Vortex lattice in type II superconductors under magnetic field
in the presence of inhomogeneities
B. Rosenstein
a,*, B.Ya. Shapiro
b, V. Zhuravlev
aaNational Center for Theoretical Studies and Electrophysics Department, National Chiao Tung University, Hsinchu 30050, Taiwan, ROC bDepartment of Physics, Bar Ilan University, Ramat Gan 52100, Israel
Accepted 30 November 2007 Available online 13 March 2008
Abstract
We investigate the structure, elastic and dynamical properties of the vortex matter in the presence of artificially created or intrinsic gradients of the critical temperature in the framework of the Ginzburg–Landau theory. The region of parameters in which vortex cores are not well separated is treated perturbatively in 1 Hc2ðT Þ=Hc2ð0Þ. Critical current for periodic pinning potential is obtained and
gen-eral expressions for elastic moduli at long wavelength are derived. We show that it is impossible to restrict the system to lowest Landau level. We use it to provide a theory of the discontinuous peak effect in critical current which appears near Hc2ðT Þ line in low Tcstrongly
type II superconductors. Influence of thermal fluctuations is also considered and we find softening of the shear modulus in the vicinity of vortex lattice melting line.
Ó 2008 Elsevier B.V. All rights reserved.
PACS: 74.60.w; 74.40.+k; 74.25.Ha; 74.25.Dw Keywords: Vortex matter; Shear modulus; Peak effect
In type II, superconductors for which the penetration depth k exceeds the correlation length n the magnetic field penetrates the sample in a form of Abrikosov vortices, which strongly interact thereby creating an elastic ‘‘vortex matter”. Impurities, always present in a sample, lead to inhomogeneities, which greatly affect the thermodynamic and especially dynamic properties of the vortex matter. Recently various experimental techniques were developed, which allow to artificially create ‘‘pinning” on the scale of up to tens of nm in an controllable way. When the inho-mogeneity is strong enough, it pins the vortex matter, resulting in dissipationless persistent current, thereby recovering an original defining property of superconductor. The pinning can be overcome if the current exceeds the crit-ical current Jcor if thermal fluctuations reduce the effect of
the inhomogeneities.
Two major theoretical simplifications are generally made. In majority of the works, the vortex matter is consid-ered as an array of elastic lines[1]. This (London) approx-imation is generally valid far from the higher critical field Hc2ðT Þ, when the vortex density is low. Critical current is
interpreted as a current at which the Lorentz force on the vortex line system overpowers the pinning force. An alter-native simplification to the vortex matter is valid far enough from the lower critical field Hc1ðT Þ. At high vortex
densities magnetic fields of many vortices overlap and the resulting magnetic inductance is nearly homogeneous and Ginzburg–Landau (GL) model at constant magnetic field can be used. It is usually supplemented by the so called lowest Landau level (LLL) approximation. Here one does not see a well separated vortices, but rather a distribution of the order parameter fields with zeroes of the order parameter marking the centers of ‘‘cores”. In most of the cases, however, the dynamics of the vortex matter is described in two steps. First, the system is treated
0921-4534/$ - see front matterÓ 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.11.091
*
Corresponding author.
E-mail address:[email protected](B. Rosenstein).
www.elsevier.com/locate/physc Physica C 468 (2008) 621–626
collectively in the ‘‘elastic medium” approximation, namely first elastic properties under external ‘‘forces” are deter-mined and then random or permanent gradients of the external parameters act of this elastic medium. Elastic properties of the vortex matter therefore become essential in understanding thermodynamic and transport properties
of the disordered vortex matter[2]. In addition, the shear
modulus was measured recently in BSCCO superconductor
using the AC response technique[3]. It demonstrates sharp
decrease at on the melting line of the vortex lattice. In par-ticular, detailed knowledge of the elasticity of the vortex lattice is required to understand the ‘‘peak effect” in critical current[4–6].
