Critical Current of Type-II Superconductors in a Broken Bose Glass State
Critical Current of Type-II Superconductors in a Broken Bose
Glass State
J. P. Rodriguez1
1Department of Physics and Astronomy,
California State University, Los Angeles, California 90032
(Dated: November 1, 2018)
Abstract
The tilt modulus of a defective Abrikosov vortex lattice pinned by material line defects is com-
puted using the boson analogy. It tends to infinity at long wavelength, which yields a Bose glass
state that is robust to the addition of weak point-pinning centers, and which implies a restoring
force per vortex line for rigid translations about mechanical equilibrium that is independent of
magnetic field. It also indicates that the Bose glass state breaks into pieces along the direction of
the correlated pinning centers if the latter have finite length. The critical current is predicted to
crossover from two dimensional to three dimensional behavior as a function of sample thickness
along the correlated pinning centers in such case. That crossover notably can occur at a film
thickness that is much larger than that expected from point pins of comparable strength. The
above is compared to the dependence on thickness shown by the critical current in certain films of
high-temperature superconductors currently being developed for wire technology.
http://arxiv.org/abs/0704.1536v6
INTRODUCTION
It well known that thin films of high-temperature superconductors exhibit larger critical
currents than their single-crystal counterparts. Thin films of the high-temperature supercon-
ductor YBa2Cu3O7−δ (YBCO) grown by pulsed laser deposition (PLD), which are actively
being developed for wire technology, achieve critical currents that are a significant fraction
of the maximum depairing current, for example[1]. Evidence exists that lines of dislocations
that run parallel to the crystalline c axis in PLD-YBCO act as correlated pinning centers for
vortex lines inside of the superconducting state[2], and thereby give rise to such high critical
currents. This is confirmed by the peak observed in the critical current of PLD-YBCO at
orientations of the c axis aligned parallel to an applied magnetic field, as well as by the dra-
matic enhancement of the former peak after more material defects in the form of nano-rods
aligned parallel to the c axis are added[3][4].
The microstructure described above for PLD-YBCO films immediately suggests that
the vortex lattice that emerges from the superconducting state in applied magnetic field
aligned parallel to the c axis is some form of Bose glass characterized by a divergent tilt
modulus[5]. In the limit of a rigid vortex lattice, to be expected at large magnetic fields,
two-dimensional (2D) collective pinning of vortex lines by the material line defects then
determines the critical current [6][7][8][9]. Recent theoretical calculations that follow this
line of reasoning find moderate quantitative agreement with the critical current measured
in films of PLD-YBCO in c-axis magnetic fields of a few to many kG, at liquid nitrogen
temperature[10]. A potential problem with the Bose glass hypothesis, however, is that the
material line defects in PLD-YBCO films can meander, or they can be of relatively short
length[3][4]. That question is addressed in this paper, where we find that the Bose glass
breaks up into pieces along the direction of the correlated pinning centers when the effective
length of the latter is less than the film thickness. In particular, the profile of the critical
current versus film thickness is predicted to reflect a crossover from two-dimensional to three-
dimensional (3D) collective pinning of the vortex lines[11]. This cross-over can occur at a
length scale that is notably much larger than that expected from point pins of comparable
strength[12]. We also find that the unbroken Bose glass state is robust to the addition of
weak point pins. Good agreement is achieved between the dependence on thickness shown
by the critical current in certain films of PLD-YBCO at applied magnetic field[12] and that
predicted for the true Bose glass state[10]. Last, the effective restoring force per vortex line
due to a rigid translation of the Bose glass about mechanical equilibrium, which is gauged
by the Labusch parameter[13], is found to depend only weakly on applied magnetic field.
This prediction agrees with recent measurements of the microwave surface resistance on
PLD-YBCO films with nano-rod inclusions[14].
TILT MODULUS OF BOSE GLASS
Material line defects in thin enough films of PLD-YBCO can be considered to be per-
fectly parallel to the c-axis. They notably arrange themselves in a manner that resembles a
snapshot of a 2D liquid[2], as opposed to a gas[15][16]. In particular, such correlated pin-
ning centers do not show clusters or voids. A defective vortex lattice that assumes a hexatic
Bose glass state will then occur for external magnetic fields aligned in parallel to such a
microstructure, in the limit of weak correlated pinning. It is characterized by parallel lines
of edge dislocation defects that are injected into the pristine vortex lattice in order to relieve
shear stress due to the correlated pins. The former do not show any intrinsic tendency to
arrange themselves into grain boundaries, however, due to the absence of clusters and voids
in the “liquid” arrangement of linear pinning centers. (Cf. refs. [15] and [16].) This results
in a vortex lattice whose translational order is destroyed at long range by the isolated lines
of edge dislocations, but which retains long-range orientational order. Collective pinning
of the dislocation defects by the correlated pins then results both in an elastic response
to shear and in a net superfluid density. Theoretical calculations[17] and numerical Monte
Carlo simulations (see fig. 1 and caption) of the corresponding 2D Coulomb gas ensemble[18]
confirm this picture.
