Thermally activated depinning in an organic superconductor

Thermally activated depinning in an organic superconductor

PHYSICA Physica C 183 ( 1991 ) 345-354 North-Holland Thermally activated depinning in an organic superconductor Universal behavior of the flux-line ...

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Physica C 183 ( 1991 ) 345-354 North-Holland

Thermally activated depinning in an organic superconductor Universal behavior of the flux-line lattice in layered superconductors

Y. K o p e l e v i c h :, A. G u p t a a n d P. E s q u i n a z i Physikalisches Institut, Universitlit Bayreuth, I41-8580Bayreuth, Germany C.-P. H e i d m a n n a n d H. Miiller Walther-Meissner-lnstitut, Walther-Meissner-Str. 8, W-8046 Garching, Germany Received 19 September 1991

With the vibrating reed technique we have measured the field dependence of the depinning temperature of the flux-line-lattice (depinning lines) in the organic superconductor K-(BEDT-TTF)2Cu(NCS)2 for different magnetic field orientations. In a reduced temperature scale the depinning lines are similar to those measured in the high temperature superconductor Bi2Sr2CaCu2Os in spite of a factor ten difference in the critical temperature. We find that the effective activation barriers for flux motion divided by the thermal energy at the critical temperature are similar for both superconductors. The results can be understood with the thermally activated flux flow model and suggest a rather universal behavior of the thermally activated depinning in layered superconductors. We further show that a small misalignment of the superconducting crystal with the applied magnetic field gives rise to two dissipation peaks; this effect can be quantitatively understood taking into account two diffusivity modes.

1. Introduction Since the discovery o f high temperature superconductivity the properties o f the flux-line lattice ( F L L ) in superconductors has attracted very much attention. Different experimental and theoretical work strongly indicates that the anisotropy o f the high-To superconductors (HTSC), e.g. YBa2Cu307 ( Y ( 1 2 3 ) ) and Bi2Sr2CaCu:Os ( B i ( 2 2 1 2 ) ) , and a small coherence length are responsible for their weak flux pinning potential or activation barriers for flux motion. These small pinning barriers together with the large thermal energy are the reason for thermally activated depinning which leads to a finite resistivity when a magnetic field is applied. At low currents, the resistivity shows approximately an exponential temperature dependence and equals the current independent diffusivity D o f the vortices . Within the thermally assisted flux flow ( T A F F ) model [ 1-4] and with the knowledge o f the field and temperature On leave from: A.F. loffe Physicotechnical Institute, 194021 Saint Petersburg, USSR.

dependence o f the resistivity (or diffusivity) one is able to understand the effects observed in the damping and resonance frequency o f mechanical oscillators as well as different experimental results from I V curves, penetration depth measurements and AC and DC susceptibilities [ 5,6]. The pinning characteristics and the magnetic field (Ba) dependence o f the effective activation barrier Ub (or a distribution o f barriers [7 ] ) which is extracted from experiments still demands further research [8,9]. Whether the FLL has "melted" or not in the thermally activated regime is still an open question [6]. Due to its large anisotropy and the weak coupling between [CuO2] planes, the Bi(2212) HTSC behaves as a two-dimensional ( 2 D ) superconductor up to ~ 0 . 9 8 o f the critical temperature Tc [ 10,11 ]. Experimental evidence from resistivity, magnetization [ 10 ] and mechanical measurements [ 12 ] indicate that all dissipation effects due to FLL motion can be explained taking into account exclusively the magnetic field component perpendicular to the conducting [CuO2] planes. The importance o f the anisotropy in the response

0921-4534/91/$03.50 © 1991 Elsevier Science Publishers B.V. All fights reserved.