In this paper, we consider the vortex lattice not far from Hc2ðT Þ in the presence of small gradients of the critical
tem-perature TcðrÞ using the GL approach. First, the
inhomo-geneity is considered as a perturbation. The changes in distribution of both the superfluid density and the super-current are obtained. The pinned state is described as a sta-tic state which carries a net current. Crista-tical current for periodic pinning potential is obtained and general expres-sions for elastic moduli at long wavelength are derived. The value for the shear modulus near Hc2ðT Þ is larger than
calculated before[7]restricting the strained system to LLL. We show that it is impossible to restrict the pinned system to LLL: contribution of the first Landau level (LL) is cru-cial for both pinning and elastic deformations. We then argue that the discontinuous peak effect in critical current which appears near Hc2ðT Þ line in low Tc strongly type II
superconductors can be understood using the elastic mod-uli. Influence of thermal fluctuations is also considered and we find softening of the shear modulus in the vicinity of vortex lattice melting line. The softening of the shear
mod-ulus was directly measured recently[3]. We focus on a
par-ticular case of strongly type II superconductors for which
the ratio j¼ k=n is very large (for high Tc cuprates and
most of the widely used and studied low Tc type II
super-conductors j is ranging between 10 and 100). Qualitatively for large j the compression and the tilt moduli are practi-cally the same as those of magnetic field in vacuum (except
the dispersion[7]) and the most important modulus id the
shear which is much smaller (by a factor of 1=j2).
Our starting point is the GL Gibbs energy of a
super-conductor in homogeneous magnetic field H¼ ð0; 0; H Þ:
G¼ Z dr h 2 2mjDaWj 2 þ h 2 2m z DzW j j2þ a0jWj2 þb 0 2jWj 4 þðB HÞ 2 8p # ; ð1Þ where a¼ x; y; m and m
z ¼ c2m are effective masses in
directions perpendicular and parallel to that of magnetic field and c is the anisotropy parameter. We assume for sim-plicity a0¼ aðT
c T Þ. It will be convenient to use
n2¼ h2 2maT
c as unit of length in the x–y plane, and nz¼ cn
in the field direction, gGL¼
H2 c2
8pj2 as unit energy density
and rescale the order parameter field W2¼2aTc
b0 w 2
. The
upper critical field Hc2¼2pnU02, is a unit of magnetic field,
so that h¼ H =Hc2 describes the external field and
b¼ B=Hc2. In these units, the dimensionless GL energy
takes a form: g¼ G 4n2ncgGL ¼ Z r 1 2jDiwj 2 1 t 2 w rð Þ j j2 þ1 2jw rð Þj 4 þj 2 4 ðb hÞ 2 ; ð2Þ
where i¼ x; y; z and Di¼oxoi iAi are covariant
derivatives.
‘‘Force” acting on the vortex matter can be physically realized in a variety of ways[8]. Let us consider a supercon-ductor with spatially inhomogeneous critical temperature
TcðrÞ. Within the framework of the GL theory it is
described by a potential:
a0¼ a T T½ cð Þr ¼ a T Tf c½1þ U rð Þg: ð3Þ
This model of generally used to describe the dTcpinning
[1]. This additional term naturally results in an elastic
deformation of the vortex lattice. In the absence of the potential term, the solution is the Abrikosov lattice
solu-tion minimizing the funcsolu-tional Eq. (2) is given by a well
defined expansion in two small parameters j2 and a
h [9]: wmfð Þ ’r ffiffiffiffiffiffia h bA r /0þ ah/cþ O a 2 h þ O j 2 bmfð Þ ’ h þ jr 2bcþ O j4 ; ð4Þ where bc¼ hhbah Aj/0j 2 þ Oða3
hÞ and /0 is the Abrikosov
wave function for hexagonal lattice: /0¼ 3 1=8h1=2 X 1 l¼1 exp ihxyþ:il pl 2 þ 3 1=4 p1=2h1=2x 1 2ðh 1=2y 31=4 p1=2lÞ2 ; ð5Þ
normalized to unit superfluid density. The correction /c
contains higher LLs. The currents pattern is simple:
vorti-ces around positions of zeroes at rn¼ ðaðn1þ n2=2Þ;
2p=an2Þ with n1; n2 integers.
