On temperature versus doping phase diagram of high critical temperature superconductors
An attempt to describe the bell-shape dependence of the critical temperature of high-T<sub>c</sub> superconductors on the charge carrier density is made. It is proposed to explain its linear increase in the region of small densities (underdoped regime) by the role of the order para...
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| Zitieren: | On temperature versus doping phase diagram of high critical temperature superconductors / V.M. Loktev, V.M. Turkowski // Condensed Matter Physics. — 1998. — Т. 1, № 1(13). — С. 113-124. — Бібліогр.: 29 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1188272025-02-10T00:51:50Z On temperature versus doping phase diagram of high critical temperature superconductors До фазової діаграми “температура -легування” надпровідників з високою критичною температурою Loktev, V.M. Turkowski, V.M. An attempt to describe the bell-shape dependence of the critical temperature of high-T<sub>c</sub> superconductors on the charge carrier density is made. It is proposed to explain its linear increase in the region of small densities (underdoped regime) by the role of the order parameter phase 2D fluctuations which become less at this density growth. The critical temperature suppression in the region of large carrier densities (overdoped regime) is connected with the appearance (because of doping) of an essential damping of long-wave bosons which, within the framework of the model proposed, define the mechanism of indirect inter-fermion attraction. Зроблена спроба опису дзвоноподібної залежності критичної температури високотемпературних надпровідників від густини носіїв заряду. Пропонується пояснити її лінійне зростання в області малих густин (недолегований режим) роллю 2D флуктуацій фази параметра порядку, які стають меншими при зростанні густини. Подавлення критичної температури в області великих густин носіїв (перелегований режим) пов’язаний з появою (завдяки легуванню) суттєвого затухання довгохвильових бозонів, які в рамках запропонованої моделі визначають механізм непрямого між-ферміонного притягання. We are glad to dedicate this paper to a well-known Ukrainian theorist, corresponding member of the National Academy of Sciences of Ukraine, Prof. I.V.Stasyuk whose numerious critical remarks are always interesting and helpful for us. 1998 Article On temperature versus doping phase diagram of high critical temperature superconductors / V.M. Loktev, V.M. Turkowski // Condensed Matter Physics. — 1998. — Т. 1, № 1(13). — С. 113-124. — Бібліогр.: 29 назв. — англ. 1607-324X DOI:10.5488/CMP.1.1.113 PACS: 67.20.+k, 74.20.-z, 74.25.-q, 74.72.-h https://nasplib.isofts.kiev.ua/handle/123456789/118827 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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English |
| description |
An attempt to describe the bell-shape dependence of the critical temperature
of high-T<sub>c</sub> superconductors on the charge carrier density is made. It is
proposed to explain its linear increase in the region of small densities (underdoped
regime) by the role of the order parameter phase 2D fluctuations
which become less at this density growth. The critical temperature suppression
in the region of large carrier densities (overdoped regime) is connected
with the appearance (because of doping) of an essential damping
of long-wave bosons which, within the framework of the model proposed,
define the mechanism of indirect inter-fermion attraction. |
| format |
Article |
| author |
Loktev, V.M. Turkowski, V.M. |
| spellingShingle |
Loktev, V.M. Turkowski, V.M. On temperature versus doping phase diagram of high critical temperature superconductors Condensed Matter Physics |
| author_facet |
Loktev, V.M. Turkowski, V.M. |
| author_sort |
Loktev, V.M. |
| title |
On temperature versus doping phase diagram of high critical temperature superconductors |
| title_short |
On temperature versus doping phase diagram of high critical temperature superconductors |
| title_full |
On temperature versus doping phase diagram of high critical temperature superconductors |
| title_fullStr |
On temperature versus doping phase diagram of high critical temperature superconductors |
| title_full_unstemmed |
On temperature versus doping phase diagram of high critical temperature superconductors |
| title_sort |
