Alexandrov A.S.,Theory of Superconductivity - From Weak to Strong Coupling

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196

Normal state of cuprates

planes only due to a simultaneous two-particle tunnelling, which is much less

probable than a single-polaron tunnelling (chapter 4). Along the planes, polarons

and inter-site bipolarons propagate with about the same effective mass. Hence,

the polaron contribution to the in-plane transport is small at low temperatures due

to their low density compared with the bipolaron density. As a result, we have a

mixture of non-degenerate quasi-two-dimensional bosons and thermally excited

fermions, which are capable of propagating along the c-axis. Polarons mainly

contribute to c-axis transport, if the temperature is not very low, which leads to a

fundamental relation between anisotropy and magnetic susceptibility [186].

To illustrate the point, let us consider a system containing only singlet

pairs and thermally excited polarons. Quite generally the in-plane bipolaron

and c-axis polaron, conductivities (σ

and σ

respectively) and the uniform spin

susceptibility (χ

) are expressed as follows:

(T , x) =−4



∞

dE σ

(E)

∂ f

∂ E

(6.34)

(T , x) =−2



∞

dE σ

(E)

∂ f

∂ E

(6.35)

(T , x) =−2µ



∞

dEN

(E)

∂ f

∂ E

(6.36)

where f

=[y

−1

exp(E/T ) − 1]

−1

and f

=[y

−1/2

exp(E/T +/2T ) + 1]

−1

are the bipolaron and polaron distribution functions, respectively, and µ

is the

Bohr magneton. Polarons are not degenerate at any temperature neither are

bipolarons degenerate above T

,sothat

≈ y exp



−



(6.37)

and

≈ y

1/2

exp



−

E + /2



. (6.38)

If the scattering mechanism is the same for polarons and bipolarons, the ratio

of the differential bipolaron and polaron conductivities (σ

(E) and σ

(E),

respectively) is independent of the energy and doping

(E)

≡ A = constant. (6.39)

There is a large difference in the values of the ppσ and ppπ hopping

integrals between different oxygen sites (section 5.4). Therefore, we expect

a highly anisotropic polaron energy spectrum with a quasi-one-dimensional

polaron density of states as also observed in high-resolution ARPES experiments

(section 8.4):

(E) 

4πda



∗



1/2

(6.40)

-axis transport and charge pseudogap

197

Figure 6.5. (a) Experimental anisotropy [214] compared with equation (6.41) and (b)

using experimental magnetic susceptibilities of La

2−x

CuO

[215].

where a is the in-plane lattice constant. Then the anisotropy and the spin

suceptibility are expressed as

(T , x)

= 2 Ay

1/2

exp







(6.41)

and

(T , x) =

2da



∗

2π T



1/2

exp



−





. (6.42)

The chemical potential, y = 2π n

(T , x)/(Tm

∗∗

), is calculated taking

into account the localization of bipolarons in a random potential (section 5.5).

The bipolaron density (n

(T , x) per cm

) is linear in temperature and doping

198

Normal state of cuprates

(T , x) ≈ xT/γ a

) in a wide temperature range above T

which ﬁts

the temperature and doping dependence of the Hall ratio R

 1/(2en

)

(section 6.2). As a result, we ﬁnd the temperature-independent y

x and

(T , x)

1/2

(T , x)

(6.43)

(T , x)





1/2

exp



−





. (6.44)

We expect some dependence of the binding energy on doping because of

screening. Bipolarons are heavy non-degenerate particles, which well screen the

electron–phonon interaction with low-frequency phonons. In a wide range of

doping near the optimum value (x = 0.15), different experiments are consistent

with

 =



(6.45)

where 

is doping independent.

One can describe all qualitative features of c-axis resistivity and magnetic

susceptibility of La

2−x

CuO

using equations (6.43), (6.44) and (6.45) without

any ﬁtting parameters. Anisotropy is quantitatively described by equation (6.43)

using the experimental values of χ

(T , x), as shown in ﬁgure 6.5. The

temperature-independent anisotropy of a heavily doped sample, x = 0.3in

ﬁgure 6.5 is explained by the contribution of polarons to the in-plane transport.

