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Third harmonic ac susceptibility measurements in MgB2 bulk: frequency behavior of IL and 3D glass pinning analysis
Physica C 408–410 (2004) 120–122 www.elsevier.com/locate/physc
Third harmonic ac susceptibility measurements in MgB2 bulk: frequency behavior of IL and 3D glass pinning analysis D. Di Gioacchino a
a,*
, P. Tripodi a,b, U. Gambardella a, V. Sandu c, S. Popa c, L. Miu c, D. Vinko b
INFN-LNF, National Institute of Nuclear Physics, National Laboratory of Frascati, Via Enrico Fermi 40, 00044 Frascaty, Italy b HERA––Hydrogen Energy Research Agency, Corso della, Repubblica 448, 00049 Velletri, Italy c National Institute of Material Physics, POB-MG-7, Bucharest-Margurele R-76900, Romania
Abstract The third harmonics of the ac susceptibility v3 of MgB2 bulk samples have been measured in the frequency range 61– 1070 Hz as a function of the temperature for different applied dc magnetic fields. The irreversibility line (Tirr versus Hdc ) deduced from the temperature onsets of v3 for each field shows a frequency dependence: The onset temperature increases as the frequency increases. This frequency dependence has been analyzed using the 3D vortex-glass model. Moreover the real versus imaginary components of the v3 for each dc magnetic field have been interpreted with a 3D glass creep using a numerical analysis based on a non-linear diffusion equation of the magnetic field. 2004 Elsevier B.V. All rights reserved. Keywords: MgB2 ; Flux dynamics; Higher harmonics of ac susceptibility
1. Introduction The studies of the flux dynamics in MgB2 near the irreversibility line (IL) and in the irreversible regime, are important as to know the dimensionality of the flux response with respect to the temperature, applied magnetic fields and electrical current induced in the sample [1]. In this way, it is possible to correlate pinning disorder and thermal fluctuations with the lattice elastic flux motion [2,3]. The ac multi-harmonic susceptibility investigation, in particular the third harmonic (v3 ¼ v03 þ iv003 ), is an effective and good tool to approach these studies. In fact, the high harmonic signals sign a non-linear response in the flux dynamics: The flux redistribution in
* Corresponding author. Tel.: +39-6-9403-2757; fax: +39-69403-2427. E-mail address: [email protected] (D. Di Gioacchino).
the superconducting state during the ac cycles is dominated by an irreversible state. In this paper we present the MgB2 IL and its frequency behavior for a bulk sample, deduced from the onset of v3 measurements in the 61 Hz < f < 1070 Hz, 0 T < Hdc < 5 T and 4 K < T ðKÞ < 40 K ranges. Moreover, a polar plot v03 versus v003 at 5 T describing the behavior in the irreversible regime is analyzed. The 3D glass model to study the IL and the irreversible state in MgB2 bulk sample, has been used.
2. Results and discussion The ac magnetic susceptibility measurements have been performed on a MgB2 bulk sample (1 · 2 · 5.6 mm3 ) prepared by a sintered reaction [4]. The measurements are carried out using a double coil susceptometer [3,5] versus temperature, T ðKÞ, at different values of the dc magnetic fields, 0 < Hdc ðT Þ < 5 and frequencies 61 < f ðHzÞ < 2070. The amplitude of the
0921-4534/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.02.062
D. Di Gioacchino et al. / Physica C 408–410 (2004) 120–122
ac driving magnetic field is 6 G and Hdc is parallel to the sample length. In Fig. 1 measurements of the real components v03 versus T ðKÞ and f ðHzÞ are shown at Hdc ¼ 5 T (imaginary component v003 is not shown). To study the IL behavior, the onsets of v3 versus f ðHzÞ and Hdc are plotted in Fig. 2. These slopes are analyzed in accordance with the vortex-glass approach where Tirr ðH ; f Þ is defined as [2,6] Tirr ðH ; f Þ ¼ Tg ðH Þ þ AðH Þ f 1=mðzþ2DÞ ;
ð1Þ
where m > 0 and z > 0 are the static and dynamic exponents [2]. In fact at the glass temperature transition Tg of the liquid/glass phases, the relevant vortex-glass correlation length n scales as, n / 1=jT Tg jm [2,7]. The flux dynamics diverges with time s, as s / 1=jT Tg jzm [2]. Moreover expliciting the system dimensionality D, the s equation can be inverted and Eq. (1) is found. From the fits of the data (Fig. 2), a small increase of the Tirr versus f ðHzÞ is explained. The best fit gives a 3D behavior slightly depending on the magnetic field with the following values: 1:7 > m > 1:4 and 2:8 > z > 2:5.
