Electronic structure of kinetic energy driven superconductors in the presence of
bilayer splitting
Yu Lan,1 Jihong Qin,2 and Shiping Feng1
Department of Physics, Beijing Normal University, Beijing 100875, China
Department of Physics, Beijing University of Science and Technology, Beijing 100083, China
(Dated: November 17, 2018)
Within the framework of the kinetic energy driven superconductivity, the electronic structure
of bilayer cuprate superconductors in the superconducting state is studied. It is shown that the
electron spectrum of bilayer cuprate superconductors is split into the bonding and antibonding
components by the bilayer splitting, then the observed peak-dip-hump structure around the [π, 0]
point is mainly caused by this bilayer splitting, with the superconducting peak being related to
the antibonding component, and the hump being formed by the bonding component. The spectral
weight increases with increasing the doping concentration. In analogy to the normal state case, both
electron antibonding peak and bonding hump have the weak dispersions around the [π, 0] point.
PACS numbers: 74.20.Mn, 74.20.-z, 74.25.Jb
I. INTRODUCTION
The parent compounds of cuprate superconductors are
the Mott insulators with an antiferromagnetic (AF) long-
range order (AFLRO), then via the charge carrier doping,
one can drive these materials through a metal-insulating
transition and enter the superconducting (SC) dome1,2,3.
It has become clear in the past twenty years that cuprate
superconductors are among the most complex systems
studied in condensed matter physics1,2,3. The compli-
cations arise mainly from (1) a layered crystal structure
with one or more CuO2 planes per unit cell separated by
insulating layers which leads to a quasi-two-dimensional
electronic structure, and (2) extreme sensitivity of the
physical properties to the compositions (stoichiometry)
which control the carrier density in the CuO2 plane
1,2,3.
As a consequence, both experimental investigation and
theoretical understanding are extremely difficult.
By virtue of systematic studies using the angle-resolved
photoemission spectroscopy (ARPES), the low-energy
electronic structure of cuprate superconductors in the
SC state is well-established by now2,3, where an agree-
ment has emerged that the electronic quasiparticle-like
excitations are well defined, and are the entities par-
ticipating in the SC pairing. In particular, the lowest
energy states are located at the [π, 0] point of the Bril-
louin zone, where the d-wave SC gap function is max-
imal, then the most contributions of the electron spec-
tral function come from the [π, 0] point2,3. Moreover,
some ARPES experimental results unambiguously estab-
lished the Bogoliubov-quasiparticle nature of the sharp
SC quasiparticle peak near the [π, 0] point4,5, then the
SC coherence of the quasiparticle peak is described by
the simple Bardeen-Cooper-Schrieffer (BCS) formalism6.
However, there are numerous anomalies for different fam-
ilies of cuprate superconductors, which complicate the
physical properties of the electronic structure2,3. Among
these anomalies is the dramatic change in the spectral
lineshape around the [π, 0] point first observed on the bi-
layer cuprate superconductor Bi2Sr2CaCu2O8+δ, where
a sharp quasiparticle peak develops at the lowest bind-
ing energy, followed by a dip and a hump, giving rise to
the so-called peak-dip-hump (PDH) structure in the elec-
tron spectrum7,8,9. Later, this PDH structure was also
found in YBa2Cu3O7−δ
10 and in Bi2Sr2Ca2Cu3O10+δ
Furthermore, although the sharp quasiparticle peaks are
identified in the SC state along the entire Fermi surface,
the PDH structure is most strongly developed around the
[π, 0] point2,7,8,9,10,11.
The appearance of the PDH structure in bilayer
cuprate superconductors in the SC state is the mostly
remarkable effect, however, its full understanding is still
a challenging issue. The earlier works2,12 gave the main
impetus for a phenomenological description of the single-
particle excitations in terms of an interaction between
quasiparticles and collective modes, which is of fun-
damental relevance to the nature of superconductivity
and the pairing mechanism in cuprate superconductors.
However, the different interpretive scenario has been
proposed2,13. This followed from the observation of the
bilayer splitting (BS) for both normal and SC states
in a wide doping range14,15,16. This BS of the CuO2
plane derives the electronic structure in the bonding and
antibonding bands due to the present of CuO2 bilayer
blocks in the unit cell of bilayer cuprate superconduc-
tors, then the main features of the PDH structure is
caused by the BS13,14,15,16, with the peak and hump
corresponding to the antibonding and bonding bands,
respectively. Furthermore, some ARPES experimental
data measured above and below the SC transition tem-
perature show that this PDH structure is totally unre-
lated to superconductivity14. The recent ARPES exper-
imental results reported by several groups support this
scenario, and most convincingly suggested that the PDH
structure originates from the BS at any doping levels17.
To the best of our knowledge, this PDH structure in bi-
layer cuprate superconductors has not been treated start-
ing from a microscopic SC theory.
Within the single layer t-t′-J model, the electronic
structure of the single layer cuprate superconductors in
the SC state has been discussed18 based on the frame-
work of the kinetic energy driven superconductivity19,
and the main features of the ARPES experiments on
the single layer cuprate superconductors have been repro-
duced, including the doping and temperature dependence
of the electron spectrum and quasiparticle dispersion. In
http://arxiv.org/abs/0704.0825v2
this paper, we study the electronic structure of bilayer
cuprate superconductors in the SC state along with this
line. Within the kinetic energy driven SC mechanism19,
we employed the t-t′-J model by considering the bilayer
interaction, and then show explicitly that the BS occurs
due to this bilayer interaction. In this case, the elec-
tron spectrum is split into the bonding and antibond-
ing components by this BS, then the SC peak is closely
related to the antibonding component, while the hump
is mainly formed by the bonding component. In other
words, the well pronounced PDH structure in the electron
spectrum of bilayer cuprate superconductors is mainly
caused by the BS. Furthermore, the spectral weight in
the [π, 0] point increases with increasing the doping con-
centration. In analogy to the normal-state case14,20,21,22,
both electron antibonding peak and bonding hump have
the weak dispersions around the [π, 0] point, in qualita-
tive agreement with the experimental observation on bi-
layer cuprate superconductors in the SC state2,7,8,9,10,11.
The paper is organized as follows. The basic formal-
ism is presented in Sec. II, where we generalize the
kinetic energy driven superconductivity from the previ-
ous single layer case18,19 to the bilayer case, and then
evaluate explicitly the longitudinal and transverse com-
ponents of the electron normal and anomalous Green’s
functions (hence the bonding and antibonding electron
spectral functions). Within this theoretical framework,
we discuss the electronic structure of bilayer cuprate su-
perconductors in the SC state in Sec. III. It is shown that
the striking PDH structure in bilayer cuprate supercon-
ductors is closely related to the BS. Finally, we give a
summary and discussions in Sec. IV.
II. FORMALISM
It has been shown from the ARPES experiments2,23
that the two-dimensional t-t′-J model is of particular
relevance to the low energy features of cuprate super-
conductors. For discussions of the physical properties of
bilayer cuprate superconductors, the t-t′-J model can be
expressed by including the bilayer interactions as,
H = −t
iη̂aσ
iaσCi+η̂aσ + t
iτ̂aσ
iaσCi+τ̂aσ
t⊥(i)(C
i1σCi2σ +H.c.) + µ
iaσCiaσ
Sia · Si+η̂a + J⊥
Si1 · Si2, (1)
supplemented by an important on-site local constraint
iaσCiaσ ≤ 1 to avoid the double occupancy, where
η̂ = ±x̂,±ŷ representing the nearest neighbors of a given
site i, τ̂ = ±x̂± ŷ representing the next nearest neighbors
of a given site i, a = 1, 2 is plane index, C
iaσ (Ciaσ)
is the electron creation (annihilation) operator, Sia =
iaσCia/2 is the spin operator with the Pauli matrices
σ = (σx, σy, σz), µ is the chemical potential, and the
interlayer coherent hopping has the form,
t⊥(k) =
(cos kx − cos ky)
2, (2)
which is strongly anisotropic and follows the theoret-
ical predictions24. In particular, this momentum de-
pendent form (2) has been experimentally verified14,15.
For this t-t′-J model (1), it has been argued that cru-
cial requirement is to impose the electron single occu-
pancy local constraint for a proper understanding of the
physical properties of cuprate superconductors. To in-
corporate the electron single occupancy local constraint,
the charge-spin separation (CSS) fermion-spin theory has
been proposed25, where the constrained electron opera-
tors are decoupled as, Cia↑ = h
ia and Cia↓ = h
with the spinful fermion operator hiaσ = e
−iΦiaσhia rep-
resents the charge degree of freedom together with some
effects of the spin configuration rearrangements due to
the presence of the doped hole itself (dressed holon),
while the spin operator Sia represents the spin degree
of freedom, then the bilayer t-t′-J Hamiltonian (1) can
be expressed in this CSS fermion-spin representation as,
H = t
i+η̂a↑hia↑S
i+η̂a + h
i+η̂a↓hia↓S
i+η̂a)
i+τ̂a↑hia↑S
i+τ̂a + h
i+τ̂a↓hia↓S
i+τ̂a)
t⊥(i)(h
i2↑hi1↑S
i2 + h
i1↑hi2↑S
i2↓hi1↓S
i2 + h
i1↓hi2↓S
i1)− µ
iaσhiaσ
+ Jeff
Sia · Si+η̂a + Jeff⊥
Si1 · Si2, (3)
where Jeff = J(1 − δ)
2, Jeff⊥ = J⊥(1 − δ)
2, and δ =
iaσhiaσ〉 = 〈h
iahia〉 is the doping concentration. It has
been shown that the electron single occupancy local con-
straint is satisfied in analytical calculations within this
CSS fermion-spin theory, and the double spinful fermion
occupancy are ruled out automatically25. Although in
common sense hiaσ is not a real spinful fermion, it be-
haves like a spinful fermion25. As in the single layer
case18, the kinetic energy terms in the bilayer t-t′-J
model have been transferred as the dressed holon-spin
interactions, which can induce the dressed holon pair-
ing state (hence the electron Cooper pairing state) by
exchanging spin excitations in the higher power of the
doping concentration. Before calculation of the electron
normal and anomalous Green’s functions of the bilayer
system in the SC state, we firstly introduce the SC order
parameter. As we have mentioned above, there are two
coupled CuO2 planes in the unit cell, and in this case,
the SC order parameters for the electron Cooper pair is
a matrix ∆ = ∆L + σx∆T , with the longitudinal and
transverse SC order parameters in the CSS fermion-spin
theory can be expressed as,
∆L = 〈C
i+η̂a↓ − C
i+η̂a↑〉
= 〈hia↑hi+η̂a↓S
i+η̂a − hia↓hi+η̂a↑S
i+η̂a〉
= −χ1∆hL, (4a)
∆T = 〈C
i2↓ − C
= 〈hi1↑hi2↓S
i2 − hi1↓hi2↑S
= −χ⊥∆hT , (4b)
respectively, where the spin correlation functions χ1 =
〈S+iaS
i+η̂a〉 and χ⊥ = 〈S
i2〉, and the longitudinal
and transverse dressed holon pairing order parameters
∆hL = 〈hi+η̂a↓hia↑ − hi+η̂a↑hia↓〉 and ∆hT = 〈hi2↓hi1↑ −
hi2↑hi1↓〉.
