Magnetic resonance in high-temperature superconductors

PHYSICS REPORTS (Review Section of Physics Letters) 200, No. 2 (1991) 51—82. North-Holland

MAGNETIC RESONANCE IN HIGH-TEMPERATURE SUPERCONDUCTORS

K.N. SHRIVASTAVA School of Physics, University of Hyderabad, P.O. Central University, Hyderabad 500134, India Editor: E.R. Andrew

ReceIved September 1989

Contents: 1. Introduction 2. Microwave absorption 2.1. Flux-quantized eigenvalues 2.2. Power spectrum 2.3. Parallel pumping 2.4. Giant moments

53 57 60 63 65 67

3. Electron paramagnetic resonance 2~ion 3.1. Cu 4. Nuclear magnetic resonance 5. Nuclear relaxation 6. Conclusions References

68 68 72 72 75 77

Abstract: Since the discovery of high-temperature superconductors a new microwave absorption has been found which is reviewed here in detail. The microwaves are absorbed in the flux-quantized eigenstates so that the absorption is proportional to the Josephson current and hence it varies as the gap of the superconductor. This absorption is found in the electron paramagnetic resonance, in the parallel pumping and in the ordinary microwave absorption experiments. The flux-quantized fields are found in small domains of the size of iO~cm. A giant moment is found to occur. The necessary theory as well as experiments in YBa 2Cu3O7_5 type of compounds are described. The Cu* electron paramagnetic resonance gives an anisotropic exchange narrowed line with anisotropic g-values with g-shifts proportional to the susceptibility. The symmetry of the g-value also reflects the symmetry of the superconducting gap. The nuclear magnetic resonance of various nuclei has been reviewed. It is pointed out that the NMR line widths are determined by the vortex—lattice spacing and the London penetration depth. Nuclear relaxation rates measured in 3Cu the superconducting nuclear relaxationstate rategive in Lathe gap energy of the superconductor. The ratio of the gap energy to the transition temperature found from the ° 1 83Sr0 17CuO4 is in reasonable agreement with the BCS theory confirming the pair structure of superconductors. The phase diagrams of La2_5Ba1CuO4 and YBa2Cu3O7_5 have been described.

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© 1991

—

Elsevier Science Publishers B.V. (North-Holland)

:

MAGNETIC RESONANCE IN HIGH-TEMPERATURE SUPERCONDUCTORS

K.N. SHRIVASTAVA School of Physics, University of Hyderabad, P.O. Central University, Hyderabad 500134, India

ii

NORTH-HOLLAND

K.N. Shrivastava, Magnetic resonance in high-temperature superconductors

53

1. Introduction In 1986, Bednorz and Muller [1], published the resistivity of an oxide of La, Ba and Cu which first reduces upon cooling until a minimum value is reached. Upon further cooling the resistivity increases until a peak develops below which it rapidly falls to very low values. Bednorz and Muller [1] announced that at the peak temperature there probably occurs a transition to the superconducting state. This transition temperature of about 36 K was higher by at least 12 K, than the highest temperature known at that time. Anderson [2] suggested that the pairs of electrons emerged directly from the insulating state so that there occurred an insulator to a superconductor transition as distinguished from the BCS theory [3] which predicts only metal to superconducting transition. We have found [4] that pairs may occur in the valence as well as in the conduction band but then there are several order parameters leading to several transition temperatures and the pairs in the valence band are nothing but two-particle states. It was found that compounds of the formula RBa2Cu3O7~(R = Y, Nd, Sm, Eu, Dy, Ho, Er, Tm) have a superconducting phase with transition temperature, T~ 90 K. The compounds containing Bi and TI have even higher transition temperatures —100—110 K. Tl2CaBa2Cu2O8~8has a T~ 105 K while T12Ca2Ba2Cu3O10 has a T~of —125 K. The compounds of the general formula (Bi/Tl)2Ca~1 (Sr/Ba)2Cu~O4÷2~ (n = 1, 2, 3,...) have a superconducting transition temperature which seems to increase with increasing number of layers of Cu atoms but the combination of other atoms is also important. The superconducting transition temperatures [5—11]of a number of compounds are given in table 1. The crystal structures of La2~Sr~CuO4, YBa2Cu3O7, Bi2Sr2Ca2Cu3Oio+a are displayed in the figs. 1—3 showing the importance of the number of layers of Cu atoms; when YBa2Cu3O7 samples were analysed it was found that another structure with about one-half layer of Cu more than YBa2Cu3O7~ existed. Its formula is Y2Ba4Cu7O~4aas shown in fig. 4 but its T~is only about 40 K. Single YBa2Cu3O6+~type blocks containing single corner sharing Cu04 square planar chains alternating with —

Table 1 Some of the superconducting compounds with their transition temperatures T~and the number of Cu—O layers, n Formula Bi2Sr2CuO6 La2~SçCuO4 TICaBa2Cu2O7 Bi2CaSr2Cu2O5 YBa5Cu3O7 YSrCaCu3O~ YSrCu3O~ TI2CaBa2Cu2O8 i2Sr2 a2 u3O~ TI2Ca2Ba2Cu3O10

T~(K) 10 36 60 85 92 80 80 1 2 1 5

n

ref.

1 1 2 2 2 2 2 2 3

[51 [1]

151 16—81 [9, 17]

[51

[51

• La, Ba; .Cu, Oo Fig. 1. The tetragonal structure of La1 85Ba,15CuO4. The atoms La and Ba are shown by large circles. The Ba atoms occupy the La sites. The Cu atoms are shown by small filled circles and the oxygen by open circles. One unit cell is shown along with oxygen atoms ofother cells to display the six-fold coordination ofthe Cu—O atoms. The unit cell dimensions are a = 3.787 A, c = 13.288 A. This structure belongs to n = I in table I and the transition temperature is about 36 K.

:

K.N. Shrivastava, Magnetic resonance in high-temperature superconductors

54

0

£ Bi •Sr • Cu

_____

-o

0

j

• ~ T

__

T~ ~ tO~°

I

__

____

to~

.

0.0

F— a

—1

•Y,Ba;.Cu;Oo

Fig. 2. The unit cell of YBa

2Cu3O7. The large filled circles in the centre are Y—Ba—Y, small filled circles represent Cu and unfilled circles are the oxygen atoms. The dimensions of the orthorhombic cell are a = 3.89 A, b = 11.69 A and c = 3.82 A. It belongs to n = 2 in table 1 and has a T~=90K.

