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H–T phase diagram of the vortex lattice in superconductors with pinning
Physica C 404 (2004) 61–68 www.elsevier.com/locate/physc
H –T phase diagram of the vortex lattice in superconductors with pinning E.H. Brandt b
a,*
, G.P. Mikitik
b
a Max-Planck-Institut f€ur Metallforschung, Heisenbergstr. 3, D-70506 Stuttgart, Germany B. Verkin Institute for Low Temperature Physics and Engineering, Ukrainian Academy of Sciences, Kharkov 61103, Ukraine
Abstract The vortex-lattice melting line in three-dimensional type II superconductors with pinning is derived by equating the free energies of the vortex system in the solid and liquid phases. We account for the elastic and pinning energies and the entropy change that originates from the disappearance of the phonon shear modes in the liquid. The pinning is assumed to be caused by point defects and to be not too strong so that the melting line lies inside the so-called bundle-pinning region. The derived equation for the melting line may be expressed in the form of some Lindemann criterion which now includes a characteristic vortex displacement caused by the pinning. Estimating the effect of pinning on the entropy jump at melting, we find the upper critical point of the melting line from the condition that this jump vanishes. Some recent experimental results on the T –H phase diagrams of type II superconductors are discussed. Ó 2004 Elsevier B.V. All rights reserved. PACS: 74.25.Qt; 74.72.Bk Keywords: Vortex-lattice melting; Pinning; Phase diagram; Order–disorder transition
1. Introduction In three-dimensional high-Tc superconductors with pinning, two phase transition lines are known to exist in the magnetic field H –temperature T plane [1–5]: The line Hm ðT Þ where a quasiordered Bragg glass [6,7] thermally melts into a flux-line liquid, and the order–disorder transition line Hdis ðT Þ separating the Bragg glass from an amorphous vortex state. The melting is caused by thermal vibrations of the lattice, while the order–
*
Corresponding author. Tel.: +49-711-6893575; fax: +49711-6891932. E-mail address: [email protected] (E.H. Brandt).
disorder transition is induced by quenched disorder in the vortex system. These two lines merge at some point in the H –T plane. Although both transitions are accompanied by a proliferation of dislocations in the vortex lattice, it was argued [8] that the dislocation density q is essentially different in these cases: q a2 for melting, and q R2 a for the order–disorder transition. Here a is the spacing between flux lines, and Ra is the so-called positional correlation length [9] within which the relative vortex displacements caused by the quenched disorder are of the order of a. In fact, an intersection of these two different phase transition lines occurs in this scenario; the order–disorder line terminates at the intersection point while the melting line continues for some distance to higher
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2003.10.037
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E.H. Brandt, G.P. Mikitik / Physica C 404 (2004) 61–68
H and terminates only at its upper critical point. Within this physical picture, the existence of the so-called slush phase [10] can be naturally explained. Recent experiments [11–15] for YBaCuO seem to support this scenario. In our paper [4] it was shown that the H –T phase diagrams of many low-Tc and high-Tc superconductors with pinning caused by point defects can be described in a universal manner and are determined by two parameters: the Ginzburg number Gi which characterizes the magnitude of the thermal fluctuations, and the ratio D 1=2 ðjc =j0 Þ which describes the strength of the pinning. Here jc is the critical current density in the single vortex pinning regime [9], and j0 is the depairing current density, both taken at T ¼ 0. Phase diagrams with various Gi and D were analyzed in Ref. [4], based on the above-mentioned scenario. In this analysis the melting line was calculated neglecting the effect of the pinning on it. This approximation was justified by the experimental finding [16–19] that an increase of the pinning leads only to a noticeable shift of the upper critical point on Hm ðT Þ but practically does not change the melting line itself (the increase only slightly pushes Hm ðT Þ downwards in the T –H plane). Of course, this approximation did not enable us to calculate the upper critical point of Hm ðT Þ and its shifts with changing D and Gi. The effect of point-defect pinning on the melting line was considered in Refs. [2,5,20,21]. In our recent paper [21] we found the melting line and its upper critical point by considering the balance of three energies: the elastic energy caused by the proliferation of dislocations at melting, the entropy gain in the free energy associated with the disappearance of the shear-phonon modes in the liquid, and the pinning energy. In this analysis we allowed for a mutual ‘‘influence’’ of the pinning and the thermal fluctuations: the entropy gain was estimated, taking into account the effect of pinning on it (just this permits us to find the upper critical point), while the pinning energy was calculated accounting for the smoothing effect of thermal fluctuations on the pinning potential, i.e., the socalled thermal depinning [9,22]. In estimating this thermal depinning we used the recipe of Ref. [9] obtained for the flux-line lattice (the Bragg glass).
However, the effect of thermal fluctuations on the pinning energy is expected to be different for the lattice and for the liquid, and in the present paper we generalize the results of our paper [21] by taking into account this effect in the liquid. Below we consider only magnetic fields H exceeding considerably the lower critical field Hc1 , and thus we put B ¼ l0 H . Besides this, we deal only with uniaxial anisotropic three-dimensional superconductors with anisotropy ¼ kab =kc < 1, neglecting completely the decoupling of the superconducting layers. Here kab and kc are the London penetration depths in the plane ab perpendicular to the anisotropy axis and along this axis, respectively. The magnetic field is assumed to be directed along the anisotropy axis. The quenched disorder in the flux-line lattice is caused by point defects and is not too strong such that the melting line lies entirely in the so-called bundle pinning region, where the transverse collective pinning length Rc [9] is greater than a.