on temperature versus doping phase diagram of high critical temperature superconductors |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
1998 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/118827 |
| citation_txt |
On temperature versus doping phase diagram of high critical temperature superconductors / V.M. Loktev, V.M. Turkowski // Condensed Matter Physics. — 1998. — Т. 1, № 1(13). — С. 113-124. — Бібліогр.: 29 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT loktevvm ontemperatureversusdopingphasediagramofhighcriticaltemperaturesuperconductors AT turkowskivm ontemperatureversusdopingphasediagramofhighcriticaltemperaturesuperconductors AT loktevvm dofazovoídíagramitemperaturaleguvannânadprovídnikívzvisokoûkritičnoûtemperaturoû AT turkowskivm dofazovoídíagramitemperaturaleguvannânadprovídnikívzvisokoûkritičnoûtemperaturoû |
| first_indexed |
2025-12-02T07:30:48Z |
| last_indexed |
2025-12-02T07:30:48Z |
| _version_ |
1850380803708551168 |
| fulltext |
Condensed Matter Physics, 1998, Vol. 1, No 1(13), p. 113–124
On temperature versus doping phase
diagram of high critical temperature
superconductors
V.M.Loktev 1 , V.M.Turkowski 2
1 Bogolyubov Institute for Theoretical Physics of the National Academy of
Sciences of Ukraine, 14-b Metrologichna St. UA-252143 Kyiv, Ukraine
2 Shevchenko Kyiv University, 6 Acad. Glushkova Av., UA-252127 Kyiv,
Ukraine
Received March 6, 1998
An attempt to describe the bell-shape dependence of the critical tempera-
ture of high- Tc superconductors on the charge carrier density is made. It is
proposed to explain its linear increase in the region of small densities (un-
derdoped regime) by the role of the order parameter phase 2D fluctuations
which become less at this density growth. The critical temperature sup-
pression in the region of large carrier densities (overdoped regime) is con-
nected with the appearance (because of doping) of an essential damping
of long-wave bosons which, within the framework of the model proposed,
define the mechanism of indirect inter-fermion attraction.
Key words: high-temperature superconductors, doping, critical
temperature, phase diagram, pseudogap
PACS: 67.20.+k, 74.20.-z, 74.25.-q, 74.72.-h
1. Introduction
In spite of the intensive study of high-Tc superconductors (HTSCs), the com-
prehension of many regularities which define their physical behaviour has not been
achieved. First of all, it is put down to the HTSC normal state properties, the de-
scription of which within the framework of the standard Fermi-liquid theory proves
to be impossible if the carrier concentration nf in the corresponding samples is
such that the resulting critical temperature Tc in them is less than some optimal
value Tmax
c characteristic of every compound. The latter appears in all (practically,
without exception) HTSC copper oxides because, as it is well-known (see, for ex-
ample, review [1]), the conductivity (consequently - superconductivity) of these
initially quasi-2D antiferromagnetic insulators results from their doping by donor
(Nd, Pr) or acceptor (Sr, Ba, O) ions.
c© V.M.Loktev, V.M.Turkowski 113
V.M.Loktev, V.M.Turkowski
Then, temperature Tc being formed within the increase of the carrier density
(or, in other words, Fermi energy) in a system also rises. This growth, however,
rather quickly stops and, after some (not large upon nf values) part of “satura-
tion”, a drop to zero of the function Tc(nf) follows, which thus acquires a bell-shape
form. As a result, HTSC compounds with a relatively large concentration of itiner-
ant carriers (the so-called overdoped regime) become non-superconducting metals
the behaviour of which to a certain degree can be explained on the basis of the
conventional theory of Fermi liquid [2]. The corresponding carrier concentration
values for the underdoped regime are such that Tc(nf ) increase is observed for
nf from ≃ 0.04 − 0.08 carrier per CuO2 layer cell (i.e. from the threshold of the
insulator-metal transition) to Tmax
c ≃ Tc(0.15−0.18); temperature Tc(nf ) becomes
zero when nf ≃ 0.25 − 0.28 [3–5]. If the initial part of function Tc(nf), where
dTc(nf)/dnf ≃ const > 0 [6,7], can be (at least qualitatively) interpreted (see [8–
10] and review [11]) proceeding from the theory of crossover from the Bose-Einstein
condensation of separate composite bosons (local pairs) and their superfluidity to
the superconductivity of the BCS type, then the reason of Tc(nf) suppression in
the BCS carrier concentration region remains, in fact, unknown yet.
