On the theory of high-Tc superconductivity of doped cuprates
A theoretical analysis is presented on possible effects of disorder by dopants in high-temperature superconducting cuprate perovskites, to define their basic spectra of spin and electronic excitations, and the subsequent observable properties, especially doping dependence of superconducting order p...
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irk-123456789-1574572019-06-21T01:28:30Z On the theory of high-Tc superconductivity of doped cuprates Pogorelov, Y.G. Loktev, V.M. A theoretical analysis is presented on possible effects of disorder by dopants in high-temperature superconducting cuprate perovskites, to define their basic spectra of spin and electronic excitations, and the subsequent observable properties, especially doping dependence of superconducting order parameter. The central point in the proposed physical picture is formation of specific impurity subband within the insulating bandgap of initial undoped material, serving as a source for the system metallization and further transition into superconducting state with anisotropic order parameter. Здiйснено теоретичний аналiз можливих ефектiв, спричинених у високотемпературних надпровiдних мiдних перовскiтах присутнiстю допантiв. Розрахованi основнi спектри спiнових та електронних збуджень i спостережуванi властивостi купратiв, насамперед поведiнка параметра порядку залежно вiд концентрацiї допантiв. Ключова особливiсть запропонованої фiзичної картини полягає у формуваннi всерединi дiелектричної щiлини вихiдного недопованого матерiалу специфiчної домiшкової пiдзони, яка слугує джерелом наступних металiзацiї i переходу у надпровiдний стан з анiзотропним параметром порядку. 2018 Article On the theory of high-Tc superconductivity of doped cuprates / Y.G. Pogorelov, V.M. Loktev // Condensed Matter Physics. — 2018. — Т. 21, № 3. — С. 33704: 1–20. — Бібліогр.: 41 назв. — англ. 1607-324X PACS: 71.30.+h, 71.35.-i, 74.0.-z, 74.20.Rp, 74.62.Dh, 74.72.-h DOI:10.5488/CMP.21.33704 arXiv:1807.10428 http://dspace.nbuv.gov.ua/handle/123456789/157457 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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A theoretical analysis is presented on possible effects of disorder by dopants in high-temperature superconducting cuprate perovskites, to define their basic spectra of spin and electronic excitations, and the subsequent
observable properties, especially doping dependence of superconducting order parameter. The central point in
the proposed physical picture is formation of specific impurity subband within the insulating bandgap of initial
undoped material, serving as a source for the system metallization and further transition into superconducting
state with anisotropic order parameter. |
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Article |
| author |
Pogorelov, Y.G. Loktev, V.M. |
| spellingShingle |
Pogorelov, Y.G. Loktev, V.M. On the theory of high-Tc superconductivity of doped cuprates Condensed Matter Physics |
| author_facet |
Pogorelov, Y.G. Loktev, V.M. |
| author_sort |
Pogorelov, Y.G. |
| title |
On the theory of high-Tc superconductivity of doped cuprates |
| title_short |
On the theory of high-Tc superconductivity of doped cuprates |
| title_full |
On the theory of high-Tc superconductivity of doped cuprates |
| title_fullStr |
On the theory of high-Tc superconductivity of doped cuprates |
| title_full_unstemmed |
On the theory of high-Tc superconductivity of doped cuprates |
| title_sort |
on the theory of high-tc superconductivity of doped cuprates |
| publisher |
Інститут фізики конденсованих систем НАН України |
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2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157457 |
| citation_txt |
On the theory of high-Tc superconductivity of doped cuprates / Y.G. Pogorelov, V.M. Loktev // Condensed Matter Physics. — 2018. — Т. 21, № 3. — С. 33704: 1–20. — Бібліогр.: 41 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT pogorelovyg onthetheoryofhightcsuperconductivityofdopedcuprates AT loktevvm onthetheoryofhightcsuperconductivityofdopedcuprates |
| first_indexed |
2025-07-14T09:53:04Z |
| last_indexed |
2025-07-14T09:53:04Z |
| _version_ |
1837615583684722688 |
| fulltext |
Condensed Matter Physics, 2018, Vol. 21, No 3, 33704: 1–20
DOI: 10.5488/CMP.21.33704
http://www.icmp.lviv.ua/journal
On the theory of high-Tc superconductivity of doped
cuprates
Y.G. Pogorelov1, V.M. Loktev2,3
1 IFIMUP-IN, Departamento de Física e Astronomia, Universidade do Porto, Porto, Portugal
2 Bogolyubov Institute for Theoretical Physics of the National Academy of Sciences of Ukraine,
14b Metrologichna St., 03143 Kyiv, Ukraine
3 National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”,
37 Peremohy Ave., 03056 Kyiv, Ukraine
Received June 12, 2018, in final form July 23, 2018
A theoretical analysis is presented on possible effects of disorder by dopants in high-temperature supercon-
ducting cuprate perovskites, to define their basic spectra of spin and electronic excitations, and the subsequent
observable properties, especially doping dependence of superconducting order parameter. The central point in
the proposed physical picture is formation of specific impurity subband within the insulating bandgap of initial
undoped material, serving as a source for the system metallization and further transition into superconducting
state with anisotropic order parameter.
Key words: dopant concentration, cuprate magnetic and electronic spectra, superconductivity,
superconducting gap and transition temperature
PACS: 71.30.+h, 71.35.-i, 74.0.-z, 74.20.Rp, 74.62.Dh, 74.72.-h
1. Introduction
The phenomenon of high-Tc superconductivity discoveredmore than 30 years ago in layered perovskite
cuprates [1, 2] and then found in many other materials [3–5] still presents a challenge for the condensed
matter theory. Besides its main distinctive feature of critical temperature Tc for superconducting (SC)
order to exceed the theoretical limits established earlier for “classical” superconductors [6], it reveals
a lot of other differences from the canonical Bardeen-Cooper-Schrieffer (BCS) picture [7] such as: an
extremely short SC coherence length ξ and respective huge values of the Ginzburg-Landau parameter
κ, unusual d-wave symmetry of the SC order parameter, onset of the SC order under doping of initially
insulating materials (as La2CuO4 → La2−cSrcCuO4) and then a non-monotonous Tc variation with
growing doping rate c, rising from zero at the metallization threshold cmet to reach a maximum and then
decaying to vanish at some upper limit ccr [8]. These exotic properties were deeply studied in various
untraditional theoretical ways, with the main accent on strong electronic correlations at the transition
from insulating to metallic state, beginning from the Anderson resonating valence bond model [9–11],
Zhang-Rice singlet model [12], Emery-Kivelson phase fluctuation model [13] and many others. For
instance, a recent study by this approach suggested an enhanced stability of the SC order in high-Tc
cuprates under the doping induced disorder [14]. Nevertheless, some issues in this field still remain out of
scope of the existing approaches, for instance recent experimental findings of anomalously long magnetic
penetration depth λ and hence anomalously low superfluid density ns in the overdoped (Tc decaying) SC
range [15], in a striking contrast with the BCS theory predictions [16].
