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3d X-Y scaling of the specific heat of YBa2Cu3O7−δ single crystals
Phys~ea C 235-240 (1994) 1769-1770 North-Holland
PHYSICA
3d X - Y S c a l i n g of the Specific Heat of Y B a 2 C u 3 0 7 . 5 S i n g l e Crysta|s NeU Overend, Mark A Howson and Ian D Lawrie Department of Physics, University of Leeds, Leeds LS29JT, UK The specific heat of a single crystal sample of YBa2CU3OT_s (YBCO) has been measured with magnetic fields up to 8 Tesla applied parallel to the c-axis of the crystal. We analyse the data in terms of the Lowest Landau Level (LLL) and critical scaling theories. Recent measurements of magnetisation and resistivity have been unable to distinguish between the critical scaling and LLL regimes. We find that the data does not scale using the U.J., theory and therefore conclude that the LLL theory is not applicable in magnetic fields less than 8 Tesla. However, we find that the data does scale using the 3-dimensional X-Y model. This provides strong evidence for the existence of a critical regime within which there is sealing behaviour characteristic of the 3-dimensional X-Y model with critical exponents consistent with those observed in superfluid 4He. There is growing evidence that YBCO exhibits critical fluctuations characteristic of the 3-dimensional XY modeP. This is the same universality class as the superfluid transition in liquid 4He. If this is so then the phase diagram in the magnetic field-temperature (B-T) plane should possess a critical point at T=T c a~_dB=0 and critical fluctuations should be observed in a region close to this point. In this region, physical properties are expected to exhibit single-parameter scaling with the sealing variable 2 tB IB m*, where v is the critical exponent describing the divergence of the coherence length, ta is the field dependent reduced temperature (T-T,(B))IT,(B) where Tc(B) is given by Tc(0)+ B/(dBc.z I dT). In particular, the fluctuation contribution to the specific heat (C~) is expected to have the scaling form ~'~ (assuming ot < 0)
C I = C o - ffot':~f(t a I B ~'2")
(1)
and f(x) is the critical scaling function. The specific heat therefore has a cusp of height Co at ra = 0 and z~= u. tne value of the critical exponent or= 2 - d r (for a system of spatial dimensionality d) is estimated theoretically for d=3 as -0.007+0.0063, while its measured value for 4lie is -0.013+0.0034 . The broadening of the superconducting transition in a magnetic field has recently been discussed in terms of the lowest Landau level (LLL) approximations Within this approximation the
fluctuation contribution to the specific heat is predicted to scale as s Cf
...
(2)
where g(x) is the LLL scaling function and zxC ~s the mean field specific heat discontinuity. There has been some confusion over the exact form of AC. Solae workers6, including ourselves7, have used AC ~ T and have found that they were unable to scale their specific heat data. Zhou et al.s have scaled their data on LuBaCuO. However, they normalise their data using the temperature dependent mean field BCS specific heat such that AC=yT(I +bt) (where the normal state electronic specific heat is given by yT). They let b=3.82 and also use a magnetic field dependent "1'to account for drifts in their experimental data 9. While it is not correct this has the effect of increasing the peak height for the higher fields and therefore improves their scaling. In the origional papers by Thouless~° and by Bray u the correct normalisatiou is
0921-4534/94/$07 00 © 1994 - Elsevier Science B V. All rights reserved SSDI 0921-4534(94)01451-5
independent. We have measured the specific heat of a single crystal sample of YBCO with magnetic fields up to 8 Tcsla applied parallel to the c-axis of the crystal. We use a high resolution ac. method detailed elsewhere ~:. Figure 1 shows the specific heat in several applied magnetic fields.
N Ouerend et al /Physica C 235-240 (1994) 1769-1770
1770
in all applied magnetic fields except for a limited temperature range in fields of 6 and 8 tesla. We would therefore expect that at fields of ==10 tesla, and larger, LLL scaling may begin to work and critical scaling would begin to fail. This value of magnetic field is consistent with the theoretical estimate by Lawrie in reference 2. 0 ~5
T(K) Figure 1. Specific heat ((2) vs. temperature in magnetic fields (from top to bottom) of 0, 0.1, 0.2, 0.5, 1, 2, 4, 6 and 8 tesla.