Fig. 1. Real v03 versus T ðKÞ and f ðHzÞ of MgB2 bulk at Hdc ¼ 5 T.
121
The insert in Fig. 2 shows the IL where Tg ðH Þ is calculated from Eq. (1) and corresponds to the zero frequency limit for each Hdc . In this case the amplitude of Hdc on the y-axis is called irreversibility field, Hirr , and on x-axis the reduced glass temperature tglass ¼ Tg ðH Þ=Tg ð0Þ is shown. The harmonic components of the susceptibility are useful to detect small multi-phase transitions in a superconducting sample and in a granular bulk sample intra/intergranular phases can be shown. All jv3 j plots versus T ðKÞ for Hdc and f ðHzÞ present only one peak (these plots are not shown for brevity). This means that only one superconducting phase is present and the granularity is hidden. This shows that the vortex-glass behavior used in the analysis data is an intrinsic characteristic. To investigate the MgB2 irreversible regime, v03 versus v003 polar plots at Hdc ¼ 5 T are shown in Fig. 3. These behaviors were studied with the diffusion equation [3,8]: oB=ot ¼ o=ox½ðq0;cr ðB; J Þ=l0 ÞoB=ox:
ð2Þ
qcr is a 3D glass creep resistivity [3,9]. The mathematical form for qcr is qcr / q0 exp U ðB; J ; T Þ=KT , where for UðJ Þ dependence, the glass behavior has been used described by ðJ =Jc Þl . l characterizes the low temperature vortex-glass state [10]. For U ðT ; BÞ behavior, the C44 and C66 elastic moduli dependences in T ðKÞ and magnetic field, have been used [11]. The solutions of Eq. (2) using sinusoidal boundary condition have been developed as Fourier coefficients and are proportional to vn [8]. For each f ðHzÞ the best fit has been done and a well-defined l has been found. The observed behaviors in Fig. 3 are well described with l values that increase as the f ðHzÞ increases. The ranges are 0:2 < l < 1 and 107 < f ðHzÞ < 2070. Increasing the f ðHzÞ the induced voltage in the sample rises and consequently also the electric current determined by I–V characteristics. These
1
107Hz , µ=0.2 503Hz , µ=0.4
0
55 Tirr (K)
50 45 40
0T 0.1T 0.5T 1T 2T 3T 5T
H irr (T)
60 6 5 4 3 2 1 0
1.24
-1
Hirr( T )=14(1-t)
0.5
0.6
0.7 0.8 tglass
0.9
1
-2 Hdc =5T
35 30
-3 -2.5
25 20 10
100 1000 frequency (Hz)
-1.25
1070Hz , µ =0.5 2070Hz, µ=1 0.0
1.25
2.5
4
10
Fig. 2. Tirr versus f ðHzÞ deduced by the v3 onset measurements. Data analyzed with vortex glass/liquid equation (1).
Fig. 3. Imaginary v003 versus real v03 for different f ðHzÞ at Hdc ¼ 5 T. Lines are fits based on numerical analysis with 3D glass creep qcr .
122
D. Di Gioacchino et al. / Physica C 408–410 (2004) 120–122
results show that l depends on the electric current value induced in the sample [10]. In conclusion, these frequency responses of the MgB2 bulk sustain a 3D vortexglass flux dynamics. References [1] L.F. Cohen, H.J. Jensen, Rep. Prog. Phys. 60 (1997) 1581. [2] D. Fisher, M. Fisher, D. Huse, Phys. Rev. B 43 (1991) 130. [3] D. Di Gioacchino et al., Supercond. Sci. Tech. 16 (2003) 534.
[4] Edison S.P.A. patent pending. [5] INFN-LNF European Facility TARI, HPRI-CT-199900088. [6] Y. Wolfus et al., Physica C 224 (1994) 213. [7] A. Rydh et al., Phys. Rev. Lett. 83 (1999) 1850. [8] D. Di Gioacchino et al., Phys. Rev. B 59 (1999) 11539. [9] D. Di Gioacchino et al., IEEE Trans. Appl. Supercond. 11 (2001) 3924. [10] C. Dekker et al., Phys. Rev. Lett. 68 (1992) 3347. [11] E.H. Brandt, Phys. Rev. B 34 (1986) 6514.