It should be noted here that all the abovementioned theoretical pictures of SC state, either canonical or
strong-correlational, consider many-body processes in high-Tc materials, even in presence of randomly
located dopants, as in uniform crystalline systems with full translational invariance, implicitly disre-
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
33704-1
https://doi.org/10.5488/CMP.21.33704
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
Y.G. Pogorelov, V.M. Loktev
garding the effects of static disorder. On the other hand, such effects can qualitatively modify spectral
and related observable properties of various crystalline systems with disorder as was shown by many
researchers, beginning from the seminal studies by Lifshitz on emergence of local modes and impurity
bands [17, 18] and by Mott on mobility thresholds between localized and band-like states [19, 20]. In
particular, restructuring of excitation spectra in high-Tc materials under perturbations from both dopants
and isovalent impurities was discussed in detail by the authors [21, 22], though leaving aside the role of
these perturbations for the very formation of the SC order. The present work is just aimed at extension
of our earlier theoretical approaches to this latter aspect of the high-Tc problem, in order to clarify
the decisive importance of disorder effects for understanding the physical difference between BCS and
high-Tc types of superconductivity. This task is being done here with parallel analysis of two different
kinds of elementary excitations in the perovskite compound: fermion quasiparticles by electron hopping
between neighbor Cu sites and boson-like spin excitations at strong enough indirect antiferromagnetic
(AFM) superexchange between these sites via intermediate O sites. Their interplay is found to define the
insulating state of the undoped material and then, introducing perturbations from dopant atoms on both
excitation modes, the possibility for its transition to a specific metallic state with strong disorder effects
and anomalously narrow conduction band is established. The latter specifics is further analyzed for its
influence, together with the fermion-boson interplay, on the formation and properties of the SC state, in
particular, the non-monotonous dependence of the SC order parameter on the doping level.
The following consideration starts in section 2 from definition of the basic Green functions, specific
for the analysis of different types of elementary excitations in a disordered crystalline system. Section 3
explains the choice of effective Hamiltonians for spin and charge degrees of freedom, including their
perturbations due to dopants. Then, we find the excitation spectra in this system, with a special emphasis
on the effects of spin dynamics on charge dynamics, either in the initial state or under finite doping. In
the latter case, the important doping effects for decay of long range spin order and for metallization of
charge carrier spectrum are analyzed in section 4. The resulting excitation spectra also contain, besides
the band-like modes described by their dispersion laws, possible localized modes near the dopant atoms,
all resumed in the corresponding density of states (DOS) function. In particular, this function permits to
define the position of Fermi level under doping, one of central issues for the onset of SC order. Then,
the characteristics of the resulting SC state as functions of doping and its related disorder are obtained in
section 5, permitting to better understand the experimentally measured anomalies in this state, compared
to the conventional BCS theory predictions. Finally, some discussion and conclusions on the developed
SC model are presented in section 6.
This work is dedicated to the 80th birthday of Prof. I.V. Stasyuk, a renowned Ukrainian theorist,
recognizing his very valuable contribution to the theory of narrow-bandmetals and pseudo-spin operators
in the field of phase transitions in solids [23], especially of superconducting transitions [24].
2. Green functions and observable values
The spin and charge dynamics in this system are analyzed below through the standard techniques of
two-time Green functions (GF’s) [25, 26]:
〈〈A|B〉〉±,ε =
i
π
∞∫
0
ei(ε+i0)t 〈[A(t), B(0)]±〉dt, (1)
where 〈. . .〉 = (Tr e−βH )−1(Tr e−βH . . .) is the quantum statistical average of any operator in the system
with full Hamiltonian H at inverse temperature β = 1/(kBT), the commutator [A, B]− or anticommutator
[A, B]+ is chosen respectively for the operators A, B of Bose or Fermi type (implying respective types
of thermal averaging for them), while positive infinitesimal imaginary part i0 in the energy argument of
Fourier transform corresponds to the retarded GF type. These functions obey the equation of motion:
ε〈〈A|B〉〉±,ε = 〈[A, B]±〉 + 〈〈[A,H]− |B〉〉±,ε , (2)
permitting their consequent calculation with given H (in what follows, the explicit symmetry and energy
indices of the GFs are typically omitted). The basic GF advantage is to provide directly the system
33704-2
On the theory of high-Tc superconductivity of doped cuprates
observable values from the spectral theorem:
〈AB〉 =
1
π
∞∫
0
Im 〈〈B |A〉〉εdε. (3)
It should be noted that the two-time GF’s defined by equations (1)–(3), though being most suitable to
analyze the effects by doped charges and associated disorder from dopant atoms (of central importance
in the present work), but are only justified in the zero temperature limit and they are mostly employed
in this limit below. Otherwise, the case of finite temperatures should be treated, strictly speaking, with
more specialized Matsubara GF’s [27, 28], though producing some technical problems for their use in
disordered systems. Nevertheless, some qualitative estimates of critical temperatures for both AFM and
SC ordering under doping are possible even from the simpler two-time GF formalism, so we choose to
stay with it for all the following study. To distinguish between the two different types of excitation spectra
in the present system, we shall denote the GF energy argumentω for Bose-type spin states and ε (referred
to the chemical potential µ) for Fermi-type charge carrier states.
3. Model Hamiltonians and excitation spectra
The relevant structure for superconductivity in perovskite cuprates is the CuO2 plane. In the absence
of doping, all the oxygens are in the O2− ionization state with fully occupied p-shell and all the coppers
are in the Cu2+ state with a single non-compensated d-electron spin (see figure 1), while the hole charge
carriers by doping are expected to reside on the cation sites changing their state for Cu3+ [29]. Then, the
spin and charge dynamics of these d-electrons are supposed to be mostly responsible for the unordinary
physical properties of cuprate materials, in conformity with various known approaches [24, 30], but the
main accent of the present work is on a considerable restructuring of both types of excitation spectra
under random perturbations from dopants.
Figure 1. (Colour online) An elementary cell (dashed lines) in the CuO2 plane with AFM ordered Cu2+
spins (shown by arrows). Indirect hoppings (permitted at presence of doped holes), t between nearest
neighbor Cu d-orbitals and t ′ between next nearest neighbors, are realized through intermediate O2−
p-orbitals. The hatched circle indicates a possible projection of an external dopant onto this plane. Inset:
small deviations of two kinds of spins from antiparallelism in the xz-plane due to weak Dzyaloshinskii-
Moriya exchange.
33704-3
Y.G. Pogorelov, V.M. Loktev
3.1. Spin dynamics
For treating the spin order and spin excitations in the initial undoped structure, we use the simplest
model Hamiltonian involving the strong AFM superexchange J between nearest neighbor Cu spins (to
impose their antiparallel orientation) as well as the weak terms of tetragonal anisotropy ∆Jt and of
Dzyaloshinskii-Moriya antisymmetric exchange D (to provide stability of the AFM order):
Hs =
∑
n,δ
[
Jsn · sn+δ − ∆Jts
y
n syn+δ − D
(
syn szn+δ − sznsyn+δ
)]
, (4)
with the vectors δ of separation between nearest neighbor Cu sites.
Under the strongest J effect only, the spin ground state would have two enclosed square sublattices
with strictly opposite spin orientations but unstable because of their free absolute orientation. Adding
the weak ∆Jt effect, restricts this freedom to the xz-plane and, finally, the weak D effect fixes the
sublattice quantization axes almost antiparallel along the z-axis and with slight symmetric deviations
from z- towards y-axis by an angle θ (see inset in figure 1). This implies the following expressions of
spin components on j-th sublattice sites in n-th unit cell through the local spin-flip operators bn j :
szn1 =
(
1
2
− b†n1 bn1
)
cos θ +
bn1 − b†n1
2i
sin θ, szn2 =
(
b†n2 bn2 −
1
2
)
cos θ +
bn2 − b†n2
2i
sin θ,
syn1 =
(
b†n1 bn1 −
1
2
)
sin θ +
bn1 − b†n1
2i
cos θ, syn2 =
(
b†n2 bn2 −
1
2
)
sin θ −
bn2 − b†n2
2i
cos θ,
sxn1 =
bn1 + b†n1
2
, sxn2 =
bn2 + b†n2
2
. (5)
Being inserted in equation (4), they generate the terms up to fourth order in b-operators but the ground
state stability needs the linear terms to vanish (together with the third order terms) and this requires the
deviation angle to be:
θ =
1
2
arctan
2D
2J − ∆Jt
� 1. (6)
Next, the quartic terms can be neglected for small excitation numbers, 〈b†n j
bn j 〉 � 1, when bn j can be
seen as Bose operators (the Holstein-Primakoff ones) and the spin Hamiltonian turns bilinear in their
Fourier-transforms bj,k = N−1/2 ∑
n eikn j bn j (summed over N unit cells):
Hs =
∑
k
[
A
(
b†1,kb1,k + b†2,kb2,k
)
+ Bγk
(
b1,kb2,−k + b†1,kb†2,−k
)
+ Cγk
(
b†1,kb2,k + b†2,kb1,k
)]
,
A = 2J cos 2θ + 2∆Jt sin2 θ + 2D sin 2θ, B = J(1 + cos 2θ) − ∆Jt cos2 θ + D sin 2θ,
C = J(1 − cos 2θ) + ∆Jt cos2 θ − D sin 2θ, γk = cos
kx
2
cos
kz
2
(7)
(with wave-vectors k measured in units of inverse lattice parameter a−1). We notice that the difference
between the big terms A ≈ 2J + D2/J and B ≈ 2J − ∆Jt + D2/2J is of the order of the small term
C ≈ ∆Jt − D2/2J.