We extract the fluctuation specific heat from the measured specific heat using the procedure aetailed in ref. 10. Figure 2 shows the data scaled using the 3d XY model (eqn. 1) with the exponents ~t =-0.013 and v = 0.669 (as in 4He), and Tc(0) = 92.00K and dBc21 d T = - 5 T K -t
-e2 O
"D
E
v
-=6 O
O
v
.
.
.
-00,5
.
.
•
.
.
.
.
.
0
.
005
tB/B1/2v (T-O747) Figure 2. 3d XY scaling of the fluctuation specific heat in magnetic fields of 0.1, 0.2, 0.5, 1, 2, 4, 6 and 8 tesla. It is evident from figure 2 that the data exhibits single parameter critical scaling over a 20 K temperature range extending to fields of the order of 8 tesla. Figure 3 shows the data scaled using the LLL scaling form (eqn. 2) with Tc(0 ) = 92.00K and dBc: / dT= -2TK -t . The LLL scaling is found to fail
~0
0
-oo2
'
.o 2
tB/(TB) 2/3 (K-2/3T -2/3) Figure 3. LLL scaling of the fluctuation specific heat in magnetic fields (from top to bottom) of 0.1, 0.2, 0.5, 1, 2, 4, 6 and 8 tesla. References 1. Salamon MB et al. Phys. Rev. B47 5520 (1993) 2. Lawrie ID (submitted to Phys. Rev. B) 3. Le Guillou JC and Zinn-Justin J Phys. Rev. B21 3976 (1980) 4. Lipa JA and Chui TCP Phys.Rev.Lett. 51 2291 (1983) 5. UUah S and Dorsey AT Phys. Rev. 1~44 262 (1991) Wilkin NK and Moore MA Phys. Rev. B48 3464 (1993) 6. Welp U et al. Phys. Rev. LetL ,J7 3180 (1991) Junod Aet al. Physica C211 304 (1993) 7. Overend et al. Phys. Rev. Lett. 72 3238 (1994) 8. Zhou et al. Phys Rev B47 11631 (1993) 9. Private communication. 10.Thouless DJ Phys Rev Lett (1975) 11.Bray AJ Phys Rev B 9 4752 (1974) i2.Overend et ai. (submitted to Play Rev B)
PHYSICA
3d X - Y S c a l i n g of the Specific Heat of Y B a 2 C u 3 0 7 . 5 S i n g l e Crysta|s NeU Overend, Mark A Howson and Ian D Lawrie Department of Physics, University of Leeds, Leeds LS29JT, UK The specific heat of a single crystal sample of YBa2CU3OT_s (YBCO) has been measured with magnetic fields up to 8 Tesla applied parallel to the c-axis of the crystal. We analyse the data in terms of the Lowest Landau Level (LLL) and critical scaling theories. Recent measurements of magnetisation and resistivity have been unable to distinguish between the critical scaling and LLL regimes. We find that the data does not scale using the U.J., theory and therefore conclude that the LLL theory is not applicable in magnetic fields less than 8 Tesla. However, we find that the data does scale using the 3-dimensional X-Y model. This provides strong evidence for the existence of a critical regime within which there is sealing behaviour characteristic of the 3-dimensional X-Y model with critical exponents consistent with those observed in superfluid 4He. There is growing evidence that YBCO exhibits critical fluctuations characteristic of the 3-dimensional XY modeP. This is the same universality class as the superfluid transition in liquid 4He. If this is so then the phase diagram in the magnetic field-temperature (B-T) plane should possess a critical point at T=T c a~_dB=0 and critical fluctuations should be observed in a region close to this point. In this region, physical properties are expected to exhibit single-parameter scaling with the sealing variable 2 tB IB m*, where v is the critical exponent describing the divergence of the coherence length, ta is the field dependent reduced temperature (T-T,(B))IT,(B) where Tc(B) is given by Tc(0)+ B/(dBc.z I dT). In particular, the fluctuation contribution to the specific heat (C~) is expected to have the scaling form ~'~ (assuming ot < 0)
C I = C o - ffot':~f(t a I B ~'2")
(1)
and f(x) is the critical scaling function. The specific heat therefore has a cusp of height Co at ra = 0 and z~= u. tne value of the critical exponent or= 2 - d r (for a system of spatial dimensionality d) is estimated theoretically for d=3 as -0.007+0.0063, while its measured value for 4lie is -0.013+0.0034 . The broadening of the superconducting transition in a magnetic field has recently been discussed in terms of the lowest Landau level (LLL) approximations Within this approximation the
fluctuation contribution to the specific heat is predicted to scale as s Cf
...
