Spin excitations in layered cuprates: a Fermi-liquid approach
We review some important aspects of the doping dependence of many physical properties of the high-Tc cuprates based on a Fermi liquid-like approach. In particular, we show that the spin fluctuation-mechanism of superconductivity on the basis of a microscopic Eliashberg approach supports the idea...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2006
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| Cite this: | Spin excitations in layered cuprates: a Fermi-liquid approach / I.Eremin, D. Manske // Физика низких температур. — 2006. — Т. 32, № 6. — С. 683–699. — Бібліогр.: 45 назв. — англ. |
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irk-123456789-1202092017-06-12T03:04:47Z Spin excitations in layered cuprates: a Fermi-liquid approach Eremin, I. Manske, D. Специальный выпуск superconductivity: XX years after the discovery We review some important aspects of the doping dependence of many physical properties of the high-Tc cuprates based on a Fermi liquid-like approach. In particular, we show that the spin fluctuation-mechanism of superconductivity on the basis of a microscopic Eliashberg approach supports the idea that the symmetry of the superconducting order parameter is of dx₂₋y₂ - wave type. Furthermore, the renormalization of the quasiparticle spectrum caused by the scattering on spin fluctuations results in the so-called kink feature seen in ARPES. The peculiar momentum dependence of the spin fluctuations will result in a strong anisotropy of the renormalization at different parts of the first Brillouin zone and thus will lead to a strong anisotropy of the kink. Another important achievement of the microscopic Eliashberg approach is that the spin excitations spectrum renormalizes strongly below Tc due to occurrence of superconductivity with a dx₂₋y₂ -wave order parameter which yields to a formation of the so-called resonance peak that can be viewed as a spin exciton. The topology of the Fermi surface and the momentum dependence of the superconducting. 2006 Article Spin excitations in layered cuprates: a Fermi-liquid approach / I.Eremin, D. Manske // Физика низких температур. — 2006. — Т. 32, № 6. — С. 683–699. — Бібліогр.: 45 назв. — англ. 0132-6414 PACS: 74.72.–h, 74.20.Mn, 74.25.–q http://dspace.nbuv.gov.ua/handle/123456789/120209 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Специальный выпуск superconductivity: XX years after the discovery Специальный выпуск superconductivity: XX years after the discovery |
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Специальный выпуск superconductivity: XX years after the discovery Специальный выпуск superconductivity: XX years after the discovery Eremin, I. Manske, D. Spin excitations in layered cuprates: a Fermi-liquid approach Физика низких температур |
| description |
We review some important aspects of the doping dependence of many physical properties of the
high-Tc cuprates based on a Fermi liquid-like approach. In particular, we show that the spin fluctuation-mechanism
of superconductivity on the basis of a microscopic Eliashberg approach supports
the idea that the symmetry of the superconducting order parameter is of dx₂₋y₂ -
wave type.
Furthermore, the renormalization of the quasiparticle spectrum caused by the scattering on spin
fluctuations results in the so-called kink feature seen in ARPES. The peculiar momentum dependence
of the spin fluctuations will result in a strong anisotropy of the renormalization at different
parts of the first Brillouin zone and thus will lead to a strong anisotropy of the kink. Another important
achievement of the microscopic Eliashberg approach is that the spin excitations spectrum
renormalizes strongly below Tc due to occurrence of superconductivity with a dx₂₋y₂ -wave order
parameter which yields to a formation of the so-called resonance peak that can be viewed as a spin
exciton. The topology of the Fermi surface and the momentum dependence of the superconducting. |
| format |
Article |
| author |
Eremin, I. Manske, D. |
| author_facet |
Eremin, I. Manske, D. |
| author_sort |
Eremin, I. |
| title |
Spin excitations in layered cuprates: a Fermi-liquid approach |
| title_short |
Spin excitations in layered cuprates: a Fermi-liquid approach |
| title_full |
Spin excitations in layered cuprates: a Fermi-liquid approach |
| title_fullStr |
Spin excitations in layered cuprates: a Fermi-liquid approach |
| title_full_unstemmed |
Spin excitations in layered cuprates: a Fermi-liquid approach |
| title_sort |
spin excitations in layered cuprates: a fermi-liquid approach |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2006 |
| topic_facet |
Специальный выпуск superconductivity: XX years after the discovery |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120209 |
| citation_txt |
Spin excitations in layered cuprates: a Fermi-liquid
approach / I.Eremin, D. Manske // Физика низких температур. — 2006. — Т. 32, № 6. — С. 683–699. — Бібліогр.: 45 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT eremini spinexcitationsinlayeredcupratesafermiliquidapproach AT mansked spinexcitationsinlayeredcupratesafermiliquidapproach |
| first_indexed |
2025-07-08T17:26:45Z |
| last_indexed |
2025-07-08T17:26:45Z |
| _version_ |
1837100551652769792 |
| fulltext |
Fizika Nizkikh Temperatur, 2006, v. 32, No. 6, p. 683–699
Spin excitations in layered cuprates: a Fermi-liquid
approach
I. Eremin1,2 and D. Manske3
1 Max-Planck Institut f�r Physik Komplexer Systeme, Dresden D-01187, Germany
E-mail: ieremin@mpipks-dresden.mpg.de
2 Institut f�r Mathematische/Theoretische Physik,
Technische Universit�t Carolina-Wilhelmina zu Braunschweig, Braunschweig D-38106, Germany
3 Institut f�r Theoretische Physik, ETH Z�rich, H�nggerberg, CH-8093 Z�rich, Switzerland
E-mail: manske@itp.ethz.ch
Received January 5, 2006
We review some important aspects of the doping dependence of many physical properties of the
high-Tc cuprates based on a Fermi liquid-like approach. In particular, we show that the spin fluc-
tuation-mechanism of superconductivity on the basis of a microscopic Eliashberg approach sup-
ports the idea that the symmetry of the superconducting order parameter is of dx y2 2�
-wave type.
Furthermore, the renormalization of the quasiparticle spectrum caused by the scattering on spin
fluctuations results in the so-called kink feature seen in ARPES. The peculiar momentum depend-
ence of the spin fluctuations will result in a strong anisotropy of the renormalization at different
parts of the first Brillouin zone and thus will lead to a strong anisotropy of the kink. Another im-
portant achievement of the microscopic Eliashberg approach is that the spin excitations spectrum
renormalizes strongly below Tc due to occurrence of superconductivity with a dx y2 2�
-wave order
parameter which yields to a formation of the so-called resonance peak that can be viewed as a spin
exciton. The topology of the Fermi surface and the momentum dependence of the superconducting
gap explains the peculiar dispersion of the resonance peak in good agreement with experiments.
PACS: 74.72.–h, 74.20.Mn, 74.25.–q
Keywords: high-Tc cuprates, Fermi-liquid approach, Brillouin zone.
1. Introduction
1.1. Generic phase diagram
In the last decade significant achievements in the
understanding of the high-Tc superconductivity phe-
nomenon have been made on the basis of spin fluctua-
tion-mediated Cooper pairing. Over the years a large
amount of experimental data has been collected which
suggest that an anisotropic superconducting order pa-
rameter is realized in the cuprate superconductors.
Moreover, there are strong experimental indications
that the superconducting order parameter is off d-
wave type. A natural explanation of this symmetry
originates from the concept of Cooper pairing due to
exchange of antiferromagnetic spin-fluctuations [1].
This hypothesis is supported by the fact that the par-
ent compounds of cuprate superconductors (like
La CuO2 4 or YBa Cu O2 3 6) are quasi two-dimensional
antiferromagnetic insulators. Doping these systems
with holes (for example, La CuO2 4 with strontium or
YBa Cu O2 3 6 with oxygen) results, first, in a transi-
tion into the metallic phase and, second, in the for-
mation of unconventional superconductivity below
a certain transition temperature Tc that increases with
increasing doping content towards its maximum
value. Further doping of the system yields, however,
in a decrease of Tc and finally to the disappearance of
superconductivity at some high doping level. Despite
of the characteristic differences between the cor-
responding phase diagrams of various hole-doped
cuprates, one can map all of them onto the so-called
generic phase diagram. This is possible if one assumes
© I. Eremin and D. Manske, 2006
that the processes relevant for the Cooper pairing and
normal state dynamics already occur in a single
CuO2 plane which is an important element of the
crystal structure of the cuprates. In such a picture, the
physical properties of each cuprate compound depend
on the number of carriers within the CuO2 plane.