Within the t-J type model, robust indications
of superconductivity with the d-wave symmetry in
doped cuprates have been found by using numerical
techniques26. On the other hand, it has been argued that
the SC transition in doped cuprates is determined by the
need to reduce the frustrated kinetic energy27. Although
it is not necessary for the strong coupling of the electron
quasiparticles and a pairing boson in their arguments27,
a series of the inelastic neutron scattering experimental
results provide a clear link between the electron quasi-
particles and magnetic excitations28,29. In particular, an
impurity-substitution effect on the low energy dynamics
has been studied by virtue of the ARPES measurement30,
this impurity-substitution effect is a magnetic analogue
of the isotope effect used for the conventional super-
conductors. These experimental results30 reveal that
the impurity-induced changes in the electron self-energy
show a good correspondence to those of the magnetic
excitations, indicating the importance of the magnetic
fluctuation to the electron pairing in cuprate supercon-
ductors. Recently, we19 have developed the kinetic en-
ergy driven SC mechanism based on the CSS fermion-
spin theory25, where the dressed holons interact occur-
ring directly through the kinetic energy by exchanging
spin excitations, leading to a net attractive force between
dressed holons, then the electron Cooper pairs originat-
ing from the dressed holon pairing state are due to the
charge-spin recombination, and their condensation re-
veals the SC ground-state. Within this SC mechanism19,
the doping and temperature dependence of the electron
spectral function of the single layer cuprate supercon-
ductors in the SC state has been discussed18. In this
section, our main goal is to generalize these analytical
calculations from the single layer case to the bilayer sys-
tem. As in the case for the SC order parameter, the full
dressed holon normal and anomalous Green’s functions
can also be expressed as g(k, ω) = gL(k, ω) + σxgT (k, ω)
and ℑ†(k, ω) = ℑ
L(k, ω) + σxℑ
L(k, ω), respectively. We
now can follow the previous discussions for the single
layer case18,19, and evaluate explicitly these correspond-
ing longitudinal and transverse parts of the full dressed
holon normal and anomalous Green’s functions as [see
the Appendix],
gL(k, ω) =
ν=1,2
U2hνk
ω − Ehνk
V 2hνk
ω + Ehνk
, (5a)
gT (k, ω) =
ν=1,2
(−1)ν+1Z
U2hνk
ω − Ehνk
V 2hνk
ω + Ehνk
, (5b)
L(k, ω) = −
ν=1,2
2Ehνk
ω − Ehνk
ω + Ehνk
, (5c)
T (k, ω) = −
ν=1,2
(−1)ν+1Z
2Ehνk
ω − Ehνk
ω + Ehνk
, (5d)
where the dressed holon quasiparticle coherence fac-
tors U2hνk = [1 + ξ̄νk/Ehνk]/2 and V
hνk = [1 −
ξ̄νk/Ehνk]/2, the dressed holon quasiparticle disper-
sion Ehνk =
[ξ̄νk]2+ | ∆̄
(k) |2, the renormalized
dressed holon excitation spectrum ξ̄νk = Z
ξνk, with
the mean-field (MF) dressed holon excitation spectrum
ξνk = Ztχ1γk − Zt
k − µ+ (−1)
ν+1χ⊥t⊥(k), where
the spin correlation function χ2 = 〈S
i+τ̂a〉, γk =
(1/Z)
eik·η̂, γ′k = (1/Z)
eik·τ̂ , Z is the num-
ber of the nearest neighbor or next nearest neighbor
sites, the renormalized dressed holon pair gap func-
tion ∆̄
(k) = Z
[∆̄hL(k) + (−1)
ν+1∆̄hT (k)], with
ν = 1 ( ν = 2) for the bonding (antibonding) case,
where ∆̄hL(k) = Σ
2L (k, ω) |ω=0= ∆̄hLγ
, with γ
(coskx − cosky)/2, ∆̄hT (k) = Σ
2T (k, ω) |ω=0= ∆̄hT , the
dressed holon quasiparticle coherent weights Z
(1)−1
hF1 − Z
hF2, Z
(2)−1
= Z−1
hF1 + Z
hF2, with Z
hF1 =
1 − Σ
1L (k0, ω) |ω=0, and Z
hF2 = Σ
1T (k0, ω) |ω=0 ,
where k0 = [π, 0], Σ
1L (k, ω) and Σ
1T (k, ω) are the cor-
responding antisymmetric parts of the longitudinal and
transverse dressed holon self-energy functions Σ
1L (k, ω)
and Σ
1T (k, ω), while the longitudinal and transverse
parts of the dressed holon self-energy functions Σ
1 (k, ω)
and Σ
2 (k, ω) have been evaluated as,
1L (k, iωn) =
p+q+k
gL(p+ k, ipm + iωn)ΠLL(p,q, ipm)
p+q+k
gT (p+ k, ipm + iωn)ΠTL(p,q, ipm)], (6a)
1T (k, iωn) =
p+q+k
gT (p+ k, ipm + iωn)ΠTT (p,q, ipm)
p+q+k
gL(p+ k, ipm + iωn)ΠLT (p,q, ipm)], (6b)
2L (k, iωn) =
p+q+k
L(−p− k,−ipm − iωn)ΠLL(p,q, ipm)
p+q+k
T (−p− k,−ipm − iωn)ΠTL(p,q, ipm)], (6c)
2T (k, iωn) =
p+q+k
(−p− k,−ipm − iωn)ΠTT (p,q, ipm)
p+q+k
L(−p− k,−ipm − iωn)ΠLT (p,q, ipm)], (6d)
where R
= [Z(tγk − t
′γ′k)]
2 + t2⊥(k), R
= 2Z(tγk −
)t⊥(k), and the spin bubbles Πη,η′(p,q, ipm) =
(1/β)
η (q, iqm)D
η′ (q + p, iqm+ ipm), with η =
L, T and η′ = L, T , and the MF spin Green’s function
D(0)(k, ω) = D
(k, ω) + σxD
(k, ω), with the cor-
responding longitudinal and transverse parts have been
given by22,
L (k, ω) =
ν=1,2
ω2 − ω2
, (7a)
(k, ω) =
ν=1,2
(−1)ν+1
ω2 − ω2
, (7b)
where Bνk = λ(A1γk−A2)−λ
′(2χz2γ
k−χ2)−Jeff⊥[χ⊥+
2χz⊥(−1)
ν ][ǫ⊥(k)+(−1)
ν ], A1 = 2ǫ‖χ
1+χ1, A2 = ǫ‖χ1+
2χz1, λ = 2ZJeff , λ
′ = 4Zφ2t
′, ǫ‖ = 1+2tφ1/Jeff , ǫ⊥(k) =
1 + 4φ⊥t⊥(k)/Jeff⊥, the spin correlation functions χ
〈SziaS
i+η̂a〉, χ
2 = 〈S
i+τ̂a〉, χ
⊥ = 〈S
i2〉, the dressed
holon particle-hole order parameters φ1 = 〈h
iaσhi+η̂aσ〉,
φ2 = 〈h
iaσhi+τ̂aσ〉, φ⊥ = 〈h
i1σhi2σ〉, and the MF spin
excitation spectrum,
ω2νk = λ
A4 − αǫ‖χ
1γk −
αǫ‖χ1
(1 − ǫ‖γk) +
αχz1 − αχ1γk
(ǫ‖ − γk)
Z − 1
γ′k +
+ λλ′α
χz1(1 − ǫ‖γk)γ
k − C2)(ǫ‖ − γk)
+ γ′k(C
2 − ǫ‖χ
2γk)−
ǫ‖(C2 − χ2γk)
+ λJeff⊥α
ǫ⊥(k)(ǫ‖ − γk)[C⊥ + χ1(−1)
+ (1− ǫ‖γk)[C
⊥ + χ
1ǫ⊥(k)(−1)
ν ] + [ǫ⊥(k) + (−1)
ǫ‖(C⊥ − χ⊥γk) + (C
⊥ − ǫ‖χ
⊥γk)(−1)
+ λ′Jeff⊥α
γ′k[C
⊥ + χ
2ǫ⊥(k)(−1)
ǫ⊥(k)[C
⊥ + χ2(−1)
k − C
⊥) + χ
k(−1)
[ǫ⊥(k) + (−1)
J2eff⊥[ǫ⊥(k) + (−1)
ν ]2, (8)
where A3 = αC1 + (1− α)/2Z, A4 = αC
1 + (1− α)/4Z,
A5 = αC3 + (1 − α)/2Z, and the spin correla-
tion functions C1 = (1/Z
η̂η̂′
i+η̂aS
i+η̂′a
C2 = (1/Z
i+η̂aS
i+τ̂a〉, C3 =
(1/Z2)
τ̂ τ̂ ′
i+τ̂aS
i+τ̂ ′a
〉, Cz1 =
(1/Z2)
η̂η̂′
〈Szi+η̂aS
i+η̂′a
〉, Cz2 =
(1/Z2)
〈Szi+η̂aS
i+τ̂a〉, C⊥ = (1/Z)
〈S+i1S
i+η̂2〉,
C′⊥ = (1/Z)
〈S+i1S
i+τ̂2〉, C
⊥ = (1/Z)
〈Szi1S
i+η̂2〉,
and C′
⊥ = (1/Z)
〈Szi1S
i+τ̂2〉. In order to satisfy the
sum rule of the spin correlation function 〈S+iaS
ia〉 = 1/2
in the case without AFLRO, the important decoupling
parameter α has been introduced in the above calcu-
lation as in the single layer case18,19,22, which can be
regarded as the vertex correction.