From the very beginning the above-mentioned behaviour of Tc(nf ) was ascribed
to strong electron-electron correlations (see, for example, book [12]) and to the
filling of usually narrow Hubbard subbands by itinerant carriers. This question
was intensively investigated by Prof. I.V.Stasyuk and his collaborators [13–16].
However, a little bit later it becomes evident that carrier concentrations corre-
sponding to the disappearance of superconductivity are so small that such a filling
of the whole valent (in fact, conduction) band in HTSCs is impossible. The alter-
native version of the HTSC effect disappearance most consistently considered in
[17,18] (see also [1,2]) consists in the assumption that a bare Fermi level in HTSCs
proves to be in the vicinity of the extended saddle point of the electronic spectrum.
According to this point of view, at nf growth the Fermi energy ǫF ≃ k2
F/2m (kF
– Fermi momentum, m – effective mass) passes the van Hove singularity in the
conduction band density of states and, hence, Tc(nf ) reveals its maximum. If it
were so, function Tc(nf) being similar to a BCS one, would have an exponential
dependence on nf far from Tmax
c which, as it is seen from the experiments and
has been already mentioned, is not confirmed by observations. On the other hand,
the very appearance of the van Hove spectrum singularity requires fine-tuning
(for example, because of a rather definite ratio between the hole next-near and
the nearest hopping constants) which seems to be hard to achieve even in one
compound, to say nothing of many.
At the same time less attention was paid to the idea that the weakness of
superconductivity can be caused by a “feedback” effect, or the changes in the
spectrum ω(k) of intermediate bosons, the exchange of which results in fermion
pairing. These bosons in HTSCs can be attributed to phonons (similarly to the
BCS-Eliashberg model) the role of which is actively advocated by Ginzburg (see,
for example, his review [19]), magnons, or spin fluctuations, [20,21], quadripole
dd-excitons (the Gaididei-Loktev-Weber mechanism (see [1])), plasmons [22], etc.
114
On temperature versus doping
It must be noted, however, that, if a phonon spectrum does not at all (or very
weakly) depend on doping, then long-wave excitations of the magnetic subsystem
(dd-excitons, as excitations over a magnetic background, can also be considered
here) are strongly suppressed because of the long-range magnetic order destruction
in the metallic phase of HTSCs [1,12].
Namely, spin waves are subjected to a most appreciable doping effect; their
low-frequency region (for the wave vectors less than some characteristic value
kmin) acquires a diffusion form, or becomes overdamped. In the insulating phase of
HTSCs, as it is shown in [23], kmin ∼ nf ; in their metallic phase the magnon damp-
ing becomes even stronger and kmin ≃ 2kF ∼ n
1/2
f [24,1], which is completely in line
with the degradation of the correlation magnetic length ξmag ∼ n
−1/2
f measured in
HTSCs [25]. In that way the long-wave damping γ(k) increases up to such high
values (γ(k) ≫ ω(k)) that corresponding intermediate bosons (most probably,
spin fluctuations) cannot participate in interaction transfer, “being out”.
The solution of the self-consistent magneto-electronic problem as a whole (i.e.
the explicit solution of the equation of superconductivity similar to the Eliash-
berg equation) is scarcely possible now. Therefore, an attempt is made below to
consider the simplest model with an indirect inter-fermion attraction provided by
intermediate massive bosons with a long-wave cut which is proportional to kF. The
carriers are supposed to appear in the system due to doping which corresponds
to a generally accepted scenario of metallization of copper oxides. For the sake of
simplicity we shall omit any other dampings (in particular, the carrier damping
because of the disorder the effect of which was analysed, for example, in [26]).
Following [9,11], we shall also suppose that in a model 2D system (in fact,
all HTSCs can be with good accuracy referred to this kind), the superconducting
condensate is formed in a way principally different from the ordinary one. In such
a case one has to distinguish the order parameter formation temperature from
the real (observable) critical temperature in a sense that the absolute value of the
former does not become zero at and above Tc.