In Fig. 1 we show an illustration of the correspond-
ing generic phase diagram for hole-doped cuprates
that reveals the main features of the cuprates. One
sees that superconductivity occurs in cuprates in a
close vicinity to antiferromagnetism. Although anti-
ferromagnetism and superconductivity seem not to co-
exist in cuprates, several experimental techniques
[2–4] indicate the presence of short-range antiferro-
magnetic spin fluctuations even at optimal doping
(i.e. maximum of Tc) and in overdoped samples. This
suggests their possible importance for the Cooper
pairing as well as for the anomalous normal state
properties, in particular, close to antiferromagnetic
transition. Certainly Landau’s simple Fermi-liquid
theory cannot be applied in the underdoped region.
Moreover, there are two other curves on the phase dia-
gram that do not refer necessarily to real phase transi-
tions, but rather represent an anomalous crossover be-
havior of the cuprates in these regions. The first curve,
T*( )x , refers to the so-called pseudogap formation
temperature. As indicated in many experiments [5],
below T* the spectral weight at the Fermi level
strongly decreases with decreasing temperature which
is represented in gap-like features in the charge and
spin responses. Another characteristic temperature,
Tc*, represents some experimental findings interpreted
in terms of the so-called formation of (local) Cooper
pairs without phase coherence [6]. There one further
reports on a strong Nernst signal which is interepreted
in terms of existing vortices [7]. Note, the behavior of
the normal state spin and charge dynamics in the
underdoped regime are far from being understood af-
ter twenty years of intensive research.
At the same time, on the other part of the phase di-
agram (so-called overdoped regime) the behavior of
cuprates in the normal state can be well described
within the concept of the Fermi liquid. This indicates
that the system is characterized by a presence of the
well-defined quasiparticles well above Tc, presence of
a rigid band which width does not change signifi-
cantly upon the changing of the dopant concentration.
In such a situation the interaction between quasi-
particles can be to a large extent described within per-
turbation theory. Moreover, the relative simplicity of
the normal state seems to make an understanding of
the superconducting properties of cuprates easier both
theoretically and experimentally in the region of
overdoped cuprates.
1.2. Superconducting state
Symmetry of the order parameter. In Fig. 1 we also
show the T xc( ) curve which has a characteristic shape
T x T xc c( ) ( . )max� � � �1 016 2 as have been pointed
out in Ref. 8. Below Tc superconductivity occurs
and it is believed to possess dx y2 2�
-wave symmetry
of the superconducting order parameter, i.e. � k �
� � �� 0 2( cos cos )k kx y . The evidence for dx y2 2�
-
wave pairing comes from several sources, and in par-
ticular from phase sensitive measurements [9]. An-
other remarkable feature about the superconducting
state of high-Tc cuprates is that they differ from con-
ventional superconductors by having a small coher-
ence length �. This length is usually associated with
the average size of a Cooper pair, which is for conven-
tional superconductors � 500 �. Therefore, the size of
684 Fizika Nizkikh Temperatur, 2006, v. 32, No. 6
I. Eremin and D. Manske
T
e
m
p
e
ra
tu
re
TN
Pseudogap T �
“Spin fluctuations”
(paramagnons)
Conventional
Fermi liquidT�c
AF
Doping x
underdoped overdoped
0 0.1 0.2 0.3
SC d-waveTc
Fig. 1. Schematic phase diagram of hole-doped cuprates.
High-Tc superconductivity occurs in the vicinity of an
antiferromagnetic phase transition. The corresponding
superconducting order parameter below the superconduct-
ing transition temperature Tc is of d-wave symmetry. In
the overdoped region, i.e. x > 0.15, the cuprates behave
like a conventional Fermi liquid, whereas in the
underdoped regime below the pseudogap temperature T*
one finds strong antiferromagnetic (AF) correlations. As
we will discuss below, Cooper pairing can mainly be de-
scribed by the exchange of AF spin fluctuations (often
called paramagnons). The doping region between Tc and
Tc
* (shaded region) may be due to local phase-incoherent
Cooper-pair formation. Only below Tc these pairs become
phase coherent.
the Cooper pair is larger than the average distance be-
tween pairs resulting in a strong overlap of the corre-
sponding wave functions. On the other hand, super-
conducting cuprates only have � � 12–15 �. Thus it is
possible to find local pairs without long-range phase
coherence. Another unusual feature relates to the ratio
of 2 50� � �k TB c that is larger than the BCS ratio of
3.52. At the same time, such a ratio points towards a
possible applicability of the strong-coupling
Eliashberg-like theories for high-Tc cuprates, at least
in the overdoped and optimally-doped cuprates. How-
ever, the unconventional symmetry of the supercon-
ducting order parameter requires another mechanism
than the exchange of phonons for the Cooper pairing.
Magnetic excitations. The idea of a spin-fluctua-
tions mechanism of superconductivity conceptually is
not new and has been originally proposed to describe
the superfluid properties of 3He [10]. In application
to cuprates it has been first developed on the basis of a
phenomenological approach which is based on the
analysis of the experimental data [11,12]. In fact, the
phenomenological approach has been proposed before
the full development of the microscopic description of
high-Tc superconductivity, a description based on
self-consistently solving the equations of the electron
self-energy and the magnetic spin susceptibility in the
frame of the Hubbard model was made. A detailed
study of the temperature behavior of nuclear magnetic
resonance (NMR) characteristics, such as the Knight
shift and the spin-lattice relaxation time, and the in-
elastic neutron scattering (INS) intensity has shown
that the detected anomalies can be explained by the
presence of strong antiferromagnetic fluctuations near
the antiferromagnetic wave vector Q � ( , )� � . Ori-
ginally, Millis, Monien, and Pines [12] proposed a
so-called Ornstein—Zernike form for the low-fre-
quency magnetic susceptibility,
( , )q , in the normal
state of hole-doped cuprates that allows for a sharp en-
hancement of fluctuations near Q:
�
( , )
( )
q
q Q
Q
�
� � � �1 2 2 i sf
. (1)
Here, Qis the value of the static spin susceptibility
at the wave vector Q, � is the magnetic correlation
length, and
sf is the characteristic frequency of spin
fluctuations. For different materials these parameters
have been found by fitting experimental data on
NMR, since the general relationship relating the data
to the dynamical spin susceptibility are well-known.
The susceptibility determined in this manner was
then used to calculate various quantities in the super-
conducting and normal phases of this compound, and
the results of such calculations can be compared with
the experimental data. In this way the main idea that
antiferromagnetic spin fluctuations are important for
the pairing process can be verified.
In the past, one of the biggest achievement of the
spin-fluctuation mediated Cooper-pairing mechanism
was a successful prediction of a correct symmetry of
the superconducting order parameter in cuprates. In
particular, in the phenomenological approach one first
sets up an effective Hamiltonian. It must consist of
two parts: H H H� �0 int , where
H C C0 � ��
� �
�
( )k k k
k
is the bare electron (hole) spectrum in the Cu–O
plane, andHint represents the interaction of electrons
and spin fluctuations:
H g s Sint � �� ( ) ( ) ( )q q q
q
. (2)
Here s( )q is the electron spin operator,
s C C( )q k q k
k
�
�
�
� �
�
��1
2 � �� �
��
� , (3)
where � represents the Pauli matrices, and S( )q is
the spin fluctuation operator, whose properties are
determined by the spin-spin correlation function
[the dynamical spin susceptibility tensor, defined by
Eq. (1)]. The interaction is assumed to be of
short-range type, with the result that g( )q is weakly
momentum dependent and may be set to a constant.
In the phenomenological theory by Pines and co-
workers, the coupling constant remains a free parame-
ter of the theory and can be fixed by a comparison
with experiment. In second order perturbation the-
ory, the interaction of the electrons and spin fluctua-
tions leads to the following effective electron-electron
interaction shown in Fig. 5 and the corresponding
expression for the self-energy of the one-particle
Green’s function in the normal state:
�( , ) ( , ) ( , ),k k k k
k
i g T i i G in n m m
m
� � � � �
�
��2
(4)
and superconducting state
�( , ) ( , )
( , ) ( ,
k k k
k k
k
i g T i i
G i G
n n m
m
m
� � � � � �
� � � �
�
��2
� �i im m
) ( , ) ,� k (5)
Eqs. (4) and (5) have to be combined with the Dyson
equation,
G i
i in
n n
( , )
( ) ( , )
k
k k
�
�
� � �
1
�
, (6)
which produces a closed system of equations for find-
ing the self-energy corrections for the normal and
Spin excitations in layered cuprates: a Fermi liquid approach
Fizika Nizkikh Temperatur, 2006, v. 32, No. 6 685
superconducting state. Here,
( , )q i l is the Matsu-
bara Green’s function of spin fluctuations which is re-
lated to the dynamic magnetic susceptibility by
�
( , )
Im ( , )
k
q
i
d
il
l
� �
�
��
��
� . (7)
As one could see within this approach the dynami-
cal spin susceptibility determines the properties of the
normal and superconducting state. In order to discuss
the symmetry of the superconducting order parameter
it is enough to look on the weak-coupling limit,
namely BCS gap equation which has the form at T � 0
( )Ek k k� ��2 2
:
� �( , )
( , )
( , )k
k k
k
kk
� �
� �
�
�
�
��g
E
m
2
2
. (8)
On of the immediate consequence of the Cooper pair-
ing due to spin fluctuations is their repulsive nature
in the singlet s-wave channel. Therefore, to overcome
the minus in front of Eq. (8), the order parameter has
to change sign. In high-Tc cuprates the dynamical
spin susceptibility is peaked at the antiferromagnetic
wave vector, Q, which connects the electronic states
at the Fermi surface close the (��,0), (0,��) in the
first Brillouin zone (BZ) (see Fig. 2 for the illust-
ration).