To first-order in U the correction to the wave function
hðrÞ can be expanded in LLs basis /N kðrÞ, where k is
qua-simomentum and N LL: h rð Þ ¼ b1a1=2h b1=2A X N k UN k/N kð Þ;r ð6Þ where UN k¼ R
r/N kUðrÞ/0and higher orders in ahwere
ne-glected. The most important contribution to the current density JðrÞ ¼i
2ðw
Dw wDwÞ comes from the first LL,
since the covariant derivatives in JðrÞ contains one ‘‘ras-ing” operator,iDx/0¼ Dy/0¼ ð2bÞ
1=2
dJx¼ ah bAb 2b ð Þ1=2X k /1kð ÞUr 1k/0ð Þ þ ccr dJy¼ iah bAb 2b ð Þ1=2X k /1kð ÞUr 1k/0ð Þr cc : ð7Þ
As an example, let us consider potential which is periodic
(matching field H). One observes (see Fig. 1) that when
minima of the potential do not coincide with the zeroes of the order parameter, the correction to current integrated
over a certain volume (the unit cell for example), dI¼
R
rdJðrÞ, is nonzero and can be defined as the Lorentz force,
FL¼ c1½dI b: FLyþ iFLx¼ ahb1A ðb=2Þ 1=2
U10. The
Lorentz force is balanced by ‘‘electric” pinning force, Fpin¼
R
rj wj 2
$U as can be explicitly seen. The current
per vortex calculated that way cannot exceed a certain value, thus determining the critical current. The above con-sideration demonstrates that an equilibrium state in the presence of the pinning force inevitably has mesoscopic
supercurrents and in view of Eq. (7) cannot be treated in
framework of LLL only.
One can determine perturbatively in U the new positions of zeroes of the order parameters (‘‘centers” of the vorti-ces). Before the perturbation was applied they were located at rn¼ ðaðn1þ n2=2Þ; 2p=an2Þ. New positions of the zeroes
are found by demanding wðrnþ unÞ ¼ 0. Since
displace-ment un is first-order in U, expanding this relation to the
first-order gives:
a1=2h b1=2A oa/0ð Þurn an¼ h rð Þ;n ð8Þ
determining all the displacements. If one considers a con-figuration possessing the hexagonal symmetry generally the elastic energy can be written as:
gel¼1 2 X k c11 kxu0xþ kyu0y 2 h þc66 kxu0y kyu0x 2 þc44k2z u 2 0xþ u 2 0y i : ð9Þ
The basic relation between stress and strain for potential oaFb¼ vba¼ cabcdocud with Fb ¼ obU one obtains the
following expression for the shear modulus:
c66¼
hah
2bA
; ð10Þ
in units of H2c2n2nc=ð2pÞ.
In solid state, physics one can conveniently consider elas-tic stress as a force acting on pointlike atoms. However, from the earliest times electromagnetic fields can also be considered as a kind of ‘‘elastic medium”. In particular, a constant uniform magnetic field H in vacuum has well
defined compression and tilt moduli C11¼ C44¼H
2 4p, while
the shear modulus vanishes[1]. The elasticity is associated with field deformation within the volume containing the field. In electromagnetically, active media generally addi-tional fields describing matter like magnetization in mag-nets, polarization vector in ferroelectrics, etc., contribute to elastic properties. Generally for any system of fields one can obtain an expression for elastic moduli in a same way one derives an expression for angular momentum (rotation modulus). Qualitative picture behind theoretical approach to elasticity was that magnetic field penetrates the material as a system of Abrikosov vortices (fluxons). Topological argument within the simplest GL model of a one component superconductor implies that if n elementary
flux units U0penetrate the sample, then the order parameter
whas exactly n zeroes. This determines unambiguously
cen-ters of vortices surrounded by normal cores of size of coher-ence length. When cores of the vortices are well separated, one can reduce the problem to the elasticity of a collection of linelike (without internal structure) objects. This descrip-tion becomes problematic near the upper critical field Hc2ðT Þ, when vortices are poorly separated and their
inter-nal structure is of importance. Moreover, there are impor-tant cases in physics when fluxons do not possess a core at all, see for example p-wave superconductors (similar to
those in superfluid He3). This does not imply that these
materials do not have a well defined elastic properties. In these, cases one should not rely on location of zeroes, but rather return to the original field theoretical description.