2. Model and main equations
The model Hamiltonian density of the electron-phonon system can be written
in the well-known form:
H(x) = −ψ†
σ(x)
(
∇2
2m
+ µ
)
ψσ(x) +Hph(ϕ(x))
+gphψ
†
σ(x)ψσ(x)ϕ(x), x = r, t, (2.1)
where Hph is the Hamiltonian of free phonons which will be described more pre-
cisely below; ψσ(x) and ϕ(x) are the fermion and boson field operators, respec-
tively; m is the effective mass of the Fermi-particles, σ =↑, ↓ - their spin variable,
and gph is the electron-phonon coupling constant; µ in (2.1) is the chemical poten-
tial of fermions which fixes their average density in the system; we put h̄ = kB = 1.
115
V.M.Loktev, V.M.Turkowski
As it was mentioned in the Introduction, we shall model a boson-exchange
interaction taking into account the dependent on doping saturation of long-wave
bosons. Under the above mentioned assumptions about a gradual non-participation
of a part of the bosons in the attracting interaction formation, the simplest way
to describe it is by using a free phonon propagator in the form:
D(ω,k) = − 2ω(k)
ω2 − ω2(k) + iδ
θ(k − kmin), δ → 0, (2.2)
where ω(k) is the boson dispersion law, and θ(k) is the step function. As it was
pointed out above, kmin in (2.2) is some characteristic wave vector which sepa-
rates the region of overdamped (k < kmin) and long-lived (k > kmin) intermediate
bosons (here: phonons by definition). Although the magnetic correlation length
measurements result in kmin = 2kF, we shall adopt a more general (or soft) rela-
tion supposing that kmin = αkF ≡ α
√
2mǫF where α is some free parameter.
It is very important that the Hamiltonian (2.1) is invariant with respect to the
global symmetry transformations
ψσ(x) → ψσ(x)e
iθ, ψ†
σ(x) → ψ†
σ(x)e
−iθ, (2.3)
which in a 2D case (unlike a 3D one) remain unbroken and the phase transition is
here accompanied by a change in the correlation function behaviour only.
The T − nf phase diagram of the system can be calculated by using the
Hubbard-Stratonovich method generalized by the case of a non-local (indirect)
interaction (the so-called auxiliary bilocal field method). For finding the grand
partition function Z it is useful to pass to Nambu spinors: Ψ† = (ψ†
↑, ψ↓) and its
conjugated one. After performing an integration over bosonic fields it is easy to
obtain the Lagrangian
L = Ψ†(x)[−∂τ + (
∇2
2m
+ µ)τz]Ψ(x)
−1
2
Ψ(x1)Ψ
†(y1)τzK(x1, y1; x2, y2)Ψ(x2)Ψ
†(y2)τz (2.4)
of the system where an integration over repeated indices is supposed. The kernel
K in (2.4) will be defined below.
Let us introduce the pairing order parameter
φ(x1, y1) = K(x1, y1; x2, y2)Ψ(x2)Ψ
†(y2)τz
≡ τ+φ(x1, y1) + τ−φ
∗(x1, y1), (2.5)
where τ+ = 1
2
(τx + iτy), τ− = 1
2
(τx − iτy) (and τz in (2.4)) are the Pauli matrices.
Then, adding a zero term to the Lagrangian L
1
2
[φ(x1, y1)−K(x1, y1; x
′
1, y
′
1)Ψ(x′1)Ψ
†(y′1)τz]K
−1(x1, y1; x2, y2)[φ(x2, y2)
−K(x2, y2; x
′
2, y
′
2)Ψ(x′2)Ψ
†(y′2)τz]
116
On temperature versus doping
for the purpose of cancelling the four-fermion interaction, one comes to the ex-
pression
L(x1, y1; x2, y2) = Ψ†(x1)
[
−∂τ +
(
∇2
2m
+ µ
)
τz − τ+φ(x1, y1)
−τ−φ∗(x1, y1)]Ψ(y1) +
1
2
φ(x1, y1)K
−1(x1, y1; x2, y2)φ(x2, y2) (2.6)
for the Lagrangian needed. The Fourier transformation of K can be written as
K(x1, y1; x2, y2) =
∫
d3Pd3p1d
3p2
(2π)9
KP (p1; p2) exp
[
−iP
(
x1 + y1
2
− x2 + y2
2
)
−ip1(x1 − y1)− ip2(x2 − y2)] ,
(pi = (pi, ωi) where i = 1, 2 and P = (P , ω) are the relative and the centre of mass
momenta, respectively). Supposing now that KP (p1; p2) is P - independent we pass
to the standard kernel form
K(p1; p2) = g2phD(p1 − p2), (2.7)
which corresponds to the indirect inter-fermion interaction.
