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Flux flow resistivity and critical fields in amorphous superconductors
Journal of Non-Crystalline Solids 61 & 62 (1984) 1155-1160 North-Holland, Amsterdam
1155
FLUX FLOW RESISTIVITY AND CRITICAL FIELDS IN AMORPHOUS SUPERCONDUCTORS
S. J. POON Department of Physics, University of Virginia, Charlottesville, Virginia 22901, USA
Flux flow resistivity pf, lower critical field Hcl, and upper critical field Hc2 in extreme type II (K - 70-80) amorphous bulk superconductors with minimal flux pinning force have been studied. Results on HcI(T ) and Hc2(T ) are found to depend on sample conditions. The low-field (Hcl < H << Hc2) and high-field (H = Hc2 ) flux flow resistivity can only be accounted for if both normal dissipation terms and anomalous terms in the time-dependent microscopic theory are included. The lower critical fields follow the theoretical predictions of Maki. The different trends in the critical fields of homogeneous and inhomogeneous samples are presented and discussed.
i. INTRODUCTION Studies of amorphous type-ll superconductors with large Ginzburg-Landau parameter < and ideal properties (weak flux pinning force, reversibility in magnetization, sharp transitions in zero field) are suitable for comparison with standard 'dirty' type-ll theories. I
In this paper, we summarize results on flux
flow resistivity, lower and upper critical fields in amorphous Zr-base superconductors. 2. EXPERIMENTAL PROCEDURE Sample preparation, analysis, superconducting measurements were given in reference 2.
Flux flow resistivity measurement was obtained by tracing voltage
versus current plots as discussed in reference 3. measured inductively.
Lower critical fields were
Upper critical fields were measured resistively with
current parallel to the applied field. 3. RESULTS Upper critical field Hc2 In figure i are shown the Hc2 curves for homogeneous and inhomogeneous (obtained at 320 C) Zr3Ni.
The difference in Hc2(T ) behavior is obvious.
The
homogeneous sample can be described by the 'dirty' type-ll theory of WerthamerHelfand-Hohenberg (WHH) .4
On the contrary, inhomogeneous samples exhibit en-
hancement in Hc2(T) above the ~so = ~ curve.
The effect of inhomogeneities on
Hc2(T) must depend on the relative dimensions of the coherence length and the scale of inhomogeneity.
Theoretically, this problem can be treated by the
0022-3093/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
S.J. Pooh / Flux flow resistivity and critical fields
1156
80
s
A
60
o
o _.=4O z B
x
2C
1
2
3
TIK) FIGURE 1 Upper critical field data taken at iA/cm 2 on (A) Zr75Ni25 quenched at 250 C and (B) Zr3Ni quenched at 320 C. Solid lines are fits to the WHH theory. Inset shows inductive transition curves. Vertical arrows indicate 0% and 90% points on the resistive curve.
proximity effect model.
Recently, this has been attempted by Nabatovskii and
Shapiro 5 for special geometries of inhomogeneities.
It is obvious that
anomalous Hc2(T) behavior can occur if the two scales mentioned earlier become comparable.
On the other hand, if the size of inhomogeneity is larger than the
coherence length (50 - i00 ~), two cases result.
If the T c of the inhomogeneity
is much lower than that of the host, then Hc2(T ) is governed by the host. Otherwise, an upturn in Hc2(T ) can occur if the inhomogeneity has a higher critical field gradient.
Recent measurement on the B-phase of Zr-Pd with ~-Zr 6 (very low Tc) embedded in it shows only 'normal' Hc2(T ) behavior. Lower critical field Hcl Similar to Hc2, the lower critical field can also be affected by the presence
of inhomogeneities.
In this case dealing with flux entry field, the effect is
due to flux pinning at the grain boundaries.
In figure 2, we show HcI(T ) of
'powdered' homogeneous Zr7oNi30 and Zr75Cu25 fitted to the Maki curve.
The
values obtained here when combined with their Hc2 values leads to a reasonable
1157
S.J. Poon/ Flux flow resistivity and critical fields
estimate
of the gap value
ment on parallel This
is due to surface
ment
in different
the effect
2Ao/kBT , in agreement w i t h
trips show anomalously flux pinning,
the BCS value. 7
(almost
as evidenced
current-field-sample
ls less sensitive
high
Measure-
linear in T) Hcl values.
from critical
orientations
(figure
current measure-
3),
In this case,
to the size of inhomogeneities.
20
15
40 30 20
\\" @
.
~
ZrToNi30
°io
•
~"
o
o
o
~
.