The Hamiltonian, equation (7), is readily diagonalized by a standard unitary transformation of the
sublattice bj,k operators into the β±,k operators of eigen-modes (symmetric or antisymmetric in the
sublattice indices). This can be compactly expressed in terms of 4-spinors transformed by a 4×4 matrix
as bk = Ûkβk, or in their components:
bk =
©«
b1,k
b†1,−k
b2,k
b†2,−k
ª®®®®¬
, Ûk =
1
√
2
©«
cosh θ+,k sinh η+,k cosh η−,k − sinh η−,k
sinh η+,k cosh η+,k − sinh η−,k cosh η−,k
cosh η+,k sinh η+,k − cosh η−,k sinh η−,k
sinh η+,k cosh η+,k sinh η−,k − cosh η−,k
ª®®®¬ , βk =
©«
β+,k
β†
+,−k
β−,k
β†
−,−k
ª®®®®¬
.
(8)
33704-4
On the theory of high-Tc superconductivity of doped cuprates
Here, the 4×4 matrices can be expressed as direct products of two kinds of Pauli matrices, σ̂j in spin
indices (e.g., between bj,k and b†
j,k) and τ̂j in sublattice/mode indices (e.g., between b1,k and b2,k). So,
the above U-matrix is presented in this way as:
Ûk =
[
eη+,kσ̂1 ⊗ (τ̂1 − iτ̂2 + τ̂3 + 1) + e−η−,kσ̂1 ⊗ (τ̂1 + iτ̂2 + τ̂3 − 1)
]
/
√
8 .
This permits to reduce multiplication of 4×4 matrices to that of 2×2 ones by the rule: [ f (σ̂j) ⊗
g(τ̂k)][ f ′(σ̂l) ⊗ g
′(τ̂m)] = [ f (σ̂j) f ′(σ̂l)] ⊗ [g(τ̂k)g
′(τ̂m)], for any functions f , f ′, g, and g′.
Then, with the choice of hyperbolic rotation arguments as:
η±,k =
1
2
arctanh
Bγk
A ± Cγk
, (9)
the diagonal Hs form is reached:
Hs =
∑
j=±
∑
k
ωj,kβ
†
j,kβj,k , with ω±,k =
√
(A ± Cγk)
2 − B2γ2
k . (10)
In the most important low-energy range, where γ2
k ≈ 1 − k2/4, the dispersion laws are approximated as:
ωj,k ≈
√
ω2
j + J2k2 , (11)
with the spectrum gaps ω− = 2D and ω+ = 2
√
2J∆Jt. These values just testify that the rigidity of the
antisymmetric mode (in-plane rotations generated mainly by the spin x-components) comes from the
in-plane D anisotropy and that of the symmetric mode (out-of-plane breathings by the y-components)
does from the out-of-plane ∆Jt anisotropy. The spin dynamics of this spin system is contained in the GF
matrix Ĝk = 〈〈βk |β
†
k〉〉. In particular, DOS of spin excitations is defined as:
ρ(ω) =
1
πN
Im Tr
∑
k
Ĝk , (12)
where the sum over 2D Brillouin zone, as usually, can be substituted by integration according to the rule:
1
N
∑
k
f (k) = 1
4π2
π∫
−π
dkx
π∫
−π
dkz f (k). (13)
For the unperturbed Hamiltonian, equation (10), the GF matrix takes the simplest diagonal form, Ĝk ⇒
ĝk, with its non-zero elements:
〈〈βj,k |β
†
j,k〉〉 =
1
ω − ωj,k
, 〈〈β†
j,k |βj,k〉〉 = −
1
ω + ωj,k
.
Then, using the equation (11) provides a simple analytic approximation for the relevant integrals (also
called the locator functions) in the low-energy range:
gj(ω) = N−1
∑
k
1
ω − ωj,k
≈ −
1
J
(
1 +
ω
2J
ln
2J
ωj − ω
)
, (14)
and hence the low-energy approximation for DOS:
ρ(ω) ≈
ω
4J2 [θ(ω − ω−) + θ(ω − ω+)] . (15)
The AFM order stability is controlled by the spectrum gaps ω±, limiting thermal occupation of
excited states over the fully ordered ground state. Eventually, at reaching the Néel temperature TN, the
mean sublattice magnetization, as, e.g.,
s = 〈szn1〉 =
1
2
−
1
N
∑
k
〈b†1,kb1,k〉 ,
33704-5
Y.G. Pogorelov, V.M. Loktev
should vanish. This average is expressed with the use of the spectral theorem, equation (3), and the
unitary transformation, equation (8), as:
1
N
∑
k
〈b†1,kb1,k〉 ≈
1
2N
∑
j,k
(
cosh2 ηj,k〈β
†
j,kβj,k〉 + sinh2 ηj,k〈βj,kβ
†
j,k〉
)
≈
J
N
∑
j,k
2nB
(
ωj,k
)
+ 1
ωj,k
−
1
2
, (16)
involving the Bose occupation function: nB(ω) = (eβω−1)−1. Since the temperature dependent part of the
sum in equation (16) is mainly due to the low-energy spectrum, this can be approximated by equation (11)
and the radial integration limit can be safely extended to infinity, giving for jth subband:
2J
N
∑
k
nB
(
ωj,k
)
ωj,k
≈
J
π
∞∫
0
nB
(√
ω2
j + J2k2
)
kdk√
ω2
j + J2k2
=
1
πJ
∞∫
ω j
nB(ω)dω =
ln
[
1 + nB(ωj)
]
πβJ
.
For temperatures much higher of the bandgaps, βωj � 1, the latter logarithm is further simplified just
to ln
[
1/(βωj)
]
.
The temperature independent part in equation (16) (presenting spin zero-point fluctuations) is con-
tributed by the whole spectrum by equation (10) and its numeric value α = N−1 ∑
j,k J/ωj,k−1/2 ≈ 0.19
is almost insensitive to the small parameters ∆Jt/J and D/J. Then, the Néel temperature is found from
the equation:
TN ≈
(1/2 − α)πJ
kB ln
[
T2
N/(ω+ω−)
] , (17)
and, for the choice of parameters: J = 135 meV (as measured in La2CuO4 [31] and supported by ab
initio calculations [32, 33] for all CuO2 planes) and ω−ω+ = 6.5 meV2, we have from equation (17) this
temperature as TN ≈ 320 K, in a good agreement with its experimental value for undoped La2CuO4 (and
actually much higher of the chosen geometric average of bandgaps √ω+ω−/kB ≈ 30 K).