A more powerful general approach to elasticity is geo-metrical in nature[12]. Elastic moduli describe the rigidity with respect to local translations. For our purposes it is suf-ficient to consider displacements in the plane perpendicular to external magnetic field uðrÞ ¼ ðua;0Þ. The corresponding
transformations of a scalar and a vector fields are: w0ðrÞ ¼ wðr þ uÞ wðrÞ þ uawa;
A0iðrÞ AiðrÞ þ ubAa;bþ ubaAb;
ð11Þ where a short notation for derivatives, e.g., ubj ¼ou
b orj is used and i¼ 1; 2; 3. Considering the displacement, ua;b¼ ua
0k b
,
and expanding in powers k, one observes that to order k0
the contributions cancel. This is just the Goldstone theo-rem, which asserts that when a continuous symmetry (glo-bal translations in the present case) is spontaneously broken, there appears a ‘‘soft” mode. Terms linear in k vanish due to reflection symmetry of the Abrikosov lattice configuration, while the terms quadratic in k determine the elastic moduli.
Fig. 1. Current distribution of the pinned vortex lattice. Pins are shown as blobs. Left: unperturbed current distribution. Center: the perturbed distribution by a localized periodic potential. Right: The current distri-bution with the unperturbed subtracted. Since the unperturbed configu-ration does not carry net current, the right picture demonstrate the persistent current.
Calculating the second functional derivatives and
substi-tuting the mean field solution of Eq.(4), one obtains
dg¼ 1
2vol Z
k;r
u0au0bkikj j2 dijAl;aAl;bþ Aa;jAi;b
þdij Dbw D awþ DbwDaw i : ð12Þ
Now we proceed to calculate the moduli in the Abrikosov lattice configuration neglecting thermal fluctuations. As mentioned in Introduction, the case of large j is of special interest. It is natural therefore to expand a physical quanti-ties in powers of j1. Fortunately the expression for the most
important shear modulus is a regular function of both j2
and the wave vector k, (the other two moduli are not[7]).
The contribution to elastic moduli to the leading order in
j2comes solely from the magnetic energy term.
Substitut-ing Aa¼12heabrb (symmetric gauge), one obtains that the
contributions to the compression and the tilt moduli are equal,
c011¼ c0 44¼
h2j2
4 ; ð13Þ
and consistent with the j2term in expansion of Eq.(12)at
k¼ 0. The contribution to the next to leading order in j2,
c1
ijab comes both from magnetic and the order parameter
terms. The magnetic term is of order a2
h: c1A ijab¼ 1 vol Z r dijA0l;aA c l;bþ A c a;jA 0 i;bþ a $ bð Þ ¼ hah 2bA 1þ 2ahd0 ð Þdaidbj:
The order parameter term (in Eq.(12)) contribution is
pro-portional to c1wijab¼ dijsab sab¼ 1 vol Z r DbwmfDawmfþ cc ¼ dabs; ð14Þ where s¼hah
2bAð1 þ 2ahd0Þ. The fact that symmetric tensor
sabis proportional to dabfollows from hexagonal symmetry
of the mean field solution. One therefore has the following contributions to compression, tilt and shear moduli c1 11¼ d0 a2 hh bA ; c 1 44¼ c 1 66¼ hah 2bAð1þ 2ahd0Þ: ð15Þ
Note that the order ah contributions to the compression
modulus from the magnetic term and the order parameter term cancel. Also the correction to the tilt modulus leads to
the known result c0
44¼hbj 2 4 .
