Mourachkine A. Room-Temperature Superconductivity

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184 ROOM-TEMPERATURE SUPERCONDUCTIVITY

stripes located between charge stripes sets in at T

. So, all these clusters

and matrix are spatially separated. Four, the real doping dependences T

(p),

(p), T

(p) and T

pair

(p) are most likely not linear but quasi-linear. Five,

in the literature, there is discrepancy among phase diagrams inferred from dif-

ferent sets of experimental data. This is because the phase diagram of every

superconducting cuprate consists of several energy/temperature scales (at least,

seven) which have different origins. Since different experimental techniques

are sensitive to different types of correlations and have different resolutions

and different characteristic times, one can then understand why there is a dis-

crepancy among phase diagrams that can be found in the literature.

Finally, consider an interesting question: what type of clusters does the

(p) temperature scale originate from? One may immediately suggest that

the T

(p) temperature scale originates from the nanoscale clusters with vertical

stripes. The answer is yes and no. As was already emphasized, the phase dia-

gram in Fig. 6.11 is a superposition of different temperature scales originating

from three types of clusters (or matrix) depicted in Fig. 6.12. The spin-glass

temperature scale T

(p) has no direct relation with superconductivity; the other

temperature scales in Fig. 6.11 are presented in two plots, according to their

origin, as sketched in Fig. 6.13. One can see in Fig. 6.13a that the T

(p) tem-

perature scale is attributed mainly, but not solely, to the matrix (clusters) with

the local antiferromagnetic order. Why? The location of a quantum critical

point at p = 0.19, where magnetic ﬂuctuations are the strongest, is the reason

for this. In the CuO

planes, the existence of the matrix (clusters) with local

antiferromagnetic order as well as the T

temperature scale is determined by

the location of a quantum critical point. In a sense, the superconducting phase

is “attracted” at low temperature by a quantum critical point. Of course, the

charge-stripe ﬂuctuations in clusters with vertical charge stripes generate spin

excitations that mediate the long-range phase coherence. However, in order

to propagate, the spin excitations must use a magnetic order, local or not. A

quantum critical point provides the occurrence of magnetic order.

3.5 Pseudogap

The pseudogap is a depletion of the density of states above the critical tem-

perature. Below a certain temperature, the cuprates do have a connected Fermi

surface that appears to be consistent with conventional band theory. Above T

the pseudogap dominates the normal-state low-energy excitations. The pseu-

dogap was observed for the ﬁrst time in NMR measurements and therefore

mistakenly interpreted as a spin gap [8]. Later, ARPES, tunneling, Raman,

speciﬁc-heat and infrared measurements also provided evidence for a gap-like

structure in electronic excitation spectra. Thus, it became clear that the pseudo-

gap is not a spin gap but a gap to both spin and charge excitations; alternatively,

there are two spatially separated pseudogaps: one is a spin gap, and the second

Third group of superconductors: Mechanism of superconductivity 185

Temperature

0.05

0.19

0.27

(a)

Temperature

0.05

0.27

(b)

QCP

Figure 6.13. Temperature scales from Fig. 6.11: (a) the T

(p) scale is determined by a

quantum critical point (QCP) associated with antiferromagnetic phase, while (b) the T

(p),

(p) and T

pair

(p) scales originate from the striped phase shown in Fig. 6.2. The T

(p)

scale is a mutual temperature scale caused, ﬁrst of all, by the quantum critical point and also by

spin excitations present in striped-phase clusters.

is a charge gap. The magnitude of pseudogap(s) is large in the underdoped

region and decreases as the doping level increases.

So, what is the pseudogap in cuprates? There is no straight answer to this

question because the answer depends upon the technique by which it was ob-

served and upon the temperature at which it was seen. Above T

, one may

observe four different gaps in the cuprates: (i) below T

, a charge gap on the

charge stripes, ∆

, (ii) below T

, a gap having the magnetic origin, ∆

NMR

(iii) it is possible that a spin gap ∆

sets in below T

in hole-poor stripes in

the striped phase shown in Fig. 6.2, and (iv) below T

pair

, the pairing gap ∆

So, generally speaking, the pseudogap in the cuprates, ∆

,is

∆



∆

+∆

NMR

+∆

. (6.10)

However, there is no experimental technique able to detect all of these gaps

at once. The magnitudes of these four gaps are not affected much on cooling

through T

, but the pairing gap ∆

becomes a part of the superconducting gap,

∆



∆

+∆

, predominant below T

. For example, in the case of ∆

below T

there is a renormalization of low-energy excitations; thus, excitations

inside the charge gap. Let us consider this renormalization.