The partition function can be written as
Z =
∫
DΨ†DΨDφDφ∗ exp
[
−β
∫
L(Ψ†,Ψ, φ∗, φ)dxdy
]
≡
∫
DφDφ∗ exp(−βΩ[G]), β ≡ 1/T,
In the last expression Ω[G] is a thermodynamic potential which in the “leading
order” on gph has the form:
βΩ[G] = −Tr
[
LnG−1 +
1
2
(φK−1φ)
]
, (2.8)
in which
G−1 = −
[
∂τ −
(
∇2
2m
+ µ
)
τz − τ+φ− τ−φ
∗
]
(2.9)
is a full fermion Green function. After the direct minimization of the potential
(2.8) it is easy to obtain an equation for the auxiliary φ-field (or order parameter):
δΩ/δφ = 2φ− tr
∫
d2kdω
(2π)3
K(p;k, ω)G(k, ω)τx = 0. (2.10)
Using (2.10) one can easily arrive at the well-known Cornwell-Jackiw-Tombou-
lis formula for the effective action in the one-loop approximation [27]:
βΩ(G) = −TrLnG−1 +
1
2
TrGKG,
117
V.M.Loktev, V.M.Turkowski
or, taking into account (2.9) (or (2.10)),
βΩ(G) = −Tr
[
LnG +
1
2
(GG−1
0 − 1)
]
. (2.11)
To investigate the possibility of the condensate formation in a 2D system it is
convenient, in accordance with [9], to pass to a modulus-phase parametrization of
the order parameter (cf. (2.3), where θ = const):
φ(x, y) = ρ(x, y) exp[−i(θ(x) + θ(y))/2] (2.12)
with a simultaneous Nambu spinor transformation
Ψ†(x) = χ†(x) exp[iθ(x)τz/2], (2.13)
corersponding to “separation” of bare fermions on their neutral χ(x) and charge
θ(x) parts (fermi- and bose-ones, respectively).
In the approximation that ρ(x, y) = ρ = const (see [28]) and spatial θ-fluctua-
tions are small, one can (using (2.12) and (2.13)) obtain the following expressions
for G and Ω which are defined in (2.9) and (2.11):
G−1 = −
[
∂τ − τz
(
∇2
2m
+ µ
)
+ iτxρ
−τz
(
∂τθ +
∇θ2
2m
)
− i
(
∇2θ
2m
+
∇θ∇
m
)]
≡ G−1(ρ)− Σ(∂θ) (2.14)
and Ω = Ωkin(ρ,∇θ) + Ωpot(ρ) with Ωpot(ρ) which is defined by (2.11) at ∇θ = 0
and
βΩkin(ρ,∇θ) = Tr[GΣ−G0Σ +
1
2
GΣGΣ− 1
2
G0ΣG0Σ
+τx
1
2
iρG(GΣ+GΣGΣ)] =
T
2
∫ β
0
dτ
∫
d2rJ(µ, T, ρ(µ, T ))(∇θ)2, (2.15)
where the effective neutral fermion stiffness
J(µ, T, ρ(µ, T )) =
1
8π
(
√
µ2 + ρ2 + µ+ 2T ln
[
1 + exp
(
−
√
µ2 + ρ2
T
)
)])
− T
4π
[
1− ρ2
4T 2
∂
∂(ρ2/4T 2)
]
∫ ∞
−µ/2T
dx
x+ µ/2T
cosh2
√
x2 + ρ2/4T 2
(2.16)
was introduced.
The evident analogy with the XY-model (two-component order parameter in
2D space) gives an equation for temperature TBKT of the Berezinskii-Kosterlitz-
Thouless phase transition in the system, namely, (see Chapter 15 in book [29]):
π
2
J(µ, TBKT, ρ(µ, TBKT)) = TBKT. (2.17)
118
On temperature versus doping
(Remember that temperature TBKT plays the role of a critical one in 2D metals.)