2. Melting of the vortex lattice with quenched disorder At melting, proliferation of dislocations occurs in the vortex lattice. These dislocations create a network in the lattice composed of edge and screw dislocations, with the mean dimensions of a unit cell of the network being of the order of a. (One has l? a for the dimension l? transverse to H , and lk a for the longitudinal dimension lk , see details in Refs. [8,21].) In the vortex system without pinning the cost in the elastic energy due to the proliferation of the dislocations is balanced by the entropy gain in the free energy of the flux-line liquid. We shall estimate this gain, assuming that the main contribution to it results from the disappearance of the shear-phonon modes in the liquid (i.e., from their transformation into some other degrees of freedom). The mechanism of this disappearance is as follows: A shear stress exerts so-called Peach–K€ ohler forces [23,24] on the dislocations, which then move and thus relax the shear stress. The above-mentioned energy balance enables one to derive the equation for the melting line of
E.H. Brandt, G.P. Mikitik / Physica C 404 (2004) 61–68
the ideal vortex lattice [9,21]. The elastic energy per unit cell of the dislocation network, Eel , associated with the proliferation of the dislocations is 1=2 proportional to ðc44 c66 Þ a3 where c44 and c66 are the tilt and shear moduli [25,26] of the flux-line lattice (the nonlocal tilt modulus should be taken at wave vectors k of the order of 1=a), e0 2 c66 2 ð1 hÞ ; 4a ð1Þ 2 e0 c44 2 ð1 hÞ: a Here h ¼ H =Hc2 ðtÞ, t ¼ T =Tc , e0 ¼ ðU0 =4pkab Þ2 , U0 is the flux quantum, and Hc2 ðtÞ is the upper critical field in the mean-field theory. The factors containing ð1 hÞ take into account the softening of the vortex lattice near the Hc2 ðtÞ line [26]. As to the entropy gain, in the unit volume of the lattice there 2 are l1 k l? shear modes with wave-lengths greater than lk , l? , and each mode contributes about T to this gain. Thus, the gain per cell of the dislocation 2 2 network is T l1 k l? lk l? T . As a result, the energy balance applied to a mean unit cell of the dislocation network in the vortex system without pinning yields: Cðc44 c66 Þ1=2 a3 T ¼ 0;
ð2Þ
where the constant C absorbs all unknown numerical factors in Eel and in the entropy gain. Inserting formulas pffiffiffi (1) into this equation and expressing C as 4 pc2L where cL is the Lindemann constant, one finds an equation for the melting line which coincides with that derived from the Lindemann criterion, u2T ¼ c2L a2 ;
ð3Þ
or from the analysis based on the Ginzburg–Landau (or London) Hamiltonian [21]. Here uT is the magnitude of the thermal displacements of the vortex lattice with respect to its equilibrium position. When pinning exists in the vortex system, the adjustment of the vortices to the pinning potential decreases the total energy of the system, and the amount of this decrease is the so-called pinning energy [9]. In the liquid, vortices can adjust themselves to the pinning potential not only via their elastic deformations as in the lattice, but also
63
via the plastic vortex-lattice displacements generated by dislocations. This additional adjustment mechanism increases the pinning energy in the liquid, and thus at melting there is a gain in pinning energy, Epin , which should be included in the energy balance. Besides this, pinning influences the entropy gain. As a result, Eq. (2) is modified by pinning as follows: Cðc44 c66 Þ
1=2 3
a T DS Epin ¼ 0;
ð4Þ
where the factor DS takes into account the effect of pinning on the entropy gain. In estimating Epin , it is necessary in general to take into account the thermal depinning [9,22] since the magnitude of the thermal displacements of the lattice, uT , is sufficiently large at the melting, uT cL a. However, the results of Ref. [21] lead to the following conclusion: If one neglects the thermal depinning, derives the equation for the melting line from the energy balance, and then rewrites this equation in the form of the Lindemann criterion, the thermal depinning can be accounted for by the simple replacement of the coherence length n by 1=2 the effective radius rp ¼ ðn2 þ u2T Þ of the pins. Using this observation, we first carry out the estimates neglecting depinning and then account for it by the above-mentioned replacement. To estimate the pinning energy per dislocation cell, Epin , it is necessary to take into account that the displacements generated by the dislocation network are essentially larger than the displacements uðl? ; 0Þ uð0; lk Þ existing in the Bragg glass at the same temperature and magnetic field within the scales of the cell, l? a; 1=2 c44 a : lk l? 1=2 c66 ð1 hÞ
ð5Þ
[The factor ð1 hÞ1=2 originates from the softening of the vortex lattice near Hc2 ðtÞ.] The displacement uðl? ; 0Þ (or uð0; lk Þ) is the rms relative shift of two line elements in the vortex lattice separated by a distance l? transverse to the magnetic field (or by a distance lk along the field). This shift is caused by the random point defects and can be expressed in terms of the transverse collective pinning length Rc at which the relative
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E.H. Brandt, G.P. Mikitik / Physica C 404 (2004) 61–68
displacements of points in the lattice are of the order of n. In particular, if the small bundle pinning regime occurs, i.e., a < Rc < kab , one has [9] u2 ða; 0Þ
n2 : 1 þ lnðR2c =a2 Þ
ð6Þ
Thus, in the lattice without dislocations, one has uða; 0Þ < n, while in the liquid the dislocations lead to displacements of about a > n at the boundary of the dislocation cell. These inequalities justify the above statement that in the liquid the plastic deformations associated with the dislocations dominate over the elastic deformations caused by the point defects. Using the results of the collective pinning theory [9], we then obtain the estimate for 0 the pinning energy Epin calculated without allowing for the thermal fluctuations, 0 Epin ðWlk l2? Þ
1=2
ð7Þ
n;
2 nn2 =a2 , fpin is the mean elementary where W ¼ fpin pinning force exerted by one point defect, n is the concentration of the defects, and the expression 1=2 ðWlk l2? Þ is the mean pinning force per cell. Note that although the displacements generated by the dislocations are large and exceed n, we multiply the mean pinning force per cell only by n in formula (7) since n is the effective range of the elementary pinning force fpin . It is convenient to rewrite expression (7) in terms of Lc , the single-vortex collective pinning 2 length, using the relation [9], fpin n 4 e20 =L3c . Then 0 one arrives at the formula for Epin : 0 e0 a½Dg0 ðtÞ Epin
3=2 1=4
h
ð1 hÞ
3=4
;
ð8Þ
where we have inserted the estimates (5) for l? , lk , have taken into account that the quantity e0 / k2 ab should lead to an additional factor 1 h when h ¼ H =Hc2 ðtÞ tends to unity [25], and have used the notation [4]: nðtÞ Dg0 ðtÞ Lc ðtÞ
ð9Þ
with nð0Þ=Lc ð0Þ ¼ D. This D is just the parameter introduced above, D ðjc =j0 Þ1=2 ; it characterizes the pinning strength [9]. The function g0 ðtÞ is given [4] by
g0 ðtÞ ¼ ð1 t2 Þ
1=2
ð10Þ
for dl pinning, and for dTc pinning by g0 ðtÞ ¼ ð1 t2 Þ
1=6
:
ð11Þ
For definiteness, we imply in Eqs. (10) and (11) and throughout below that n2 ðtÞ ¼ n2 ð0Þ=ð1 t2 Þ. As was mentioned above, the thermal vibrations of the vortex system lead to a smoothing of the pinning potential and thereby affect the pinning [9,22]. This thermal depinning occurs both in the lattice and in the liquid, and we shall take it into account later. However, in the flux-line liquid there is an additional mechanism that affects the pinning energy and is associated with the thermal 0 fluctuations. The above Epin gives a characteristic pinning energy per dislocation cell and is due to the dislocation adjustment of the vortices to the pinning potential. But the dislocations in the liquid can have different configurations, and for some configurations the vortex system indeed adjusts itself to the pinning potential and hence there is a gain in the pinning energy, while some other configurations lead to a cost in this energy. Thus, one should carry out the thermodynamic averaging of the pinning energy, taking into account that different dislocation configurations can exist in the liquid. When the temperature is so low that 0 Epin T , only the configurations with a gain in the pinning energy have a noticeable probability to 0 occur, and this gain is given by the above Epin , i.e., 0 0 Epin Epin . But if Epin T , the probabilities of various dislocation configurations are almost equal, and the pinning energy is of the order of 0 2 ½Epin =T . We shall use the following simple estimate: ! 0 Epin 0 Epin Epin tanh ; ð12Þ T which describes both limiting cases. This estimate is obtained if one carries out the thermodynamic averaging, assuming that only two states with the 0 0 and Epin can occur. Note that the energies Epin described reduction of the pinning energy by thermal fluctuations is specific for the liquid, where a dense dislocation network exists, and the volume lk l2? of the dislocation cell is sufficiently small.
