Freezing of the vortex liquid in the high-Tc superconductors

Solid State Communications, Vol. 90, No. 8, pp. 479-482, 1994 Elsevier Science Ltd Primed in Great Britain 0038-I098(94)$7.00+ .00 0038- I098(94)E0225-2

Pergamon

FREEZING OF T H E V O R T E X LIQUID IN T H E HIGH-T~ S U P E R C O N D U C T O R S D. J. C. Jackson and M. P. Das Department of Theoretical Physics, Research SchooLof Physical Sciences and Engineering, The Australian National University, Canberra, ACT 0200, Australia (Received 2 February 1994,acceptedfor publication7 March 1994 by A.H.MacDonald) We discuss the freezing transition of a vortex liquid in the high temperature superconductors by using the density functional approach. We calculate the freezing line in the B - T plane for the HgBa~Cu04+~ superconductor. PACS number: 2460E Keywords: vortex liquid, freezing, density functionM, phase diagram

There has been much interest of late in the magnetic phase diagram for the new high temperature superconductors (HTSC). This has many similarities to the phe~e diagram for conventional type-II superconductors. The main points of interest in the conventional superconductors phase diagram are the upper and lower critical field lines, given by I H~

increases, the lattice spacing decreases until the cores of the flux lines start to overlap. At the upper critical field, the normal cores overlap sufficiently to cause the material to revert back to its normal state, destroying the superconducting properties. The phase between the two critical fields remains superconducting in the presence of these flux lines for the following reason. When a current is applied to a superconductor, the interaction between thh current and the flux lines cause the lines to move under the Lorentz force. This means that the lattice would he free to move through the sample, and as it did so it would be able to scatter the electrons, generating a finite resistance. In normal materials this is generally not the case, as the lattice is pinned by defects in the crystal structure. Once pinned, the electrons are then able to flow around the lattice, and so the sample remains superconducting. With the discovery of high temperature superconductors (HTSC) by Bednorz and Mfdler,s the magnetic phase diagram for these new materials was found to differ from that for conventional superconductors. One of the main differences was the appearance of a line between He1 and Hca. This line has been christened the irreversible line. This name stems from the fact that below this line, the magnetic properties of the material are reversible, while above it, the material exhibits irreversible behavoir. This effect is thought to be due in some part to the melting of the flux line lattice. This melting into a flux line liquid has many important consequences for the HTSC materiah, mcet notably of which is its effect on the critical current. One method available to study the melting/freezing transition in these materials is density functionai theory (DFT). This is a mean field examination of the melting

~0 ~Ina¢

(1) H~=

2~r~(T)'

where A is the penetration depth, ~ the coherence length

and

s =

~/~.

The critical field lines can he thought of as markers for the different phases for the flux line lattice. Below He1 there is no magnetic filed penetration of the sample, and the material is in the MeiMner state. In this state, all magnetic flux is expelled from the interior of the sample. Above He1 the magnetic field starts to penetrate in the form of discrete flux quanta. Theses quanta all have the same universal strength given by the elemental flux quantum hc = --. 2e

(2)

In the core of the flux line, the material is nonsuperconducting, while the rest of the material remains superconducting. As the applied magnetic field increaa~ toward Hca, the flux lines become forced closer together. The effect of the external field, and the mutual repulsion produce a 2D Abriko6ov lattice. 2 As the field 479

FREI~.TJNG OF THE VORTEX LIQUID

480

transition, and is applied to study the newer mercury compound. However, before examining this transition in the new HTSC materials, it will be constructive to look first at conventional superconductors. These materials also have an irreversible line in their phase diagram, however it will be shown below that it lies so dose to H ~ as to be experimentally indistinguishable. A very simple way to examine the melting transition is to use the Lindemann criterion. 4 This is based on the idea that the elements that make up a lattice have some inherent motion about their mean positions. It is then assumed that once this motion exceeds some fraction of the lattice spacing, generally taken to be a round 10% for a classical system, the lattice will melt. This can be written as (u2(T.)),h ~ ~ a 2,

(3)

where a is the lattice spacing, cL is the Lindemann number, and u(Tm) is the amplitude of the motion about the lattice site at the melting temperature. Using this equation one can gain a rough estimate of the melting transition as follows. The expression for the lattice spacing in a superconductor is given by

where B is the value of the applied magnetic field. Other typical parameters for these materials are ~ ~ 20, Tc ~ 20K and H..~(0) ~ 20Tesl,. Substituting these values into the Lindemmm expression at the melting temperature gives,

B,(T) ~ /~, ( ~ ) H~(O)(1- t)', where tim ~ 10, t -ber, given by s

T/Tc and Gi is the

(4)

Ginzburg num-

T~

G, 2(~(0)¢,(0)p. =

Using this in equation (4), it can be shown that

B,~

£ H,~(T).

