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Magnetic field dependent microwave absorption evidence for a mixed liquid-solid vortex phase in Bi2Sr2CaCu2O8 + x
J. Phys. Chem. Solids Vol. 55. No. 2, pp. 167-170, 1994 Elsevier Science Ltd Printed in Goal Britain. 0022-3697/94 $6.00 + 0.00
Pergamon
MAGNETIC FIELD DEPENDENT MICROWAVE ABSORPTION EVIDENCE FOR A MIXED LIQUID-SOLID VORTEX PHASE IN Bi, Sr, CaCu, 0, +x F. J. OWENS Armament Research & Development Ctr, Picatinny, NJ 07806, U.S.A., and Dept. of Physics, Hunter College, City University of New York, 1095, Park Ave., NY 10021, U.S.A. (Received
16 August 1993; accepted
14 October 1993)
Abstract-The depinning of vortices can be observed as a change in the slope of the dependence of the surface resistance on B”*. However, close to the transition magnetic field it is observed that there is a departure from the E w dependence. It is shown that this deviation is inconsistent with second order behavior associated with a vortex glass. The deviation, however, can be accounted by the existence of a mixture of solid and vortex liquid phases resulting from a magnetic field dependent distribution of pinning barriers. Keyworris: Superconductivity,
microwave absorption, vortices, solid vortex to liquid transition.
INTRODUCTION
The application of a d.c. magnetic field to a type II superconductor below T, in the presence of microwave radiation results in an increase in the microwave energy absorbed by the sample. This increase in the surface resistance is a rest& of dissipation due to oscillation of vortices caused by the force on the vortices arising from the microwave current. The dynamical response of the vortices to the microwave current depends on whether the vortices are in the solid or liquid phase. It has been shown that there is a change in the magnetic field dependence of the surface resistance when the pinning to depinning transition occurs [l-5]. At constant temperature the magnetic field dependent surface resistance, R,(B) - R,(O) depends linearly on B I,‘* but has a different slope in the liquid and solid phase. In the liquid phase the vortices are disordered and not arranged in a lattice array. All of the studies to date appear to show that the transition is abrupt suggestive of a first order transition [l-5]. In this work the dependence of R,(B) - R,(O) on magnetic field is examined in single crystals of Bi, Sr, CaCu, 0, + _~at much smaller magnetic field intervals than previously studied in the depinning to pinning transition region in order to address the question of the nature of the transition. Of particular interest is whether the transition is first or second order. A second order nature to the transition would provide support for the existence of a vortex glass state [6,7].
EXPERIMENTAL
The relative change in the surface resistance is measured as a function of d.c. magnetic field by measuring the reflection coefficient r (B, T) of a resonant cavity containing the sample and located between the poles of an electromagnet. The cavity, which resonates at 9.2 GHz in the TEloz mode, is in one arm of a microwave bridge. When the sample absorbs, the microwave energy reflected from the cavity to the arm of the bridge containing the diode detector changes. The change in the diode current as a function of changing d.c. magnetic field is proportional to the change in the surface resistance. In order to measure that part of the absorption due to flux dissipation r (B, T) - r (0, T) which is proportional to R,(B, T) - R,(O, T) is measured. Because the change in the surface resistance is small for changes in the d.c. magnetic field of the order of 20G, the mi~roammeter is interfaced to a PC computer allowing collection of a 1000 readings at each magnetic field and averaging. This allows separation of the data from the noise. The sample in the cavity is contained in a double walled quartz glass finger Dewar which is part of an ADP heli-tran system. The temperature is controlled by the flow of cold helium or nitrogen gas over the sample and measured with a diode in contact with the sample. Further details of the experimental method have been described previously [ZJ. The measurements are made on single crystals of Bi,Sr,CaCu,O,+ 1having a T, of 85 K. The details of 167
F. J. OWENS
168
the crystal growth and characterization elsewhere [8].
