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Measurements of vortex-transport entropy in epitaxial Nd1.85Ce0.15CuO4−σ films evidence for quasiparticle bound state quantization in the vortex core
PHYSICA ELSEVIER
PhysicaC254 (1995) 175-180
Measurements of vortex-transport entropy in epitaxial Nd1.85Ce0.15CuO4_ ~ films Evidence for quasiparticle bound state quantization in the vortex core Xiuguang Jiang, Wu Jiang, S.N. Mao, R.L. Greene, T. Venkatesan, C.J. Lobb * Center for Superconductivity Research, Department of Physics, University of Maryland, College Park, MD 20742, USA Received 12 May 1995; revised manuscript received 23 August 1995
Abstract
We report measurements of the vortex-transport entropy in the high-temperature superconductor Nd].85CeoAsCuO4_~. The vortex transport entropy, determined from flux-flow Nernst effect and resistivity measurements, is about 100 times smaller than Maki's theoretical prediction. Using a simple microscopic model, we show that the entropy decrease is a result of the large energy level spacing between quasiparticle bound states in the vortex core, providing strong evidence for the discreteness of these states. A vortex in a superconductor consists of a circulating supercurrent around a small core of size comparable to the coherence length ~(T). Although this core usually is treated as a region with normal-metal density of states, Caroli et al. [1] showed theoretically in 1964 that the quasiparticle states are quantized due to confinement in the core. It was only a recent paper by Karra'i et al. which reported observing discrete states by a far-infrared resonance technique [2]. Because of the closeness between the observed energy level spacing and the superconducting gap [2], and the complication of anisotropic properties of the materials, there is still ambiguity regarding the interpretation of their data. The discrete nature of the energy levels in the vortex core should affect other physical properties as
* Corresponding author.
well. One particularly interesting property is the entropy carried by a moving vortex, which is primarily a measure of how many quasiparticle bound states exist in the core. These bound states result from local suppression of the superconducting gap in the core, which leads to a cone-shaped quantum well [3]. The standard Maki expression [4] for this vortex transport entropy implicitly treats the bound states in the core as continuous, and thus should overestimate the entropy when the level spacing ~E becomes comparable to or greater than the thermal energy kT. Because of their small coherence lengths, the level spacing between bound quasiparticle states in high-Tc superconductors is expected to be so large that it may not be treated as continuous. Thus, measurement of vortex-transport entropy in high-Tc materials provides important information on the nature of the bound quasiparticle states. In this paper, we report measurements of the
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X. Jiang et al. / Physica C 254 (1995) 175-180
flux-flow Nemst effect in epitaxial Ndl.85Ce0.15CuO 4_ ~ (NCCO) films. This experiment measures the vortex-transport entropy. As qualitatively predicted by the Maki expression, the measured vortextransport entropy S,b decreases rapidly as the magnetic field or temperature approaches Hc: or T~, respectively. Quantitatively, however, Maki's prediction is nearly two orders of magnitude larger than our measured values. We show that this large disagreement between experiment and theory is a consequence of the quasiparticle bound state quantization in the vortex core. To make a quantitative comparison, we calculate the entropy of a vortex with discrete states using statistical mechanics. These calculations are in good agreement with experiment. As a further test of our model, we calculate the transport entropy of a conventional superconductor Pb0.4In0.6 with a high density of core states, and get good agreement with both the measured value and the Maki prediction. The epitaxial c-axis oriented NCCO films (1200 .~) employed in this experiment are made by a pulsed-laser deposition technique [5]. The typical transition temperature at mid-point is 21.5 K, and the upper critical field at T = 0 K is about 6.0 T. The sample is cut into a rectangular shape of 2 × 7 mm 2, with a thermal gradient VT established along the longer side of the sample, as shown in the inset to Fig. l(a). (For comparison, we also show data from Sample #2, a patterned sample of 2 × 0.7 mm 2 with VT along its shorter side and from a single crystal, Sample #3.) The sample is mounted like a diving board with one end heat-sunk to a copper block and the rest stretching out in vacuum. The heater, Nemst voltage leads and thermocouple leads are attached to the sample in the way shown in the inset to Fig. 1. The external field is always perpendicular to the sample. The typical temperature gradient in our measurements is about 2 K / c m . Once a temperature gradient is established across the sample, vortices are moved by the thermodynamic force -S6VT from the hotter to the cooler end of the sample, where $6 is the vortex transport entropy. The Nernst electric field E, the vortex velocity Vv, and the magnetic field B are related by E= -Vv xB. Fig. l(a) shows normalized Nernst voltage, E/(dT/dx), as a function of temperature at four
"-" k~
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T (K) Fig. 1. (a) Normalized Nernst voltage as a function of temperature at B = 0.5, 1, 2, and 3 T, respectively. (b) Vortex-transport entropy as a function of temperature at fields listed above. Symbols both in (a) and (b) depict measured values while the solid lines are guides to the eye. The inset shows our experimental arrangement.
