc2 (T ) in MgB2 beyond the linear Tc ? T term [87, 88] have a rather limited applicability, since a1 (T ) and a2 (T ) change signs at very di?erent temperatures T1 and T2 . For λmn of Ref. [77], T1 ? 0.9Tc0 and T2 ? 0.1Tc0 so higher order gradient terms (automatically taken into account in the Eliashberg/Eilenberger/Usadel based theories) become important. For example, at T ≈ T1 where a2 (T1 ) ? a1 (T ), retaining the ?rst gradient term ∝ c2 requires taking into account a next order term ? H 2 in the ?rst brackets in Eq. (61), which is beyond the GL accuracy. Thus, applying the GL theory in a wider temperature range [87, 88] makes it a procedure of unclear accuracy, which can result in a spurious upward curvature in Hc2 (T ) not always present in a more consistent theory (for example, in the dirty limit at D1 ? D2 ). In addition, the anisotropy of D1 (α) may further limit the applicability of the GL theory for H||ab, as for c1 ? c2 higher order gradient terms in the π band become important [85].

where T1 = Tc0 exp[?(λ0 ? λ? )/2w], and T2 = Tc0 exp[?(λ0 + λ? )/2w]. The GL equations are obtained by varying F dV . I would like to point out the misprints with wrong signs of ai , c1 and c2 in Eqs. (13), (14) and (20) in Ref. [38] (see also Ref. [86]). The ?rst two lines in Eq. (49) are the GL intraband free energies and the term ai Re(Ψ1 Ψ? ) describes 2 the Josephson coupling of Ψ1 and Ψ2 . Interband scattering increases a1 and a2 , and the interband coupling constant ai . The net result is the reduction of Tc determined by the equation 4a1 (Tc )a2 (Tc ) = a2 , which reproi duces Eq. (16). Besides the renormalization of am , bm and cm , interband scattering produces new terms, which describe the mixed gradient coupling and the nonlinear quatric interaction of Ψ1 and Ψ2 . Similar terms were introduced in the GL theories of heavy fermions [12] and borocarbides [89], and phenomenological models of Hc2 in MgB2 [87]. These terms result from interband scattering, so both ci and bi vanish in the clean limit [86]. The mixed gradient terms in Eq. (49) produce interference terms in the current density J = ?cδF/δA: J = ?[(2c1 ?2 + ci ?1 ?2 cos θ)Q1 + 1 (2c2 ?2 + ci ?1 ?2 cos θ)Q2 + 2 ci (?2 ??1 ? ?1 ??2 ) sin θ]2πc/φ0

DISCUSSION

(59)

where Qm = ?θm + 2πA/φ0 , and θ = θ1 ? θ2 . Here J is no longer the sum of independent contributions of two bands, because phase gradients in one band produce currents in the other. Moreover, J acquires new cos θ terms and the peculiar sin θ interband Josephsonlike contribution for inhomogeneous gaps. For currents well below the depairing limit, both bands are phaselocked (θ = 0), and Eq. (59) de?nes the London penetration depth Λ2 = cφ0 Q/8π 2 |J|: Λ = φ0 /4π[2π(c1 ?2 + ci ?1 ?2 + c2 ?2 )]1/2 1 2 (60)

where c1 , c2 and ci depend on the ?eld orientation according to Eq. (44). Eq. (47) can be used to calculate ⊥ Hc2 (T ) from the linearized GL equations, which give Hc2 as a solution of the quadratic equation [89] 4 2πc1 H + a1 φ0 2πci H 2πc2 H + a2 = ai + φ0 φ0
2

(61)

The remarkable ten-fold increase of Hc2 (T ) in C-doped MgB2 ?lms [25, 26, 27, 28, 29] has brought to focus new and largely unexplored physics and materials science of two-gap superconducting alloys. Moreover, the observations of Hc2 close to the BCS paramagnetic limit poses the important question of how far can Hc2 be further increased by alloying. This possibility may be naturally built in the band structure of MgB2 , which provides weak interband coupling and weak interband scattering, thus allowing MgB2 to be alloyed without strong suppression of Tc . For example, for the C-doped MgB2 ?lm shown in Fig. 1, ρn was increased from ? 0.4??cm to 560??cm, yet Tc was only reduced down to 35K [28]. It is the weakness of interband scattering, which apparently makes it possible to take advantage of very dirty π band to significantly boost Hc2 in carbon-doped ?lms which typically have Dπ ? 0.1Dσ . The reasons why scattering in the π band of C-doped MgB2 ?lms is so much stronger than in the σ band has not been completely understood, but another immediate bene?t for high-?eld magnet applications [37] is that carbon alloying signi?cantly reduces the anisotropy of Hc2 down to Γ(T ) ? 1 ? 2. Despite many yet unresolved issues concerning the twogap superconductivity in MgB2 alloys, Hc2 of C-doped MgB2 has already surpassed Hc2 of Nb3 Sn (see Fig. 1). Given the intrinsic weakness of interband scattering, which enables tuning MgB2 by selective atomic substitutions on Mg and B sites, there appear to be no fundamental reasons why Hc2 of MgB2 alloys cannot be pushed further up toward the strong-coupling paramagnetic limit (1). Thus, understanding the mechanisms of intra and interband impurity scattering in carbon-doped MgB2 , and

9 the competition between scattering and doping e?ects becomes an important challenge for the computational physics. For instance, it remains unclear why the multiphased C-doped HPCVD grown ?lms [32] exhibit higher Hc2 and weaker Tc suppression [28] than uniform carbon solid solutions [26, 69, 70]. This unexpected result may indicate other extrinsic mechanisms of Hc2 enhancement, which are not accounted by the simple two-gap theory presented here. Among those may be e?ects of electron localization or strong lattice distortions in multiphased C-doped ?lms which can manifest themselves in the buckling of the Mg planes observed in the dirty ?ber-textured MgB2 ?lms shown in Fig. 2 [25]. Such buckling may enhance scattering in the π band formed by out-of-plane pz boron orbitals. Recently signi?cant enhancements of vortex pinning and critical current densities Jc in MgB2 [90, 91, 92, 93, 94, 95, 96] has been achieved, particularly by introducing SiC [92] and ZrB2 [95] nanoparticles. Given these promising results combined with weak current blocking by grain boundaries [97], the lack of electromagnetic granularity [98], and very slow thermally-activated ?ux creep [99, 100], it is not surprising that MgB2 is being regarded as a strong contender of traditional high-?eld magnet materials like NbTi and Nb3 Sn. Despite these achievements, a detailed theory of pinning in MgB2 is trill lacking. Such theory should take into account a composite structure of the vortex core, which consists of concentric regions of radius ξσ and ξπ where ?σ (r) and ?π (r) are suppressed [101, 102, 103, 104, 105, 106]. For example, in MgB2 single crystals the larger vortex cores in the π band start overlapping above the ”virtual up2 per critical ?eld” Hv = φ0 /2πξπ ? 0.5T , causing strong overall suppression of ?π well below Hc2 [102, 103]. This e?ect can reduce Jc at H > Hv , however both Hv and Hc2 can be greatly increased by appropriate enhancement of impurity scattering in MgB2 alloys. Recently there has been an emerging interest in microwave response of MgB2 [107, 108, 109] and a possibility of using MgB2 in resonant cavities for particle accelerators [110, 111]. These issues require understanding nonlinear electrodynamics and current pairbreaking in two-gap superconductors [112, 113], in particular, band decoupling and the formation of interband phase textures at strong rf currents [16, 17]. This work was partially supported by in-house research program at NHMFL. NHMFL is operated under NSF Grant DMR-0084173 with support from state of Florida.
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