3.5.1 Charge gap on the stripes

The pseudogap which can be observed in tunneling and ARPES measure-

ments is a charge gap on charge stripes, manifesting itself in the form of wide

humps present in the spectra at high bias. Figure 6.14a shows an ARPES

186 ROOM-TEMPERATURE SUPERCONDUCTIVITY

0.8

-200 -100 0 100 200

Normalized dI/dV

Bias (mV)

T = 122 K

4∆

(a) (b)

= 88 K

Bi2212

400

300

200

100

Binding Energy (meV)

Intensity

Bi2212

T < T

Figure 6.14. (a) ARPES spectrum obtained in a slightly overdoped Bi2212 single crystal with

=91KatT  T

; and (b) Tunneling pseudogap in slightly overdoped Bi2212 obtained in

a SIS junction at T>T

(see references in [19]).

spectrum obtained below T

in Bi2212. In the plot, the position of main

quasiparticle peaks measures the magnitude of ∆

, while the hump at a bias

of V

hump

 3∆

/e is caused by the charge gap on the stripes. The ratio

hump

peak

 3, inferred from either ARPES or superconductor-insulator-

normal metal (SIN) tunneling spectra, is independent of doping level. This

ratio is determined by the ratio ∆

/∆

 3 in Eq. (6.6).

In superconductor-insulator-superconductor (SIS) tunneling conductances,

however, the bias ratio is about V

hump

peak

 2 because the humps in SIS

conductances do not correspond directly to ∆

but are a product of tunneling

between genuine humps and quasiparticle peaks. Thus at T  T

, tunneling

dI(V )/dV and I(V ) characteristics as well as ARPES spectra consist of two

contributions caused by the superconducting condensate (bisolitons) and by

the charge gap on the stripes. In SIN conductances, these two contributions are

superimposed linearly while in SIS conductances, nonlinearly. This issue was

discussed in detail in Chapter 12 of [19].

Figure 6.14b depicts the pseudogap in a tunneling conductance obtained

by a break-junction technique in Bi2212 above T

. In the measurements, the

temperature of 122 K is above T

pair

 120 K in slightly overdoped Bi2212

with p  0.19. The gap-like structure in tunneling conductances disappears

upon increasing the temperature.

Consider now the charge-gap renormalization at T

. Let us start with tun-

neling measurements. Figure 6.15a sketches the charge gap above and deep

below T

. Above the critical temperature, there are quasiparticle excitations

at the Fermi level (at zero bias in Fig. 6.15a). As the temperature is lowered

through T

, quasiparticle excitations at the Fermi level, thus inside the gap,

are renormalized: the charge gap deepens at low bias, as shown in Fig. 6.15a.

Third group of superconductors: Mechanism of superconductivity 187

However, the maximum magnitude of the gap, in a ﬁrst approximation, is un-

changed. Analysis of ARPES data even suggests that a full gap opens up at

the Fermi level, as shown in Fig. 6.15b. The magnitude of this gap is slightly

smaller than the magnitude of ∆

, and its temperature dependence is reminis-

cent of the BCS temperature dependence (see references in [19]). If this is the

case, then the charge gap at T  T

has an anisotropic s-wave symmetry with

an anisotropy ratio of about 3 because ∆

cg,min

≈ ∆

and ∆

cg,max

 3∆

Thus, the ratio is ∆

cg,max

/∆

cg,min

≈ 3∆

/∆

= 3.

From Eqs. (6.5)–(6.7), the gap ratio 2∆

/(k

)  6 is sufﬁciently

large, indicating a strong coupling between the lattice and holes. As noted

above, the charge stripes represent self-trapped electronic states in the CuO

planes. In crystals, self-trapped states occur usually due to an interaction of

quasiparticles with acoustic phonons. The electron-phonon interaction in the

CuO

planes will be discussed separately.