With the help of the longitudinal and transverse parts
of the full dressed holon normal and anomalous Green’s
functions in Eq. (5) and MF spin Green’s function
in Eq. (7), we now can calculate the electron nor-
mal and anomalous Green’s functions G(i − j, t − t′) =
〈〈Ciσ(t);C
′)〉〉 = GL(i− j, t− t
′) + σxGT (i− j, t− t
and Γ†(i−j, t−t′) = 〈〈C
i↑(t);C
′)〉〉 = Γ
L(i−j, t−t
T (i−j, t−t
′), where these longitudinal and transverse
parts are the convolutions of the corresponding longitudi-
nal and transverse parts of the full dressed holon normal
and anomalous Green’s functions and MF spin Green’s
function in the CSS fermion-spin theory, and can be eval-
uated explicitly as,
GL(k, ω) =
L(1)µν (k,p)
U2hµp−k
ω + Ehµp−k − ωνp
V 2hµp−k
ω − Ehµp−k + ωνp
+ L(2)µν (k,p)
U2hµp−k
ω + Ehµp−k + ωνp
V 2hµp−k
ω − Ehµp−k − ωνp
, (9a)
GT (k, ω) =
(−1)µ+νZ
L(1)µν (k,p)
U2hµp−k
ω + Ehµp−k − ωνp
V 2hµp−k
ω − Ehµp−k + ωνp
+ L(2)µν (k,p)
U2hµp−k
ω + Ehµp−k + ωνp
V 2hµp−k
ω − Ehµp−k − ωνp
, (9b)
L(k, ω) =
(p− k)
2Ehµp−k
L(1)µν (k,p)
ω − Ehµp−k + ωνp
ω + Ehµp−k − ωνp
+ L(2)µν (k,p)
ω − Ehµp−k − ωνp
ω + Ehµp−k + ωνp
, (9c)
(k, ω) =
(−1)µ+νZ
(p− k)
2Ehµp−k
L(1)µν (k,p)
ω − Ehµp−k + ωνp
ω + Ehµp−k − ωνp
+ L(2)µν (k,p)
ω − Ehµp−k − ωνp
ω + Ehµp−k + ωνp
, (9d)
where L
µν (k,p) = [coth(βωνp/2) − th(βEhµp−k/2)]/2
and L
µν (k,p) = [coth(βωνp/2) + th(βEhµp−k/2)]/2,
then the longitudinal and transverse parts of the
electron spectral function AL(k, ω) = −2ImGL(k, ω)
and AT (k, ω) = −2ImGT (k, ω) and SC gap func-
tion ∆L(k) = (1/β)
L(k, iωn) and ∆T (k) =
(1/β)
(k, iωn) are obtained as,
AL(k, ω) = π
{L(1)µν (k,p)[U
hµp−kδ(ω + Ehµp−k − ωνp) + V
hµp−kδ(ω − Ehµp−k + ωνp)]
+ L(2)µν (k,p)[U
hµp−kδ(ω + Ehµp−k + ωνp) + V
hµp−kδ(ω − Ehµp−k − ωνp)]}, (10a)
AT (k, ω) = π
(−1)µ+νZ
{L(1)µν (k,p)[U
hµp−kδ(ω + Ehµp−k − ωνp) + V
hµp−kδ(ω − Ehµp−k + ωνp)]
+ L(2)µν (k,p)[U
hµp−kδ(ω + Ehµp−k + ωνp) + V
hµp−kδ(ω − Ehµp−k − ωνp)]}, (10b)
∆L(k) = −
p,µ,ν
(p− k)
Ehµp−k
βEhµp−k]coth[
βωνp], (10c)
∆T (k) = −
p,µ,ν
(−1)µ+νZ
(p− k)
Ehµp−k
βEhµp−k]coth[
βωνp]. (10d)
With the above longitudinal and transverse parts of the
SC gap functions in Eqs. (10c) and (10d), the corre-
sponding longitudinal and transverse SC gap parameters
are obtained as ∆L = −χ1∆hL and ∆T = −χ⊥∆hT ,
respectively. In the bilayer coupling case, the more ap-
propriate classification is in terms of the spectral func-
tion and SC gap function within the basis of the an-
tibonding and bonding components13,14,15,16,17. In this
case, the electron spectral function and SC gap parame-
ter can be transformed from the plane representation to
the antibonding-bonding representation as,
A(a)(k, ω) =
[AL(k, ω)−AT (k, ω)], (11a)
A(b)(k, ω) =
[AL(k, ω) +AT (k, ω)], (11b)
∆(a) = ∆L −∆T , (11c)
∆(b) = ∆L +∆T . (11d)
respectively, then the antibonding and bonding parts
have odd and even symmetries, respectively.
III. ELECTRON STRUCTURE OF BILAYER
CUPRATE SUPERCONDUCTORS
We now begin to discuss the effect of the bilayer in-
teraction on the electronic structure in the SC state. We
first plot, in Fig. 1, the antibonding (solid line) and
bonding (dashed line) electron spectral functions in the
[π, 0] point for parameters t/J = 2.5, t′/t = 0.3, and
t⊥/t = 0.35 with temperature T = 0.002J at the doping
concentration δ = 0.15. In comparison with the single
layer case18, the electron spectrum of the bilayer system
has been split into the bonding and antibonging compo-
nents, with the bonding and antibonding SC quasipar-
ticle peaks in the [π, 0] point are located at the differ-
ent positions. In this sense, the differentiation between
the bonding and antibonding components of the electron
spectral function is essential. The antibonding spectrum
consists of a low energy antibonding peak, corresponding
to the SC peak, and the bonding spectrum has a higher
energy bonding peak, corresponding to the hump, while
the spectral dip is in between them, then the total con-
tributions for the electron spectrum from both antibond-
ing and bonding components give rise to the PDH struc-
ture. Although the simple bilayer t-t′-J model (1) can-
not be regarded as a comprehensive model for a quanti-
tative comparison with bilayer cuprate superconductors,
our present results for the SC state are in qualitative
agreement with the major experimental observations on
bilayer cuprate superconductors2,7,8,9,10,11,16.
We now turn to discuss the doping evolution of the
electron spectrum of bilayer cuprate superconductors in
( )/J
-1.0 -.5 0.0 .5
Bonding
Antibonding
FIG. 1: The antibonding (solid line) and bonding (dashed
line) electron spectral functions in the [π, 0] point for t/J =
2.5, t′/t = 0.3, and t⊥/t = 0.35 with T = 0.002J at δ = 0.15.
( )/J
-1.0 -.5 0.0 .5
FIG. 2: The electron spectral functions at [π, 0] point for
t/J = 2.5, t′/t = 0.3, and t⊥/t = 0.35 with T = 0.002J at
δ = 0.09 (solid line), δ = 0.12 (dashed line), and δ = 0.15
(dotted line).
the SC state. We have calculated the electron spec-
trum at different doping concentrations, and the result
of the electron spectral functions in the [π, 0] point for
t/J = 2.5, t′/t = 0.3, and t⊥/t = 0.35 with T = 0.002J
at δ = 0.09 (solid line), δ = 0.12 (dashed line), and
δ = 0.15 (dotted line) are plotted in Fig. 2. In compari-
son with the corresponding ARPES experimental results
of the bilayer cuprate superconductor Bi2Sr2CaCu2O8+δ
in the SC state in Ref.12, it is obviously that the doping
evolution of the spectral weight of the bilayer supercon-
Bonding Band
Antibonding Band
(-0.2 , ) (0, ) (0.2 , )
FIG. 3: The positions of the antibonding peaks and bonding
humps in the electron spectrum as a function of momentum
along the direction [−0.2π, π] → [0, π] → [0.2π, π] with T =
0.002J at δ = 0.15 for t/J = 2.5, t′/t = 0.3, and t⊥/t = 0.35.
ductor Bi2Sr2CaCu2O8+δ is reproduced. With increas-
ing the doping concentration, both SC peak and hump
become sharper, and then the spectral weights increase
in intensity. Furthermore, we have also calculated the
electron spectrum with different temperatures, and the
results show that the spectral weights of both SC peak
and hump are suppressed with increasing temperatures.
Our these results are also qualitatively consistent with
the ARPES experimental results on bilayer cuprate su-
perconductors in the SC state2,9,12.
To better perceive the anomalous form of the antibond-
ing and bonding electron spectral functions as a function
of energy ω for k in the vicinity of the [π, 0] point, we
have made a series of calculations for the electron spec-
tral function at different momenta, and the results show
that the sharp SC peak from the electron antibonding
spectral function and hump from the bonding spectral
function persist in a very large momentum space region
around the [π, 0] point. To show this point clearly, we
plot the positions of the antibonding peak and bonding
hump in the electron spectrum as a function of momen-
tum along the direction [−0.2π, π] → [0, π] → [0.2π, π]
with T = 0.002J at δ = 0.15 for t/J = 2.5, t′/t = 0.3,
and t⊥/t = 0.35 in Fig. 3. Our result shows that there
are two branches in the quasiparticle dispersion, with
upper branch corresponding to the antibonding quasi-
particle dispersion, and lower branch corresponding to
the bonding quasiparticle dispersion. Furthermore, the
BS reaches its maximum at the [π, 0] point. Our present
result also shows that in analogy to the two flat bands ap-
peared in the normal state22, both electron antibonding
peak and bonding hump have a weak dispersion around
the [π, 0] point, in qualitative agreement with the ARPES
experimental measurements on bilayer cuprate supercon-
ductors in the SC state2,7,8,9,10,11,14.
In the above calculations, we find that although the
antibonding SC peak and bonding hump have different
dispersions, the transverse part of the SC gap param-
eter ∆T ≈ 0. To show this point clearly, we plot the
antibonding and bonding gap parameters in Eqs. (11c)
00.00 0.05 0.10 0.15 0.20 0.25 0.30
00.00
FIG. 4: The antibonding (solid line) and bonding (dashed
line) gap parameters as a function of the doping concentration
with T = 0.002J for t/J = 2.5, t′/t = 0.3, and t⊥/t = 0.35.
and (11d) as a function of the doping concentration with
T = 0.002J for t/J = 2.5, t′/t = 0.3, and t⊥/t = 0.35 in
Fig. 4. As seen from Fig. 4, both antibonding and bond-
ing gap parameters have the same d-wave SC gap mag-
nitude in a given doping concentration, i.e., ∆a ≈ ∆b.
This result shows that although there is a single elec-
tron interlayer coherent hopping (2) in bilayer cuprate
superconductors in the SC state, the electron interlayer
pairing interaction vanishes. This reflects that in the
present kinetic energy driven SC mechanism, the weak
dressed holon-spin interaction due to the interlayer co-
herent hopping (2) from the kinetic energy terms in Eq.
(3) does not induce the dressed holon interlayer pair-
ing state by exchanging spin excitations in the higher
power of the doping concentration. This is different from
the dressed holon-spin interaction due to the intralayer
hopping from the kinetic energy terms in Eq. (3), it
can induce superconductivity by exchanging spin excita-
tions in the higher power of the doping concentration19.
Our this result is also consistent with the ARPES ex-
perimental results of the bilayer cuprate superconductor
Bi(Pb)2Sr2CaCu2O8+δ
14,16, where the SC gap separately
for the bonding and antibonding bands has been mea-
sured, and it is found that both d-wave SC gaps from
the antibonding and bonding components are identical
within the experimental uncertainties.
To our present understanding, two main reasons why
the electronic structure of bilayer cuprate superconduc-
tors in the SC state can be described qualitatively in
the framework of the kinetic energy driven supercon-
ductivity by considering the bilayer interaction are as
follows. Firstly, the bilayer interaction causes the BS,
this leads to that the full electron normal (anomalous)
Green’s function is divided into the longitudinal and
transverse parts, respectively, then the bonding and an-
tibonding electron spectral functions (SC gap functions)
are obtained from these longitudinal and transverse parts
of the electron normal (anomalous) Green’s function,
respectively. Although the transverse part of the SC
gap parameter ∆T ≈ 0, the antibonding peak around
the [π, 0] point is always at lower binding energy than
the bonding peak (hump) due to the BS. In this sense,
the PDH structure in the bilayer cuprate superconduc-
tors in the SC state is mainly caused by the BS. Sec-
ondly, the SC state in the kinetic energy driven SC
mechanism is the conventional BCS like as in the sin-
gle layer case18,19. This can be understood from the
electron normal and anomalous Green’s functions in Eq.
(9). Since the spins center around the [π, π] point in
the MF level18,19,22, then the main contributions for the
spins comes from the [π, π] point. In this case, the
longitudinal and transverse parts of the electron nor-
mal and anomalous Green’s functions in Eq. (9) can
be approximately reduced in terms of ωνp=[π,π] ∼ 0 and
one of the self-consistent equations22 1/2 = 〈S+iaS
ia〉 =
1/(4N)
(Bνk/ωνk)coth[(1/2)βωνk] as,
GL(k, ω) ≈
ν=1,2
ω − Eνk
V 2νk
ω + Eνk
(12a)
GT (k, ω) ≈
ν=1,2
(−1)ν+1Z
ω − Eνk
V 2νk
ω + Eνk
(12b)
(k, ω) =
ν=1,2
z (k)
ω − Eνk
ω + Eνk
(12c)
T (k, ω) =
ν=1,2
(−1)ν+1Z
z (k)
ω − Eνk
ω + Eνk
, (12d)
where the electron coherent weights Z
FA = Z
/2, the
electron quasiparticle coherence factors U2νk ≈ V
hνk−kA
and V 2νk ≈ U
hνk−kA
, the SC gap function ∆̄
z (k) ≈
(k − kA) and the electron quasiparticle spectrum
Eνk ≈ Ehνk−kA , with kA = [π, π]. As in the sin-
gle layer case18,19, this reflects that the hole-like dressed
holon quasiparticle coherence factors Vhνk and Uhνk and
hole-like dressed holon quasiparticle spectrum Ehνk have
been transferred into the electron quasiparticle coher-
ence factors Uνk and Vνk and electron quasiparticle spec-
trum Eνk, respectively, by the convolutions of the corre-
sponding longitudinal and transverse parts of the MF
spin Green’s function and full dressed holon normal
and anomalous Green’s functions due to the charge-spin
recombination27. As a result, these electron normal and
anomalous Green’s functions in Eq. (12) are typical bi-
layer BCS like6. This also reflects that as in the single
layer case18,19, the dressed holon pairs condense with the
d-wave symmetry in a wide range of the doping concen-
tration, then the electron Cooper pairs originating from
the dressed holon pairing state are due to the charge-
spin recombination, and their condensation automati-
cally gives the electron quasiparticle character. These
are why the basic bilayer BCS formalism6 is still valid in
discussions of SC coherence of the quasiparticle peak and
hump, although the pairing mechanism is driven by the
intralayer kinetic energy by exchanging spin excitations,
and other exotic magnetic scattering28,29 is beyond the
BCS formalism.