Bc

°

OCo

V

Cu 0

•o.~

Fig. 3. Structure of Bi2Sr2Ca2Cu3O10~belonging to the n = 3 class in table 1. Its long dimension c 37.1 A is due to multilayer structure [2491

‘Y’2BO4Cu7014_ö

Fig. 4. The structure of Y2Ba4Cu,O14,~showing one more layer of Cu than YBa2Cu3O7~.Its T~is about 40K. It has one unit of 06 type attached to one unit of 08 type. Hence it belongs to n = 2.5. Although it has n larger than 2, its T~is lower [2491.

single YBa2Cu4O8 type blocks containing double edge sharing Cu04 square planar chains are found. Therefore it belongs to n 2.5 but its T~is smaller than for n = 2. According to the BCS theory, the superconductivity arises from the formation of pairs of electrons of oppositely directed spins and oppositely directed momenta, (ki’, k~)so that there is a uniform mode of zero spin which accounts for diamagnetism; hence from the Maxwell equations and the Ohm’s law, there results a zero resistivity. In metals, the usual electron—phonon interaction leads to an attractive —

—

K N Shrivastava Magnetic resonance in high temperature superconductors

55

interaction between electrons when the electronic energies are small compared with the phonon frequencies. In a single conduction band, the Hamiltonian of the system is given by ~ Dc~,c~(a,~ +a_k)~(k+kk), kr

(11)

kk’k”o

where e,~is the kinetic energy of the electrons with ~ as the Fermi energy. c~ff(ckff) are the creation (annihilation) operators for electrons of wave vector k and spin a~.They obey the anticommutators. D is the energy of the interaction of electrons with phonons and a~.(ak.) are the phonon creation (annihilation) operators obeying Bose statistics. The pairs of the electrons are formed from 1~~ type of operators which explains the gap of the single-particle dispersion relation and hence the infrared peak. The bound state of two electrons is described by the Bethe—Salpeter equation as in fig. 5. The transition temperature determined from the bound state condition is given by -

k8T~ 1.l3hw exp[—1/N(0)V],

(1.2)

which in general is quite small. The density of states near the Fermi surface, N(0), the attractive coupling constant of the second-order electron—phonon interaction, V and the phonon frequencies are in general so adjusted that the transition temperature is less than 20 K. For a pure metal, the highest transition temperature is only 9 K for Nb, and for alloys the highest known value of Nb3Ge is about 23 K. Thus the BCS theory may not be adequate to explain T~ 100 K but its essential structure should remain unchanged. Many new theories have been proposed in the literature. We have considered the random distribution of electrons in the lattice interacting via phonons and a hybrid interaction which connects the valence band with the conduction band. In this theory the energy range in which the electrons are attractive is more than in the BCS theory and hence the transition temperature for the paired state is higher than in the pure BCS case. We consider the mixed-valence phenomenon so that the Hamiltonian of our [4] system is, —

+~

e~C~C~+

~ Ii Wqf3~qf3q+ ~

V(c,~c~ + c~c~~)

+ ~ Ajjqg[C~gt~ja(f3q+ 13~q)+ h.c.],

(1.3)

where the first term describes the kinetic energy of electrons localized at the ith site and s,.~is the single-electron energy of spin o~,c~ (c1) are the creation (annihilation) operators for electrons in the impurity valence band. The second term describes a conduction band with single particle energy given by ~ for the wave vector k and spin 0 which does not overlap with the ith localized band. The third term describes the phonon continuum with /Ioq as the single-phonon energy and /3 (f3q) as the phonon creation (annihilation) operators with wave vector q. The next term is a hybrid interaction containing creation and annihilation operators for electrons in the conduction band so that the electrons from the ~

:

+

+

)

)

~.

+ ii.

L ~4

Fig. 5. Two-electron Bethe—Salpeter equation as in BCS theory. The coiled line indicates the phonon and the straight line describes the electrons.

56

K.N. Shriva.stava, Magnetic resonance in high-temperature superconductors

sites can be promoted to the conduction band. Such an interaction has been found useful in the study of the mixed-valence phenomenon. The next term in eq. (1.3) is a phonon-induced intersite coupling which permits the hopping of electrons by phonons. Suitable canonical transformations performed on eq. (1.3) show that the electrons become attractive in the conduction band but pairs occur both in the conduction as well as in the valence band. The two-particle Bethe—Salpeter equation in this case is shown in fig. 6. It is seen that the pairing and spin structure of the BCS is preserved but there are intermediate states for the pairs in the valence band which is not the case in the BCS theory. The transition temperature is slightly enhanced. In 1985 Muller et a!. [12] were studying the lattice-dynamic shift [13—16]of the electron-paramagnetic resonance lines of Cr3~:BaTiO3.Since the discovery of high-temperature superconductivity by Bednorz and Muller [1] is a development from the 1985 paper of Muller et a!. [12], it may be that the orbit—lattice interaction [14—16]is important, in which case the phonon is believed to be emitted from one site and absorbed by another site to lead to an attractive interaction between two atoms. Then a hybrid interaction which has operators for creation and annihilation of electrons at the sites as well as in the conduction band, can give rise to a paired state which is superconducting. In 1987 Shrivastava [17] and Blazey et at. [18, 19] reported a microwave absorption at low fields in the high-temperature superconductors. We continued working on the problem of flux-quantized eigenvalues until all its essential features were found [20—23].Vesnin et at. [24]have discussed that the microwave absorption may be resonant. As we have pointed out, there are resonances superimposed on the broad nonresonant absorption. Actually all of the superconductors have a strong microwave absorption which indicates that the flux is quantized with the elementary charge of 2e. It was discovered while doing the ordinary electron paramagnetic resonance absorption experiment 2~was with YBa2Cu3O7~ seen. As the pellet samples. The usual exchange narrowed line with g 2 corresponding to Cu g-value is anisotropic both the g 11 and g~components are visible. As the magnet is turned on there is a signal at very low fields which shows hysteresis. The signal amplitude also depends on whether the sample is cooled in zero field or in a finite field. This signal corresponds to flux-quantized eigenvalues which is another way of saying that there is flux quantization. As the current and voltage characteristics of a Josephson junction show steps upon illumination with microwaves so also here we find that when the absorption is recorded as a function of frequency, flux-quantized oscillators occur. Thus the method of observation of the microwave signal is the same as that of the electron—paramagnetic resonance. Therefore a complete study of the microwave effect requires that we include in our discussion the EPR absorption also. The type-I superconductors arise from the pairing of electrons which have zero spin and hence exhibit the Meissner effect, which is the exclusion of the magnetic field flux from the superconductor below a critical temperature. In the presence of magnetic atoms, the Meissner effect does not occur but instead, the field is allowed to enter the sample within the vortices of a small cross section so that if a

nickel smoke is deposited on the sample viewed from a microscope then vortices can be seen. Thus

:~:~i~T~I: ÷~3)j3 +

.

Fig. 6. The two-particle Bethe—Salpeter equation for a pair of electrons when the phonons interact with electrons at random sites and a hybrid interaction promotes them to the conduction band. The zig—zag line belongs to electrons in the valence band, the coiled lines give the phonons and the straight lines describe the electrons.