The parameters µ and ρ in (2.17), being dependent on T , are still unknown;
therefore, it is necessary to obtain equations which connect them with carrier
density nf . The first one follows from (2.10) for ρ 6= 0:
1 = T
∞
∑
m=−∞
∫ d2k
(2π)2
K(ωm)
ω2
m + ξ2(k) + ρ2
, (2.18)
where ωn = (2n+1)πT is the Matsubara fermion frequencies, ξ(k) = k2/2m−µ and
the Einstein model was used for phonon dispersion: ω(k) = ω0. The dependence
on parameter α in equation (2.18) is preserved through kernel K (see (7) and (2)).
The second one is defined by the condition V −1∂Ωpot(ρ)/∂µ = −nf (V is the
volume of the system) which results in the well-known number equation:
√
µ2 + ρ2 + µ+ 2T ln
[
1 + exp
(
−
√
µ2 + ρ2
T
)]
= 2ǫF, (2.19)
where the equality ǫF = πnf/m was used; it is correct for free 2D fermions with
the quadratic dispersion law.
Thus, we have obtained a self-consistent set of equations (17)-(19) needed to
investigate the phase diagram of a 2D metal with an arbitrary carrier density. The
last parameter ρ defines such metal superconducting properties.
3. Phase diagram of a system
As it can be seen from the previous Section (see also [9,11]), there exist two
characteristic temperatures in a system: Tρ, where formally the complete order
parameter given by equation (2.5) arises, but its phase is a random quantity, i.e.
〈φ(x, y)〉 = 0, and another one, TBKT < Tρ, where the phase of the order parameter
becomes ordered, so that 〈φ(x, y)〉 6= 0. It must be, however, stressed that tem-
perature Tρ is not a real critical temperature; it only denotes the characteristic
region where the modulus of the order parameter achieves its maximal growth at
T decreasing. Unlike Tρ, temperature TBKT does correspond to the phase transi-
tion when correlators 〈exp[iθ(r, τ)] exp[iθ(r′)]〉 as functions of |r−r′| change their
behaviour [29].
Let us find the nf -dependence of temperatures Tρ and TBKT. “Effective” tem-
perature Tρ can be estimated from the set (2.16)-(2.19) in the mean-field approx-
imation by putting ρ = 0. Another temperature TBKT follows from the equations
(17) and (19).
It is impossible to solve the equations obtained analytically, so we shall do that
by numerical calculation. Nevertheless, some asymptotical expressions for these
temperatures as functions of nf can be found in the analytical form:
i) at ǫF/ω0 → 0 one obtains TBKT = ǫF/8, and Tρ satisfies the simple mean-field
equation Tρ ln(Tρ/ǫF) = ω0 exp(−2/λ), where λ = g2phm/2π is the dimensionless
coupling constant.
119
V.M.Loktev, V.M.Turkowski
ii) ǫF → ǫcrF ; the critical point ǫF = ǫcrF (or nf = ncr
f at which Tρ = TBKT = 0
can be found from equations (17)-(19). This unknown energy is the solution of the
equation
1 =
λ
2
ln
(W − ǫF)[(α− 1)ǫF + ω0]
(α− 1)ǫF(W − ǫF + ω0)
→ λ
2
ln
(α− 1)ǫF + ω0
(α− 1)ǫF
|W→∞. (3.1)
(W is a conduction bandwidth determined by the evident condition: W = k2
B/2m,
where kB is the Brillouin wave vector). In other words, it follows from (2.18) that
because the long waves phonons begin to quit the interaction transfer, there exists
(at α > 1 only), a point where both Tρ and TBKT temperatures become zero,
which means that superconductivity (but not conductivity) is suppressed. Near
this point the temperatures have the following behaviour (W → ∞):
Tρ = ω0/ ln
4ǫcrF [(α− 1)ǫcrF + ω0]
λω0(ǫcrF − ǫF)
;
TBKT = ǫcrF
9
144
(α− 1)4
(
λω0(ǫ
cr
F − ǫF)
ǫcrF [(α− 1)ǫcrF + ω0]
)2
1/5
.