IV. SUMMARY AND DISCUSSIONS
We have studied the electronic structure of bilayer
cuprate superconductors in the SC state based on the
kinetic energy driven SC mechanism19. Our results show
that the electron spectrum of bilayer cuprate supercon-
ductors is split into the bonding and antibonding com-
ponents by the BS, then the observed PDH structure
around the [π, 0] point is mainly caused by this BS, with
the SC peak being related to the antibonding compo-
nent, and the hump being formed by the bonding com-
ponent. The spectral weight increases with increasing
the doping concentration. In analogy to the two flat
bands appeared in the normal state, the antibonding
and bonding quasiparticles around the [π, 0] point dis-
perse weakly with momentum, in qualitative agreement
with the experimental observation on the bilayer cuprate
superconductors2,7,8,9,10,11. Our these results also show
that the bilayer interaction has significant contributions
to the electronic structure of bilayer cuprate supercon-
ductors in the SC state.
It has been shown from the ARPES experiments2,14
that the BS has been detected in both normal and SC
states, and then the electron spectral functions display
the double-peak structure in the normal state and PDH
structure in the SC state. Recently, we22 have studied the
electron spectrum of bilayer cuprate superconductors in
the normal state, and shown that the double-peak struc-
ture in the electron spectrum in the normal state is dom-
inated by the BS. On the other hand, although the anti-
bonding and bonding SC peaks have different dispersions,
the antibonding and bonding parts have the same d-wave
SC gap amplitude as mentioned above. Incorporating our
previous discussions for the normal state case22 and the
present studies for the SC state case, we therefore find
that the one of the important roles of the interlayer co-
herent hopping (2) is to split the electron spectrum of the
bilayer system into the bonding and antibonding compo-
nents in both normal and SC states. As a consequence,
the well pronounced PDH structure of bilayer cuprate su-
perconductors in the SC state and double-peak structure
in the normal state are mainly caused by the BS.
Acknowledgments
The authors would like to thank Dr. H. Guo and Dr.
L. Cheng for the helpful discussions. This work was sup-
ported by the National Natural Science Foundation of
China under Grant No. 90403005, and the funds from
the Ministry of Science and Technology of China under
Grant Nos. 2006CB601002 and 2006CB921300.
APPENDIX A: DRESSED HOLON BCS TYPE
NORMAL AND ANOMALOUS GREEN’S
FUNCTIONS IN BILAYER CUPRATE
SUPERCONDUCTORS
In the single layer case, it has been shown19 that
the dressed holon-spin interactions from the kinetic en-
ergy terms of the t-t′-J model are quite strong, and
in the case without AFLRO, these interactions can in-
duce the dressed holon pairing state (then the electron
Cooper pairing state) by exchanging spin excitations in
the higher power of the doping concentration. Following
their discussions18,19, we obtain in terms of Eliashberg’s
strong coupling theory31 that the self-consistent equa-
tions that satisfied by the full dressed holon normal and
anomalous Green’s functions in the bilayer system in the
SC state as,
g(k, ω) = g(0)(k, ω) + g(0)(k, ω)
1 (k, ω)g(k, ω)
2 (−k,−ω)ℑ
†(k, ω)
, (A1a)
ℑ†(k, ω) = g(0)(−k,−ω)
1 (−k,−ω)ℑ
†(−k,−ω)
2 (−k,−ω)g(k, ω)
, (A1b)
respectively, where the MF dressed holon normal Green’s
function22 g(0)(k, ω) = g
(k, ω) + σxg
(k, ω), with
the longitudinal and transverse parts are evaluated as
L (k, ω) = (1/2)
ν=1,2(ω − ξνk)
−1 and g
T (k, ω) =
(1/2)
ν=1,2(−1)
ν+1(ω − ξνk)
−1, respectively, while
the dressed holon self-energy functions Σ
1 (k, ω) =
1L (k, ω) + σxΣ
1T (k, ω) and Σ
2 (k, ω) = Σ
2L (k, ω) +
2T (k, ω), with the corresponding longitudinal and
transverse parts have been given in Eq. (6).
In the previous discussions of the electronic struc-
ture for the single layer cuprate superconductors in the
SC state18, it has been shown the self-energy function
2 (k, ω) describes the effective dressed holon pair gap
function, while the self-energy function Σ
1 (k, ω) de-
scribes the quasiparticle coherence. Since Σ
2 (k, ω) is
an even function of ω, while Σ
1 (k, ω) is not, therefore
for the convenience, the self-energy function Σ
1 (k, ω)
can be broken up into its symmetric and antisymmetric
parts as, Σ
1 (k, ω) = Σ
1e (k, ω)+ωΣ
1o (k, ω), then both
1e (k, ω) and Σ
1o (k, ω) are even functions of ω. Now
we can define the dressed holon quasiparticle coherent
weights in the present bilayer system as Z−1
hF1(k, ω) =
1 − Σ
1L (k, ω) and Z
hF2(k, ω) = Σ
1T (k, ω). As in the
single layer case18, we only discuss the low-energy behav-
ior of the electronic structure of bilayer cuprate super-
conductors, which means that the effective dressed holon
pair gap functions and quasiparticle coherent weights
can be discussed in the static limit, i.e., ∆̄h(k) =
2 (k, ω) |ω=0= ∆̄hL(k) + σx∆̄hT (k), Z
hF1(k) = 1 −
1L (k, ω) |ω=0 and Z
hF2(k) = Σ
1T (k, ω) |ω=0. As in
the single layer case18, although ZhF1(k) and ZhF2(k)
still are a function of k, the wave vector dependence
may be unimportant. This followed from the ARPES
experiments2 that in the SC-state of bilayer cuprate su-
perconductors, the lowest energy states are located at
the [π, 0] point, which indicates that the majority con-
tribution for the electron spectrum comes from the [π, 0]
point. In this case, the wave vector k in ZhF1(k) and
ZhF2(k) can be chosen as Z
hF1 = 1−Σ
1L (k) |k=[π,0] and
hF2 = Σ
1T (k) |k=[π,0]. With the help of the above dis-
cussions, the corresponding longitudinal and transverse
parts of the dressed holon normal and anomalous Green’s
functions in Eqs. (A1a) and (A1b) now can be obtained
explicitly as,
gL(k, ω) =
ν=1,2
U2hνk
ω − Ehνk
V 2hνk
ω + Ehνk
, (A2a)
gT (k, ω) =
ν=1,2
(−1)ν+1Z
U2hνk
ω − Ehνk
V 2hνk
ω + Ehνk
, (A2b)
L(k, ω) = −
ν=1,2
2Ehνk
ω − Ehνk
ω + Ehνk
, (A2c)
T (k, ω) = −
ν=1,2
(−1)ν+1Z
2Ehνk
ω − Ehνk
ω + Ehνk
, (A2d)
with the dressed holon effective gap parameters and
quasiparticle coherent weights satisfy the following four
equations,
∆̄hL = −
k,q,p
ν,ν′,ν′′
k−p+qCνν′′ (k+ q)
(ν′′)
Bν′pBνq
ων′pωνq
(ν′′)
νν′ν′′(q,p) + F
νν′ν′′(k,q,p)
[ων′p − ωνq]2 − E
hν′′k
νν′ν′′(q,p) + F
νν′ν′′(k,q,p)
[ων′p + ωνq]2 − E
hν′′k
, (A3a)
∆̄hT = −
k,q,p
ν,ν′,ν′′
(−1)ν+ν
′+ν′′+1Cνν′′(k+ q)
(ν′′)
Bν′pBνq
ων′pωνq
(ν′′)
νν′ν′′(q,p) + F
νν′ν′′(k,q,p)
[ων′p − ωνq]2 − E
hν′′k
νν′ν′′(q,p) + F
νν′ν′′(k,q,p)
[ων′p + ωνq]2 − E
hν′′k
, (A3b)
= 1 +
ν,ν′,ν′′
[1 + (−1)ν+ν
′+ν′′+1]Cνν′′ (p+ k0)
(ν′′)
Bν′pBνq
ων′pωνq
νν′ν′′(q,p)
[ων′p − ωνq + Ehν′′p−q+k0 ]
νν′ν′′(q,p)
[ων′p − ωνq − Ehν′′p−q+k0 ]
νν′ν′′(q,p)
[ων′p + ωνq + Ehν′′p−q+k0]
νν′ν′′ (q,p)
[ων′p + ωνq − Ehν′′p−q+k0 ]
, (A3c)
= 1 +
ν,ν′,ν′′
[1− (−1)ν+ν
′+ν′′+1]Cνν′′ (p+ k0)
(ν′′)
Bν′pBνq
ων′pωνq
νν′ν′′(q,p)
[ων′p − ωνq + Ehν′′p−q+k0 ]
νν′ν′′(q,p)
[ων′p − ωνq − Ehν′′p−q+k0 ]
νν′ν′′(q,p)
[ων′p + ωνq + Ehν′′p−q+k0]
νν′ν′′ (q,p)
[ων′p + ωνq − Ehν′′p−q+k0 ]
, (A3d)
where Cνν′′ (k) = [Z(tγk − t
′γ′k) + (−1)
ν+ν′′t⊥(k)]
νν′ν′′
(q,p) = nB(ωνq)+nB(ων′p)+2nB(ωνq)nB(ων′p),
νν′ν′′(k,q,p) = [2nF (Ehν′′k) − 1][ων′p −
ωνq][nB(ωνq) − nB(ων′p)]/Ehν′′k, F
νν′ν′′(q,p) =
1 + nB(ωνq) + nB(ων′p) + 2nB(ωνq)nB(ων′p),
νν′ν′′
(k,q,p) = [2nF (Ehν′′k) − 1][ων′p + ωνq][1 +
nB(ωνq) + nB(ων′p)]/Ehν′′k, H
νν′ν′′(q,p) =
nF (Ehν′′p−q+k0)[nB(ων′p) − nB(ωνq)] + nB(ωνq)[1 +
nB(ων′p)], H
νν′ν′′(q,p) = nF (Ehν′′p−q+k0)[nB(ωνq) −
nB(ων′p)] + nB(ων′p)[1 + nB(ωνq)], H
νν′ν′′(q,p) =
[1 − nF (Ehν′′p−q+k0)][1 + nB(ωνq) + nB(ων′p)] +
nB(ωνq)nB(ων′p), H
νν′ν′′
(q,p) = nF (Ehν′′p−q+k0)[1 +
nB(ωνq)+nB(ων′p)]+nB(ωνq)nB(ων′p), and k0 = [π, 0].