K N Shrivastava Magnetic resonance in high temperature superconductors

57

T

N

10

Fig. 7. The mean-field phase diagram as a function of temperature T, magnetic field H and the bond occupation probability p near percolation threshold p~exhibiting: Meissner (M), spin-glass (SG), and Abrikosov vortex (A) phases in a randomly diluted granular superconductor.

magnetic atoms destroy the Meissner state but the resistivity remains zero in spite of the vortices. This is called the type-Il superconductivity. The presence of Cu2~ions leads to type-Il superconductivity in YBa 2Cu3O7~type samples where & arises from the mixed valence. The boundary between the Meissner state and Abrikosov’s vortex state [25] in the case of a randomly diluted granular superconductor near percolation in the low temperature limit [26]is shown in fig. 7. It has a spin-glass phase in addition to the familiar Meissner phase and an Abrikosov vortex lattice phase. The g-value measured in the EPR of type-Il superconductors describes the symmetry of the band structure. 2~ Although it isbereported in the literature thatmetallic some samples do signals not give EPR ions, it may that some samples are very so that the arethe difficult characteristic of Cu to analyse and identify. From the XAFS study it is believed that both the Cu2~and the Cu~ions are present in the samples. In any case the source of the electrons is to be found in the mixed-valence phenomenon. Similarly, the NMR and NQR can be used to measure the energy gap which arises due to pairing in the single-particle dispersion relation. Therefore, we present here a comprehensive account of the microwave effect, EPR, NMR, NQR and relaxation in the high-temperature superconductors.

2. Microwave absorption The samples of superconducting materials such as Nb, YRh 4B4, RBa2Cu3O7~(R = Y, Gd, La, etc.) in a microwave cavity show a strong microwave absorption upon cooling below the transition temperature. The amplitude of this absorption varies with the magnetic field and with temperature. It was found [17] that it is proportional to the Josephson current in the sample, and hence its intensity varies as the gap of the superconductor [21]. In the case of a powdered sample containing a large number of randomly oriented grains, there is a considerable noise due to flux-quantized oscillators which develop into well resolved fine lines in a single crystal [21]. Equally spaced resonances are also seen superimposed on a broad resonance [22].

58

K.N. Shrivastava, Magnetic resonance in high-temperature superconductors

Some authors have thought the absorption to be nonresonant while others have interpreted it to be resonant [24]. The well resolved structure clearly favours a resonant interpretation. Blazey et a!. [18, 27—30] have reported a detailed study of the microwave absorption and fluxon nucleation in several superconductors. This effect has also been seen by many other workers [31—107].The most popular compound in which the effect has been seen is YBa2Cu3O7~but it is also seen in Bi2CaSr2Cu2O~[51, 101] and T15Ba2Ca2Cu3O105 [51].At low fields, the signal in one of the spectra of ErBa2Cu3O75 at 9.2 GHz is shown in fig. 8. Actually, there is some absorption at the lowest of the fields so that one has to redesign the scanning magnet. The Varian magnets have a large residual field. In fact a small Helmholtz coil is good enough to provide the required field. In some of the samples, the absorption occurs even below 0.1 G. In fact, then one has to be careful about the orientation of the magnet with respect to the earth’s magnetic field. The absorption can be seen even in the earth’s magnetic field. In addition to a broad absorption, there are sharp lines superimposed on the broad line. As the field is reduced, the spectra do not reproduce themselves so that it may be concluded that there is hysteresis. The intensity of the broad line has a strong temperature dependence vanishing at T~ as shown in fig. 9. As the sample is cooled below 10 K, there occurs a lot of noise [23]as seen in fig. 10, which is reproducible like a signal. In the case of a single crystal, the noise occurs in the form of a large number of very sharp equally spaced lines displayed in fig. 11. In a multigrain sample the equally spaced lines of a crystal can be converted into a noise signal because of the angular dependence of the line spacings [108]. There is some difference between the zero-field cooled and the field cooled signals as also measured by Pakulis [81]and shown in fig. 12 from our work. We also found that the intensity of the broad microwave signal is proportional to the surface area of the sample as it should be if it is proportional to a current as shown in fig. 13. As the system is cooled the peak shifts towards higher fields (fig. 14).

100

50

100

150

1500

2006

Fig. 8. Low-field microwave absorption in ErBa2Cu,078 at 9.2 GHz. The signal shows hysteresis and resonances with a separation of 12.5 G as given in ref. [22]. An electron paramagnetic resonance spectrometer operating at x-band microwave frequencies with Helmholtz coils was used.

K.N. Shrivastava, Magnetic resonance in high-temperature superconductors

-;;

59

-

.~10

2

~ C 01 4C 01 0 3 0

C

-4 10

I

30

60

90K

Temperoture Fig. 9. The intensity of the broad line of fig. 8 as a function of temperature varying as the gap of the superconductor.

dx” dH IC

50

100 150 Magnetic Field

Fig. 10. The low-field signal in the electron paramagnetic resonance of a pellet sample of YBa intensity of the noise signal ‘N has T~ = 10 K whereas the intensity of the broad signal has TC

2006

2Cu3O,_5 showing noise at low temperatures. The 96 K (see ref. [21]).

60

K. N. Shrivastava, Magnetic resonance in high-temperature superconductors

ax’ dH

I

0

I

I

I

2

4

66

Fig. 11. Sharp lines in the microwave absorption of a single crystal of ErBa

2Cu3O7_8 with field normal to the crystal surface.

C

‘0.0

~~1:1~0t3Ti

Zero-field cooled

20

T(K) Fig. 12. Remanent microwave absorption from a sample of ErBa2Cu3O7_5 for the field cooled from H = 3 kG and the zero-field cooled cases.

2.1. Flux-quantized eigenvalues We now prove that the low-field peak as shown in figs. 8 and 10 corresponds to the Josephson current and its intensity shown in fig. 9 is related to the gap of the superconductor. The Josephson current is given by J=Josin(yi_y2_~JA.d1),

(2.1)

where Yi and y2 are phase factors, A is the vector potential of the electromagnetic wave and the integral is extended over the width of the Josephson junction. The maximum value is given by ‘max =

2J~sin(~/~0)(ir~l~0)’ cos(n~/q50),

(2.2)

61

K.N. Shriva.stava, Magnetic resonance in high-temperature superconductors

1.0

0.5

0.05

0.10 Surface area

0.15

0.20

cm2

Fig. 13. The intensity of the microwave signal such as that of fig. 8, as a function of surface of the plate like sample chips.

6 20

IS Hm 10

S

0

20

40

60 lemperature

80

lOOK

Fig. 14. Shift of the centre of the signal of fig. 8 upon cooling.

where = H(2A + t)w with A as the London penetration depth, t the thickness of the normal barrier, w the width, 4 = nh/2e and ~ = hI2e. Since A = H X r, the x =0 point, in the argument of the spherical Bessel function (sin x) Ix, where the maximum current occurs, is given by H 0w(t + 2A) = n~0.

