Dionne G.F. Magnetic Oxides
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452 8 Spin Transport Properties
which gives values of 67 for La
1:8
Sr
0:2
CuO
4
and63forYBa
2
Cu
3
O
7
, both in good
agreement with the value of 62 derived from experiment [67]. These results serve
to confirm the expectation that the high-T
c
perovskites behave as natural type-II
superconductors, with
L
=Ÿ
0
1. (Metals with the large K values are “Pippard”
superconductors, with
L
=Ÿ
0
1.) Although is temperature independent, in ac-
cord with traditional theory [104], both
L
and
0
derive individual temperature
dependences through the supercarrier density; as T ! T
c
,
L
and
0
!1through
their common dependence on
n
e
s
1=2
from (8.44). Although Pippard’s uncertainty
principle arguments were used in these derivations of coherence length, the CET for-
malism gives the above temperature dependence of
0
that is more characteristic of
the GL definition, which also produces a temperature-insensitive D .T /=.T /
[105]. If Pippard’s use of the carrier mean-free-path (here replacing the symbol `
from Sect. 8.4.1 with d , which could represent the thickness of a film or the diam-
eter of a fine wire) is adopted to define a reduced effective coherence length ,we
obtain
1= D 1=
0
C 1=d; (8.74)
for which ! d where
0
d .Ifd represents the radius of a fluxoid domain
in this model, i.e., 2d is the spacing between nonsuperconducting regions, a de-
cline in superconduction efficiency can set in as d becomes increasingly smaller.
Microscopic-scale d values can lower n
s
by raising the polaron dispersal parameter
ˇ, and thereby increasing W to cause a lower T
c
. Another effect of short mean-
free-paths would occur in the size of the large-polaron radius r
p
[106]. Since r
p
determines n
0
(and hence n
t
/, the nominal threshold density for superconduction,
any decrease in its value would also increase W .
At this point, we can compare the physical meanings and relationships of the
large-polaron radius and the coherence length. The cell radius of an isolated polaron
r
p
is a normal-state parameter that determines the minimum density for which a po-
laron chain may condense to the superconduction state, and is directly dependent
on the transfer integral b
p
. The coherence length
0
, on the other hand, emerges
from the uncertainty principle as applied to the wave mechanical nature of the
superconducting state, i.e.,
0
is larger for smaller momentum values. Since it varies
as
n
e
s
1=2
according to (8.70), it is dependent on the ˇ dispersal parameter and
would be generally smaller for oxides where x
s
<0:1.
8.5.7 Type-II Superconductors
As introduced in Sect. 8.4.1, the three general categories of superconductors are
distinguished by the relations between
0
and
L
. In reality, only type-I materials
(“pure” superconductors with
0
L
) feature a single critical magnetic field,
in which the flux expulsion takes place completely once the thermodynamic H
c
threshold is reached (in a bulk specimen). In materials of practical importance,
however, flux expulsion occurs over a range of fields, with the limits defined by
8.5 Supercurrents and Magnetic Fields 453
Fig. 8.42 Variation of H and
s
at the fluxoid/superconductor interface for 1 (type I) and
1 (type II) [9]
H
c1
H
c
H
c2
. As indicated by the spatial variation in H
c
and
s
at a nor-
mal/superconductor interface sketched in Fig. 8.42, type-II superconductors with
1 have interface regions set by narrow enough and
L
large enough to permit
the bulk volume of the material to divide into islands of “fluxoid” domains that har-
bor magnetic flux in their “normal” cores at a field less than H
c
. With the presence
of fluxoid domains, the thermodynamic critical field H
c
can no longer be measured
directly. Since the invasion of flux reduces the volume of material from which flux
is expelled, a larger field H
c2
will be required to offset the condensation energy
defined in (8.50). As analyzed in detail by Abrikosov [107] and discussed by Kittel
[108] and Tinkham [109] the critical magnetic fields for “clean” highT
c
perovskite
superconductors with
0
and
L
may be approximated by
H
c1
=H
c
Œ.
p
2/ k
1
ln k (8.75)
and
H
c2
=H
c
.
p
2/ : (8.76)
Since is temperature independent according to (8.75)and(8.76), H
c1
and H
c2
should both track with H
c
as functions of temperature. As shown in Fig. 8.43,a
universal curve with W D 1 is fitted to the H
c2
data [110] from bulk polycrystalline
(Dy,Eu)Ba
2
Cu
3
O
y
.
