- Email: [email protected]
Josephson plasma edge in La1.85Sr0.15CuO4
Physica B 211 (1995) 260-264
ELSEVIER
Josephson plasma edge in La1.85Sr0.15CuO4 P.J.M. van Bentum a'*, A.M. Gerrits a, M.E.J. Boonman a, A. Wittlin a, V.H.M. Duijn b, A.A. Menovsky b a High Field Magnet Laboratory and Research Institute for Materials, University ofN(imegen, Toernooiveld, NL-6525 ED Nijmegen, The Netherlands b Van der Waals Zeeman Laboratory, University o f Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands
Abstract
The far-infrared dielectric c-axis response for Lal.asSro.l 5CUO4 shows a pronounced plasma edge below the BCS superconductive gap frequency. The low frequency of this plasmon was previously interpreted as an indication for anomalous mass enhancement for the c-axis transport. However, a natural explanation for this low-frequency plasmon can be given in terms of Josephson plasma oscillations of the order parameter phase difference between adjacent planes in the layered high-To materials. In this contribution we concentrate on the effect of a large magnetic field parallel to the planes of the superconducting layers. We will demonstrate that the spectroscopic measurement of the plasma frequency and damping as a function of magnetic field yields direct information about the c-axis critical current density and the vortex dynamics for this configuration. In particular, we find for Lal.asSro.15CuO~ that the vortices for B parallel to the planes are not pure Josephson vortices but rather in between the Abrikosov and Josephson limits.
1. Introduction
The high-To superconductors are generally treated as quasi two-dimensional layered materials. Nevertheless, the nature of the vertical (c-axis) transport in the normal state and the coupling mechanism for the layers in the superconductive state are still unclear. In general the materials are much more anisotropic than expected from bandstructure calculations and the low c-axis conductivity is sometimes interpreted as an indication for non Fermi-liquid behaviour. In the superconducting state, measurements of the upper critical field indicate a c-axis coherence length of the order of the interplanar distance. This suggests that one can approximate these highly anisotropic materials as Josephson-coupled stacks of 2-D superconducting building blocks. In the mixed state it * Corresponding author.
has for example become fashionable to describe the vortices as Abrikosov pancakes weakly coupled by Josephson strings. Direct observation of Josephson-like interplane coupling in several different high-To compounds with double and triple CuO2 layers, was for example reported in the work by Kleiner and Miiller [1]. The interpretation of the ab-plane optical properties remains highly controversial and a clear assessment of the energy gap is not possible, partly due to the cleanlimit electrodynamics in the planes. For the c-axis optical properties the situation is reversed and one has to deal with the extreme dirty limit where the conductivity is in fact below the metallic limit and the mean free path is of the order of atomic distances. With the progress in crystal growth it has recently become possible to study the full anisotropic far-infrared response for high-quality single crystals of sufficient size. Tamasaku et al. I-2] reported a very prominent change from a poor incoherent
0921-4526/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 4 ) 0 1 0 0 2 - 1
261
P.J.M. van Bentum et al. / Physica B 211 (1995) 260-264
conductivity in the normal state to a coherent transport with a well-defined plasma edge. This edge was attributed to carriers in the superconductive condensate using a phenomenological 2-fluid model. In this contribution we will present new data of the optical reflectivity as a function of temperature and for magnetic fields up to 25 T. We will show that the zerofield reflectance edge in the c-axis optical response of La ~.ssSro. ~5CUO4 can be understood in terms of a simple Josephson model. The response in a magnetic field can be understood if both pairbreaking and interactions with the vortex lattice are taken into account. It turns out that far-infrared spectroscopy can be a useful technique to determine 'practical' properties such as the DC-critical current and the vortex pinning force and viscosity.