(2.3)

Here n is an integer. Since in the electron—paramagnetic resonance spectrometer, we measure d~”IdH,

K.N. Shrivastava, Magnetic resonance in high-temperature superconductors

62

the field given by (2.3) corresponds to the centre of the absorption in figs. 8 and 10 for n = 1. For the argument of the sine in (2.2), x = 0, n = I gives the peak, x = n~rgives the zeros, for any other value of n. Thus only the n = 1 is a maximum and all other values of n give zeros of the current. Since JO=irJiI2SRN

(2.4)

with 4 the gap of the superconductor, RN the normal-state resistivity of the insulating region and S the surface area, the intensity of the signal of the microwave absorption of figs. 8 and 10, varies as the gap, 4, of the superconductor which explains the result of fig. 9. The zero of the 4 determines the transition temperature T~. In the case of type-Il superconductors, the Abrikosov vortex state has two critical fields, H~1= 4~I 2 and H~ 2which are related to two characteristic distances in the superconductor, the irA 5= 40fir~ London penetration depth A and the coherence length The domain area w(t + 2A) in eq. (2.3) is independent of the critical fields I-I~~ or H~ 2as the zeros of eq. (2.1) are not critical. Finite current occurs as the phase moves away from the zeros of eq. (2.1) by arbitrarily small amounts. The centre of the line in fig. 8 moves towards higher fields upon coolinginasthe shown in critical fig. 14. field. This field corresponds to 2 occurring lower theThere area w(t + 2A) compared with the area A is hysteresis in fig. 8 which shows that the response of the system to excitation by microwaves is a multivalued function of the magnetic field as there are stable solutions via unstable ones [1091.The response of the system depends on whether the cooling is done first or the magnetic field is applied first as it traces the hysteresis loop. The response also depends on the scanning rate. The supercurrent at E = 0 is given by ~.

(a/at)AJ 5 1/A

=

—

E = 0,

(2.5)

pe~x5’~Imi.

(2.6)

Here ~ is the coherence length, I the mean free path of the electron in the normal state, p the charge density and m is the mass of the electron. The response of the sample subjected to microwaves is then given by 2 (~fl)2], (2.7) x” = 1 ~[1/A~0[(w w~) which peaks at w = w,, with ~fl as the line width. Since ~ = h2k~I2m4,the Fermi wave vector kF = (3ir2p)113 and the mean free path I = ti.rkF/m, the imaginary part of the susceptibility has a linear dependence on the normal-state life time of the electron. The energy levels E = /1w~are equally spaced as required by the equally spaced lines seen in figs. 8 and 11. It also explains the noise seen in the signals from pellet samples, e.g., in fig. 10, as there are many randomly oriented crystallites in the sample with strongly angular dependent h ai,,. Since the quantity hw~ I4~has the dimensions of the charge per unit time, we introduce the modulus of a current J such that, —

—

E~hw~=nJSçb 0,

(28)

where S is the area of cross section of the current flow We thus understand the equally spaced levels as

63

K N Shrivastava Magnetic resonance in high temperature superconductors

flux quantized eigenvalues. Our expression (2.8) is also consistent with the quantization of the force on the flux bundles, f = J J x B dr. The penetration of flux lines into the superconductor causes a pinning force on the flux bundles which opposes the electromagnetic force [110]so that the rate of creep of the magnetic field is determined from quantized energies 22 Power spectrum We consider a loop of N identical superconducting grains with intergranular separation a and coupling energy J. The total flux through the loop is f(t) = f~ f~ cos wt in which both the f0 = SH01410 —

and f1 = SF11 141~are quantized where S is the loop area [111].As the flux 4’ through the loop varies, the phase slips when the flux crosses one of the cusps of the energy diagram, i.e., when 41 = n410 orf= n + ~ (n is an integer) and two cusps at f = ±~are traversed in a single cycle. Phase jumps occur when either f0 + f,~or f0 f~f1 j where f~ = 0.5 corresponds to the first energy cusps. The phase difference 41(t) between two neighbouring grains can be written as follows: —

41(t)

=

—f1 cos wt —1

(2rr/N)[f0

+ 0(t

t1) +

—

0(t

—

t2)

—

0(t

—

T+

t2)

—

0(t

—

T + t3]0(f1

—f~)~

(°~f0
(2.9)

—f1 cos wi + ~(t

41(t) = (2irIN)[f0

—

t2)

—

0(t

—

T+

t2)],

1-f1)>f0If~-f1I,

(11w) cos~[(f0 —f~)/f1]

t2

,

=

(11w) cos~[(f0 —f~)1f1]

(2.10)

.

T = 2rrlw is the time period of the a.c. field, 0(x) is the step function and min(a,b) means the smaller of the quantities a and b. The magnetic moment ~ of the loop is = ~ (X~ x 1~1x~),

(2.11)

~—

(i,j)

1,1=1~sin(41~—411—A~1).

(2.12)

(x1 + x)/2 is the vector joining the origin to the midpoint between grains i and j, and The time dependent magnetic moment is given by =

p..(t)

=

p~sin 41(t)

=

—

x1.

(2.13)

,

where 41(t) is given by (2.9), and SI~sin(2ir/N)[c(2irIN)]~

=

Expanding ~(t)

~(t)

=

.

(2.14)

in a Fourier series, for 2ir/N ~ 1, we write

,.Lo(~ao+

,,=~

a~CoS(nwt)),

(2.15)

64

K. N. Shrivastava, Magnetic resonance in high-temperature superconductors

a0

=

—(41N)[a1 + a2

=

—(4InN)[sin(na1) + sin(na2) + ~irf13~1],

—

ir(f0 + 1)],

(2.16) (2.17)

where a~ wt1, a2 = wt2 and n = 1,2,3,... When the d.c. field is zero, H0 = 0 or f0 = 0, we obtain, a1

=

cos~(fC/fl),

a2

cos(f~/f1)=

iT

—

a1

(2.18)

.

The magnetic moment of (2.15) is then non-zero only for odd harmonics, since ~(t) is proportional to the current, 1(t), circulating in the ioop, the power is proportional to the square of the current and hence P(t) p~(t) so that —

.—

22 P(nw) = a~~tt0.

(2.19)

From (2.14), (2.17) and (2.19), we plot the power spectrum as shown in fig. 15 with zero d.c. field, H0 = 0, so that all even harmonics are zero and only the odd harmonics (n = 1, 3, 5,. .) up to a large value of n are generated. The envelope of the spectrum is 1 /(2n + 1)2 which falls off very slowly with increasing n. There are oscillations in the amplitudes as a function of n because of the term sin[(2n + 1)ail. For finite fields, H0, even harmonics appear in addition to the odd ones as shown in fig. 16. In the case of a Josephson junction, the voltage in terms of fluctuations of the phase is given by .

V(t) = —(h/2e) d4’/dt.

(2.20)

Substituting (2.9) in (2.20) gives the voltage in terms of Dirac ô-functions in time, but the divergences can be removed by replacing the s-function by =

(1 /2r) exp(—~t~ /7),

(2.21)

a

INO —..

—2

0

~

-3

01357

9

11

13

15

17

19

21

23

2, H, = 14G. In zero d.c. field, H Fig. 15. Power spectrum for the area S = iO~cm

0 = 0, only odd harmonics are generated liii].