Y. Kopelevich et al. / Thermally activated depinning in an orfanic sut~erconductor

of the FLL to transport currents and the broadening of the resistivity transition has been further demonstrated through studies on artificially grown copper-oxide superlattices [ 13 ]. We are, however, becoming acquainted with different observations on low-T¢ superconductors like organic superconductors (T¢ ~ 10 K) [ 14-21 ] and Mo77Ge23/Ge multilayers ( Tc ~ 5 K) [22 ], which resemble the behavior observed in HTSC. In particular, the work on torque magnetometry [ 14], resistivity [ 16,15,19], magnetic susceptibility [ 19,20 ] and microwave-loss [ 17 ] indicate similarities between the behavior of the FLL of the organic superconductor ~:-(BEDTTTF)2Cu(NCS)2 (or an analog like r-(BEDTT T F ) 2 C u [ N ( C N ) 2 ] B r ) and that of the Bi-based HTSC (BEDT-TTF is short for bis (ethylenedithio) tetrathiafulvalene), though somehow contradictory DC magnetization measurements [23,21,24]. These organic superconductors are also layered materials with a stack of conducting layers [16]. Crystal structure analysis and resistivity measurements [ 16,25 ] indicate a high degree of anisotropy which leads to the 2D behavior observed in different experiments and confirmed by the measurement of the frequency of the Shubnikov-de Haas oscillations as a function of the angle between the applied magnetic field and the a*-axis [26,27] and from the cyclotron effective mass (de Haas-van Alphen oscillations) [28]. (To avoid confusion with the axis in HTSC we will describe the relative position of the crystal with respect to the field taking the conductin8 planes as reference). The nearly two-dimensional character of the organic superconductors [25,28 ] appears to be an evident cause for the observed similarity with HTSC. Since in the organic superconductors the relevant temperature scale for flux motion is 10 times lower than in HTSC, it is important to know whether the TAFF model and the diffusion picture still applies. In this paper we present a systematic study of the mechanical response of the organic superconductor t:-(BEDT-TTF)2Cu(NCS)2 and compare it with results obtained on a Bi(2212) HTSC single crystal.

pects needed to understand the effects observed in vibrating superconductors and the influence of thermally activated depinning. For more details on this subject the reader is referred to refs. [29-31,6]. Figure 1 shows the three different geometries used in this work. In the arrangement depicted in fig. 1 (a) the crystal is glued on the main surface of a (dielectric) polymer reed and the magnetic field is parallel to the main area of the crystal (or sample length l), i.e. parallel to its conducting planes. The reed vibrates perpendicular to the applied field direction. For applied fields Ba >> B¢~ we can neglect the magnetization of the superconductor. If the FLL is pinned in the superconducting sample and the vibration amplitude is small (typical tilt angle ¢ < I0-4), it tilts with the sample affecting the external magnetic field around the reed. In this case very small shielding currents on both surfaces of the sample drive the vortices and add a magnetic restoring force which increases the resonance frequency of the crystal-reed. For typical crystal dimensions we have width w >> thickness d; then the resulting stray field causes a line tension P = ( x w 2 / 4 ) B 2 / p . o . In the arrangement depicted in fig. 1 (b) the crystal is glued on the top of the polymer reed with its main area transversal to the direction of the field. In this case the shielding described above does not occur since the small field component Ba¢ is parallel to the sample. Still a restoring force will be observed which comes from the tilt modulus of the FLL c44~-B2/p.o. However, if pinning is weak, i.e. if the penetration depth for tilt waves 444 = (c44/a) ~/2 is larger than the sample thickness d, the FLs are bound elastically to the pinning sites and only the elastic

2. Vibrating superconductors and flux diffusion

Fig. 1. Three different arrangements used for the experiments: The reed was made of a non magnetic polymerwith 20 nm Au sputtered on both surfaces. I(w) is the sample length (width) and ~> 85 °.