ible with a d-wave gap. It is not possible to determine the (anisotropic) gap due to the very strong contribution of the phonons at higher frequencies. For the study of the effect of a magnetic field, the samples were mounted in a magnet cyrostat, and the unpolarized reflectance was measured at 1.2 K as a function of frequency for magnetic fields up to 17.3 T oriented parallel to the ab-plane of the sample. In a magnetic field the plasma edge shift,, slightly to lower frequencies, while at the same time tl~e damping increases, as indicated in Fig. 2. The magne :ic field dependence of the plasma frequency is much s nailer than anticipated in the literature. We will come back to a more quantitative comparison in the next section. The plasma frequency rapidly drops to zero when approaching the critical temperature, and a F )urier spectrometer is not the best instrument to study the very
2. Experimental results
For the far-infrared measurements we used various single crystals of La~ -~SrxCuO4 grown by the traveling solvent floating zone method. In this paper we will concentrate on the samples that are close to optimal doping (x = 0.15, T~ = 34 K). In this experiment we used various spectrometers. The zero magnetic field measurements were done with a Bruker IFS 113v Fourier Transform spectrometer coupled to a variable temperature optical cryostat. The low-temperature wide band reflectivity in magnetic fields up to 17.5 T was measured with the same instrument coupled to a magnet cryostat with 3He cooled detectors. The temperature-dependent low-frequency reflectivity was measured in the Nijmegen I Hybrid Magnet for fields up to 25 T using both an optically pumped molecular gas FIR laser and a tunable milliwave vector network analyser with heterodyne detection. In Fig. 1 we show the polarized c-axis reflectance below 80 cm-~ at various temperatures between 10 and 36 K. The data confirm the previous results of Tamasaku et al. [2]. Above T¢ the spectrum resembles that of an insulator whereas below T ¢ a weakly damped plasma edge appears. The inset clearly shows the characteristic feature of a plasmon edge related to the zero-crossing of the real part of the dielectric function. The real part of the conductivity a~(to) decreases substantially at low frequencies when cooling below T¢. This is consistent with the opening of a gap. However, the conductivity does not vanish completely, which explains why the damping of the plasmon is low but not zero. Part of the background conductivity inside the gap could be due to inhomogeneities or off-stochiometric inclusions. For example, the shoulder in the reflectivity just above the plasma frequency can be well-explained in terms of an effective medium response with non-superconducting inclusions [3]. On the other hand, the conductivity is not incompat-
1.0
2o.o ~ - 1o.o o.o
\
......
\ \ \\ \
\
\
~- - - -:
3oo4oo~oo~,o~oo
\
Frequency (cm ~)
10K 0.0 21).0 30.0 40.0 5 0 . 0 6 0 . 0 70.0 80.0 Frequency (cm 1) Fig. I. Reflectance of Lal.ssSro.15CuO4 at various temperatures between 36 and 10 K. The inset shows the r}al part of the dielectric function e~(~o)deduced from a Kramers4Kronig analysis.
42.0
• qi
.
.
.
.
"
11.0
"
9.0
41.0 40.0
. ~
39.(/
,~
7.0 ~ -
38.0 •
37.0 0.0
' ' 5.0 10.0 15.0 Magnetic field [T]
5.0 0.0
Fig. 2. Plasma frequency ~op and damping 7 as~ function o f magnetic field, deduced from a fit to the simple Josephson model. Solid line: fit to Eq. (see text).
P.J.M. van Bentum et al./ Physiea B 211 (1995) 260-264
262
low-frequency spectrum. For this reason we measured the reflectance at fixed frequencies in the far infrared and microwave region as a function of temperature for various values of the magnetic field using a 25 T hybrid magnet. In Fig. 3 we show the reflectance at zero
1.8
8
1.4
,~ l.O
3.Theory
0.6
0.2 15.0
25.0 Temperature [K]
35.0
Fig. 3. (a) Reflectance as a function of temperature for ~0 = 6.5 (top), 17.5 (center) and 32.7 cm- ~(bottom curve). (b) Simulation for the same frequencies within a BCS context (see text).
32.0
ls(n,n + 1) = / o s i n ( x , - ~,+1) = lc sin(g,),
]
~. l 7 280~.....,,__~
t
26.0 '
1.8
h d~o/dt. V=2e
tl,
24.0
~
o0
,o.o
2o.0
300
B [T]
8~ 6.5crn'
1
1.0 18crn
0.2 15.0
In this section we will describe a simple model to describe the observed plasmon effects. Let us assume that the layered structure can be treated as a stack of quasi 2-dimensional superconductors at a periodic distance d, separated by Josephson junctions of either a weak metallic or dielectric nature. Then the macroscopic wave function of layer n can be written as: ~ , = Aoeix° and the current and potential difference between adjacent layers are connected by the Josephson relations:
•
30.0~
2.2
1.4
magnetic field as a function of temperature for three characteristic frequencies. In the microwave region, the reflectance changes nearly stepwize at a temperature very close to To. At higher frequencies, the reflectance first goes through a m i n i m u m and then gradually approaches unity. The effect of a magnetic field is depicted in Fig. 4. The top panel shows the reflectance vs. temperature at 6.5 c m - 1 (195 GHz), the bottom for 18.1 c m - 1. The successive curves for B = 0, 8, 20, and 25 T are shifted downward for clarity. F r o m these curves it is immediately clear that the main effect of the field at low frequencies is an increase of the damping and a decrease of the effective plasma frequency. The inset shows the magnetic field dependence of the 'midpoint' temperature T* for both frequencies. F r o m the low-frequency data it is clear that the magnetic field has only a very small pairbreaking effect on T~.
