Poole Ch.P., Jr. Handbook of Superconductivity

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t.,o

Table 9.1.

Coherence lengths ~ and penetration depths 2 for various isotropic superconductors. Values for the critical temperature T c and the dimensionless Ginzburg-Landau

parameter x = ),/~ are given, when known. References are identified by first author and year.

Material T e r 2 x Comments Reference

(K) (nm) (nm) (2/~)

Zn 0.535 29 Ultrasonic Almond (1975)

Cd 0.56 760 110 0.14 Meservey (1969)

Cd 0.875 42 Ultrasonic Almond (1975)

AI* 46-51 10% anisotropy Doezema (1986)

AI* 1.18 1550 45 0.03 Donnelly (1989), Poole (1995)

In* 3.41 360 40 0.11 Poole (1995)

Sn* 3.72 180 42 0.23 Poole (1995)

Pb* 96 30.5 0.32 Gasparovic (1970)

Pb* 7.20 87 39 0.48 Donnelly (1989), Poole (1995)

Nb* 9.25 39 52 1.3 Donnelly (1989), Poole (1995)

PbIn 7 30 150 5 Donnelly (1989)

PbBi 8.3 20 200 10 Donnelly (1989)

NbTi 9.6 3.8 130 27 Polycrystal Hampshire (1998)

200-400 Filament diameter 50-500nm; see Jc table Cave (1989)

9.5 4 300 75 Donnelly (1989)

NbN 16 5 200 40 Donnelly (1989)

Nb0.TaGe0.26 3.4 6.5 802 76 Amorphous film Berghuis (1993)

Nb0.65Ge0.35 2.9 7.2 913 77 Amorphous film Berghuis (1993)

PbMo6S 8 Chevrel 14 2.2 215 98 Polycrystal Hampshire (1998)

SnMo6S 8 Chevrel 14 3.3 240 73 Polycrystal Hampshire (1998)

V3Ga A15 15.3 2-3 90 ~36 Donnelly (1989)

V 3 Si A 15 16.3 3 60 20 Donnelly (1989)

17 145

Gross-Alltag (1991)

Nb3Sn A15 16 2.8 93 33 Polycrystal Hampshire (1998)

Nb3Sn A15 18.0 3 65 22

Nb 3 Ge A 15 23.2 3 90 30

Mo4Si 7.2 4.7 500 65

UBel3 0.88 7 400 60

0.86 1050

UPt 3 0.5 1900

< 1 11.1 600 54

YNi2B2C 15 8.1 103 12.7

BaxC6o 6.8 11.6

K3C6o 17.8 4.4

19.4 2.8 240 92

K2CsC6o 24 3.4

23 4.0 200 50

Rb3C6o 30.7 2.6

28.5 2.2 157 76

30.7 3.2 186 58

29.6 2.0 247 124

RbzCsC6o 32.5 2.5

(Lao.925 Sro.075)2CuO 4 37 2.0 200 100

YBa2Cu307_ 6 2.1

91 1.65 156 95

Bi2 Sr2CaCu20 x 89 1.8 250 139

BaPbo.75Bio.2503 7

HgBa 2Ca 2Cu 30 x 129 2.3 100

Thin film

Heavy electron compound

Heavy electron, c-magnetization

Heavy electron material

2~(T c-TfnearT cwithfl~ 1

Mass enhancement

m*/m o = 9.4; see B e table

see B c table

~GL given, 40 = 15

4GL given, 4o = 5.0

4(0) -- 2.1, 4o -- 2.5 nm

Polycrystal

2 and the pairing state

Donnelly (1989)

Orlando (1991)

W6rdenweber (1989)

Alekseevskii (1986), Signore (1995)

Gross-Alltag (1991)

Kleiman (1992)

Koziol (1991 )

Cywinski (1994)

Baenitz (1995), Lfiders (1997)

Heinze (1996), Holczer (1991)

G/irtner (1992)

Heinze (1996)

Baenitz (1998)

Heinze (1996)