If the binding energy  is below 100 K, the polaron contribution ﬂattens

the temperature dependence of the anisotropy. An exponential temperature

dependence for the c-axis resistivity and anisotropy was also interpreted

within the framework of the bipolaron model in other cuprates, in particular

in Bi

CaCu

8+δ

[197], YBa

6+x

[213] and HgBa

CuO

4+δ

[212].

Importantly, the uniform magnetic susceptibility above T

increases with doping

(ﬁgure 6.5(b)). It proves once again that cuprates are doped insulators, where low-

energy charge and spin degrees of freedom are due to holes doped into a parent

insulating matrix with no free carriers and no free spins. A rather low magnetic

susceptibility of parent insulators in their paramagnetic phase is presumably due

to a singlet pairing of copper spins as discussed in section 5.5.

6.6 Infrared conductivity

Studies of photoinduced carriers in dielectric ‘parent’ compounds like La

CuO

YBa

, and others demonstrated the formation of self-localized small

polarons or bipolarons. In particular, Mihailovic et al [170] described the

spectral shape of the photoconductivity with the small polaron transport theory.

They also argued that a similar spectral shape and systematic trends in both

the photoconductivity of optically doped dielectric samples and the infrared

conductivity of chemically doped high-T

oxides indicate that carriers in a

Infrared conductivity

199

Figure 6.6. (a) Photoinduced infrared conductivity in the insulator precursors for

CaCu

(top), YBa

(middle) and La

2−x

CuO

(bottom) compared

with the polaron infrared conductivity (broken curve); (b) the infrared conductivity for

superconducting samples (broken curves) compared with (a).

concentrated (metallic) regime retain much of the character of carriers in a dilute

(photoexcited) regime. The measured photoinduced infrared conductivity σ(ν)

(full curves) in insulating parent compounds is shown in ﬁgure 6.6(a). Its

characteristic maximum is due to the incoherent spectral weight of the polaron

Green function (section 4.5) as follows from a comparison with a theoretical

small-polaron infrared conductivity (broken curves) in ﬁgure 6.6(a). The infrared

conductivity of high-T

superconductors (broken curves) is compared with the

photoinduced infrared conductivity (full curves) in respective insulator precursors

in ﬁgure 6.6(b). One of the qualitative observations, which follow from a

comparison of the infrared conductivities of insulating and superconducting

materials is that in all perovskites exhibiting superconductivity, the peak

frequency shifts toward lower values as the T

of the material increases. The

peak position in σ(ν) is related to the polaron level shift and, therefore, to the

(bi)polaron mass. The critical temperature of the superconducting transition is

proportional to 1/m

∗

in bipolaron theory (section 4.7.5). A lower peak frequency

corresponds to a lower mass of bipolaronic carriers and, therefore, to a higher T

Chapter 7

Superconducting transition

7.1 Parameter-free description of T

An ultimate goal of the theory of superconductivity is to provide an expression for

as a function of some well-deﬁned parameters characterizing the material. In

the framework of BCS theory, the Eliashberg equation (3.74) for the gap function

properly takes into account a realistic phonon spectrum and retardation of the

electron–phonon interaction. T

is fairly approximated by McMillan’s formula

(3.96), which works well for simple metals and their alloys. But applying a theory

of this kind to high-T

cuprates is problematic. Since bare electron bands are

narrow, strong correlations result in the Mott insulating state of undoped parent

compounds. As a result, µ

∗

and λ are ill deﬁned in doped cuprates and polaronic

effects are important as in many doped semiconductors [13]. Hence, an estimate

of T

in cuprates within BCS theory appears to be an exercise in calculating µ

∗

rather than T

itself. Also, one cannot increase λ without accounting for a polaron

collapse of the band (chapter 4). This appears at λ  1 for uncorrelated electrons

(holes) and even at a smaller value of bare electron–phonon coupling in strongly

correlated models [216].