It can be yet noted that the stabilizing anisotropy D factor for theAFMordermay also remain important
beyond this order range, for instance, defining the exotic pseudogap feature in the electronic spectrum near
its Fermi level in the temperature range from TN to the much higher structural tetragonal-orthorhombic
transition Tt-o [35, 36], but this issue is left out of the present consideration.
3.2. Charge dynamics
Another kind of dynamics results from much faster charge carrier hopping over Cu sites. Aiming to
define the Fermi level position from the resulting spectrum, we start with the model hopping Hamiltonian
referred to the atomic Cu d-term:
Hhop = −
∑
k,σ
[
tk(a
†
1,σ,ka2,σ,k + a†2,σ,ka1,σ,k) + t ′k(a
†
1,σ,ka1,σ,k + a†2,σ,ka2,σ,k)
]
. (18)
Here, aj,σ,k = N−1/2 ∑
n eikn j an j,σ are the Fourier transforms of the Fermi operators an,σ for an electron
with the given spin value σ on n-th site. The hopping factors tk = 4t cos kx
2 cos kz
2 (for nearest neighbors)
and t ′k = 2t ′(cos kx + cos kz) (next nearest neighbors) involve the hopping parameters t and t ′ (both taken
with negative sign for their indirect realization through the excited O levels [29]). This model can be seen
as a simplified version of the much more detailed LDA Hamiltonian [34].
Actually, the spin operators (in neglect of small deviations θ from z-axis, inessential for the charge
dynamics) can be also presented in terms of the Fermi operators as:
szn =
1
2
(
a†n,↑an,↑ − a†n,↓an,↓
)
, s+n = a†n,↑an,↓ , s−n = a†n,↓an,↑ . (19)
33704-6
On the theory of high-Tc superconductivity of doped cuprates
Then, the dominant purely exchange part of the spin Hamiltonian, equation (4), obtains in these terms a
four-fermion structure:
Hex = J
∑
n,δ
[
1
4
(
a†n1,↑
an1,↑ − a†n1,↓
an1,↓
) (
a†n1+δ,↑
an1+δ,↑ − a†n1+δ,↓
an1+δ,↓
)
+ 1
2
(
a†n1,↑
an1,↓a
†
n1+δ,↓
an1+δ,↑ + a†n1,↓
an1,↑a
†
n1+δ,↑
an1+δ,↓
)]
, (20)
or, passing to the band operators:
Hex =
J
N
∑
k,k′,q
[(
a†1,↑,ka1,↑,k′ − a†1,↓,ka1,↓,k′
) (
a†2,↑,k+qa2,↑,k′−q − a†2,↓,k+qa2,↓,k′−q
)
+ 2γk−k′
(
a†1,↑,ka1,↓,k′a
†
2,↓,q−ka2,↑,q−k′ + a†1,↓,ka1,↑,k′a
†
2,↑,q−ka2,↓,q−k′
)]
. (21)
This can be also reduced to a bilinear form, like that in equation (18), by a common separation of
some non-zero pair averages beside the resting operator pair products (and neglecting as usually small
fluctuations beside the averages). We note that this known procedure involves here either “normal”,
〈a†a〉-type, and “anomalous”, 〈aa〉- or 〈a†a†〉-type averages. They will express respectively the spin
exchange effects on the normal electronic spectrum and on its SC restructuring.
Particularly, the “normal” averages relate to the local spin values and they are selected from the first,
longitudinal part under the sum in equation (20) as:
〈a†n1,↑
an1,↑ − a†n1,↓
an1,↓〉 = −〈a
†
n2,↑
an2,↑ − a†n2,↓
an2,↓〉 = 2s, (22)
where the s value can be evaluated (in the function of temperature and doping) from the above considered
Bose-type GFs. Otherwise, the “anomalous” averages are due to delocalized pairs of electrons with
opposite spins and momenta and they follow from both parts in equation (21) (to be detailed later).
Now, we compose the effective two-fermion Hamiltonian for normal electrons: Hn = Hhop + H(2)n ,
involving the “normally” averaged spin exchange part:
H(2)n = J
∑
k
(
a†1,↓,ka1,↓,k + a†2,↑,ka2,↑,k − a†1,↑,ka1,↑,k − a†2,↓,ka2,↓,k
)
, (23)
which can be seen as a spin-Hubbard repulsion, favouring an electron with the given spin to occupy the
sites from its proper sublattice (1 for ↑ and 2 for ↓).
Similarly to the above boson case, we construct fermion 4-spinors in sublattice/spin indices:
a†k = (a
†
1,↑,k, a
†
2,↑,k, a
†
2,↓,k, a
†
1,↓,k), and transform them into such spinors in mode/spin indices: α†k =
(α†
+,↑,k, α
†
−,↑,k, α
†
+,↓,k, α
†
−,↓,k) by a unitary rotation:
ak = eiλkτ̂2 ⊗ 1̂αk , (24)
(with the direct product of τ̂-matrices in mode indices and unit matrix in spin indices). Then, the choice
of this rotation argument λk =
1
2 arctan tk/J brings the effective two-fermion Hamiltonian to its diagonal
form:
Hn =
∑
j,σ,k
εj,kα
†
j,σ,kαj,σ,k , (25)
with 4 composite eigen-modes, of spin-majority ( j = +) or spin-minority ( j = −) composition [not to
be mixed up with the symmetric/antisymmetric boson modes in equation (10)] for each σ value. The
corresponding eigen-energies:
ε±,k = t ′k ∓
√
J2 + t2
k , (26)
(note their double degeneracy in spin index) display the possibility for an energy gap to open between
the top of the majority subband, ε+,max = 4t ′ − J reached at the BZ M-points, k = (±π,±π), and the
33704-7
Y.G. Pogorelov, V.M. Loktev
bottom of minority bands, ε−,min = J at the X-points, k = (0,±π), or Y-points, k = (±π, 0). This most
important spin-exchange effect appears under the condition J > 2t ′, realistic for strong AFM coupling J
in cuprates. In the absence of doping, it defines the system insulating state with fully occupied majority
bands and empty minority bands, the Fermi energy (the chemical potential at T = 0) being located
at the gap middle point: µ = 2t ′. In this Fermi energy reference, the subband edges are written as:
ε−,min = J − 2t ′ = −ε+,max. More details of this spectrum are seen from its DOS function:
ρ(ε) =
1
πN
∑
j,σ,k
Im 〈〈αj,σ,k |α
†
j,σ,k〉〉 =
2
N
∑
j,k
δ
(
ε − εj,k
)
, (27)
where the corresponding integration can be simplified, passing from the variables kx and kz to u =
cos kx
2 cos kz
2 and v = 1 + (cos kx + cos kz)/2 and using the Jacobian:
D(u, v)
D
(
kx, ky
) = 1
4
sin kx
2 sin kz
2 (cos kz − cos kx) .
The latter is then expressed in the function of u, v:
D(u, v)
D (kx, kz)
= −
1
2
√(
u2 + 1 − v
) (
v2 − 4u2) .
Also conversion of the DOS integrals by equation (27) to the u, v variables requires to express the
dispersion laws as
ε±,k ≡ ε±(u, v) = −4t ′v ∓
√
J2 + 16t2u2
and to modify the integration limits.
The relevant integration area between BZ points Γ, X, and M is delimited in the kx, kz variables
[figure 2(a)] by the lines: kz = 0 at 0 6 kx 6 π (Γ-X), kz = kx at 0 6 kx 6 π (Γ-M), and kx = π at
0 6 kz 6 π (X-M). In the u, v variables [figure 2(b)] they are converted to the lines: u = 0 at 0 6 v 6 1
(M-X), v = 2u at 0 6 u 6 1 (M-Γ), and v = 1 + u2 at 0 6 u 6 1 (X-Γ). Eventually, the DOS integrals in
these variables are written as:
ρ±(ε) =
1
2π2
1∫
0
du
1+u2∫
2u
dv
δ
(
ε − 4t ′v ±
√
J2 + 16t2u2
)
√(
u2 + 1 − v
) (
v2 − 4u2) ,
and, after the trivial δ-function integration in v, they are reduced to one-dimensional u-integrals:
ρ±(ε) =
1
8π2 |t ′ |
1∫
0
θ
(
u2 + 1 − v±
)
θ (v± − 2u)√(
u2 + 1 − v±
) (
v2
± − 4u2) du , (28)
Figure 2. a) The BZ relevant integration area (shadowed) in the kx, kz variables (in a−1 units). b) The
same in the u, v-variables.