applied fields obtained from Sample #1. At each fixed B, the Nernst voltage increases first as temperature decreases. As the temperature decreases further, the Nemst voltage eventually goes to zero, an indication of flux pinning at lower temperature. Fig. l(b) shows transport entropy as a function of temperature from Sample #1. The vortex transport entropy is obtained from the Nernst effect and fluxflow resistivity data, using the following equation [61: ~0 = --
E
pf~ [ d T / d x [
,
(1)
where E is the Nernst electric field perpendicular to the thermal gradient dT/dx, and Pn and ~0 are the flux-flow resistivity and the flux quantum, respectively. As shown in Fig. l(b), the vortex-transport entropy increases as temperature decreases at fixed magnetic field. When the magnetic field increases, the transport entropy decreases at a fixed temperature. This reflects the fact that the transport entropy goes to zero as B ~ Be2. Since there are no vortices
177
X. Jiang et al./Physica C 254 (1995) 175-180 0.7
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5
B(tesla) Fig. 2. (a) Normalized Nernst voltage as a function of magnetic field from Sample #1. Inset: Data from an alternatively patterned film, Sample # 2 (O), and a single crystal, Sample #3 (O), see text. (b) Vortex-transport entropy as a function of magnetic field for Sample #1. The filled symbols ( ( 0 ) , ( • ) , ( • ) ) connected by dashed lines depict measured results, while the solid lines represent plots of the Maki expression (Eq. (3)).
when the sample is normal, the nearly temperatureindependent values for B = 2 T and 3 T in Fig, 1(a) at higher temperatures result from the normal-state Nemst effect. Fig. 2(a) shows the normalized Nernst voltage as a function of magnetic field at various temperatures obtained from Sample #1. At each fixed temperature, the Nernst voltage increases rapidly as the magnetic field increases from B = 0. It then starts to decrease after it peaks around H = H c 2 / 2 and reaches a minimum near H~2. For comparison, data from Sample #2, a patterned thin film of 2 X 0.7 mm 2 (with the temperature gradient established along the shorter side) of Tc = 22 K, measured at T = 17 K, and Sample #3, a single crystal of T~ = 23 K, measured T = 18.1 K are shown in the inset of Fig. 2(a). Evidently, at about the same normalized temperature T/T¢ -- 0.78, the Nemst voltages from the three samples are about the same. In addition, the magnitude of the normal-state Nernst effect (,-, 0.05 IxV/K, T near T~) is surprisingly large [7] compared with YBCO ( ~ 0.005 I~V/K, T near Tc). Fig. 2(b) shows transport entropy as a function of magnetic field for Sample #1. The transport-entropy data shown here were extracted by using the Nernst
where t¢ = A/~ is the Ginzburg-Landau parameter, Bc2(T) is the upper critical field, and L(T/Tc) depends upon temperature and l/~ [4] with l being the electron mean free path. The actual function of L(T/T¢) is rather complicated [5]. Numerical calculation shows that L(T/Tc) decreases as temperature decreases and becomes zero at T = 0. In the clean limit (l/~ >> 1) [4] such as in YBCO, L( ~ 1) = l/E, while in the dirty limit [4], L ( ~ 1) = 1. Specifically, I/~ is = 1 for NCCO and --- 10 for YBCO. Other parameters in the Maki expression for NCCO such a s BeE(T) , ~ ( 0 ) - - - 7 5 /~, and Tc are determined by our measurements. The upper critical field was obtained from a resistivity versus B measurement, the coherence length was obtained from Hc2 = ~b0/27r~ 2, T~ was the temperature at which the resistance is half of its onset value, and the penetration depth A(0)= 1300 .& is from a microwave measurement [8]. The measured transport entropy and Maki's prediction shown in Fig. 2(b) are dearly in strong disagreement. At T = 16.75 K and B = 1.5 T, for instance, the predicted value is S,~ = 0 . 6 X 10 -13 J / m K, using Be2 = 2:05 T and K = 18. The measured value, however, is S,~ = 0.5 X 10-]5 J / m K, more than 100 times smaller than the predicted one. The disagreement between theory and experiment varies with temperature and magnetic field. Nonetheless, as shown in Fig. 2(b), the Maki expression overestimates the vortex-transport entropy in NCCO by almost two orders of magnitude for nearly all temperatures and magnetic fields. To force Eq. (2) to agree with the experiment, A(0) would have to be 1.5 tJ,m, 11 times larger than the measured value [8]. Although the penetration depth varies slightly from
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X. Jiang et al. /Physica C 254 (1995) 175-180