G/irtner (1992)

Baenitz (1998)

Spare (1992)

Heinze (1996)

Poole (1998)

Kes (1989)

Hampshire (1998)

Hampshire (1988)

Takagi (1990)

Gao (1993), Schilling (1994)

Annett (1990)

* To obtain perspective on the accuracy of reported 4 and 2 values see Poole

et al. (1995), p. 271.

ta, a

-,q

438 Chapter 9: Characteristic Parameters

Coherence Lengths and Penetration Depths

In the Ginzburg-Landau theory the coherence length ~ is given by

- h/(Zm*la]) 1/2,

(13)

where m*- 2m e is twice the electron mass, and a is the coefficient of the

quadratic term ar162 in the GL free energy Gs(~b ) of the superconducting state

with the temperature dependence

a(T) ~ ao[t-

1] (14)

in the neighborhood of T below Tc, where

t- T/T c

is the reduced temperature

and

a(T)

is negative below T c. The BCS theory provides the approximation

~0 ~ hvF/~A(O) (15)

for the coherence length ~0 at absolute zero temperature in terms of the Fermi

velocity

vv,

where Eg -- 2A is the energy gap. In the Pippard approximation the

coherence length of a superconducting metal depends on the electron mean free

path I through the expression

1/~- 1/~ 0 +

1/l.

(16)

Since the mean free path of an ordinary conductor such as copper becomes very

long when the impurity level is low, the limit l >> ~0 in Eq. (16) is called the clean

limit with ~ ~ ~0. In like manner, l < ~0 is called the dirty limit with ~ < ~0, and

the case ~ << ~0 with ~ ~ l is called the extremely dirty limit. Table 9.1 provides

coherence lengths for a number of isotropic superconductors.

The London theory gives for the penetration depth

~'L --

(m/~onse2) 1/2,

(17)

an expression which also appears naturally in the GL theory, where n s is the

density of superconducting charge carriers, that is, Cooper pairs. The penetration

depth has the value 2(0)- 2 o at absolute zero and becomes very large in the

neighborhood of T c, with the following temperature dependence predicted by the

two-fluid model:

2(T) - 20(1

- t4) -1/2.

(18)

Figure 9.1 compares this temperature dependence (dashed line) with three

limiting cases of the BCS theory, namely the pure sample in the local limit

2L(T), the pure sample in the extreme anomalous limit 2~(T), and the dirty local

limit 2efr(T ). The book by Tinkham (1996, p. 103) should be consulted for the

definitions of these limits. Tables 9.1 and 9.2 list some alternative temperature

dependences.

Table 9.2.

Coherence lengths ~ and penetration depths 2 perpendicular to (~ob,

~'ab),

and parallel to (~c, 2r the c-axis for various anisotropic superconductors. Values for the

critical temperature T C and the anisotropy ratio F =

(mc/mab) 1/2 --- ~ab/~c --- )~c/)Cab

are given, when known. References are identified by first author and year.

.......................

Material

~ab ~c ~ab ~c I"a

Comments Reference

(K) (nm) (nm) (nm) (nm)

(~ab/~c,

2c/2ab) (t =

T/Tc)

t.,a

Nbge 2 7.7 2.3 69

K0.33 (H20)o.66TaS2

Nb3S 4

CsKHg

CsRbHg

UPt 3

YNi2B2C

(BEDT-TTF)2Cu(NCS)2

K-(ET)zCu[NCS]2

K-(ET)2Cu[N(CN)]2B1

(TMSF)2C104

La2_xCal+xCu206_x

(Lal_xSrx)zCu04

(Lao.9 Sro.l)2CuO4

(Lao.91Sro.09)2CuO4

(La0.925 81"0.075)2 CuO4

(Nd0.925 Ce0.075)2 CuO4

(Ndo.9Ceo. 1 )2 CuO4

YBa2Cu3 O6.9

YBa2Cu307_6

1.88

1.41

0.46

11.4

11.6

1.1

21.5

230 3.2

496 124

30 0.9 3,400 71,000 Intercalated compound

2.9 0.57 Bibersacher (1980)

