Crystal Growth and Physical Properties of T*- Phase SmLa1-xSrxCuO4-d and T-Phase La1.6-xNd 0.4Sr xCuO 4- d - Appendix B

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Crystal Growth and Physical Properties of T*- Phase SmLa1-xSrxCuO4-d and

T-Phase La1.6-xNd 0.4Sr xCuO 4- d

Sutjahja, I.M.

Publication date

2003

Link to publication

Citation for published version (APA):

Sutjahja, I. M. (2003). Crystal Growth and Physical Properties of T*- Phase

SmLa1-xSrxCuO4-d and T-Phase La1.6-xNd 0.4Sr xCuO 4- d.

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Thiss appendix is an attempt to present the general substance of the XY model and the Ginzburg-Landauu - lowest-Landau-level (GL-LLL) theory for describing the fluctuationn properties of the high-temperature superconductors (HTSC's).

Thee fluctuating behavior in HTSC's in the vicinity of the superconducting critical temperature,, TC(H), or of the upper critical field curve, Hc2(T), has been a subject of intensee experimental [1-6] and theoretical [7-9] studies. The temperature interval aroundd Tc in which critical fluctuations are important is approximately given by TcGi

[10],, in which the Ginzburg number, Gi, is expressed by:

Gi-yGi-y22 vrjHlioy^l^A /2 In this expression, / is the anisotropy parameter and HHcc(0)(0) is the zero-temperature thermodynamic critical field; %ab and <£ are the in-plane andd out-of-plane coherence lengths, respectively. Typical values for Gi are ~ 10" for HTSC'ss and ~ 10"8 for conventional superconductors [10].

AA useful tool in the analysis of the nature of critical fluctuation is the dimension-dependentt scaling of the various thermodynamic and transports properties off the system. The scaling hypothesis implies that a dimensionless form of a physical quantityy can be written as a function of a certain (dimensionless) scaling variable. Bothh the scaling variable and the dimensionless form of the particular quantity depend uponn the theoretical model and upon the dimensionality of system. However, we stress that,, while the scaling variable is known, the function does not need to be known, and typicallyy is not.

150 0 Appendixx B

HHc2c2(T) (T)

Figuree A.l: A schematic sketch of the region of interest in the H — T phase diagram

atat temperatures around Tc. The isotropic XY scaling applies for fields H < H , while the LLL approximationapproximation is valid above the field H and around the Hc2 line. From Ref [14].

Dependingg on the magnetic field strength, two well-known models have been introducedd for explaining the critical behavior in the HTSC's, namely the XY model [11]] and the GL-LLL model [7, 8, 12], It is worth noting that the term "Landau levels" (LL's)) refers to the quantized BCS-pair energies, obtained by solving the first Ginzburg-Landauu (GL) differential equation for the order parameter [13], Fig. A.l illustratess the validity region of the XY model and the GL-LLL theory in the H - T phasee diagram [14].

Itt is clearly seen from this figure that the XY model is valid for low fields, while thee GL-LLL approximation is adequate for higher fields. The field HXY below which thee XY model can be justified to be valid may be estimated by realizing that the magneticc field breaks the XY symmetry if the correlation length £AT (= § ƒ ' , with

// = | TITC - 1 I and v = vxy ~ 2/3) exceeds the magnetic length scale ^ / ®0/ ( ; r f / ) .

Onn the other hand, the field HLLL above which the LLL approximation should be valid iss usually estimated to be H* = HLLl = (G//16) (Tc/Tc0) Hc2(0) [15], where 7^ is the mean-fieldd transition temperature at zero field. Naively, one might think that, forr sufficiently small magnetic fields, the fluctuating Cooper pairs occupy many LL's withh significant intra- and inter-LL interactions. At increasing the magnetic field, the fluctuatingg pairs occupy a smaller number of the lower LL's with less inter-level

mixing.. By further increasing the field up to a certain value H , only the lowest Landau levell is occupied and only intra-level interactions are important. This rough estimate yields:: Z/LLL ~ 1 - 3 T for REBaiC^O?^ (RE = Y, Lu) single-crystalline samples [14,, 16], //LLL ~ 1 T for (Bi,Pb)2Sr2Ca2Cu3Oy aligned bulk samples [17] and for

Bi2Sr2CaCu208+ss single crystals [18], and HLLL * 400 Oe for a T - phase SmLao.gSro.2Cu04.55 single crystal [19].

Focusingg on the thermodynamic quantity of the magnetization, the scaling formulass based on the XY model are given as follows [11, 20]:

(4TTM)(4TTM)2D2D = F: f f 2D-XY 2D-XY

T-T T-T

= H

F,

T-T. T-T.