sample to sample, it is unlikely to change by a factor of 11 [91. The disagreement between the Maki expression and experiment is not unique to NCCO. Although the results of vortex-transport entropy in high-T¢ materials [10-14] measured by different groups scatter somewhat, there is a common behavior: the measured values of transport entropy near He2 are consistently smaller than predicted by the Maki expression. The ratio between predicted values from the Maki expression (in the clean limit) and measured values near H¢2 is between 10 to 30 in YBCO [11,12,15]. This is in sharp contrast with the excellent agreement between theory and experiment in conventional superconductors [16,17]. If the Nernst effect were from thermally assisted vortex motion, while the flux-flow resistivity pff in Eq. (1) were from simple Bardeen-Stephen flux flow [18], the entropy extracted would be much smaller than its real value. This cannot be the case because the flux-flow resistivity pff changes little when the transport current is reduced. Moreover, our measurements extend to H---HoE, where pinning is negligible. We propose that the large discrepancy between measured vortex entropy and Eq. (2) results from the size quantization of quasiparticle bound states in the vortex core. Because of small coherence lengths in high-Tc cuprates, the energy difference between two adjacent states (of k z = 0) becomes comparable to or even greater than the thermal energy kT. Fewer available energy levels implies a reduction of entropy. Therefore, the Maki expression, which implicitly assumes bulk quasiparticle bound states in the vortex core, overestimates the vortex transport entropy in high-T~ materials. Physically, the vortex-transport entropy is the entropy difference between a superconductor with a single vortex and the superconductor in the Meissner state at the same field and temperature [19]. The main contribution to S,t, comes from bound quasiparticles in the vortex core. Other contributions, such as effects from the scattering of incoming quasiparticles by the core potential, are much smaller. We estimate the vortex transport entropy in the following way. First we use the standard Fermi statistics formula for the entropy [20]. In order to calculate the transport entropy, we restrict the sum to
those states which are transported with the vortex, the localized states in the vortex core. Thus, we consider only the states which are within + ,4 of the gap,
S~ -~ -~
-
]
I ~ l k Y + 1 + kT ln(1 + e - ~ / k r ) , 8=--AK~
(3) where the energy ~ is measured relative to the Fermi energy. Note that Eq. (3) includes the effects from both magnetic field and circulating current near the vortex core because of the self-consistency associated with the order parameter, magnetic field and circulating current near the vortex core. In a clean isotropic BCS superconductor, with many states in the core, the energy levels are given by [1] ,4 2
E~k, , =/a'Ev"1~/ _
k z2/ k 2v
'
(4)
for/~ << E F / A , where /~ = + ½, + -~. . . . + (n + ½), and hk z is the z direction momentum and k F = xrmA(O)~(O)/h 2. By using the above equations, we calculate the transport entropy of a vortex line for Pb0.4In0.6 alloy [16] with Tc --7 K, K = 5, and He2(0)= 0.7 T. At T = 4.5 K and B = 0.1 T, for instance, the predicted value by our model is S,~= 0.36 × 10 -13 J / m K, while the Maki prediction (in the dirty limit) is S~ = 0.66 × 10 -13 J / m K, both consistent with the measured value of S~ = 0.55 × 10 -13 J / m K. In high-To superconductors, however, Eq. (4), as well as some other expressions obtained in the clean isotropic limit, is not expected to hold. By using the BCS relation, k F = xrmA(O)~(O)/h 2, for instance, one derives a Fermi energy E F = 50 meV for NCCO, much smaller than the value 0.3 eV determined from photoemission measurements [21]. The reason for this disagreement is not quantitatively understood. We note here, however, that NCCO is highly anisotropic, and only moderately clean; thus it may not be described by the isotropic weak-coupling BCS theory. Instead of using Eq. (4), we propose a more general expression [2] based on the approximate
X. Jiang et al. /Physica C 254 (1995) 175-180
energy levels of particles confined to a vortex core of size r e , h2
Euk. z = tZrnr-" __ k z2/ k F 2
(5)
The parameter rc here is the effective size to which the bound quasiparticles are confined. By choosing rc(0) ~ ~(0), we estimate the level spacing 8 E - E1/2, 0 - E 1 / 2 , o --16.5 K. Thus, the only allowed 5 states are / z = + ½, + 3, and + 7We first calculate the transport entropy using Eqs. (3) and (5) with e=Ez.k~ The parameter r c is chosen to be the same as the coherence length ~(T) -=- SC(0)[1 - (T/Tc)2] -1/2. The temperature dependence of the energy gap is A ( T ) - - - A ( 0 ) [ 1 (T/T~)2] 1/2. At T = 16.75 K and B --- 0 T, the calculated transport entropy is 1.3 X 10 -14 J / m K. To compare the calculation with the measured value, we extrapolate the calculated entropy at magnetic field of B = 0 T to B = 1.5 T. We assume that transport entropy varies in magnetic field as S6(H, T)-~ S~(O, T)(1 - H / H c 2 ) , which is similar to the Maki expression. The extrapolated value at T = 16.75 K and B = 1.5 T is 3 × 10 -15 J / m K, within a factor of 6 of the measured value. Considering the simplicity of our calculation, the agreement between the