One observes that, while the compression and the tilt moduli are the same as in thermodynamically calculated [7], the value of the shear modulus is different. The
thermo-dynamic and the LLL calculation gives near Hc2ðT Þ a value
of c66’b0:242 Aj2
a2
h, smaller than in Eq. (15). We discuss this
below.
The mean field result of the previous subsection is
mod-ified in strongly fluctuating superconductors like high Tc
materials by thermal fluctuations. Generally it results is softening of elastic moduli. In the extreme case, when the
system approaches the overheated crystal spinodal line the shear modulus vanishes making the crystalline state unstable. Below this temperature, however, the melting transition into the vortex liquid state takes place. Thermal fluctuations on the mesoscopic scale are accounted for by averaging over all the configurations of fields with corre-sponding Boltzmann factor. Generally for strongly type II materials fluctuations of the magnetic field are negligibly small. Since the compression and the tilt moduli remain finite even in the liquid state, the only important modulus is the shear. Therefore we concentrate on calculation of corrections to the mean field for the quantity
sab¼ 1 vol Z r DawDbw þ D aw D bw th ¼Z 1 vol Z w;w Z r DawDbw þ D aw D bw n o eGf gTw; ð16Þ where Z¼Rw;w exp½ gfwg p2pffiffiffiffiffi2Git.
Assuming hexagonal symmetry, this symmetric tensor simplifies sab¼ dabs, so that c66¼ vol1
R
rhw D2wi
th. When
thermal fluctuations are not very large, one can apply the low temperature perturbation theory around the field solu-tion w¼ wmfþ v. In the leading order, in ahone can neglect
thermal fluctuations of the higher Landau harmonics, restricting the field to LLL: D2
wLLL¼ hwLLL. In this case,
the modulus and the Gibbs energy simplify significantly: c66¼h2vol1 Rrhj wj2ith, gLLL ¼ R r½ahjwj 2 þ1 2jwj 4 . Therefore within this approximation the shear modulus is propor-tional to superfluid density. In the low temperature, expan-sion it was calculated in[9]
c66¼ hah 2bA 0:55 ffiffiffiffiffi Gi p th2 a1=2h : ð17Þ
The softening intensifies upon approaching the mean
field transition line ah¼ 0. However, well below this line
the perturbation theory breaks down.
The shear modulus softening just below Hc2ðT Þ plays a
crucial role in explaining the ‘‘peak effect” in the critical
current [13]. The peak generally appears just before the
‘‘melting” of the Abrikosov lattice due to thermal fluctua-tions. Within the collective pinning theory[1], the critical current is estimated from the balance of the pining force
on Larkin domain and the Lorentz force JcB. Size of the
Larkin domains can be estimated via relevant elastic mod-uli leading to
Jc¼
A bc44c266
: ð18Þ
The constant A is dependent on ah and gets smaller near
Hc2ðT Þ, although the exact dependence is not known.
How-ever, since c66of the thermodynamical argument is
propor-tional to a2
h; it was argued that one obtains a gradual
increase in Jc approaching Hc2ðT Þ since ‘‘softening” of
the vortex lattice overcomes decrease of the pinning force. This corresponds to an ‘‘old” view on the ‘‘peak effect”, when this increase was thought to be followed by an abrupt
jumps of the critical current to zero at the melting point (in practice might be smeared out by sample inhomogeneities). The recent view, supported by experiments in which Corbi-no geometry or width dependence were used to minimize or
subtract the edge effects [4,5], attributes the peak to the
amorphous homogeneous state. Critical current actually monotonically decreases with field and then jumps from a relatively low value in the crystalline state to a very high value in the vortex glass (this was noticed early on in [10]). Qualitatively this is due to the fact that it is easier to pin a disordered homogeneous state than a rigid crystal-line one. The continuous rise of the critical current ob-served in numerous earlier experiments was caused by poor resolution due to overheating of the solid and overco-oling of the homogeneous states. The critical current in the amorphous phase rapidly drops aspffiffiffiffiffiffiffiffiffiffiffiffiffiffiT Tg
,when temper-ature approaches the glass tempertemper-ature[14]Tg. Thus tradi-tional picture predicts a gradual increase with subsequent drop of the critical current, while modern picture predicts a sudden increase followed by a fast but continuous de-crease. If one uses the larger value of the shear modulus
ob-tained here, Eq. (10), one indeed obtains a monotonic
decrease to a constant value since both pinning force and softening drop with similar rate in Eq.(18).