K.N. Shrivastava, Magnetic resonance in high-temperature superconductors

65

r~i0 —..

0~

—3

9-4

L1~litH11H1

Fig. 16. Both odd and even harmonics appear in a finite field, H0 = I G, in the microwave power spectrum [111].

where

7

is a characteristic relaxation time. The absorption per unit time is given by

2(t)) (1/R~)(V where average is taken over one period. From (2.9), (2.20) and (2.22) we calculate

x”

(2.22)

=

dr” dH0

2V~w2S 2(TIT)2] irR~410[1 + (2ir)

(

(f,—f [f~ (f~-f0)2]1’2 [f~ (f~±f0) (f~+f 2]”2)‘t~ ~1 0) 0) —

~

-

-

—

~‘

(2.23) d~

2

2V

2

0wS(f~—f0) 2(TIT)2][f~ (f 2]t’2’ dH0 irR~qS0[1 + (2ir) 0 —f~) min(f~+f 1, f~+ 1-f1)>f01f1 -f~I. —

224

In the limit T —*0, (2.24) becomes independent of ‘r. We plot d~”IdH0in fig. 17 as a function of d.c. field for fixed amplitude of the a.c. field and a given loop size. The shape of the curve in this sample model depends on the area of the loop, and on the shunt resistance R~and in general appears to be in qualitative agreement with the experiments. We shall see later that the spherical harmonics can be used more effectively than the Fourier series. 2.3. Parallel pumping Jeffries et al. [112—114]have studied the same problem as above except that the oscillating field is applied parallel to the d.c. field whereas in the work of Blazey et al. [18, 27—30J and in our work [17,

66

K.N. Shrivastava, Magnetic resonance in high-temperature superconducto~s

H(G) 2,

2Sf(irR~ç6

Fig. 17. The derivative of the absorption d~”/dHin units of V~u H, =0.04G and r=0.O1T liii].

0) versus d.c. field for fixed amplitude of a.c. field for S = 1O~ cm

20—23] the oscillating field was perpendicular to the d.c. field as in the electron paramagnetic resonance spectrometer. Jeifries’ configuration is similar to that of parallel pumping which we describe below. Instead of (2.12) the Josephson current—phase relation may be written as 1(t) = I~sin y,

y

(2irS 0/410)(H0 + H1 sin cot),

(2.25)

where H1 is the amplitude of the oscillating field parallel to H0. For a single loop the Fourier components of the current are I~(t)-~-~J~(2irf1)sin(2irf0) cOs(nwt),

n even,

1~(t) J~(2irf1)cos(2irf0) sin(nwt),

n odd,

(2.26) —

where J~ (2 irf1) is the Bessel function of integer order n. If the loop areas S are characterized by a distribution function F(A), then averaging over all the loops, the signal amplitude of the nth harmonic becomes, (Va)

=

~

f J

AJ~(2irf1A)cos(2irAf0) F(A) dA,

n odd, (2.27)

0=

(va)

=

AJ~(2irf1A)sin(2irf0A) F(A) dA,

n even,

where G = f AF(A) dA is a normalizing factor. If the signal has only one phase, the harmonic power is P(nw) =20log10j(V,)~in dB units. Jeifries et al. [114] find that the power P(2w) agrees with that expected from the Josephson relation [115] =

[ir4(T)/2eR~] tanh[4(T)I2k~T],

(2.28)

:

K N Shrivastava Magnetic resonance in high temperature superconductors

67

where 4(T)14(0) = 1.74(1 T1T~)1’2with T~ 88.5 K. This result is analogous to that of the perpendicular case [21]. In all cases the data suggest that the samples are ensembles of flux-quantized loops with weak links. Poiner et al. [116] have studied the microwave absorption from a single crystal of YBa 2Cu3O7_5 at 16.8 0Hz, with the magnetic field parallel and perpendicular to the crystallographic c-axis. The variation of the shift of the resonance frequency is measured as a function of temperature. The static magnetic field (0—8.5 T) was applied perpendicular to the microwave field. The shift is slightly negative and constant for temperatures higher than 90 K, but at this temperature it drops abruptly to large negative values and saturates below 40 K. The negative shift is consistent with the diamagnetic character of the superconducting state. —

2.4. Giant moments We have seen that in the microwave absorption there are equally spaced lines. The separation of these lines can be described by H~=H0±&H(n+fl,

(2.29)

where n is an integer (n =0, 1, 2,. . .). Here the positive sign gives the lines on the right-hand side of the centre and the negative sign gives those on the left. We look for a hamiltonian to describe the pattern of lines in the spectra [21, 29]. The major terms are of the form, =

11 g5~c,HS5+ ~ hcu~a~a1 + ~ g~,,jçHo~ + ~ g~p.5~H5o-2S2(a~+ a~~)(a1 + a 1),

(2.30)

where g5p~is the effective magnetic moment of the object in pattern, S~is the z-component of its spin, ~ is one more spin component. We assume that ~ g~,j.s,,Hcr~,is negligibly small as in a glassy domain with completely random orientations of ~.ç. For S = = the above has the eigenvalues, E5~~given as follows: ~,

E÷~

~

-n

=

+ n/lw + ~‘g5p~H(n +

~gjL5H+nhw

—

~gjL5&H (n

~)

+

1).

(2.31) E_~~=—~g,p.5H+n/lw+ ~ -

n

=

~g,/~L5H + n/lw

—

~g,js,

~H(n

+

~),

8~~ with n as the Bose where used nthe exact correlation function (a, + a~1)(a1 + a~1)= (2n + l) numberwe density = (a~a 1).Taking the energy differences, E~÷~ E__~=g,ji~H+g~p..5~H(n+ ~), —

(2.32) E÷ ~—E_~~ =g5p.,H—g~j.~,bH(n+ ~),

68

with n

K.N. Shrivastava, Magnetic resonance in high-temperature superconductors =

0, 1, 2,. . . in these two relations, we find the energies ~ ~H),

g5p.5(H±~ ~H),

g~p~0(H± ~~H),.

.

where the first relation in (2.32) has positive sign and hence describes the spectral lines on the right-hand side of the centre, while the second relation has negative sign and hence lines on the left of the centre. Using the relation hv = g,p.5H0, we find for H0 = 3.8 G and v = 9.4 0Hz, = 16.4 >‘ 1018 erg/G which compared with the Bohr magneton, p~,is equal to ~ = 880g5~. This is the giant moment which may belong to a grain connected to a glassy mass by oscillations. If we put g0 = g~, then ~ = 880~.If ~ is composed of many occupied sites of electrons, then this means that 880 sites are needed to explain p~.Assuming that these sites are in a square plane, the number of sites in a line would amount to 30. Since the cell dimension is 3.85 A, these sites would cover a distance of 115 A, whereas the London penetration depth in this compound is estimated to be about 200 A. Thus the dimensions of the object leading to the giant moment are of the same order of magnitude as the penetration depth. The object is thus a vibrating cluster which resonates creating bosons. As the crystal is rotated away from the [110] direction, the lines spread apart; there is considerable dependence of the line spacings on the orientation of the crystal with respect to the magnetic field as may be described by spherical harmonics. The spacings also depend on temperature and r.f. power. Our finding of the giant moment is consistent with the suggestion of g >200 given by Vesnin et al. [24].