.
.....
"~
Zr75Cu25 ... "' 1 ..'
.."
0
I~3
• Zr75Ni25 • (Quenched 320C)
•
I
30
20
H (kOe)
0
1
2
FIGURE 3 Flux pinning force as a function of applied field at reduced temperature t = 0.7 for Zr70Ni30 and Zr75Ni25 (320 C) samples. Sample strip - field current orientations are as shown.
3
T(K)
FIGURE 2 Lower critical field as a function of temperature for homogeneous and 'powdered' samples of Zr70Ni30 and Zr75Cu25 alloys. Solid lines are fits to the Maki theory.
Flux flow resistivity
pf
To explain with success conductors, equations
results
the flux flow resistivity
obtained
must be employed•
(H << Hc2) 8 and high field
in dirty type-ll
from the time-dependent-Ginzburg-Landau Solutions
are obtained
(H ~ Hc2 )9'I0 limits.
super(TDGL)
in both the low field
115 8
S.J. Poon/ Flux flow resistivity and critical fields
In figure 4 is shown the low-field viscosity coefficient i/b(t) versus i/(l-t) for four samples.
It is necessary to include both the normal dissipa-
tion terms (Bardeen-Stephen BS and Tinkham T; or Kupriyanov-Likharev KL) and the anomalous term (Gor'kov-Kopnin GK) obtained from the time-dependent theory in order to explain the results satisfactorily (figures 4 and 5).
• , , i .... i • Zr3Rh (AS QUENCHED) Zr3Rh (ANNEALED) X Zr3Ni (AS QUENCHED) ~o~>.~ 0 Zr3Ni(ANNEALED) ~ ,
~ . ~
G
~. ~ L
.S~S~-~
GKKL
.L t=066 /
t /
/
,
I
. . . .
5
I
t=095
~,,
,
IO
50
.... 5
,
~
,
~0
5O
ll(l-t)
1/(l-t)
FIGURE 4 Viscosity coefficient I/b(t) as a function of i/(l-t) on log-log scale. Theoretical curves and their notations are discussed in the text.
FIGI~E 5 Same as in figure 4. Theoretical curves are obtained by combining those of figure 4.
At high field, one compares the parameter ~* = (1/@n)(dpf/dh) at reduced field h = l
with the time-dependent theories.
Figure 6 illustrates the values
of c~* and theoretical curves for strong-coupling (Thompson, Takayama-Ebisawa) and weak-coupling (Caroli-Maki) superconductors.
Close to Tc, pair-breaking
effect has to be included leading to a decreased ~*.
Thus, the results are
in good agreement with the general results (T,T-E). D
• Zr3Rh (AS QUENCHED)
Zr3Rh(ANNEALED) x Zr3Ni (ASQUENCHED)
4
o Zr3 Ni (ANNEALED)
~
. . . . -*
•~- ~ - - ~ 3 FAKAYAMA ;BISAWA ~
.~
~~
\
~'\ o\ \~ \~
2
1
o
o!2
o'.6
o!4
0:8
,.o
t
FIGURE 6 The gradient parameter ~* versus reduced temperature. trends followed by data points.
Dashed lines indicate
S.ZPoon/Flux flow resistivity and critical fields
1159
At intermediate fields, a universal behavior in Pf/Pn as a function of h and t can be seen.
Therefore,
the present results provide an additional test of the
time-dependent theories using 'ideal' type-II superconductors.
ACKNOWLEDGEMENT This work is sponsored by the National Science Foundation under Grant DMR83-02624. REFERENCES i) D. Saint-James, G. Sarna, and E.J. Thomas, "Type II Superconductivity" (Pergamon, Oxford, 1969). 2) S.J. Poon, Phys. Rev. B2_~7, 5519 (1983). 3) S.J. Pooh and K.M. Wong, Phys. Rev. B27, 6985 (1983). 4) N.R. Werthamer, E. Helfand, and P.C. Hohenberg, Phys. Rev. 147, 295 (1966). 5) V.M. Nabutovskii and B. Ya. Shapiro, J. Low Temp. Phys. 49, 465 (1982). 6) S.J. Poon, K.M. Weng, and S.E. Anderson, unpublished results. 7) S.J. Pooh and P.L. Dunn, to be published in J. Low Temp. Phys.
(1984).
8) L.P. Gor'kov and N.B. Kopnin, Sov. Phys.-JETP 38, 195 (1974), and references cited therein. 9) R.S. Thompson, Phys. Rev. B!, 3.27 (1970). i0) H. Takayama and H. Ebisawa, Prog. Theor. Phys. 44, 1450 (1970).