Using the GL theory under the assumption that the sys-tem under a stress remains constrained to LLL, Brandt
derived[7]expressions for the softest modulus, the shear,
of the vortex lattice. At large j and near the mean field line C66 is proportional to a2h. The modulus is consistent with
thermodynamic derivation in which lattice energies of dif-ferent symmetries were compared. The expression for shear modulus and other moduli are used in numerous theoreti-cal descriptions of phenomena as different as vortex lattice melting[11]and critical current[1,7]of the pinned lattice. As we demonstrate below the LLL assumption does not follow directly from the ‘‘thermodynamic” argument. The stress necessarily transforms the LLL equilibrium state into a state containing significant contribution of higher LLs near the Hc2ðT Þ line. Qualitatively it reflects the fact that
shear for example changes the shape of the order parameter spatial distribution into sheared one (ellipsoidal one rather than round). The location of zeroes is exactly the same, but internal structure of the vortex becomes important.
The sheared state is not a ground state of the Abrikosov lattice with the same symmetry. The later is explicitly
con-structed in the symmetric gauge by Brandt[7]. Restricting
the shear transformations to LLL, he effectively retained the notion of a single state for a given lattice symmetry. Physically it is equivalent to an assumption that the degrees of freedom related to shape of the vortices can ‘‘relax” to their positions with minimal energy. It would mean that the system returns to an LLL state upon this relaxation. There is a popular belief that all degrees of freedom can be divided in two sets: ‘‘slow” and ‘‘fast”. ‘‘Slow” variables are the locations of vortices determined, for example, by
the vortex center positions (where w¼ 0), and ‘‘fast”
vari-ables which contain all the other degrees of freedom related
to the shape of the vortices. While near Hc1 one can argue
that the internal degrees of freedom are very costly energet-ically, near Hc2this is not correct. To our knowledge, there
are no works which establish in what field range the sepa-ration between two set of degrees of freedom becomes pos-sible. The GL energy does not contain an evident small parameter or ‘‘energy gap” which allows such a separation. Correspondingly, in dynamics based on the time-dependent GL equation there is no separation of time relaxation of different degrees of freedom. Mathematically the shear
transformation takes the ground state /0 out of the LLL
sector since it does not commute with the ‘‘Hamiltonian” 1
2D 2þh
2. Although in our calculation magnetic induction
deviates slightly from the external field h, the reason for a significant increase of the shear modulus compared to ear-lier estimates is not related to this.
To summarize, we considered elastic response of the vor-tex lattice to inhomogeneity near the second critical field
Hc2ðT Þ using the GL approach and showed that in the
pin-ned state the system is necessarily excited to states outside of the LLL. This reflects the deformation of the current distri-bution profile under stress. As a result the shear modulus is
much larger (of order 1 T =Tc H =Hc2) than that found
by considering minimal energies of configurations with symmetries corresponding to sheared lattice, leading to ð1 T =Tc H =Hc2Þ
2
The obtained shear modulus leads to a monotonic decrease of the bulk contribution to the crit-ical current in the crystalline phase before it discontinuously jumps to a much higher value in vortex glass. Such a behav-ior was obtained experimentally recently when the edge contributions were minimized.
Acknowledgement
It is a pleasure to thank V. Vinokur, E. Zeldov, E. An-drei, E. Sonin, P. Kes, B. Shapiro, D.P. Li, N. Kokubo, T. Maniv for discussions and sharing unpublished data with us. We are especially grateful to E.H. Brandt for numerous illuminating discussions during his two visits. The work is supported by NSC of R.O.C. No. 952112M009048 and the MOE ATU Program.
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