E.H. Brandt, G.P. Mikitik / Physica C 404 (2004) 61–68
Let us now estimate the entropy term in the energy balance (4). In the vortex system with pinning a dislocation cannot move if the pinning 1=2 force per dislocation cell, ðWlk l2? Þ , exceeds the Peach–K€ ohler force ac66 uxy lk exerted on a dislocation segment of length lk where uxy is the shear deformation of the vortex system in the plane normal to the magnetic field. Thus, for the dislocation relaxation of the shear stress to occur, the deformation uxy must exceed the critical value ucr xy ¼
ðWlk l2? Þ1=2 ac66 lk
½Dg0 ðtÞ
3 a ð1 hÞ3=2 u ða; 0Þ n ðtÞ Lc 2
2
¼ ð2pÞ3=2 n2 ðtÞ½Dg0 ðtÞ3 h3=2 ð1 hÞ3=2 ; ð16Þ where h ¼ H =Hc2 ðtÞ, t ¼ T =Tc , Hc2 ðtÞ ¼ U0 =2pn2 . For uT we use the formula [21], u2T n2 ðtÞFT ðtÞh1=2 ð1 hÞ
3=2
;
ð17Þ
where
3=2 1=4
h
3=4
ð1 hÞ
:
ð13Þ
Here again we have inserted the relations used in deriving Eq. (8). In the volume with dimensions L? and Lk , thermal fluctuations generate shear deformations uxy that can be estimated from c66 u2xy Lk L2? T :
ð14Þ
Hence, for large L? and Lk , when Lk L2? > ½Lk L2? cr
T c66 ðucr xy Þ
2
;
the deformations are less than the critical value; the relaxation of the shear stress in the liquid does not occur, and these modes do not contribute to the entropy gain (they do not differ essentially from the appropriate modes in the Bragg glass). Thus, the entropy gain is decreased with increasing pinning, and we obtain the factor DS in Eq. (4): DS ¼ 1 P ðt; hÞ, where P ðt; hÞ ¼
65
3 1=2 lk l2? ½Dg0 ðtÞ ð1 t2 Þ : tGi1=2 h ½Lk L2? cr
ð15Þ
Insertion of formulas (8), (12) and (15), multiplied by some numerical factors, into Eq. (4) gives the equation for the melting line in the bundlepinning region. We write it in the form of a Lindemann criterion, expressing all the terms of the equation via the magnitude of thermal vibrations of the lattice, uT , and the characteristic displacement uða; 0Þ caused by pinning. Without allowing for thermal depinning, this uða; 0Þ is given by [9,21]
Gi FT ðtÞ 2t 1 t2
1=2 ;
ð18Þ
and Gi is the Ginzburg number both pffiffiffi [9]. Dividing 1=2 sides p offfiffiffi Eq. (4) by 4 pðc44 c66 Þ a3 , putting C ¼ 4 pc2L , and taking into account formulas (16) and (17), we find the equation for the melting line in the form of a Lindemann criterion. In the equation thus obtained we replace n by rp ¼ ðn2 þ u2T Þ1=2 to account for thermal depinning, 1 and eventually we arrive at aA1 rp uða; 0Þ 2 2 uT B1 u ða; 0Þ þ A1 rp uða; 0Þ tanh u2T ¼ c2L a2 ;
ð19Þ
where A1 , B1 , a are some unknown numerical factors. This equation generalizes the appropriate 0 result of Ref. [21] where the factor tanhðEpin =T Þ was not taken into account in estimating the pinning energy. Note that since the factor 1 P ¼ DS is equal to 1 B1 ½u2 ða; 0Þ=u2T , Eq. (19) is valid when u2T P B1 u2 ða; 0Þ, and the condition u2T ¼ B1 u2 ða; 0Þ
ð20Þ
determines the position of the upper critical point on the melting line. At this point the entropy jump in the vortex system at melting vanishes. Some
1 Of course, the quantity uða; 0Þ is now described by a slightly modified formula [9] as compared with Eq. (16). To account for thermal depinning, the right hand side of Eq. (16) has to be multiplied by ðn=rp Þ4 .
66
E.H. Brandt, G.P. Mikitik / Physica C 404 (2004) 61–68 1
1
0.9
0.9
H / Hc2(0)
0.7
2 1/2
Hc2
0.6 0.8
0.6
cL= 0.25
0.7
dp
A1= 0.70 2 B1= 0.5*A1
1.2 0.4
0.6 0.5
D/cL= 0.8 Hdis
H
sv
0.1
1.8 0.2
Hm
0 0 1
0.1
0.1
0.2
0.3
0.4
H / Hc2(0)
0.7 0.6
0.3
0.4
0.5
0.6
0.7
t = T/Tc
0.4
0.8
0.9
0.8
δTc pinning:
0.7
2 1/6 g = (1–t )
Hc2
0
cL= 0.25
0.6
Gi = 0.01
Hdp
0.8
0.8
0.9
δl pinning: 2 1/2 g0=(1–t ) H
D/c = 1.3
c2
L
0.6
A1=0.70 2 B =0.5*A
0.5
1
Hsv
0.4
1
Hm
Hdis
0.1 0 0
Hm
0.3
0.7
0.2
1.2
0.4
Hsv
0.3
A1= 0.70 2 B1= 0.5*A1
1
0.5
0.6
Gi = 0.01
H / Hc2(0)
0.9 0.8
0.2
D/cL= 0
0.5
t = T/Tc
0.9 0.1
A1= 0.70 2 B1= 0.5*A1
Hm
0.2
1.6
0 10
δl pinning: 2 1/2 g0=(1–t )
Hc2
0.4 0.3
1.4
0.3
Hsv
Gi = 0.01
Gi = 0.01
H 1
0.5
g0= (1–t )
0.8
H / Hc2(0)
0.8
δl pinning:
D/c = 0 L 0.4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t = T/Tc
1.4
0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t = T/T
c
Fig. 1. The melting line Hm ðtÞ for the vortex lattice with quenched disorder, from Eq. (19), for Gi ¼ 0:01 and D=cL ¼ 0; 0:4; 0:6; . . . ; 1:8 in the case of dl pinning (top) and D=cL ¼ 0; 0:4; . . . ; 1:4 in the case of dTc pinning (bottom). Here cL ¼ 0:25, A1 ¼ 0:7, B1 ¼ 0:5A21 , a ¼ B1 =A21 ¼ 0:5. The upper line D ¼ 0 is the melting line of the ideal lattice. The dashed line gives the depinning line Hdp where uT ¼ n, while the dotted line shows Hc2 ðtÞ=Hc2 ð0Þ ¼ 1 t2 . Compare with Figs. 2 and 3 of Ref. [21], obtained from Eq. (19) without factor tanhð Þ and for A1 ¼ 1, B1 ¼ 0:8A21 .
melting lines obtained from Eq. (19) are shown in Figs. 1–3.