(2)
where g(x) is the LLL scaling function and zxC ~s the mean field specific heat discontinuity. There has been some confusion over the exact form of AC. Solae workers6, including ourselves7, have used AC ~ T and have found that they were unable to scale their specific heat data. Zhou et al.s have scaled their data on LuBaCuO. However, they normalise their data using the temperature dependent mean field BCS specific heat such that AC=yT(I +bt) (where the normal state electronic specific heat is given by yT). They let b=3.82 and also use a magnetic field dependent "1'to account for drifts in their experimental data 9. While it is not correct this has the effect of increasing the peak height for the higher fields and therefore improves their scaling. In the origional papers by Thouless~° and by Bray u the correct normalisatiou is
0921-4534/94/$07 00 © 1994 - Elsevier Science B V. All rights reserved SSDI 0921-4534(94)01451-5
independent. We have measured the specific heat of a single crystal sample of YBCO with magnetic fields up to 8 Tcsla applied parallel to the c-axis of the crystal. We use a high resolution ac. method detailed elsewhere ~:. Figure 1 shows the specific heat in several applied magnetic fields.
N Ouerend et al /Physica C 235-240 (1994) 1769-1770
1770
in all applied magnetic fields except for a limited temperature range in fields of 6 and 8 tesla. We would therefore expect that at fields of ==10 tesla, and larger, LLL scaling may begin to work and critical scaling would begin to fail. This value of magnetic field is consistent with the theoretical estimate by Lawrie in reference 2. 0 ~5
T(K) Figure 1. Specific heat ((2) vs. temperature in magnetic fields (from top to bottom) of 0, 0.1, 0.2, 0.5, 1, 2, 4, 6 and 8 tesla.
We extract the fluctuation specific heat from the measured specific heat using the procedure aetailed in ref. 10. Figure 2 shows the data scaled using the 3d XY model (eqn. 1) with the exponents ~t =-0.013 and v = 0.669 (as in 4He), and Tc(0) = 92.00K and dBc21 d T = - 5 T K -t
-e2 O
"D
E
v
-=6 O
O
v
.
.
.
-00,5
.
.
•
.
.
.
.
.
0
.
005
tB/B1/2v (T-O747) Figure 2. 3d XY scaling of the fluctuation specific heat in magnetic fields of 0.1, 0.2, 0.5, 1, 2, 4, 6 and 8 tesla. It is evident from figure 2 that the data exhibits single parameter critical scaling over a 20 K temperature range extending to fields of the order of 8 tesla. Figure 3 shows the data scaled using the LLL scaling form (eqn. 2) with Tc(0 ) = 92.00K and dBc: / dT= -2TK -t . The LLL scaling is found to fail
~0
0
-oo2
'
.o 2
tB/(TB) 2/3 (K-2/3T -2/3) Figure 3. LLL scaling of the fluctuation specific heat in magnetic fields (from top to bottom) of 0.1, 0.2, 0.5, 1, 2, 4, 6 and 8 tesla. References 1. Salamon MB et al. Phys. Rev. B47 5520 (1993) 2. Lawrie ID (submitted to Phys. Rev. B) 3. Le Guillou JC and Zinn-Justin J Phys. Rev. B21 3976 (1980) 4. Lipa JA and Chui TCP Phys.Rev.Lett. 51 2291 (1983) 5. UUah S and Dorsey AT Phys. Rev. 1~44 262 (1991) Wilkin NK and Moore MA Phys. Rev. B48 3464 (1993) 6. Welp U et al. Phys. Rev. LetL ,J7 3180 (1991) Junod Aet al. Physica C211 304 (1993) 7. Overend et al. Phys. Rev. Lett. 72 3238 (1994) 8. Zhou et al. Phys Rev B47 11631 (1993) 9. Private communication. 10.Thouless DJ Phys Rev Lett (1975) 11.Bray AJ Phys Rev B 9 4752 (1974) i2.Overend et ai. (submitted to Play Rev B)