are described
RESULTS AND DISCUSSION
Figure 1 is a plot of R,(B) - R,(O) vs the square root of the magnetic field at 77 K for both increasing and decreasing field in single crystals of BiZSr,CaCu,O,+,. The data show that R,(B) - R,(O) depends on B “* but changes slope at a field B*. Above B* the absorption is reversible in that the magnitude of the absorption does not depend on the direction of the magnetic field sweep while below B* the absorption is dependent on the direction of the sweep This change in the reversibility indicates that the B* corresponds to the pinning to depinning transition. These results are in general agreement with the theory of Coffey and Clem [9] for dissipation by vortex oscillation which predicts a B ‘I2 dependence for the surface resistance. The theory, however, predicts a larger slope in the liquid phase contrary to the experimental observations in many copper oxide superconductors. The reason for this discrepancy is not entirely clear. One possibility is that in the liquid phase there is considerably more misalignment of the vortices from the direction of the applied d.c. magnetic field. The data in Fig. 1 appear to show that the transition is abrupt, however the data points are separated by IOOG. When measurements are made at much smaller intervals using computer averaging as described above, deviation from the B Ii2 dependence is observed as shown in Fig. 2. If this deviation were a result of the existence of a vortex glass the surface resistance could be expected to depend on B as (B - B,)Q, where Bg is the magnetic field where the glass transition starts. The data in Fig. 2 cannot
0.34
0.38
0.34
Fig. 2. Surface resistance in arbitrary units vs B ‘I*measured at 20 G intervals in the magnetic field region of the depinning transition
of Bi-Sr-Ca-Cu-O.
at 77 K showing the deviation dependence.
from the B ‘O
however, be described by this dependence for positive values of Q. The presence of a vortex glass is indicated by a current dependent resistivity [6,7]. In the case of Y Ba, Cu, O6 + _~the resistivity has been shown to depend on the current as
(1)
ew(-J&Y,
where u = 0.19 at low current densities and 0.94 at higher current densities [6]. It is possible to measure the dependence of the surface resistance on microwave current at constant applied d.c. magnetic field. The microwave power absorbed per unit volume, P,, is (R,(B) - R,(O))J*, where J is the microwave current density which is proportional to P I’*, where P, is the incident microwave power per unit volume.
0-
0.4
B”’ (Tesla)
crystals
0.38
Bin (Terlr)
0.42
Fig. 1. The surface resistance in arbitrary units vs B ‘I2for increasing and decreasing magnetic field at 77 K in single
0.36
0.8
1.2
Incident power (MW) Fig.
3. Plot of microwave power absorbed vs incident microwave power at 77 K. The units are arbitrary.
Liquid-solid vortex phase in Bi, Sr, CaCu, O8+ x U’
169
magnetic field region the microwave absorption be given by R, = C (S) (1 -f)B
I’*+ C (L )j-S I’*,
will
(1)
where C(S) and C(L. ) are the slopes of R, vs B ‘/* in the solid and liquid phase, respectively, and f the fraction of vortices which has melted at a given magnetic field given by, “’ N (u)du Fraction melted at B
s
f= ",
Fig. 4. Illustration of formation of a mixed liquid-solid vortex phase due to a distribution of pinning barriers.
(2)
N (u)du
s0 for a Gaussian distribution
f has the form,
f = Aerf((u * - uo)/A), A measurement of Pa as a function of Pi (shown in Fig. 3) is a straight line indicating that the surface resistance, R,(B) - R,(O), is independent of the microwave current used in this experiment. Magnetic relaxation experiments by Van der Beek et al. [7] have suggested the possibility of a vortex glass in Bi, Sr, CaCu, OS+ .~. At the levels of microwave current, where the anomaly in the field dependence (Fig. 2) is observed no dependence of the surface resistance on microwave current is detected. Thus if eqn (1) is applicable to the case of high frequency a.c. current, these results would suggest that no vortex glass exists at the levels of microwave current of the experiment. The deviation of the surface resistance from the B1jZ dependence in the magnetic field region in which the depinning transition occurs may be accounted for by a magnetic field dependent distribution of pinning barriers and the existence of a critical pinning barrier, u* below which the vortices are not pinned. Those vortices arriving at sites having barriers less than u* at a given temperature will be in the fluid phase while those with barriers greater than u* will be in the solid phase. The possibility of a distribution of pinning barriers has been indicated by a number of experiments such as magnetic relaxation studies [IO]. For the purposes of analysis it will be assumed that the distribution of pinning barriers, N(u), is Gaussian. The effect of a field dependent distribution of pinning barriers on the depinning process is illustrated in Fig. 4. As the magnetic field increases the center of the distribution shifts to lower values of the pinning barrier and some fraction of the vortices denoted by A and indicated by the shaded area in the figure will have pinning barriers less than u*. Vortices in this region will have melted. Thus there will be a small magnetic field region where there will be a mixture of the solid and molten phase. In this
(3)
where A is the width of the distribution which will be assumed to be independent of the magnetic field and u, is the pinning barrier at the center of the distribution which is taken to have a magnetic field dependence k/B from the Anderson-Kim mode1 (111. For values of (u* - u,,)/A less than 0.5 the error function can be approximated as a straight line in (u * - u&A. This results in the field dependence of the surface resistance in the mixed phase region having the form R,=C(1)B”2-C(2)/B”2,
(4)
where C (1) and C(2) are constants. This equation describes the magnetic field dependence of the surface resistance in the transition region well, as demonstrated by the straight line that is obtained when R,/B Ii2 is plotted vs B -‘, as shown in Fig. 5. This analysis suggests that the observed deviation in the d.c. magnetic field dependence from B’j2 at constant temperature in the region of the depinning magnetic field is a result of a mixture of a solid and
211 276 215 c
274
“m $
213
212 271 270 1.6
1.8
l/B (Tesla) Fig. 5. Plot of
R,/B”’
in arbitrary units vs
B-l.