The correlation length Lc(|) for order along the magnetic field direction is infinite for
a Bose glass state. In the limit of weak correlated pins, the density of dislocation defects
that thread the corresponding vortex lattice can then be obtained by applying the theory
of 2D collective pinning[6][7][8]. Each line of unbound edge dislocations is in one-to-one
correspondence with a well-ordered bundle of vortex lattice, or Larkin domain, that has
dimensions Rc(|) × Rc(|) in the directions transverse to the magnetic field. The injection
of the dislocation lines into the pristine vortex lattice then results in plastic creep of each
Larkin domain by a Burgers vector[19]. The transverse Larkin scale is hence obtained by
minimizing the sum of the elastic energy cost due to the edge dislocations with the gain in
pinning energy due to the translation of a Larkin domain by an elementary Burgers vector
of the triangular vortex lattice, b = a△. This yields the estimate[6][7][8]
Rc(|)−2 = C20np(fp/c66b)2, (1)
for the density of Larkin domains, which coincides with the density of lines of unbound
dislocations. Here np denotes the density of pinned vortex lines, fp denotes the maximum
pinning force along a material line defect per unit length, and c66 denotes the elastic shear
modulus of the pristine vortex lattice. The prefactor above is of order[8] C0 ∼= π/ln(Rc/a′df)2,
where a′df is the core diameter of a dislocation in the vortex lattice. Consider now the
limit of weak pinning centers that do not crowd together: fp → 0 and πr2p · nφ ≪ 1,
respectively, where nφ denotes the density of material line defects, and where rp denotes
their range. Simple considerations of probability then yield the identity np/nφ = nB · πr2p
between the fraction of occupied pinning centers and the product of the density of vortex
lines, nB, with the effective area of each pinning center. Substituting it plus the estimate
c66 = (Φ0/8πλL)
2nB for the shear modulus of the pristine vortex lattice[21] into Eq. (1)
then yields the result Rc(|)−2 ∼= (
3π/2)C20(4fprp/ε0)
2nφ for the density of Larkin domains,
which depends only weakly on magnetic field. Here, λL denotes the London penetration
depth and ε0 = (Φ0/4πλL)
2 is the maximum tension of a flux line in the superconductor. All
of the above is valid in the 2D collective pinning regime that exists at perpendicular magnetic
fields beyond the threshold Bcp = C
3/2)(4fp/ε0)
2Φ0, in which case many vortex lines are
pinned by material line defects within a Larkin domain of transverse dimensions Rc(|)×Rc(|)
[10].
We will now exploit the boson analogy for vortex matter in order to compute the uniform
tilt modulus of the hexatic Bose glass in the absence of point pinning centers[5]. It amounts
to a London model set by the free-energy density
gB({r}, z) =
V0(ri, rj) +
Vp(ri) (2)
for vortex lines located at transverse positions {ri(z)}, at a coordinate z along the field
direction. Here ε̃l denotes the tension of an isolated vortex line, while the pair potential
V0(r, r
′) describes the interaction between vortex lines at the same longitudinal coordinate
z. The energy landscape for the correlated pinning centers is described by the potential
energy Vp(r), which is independent of the coordinate z along the field direction. The ther-
modynamics of this system in the presence of an external tilt stress nBa is then set by the
partition function
ZB[a] = (Πi
D[ri(z)])exp[−(kBT )−1
dz[gB({r}, z)−
d2r jB · a]], (3)
which under periodic boundary conditions, ri(z+Lz) = ri(z), is equivalent to a system of 2D
bosons. Above, jB(r, z) =
(2)[r − ri(z)](dri/dz) is the current density within the boson
analogy. Observe now that the kernel Πµ,ν(ω) of the uniform electromagnetic response for
alternating current (AC) is connected to this partition function through the proportionality
relationship
ZB[a] ∝ exp
(kBT )
a ·Π · a
in the limit that the corresponding uniform tilt stress vanishes, a → 0. Above, the Matsubara
frequencies iωn are given by the allowed wavenumbers along the magnetic field, qz = 2πn/Lz,
and V = LxLyLz is the volume of the system. The fact that the tilt stress is given by nBa
then yields the identity
C44(qz) = n
B/Π⊥(ωn) (5)
between the uniform tilt modulus and the uniform AC electromagnetic response of the
2D Bose glass. The subscript “⊥” above is a tag for the pure shear component of the
electromagnetic kernel Πµ,ν (Cf. ref. [5]).