However, the bipolaron theory provides a parameter-free expression for

[217], which ﬁts the experimentally measured T

in many cuprates for any

level of doping. T

is calculated using the density sum rule as the Bose–Einstein

condensation (BEC) temperature of 2e charged bosons on a lattice. Just before

the discovery [1], we predicted T

would be as high as ∼100 K using an estimate

of the bipolaron effective mass [105]. Uemura [218] established a correlation

of T

with the in-plane magnetic ﬁeld penetration depth measured by the µsR

technique in many cuprates as T

1/λ

. The technique is based on the

implantation of spin-polarized muons. It monitors the time evolution of the

muon spin polarization. He concluded that cuprates are neither BCS nor BEC

superﬂuids but that they exist in the crossover region from one to the other,

because the experimental T

was to be found about three or more times lower

than the BEC temperature.

200

Parameter-free description of

201

Here we calculate the T

of a bipolaronic superconductor taking properly

into account the microscopic band structure of bipolarons in layered cuprates as

derived in section 5.4. We arrive at a parameter-free expression for T

, which in

contrast to [218] involves not only the in-plane, λ

, but also the out-of-plane, λ

magnetic ﬁeld penetration depth and a normal state Hall ratio R

just above the

transition. It describes the experimental data for a few dozen different samples

clearly indicating that many cuprates are in the BEC rather than in the crossover

regime.

The energy spectrum of bipolarons is twofold degenerate in cuprates

(section 5.4). One can apply the effective mass approximation at T  T

equation (5.22), because T

should be less than the bipolaron bandwidth. Also a

three-dimensional correction to the spectrum is important for BEC (appendix B)

which is well described by the tight-binding approximation:

x,y

∗∗

y,x

∗∗

+ 2t

⊥

[1 − cos(K

d)] (7.1)

where d is the interplane distance and t

⊥

is the inter-plane bipolaron hopping

integral. Substituting the spectrum (equation (7.1)) into the density sum rule,



K,i=(x,y)

[exp(E

) − 1]

−1

= n

(7.2)

one readily obtains T

(in ordinary units):

= f



⊥



3.31

/2)

2/3

∗∗

)

1/3

(7.3)

where the coefﬁcient f (x) ≈ 1 is shown in ﬁgure 7.1 as a function of the

anisotropy, t

⊥

/(k

) and m

∗∗

/(2|t

⊥

This expression is rather ambiguous because the effective mass tensor as well

as the bipolaron density n

are not well known. Fortunately, we can express the

band-structure parameters via the in-plane magnetic ﬁeld penetration depth



∗∗

8πn

∗∗

+ m

∗∗

)



1/2

and out-of-plane penetration depth,



∗∗

16πn



1/2

(we use c = 1). The bipolaron density is expressed through the in-plane Hall ratio

(above the transition) as

2en

∗∗

+ m

∗∗

)

(7.4)

202

Superconducting transition

Figure 7.1. Correction coefﬁcient to the three-dimensional Bose–Einstein condensation

temperature as a function of anisotropy.

which leads to

= 1.64 f



⊥







1/3

. (7.5)

Here T

is measured in kelvin, eR

in cm

and λ in cm. The coefﬁcient f is

about unity in a very wide range of t

⊥

/(k

) ≥ 0.01, ﬁgure 7.1. Hence, the

bipolaron theory yields a parameter-free expression, which unambiguously tells

us how near the cuprates are to the BEC regime:

≈ T

(3D) = 1.64





1/3

. (7.6)

