0 Brazilian Journal of Physics, vol. 33, no. 4, December, 2003
Anomalous Effects of Two Gap Superconductivity in M gB2
Applied Superconductivity Center, University of Wisconsin, Madison, Wisconsin 53706, USA
Received on 23 May, 2003.
In this paper a brief overview of anomalous behavior resulting from the two-gap superconductivity in M gB2
is given. We focus on two characteristic effects: an anomalous enhancement of the upper critical ﬁeld by
nonmagnetic impurities and nonequilibrium interband phase textures which appear as a result of interband
breakdown caused by electric ﬁeld. Both effects distinguish M gB2 from the existing low-Tc and high-Tc
1 Introduction ⊥
ﬁelds Hc2 (0) 3 − 5T and Hc2 (0) 15 − 19T of M gB2
single crystals [3, 10], where ⊥ and || correspond to H per-
The discovery of the two-gap superconductivity in M gB2 [1, pendicular and parallel to the ab plane. As far as Hc2 is
2, 3] (and perhaps in N bSe2 ) has brought to focus new concerned, it can be increased by nonmagnetic impurities,
effects of unconventional pairing and multicomponent order following the well-known route for dirty one-gap supercon-
parameters ψ with internal degrees of freedom [5, 6]. In ductors in which the zero-temperature Hc2 (0) and the slope
particular, M gB2 has two different s-wave superconducting Hc2 = dHc2 /dT at Tc are increased proportionally to the
gaps ∆σ (0) ≈ 7.2mV and ∆π (0) ≈ 2.3mV residing on dis- normal state residual resistivity ρ:
connected sheets of the Fermi surface (FS), which comprises
nearly cylindrical 2D parts formed by in-plane σ antibond- Hc2 (0) = 0.69Tc Hc2 , Hc2 = 4eckB NF ρ/π, (1)
ing pxy orbitals of B, and a more isotropic 3D tubular net-
work formed by out-of-plane π bonding and antibonding pz where NF is the density of states at the FS and −e is the
orbitals of B. For two weakly coupled s-wave order param- electron charge. The same approach has also been applied
eters ψ1 = ∆1 eiθ1 and ψ2 = ∆2 eiθ2 , the internal degree of to M gB2 in which scattering was introduced by irradiation
freedom is the interband phase difference θ(r, t) = θ1 − θ2 . or atomic substitutions on both B and Mg sites . For
In this case, in addition to the phase-locked states (θ = 0, π), instance, in c-axis oriented M gB2 ﬁlms , ρ was in-
peculiar phase textures θ(r, t) and collective modes  oc- creased from ∼ 1µΩcm to more than 200µΩcm, resulting
cur. in Hc2⊥ ≈ 1T/K and Hc2 ≈ 1.8T/K, while reducing Tc
This paper addresses new electromagnetic effects, which down to ≈ 31K. Based on these numbers, the extrapola-
principally result from the two-band superconductivity, tion (1) gives Hc2 (0) ≈ 20T, still below Hc2 (0) ≈ 30T of
making M gB2 unique among the existing superconductors. N b3 Sn. However, Eq. (1) signiﬁcantly underestimates the
Such effects manifest themselves in the following areas: 1. actual Hc2 in two-gap superconductors, thus Hc2 of M gB2
High-ﬁeld superconductivity in dirty two-gap superconduc- can exceed Hc2 (0) of N b3 Sn even for Hc2⊥ ≈ 1T/K which
tors due to their anomalous response to nonmagnetic im- have already been achieved[11, 8].
purities.  This makes it possible to greatly increase the The Fermi surface of M gB2 provides three different im-
upper critical ﬁeld Hc2 by alloying M gB2 and optimizing purity scattering channels: intraband scattering within σ and
the ratio of intraband scattering rates, as has already been π FS sheets, and interband scattering. Intraband scattering
observed.  2. Interband tunneling and intrinsic Joseph- reduces the intrinsic anisotropy of ∆σ and ∆π with no effect
son effect, which give rise to dislocation-like phase textures of Tc , while the pairbreaking effect of interband scattering
in the order parameter, and interband breakdown caused by is weak due to orthogonality of σ and π orbitals. The
the electric ﬁeld.  These textures manifest themselves in multiple scattering channels provide the essential ﬂexibility
new effects in nonlinear electromagnetic response. to increase the Hc2 of M gB2 to a much greater extent than