~
The DC-Josephson relation for the supercurrent remains valid for finite frequencies co ,~ A. At low current densities this relation can be linearized by 1,(0 = lc sin(q~(t)) lc~o(t). The total AC current density can now be written as j(t) = j,(t) + ja(t) + jqp(t), wherejdt) is the pair Josephson current, ja(t) is the dielectric displacement current, and jqp(t) is the dissipative single particle current due to excited quasiparticles. Using the Josephson equations we can write Js = jcq~,
ja=cdV 25.0 Temperature [K]
35.0
Fig. 4. Reflectance as a function of temperature for 6.5 and 17.5 cm- ~ for (from top to bottom) B = 0, 8, 20 and 25 T. Inset: Magnetic field dependence of the 'midpoint' temperature T* for both frequencies.
Jqp
V p
d ( h dq~) hd2q~ dt = C d - t 2 e ~ - = C 2 e d t 2' hd~p 2ep dr'
where the quasiparticle resistivity p is a function of both temperature and frequency. With the ansatz
P.J.M. van Bentum et al./ Physica B 211 (1995) 260 264 j ( t ) = j o e - i,~t q> = q~1e - i,,~,we obtain
( j(t)=j~
V(t) =
d2q9 Ldq>) tp+LC-d~-+p-~, - itoL
1 - toZLC - itoL/p
jo e-i~'t,
where we have introduced the usual nonlinear inductance L = h/2ejc. Using j = a / E , E = V / d , C = e/d, it is straightforward to deduce the complex conductivity a(to) or the dielectric response e(co) = io/to, 1 - t o z L C - icoL/p a(to) = ~o
- icoLC
and ~(to)=eo
1 - - "(/)2 P+
,
where cop = 1 / x / - ~ and y- 1 = pC. From this derivation it is clear that any layered superconductor with a sufficiently low (Josephson) coupling between the planes will have a plasma edge at frequencies below the energy gap, and the electrodynamic properties are completely analogous to the classical results derived for extended single junctions, with the exception that the effective width of the junction d + 2 . 2 is replaced by the interlayer distance. An even simpler way to look at this problem is to use the Kramers-Kronig relations which state that causality requirements closely link the real and imaginary parts of the conductivity spectrum. If the real part is known for all frequencies then the imaginary part can be calculated and vice versa. If there is a dissipationless DC supercurrent, described by a delta function at zero frequency in cq (co) then it follows from the Kramers-Kronig relations that there should be a complimentary contribution to the imaginary part o2(co) proportional to 1/to. With e(to) = e + ia(co)/co this directly gives a plasmon-type of dielectric function. This also shows that the presence of a plasma edge is not restricted to pure Josephson coupled structures. In the classical bulk superconductors the same argument applies, but because the metallic conductivity is usually much larger a pole would occur at frequencies far above the gap, and the response in the superconducting state is very similar to that of the normal state. Note that in this case the plasma edge is due to a resonance in the charge density oscillation, in contrast to the Josephson case where the plasma edge is due to resonant oscillations of the local phase of the order parameter. Ferrell showed that the Josephson relations, including the so-called self-capacitive effects can be derived self-
263
consistently from the BCS theory [4]. This provides an elegant framework to extend the above analysis to frequencies comparable to the energy gap and !nclude the dephasing effects of an applied DC-current ld¢ = I~ sin(0). The basic result given by Ferrgll, valid at T = 0 and for a BCS s-wave gap A = 1.76kTc is a(to, ~) or,
i [(K(co) + n/2)cos(~b) 2co + 2E(to) -- K(CO) -- n / 2 - 2Coto2],
where Co = togC/cr, (C is the geometrical capacitance as introduced above and cog is the gap frequency)and K(to) and E(co) are the complete elliptic integrals ]of the first and second kind. For Lal.8sSro.lsCuO4 (top ~ co8/2) with no DC current (4> = 0) the result is virtually identical to ihe result of the simple Josephson model. From the considerations given above it follows that the plasma frequency is directly related to thei magnitude of the DC critical current density. The absolute value of the i measured jc(0) ~ 8 X 109 A/m 2 agrees well with the B ( ~ prediction Jc=(nA(T)/2ep)tanh(A(T)/2kT) for A(0) =l.76kTc and the optically determined a, = 8 f2-1 cm - !. However, we do find a slightly better fit to the temperature dependence if we assume a larger gap A(0) ~ 2 . 