Prom this it can be seen that the irreversible line does exist for conventional superconductors, but lies very close to the upper critical field line. The.same calculation can also be carried out for the HTSC, sad the difference is quite surprising. The main reason behind the difference in values is due to the difference in the input parameters. For the HTSC material, the average in-plane values are

~R >

100

H..,(0) > 100T. When these values are used in the Lindemann expres-

Vol. 90, No. 8

sion, they produce a value for the irreversible line that is very much smaller than the upper critical field, making the line more accessible to experimental obser~tion. HTSC materials also suffer from another important effect, and that is quantum fluctuations. The strength of this elfect is measured using the resistance ratio

e2pN where d is the interlayer spacing and pN is the normal state rmistivity. It has been shown that in this quantum regime, the 2D melting transition can produce much larger Lindemann numbers, s Because of the importance of this new line in the phase diagram for the HTSC materials, there has been a lot of experimental interest in its measurement. Decoration experiments 7 can be used to map out the magnetic flux line structure on the surface d a sample. However this gives no information about what is happening inside the munple. Mechanics] mewmrements s can be used to plot out the melting temperature in the H - T plane, and thus generate the irreversible line. This can also be achieved using measurements of resistivity versus temperature, s and by using the variation d the critical current, both intra-grain and inter-grain, with applied magnetic field, l° There are several theoretical models used to explain the irreversibility line. Malc~emoff11 uses an idea based on the Andermn-Kim model, l~ In his model, the flux jumps in larger groups of many individual flux lines, leading to "giant flux creep ~. Nelson ~s puts forward an idea of flux cutting and entanglement, this can act so as to restrict flux flow inside the material, and introduces extra phases on the phase diagram. A slightly different idea is that of the glassy solid put forward by Fisher et o/. 14 which has its basis in the model developed for spin-giasses. Larkin and Ovchinnikovis have examined the phenomenon of collective flux creep, a process that can occur in the weak pinning regime, and is a similar idea to giant flux creep. The discussion of this ides has been extended to the HTSC materials. ~s By using the Lindemann criteria (3), others 17-1° have studied the melting line in the HTSC. The melting line calculated in this method is related to the choice of the Lindemmm number eL. We shall present an argument elsewhere 2° that the melting line in s real material is in fact different from the freezing line due to strong hysteresis, and so these two approaches will tend to yield different results. In the remainder of this paper, the freezing transition in the HTSC materials will be examined. The choice of freezing over melting is simply that the liquid state is a homogeneous one, and the correlations are short ranged. This choice helps with the physical understanding, as well as making the calculation slightly simpler. To study this transition, we will use DFT. In this model the important quantity is the difference in free energy between the solid and liquid phases. Points of coexistence of the liquid and solid phases, marked by

Vol. 90, No. 8

FREEZING OF THE VORTEX LIQUID

the difference in free energy as being zero, are plotted out in the p-T plane. This is equivalent to the phase diagram discussed above, enabling the irreversible line to be plotted. The expressions for the solid and liquid densities are,

pl(r)

=

p,

p,(r)--_ ~"~pk e'k'r k with Ap(r) = p,(r) - p,.

