EFFECT OF VORTEX PINNING BY POINT DEFECTS ON THE LOWER CRITICAL FIELD IN LAYERED SUPERCONDUCTORS
G. P. Mikitik*
Verkin Institute for Low Temperature Physics and Engineering, Ukrainian Academy of Sciences
61103, Kharkov, Ukraine
Received March 2, 2014
The lower critical field Hci in layered superconductors is calculated under the assumption that vortex pinning by point defects is strong in these materials. We consider the case of a purely electromagnetic coupling of vortex pancakes and the case of both the electromagnetic and Josephson couplings of the pancakes in a vortex line. In the latter case, singularities in the temperature dependence of Hci are predicted at certain characteristic temperatures.
DOI: 10.7868/S0044451014090168
1. INTRODUCTION
Effects of thernial fluctuations of vortices on the lower critical field Hci and on the magnetization of type-II superconductors were considered in a number of papers fl 6]. It was shown that the fluctuations lead to a renormalization of the temperature dependence of Hcl. In addition, effects of flux-line pinning on the equilibrium magnetization M of superconductors were analyzed for the cases of pinning by point [7, 8] and columnar [9, 10] defects. In this paper, we consider the effect of pinning by point defects on the lower critical field in layered superconductors, leaving aside the analysis of this effect for three-dimensional superconducting materials.
In layered superconductors like Bi-iSr-iCaCu-iOg+rf. a vortex is the stack of vortex pancakes (VPs) localized in superconductive layers, and the vortex elasticity e/ displays two features that, as we see in what follows, result in a noticeable effect of vortex pinning by point defects on Hci. Both these features are caused by large anisotropy of these superconductors. The first feature is that the elasticity is relatively small, and this small-liess leads to the Larkin length Lc that does not exceed the interlayer spacing d. In other words, the characteristic pinning energy of a vortex pancake is larger than its characteristic elastic energy, and hence pinning of the VPs is strong in these superconductors at least
E-mail: mikitik'flilt.kharkov.ua
for not too high temperatures T fll 13]. The second feature is that in contrast to the practically constant El in three-dimensional superconductors, the elasticity-ill layered superconductors essentially depends on the scale of the vortex distortion, i.e., on the wave vector k- along the vortex fll 14]. This function £i(k~) results from an interplay of the electromagnetic and Josephson couplings of the VPs in a vortex line.
In the experiments in [15, 16], the temperature dependences of the magnetization M were measured at various magnetic inductions B in Bi-iSr-iCaCu-iOg+rf crystals, and a second-order phase transition line Bg(T) was observed in the vortex system of these superconductors at moderate temperatures of the order of 40 Iv. Since a second-order phase transition line cannot have a critical point similar to that of the first-order phase transition line between a liquid and its vapor [IT], the end of the curve Bg(T) in the T B plane should lie on the line B = 0 that corresponds to the lower critical field. Hence, the experimental data in [15, 16] indirectly suggest that the Bi-based superconductors may exhibit a singular behavior of Hci(T) near a temperature close to 40 Iv.
This paper is organized as follows. In Sec. 2, a simple model for the vortex elasticity ei(k-) in layered superconductors is formulated, and in Sec. 3 the main formulas describing strong pinning of the VPs are presented. In Sec. 4, the field Hci is studied in the case where purely electromagnetic coupling of the VPs in a vortex line occurs. In this situation, Hci(T) is renor-malized both by thermal fluctuations of the vortex pan-
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cakos and by their pinning. The lower critical field in the case of both the Josephson and electromagnetic couplings of the VPs is considered in Sec. 5. In this case, the renornialization of Hci(T) is accompanied by singularities in the Independence of Hci at certain temperatures. The results of the paper are briefly summarized in Sec. 6.
2. ELASTICITY OF A VORTEX LINE
In a layered superconductor, pinning forces and thermal fluctuations shift the VPs comprising a vortex line away from its axis, and the line is distorted. Below, we deal with the distortions with large wave vectors k- of the order of ir/d, where d is the interlayer spacing. For such k~, the elasticity £i(k~) of a vortex line in a layered superconductor has the form fll 14]
where e0 = (^o/^ttA)2, A is the planar London penetration depth, is the flux quantum, e -C 1 is the anisotropy parameter of the superconductor, and u is the amplitude of the vortex pancake displacements. It is taken into account in Eq. (1) that in the case of strong pinning, the displacement u can be large, ttk- > 1. The first term in formula (1) describes the Josephson coupling of the VPs, and the second term is due to their electromagnetic interaction. The parameter eXk- characterizes the relative roles of the Josephson and electromagnetic couplings of the VPs in the elasticity of the vortex line.
