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Precursor magnetism for high critical temperature superconductors
Physica C 408–410 (2004) 441–442 www.elsevier.com/locate/physc
Precursor magnetism for high critical temperature superconductors E.V.L. de Mello *, J.L. Gonz alez Departamento de Fısica, Universidade Federal Fluminense, Niteroi, RJ 24210-340, Brazil
Abstract Most of the high critical temperature superconductors (HTSC) have an intrinsic inhomogeneity in their doping level inside the samples. As one of the consequences a large diamagnetic signal above the critical temperature Tc has been measured. Here we use the critical-state model to the local superconducting domains between the pseudogap temperature T and Tc and show that the resulting diamagnetic signal is in agreement with the experimental results. Ó 2004 Elsevier B.V. All rights reserved. PACS: 74.20.)z; 74.25.)q; 74.25.Ha; 74.72.)h Keywords: Diamagnetism of normal state; Disordered cuprates; Critical-state model
1. Introduction Recent magnetic imaging through a scanning superconducting quantum device (SQUID) microscopy has displayed a static Meissner effect at temperatures as large as three times the Tc of an underdoped LSCO film [1]. Following up SQUID magnetization measurements on YBCO and LSCO crystals [2,3] have shown a rather high magnetic response which, due to its large signal, cannot be attributed to the fluctuating superconducting magnetization. This is an unconventional behavior and the interpretation was that such strong magnetic response was due to the formation of superconducting islands between T and Tc [2,3]. Based on the STM/S experiments [1], we have recently proposed a bi-modal distribution of the charges inside a given compound in order to model the charge distribution for a family of HTSC compounds [4]. The local variations in the doping level yields a distribution of local Tc ’s. We assume that the maximum Tc is exactly
*
Corresponding author. Tel.: +55-21-620-7729; fax: +55-21620-3881. E-mail address: [email protected]ff.br (E.V.L. de Mello).
T for a compound with doping level nm . Here we show that the diamagnetic signal measured above Tc by several groups [1–3] and which was found not to follow the fluctuation magnetization for a layered superconductor can be understood within this picture of a superconductor formed by static domains or droplets with a spatial varying Tc ðrÞ. Furthermore the droplets follow the critical-state model (CSM) [5] to the magnetization response under the applied fields. We demonstrate that this procedure is able to explain and reproduce the precursor diamagnetism behavior as recently measured [2,3].
2. The model In order to estimate the MðBÞ we follow the ideas of the CSM to each superconducting droplet between Tc and T . Upon applying an external magnetic field, a critical superconducting current ðJc Þ is established of the form Jc ðBÞ ¼ aðT Þ=B. For simplicity we take these superconducting droplets as cylinders of radius R, which is sufficient small in order to have a constant charge density n and consequently the critical temperature Tc ðnÞ is the same within such cylinder region. The CSM approach [5] leads to:
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.03.028
442
0.000 3
M2 ðBÞ ¼
0.005
ð1Þ
for B 6 Bc1
B 4B 8B þ ; l0 5l0 B2 15l0 B4
B 4B M3 ðBÞ ¼ l0 15l0
-0.005
5
Bc1 6 B 6 B
ð2Þ
2 5=2 ! B5 B3 B 2 5 5 3 2 2 1 B B B
for B 6 B 6 Bc2
ð3Þ
-0.015
nc
where P ðnÞ is the distribution function for the charge distribution and nc is the onset of superconductivity ðnc ¼ 0:05Þ.
0.0000
-0.020
27.8 K 28.0 K
-0.0001
-0.025 -0.0002
-0.030 -0.035
where B is the applied external field to produce full penetration inside a cylindrical superconducting droplet. The dependence of the local critical temperatures Tc ðnÞ is that of the T ðnm Þ ðTc ðnÞ ¼ T0 b ðn nc ÞÞ. Since Tc ðnÞ is constant inside a superconducting cylindrical droplet, the critical fields (Bc1 and Bc2 ) will have their temperature dependence according to the Ginzburg–Landau theory, e.g., Bc1 ðT Þ ¼ Bc1 ð0Þ½ð1 T =Tc ðnÞ and Bc2 ðT Þ ¼ Bc2 ð0Þ½ð1 T =Tc ðnÞ. Taking into account the dependence of Tc ðnÞ on n, we arrive at the expressions for the critical fields Bc1 ðT ; nÞ and Bc2 ðT ; nÞ. In the low field regime the superconducting clusters will contribute to the magnetization of the sample in three forms: there are some clusters, which are not penetrated by the field B, that is B 6 Bc1 ðT ; nÞ and they contribute to the magnetization with perfect diamagnetism (Eq. (1)). The second group of clusters have their Bc1 ðT ; nÞ lower than the applied field but B is also lower than B ðn; T Þ. This group is partially penetrated by the field and they contribute to MðBÞ according Eq. (2). Lastly, there are some superconducting granules for which the applied field is higher than B ðT ; nÞ but also lower than Bc2 ðT ; nÞ. Only the droplets with n bigger than nmax do not contribute to the sample’s magnetization because their superconductivity is destroyed by the field B. Thus we expect that Z nmax ðBc2 Þ MðT ; BÞ ¼ P ðnÞMðB; T ; nÞ dn ð4Þ
T c =26.86 K
-0.010 3
B l0
M(emu/cm )
M1 ðBÞ ¼
E.V.L. de Mello, J.L. Gonzalez / Physica C 408–410 (2004) 441–442
-0.040
27.0 27.2 27.6 28.0
-0.045 0.000
K K K K
-0.0003
-0.0004
0.000
0.001
0.002
0.001
0.003
0.002
0.003
0.004
0.004
0.005
0.005
B(T)
Fig. 1. Magnetization for appropriated parameters to the overdoped LSCO as calculated from Eqs. (7)–(11). This result is to be compared with the measurements from Ref. [3].