Then, one could see that the natural solution of the
gap equation for the particular topology of the Fermi
surface and effective pairing interaction is the
d
x y2 2
�
-wave one. In particular, due to the change of
sign of the order parameter the total minus in front of
the gap equation is overcome and the solution of the
gap equation is possible. Moreover, based on this
phenomenological form of the spin susceptibility one
can further determine Tc and the phase diagram which
to a large extent agrees with experiment. Note, such a
solution does not exclude a possible sub-dominant
contribution of the electron-phonon interaction if the
latter is peaked at momenta (�0 4. �,0), or (0,�0 4. �),
for example. The sign of the order parameter for elec-
tronic states in the first BZ connected by those vectors
still will be the same and thus, a further enhancement
of Tc is possible.
Despite the success of this phenomenological form
of the spin susceptibility in describing the normal
state spin dynamics it fails to explain the changes in
the spin excitation spectrum in the superconducting
state. In particular, in Fig. 3 we show the INS results
for slightly underdoped YBa Cu O2 3 6.7 [13]. The in-
tensity in INS is proportional to the magnetic struc-
ture factor that contains the imaginary part of the dy-
namical spin susceptibility. Thus, INS provides a
direct probe of the spin susceptibility. As one sees, at
the antiferromagnetic wave vector Q the normal state
spin dynamics follows the Ornstein-Zernike behavior
reflected by the rapid and linear increase of Im
( , )Q
and a maximum at
sf . However, the behavior of
Im ( , )
Q in the superconducting state is much dif-
ferent. The spin excitation spectrum strongly renor-
malizes and, first, becomes gapped at lower frequen-
cies and then a sharp develops at
res � 33 meV (it
shifts to 41 meV in optimally doped cuprates). The ap-
pearance of the resonance peak firstly reported in Ref.
14 suggests a strong renormalization of the spin fluc-
tuation spectrum in the superconducting state that
cannot be accounted by the simple phenomenological
form of the spin susceptibility. In a recent experimen-
686 Fizika Nizkikh Temperatur, 2006, v. 32, No. 6
I. Eremin and D. Manske
12 K
E, meV
IN
S
in
te
n
si
ty
200
200
0
0
0 30 60 90
200 K
YBa Cu O2 3 6.7
Fig. 3. INS results for optimally-doped YBa Cu O2 3 6.7
taken from Ref. [13] at the antiferromagnetic wave vector
Q = (�,�) for the normal and superconducting state.
0
–�
�
+ +
–
–
Q
(
,
)
AF
�
�
�
0 �
kx
ky
Fig. 2. Depicted Fermi surface for the layered cuprates in
the first BZ. The arrows show the electronic states con-
nected by the antiferromagnetic wave vector QAF. The
dotted lines and signs refer to the dx y2 2�
-wave symmetry
of the superconducting gap.
tal work [15] an INS measurements have found also a
resonant spin excitation in the overdoped and opti-
mally-doped Bi2Sr2CaCu2O8+y (BSCCO) with mode
energies Eres � 38 and 43 meV, respectively. This ob-
servation, together with the previous measurements in
underdoped phase of YBCO indicates an approxi-
mately constant ratio E /k TB cres = 5.4 through the
whole doping range. Very recently, the resonance
peak emerged as a key factor in the phenomenology
of the copper oxide superconductors and, in particu-
lar, of a single CuO2 plane due to the observation
of the resonance like feature in the superconducting
state of single CuO2 layer cuprate superconductor
Tl Ba CuO2 2 6�x near optimal doping [16].
One of the most appealing results concerning the
resonant spin excitations is their dispersion away from
(�,�). In particular, as one moves away from Q, the
peak disperses downwards nearly quadratically and its
intensity decreases rapidly, vanishing around
Q0 0 8 0 8� ( . , . )� � . Recent INS experiments in the
superconducting state of YBCO [17–19] have de-
tected a new resonant magnetic excitation at incom-
mensurate momenta, but at frequencies larger than
res ( )Q . Moreover, the high and low-energy excita-
tions are separated from each by a region (so-called
«silent bands») where no intensity has been observed.
The whole dispersion of the resonant excitations is
shown in Fig. 4.
Furthermore, recent experiments on the untwinned
YBCO reveal a strong anisotropy of the resonant exci-
tations below
res ( )Q . In particular, in the momen-
tum space with a fixed energy, the resonant excitation
form a so-called ring structure with four pronounced
peaks along the bond directions. In the untwinned
samples two of these peaks are suppressed, while two
others remain [20]. All recent experimental facts are
confirmed by various groups, and the observation in a
different family of cuprate compounds indicate the ne-
cessity of the correct analysis of the spin fluctuations
spectrum and its doping dependence in high-Tc cup-
rates. This is a significant ingredient of a microscopic
theory of cuprates must contain.
Elementary excitations. Another important ques-
tion which one has to understand concerns the behav-
ior of the elementary excitations in a CuO2 plane. As-
suming the presence of strong antiferromagnetic spin
fluctuations one expects, following Eqs. (4)–(7), a
strong renormalization of the quasiparticle scattering
rate and the corresponding energy dispersion. (The
corresponding self-energy is illustrated in Fig. 5.) The
effects of the scattering of the quasiparticles by the
spin excitations can be seen by various experiments
such as resistivity, optical conductivity, angle re-
solved photoemission (ARPES), tunneling and some
others [21]. In particular, these days the development
of ARPES experiment allows to get detailed informa-
tion about the elementary excitations in the cuprates.
This, for example, concerns a so-called «kink» fea-
ture, observed recently in various cuprate compounds
[22–25]. Let us consider these experiments in more de-
tail. Under well-defined assumptions it is believed
that ARPES measures the spectral density of the elec-
trons in the CuO2 planes:
A( , )
Im ( , )
( ( ) Re ( , )) (Im ( , ))
k
k
k k k
�
�
�
� � � �
� �
� � �
1
2 2
�
� �
,
(9)
where �( ) (cos cos ) cos cosk � � � � � �2 4t k k t k kx y x y
� � � � �4 2 2t k kx y(cos cos ) is the unrenormalized
tight-binding energy dispersion with hoping integrals
between nearest, next-nearest, and so on sites. Due to
Spin excitations in layered cuprates: a Fermi liquid approach
Fizika Nizkikh Temperatur, 2006, v. 32, No. 6 687
E
n
e
rg
y,
m
e
V
70
60
50
40
30
20
(H,H)
0.3 0.4 0.5 0.6 0.7
Fig. 4. Measured dispersion of the resonant excitations away
from the antiferromagnetic wave vector QAF = 0.5(H,H)
along the diagonal of the first BZ as taken from Ref. 17.
The vertical dashed region indicates the position of the silent
bands.
� �
�q
G0 U G Ueff G
Fig. 5. Cooper-pairing in the cuprates due to coupling of
carriers (holes or electrons) in CuO2 planes to
antiferromagnetic spin fluctuations characterized by the
spin-susceptibility � �( , )q (G refers to matrix Green’s func-
tion of quasi-particles).
recent developments in ARPES, A( , )k
can be stud-
ied with high accuracy versus frequency at fixed mo-
mentum (so-called EDC) as well as function of mo-
mentum at fixed frequency (so-called MDC). One of
the most important information that one gets by ana-
lyzing MDC and EDC is the renormalized energy dis-
persion
k. In particular, the latter is obtained by
studying the maximum of Eq. (9) which is satisfied if
� � �( ) Re ( , )k k� 0 . (10)
Note that MDC curves suit the purpose of defining of
the energy dispersion much better due to a weaker
momentum than frequency dependence of the self-en-
ergy �( , )k
.
In Fig. 6 we reproduce the results of Lanzara et al.