calculation and the measurement is remarkable. Since the measured transport entropy is about six times smaller than our calculation, it suggests that the energy level spacing may be even greater than that found from Eq. (5). This is equivalent to saying that the effective core size is smaller than that found from G - L theory. By using the measured entropy and Eqs. (3) and (5), we can estimate this effective core size. At T = 16.75 K, foro instance, the parameter r c is found to be about 56 A, about half the value of SC(T)- 129 ,~ derived from the measured H¢2. When the core size is calculated self-consistently [3] or the Coulomb interaction is accounted for [22], the energy level spacings in the vortex core become larger, thus leading to a smaller effective core size r c. In addition, near H¢2 core quasiparticel states begin to overlap, and thus are no longer localized, further reducing the transport entropy. Thus, our calculation somewhat overestimates the transport entropy, since we assume vortices are isolated. Further, our model should not be used when
179
T--- To, since not only Eq. (5) is no longer true, but also other contributions [23] to the entropy have to considered as well. We find that the broadening of energy states in the vortex core due to a finite quasiparticle lifetime has little effect on the transport entropy. Physically, this is because the broadening does not change the total number of available states in the vortex core. By assuming that the energy level broadening is Gaussian and the broadening 6 equals the difference between adjacent energy levels, we calculated the transport entropies using discrete energy levels and broadened energy levels. The two results differ by less than 8%. We now calculate the transport entropy in the isotropic clean BCS limit using Eqs. (3) and (4). Here, the energy gap A(T) varies with temperature as given above. By using the parameters E v = 0.3 eV [21] and A(0) = 2.1 kTc [8] at T = 16.75 K and B = 1.5 T, we calculate S,~ = 0.9 X 10 -13 J / m K, a factor of ~- 100 greater than the measured value. On the other hand, this value is consistent with the value of -- 0.6 × 10 -13 J / m K obtained from Eq. (2) for B = 1.5 T. The agreement between the Maki expression and the entropy calculated from Eqs. (3) and (4) is not coincidence: as pointed out in Ref. [1], the effective density of states for many-level vortex cores is equivalent to that in a volume of normal metal of size ~. With a similar argument, we can also explain the large difference between the measured vortex-transport entropy in YBCO and Maki's prediction. By using E F -- 0.5 eV, m* = 2me, rc(0) -- SCab(0)= 16 A, Tc = 90 K, and A(0) = 2.5 kT~ [24], we find that the transport entropy calculated in the isotropic limit (expressed by Eq. (4) in the clean limit) is about 20 times larger than that Calculated using Eq. (5). This is consistent with the fact that the vortex-transport entropy predicted by the Maki expression is 10 to 30 times greater than the measured values. Previous reports [12] on YBCO report agreement between measured transport entropy and the Maki expression. This agreement was obtained by incorrectly using the dirty-limit formula. Our analysis of the YBCO data using the correct clean-limit formula indicates that the measured entropy is more than 10 times smaller than predicted. It is puzzling that YBCO, with smaller coherence
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X. Jiang et al. /Physica C 254 (1995) 175-180
length, has a smaller disagreement with the Maki expression than NCCO. Part of this puzzle may be caused by the higher temperatures (T/Tc) used in YBCO. In addition, we speculate that the proposed d wave order parameter in YBCO might further modify the vortex entropy. In summary, we have shown that the measured vortex-transport entropy of NCCO is nearly 100 times smaller than the theoretical prediction by Maki, and that other high-To superconductors have smaller, but still substantial, disagreement with the theory. We have quantitatively shown that the size quantization of the quasiparticle states in the vortex core causes this discrepancy.
Acknowledgements The authors gladly acknowledge useful conversations with S. Das Sarma, H.D. Drew, H. Hao, T. Hsu, and F. Zhang. This work is supported in part by NSF Grants DMR-9510464 and DMR-9510475.
References [1] C. Caroli, P.G. De Gennes and J. Matricon, Phys. Lett. 9 (1964) 307. [2] K. Karra'i, E.J. Choi, F. Dunmore, S. Liu, H.D. Drew, Qi LI, D.B. Fenner, Y.D. Zhu and F.-C. Zhang, Phys. Rev. Lett. 69 (1992) 152. [3] F. Gygi and M. Shliiter, Phys. Rev. B 43 (1991) 7609 and references therein. [4] K. Maki, J. Low T. Phys. 1 (1969) 45; idem, Prog. Theor. Phys. 41 (1969) 902. [5] S.N. Mao, X.X. Xi, Qi Li, T. Venkatesan, D. Prasad Beesabathina, L. Salamanca-Riba and X.D. Wu, Appl. Phys. Lett. 61 (1992) 2353. [6] R.P. Huebener, Magnetic Flux Structures in Superconductors (Springer, Berlin, Heidelberg, 1979).