5 k T L The critical current density derived from magnetization hysteresis is about an order of magnitud~ lower [5]. This indicates that the spectroscopic deterrhination of Jc is more intrinsic in the sense that it is a n~ar-equilibrium technique with very small amplitudes oflhe probing (AC) current. In transport measurements self,field effects and possible local heating cannot be avoide~. In a magnetic field the transport jr is limited by the cmset of flux motion. Within this model it is possible to c~lculate the electrodynamic response for arbitrary frequency and temperature in a self-consistent way. In Fig. ~(b) we have calculated the reflectance versus temperatule for three representative frequencies, assuming a BCS gap, a Mattis-Bardeen conductivity ol(co)/a . with ~ frequency independent a,(to) and a small subgap conduOtance taken from a fit to the Kramers-Kronig data at T = 35 K. The agreement with the experimental data is quite satisfactory. We now turn to the influence of an external m~tgnetic field. First we consider the pairbreaking effect of the magnetic field. The exact value of the upper critical ]field is not known, but from extrapolations of the low-fi~ld data we estimate B¢2(Bllab) to be well above 100T.)In the Abrikosov-Gorkov theory this would lead to i about 3% decrease of T~ and 2% drop in A(0), or le~s than 1% change in the plasma frequency for a magr~etic field of
264
P.J.M. van Bentum et aL /Physica B 211 (1995) 260-264
25 T. The change in Tc is indeed of this order of magnitude, but the drop in oJp is obviously much larger, as indicated in Fig. 2. To resolve this discrepancy one has to include the interaction of the induced AC currents with the vortices. This problem is well-studied in the context of classical Josephson junctions [6]. For ideal Josephson vortices the equation of motion is governed by the sine-Gordon equation, which shows that the vortices move essentially as ballistic objects or solitons. In this case the dispersion relation for propagating electromagnetic waves becomes linear with a characteristic gap if the k-vector of the light matches the q-vector for the vortex lattice. Since the infrared reflectivity is essentially a k = 0 process this would indicate a rapid collapse of the plasma frequency with applied magnetic field. A comparison with the experimental data immediately shows that this is not the case, and the vortices must have a somewhat intermediate character between the Josephson and the Abrikosov limits. The popular description of the B II c transport properties in terms of 'pancake' vortices might be applicable for the Bi and TI high Tc superconductors, but is not correct for Lal.ssSro.lsCuO4. A detailed description for the electrodynamic response of an anisotropic superconductor in the presence of magnetic fields was given very recently by Tachiki et al. [7]. F o r the parallel magnetic field case, their basic result (in SI units) is given by e(~) _ 1 e
COp 2 [ ~1o+B o- 1 1-1 ' t~(co + i0 +) #022 Xp - kor/-- Mo) 2
where ~o is the unit flux, Bo the flux density, the xp and r/the pinning force constant and viscous drag-force coefficient, respectively. M is the vortex inertial mass per unit length. In their model the value for the plasma frequency is taken from an empirical 2-fluid description of the superconducting state. However, the interaction with the vortices is based on a quite general theory and should be applicable in the region intermediate between the Josephson and Abrikosov limits. The solid line in Fig. 2 is a fit according to this equation, for Be2 = 120 T, 2¢ = 2 x 10-s m, Xp = 500(Pa),
r / = l . 5 x l 0 - 1 ° ( P a s ) and M = 0 . These values are quite realistic when compared with transport results. Using Jc, tr = 5-10 s A/m 2 [5] and the relation jc.tr = Xp¢ab/~bo we find for the effective pinning length s c a l e Cab = 2 nm, in good agreement with literature data for the coherence length in the ab-plane. The viscosity can be estimated from the Bardeen-Stephen model: r / = ~boBc2 fin. With the Kramers-Kronig result tr, = 8 0 0 [ ~ - ~ m -1] and B¢2 = 120T this yields ~/= 1.9x 10-1°(Pas).
4. Conclusions
The observed low-frequency plasma edge in the c-axis dielectric response for Lal.ssSro.lsCuO4 can be attributed to a Josephson type of phase oscillations of the superconducting order parameter. The vortices introduced by a magnetic field parallel to the planes couple strongly to the electrodynamic response. The upper limit for the critical current is in good agreement with BCS predictions, while the pinning of the vortices indicates that these are intermediate between the Josephson and Abrikosov limits.
References
[1] R. Kleiner and P. Miiller, Phys. Rev. B 49 (1994) 1327. I-2] K. Tamasaku, Y. Nakamura and S. Uchida, Phys. Rev. Lett. 69 (1992) 1455. 1-3] A.M. Gerrits, A. Wittlin, V.H.M. Duyn, A.A. Menovsky, J.J.M. Franse and P.J.M. van Bentum, Physica C 235-240 (1994) 1117. [4] R.A. Ferrell, Phys. C 152 (1988) 10. [5] K. Kishio, Y. Nakayama, N. Motohira, T. Noda, T. Kobayashi, K. Kitazawa, K. Yamafuji, I. Tanaka and H. Kojima, Supercond. Sci. Technol. 5 (1992) S1, $69. [6] K.K. Likharev, Dynamics of Josephson Junctions and Circuits (Gordon and Breach, New York, 1986). 1-7] M. Tachiki, T. Koyama and S. Takahashi, Phys. Rev. B 50 (1994) 7065.