[22] for the measured
k along (0,0) � (�,�) direc-
tion in the first Brillouine zone. One clearly sees that
the curve shows change in its slope around
(50�15) meV. This is a so-called kink feature which
was observed by several groups. As was argued in
Ref. 26, the kink feature is seen along all directions in
the BZ although it is doping and temperature depend-
encies are different. Originally, the kink feature was
investigated along (0,0) � ( , )� � direction. This is
connected to the fact that along (0,0) � ( , )� 0 direc-
tion there are additional effects such as matrix ele-
ments and bilayer splitting which complicate the anal-
ysis of the experimental data. At the same, as we will
argue, the main kink feature which could be related to
the feedback of superconductivity occurs along
( , ) ( , )0 � � �� due to the coupling to the resonant spin
excitations. Originally the kink feature was attributed
to a coupling of the itinerant carriers to phonons, in
particular, to a longitudinal optical phonon mode at
60 meV which was found to behave anomalously in
several experiments [22]. On the other hand, the
changes in the elementary excitations spectrum due to
coupling of the carriers to spin fluctuations are also
expected. Moreover, spin fluctuations would result in
rather different frequency and momentum dependen-
cies of the self-energy that will be reflected in the ex-
periment than in phonon case. The latter seems to be
confirmed by recent experimental results [26].
To conclude our Introduction, the intensive studies
of high-Tc cuprates in recent years have indicated that
there is a strong interplay between strong electronic
correlations and superconductivity in these materials
which changes upon doping. In particular, a crossover
from Fermi-liquid towards non-Fermi-liquid behavior
occurs if one goes from the over- to underdoped regime
and also an asymmetry with respect to hole or electron
doping is found. Superconducting properties in cup-
rates reveal the d-wave superconducting order parame-
ter and cannot be described in a frame of the standard
BCS theory. Despite of the variety of theoretical and
experimental problems that exist at the moment, in
our review we will concentrate on the study of the
spin excitations in the high-Tc cuprates and their feed-
back on the elementary excitations. We will show that
spin fluctuation-induced Cooper pairing allows to get
a consistent and self-contained explanation of many
experimental facts and to describe qualitatively the
phase diagram of cuprates upon doping.
2. Theory: generalized Eliashberg equations
As we have mentioned in the Introduction the
phenomenological approach based on the Cooper pair-
ing via exchange of antiferromagnetic spin fluctua-
tions explains some important facts on superconduct-
ing properties of cuprates. However, the complete
picture requires a developing of a microscopic theory.
A successful attempt in this direction was done inde-
pendently by several groups [27–30] in the framework
of a so-called FLEX approximation. Since the details
of this method were already discussed in several re-
views [1,31,32] let us just briefly discuss the theoreti-
cal aspects of FLEX.
The simplest model that may describe the physics
of a single CuO2 plane is a two-dimensional one-band
Hubbard model which is an appropriate starting point
of our calculations:
688 Fizika Nizkikh Temperatur, 2006, v. 32, No. 6
I. Eremin and D. Manske
E
n
e
rg
y,
m
e
V
0
–100
–200
20 K
100 K
� = 0.15
LSCO
0 1
k’
Fig. 6. ARPES results for the renormalized energy disper-
sion along (0,0) � (�,�) direction of the first Brillouine
zone taken from Ref. 22. The horizontal axis shows the
distance (k – kF) in the normalized units k k kF� � � .
H c c
U
N
c c c c
kk q
� �� �
� �
�
�� � �
�
� �
� �
�
� � � �
�
k k k
k
k k k q k q2 , , , , .
(11)
Here, ck�
� (ck�) are creation (annihilation) electron or
hole operators on site i with spin �, U corresponds to
the on-site effective Coulomb repulsion and
k is the
tight-binding energy dispersion of the form
�k x y x yt k k t k k� � � � � �2 4( cos cos ) cos cos ,
(12)
were t and t’ refers to the hopping of a hole(electron)
between nearest, next-nearest sites on the square lat-
tice and � is the chemical potential that defines the
doping. Here and in the following we set the lattice
constant to unity. The description of the electron-
and hole-doped cuprates within a one-band approxi-
mation is possible if one takes into account different
quasiparticle dispersion as observed in the experi-
ment [33].
In the one-band model, the same electrons partici-
pate in the formation of antiferromagnetic fluctua-
tions and in the Cooper pairing due to the exchange of
such fluctuations. In this many-body problem both the
magnetic susceptibility and the quasiparticle self- en-
ergy must be calculated self-consistently. We begin by
writing the electron self-energy in the normal state,
�( , )k i n
. In the Nambu representation, the 2�2 ma-
trix �( , )k i n
is given by the following expressions:
�( , ) [ ( , ) ( , )]
,
k k k k k
k
i V i U in s m s m
i n
�
�
�
�
� � � � � �
� �
� 2
0
0 0 1 1G i V i G in c m n
i n
( , ) ( , ) ( , ) .
,
k k k k
k
� � � � �� �
� �
�� �
� � �
� (13)
Here, � i (i = 0, 1, 2, 3) are the Pauli matrices,
G i n( , )k� �
is an electron Greens function, and
i i in n n�
� � � . The term U is m
2
0 �( , )k k� � re-
moves a double counting of diagrams in second order.
Vs and Vc are the matrix elements of the elec-
tron-electron interaction due to spin and charge fluc-
tuations:
V i U i
V i U i
s m s m
c m c m
( , ) ( , ) ,
( , ) ( , ),
q q
q q
� �
� �
�
�
3
2
1
2
2
2
(14)
where �c ni( , )q and �s ni( , )q are the dynamical
parts of the charge and spin susceptibility, respec-
tively. In the random phase approximation (RPA)
they are given by well-known diagrams corresponding
to summation of loop diagrams:
�
�
�
�
s m
s m
s m
c m
c
i
i
U i
i
( , )
( , )
( , )
,
( , )
( ,
q
q
q
q
q
�
�
�
0
0
0
1
i
U i
m
c m
�
�
)
( , )
.
1 0� q
(15)
Here �s mi0( , )q and �c mi0( , )q are irreducible parts
which in the single-loop approximation are given by
the following formulas:
�
� �
�
s m
n m n
k i
i
T G i i G i
0
0 0
1
2
( , )
[ ( , ) ( , ) ] ,
,
q
k q k
�
� � � �tr
�n
�
�
� �
�
�
c m
n m n
k i
i
T G i i G i
0
3 3
1
2
( , )
[ ( , ) ( , ) ]
,
q
k q k
�
� � � �tr
n
� ,
(16)
(17)
where tr denotes the trace of a two-row matrix.
The Green’s function G i n( , )k
and the self-energy
�( , )k i n
are connected by Dyson’s equation:
G i G i in n n( , ) [ ( , ) ( , )]k k k
� �� �
0
1 1� , (18)
where in the case of singlet pairing the zeroth-order
Green’s function is given by
G i in n0
1
0 3
� � �( , )k k
�
� . (19)
We expand the self-energy matrix in the standard
form:
�
�
( , ) [ ( , )]
( , ) ( , ) .
k k
k k
i i Z i
i i
n n n
n n
�
�
�
�
� � �
� �
1 0
3 1 (20)
Here Z is the mass renormalization factor that deter-
mines the renormalization frequency
~( , ) ( , )
k ki Z in n n� , (21)
�
( , )k i n specifies the energy shift due to interaction
of quasiparticles, and the function �( , )k i n
deter-
mines the superconducting gap:
�
�
( , )
( , )
( , )
k
k
k
i
i
Z in
n
n
� . (22)
Equations (13) and (20) determine simultaneously
three coupled nonlinear equations for these three
quantities:
~( , ) [ ( , )
( ,
,
k k k
k k
k
i V i i
V i i
n n s n m
m
c n
� � � � � �
� � � �
�
�
m
m
m
i
D i
)]
~( , )
( , )
,
k
k
�
�
(23)
Spin excitations in layered cuprates: a Fermi liquid approach
Fizika Nizkikh Temperatur, 2006, v. 32, No. 6 689
�
( , ) [ ( , )
( ,
,
k k k
k k
k
i T V i i
V i i
n s n m
m
c n m
� � � � � �
� � � �
�
�
)]
( , )
( , )
,
�
k k
k
� � �
�
i
D i
m
m
(24)
�( , ) [ ( , )
( ,
,
k k k
k k
k
i T V i i
V i i
n s n m
m
c n m
� � � � � �
� � � �
�
�
)]
( , )
( , )
,
� k
k
�
�
i
D i
m
m
(25)
where D i m( , )k�
is the denominator of the matrix
electron Green’s function:
D i i
i i
m m
m m
( , ) [~( , )]
[ ( , )] [ ( , )]
k k
k k
� � � �
� � � �
�
2
2 2� . (26)
These three equations must be augmented by an equa-
tion for the chemical potential
n T
i
i
k m
mn
� �
� �
���
�
1
2
2
0
�
( , )
( , )
k
k
k
�
. (27)
Equations (23)–(27) and (14)–(17) constitute a sys-
tem of coupled equations for the electron Green’s
function and the dynamic susceptibilities of the su-
perconductor and must be solved numerically for
given values of the parameters t, T, U and n by itera-
tions. For instance, we can plug the approximate
values of the magnetic and dielectric susceptibilities
into Eqs. (23)–(27) and calculate the values of
~( , )
k� i m , �
( , )k i n , and �( , )k i n
which are func-
tions that characterize the electron self-energy. Then,
inserting these values into Eqs. (14)–(17), we calcu-
late new values of the susceptibilities and plug them
into Eqs. (23)–(27) being kernels of integral equa-
tions, which are then used to calculate new values of
~( , )
k� i m , �
( , )k i n , and �( , )k i n
. This procedure is
repeated many times, as long as it is necessary to ob-
tain a stable result. This scheme allows to calculate
all properties of the system self-consistently such as
the superconducting phase diagram, superconducting
order parameter, energy dispersion and dynamical
spin susceptibility which then have to be compared
with experiment. An important remark which we
would like to stress is that in contrast to the elec-
tron-phonon interaction, the spin-fluctuation-medi-
ated pairing interaction is repulsive (in k-space) and
strongly momentum dependent. Therefore, as argued
in connection with Eq. (8) the symmetry of the
superconducting order parameter will be dx y2 2�
-wave
type. As already been mentioned the presented above
theory can explain the symmetry of the superconduct-
ing gap and also main parts of the phase diagram of
hole- and electron-doped cuprates. Thus, in the fol-
lowing we will mainly analyze the behavior of the
spin excitations and their feedback on the quasi-
particle dispersion.