200 10 20 Effect of pressure

200 10 20

6.0 5.5

3.7

54-60

3.3

3.2

4.6

3.3

3.2

0.4

1.2

0.055

0.3

3.0

0.27

782 707

980

650

290 450

283

142

260

2 x 105 ~250

100

> 700

> 200

9.3

2.8

1.5

1.3

~2.5

Heavy electron compound

Thin film c-axis oriented

Josephson 2

~a =

60,

~b --- 54 nm

Monocrystal

Values are for x -- 0.08.

Values are for x = 0.3

Magnetic measurements

Monocrystal thin films

~,(t) = 2(0)[1 -- t4] -1/2

Salamon (1989)

Finley (1980)

Gygax (1982)

Iye (1982)

lye (1982)

Broholm (1990)

Arisawa (1995)

Mansky (1994)

Harshman (1990)

Farrell (1990)

Lang (1992)

Kwok (1990)

Murata (1982)

Watanabe (1991)

Suzuki (1991)

Cobb (1994)

Wu (1993)

Suzuki (1989)

O and Markert (1993)

Harshman (1989)

Lee (1991)

(continued)

Table 9.2.

(continued)

Material

Tc ~ab ~c "~ab

(K) (rim) (rim) (rim)

~'c I-'a

(nnl) (~ab/~c, ~c/'~ab)

Comments

(t = 7"lTc)

Reference

YBa2Cu3 O7_~

YBa2Cu408

EuBa2Cu3 O7_ 6

92.4

1.6

2.8

1.5

2.5

3.4

4.3

1.2

1.3

2.65

12.9

2.7

0.3

0.08

0.3

140 610

0.8 3.3

0.7 36 125 5

0.7 27 180 6.4

0.3 89 550 ~ 5

0.2 130 450 ~ 5

140 1040

0.09

4.0 800

130

140

139

198

0.6

2600 3

650'

610

700

990

4.4

4.5

Oriented powder

Demagnetization

Film, pulsed fields

Fluctuation conductivity and

magnetoresistance

Co and Zn doping studied also

Magnetoresistance

Scheidt (1989)

Welp (1989)

Staghun (1989)

Pagnon (1991)

Chaudhari (1987)

Worthington (1987)

Gallagher (1988)

Salamon (1989)

Krusin-Elbaum (1989)

Porch (1993)

Gob (1994)

Microwave measurements at 86.5 K; Jiang (1993, 1994)

2ab(O )

-'- 180nm

2(0 = 2(0)[1 - t4] -1/2 Pumpin (1990a,b)

BCS temperature-dependent, oriented Scheidt (1989)

grains; see B e table

Polycrystals Schilling (1990)

Hikita (1987)

EuBa2Cu307_ 6 94 3.5

TmBa2Cu3 O7_ 6 86 7.4

Y0.8Pro.2 Cu3 O7+,~ 73

Pb 2Sr 2(YCa)Cu 308+ ~ 75

Bi2 Sr2CaCu20 8 109

Bi2 Sr2CaCu208 80

2.4

1.42

0.38

0.9

0.78

Bi2Sr2Ca2Cu3Olo 109 2.9 0.09

111 1.0 0.02

2.9 0.57

Bi3.

9 Sr3. 3 Cal. 3 (Cu0.961 Fe0.039)30x

(Bi,Pb)2 Sr 2 CaCu 2 08 91 2.0

(Bi,Pb)2 Sr2 Ca2 Cu30lo_ ~ 103 2.04 0.037

T12 Ba 2 CaCu 2 O8_ ~ 100

TlzBazCazCu3Olo 123

TlzBazCa2Cu3Olo 100

T12BazCaCu2 Olo 103

TlzBazCazCu3Olo 100

T1BaCaCuO 2.8 0.08

HgBazCuO4+~ 93 2.1

HgBa2 Ca2 Cu3 O8+~ 133 1.3

1.5 0.19

258

500

643

200

178

182

173 480

182 1960

175 2350

117

130 3500

9.2

8.2

3.1

2.5

10 3

2.8

>~ 300

10.7

13.4

~> 64

Tajima (1988)

