Optically detected flux lattice melting above the irreversibility line in Bi2Sr2CaCu2O8

PhysicaC 202 (1992) 219-233 North-Holland

Optically detected flux lattice melting above the irreversibility line in Bi2Sr2CaCu208 C.Testelin and R.S. Markiewicz Physics Department and Barnett Institute, Northeastern University, Boston, MA 02115, USA Received 21 July 1992

We calculate the infrared reflectivity associated with the flux lattice in a superconductor. We find that a recently observed reflectivity step can be caused by a divergence of the Campbell length (vanishing of the critical current). The step cannot be explained in terms of thermal activation of vortices, or helicon wave formation, or inertial effects associated with a finite vortex mass.

1. Introduction: gap or flux-line transition? There is still considerable uncertainty about the magnitude of the superconducting gap in the new high-T, cuprate superconductors, with various experiments measuring values in the range 2A( 0) / keTcz 3-8. Recent experiments seem to be converging on the larger value, including photoemission [ 11, tunneling [ 21, and optical measurements on YBazCuj0,_6 [ 31. On the other hand, optical measurements of BilSrzCaCuzOs (Bi-22 12) [ 41 find a sharp drop in reflectivity at a frequency which would correspond to a small gap value, U(O)/k,T,-3.5. However, measurements of the magnetic field dependence of this reflectivity anomaly [ 4,5] reveal that it does not behave like a conventional gap - that is, the transition occurs in a field B* which is much smaller than the critical field B,,, and corresponds more closely to the crossover from flux creep to flux flow. It is the purpose of the present paper to point out that since the low-frequency reflectivity measures the optical conductivity, it should also be sensitive to any kind of “depinning” transition of the vortices which leads to dissipation - i.e., to the “irreversibility line” (for a review see ref. [ 6 ] ) . We present a quantitative analysis of the reflectivity data, extracting values of the pinning energy in good agreement with direct transport measurements, and show that the in-field

optical transition occurs at a field, B*, which is considerably less than Bc2,but which lies higher than the irreversibility line, Bti. Moreover we find that this line can be theoretically modeled as a flux-lattice melting transition in the vortex system. This paper is organized as follows. The calculation of the reflectivity is summarized in section 2, while sections 3 and 4 discuss the field and temperature dependences of parameters entering into the calculation (pinning energy, conductivity). In particular, we suggest that there is an effective pinning potential proportional to the vortex lattice shear modulus, which vanishes in the fluid phase. In section 5, the reflectivity is calculated and compared to experiment, and it is shown that only the effective potential can reproduce the observed step in reflectivity. The derived flux lattice parameters are shown in section ‘6 to be comparable to those found in transport measurements. A number of features left out of our calculations could in principle also produce a reflectivity step. Additional calculations are presented in the following sections to rule out these possibilities. Thus, section 7 includes a finite vortex mass, section 8 helicon waves, and section 9 thermal activation of vortices. None of these can generate the observed step. In section 10, some consequences of the present results are discussed; in particular, the relation between flux lattice melting and the divergence of the Campbell length, and a possible interpretation of the

0921-4534/92/$05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved.

220

C. Testelin, R.S. Markiewicz /Optically detected melting

irreversibility line. Conclusions are given in section 11.

locity c, with e. the dielectric constant of vacuum. One can deduce the reflectance from the complex penetration depth and the surface impedance: R_Zf-l2 -lz’+l I ’

2. Reflectivity and pinning In a type-II superconductor, pinning has a much stronger influence on DC conductivity than on highfrequency conductivity, since vortices need not be depinned to contribute to high-frequency dissipation. An electromagnetic field induces an alternating current and a vibration of pinned flux tubes, in the mixed state. This effect is used in radio-frequency experiments, and for a microwave frequency current, Gittleman and Rosenblum [ 7 ] described the motion of vortices and the power dissipation, in a flux flow model including pinning. Recently, Coffey and Clem [ 81 have presented a new theoretical calculation of the radio-frequency surface impedance and penetration depth, over a large range of frequencies o, magnetic inductions B and temperatures T. In this paper, we interpret reflectivity experiments [4] in the far-infrared region (w= 1012-3x lOI Hz) in terms of this flux motion, following the derivation of Coffey and Clem [ 8 1. In order to calculate the reflectivity on the superconducting surface, we include the effects of the dielectric constant and the displacement current. Effects due to finite vortex mass and to helicon waves are also discussed. We consider a planar geometry: the superconducting surface is perpendicular to the x-axis and occupies the half-space x> 0. A static magnetic filed, B, parallel to the x-axis, establishes the vortex state. A radio-frequency or far-infrared radiation reaches the superconducting surface with the value b= b,,e’“‘z at x=0. This radiation generates an AC current and a Lorentz force acting on the vortices. In the superconductor, the vortex motion and the AC field are described by the complex penetration depth g a function of B, w and T. The surface impedance 2 and the dielectric constant e = cOe’are related to this penetration depth:

(la) assuming a local permeability equal to A, light ve-

(lb)

where z

1

z’=E=J;;*

(lc)

In the mixed state (B,, c B-c Bc2;B,, and B,, are the lower and the upper critical fields), the pinning of a vortex can be simply modeled by a periodic potential:

(2) where U is the pinning energy of one vortex, L is the characteristic length of the vortex along the c-axis, a0 is the lattice parameter, and u is the vortex displacement. We have chosen L=& the coherence length along the c-axis, as suggested in ref. [ 9 1. In highly anisotropic superconductors, the flux line lattice is quasi-two-dimensional at sufficiently high fields [ 10,111 (u,, is smaller than the interlayer Josephson length sm with s the interlayer distance and M/m the mass anisotropy; for Bi-2212, this condition leads to Bb 0.3 T). In this regime, other choices for L are possible, such as the layer thickness or half the unit cell length [ 111. For our analysis, the particular definition chosen is not important. Since the melting temperature is far from the critical temperature, the coherence length is only weakly temperature dependent. Therefore, any change in the choice of L will only lead to the renormalization of U(B, T) but the B and T dependence of the pinning energy will be the same. This energy leads to a force F on each vortex. For a current below J,, the vortex motion is weak and F may be written: (3) go = hc/2e is the flux quantum, a0 = m is the periodicity of the pinning potential. Therefore, the vortex equation of motion is (per unit length) mri’+qu+ug=JA@o,