In this section we summarize some theoretical as-



Po .r_dJ

Y. Kopelevichet al. / Thermally activated depinning in an organicsuperconductor pinning contributes to the restoring force (c~ is the coupling constant between the FLL and the superconductor which depends on field and temperature) [ 31 ]. For finite coupling ot an increase in the resonance frequency from its B~ = 0 value is observed. Because the pinning of the FLL is not ideal, the FLs move relative to the atomic lattice and a dissipation is always measured. Depending on the relative displacement of the FLL which depends on pinning and the reed amplitude, hysteretic (amplitude and frequency dependent) or a viscous damping is observed. Particularly at low vibration frequencies the hysteretic dissipation exceeds the linear viscous contribution. If at finite temperatures thermally activated depinning of the FLs occurs, it influences the response of a vibrating superconductor. The depinning by thermal activation depends mainly on the ratio of an effective magnetic field and temperature dependent activation energy over the thermal energy: Ub(B~, T) / kBT. In the TAFF regime the diffusivity D of the FLs equals the thermally activated flux flow resistivity PTAFF:

D( Ba,T) =PTAFF(Ba,T) /gO PFr e x p ( - Ub(Ba,T)/kBT),


where PFF is the flux flow resistivity [ 32 ]. Due to thermally activated depinning of the FLL the dissipation measured on a vibrating superconductor (or in AC susceptibility) shows a maximum at TD when the frequency to of the AC field equals a characteristic diffusion time z: to~- z -1 =D(Ba,TD) ~2/1"2,


where l* is a length proportional to a particular dimension of the vibrating sample and will be discussed in detail below. Equation 2 defines what is called a "depiniaing line" (DL) TD(B,) [2,3]. Above the DL pinning becomes ineffective and the FLs move freely. Clearly, the DL depends on the frequency of the measurement, a characteristic sample dimension and, of course, on the ratio Ub/knT. Since usually to and 1" remain practically constant when the measurement is performed a s a function of temperature at constant field, a DL coincides with a line of constant resistivity. The strong temperature dependence OfprAvF


is the cause for the sharpness of the maximum in the dissipation. Due to the sensitive measurement of the reed amplitude, with a mechanical oscillator at frequencies of about 1 kHz one can measure lines of constant resistivity as low as l0 - s ~ cm even in isolated of 40 ~tm diam. grains [ 31 ]. For the sample-reed arrangement shown in fig. 1 (a) the maximum dissipation occurs when the "skin depth" for flux diffusion defined as k - l = (D~2/to) 1/2_ l*--- l. That the characteristic sample dimension for diffusion is proportional to the sample length can be seen if we take into account that after tilting the reed the FL distortion diffuses along the length of the sample (from both ends). In other words, near the maximum in the damping the small transversal AC field (or any change of the applied field produced by the tilting of the reed) penetrates completely the superconductor in the time scale of the measurement given by t o - 1. For the geometry of fig. l (b) the characteristic diffusion length is now proportional to the sample thickness. This geometry is equivalent to having a small AC field parallel to the main area of the sample and transversal to the applied field. A small rotation of the superconducting sample with respect to the applied field is enough to generate a measurable dissipation in a mechanical oscillator as for example in the geometry of fig. 1 (b). One should be aware of this sensitivity if the applied field has an arbitrary direction with respect to some principal axis of the crystal. If thermally activated depinning is important, depending on the vibration mode and geometry of the sample-field arrangement, it is possible to measure a broadening of the main peak or even observe two (or more) peaks since different diffusion lengths can satisfy eq. (2) at different temperatures. As an example, we have studied the influence of a small sample misalignment on the response of the mechanical oscillator. In the geometry of fig. 1 (c) the crystal is glued on the top of the polymer reed with an angle q~> 85 degrees with respect to the direction of the applied field. In this case the vibration of the crystal is composed of two vibration modes. One mode is equivalent to that shown in fig. 1 (a) but with a diffusion length w. The other one is equivalent to that of fig. 1 (b). Therefore, for the same frequency, we expect to see two well-de-


Y. Kopelevich et aL / Thermally activated depinning in an organic superconductor

fined damping peaks corresponding to diffusivities (Eq. ( 2 ) ) proportional to d E and w 2.