in one-gap superconductors not only by the usual increase of
ρ, but also by optimizing relative weights of σ and π scatter-
2 High-ﬁeld superconductivity ing rates by selective atomic substitution on B and Mg sites.
This follows from recent calculations of Hc2 from the Us-
So far all attempts to increase Tc of M gB2 by doping have adel equations in which all scattering channels in M gB2 are
been unsuccessful, while the signiﬁcant potential of M gB2 accounted for via the electron diffusivity tensors Dm for
for applications is still limited by rather low upper critical each m-th FS sheet and the interband scattering rates γmm .
A. Gurevich 1
The Usadel equations for two-gap superconductors are:  measurements of Hc2 (T ) on resistive 220µΩcm c-axis ori-
ented ﬁlm , which has very high Hc2 (T ) exceeding Hc2
1 αβ of N b3 Sn. The ﬁt in Figs. 2 and 3 also revealed that the π
ωf1 − D1 [g1 Πα Πβ f1 − f1 α β g1 ] =
2 band is this ﬁlm is much dirtier (Dπ 0.1Dσ ) than the σ
ψ1 g1 + γ12 (g1 f2 − g2 f1 ) (2) band, which may be due to distorted and buckled Mg sub-
1 αβ lattice .
ωf2 − D2 [g2 Πα Πβ f2 − f2 α β g2 ] =
ψ2 g2 + γ21 (g2 f1 − g1 f2 ), (3)
Eqs. (2) and (3) are supplemented by the self-consistency
equations for the order parameters ψm = ∆m exp(iθm ),
220 µΩ cm
H ⊥ ab
ψm = 2πT λmm fm (r, ω), (4) 25
H⊥ , Tesla
Here |fm |2 + gm = 1, the band index m runs from 1 and
2, Nm is the partial density of states, Π = + 2πiA/φ0 ,
A is the vector potential, φ0 is the ﬂux quantum, and ω =
πT (2n + 1), n = 0, ±1, ..., and the matrix elements of the MgB2
BCS coupling constants λmm are given by λσσ ≈ 0.81, 7 µΩ cm
λππ ≈ 0.285, λσπ ≈ 0.119, and λπσ ≈ 0.09  (the in-
dices 1 and 2 correspond to σ and π bands, respectively). MgB2 single crystal
The Usadel equations were recently used to calculate vor-
tices in M gB2 . The values of γmm and Dm can be 0 10 20 30 40
either calculated from ﬁrst principles or extracted from the T, Kelvin
observed Hc2 (T ) and ρ(T ) curves . For the 2D σ band,
the principal value Dσ along the c-axis is much smaller ⊥
Figure 1. Temperature dependence of Hc2 (T ). The data
(a) (b) αβ
than the in-plane Dσ and Dσ , but the anisotropy in Dπ points show experimental data for dirty 220µΩcm ﬁlm and
for the 3D π-band is much weaker. epitaxial M gB2 ﬁlm , and the solid curve is calculated
Solving Eqs. (2)-(4) [7, 14] for γmm = 0, yields the from Eq. (5) with Dπ = 0.12Dσ .