F. J.
170
liquid vortex phases resulting from a field dependent distribution of pinning barriers.
REFERENCES I. Owens F. J., Phys. tirt. Af66, 61 (1992). 2. Owens F. J., Physkz Cl%, 2.55 (1992). 3. Owens F. J., Solid Stare Commun. 81, 97 (1992). 4. Sridhar S., Maheswaran B., Willemsen B. A. and Wu D. H., Phys. Rev. Lat. 68, 220 (1992). 5. Owens F. J., Physica C178, 456 (1991).
OWENS
6. Decker C., EideIioth W. and Koch R. H., P&S. Rev. Lett. 68, 334 (1992); Phys. Rev. Lat. 63, 1512 (1989). 7. van der Beek C. J., Kees P. H., Maley M. P., Menken M. J. V. and Menovskv A. A.. Phvsica Cl%. 307 _ (1992). 8. Owens F. J., Iqbaland Z. and Wolf E., J. Phys. C4, 205 (i992). 9. Co&y MI W. and Ciem J. R., Phy.x 1peo. LN. 67, 386 (1991). IO. Reissner M. and Steiner W., Supercond. Sci. Technol. 5, 5367 (1992). 11. Anderson P. W. and Kim Y. B., Rev. Mod Phys. 36, 39 (1964).
Pergamon
MAGNETIC FIELD DEPENDENT MICROWAVE ABSORPTION EVIDENCE FOR A MIXED LIQUID-SOLID VORTEX PHASE IN Bi, Sr, CaCu, 0, +x F. J. OWENS Armament Research & Development Ctr, Picatinny, NJ 07806, U.S.A., and Dept. of Physics, Hunter College, City University of New York, 1095, Park Ave., NY 10021, U.S.A. (Received
16 August 1993; accepted
14 October 1993)
Abstract-The depinning of vortices can be observed as a change in the slope of the dependence of the surface resistance on B”*. However, close to the transition magnetic field it is observed that there is a departure from the E w dependence. It is shown that this deviation is inconsistent with second order behavior associated with a vortex glass. The deviation, however, can be accounted by the existence of a mixture of solid and vortex liquid phases resulting from a magnetic field dependent distribution of pinning barriers. Keyworris: Superconductivity,
microwave absorption, vortices, solid vortex to liquid transition.
INTRODUCTION
The application of a d.c. magnetic field to a type II superconductor below T, in the presence of microwave radiation results in an increase in the microwave energy absorbed by the sample. This increase in the surface resistance is a rest& of dissipation due to oscillation of vortices caused by the force on the vortices arising from the microwave current. The dynamical response of the vortices to the microwave current depends on whether the vortices are in the solid or liquid phase. It has been shown that there is a change in the magnetic field dependence of the surface resistance when the pinning to depinning transition occurs [l-5]. At constant temperature the magnetic field dependent surface resistance, R,(B) - R,(O) depends linearly on B I,‘* but has a different slope in the liquid and solid phase. In the liquid phase the vortices are disordered and not arranged in a lattice array. All of the studies to date appear to show that the transition is abrupt suggestive of a first order transition [l-5]. In this work the dependence of R,(B) - R,(O) on magnetic field is examined in single crystals of Bi, Sr, CaCu, 0, + _~at much smaller magnetic field intervals than previously studied in the depinning to pinning transition region in order to address the question of the nature of the transition. Of particular interest is whether the transition is first or second order. A second order nature to the transition would provide support for the existence of a vortex glass state [6,7].