The hexatic Bose glass state is clearly a 2D dielectric insulator within the boson analogy.
Its electromagnetic response can therefore be modeled by the kernel
Π⊥(ω) = (nB/ε̃l)ω
2/(ω2 − ω20), (6)
which is dielectric in the low-frequency limit, and which conserves charge by satisfying the
oscillator f-sum rule. Above, ω0 is the natural frequency of the electric dipole degrees of free-
dom in the Bose glass. The latter correspond to the 2D Larkin domains in reality, since they
represent the smallest units of well-ordered vortex lattice that can respond independently
to an applied force. We then have that the above natural frequency is of order the resonant
frequency for transverse sound inside a Larkin domain of the 2D Bose glass: ω0 ∼ γ/Rc(|),
where γ = (c66/c44)
1/2 is the effective mass anisotropy parameter equal to the transverse
sound speed within the boson analogy. Here c44 = nBε̃l is the tilt modulus due to isolated
flux lines. After substitution into Eqs. (6) for the AC response, the identity (5) then yields
a divergent tilt modulus for the Bose glass at long wavelength
C44(qz) = nBε̃l[1 + (qzL∗)
−2], (7)
with a longitudinal scale L∗ = ω
0 that is related to the transverse Larkin length by the
anisotropic scale transformation L∗ ∼ Rc(|)/γ.
Expression (7) for the uniform tilt modulus of a Bose glass is the central result of the
paper. We shall first extract the Labusch parameter[13] from the singular behavior that it
shows at long wavelength. In particular, observe that the elastic energy density for a periodic
tilt of the Bose glass by a displacement u0 at long wavelength,
C44(qz)q
0, acquires a
contribution of the form 1
0 from this divergence, with k0 = c66/Rc(|)2. The latter is
simply the spring constant per unit volume of the restoring force for a rigid translation of
the Bose glass state about mechanical equilibrium. Using the estimate c66 =
ε0nB for the
shear modulus of the vortex lattice[21] yields an effective spring constant per vortex line due
to 2D collective pinning limited by plastic creep, k0/nB, that is given by kp =
ε0/Rc(|)2.
Notice that kp depends only weakly on magnetic field. The Labusch parameter extracted
from the microwave surface resistance on PLD-YBCO films with nano-rod inclusions also
exhibits only a weak dependence on external magnetic field aligned parallel to the nano-rods
(or c-axis)[14]! The above should be compared to the corresponding Labusch parameter due
to point pins[13], which depends strongly on magnetic field. Indeed, given the conjecture
kp = (c66/R
c + c44/L
c)/nB for the Labusch parameter due to 3D collective-pinning implies
that it decays with increasing magnetic field instead as 1/B2 in such case. Here we have
assumed anisotropic scaling, and we have used[9] Rc ∝ B.
We shall next use Eq. (7) for the uniform tilt modulus to test how robust the hexatic Bose
glass is to the addition of point pinning centers. The hexatic Bose glass shows long-range
orientational order in all directions (see fig. 1). The addition of point pins will therefore
break it up into Larkin domains, of dimensions R′c × R′c × Lc transverse and parallel to the
magnetic induction, that tilt in the transverse direction by a distance of order the size of
the vortex core, ξ. Such a break-up then has an elastic energy cost per unit volume and a
pinning energy gain per unit volume that sum to[6][9]
R′2c Lc
f ′0ξ. (8)
Here C66 denotes the shear modulus of the hexatic Bose glass, which can be approximated
by c66 in the limit of weak correlated pinning, while the corresponding tilt modulus C44 is
given by expression (7) evaluated at wavenumber qz = 1/Lc. Also, n
0 denotes the density of
point pins, while f ′0 denotes the magnitude of their characteristic force. Minimizing δu with
respect to the dimensions of the Larkin domains then yields standard results for these[9]:
Lc = Lc(·)[1 + (qzL∗)−2] and R′c = Rc(·)[1 + (qzL∗)−2]1/2, where Lc(·) = 2c44c66ξ2/n′0f ′20 and
Rc(·) = 21/2c1/244 c
2/n′0f
0 are respectively the longitudinal Larkin scale and the transverse
Larkin scale in the absence of correlated pinning centers. The first equation is quadratic in
terms of the variable L−1c , and it has a formal solution L
c = (2Lc(·))−1+[(2Lc(·))−2−L−2∗ ]1/2.