~'~ 0.20

B= .5T ~

t v

r 0.15

2.00 r_ I~

3. Experimental details and results

c:) z

~_ O.lO

High quality single crystals of ~:-(BEDTTTF) 2Cu (NCS) 2 were grown electrochemically according to the synthesis procedure described in detail elsewhere [33,34]. Two crystals from the same batch with Tc=9.5 K (determined by the onset of dissipation of the vibrating reed) have been measured. The dimensions of crystal ! (II) were: length l=0.10 cm (0.22 cm), width w=0.60 mm (0.46 ram) and thickness d = ( 2 0 + 2 ) ~tm ( 1 8 + 2 ~tm). The dissipation measurements as a function of temperature at constant applied magnetic field have been done with the vibrating reed technique [ 35 ]. The single crystals were glued to a non-magnetic polymer reed with 20 nm Au-film sputtered on both surfaces. The dissipation of the polymer reed without the crystals showed a nearly temperature independent behavior below l0 K as we expect for an amorphous material; no magnetic field dependence has been detected within experimental error in the measured temperature range. The resonance frequency of the reed was 0.8 kHz. The experiments have been performed in two cryostats. For the measurements on crystal I and at T> 4 K a He a cryostat with an 8 T solenoid has been used. Crystal II has been measured in a dilution cryostat with a superconducting solenoid up to a field of 0.6 T. Figure 2 (a) shows the damping F as a function of temperature for a fixed magnetic field Ba = 0.5 T for the three geometries shown in fig. I. Each peak in the damping is accompanied with a decrease in the resonance frequency indicating a decrease in the coupling (depinning) between the FLL and the atomic lattice, see fig. 2(b). From fig. 2 we note: ( 1 ) the peak in the damping (and the decrease of the resonance frequency) for the applied field parallel to the conducting planes (fig. 1 ( a ) ) is shifted ,,,4 K to higher temperatures with respect to the dissipation peak when the field is applied perpendicular (fig. l (b)). (2) A double peak and a double step in the resonance frequency are measured with the geometry of


1.00 £3 <



0.00 b

3oo \



10.00 ,._,, t~ 0.0~.^ 0

2. 0

4. O"




TEMPERATURE (K) Fig. 2. (a) Damping o f the organic superconductor t~-(BEDTTTF) 2Cu (NCS) 2 as a function of temperature for a field of 0.5 T (0.56 T for the applied field parallel to the conducting planes, right scale). ( , ) : geometry of fig. 1 (b), crystal II; ( • ) : geometry of fig. l (c), crystal II, ( A ): geometry o f fig. 1 (a), crystal I. (b) The same as (a) but for the resonance frequency as a function of temperature (v(0) = 0.8 kHz). B I (B±) means applied field parallel (perpendicular) to the conducting planes. The continuous lines are only a guide to the eye.

fig. 1 (c). Total depinning of the FLL is reached only above the second peak when the resonance frequency approaches its ~o(Ba=0) value. (3) The damping peaks located at higher temperatures (for the geometries depicted in figs. 1 (a) and (c) ) are larger than those at lower temperatures (geometries figs. 1 (b) and (c)). Below, it will become clear that these results are consistent with thermally activated depinning and FLL diffusion in a quasi-2D superconductor if we take into account different diffusion lengths and the effective magnetic field perpendicular to the conducting planes. We define the experimental depinning temperature at the damping peak. Its field dependence defines a DL. The DLs corresponding to the different geometries can be seen in figs. 3 and 4. In the same figures we have plotted the DLs measured in

Y. Kopelevich et al. / Thermally activated depinning in an organic superconductor



', Bi2 St= Caeu208 /




123 d w •





k-(BEDT-I-rF)2 Cu(NCS) 2 O.8kHz Bx

< -2




6 46 . . . . . .




6;@6 . . . . . .

[email protected] . . . . . .