[7] X. Jiang, W. Jiang, S.-N. Mao, X.X. Xi, R.L. Greene, T. Venkatesan and C.J. Lobb, Physica B 194-196 (1994) 2305. A more detailed discussion will be given in a future paper. [8] D. Ho Wu, J. Mao, S.-N. Mao, J.L. Peng, X.X. Xi, T. Venkatesan, R.L. Greene and S.M. Anlage, Phys. Rev. Lett. 70 (1993) 85. [9] C. Almasan M.C. de Andrade, Y. Dalichaouch, J.J. Neumeier, C.L. Seaman, M.B. Maple, R.P. Guertin, M.V. Kuric and J.C. Garland, Phys. Rev. Lett. 69 (1992) 3812. [10] T.T.M. Palstra, B. Batlogg, L.F. Schneemeyer and J.V. Weszczak, Phys. Rev. l_~tt. 64 (1990) 3090. [11] S.J. Hagen, C.J. Lobb, R.L. Greene, M.G. Forrester and L Talvacchio, Phys. Rev. B 42 (1990) 6777. [12] M. Zeh, H.-C. Ri, F. Kober, R.P. Huebener, A.V. Ustinov, J. Mannhart, R. Gross and A. Gupta, Phys. Rev. Lett. 64 (1990) 3195; M. Oussena, R. Gagnon, Y. Wang and M. Aubin, Phys. Rev. B 46 (1992) 528; H.-C. Ri, R. Gross, F. Gollnik, A. Beck and R.P. Huebener, Phys. Rev. B 50 (1994) 3312 and references therein. [13] N.V. Zavaritsky, A.V. Samoilov and A.A. Yurgens, Physica C 180 (1991) 417. [14] G.Y. Logvenov, M. Hartman and R.P. Huebener, Phys. Rev. B 46 (1993) 11102. [15] In Ref. [11], when calculating the Nemst electric field from measured voltages, the authors mistakenly used a distance 30 times too small; the correct result should be 30 times smaller. [16] P.R. Solomon and F.A. Otter Jr., Phys. Rev. 164 (1967) 608. [17] F. Vidal, Phys. Rev. B 8 (1973) 1982 and references therein. [18] M.N. Kunchur, D.K. Christen and J.M. Philips, Phys. Rev. Lett. 70 (1993) 998 and references therein. [19] A.L. Fetter and P.C. Hohenberg, Superconductivity, ed. R.D. Parks (Marcel Dekker, New York, 1969). [20] R.K. Pathria, Statistical Mechanics (Pergamon, Oxford, New York, 1972). [21] D.M. King, Z.-X. Shen, D.S. Dessau, B.O. Wells, W.E. Spicer, A.J. Arko, D.S. Marshall, J. DiCarlo, A.G. Loeser, C.H. Park, E.R. Rather, J.L. Peng, Z.Y. Li and R.L. Greene, Phys. Rev. Lett. 70 (1993) 3159. [22] T. Hsu, private communication. [23] J. Bardeen, R. Kiimmel, A.E. Jacobs and L. Tewordt, Phys. Rev. 187 (1969) 556. [24] G. Bums, High-Temperature Superconductivity (Academic Press, San Diego, 1992).
PhysicaC254 (1995) 175-180
Measurements of vortex-transport entropy in epitaxial Nd1.85Ce0.15CuO4_ ~ films Evidence for quasiparticle bound state quantization in the vortex core Xiuguang Jiang, Wu Jiang, S.N. Mao, R.L. Greene, T. Venkatesan, C.J. Lobb * Center for Superconductivity Research, Department of Physics, University of Maryland, College Park, MD 20742, USA Received 12 May 1995; revised manuscript received 23 August 1995
Abstract
We report measurements of the vortex-transport entropy in the high-temperature superconductor Nd].85CeoAsCuO4_~. The vortex transport entropy, determined from flux-flow Nernst effect and resistivity measurements, is about 100 times smaller than Maki's theoretical prediction. Using a simple microscopic model, we show that the entropy decrease is a result of the large energy level spacing between quasiparticle bound states in the vortex core, providing strong evidence for the discreteness of these states. A vortex in a superconductor consists of a circulating supercurrent around a small core of size comparable to the coherence length ~(T). Although this core usually is treated as a region with normal-metal density of states, Caroli et al. [1] showed theoretically in 1964 that the quasiparticle states are quantized due to confinement in the core. It was only a recent paper by Karra'i et al. which reported observing discrete states by a far-infrared resonance technique [2]. Because of the closeness between the observed energy level spacing and the superconducting gap [2], and the complication of anisotropic properties of the materials, there is still ambiguity regarding the interpretation of their data. The discrete nature of the energy levels in the vortex core should affect other physical properties as
* Corresponding author.
well. One particularly interesting property is the entropy carried by a moving vortex, which is primarily a measure of how many quasiparticle bound states exist in the core. These bound states result from local suppression of the superconducting gap in the core, which leads to a cone-shaped quantum well [3]. The standard Maki expression [4] for this vortex transport entropy implicitly treats the bound states in the core as continuous, and thus should overestimate the entropy when the level spacing ~E becomes comparable to or greater than the thermal energy kT. Because of their small coherence lengths, the level spacing between bound quasiparticle states in high-Tc superconductors is expected to be so large that it may not be treated as continuous. Thus, measurement of vortex-transport entropy in high-Tc materials provides important information on the nature of the bound quasiparticle states. In this paper, we report measurements of the
0921-4534//95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 3 4 ( 9 5 ) 0 0 5 3 1 - 5
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X. Jiang et al. / Physica C 254 (1995) 175-180
flux-flow Nemst effect in epitaxial Ndl.85Ce0.15CuO 4_ ~ (NCCO) films. This experiment measures the vortex-transport entropy. As qualitatively predicted by the Maki expression, the measured vortextransport entropy S,b decreases rapidly as the magnetic field or temperature approaches Hc: or T~, respectively. Quantitatively, however, Maki's prediction is nearly two orders of magnitude larger than our measured values. We show that this large disagreement between experiment and theory is a consequence of the quasiparticle bound state quantization in the vortex core. To make a quantitative comparison, we calculate the entropy of a vortex with discrete states using statistical mechanics. These calculations are in good agreement with experiment. As a further test of our model, we calculate the transport entropy of a conventional superconductor Pb0.4In0.6 with a high density of core states, and get good agreement with both the measured value and the Maki prediction. The epitaxial c-axis oriented NCCO films (1200 .~) employed in this experiment are made by a pulsed-laser deposition technique [5]. The typical transition temperature at mid-point is 21.5 K, and the upper critical field at T = 0 K is about 6.0 T. The sample is cut into a rectangular shape of 2 × 7 mm 2, with a thermal gradient VT established along the longer side of the sample, as shown in the inset to Fig. l(a). (For comparison, we also show data from Sample #2, a patterned sample of 2 × 0.7 mm 2 with VT along its shorter side and from a single crystal, Sample #3.) The sample is mounted like a diving board with one end heat-sunk to a copper block and the rest stretching out in vacuum. The heater, Nemst voltage leads and thermocouple leads are attached to the sample in the way shown in the inset to Fig. 1. The external field is always perpendicular to the sample. The typical temperature gradient in our measurements is about 2 K / c m . Once a temperature gradient is established across the sample, vortices are moved by the thermodynamic force -S6VT from the hotter to the cooler end of the sample, where $6 is the vortex transport entropy. The Nernst electric field E, the vortex velocity Vv, and the magnetic field B are related by E= -Vv xB. Fig. l(a) shows normalized Nernst voltage, E/(dT/dx), as a function of temperature at four
"-" k~
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, . , • , B@ B..n,t volt.a. H,,,t,, 1T
0.4
~'%
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m
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0.0 100
. . . . . . . . .