Another comparison with experiment may be carried out with the values of criti-
cal fields estimated from measurements of @H
c
.T /=@T at T D T
c
.ForW D 1,the
slope from (8.56)issimplyH
c
.0/=T
c
, and the results of Finnemore et al. [111],
@H
c
.T /=@T D165 Oe K
1
for YBa
2
Cu
3
O
7
conform closely to the calculated es-
timate of 155 Oe K
1
, with H
c
.0/ D 14 kOe [computed beneath (8.52)] and T
c
D
91 K. If (8.75)and(8.76) are used with the above approximations, and with D 63
from (8.73), the ratio Œ@H
c2
.T /=@T Œ@H
c1
.T /=@T 2
2
= ln D 1; 920,which
matches closely the result of the Cava et al. [67]:
13 kOe K
1
7 Oe K
1
D
1; 860. It also follows from (8.75)and(8.76) that for H
c
D 14 kOe, H
c1
660 Oe
and H
c2
1; 250 kOe.
The question of the “dirty” superconductor may now be examined in the above
context. Traditionally, a type-II superconductor is a metal with a reduced to values
454 8 Spin Transport Properties
Fig. 8.43 CET-calculated curve of H
c2
.
t
/
=H
c2
.0/ vs. t compared with data of Yamagishi et al.
[110]forW D 0:8 [9]
below
L
by the presence of inhomogeneities – “normal” regions that act as nucle-
ation centers for the fluxoid lattice, in a manner similar to the formation of reverse
magnetic domains about pores and grain boundaries in a ferromagnet. According to
(8.75), the situation for which
L
=
L
=d 1 occurs in metals where d
0
,
thus limiting the coherence length to a mean-free path. With the discovery of the
high-temperatureoxide superconductorswith typical
0
30
˚
A, the
L
= 1 con-
dition appears to exist naturally even in macroscopically homogeneous specimens.
The effect of local or dispersed inhomogeneities in the form of impurities or lat-
tice defects has somewhat different implications for a metal and for an insulator. In
metals, with the greatest charge screening effects,
0
L
, and the effect of inho-
mogeneities is significant in reducing to set up the condition for establishing the
fluxoid lattice, which can occur where d
L
. In effect, the impurities in a metal
are required for the creation of a type-II superconductor through the nucleation and
pinning of fluxoid regions at the impurity centers. In superconducting oxides with
finite (measurable) dielectric constants,
0
L
, and the fluxoid lattice forms in
the absence ofimpurities. Without pinning centers, they may be described as natural
type-II superconductors.
Since the fluxoids are mobile, the inhomogeneities will only affect the fluxoid
lattice if d
0
at which point the impurity regions would then serve as pin-
ning centers for the fluxoids. The use of induced inhomogeneities as a practical
design strategy for increased magnetic field and supercurrent capabilities of high-
T
c
materials may therefore be viewed as a stabilization of the fluxoid lattice due
to the “pinning” of domains about the inhomogeneities. Without these centers, the
fluxoid structure is likely to be fluid, and with the merging and collapsing of domains
Appendix 8A 455
Table 8.2 Superconductor parameter values and their dependence on carrier density
Theory
Parameter n
e
s
Dependence prediction Experiment
G.0/
n
e
s
2
8 10
6
10
6
–10
7
erg cm
3
v
s
.0/
n
e
s
1=2
8:4 10
6
cm s
1
i
c
.0/ Film
n
e
s
3=2
6 10
8
4 10
7
Acm
2
Bulk
n
e
s
L
.0/
n
e
s
1=2
1,740 1,500–2,000
˚
A
0
.0/
n
e
s
1=2
27 20–35
˚
A
.
D
L
=
0
/
– 63 43–100
H
c
.0/ n
e
s
14 10–20 kOe
T
c
.0/ n
e
s
.0/ ––(K)
@H
c
=@T
.
T D T
c
/
H
c
.0/=T
c
.
W D 1
/
155 165 Oe K
1
.
@H
c2
=@T
/
C
.
@H
c1
=@T
/
2›
2
= ln › 1,920 1,860
R
S
.
T D 77 K
/
n
n
=
n
e
s
3=2
2 10
5
10
5
increasing as T ! T
c
.Since
L
.T / increases more rapidly with T for small val-
ues of W , the melting of the fluxoid lattice at temperatures well below the critical
temperature may occur even in materials with d>
L
.0/.
To summarize the results of the above analyses, it has been shown that the magni-
tudes of
L
, H
c
, i
c
,and
0
are all determined by the effective density of supercarriers
n
e
s
.T /. The various relationships are organized in Table 8.2, together with the param-
eter values determined in the foregoing analyses and their projected limits where
applicable.