( TD /

Tc ')


Fig. 3. ( * ) Depinning line of the organic superconductor r-(BEDT-TTF )zCu (NCS)2 for magnetic fields perpendicular to the conducting planes ( geometry of fig. 1 (b) ); ( • ) DLs from the double peaks measured with the geometry of fig. 1 ( c ). The solid lines ( 1 ) and (2) correspond to constant resistivity lines p = 10 -5 pf~ cm and 10 -2 pt~ cm of the superconductor ~-(BEDT-TTF)2Cu[N(CN)2]Br for fields perpendicular to the conducting planes extrapolated from the data of ref. [ 15 ]. The error bar indicates the approximate error in the extrapolation. The dashed line corresponds to the DL measured on a Bi(2212) HTSC with the geometry of fig. 1 (b).



', f

c'~ J LLI

k-(BEDT-I-i'F) 2 Cu(NCS)2 , .i" 0.SkHz o

Bi2Sr2 CaCu208 2kHz BII

I, a





• ~ ~ . A




o_ 13._ .<

! •

k - (BEDT-'Fi'F) 2 Cu(N(CN) 2 )Br IOGHz



[] •

• i

10 0.5(3 . . . . . .

C)~() . . . . . .


()~7b . . . . . .


0~(3 ......


()~9(3' ' ' ( TD /

{:3. . . .

1.00 Tc~

Fig. 4. ( • ) Depinning line of the organic superconductor r-(BEDT-TTF)2Cu(NCS)2 for the geometry of fig. 1 (a) corresponding to magnetic fields parallel to the conducting planes. Solid line (1) represents a constant resistivity line p = 10-' ~ cm of the organic superconductor ~t-(BEDT-TTF) 2Cu [ N (CN) 2] Br obtained from ref. [ 15 ] ( field parallel to conducting planes). The dashed line corresponds to the DL measured on a Bi (2212 ) HTSC with analogous geometry. ( [] ) Data corresponding to the differential microwave-loss peaks ofr-(BEDT-TTF)2Cu[N(CN)2]Br for magnetic fields perpendicular to the conducting planes (data taken from ref. [ 17] ). The solid line (2) represents a constant resistivity linep= 102 pt~ cm taken from fig. 5. Bi2Sr2CaCu208 single crystal with Tc=90.5K from refs. [ 1 1 , 3 6 , 3 7 ] f o r t w o g e o m e t r i e s i d e n t i c a l t o t h a t s h o w n i n fig. 1 ( a ) a n d ( b ) . T h e d i m e n s i o n s o f t h e

B i - b a s e d H T S C single c r y s t a l w e r e w = 0 . 1 0 c m a n d d = 2 0 pro.




Y. Kopelevich et al. / Thermally activated depinning in an organic superconductor

4. Interpretation For a quantitative interpretation of the D i s within the diffusion picture we need to know the diffusivity or resistivity D(Ba,T) (eq. ( 1 ) ) of the organic sample. Since this data is not available for the measured sample we take the results from ref. [ 15 ] obtained for a similar organic superconductor K-(BEDTT T F ) 2Cu [ N (CN) 2 ] Br. This organic superconductor is not identical to the one studied in this work. There are differences in the crystal symmetry, the anion structure, as well as a factor ~ 4 larger resistivity at room temperature. Nevertheless, we think that the comparison with a similar organic superconductor is justified since the FLL dynamics and its pinning is mainly determined by the layered structure of the material as will become clear below. The reduced resistivity as a function of 1/T is shown in fig. 5 for applied fields parallel (fig~ 5(a) ) and perpendicular (fig. 5 ( b ) ) t o the conducting planes. For p/p(16 K) <0.2 (p(16 K) < 1031xfl cm taken from refs. [38,18] ) we observe a reasonable

BII conducting planes


o ~. ++**



-~ 10 -a ~. ~"



............... ~51'~b ............. ~i'~'~ . . . . . . . . . . . . . . . . . b B_L conducting planes

.19.5 1 2

o o~c~ .............