33704-8
On the theory of high-Tc superconductivity of doped cuprates
Figure 3. Density of states with the energy gap between spin-majority (filled) and spin-minority (empty)
electronic subbands and the van Hove singularities at their characteristic energies (see the text) calculated
from equation (28) at the choice of the model parameters: t = 0.5 eV, t ′ = 50 meV, J = 135 meV. Its
more detailed behavior in the gap range (delimited by the dashed rectangle) is shown in the inset.
with v±(ε, u) = (ε ±
√
J2 + 16t2u2)/4t ′. Then, an easy numerical calculation results in the DOS profiles
shown in figure 3 for a realistic choice of the hopping parameters: t = 0.5 eV and t ′ = 50 meV, compared
to the above chosen exchange parameter J = 135 meV. They present two types of van Hove singularities:
1) discontinuous steps at the subband edges, ε+,min = −6t ′−
√
J2 + 16t2, ε+,max = 2t ′− J, ε−,min = J,
ε−,max = 2t ′ +
√
J2 + 16t2, and also at the minority subband local maximum ε−,lmax = 2t ′ + J,
2) logarithmic peaks at the saddle points, ε+,s = −2t ′ − J and ε−,s = 2t ′ + J
√
1 − (t ′/t)2.
A notable asymmetry between the majority and minority subbands seen in figure 3, in particular, in
their bandwidth and edge DOS levels, can serve for understanding the experimentally detected differ-
ence in the cuprate behavior under hole and electron types of doping [30], but we restrict the present
consideration only to the more habitual hole type.
In the following analysis of the spectrum restructuring under such doping, into metallic and further
into SC state, a special importance pertains to the ε+,k isoenergetic lines in BZ around its M-point that
display C4 symmetry as shown in figure 4, in general similarity to the known experimental measurements
of these lines with the ARPES techniques [37]. Their maximum distance qmax from the M-point (along
the kx and kz axes) is written in the function of the energy depth δ = ε+,max − ε as:
qmax = 2 arcsin
√
δ
4t ′
, (29)
and the minimum distance (along the kx = ±kz lines) does as:
qmin = 2
√
2 arcsin
√√
t2δ(2J + δ) + (2Jt ′)2 − 2t ′(J + δ)
4(t2 − 4t ′2)
, (30)
but in the most relevant energy range, J(t ′/t)2 � δ � J, the latter is well approximated as:
qmin ≈
√
8 arcsin 4
√
δ
Ω
, (31)
with a much greater energy scale Ω = 8
(
t − 4t ′2/t
)2
/J � J. Then, the approximated dispersion law for
δ in polar coordinates around the M-point can be presented as:
δ(q, ϕ) ≈ 4t ′ sin2 q
2
cos2 ϕ +Ω sin4 q
√
8
sin2 ϕ. (32)
33704-9
Y.G. Pogorelov, V.M. Loktev
Figure 4. Isoenergetic lines q(δ, φ) for momentum relative to the M-point of BZ at different energy depths
δ below the top of the majority subband by equation (26) (their values in meV indicated by the arrows)
present the C4 symmetry in the azymuthal angle φ and are transformed with growing δ from separate
arcs to continuous lines around the BZ Γ-point. The same symmetry and topology of isoenergetic lines
is also preserved at finite doping describing the evolution of Fermi lines and associated energy levels.
Such factorization of radial q- and azymuthal ϕ- dependencies essentially simplifies the further analysis
of the system spectral characteristics under doping.
4. Doping effects
Now, we pass to the effects of doping on both kinds of excitation spectra. First of all, there are two
principal perturbation mechanisms for a non-isovalent dopant atom (located out of the relevant CuO2
plane) on the insulating host with the AFM spin order:
1) the dopant Coulomb potential, acting on in-plane Cu sites within a certain screening radius rscr,
and
2) appearance of a charge carrier (a hole), localized by this potential within a certain localization
radius rloc.
Generally, these mechanisms can influence all the parameters in the original Hamiltonians, equa-
tions (4) and (18), producing quasiparticle scattering of rather complicated structure and considerable
technical problems for its analysis.
Nevertheless, a qualitatively reasonable physical picture for such a disordered system is expected
to follow already from the simplest one-parametric Lifshitz perturbation models for each diagonalized
Hamiltonian, equations (10) or (25). Of course, strictly speaking, the Lifshitz model with a single
momentum-independent scattering amplitude should correspond to a purely point-like perturbation po-
tential in the real space, or, in more general, to a perturbation that affects a single local component of a
plane-wave excitation state (that is, its inverse Fourier transform). However, in the present system, the αk
and βk eigen-modes are composed of the original ak and bk modes, according to equations (24) and (8),
and their inverse Fourier transforms:
αn =
1
√
N
∑
k
e−iknαk =
∑
n′
f̂ (a)(n − n′)an′ , βn =
1
√
N
∑
k
e−iknβk =
∑
n′
f̂ (b)(n − n′)bn′ , (33)
are the distributions of the original point-like an and bn with some kernel matrices f̂ (a,b)(r). The latter
matrix elements are made of the inverse Fourier transforms of cos λk, sin λk (for α-modes) or cosh η±,k,
sinh η±,k (for β-modes) and they have as long decay range as r (b) ∼ J/ωj � 1 for boson β-states but
only r (a) ∼ t/J ∼ 1 for fermion α-states. Therefore, the above mentioned perturbations of ∼ rloc or ∼ rscr
range can be felt by the broadly distributed boson modes as effectively point-like and that may justify the
33704-10
On the theory of high-Tc superconductivity of doped cuprates
Lifshitz model approximation for them. As to the sharper resolved fermion modes, where a more detailed
perturbation model should be applied, fortunately it does not make essential technical problems to obtain
an effective Lifshitz approximation also for this case (to be shown later).
Now, beginning from the spin excitations, we supplement equation (10) with the Lifshitz perturbation
Hamiltonian of the form:
H ′s = −
Vs
N
∑
j,k,k′,p
ei(k−k′)pβ†
j,kβj,k′ , (34)
where p are the on-plane projections of random dopant positions (see figure 1) and the negative sign
of the single perturbation parameter Vs indicates a general effective suppression of spin exchange and
anisotropies due to the dopant potentials on Cu and O sites and to the reduction of spin density because
of hole carriers. Then, in the standard general expression for GF matrix Ĝk = (ĝ
−1
k − Σ̂k)
−1, the simplest
approximation for the self-energy matrix is: Σ̂k ≈ cT̂s, where the momentum-independent (but energy-
dependent) diagonal T-matrix for spin states:
T̂s(ω) =
[
1 + τ̂3
2
T+(ω) +
1 − τ̂3
2
T−(ω)
]
⊗ 1̂, (35)
and the subband terms T±(ω) = −Vs/[1 + Vsg±(ω)] involve the locator functions by equation (14).