1155
FLUX FLOW RESISTIVITY AND CRITICAL FIELDS IN AMORPHOUS SUPERCONDUCTORS
S. J. POON Department of Physics, University of Virginia, Charlottesville, Virginia 22901, USA
Flux flow resistivity pf, lower critical field Hcl, and upper critical field Hc2 in extreme type II (K - 70-80) amorphous bulk superconductors with minimal flux pinning force have been studied. Results on HcI(T ) and Hc2(T ) are found to depend on sample conditions. The low-field (Hcl < H << Hc2) and high-field (H = Hc2 ) flux flow resistivity can only be accounted for if both normal dissipation terms and anomalous terms in the time-dependent microscopic theory are included. The lower critical fields follow the theoretical predictions of Maki. The different trends in the critical fields of homogeneous and inhomogeneous samples are presented and discussed.
i. INTRODUCTION Studies of amorphous type-ll superconductors with large Ginzburg-Landau parameter < and ideal properties (weak flux pinning force, reversibility in magnetization, sharp transitions in zero field) are suitable for comparison with standard 'dirty' type-ll theories. I
In this paper, we summarize results on flux
flow resistivity, lower and upper critical fields in amorphous Zr-base superconductors. 2. EXPERIMENTAL PROCEDURE Sample preparation, analysis, superconducting measurements were given in reference 2.
Flux flow resistivity measurement was obtained by tracing voltage
versus current plots as discussed in reference 3. measured inductively.
Lower critical fields were
Upper critical fields were measured resistively with
current parallel to the applied field. 3. RESULTS Upper critical field Hc2 In figure i are shown the Hc2 curves for homogeneous and inhomogeneous (obtained at 320 C) Zr3Ni.
The difference in Hc2(T ) behavior is obvious.
The
homogeneous sample can be described by the 'dirty' type-ll theory of WerthamerHelfand-Hohenberg (WHH) .4
On the contrary, inhomogeneous samples exhibit en-
hancement in Hc2(T) above the ~so = ~ curve.
The effect of inhomogeneities on
Hc2(T) must depend on the relative dimensions of the coherence length and the scale of inhomogeneity.
Theoretically, this problem can be treated by the
0022-3093/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
S.J. Pooh / Flux flow resistivity and critical fields
1156
80
s
A
60
o
o _.=4O z B
x
2C
1
2
3
TIK) FIGURE 1 Upper critical field data taken at iA/cm 2 on (A) Zr75Ni25 quenched at 250 C and (B) Zr3Ni quenched at 320 C. Solid lines are fits to the WHH theory. Inset shows inductive transition curves. Vertical arrows indicate 0% and 90% points on the resistive curve.
proximity effect model.
Recently, this has been attempted by Nabatovskii and
Shapiro 5 for special geometries of inhomogeneities.
It is obvious that
anomalous Hc2(T) behavior can occur if the two scales mentioned earlier become comparable.
On the other hand, if the size of inhomogeneity is larger than the
coherence length (50 - i00 ~), two cases result.
If the T c of the inhomogeneity
is much lower than that of the host, then Hc2(T ) is governed by the host. Otherwise, an upturn in Hc2(T ) can occur if the inhomogeneity has a higher critical field gradient.
Recent measurement on the B-phase of Zr-Pd with ~-Zr 6 (very low Tc) embedded in it shows only 'normal' Hc2(T ) behavior. Lower critical field Hcl Similar to Hc2, the lower critical field can also be affected by the presence
of inhomogeneities.
In this case dealing with flux entry field, the effect is
due to flux pinning at the grain boundaries.
In figure 2, we show HcI(T ) of
'powdered' homogeneous Zr7oNi30 and Zr75Cu25 fitted to the Maki curve.
The
values obtained here when combined with their Hc2 values leads to a reasonable
1157
S.J. Poon/ Flux flow resistivity and critical fields
estimate
of the gap value
ment on parallel This
is due to surface
ment
in different
the effect
2Ao/kBT , in agreement w i t h
trips show anomalously flux pinning,
the BCS value. 7
(almost
as evidenced
current-field-sample
ls less sensitive
high
Measure-
linear in T) Hcl values.
from critical
orientations
(figure
current measure-
3),
In this case,
to the size of inhomogeneities.
20
15
40 30 20
\\" @
.
~
ZrToNi30
°io
•
~"
o
o
o
~
.
.
.....
"~
Zr75Cu25 ... "' 1 ..'
.."