5.1 T

t~:¢o ............. t~:~ .............


T-I(K -1) Fig. 5. Reduced resistivity as a function of the inverse temperature at different magnetic fields for the organic superconductor K-(BEDT-TTF) 2Cu [ N (CN) 21Br taken from ref. [ 151.

thermally activated Arrhenius-like behavior. From the slopes of fig. 5 we obtain a field dependent effective activation barrier for flux motion Ub(Ba). This function divided by the thermal energy at the critical temperature kB/Tc is plotted in fig. 6 where the data for Bi(2212) HTSC from Palstra et al. [39] is also shown. We note that the ratio Ub/ksTc is similar for both samples within factor two for Ba < 2 T. This result together with the similarities between the DLs for both field orientations in the organic and high temperature superconductors (figs. 3 and 4) is notable and suggests a diffusive FL motion in the organic superconductor. In what follows we show that with the diffusivity, i.e. the resistivity (eq. ( 1 ) ), we can understand the positions of the DLs shown in figs. 3 and 4. In section 2 we have explained that for the geometry of fig. 1 (b) the diffusion length is given by the thickness of the crystal. With the measured sample geometry and frequency and from eq. (2) we obtain that the DL should follow a constant resistivity line with p ~ 10-51xfl cm. Curve ( 1 ) in fig. 3 represents a line of constant resistivity p--- 10- 5 ix~ cm; the error bar takes into account an error in the extrapolation to lower values of the resistivity taken from fig. 5. To illustrate the frequency dependence of the DL we include in fig. 4 the DL obtained from microwave-loss data [ 17 ] for a ~:-(BEDTT T F ) 2Cu [ N (CN) 2] Br crystal for the same (static) field orientation, i.e. magnetic field perpendicular to the conducting planes. Since in the microwave-loss experiment the diffusion length is also the thickness oftbe crystal, the observed difference in the position of the DLs is an evident manifestation of the much higher frequency ( --- l0 l° Hz) [ 17 ]. Taking into account the microwave frequency we obtain a constant resistivity line p-~ l 0 +2 IJ,~'~cm. This line is shown in fig. 4 (solid line ( 2 ) ) and matches perfectly the DL measured in ref. 17. As first proposed by Kes et al. [ 10 ] the dissipation in Bi (2212) HTSC can be quantitatively understood if we assume that only the magnetic field component perpendicular to the planes leads to dissipation. Different experiments including also the angular dependent dissipation measured with mechanical oscillators can be understood based on this hypothesis. Because of the similar behavior between our organic superconductor and the Bi (2212) HTSC we assume

Y. Kopelevich et al. / Thermally activated depinning in an organic superconductor


10 2

i-..9 ~10


10 -1





, i,,i






~ i1,1,~)





, i ,

10 2


Fig. 6. Effective activation barrier divided by the thermal energy at Tc as a function of applied field for the organic r-(BEDTTTF)2Cu[N(CN)2]Br ( Q ) (from fig. 5) and Bi(2212) superconductors ( , ) taken from ref. [39].

also that no Abrikosov FLL exists parallel to the conducting planes and only the perpendicular component accounts for the measured dissipation. In this case we can also understand the DL obtained with the geometry of fig. 1(a). A small field misalignment should be enough to produce a FLL of kinks or pancakes vortices [40] distributed on the conducting planes only. For the tilt movement of the crystal (fig. l ( a ) ) the diffusion length is proportional to its length, the effective field B ~ which produces the FLL is only a fraction of the applied field. Assuming Baf~B, sin(5 ° ) and a temperature independent pinning barrier the DL taken with the geometry of fig. 1(a) can be shifted to lower fields and coincides approximately with the DL obtained from the high temperature peak with the geometry of fig. 1 (c) (note that the length of crystal I is only a factor two larger than the width of crystal II). For completeness and only for a qualitative comparison since the sample or field misalignment may be different, we have plotted in fig. 4 the constant resistivity line (solid line ( 1 ) ) obtained for the field parallel to the conducting planes (from ref. [15] ) corresponding to / 9 " 10 -1 }.t~"~cm. Now, we discuss the double peak structure seen in fig. 2 for the geometry of fig. 1 (c). As explained in section 2, the peak at lower temperatures is due to the diffusion of the FLL along the thickness of the