The hexatic Bose glass is therefore robust to the addition of weak point pins. In particular,
the longitudinal Larkin scale Lc remains divergent for Lc(·) > L∗/2, which is equivalent to
the inequality 23/2Rc(·) > Rc(|).
2D-3D CROSSOVER IN CRITICAL CURRENT
The critical current density of the above hexatic Bose glass, which is robust to the addition
of point pins, can be obtained by applying 2D collective pinning theory[10]. All vortex lines
can be considered to be rigid rods. Balancing the Lorentz force against the collective pinning
force over a Larkin domain[6][9] yields the identity JcB/c = [(npf
p + n
p )/R
1/2 for the
product of the critical current density along the film, Jc, with the perpendicular magnetic
field, B, aligned parallel to the material line defects[10]. Here np and n
p denote the density of
vortex lines pinned by material line defects and the density of interstitial vortex lines pinned
by material point defects, while fp and f
p are the maximum force exerted on the respective
vortex lines per unit length. Again, it is important to observe that the critical current
is limited by plastic creep of the vortex lattice due to slip of the quenched-in lines of edge
dislocations along their respective glide planes[19]. Minimization of the sum of the elastic and
pinning energy densities then yields a higher density of Larkin domains in the hexatic Bose
glass with material point defects added by comparison (1): R−2c = C
0 (npf
p )/(c66b)
This reflects the injection of extra lines of edge dislocations that relieve shear stress caused
by point pins. Substitution in turn yields a critical current density, Jc = jc + j
c, that has a
component due to correlated pins set by the identity
jcB/c = C0npf
p/c66b, (9)
and that has a component due to point pins set by the ratio j′c/jc = n
p /npf
p . The
critical current density notably varies as Jc ∝ B−1/2 with magnetic field in the limit of weak
pinning[10].
Consider now a film geometry of thickness τ along the axis of the material line defects.
The forces due to point pins add up statistically along a rigid interstitial vortex line. The
effective pinning force per unit length experienced by an interstitial vortex line is then given
by[7] f ′p = f
1/2 at film thicknesses τ that are much greater than the average separation
τ ′p between such pins along the field direction. Again, f
0 denotes the maximum force exerted
by a point pin. The relative contribution by point pins to the critical current density is then
predicted to show an inverse dependence on film thickness, j′c/jc = τ0/τ , that is set by the
scale τ0 = (n
p/np)(f
p). We thereby obtain the linear dependence on film thickness
I(2D)c = (τ0 + τ)jc for the net critical current per unit width, τJc. Last, comparison of Eq.
(1) with Eq. (9) yields the useful expression
L∗ = γ
−1[(35/4/27/2C0)(ξ · avx)(j0/jc)]1/2 (10)
for the longitudinal scale characteristic of the Bose glass as a function of the bulk critical
current density, jc. Here avx = n
B is the average distance between vortex lines and
j0 is the depairing current density (see ref. [9]). Figure 2 shows a fit to data for the
critical current versus thickness obtained from a thin film of PLD-YBCO at liquid-nitrogen
temperature in 1T magnetic field aligned parallel to the c axis[12]. A bulk critical current
density jc = 0.22MA/cm
is extracted from it. Using a value of j0 = 36MA/cm
for the
depairing current (ref. [12]), of ξ = 11 nm for the coherence length, of γ = 7 for the mass
anisotropy parameter, and setting C0 = 1 yields a longitudinal scale L∗ = 24 nm.