These four equations must be solved self-consistently
in combination with other equations as in the single
layer case18,19, then all order parameters, decoupling
parameter α, and chemical potential µ are determined
by the self-consistent calculation.
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http://arxiv.org/abs/cond-mat/0205385
http://arxiv.org/abs/cond-mat/0107002

  Within the framework of the kinetic energy driven superconductivity, the
electronic structure of bilayer cuprate superconductors in the superconducting
state is studied. It is shown that the electron spectrum of bilayer cuprate
superconductors is split into the bonding and antibonding components by the
bilayer splitting, then the observed peak-dip-hump structure around the
$[\pi,0]$ point is mainly caused by this bilayer splitting, with the
superconducting peak being related to the antibonding component, and the hump
being formed by the bonding component. The spectral weight increases with
increasing the doping concentration. In analogy to the normal state case, both
electron antibonding peak and bonding hump have the weak dispersions around the
$[\pi,0]$ point.

Electronic structure of kinetic energy driven superconductors in the presence of
bilayer splitting
Yu Lan,1 Jihong Qin,2 and Shiping Feng1
Department of Physics, Beijing Normal University, Beijing 100875, China
Department of Physics, Beijing University of Science and Technology, Beijing 100083, China
(Dated: November 17, 2018)
Within the framework of the kinetic energy driven superconductivity, the electronic structure
of bilayer cuprate superconductors in the superconducting state is studied. It is shown that the
electron spectrum of bilayer cuprate superconductors is split into the bonding and antibonding
components by the bilayer splitting, then the observed peak-dip-hump structure around the [π, 0]
point is mainly caused by this bilayer splitting, with the superconducting peak being related to
the antibonding component, and the hump being formed by the bonding component. The spectral
weight increases with increasing the doping concentration. In analogy to the normal state case, both
electron antibonding peak and bonding hump have the weak dispersions around the [π, 0] point.
PACS numbers: 74.20.Mn, 74.20.-z, 74.25.Jb
I. INTRODUCTION
The parent compounds of cuprate superconductors are
the Mott insulators with an antiferromagnetic (AF) long-
range order (AFLRO), then via the charge carrier doping,
one can drive these materials through a metal-insulating
transition and enter the superconducting (SC) dome1,2,3.
It has become clear in the past twenty years that cuprate
superconductors are among the most complex systems
studied in condensed matter physics1,2,3. The compli-
cations arise mainly from (1) a layered crystal structure
with one or more CuO2 planes per unit cell separated by
insulating layers which leads to a quasi-two-dimensional
electronic structure, and (2) extreme sensitivity of the
physical properties to the compositions (stoichiometry)
which control the carrier density in the CuO2 plane
1,2,3.
As a consequence, both experimental investigation and
theoretical understanding are extremely difficult.
By virtue of systematic studies using the angle-resolved
photoemission spectroscopy (ARPES), the low-energy
electronic structure of cuprate superconductors in the
SC state is well-established by now2,3, where an agree-
ment has emerged that the electronic quasiparticle-like
excitations are well defined, and are the entities par-
ticipating in the SC pairing. In particular, the lowest
energy states are located at the [π, 0] point of the Bril-
louin zone, where the d-wave SC gap function is max-
imal, then the most contributions of the electron spec-
tral function come from the [π, 0] point2,3. Moreover,
some ARPES experimental results unambiguously estab-
lished the Bogoliubov-quasiparticle nature of the sharp
SC quasiparticle peak near the [π, 0] point4,5, then the
SC coherence of the quasiparticle peak is described by
the simple Bardeen-Cooper-Schrieffer (BCS) formalism6.
However, there are numerous anomalies for different fam-
ilies of cuprate superconductors, which complicate the
physical properties of the electronic structure2,3. Among
these anomalies is the dramatic change in the spectral
lineshape around the [π, 0] point first observed on the bi-
layer cuprate superconductor Bi2Sr2CaCu2O8+δ, where
a sharp quasiparticle peak develops at the lowest bind-
ing energy, followed by a dip and a hump, giving rise to
the so-called peak-dip-hump (PDH) structure in the elec-
tron spectrum7,8,9. Later, this PDH structure was also
found in YBa2Cu3O7−δ
10 and in Bi2Sr2Ca2Cu3O10+δ
Furthermore, although the sharp quasiparticle peaks are
identified in the SC state along the entire Fermi surface,
the PDH structure is most strongly developed around the
[π, 0] point2,7,8,9,10,11.
The appearance of the PDH structure in bilayer
cuprate superconductors in the SC state is the mostly
remarkable effect, however, its full understanding is still
a challenging issue. The earlier works2,12 gave the main
impetus for a phenomenological description of the single-
particle excitations in terms of an interaction between
quasiparticles and collective modes, which is of fun-
damental relevance to the nature of superconductivity
and the pairing mechanism in cuprate superconductors.
However, the different interpretive scenario has been
proposed2,13. This followed from the observation of the
bilayer splitting (BS) for both normal and SC states
in a wide doping range14,15,16. This BS of the CuO2
plane derives the electronic structure in the bonding and
antibonding bands due to the present of CuO2 bilayer
blocks in the unit cell of bilayer cuprate superconduc-
tors, then the main features of the PDH structure is
caused by the BS13,14,15,16, with the peak and hump
corresponding to the antibonding and bonding bands,
respectively. Furthermore, some ARPES experimental
data measured above and below the SC transition tem-
perature show that this PDH structure is totally unre-
lated to superconductivity14. The recent ARPES exper-
imental results reported by several groups support this
scenario, and most convincingly suggested that the PDH
structure originates from the BS at any doping levels17.
To the best of our knowledge, this PDH structure in bi-
layer cuprate superconductors has not been treated start-
ing from a microscopic SC theory.
Within the single layer t-t′-J model, the electronic
structure of the single layer cuprate superconductors in
the SC state has been discussed18 based on the frame-
work of the kinetic energy driven superconductivity19,
and the main features of the ARPES experiments on
the single layer cuprate superconductors have been repro-
duced, including the doping and temperature dependence
of the electron spectrum and quasiparticle dispersion. In
http://arxiv.org/abs/0704.0825v2
this paper, we study the electronic structure of bilayer
cuprate superconductors in the SC state along with this
line. Within the kinetic energy driven SC mechanism19,
we employed the t-t′-J model by considering the bilayer
interaction, and then show explicitly that the BS occurs
due to this bilayer interaction. In this case, the elec-
tron spectrum is split into the bonding and antibond-
ing components by this BS, then the SC peak is closely
related to the antibonding component, while the hump
is mainly formed by the bonding component. In other
words, the well pronounced PDH structure in the electron
spectrum of bilayer cuprate superconductors is mainly
caused by the BS. Furthermore, the spectral weight in
the [π, 0] point increases with increasing the doping con-
centration. In analogy to the normal-state case14,20,21,22,
both electron antibonding peak and bonding hump have
the weak dispersions around the [π, 0] point, in qualita-
tive agreement with the experimental observation on bi-
layer cuprate superconductors in the SC state2,7,8,9,10,11.
The paper is organized as follows. The basic formal-
ism is presented in Sec. II, where we generalize the
kinetic energy driven superconductivity from the previ-
ous single layer case18,19 to the bilayer case, and then
evaluate explicitly the longitudinal and transverse com-
ponents of the electron normal and anomalous Green’s
functions (hence the bonding and antibonding electron
spectral functions). Within this theoretical framework,
we discuss the electronic structure of bilayer cuprate su-
perconductors in the SC state in Sec. III. It is shown that
the striking PDH structure in bilayer cuprate supercon-
ductors is closely related to the BS. Finally, we give a
summary and discussions in Sec. IV.
II. FORMALISM
It has been shown from the ARPES experiments2,23
that the two-dimensional t-t′-J model is of particular
relevance to the low energy features of cuprate super-
conductors. For discussions of the physical properties of
bilayer cuprate superconductors, the t-t′-J model can be
expressed by including the bilayer interactions as,
H = −t
iη̂aσ
iaσCi+η̂aσ + t
iτ̂aσ
iaσCi+τ̂aσ
t⊥(i)(C
i1σCi2σ +H.c.) + µ
iaσCiaσ
Sia · Si+η̂a + J⊥
Si1 · Si2, (1)
supplemented by an important on-site local constraint
iaσCiaσ ≤ 1 to avoid the double occupancy, where
η̂ = ±x̂,±ŷ representing the nearest neighbors of a given
site i, τ̂ = ±x̂± ŷ representing the next nearest neighbors
of a given site i, a = 1, 2 is plane index, C
iaσ (Ciaσ)
is the electron creation (annihilation) operator, Sia =
iaσCia/2 is the spin operator with the Pauli matrices
σ = (σx, σy, σz), µ is the chemical potential, and the
interlayer coherent hopping has the form,
t⊥(k) =
(cos kx − cos ky)
2, (2)
which is strongly anisotropic and follows the theoret-
ical predictions24. In particular, this momentum de-
pendent form (2) has been experimentally verified14,15.
For this t-t′-J model (1), it has been argued that cru-
cial requirement is to impose the electron single occu-
pancy local constraint for a proper understanding of the
physical properties of cuprate superconductors. To in-
corporate the electron single occupancy local constraint,
the charge-spin separation (CSS) fermion-spin theory has
been proposed25, where the constrained electron opera-
tors are decoupled as, Cia↑ = h
ia and Cia↓ = h
with the spinful fermion operator hiaσ = e
−iΦiaσhia rep-
resents the charge degree of freedom together with some
effects of the spin configuration rearrangements due to
the presence of the doped hole itself (dressed holon),
while the spin operator Sia represents the spin degree
of freedom, then the bilayer t-t′-J Hamiltonian (1) can
be expressed in this CSS fermion-spin representation as,
H = t
i+η̂a↑hia↑S
i+η̂a + h
i+η̂a↓hia↓S
i+η̂a)
i+τ̂a↑hia↑S
i+τ̂a + h
i+τ̂a↓hia↓S
i+τ̂a)
t⊥(i)(h
i2↑hi1↑S
i2 + h
i1↑hi2↑S
i2↓hi1↓S
i2 + h
i1↓hi2↓S
i1)− µ
iaσhiaσ
+ Jeff
Sia · Si+η̂a + Jeff⊥
Si1 · Si2, (3)
where Jeff = J(1 − δ)
2, Jeff⊥ = J⊥(1 − δ)
2, and δ =
iaσhiaσ〉 = 〈h
iahia〉 is the doping concentration. It has
been shown that the electron single occupancy local con-
straint is satisfied in analytical calculations within this
CSS fermion-spin theory, and the double spinful fermion
occupancy are ruled out automatically25. Although in
common sense hiaσ is not a real spinful fermion, it be-
haves like a spinful fermion25. As in the single layer
case18, the kinetic energy terms in the bilayer t-t′-J
model have been transferred as the dressed holon-spin
interactions, which can induce the dressed holon pair-
ing state (hence the electron Cooper pairing state) by
exchanging spin excitations in the higher power of the
doping concentration. Before calculation of the electron
normal and anomalous Green’s functions of the bilayer
system in the SC state, we firstly introduce the SC order
parameter. As we have mentioned above, there are two
coupled CuO2 planes in the unit cell, and in this case,
the SC order parameters for the electron Cooper pair is
a matrix ∆ = ∆L + σx∆T , with the longitudinal and
transverse SC order parameters in the CSS fermion-spin
theory can be expressed as,
∆L = 〈C
i+η̂a↓ − C
i+η̂a↑〉
= 〈hia↑hi+η̂a↓S
i+η̂a − hia↓hi+η̂a↑S
i+η̂a〉
= −χ1∆hL, (4a)
∆T = 〈C
i2↓ − C
= 〈hi1↑hi2↓S
i2 − hi1↓hi2↑S
= −χ⊥∆hT , (4b)
respectively, where the spin correlation functions χ1 =
〈S+iaS
i+η̂a〉 and χ⊥ = 〈S
i2〉, and the longitudinal
and transverse dressed holon pairing order parameters
∆hL = 〈hi+η̂a↓hia↑ − hi+η̂a↑hia↓〉 and ∆hT = 〈hi2↓hi1↑ −
hi2↑hi1↓〉.