(4)

221

C. Testelin, R.S. Markiewicz /Optically detected melting

where q= u,,&,B,~is the flow viscosity per unit length and a,, is the normal AC conductivity. Coffey and Clem [ 8 ] have derived fi for any static field B.Using their derivation, we have included the displacement current and the dielectric constant. For a magnetic field along the x-axis, that yields nZ=

n2-i&2/2 1+ 2iP/s; - #ll&o2~2 -

(5)

Here 1 is the London penetration length, which we take as

Other lengths are defined:

&= (6f2-i(A;2

and the coherence length along the c-axis, we will take A(O)=3000 A [ 141 and t;=&(O)/,/-, with C(O)-0.5 A [ 151. We will assume that the vortex mass is negligible at the frequency considered. This assumption will be discussed in section 6. In ref. [4], a sharp decrease in the reflectivity is observed when the magnetic field increases at faed temperature. We have attributed the decrease of R to a rapidly vanishing pinning potential (the bottom of the feature corresponding to U= 0). We used this condition to determine the normal conductivity law and defined an instability line T,(B) , associated to the condition U( B, T) = 0. The experimental results of ref. [ 4 ] were obtained on a Bi-22 12 film with the static magnetic field perpendicular to the surface and along the crystallographic c-axis. We will choose physical parameters appropriate for this compound.

-6;2)/2)-1/2, 3. The pinning energy

AC is the Campbell penetration

depth [ 121. The

equations ( 1) yield -P=

l/e/l&2.

U(B, (6)

Finally, the complex penetration depth is L(

1;;$/2*:Jf2

We find that the precise form of the temperature dependence of the pinning potential U plays an essential role in determining the sharpness of the reflectivity step. In many previous studies, it was assumed that the pinning potential could be fitted to the form

(7)

and the dielectric constant is e= - E~(w~/c)~. In order to calculate the reflectance of the superconducting surface, it is necessary to estimate the field, frequency, and temperature dependences of various parameters entering into the above equations. The pinning energy, U( B, T) will be discussed in section 3, and the normal state conductivity in section 4, before the calculations are applied to the experimental data [ 41 on Bi-2212, section 5. For B,, ( T), we assume the behavior described by Werthamer et al. [ 13 1, assuming a value Bc2(0) = 50 T along the c-axis. For the London penetration length

T)=U,y

,

(8)

where t= T/T, with T, the critical temperature ( T,= 87 K), and n and q are two exponents. Typically, n is found to be approximately 1.5, and q- 1 [ 16 1. With this form, U typically becomes of order k,T, slightly above the irreversibility line, so that transport measurements become insensitive to the precise value of U. However, in fitting the reflectivity we find that a potential of the form of eq. (8) varies too slowly to produce a sharp structure in the reflectivity. Thus, we take up an earlier suggestion [ 16,17 ] that most of the vortices are held in place not by direct pinning, but by the presence of the remainder of the vortex lattice. In this case, there will be an effective pinning potential Uaac66, where c66is the flux lattice shear modulus. In contrast to eq. (8), this pinning would zxutish within a fluid phase, for which

222

C. Testelin, R.S. Markiewicz /Optically detected melting

cs6= 0. If there is a well-defined melting transition at a temperature T,(B) < T,, then one might postulate U&B,

T)=U,,(‘;;*)“,

where t* = T/T,,,(B) , and the exponents n and q could take on different values from those of eq. (8 ). This effective potential is further discussed in appendix A. U(B, T) depends on three parameters: U,, n and q. For the magnetic field exponent, we chose q= 0.3; in this magnetic field range, this value of q is comparable to those obtained in bismuth compounds by transport experiments: q=O.2 and 0.46 [ 181, 0.3 [ 191, 0.25 [20], 0.17 and 0.59 [21]. The choice of n is discussed in appendix B. The parameter U, will be deduced from the reflectivity analysis.

This value for r0 may be compared with the value found in ref. [25], r,,rrO.5, for Bi-2212. Using the above parameters to calculate the reflectivity produces a field dependence of R at fields above the step, which is not observed in the experiments [ 41, see figs. 1 and 2. The reasons for this discrepancy are not clear. However, the variation amounts to only N l/2% in R, and may be near the limits of the measurement accuracy. Moreover, the theoretical reflectivity curves find that the size of the step varies with T, in the range m~O.04 f 0.009. This variation of AR with the temperature suggests that our model for the normal state reflectivity may be oversimplified. This may be due to neglect of the “mid-infrared feature” [ 23,261.

5. Reflectivity

4. Normal conductivity Thomas et al. [ 221 and Reedyk et al. [ 231 have studied the normal state optical conductivity of highT, superconductors and have observed a modified Drude law:

G(W, T)=

&&.

(10)

In the frequency range considered (ho< 25 meV- 200 cm-‘), it has been shown that r is a function of w and T [24],

(11) The reflectivity data [4] can be directly used to determine a first relation between the Drude parameter w, and r,, if it is assumed that U vanishes at fields above the melting line (section 3). Thus, in ref. [ 41, the variation of R across the step is about 0.04, for different temperatures and fiw= 10.4 meV. Assuming U=O when R=0.96, and knowing the normal conductivity at 120 K (~~~3.6 lo3 SZ-’ cm- l ), we obtain values for w, and r, for each reflectivity curve and each temperature. We have chosen their average: 0~~4.8 r, =0.40

lo3 cm-’ , .