EXPERIMENTAL
The relative change in the surface resistance is measured as a function of d.c. magnetic field by measuring the reflection coefficient r (B, T) of a resonant cavity containing the sample and located between the poles of an electromagnet. The cavity, which resonates at 9.2 GHz in the TEloz mode, is in one arm of a microwave bridge. When the sample absorbs, the microwave energy reflected from the cavity to the arm of the bridge containing the diode detector changes. The change in the diode current as a function of changing d.c. magnetic field is proportional to the change in the surface resistance. In order to measure that part of the absorption due to flux dissipation r (B, T) - r (0, T) which is proportional to R,(B, T) - R,(O, T) is measured. Because the change in the surface resistance is small for changes in the d.c. magnetic field of the order of 20G, the mi~roammeter is interfaced to a PC computer allowing collection of a 1000 readings at each magnetic field and averaging. This allows separation of the data from the noise. The sample in the cavity is contained in a double walled quartz glass finger Dewar which is part of an ADP heli-tran system. The temperature is controlled by the flow of cold helium or nitrogen gas over the sample and measured with a diode in contact with the sample. Further details of the experimental method have been described previously [ZJ. The measurements are made on single crystals of Bi,Sr,CaCu,O,+ 1having a T, of 85 K. The details of 167
F. J. OWENS
168
the crystal growth and characterization elsewhere [8].
are described
RESULTS AND DISCUSSION
Figure 1 is a plot of R,(B) - R,(O) vs the square root of the magnetic field at 77 K for both increasing and decreasing field in single crystals of BiZSr,CaCu,O,+,. The data show that R,(B) - R,(O) depends on B “* but changes slope at a field B*. Above B* the absorption is reversible in that the magnitude of the absorption does not depend on the direction of the magnetic field sweep while below B* the absorption is dependent on the direction of the sweep This change in the reversibility indicates that the B* corresponds to the pinning to depinning transition. These results are in general agreement with the theory of Coffey and Clem [9] for dissipation by vortex oscillation which predicts a B ‘I2 dependence for the surface resistance. The theory, however, predicts a larger slope in the liquid phase contrary to the experimental observations in many copper oxide superconductors. The reason for this discrepancy is not entirely clear. One possibility is that in the liquid phase there is considerably more misalignment of the vortices from the direction of the applied d.c. magnetic field. The data in Fig. 1 appear to show that the transition is abrupt, however the data points are separated by IOOG. When measurements are made at much smaller intervals using computer averaging as described above, deviation from the B Ii2 dependence is observed as shown in Fig. 2. If this deviation were a result of the existence of a vortex glass the surface resistance could be expected to depend on B as (B - B,)Q, where Bg is the magnetic field where the glass transition starts. The data in Fig. 2 cannot
0.34
0.38
0.34
Fig. 2. Surface resistance in arbitrary units vs B ‘I*measured at 20 G intervals in the magnetic field region of the depinning transition
of Bi-Sr-Ca-Cu-O.
at 77 K showing the deviation dependence.
from the B ‘O
however, be described by this dependence for positive values of Q. The presence of a vortex glass is indicated by a current dependent resistivity [6,7]. In the case of Y Ba, Cu, O6 + _~the resistivity has been shown to depend on the current as
(1)
ew(-J&Y,
where u = 0.19 at low current densities and 0.94 at higher current densities [6]. It is possible to measure the dependence of the surface resistance on microwave current at constant applied d.c. magnetic field. The microwave power absorbed per unit volume, P,, is (R,(B) - R,(O))J*, where J is the microwave current density which is proportional to P I’*, where P, is the incident microwave power per unit volume.
0-
0.4
B”’ (Tesla)
crystals
0.38
Bin (Terlr)
0.42
Fig. 1. The surface resistance in arbitrary units vs B ‘I2for increasing and decreasing magnetic field at 77 K in single
0.36
0.8
1.2
Incident power (MW) Fig.