Within the t-J type model, robust indications
of superconductivity with the d-wave symmetry in
doped cuprates have been found by using numerical
techniques26. On the other hand, it has been argued that
the SC transition in doped cuprates is determined by the
need to reduce the frustrated kinetic energy27. Although
it is not necessary for the strong coupling of the electron
quasiparticles and a pairing boson in their arguments27,
a series of the inelastic neutron scattering experimental
results provide a clear link between the electron quasi-
particles and magnetic excitations28,29. In particular, an
impurity-substitution effect on the low energy dynamics
has been studied by virtue of the ARPES measurement30,
this impurity-substitution effect is a magnetic analogue
of the isotope effect used for the conventional super-
conductors. These experimental results30 reveal that
the impurity-induced changes in the electron self-energy
show a good correspondence to those of the magnetic
excitations, indicating the importance of the magnetic
fluctuation to the electron pairing in cuprate supercon-
ductors. Recently, we19 have developed the kinetic en-
ergy driven SC mechanism based on the CSS fermion-
spin theory25, where the dressed holons interact occur-
ring directly through the kinetic energy by exchanging
spin excitations, leading to a net attractive force between
dressed holons, then the electron Cooper pairs originat-
ing from the dressed holon pairing state are due to the
charge-spin recombination, and their condensation re-
veals the SC ground-state. Within this SC mechanism19,
the doping and temperature dependence of the electron
spectral function of the single layer cuprate supercon-
ductors in the SC state has been discussed18. In this
section, our main goal is to generalize these analytical
calculations from the single layer case to the bilayer sys-
tem. As in the case for the SC order parameter, the full
dressed holon normal and anomalous Green’s functions
can also be expressed as g(k, ω) = gL(k, ω) + σxgT (k, ω)
and ℑ†(k, ω) = ℑ
L(k, ω) + σxℑ
L(k, ω), respectively. We
now can follow the previous discussions for the single
layer case18,19, and evaluate explicitly these correspond-
ing longitudinal and transverse parts of the full dressed
holon normal and anomalous Green’s functions as [see
the Appendix],
gL(k, ω) =
ν=1,2
U2hνk
ω − Ehνk
V 2hνk
ω + Ehνk
, (5a)
gT (k, ω) =
ν=1,2
(−1)ν+1Z
U2hνk
ω − Ehνk
V 2hνk
ω + Ehνk
, (5b)
L(k, ω) = −
ν=1,2
2Ehνk
ω − Ehνk
ω + Ehνk
, (5c)
T (k, ω) = −
ν=1,2
(−1)ν+1Z
2Ehνk
ω − Ehνk
ω + Ehνk
, (5d)
where the dressed holon quasiparticle coherence fac-
tors U2hνk = [1 + ξ̄νk/Ehνk]/2 and V
hνk = [1 −
ξ̄νk/Ehνk]/2, the dressed holon quasiparticle disper-
sion Ehνk =
[ξ̄νk]2+ | ∆̄
(k) |2, the renormalized
dressed holon excitation spectrum ξ̄νk = Z
ξνk, with
the mean-field (MF) dressed holon excitation spectrum
ξνk = Ztχ1γk − Zt
k − µ+ (−1)
ν+1χ⊥t⊥(k), where
the spin correlation function χ2 = 〈S
i+τ̂a〉, γk =
(1/Z)
eik·η̂, γ′k = (1/Z)
eik·τ̂ , Z is the num-
ber of the nearest neighbor or next nearest neighbor
sites, the renormalized dressed holon pair gap func-
tion ∆̄
(k) = Z
[∆̄hL(k) + (−1)
ν+1∆̄hT (k)], with
ν = 1 ( ν = 2) for the bonding (antibonding) case,
where ∆̄hL(k) = Σ
2L (k, ω) |ω=0= ∆̄hLγ
, with γ
(coskx − cosky)/2, ∆̄hT (k) = Σ
2T (k, ω) |ω=0= ∆̄hT , the
dressed holon quasiparticle coherent weights Z
(1)−1
hF1 − Z
hF2, Z
(2)−1
= Z−1
hF1 + Z
hF2, with Z
hF1 =
1 − Σ
1L (k0, ω) |ω=0, and Z
hF2 = Σ
1T (k0, ω) |ω=0 ,
where k0 = [π, 0], Σ
1L (k, ω) and Σ
1T (k, ω) are the cor-
responding antisymmetric parts of the longitudinal and
transverse dressed holon self-energy functions Σ
1L (k, ω)
and Σ
1T (k, ω), while the longitudinal and transverse
parts of the dressed holon self-energy functions Σ
1 (k, ω)
and Σ
2 (k, ω) have been evaluated as,
1L (k, iωn) =
p+q+k
gL(p+ k, ipm + iωn)ΠLL(p,q, ipm)
p+q+k
gT (p+ k, ipm + iωn)ΠTL(p,q, ipm)], (6a)
1T (k, iωn) =
p+q+k
gT (p+ k, ipm + iωn)ΠTT (p,q, ipm)
p+q+k
gL(p+ k, ipm + iωn)ΠLT (p,q, ipm)], (6b)
2L (k, iωn) =
p+q+k
L(−p− k,−ipm − iωn)ΠLL(p,q, ipm)
p+q+k
T (−p− k,−ipm − iωn)ΠTL(p,q, ipm)], (6c)
2T (k, iωn) =
p+q+k
(−p− k,−ipm − iωn)ΠTT (p,q, ipm)
p+q+k
L(−p− k,−ipm − iωn)ΠLT (p,q, ipm)], (6d)
where R
= [Z(tγk − t
′γ′k)]
2 + t2⊥(k), R
= 2Z(tγk −
)t⊥(k), and the spin bubbles Πη,η′(p,q, ipm) =
(1/β)
η (q, iqm)D
η′ (q + p, iqm+ ipm), with η =
L, T and η′ = L, T , and the MF spin Green’s function
D(0)(k, ω) = D
(k, ω) + σxD
(k, ω), with the cor-
responding longitudinal and transverse parts have been
given by22,
L (k, ω) =
ν=1,2
ω2 − ω2
, (7a)
(k, ω) =
ν=1,2
(−1)ν+1
ω2 − ω2
, (7b)
where Bνk = λ(A1γk−A2)−λ
′(2χz2γ
k−χ2)−Jeff⊥[χ⊥+
2χz⊥(−1)
ν ][ǫ⊥(k)+(−1)
ν ], A1 = 2ǫ‖χ
1+χ1, A2 = ǫ‖χ1+
2χz1, λ = 2ZJeff , λ
′ = 4Zφ2t
′, ǫ‖ = 1+2tφ1/Jeff , ǫ⊥(k) =
1 + 4φ⊥t⊥(k)/Jeff⊥, the spin correlation functions χ
〈SziaS
i+η̂a〉, χ
2 = 〈S
i+τ̂a〉, χ
⊥ = 〈S
i2〉, the dressed
holon particle-hole order parameters φ1 = 〈h
iaσhi+η̂aσ〉,
φ2 = 〈h
iaσhi+τ̂aσ〉, φ⊥ = 〈h
i1σhi2σ〉, and the MF spin
excitation spectrum,
ω2νk = λ
A4 − αǫ‖χ
1γk −
αǫ‖χ1
(1 − ǫ‖γk) +
αχz1 − αχ1γk
(ǫ‖ − γk)
Z − 1
γ′k +
+ λλ′α
χz1(1 − ǫ‖γk)γ
k − C2)(ǫ‖ − γk)
+ γ′k(C
2 − ǫ‖χ
2γk)−
ǫ‖(C2 − χ2γk)
+ λJeff⊥α
ǫ⊥(k)(ǫ‖ − γk)[C⊥ + χ1(−1)
+ (1− ǫ‖γk)[C
⊥ + χ
1ǫ⊥(k)(−1)
ν ] + [ǫ⊥(k) + (−1)
ǫ‖(C⊥ − χ⊥γk) + (C
⊥ − ǫ‖χ
⊥γk)(−1)
+ λ′Jeff⊥α
γ′k[C
⊥ + χ
2ǫ⊥(k)(−1)
ǫ⊥(k)[C
⊥ + χ2(−1)
k − C
⊥) + χ
k(−1)
[ǫ⊥(k) + (−1)
J2eff⊥[ǫ⊥(k) + (−1)
ν ]2, (8)
where A3 = αC1 + (1− α)/2Z, A4 = αC
1 + (1− α)/4Z,
A5 = αC3 + (1 − α)/2Z, and the spin correla-
tion functions C1 = (1/Z
η̂η̂′
i+η̂aS
i+η̂′a
C2 = (1/Z
i+η̂aS
i+τ̂a〉, C3 =
(1/Z2)
τ̂ τ̂ ′
i+τ̂aS
i+τ̂ ′a
〉, Cz1 =
(1/Z2)
η̂η̂′
〈Szi+η̂aS
i+η̂′a
〉, Cz2 =
(1/Z2)
〈Szi+η̂aS
i+τ̂a〉, C⊥ = (1/Z)
〈S+i1S
i+η̂2〉,
C′⊥ = (1/Z)
〈S+i1S
i+τ̂2〉, C
⊥ = (1/Z)
〈Szi1S
i+η̂2〉,
and C′
⊥ = (1/Z)
〈Szi1S
i+τ̂2〉. In order to satisfy the
sum rule of the spin correlation function 〈S+iaS
ia〉 = 1/2
in the case without AFLRO, the important decoupling
parameter α has been introduced in the above calcu-
lation as in the single layer case18,19,22, which can be
regarded as the vertex correction.