Figure 1(a) shows our attempts to describe the reflectivity step assuming the pinning potential to be of the (non-critical) form, eq. (8), by varying CL-, with fixed n= 1.5, q= 1. We find that while the the-. ory describes a gradual crossover in the vicinity of the observed step, the only way to reproduce a sharp step is to choose value for n which are unphysically large (n sz4), and which have the effect of forcing U-0 for T> T,,,(H). By contrast, the “critical” pinning, eq. (9 ), provides a natural explanation of the step, fig. 1(b). In order to calculate the reflectivity, we have chosen n= 1.52 (appendix B) nd qGO.3, and varied U, to optimize the fit. Figure 1(b) compares the experimental curve (for T= 40 K and fiw = 10.4 meV ) with the reflectivity calculated for different values of U,. This analysis yields the value U,,= 2000 K. With pinning energy given by eq. (9 ), the reflectivity as a function of the magnetic field for different temperatures has been derived. For a fmed value of U, ( = 2000 K), at the frequency fiw = 10.4 meV, these curves are presented in fig. 2, with the experimental results of ref. [ 41. A good agreement is obtained for the investigated range of temperatures (T=25-60 K) and magnetic fields (8=0-20 T), which confvms that our model is able to reproduce the optical reflectivity of Bi-22 12 in the far-infrared. U, has been chosen independent of T, in order to simplify our study, but by introducing a temperature dependence

223

C. Testelin, R.S. Markiewicz /Optically detected melting

(4

-

I”“l”“I”“-

--------’

0.9

p:

0.a

0.7 0

15

5 B

20

;;,

Fig. 2. Temperature dependence of the optical reflectivity of a Bi22 12 film vs. magnetic field ( fiw= 10.4 meV) . The dashed lines are the experimental curves of ref. [ 41 and the solid lines correspond to our theoretical model (eq. ( 9)) C&= 2000 K) . The temperature increases from the upper curve to the lower one ( T= 25, 30,35,40,45 and 60 K). The origin is shifted for each temperature, the reflectivity being equal to 1 at B=O T. 0.98

-

0.96

-

!x

0.94

:

2

4

6

a

10

B (‘0 Fig. 1. Optical reflectivity of a Bi-22 12 film vs. magnetic field (Ro= 10.4 meV), at T=40 K. The dashed line is the experimental curve extracted from ref. [ 41. The solid lines are calculated using(a)thepinningenergylaw (8)withU0=500(lowercurve), 1000 and 2000 K (upper curve), or (b) the pinning energy law (9)witbU,,=750(lowercurve),2OOOand35OOK(uppercurve). The dot-dashed lines are calculated including thermally activated hopping (section 9), using (a) eq. (8) witb &=2000 (lower curve) and 4000 K, (b) eq. (9) with UO=lOOO (lower curve), 2000 and 4000 K (upper curve).

U,(T) it is possible to improve the agreement between experimental results and the theoretical reflectivity. The derived melting phase diagram is presented in

0

20

40 T (K)

6fO

80

Fig. 3. Irreversibility and melting tempemtures vs. magnetic field. The open squares are calculated values of & The open circles represent the melting curve T, derived from the data of ref. [ 41, while the black squares and circles are the “irmversibility curves” defined by e-““= 10m9or 10-r’, respectively, for the critical pinning law of eq. (9) (U0=2000 K and n=1.52). These are compared to the experimentally observed irreversibility lines of ref. [27] (open triangles), ref. [ 191 (+s corresponding to e-“I == 10m9), and ref. [ 291 ( x s from high-Q mechanical oscillator measurements).

224

C. Testelin. R.S. Markiewicz /Optically detected melting

fig. 3,and compared to both the expected form of Bc2and several experimentally measured irreversibility lines, obtained when the critical current J,+O. In particular, the data of ref. [27] were made on a thin film of Bi-22 12, which was synthesized by the same technique as the sample of ref. [ 41, and exhibits roughly the same resistivity at T,.We may expect the same behavior of U(B, T) for these two films. Despite this fact, the melting curve is seen to lie clearly above the irreversibility line, and hence to be associated with a different physical phenomenon. The irreversibility line will be further discussed in sections 6 and 10.2. Therefore, the step in the reflectivity can only be understood by postulating a pinning energy which vanishes at a well defined T,(B).This is an important consequence of this study, and in sections 7-9, we will investigate the possibility of an alternative interpretation for the step. We note that the power law form of eq. (9), U- (1 -r*)’ is suggestive of a second-order phase transition. However, a hysteresis of the reflectivity step was observed in ref. [ 41, suggestive of a first-order transition. The pinning law of eq. (8 ) has already been considered with some success in YBaCuO compounds [ 16,28 1. However, one must remember that in these superconductors, the irreversibility lien and the critical field Bc2(T) are close, so that it is difficult to distinguish eqs. (8) and (9). (In our range of magnetic field, t u t*. ) The situation is very different for Bi-22 12 because the irreversibility line is far from Bc2(T).Thanks to this property, the pinning energies given by eqs. ( 8 ) and ( 9 ) are easily distinguishible, and eq. (9) is clearly favored.

6. Comparison with transport measurements and irreversibility line The usual way to study the pinning energy is by transport measurement. In the flux creep regime, the resistivity follows an Arrhenius law p=poe- “IT from which U( B, T) canbe deduced [ 18,19 1. In order to compare our pinning energy law with those obtained by this technique, we plot in figs. 4 and 5 the pinning potential U(B, T) versus T and the ratio U(B, T)/ Tversus 1/ T for several values of the magnetic field. Batlogg et al. [ 281 and Palstra et al. [ 191 have de-

1000 8 800 F: g 600 5

400 200 0 0

20

r

40

60

80

T 6) Fig. 4. Pinning energy vs. temperature for several magnetic fields. These curves can be compared with results of ref. [ 28 1. 400

300

’ q 3T n 9T

05T 0 14 T

t

0

l/T Fig. 5. Ratio U(B, T) / T vs. 1/T. A linear behavior is observed asinrefs. [18] and [19].

duced, from resistance measurements as a function of temperature and magnetic field, the same kind of law for the pinning energy, which confiis the choice of the U(B, T) expression. Furthermore, we have calculated the slope U, (B) of the curve U(B, T)/T verws 1/T and compare our results and those of other authors in fig. 6. Once again, the parameters we have chosen are in the usual experimental range of results. It may further be asked whether our derived pinning energies are consistent with the experimentally measured irriversibility lines. These curves are associated with extremely low resistivities. For instance, in ref. [ 271, the sensitivity is smaller than 1Om6p,, with p,, the normal resistivity. In ref. [ 19 1,

225

C. Testelin,R.S. Markiewicz/Opticallydetectedmelting

10000

\\__-i

0.96

100

I

0.1

,,..I

. . . ..I

1

10

I

100

b (T)

Fig. 6. Pinning potential U,(B) vs. magnetic field in Bi-2212

compounds. The solid squares correspond to our calculation. The open circles are the results from ref. [ 191; the triangles and the diamonds are the results from ref. [ 181.