Fig. 2. The phase diagram in the case of dl pinning, g0 ðtÞ ¼ ð1 t2 Þ1=2 . Here A1 ¼ 0:7, B1 ¼ 0:5A21 , a ¼ B1 =A21 ¼ 0:5, cL ¼ 0:25, Gi ¼ 0:01, and D=cL ¼ 0:8 (top) or D=cL ¼ 1:3 (bottom). The melting line Hm ðtÞ from Eq. (19) and the order–disorder line Hdis ðtÞ from Eqs. (B8), (B11) of Ref. [21] are shown as solid lines, while the dashed line marks the boundary of the single vortex pinning region, Hsv ðtÞ, see Eqs. (B1), (B2) of our paper [21]. The dotted line shows Hc2 ðtÞ=Hc2 ð0Þ ¼ 1 t2 . The upper critical point (Tup , Hup ) is marked by a dot and the intersection point (Ti , Hi ) by a circle. Compare with Fig. 5 of Ref. [21], obtained without factor tanhð Þ in Eq. (19) and for A1 ¼ 0:66, B1 ¼ 0:8A21 .
ment of the hyperbolic tangent in Eq. (19) is small, and the sum of the second and third terms in this equation reduces to ! r2 2 p aA1 2 B1 u2 ða; 0Þ: uT
3. Discussion When the temperature is so high that T > Tdp ðH Þ where Tdp ðH Þ is the so-called depinning line [9] defined by the condition uT n in the T –H plane, one has uT rp uða; 0Þ, the argu-
If one assumes that a B1 =A21 , then the sum in this depinning region becomes very small 2 ½ B1 u2 ða; 0Þðn=uT Þ u2T whatever the constants A1 and B1 , and Eq. (19) practically coincides with the Lindemann criterion for the ideal lattice, Eq.
E.H. Brandt, G.P. Mikitik / Physica C 404 (2004) 61–68 1
δl pinning: 2 1/2 g0=(1–t )
0.9
Hsv
0.8
H
H / Hc2(0)
D/cL= 0.8
c2
0.7
Gi = 0.0001 A1= 0.70 2 B1= 0.5*A1
0.6
H
dis
0.5 0.4
Hm
0.3
H
sv
0.2 0.1 0 0 1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
t = T/Tc
0.9 0.8
H
c2
0.9
D/c = 1.3 L
0.7
H / Hc2(0)
0.8
δl pinning: 2 1/2 g0=(1–t ) Gi = 0.0001 A1= 0.70 B = 0.5*A2 1 1
0.6
Hsv
0.5 0.4 0.3
Hdis
0.2
0 0
Hm
Hsv
0.1 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
67
at different pinning strengths D for the cases of dl and dTc pinning. In a qualitative sense, these lines look like those presented in Figs. 2 and 3 of Ref. [21], and so the main results of that paper do not change, but now the temperature region where the melting lines practically coincide with the Hm ðT Þ of the ideal vortex lattice becomes wider. It was found recently [12,15] that the upper critical point of the melting line and the intersection point of this line with the order–disorder line coincide in overdoped YBa2 Cu3 Oy crystals (y > 6:92) for which this point lies at sufficiently high magnetic fields. On the other hand, for optimally dopped YBa2 Cu3 Oy crystals (y 6:92) it was discovered [11–15] that although the upper critical point shifts to lower fields and higher temperatures, it lies at larger magnetic field than the intersection point. Interestingly, this property of the phase diagrams can be qualitatively reproduced 3 for not too small Gi, see Figs. 2 and 3, if one uses Eq. (19) for the melting line and the results of Refs. [4,21] for the order–disorder line.
t = T/Tc
Fig. 3. As Fig. 2, but for Gi ¼ 0:0001. Compare with Fig. 6 of Ref. [21].
(3). Hence, we see that under this assumption 2 the pinning almost has no effect on the melting line in this region of the T –H plane (it only negligibly pushes the line downwards), in agreement with the experimental data [16–19]. When the melting line leaves the depinning region, the hyperbolic tangent in Eq. (19) tends to unity, and this equation reduces to that considered in Ref. [21]. In this case pinning starts to influence the shape of the melting line, and eventually, when condition (20) is fulfilled, it leads to the termination of Hm ðT Þ at the upper critical point. As was point out before [21], the properties of the upper critical point and the melting line observed in pffiffiffi experiments [16–19] are reproduced if B1 , A1 1 and B1 < A21 (the shape of the melting line is mainly determined by the ratio B1 =A21 ). Assuming such A1 and B1 , in Fig. 1 we show the melting line
2 If a > B1 =A21 , the effect of pinning on the melting line is larger, but it is still small in this depinning region.
Acknowledgements This work was supported by the German Israeli Research Grant Agreement (GIF) no. G-70550.14/01, by the European INTAS project no. 012282, and by the ESF Vortex Programme.
References [1] D. Ertasß, D.R. Nelson, Physica C 272 (1996) 79. [2] T. Giamarchi, P. Le Doussal, Phys. Rev. B 55 (1997) 6577. [3] V. Vinokur, B. Khaykovich, E. Zeldov, M. Konczykowski, R.A. Doyle, P.H. Kes, Physica C 295 (1998) 209. [4] G.P. Mikitik, E.H. Brandt, Phys. Rev. B 64 (2001) 184514.
3 In the construction of all figures, we fix the ratio B1 =A21 < 1 determining the shape of Hm ðT Þ and choose the value of A1 such that the upper critical point and the intersection point coincide at some fixed values of D and Gi (we take D=cL ¼ 0:7, Gi ¼ 0:01 as these values). We find that the value of A1 thus obtained is almost independent of the specific choice of the fixed D and Gi if the upper critical point at these D and Gi lies in the high-field region.
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E.H. Brandt, G.P. Mikitik / Physica C 404 (2004) 61–68
[5] Y. Radzyner, A. Shaulov, Y. Yeshurun, Phys. Rev. B 65 (2002) 100513(R); Y. Radzyner, A. Shaulov, Y. Yeshurun, I. Felner, K. Kishio, J. Shimoyama, Phys. Rev. B 65 (2002) 100503(R). [6] T. Giamarchi, P. Le Doussal, Phys. Rev. B 52 (1995) 1242. [7] T. Nattermann, S. Scheidl, Adv. Phys. 49 (2000) 607. [8] J. Kierfeld, V. Vinokur, Phys. Rev. B 61 (2000) 14928. [9] G. Blatter, M.V. FeigelÕman, V.B. Geshkenbein, A.I. Larkin, V.M. Vinokur, Rev. Mod. Phys. 66 (1994) 1125. [10] T.K. Worthington, M.P.A. Fisher, D.A. Huse, J. Toner, A.D. Marwick, T. Zabel, C.A. Feild, F. Holtzberg, Phys. Rev. B 46 (1992) 11854. [11] D. Pal, S. Ramakrishnan, A.K. Grover, D. Dasgupta, B.K. Sarma, Phys. Rev. B 63 (2001) 132505. [12] N. Kobayashi, T. Nishizaki, K. Shibata, T. Sato, M. Maki, T. Sasaki, Physica C 362 (2001) 121. [13] F. Bouquet, C. Marcenat, E. Steep, R. Calemczuk, W.K. Kwok, U. Welp, G.W. Crabtree, R.A. Fisher, N.E. Phillips, A. Schilling, Nature (London) 411 (2001) 448. [14] T. Nishizaki, K. Shibata, M. Maki, N. Kobayashi, J. Low Temp. Phys. 131 (2003) 931. [15] K. Shibata, T. Nishizaki, T. Sasaki, N. Kobayashi, Phys. Rev. B 66 (2002) 214518.