Appendix 8A Magnetic Levitation
Magnetic levitation is the most visually dramatic manifestation of superconductiv-
ity, occurring when the inducement of a diamagnetic supercurrent in a magnetic
field attends the expulsion of flux from the interior of the specimen. An important
distinction should now be recalled: a material qualifies as a superconductor where
n
s
n
t
(requiring that T T
c
); however, it does not become superconducting until
the carrier chain condenses to form a Meissner supercurrent for H H
c
. Thus, the
supercarrier density is controlled by temperature, but the supercurrent by magnetic
field. Analogous to the concept of an image charge representing the effects of a
metal plane beneath a real electric charge of opposite sign, the induced supercurrent
in a specimen may be represented by a magnetic dipole, as sketched in Fig. 8.44.
Since the current produces a diamagnetic moment, the dipole moment is a mirror
image of the external magnet, and the force between them is repulsive.
As pointed out earlier, the spatial ordering constraint of the covalent bond fur-
nishes the eigenstate rigidity required for the r
s
D 0 condition. In this context,
456 8 Spin Transport Properties
Fig. 8.44 Diagrammatic representation of the Meissner flux expulsion/levitation effect from
induced diamagnetism in superconductor [9]
the celebrated levitation property of the Meissner effect can be appreciated in the
context of a magnetomotive force imparted to a current-carrying coil in an inho-
mogeneous magnetic field. In a superconductor, however, the current loop will be
established on the specimen according to the disposition of the external field relative
to the specimen. Similar to the coil, a superconducting object will assume an appro-
priate equilibrium position and attitude in an external field; unlike the coil, however,
there are no restoring forces acting on the specimen when it is disturbed from its
“equilibrium” position. Since the current path will then adopt a different chain of
covalent orbital lobes, a new equilibrium state may be established within the same
specimen orientation by altering either the position of the specimen or the magnetic
field conditions.
Without external mechanical assistance to alter the carrier condensation energy
(or extent of polaron alignment) G
D H
2
c
=8
, the spatial relation between the
specimen and magnet is fixed, except for rotation about the magnetic field axis.
Rotation about an axis perpendicular to the magnetic field, or any translational
adjustment, requires an energy input to establish a new equilibrium current con-
dition (thus precluding any restoring-force effects). The mechanical aspects of the
Meissner effect lend further credence to the conclusion that superelectrons are part
of the binding forces of the lattice, and do not exist independently as unbound elec-
trons in the superconducting state.
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Index
A
Activation energy
in double exchange, 119, 121, 124, 389,
410, 412
hopping electron, 125, 389, 400, 405, 409,
421
from polaron trap, 396
Anderson, E.E., 161, 312
Anderson, P.W., 116, 118, 127
Angular momentum, L; S; J; 9–11, 25, 37, 42,
50, 52, 55, 63, 64, 71, 88, 187, 188,
222, 224, 289, 356, 358, 362, 371
Anion-cation, 24, 107, 113, 122, 127, 137,
193, 438
Anisotropy
crystal-field, 201, 202, 204, 205, 207–213,
216, 220–222, 224–227, 237, 240,
241, 244
magnetocrystalline, 7, 37, 52, 133, 174,
176, 201, 204, 226, 228–250, 254,
274, 294, 309, 406–407, 410
shape, 201, 249, 251, 252, 258, 259, 265,
293, 295, 298, 309
Approach to saturation, 234, 256–257,
299
Aufbau occupancy diagrams
high and low-spin states, 75–79, 89, 108,
217
J-T and S-O stabilized, 84–86, 217, 219,
221
octahedral site, 97, 98, 109, 123, 129,
217
tetrahedral site, 97, 129
B
Ballhausen, C.J., 64, 69, 91, 113, 243, 263,
264
Bardeen, J., 418
BCS ensemble wavefunction, 418, 419, 444,
445, 447
Bi
3C
Fe
3C
MO hybrid excited state,
371–375