481

where p is the density, ~(k) (~(k)) is the three dimensional Fourier transform of I/.(r) (c~(r)) and Vn(r) is the interaction potential for the n 'h layer. The correlation function is therefore governed by the choice of the interaction potential. The potential chosen is that due to Feigel'man at a/. z4 This potential contadns a full three dimension interaction between the layers, and an in-plane interaction of the form of that for conventional type-II superconductors. In Fourier space, taking k, to vary like rid, with n being the layer index and d being the layer separation, the potential can be written as

The change in free energy between the solid and liquid phases is given by

/~AF = - rl_. dr Ap(,) + "[ dr p,(r)In l|#o(r)| [ t Pi J JV

J

-~- f drdr'

c/,(l

r - r' 1,p,) Ap(r)~p(r') (5)

-~-.o°,

where cr is the direct correlation function, in a liquid defined by _#

cm(n--

=

b~F~,~ 6p(n)6p(r

)"

The second term in equation (5) can be handled more easily by going over to Fourier space. This changes the integral over all space, to a sum over all reciprocal lattice vectors. One then has a choice on the number of order parameters to keep. The one order parameter theory was originally developed by Rarnakrishnan and Yussoutf, 21 this has been extended to many order parameters, for example see Hsymet. ~ There are many arguments over the number of order parameters that should be kept, and more details on this can be found in the references. In this simple theory, one order parameter will be used. Another approximation that is made is to use a Gaussian form for the crystal density. Once this approximation is used, the integral in the first term can be done analytically. If the Gaussian approximation is not used, then another method is available. This method uses a Fourier expansion of the density. While this method is slightly more accurate, in that the solid density is not considered to be homogeneous , it is complicated by the fact that every Fourier component becomes a variational parameter. This makes the calculation much more intensive, for what is generally very little gain. The change in free energy ahto requires knowledge of the two particle direct correlation function. This function is related to the interaction potential of the system. It can be numerically calculated using the hypernetted chain approximation. This requires the iteration of the following set of equations, ~

where P is a dimensioule~ strength parameter and b~ = 1/k~T. The Fourier transform of the potential is then substituted into the HNC equations to obtain the two particle correlation function. Then the expression for the change in free energy is minimised with respect to any free parameters. The input parameters are then altered until a set of values are found that give A F ----0, and one coexistence point is found. Either the applied field or temperature are changed, and the whole process is repeated until the phase diagram is plotted. There are other ways to formulate DFT, to try to improve the accuracy. Details of some of these other methods, such as the weighted density approximation zs (WDA) and the effective liquid approximation ze (ELA) can be found in a review by Lutsko and BansYt The calculation was carried out initially for a 2D system. The result for the this system is a freezing temperature, T~2~,that remains constant for all applied magnetic field. The calculation was then also performed for the ful] 3D case for the HgBa, CuO4+. compound, and the result is shown in figure I. As the applied magnetic field is increased, the freezing temperature decreases, and eventually approaches the value for a 2D system. This would tend to show that as the field is increased, the interlayer interaction

I

0

c~(r) - e-~v-(r) + exp (y.(r)) - y.(r) ~(k) = I

-

~(k) p~(k)'

-

I

. . . . 10

,

. . . .

t 20

. . . . . . . . .

, 30

. . . . . . . . .

i

. . . . . . . .

40

FIG. 1. Plot of freezing temperatures for Hg (1201)

5O

FREEZING OF THE VORTEX LIQUID

482

starts to decrease. The vortices tend to become pancake like, moving in a layer of thickness d. However, the results for mercury, while being of the correct form, lle quite far away from the experimental values. The melting line has recently been measured l°,~s for the mercury compounds, including Hg (1201). These results show the melting line to be of the same form, with the temperature starting to approach a constant as the field is increased, but the actual melting temperature is much higher. It should be noted that these two resuits are not in exact agreement, due to the samples not being well characterised. The reason for this discrepancy between experiment and theory is to do with the DFT method. If a similar calculation is performed for the BSCCO material, 2s then the results compare much more favourably with experiment. Why is this? The DFT does not take any pinning in the system into account. It would appear that this is a good approx-

Vol. 90, No. 8

imation for the BSCCO system, which would point to weak pinning in these materials. It is possible that in the real mercury system, pinning plays a much larger role. The pinning would act so as to restrict the movement of the flux lines, which would lead to a higher freezing temperature. This is because more energy would be needed to knock the flux lines out from their pinning sites. While this explanation is very simple, it does help explain the discrepancy in the results. There are probably other effects occurring in these materials, hut pinning is likely to be the most important in this case. In summary, we have discussed the liquid-solid transition in the vortex phase of an extreme type-II superconductor. DFT has been used to obtain a plot of the irreversible line for the Hg (1201) compound. It is hoped that if the effect of the pinning sites can somehow be included in the density functional method, then the calculations would give a better agreement with experiment.

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