Third harmonic ac susceptibility measurements in MgB2 bulk: frequency behavior of IL and 3D glass pinning analysis D. Di Gioacchino a
a,*
, P. Tripodi a,b, U. Gambardella a, V. Sandu c, S. Popa c, L. Miu c, D. Vinko b
INFN-LNF, National Institute of Nuclear Physics, National Laboratory of Frascati, Via Enrico Fermi 40, 00044 Frascaty, Italy b HERA––Hydrogen Energy Research Agency, Corso della, Repubblica 448, 00049 Velletri, Italy c National Institute of Material Physics, POB-MG-7, Bucharest-Margurele R-76900, Romania
Abstract The third harmonics of the ac susceptibility v3 of MgB2 bulk samples have been measured in the frequency range 61– 1070 Hz as a function of the temperature for different applied dc magnetic fields. The irreversibility line (Tirr versus Hdc ) deduced from the temperature onsets of v3 for each field shows a frequency dependence: The onset temperature increases as the frequency increases. This frequency dependence has been analyzed using the 3D vortex-glass model. Moreover the real versus imaginary components of the v3 for each dc magnetic field have been interpreted with a 3D glass creep using a numerical analysis based on a non-linear diffusion equation of the magnetic field. 2004 Elsevier B.V. All rights reserved. Keywords: MgB2 ; Flux dynamics; Higher harmonics of ac susceptibility
1. Introduction The studies of the flux dynamics in MgB2 near the irreversibility line (IL) and in the irreversible regime, are important as to know the dimensionality of the flux response with respect to the temperature, applied magnetic fields and electrical current induced in the sample [1]. In this way, it is possible to correlate pinning disorder and thermal fluctuations with the lattice elastic flux motion [2,3]. The ac multi-harmonic susceptibility investigation, in particular the third harmonic (v3 ¼ v03 þ iv003 ), is an effective and good tool to approach these studies. In fact, the high harmonic signals sign a non-linear response in the flux dynamics: The flux redistribution in
* Corresponding author. Tel.: +39-6-9403-2757; fax: +39-69403-2427. E-mail address: [email protected] (D. Di Gioacchino).
the superconducting state during the ac cycles is dominated by an irreversible state. In this paper we present the MgB2 IL and its frequency behavior for a bulk sample, deduced from the onset of v3 measurements in the 61 Hz < f < 1070 Hz, 0 T < Hdc < 5 T and 4 K < T ðKÞ < 40 K ranges. Moreover, a polar plot v03 versus v003 at 5 T describing the behavior in the irreversible regime is analyzed. The 3D glass model to study the IL and the irreversible state in MgB2 bulk sample, has been used.
2. Results and discussion The ac magnetic susceptibility measurements have been performed on a MgB2 bulk sample (1 · 2 · 5.6 mm3 ) prepared by a sintered reaction [4]. The measurements are carried out using a double coil susceptometer [3,5] versus temperature, T ðKÞ, at different values of the dc magnetic fields, 0 < Hdc ðT Þ < 5 and frequencies 61 < f ðHzÞ < 2070. The amplitude of the
0921-4534/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.02.062
D. Di Gioacchino et al. / Physica C 408–410 (2004) 120–122
ac driving magnetic field is 6 G and Hdc is parallel to the sample length. In Fig. 1 measurements of the real components v03 versus T ðKÞ and f ðHzÞ are shown at Hdc ¼ 5 T (imaginary component v003 is not shown). To study the IL behavior, the onsets of v3 versus f ðHzÞ and Hdc are plotted in Fig. 2. These slopes are analyzed in accordance with the vortex-glass approach where Tirr ðH ; f Þ is defined as [2,6] Tirr ðH ; f Þ ¼ Tg ðH Þ þ AðH Þ f 1=mðzþ2DÞ ;
ð1Þ
where m > 0 and z > 0 are the static and dynamic exponents [2]. In fact at the glass temperature transition Tg of the liquid/glass phases, the relevant vortex-glass correlation length n scales as, n / 1=jT Tg jm [2,7]. The flux dynamics diverges with time s, as s / 1=jT Tg jzm [2]. Moreover expliciting the system dimensionality D, the s equation can be inverted and Eq. (1) is found. From the fits of the data (Fig. 2), a small increase of the Tirr versus f ðHzÞ is explained. The best fit gives a 3D behavior slightly depending on the magnetic field with the following values: 1:7 > m > 1:4 and 2:8 > z > 2:5.