Noel (1987)

2a:2b:2 c -- 1 : 1.7:4.4; see B c table Moshchalkov (1990)

Jia (1992)

Temperature dependence Reedyk (1991)

Maeda (1992 )

Thin film; Nernst, Seebeck and Hall Ri (1994)

measurements

2(T) -- 2(0)[1 - t4] -1/2, muon spin

rotation

F = 2c/2,b

Magnetization measurement

2(T) = 2(0)[1 +

aT + bT 2 + cT 3]

Oriented film; see B c table

Grebinnik (1990)

Matsubara (1992 )

Li (1992)

Babic (1995)

vom Hedt (1995)

Zhang (1992)

Lee (1991)

Ning (1992)

Thompson (1990 )

Thompson (1993)

Ning (1992)

Staguhn (1989)

Thompson (1993)

Schilling (1994)

a Values of F

~ab/~c --

2c/)~ab

are listed as approximate when the ratios

~ab/~r

and

)~c/)~ab

do not agree.

442

Chapter 9: Characteristic Parameters

Fig. 9.1.

~2(o)

~.2(0

1.0

0.8

0.6

0.4

0.2

_ t 4

0 0.2 0.4 0.6 0.8 1.0

I ~ ---

Comparison of the predicted temperature dependence of 1/22 for the two-fluid model ( .... )

and three limiting cases of the BCS theory (~), as explained in the text. (From Tinkham,

1996, p. 104.)

Until now we have not taken anisotropy into account. The tetragonal cuprates are

axially symmetric, and the orthorhombic ones are very close to axial, with

coherence lengths that are related to their respective penetration depths through

the expression

~ c ~ ~ a b ~ i~ a b ~ 2c,

(19)

with the anisotropy factor F given by

r - (mc/mab) 1/2 -- ~ab/~c -- ~c/~ab,

(20)

where

mab

and m C are effective masses. The validity of this expression (19) for many

anisotropic materials is clear from the experimental values listed in Table 9.2.

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Superconducting Gap Parameters

John E Zasadzinski and Roberta K. Zasadzinski

Below the transition temperature of a superconductor, an energy gap A- 89

develops in the electronic density of states that is exhibited in a number of

experimental probes, including specific heat, optical reflectivity, and ultrasonic

attenuation. The most direct probe of the gap is from tunneling spectroscopy.

Specifically, the tunneling conductance,

dI/dV

vs. V, in normal metal-insulator-

superconductor (NIS) junctions is proportional to the electronic density of states

in the superconductor. Table 9.3 presents energy gaps which have been deter-

mined by tunneling. One exception is the fullerene Rb3C60 where the energy gap

was determined by muon spin rotation. The text by E.L. Wolf (1985) provides a

thorough description of the tunneling experiment and includes a comprehensive

list of gap values of low-T c superconductors.

For high-T c cuprates and other oxides, there are lengthy reviews (Hasegawa

et al.,

1992; Ekino and Akimitsu, 1992) and general articles (Zasadzinski

et al.,

1996) that cover most of the important phases and describe the problems

associated with tunneling in these exotic materials. Energy gaps of individual

compounds can vary considerably among different tunneling experiments and an

attempt has been made to select the best values. There is considerable evidence

that the gap in high-T c cuprates is highly anisotropic within the

a-b

plane and is

most likely of dx2_y2 symmetry. Thus, the energy gap is decribed by

A(k) - 1/2Ao(cos

kxa -

cos

kya),

(21)

where

k x, ky

are the components of the wave vector within the plane. This

symmetry has nodal regions where the gap is zero. The parameters listed for high-

T c cuprates are the maximum values A 0 of the gap. For electron-doped cuprates

such as (Ndl_xCex)zCuO 4 which have relatively low Tc, the experimental