With the help of the longitudinal and transverse parts
of the full dressed holon normal and anomalous Green’s
functions in Eq. (5) and MF spin Green’s function
in Eq. (7), we now can calculate the electron nor-
mal and anomalous Green’s functions G(i − j, t − t′) =
〈〈Ciσ(t);C
′)〉〉 = GL(i− j, t− t
′) + σxGT (i− j, t− t
and Γ†(i−j, t−t′) = 〈〈C
i↑(t);C
′)〉〉 = Γ
L(i−j, t−t
T (i−j, t−t
′), where these longitudinal and transverse
parts are the convolutions of the corresponding longitudi-
nal and transverse parts of the full dressed holon normal
and anomalous Green’s functions and MF spin Green’s
function in the CSS fermion-spin theory, and can be eval-
uated explicitly as,
GL(k, ω) =
L(1)µν (k,p)
U2hµp−k
ω + Ehµp−k − ωνp
V 2hµp−k
ω − Ehµp−k + ωνp
+ L(2)µν (k,p)
U2hµp−k
ω + Ehµp−k + ωνp
V 2hµp−k
ω − Ehµp−k − ωνp
, (9a)
GT (k, ω) =
(−1)µ+νZ
L(1)µν (k,p)
U2hµp−k
ω + Ehµp−k − ωνp
V 2hµp−k
ω − Ehµp−k + ωνp
+ L(2)µν (k,p)
U2hµp−k
ω + Ehµp−k + ωνp
V 2hµp−k
ω − Ehµp−k − ωνp
, (9b)
L(k, ω) =
(p− k)
2Ehµp−k
L(1)µν (k,p)
ω − Ehµp−k + ωνp
ω + Ehµp−k − ωνp
+ L(2)µν (k,p)
ω − Ehµp−k − ωνp
ω + Ehµp−k + ωνp
, (9c)
(k, ω) =
(−1)µ+νZ
(p− k)
2Ehµp−k
L(1)µν (k,p)
ω − Ehµp−k + ωνp
ω + Ehµp−k − ωνp
+ L(2)µν (k,p)
ω − Ehµp−k − ωνp
ω + Ehµp−k + ωνp
, (9d)
where L
µν (k,p) = [coth(βωνp/2) − th(βEhµp−k/2)]/2
and L
µν (k,p) = [coth(βωνp/2) + th(βEhµp−k/2)]/2,
then the longitudinal and transverse parts of the
electron spectral function AL(k, ω) = −2ImGL(k, ω)
and AT (k, ω) = −2ImGT (k, ω) and SC gap func-
tion ∆L(k) = (1/β)
L(k, iωn) and ∆T (k) =
(1/β)
(k, iωn) are obtained as,
AL(k, ω) = π
{L(1)µν (k,p)[U
hµp−kδ(ω + Ehµp−k − ωνp) + V
hµp−kδ(ω − Ehµp−k + ωνp)]
+ L(2)µν (k,p)[U
hµp−kδ(ω + Ehµp−k + ωνp) + V
hµp−kδ(ω − Ehµp−k − ωνp)]}, (10a)
AT (k, ω) = π
(−1)µ+νZ
{L(1)µν (k,p)[U
hµp−kδ(ω + Ehµp−k − ωνp) + V
hµp−kδ(ω − Ehµp−k + ωνp)]
+ L(2)µν (k,p)[U
hµp−kδ(ω + Ehµp−k + ωνp) + V
hµp−kδ(ω − Ehµp−k − ωνp)]}, (10b)
∆L(k) = −
p,µ,ν
(p− k)
Ehµp−k
βEhµp−k]coth[
βωνp], (10c)
∆T (k) = −
p,µ,ν
(−1)µ+νZ
(p− k)
Ehµp−k
βEhµp−k]coth[
βωνp]. (10d)
With the above longitudinal and transverse parts of the
SC gap functions in Eqs. (10c) and (10d), the corre-
sponding longitudinal and transverse SC gap parameters
are obtained as ∆L = −χ1∆hL and ∆T = −χ⊥∆hT ,
respectively. In the bilayer coupling case, the more ap-
propriate classification is in terms of the spectral func-
tion and SC gap function within the basis of the an-
tibonding and bonding components13,14,15,16,17. In this
case, the electron spectral function and SC gap parame-
ter can be transformed from the plane representation to
the antibonding-bonding representation as,
A(a)(k, ω) =
[AL(k, ω)−AT (k, ω)], (11a)
A(b)(k, ω) =
[AL(k, ω) +AT (k, ω)], (11b)
∆(a) = ∆L −∆T , (11c)
∆(b) = ∆L +∆T . (11d)
respectively, then the antibonding and bonding parts
have odd and even symmetries, respectively.
III. ELECTRON STRUCTURE OF BILAYER
CUPRATE SUPERCONDUCTORS
We now begin to discuss the effect of the bilayer in-
teraction on the electronic structure in the SC state. We
first plot, in Fig. 1, the antibonding (solid line) and
bonding (dashed line) electron spectral functions in the
[π, 0] point for parameters t/J = 2.5, t′/t = 0.3, and
t⊥/t = 0.35 with temperature T = 0.002J at the doping
concentration δ = 0.15. In comparison with the single
layer case18, the electron spectrum of the bilayer system
has been split into the bonding and antibonging compo-
nents, with the bonding and antibonding SC quasipar-
ticle peaks in the [π, 0] point are located at the differ-
ent positions. In this sense, the differentiation between
the bonding and antibonding components of the electron
spectral function is essential. The antibonding spectrum
consists of a low energy antibonding peak, corresponding
to the SC peak, and the bonding spectrum has a higher
energy bonding peak, corresponding to the hump, while
the spectral dip is in between them, then the total con-
tributions for the electron spectrum from both antibond-
ing and bonding components give rise to the PDH struc-
ture. Although the simple bilayer t-t′-J model (1) can-
not be regarded as a comprehensive model for a quanti-
tative comparison with bilayer cuprate superconductors,
our present results for the SC state are in qualitative
agreement with the major experimental observations on
bilayer cuprate superconductors2,7,8,9,10,11,16.
We now turn to discuss the doping evolution of the
electron spectrum of bilayer cuprate superconductors in
( )/J
-1.0 -.5 0.0 .5
Bonding
Antibonding
FIG. 1: The antibonding (solid line) and bonding (dashed
line) electron spectral functions in the [π, 0] point for t/J =
2.5, t′/t = 0.3, and t⊥/t = 0.35 with T = 0.002J at δ = 0.15.
( )/J
-1.0 -.5 0.0 .5
FIG. 2: The electron spectral functions at [π, 0] point for
t/J = 2.5, t′/t = 0.3, and t⊥/t = 0.35 with T = 0.002J at
δ = 0.09 (solid line), δ = 0.12 (dashed line), and δ = 0.15
(dotted line).
the SC state. We have calculated the electron spec-
trum at different doping concentrations, and the result
of the electron spectral functions in the [π, 0] point for
t/J = 2.5, t′/t = 0.3, and t⊥/t = 0.35 with T = 0.002J
at δ = 0.09 (solid line), δ = 0.12 (dashed line), and
δ = 0.15 (dotted line) are plotted in Fig. 2. In compari-
son with the corresponding ARPES experimental results
of the bilayer cuprate superconductor Bi2Sr2CaCu2O8+δ
in the SC state in Ref.12, it is obviously that the doping
evolution of the spectral weight of the bilayer supercon-
Bonding Band
Antibonding Band
(-0.2 , ) (0, ) (0.2 , )
FIG. 3: The positions of the antibonding peaks and bonding
humps in the electron spectrum as a function of momentum
along the direction [−0.2π, π] → [0, π] → [0.2π, π] with T =
0.002J at δ = 0.15 for t/J = 2.5, t′/t = 0.3, and t⊥/t = 0.35.
ductor Bi2Sr2CaCu2O8+δ is reproduced. With increas-
ing the doping concentration, both SC peak and hump
become sharper, and then the spectral weights increase
in intensity. Furthermore, we have also calculated the
electron spectrum with different temperatures, and the
results show that the spectral weights of both SC peak
and hump are suppressed with increasing temperatures.
Our these results are also qualitatively consistent with
the ARPES experimental results on bilayer cuprate su-
perconductors in the SC state2,9,12.
To better perceive the anomalous form of the antibond-
ing and bonding electron spectral functions as a function
of energy ω for k in the vicinity of the [π, 0] point, we
have made a series of calculations for the electron spec-
tral function at different momenta, and the results show
that the sharp SC peak from the electron antibonding
spectral function and hump from the bonding spectral
function persist in a very large momentum space region
around the [π, 0] point. To show this point clearly, we
plot the positions of the antibonding peak and bonding
hump in the electron spectrum as a function of momen-
tum along the direction [−0.2π, π] → [0, π] → [0.2π, π]
with T = 0.002J at δ = 0.15 for t/J = 2.5, t′/t = 0.3,
and t⊥/t = 0.35 in Fig. 3. Our result shows that there
are two branches in the quasiparticle dispersion, with
upper branch corresponding to the antibonding quasi-
particle dispersion, and lower branch corresponding to
the bonding quasiparticle dispersion. Furthermore, the
BS reaches its maximum at the [π, 0] point. Our present
result also shows that in analogy to the two flat bands ap-
peared in the normal state22, both electron antibonding
peak and bonding hump have a weak dispersion around
the [π, 0] point, in qualitative agreement with the ARPES
experimental measurements on bilayer cuprate supercon-
ductors in the SC state2,7,8,9,10,11,14.
In the above calculations, we find that although the
antibonding SC peak and bonding hump have different
dispersions, the transverse part of the SC gap param-
eter ∆T ≈ 0. To show this point clearly, we plot the
antibonding and bonding gap parameters in Eqs. (11c)
00.00 0.05 0.10 0.15 0.20 0.25 0.30
00.00
FIG. 4: The antibonding (solid line) and bonding (dashed
line) gap parameters as a function of the doping concentration
with T = 0.002J for t/J = 2.5, t′/t = 0.3, and t⊥/t = 0.35.
and (11d) as a function of the doping concentration with
T = 0.002J for t/J = 2.5, t′/t = 0.3, and t⊥/t = 0.35 in
Fig. 4. As seen from Fig. 4, both antibonding and bond-
ing gap parameters have the same d-wave SC gap mag-
nitude in a given doping concentration, i.e., ∆a ≈ ∆b.
This result shows that although there is a single elec-
tron interlayer coherent hopping (2) in bilayer cuprate
superconductors in the SC state, the electron interlayer
pairing interaction vanishes. This reflects that in the
present kinetic energy driven SC mechanism, the weak
dressed holon-spin interaction due to the interlayer co-
herent hopping (2) from the kinetic energy terms in Eq.
(3) does not induce the dressed holon interlayer pair-
ing state by exchanging spin excitations in the higher
power of the doping concentration. This is different from
the dressed holon-spin interaction due to the intralayer
hopping from the kinetic energy terms in Eq. (3), it
can induce superconductivity by exchanging spin excita-
tions in the higher power of the doping concentration19.
Our this result is also consistent with the ARPES ex-
perimental results of the bilayer cuprate superconductor
Bi(Pb)2Sr2CaCu2O8+δ
14,16, where the SC gap separately
for the bonding and antibonding bands has been mea-
sured, and it is found that both d-wave SC gaps from
the antibonding and bonding components are identical
within the experimental uncertainties.
To our present understanding, two main reasons why
the electronic structure of bilayer cuprate superconduc-
tors in the SC state can be described qualitatively in
the framework of the kinetic energy driven supercon-
ductivity by considering the bilayer interaction are as
follows. Firstly, the bilayer interaction causes the BS,
this leads to that the full electron normal (anomalous)
Green’s function is divided into the longitudinal and
transverse parts, respectively, then the bonding and an-
tibonding electron spectral functions (SC gap functions)
are obtained from these longitudinal and transverse parts
of the electron normal (anomalous) Green’s function,
respectively. Although the transverse part of the SC
gap parameter ∆T ≈ 0, the antibonding peak around
the [π, 0] point is always at lower binding energy than
the bonding peak (hump) due to the BS. In this sense,
the PDH structure in the bilayer cuprate superconduc-
tors in the SC state is mainly caused by the BS. Sec-
ondly, the SC state in the kinetic energy driven SC
mechanism is the conventional BCS like as in the sin-
gle layer case18,19. This can be understood from the
electron normal and anomalous Green’s functions in Eq.