following equation for Hc2 :
a0 [ln t + U (h/t)][ln t + U (ηh/t)] +
a2 [ln t + U (ηh/t)] + a1 [ln t + U (h/t)] = 0, (5) (b)
where a1 = 1 + λ− /λ0 , a2 = 1 − λ− /λ0 , a0 = 2w/λ0 , 220 µΩ cm H || ab
λ0 = (λ2 + 4λ12 λ21 )1/2 , λ± = λ11 ± λ22 , w = λ11 λ22 −
λ12 λ21 , η = D2 /D1 , h = Hc2 D1 /2φ0 Tc , t = T /Tc , 35
and ψ(x) is the di-gamma function. For equal diffusivities, 30
H|| , Tesla
η = 1, Eq. (5) reduces to the one-gap Maki-deGennes equa-
tion ln t + U (h/t) = 0. To account for the dependence of 25
Hc2 (θ) on the angle between H and the c-axis, D1 and D2 20 7 µΩ cm
in Eq. (5) should be replaced by the angular dependent dif-
fusivities D1 (θ) and D2 (θ) for both bands :
10 MgB2 single crystal
Dm (θ) = [Dm cos2 θ + Dm Dm sin2 θ]1/2
(a)2 (a) (c)
Eqs. (5) and (6) describe a rather anomalous behavior, 0
0 10 20 30 40
depending on the material parameter η = D1 /D2 which T, Kelvin
can be varied by disordering either B or Mg sublattices. In
the case of large difference between D1 and D2 , the de-
pendence Hc2 (T ) can exhibit a signiﬁcant upward curva- Figure 2. Temperature dependence of Hc2 (T ). The data
ture, because the slope Hc2 at Tc is inversely proportional to points show experimental data for dirty 220µΩcm ﬁlm
the maximum diffusivity, while Hc2 (0) is inversely propor- and epitaxial M gB2 ﬁlm , and the solid curve is cal-
tional to the minimum diffusivity. Thus, Hc2 (0) can be much culated from Eqs. (5) and (6) with [Dπ Dπ ]1/2 =
higher than the one-gap extrapolation (1) suggests. Figs. 1 0.2[Dσ Dσ ]1/2 .
and 2 show good ﬁt of Eqs. (5) and (6) to pulse high-ﬁeld
2 Brazilian Journal of Physics, vol. 33, no. 4, December, 2003
6 For M gB2 , the tensor Λ−2 is a sum of the diffusivities D1
5.5 1 and D2 with markedly different anisotropies and absolute
values. Thus, Λαβ is always limited by the cleanest band
with the maximum diffusivity, so the ratio Hc1 (θ)/Hc2 (θ)
4.5 not only becomes dependent on the ﬁeld orientation, but its
4 angular dependence turns out to be different at different T.
The two-band superconductivity in M gB2 provides a
new way to boost Hc2 , because a higher Hc2 (0) is possible
3 for a given slope Hc2 at Tc . For example, if Hc2 = 1T /K
and Tc = 40 K, the theory predicts Hc2 (0) >40 Tesla,
which exceeds Hc2 (0) of N b3 Sn, even though Hc2 is still
2 2 smaller than 2 T/K characteristic of many low-Tc and high-
1.5 Tc materials. For Hc2 = 1T /K, the shortest GL in-plane
coherence length ξ(0) = [φ0 /2πTc Hc2 ]1/2 ≈ 3 nm for the
0 0.2 0.4 0.6 0.8 1 σ band is still large enough to ensure no signiﬁcant mag-
T/Tc netic granularity and weak link behavior at grain boundaries.
Thus, there are no inherent limitations to further increase
Figure 3. Temperature dependence of the anisotropy pa- of Hc2 toward the high-Tc level of 2 T/K by proper alloy-
rameter Hc2 (T )/Hc2 (T ). Solid squares and empty triangles ing or by quenched-in lattice disorder in M gB2 with the
correspond to the dirty 220µΩcm and the epitaxial M gB2 account of its complex substitutional chemistry.  For
ﬁlm, respectively. The curve 1 is calculated from Eqs. (5) Hc2 2T /K, the ﬁeld Hc2 (0) would approach the param-
(ab) (c) (ab) (ab)
and (6) with Dσ = 36Dσ , and Dπ = 5Dσ , and agnetic limit of 70 Tesla, in which case a more general
(ab) (ab) (ab) (c)
curve 2 for Dπ = 0.09Dσ , and Dσ = 3Dσ Eliashberg theory should be used to include strong coupling
and spin effects.