ELSEVIER
Josephson plasma edge in La1.85Sr0.15CuO4 P.J.M. van Bentum a'*, A.M. Gerrits a, M.E.J. Boonman a, A. Wittlin a, V.H.M. Duijn b, A.A. Menovsky b a High Field Magnet Laboratory and Research Institute for Materials, University ofN(imegen, Toernooiveld, NL-6525 ED Nijmegen, The Netherlands b Van der Waals Zeeman Laboratory, University o f Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands
Abstract
The far-infrared dielectric c-axis response for Lal.asSro.l 5CUO4 shows a pronounced plasma edge below the BCS superconductive gap frequency. The low frequency of this plasmon was previously interpreted as an indication for anomalous mass enhancement for the c-axis transport. However, a natural explanation for this low-frequency plasmon can be given in terms of Josephson plasma oscillations of the order parameter phase difference between adjacent planes in the layered high-To materials. In this contribution we concentrate on the effect of a large magnetic field parallel to the planes of the superconducting layers. We will demonstrate that the spectroscopic measurement of the plasma frequency and damping as a function of magnetic field yields direct information about the c-axis critical current density and the vortex dynamics for this configuration. In particular, we find for Lal.asSro.15CuO~ that the vortices for B parallel to the planes are not pure Josephson vortices but rather in between the Abrikosov and Josephson limits.
1. Introduction
The high-To superconductors are generally treated as quasi two-dimensional layered materials. Nevertheless, the nature of the vertical (c-axis) transport in the normal state and the coupling mechanism for the layers in the superconductive state are still unclear. In general the materials are much more anisotropic than expected from bandstructure calculations and the low c-axis conductivity is sometimes interpreted as an indication for non Fermi-liquid behaviour. In the superconducting state, measurements of the upper critical field indicate a c-axis coherence length of the order of the interplanar distance. This suggests that one can approximate these highly anisotropic materials as Josephson-coupled stacks of 2-D superconducting building blocks. In the mixed state it * Corresponding author.
has for example become fashionable to describe the vortices as Abrikosov pancakes weakly coupled by Josephson strings. Direct observation of Josephson-like interplane coupling in several different high-To compounds with double and triple CuO2 layers, was for example reported in the work by Kleiner and Miiller [1]. The interpretation of the ab-plane optical properties remains highly controversial and a clear assessment of the energy gap is not possible, partly due to the cleanlimit electrodynamics in the planes. For the c-axis optical properties the situation is reversed and one has to deal with the extreme dirty limit where the conductivity is in fact below the metallic limit and the mean free path is of the order of atomic distances. With the progress in crystal growth it has recently become possible to study the full anisotropic far-infrared response for high-quality single crystals of sufficient size. Tamasaku et al. I-2] reported a very prominent change from a poor incoherent
0921-4526/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 4 ) 0 1 0 0 2 - 1
261
P.J.M. van Bentum et al. / Physica B 211 (1995) 260-264
conductivity in the normal state to a coherent transport with a well-defined plasma edge. This edge was attributed to carriers in the superconductive condensate using a phenomenological 2-fluid model. In this contribution we will present new data of the optical reflectivity as a function of temperature and for magnetic fields up to 25 T. We will show that the zerofield reflectance edge in the c-axis optical response of La ~.ssSro. ~5CUO4 can be understood in terms of a simple Josephson model. The response in a magnetic field can be understood if both pairbreaking and interactions with the vortex lattice are taken into account. It turns out that far-infrared spectroscopy can be a useful technique to determine 'practical' properties such as the DC-critical current and the vortex pinning force and viscosity.