The logarithmic factors in formula (1) are of the same order of magnitude when A ~ d/e. This situation just occurs in Bi-based superconductors at not too high temperatures (e.g., at A = 0.2/rni, d = 1.5 11111, and e = 1/200, we obtain eX/d « 0.7). Hereafter, we replace the logarithmic factors by the quantity q = 0.5In(k2£2/{u2)), where k = A/£ is the Ginzburg Landau parameter, £ is the planar coherence length, and (u2) gives the averaged value of u2 for the VPs in the case of strong pinning. The explicit value of this quantity q is given below (see formula (23)). To simplify our analysis further, we use the following model dependence for £i(k~) that reproduces the main features of Eq. (1):
£,(k-_) = eoqe2, k'J-"-r > k, > kx. (2)
SqQ
(3)
This model is similar to that used in Refs. [IS, 19] (in those papers, q = 1). Here, klnar = ir/d is the maximum value of k~, and k\ = (eA)_1. Formula (2) describes the Josephson coupling of the VPs, and Eq. (3) corresponds to their electromagnetic coupling.
To characterize the type of the coupling in a vortex, we define the parameter p as
When p < 1, the region of the Josephson coupling is absent for all k~. In this case, the elastic energy of a vortex
rr/d
E(,= J '^ei(k,)k*\u(k,)\2 (5)
"o
can be represented in the form [12]
Ed = (6)
where i/j is the displacement of the VP in the ¿tli layer of the superconductor,
u(k-) d^ u, exp( —ik-Zi) i
is the corresponding Fourier transform, and Ecm = eo'iC2/A2. Formula (C) shows that the VPs in different layers can be regarded as independent "particles" in an effective mean-field harmonic potential generated by all other VPs of the vortex line [12].
When p > 1, the elastic energy Ec[ consists of two parts, Eci = + Eft. The Josephson coupling of the VPs comprising the vortex line occurs for the vibration modes of the vortex with k- in the interval k'rax > k- > kx- and the elastic energy of these modes
k™a'
kx
On tlie otlier hand, tlie vibrating modos witli k- < ka load to an uncorrclated niotion of vortex segments of tlie length L\ = (k'rax/k\) d = ?reA, and tlie olastic energy of tiloso longwave modos is givon by an expression similar to Eq. (C),
^E^ttÍ- (8)
i
wliero üj is tlie displacoment of tlio jth segmont as a whole.
In Socs. 3 and 4, tlie caso of puroly eloctromagnetic coupling of tlio VPs (p < 1) is considerod, wheroas tlie caso p > 1 is discussod in Sec. 5.
3. STRONG PINNING OF THE VORTEX PANCAKES
Strong pinning of the VPs was analyzed in Refs. fll 13]. Here, using somewhat different approach, we derive the appropriate formulas again and present them in the form that permits us to use the obtained equations at realistic values of the vortex elasticity and pinning.
We consider an individual VP in a pinning potential generated by point defects. The distribution w(E) of its potential energies is Gaussian1^ fll]:
w(E) =
1
■ exp
ËL
u2
(9)
where the parameter Up is of the order of Up = = C(/p%C2ii)1/2, the characteristic pinning energy of the VPs; fp is the mean pinning force caused by a point pinning center; and np is the density of these centers. For low B and T, we have U°,UP Ecm for Bi-based superconductors fll 13]. As in Refs. fll 13], we assume that for the unit area of a superconducting layer containing the vortex pancake, the number of the pinning-potential extrema lying below an energy E is given by
n(E) =
1
7tÇ2
e
dE'w(E') =
1 + erf (E/U, 2tt£2
(10)
where 1/ttC2 is the density of these extrema, i.e., of pinning wells and humps, and orf(x) is the probability-integral [20],
orf(x)
dt exp(—t2
(ID
We now consider a VP in the vortex line. Its total energy is the sum of its energy in the pinning potential and of its elastic energy. The pinning potential "stimulates" the pancake to seek the deepest minimum of this potential in the appropriate layer. On the other hand, the displacement u of the vortex pancake from the vortex-line axis leads to an increase in its elastic energy Eci(u) = Ecmqu2 / £2. At T = 0, in each layer, the appropriate vortex pancake occupies the energy minimum with the lowest total energy, i.e., the absolute
11 A uniform distribution of point defects leads to a renormal-ization of A and hence of Hc i. This renormalization of Hc i is proportional to the mean density of the defects, np, and is not considered here. The pinning potential is generated by spatial fluctuations of the density around np, and hence the mean energy for distribution (9) is zero.
energy minimum in the layer. To proceed with the analysis of this absolute minimum, we first estimate the distribution of the local energy minima in the layer in the case of strong collective pinning of the VPs by point defects. This strong pinning occurs when the characteristic scale of the pinning potential, Up, is essent
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