3. Results and discussion
3.1. Spin excitations
The investigation of the spin fluctuations spectrum
is a key problem for understanding the physics of
cuprates. Many researchers believe that the most sig-
nificant problem which one has to clarify is the feed-
back of superconductivity on the spin excitation spec-
trum. This is of general interest, since in contrast to a
phonon spectrum in conventional superconductors,
spin excitations are strongly renormalized in the su-
perconducting state that allows to identify «finger-
prints» of the spin fluctuation scenario of supercon-
ductivity. For example, in many heavy fermions
compounds and also recently discovered ferromagnetic
superconductors the underlying pairing mechanism is
not known, but very likely it is due spin fluctuations.
Thus the knowledge about the fingerprints of a spin
fluctuations scenario in cuprates is highly desirable.
Non-interacting part of the spin susceptibility. Let
us first analyze the susceptibility of the non-interact-
ing electron gas on a square lattice when we switch off
the interaction between quasiparticles. To recall the
behavior of the bare (non-interacting) susceptibility
in the normal state we have to mention that in general
on the square lattice there are eight characteristic
scattering processes from k to k� in the first BZ if a
tight-binding band is used. The momenta k involved
are those in which the boundary of the magnetic zone
(defined by cos cosk kx y� � ) crosses the Fermi sur-
face (FS). Two of these scattering processes are direct
690 Fizika Nizkikh Temperatur, 2006, v. 32, No. 6
I. Eremin and D. Manske
(3)
(2) (2)
(1)
Q
k’
�
( , )� �
k
Fig. 7. Calculated Fermi surface for the for the first few
BZ using a tight-binding energy dispersion from Eq. (12)
with t/t �0.4 and optimal doping. The arrows indicate
three different scattering processes as described in the text.
involving momenta k k Q� � � , two involve umklapp
scattering with k k Q� � � � ( , )2 2� � and four involve
umklapp scattering with k k Q� � � � ( , )2 0� and
k k Q� � � � ( , )0 2� . All these processes are degenerate
at the antiferromagnetic wave vector and, thus, all
eight processes equally contribute to the spin suscepti-
bility yielding Im
�
0( , )Q � � i where � is the Lan-
dau damping rate. These transitions are illustrated in
Fig. 7. In the superconducting state the susceptibility
expression has a standard BCS form:
� �
� �
0
1
2
1
( , )
(
q
k k q k k q
k k q
k
�
� �
��
�
�
�
�
�
�
�
�
�
�
��
� �
�
�� �
E E
f E q k
k q kk
k k q k k q
k k q
) ( )
( )
�
� � �
�
� �
��
�
� �
�
�
f E
E E i
E E
� �
� �1
4
1
� �
�
�
�
�
�
�
�
� �
� � �
�
� �
�
�
�
�
1
1
4
1
f E f E
E E i
( ) ( )
( )
k q k
k q k
k k q
� �
� � �k k q
k k q
k q k
k q k
� �
�
�
�
�
�
�
�
�
�
�
�
�
� � �
�
�
�
�E E
f E f E
E E i
( ) ( )
( )� ��
.
(28)
The imaginary part at T � 0 and positive frequen-
cies reads:
Im ( , )
(
�
0
1
4
1
q
k k q k k q
k k q
k
�
� �
��
!
!
"
#
$
$
� �
� �
�
� �
E E
E Ek
k
�
%
&
'
('
)
*
'
+'
� q) .
(29)
Here, f E( )k denotes the Fermi function and
Ek k k
� �
2 2� is the dispersion of the Cooper pairs in
the superconducting state. As one sees from Eq. (29),
the imaginary part of the BCS spin susceptibility is
zero below the characteristic frequency
, � �c( ) | | | |q k k q� � � which is the same for all eight
scattering channels at Q. Moreover, due to a
dx y2 2�
-wave symmetry of the superconducting gap
one finds for the antiferromagnetic wave vector
� �k Q k� � � . As a result, the anomalous coherence
factor gets enhanced and Im
0( , )Q possesses a dis-
continuous jump at this frequency as shown in Fig. 8.
The height of the jump depends on the coherence
factor which is equal to 2 if perfect nesting
( )
k Q k� � � would be present. Note this is in strong
contrast to the conventional superconductors with
s-wave symmetry of the superconducting gap. The
latter will show a continuous enhancement of
Im ( , )
0 Q above the threshold frequency, but not a
discontinuous jump. The jump in d-wave super-
conductors is important because simultaneously
Re ( , )
0 Q will display a logarithmic singularity at
the threshold frequency and, moreover, will be en-
hanced in comparison to the normal state behavior.
RPA corrections. As a next step, within a conven-
tional RPA, the imaginary part of the spin susceptibil-
ity can be written as:
Im ( , )
Im ( , )
( Re ( , )) (Im ( , ))
.
Q
Q
Q Q
�
� �
0
0
2 2
0
21 U
(30)
One can immediately see that for any positive U, a
resonance condition is fulfilled if 1 0�U Re ( , )
Q ,
i.e. there is a pole in the RPA spin susceptibility
which occurs for energies lower than the threshold.
Therefore, a so-called spin exciton (or spin density
collective mode) forms. With increasing U the reso-
nance shifts towards lower values of energy and in-
creases its intensity as it is shown in Fig. 9.
Spin excitations in layered cuprates: a Fermi liquid approach
Fizika Nizkikh Temperatur, 2006, v. 32, No. 6 691
0.6
0.4
0.2
0 0.04 0.08 0.12
, eV
Im
(Q
,
)
0
Fig. 8. Frequency dependence of Im � �0( , )Q at the
antiferromagnetic wave vector q Q� . It reveals a jump be-
cause of the d-wave symmetry of the superconducting gap.
T = 10 K
2 = 58 meV�0
YBCO
Resonance
frequency
res
Im
(Q
,
)
14
12
10
8
6
4
2
0 10 20 30 40 50 60 70 80
, meV
Fig. 9. Numerical results for the resonance peak in
weak-coupling limit from Ref. 34 for optimal doping.
Imaginary part of the RPA spin susceptibility (in units of
states/eV) at wave vector Q � ( , )� � for U/t � 1, 2, 3, and
4 (from bottom to top). The resonance frequency was
found at �res � 41 meV.
The self-consistency effects within the generalized
Eliashberg formalism do not change the picture signif-
icantly, although deeper knowledge on the pairing in-
teraction and its frequency dependence can be
achieved. In Fig. 10 we demonstrate the results for Im
RPA( , )Q calculated within the FLEX approxima-
tion. In the superconducting state the resonance peak
forms similar to the weak-coupling results. However,
one has to remember here that the superconducting
gap, �( , )k
, has been obtained in a self-consistent
manner. For example one notices that the resonance
peak evolves very fast below Tc which agrees well
with experimental data. The reason behind is that
within Eliashberg theory the superconducting gap in-
crease rapidly below Tc and stabilizes the supercon-
ducting phase. Furthermore, a self-consistent scheme
allows also to analyze the behavior of the resonance
peak position with doping. In Fig. 11 we show the
doping dependence of resonance peak position for var-
ious doping concentrations. One finds that
res - k TB c in the optimally and overdoped region in
good agreement with experiment. Another interesting
fact is a comparison with the normal state. Note, in
the normal state the spin excitations spectrum within
FLEX is characterized by a broad maximum at the en-
ergies of about
sf t� 01. at the antiferromagnetic
wave vector Q. Therefore, the spin excitations are
paramagnons, i.e. overdamped spin waves with a char-
acteristic upward dispersion ~ q2. In the supercon-
ducting state one finds that the paramagnon feature
disappears and the whole spectrum will be dominated
by the dependence of the superconducting gap as a
function of q on the Fermi surface. We will analyze
this further by discussing the dispersion of the reso-
nant excitations.