It is well known for disordered systems that zeroes of T-matrix denominators beyond the band
spectrum define localized impurity levels in the diluted limit of doping and those can produce new
impurity subbands at higher doping levels. Since each gj(ω) is negative and logarithmically diverging near
the respective subband edge ωj , such levels ωj,loc are formally produced by both terms in equation (35),
but of extremely shallow depth:
ωj − ωj,loc ≈ 2J exp
[
−
2J(J − Vs)
ωjVs
]
, (36)
due to the huge∼ J/ωj � 1 factor in the negative exponent (except for an improbable closeness ofVs to J
within to∼ ωj), so they are practically stuck to those edges.Moreover, the fast decay of logarithms in g±(ω)
by equation (14) beyond an exponentially narrow vicinity of the edge, ωj −ω ∼ J(ωj −ωj,loc)/ωj � ωj ,
makes them negligible besides the resting constant and then the only doping effect in the spin spectrum
remains in equal downshifts of the subband edges as:
ωj(c) = ωj −
cVsJ
J − Vs
. (37)
This immediately defines the concentrational decay of the Néel temperature, TN(c), calculated again
from equation (17) but with the change of ωj for ωj(c). In particular, the critical condition when the
smaller gap ωmin(c) = minj ωj(c) reaches zero [and so TN(c), with no more long-range AFM order in the
system] defines the critical doping value
ccr = ωmin(0)
(
1
Vs
−
1
J
)
. (38)
At c < ccr, for a given product ω+ω− [e.g., fitted to the known TN = TN(0) as in section 3.1], the fastest
decay of TN(c) is evidently attained for the case of equal gaps, ω+ = ω− = ω±, and the corresponding
behavior is presented in figure 5, at the choice of perturbation parameter asVs = ω±/(ccr+ω±/J). Notably,
with the use of the experimental value of ccr ≈ 0.02 in La2CuO4, this parameter results Vs = 65.5 meV
which is safely far from the adopted value of J = 135 meV, justifying the above general conclusions on
the magnon spectrum reordering under doping.
Now, passing to the charge carrier spectrum, we consider the perturbation to the Hamiltonian,
equation (25) from equal upshifts of atomic energy levels by Vc on nearest neighbor Cu sites to a dopant
(due to its repulsive potential for electrons). Then, in the presence of the AMF spin order (at least, local)
in the lattice, there exist two types of dopant positions, with the up-spin neighbor sites along the x- or
33704-11
Y.G. Pogorelov, V.M. Loktev
Figure 5. Doping dependence of the Néel temperature calculated from equation (17) at parameter values
ω+ = ω− = 2.55 meV, Vs = 65.5 meV.
z-axes (the latter case just presented in figure 1). The corresponding perturbation Hamiltonian takes the
form:
H ′hop =
1
N
∑
j=x,z
∑
p j,k,k′
ei(k−k′)pα†kV̂ (j)k,k′αk′ , (39)
with the scattering matrices:
V̂ (x)k,k′ =
Vc
2
eiλkτ̂2
[
(1 + τ̂3) cos
kx − k ′x
2
+ (1 − τ̂3) cos
kz − k ′z
2
]
e−iλk′ τ̂2 , (40)
and V̂ (z)k,k′ only differing from the above by the interchange of cosine factors at the 1 ± τ̂3 matrices. Then,
within the T-matrix approximation, these two types of dopants give equal additive contributions to the
carrier self-energy and we can do the further consideration only for one of them, e.g., for the x-type by
equation (40), multiplying the result by the total number of dopants.
In this course, we focus on the energy gap ε+,max 6 ε 6 ε−,min between the subbands and specifically
on the possibility for in-gap localized levels to appear here and to subsequently generate narrow impurity
bands. The general observation is that, for positive Vc value, such levels can be only created by the
majority subband whose corresponding locator function f+(ε) = N−1 ∑
k(ε − ε+,k)
−1 is positive within
the gap, but not by the minority one with f−(ε) < 0 here. So, the most important momentum range for
the majority locator is the vicinity of M-point in BZ, where both scattering matrices tend to a simple
constant: V̂ (x,z)k,k′ ≈ Vc
[
1 +O(q2
+)
]
, thus justifying the Lifshitz approximation also for this case.
In the framework of this approximation, the T-matrix for carrier states is obtained in an analogy to
the boson case, equation (35):
T̂c(ε) =
[
1 + τ̂3
2
T+(ε) +
1 − τ̂3
2
T−(ε)
]
⊗ 1̂, (41)
but with the subband terms T±(ε) = Vc/[1 − Vc f±(ε)]. Here, the majority locator f+ (namely, its relevant
real part) can be well approximated using the expansion of equation (32) within its ∼ q2 term:
f+(ε) ≈
1
4πt ′
ln
4πt ′
ε − ε+,max
(42)
33704-12
On the theory of high-Tc superconductivity of doped cuprates
[to be compared with equation (14) for boson g-functions], and this readily defines the position of
localized in-gap level:
εloc = ε+,max + 4πt ′e−4πt′/Vc , (43)
whose separation from the band edge ε+,max occurs sizable compared to the band gap. Thus, choosing
the perturbation parameter Vc = 0.22 eV and the previous parameters of the host system, we obtain this
separation εloc − ε+,max ≈ 36 meV, nearly half of the gap, ε−,min − ε+,max = 70 meV.
At finite dopant concentration c, the renormalized dispersion equation for the majority subband (in
T-matrix approximation):
ε − ε+,k − c ReT+(ε) = 0 , (44)
describes a slightly modified main band and also a new impurity subband that arose near εloc, as seen in
their modified DOS: ρ̃(ε) = ρ+[ε − c ReT+(ε)] presented in figure 6. This function permits to evaluate
the necessary precondition for the onset of superconductivity in the doped system, its transition from
original insulating to metallic state with the growing doping. First of all, this requires that the Fermi level,
εF, (located within the bandgap in the absence of doping) should occur within the continuous spectrum,
and we can check this condition through analysis of the DOS in figure 6. It is very important that the full
weight of the impurity subband:
wimp =
εimp,max∫
εloc
ρ̃(ε)dε , (45)
is found notably smaller than the total weight c of the doped holes (for instance, wimp ≈ 0.34c for the
case in figure 6), due to a certain weight transfer from the impurity subband to its generating majority
subband. Thus, the resting c − wimp holes should occupy a certain range from ε+,max to εF within the
majority subband, according to:
ε+,max∫
εF
ρ̃(ε)dε =
c − wimp
2
(46)
(taking into account the spin degeneracy of the majority subband), and the numeric solution of this
equation with respect to εF gives the Fermi level depth δF in function of doping c, shown by the solid
Figure 6. Density of carrier states near the band gap in presence of dopants with concentration c = 0.1
and perturbation potential Vc = 0.22 eV. The impurity subband (hatched) formed above the localized
impurity level εloc has the weight of only wimp ≈ 0.34c when the rest 0.66c of the doped holes occupy
the top of majority subbands (with σ =↑, ↓) above the Fermi level. Besides the overall spectrum shift to
the εF reference, its certain deformation vs the undoped case in figure 3 is seen, e.g., in the van Hove ε+,s
peak getting closer to the band top ε+,max while the initial bandgap (ε+,max, ε−,min) getting wider.
33704-13
Y.G. Pogorelov, V.M. Loktev
Figure 7.Doping dependent Fermi level depth δF, obtained from equations (45), (46), in comparison with
that for the Mott mobility edge, δc , estimated from equation (47).
line in figure 7. From comparison with the profiles in figure 3, this level defines the Fermi line in BZ
as a set of disjoint Fermi arcs, in correspondence with the experimental observations in underdoped
perovskites [38].