3. Electron paramagnetic resonance The electron paramagnetic high-temperature studied exten2~andresonance Gd3~ionsof have been reported.superconductors The Cu2~giveshas an been exchange narrowed sively [117—180]. The Cu line with anisotropic g-value. The Gd3~ion also gives a similar exchange narrowed line. The Ag-doped systems have been reported [173—174].The Eu-containing systems have also been studied [175,176]. The line broadens upon cooling in the superconducting state. In fact it has a liquid-like temperature dependence with the minimum width occurring at the transition temperature. The anisotropy of the g-value reflects the symmetry of the superconducting gap. Usually the conduction electrons give a line with Dysonian shape with an anisotropic g-value. However, in the superconducting state the spin-up and spin-down electrons are so paired that the spin of the Cooper pairs is zero, so that the intensity of the conduction electron spin resonance vanishes. In the case of type-Il superconductors, there are vortices where paramagnetic atoms are trapped so that a finite field reaches the spins leading to a paramagnetic resonance signal. In the case of dense paramagnetic materials only type-Il superconductivity is possible so that only the exchange narrowed lines are predicted. We describe below the results of electron paramagnetic resonance of several high-temperature superconductors.

3.1. Cu2~ion The compounds ReBa 2~(3d9)ions with 2Cu3Oi_a withof Re or a rare earth, have fluctuating valence. However, some the= Y samples have onlyusually Cu + (3d ‘°)only and Cu hence do not give paramagnetic resonance signals The compounds ReBa 2Cu3O6 are semiconductors as their resistivity

69

K N Shrivasiava Magnetic resonance in high temperature superconductors

increases upon cooling at low temperatures but ReBa2Cu3O6 ~superconducts with T~ 90 K. In fact, all the values of 8 0.55 give superconducting phases with varying transition temperatures as we shall see later. Changing of Re produces only small changes —2 K in the transition temperature. Almost all 2~ions. In one of the the samples give an anisotropic exchange narrowed line corresponding to Cu pellets of YBa 2Cu3O7_5 (6 ~0.2), at room temperature (RT), we find, —~

—~

g1~=22167±00010,

g1 =20475±00010,

(31)

which upon cooling remains nearly constant till near the transition temperature in the paramagnetic phase between 98 K and11with 94 K as fig. means 18, where the whereas g11 showsy =a 1.3 mean field exponent, ~ shown = 91 K in which y = 1.1, is expected for real g1~(RT) g~1=(T— T~1) three-dimensional systems [17].The g-values are obtained from the analysis of a powder-type line shape which occurs for a pellet sample. Although the largest possible errors are given the values are systematically correct in which the accuracy is limited by the line width. In going from one temperature to the other, the error is much smaller than the quoted value so that it is possible to find the exponent. In view of the errors, the dynamical exponent cannot be determined but the measured value is consistent only with the mean field theory. Below T~ 1we find that the g-vaiue reduces in magnitude. The quantity g~(RT) g1 increases from 0 to about 0.005 in going from 90 K to 79 K. This means that the g-value reduces by about 0.005 per Kelvin as shown in fig. 19. This reduction is possible if we assume that there is a contribution to the g-value proportional to the susceptibility. As the sample is cooled below T~1,it becomes superconducting so that the susceptibility reduces owing to negative diamagnetic contribution leading to the reduced —

—

0.006

I

I

105

110

0.005

0.004 Dl

0.003 Dl

0.002 0.001

90

95

100

115

T (K) Fig. 18. The temperature variation of g~1(RT)— g11 as a function of temperature showing paramagnetic behaviour. This is also the normal phase above T~in a pellet sample of YBa5Cu,07_,.

70

K.N. Shrivastava, Magnetic resonance in high-temperature superconductors

2.048 I

I

2.046

2.044

2.042

2.040

I

I

78

80

82

84

86

88

I (K)

2~in YBa

Fig. 19, Variation in g~of Cu

I

2Cu3O7_8 (b

=

0.2) pellet sample in the superconducting phase below T~showing negative susceptibility

contribution owing to the onset of diamagnetism.

g-value. Thus we are able to understand the g-shift in terms of diamagnetic susceptibility associated with superconductivity. The form g11 is comparatively lessunit sensitive to this phasea change. 2~—O2 ions planes with the cell dimensions, = 3.89 A, b = 3.82 A and As the Cu c = 11.69 A we expect the d-orbitals to be more susceptible to the changes in the ab-plane than in the c-direction. The g-shift, (3.2) being proportional to the susceptibility, reflects the symmetry of the superconducting state. In the ab-plane, below T~,the susceptibility becomes negative so that g~reduces, while g 1~ remains nearly constant. This means that the d-orbital is aligned along the c-direction and the superconducting electrons are mostly in the ab-plane. If that is the case, we expect the superconducting state to be occurring mostly in the ab-plane while tunneling is expected along the c-direction. The symmetry of the g-value is such that g1 notices the effect of the negative susceptibility much more than g11. Kojima et a!. [181] have studied theCu2~ions ESR and susceptibility of Y2BaCuO5, while Mehran et al. [150] 3~and in the antiferromagnetic insulating green compounds have studied the EPR of Gd Gd 2BaCuO5 and Y2BaCuO5. These compounds upon cooling do not exhibit a superconducting phase. The black compounds GdBa2Cu3O7_~and YBa2Cu3O7~have high temperature superconducting phases. It has been reported that small amounts of green phases are usually present in nominally black phases. The insulating green part of2~the sample does superconduct uponparts cooling. exists oniy in thenot non-superconducting of the sample. In the al. [95]suggest thathave Cu fluctuating valence bonds. In fact, we can measure the number of restDurny of theetsample the atoms

K.N. Shrivastava, Magnetic resonance in high-temperature superconductors

71

Cu2~ions in the superconducting phase as a function of temperature by measuring the intensity of the EPR lines and locating its zero which gives the temperature at which there is perfect diamagnetism [17]. The mixed valence phenomenon is believed to give rise to EPR signals. However, some of the samples give signals which are not easily identified and signals may arise from inclusions of the semiconducting phase. According to Mehran et al. there is a small positive g-shift in Gd 2BaCuO5 at high temperatures. This positive g-shift is consistent with our interpretation that the shift is proportional to the susceptibility and hence positive due to the paramagnetism of electrons. In the case of the superconducting sample GdBa2Cu3O7_~,the shift is negative as we expect for the diamagnetic susceptibility. Shaltiel et al. [118] have reported the electron paramagnetic resonance spectra of superconducting YBa2Cu3O7_a which have Dysonian line shape and hence indicate semiconducting behaviour. The EPR of Ag, Eu and Gd doped superconductors hashave beend~ reported by Lue [173, 174] but in these samples the 2~which lines are believed to arise from Cu 2~atoms with spin2~2 S =as0the are ground formed.state. In some of the samples “EPR It is also thought that pairs of Cu silence” has been reported [151, 178] indicating that Cu2~ions do not exist. It may be that they resonate to form Cu~(3&°) which has spin zero and hence cannot give an E~’Rsignal. It is also possible that pairs of atoms such as Cu2~—Gd3~ are formed. EuBa 2Cu3O9~gives a single line [175]the width of which varies with temperature with a minimum at about 15 K below which it becomes very large. It may be that motions which narrow the line are frozen at low temperatures resulting into broadening. A measurement [175] of the line width with several different dopants shows that the minimum width occurs in all samples as shown in fig. 20.