0.94

\

2

4

B

(6T)

B

(BT)

6

10

6

10

Palstra et al. have found p. u 1000 p,,. These conditions correspond to e- “IT= 10S9. While our model does not contain the physics of the irreversibility line, nevertheless the line corresponds to the appropriate resistivity range found by extrapolating our U-values to lower temperatures. For example, in fig. 3 we show curves for which e-“lT= 10m9or 10-i’; these are in good agreement with the experimental irreversibility lines of refs. [ 19,27,29].

7. Influence of the vortex mass In the microwave ( N 1O6Hz ) and radio-frequency ( w lo9 Hz) range, the vortex mass and the inertial term mm2 can be neglected, but in the far-infrared range ( M 10” Hz), does this assumption still hold? The vortex mass per unit length is [ 301:

2

4

Fig. 7. Effect of finite vortex mass on optical reflectivity of Bi22 12. The experimental data (dashed line) are the same as in fig. 1, with the calculated curves for m=O (solid line) and m=3x108m. (dotted line). (a) U(E, T) follows eq. (8), C&2000 K, (b) U(B, T) followseq. (9), U,,=2OOOK.

(12) where Ef is the Fermi energy, and m, is the electronic mass. In the cuprates, the Fermi energy has been found equal to -0.3-0.4 eV in YBa2Cu30, and La1_$Sr,CuO, [ 3 1 ] and to 0.13 eV in Bi-2223 [ 32 1. Using the lowest vahte and taking I = 3000 A, one has m N l-2 x lo%, per unit length. In fig. 7 (a,b), the reflectivity curve calculated for a vortex mass m= 3 x 108m, is compared to the

equivalent calculation with a vanishing mass at T= 40 K, for the two pinning energy laws (8) and (9). No significant changes are observed between the two curves. Therefore, in spite of the high frequency, the vortex mass remains negligible and cannot explain the step in the reflectivity in the far-infrared range. For larger values of the vortex mass, we find anomalous non-monotonic variations in R, but have not been able to reproduce a step-like feature.

226

C. Testelin,R.S. Markiewicz /Optically detectedmelting

8. Helicon waves

(15) In a magnetic field, the conductivity, eq. ( lo), should be corrected for the effects of magnetoresistante and Hall effect. Magnetoresistance is known to be negligible for the field range of interest, but the Hall effect could in principle lead to unusual effects in the reflectivity. In their study of the vortex motion in superconductors, de Gennes and Ma&icon [ 331 calculated the vibration modes of flux lines, named helicon waves, and showed that the penetration length may differ for different circular polarization of the incident radiation. For a thin sample, the superconducting medium could become transparent for one sign of circular polarization and not for the polarization of opposite sign. However, calculation of helicon waves in a superconductor is subject to the same problem that arises in the study of the usual Hall effect: the Hall coefficient is experimentally observed to change sign upon entering the superconducting state. While the origin of this phenomenon is by no means agreed upon, we will follow the analysis of Hagen et al. [ 341, to determine whether helicon waves could somehow give rise to the observed reflectivity step. In superfluid 4He, the relative motion between the vortices and the superfluid produces a Magnus force whose analog term in superconductors has been developed by Nozibres and Vinen [ 351: F=n,e&(YS -vi) AB,

(13)

n, is the superfluid electron density, v,= J,/n,e is the superfluid velocity, y is the vortex velocity, and LB/B is a unit vector. Recently, Hagen et al. [34] have interpreted the Hall effect in superconductors by including drag forces in the Nozitres-Vinen model. They proposed a generalized drag force of the form:

f= -tp,-?@Av,,

where n”, is associated with a circular polarization b,,= f ib,, and

where S,= (2B@0/~~,w)‘/2,

q,=q’-nn,e&,,

S’,

=

a$

=aL+ia&, and =wG/Po~, ah (pk) is the longitudinal conductivity (resistivity), and &,(p&,) is the perpendicular conductivity (resistivity ) . The index k= n (resp. s) refers to the normal (resp. superconducting) state of the medium. According to Hagen et al. [ 341: 2/pQwo”+

This ratio can be estimated from the Hall effect measurement by Iye et al. [ 381. As pJ&/p& *: 1, a&= u,(w, T) and ~$==P$/(P&)~=~J,(w, T)P$JPL. We have calculated the reflectivity for each circularly polarized radiation, in the presence of this drag force, for the parameter values deduced from the Hall effect measurements [ 381. Figures 8(a,b) give the average of these two reflectivity coeffkients, which is the reflectance of linearly polarized radiation. For the considered parameter values, inclusion of the hall effect leads to only minor changes in the reflectivity, 1, -I_ . For comparison, the reflectivity is also shown on the assumption of an effective viscosity tte 30 (or 100) times larger than expected - even in this case, the modifications in R are small. Thus, the presence of vibration modes cannot explain the behavior of the reflectivity observed in the Bi-22 12 film of ref. [ 41, which enforces the result of section 5.

(14)

which had been introduced by Hall and Vinen [ 361 and Ambegaokar et al. [ 371 to interpret the hydrodynamics of superfluid 4He. In order to consider the effect of the vortex vibration modes, we have derived the complex penetration length when the Magnus force replaces the Lorentz force and when a perpendiculandrag force, proportional to rf, appears. The derivation of fiis given in appendix 3. The final result is (m = 0)

9. Thermal activation Thermal activation of vortex motion may also be taken into account by adding a Langevin force which induces Brownian motion [ 8 1. When the vortex mass is negligible, it is possible to derive an analytic expression for the complex conductivity:

C. Testelin, R.S. Markiewicz /Optically detected melting

tl

1.00

\

\

\

0.96

\

‘=~z&)z~(v)’

(19a)

1 P=ra(y)’

(19b)

with v = U/2kBT; and Z, is the modified Bessel function of the first kind of order p. The complex penetration depth is deduced from eq. (7) by replacing 6, by:

0.98 p:

G(V)-1

221

\_____

(20)

0.94 2

I, 4

I

I

I

I

I

B

(BT)

B

&I

I

I

I

I 6

I

I

I

I

10

1.00

0.96 lx

0.96

0.94 2

4

6

10

Fig. 8. Effect of helicon wave absorption on optical reflectivity of Bi 22 12. The experimental data (dashed line) are the same as in fig. 1, with the calculated cm-ves for qO=O (solid line), qO=qb (dottedline),and~O=lOO~~ (fw (a))or30$$ (fig.(b)) (dotdashed line), where ~5 is equal to the experimental value deduccdfmmref. [34]. (a) U(B, T) followseq. (8), U0=2000K; (b) U(E, T) follows eq. (9), U0=2000 K.