[16] B. Khaykovich, M. Konczykowski, E. Zeldov, R.A. Doyle, D. Majer, P.H. Kes, T.W. Li, Phys. Rev. B 56 (1997) R517. [17] J.A. Fendrich, W.K. Kwok, J. Giapintzakis, C.J. van der Beek, V.M. Vinokur, S. Fleshler, U. Welp, H.K. Viswanathan, G.W. Crabtree, Phys. Rev. Lett. 74 (1995) 1210. [18] L.M. Paulius, W.-K. Kwok, R.J. Olsson, A.M. Petrean, V. Tobos, J.A. Fendrich, G.W. Crabtree, C.A. Burns, S. Ferguson, Phys. Rev. B 61 (2002) R11910. [19] T. Nishizaki, T. Naito, S. Okayasu, A. Iwase, N. Kobayashi, Phys. Rev. B 61 (2000) 3649. [20] A.I. Larkin, V.M. Vinokur, Phys. Rev. Lett. 75 (1995) 4666. [21] G.P. Mikitik, E.H. Brandt, Phys. Rev. B 68 (2003) 054509. [22] M.V. FeigelÕman, V.M. Vinokur, Phys. Rev. B 41 (1990) 8986. [23] L.D. Landau, E.M. Lifshitz, Theory of Elasticity, Course in Theoretical Physics, vol. 7, Pergamon Press, London, 1959. [24] M.C. Marchetti, D.R. Nelson, Phys. Rev. B 41 (1990) 1910. [25] E.H. Brandt, Rep. Prog. Phys. 58 (1995) 1465. [26] E.H. Brandt, J. Low Temp. Phys. 26 (1977) 709, 735; E.H. Brandt, J. Low Temp. Phys. 28 (1977) 263, 291.
H –T phase diagram of the vortex lattice in superconductors with pinning E.H. Brandt b
a,*
, G.P. Mikitik
b
a Max-Planck-Institut f€ur Metallforschung, Heisenbergstr. 3, D-70506 Stuttgart, Germany B. Verkin Institute for Low Temperature Physics and Engineering, Ukrainian Academy of Sciences, Kharkov 61103, Ukraine
Abstract The vortex-lattice melting line in three-dimensional type II superconductors with pinning is derived by equating the free energies of the vortex system in the solid and liquid phases. We account for the elastic and pinning energies and the entropy change that originates from the disappearance of the phonon shear modes in the liquid. The pinning is assumed to be caused by point defects and to be not too strong so that the melting line lies inside the so-called bundle-pinning region. The derived equation for the melting line may be expressed in the form of some Lindemann criterion which now includes a characteristic vortex displacement caused by the pinning. Estimating the effect of pinning on the entropy jump at melting, we find the upper critical point of the melting line from the condition that this jump vanishes. Some recent experimental results on the T –H phase diagrams of type II superconductors are discussed. Ó 2004 Elsevier B.V. All rights reserved. PACS: 74.25.Qt; 74.72.Bk Keywords: Vortex-lattice melting; Pinning; Phase diagram; Order–disorder transition
1. Introduction In three-dimensional high-Tc superconductors with pinning, two phase transition lines are known to exist in the magnetic field H –temperature T plane [1–5]: The line Hm ðT Þ where a quasiordered Bragg glass [6,7] thermally melts into a flux-line liquid, and the order–disorder transition line Hdis ðT Þ separating the Bragg glass from an amorphous vortex state. The melting is caused by thermal vibrations of the lattice, while the order–
*
Corresponding author. Tel.: +49-711-6893575; fax: +49711-6891932. E-mail address: [email protected] (E.H. Brandt).
disorder transition is induced by quenched disorder in the vortex system. These two lines merge at some point in the H –T plane. Although both transitions are accompanied by a proliferation of dislocations in the vortex lattice, it was argued [8] that the dislocation density q is essentially different in these cases: q a2 for melting, and q R2 a for the order–disorder transition. Here a is the spacing between flux lines, and Ra is the so-called positional correlation length [9] within which the relative vortex displacements caused by the quenched disorder are of the order of a. In fact, an intersection of these two different phase transition lines occurs in this scenario; the order–disorder line terminates at the intersection point while the melting line continues for some distance to higher
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2003.10.037
62
E.H. Brandt, G.P. Mikitik / Physica C 404 (2004) 61–68
H and terminates only at its upper critical point. Within this physical picture, the existence of the so-called slush phase [10] can be naturally explained. Recent experiments [11–15] for YBaCuO seem to support this scenario. In our paper [4] it was shown that the H –T phase diagrams of many low-Tc and high-Tc superconductors with pinning caused by point defects can be described in a universal manner and are determined by two parameters: the Ginzburg number Gi which characterizes the magnitude of the thermal fluctuations, and the ratio D 1=2 ðjc =j0 Þ which describes the strength of the pinning. Here jc is the critical current density in the single vortex pinning regime [9], and j0 is the depairing current density, both taken at T ¼ 0. Phase diagrams with various Gi and D were analyzed in Ref. [4], based on the above-mentioned scenario. In this analysis the melting line was calculated neglecting the effect of the pinning on it. This approximation was justified by the experimental finding [16–19] that an increase of the pinning leads only to a noticeable shift of the upper critical point on Hm ðT Þ but practically does not change the melting line itself (the increase only slightly pushes Hm ðT Þ downwards in the T –H plane). Of course, this approximation did not enable us to calculate the upper critical point of Hm ðT Þ and its shifts with changing D and Gi. The effect of point-defect pinning on the melting line was considered in Refs. [2,5,20,21]. In our recent paper [21] we found the melting line and its upper critical point by considering the balance of three energies: the elastic energy caused by the proliferation of dislocations at melting, the entropy gain in the free energy associated with the disappearance of the shear-phonon modes in the liquid, and the pinning energy. In this analysis we allowed for a mutual ‘‘influence’’ of the pinning and the thermal fluctuations: the entropy gain was estimated, taking into account the effect of pinning on it (just this permits us to find the upper critical point), while the pinning energy was calculated accounting for the smoothing effect of thermal fluctuations on the pinning potential, i.e., the socalled thermal depinning [9,22]. In estimating this thermal depinning we used the recipe of Ref. [9] obtained for the flux-line lattice (the Bragg glass).