Dispersion of the resonance excitations. In the
superconducting state away from (�,�) the degeneracy
in Im
0( , )Q i is lifted. In particular one has three
critical frequencies, , c
i( )( )q (i = 1, 2, 3). , c
( )( )1 q is
associated with direct process, , c
( )( )2 q with umklapp
process q — ( , )2 0� , and q — ( , )0 2� , and , c
( )( )3 q with
an umklapp process q — ( , )2 2� � . For all of these pro-
cesses the gap is still changing sign and, therefore,
Im ( , )
0 Q i demonstrates three discontinuous jumps
as illustrated in Fig. 12 and Re
0( , )Q i yields a lo-
garithmic singularity.
However, Im
0( , )Q i is still zero only below the
lowest jump, therefore the true resonance is only pos-
sible below the lowest jump which is in this case de-
scribe the direct scattering. Approaching
Q0 0 8 0 8� ( . , . )� the jump due to the direct scattering
approaches zero, because the states close the diagonal
of the BZ are connected. The superconducting gap is
zero there and, thus, Im
0( , )Q i is gapless there
(see Fig. 12,c). Therefore, away from ( , )� � the reso-
692 Fizika Nizkikh Temperatur, 2006, v. 32, No. 6
I. Eremin and D. Manske
U = 4t
x = 0.15
T = 1.5T
T = 0.9T
T = 0.75T
c
c
c
/t
Im
(Q
,
)
16
14
12
10
8
6
4
2
0 0.1 0.2 0.3 0.4 0.5
Fig. 10. Imaginary part of the RPA spin susceptibility at
Q � ( , )� � calculated within the FLEX approximation
taken from Ref. 34 for optimal doping. For the normal
state one gets �sf t� 01. and in the superconducting state
�res � 015. t. Assuming t � = 250 meV one finds that
�res � 41 meV.
U = 4t
T = 10 K
re
s
, m
e
V
50
40
30
20
10
Doping x
0.05 0.10 0.15 0.20 0.25 0.30
Fig. 11. Calculated results taken from Ref. 34 for the res-
onance frequency �res versus doping. In the overdoped re-
gime one finds a constant ratio of �res � Tc.
Im Im
Im
(2)
c
(2)
c
(1)
c
(1)
c
(2)
c
(3)
c
(3)
c
(3)
c
Q mode
Q mode
Q mode�
0.8
0.8
0.8
1 1a b
c d
Fig. 12. Schematic behavior of Im � �0( , )Qi and within
RPA, Im � �( , )Qi , for various momenta q � � �( , ).
nance will disperse downwards following the momen-
tum dependence of the superconducting gap at the
Fermi surface. Due to the nodes of the superconduct-
ing gap, the downward dispersion should completely
eliminate for the momenta close to Q0 0 8 0 8� ( . , . )�
which explains the formation of the silent bands in the
experiments. However, for the momenta smaller
q Q. 0 the nodal points cannot be connected and a di-
rect gap opens for the excitations into the p–h contin-
uum. This gap is not related to the superconducting
gap and is determined by , c
d
F( ) ( )q v q Q� � 0 . Due
to the large Fermi velocity along the diagonal of the
BZ the direct gap reaches fast the position of the sec-
ond jump in Im
0( , )Q i which occurs due to
umklapp processes. As a result the damping is re-
moved and a true pole in the Eq. (32) is again possi-
ble, yielding a new resonance formation separated
from the original resonance excitations by the silent
band. This is clearly visible from Fig. 13 where we
present our results for Im
( , )Q i within RPA. It is
perhaps interesting to note that away from Q0 the sec-
ond «umklapp» resonance is again possible due to the
fact that a coincidence of the both the direct gap and a
second jump is required in order to see the resonance.
Furthermore, both modes are well separated in energy
and momentum. We would like to stress that within a
spin exciton and Fermi liquid-like scenario for the res-
onance peak for a given momentum only one single
real pole is possible. Moreover, the intensity of the
umklapp resonance is about 5 times smaller that the
original resonance at ( , )� � which is due to the fact
that the new resonance excitations lie closer to the
continuum than the original resonance at wavevector
Q. Furthermore, the dispersion of the umklapp reso-
nance is nearly vertical and disappers upon reaching
the continuum. The origin of such a steep dispersion is
governed by the Fermi velocity along the nodal direc-
tion.
Another interesting fact which further separates
both resonance excitations is their intensity patterns
in the momentum space for fixed frequencies is the fol-
lowing: In Fig. 14 we show the intensity pattern
for the
� 30 meV (
. res ( )Q ) and
� 57 meV
( ( ))
/ res Q . While for the original resonant excita-
tions (so-called Q-mode in notation of Ref. 35) the
maxima are certainly along the bond directions, for
the novel mode (so-called Q *-excitations) are along
the diagonal of the BZ. The rotation of the maxima re-
flects the qualitative difference in the origin of both
modes. In the first case the electrons which scatter
along the bonds are located further away from the
nodes than those which scatter along the diagonal.
Thus, the intensity of the resonance is stronger for the
bond direction. On the contrast, the resonance arises
due to a rapid opening of the direct gap in the p–h
continuum and its coincidence with the discontinuous
jump due to umklapp scattering. The latter is only
possible along the diagonal of the BZ. In Ref. 35 it
also has been found that the intensity pattern for the
«upward» dispersion of the resonance remains always
along the diagonal direction. However, the situation
may change if for some reason the structure of the p–h
continuum modifies in a certain way. In particular, it
may happen that for some specific band structure pa-
rameters the border of the p–h continuum along the
bonds lies higher than the diagonal direction. Then,
the resonance excitations for theQ *-mode will also be
along the bond directions. However, this occurs only
very close to the border of p–h continuum.
Influence of orthorhombic distortions. One of the
interesting question is: what will happen to the reso-
nant excitations in the presence of orthorhombic dis-
tortions. This is, in particular, intriguing due to the
recent INS experiments on untwined YBCO [20].
Within a Fermi-liquid description the influence of the
orthorhombic distortion can be taken into account via
Spin excitations in layered cuprates: a Fermi liquid approach
Fizika Nizkikh Temperatur, 2006, v. 32, No. 6 693
Im
0
, eV
300
200
100
0
1.4
1.2
1.0
0.8
0.6 0
0.2
0.4
0.6
0.8
Fig. 13. RPA results for magnetic excitations in a
dx y2 2�
-wave superconductor. Im � �( , )q is obtained from
Eq. (36) as a function of momentum (along the diagonal
q � � �( , )) and frequency in the superconducting state taken
from Ref. 35. The arrows indicate the position of the reso-
nance modes.
q /y � q /y �
q
/ x
�
q
/ x
�
1.5 1.5
1.0 1.0
0.5 0.5
1.5 1.51.0 1.00.5 0.5
a b
Fig. 14. Intensity patterns of the resonance excitations for
� � 30 meV (a) and � � 57 meV (b) taken from Ref. 35.
the modification of the tight-binding parameters of
the model and an additional s-wave contribution to
the superconducting gap. Most importantly, for the
hopping along the bond directions one has to take
t tx y1 . In YBCO one finds due to the presence of the
underlying CuO-chains that the lattice constant b a/ .
Naively, one would expect that the hopping matrix el-
ement along the chains should be smaller than perpen-
dicular to them. This will be the case for the pure
two-dimensional model. However, in reality the plane
states are hybridized with the chain states. The latter
are metallic and thus, enhance the conductivity along
the chain direction in the CuO2 plane. This assump-
tion is further supported by experimental data [36].
Therefore, in order to model correctly the tight-bind-
ing parameters in the two-dimensional model Ha-
miltonian one has to assume t ta b. . This will produce
the main effect in the calculations. Since the reso-
nance excitations are sensitive to the topology of the
Fermi surface, its change will affect the intensity of
the peaks along the bond direction as shown in
Fig. 15. In particular, due to the change of the hop-
ping parameter, the Fermi surface around (��,0)
points will shift away and (0,��) shrinks towards
these points, leading in the extreme case even towards
closing of the Fermi surface around these points.
However, this necessarily should not take place. The
important results, however, is that due to the change
of the topology of the Fermi surface, the phase space
along x and along the y-direction is changing. In addi-
tion, the superconducting gap will not be pure d-wave
and an inclusion of the small admixture of the s-wave
component is required.