A

1500

.

I

.

0~

~

::

~ T(K)

Fig. 20. The peak to peak line width in the electron-paramagnetic resonance of EuBa2(Cu1~2M,)3O9~ with M=Cr, Mn, Co, Ni or Zn 11751, showing the freezing of motions below a temperature of —‘15 K. The triangular points showy = 0.01 of Co, the filled circles indicate y = 0.15 of Mn, the crosses showy = 0.08 of Zn, the open squares y = 0.037 of Cr, the filled squares showy = 0.05 of Ni and the open circles showy = 0.01 of Mn and the continuous line belongs to M = 0 (undoped) EuBaCuO sample.

—

72

K. N. Shrivastava, Magnetic resonance in high-temperature superconductors

4. Nuclear magnetic resonance [186], t3Cu [187—201],89Y The nuclear magnetic resonance of ‘H [182—184],‘4N [185], ~ [202—206],‘39La [207] and 205 Ti [208] nuclei has been studied extensively. The nuclear quadrupole resonance [187,188, 209—227] experiments have also been performed. The best results are found by the use of spin echo, relaxation and pulsed techniques [228—240].Several results of general interest have been reported [240—255]. The nuclear magnetic resonance of ‘H in YBa 2Cu3O7_a H0 2 shows that the line width becomes a minimum at the critical temperature widening upon cooling [183]so it appears that there are motions in the 2~ normal in the superconducting state.narrowing The exchange between ions is state foundwhich to be freeze about ~‘eff = 1100 cm which causes of the interaction EPR lines. [191] However, the Cu line broadening may be associated with dipole—dipole interaction. The study of 89Y nuclear NMR magnetic resonance suggests [202] that the line width may be used to calculate the vortex—lattice spacing as, ‘

2(161T3)”2[1 + (d/21TA)2]~’’2, 0A where 4i~= hcl2e. For ~H = 6.4 G, the vortex—lattice spacing is calculated to be d (~H2)’’2= ~

London penetration depth is found to be A

(4.1) 170 A and the

3800 A.

5. Nuclear relaxation In the normal state, the spin—lattice relaxation rate is proportional to the electronic density of states at the Fermi surface and hence it could be used to measure the metallic character of the material. It also has the square of the hyperfine field at the site of the nucleus and hence we write the relaxation rate as lIT,

=

(5.1)

2hkBTy~[Hhf(i)]2I3~p2(EF),

where y,, is nuclear gyromagnetic ratio, f3~ 1 is a numerical factor which depends on the mechanism of relaxation, Hhf is the magnetic field at the nucleus [229]. In a normal metal, the single-particle dispersion relation is isotropic but in the paired state there is a gap in the dispersion relation. Therefore, the above expression (5.1) is modified in the superconducting state. The ratio of the nuclear spin—lattice relaxation time in the superconducting state to that in the normal state is given by —~

=

dw f(w)[1 _f(w)][(Re

[w2

-

~2(k)J’ 2)

+

(Re

[w2

~2(k)]’

2)]

(52)

for T ~ T~,the Fermi function f(w) selects quasiparticles that contribute to the integral in a direction where the gap is small, leading to the Arrhenius behaviour, (1ITl)ccexp(_~iIkBT), T<
(5.3)

The gap energy z~can be interpreted to be the minimum gap In the case of La,

63Cu 8~Sr0,7CuO4, the

K.N. Shrivastava, Magnetic resonance in high-temperature superconductors

73

nuclear spin—lattice relaxation rate gives [229] in the normal state, T, T = 14 ± 1 Ks. In the type-Il superconductors, the London penetration depth is much larger than the coherence length A ~ ~, so that in that case it is found that AI~ K 50 and the carrier density is —5 x 1021 cmT with ~ 20 A and A 1000 A. The large value of the Korringa constant suggests strong d-character of the conduction electrons. From eq. (5.3) the measured gap is found to be A 24 ±4 K, suggesting that the ratio of the gap to the transition temperature is 2z1/kBTC = 1.3 ±0.2, whereas a value of 3.52 is expected from the BCS theory. However, there are approximations in the BCS theory which require that ratio to be replaced by proper numerical solutions [17]. Therefore, the order of magnitude of the measured value may be regarded to be correct. The nuclear spin—lattice relaxation measurements on ti3Cu have also been performed with a single crystal of YBa 2CU3O7_a in the superconducting state [230]. The temperature dependence of the relaxation rate for Cu nuclei in the ab-plane gives A = 9.6 ±0.2 meV while those along the c-axis give A = 32.8 ±0.8 meV. Thus 2AIkBTC = 8.3 for the nuclei along the c-axis and 2AIkBTC = 2.4 for those in the ab-plane for T~ 92 K. The relaxation rates of the planar nuclei are described by a constant term and an activated term (5.3). Therefore, it may be concluded that there are several mechanisms of relaxation. The dipole—dipole interaction may be important in addition to the direct process. In the case 139La in La of 1 8Sr0 2CuO4_~the relaxation rates obey (5.3) with A = 11.5 meV so that 2zIIkBTC ~~o7.1is found [238] for T~ 38 K. Thus the measured values of 2A IkB T~are qualitatively in accord with the BCS theory. The phase diagram of YBa2Cu3O7_a as a function of transition temperature and concentration, 6, is shown in fig. 21. It is based on the neutron scattering [255],~iSR[256, 257] and Y Knight shift [205] studies. Tranquada et al. have done considerably extensive XAFS studies which determine 8 quite accurately [255]. For 6 <0.6 the samples are metallic while for 1 > 3 >0.6 the system is insulating [258]. The highest transition temperature for the superconducting state of YBa2Cu3O7~found by us is about 96 K. There is no boundary between insulating and superconducting regions. Tozer et al. [259] and Penney et al. [2601 have published the resistivity of a single crystal of YBa2Cu3O7_a which in the ab-plane shows a transition from a metal to a superconductor. The 89

500 YBn2Cu3O7_fi

400 i-.

300

_~-~-I

OrthorhombiC I

L. ~

METAL

Ictrogoflot ANTIFERR— MAGNETIC

,rIrI

E I—

,

‘

INSULATOR

100 R~NOUCTIN’~’~D\+

0.8

1.0

Fig. 21. The phase diagram of the high-temperature superconductor YBa2Cu3O7_0 showing various phases as a function of the transition temperature and the concentration ~S12581.