&-

1 1+ (or)2 pf fl+(wr)*+i(l-p)wr’

where pf=p,,B/Bcz is the flux-flow resistivity,

(18)

As in section 5, we have calculated the reflectivity curves for the two pinning energy laws (8) and (9). The temperature exponent is n ~0.84 in eq. (9). We repeated the calculation for several values of U, (cf figs. 1(a,b) ). Note that while inclusion of thermal activation processes changes the parameters of the tit (e.g., the value of n), the resulting reflectivity curves are similar to those found in figs. 1. As previously, the “critical” pinning law (eq. (9) ) provided a good agreement between the experimental and theoretical curves, fig. 1 (b), with U,= 2000 K. For a pinning energy vanishing at T,, it is not possible to reproduce the sharp transition in the reflectivity curve (cf. fig. 1(a) ). In conclusion, we have considered the effects of the viscosity force, the vortex mass, the helicon wave and the thermal activation, and have found that none of these effects can reproduce the sharp step in the reflectivity observed experimentally [ 41. Hence, we conclude that the reflectivity behavior in the superconductor is described by a pinning energy vanishing at T,,,c T,.

10. Discussion IO.1. Melting and the Campbell length

From the time that the irreversibility line was first recognized, there has been considerable divergence in its interpretation, with some interpreting it as a sign of thermally activated depinning, others as a melting transition, either from a flux lattice or a vortex glass. More recently, a second transition has been identified in YBCO, associated with a knee [ 39 ] in

228

C. Testelin, R.S. Markiewicz /Optically detected melting

the resistivity versus temperature curve and taken as evidence for a crossover from flux creep to flux flow. A number of recent experiments [ 40-421 have found evidence for a melting transition which lies above the irreversibility line, but which may coincide with the creep-flow crossover [ 4 11. Raffy et al. [ 27 ] have also detected a second transition in Bi-22 12, and have shown that it approximately coincides with the T,,, line found optically, although they offer a different interpretation of the transition. The present analysis strongly supports this interpretation of the upper transition. As pointed out earlier [ 221, the thermally-activated depinning model could provide evidence for either the flux creep model or the melting model, depending on the nature of the temperature-dependence of the effective pinning potential, U. In the absence of a melting transition, U can vanish only at T,(B) , when the superconductivity itself vanishes - this form explicitly arises in any model calculations of U, since U-B: or B,,. However, if there is a well-defined melting transition, direct pinning of vortices is weak, and most are held in place by the vortex lattice’s resistance to shear flow. In this case, there will be an effective pinning potential U,- c66 (for details, see appendix 1) . Above the melting point, c66 and U,, both vanish. For transport measurements, the resistivity is p N exp ( - U/ kBT), and there is very little difference between UE k,T and U= 0, although the analysis of Palstra et al. [ 191 clearly indicated U-+0 at T< T,. On the other hand, the optical reflectivity is very sensitive to the difference between small and vanishing U, as seen in fig. 1. Only the vanishing of U can explain the observed step in the reflectivity. We have fit the step by assuming that U, or c66vanishes as a power law in t* - i.e., that the melting transition is second order. However, we cannot rule out a weakly first-order transition, and indeed the experimental reflectivity data show hysteresis in the transition. The crossover from creep to flow follows naturally within the present model. In conventional superconductors, such a crossover is usually observed as a function of applied current. As the current is increased, the Lore& force overcomes the pinning force, driving the transition to vortex flow. In the present case, the transition occurs at low applied currents, as a function of temperature, suggesting that

resistance to flow has completely vanished above T,, as would happen if U,eO. It can be shown that the vanishing of U corresponds to the divergence of the Campbell length of the vortex lattice [ 12 ] :

The significance of the Campbell length is that it is proportional to 1/A, and hence the divergence of the Campbell length is equivalent to the vanishing of the critical current, Hence, divergence of I, would seem to characterize the creep-flow crossover: in the creep regime, there is still pinning, but there is a finite resistivity due to thermal activation; in the flow regime pinning is entirely negligible. It is useful to recall the situation in conventional superconductors. ( 1) A creep-flow crossover is often observed, but usually as a function of applied current: there is a critical current above which the Lorentz force overcomes the pinning force, and the vortices can flow freely. The remarkable feature about high-Tc superconductors is that this critical current vanishes above a given temperature, so that arbitrarily small Lorentz forces lead to flux flow (absence of pinning force ) . (2 ) Sharp changes in microwave dissipation in superconductors have been related to the divergence of the Campbell length [ 43 1, but this has been assumed to occur at Bc2. (For conventional, low T, superconductors, the difference between the flux lattice melting temperature and T, is probably very small.) 10.2. The irreversibility line as a depinning transition? What about the irreversibility line itself! The present model does not provide any direct evidence as to the nature of the irreversibility line; indeed, the model would be consistent with the irreversibility line simply corresponding to the point at which the creep resistivity becomes immeasurably small, in the absence of any sharp transition (see section 6 ) . However, a number of recent experiments [ 441 have provided evidence that there is a sharp transition at Bin, and our analysis can provide some indirect evidence as to the nature of the transition.