However, the non-zero Fermi depth is not yet sufficient for the system metallization. It is important
to take in mind that the dispersion equation, equation (44), only makes sense for truly band-like states
with well defined momentum values, satisfying the known Ioffe-Regel-Mott (IRM) criterion [19, 39]:
q+ · ∇ε+,k � Γ(ε) , (47)
with the quasiparticle inverse lifetime Γ(ε) = c ImT+(ε). The energy range where this criterion fails
is occupied by randomly localized states, not contributing either to metallic conductivity or to SC
condensate, though the above estimates of their integrated DOS in equations (45), (46) with ρ̃(ε) are not
sensitive to this. The threshold between the band-like and localized ranges, the Mott mobility edge εc ,
is estimated from equation (47) at � changed for ∼. Then, a sufficient condition of metallization just
consists in the Fermi level occurring within the band-like range, εF < εc .
Now, the straightforward use of equations (28), (42) in equation (47) gives an estimate for the mobility
edge depth δc in function of doping (the dashed line in figure 6). Its comparison with the Fermi depth
δF function shows that the two should be very close until the doping value ≈ 0.05 and then a growing
dominance of δF after this value, indicating it as the critical value cmet for metallization in the considered
model, in agreement with its experimental observation in doped perovskite La2CuO4.
Moreover, it should be noted that the effective weight of conductive states in such a metallic system
results anomalously low:
wmet(c) =
εF∫
εc (c)
ρ(ε)dε � c, (48)
with respect to the total c of the carriers doped into the system, as seen from the numerical calculation
by equation (45) shown in figure 8, this deficit being at the cost of the impurity subband and of the
localized range in the main subband. At last, there is a progressive deformation of the overall spectrum
with doping, as seen in some compression of the upper part of the majority subband and in extension of
the gap between the majority and minority subbands in figure 6 compared to figure 3.
Now, we are in a position to discuss the onset and properties of SC state in the doped perovskite
system with the above defined normal state properties.
5. Superconductivity
As it was already referred, the unusual features of SC state in doped perovskite compounds are mainly
treated in literature within different scenarios of strongly correlated electrons in a uniform medium, also
33704-14
On the theory of high-Tc superconductivity of doped cuprates
Figure 8. Doping dependence of impurity subband weight from equations (45), (46), and the mobile
carriers weight wmet, compared to the total weight c of the doped carriers.
incorporating the doping effects into this framework [40]. Here, we instead consider the narrow J-
induced gap and explicit localization by dopant potential as effective factors of such strong correlations.
A similar approach was recently used for a description of destructive effects by magnetic impurities
on high-Tc SC order [41], [22] but now we focus on its relevance for establishing of such an order.
Namely, the possibility for SC coupling between charge carriers in the considered system is brought by
the “anomalously” averaged part of its four-fermion Hamiltonian, equation (21). Then, using the unitary
transformation, equation (24), the eigen-modes from the relevant majority subband (for brevity, this
subband index is omitted in what follows) are extracted from both longitudinal and transversal terms in
equation (21), enabling the common formation of Cooper pairs. Thus, we get the anomalous part of the
SC Hamiltonian as:
H(sc)an =
∑
k
(
∆kα↑,kα↓,−k + ∆
∗
kα
†
↓,kα
†
↑,−k
)
, (49)
with the gap function:
∆k =
1
N
∑
k′
V (sc)k,k′ 〈α
†
↑,k′α
†
↓,−k′〉 (50)
and the SC coupling function:
V (sc)k,k′ = Jγk−k′
(
1 + cos2 λk + cos2 λk′ − 2 cos2 λk cos2 λk′
)
. (51)
Then, taking the normal part of this Hamiltonian from equation (25) as:
H(sc)n =
∑
σ,k
ξkα
†
σ,kασ,k, (52)
with the dispersion law ξk = εk − εF, the usual diagonal form of the SC Hamiltonian:
H(sc) = H(sc)n + H(sc)an =
∑
σ,k
Ekα̃
†
σ,kα̃σ,k, (53)
contains the BSC spectrum Ek =
√
ξ2
k + |∆k |2 and the Bogolyubov quasiparticle operators:
α̃σ,k = cos φkασ,k + sin φkα
†
−σ,−k,
with φk = arctan Ek/|∆k |.
As known, the BCS standard routine for common metals expresses the anomalous average in the
gap function, equation (50), through the spectral theorem, equation (3), where the imaginary part of
anomalous GF is:
1
π
Im 〈〈α†
↓,−k |α
†
↑,k〉〉 = ∆kδ
(
ε2 − E2
k
)
tanh
βε
2
. (54)
33704-15
Y.G. Pogorelov, V.M. Loktev
Then, its integration in ε results in the general gap equation:
∆k =
1
2N
∑
k′
′V
(sc)
k,k′∆k′
Ek′
tanh
βEk′
2
, (55)
where the summation range in
∑′
k′ is delimited by isoenergetic surfaces of equal energy distance ωD (the
Debye energy) up and down from εF (the BCS shell), and the SC coupling function is usually taken in
the factorized form, V (sc)k,k′ = V (sc) fk fk′ , with the SC coupling parameter V (sc) and the SC symmetry factor
fk = f (ϕ)θ(ω2
D − ξ
2) restricted to the BCS shell and admitting some angular dependence. Then, the gap
function is expressed as ∆k = ∆ fk where the gap parameter ∆ is found from the BCS gap equation:
1
λ
=
1
2π
2π∫
0
dϕ f 2(ϕ)
ω∫
0
dξ√
ξ2 + ∆2 f 2(ϕ)
tanh
β
√
ξ2 + ∆2 f 2(ϕ)
2
(56)
with the dimensionless SC coupling constant λ = V ρF, and this should define all the SC state properties.
However, the considered case of doped and disordered metal differs from the above classical scenario
in some essential conditions:
I) the band-like states possess a finite linewidth Γ(ε)whose characteristic value near the Fermi energy
ΓF = Γ(εF) ∼ δc [by equation (47) at the ∼ relation] can be comparable to or even greater than the
expected SC spectrum gap ∆,
II) the inner limit for
∑′ around the M-point is defined by the mobility edge εc , since there are no
mobile carriers for εσ,k > εc , so the BCS shell upper width (above εF), wu(c) = δF − δc , only makes
sense at c > cmet and then it grows with c as shown in figure 7,
III) the outer
∑′ limit is defined by the decay of SC coupling function and it is deduced from
equation (51) to be near the isoenergetic line of ∼ J, which is doping independent, but the BCS shell
lower width (below εF): wl(c) = J − δF, decreases with c from a high initial value and vanishes at
certain ccr.
The condition I) modifies equation (54) by changing there the delta-function for a Lorentzian:
2εΓF(
ε2 − E2
k − G2
F
)2
+ 4ε2Γ2
F
,
and, after the spectral theorem integration in ε (at T = 0), this introduces to equation (55) the factor:
1 −
1
π
arctan
2Ek′ΓF
E2
k′ − Γ
2
F
,
besides E−1
k′ , which notably differs from unity at Ek′ ∼ ΓF.
The conditions II) and III) raise the question of restoring the particle-antiparticle symmetry (ξ ↔ −ξ)
of the SC spectrum, equation (49), at the original asymmetry of inner and outer limits in the gap equation,
this asymmetry yet varying with c. The simplest remedy here can be sought in the choice of symmetric
but c-dependent BCS shell of the energy width:
w(c) = min {wu(c), wl(c)} , (57)
as shown in figure 9.
Next, though there is no exact factorization in the full functional form by equation (51), it appears in
the dominant part of its γ-term for k and k′ near the M-point:
γk−k′ ≈ sin kx
2 sin kz
2 sin k′x
2 sin k′z
2 ,
while all the cos2 λ terms in this k, k′ range are close to unity within O(q4) � 1 precision, defining the
approximate factor function:
fk = sin kx
2 sin kz
2 , (58)
33704-16
On the theory of high-Tc superconductivity of doped cuprates
Figure 9. Doping dependent BCS-like energy shell around the Fermi level obtained by symmetrization
of two limiting functions wu(c) and wl(c) (with their mirror images) and the characteristic concentration
values to produce the bell-shaped ∆(c) behavior.
besides the coupling parameter V (sc) = J. Then, it should be noted that the gap function ∆k resulting
from the choice of factor by equation (58) automatically presents the d-wave symmetry with respect to
the BZ Γ-point, since this fk changes its sign at different Fermi arcs (near different C4 replicas of the
M-point) as seen in figure 10. Thus, this is straightly defined by the CuO2 lattice geometry and by strong
AFM spin correlations, getting rid of a more involved analysis on equation (55) with several competing
candidate symmetries of the SC order.