74

K.N. Shrivastava, Magnetic resonance in high-temperature superconductors

insulating region is not found [261].In the resistivity along the c-direction there is a region as shown in fig. 22 in which the resistivity increases upon cooling till a peak is found below which it superconducts. However, in this case there is Josephson type tunneling along the c-direction and the rise in resistivity, p A IT, is caused by this tunneling. The superconducting quasiparticles are found in the ab-plane and there is an insulating layer between two ab-planes with interlayer tunneling. Thus the superconducting and the insulating quasiparticles are kept separately. The insulating material in the interlayer spacing does not transform into a superconductor but the pairs from the superconducting layer can pass through the insulating atoms along the c-direction. As 6 increases the transition temperature of YBa2Cu3O7~reduces as in fig. 21. This decrease is consistent -~

with the Abrikosov—Gorkov theory which describes the reduction of the transition temperature with respect to the BCS value due to the scattering of electrons by magnetic atoms, =

T~CS —

1ThI(4kBr,)

(5.4)

where r~is the electron scattering life time and kB is the Boltzmann constant. The phase diagram of La2~Ba~CuO4...~ is shown in fig. 23. For 6<0.05 there is an antiferromagnetic phase marked by a Néel line [262]while for 6 >0.05 there is superconducting phase. A I;

7- 20 3

Qcm

x10 Qcm

I

o”

/ 0 C

—p

/ -

15

/

3 C,

2-

~0

0

0”

/ -S

0

100

200

300K

Temperature Fig 22 The resisttvity of a single crystal of YBa 2Cu,07 showing interlayer tunneling of pairs.

along the a direction showing metal to superconductor transition and along c direction

K N Shrivastava Magnetic resonance in high temperature .superconduc:ors

75

Lo2_6Bo5CuO4_~ 200

:

3.90 ;U

AF\~J~~

0.02

0.04

0.06

0.08

OJO

~~_~~//fl////”

lOOK

50

T~ Fig. 23. The phase diagram of La5., Ba,CuO4_~showing superconducting normal and antiferromagnetic phases 11961.

Fig. 24. The superconducting critical temperature as a function of Cu—Cu distance in several systems (see ref. 12681). 1. BiSrCuO, 2. LaSrCuO, 3. LaCaCuO, 4. LaSrCuO, 5. BiCaSrCuO, 6. BiCaSrCuO,

7. ErBaCuO, 8. YBaCuO, 9. TIBaCaCuO.

similar phase diagram for (La1_5Sr5)2CuO4 is given by Kitazawa et a!. [263] and by Weidinger et al. [264]. Shrivastava and Jaccarino [265]have studied the dependence of the exchange interaction on the internuclear distance. Although the exchange interaction decreases with increasing distance, there is a phonon contribution [266—269]which may be antiferromagnetic. Hence in antiferromagnetic coupling the net exchange can appear to increase by increasing distances. The effect of the magnetic exchange interaction [269] is to reduce the superconducting critical temperature owing to its repulsive character. In fig. 24 we show the superconducting transition temperature of several compounds as a function of the Cu—Cu distance. In the neighbourhood of T~ 100 K, T~increases with increasing distance corresponding to decreasing exchange interaction showing the importance of static (non-phonon) mechanisms. However, in the neighborhood of T~ 30 K, T~increases with decreasing distance, indicating that a phonon-induced exchange may be important. The data for fig. 24 are taken from the work of Morosin et al. [270] and are based on the work of several papers [271—274].Morosin et al. emphasize that when the external pressure is varied from lkbar to about 6kbar, T~increases from —106 K to 107.5 K in Tl5Ba5Ca2Cu6O~.

6. Conclusions The microwave absorption of the superconductors arises from the flux-quantized eigenvalues, which is another way of saying that the flux is quantized. There are domains of size of ~~10_6 m and hence there are giant moments in the superconductors. There are oscillations and there are glassy regions.

76

K.N. Shrivastava, Magnetic resonance in high-temperature superconductors

The electron paramagnetic resonance experiments are found to be useful for the observation of such effects in the type-Il superconductors. The g-value gives the symmetry of the susceptibility and hence negative shifts are found. The NMR and NQR measurements are used to detect the phase transitions. The Knight shift gives useful information about the conduction electron susceptibility. The nuclear spin—lattice relaxation rate measured as a function of temperature yields the energy gap in the single-particle dispersion of the superconductor and thus helps in the understanding of the gap to critical temperature ratio. The measured values are qualitatively in accord with the pairing theory. The phase diagram of a superconductor as a function of the coupling constant, D of the electron— phonon interaction and the transition temperature is given by Chakraverty [2751.It was found that for small values of D the system is repulsive. As D increases in comparison with electron—electron Coulomb interaction, at some value D = A,, the interaction becomes attractive so that there is a metal to superconductor transition. For still larger values of D there is a bipolaronic insulator, cooling of which also produces a superconducting state. This means that there are two superconducting phases, one approaching from the metal and the other from the insulator. These two states must differ by a similarity transformation otherwise there must be one more phase boundary separating the two types of superconducting states. Such a phase diagram is shown in fig. 25 with the only difference that we have included an antiferromagnetic phase also. For a small electron—phonon interaction this diagram resembles that of YBa2Cu3O7_8 but for D > A2 it does not. It seems that a bipolaronic insulator to superconductor line is not yet found. In the real systems shown in figs. 22 and 23, the evidence for the normal metallic phase comes from the Knight shift. If this is accepted then we have a metal to superconductor transition but then the insulator to the superconductor transition is not found. However, at T = 0, upon varying the concentration, there is an insulator to a superconductor transition. At finite temperatures, the theory has yet to be worked out and new ideas [276] may be needed. The BCS theory describes the metal to superconductor transition by pairing of electrons via the electron—phonon interaction at low temperatures but the transition temperatures of the present materials are too high to be easily understood even when the multiple scattering [277] of the two-particle bound state given by the diagrams of fig. 26 is

/

// / /

/B~OLARON

/

METAL

INSULATOR

LI

~

zo,

~--~-~

SUPERCONDUCTOR ELECIRON—PHONON

INTERACTION

MATRIX ELEMENT

Fig. 25. Phase diagram showing the metal to superconductor transition, and the antifenomagnetic phase by continuous lines. The dashed lines show the metal to bipolaronic insulator and bipolaronic insulator to superconductor transitions.

K. N. Shriva.stava, Magnetic resonance in high-temperature superconductors

77

Fig. 26. Diagrams describing the contribution of pair fluctuations to the thermodynamic potential. The coiled line describes the interaction via a phonon. The first concentric circles represent a non-interacting two-particle state to which a geometric series, the summation of which is known, is shown to describe correlations [277].

taken into account. In the Bose condensation the transition temperature varies as ~2/d where n is the particle density and d is the dimensionality of the system but then the bosons cannot carry the charge. In any case, the magnetic resonance technique can be used effectively to obtain useful information about the superconductors.

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