Eqs. (5) and (6) also describe an unusual temperature
dependence of the anisotropy parameter γ(T ) = Hc2 /Hc2
different from the predictions of the anisotropic one-gap GL
theory in which γ(T ) =const. Because the 2D σ band in 3 Intrinsic Josephson effect and inter-
(c) (a) (c) (a)
M gB2 results in Dσ /Dσ Dπ /Dπ , γ(T ) can ei- band phase textures
ther increase as T decreases if Dπ > Dσ , or decrease as
T decreases if Dπ Dσ . The ﬁrst case is characteristic of To calculate the interband phase textures θ(r, t), we derive
cleaner samples , whereas the second case was observed the equations of motion for θ and the electric ﬁeld E at
on dirty ﬁlms, as shown in Fig 3. The anisotropy of the lower T ≈ Tc from the time-dependent Ginzburg-Landau (TDGL)
critical ﬁeld Hc1 (T ) is different from that of Hc2 (T ) , ∗
equations, Γm (∂t − 2πciϕ/φ0 )ψm = −δF/δψm . Here ϕ
as evident from the London penetration depth tensor Λαβ in is the electric potential, Γm are damping constants, and the
the dirty imit : free energy F = d3 r(f1 + f2 + fm + fint ) contains the
magnetic part fm = | × A|2 /8π, the GL intraband part
4π 2 e2 αβ ∆1 αβ ∆2
N1 D1 ∆1 th + N2 D2 ∆2 th . (7) fm , and the interband energy fint :
c 2T 2T
fm = αm |ψm |2 + |ψm |4 + gm + A ψm , (8)
fint = γ(ψ1 ψ2 + ψ1 ψ2 )/2 = γ∆1 ∆2 cos θ, (9)
The current density J is a sum of supercurrent and the nor- For weak interband coupling, γ α1,2 , the gaps ∆1,2
mal current, are not affected by the phase textures, in which case the
equation of motion for θ = θ1 − θ2 become 
Js = −8π 2 c(g1 ∆2 Q1 + g2 ∆2 Q2 )/φ2 + σE,
1 2 0 (10)
where Qm = A − φ0 θm /2π, σ is the normal conductiv- ˙
τ θ θ = L2 2
θ + sign(γ) sin θ + αθ divJs , (11)
ity, and the supercurrent is a sum of independent intraband
contributions. Static Eqs. (8)-(10) were also derived from where the relaxation time τθ , the decay length Lθ , and the
the microscopic Usadel equations . charge coupling parameter αθ are given in Ref. . As
A. Gurevich 3
follows from Eq. (11), the θ-mode does not contribute to breakdown causes periodic generation of θ-solitons near the
the static magnetic response, since divJs = 0 for any dis- current leads and penetration of phase textures in the bulk,
tribution of bulk supercurrents. However, the θ-mode inter- as shown in Figs. 5,6. Here Jt = 2Lθ /αθ for Lθ Le , and
acts with a nonuniform electric ﬁeld due to nonequilibrium Jt = Le /αθ tanh(a/Le ) for Lθ Le .
charge imbalance, divJs = −σdivE. This happens near the
normal current leads, where the difference in the injected in-
traband charge densities provides the driving term αθ divE
in Eq. (11) due to the bands asymmetry, Γ1 g2 = Γ2 g1 .
Static distributions θ are described by the sine-Gordon equa- 80
tion L2 2 θ = sign(−γ) sin θ, which has a single-soliton
or staircase solutions similar to the vortex solutions in long
Josephson contacts . However, these θ-solitons differ
from the Josephson vortices, because they do not carry mag- 20
netic ﬂux and do not interact with magnetic ﬁelds and su- 0
percurrents, but can be driven by a nonequilibrium charge 1
density injected from normal electrodes. Thus, equilibrium
Time 500 0.5
nonuniform solutions θ(x) are always energetically unfa- 0 0 Distance
vorable as compared to the phase-locked states, θ = 0 for
γ < 0, or π for γ > 0, yet dynamic or quenched phase Figure 5. Formation of a soliton chain in the right half
textures can be generated during current-induced interband (0 < x < a) of the bridge of length 2a after J(t) was turned
breakdown. on from 0 to 1.025Jt at t = 0, and Le = a/10, Lθ = 0.1Le .
Times and distances are normalized to τθ and a, respectively.