sample. The corresponding DL coincides reasonably with the one obtained for the geometry of fig. 1 (b); the small shift might be attributed to different background contribution from the peak at higher temperatures. The DL corresponding to the peak located at higher temperatures is related to the diffusion of the FL distortion parallel to the conducting planes with diffusion length l* = w >> d. With the same procedure as used before we obtain the constant resistivity line p--- 10- 2 ~tfl cm (curve (2) ) shown in fig. 3. The agreement is excellent taking into account the uncertainty in the extrapolation of the resistivity and that we are comparing different organic superconductors. It should be mentioned here, that a similar double peak structure has also been observed in HTSC Bi(2212) crystal by Dur~in et al. [12] using a high(2 oscillator and was interpreted as "two step melting" of the FLL. However, those results and the AC susceptibility data for two directions of the AC field obtained by the same authors [41 ] can be quantitatively understood taking into account two diffusivity modes as described above. The positions of the damping peaks are easier to understand than their height. The calculation of the dissipation in vibrating superconductors is a difficult task mainly due to the contribution of hysteretic (non-linear) movement, i.e. jumps of the FLs. The


Y. Kopelevich et al. / Thermally activated depinning in an organic superconductor

hysteretic contribution has been observed in HTSC [ 3 ] and also in amorphous superconductors [ 42 ]. For very small grains embedded in a polymer matrix [31 ] only the viscous contribution has been observed since the FL displacement inside the superconducting grains is very small. It is possible that the different height of the peaks, see fig. 2(a), is due to different excursions of the vortices along the sample.

5. Predictions of melting theories

In this section we want to discuss the predictions of theories of melting of the FLL which have been proposed by several authors in the last years [ 6 ]. It seems reasonable to believe that thermal fluctuations would grow continuously with temperature, adding more disorder to an already disordered FLL, therefore no sharp transition should be expected. The "melting" temperatures calculated below should be understood as reference temperatures indicating a probable state of the FLL. Evidently, a comparison of the predictions of the theories of melting of a 3Dor 2D-FLL when applied to the organic and Bi (2212) superconductors is of interest. Firstly, we calculate the 3D thermal fluctuations (u2 > of the FL-positions. Following Brandt [ 43 ] we have:


u 2 ,,~kaT 4riB


>( \Bc2__Ba, ]


(3) where 7 2 is the anisotropy mass ratio, ~o the flux quantum and the shear modulus is given by c66 ~ Ba ~o ( 1 - b)2/( 16 n2~/~o).


Here 2 II means the penetration depth parallel to the conducting planes. From torque experiments [ 14 ], and muon spin resonance measurements [44] we have 2 n(4.5 K) ~ 103 nm. Inserting this value in eq. (4) for a field of 0.5 T (reduced field b < 0 . 2 ) we obtain c66~ 16 N / m E. With this value and 7---200 [ 14 ] we obtain


---0.5, ao


with ao the FL-spacing calculated at 0.5 T. From a

comparison with the Lindemann criterion for 3Dmelting (u2>l/2/ao=O.l-0.2 we would expect to have the FLL already "melted" in the whole measured temperature range. Note that the difference of 7 and penetration depth ;tim between the Bi(2212) HTSC and the organic superconductor (7(Bi(2212))/7(organic) ~0.2 and 21)Bi(2212))/ 2 ii(organic) ~ 0.2) overwhelms the factor ten difference in temperature. However, when the thermal energy exceeds the interaction of point (pancake [40 ] ) vortices between the [CUP2] layers, the vortex lattice in adjacent layers decouples. Then, for layered superconductors a 2D-FLL would melt at finite fields in each layer via, for example, dislocations [45]. The field independent melting temperature T~o is given by the equation [46]: T2D t2b02S m = 64n2/ZOka2 ~"