C. Testelin, R.S. Markiewicz /Optically detected melting

First, since it lies below the melting transition, this transition must be between two solid phases of the vortex lattice. This immediately rules out such interpretations as a transition between a fluid phase and a lattice or glassy phase. The only alternative possibility is that the melting transition is two phase e.g., solid-rhexatic+isotropic fluid. Such a double transition would be consistent with the present analysis, in that ( 1) C66 remains finite in the hexatic phase [ 451, vanishing only at the hexatic-isotopic crossover; and (2 ) cs6 has a sharp increase at the lattice-hexatic crossover, which would cause a sudden increase in the apparent pinning strength, suggestive of a phase transition at Bin. However, it seems likely that pinning in these materials would smear out the double transition. A careful consideration of the role of pinning in these materials suggests an alternative interpretation of the irreversibility line, as a depinning transition. Larkin and Ovchinnikov [ 46 ] showed that pinning is a relevant parameter in the flux lattice - i.e., that pinning introduces a finite correlation length in the flux lattice, thereby precluding the existence of diverging correlation lengths, characteristic of a true phase transition. However, the likely candidates for point pinning in the cuprates, such as oxygen vacancies, would individually produce a pinning energy smaller than thermal energies. Any net pinning effect near Hi, must be a form of collective pinning - with a net pinning produced by (statistically) summing up the effects of many, individually weak pins. Such collective pinning would lead to a phase transition of a very different sort from the present model of melting. In particular, the flux lattice correlation length is fixed by balancing the effects of collective pinning against those of flux lattice elasticity. As C,, decreases, pinning should cause the correlation length to decrease and j, to increase, leading ultimately to a “peak effect”. Non-observation of such a peak effect in these cuprates suggests that, as assumed in the present model, there is no collective pinning due to point pins, but only pinning by extended two-dimensional pins (grain boundaries or, in YBCO, twin boundaries). Why has the point pinning completely vanished? It seems likely that, when the energy per pin is extremely small compared to k,T, pinning is negligible

229

and it no longer makes sense to add the pins collectively. There does not, however, seem to be any way of smoothly crossing over between this regime and the regime in which the energy per pin is smaller than k,T, but collective pinning makes sense. For instance, in a “Gedanken” experiment, the energy per pin and the number of pins can be simultaneously varied, in such a way that the collective pinning energy remains constant, even though the energy per pin+O. In such a system, a depinning transition could arise only through a (presumably first-order) phase transition. Indeed, recent Monte Carlo calculations [ 471 of flux lattices at finite temperatures have revealed a double transition, wherein the lower transition has many characteristics of a first-order depinning transition. That is, the vortex coherence length changes discontinuously at the transition, and is smaller in the low-temperature phase, suggestive of a decreased role of pinning above the phase transition. There is less clear evidence for the nature of the upper transition, but it appears to be consistent with the fluxlattice melting transition described herein.

11. Conclusions In this paper, we have demonstrated that ( 1) the optical reflectivity of Bi-22 12 in a field shows a step at a line of pinning collapse ( U-+0), where there is a large change in the optical conductivity due to depinning of vortices, which may be associated with flux-lattice melting; (2) this line of collapse is distinct from the usual irreversibility line, but appears to correspond to the “knee” in the resistive transition, and with the crossover from creep to flow of the flux lines; (3) the pinning energies so derived are in good agreement with earlier determinations; (4) this pinning collapse line is the locus of fields at which the vortex pinning energy smoothly goes to zero, indicative of a second-order (or weakly firstorder) phase transition. However, the observed hysteresis [ 41 is suggestive of a first-order transition. Hence, at least for the field-dependent data, we have resolved the problem of the low values derived for B,,; the experiments measure B”, not Bc2. What about the problem of the low gap value derived from

230

C. Testelin, R.S. Markiewicz /Optically detected melting

zero-field measurements? In principle, a similar model should apply. Since the superconductors are nearly two-dimensional, the thermal energy required to nucleate a vortex pair is expected to be low, and indeed it has been suggested that the phase transition at B= 0 is Kosterlitz-Thouless-like [ 48 1, associated with the unbinding of thermal vortex pairs. In this case, a high-frequency measurement should detect dissipation associated with vibrations of thermal vortices at energies considerably below the superconducting gap.

Acknowledgements Wewould like to thank J. Clem, S. Sridhar and J.V. Jose for stimulating conversations and for providing preprints of their work, and T.K. Worthington for pointing out the importance of the Campbell length.

creep rate would be expected to diverge at the melting transition. In the critical state, there will always be a significant number of dislocations present in the vortex lattice. Indeed, any field gradient must be made up as an array of dislocations. Hence, the actual dislocation structure is likely to be highly history dependent, which suggests that it should be possible to study the melting transition as a function of dislocation density. In this interpretation, B* represents the flux-lattice melting transition. Below B*, there is ordinary flux creep, greatly enhanced by the strong temperature dependence of Uccc66.It is thus very interesting to note that B* agrees with the “knee” in the resistive transition: the point at which the resistivity crosses over form flux creep to flux flow.

Appendix B. Estimation of the critical exponent n For each curve of reflectivity, we have determined

Appendix A. Effective pinning potential in a flux lattice The relation U,ewcs6 can be derived if certain conditions are met by th flux lattice. Pinning must be dominated by a few strong pins, while any bulk point pinning centers (e.g., oxygen vacancies) must have a negligible effect. This indeed seems to be the case in the cuprates, particularly the more two-dimensional materials. Strong pinning is associated with grain boundaries or twin boundaries in YBCO. In Bi-2212 single crystals, there is evidence that, at higher temperatures pinning exists only at the crystal surfaces, and not in the bulk [ 491. In this case, the vortices between these strong pinning centers will form nearly perfect arrays. In the presence of an applied current, these arrays will not move (no dissipation), because the unpinned vortices would have to shear past the vortices which are strongly pinned. This model was introduced by Tinkham. Hence, there will be an effective pinning potential Uerrwc66,the shear modulus. When the lattice melts, cs6=0, and the unpinned vortices can freely flow around the pinned vortices. Below the melting transition, there will be dissipation due to creep, motion of dislocations in the vortex array. The

U(B, T) at the four points where the reflectivity

variation is equal to A&/4, A&/2, 3AR,/4 and AR,, with AR, the size of the step. The fourth point corresponds to U(B, T) =O and gives a point of the melting curve. Moreover, we have assumed that, on each curve, for a fixed temperature T= T,,, (B,), these four points are equidistant (B=B,-pAB( T), p=O, 1,2,3 ). At the temperature T= T,,, (B,,) and near B,,, one can write: 1-t*=

Tm(B)- Tm(Bo) T,(B) &(B-BO) pm(T).