Finally, it is worth to mention that in the ARPES spectrum shown in Fig.

6.14a, the dip that naturally occurs from a superposition of the quasiparticle

peak and the hump has no physical meaning. In ARPES and SIN-tunneling

spectra, this dip is situated at e|V |2∆

(p), while in SIS tunneling conduc-

tances, it appears at e|V |≈3∆

(p).

3.5.2 T

(p) temperature scale

The T

temperature scale shown in Fig. 6.11 manifests itself in resistivity,

NMR and speciﬁc-heat measurements. It originates from local antiferromag-

netic ordering in hole-poor matrix (clusters) depicted in Fig. 6.12. Therefore,

the energy scale related to the T

temperature scale, ∆

NMR

, has also the

magnetic origin. Figure 6.16 illustrates how the temperature T

is deter-

mined in resistivity measurements: T

is the temperature at which, upon

cooling, the resistivity deviates from a linear temperature dependence.

(a)

dI/dV

−∆

∆

(b)

−∆

∆

dI/dV

−∆

∆

T > T

T < T

Figure 6.15. Gap on the charge stripes above and deep below T

, inferred from (a) tunneling

and (b) ARPES measurements (see references in [19]). In both plots, the solid lines are slightly

shifted up for clarity.

188 ROOM-TEMPERATURE SUPERCONDUCTIVITY

300

(T)/

(300)

Temperature (K)

Figure 6.16. Pseudogap in resistivity measurements: T

is the temperature at which the

resistivity deviates on cooling from a linear dependence.

Why does the resistivity drop at T

? The temperature T

is the onset

temperature of local antiferromagnetic order in either the insulating matrix, if

p 0.19, or insulating clusters, if p → 0.19. In nanoscale clusters with charge

stripes, holes can move easily along the stripes. To sustain electrical current

on a macroscopic scale, quasiparticles must tunnel through the insulating ma-

trix (clusters). It is mistakenly believed that an antiferromagnetic ordering

blocks electron transport. The opposite is true: electrons can tunnel through a

thin layer of NiO

ordered antiferromagnetically about ﬁve times easier than

through a thin layer of non-magnetic metal oxides [46]. This is why the tem-

perature dependence of in-plane resistivity in the cuprates has a kink at T

the resistivity simply drops at T

3.6 Soliton-like excitations on charge stripes

Depending on temperature, quasiparticles in superconducting cuprates are

soliton-like excitations or pairs of these excitations, observed experimentally

in tunneling measurements [16–19]. In the CuO

planes, the soliton-like ex-

citations appear on the charge stripes immediately after the formation of the

stripes, thus, somewhat below T

. They become paired at T

pair

shown in the

phase diagram in Fig. 6.11. Consider ﬁrst the elementary excitations; the pairs

will be discussed in the following subsection.

In the cuprates the soliton-like excitations propagate in the middle of the

charge gap present on the stripes, as depicted in Fig. 6.17. Since these exci-

tations have the quantum numbers of an electron, they are fermions. Then, a

question naturally rises: How can the fermions occupy the same energy level?

There are two possible explanations to this fact: Either there is the spin-charge

Third group of superconductors: Mechanism of superconductivity 189

separation on the stripes or, between T

and T

pair

, there is no connected

Fermi surface in the cuprates yet. Theoretically the spin-charge separation is

indeed possible in one-dimensional systems, leading to the formation of the so-

called Luttinger liquid. However, an experimental veriﬁcation of this hypothe-

sis, carried out in underdoped YBCO, showed that, within the resolution limits

of the experiment, there is no spin-charge separation in YBCO [47]. Hence, it

is most likely that, at T

pair

<T <T

, there is no connected Fermi surface

in the cuprates. This means that, at high temperatures, the charge stripes in the

CuO

planes exist independently of each other. For example, in LSCO, upon

cooling the chemical potential literally jumps at about 250 K. Thus, if this is

the case, it is then possible that the soliton-like excitations become paired at

pair

because they are required to. When the Fermi surface in the cuprates be-

comes connected somewhat above T

pair

, the soliton-like excitations being at

the same energy level must either disappear, except for two, or be paired. The

paired quasiparticles are bosons, therefore, the Pauli exclusion principle is no

longer applicable to them.