ible with a d-wave gap. It is not possible to determine the (anisotropic) gap due to the very strong contribution of the phonons at higher frequencies. For the study of the effect of a magnetic field, the samples were mounted in a magnet cyrostat, and the unpolarized reflectance was measured at 1.2 K as a function of frequency for magnetic fields up to 17.3 T oriented parallel to the ab-plane of the sample. In a magnetic field the plasma edge shift,, slightly to lower frequencies, while at the same time tl~e damping increases, as indicated in Fig. 2. The magne :ic field dependence of the plasma frequency is much s nailer than anticipated in the literature. We will come back to a more quantitative comparison in the next section. The plasma frequency rapidly drops to zero when approaching the critical temperature, and a F )urier spectrometer is not the best instrument to study the very
2. Experimental results
For the far-infrared measurements we used various single crystals of La~ -~SrxCuO4 grown by the traveling solvent floating zone method. In this paper we will concentrate on the samples that are close to optimal doping (x = 0.15, T~ = 34 K). In this experiment we used various spectrometers. The zero magnetic field measurements were done with a Bruker IFS 113v Fourier Transform spectrometer coupled to a variable temperature optical cryostat. The low-temperature wide band reflectivity in magnetic fields up to 17.5 T was measured with the same instrument coupled to a magnet cryostat with 3He cooled detectors. The temperature-dependent low-frequency reflectivity was measured in the Nijmegen I Hybrid Magnet for fields up to 25 T using both an optically pumped molecular gas FIR laser and a tunable milliwave vector network analyser with heterodyne detection. In Fig. 1 we show the polarized c-axis reflectance below 80 cm-~ at various temperatures between 10 and 36 K. The data confirm the previous results of Tamasaku et al. [2]. Above T¢ the spectrum resembles that of an insulator whereas below T ¢ a weakly damped plasma edge appears. The inset clearly shows the characteristic feature of a plasmon edge related to the zero-crossing of the real part of the dielectric function. The real part of the conductivity a~(to) decreases substantially at low frequencies when cooling below T¢. This is consistent with the opening of a gap. However, the conductivity does not vanish completely, which explains why the damping of the plasmon is low but not zero. Part of the background conductivity inside the gap could be due to inhomogeneities or off-stochiometric inclusions. For example, the shoulder in the reflectivity just above the plasma frequency can be well-explained in terms of an effective medium response with non-superconducting inclusions [3]. On the other hand, the conductivity is not incompat-
1.0
2o.o ~ - 1o.o o.o
\
......
\ \ \\ \
\
\
~- - - -:
3oo4oo~oo~,o~oo
\
Frequency (cm ~)
10K 0.0 21).0 30.0 40.0 5 0 . 0 6 0 . 0 70.0 80.0 Frequency (cm 1) Fig. I. Reflectance of Lal.ssSro.15CuO4 at various temperatures between 36 and 10 K. The inset shows the r}al part of the dielectric function e~(~o)deduced from a Kramers4Kronig analysis.
42.0
• qi
.
.
.
.
"
11.0
"
9.0
41.0 40.0
. ~
39.(/
,~
7.0 ~ -
38.0 •
37.0 0.0
' ' 5.0 10.0 15.0 Magnetic field [T]
5.0 0.0
Fig. 2. Plasma frequency ~op and damping 7 as~ function o f magnetic field, deduced from a fit to the simple Josephson model. Solid line: fit to Eq. (see text).
P.J.M. van Bentum et al./ Physiea B 211 (1995) 260-264
262
low-frequency spectrum. For this reason we measured the reflectance at fixed frequencies in the far infrared and microwave region as a function of temperature for various values of the magnetic field using a 25 T hybrid magnet. In Fig. 3 we show the reflectance at zero
1.8
8
1.4
,~ l.O
3.Theory
0.6
0.2 15.0
25.0 Temperature [K]
35.0
Fig. 3. (a) Reflectance as a function of temperature for ~0 = 6.5 (top), 17.5 (center) and 32.7 cm- ~(bottom curve). (b) Simulation for the same frequencies within a BCS context (see text).
32.0
ls(n,n + 1) = / o s i n ( x , - ~,+1) = lc sin(g,),
]
~. l 7 280~.....,,__~
t
26.0 '
1.8
h d~o/dt. V=2e
tl,
24.0
~
o0
,o.o
2o.0
300
B [T]
8~ 6.5crn'
1
1.0 18crn
0.2 15.0
In this section we will describe a simple model to describe the observed plasmon effects. Let us assume that the layered structure can be treated as a stack of quasi 2-dimensional superconductors at a periodic distance d, separated by Josephson junctions of either a weak metallic or dielectric nature. Then the macroscopic wave function of layer n can be written as: ~ , = Aoeix° and the current and potential difference between adjacent layers are connected by the Josephson relations:
•
30.0~
2.2
1.4
magnetic field as a function of temperature for three characteristic frequencies. In the microwave region, the reflectance changes nearly stepwize at a temperature very close to To. At higher frequencies, the reflectance first goes through a m i n i m u m and then gradually approaches unity. The effect of a magnetic field is depicted in Fig. 4. The top panel shows the reflectance vs. temperature at 6.5 c m - 1 (195 GHz), the bottom for 18.1 c m - 1. The successive curves for B = 0, 8, 20, and 25 T are shifted downward for clarity. F r o m these curves it is immediately clear that the main effect of the field at low frequencies is an increase of the damping and a decrease of the effective plasma frequency. The inset shows the magnetic field dependence of the 'midpoint' temperature T* for both frequencies. F r o m the low-frequency data it is clear that the magnetic field has only a very small pairbreaking effect on T~.