sured by Lascialfari et al. [3]. The qualitative features of the measurements are entirely reproduced and are simply explained by the CSM: at low field the perfect diamagnetism is expected for fields lower than Bc1 . By the same token, the penetrating field B should not be very strong what decreases rapidly the overall diamagnetic signal for field much weaker than Bc2 . As the applied field increases, the magnetic response dies off and is reduced to the fluctuations. This is the reason why MðBÞ has a minimum ðBup Þ at very low applied fields [2,3] as it is shown in Fig. 1.
4. Conclusion The CSM results for a HTSC under an applied field demonstrated that, as concluded also from other different calculations [2,3], the measured magnetization curves, the Bup fields and the STM magnetic imaging results are interpreted very simply through the formation of static superconducting islands.
References 3. Results and discussion The above theory was developed to model the measured magnetization curves of the La1x Srx CuO4 family of compounds. In Fig. 1 we plot the results of the CSM with the parameter which corresponds to a nm ¼ 0:2 and Tc ¼ 26:86 K. The values of Bc1 ð0Þ and B ð0Þ are not found in the literature and were chosen in order to obtain a MðBÞ curve with the same features as those measured on an underdoped LSCO compound mea-
[1] I. Iguchi, I. Yamaguchi, A. Sugimoto, Nature 412 (2001) 420. [2] A. Lascialfari, A. Rigamonti, L. Romano, P. Tedesco, A. Varlamov, D. Embriaco, Phys. Rev. B 65 (2002) 144523. [3] A. Lascialfari, A. Rigamonti, L. Romano, A. Varlamov, I. Zucca, Phys. Rev. B 68 (2003) 100505(R). [4] E.V.L. de Mello, E.S. Caixeiro, J.L. Gonzalez, Phys. Rev. B 67 (2003) 024502. [5] Michael Tinkham, Introduction to superconductivity, McGraw-Hill Inc., New York, 1975.
Precursor magnetism for high critical temperature superconductors E.V.L. de Mello *, J.L. Gonz alez Departamento de Fısica, Universidade Federal Fluminense, Niteroi, RJ 24210-340, Brazil
Abstract Most of the high critical temperature superconductors (HTSC) have an intrinsic inhomogeneity in their doping level inside the samples. As one of the consequences a large diamagnetic signal above the critical temperature Tc has been measured. Here we use the critical-state model to the local superconducting domains between the pseudogap temperature T and Tc and show that the resulting diamagnetic signal is in agreement with the experimental results. Ó 2004 Elsevier B.V. All rights reserved. PACS: 74.20.)z; 74.25.)q; 74.25.Ha; 74.72.)h Keywords: Diamagnetism of normal state; Disordered cuprates; Critical-state model
1. Introduction Recent magnetic imaging through a scanning superconducting quantum device (SQUID) microscopy has displayed a static Meissner effect at temperatures as large as three times the Tc of an underdoped LSCO film [1]. Following up SQUID magnetization measurements on YBCO and LSCO crystals [2,3] have shown a rather high magnetic response which, due to its large signal, cannot be attributed to the fluctuating superconducting magnetization. This is an unconventional behavior and the interpretation was that such strong magnetic response was due to the formation of superconducting islands between T and Tc [2,3]. Based on the STM/S experiments [1], we have recently proposed a bi-modal distribution of the charges inside a given compound in order to model the charge distribution for a family of HTSC compounds [4]. The local variations in the doping level yields a distribution of local Tc ’s. We assume that the maximum Tc is exactly
*
Corresponding author. Tel.: +55-21-620-7729; fax: +55-21620-3881. E-mail address: [email protected]ff.br (E.V.L. de Mello).
T for a compound with doping level nm . Here we show that the diamagnetic signal measured above Tc by several groups [1–3] and which was found not to follow the fluctuation magnetization for a layered superconductor can be understood within this picture of a superconductor formed by static domains or droplets with a spatial varying Tc ðrÞ. Furthermore the droplets follow the critical-state model (CSM) [5] to the magnetization response under the applied fields. We demonstrate that this procedure is able to explain and reproduce the precursor diamagnetism behavior as recently measured [2,3].