In Fig. 16 we show the results for the RPA spin sus-
ceptibility as a function of the momentum for a con-
stant energy of 35 meV. On can see that inclusion of
the orthorhombic distortions results in the suppression
of two peaks along y-axis. This is a direct consequence
of the reduction of the phase space illustrated in
Fig. 15. Note this agrees well with the available exper-
imental data [20].
Comparison with experiment. Finally we would
like to note that surprisingly a simple Fermi liquid ap-
proach works rather well in explaining many features
of the spin excitations in the superconducting state
like the formation of the resonance peak at the
antiferromagnetic wave vector Q � ( , )� � and its pecu-
liar dispersion including silent bands observed away
from ( , )� � . At the same time one should note certain
difficulties. In particular, within the spin exciton sce-
nario of the resonance peak there is always only one
single pole at a given momenta as a function of fre-
quency which yields a resonance peak formation.
Therefore, the resonance dispersion does not form a
so-called X-shape which is often claimed to be ob-
served in the experiment [17]. This question has to be
further clarified experimentally and theoretically. An-
other problem concerns the behavior of the resonant
excitations in the pseudogap region of underdoped
cuprates. There the spin excitations develop well
above Tc and exhibit slight renormalization in the
superconducting state. Moreover, there is a surprising
similarity of the spin excitations in the superconduct-
ing cuprates [17–19] and their non-superconducting
«striped» counterpart [38] indicating a certain
cross-over between the two systems. These aspects
cannot be covered within a Fermi-liquid type theory
developed here and is a subject of future studies.
3.2. Elementary excitations
For understanding the high-Tc cuprates their
low-energy excitations are of central significance.
Thus we start the discussion of theoretical results by
694 Fizika Nizkikh Temperatur, 2006, v. 32, No. 6
I. Eremin and D. Manske
k x
ky
Qx
Qy
Fig. 15. Illustration of the possible drastic changes of the
Fermi surface due to various parameters of the orthorhom-
bicity,
0 (t ta � �( )1 0
, t tb � �( )1 0
) as adopted from
Ref. 37. The arrows show the change of the phase space
for the bond scattering.
q
/ y
�
q
/ y
�
q /x �q /x �
1.5 1.5
1.0 1.0
0.5 0.5
1.5 1.51.0 1.00.5 0.5
a b
Fig. 16. Calculated normalized two-dimensional intensity
plot of Im � �RPA( , )q for constant energy of 35 meV taken
from Ref. 37, (a) without, (b) with inclusion of
orthorhombicity (
0 003� � . ). Note that two of the four in-
commensurate peaks are suppressed.
analyzing the spectral density in the normal state and
compare it with available experimental data on
ARPES that study in details the spectral density and
in particular the renormalized energy dispersion. One
general grounds one expects that antiferromagnetic
spin fluctuations exist relatively close to the Fermi en-
ergy. Therefore, if one goes below the Fermi level the
quasiparticles below some energy will not feel the ef-
fect of the scattering and will behave more like in a
conventional Fermi-liquid picture. A similar effect is
expected for electron-phonon interaction, but the be-
havior of the self-energy in the case phonons and spin
fluctuations qualitatively differ from each other. This
may help to see that the main effects arises from spin
fluctuations.
In Fig. 17 we show results of Ref. 39 for the fre-
quency dependence of the spectral density for differ-
ent momentum along ( , ) ( , )0 0 0� � and ( , ) ( , )0 0 � � �
directions using the parameters for hole-doped
cuprates and calculated as described in the previous
section. The curves in Fig. 17 are analogous of a
so-called EDC curves measured in the experiment.
One clearly sees that both the width of the curves
(which is proportional to Im �( , )k
) and position of
the maximum (that represents a renormalized disper-
sion) are changing approximately at (65�15) meV.
Moreover, one could also notice that the changes
along ( , ) ( , )0 0 0� � are more pronounced reflecting
stronger nesting of the Fermi surface in this direction.
Remarkably this is also the case in the experiment as
one sees on Fig. 6. In order to see the «kink» structure
in the dispersion we show in Fig. 18 the positions of
the peaks shown in Fig. 17 as a function of (k k� f )
for different temperatures above and below Tc. The
slope of all curves changes at (65�15) meV reflecting
the changes in the self-energy and, most importantly,
weakly depend on temperature. These results are also
in good agreement with experimental data, however,
note that experimentally the kink feature is more of-
ten and easier extracted from the MDC curves.
However, this difference relates mainly just to the
technical aspects in the experiment, since it is much
easier to analyze the intensity of MDC Lorentzian
curves than the EDC ones, in both cases the resulting
dispersion should not differ. In the superconducting
state only small changes in the dispersion are obtained
(see inset) which is also in agreement with experiment
[41]. It is remarkable that for electron-doped super-
conductors with a different dispersion we get no kink
feature up to frequencies about 100 meV. The reason
behind this is that spin fluctuations spectrum in elec-
tron-doped cuprates has no pronounced structures at
low frequencies and fewer density of states.
One of the main experimental results is that the
kink feature is more pronounced along ( , ) ( , )0 0 0� �
direction, i.e. the change of the slope in the disper-
sion, is larger than along ( , ) ( , )0 0 � � � direction
[25,26]. However, the position is almost the same in
all direction of the Brillouine zone. In the case of scat-
tering of quasiparticles on antiferromagnetic spin fluc-
tuations one expects this behavior due to the pro-
nounced nesting properties of the Fermi surface
around ( , )�� 0 points of the Brillouine zone. Thus, the
Spin excitations in layered cuprates: a Fermi liquid approach
Fizika Nizkikh Temperatur, 2006, v. 32, No. 6 695
In
te
n
si
ty
,a
rb
.u
n
its
In
te
n
si
ty
, a
rb
. u
n
its
Energy, meV
Energy, meV
Fig. 17. Calculated spectral density from Ref. 39 of the
quasiparticles in hole-doped cuprates at T � 100 K along
( , ) ( , )00 0� � and ( , ) ( , )00 � � � (inset) directions of the first
BZ. The dashed line denotes the unrenormalized chemical
potential. In both directions at energies approximately (65
�15) meV a kink occurs since the velocity of quasiparticles
changes.
0 0.04 0.08
k – k ,F �
–1
k – k ,F �
–1
E
n
e
rg
y,
m
e
V
E
n
e
rg
y,
m
e
V
8
4
0
–4
0
–50
–100
–150
–200
–250
0 0.02 0.04 0.06 0.08
T = 2T
T = 1.5T
T = 0.95T
T = 0.5T
c
c
c
c
Fig. 18. Positions of the peaks in the spectral density
A( , )k � versus ( )k k� f (energy dispersion) along the
( , ) ( , )00 0� � direction of the BZ calculated in Ref. 39 for
optimally hole-doped cuprates. Inset: change in the peak
positions of A( , )k � in the superconducting state. Note, in
underdoped cuprates the kink feature shifts to lower fre-
quencies due to decreasing of �sf . In the overdoped regime
the kink feature should become less pronounced and iso-
tropic in different direction of the BZ.
changes of the self-energy and correspondingly in the
quasiparticle damping will be stronger there. In order
to see this in more details we show in Fig. 19 the evo-
lution of the
Im ( , )Z k versus frequency for differ-
ent doping and temperatures at different points of the
BZ calculated within the FLEX approximation in the
normal and superconducting state. Note the strong
anisotropy for
� 0 for optimal doping between the
nodal point ( . , . )0 41 0 41� � and the so-called antinodal
point ( . , )015� � of the BZ. In particular, in the normal
state the behavior of the quasiparticle damping is
linear at the antinodal point and more quadratic
Fermi-liquid like at the nodal direction. This further
enhances with decreasing temperature. Therefore, the
kink should be more pronounced along ( , ) ( , )0 0 0� �
direction. Moreover, the important change happens in
the superconducting state. While along the nodal di-
rection the effect of superconductivity is relatively
weak, the change at the antinodal point is more pro-
nounced. Therefore, an additional structure is
expected in the dispersion along ( , ) ( , )0 0 0� � direc-
tion of the BZ which was found recently [25]. This
further confirms that the kink results from spin fluctu-
ations. Other important changes occurs if one goes to-
wards overdoped compounds as we show in Fig. 19,b.
In this case the anisotropy between nodal and
anti-nodal direction almost disappears. Therefore, the
kink feature will be the same for the overdoped re-
gime. This can be further studied in ARPES experi-
ments. In contrast to the overdoped case, in the
underdoped regime the anisotropy of the kink should
become more visible and also the Fermi velocity
should change along the Fermi line. Moreover, due
close vicinity to an antiferromagnetism one expects
the shift of the kink position towards lower values as
was observed in the experiment [25,26].