Finally, consider again an arrangement of material line defects that are aligned along the
c axis, that show no voids or clusters in the transverse directions, but that are broken up
into relatively long rods of length L0 ≫ L∗. It can be realized by meandering material line
defects[1][2], where L0 is the correlation length for alignment along the c-axis, or by artificial
nano-rod defects[3][4]. Expression (7) for the divergent tilt modulus indicates proximity to
the regime of 2D collective pinning, where both Lc and L0 are divergent. It hence indicates
that Lc is also large compared to L∗ here. Second, observe that L∗ ∼ Rc(|)/γ coincides
with the longitudinal Larkin scale if the rods are considered to be point defects. The
previous ultimately implies the chain of inequalities L0 ≥ Lc ≫ L∗. They are consistent
with a broken Bose glass state for the vortex lattice, which is threaded by isolated lines
of edge dislocations of length Lc along the c-axis that are connected together by lines of
screw-dislocations[19] along the transverse directions[11]. As in the limiting case of the true
Bose glass state (fig. 1), the lines of edge dislocations are injected into the pristine vortex
lattice in order to relieve shear stress due to the correlated pins. Larkin domains hence
are finite volumes[6][9]. In contrast to their transverse dimensions, however, Larkin domains
exhibit well defined boundaries along the direction parallel to the correlated pinning centers,
across which the vortex lattice slips by a Burger’s vector due to the presence of the screw
dislocations[19]. The critical current density expected from this peculiar example of 3D
collective pinning therefore coincides with that of a Bose glass of thickness Lc: Jc = jc + j
with a bulk component due to interstitial vortex lines j′c/jc = τ0/Lc. The critical current
per unit width then varies with film thickness as I(3D)c = τJc, showing no offset. Figure 2
depicts the predicted dependence on film thickness for the critical current of such a broken
Bose glass. It notably exhibits 2D-3D cross-over at film thicknesses in the vicinity of Lc [11].
CONCLUDING REMARKS
The dependence of the critical current on thickness τ shown by certain films of PLD-
YBCO is in fact consistent with dimensional cross-over at τ ∼ 1µm. A previous attempt
by Gurevich to account for such behavior by collective pinning of individual vortex lines at
point defects yields a longitudinal Larkin scale Lc ∼ 10 nm that is too small, however[12].
We find here, on the other hand, that pinning due to correlated material defects of length L0
yields a Larkin scale Lc that is much longer than that [L∗ = 24 nm, see Eq. (10)] expected
from point pins of comparable strength if L0 is much longer than that scale as well. Indeed,
we predict here that the film thickness at which the critical current crosses over from 2D
to 3D behavior is of order the effective length of the correlated pinning centers when that
length satisfies L0 ≫ L∗.
The author thanks Leonardo Civale, Chandan Das Gupta and Sang-il Kim for discussions.
This work was supported in part by the US Air Force Office of Scientific Research under
grant no. FA9550-06-1-0479.
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FIG. 1: Shown is a Delaunay triangulation of a low-temperature groundstate made up of 2016
vortices that interact logarithmically over a 336 × 336 grid with periodic boundary conditions,
and that experience an equal number of identically weak δ-function pinning centers arranged in
a “liquid” fashion. (See ref. [18].) The state shows macroscopic phase coherence and long-range
hexatic order, with a supefluid density and a hexatic order parameter, respectively, that are 29%
and 58% of the maximum possible values attained by the perfect triangular vortex lattice. It was
obtained after simulated annealing from the liquid state down to low temperature, which resulted
in 371 pinned vortices. A red and green pair of disclinations forms a dislocation (see ref. [19]).
The above Monte Carlo simulation results are consistent with theoretical predictions of a hexatic
vortex glass state in two dimensions, in the zero-temperature limit, for pinning arrangements that
do not show any clusters or voids (ref. [17]). Josephson coupling between layers then produces a
Bose glass transition above zero temperature as long as the 2D glass transtion is second order (ref.
[20]).
-0.1 0 0.1 0.2 0.3 0.4 0.5
film thickness (microns)
PLD-YBCO in liq. N @ H=1T  (ref. 12)
2D COLLECTIVE PINNING
3D COLLECTIVE PINNING
2D-3D CROSSOVER
FIG. 2: Shown is the dependence on film thickness predicted for the critical current of a defective
vortex lattice found in a broken Bose glass state (solid line). The dashed and dotted lines are
extrapolated from the 2D and 3D behaviors, respectively. Measurements made by Sang Kim
(circles, ref. [12]) of the critical current on a thin film of PLD-YBCO at liquid nitrogen temperature
subject to 1T magnetic field aligned parallel to the c-axis are fit to the straight dashed line
predicted by 2D collective pinning (ref. [10]). This yields an intercept −τ0 = −69 nm and a
slope jc = 0.22MA/cm
. Although the value of Lc shown here is indeed larger than the lower
bound L∗ = 24nm [see Eq. (10)] and is consistent with the fit to 2D collective pinning, it is only
hypothetical.
	Introduction
	Tilt modulus of Bose glass
	2D-3D crossover in critical current
	Concluding remarks
	Acknowledgments
	References