We compare the last two expressions with the experimental T

of more than 30

different cuprates, for which both λ

and λ

have been measured along with

+ 0) in table 7.1 and ﬁgure 7.2. The Hall ratio has a strong temperature

dependence above T

(section 6.1). Therefore, we use the experimental Hall ratio

just above the transition. In a few cases (mercury compounds), where R

+0)

is unknown, we take the inverse chemical density of carriers (divided by e)as

. For almost all samples, the theoretical T

ﬁts experimental values within an

experimental error bar for the penetration depth (about ±10%). There are a few

Zn-doped YBCO samples (ﬁgure 7.2) whose critical temperature is higher than

the theoretical one. If we assume that the degeneracy of the bipolaron spectrum is

removed by the random potential of Zn, then the theoretical T

would be almost

the same as the experimental values for these samples as well.

Parameter-free description of

203

Table 7.1 . Experimental data on T

(K), ab and c penetration depth (nm), Hall coefﬁcient

(10

−3

(cm

−1

)), and calculated values of T

, respectively, for La

2−x

CuO

(La),

YBaCuO (x%Zn) (Zn), YBa

7−x

(Y) and HgBa

CuO

4+x

(Hg) compounds.

Compound T

exp

(3D) T

La(0.2) 36.2 200 2 540 0.8 38 41 93

La(0.22) 27.5 198 2 620 0.62 35 36 95

La(0.24) 20.0 205 2 590 0.55 32 32 88

La(0.15) 37.0 240 3 220 1.7 33 39 65

La(0.1) 30.0 320 4 160 4.0 25 31 36

La(0.25) 24.0 280 3 640 0.52 17 19 47

Zn(0) 92.5 140 1 260 1.2 111 114 172

Zn(2) 68.2 260 1 420 1.2 45 46 50

Zn(3) 55.0 300 1 550 1.2 35 36 38

Zn(5) 46.4 370 1 640 1.2 26 26 30

Y(0.3) 66.0 210 4 530 1.75 31 51 77

Y(0.43) 56.0 290 7 170 1.45 14 28 40

Y(0.08) 91.5 186 1 240 1.7 87 88 98

Y(0.12) 87.9 186 1 565 1.8 75 82 97

Y(0.16) 83.7 177 1 557 1.9 83 89 108

Y(0.21) 73.4 216 2 559 2.1 47 59 73

Y(0.23) 67.9 215 2 630 2.3 46 58 73

Y(0.26) 63.8 202 2 740 2.0 48 60 83

Y(0.3) 60.0 210 2 880 1.75 43 54 77

Y(0.35) 58.0 204 3 890 1.6 35 50 82

Y(0.4) 56.0 229 4 320 1.5 28 42 65

Hg(0.049) 70.0 216 16 200 9.2 23 60 115

Hg(0.055) 78.2 161 10 300 8.2 43 92 206

Hg(0.055) 78.5 200 12 600 8.2 28 69 134

Hg(0.066) 88.5 153 7 040 6.85 56 105 229

Hg(0.096) 95.6 145 3 920 4.7 79 120 254

Hg(0.097) 95.3 165 4 390 4.66 61 99 197

Hg(0.1) 94.1 158 4 220 4.5 66 105 216

Hg(0.101) 93.4 156 3 980 4.48 70 107 220

Hg(0.101) 92.5 139 3 480 4.4 88 127 277

Hg(0.105) 90.9 156 3 920 4.3 69 106 220

Hg(0.108) 89.1 177 3 980 4.2 58 90 171

One can argue that, due to a large anisotropy, cuprates may belong to a two-

dimensional ‘XY’ universality class with the Kosterlitz–Thouless (KT) critical

temperature T

of preformed bosons [219, 220] or Cooper pairs [164]. Should

this be the case, one could hardly discriminate the Cooper pairs with respect to

bipolarons. The KT critical temperature can be expressed through the in-plane

204

Superconducting transition

Figure 7.2. Theoretical critical temperature compared with the experiment (the theory

is exact for samples on the straight line) for LaSrCuO compounds (squares), for

Zn-substituted YBa

1−x

(circles), for YBa

7−δ

(triangles), and for

HgBa

CuO

4+δ

(diamonds). Experimental data for the London penetration depth are taken

from Xiang T et al 1998 Int. J. Mod. Phys. B 12 1007 and Janossy B et al 1991 Physica