As we have already mentioned, a very naive expla-
nation for the kink feature is that the quasiparticles
below the Fermi level will not experience the scatter-
ing on spin fluctuations. In other words the
antiferromagnetic wave vector Q pair will not connect
these holes on different Fermi arcs with each other.
Then, these holes will behave in according to the stan-
dard Fermi liquid theory. There is a wide discussion
whether or not layered cuprate superconductors be-
have like conventional Fermi liquids. Earlier experi-
ments reveal non-Fermi liquid properties, in particu-
lar a linear resistivity 2( )T for optimal doping, non
well-defined quasiparticle peaks above the supercon-
ducting transition temperature Tc, and a strong tem-
perature dependence of the uniform spin susceptibility
observed by nuclear magnetic resonance [42]. Ori-
ginally, the phenomenological concepts of marginal
Fermi liquid (MFL) and nested Fermi liquid (NFL)
have been introduced in order to explain the devia-
tions in the normal state from the Fermi liquid theory
[43,44]. Our results shed more light on this question.
In agreement with picture of Ruvalds and co-workers
we obtain the
- and T-dependence of the self-energy
mainly due to scattering of the quasiparticles on spin
fluctuations which is strongest for a nested Fermi to-
pology. In order to investigate the effect of the self-en-
ergy �( , )k
on the dispersion
k we show in Fig. 20
results for Im ( , )� kn
at the wave vector along the
nodal line as in the previous paragraph. In the normal
state one clearly sees the transition from
��� -( , )k
2 to ��� -( , )k
for low-lying frequen-
cies for various temperatures. The deviation from Lan-
dau’s theory results in our picture from strong scatter-
ing of the quasiparticles on the spin fluctuations and
is expected to disappear at temperatures T � 0, see
inset of Fig. 20. In particular, the changes in the
696 Fizika Nizkikh Temperatur, 2006, v. 32, No. 6
I. Eremin and D. Manske
x = 0.15a
b
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
/t
/t
x = 0.22
Im
Z
(k
,
)
Im
Z
(k
,
)
Fig. 19. Comparison of the quasiparticle damping
� � �Im ( , )Z k in the normal state for (a) optimal doping
(x � 015. ) and for (b) the overdoped case (x � 022. ) taken
from Ref. 40. The solid curves are calculated at T Tc� 2
and the dashed curves refer to T Tc� 105. . The upper curves
correspond to the antinodal wavevector ka � ( . , )015� �
whereas the lower curves correspond to the nodal one
kn � ( . , . )041 041� � both on the Fermi line. For a compari-
son, we show in (a) also results for the superconducting
state (T Tc� 075. , dashed-dotted curve). Note that for both
doping concentrations a linear behavior for larger frequen-
cies and a vanishing anisotropy for �� 0 in the
overdoped case.
velocity of quasiparticles are determined in EDC as
u u d dF F k
* ( )� � � � �1 �
versus frequency. At frequen-
cies around 65 meV, the real part of �( , )k
shows a
flattening as can be seen via a Kramers-Kronig analy-
sis of Im �( , )k
. Therefore, at this frequency the ef-
fect of the scattering on spin fluctuations starts to re-
duce. Then, a more Fermi-liquid-like behavior can be
found.
In Fig. 21 we also demonstrate the feedback of su-
perconductivity on �( , )k
. We expect that it should
be strongest for the antinodal direction of the BZ
where the superconducting gap is maximal. Note that
the superconducting gap �( , )k
induces changes in
the self-energy. Therefore, the superconductivity will
induce an extra structure in the kink feature (a
so-called s-shape) which should be further verified in
the experiment.
To summarize this subsection we have calculated
the pronounced momentum and frequency dependence
of the quaspiarticle self-energy for various doping con-
centrations in hole-doped cuprates and find that this
results in a kink feature in the renormalized energy
dispersion which agrees well with recent ARPES ex-
periments. The origin of this is the coupling of the
quasiparticles to spin fluctuations that seems to be the
case for cuprates. Another reason for the kink struc-
ture is a the change in the frequency dependence of the
self-energy from non-Fermi-liquid to a Fermi-liq-
uid-like. Due to a different spectrum of the spin fluc-
tuations and fewer density of states the electron-doped
cuprates do not exhibits a kink feature.
4. Conclusions
In our short review we have covered some impor-
tant aspects of the doping dependence of many physi-
cal properties of cuprates based on a Fermi liquid-like
approach. In particular, we have shown that the spin-
fluctuation mechanism of superconductivity on the ba-
sis of a microscopic Eliashberg approach supports the
idea that the symmetry of the superconducting order
parameter is of dx y2 2�
-wave type. It has been found
that the renormalization of the quasiparticle spectrum
caused by the scattering on spin fluctuations results in
the so-called kink-feature seen in ARPES. Moreover,
the spin fluctuations will result in a strong anisotropy
of the renormalization at different parts of the BZ and
thus will lead to a strong anisotropy of kink feature
which has to be further studied in experimentally and
theoretically. Another important result is that the
non-Fermi liquid behavior observed in the experiment
that led to the suggestion of marginal Fermi-liquid
approach can be successfully explained on the basis of
a microscopic theory that takes into account the
strong scattering of quasiparticles on spin fluctua-
tions.
Another important achievement of the microscopic
Eliashberg approach is that the spin excitations spec-
trum renormalizes below Tc due to occurrence of su-
perconductivity with d-wave order parameter and
Spin excitations in layered cuprates: a Fermi liquid approach
Fizika Nizkikh Temperatur, 2006, v. 32, No. 6 697
0.6
0.4
0.2
0 50
50
100
100
150
150
200
200
Im
(k
,
)
�
a
Im
(k
,
)
a
, meV
�, meV
0.15
0
sc
ns
Fig. 21. Calculated frequency dependence of the quasi-
particle self-energy at the anti-nodal point ka = ( . , )015� � in
the first BZ taken from Ref. 41 for optimally hole-doped
cuprates. The solid curve correspond to the normal state at
T Tc� 2 , whereas the dashed curves to the superconducting
state at T Tc� 05. . Inset: the corresponding superconduct-
ing gap function �( , )ka � versus frequency.
0.6
0.6
0.4
0.2
0
0
50 100 150 200
Im
(k
,
)
�
n
Im
(k
,
)
n
T = 0.9 K
–200 0 200
, meV
�, meV
Fig. 20. Calculated frequency dependence of the
quasiparticle self-energy at the nodal point
kn � ( . , . )041 041� � in the first BZ taken from [39] for opti-
mally hole-doped cuprates. The solid curves correspond to
the normal state at T Tc� 2 , whereas the dashed curves to
the superconducting state at T Tc� 05. . In the normal state
one clearly sees approximately at 65 meV a crossover from
Fermi liquid behavior (�� � �( , )k � �2) to a non-Fermi-liq-
uid behavior (�� � �( , )k � �) for low-energy frequencies as a
function of temperature. We show in the inset the behav-
ior of �( , )kn � calculated at very low temperatures without
superconductivity (T � 09. K, dashed curve). Note, the be-
havior of the self-energy will become more Fermi liquid
like in the overdoped regime and thus a kink feature will
be less pronounced.
yields the resonance peak. Furthermore, the topology
of the Fermi surface and the momentum dependence of
the superconducting gap explains the peculiar disper-
sion of the resonance peak observed experimentally.
The spin fluctuation scenario can also explain many of
the properties of cuprates in the underdoped and
overdoped cases due to nearness to an antiferro-
magnetic instability. In the overdoped regime the be-
havior of cuprates seems to be more Fermi-liquid-like,
since the influence of antiferromagnetic fluctuations
becomes less important and all features will mainly
depend on the superconducting gap. Finally, the most
important achievement in recent years was the under-
standing of the asymmetry of the phase diagram be-
tween hole- and electron-doped cuprates. We would
like to note, that the results of the FLEX approxima-
tion agree to a large extent with the spin-fermion
model [45].
On the other hand, one has to mention that the
whole problem of high-Tc superconductivity is far
from being resolved. This concerns, for example, the
nature of the pseudogap observed in underdoped
cuprates. We did not discuss this problem in this re-
view, since despite of several scenarios which exist at
the moment none of them is well accepted. In this con-
nection, one of the main theoretical problems that
remains to be understood is the calculation of the dy-
namical spin susceptibility near half-filling and inclu-
sion of the (localized) short-range antiferromagnetic
correlations. Even though, the concept of Cooper-
pairing due to spin fluctuations seems to be in order
for describing physics of cuprates, many further ef-
forts are needed in order to provide a complete picture
of high-Tc superconductivity and its phase diagram.
We are thankful to M. Sigrist and A. Chubukov for
helpful discussions. DM acknowledges the finan-
cial support from the Alexander von Humboldt Foun-
dation.
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