What type of excitations can occur on the charge stripes in the cuprates?

Let us consider all possible excitations on a 2k

charge stripe sketched in Fig.

6.9. In the case of hole-doped cuprates, we are exclusively interested in the

stripe excitations with a charge of +|e|, where e is the electron charge. Figure

6.18 shows schematically three types of excitations on the 2k

charge stripes:

a soliton, a kink-up and a kink-down. In Fig. 6.18, the stripe excitations are

shown at rest; however, in reality, they are dynamic and propagate along the

stripes which themselves ﬂuctuate in the CuO

planes. As suggested by Za-

valence band

conduction band

electrosoliton

∆

Figure 6.17. Energy level of soliton-like excitations that reside on charge stripes in the CuO

planes at high temperatures. ∆

is the magnitude of the charge gap on the stripes.

190 ROOM-TEMPERATURE SUPERCONDUCTIVITY

...

(a)

(b)

...

(c)

Figure 6.18. Sketch of three types of excitations on 2k

charge stripes, shown at rest: (a)

soliton; (b) kink-up, and (c) kink-down.

anen and co-workers, the kink-up and the kink-down may have different spin

orientations.

Such a description of soliton-like excitations in the cuprates is rather phe-

nomenological. In a ﬁrst approximation, they can also be described mathemat-

ically in the framework of the Davydov theory of electrosolitons and bisoli-

tons for one-dimensional systems [3, 7, 9, 10]. Let us consider brieﬂy the

electrosoliton model. As discussed in Chapter 1, solitons are nonlinear ex-

citations which are localized in space. In some systems, they represent self-

localized states (i.e. self-trapped states). In solids, if the particle-ﬁeld inter-

action (electron-phonon coupling) is strong, both the particle wavefunction

and the lattice deformation will be localized. In the three-dimensional case,

this localized entity is known as a (Holstein) polaron and, in one dimension,

as a Davydov soliton (also known as an electrosoliton or polaronic soliton).

Their integrity is maintained owing to a dynamical balance between dispersion

(exchange inter-site interaction) and nonlinearity (electron-phonon coupling).

Figure 6.19 schematically shows a self-trapped state of a particle (small po-

laron or Davydov soliton). In a self-trapped state, both the particle wavefunc-

tion and the lattice deformation are localized.

Particle wave-function

Potential well

Figure 6.19. Sketch of a self-trapped state: a particle (electron or hole) with a given wave-

function deforms the lattice inducing a potential well which in turn traps the particle.

Third group of superconductors: Mechanism of superconductivity 191

To describe the self-focusing phenomena, the nonlinear Schro¨dinger (NLS)

equation is used,



i¯h

∂

∂t

¯h

∂

∂x

+ G|ψ(x, t)|



ψ(x, t)=0. (6.11)

The equation is written in the long-wave approximation when the excitation

wavelength λ is much larger than the characteristic dimension of discreteness

in the system, i.e. under the condition ka =2πa/λ  1. The equation

describes the complex ﬁeld ψ(x, t) with self-interaction. The function |ψ|

determines the position of a quasiparticle of mass m. The second term in the

NLS equation is responsible for dispersion, and the third one for nonlinearity.

The coefﬁcient G characterizes the intensity of nonlinearity.

When the nonlinearity is absent (G = 0), the NLS equation has solutions in

the form of plane waves,

ψ(x, t)=Φ

exp[i(ka − ω(k)t)], (6.12)

with the square dispersion law ω(k)=¯hk

/2m. With nonlinearity (G = 0)

in the system having a translational invariance, the excited states move with

constant velocity v. Therefore, it is convenient to study solutions of the NLS

equation in the reference frame

ζ =(x − vt)/a, (6.13)

moving with constant velocity. In this reference frame the NLS equation has

solutions in the form of a complex function

ψ(x, t)=Φ(ζ) exp[i(kx − ωt)],k= mv/¯h, (6.14)

where the function Φ(ζ) is real.