2. The model In order to estimate the MðBÞ we follow the ideas of the CSM to each superconducting droplet between Tc and T . Upon applying an external magnetic field, a critical superconducting current ðJc Þ is established of the form Jc ðBÞ ¼ aðT Þ=B. For simplicity we take these superconducting droplets as cylinders of radius R, which is sufficient small in order to have a constant charge density n and consequently the critical temperature Tc ðnÞ is the same within such cylinder region. The CSM approach [5] leads to:
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.03.028
442
0.000 3
M2 ðBÞ ¼
0.005
ð1Þ
for B 6 Bc1
B 4B 8B þ ; l0 5l0 B2 15l0 B4
B 4B M3 ðBÞ ¼ l0 15l0
-0.005
5
Bc1 6 B 6 B
ð2Þ
2 5=2 ! B5 B3 B 2 5 5 3 2 2 1 B B B
for B 6 B 6 Bc2
ð3Þ
-0.015
nc
where P ðnÞ is the distribution function for the charge distribution and nc is the onset of superconductivity ðnc ¼ 0:05Þ.
0.0000
-0.020
27.8 K 28.0 K
-0.0001
-0.025 -0.0002
-0.030 -0.035
where B is the applied external field to produce full penetration inside a cylindrical superconducting droplet. The dependence of the local critical temperatures Tc ðnÞ is that of the T ðnm Þ ðTc ðnÞ ¼ T0 b ðn nc ÞÞ. Since Tc ðnÞ is constant inside a superconducting cylindrical droplet, the critical fields (Bc1 and Bc2 ) will have their temperature dependence according to the Ginzburg–Landau theory, e.g., Bc1 ðT Þ ¼ Bc1 ð0Þ½ð1 T =Tc ðnÞ and Bc2 ðT Þ ¼ Bc2 ð0Þ½ð1 T =Tc ðnÞ. Taking into account the dependence of Tc ðnÞ on n, we arrive at the expressions for the critical fields Bc1 ðT ; nÞ and Bc2 ðT ; nÞ. In the low field regime the superconducting clusters will contribute to the magnetization of the sample in three forms: there are some clusters, which are not penetrated by the field B, that is B 6 Bc1 ðT ; nÞ and they contribute to the magnetization with perfect diamagnetism (Eq. (1)). The second group of clusters have their Bc1 ðT ; nÞ lower than the applied field but B is also lower than B ðn; T Þ. This group is partially penetrated by the field and they contribute to MðBÞ according Eq. (2). Lastly, there are some superconducting granules for which the applied field is higher than B ðT ; nÞ but also lower than Bc2 ðT ; nÞ. Only the droplets with n bigger than nmax do not contribute to the sample’s magnetization because their superconductivity is destroyed by the field B. Thus we expect that Z nmax ðBc2 Þ MðT ; BÞ ¼ P ðnÞMðB; T ; nÞ dn ð4Þ
T c =26.86 K
-0.010 3
B l0
M(emu/cm )
M1 ðBÞ ¼
E.V.L. de Mello, J.L. Gonzalez / Physica C 408–410 (2004) 441–442
-0.040
27.0 27.2 27.6 28.0
-0.045 0.000
K K K K
-0.0003
-0.0004
0.000
0.001
0.002
0.001
0.003
0.002
0.003
0.004
0.004
0.005
0.005
B(T)
Fig. 1. Magnetization for appropriated parameters to the overdoped LSCO as calculated from Eqs. (7)–(11). This result is to be compared with the measurements from Ref. [3].
sured by Lascialfari et al. [3]. The qualitative features of the measurements are entirely reproduced and are simply explained by the CSM: at low field the perfect diamagnetism is expected for fields lower than Bc1 . By the same token, the penetrating field B should not be very strong what decreases rapidly the overall diamagnetic signal for field much weaker than Bc2 . As the applied field increases, the magnetic response dies off and is reduced to the fluctuations. This is the reason why MðBÞ has a minimum ðBup Þ at very low applied fields [2,3] as it is shown in Fig. 1.
4. Conclusion The CSM results for a HTSC under an applied field demonstrated that, as concluded also from other different calculations [2,3], the measured magnetization curves, the Bup fields and the STM magnetic imaging results are interpreted very simply through the formation of static superconducting islands.
References 3. Results and discussion The above theory was developed to model the measured magnetization curves of the La1x Srx CuO4 family of compounds. In Fig. 1 we plot the results of the CSM with the parameter which corresponds to a nm ¼ 0:2 and Tc ¼ 26:86 K. The values of Bc1 ð0Þ and B ð0Þ are not found in the literature and were chosen in order to obtain a MðBÞ curve with the same features as those measured on an underdoped LSCO compound mea-
[1] I. Iguchi, I. Yamaguchi, A. Sugimoto, Nature 412 (2001) 420. [2] A. Lascialfari, A. Rigamonti, L. Romano, P. Tedesco, A. Varlamov, D. Embriaco, Phys. Rev. B 65 (2002) 144523. [3] A. Lascialfari, A. Rigamonti, L. Romano, A. Varlamov, I. Zucca, Phys. Rev. B 68 (2003) 100505(R). [4] E.V.L. de Mello, E.S. Caixeiro, J.L. Gonzalez, Phys. Rev. B 67 (2003) 024502. [5] Michael Tinkham, Introduction to superconductivity, McGraw-Hill Inc., New York, 1975.