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,
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.
.
.
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'
T
0.5T
16
20
i
T (K) Fig. 1. (a) Normalized Nernst voltage as a function of temperature at B = 0.5, 1, 2, and 3 T, respectively. (b) Vortex-transport entropy as a function of temperature at fields listed above. Symbols both in (a) and (b) depict measured values while the solid lines are guides to the eye. The inset shows our experimental arrangement.
applied fields obtained from Sample #1. At each fixed B, the Nernst voltage increases first as temperature decreases. As the temperature decreases further, the Nemst voltage eventually goes to zero, an indication of flux pinning at lower temperature. Fig. l(b) shows transport entropy as a function of temperature from Sample #1. The vortex transport entropy is obtained from the Nernst effect and fluxflow resistivity data, using the following equation [61: ~0 = --
E
pf~ [ d T / d x [
,
(1)
where E is the Nernst electric field perpendicular to the thermal gradient dT/dx, and Pn and ~0 are the flux-flow resistivity and the flux quantum, respectively. As shown in Fig. l(b), the vortex-transport entropy increases as temperature decreases at fixed magnetic field. When the magnetic field increases, the transport entropy decreases at a fixed temperature. This reflects the fact that the transport entropy goes to zero as B ~ Be2. Since there are no vortices
177
X. Jiang et al./Physica C 254 (1995) 175-180 0.7
~" N "~
0.6 0.5 0.4
o~[ I/-%,
Is.75K
0.3 0.2
I
/\
.
/~..o.o~'~
voltage data minus the normal-state contribution, where the normal-state contribution is estimated by fitting the data for H > Hc2 (4 T ~
._
.
.
.
o.1 o.o, 1oo
7O
~,
"m.
~.
"~.
,
0.01 0
1
".. e.
12.73K
"-~/-..
S~ =
1 10.78K
".....
3
p.0 T
1.16(2K 2 - 1) + 1
'
(2)
(b
....
2
~o [B¢2(T)-B]L(T/T~)
4
5
B(tesla) Fig. 2. (a) Normalized Nernst voltage as a function of magnetic field from Sample #1. Inset: Data from an alternatively patterned film, Sample # 2 (O), and a single crystal, Sample #3 (O), see text. (b) Vortex-transport entropy as a function of magnetic field for Sample #1. The filled symbols ( ( 0 ) , ( • ) , ( • ) ) connected by dashed lines depict measured results, while the solid lines represent plots of the Maki expression (Eq. (3)).
when the sample is normal, the nearly temperatureindependent values for B = 2 T and 3 T in Fig, 1(a) at higher temperatures result from the normal-state Nemst effect. Fig. 2(a) shows the normalized Nernst voltage as a function of magnetic field at various temperatures obtained from Sample #1. At each fixed temperature, the Nernst voltage increases rapidly as the magnetic field increases from B = 0. It then starts to decrease after it peaks around H = H c 2 / 2 and reaches a minimum near H~2. For comparison, data from Sample #2, a patterned thin film of 2 X 0.7 mm 2 (with the temperature gradient established along the shorter side) of Tc = 22 K, measured at T = 17 K, and Sample #3, a single crystal of T~ = 23 K, measured T = 18.1 K are shown in the inset of Fig. 2(a). Evidently, at about the same normalized temperature T/T¢ -- 0.78, the Nemst voltages from the three samples are about the same. In addition, the magnitude of the normal-state Nernst effect (,-, 0.05 IxV/K, T near T~) is surprisingly large [7] compared with YBCO ( ~ 0.005 I~V/K, T near Tc). Fig. 2(b) shows transport entropy as a function of magnetic field for Sample #1. The transport-entropy data shown here were extracted by using the Nernst
where t¢ = A/~ is the Ginzburg-Landau parameter, Bc2(T) is the upper critical field, and L(T/Tc) depends upon temperature and l/~ [4] with l being the electron mean free path. The actual function of L(T/T¢) is rather complicated [5]. Numerical calculation shows that L(T/Tc) decreases as temperature decreases and becomes zero at T = 0. In the clean limit (l/~ >> 1) [4] such as in YBCO, L( ~ 1) = l/E, while in the dirty limit [4], L ( ~ 1) = 1. Specifically, I/~ is = 1 for NCCO and --- 10 for YBCO. Other parameters in the Maki expression for NCCO such a s BeE(T) , ~ ( 0 ) - - - 7 5 /~, and Tc are determined by our measurements. The upper critical field was obtained from a resistivity versus B measurement, the coherence length was obtained from Hc2 = ~b0/27r~ 2, T~ was the temperature at which the resistance is half of its onset value, and the penetration depth A(0)= 1300 .& is from a microwave measurement [8]. The measured transport entropy and Maki's prediction shown in Fig. 2(b) are dearly in strong disagreement. At T = 16.75 K and B = 1.5 T, for instance, the predicted value is S,~ = 0 . 6 X 10 -13 J / m K, using Be2 = 2:05 T and K = 18. The measured value, however, is S,~ = 0.5 X 10-]5 J / m K, more than 100 times smaller than the predicted one. The disagreement between theory and experiment varies with temperature and magnetic field. Nonetheless, as shown in Fig. 2(b), the Maki expression overestimates the vortex-transport entropy in NCCO by almost two orders of magnitude for nearly all temperatures and magnetic fields. To force Eq. (2) to agree with the experiment, A(0) would have to be 1.5 tJ,m, 11 times larger than the measured value [8]. Although the penetration depth varies slightly from