3. Plot of microwave power absorbed vs incident microwave power at 77 K. The units are arbitrary.
Liquid-solid vortex phase in Bi, Sr, CaCu, O8+ x U’
169
magnetic field region the microwave absorption be given by R, = C (S) (1 -f)B
I’*+ C (L )j-S I’*,
will
(1)
where C(S) and C(L. ) are the slopes of R, vs B ‘/* in the solid and liquid phase, respectively, and f the fraction of vortices which has melted at a given magnetic field given by, “’ N (u)du Fraction melted at B
s
f= ",
Fig. 4. Illustration of formation of a mixed liquid-solid vortex phase due to a distribution of pinning barriers.
(2)
N (u)du
s0 for a Gaussian distribution
f has the form,
f = Aerf((u * - uo)/A), A measurement of Pa as a function of Pi (shown in Fig. 3) is a straight line indicating that the surface resistance, R,(B) - R,(O), is independent of the microwave current used in this experiment. Magnetic relaxation experiments by Van der Beek et al. [7] have suggested the possibility of a vortex glass in Bi, Sr, CaCu, OS+ .~. At the levels of microwave current, where the anomaly in the field dependence (Fig. 2) is observed no dependence of the surface resistance on microwave current is detected. Thus if eqn (1) is applicable to the case of high frequency a.c. current, these results would suggest that no vortex glass exists at the levels of microwave current of the experiment. The deviation of the surface resistance from the B1jZ dependence in the magnetic field region in which the depinning transition occurs may be accounted for by a magnetic field dependent distribution of pinning barriers and the existence of a critical pinning barrier, u* below which the vortices are not pinned. Those vortices arriving at sites having barriers less than u* at a given temperature will be in the fluid phase while those with barriers greater than u* will be in the solid phase. The possibility of a distribution of pinning barriers has been indicated by a number of experiments such as magnetic relaxation studies [IO]. For the purposes of analysis it will be assumed that the distribution of pinning barriers, N(u), is Gaussian. The effect of a field dependent distribution of pinning barriers on the depinning process is illustrated in Fig. 4. As the magnetic field increases the center of the distribution shifts to lower values of the pinning barrier and some fraction of the vortices denoted by A and indicated by the shaded area in the figure will have pinning barriers less than u*. Vortices in this region will have melted. Thus there will be a small magnetic field region where there will be a mixture of the solid and molten phase. In this
(3)
where A is the width of the distribution which will be assumed to be independent of the magnetic field and u, is the pinning barrier at the center of the distribution which is taken to have a magnetic field dependence k/B from the Anderson-Kim mode1 (111. For values of (u* - u,,)/A less than 0.5 the error function can be approximated as a straight line in (u * - u&A. This results in the field dependence of the surface resistance in the mixed phase region having the form R,=C(1)B”2-C(2)/B”2,
(4)
where C (1) and C(2) are constants. This equation describes the magnetic field dependence of the surface resistance in the transition region well, as demonstrated by the straight line that is obtained when R,/B Ii2 is plotted vs B -‘, as shown in Fig. 5. This analysis suggests that the observed deviation in the d.c. magnetic field dependence from B’j2 at constant temperature in the region of the depinning magnetic field is a result of a mixture of a solid and
211 276 215 c
274
“m $
213
212 271 270 1.6
1.8
l/B (Tesla) Fig. 5. Plot of
R,/B”’
in arbitrary units vs
B-l.
F. J.
170
liquid vortex phases resulting from a field dependent distribution of pinning barriers.
REFERENCES I. Owens F. J., Phys. tirt. Af66, 61 (1992). 2. Owens F. J., Physkz Cl%, 2.55 (1992). 3. Owens F. J., Solid Stare Commun. 81, 97 (1992). 4. Sridhar S., Maheswaran B., Willemsen B. A. and Wu D. H., Phys. Rev. Lat. 68, 220 (1992). 5. Owens F. J., Physica C178, 456 (1991).
OWENS
6. Decker C., EideIioth W. and Koch R. H., P&S. Rev. Lett. 68, 334 (1992); Phys. Rev. Lat. 63, 1512 (1989). 7. van der Beek C. J., Kees P. H., Maley M. P., Menken M. J. V. and Menovskv A. A.. Phvsica Cl%. 307 _ (1992). 8. Owens F. J., Iqbaland Z. and Wolf E., J. Phys. C4, 205 (i992). 9. Co&y MI W. and Ciem J. R., Phy.x 1peo. LN. 67, 386 (1991). IO. Reissner M. and Steiner W., Supercond. Sci. Technol. 5, 5367 (1992). 11. Anderson P. W. and Kim Y. B., Rev. Mod Phys. 36, 39 (1964).