Bismuth Faraday anomaly, 376
Bi-YIG anomalous Faraday effect, spectral
model, 365–367
Bleaney, B., 215, 225
Bloch–Bloembergen, 7, 19, 28, 252, 295, 297,
334, 356, 385, 416
Boltzmann statistics, 11, 237
Bozorth, M., 5, 229, 256
Brillouin theory
curve shapes, 17, 18, 445
function, 13, 15, 19, 130, 131, 133, 155,
156, 165, 196–197, 202, 215, 238,
265, 289, 302, 411, 442
Button, K.J., 23, 24, 137, 192, 329, 334, 351
C
Canting
iron sublattice, 185, 190, 193, 195, 197,
221, 237–239, 246, 249, 256, 265,
302, 303, 305, 345, 365, 369, 377,
379, 386, 398, 401, 421, 433, 437,
438
rare-earth garnet effect, 184–189
Charge carriers, 142, 386, 396, 417, 422
Charge transfer probability, 386–389, 392,
393, 397–399, 403, 405, 409, 411,
413, 418, 420, 422, 426
Chemical bonding
covalent, 23, 42, 88–101, 103, 107, 109,
113, 121, 122, 124, 129, 134, 168,
174, 222, 228, 344, 388, 392, 402,
419, 424, 431, 440, 455
ionic, 40, 49, 73, 90, 92, 99, 103, 112, 387
metallic, 88, 107, 408
Chikazumi, S., 229, 236
461
462 Index
Co
2C
exchange isolation
in LiTiferrite, 176, 307
in YIG, 176, 179
Coherence length, 416–418, 424, 445,
450–452, 454
Colossal magnetoresistance (CMR), 386,
401–403, 405, 407, 409
Concentration threshold x
s
>x
t
, 428
Condensation, 86, 171, 411, 419, 420, 439,
440, 442–443, 447–451, 453, 456
Conductivity
normal vs super, 386
Cooper, L.N., 418
Correlation exchange, 121
Covalent electron transfer, 94, 122
Covalent electron transfer theory (CET), 413,
437, 444, 445, 447, 449–452, 454
Critical current density, 447–450
Critical magnetic field, 439, 445–447, 450,
452, 453
Critical rf field, 311, 337
Cross transfers, 365
Crystal, 37, 40, 43–45, 48–73, 75, 76, 80, 82,
83, 86–92, 97–99, 101, 107–109,
113–115, 118, 121, 129, 130, 132,
134–136, 138, 141–143, 147, 152,
169, 175, 177, 182, 183, 185–192,
201, 202, 204, 205, 207–213, 216,
220–222, 224–228, 231–234, 236,
237, 240, 241, 244, 254–256, 258,
261–265, 267, 268, 273, 274, 284,
286, 287, 289–292, 295, 304, 306,
307, 319, 325, 360, 372–374, 385,
386, 388, 396, 397, 402, 404, 412,
420, 431, 432, 438
Crystal sites, 360
Crystal systems, 44–45
D
Degenerate orbital states, 49, 80
Delocalization, 24, 94, 114, 120–124, 126,
379, 380, 398, 402, 413, 430, 438
Demagnetizing factor, 5, 308
Demagnetizing field, 5, 6, 234, 257–259, 273,
292, 312, 313, 315, 317, 318, 345,
411
D
4h
vs. C
3v
distortions, 420
Dillon, F., 317, 360, 381, 382
Domain wall resonance, 273, 319, 338
E
Effective linewidth in polycrystals, 299
Elastic constants, 81, 232, 233
Electric-dipole transition, 357, 358, 360, 361
Electron correlation energy, 123
Energy level diagrams, 69, 97, 226, 261, 289,
348, 357, 375
Exchange
double, 119, 121, 124, 389, 410, 412
energy, 19, 24, 113, 116, 118, 122, 143,
161, 162, 168, 186, 196–197, 216,
235, 237, 243, 252, 265, 302, 304,
320, 339, 346, 347, 350, 365, 388,
389, 394, 405, 419–420
integral, 21, 25, 92, 113, 119, 226, 375,
390, 394, 399, 424, 436
resonance, 29, 264, 344–351, 355
semicovalent, 122, 123
super, 23, 24, 107, 109, 111, 113, 115–117,
119–122, 126, 127, 129–131, 133,
134, 136, 139, 140, 142–144, 146,
152, 161–168, 173, 183, 193, 201,
222, 366, 371, 372, 375, 376, 378,
379, 385, 397–399, 401, 402, 410,
430
trap, 124, 421, 438, 439
Exchange resonance, 344–351
Extinction coefficient k; 355, 361, 381
F
Faraday, 42, 70, 193, 305, 318, 330, 332–333,
343, 353–355, 366, 367, 369,
374–376, 379–381, 413, 414
Faraday rotation, 42, 70, 193, 305, 318, 330,
332–333, 343, 353–355, 366,
374–376, 379–381
Ferrites
garnets, 155, 156, 158–161, 165, 167–170,
175–185, 189, 190, 192–194, 197,
202, 220, 238–240, 245, 246, 249,
250, 294, 298–300, 302, 307, 343,
360–366, 372, 373, 379, 380
hexagonals, 48, 153, 169, 190–193, 240,
260, 295, 322, 323
spinels, 46, 77, 153, 167–169, 175, 176,
178, 179, 216, 224, 240, 244, 249,
257, 305, 307
Ferromagnetic limit, in CMR manganites, 401,
402
G
Geller, S., 159–161, 166, 175, 177, 178,
183–185
Giant molecule concept, London’s rn
s
D 0
condition, 419