~
The DC-Josephson relation for the supercurrent remains valid for finite frequencies co ,~ A. At low current densities this relation can be linearized by 1,(0 = lc sin(q~(t)) lc~o(t). The total AC current density can now be written as j(t) = j,(t) + ja(t) + jqp(t), wherejdt) is the pair Josephson current, ja(t) is the dielectric displacement current, and jqp(t) is the dissipative single particle current due to excited quasiparticles. Using the Josephson equations we can write Js = jcq~,
ja=cdV 25.0 Temperature [K]
35.0
Fig. 4. Reflectance as a function of temperature for 6.5 and 17.5 cm- ~ for (from top to bottom) B = 0, 8, 20 and 25 T. Inset: Magnetic field dependence of the 'midpoint' temperature T* for both frequencies.
Jqp
V p
d ( h dq~) hd2q~ dt = C d - t 2 e ~ - = C 2 e d t 2' hd~p 2ep dr'
where the quasiparticle resistivity p is a function of both temperature and frequency. With the ansatz
P.J.M. van Bentum et al./ Physica B 211 (1995) 260 264 j ( t ) = j o e - i,~t q> = q~1e - i,,~,we obtain
( j(t)=j~
V(t) =
d2q9 Ldq>) tp+LC-d~-+p-~, - itoL
1 - toZLC - itoL/p
jo e-i~'t,
where we have introduced the usual nonlinear inductance L = h/2ejc. Using j = a / E , E = V / d , C = e/d, it is straightforward to deduce the complex conductivity a(to) or the dielectric response e(co) = io/to, 1 - t o z L C - icoL/p a(to) = ~o
- icoLC
and ~(to)=eo
1 - - "(/)2 P+
,
where cop = 1 / x / - ~ and y- 1 = pC. From this derivation it is clear that any layered superconductor with a sufficiently low (Josephson) coupling between the planes will have a plasma edge at frequencies below the energy gap, and the electrodynamic properties are completely analogous to the classical results derived for extended single junctions, with the exception that the effective width of the junction d + 2 . 2 is replaced by the interlayer distance. An even simpler way to look at this problem is to use the Kramers-Kronig relations which state that causality requirements closely link the real and imaginary parts of the conductivity spectrum. If the real part is known for all frequencies then the imaginary part can be calculated and vice versa. If there is a dissipationless DC supercurrent, described by a delta function at zero frequency in cq (co) then it follows from the Kramers-Kronig relations that there should be a complimentary contribution to the imaginary part o2(co) proportional to 1/to. With e(to) = e + ia(co)/co this directly gives a plasmon-type of dielectric function. This also shows that the presence of a plasma edge is not restricted to pure Josephson coupled structures. In the classical bulk superconductors the same argument applies, but because the metallic conductivity is usually much larger a pole would occur at frequencies far above the gap, and the response in the superconducting state is very similar to that of the normal state. Note that in this case the plasma edge is due to a resonance in the charge density oscillation, in contrast to the Josephson case where the plasma edge is due to resonant oscillations of the local phase of the order parameter. Ferrell showed that the Josephson relations, including the so-called self-capacitive effects can be derived self-
263
consistently from the BCS theory [4]. This provides an elegant framework to extend the above analysis to frequencies comparable to the energy gap and !nclude the dephasing effects of an applied DC-current ld¢ = I~ sin(0). The basic result given by Ferrgll, valid at T = 0 and for a BCS s-wave gap A = 1.76kTc is a(to, ~) or,
i [(K(co) + n/2)cos(~b) 2co + 2E(to) -- K(CO) -- n / 2 - 2Coto2],
where Co = togC/cr, (C is the geometrical capacitance as introduced above and cog is the gap frequency)and K(to) and E(co) are the complete elliptic integrals ]of the first and second kind. For Lal.8sSro.lsCuO4 (top ~ co8/2) with no DC current (4> = 0) the result is virtually identical to ihe result of the simple Josephson model. From the considerations given above it follows that the plasma frequency is directly related to thei magnitude of the DC critical current density. The absolute value of the i measured jc(0) ~ 8 X 109 A/m 2 agrees well with the B ( ~ prediction Jc=(nA(T)/2ep)tanh(A(T)/2kT) for A(0) =l.76kTc and the optically determined a, = 8 f2-1 cm - !. However, we do find a slightly better fit to the temperature dependence if we assume a larger gap A(0) ~ 2 . 