Further on, the factor function square, f 2
k , decays from unity in the M-point down to 1/2 at the
isoenergetic line by ≈ J below ε+,max, justifying its use for the BCS shell in the gap equation. As to the
Figure 10. (Colour online) The d-wave symmetry of the SC coupling factor fk (and so of the gap function
∆k) when referred to the central Γ-point in BZ. Red and blue areas present respectively the ranges of its
positive and negative values (in the limits defined for the doping level c = 0.15) and correspond to the
d-wave antinodal directions, while its zero values along Γ-X and Γ-Y axes do to the nodal directions.
33704-17
Y.G. Pogorelov, V.M. Loktev
Figure 11. (Colour online) a) Numerical solutions of the gap equation, equation (59), for different doping
levels: c = 0.075 (blue), c = 0.1 (green), c = 0.15 (orange), c = 0.2 (red), c = 0.22 (purple). The
suppression, due to the Fermi linewidth ΓF (downsteps in the curves), is most notable at intermediate
dopings (around the bell top, at copt) while tending to the standard BCS solutions near the end points
cmet and ccr. b) The resulting non-monotonous (bell-shaped) ∆(c) behavior.
d-wave angular dependence for this factor, it can be neglected when integrating mainly within the central
sector of one BZ quadrant (of one lobe in figure 10).
Under these simplifications, the actual gap equation (at zero temperature) is written as:
1
λsc
=
tc∫
0
dt
(
1 −
1
π
arctan
2δc cosh t
cosh2 t − δ2
c
)
, (59)
where λsc = J/t ′ resulted from the expansion: ε+,k ≈ ε+,max − t ′q2, of equation (26) near the M-point
and the integration upper limit is doping dependent: tc = arcsinh[w(c)/∆]. Its numeric solution for the
gap parameter ∆ in function of doping c presents a non-monotonous dependence (figure 11) vanishing at
the terminal values cmet and ccr, in qualitative agreement with the experimentally observed bell-shaped
∆(c) curve in doped high-Tc cuprate systems. The effect of Fermi level broadening [by the arctan term in
equation (59)] is most pronounced for intermediate c values from the whole doping range for SC.
Another important feature of the SC state corresponding to this solution is anomalously low superfluid
density ns, close to the effective weight of conductive hole states by equation (48) at not too high doping
levels and vanishing at c → ccr, in qualitative similarity with the experimental observations in the
overdoped range [15]. Moreover, an additional analysis shows that, in the system normal state, there can
be also conductive states in the impurity subband but they do not contribute to the SC condensate (this
issue staying beyond the scope of the present work).
6. Discussion and conclusions
The above model consideration does not pretend to readily give a complete description of all the
details in the multifaceted process of SC pairing in real doped cuprate perovskites (which are known
as “bad metals” [8]) and its transformations with the external parameters such as temperature, magnetic
field, current density, etc. Nevertheless, this first model approximation demonstrates that some basic
features of such a process can be really determined by the specific mechanisms introduced here:
33704-18
On the theory of high-Tc superconductivity of doped cuprates
- the spin-exchange splitting of the initial spin-independent band structure into the subbands of spin-
majority and spin-minority carriers, as valence and conductance bands for a narrow gap insulating state
of the undoped system;
- the fundamental role of the localized impurity level produced within this bandgap by the dopant
potential to give rise to a continuous impurity subband with a growing doping;
- a strong hybridization between the spin-majority and impurity subbands that produces a significant
quasiparticle weight transfer from impurity to valence subband and hence a transfer of the Fermi level to
within the spin-majority subband, near its top;
- the essential disorder effect on the states near this Fermi level, defining the necessary condition
for metallization of the initially insulating system and an anomalous closeness of this level to the Mott
mobility edge delimiting the quasiparticle weight of conductive states;
- the strong dependence of the size and shape of the Fermi surface in the Brillouin zone on the doping
level above the metallization threshold, with a tendency to be transformed from separate Fermi arcs to a
closed Fermi line;
- the specific limitation of the ranges for SC pairing on carrier states above and below the Fermi level
(an analog to the classical BCS shell), whose width is determined by the mobility edge in the normal
spectrum combined with the common decay range of the SC coupling function;
- the d-wave symmetry of the resulting SC order parameter directly related to the local symmetry
of two AFM sublattices, its non-monotonous dependence on the doping level, and an anomalously low
superfluid density of the SC condensate.
Other physical properties of the proposed model, including its behavior at finite temperatures, will
constitute the topic of further studies.
Acknowledgements
V.M.L. thanks the Physics and Astronomy Department of National Academy of Sciences of Ukraine
for supporting these researches within the projects No. 0117U00236 and No. 0117U00240.
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До теорiї високотемпературної надпровiдностi допованих
купратiв
Ю.Г. Погорєлов1, В.М. Локтєв2,3
1 IFIMUP-IN, факультет фiзики та астрономiї, Портський унiверситет, Порто, Португалiя
2 Iнститут теоретичної фiзики iм.М.М. Боголюбова НАН України,
вул.Метрологiчна, 14-б, 03143 Київ, Україна
3 Нацiональний технiчний унiверситет України «Київський полiтехнiчний iнститут iменi Iгоря
Сiкорського», пр-т Перемоги, 37, 03056 Київ, Україна
Здiйснено теоретичний аналiз можливих ефектiв, спричинених у високотемпературних надпровiдних мi-
дних перовскiтах присутнiстю допантiв. Розрахованi основнi спектри спiнових та електронних збуджень i
спостережуванi властивостi купратiв, насамперед поведiнка параметра порядку залежно вiд концентрацiї
допантiв. Ключова особливiсть запропонованої фiзичної картини полягає у формуваннi всерединi дiеле-
ктричної щiлини вихiдного недопованого матерiалу специфiчної домiшкової пiдзони, яка слугує джере-
лом наступних металiзацiї i переходу у надпровiдний стан з анiзотропним параметром порядку.
Ключовi слова: концентрацiя допантiв, магнiтнi та eлектроннi спектри купратiв, надпровiднiсть,
надпровiдна щiлина i температура переходу
33704-20
https://doi.org/10.1080/00018736700101265
https://doi.org/10.1088/0022-3719/20/21/008
https://doi.org/10.1063/1.5020892
https://doi.org/10.1016/0921-4534(89)90456-5
https://doi.org/10.1070/pu1960v003n03abeh003275
https://doi.org/10.1143/PTP.14.351
https://doi.org/10.1063/1.1943533
https://doi.org/10.1103/PhysRevB.96.115117
https://doi.org/10.1103/PhysRevB.42.10220
https://doi.org/10.1103/PhysRevB.48.9788
https://doi.org/10.1103/PhysRevB.50.13768
https://doi.org/10.1016/0022-3697(95)00269-3
https://doi.org/10.1103/PhysRevLett.112.017004
https://doi.org/10.7566/JPSJ.87.063707
https://doi.org/10.1063/1.5037550
https://doi.org/10.1038/nature05872
https://doi.org/10.1103/RevModPhys.78.17
https://doi.org/10.1103/PhysRevLett.117.257002
Introduction
Green functions and observable values
Model Hamiltonians and excitation spectra
Spin dynamics
Charge dynamics
Doping effects
Superconductivity
Discussion and conclusions
|