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X. Jiang et al. /Physica C 254 (1995) 175-180
sample to sample, it is unlikely to change by a factor of 11 [91. The disagreement between the Maki expression and experiment is not unique to NCCO. Although the results of vortex-transport entropy in high-T¢ materials [10-14] measured by different groups scatter somewhat, there is a common behavior: the measured values of transport entropy near He2 are consistently smaller than predicted by the Maki expression. The ratio between predicted values from the Maki expression (in the clean limit) and measured values near H¢2 is between 10 to 30 in YBCO [11,12,15]. This is in sharp contrast with the excellent agreement between theory and experiment in conventional superconductors [16,17]. If the Nernst effect were from thermally assisted vortex motion, while the flux-flow resistivity pff in Eq. (1) were from simple Bardeen-Stephen flux flow [18], the entropy extracted would be much smaller than its real value. This cannot be the case because the flux-flow resistivity pff changes little when the transport current is reduced. Moreover, our measurements extend to H---HoE, where pinning is negligible. We propose that the large discrepancy between measured vortex entropy and Eq. (2) results from the size quantization of quasiparticle bound states in the vortex core. Because of small coherence lengths in high-Tc cuprates, the energy difference between two adjacent states (of k z = 0) becomes comparable to or even greater than the thermal energy kT. Fewer available energy levels implies a reduction of entropy. Therefore, the Maki expression, which implicitly assumes bulk quasiparticle bound states in the vortex core, overestimates the vortex transport entropy in high-T~ materials. Physically, the vortex-transport entropy is the entropy difference between a superconductor with a single vortex and the superconductor in the Meissner state at the same field and temperature [19]. The main contribution to S,t, comes from bound quasiparticles in the vortex core. Other contributions, such as effects from the scattering of incoming quasiparticles by the core potential, are much smaller. We estimate the vortex transport entropy in the following way. First we use the standard Fermi statistics formula for the entropy [20]. In order to calculate the transport entropy, we restrict the sum to
those states which are transported with the vortex, the localized states in the vortex core. Thus, we consider only the states which are within + ,4 of the gap,
S~ -~ -~
-
]
I ~ l k Y + 1 + kT ln(1 + e - ~ / k r ) , 8=--AK~
(3) where the energy ~ is measured relative to the Fermi energy. Note that Eq. (3) includes the effects from both magnetic field and circulating current near the vortex core because of the self-consistency associated with the order parameter, magnetic field and circulating current near the vortex core. In a clean isotropic BCS superconductor, with many states in the core, the energy levels are given by [1] ,4 2
E~k, , =/a'Ev"1~/ _
k z2/ k 2v
'
(4)
for/~ << E F / A , where /~ = + ½, + -~. . . . + (n + ½), and hk z is the z direction momentum and k F = xrmA(O)~(O)/h 2. By using the above equations, we calculate the transport entropy of a vortex line for Pb0.4In0.6 alloy [16] with Tc --7 K, K = 5, and He2(0)= 0.7 T. At T = 4.5 K and B = 0.1 T, for instance, the predicted value by our model is S,~= 0.36 × 10 -13 J / m K, while the Maki prediction (in the dirty limit) is S~ = 0.66 × 10 -13 J / m K, both consistent with the measured value of S~ = 0.55 × 10 -13 J / m K. In high-To superconductors, however, Eq. (4), as well as some other expressions obtained in the clean isotropic limit, is not expected to hold. By using the BCS relation, k F = xrmA(O)~(O)/h 2, for instance, one derives a Fermi energy E F = 50 meV for NCCO, much smaller than the value 0.3 eV determined from photoemission measurements [21]. The reason for this disagreement is not quantitatively understood. We note here, however, that NCCO is highly anisotropic, and only moderately clean; thus it may not be described by the isotropic weak-coupling BCS theory. Instead of using Eq. (4), we propose a more general expression [2] based on the approximate
X. Jiang et al. /Physica C 254 (1995) 175-180
energy levels of particles confined to a vortex core of size r e , h2
Euk. z = tZrnr-" __ k z2/ k F 2
(5)