The self-trapping effect occurs when two linear systems interact with one

another. Consider an excesselectron in a quasi-one-dimensionalatomic (molec-

ular) chain. If neutral atoms (molecules) are rigidly ﬁxed in periodically ar-

ranged sites na of a one-dimensional chain, then due to the translational in-

variance of the system, the lowest energy states of an excess electron are deter-

mined by the conduction band. The latter is caused by the electron collectiviza-

tion. In the continuum approximation, the inﬂuence of a periodic potential is

taken into account by replacing the electron mass m

by the effective mass

m =¯h

/(2a

J) which is inversely proportional to the exchange interaction

energy J that characterizes the electron jump from one node site into another.

In this approximation, the electron motion along an ideal chain corresponds to

the free motion of a quasiparticle with an effective mass m and the electron

charge.

192 ROOM-TEMPERATURE SUPERCONDUCTIVITY

Taking into account small displacements of molecules of mass M ( m)

from their periodic equilibrium positions, there arises the short-range defor-

mation interaction of quasiparticles with these displacements. When the defor-

mation interaction is rather strong, the quasiparticle is self-localized. The local

displacement caused by a quasiparticle manifests itself as a potential well that

contains the particle, as schematically shown in Fig. 6.19. In turn, the quasi-

particle deepens the well.

A self-trapped state can be described by two coupled differential equations

for the ﬁeld ψ(x, t) that determines the position of a quasiparticle, and the ﬁeld

ρ(x, t) that characterizes a local deformation of the chain and determines the

decrease in the relative distance a → a − ρ(x, t) between molecules of the

chain,



i¯h

∂

∂t

¯h

∂

∂x

+ σρ(x, t)



ψ(x, t)=0, (6.15)



∂

∂t

− c

∂

∂x



ρ(x, t) −

∂

∂x

|ψ(x, t)|

=0. (6.16)

The ﬁrst equation characterizes the motion of a quasiparticle in the local de-

formation potential U = −σρ(x, t). The second equation determines the ﬁeld

of a local deformation caused by a quasiparticle. The two equations are con-

nected through the parameter σ of the interaction between a quasiparticle and

a local deformation. The velocity c

= a



k/M is the longitudinal sound ve-

locity in the chain with elasticity coefﬁcient k. In the case of one quasiparticle

in the chain, the function ψ(x, t) is normalized by

∞



−∞

|ψ(x, t)|

dx =1. (6.17)

In the reference frame ζ =(x −vt)/a moving with constant velocity v, the

following equality ∂ρ(x, t)/∂t = −v/a×∂ρ/∂ζ holds. Then, the solution for

ρ(x, t) has the form

ρ(x, t)=

k(1 − s

)

|ψ(x, t)|

, if s

= v

 1. (6.18)

Substituting the expression for ρ(x, t) into Eq. (6.15), we obtain the follow-

ing nonlinear equation for the function ψ(x, t),



i¯h

∂

∂t

¯h

∂

∂x

+2gJ|ψ(x, t)|



ψ(x, t)=0. (6.19)

where

g ≡

2k(1 − s

(6.20)

Third group of superconductors: Mechanism of superconductivity 193

is the dimensionless parameter of the interaction of a quasiparticle with local

deformation. Substituting the function

ψ(x, t)=Φ(ζ) exp[i(kx − ωt)],k= mv/¯h (6.21)

into Eq. (6.19) we get the equation

[¯hω −

− J

∂

∂ζ

+2gJΦ

(ζ)]Φ(ζ)=0, (6.22)

for the amplitude function Φ(ζ) normalized by



(ζ)dζ =1. The solution

of this equation is

Φ(ζ)=

√

g × sech(gζ/2), (6.23)

with the dispersion law

¯hω =

−

J. (6.24)

The last term in Eq. (6.24) determines the binding energy of the particle and

the chain deformation produced by the particle itself. According to Eq. (6.23),

the quasiparticle is localized in a moving reference frame

∆ζ =2π/g, (6.25)

as shown in Fig. 6.20. In this region, the ﬁeld localization is characterized by

the function

ρ(ζ)=

gσ

4k(1 − s

)

sech

(gζ/2). (6.26)

The following energy is necessary for the deformation:

W =

k(1 + s

)



(ζ)dζ =

J(1 + s

). (6.27)

(

)

Figure 6.20. Size of a self-trapped soliton is about 2π/g (compare with Fig. 2.4).