TT H

forr 2D and 3D fluctuations with the universal scaling functions F2D-XY and F3D.XY,

respectively.. It is important to note that, theoretically, the XY scaling is justified for infinitesimallyy small fields; however, the experimental results show that it is valid only forr a field range near the H = 0 critical point [16].

Similarr to the XY scaling formulas, the scaling functions of the magnetization withinn the GL-LLL theory are expressed as [6, 12, 15, 16]:

(HTf(HTf

(4*A/) )

== F.

(HT) (HT)

J-TJ-T

(H)' (H)'

(HT)(HT)

fT-TM fT-TM

(HT) (HT)

wheree A and A' are field- and temperature-independent coefficients, while F2D.LLL and

^3D-LLLL are the 2D and 3D scaling functions, respectively. The temperature region wheree the GL-LLL scaling holds, is approximated by T > Tc0 - 3H/H'c2 , with

152 2

Appendixx B

denominatorr of the above scaling function is connected with the dimensionality of the systemm D through n = (D - \)ID. Obviously, n = 1/2 for a 2D system and n = 2/3 forr a 3D system.

References s

[I]] P.L. Gammel, D.J. Bishop, G.J. Dolan, J.R. Kwo, C.A. Murray, L.F. Scheemeyer andd J.V. Waszczak, Phys. Rev. Lett. 59, 2952 (1987).

[2]] P.H. Kes, C.J. van der Beek, M.P. Maley, M.E. Mc Henry, D.A. Huse, M.J.. Menken and A.A. Menovsky, Phys. Rev. Lett. 67, 2383 (1991).

[3]] U. Welp, S. Fleshier, W.K. Kwok, R.A. Klemm, V.M. Vinokur, J. Downey, B.. Veal andG.W. Crabtree, Phys. Rev. Lett. 67, 3180 (1991).

[4]] W. C. Lee, R. A. Klemm and D. C. Johnston, Phys. Rev. Lett. 63, 1012 (1989); Phys.. Rev. B 43, 457(1991).

[5]] J.S. Urbach, W.R. White, M.R. Beasley and A. Kapitulnik, Phys. Rev. Lett. 69, 2407(1992). .

[6]] Q. Li, M. Suenaga, J. Gohng, D. K. Finnemore, T. Hikata and K. Sato, Phys.. Rev. B 46, 3195 (1992); Q. Li, M. Suenaga, T. Hikata and K. Sato, ibid. 46, 58577 (1992); Q. Li, K. Shibutani, M. Suenaga, I. Shigaki and R. Ogawa, ibid. 48, 9877 (1993). .

[7]] S. Ullah and A.T. Dorsey, Phys. Rev. Lett. 65, 2066 (1990); Phys. Rev. B 44, 262 (1991),, Phys. Rev. B 46, 5857 (1992).

[8]] R. Ikeda, T. Ohmi and T. Tsuneto, Phys. Rev. Lett. 67, 3874 (1991), J.. Phys. Soc. Jpn. 60, 1051 (1991).

[9]] Z. Tesanovic and L. Xing, Phys. Rev. Lett. 67, 2729 (1991).

[10]] G. Blatter, M.V. Feigel'man, V.B. Geshkenbein, A.I. Larkin and V.M. Vinokur, Rev.. Mod. Phys. 66, 1125 (1994).

[II]] D.S. Fisher, M.P.A. Fisher and D.A. Huse, Phys. Rev. B 43, 130 (1991).

[12]] Z. Tesanovic, L. Xing, L. Bulaevskii, Q. Li and M. Suenaga, Phys. Rev. Lett. 69, 3563(1992). .

[13]] M. Tinkham, Introduction to Superconductivity, second edition (1996), Mc. Graw-Hill,, Inc., p. 134.

[14]] O. Jeandupeux, A. Schilling, H.R. Ott and A. van Otterlo, Phys. Rev. B 53, 12 475 (1996). .

[15]] Z. Tesanovic and A.V. Andreev, Phys. Rev. B 49, 4064 (1994).

[16]] S.W. Pierson, T.M. Katona, Z. Tesanovic and O.T. Vails, Phys. Rev. B 53, 8638 (1996). .

[17]] N. Kobayashi, K. Egawa, K. Miyoshi, H. Iwasaki, H. Ikeda and R. Yoshizaki, PhysicaCC 219, 265(1994).

[18]] D. Darminto, I.M. Sutjahja, A. A. Menovsky and M. O. Tjia, Physica C 369, 286 (2002). .

[19]] I.M. Sutjahja, D. Darminto, R. Sutarto, A.A. Nugroho, M. O. Tjia, A.. A. Menovsky and J. J. M. Franse, Physica C 369, 291 (2002).

[20]] A. Junod, in Studies of High-Temperature Superconductivity, A.V. Narlikar (Ed.), Vol.. 19, Nova Science Publisher, New York, 1996.