However, the effect of thermal fluctuations on the pinning energy is expected to be different for the lattice and for the liquid, and in the present paper we generalize the results of our paper [21] by taking into account this effect in the liquid. Below we consider only magnetic fields H exceeding considerably the lower critical field Hc1 , and thus we put B ¼ l0 H . Besides this, we deal only with uniaxial anisotropic three-dimensional superconductors with anisotropy ¼ kab =kc < 1, neglecting completely the decoupling of the superconducting layers. Here kab and kc are the London penetration depths in the plane ab perpendicular to the anisotropy axis and along this axis, respectively. The magnetic field is assumed to be directed along the anisotropy axis. The quenched disorder in the flux-line lattice is caused by point defects and is not too strong such that the melting line lies entirely in the so-called bundle pinning region, where the transverse collective pinning length Rc [9] is greater than a.
2. Melting of the vortex lattice with quenched disorder At melting, proliferation of dislocations occurs in the vortex lattice. These dislocations create a network in the lattice composed of edge and screw dislocations, with the mean dimensions of a unit cell of the network being of the order of a. (One has l? a for the dimension l? transverse to H , and lk a for the longitudinal dimension lk , see details in Refs. [8,21].) In the vortex system without pinning the cost in the elastic energy due to the proliferation of the dislocations is balanced by the entropy gain in the free energy of the flux-line liquid. We shall estimate this gain, assuming that the main contribution to it results from the disappearance of the shear-phonon modes in the liquid (i.e., from their transformation into some other degrees of freedom). The mechanism of this disappearance is as follows: A shear stress exerts so-called Peach–K€ ohler forces [23,24] on the dislocations, which then move and thus relax the shear stress. The above-mentioned energy balance enables one to derive the equation for the melting line of
E.H. Brandt, G.P. Mikitik / Physica C 404 (2004) 61–68
the ideal vortex lattice [9,21]. The elastic energy per unit cell of the dislocation network, Eel , associated with the proliferation of the dislocations is 1=2 proportional to ðc44 c66 Þ a3 where c44 and c66 are the tilt and shear moduli [25,26] of the flux-line lattice (the nonlocal tilt modulus should be taken at wave vectors k of the order of 1=a), e0 2 c66 2 ð1 hÞ ; 4a ð1Þ 2 e0 c44 2 ð1 hÞ: a Here h ¼ H =Hc2 ðtÞ, t ¼ T =Tc , e0 ¼ ðU0 =4pkab Þ2 , U0 is the flux quantum, and Hc2 ðtÞ is the upper critical field in the mean-field theory. The factors containing ð1 hÞ take into account the softening of the vortex lattice near the Hc2 ðtÞ line [26]. As to the entropy gain, in the unit volume of the lattice there 2 are l1 k l? shear modes with wave-lengths greater than lk , l? , and each mode contributes about T to this gain. Thus, the gain per cell of the dislocation 2 2 network is T l1 k l? lk l? T . As a result, the energy balance applied to a mean unit cell of the dislocation network in the vortex system without pinning yields: Cðc44 c66 Þ1=2 a3 T ¼ 0;
ð2Þ
where the constant C absorbs all unknown numerical factors in Eel and in the entropy gain. Inserting formulas pffiffiffi (1) into this equation and expressing C as 4 pc2L where cL is the Lindemann constant, one finds an equation for the melting line which coincides with that derived from the Lindemann criterion, u2T ¼ c2L a2 ;
ð3Þ
or from the analysis based on the Ginzburg–Landau (or London) Hamiltonian [21]. Here uT is the magnitude of the thermal displacements of the vortex lattice with respect to its equilibrium position. When pinning exists in the vortex system, the adjustment of the vortices to the pinning potential decreases the total energy of the system, and the amount of this decrease is the so-called pinning energy [9]. In the liquid, vortices can adjust themselves to the pinning potential not only via their elastic deformations as in the lattice, but also
63
via the plastic vortex-lattice displacements generated by dislocations. This additional adjustment mechanism increases the pinning energy in the liquid, and thus at melting there is a gain in pinning energy, Epin , which should be included in the energy balance. Besides this, pinning influences the entropy gain. As a result, Eq. (2) is modified by pinning as follows: Cðc44 c66 Þ
1=2 3
a T DS Epin ¼ 0;
ð4Þ
where the factor DS takes into account the effect of pinning on the entropy gain. In estimating Epin , it is necessary in general to take into account the thermal depinning [9,22] since the magnitude of the thermal displacements of the lattice, uT , is sufficiently large at the melting, uT cL a. However, the results of Ref. [21] lead to the following conclusion: If one neglects the thermal depinning, derives the equation for the melting line from the energy balance, and then rewrites this equation in the form of the Lindemann criterion, the thermal depinning can be accounted for by the simple replacement of the coherence length n by 1=2 the effective radius rp ¼ ðn2 þ u2T Þ of the pins. Using this observation, we first carry out the estimates neglecting depinning and then account for it by the above-mentioned replacement. To estimate the pinning energy per dislocation cell, Epin , it is necessary to take into account that the displacements generated by the dislocation network are essentially larger than the displacements uðl? ; 0Þ uð0; lk Þ existing in the Bragg glass at the same temperature and magnetic field within the scales of the cell, l? a; 1=2 c44 a : lk l? 1=2 c66 ð1 hÞ
ð5Þ
[The factor ð1 hÞ1=2 originates from the softening of the vortex lattice near Hc2 ðtÞ.] The displacement uðl? ; 0Þ (or uð0; lk Þ) is the rms relative shift of two line elements in the vortex lattice separated by a distance l? transverse to the magnetic field (or by a distance lk along the field). This shift is caused by the random point defects and can be expressed in terms of the transverse collective pinning length Rc at which the relative
64
E.H. Brandt, G.P. Mikitik / Physica C 404 (2004) 61–68
displacements of points in the lattice are of the order of n. In particular, if the small bundle pinning regime occurs, i.e., a < Rc < kab , one has [9] u2 ða; 0Þ
n2 : 1 þ lnðR2c =a2 Þ
ð6Þ