The results of the numerical calculations of equations (17)-(19) are shown in
figure 1.
It is seen that due to the long-wave phonon suppression (in fact, supposed to
be strong), functions Tρ(nf ) and TBKT(nf) quickly acquire a bell-like shape at nf
variation. This “bell” proves to be non-symmetrical, and its height, width and to
some extent form depend on α(> 1). In a certain sense, such a result is surprising
because, as it is generally accepted, not long-, but short-wave intermediate bosons
play the main role in the attraction which appears in the BCS-Eliashberg model
due to the electron-phonon (or any other boson) interaction. The sensitivity of
superconducting properties of a 2D metal to a long-wave part of the intermediate
boson spectrum is rather unusual and allows one to hope that a more accurate
consideration also results in the similar effect.
4. Conclusions
The existence of two different temperatures in underdoped HTSCs is now a
well established fact. They, as it is shown in many papers (see review [11]), are
one of the consequences of two-dimensionality of their electronic and magnetic
properties when the “ordering” of the order parameter modulus and phase takes
place at different temperatures. In a pure 2D system temperature TBKT, as it was
mentioned, has to be considered as a critical one, and in the region TBKT < T < Tρ
the so-called pseudogap and also a normal phase are formed in underdoped HTSCs.
It is destroyed when T > Tρ or nf becomes so large (overdoped regime) that the
chemical potential of fermions and the Fermi energy are indistinctive (µ = ǫF).
The aim of this paper is to demonstrate that in a superconducting system with
120
On temperature versus doping
0
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
T/TMF
BCS
ǫF/ω0
α=0
0
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
T/TMF
BCS
ǫF/ω0
α=0.1
0
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
T/TMF
BCS
ǫF/ω0
α=1.1
Figure 1. The characteristic patterns of the T − nf phase diagram of 2D metal
with coupling constant λ = 1. Doted and solid lines define temperatures Tρ and
TBKT, respectively.
121
V.M.Loktev, V.M.Turkowski
an indirect attraction the role of long-wave bosons (for instance, phonons, spin
fluctuations and so on) can be crucial in the region where the standard Fermi
liquid theory becomes applicable.
In spite of some qualitative similarity between the experimental and the ob-
tained pictures, the considered model is so simple (and even rough because prop-
agator (2) is, in fact, postulated, while it must contain the boson damping only)
that any quantitative use of it is almost senseless. Therefore, it must be generalized
by taking into account such HTSC features as: quasi-two-dimensionality (which
results in real Tc), intermediate boson dispersion ω(k) (as it takes place for spin
fluctuations) and damping γ(k); pairing anisotropy, etc. These problems will be
considered separately.
Acknowledgements
We are glad to dedicate this paper to a well-known Ukrainian theorist, corre-
sponding member of the National Academy of Sciences of Ukraine, Prof. I.V.Sta-
syuk whose numerious critical remarks are always interesting and helpful for us.
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До фазової діаграми “температура -легування”
надпровідників з високою критичною температурою
В.М.Локтєв 1 , В.М.Турковський 2
1 Інститут теоретичної фізики ім.М.М.Боголюбова НАН України,
252143 м. Київ, вул. Метрологічна, 14б
2 Київський національний університет ім. Т.Шевченка
252127 м. Київ, просп, акад. Глушкова, 6
Отримано 6 березня 1998 р.
Зроблена спроба опису дзвоноподібної залежності критичної темпе-
ратури високотемпературних надпровідників від густини носіїв заря-
ду. Пропонується пояснити її лінійне зростання в області малих густин
(недолегований режим) роллю 2D флуктуацій фази параметра по-
рядку, які стають меншими при зростанні густини. Подавлення кри-
тичної температури в області великих густин носіїв (перелегований
режим) пов’язаний з появою (завдяки легуванню) суттєвого затухан-
ня довгохвильових бозонів, які в рамках запропонованої моделі ви-
значають механізм непрямого між-ферміонного притягання.
Ключові слова: високотемпературні надпровідники, легування,
критична температура, фазова діаграма, псевдограф
PACS: 67.20.+k, 74.20.-z, 74.25.-q, 74.72.-h
124
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