5 k T L The critical current density derived from magnetization hysteresis is about an order of magnitud~ lower [5]. This indicates that the spectroscopic deterrhination of Jc is more intrinsic in the sense that it is a n~ar-equilibrium technique with very small amplitudes oflhe probing (AC) current. In transport measurements self,field effects and possible local heating cannot be avoide~. In a magnetic field the transport jr is limited by the cmset of flux motion. Within this model it is possible to c~lculate the electrodynamic response for arbitrary frequency and temperature in a self-consistent way. In Fig. ~(b) we have calculated the reflectance versus temperatule for three representative frequencies, assuming a BCS gap, a Mattis-Bardeen conductivity ol(co)/a . with ~ frequency independent a,(to) and a small subgap conduOtance taken from a fit to the Kramers-Kronig data at T = 35 K. The agreement with the experimental data is quite satisfactory. We now turn to the influence of an external m~tgnetic field. First we consider the pairbreaking effect of the magnetic field. The exact value of the upper critical ]field is not known, but from extrapolations of the low-fi~ld data we estimate B¢2(Bllab) to be well above 100T.)In the Abrikosov-Gorkov theory this would lead to i about 3% decrease of T~ and 2% drop in A(0), or le~s than 1% change in the plasma frequency for a magr~etic field of
264
P.J.M. van Bentum et aL /Physica B 211 (1995) 260-264
25 T. The change in Tc is indeed of this order of magnitude, but the drop in oJp is obviously much larger, as indicated in Fig. 2. To resolve this discrepancy one has to include the interaction of the induced AC currents with the vortices. This problem is well-studied in the context of classical Josephson junctions [6]. For ideal Josephson vortices the equation of motion is governed by the sine-Gordon equation, which shows that the vortices move essentially as ballistic objects or solitons. In this case the dispersion relation for propagating electromagnetic waves becomes linear with a characteristic gap if the k-vector of the light matches the q-vector for the vortex lattice. Since the infrared reflectivity is essentially a k = 0 process this would indicate a rapid collapse of the plasma frequency with applied magnetic field. A comparison with the experimental data immediately shows that this is not the case, and the vortices must have a somewhat intermediate character between the Josephson and the Abrikosov limits. The popular description of the B II c transport properties in terms of 'pancake' vortices might be applicable for the Bi and TI high Tc superconductors, but is not correct for Lal.ssSro.lsCuO4. A detailed description for the electrodynamic response of an anisotropic superconductor in the presence of magnetic fields was given very recently by Tachiki et al. [7]. F o r the parallel magnetic field case, their basic result (in SI units) is given by e(~) _ 1 e
COp 2 [ ~1o+B o- 1 1-1 ' t~(co + i0 +) #022 Xp - kor/-- Mo) 2
where ~o is the unit flux, Bo the flux density, the xp and r/the pinning force constant and viscous drag-force coefficient, respectively. M is the vortex inertial mass per unit length. In their model the value for the plasma frequency is taken from an empirical 2-fluid description of the superconducting state. However, the interaction with the vortices is based on a quite general theory and should be applicable in the region intermediate between the Josephson and Abrikosov limits. The solid line in Fig. 2 is a fit according to this equation, for Be2 = 120 T, 2¢ = 2 x 10-s m, Xp = 500(Pa),
r / = l . 5 x l 0 - 1 ° ( P a s ) and M = 0 . These values are quite realistic when compared with transport results. Using Jc, tr = 5-10 s A/m 2 [5] and the relation jc.tr = Xp¢ab/~bo we find for the effective pinning length s c a l e Cab = 2 nm, in good agreement with literature data for the coherence length in the ab-plane. The viscosity can be estimated from the Bardeen-Stephen model: r / = ~boBc2 fin. With the Kramers-Kronig result tr, = 8 0 0 [ ~ - ~ m -1] and B¢2 = 120T this yields ~/= 1.9x 10-1°(Pas).
4. Conclusions
The observed low-frequency plasma edge in the c-axis dielectric response for Lal.ssSro.lsCuO4 can be attributed to a Josephson type of phase oscillations of the superconducting order parameter. The vortices introduced by a magnetic field parallel to the planes couple strongly to the electrodynamic response. The upper limit for the critical current is in good agreement with BCS predictions, while the pinning of the vortices indicates that these are intermediate between the Josephson and Abrikosov limits.
References
[1] R. Kleiner and P. Miiller, Phys. Rev. B 49 (1994) 1327. I-2] K. Tamasaku, Y. Nakamura and S. Uchida, Phys. Rev. Lett. 69 (1992) 1455. 1-3] A.M. Gerrits, A. Wittlin, V.H.M. Duyn, A.A. Menovsky, J.J.M. Franse and P.J.M. van Bentum, Physica C 235-240 (1994) 1117. [4] R.A. Ferrell, Phys. C 152 (1988) 10. [5] K. Kishio, Y. Nakayama, N. Motohira, T. Noda, T. Kobayashi, K. Kitazawa, K. Yamafuji, I. Tanaka and H. Kojima, Supercond. Sci. Technol. 5 (1992) S1, $69. [6] K.K. Likharev, Dynamics of Josephson Junctions and Circuits (Gordon and Breach, New York, 1986). 1-7] M. Tachiki, T. Koyama and S. Takahashi, Phys. Rev. B 50 (1994) 7065.