(B-1)

If we consider that B4 is almost constant during the variation of R, for each curve and couple of papinning the rameters (T, B,), U( T, Bo-pAB(T))=u(T, p) can be rewritten: ~(T,P) =p”, u(T, 1)

F(P) = ~

(B.2)

even if m( T) is varying with the temperature. We

231

C. Testelin, R.S. Martiewicz /Optically detected melting

have determined F(p) for a negligible vortex mass. First, we derived F(p) without thermal activation (cf. fig. 9(a)) and obtained n= 1.52. Then, including the thermal activation as modeled by Clem and Coffey [8], we derived n=0.84 (cf. fig. 9(b)). In each of sections 5 and 9, we redetermined the value of the exponent n which led to the best fit. Thus, the value of the temperature exponent is not known exactly and will depend on the shape of the step in the reflectivity curve and on the assumptions. Nevertheless, whatever the exact value of the critical exponent n, the pinning energy law of the eq. (9) is able to reproduce the reflectivity curve, which the

1

/

t

more usual law of eq. (8) fails to interpret.

Appendix C. Complex penetratlon length in the presence of helicon waves Let b be the AC magnetic field, E the AC electric field and J the AC current density (J, and J, are the normal and superconducting current densities, respectively). All these vectors are along the plane yz. We will consider a two-fluid model:

J=J, +J, .

(C.1)

The superconducting medium being in the halfspace x>O, the AC vectors will follow a law e-“/*e’“’ with 1 the penetration length. The static magnetic field is B=B,. The Faraday and Ampere laws, relating b, E and J, yield E,, = i&b*

(C.2)

E, = -id&b,,

(C.3)

bz J,,= --ao2A&

(C.4)

Pot

b J == - --A+m2xb,, 0

1

2

3

P0l

4

P

.

(C.5)

J, and J, can be calculated independently using the relation J,=aLE-a&,xAE,

(C.6)

and the London equation [ 81 VAJ,=-

&

(B,-@oW

(C.7)

;

n is a local areal density of vortices, Bl is the local magnetic field, and 1 is the London penetration length. We set B,=B+b and @oti= B+b, where b+ -VA (BA u) is the magnetic field induced by the vortex motion [ 8 1. E and b being related by eqs. (C.2), (C.3) it is easy to derive the components of J,, as functions of b,, and b,. Moreover, the vortex equation of motion, Fig. 9. Normalized pinning energy F(p) vs p. The solid line is a least-squanzfit (a) F(p)=p*.s2 (withoutthermel activation), (b) F(p) =P’.~ (with thermal activation). We have considered the values of F(p) for T= 25,30,35,40 and 45 K.

&+v&+

(q’-n,e&,)gA&+~+r=JA+~

,

(C.8)

leads to a relation between the component of the vortex motion II and J. Using the eqs. (C.7) and

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C. Testelin,R.S. Markiewicz/Opticallydetectedmelting

( C.4) and (C. 5 ) , Jsyand .I, may be related to by and b.Z-

Finally, in comparing J, given by eqs. (C.4) and (C.5) and&+&derived from eqs. (C.6) and (C.7), one obtains a linear system of equations in b,, and 6, with non-zero solutions for: (C.9) For the definition of the parameters, see text (section 8). Then, the expression ( 15 ) of P is deduced by considering eq. ( 6 ) .

References [ 1 ] J.M. Imer, F. Patthey, B. Dardel, W.D. Schneider, Y. Baer, Y. Petroff and A. Zettl, Phys. Rev. Lett. 62 (1989) 336; R’.Manzke, T. Buslaps, R. Claessen and J. Fink, Europhys. Lett. 9 (1989) 477; C.G. Olson, R. Liu, A.B. Yang, D.W. Lynch, A.J. Arko, R.S. List, B.W. Veal, Y.C. Chang, P.Z. Jiang and A.P. Paul&as, Science 245 (1989) 731. [ 2 ] T. Walsh, J. Moreland, R.H. Ono and T.S. Kalkur, Phys. Rev.Lett.66 (1991) 516. [ 31 J. Grenstein, G.A. Thomas, A.J. Millis, S.L. Cooper, D.H. Rapkine, T. Timusk, L.F. Schneemeyer and J.V. Waszczak, Phys. Rev. B 42 (1990) 6342; Z. Schlesinger, L.D. Rotter, R.T. Collins, F. Holtzberg, C. Feild, U. Welp, G.W. Crabtree, J.Z. Liu, Y. Fang and K.G. Vandervoort, Physica C 185-189 (1991) 57. [4] L.C. Brunel, S.G. Lottie, G. Martinez, S. Labdi and H. RafTy, Phys. Rev. Lett. 66 ( 199 1) 1346. [ 5 ] G. Martinez, L.C. Brunel, S.G. Lottie, S. Labdi and H. Raffy, PhysicaC 185-189 (1991) 1753. [6] D.J. Bishop, P.L. Gammel, D.A. Huse and C.A. Murray, Science 255 (1992) 165. [ 71 J.I. Gittleman and B. Rosenblum, Phys. Rev. Lett. 16 (1966) 734.. [ 8 ] M.W. Coffey and J.R. CIem, Phys. Rev. Lett. 67 ( 199 1) 386; ibid., Phys. Rev. B 45 (1992) 9872; ibid., Phys. Rev. B 45 (1992) 10527; J.R. Clem and M.W. Coffey, Physica C 185-189 (1991) 1915; and unpublished. [9] Y. Yeshurun and A.P. Malozemoff, Phys. Rev. Lett. 60 (1988) 2202. [ 10 ] M.V. Feigel’man, V.B. Geshkenbein and A.I. Larkin, Physica C 167 (1990) 177. [ 111 V.M. Vinokur, P.H. Kes and A.E. Koshelev, Physica C 168 (1990) 29. [ 121 A.M. Campbell, J. Phys. C 2 (1969) 1492; ibid., 4 (1971) 3186.