which is only valid for fields Ba>Bo, where Bo=4~o/ s272 ~ 0.1 T and s is the distance between conducting planes (layers) = 1.6 nm for the organic sample. Replacing in eq. (6) the appropriate values for the different parameters and constants we obtain T 2r' ~- 0.6 K ( _ 10 K) for the organic (Bi(2212) HTSC) superconductor. This is a striking result: the calculated melting temperatures for a 2D-FLL differs one order of magnitude between both superconductors. If taken seriously, this theoretical result would also indicate a melted FLL in the whole measured temperature range. This estimate together with our interpretation of the damping peaks and frequency change by thermally activated depinning indicate that, were the FLL melted, it is also pinned. Although, in 3D superconductors this conclusion is evident since there are many pinning centers per FL (a melted FLL is better pinned than a not melted FLL) in 2D layered superconductors it is not obvious. It remains unclear how a melting transition, if it occurs, can be experimentally confirmed.

6. Conclusions

In this work we have studied the mechanical response of the FLL in single crystals of the organic superconductor K-(BEDT-TTF)2Cu(NCS)2 with

Y. Kopelevich et al. / Thermally activated depinning in an organic superconductor

Tc = 9.5 K . The m e a s u r e m e n t o f the dissipation d o n e with the v i b r a t i n g reed m e t h o d a n d with different geometries yields the t e m p e r a t u r e for thermally activated depinning. T h e m a i n results o f this work can be interpreted in terms o f thermally assisted flux flow m o t i o n and the diffusion o f the FLL. ( l ) W i t h the electrical resistivity, resonance frequency o f the reed a n d the d i m e n s i o n o f the crystal we can u n d e r s t a n d quantitatively the position o f the DLs. ( 2 ) The field d e p e n d e n t ratio between the effective barrier for flux m o t i o n a n d t h e r m a l energy at Tc is similar for the K - ( B E D T - T T F ) E C u ( N C S ) 2 a n d for the B i ( 2 2 1 2 ) superconductors. This result agrees with the result that the position o f the depinning lines is similar in a reduced t e m p e r a t u r e scale for b o t h superconductors. The peaks in the d a m p i n g a n d the frequency change represent the thermally activated depinning o f the F L L from the sample under the t i m e scale and geometrical conditions o f the experiment. The observed similarities between Bi (2212) H T S C and K-(BEDT-TTF)ECu(NCS)2 superconductor speak in favor o f a rather universal b e h a v i o r o f the F L L in layered superconductors. The actual nature o f pinning (for e x a m p l e O-vacancies in B i ( 2 2 1 2 ) H T S C ) would affect the field ( a n d t e m p e r a t u r e ) dependence o f the effective activation barriers which lead to slightly different field d e p e n d e n c e o f the depinning t e m p e r a t u r e To. Since the 2D nature o f the F L L in layered superconductors prevails in determ i n i n g the pinning, a technologically significant increase o f the activation barriers by introducing artificial pinning centers seems rather unlikely. W h e n the crystal has a small misalignment with the a p p l i e d magnetic field a n d the vortex diffusion is c o m p o s e d o f two diffusion m o d e s we were able to observe two peaks in the dissipation which were also observed in Bi (2212 ) H T S C [ 12 ]. Within the T A F F m o d e l this b e h a v i o r is expected a n d can be quantitatively explained. The striking predictions o f the theories o f F L L melting is certainly a result which incites further research.

Acknowledgements We t h a n k H. Braun for the fruitful discussions a n d


a careful reading o f the manuscript. We are grateful to W. Lorentz for his help with the experiments below 4 K.

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