Thus, in the lattice without dislocations, one has uða; 0Þ < n, while in the liquid the dislocations lead to displacements of about a > n at the boundary of the dislocation cell. These inequalities justify the above statement that in the liquid the plastic deformations associated with the dislocations dominate over the elastic deformations caused by the point defects. Using the results of the collective pinning theory [9], we then obtain the estimate for 0 the pinning energy Epin calculated without allowing for the thermal fluctuations, 0 Epin ðWlk l2? Þ
1=2
ð7Þ
n;
2 nn2 =a2 , fpin is the mean elementary where W ¼ fpin pinning force exerted by one point defect, n is the concentration of the defects, and the expression 1=2 ðWlk l2? Þ is the mean pinning force per cell. Note that although the displacements generated by the dislocations are large and exceed n, we multiply the mean pinning force per cell only by n in formula (7) since n is the effective range of the elementary pinning force fpin . It is convenient to rewrite expression (7) in terms of Lc , the single-vortex collective pinning 2 length, using the relation [9], fpin n 4 e20 =L3c . Then 0 one arrives at the formula for Epin : 0 e0 a½Dg0 ðtÞ Epin
3=2 1=4
h
ð1 hÞ
3=4
;
ð8Þ
where we have inserted the estimates (5) for l? , lk , have taken into account that the quantity e0 / k2 ab should lead to an additional factor 1 h when h ¼ H =Hc2 ðtÞ tends to unity [25], and have used the notation [4]: nðtÞ Dg0 ðtÞ Lc ðtÞ
ð9Þ
with nð0Þ=Lc ð0Þ ¼ D. This D is just the parameter introduced above, D ðjc =j0 Þ1=2 ; it characterizes the pinning strength [9]. The function g0 ðtÞ is given [4] by
g0 ðtÞ ¼ ð1 t2 Þ
1=2
ð10Þ
for dl pinning, and for dTc pinning by g0 ðtÞ ¼ ð1 t2 Þ
1=6
:
ð11Þ
For definiteness, we imply in Eqs. (10) and (11) and throughout below that n2 ðtÞ ¼ n2 ð0Þ=ð1 t2 Þ. As was mentioned above, the thermal vibrations of the vortex system lead to a smoothing of the pinning potential and thereby affect the pinning [9,22]. This thermal depinning occurs both in the lattice and in the liquid, and we shall take it into account later. However, in the flux-line liquid there is an additional mechanism that affects the pinning energy and is associated with the thermal 0 fluctuations. The above Epin gives a characteristic pinning energy per dislocation cell and is due to the dislocation adjustment of the vortices to the pinning potential. But the dislocations in the liquid can have different configurations, and for some configurations the vortex system indeed adjusts itself to the pinning potential and hence there is a gain in the pinning energy, while some other configurations lead to a cost in this energy. Thus, one should carry out the thermodynamic averaging of the pinning energy, taking into account that different dislocation configurations can exist in the liquid. When the temperature is so low that 0 Epin T , only the configurations with a gain in the pinning energy have a noticeable probability to 0 occur, and this gain is given by the above Epin , i.e., 0 0 Epin Epin . But if Epin T , the probabilities of various dislocation configurations are almost equal, and the pinning energy is of the order of 0 2 ½Epin =T . We shall use the following simple estimate: ! 0 Epin 0 Epin Epin tanh ; ð12Þ T which describes both limiting cases. This estimate is obtained if one carries out the thermodynamic averaging, assuming that only two states with the 0 0 and Epin can occur. Note that the energies Epin described reduction of the pinning energy by thermal fluctuations is specific for the liquid, where a dense dislocation network exists, and the volume lk l2? of the dislocation cell is sufficiently small.
E.H. Brandt, G.P. Mikitik / Physica C 404 (2004) 61–68
Let us now estimate the entropy term in the energy balance (4). In the vortex system with pinning a dislocation cannot move if the pinning 1=2 force per dislocation cell, ðWlk l2? Þ , exceeds the Peach–K€ ohler force ac66 uxy lk exerted on a dislocation segment of length lk where uxy is the shear deformation of the vortex system in the plane normal to the magnetic field. Thus, for the dislocation relaxation of the shear stress to occur, the deformation uxy must exceed the critical value ucr xy ¼
ðWlk l2? Þ1=2 ac66 lk
½Dg0 ðtÞ
3 a ð1 hÞ3=2 u ða; 0Þ n ðtÞ Lc 2
2
¼ ð2pÞ3=2 n2 ðtÞ½Dg0 ðtÞ3 h3=2 ð1 hÞ3=2 ; ð16Þ where h ¼ H =Hc2 ðtÞ, t ¼ T =Tc , Hc2 ðtÞ ¼ U0 =2pn2 . For uT we use the formula [21], u2T n2 ðtÞFT ðtÞh1=2 ð1 hÞ
3=2
;
ð17Þ
where
3=2 1=4
h
3=4
ð1 hÞ
:
ð13Þ
Here again we have inserted the relations used in deriving Eq. (8). In the volume with dimensions L? and Lk , thermal fluctuations generate shear deformations uxy that can be estimated from c66 u2xy Lk L2? T :
ð14Þ
Hence, for large L? and Lk , when Lk L2? > ½Lk L2? cr
T c66 ðucr xy Þ
2
;
the deformations are less than the critical value; the relaxation of the shear stress in the liquid does not occur, and these modes do not contribute to the entropy gain (they do not differ essentially from the appropriate modes in the Bragg glass). Thus, the entropy gain is decreased with increasing pinning, and we obtain the factor DS in Eq. (4): DS ¼ 1 P ðt; hÞ, where P ðt; hÞ ¼
65
3 1=2 lk l2? ½Dg0 ðtÞ ð1 t2 Þ : tGi1=2 h ½Lk L2? cr
ð15Þ
Insertion of formulas (8), (12) and (15), multiplied by some numerical factors, into Eq. (4) gives the equation for the melting line in the bundlepinning region. We write it in the form of a Lindemann criterion, expressing all the terms of the equation via the magnitude of thermal vibrations of the lattice, uT , and the characteristic displacement uða; 0Þ caused by pinning. Without allowing for thermal depinning, this uða; 0Þ is given by [9,21]
Gi FT ðtÞ 2t 1 t2
1=2 ;
ð18Þ
and Gi is the Ginzburg number both pffiffiffi [9]. Dividing 1=2 sides p offfiffiffi Eq. (4) by 4 pðc44 c66 Þ a3 , putting C ¼ 4 pc2L , and taking into account formulas (16) and (17), we find the equation for the melting line in the form of a Lindemann criterion. In the equation thus obtained we replace n by rp ¼ ðn2 þ u2T Þ1=2 to account for thermal depinning, 1 and eventually we arrive at aA1 rp uða; 0Þ 2 2 uT B1 u ða; 0Þ þ A1 rp uða; 0Þ tanh u2T ¼ c2L a2 ;
ð19Þ
where A1 , B1 , a are some unknown numerical factors. This equation generalizes the appropriate 0 result of Ref. [21] where the factor tanhðEpin =T Þ was not taken into account in estimating the pinning energy. Note that since the factor 1 P ¼ DS is equal to 1 B1 ½u2 ða; 0Þ=u2T , Eq. (19) is valid when u2T P B1 u2 ða; 0Þ, and the condition u2T ¼ B1 u2 ða; 0Þ
ð20Þ
determines the position of the upper critical point on the melting line. At this point the entropy jump in the vortex system at melting vanishes. Some
1 Of course, the quantity uða; 0Þ is now described by a slightly modified formula [9] as compared with Eq. (16). To account for thermal depinning, the right hand side of Eq. (16) has to be multiplied by ðn=rp Þ4 .