C 181 51 in YBa

7−δ

and YBa

1−x

; from Grebennik V G et al 1990

Hyperﬁne Interact. 61 1093 and Panagopoulos C Private communication in underdoped

and overdoped La

2−x

CuO

, respectively, and from Hofer J et al 1998 Physica C 297

103 in HgBa

CuO

4+δ

. The Hall coefﬁcient above T

is taken from Carrington A et al

1993 Phys. Rev. B 48 13 051 and Cooper J R Private communication (YBa

7−δ

and YBa

1−x

) and from Hwang H Y et al 1994 Phys.Rev.Lett.72 2636

(La

2−x

CuO

penetration depth alone [164]

≈

0.9d

16πe

. (7.7)

It appears signiﬁcantly higher than the experimental values in many cases (see

table 7.1). There are also quite a few samples with about the same λ

and the

same d but with very different values of T

, which proves that the phase transition

is not the KT transition. In contrast, our parameter-free ﬁt of the experimental

critical temperature and the critical behaviour (sections 7.3 and 7.4) favour 3D

Bose–Einstein condensation of charged bosons as the mechanism for high T

rather than any low-dimensional phase-ﬂuctuation scenario.

The ﬂuctuation theory [221] further conﬁrms the three-dimensional

character of the phase transition in cuprates. However, it does not mean that

Isotope effect on

and on supercarrier mass

205

Table 7.2 . Mass enhancement in cuprates.

Compound m

La(0.2) 22.1 3558

La(0.15) 15.0 2698

La(0.1) 11.3 1909

Y(0.0) 7.2 584

Y(0.12) 8.3 600

Y(0.3) 10.6 1994

all cuprates are in the BEC regime with charged bosons as carriers. Some of

them, in particular overdoped samples, might be in the BCS or intermediate

regime, which makes the BCS–BEC crossover problem relevant. Starting from

the pioneering works by Eagles [72] and Legget [222], this problem received

particular attention in the framework of a negative Hubbard U model [223, 224].

Both analytical (diagrammatic [225], path integral [226]) and numerical [227]

studies have addressed the intermediate-coupling regime beyond a variational

approximation [223], including two-dimensional systems [227–229]. However,

in using the negative Hubbard U model, we have to realize that this model, which

predicts a smooth BCS-BEC crossover, cannot be applied to a strong electron–

phonon interaction and polaron–bipolaron crossover (section 4.5). An essential

effect of the polaron band-narrowing is missing in the Hubbard model. As

discussed in section 4.7.5, the polaron collapse of the bandwidth is responsible

for high T

. It strongly affects the BCS–BEC crossover, signiﬁcantly reducing the

crossover region.

It is interesting to estimate the effective mass tensor using the penetration

depth and the Hall ratio. These estimates for in-plane and out-of-plane boson

masses are presented in table 7.2. They well argee with the inter-site bipolaron

mass (sections 4.6 and 5.4). We note, however, that an absolute value of the

effective mass in terms of the free electron mass does not describe the actual

band mass renormalization if the bare (band) mass is unknown. Nevertheless, an

assumption [141] that the number of carriers is determined by Luttinger’s theorem

(i.e. n = 1 + x) would lead to much heavier carriers with m

∗

about 100m

7.2 Isotope effect on T

and on supercarrier mass

The advances in the fabrication of the isotope substituted samples made it possible

to measure a sizeable isotope effect, α =−dlnT

/dlnM in many high-T

oxides. This led to a general conclusion that phonons are relevant for high T

Moreover, the isotope effect in cuprates was found to be quite different from the

BCS prediction: α = 0.5 (or less) (section 2.5). Several compounds showed