Fig. 1. Real v03 versus T ðKÞ and f ðHzÞ of MgB2 bulk at Hdc ¼ 5 T.
121
The insert in Fig. 2 shows the IL where Tg ðH Þ is calculated from Eq. (1) and corresponds to the zero frequency limit for each Hdc . In this case the amplitude of Hdc on the y-axis is called irreversibility field, Hirr , and on x-axis the reduced glass temperature tglass ¼ Tg ðH Þ=Tg ð0Þ is shown. The harmonic components of the susceptibility are useful to detect small multi-phase transitions in a superconducting sample and in a granular bulk sample intra/intergranular phases can be shown. All jv3 j plots versus T ðKÞ for Hdc and f ðHzÞ present only one peak (these plots are not shown for brevity). This means that only one superconducting phase is present and the granularity is hidden. This shows that the vortex-glass behavior used in the analysis data is an intrinsic characteristic. To investigate the MgB2 irreversible regime, v03 versus v003 polar plots at Hdc ¼ 5 T are shown in Fig. 3. These behaviors were studied with the diffusion equation [3,8]: oB=ot ¼ o=ox½ðq0;cr ðB; J Þ=l0 ÞoB=ox:
ð2Þ
qcr is a 3D glass creep resistivity [3,9]. The mathematical form for qcr is qcr / q0 exp U ðB; J ; T Þ=KT , where for UðJ Þ dependence, the glass behavior has been used described by ðJ =Jc Þl . l characterizes the low temperature vortex-glass state [10]. For U ðT ; BÞ behavior, the C44 and C66 elastic moduli dependences in T ðKÞ and magnetic field, have been used [11]. The solutions of Eq. (2) using sinusoidal boundary condition have been developed as Fourier coefficients and are proportional to vn [8]. For each f ðHzÞ the best fit has been done and a well-defined l has been found. The observed behaviors in Fig. 3 are well described with l values that increase as the f ðHzÞ increases. The ranges are 0:2 < l < 1 and 107 < f ðHzÞ < 2070. Increasing the f ðHzÞ the induced voltage in the sample rises and consequently also the electric current determined by I–V characteristics. These
1
107Hz , µ=0.2 503Hz , µ=0.4
0
55 Tirr (K)
50 45 40
0T 0.1T 0.5T 1T 2T 3T 5T
H irr (T)
60 6 5 4 3 2 1 0
1.24
-1
Hirr( T )=14(1-t)
0.5
0.6
0.7 0.8 tglass
0.9
1
-2 Hdc =5T
35 30
-3 -2.5
25 20 10
100 1000 frequency (Hz)
-1.25
1070Hz , µ =0.5 2070Hz, µ=1 0.0
1.25
2.5
4
10
Fig. 2. Tirr versus f ðHzÞ deduced by the v3 onset measurements. Data analyzed with vortex glass/liquid equation (1).
Fig. 3. Imaginary v003 versus real v03 for different f ðHzÞ at Hdc ¼ 5 T. Lines are fits based on numerical analysis with 3D glass creep qcr .
122
D. Di Gioacchino et al. / Physica C 408–410 (2004) 120–122
results show that l depends on the electric current value induced in the sample [10]. In conclusion, these frequency responses of the MgB2 bulk sustain a 3D vortexglass flux dynamics. References [1] L.F. Cohen, H.J. Jensen, Rep. Prog. Phys. 60 (1997) 1581. [2] D. Fisher, M. Fisher, D. Huse, Phys. Rev. B 43 (1991) 130. [3] D. Di Gioacchino et al., Supercond. Sci. Tech. 16 (2003) 534.
[4] Edison S.P.A. patent pending. [5] INFN-LNF European Facility TARI, HPRI-CT-199900088. [6] Y. Wolfus et al., Physica C 224 (1994) 213. [7] A. Rydh et al., Phys. Rev. Lett. 83 (1999) 1850. [8] D. Di Gioacchino et al., Phys. Rev. B 59 (1999) 11539. [9] D. Di Gioacchino et al., IEEE Trans. Appl. Supercond. 11 (2001) 3924. [10] C. Dekker et al., Phys. Rev. Lett. 68 (1992) 3347. [11] E.H. Brandt, Phys. Rev. B 34 (1986) 6514.