0
I~3
• Zr75Ni25 • (Quenched 320C)
•
I
30
20
H (kOe)
0
1
2
FIGURE 3 Flux pinning force as a function of applied field at reduced temperature t = 0.7 for Zr70Ni30 and Zr75Ni25 (320 C) samples. Sample strip - field current orientations are as shown.
3
T(K)
FIGURE 2 Lower critical field as a function of temperature for homogeneous and 'powdered' samples of Zr70Ni30 and Zr75Cu25 alloys. Solid lines are fits to the Maki theory.
Flux flow resistivity
pf
To explain with success conductors, equations
results
the flux flow resistivity
obtained
must be employed•
(H << Hc2) 8 and high field
in dirty type-ll
from the time-dependent-Ginzburg-Landau Solutions
are obtained
(H ~ Hc2 )9'I0 limits.
super(TDGL)
in both the low field
115 8
S.J. Poon/ Flux flow resistivity and critical fields
In figure 4 is shown the low-field viscosity coefficient i/b(t) versus i/(l-t) for four samples.
It is necessary to include both the normal dissipa-
tion terms (Bardeen-Stephen BS and Tinkham T; or Kupriyanov-Likharev KL) and the anomalous term (Gor'kov-Kopnin GK) obtained from the time-dependent theory in order to explain the results satisfactorily (figures 4 and 5).
• , , i .... i • Zr3Rh (AS QUENCHED) Zr3Rh (ANNEALED) X Zr3Ni (AS QUENCHED) ~o~>.~ 0 Zr3Ni(ANNEALED) ~ ,
~ . ~
G
~. ~ L
.S~S~-~
GKKL
.L t=066 /
t /
/
,
I
. . . .
5
I
t=095
~,,
,
IO
50
.... 5
,
~
,
~0
5O
ll(l-t)
1/(l-t)
FIGURE 4 Viscosity coefficient I/b(t) as a function of i/(l-t) on log-log scale. Theoretical curves and their notations are discussed in the text.
FIGI~E 5 Same as in figure 4. Theoretical curves are obtained by combining those of figure 4.
At high field, one compares the parameter ~* = (1/@n)(dpf/dh) at reduced field h = l
with the time-dependent theories.
Figure 6 illustrates the values
of c~* and theoretical curves for strong-coupling (Thompson, Takayama-Ebisawa) and weak-coupling (Caroli-Maki) superconductors.
Close to Tc, pair-breaking
effect has to be included leading to a decreased ~*.
Thus, the results are
in good agreement with the general results (T,T-E). D
• Zr3Rh (AS QUENCHED)
Zr3Rh(ANNEALED) x Zr3Ni (ASQUENCHED)
4
o Zr3 Ni (ANNEALED)
~
. . . . -*
•~- ~ - - ~ 3 FAKAYAMA ;BISAWA ~
.~
~~
\
~'\ o\ \~ \~
2
1
o
o!2
o'.6
o!4
0:8
,.o
t
FIGURE 6 The gradient parameter ~* versus reduced temperature. trends followed by data points.
Dashed lines indicate
S.ZPoon/Flux flow resistivity and critical fields
1159
At intermediate fields, a universal behavior in Pf/Pn as a function of h and t can be seen.
Therefore,
the present results provide an additional test of the
time-dependent theories using 'ideal' type-II superconductors.
ACKNOWLEDGEMENT This work is sponsored by the National Science Foundation under Grant DMR83-02624. REFERENCES i) D. Saint-James, G. Sarna, and E.J. Thomas, "Type II Superconductivity" (Pergamon, Oxford, 1969). 2) S.J. Poon, Phys. Rev. B2_~7, 5519 (1983). 3) S.J. Pooh and K.M. Wong, Phys. Rev. B27, 6985 (1983). 4) N.R. Werthamer, E. Helfand, and P.C. Hohenberg, Phys. Rev. 147, 295 (1966). 5) V.M. Nabutovskii and B. Ya. Shapiro, J. Low Temp. Phys. 49, 465 (1982). 6) S.J. Poon, K.M. Weng, and S.E. Anderson, unpublished results. 7) S.J. Pooh and P.L. Dunn, to be published in J. Low Temp. Phys.
(1984).
8) L.P. Gor'kov and N.B. Kopnin, Sov. Phys.-JETP 38, 195 (1974), and references cited therein. 9) R.S. Thompson, Phys. Rev. B!, 3.27 (1970). i0) H. Takayama and H. Ebisawa, Prog. Theor. Phys. 44, 1450 (1970).