The parameter rc here is the effective size to which the bound quasiparticles are confined. By choosing rc(0) ~ ~(0), we estimate the level spacing 8 E - E1/2, 0 - E 1 / 2 , o --16.5 K. Thus, the only allowed 5 states are / z = + ½, + 3, and + 7We first calculate the transport entropy using Eqs. (3) and (5) with e=Ez.k~ The parameter r c is chosen to be the same as the coherence length ~(T) -=- SC(0)[1 - (T/Tc)2] -1/2. The temperature dependence of the energy gap is A ( T ) - - - A ( 0 ) [ 1 (T/T~)2] 1/2. At T = 16.75 K and B --- 0 T, the calculated transport entropy is 1.3 X 10 -14 J / m K. To compare the calculation with the measured value, we extrapolate the calculated entropy at magnetic field of B = 0 T to B = 1.5 T. We assume that transport entropy varies in magnetic field as S6(H, T)-~ S~(O, T)(1 - H / H c 2 ) , which is similar to the Maki expression. The extrapolated value at T = 16.75 K and B = 1.5 T is 3 × 10 -15 J / m K, within a factor of 6 of the measured value. Considering the simplicity of our calculation, the agreement between the calculation and the measurement is remarkable. Since the measured transport entropy is about six times smaller than our calculation, it suggests that the energy level spacing may be even greater than that found from Eq. (5). This is equivalent to saying that the effective core size is smaller than that found from G - L theory. By using the measured entropy and Eqs. (3) and (5), we can estimate this effective core size. At T = 16.75 K, foro instance, the parameter r c is found to be about 56 A, about half the value of SC(T)- 129 ,~ derived from the measured H¢2. When the core size is calculated self-consistently [3] or the Coulomb interaction is accounted for [22], the energy level spacings in the vortex core become larger, thus leading to a smaller effective core size r c. In addition, near H¢2 core quasiparticel states begin to overlap, and thus are no longer localized, further reducing the transport entropy. Thus, our calculation somewhat overestimates the transport entropy, since we assume vortices are isolated. Further, our model should not be used when
179
T--- To, since not only Eq. (5) is no longer true, but also other contributions [23] to the entropy have to considered as well. We find that the broadening of energy states in the vortex core due to a finite quasiparticle lifetime has little effect on the transport entropy. Physically, this is because the broadening does not change the total number of available states in the vortex core. By assuming that the energy level broadening is Gaussian and the broadening 6 equals the difference between adjacent energy levels, we calculated the transport entropies using discrete energy levels and broadened energy levels. The two results differ by less than 8%. We now calculate the transport entropy in the isotropic clean BCS limit using Eqs. (3) and (4). Here, the energy gap A(T) varies with temperature as given above. By using the parameters E v = 0.3 eV [21] and A(0) = 2.1 kTc [8] at T = 16.75 K and B = 1.5 T, we calculate S,~ = 0.9 X 10 -13 J / m K, a factor of ~- 100 greater than the measured value. On the other hand, this value is consistent with the value of -- 0.6 × 10 -13 J / m K obtained from Eq. (2) for B = 1.5 T. The agreement between the Maki expression and the entropy calculated from Eqs. (3) and (4) is not coincidence: as pointed out in Ref. [1], the effective density of states for many-level vortex cores is equivalent to that in a volume of normal metal of size ~. With a similar argument, we can also explain the large difference between the measured vortex-transport entropy in YBCO and Maki's prediction. By using E F -- 0.5 eV, m* = 2me, rc(0) -- SCab(0)= 16 A, Tc = 90 K, and A(0) = 2.5 kT~ [24], we find that the transport entropy calculated in the isotropic limit (expressed by Eq. (4) in the clean limit) is about 20 times larger than that Calculated using Eq. (5). This is consistent with the fact that the vortex-transport entropy predicted by the Maki expression is 10 to 30 times greater than the measured values. Previous reports [12] on YBCO report agreement between measured transport entropy and the Maki expression. This agreement was obtained by incorrectly using the dirty-limit formula. Our analysis of the YBCO data using the correct clean-limit formula indicates that the measured entropy is more than 10 times smaller than predicted. It is puzzling that YBCO, with smaller coherence
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X. Jiang et al. /Physica C 254 (1995) 175-180
length, has a smaller disagreement with the Maki expression than NCCO. Part of this puzzle may be caused by the higher temperatures (T/Tc) used in YBCO. In addition, we speculate that the proposed d wave order parameter in YBCO might further modify the vortex entropy. In summary, we have shown that the measured vortex-transport entropy of NCCO is nearly 100 times smaller than the theoretical prediction by Maki, and that other high-To superconductors have smaller, but still substantial, disagreement with the theory. We have quantitatively shown that the size quantization of the quasiparticle states in the vortex core causes this discrepancy.
Acknowledgements The authors gladly acknowledge useful conversations with S. Das Sarma, H.D. Drew, H. Hao, T. Hsu, and F. Zhang. This work is supported in part by NSF Grants DMR-9510464 and DMR-9510475.
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