66
E.H. Brandt, G.P. Mikitik / Physica C 404 (2004) 61–68 1
1
0.9
0.9
H / Hc2(0)
0.7
2 1/2
Hc2
0.6 0.8
0.6
cL= 0.25
0.7
dp
A1= 0.70 2 B1= 0.5*A1
1.2 0.4
0.6 0.5
D/cL= 0.8 Hdis
H
sv
0.1
1.8 0.2
Hm
0 0 1
0.1
0.1
0.2
0.3
0.4
H / Hc2(0)
0.7 0.6
0.3
0.4
0.5
0.6
0.7
t = T/Tc
0.4
0.8
0.9
0.8
δTc pinning:
0.7
2 1/6 g = (1–t )
Hc2
0
cL= 0.25
0.6
Gi = 0.01
Hdp
0.8
0.8
0.9
δl pinning: 2 1/2 g0=(1–t ) H
D/c = 1.3
c2
L
0.6
A1=0.70 2 B =0.5*A
0.5
1
Hsv
0.4
1
Hm
Hdis
0.1 0 0
Hm
0.3
0.7
0.2
1.2
0.4
Hsv
0.3
A1= 0.70 2 B1= 0.5*A1
1
0.5
0.6
Gi = 0.01
H / Hc2(0)
0.9 0.8
0.2
D/cL= 0
0.5
t = T/Tc
0.9 0.1
A1= 0.70 2 B1= 0.5*A1
Hm
0.2
1.6
0 10
δl pinning: 2 1/2 g0=(1–t )
Hc2
0.4 0.3
1.4
0.3
Hsv
Gi = 0.01
Gi = 0.01
H 1
0.5
g0= (1–t )
0.8
H / Hc2(0)
0.8
δl pinning:
D/c = 0 L 0.4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t = T/Tc
1.4
0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t = T/T
c
Fig. 1. The melting line Hm ðtÞ for the vortex lattice with quenched disorder, from Eq. (19), for Gi ¼ 0:01 and D=cL ¼ 0; 0:4; 0:6; . . . ; 1:8 in the case of dl pinning (top) and D=cL ¼ 0; 0:4; . . . ; 1:4 in the case of dTc pinning (bottom). Here cL ¼ 0:25, A1 ¼ 0:7, B1 ¼ 0:5A21 , a ¼ B1 =A21 ¼ 0:5. The upper line D ¼ 0 is the melting line of the ideal lattice. The dashed line gives the depinning line Hdp where uT ¼ n, while the dotted line shows Hc2 ðtÞ=Hc2 ð0Þ ¼ 1 t2 . Compare with Figs. 2 and 3 of Ref. [21], obtained from Eq. (19) without factor tanhð Þ and for A1 ¼ 1, B1 ¼ 0:8A21 .
melting lines obtained from Eq. (19) are shown in Figs. 1–3.
Fig. 2. The phase diagram in the case of dl pinning, g0 ðtÞ ¼ ð1 t2 Þ1=2 . Here A1 ¼ 0:7, B1 ¼ 0:5A21 , a ¼ B1 =A21 ¼ 0:5, cL ¼ 0:25, Gi ¼ 0:01, and D=cL ¼ 0:8 (top) or D=cL ¼ 1:3 (bottom). The melting line Hm ðtÞ from Eq. (19) and the order–disorder line Hdis ðtÞ from Eqs. (B8), (B11) of Ref. [21] are shown as solid lines, while the dashed line marks the boundary of the single vortex pinning region, Hsv ðtÞ, see Eqs. (B1), (B2) of our paper [21]. The dotted line shows Hc2 ðtÞ=Hc2 ð0Þ ¼ 1 t2 . The upper critical point (Tup , Hup ) is marked by a dot and the intersection point (Ti , Hi ) by a circle. Compare with Fig. 5 of Ref. [21], obtained without factor tanhð Þ in Eq. (19) and for A1 ¼ 0:66, B1 ¼ 0:8A21 .
ment of the hyperbolic tangent in Eq. (19) is small, and the sum of the second and third terms in this equation reduces to ! r2 2 p aA1 2 B1 u2 ða; 0Þ: uT
3. Discussion When the temperature is so high that T > Tdp ðH Þ where Tdp ðH Þ is the so-called depinning line [9] defined by the condition uT n in the T –H plane, one has uT rp uða; 0Þ, the argu-
If one assumes that a B1 =A21 , then the sum in this depinning region becomes very small 2 ½ B1 u2 ða; 0Þðn=uT Þ u2T whatever the constants A1 and B1 , and Eq. (19) practically coincides with the Lindemann criterion for the ideal lattice, Eq.
E.H. Brandt, G.P. Mikitik / Physica C 404 (2004) 61–68 1
δl pinning: 2 1/2 g0=(1–t )
0.9
Hsv
0.8
H
H / Hc2(0)
D/cL= 0.8
c2
0.7
Gi = 0.0001 A1= 0.70 2 B1= 0.5*A1
0.6
H
dis
0.5 0.4
Hm
0.3
H
sv
0.2 0.1 0 0 1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
t = T/Tc
0.9 0.8
H
c2
0.9
D/c = 1.3 L
0.7
H / Hc2(0)
0.8
δl pinning: 2 1/2 g0=(1–t ) Gi = 0.0001 A1= 0.70 B = 0.5*A2 1 1
0.6
Hsv
0.5 0.4 0.3
Hdis
0.2
0 0
Hm
Hsv
0.1 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
67
at different pinning strengths D for the cases of dl and dTc pinning. In a qualitative sense, these lines look like those presented in Figs. 2 and 3 of Ref. [21], and so the main results of that paper do not change, but now the temperature region where the melting lines practically coincide with the Hm ðT Þ of the ideal vortex lattice becomes wider. It was found recently [12,15] that the upper critical point of the melting line and the intersection point of this line with the order–disorder line coincide in overdoped YBa2 Cu3 Oy crystals (y > 6:92) for which this point lies at sufficiently high magnetic fields. On the other hand, for optimally dopped YBa2 Cu3 Oy crystals (y 6:92) it was discovered [11–15] that although the upper critical point shifts to lower fields and higher temperatures, it lies at larger magnetic field than the intersection point. Interestingly, this property of the phase diagrams can be qualitatively reproduced 3 for not too small Gi, see Figs. 2 and 3, if one uses Eq. (19) for the melting line and the results of Refs. [4,21] for the order–disorder line.
t = T/Tc
Fig. 3. As Fig. 2, but for Gi ¼ 0:0001. Compare with Fig. 6 of Ref. [21].
(3). Hence, we see that under this assumption 2 the pinning almost has no effect on the melting line in this region of the T –H plane (it only negligibly pushes the line downwards), in agreement with the experimental data [16–19]. When the melting line leaves the depinning region, the hyperbolic tangent in Eq. (19) tends to unity, and this equation reduces to that considered in Ref. [21]. In this case pinning starts to influence the shape of the melting line, and eventually, when condition (20) is fulfilled, it leads to the termination of Hm ðT Þ at the upper critical point. As was point out before [21], the properties of the upper critical point and the melting line observed in pffiffiffi experiments [16–19] are reproduced if B1 , A1 1 and B1 < A21 (the shape of the melting line is mainly determined by the ratio B1 =A21 ). Assuming such A1 and B1 , in Fig. 1 we show the melting line
2 If a > B1 =A21 , the effect of pinning on the melting line is larger, but it is still small in this depinning region.
Acknowledgements This work was supported by the German Israeli Research Grant Agreement (GIF) no. G-70550.14/01, by the European INTAS project no. 012282, and by the ESF Vortex Programme.
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