N N Time
0 0 Distance
Figure 6. Moving soliton shuttle in the right half of the hori-
zontal leg (0 < x < a) in the four-terminal geometry shown
Figure 4. Geometries in which the interband phase break- in Fig. 4c. J(t) was turned on from 0 to 1.012Jt at t = 0,
down could occur. Here N labels normal electrodes, gray and the rest is the same as in Fig. 5.
domains show phase solitons moving along thin arrows, and
block arrows indicate current directions. Static phase tex-
tures form in microbridges (a) and point contacts (b), while Eqs. (11) and (12) were solved numerically for the
in the four-terminal geometry (c) the solitons and antisoli- bridge (Fig. 4a) where E(x, t) and θ(x, t) are even and odd
tons continuously annihilate in the center. functions of x, respectively, E(±a, t) = E0 , E (0, t) = 0,
The equation for E has the form  θ(0) = 0, θ (±a, t) = 0, and supercurrents in both bands
vanish at the normal electrodes, J = σE. In this case θ-
˙ ˙ ˙
τe E + E − L2 graddivE + αe θ = τe J/σ, (12) solitons ﬁrst appear at the bridge edges, but for J > Jt ,
they are pushed to the bulk by the strong gradient of E(x).
where J(t) is the driving current density, Le is the electric Then the next soliton forms near the edge and the process re-
ﬁeld penetration depth, τe is the charging time constant ob- peats periodically, resulting in the propagation of two soliton
tained in Ref. , and the coupling term αe θ describes an chains from the opposite current leads as shown in Fig. 5.
electric ﬁeld caused by moving phase textures. After the ﬁrst two solitons in the chains collide in the center
Eqs. (11) and (12) which describe nonlinear electro- they stop, while new solitons keep entering the bridge. Dur-
dynamics of a two-gap superconductor at ﬁxed gaps ∆1,2 ¯
ing this soliton pileup, the mean slope θ (t) increases, reach-
were used to calculate θ(x, t) in a current-carrying micro- ¯
ing a critical value θc αθ J/L2 (for J Jt ) at which
bridge of length 2a (Fig. 4). Below the critical current the soliton generation at the edges stops and a static texture
density Jt the bridge is in a phase-locked state, except lo- ¯
forms. During the soliton penetration, t < tc ∼ τθ aθc /2π,
calized phase kinks at the edges. For J > Jt , the interband a transient resistance and voltage oscillations are generated.
4 Brazilian Journal of Physics, vol. 33, no. 4, December, 2003
A similar behavior occurs at the point contact (Fig. 4b), the rf electric ﬁleld E depends on the polarization of E: if
in which concentric soliton shells propagate into the bulk, E(t) is parallel to the sample surface, then divE = 0, thus
forming a static structure. the phase mode is not excited by the rf ﬁeld. However, the
A very different kind of soliton dynamics occurs in the θ-mode contributes to the rf impedance if the rf ﬁeld has a
4-terminal geometry (Fig. 4c), for which currents ﬂow in component perpendicular to the sample surface.
the opposite directions, making 90◦ turns around the cen- Other interesting effects could occur in the point contact
tral stagnation point (x = 0) where θ = 0 by symmetry. geometry (Fig. 4c) in which high current densities J ∼ Jt
In this case E(x) is an odd functions of x so the driving near the contact (for example, an STM tip) can be achieved.
charge density divE does not change sign along the hori- If the tip is perpendicular to the ﬁlm surface of a c-axis
zontal leg of the cross in Fig. 4c, the total charge along oriented ﬁlm, then it mainly injects current into the 3D π
the horizontal leg is compensated by the opposite charge band, because the c-axis tunneling into the 2D σ band is
distributed along the vertical leg. The asymmetry of E(x) strongly suppressed. The resulting strong charge imbalance
causes generation of solitons and antisolitons at the oppo- between σ and π bands greatly facilitates generation of con-
site current leads, which then move toward the center of the centric soliton structures, which can be used to probe the
cross where they annihilate, as shown in Fig. 6. Such con- interband breakdown with point contacts. If the currents are
tinuous soliton motion takes place if the width wy of the ver- simultaneously injected from two point contacts and drained
tical leg is greater than the width wx of the horizontal leg, into another current lead, the periodic voltage oscillations
so that the current density in the horizontal leg I/wx ex- between the contacts occur in a way similar to the above-
ceeds Jt , while the vertical leg remains in the phase-locked described oscillations in a mictobridge.
state I/wy < Jt , where I is the total sheet current. For This work was supported by the NSF MRSEC (DMR
a Le , the soliton-antisoliton annihilation in the center is 9214707), AFOSR MURI (F49620-01-1-0464).
unaffected by the charge imbalance near the current leads.
Two different dynamic states represented in Figs. 5
and 6 have clear analogs in the theory of long Josephson References
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