[ 131 N.R. Werthamer, E. Helfand and P.C. Hohenberg, Phys. Rev. 147 (1966) 295. [14]S. Mitra, J.H. Cho, W.C. Lee, D.C. Johnston and V.G. Kogan, Phys. Rev. B 40 (1989) 2674. [ 151 T.T.M.Palstra,B. Batlogg,L.F. SchneemeyerandR.J.Cava, Phys. Rev. B 38 (1988) 5102. [ 161 M. Tit&ham, Phys. Rev. Lett. 61 (1988) 1658. [ 171 R.S. Markiewicz, Physica C 171 (1990) 479.. [ 181 N. Kobayashi, H. Iwasaki, H. Kawabe, K. Watanabe, H. Yamane, H. Kurosawa, H. Masumoto, T. Hirai and Y. Muto, Physica C 159 (1989) 295. [ 191 T.T.M. Palstra, B. Batlogg, L.F. Schneemeyer and J.V. Waszczak, Phys. Rev. Lett. 61 (1988) 1662. [20] A. Gupta, P. Esquinazi, H.F. Braun, H.-W. Neumtlller, G. Ries, W. Schimdt and W. Gerhfuser, Physica C 170 ( 1990) 95. [21] T. Fumaki, T. Kamura, A.A.A. Youssef, Y. Horie and S. Mase, Physica C 159 ( 1989) 422. [ 22 ] G.A. Thomas, J. Orenstein, D.H. Rapkine, M. Capizzi, A.J. Millis, R.N. Bhatt, L.F. Schneemeyer and J.V. Waszczak, Phys. Rev. I&t. 61 (1988) 1313. [23]M. Reedyk, D.A. Bonn, J.D. Garret, J.E. Greedan, C.V. Stager and T. Timusk, Phys. Rev. B 38 ( 1989) 1198 1. [24] R.S. Markiewicz, Phys. Rev. Lett. 62 (1989) 603; M.V. Klein, S.L. Cooper,F. Slakey, J.P. Rice, E.D. Bukowski and D.M. Ginsberg, in: “Strong Correlation and Superconductivity”, Springer Series in Solid State Sciences, 89 (eds. H. Fukuyama, S. Maekawa and A.P. Malozemoff (Springer, Berlin, 1989) p. 226; C.M. Varma, P.B. Littlewood, S. Schmitt-Rink, E. Abrahams and A.E. Ruckenstein, Phys. Rev. Lett. 63 (1989) 1996. [ 25 ] I. Terasaky, S. Takebayashi, I. Tsukuda, A. Maeda and K. Uchinokura, PhysicaC 185-189 (1991) 1017. [26] D.B. Romero, C.D. Porter, D.B. Tanner, L. Forro, D. Mar&us, L. Mihaly, G.L. Carr and G.P. Williams, Phys. Rev. Lett. 68 ( 1992) 1590. [ 271 H. RalTy, S. Labdi, 0. Laborde and P. Monceacu, Physica C 184 (1991) 159. [ 281 B. Batlogg, T.T.M. Palstra, L.F. Schneemeyer and J.V. Waszczak, in: “Strong Correlation and Superconductivity”, op. cit. (ref. [23]) p. 368. [29] P.L. Gammel, L.F. Schneemeyer, J.V. Waszczak and D.J. Bishop, Phys. Rev. Lett. 6 1 ( 1988) 1666. [30] H. Suhl, Phys. Rev. Lett. 14 (1965) 226. [ 3 1 ] L.P. Gor’kov and N.B. Kopnin, Sot. Phys. Usp. 3 1 ( 1988) 850. [32] R.A. Doyle, O.L. de Lange and V.V. Gridin, Phys. Rev. B 45 (1992) 12580. [ 331 P.G. de Genes and J. Matricon, Rev. Mod. Phys. 36 (1964) 45. [34] S.J. Hagen, C.J. Lobb, R.L. Greene, M.G. Forresterand J.H. Kang, Phys. Rev. B 41 (1990) 11630. [35] P. Nozieres and W.F. Vinen, Phil. Mag. 14 (1966) 667. [ 36) H.E. Hall and W.F. Vinen, Proc. Roy. Sot. London 238 (1956) 215. [37] V.Ambegaokar, B.I. Halperin,D.RNelsonandE.D. Siggia, Phys. Rev. B 21 ( 1980) 1806.

C. Testelin, R.S. Markiewicz /Optically detected melting [38] Y. Iye, S. Nakamura and Tamegai, Physica C 159 (1989) 616. [39] T.K. Worthington, F.H. Holtzberg and C.A. Feild, Cryogenics 30 (1990) 417. [40] D.E. Farrell, J.P. Rice and D.M. Ginsberg, Phys. Rev. Lett. 67 (1991) 1165. [41] T.K. Worthington, M.P.A. Fisher, J. Toner, T.H. Zabel, C.A. Feild and F. Holtzberg, Bull. Am. Phys. Sot. 37 ( ( 1992) 592; T.K. Worthington, M.P.A. Fisher, D.A. Huse, J. Toner, A.D. Marwick, T. Z&e-l, C.A. Feild and F. Holtzbeq, unpublished. [42] H. Safar, P.L. Gammel, D.A. Huse, D.J. Bishop, J.P. Rice and D.M. Ginsberg, unpublished.

233

[43] W.J.Tomasch,H.A. Blackstead,S.T.Ruggiero,P.J.McGinn, J.R. Clem, K. Shen, J.W. Weber and D. Boyne, Phys. Rev. B 37 (1988) 9864. [ 44 ] R.H. Koch, V. Foglietti, W.J. Gallagher, G. Koren, A. Gupta andM.P.A. Fisher, Phys. Rev. I&t. 63 (1989) 1511; P.L. Gammel, L.F. Schneemeyer and D.J. Bishop, Phys. Rev. Lett. 66 (1991) 953. [ 45 ] S. Ryu, S. Doniach, G. Deutscher and A. Kapitulnik, Phys. Rev. Lett. 68 (1992) 710. [ 461 A.I. Larkin and Yu.N. Gvchinnikov, J. Low Temp. Phys. 34 (1979) 409. [47] J.V. Jose and G. Ran&z, unpublished. [48] S. Martin, A.T. Fiory, R.N. Flemming, G.P. Espinosa and A.S. Cooper, Phys. Rev. Lett. 62 ( 1989) 677. [49] A.P. Malozemoff, M. Konczykowski and N. Chikumoto, unpublished.