(9). Since the spins center around the [π, π] point in
the MF level18,19,22, then the main contributions for the
spins comes from the [π, π] point. In this case, the
longitudinal and transverse parts of the electron nor-
mal and anomalous Green’s functions in Eq. (9) can
be approximately reduced in terms of ωνp=[π,π] ∼ 0 and
one of the self-consistent equations22 1/2 = 〈S+iaS
ia〉 =
1/(4N)
(Bνk/ωνk)coth[(1/2)βωνk] as,
GL(k, ω) ≈
ν=1,2
ω − Eνk
V 2νk
ω + Eνk
(12a)
GT (k, ω) ≈
ν=1,2
(−1)ν+1Z
ω − Eνk
V 2νk
ω + Eνk
(12b)
(k, ω) =
ν=1,2
z (k)
ω − Eνk
ω + Eνk
(12c)
T (k, ω) =
ν=1,2
(−1)ν+1Z
z (k)
ω − Eνk
ω + Eνk
, (12d)
where the electron coherent weights Z
FA = Z
/2, the
electron quasiparticle coherence factors U2νk ≈ V
hνk−kA
and V 2νk ≈ U
hνk−kA
, the SC gap function ∆̄
z (k) ≈
(k − kA) and the electron quasiparticle spectrum
Eνk ≈ Ehνk−kA , with kA = [π, π]. As in the sin-
gle layer case18,19, this reflects that the hole-like dressed
holon quasiparticle coherence factors Vhνk and Uhνk and
hole-like dressed holon quasiparticle spectrum Ehνk have
been transferred into the electron quasiparticle coher-
ence factors Uνk and Vνk and electron quasiparticle spec-
trum Eνk, respectively, by the convolutions of the corre-
sponding longitudinal and transverse parts of the MF
spin Green’s function and full dressed holon normal
and anomalous Green’s functions due to the charge-spin
recombination27. As a result, these electron normal and
anomalous Green’s functions in Eq. (12) are typical bi-
layer BCS like6. This also reflects that as in the single
layer case18,19, the dressed holon pairs condense with the
d-wave symmetry in a wide range of the doping concen-
tration, then the electron Cooper pairs originating from
the dressed holon pairing state are due to the charge-
spin recombination, and their condensation automati-
cally gives the electron quasiparticle character. These
are why the basic bilayer BCS formalism6 is still valid in
discussions of SC coherence of the quasiparticle peak and
hump, although the pairing mechanism is driven by the
intralayer kinetic energy by exchanging spin excitations,
and other exotic magnetic scattering28,29 is beyond the
BCS formalism.
IV. SUMMARY AND DISCUSSIONS
We have studied the electronic structure of bilayer
cuprate superconductors in the SC state based on the
kinetic energy driven SC mechanism19. Our results show
that the electron spectrum of bilayer cuprate supercon-
ductors is split into the bonding and antibonding com-
ponents by the BS, then the observed PDH structure
around the [π, 0] point is mainly caused by this BS, with
the SC peak being related to the antibonding compo-
nent, and the hump being formed by the bonding com-
ponent. The spectral weight increases with increasing
the doping concentration. In analogy to the two flat
bands appeared in the normal state, the antibonding
and bonding quasiparticles around the [π, 0] point dis-
perse weakly with momentum, in qualitative agreement
with the experimental observation on the bilayer cuprate
superconductors2,7,8,9,10,11. Our these results also show
that the bilayer interaction has significant contributions
to the electronic structure of bilayer cuprate supercon-
ductors in the SC state.
It has been shown from the ARPES experiments2,14
that the BS has been detected in both normal and SC
states, and then the electron spectral functions display
the double-peak structure in the normal state and PDH
structure in the SC state. Recently, we22 have studied the
electron spectrum of bilayer cuprate superconductors in
the normal state, and shown that the double-peak struc-
ture in the electron spectrum in the normal state is dom-
inated by the BS. On the other hand, although the anti-
bonding and bonding SC peaks have different dispersions,
the antibonding and bonding parts have the same d-wave
SC gap amplitude as mentioned above. Incorporating our
previous discussions for the normal state case22 and the
present studies for the SC state case, we therefore find
that the one of the important roles of the interlayer co-
herent hopping (2) is to split the electron spectrum of the
bilayer system into the bonding and antibonding compo-
nents in both normal and SC states. As a consequence,
the well pronounced PDH structure of bilayer cuprate su-
perconductors in the SC state and double-peak structure
in the normal state are mainly caused by the BS.
Acknowledgments
The authors would like to thank Dr. H. Guo and Dr.
L. Cheng for the helpful discussions. This work was sup-
ported by the National Natural Science Foundation of
China under Grant No. 90403005, and the funds from
the Ministry of Science and Technology of China under
Grant Nos. 2006CB601002 and 2006CB921300.
APPENDIX A: DRESSED HOLON BCS TYPE
NORMAL AND ANOMALOUS GREEN’S
FUNCTIONS IN BILAYER CUPRATE
SUPERCONDUCTORS
In the single layer case, it has been shown19 that
the dressed holon-spin interactions from the kinetic en-
ergy terms of the t-t′-J model are quite strong, and
in the case without AFLRO, these interactions can in-
duce the dressed holon pairing state (then the electron
Cooper pairing state) by exchanging spin excitations in
the higher power of the doping concentration. Following
their discussions18,19, we obtain in terms of Eliashberg’s
strong coupling theory31 that the self-consistent equa-
tions that satisfied by the full dressed holon normal and
anomalous Green’s functions in the bilayer system in the
SC state as,
g(k, ω) = g(0)(k, ω) + g(0)(k, ω)
1 (k, ω)g(k, ω)
2 (−k,−ω)ℑ
†(k, ω)
, (A1a)
ℑ†(k, ω) = g(0)(−k,−ω)
1 (−k,−ω)ℑ
†(−k,−ω)
2 (−k,−ω)g(k, ω)
, (A1b)
respectively, where the MF dressed holon normal Green’s
function22 g(0)(k, ω) = g
(k, ω) + σxg
(k, ω), with
the longitudinal and transverse parts are evaluated as
L (k, ω) = (1/2)
ν=1,2(ω − ξνk)
−1 and g
T (k, ω) =
(1/2)
ν=1,2(−1)
ν+1(ω − ξνk)
−1, respectively, while
the dressed holon self-energy functions Σ
1 (k, ω) =
1L (k, ω) + σxΣ
1T (k, ω) and Σ
2 (k, ω) = Σ
2L (k, ω) +
2T (k, ω), with the corresponding longitudinal and
transverse parts have been given in Eq. (6).
In the previous discussions of the electronic struc-
ture for the single layer cuprate superconductors in the
SC state18, it has been shown the self-energy function
2 (k, ω) describes the effective dressed holon pair gap
function, while the self-energy function Σ
1 (k, ω) de-
scribes the quasiparticle coherence. Since Σ
2 (k, ω) is
an even function of ω, while Σ
1 (k, ω) is not, therefore
for the convenience, the self-energy function Σ
1 (k, ω)
can be broken up into its symmetric and antisymmetric
parts as, Σ
1 (k, ω) = Σ
1e (k, ω)+ωΣ
1o (k, ω), then both
1e (k, ω) and Σ
1o (k, ω) are even functions of ω. Now
we can define the dressed holon quasiparticle coherent
weights in the present bilayer system as Z−1
hF1(k, ω) =
1 − Σ
1L (k, ω) and Z
hF2(k, ω) = Σ
1T (k, ω). As in the
single layer case18, we only discuss the low-energy behav-
ior of the electronic structure of bilayer cuprate super-
conductors, which means that the effective dressed holon
pair gap functions and quasiparticle coherent weights
can be discussed in the static limit, i.e., ∆̄h(k) =
2 (k, ω) |ω=0= ∆̄hL(k) + σx∆̄hT (k), Z
hF1(k) = 1 −
1L (k, ω) |ω=0 and Z
hF2(k) = Σ
1T (k, ω) |ω=0. As in
the single layer case18, although ZhF1(k) and ZhF2(k)
still are a function of k, the wave vector dependence
may be unimportant. This followed from the ARPES
experiments2 that in the SC-state of bilayer cuprate su-
perconductors, the lowest energy states are located at
the [π, 0] point, which indicates that the majority con-
tribution for the electron spectrum comes from the [π, 0]
point. In this case, the wave vector k in ZhF1(k) and
ZhF2(k) can be chosen as Z
hF1 = 1−Σ
1L (k) |k=[π,0] and
hF2 = Σ
1T (k) |k=[π,0]. With the help of the above dis-
cussions, the corresponding longitudinal and transverse
parts of the dressed holon normal and anomalous Green’s
functions in Eqs. (A1a) and (A1b) now can be obtained
explicitly as,
gL(k, ω) =
ν=1,2
U2hνk
ω − Ehνk
V 2hνk
ω + Ehνk
, (A2a)
gT (k, ω) =
ν=1,2
(−1)ν+1Z
U2hνk
ω − Ehνk
V 2hνk
ω + Ehνk
, (A2b)
L(k, ω) = −
ν=1,2
2Ehνk
ω − Ehνk
ω + Ehνk
, (A2c)
T (k, ω) = −
ν=1,2
(−1)ν+1Z
2Ehνk
ω − Ehνk
ω + Ehνk
, (A2d)
with the dressed holon effective gap parameters and
quasiparticle coherent weights satisfy the following four
equations,
∆̄hL = −
k,q,p
ν,ν′,ν′′
k−p+qCνν′′ (k+ q)
(ν′′)
Bν′pBνq
ων′pωνq
(ν′′)
νν′ν′′(q,p) + F
νν′ν′′(k,q,p)
[ων′p − ωνq]2 − E
hν′′k
νν′ν′′(q,p) + F
νν′ν′′(k,q,p)
[ων′p + ωνq]2 − E
hν′′k
, (A3a)
∆̄hT = −
k,q,p
ν,ν′,ν′′
(−1)ν+ν
′+ν′′+1Cνν′′(k+ q)
(ν′′)
Bν′pBνq
ων′pωνq
(ν′′)
νν′ν′′(q,p) + F
νν′ν′′(k,q,p)
[ων′p − ωνq]2 − E
hν′′k
νν′ν′′(q,p) + F
νν′ν′′(k,q,p)
[ων′p + ωνq]2 − E
hν′′k
, (A3b)
= 1 +
ν,ν′,ν′′
[1 + (−1)ν+ν
′+ν′′+1]Cνν′′ (p+ k0)
(ν′′)
Bν′pBνq
ων′pωνq
νν′ν′′(q,p)
[ων′p − ωνq + Ehν′′p−q+k0 ]
νν′ν′′(q,p)
[ων′p − ωνq − Ehν′′p−q+k0 ]
νν′ν′′(q,p)
[ων′p + ωνq + Ehν′′p−q+k0]
νν′ν′′ (q,p)
[ων′p + ωνq − Ehν′′p−q+k0 ]
, (A3c)
= 1 +
ν,ν′,ν′′
[1− (−1)ν+ν
′+ν′′+1]Cνν′′ (p+ k0)
(ν′′)
Bν′pBνq
ων′pωνq
νν′ν′′(q,p)
[ων′p − ωνq + Ehν′′p−q+k0 ]
νν′ν′′(q,p)
[ων′p − ωνq − Ehν′′p−q+k0 ]
νν′ν′′(q,p)
[ων′p + ωνq + Ehν′′p−q+k0]
νν′ν′′ (q,p)
[ων′p + ωνq − Ehν′′p−q+k0 ]
, (A3d)
where Cνν′′ (k) = [Z(tγk − t
′γ′k) + (−1)
ν+ν′′t⊥(k)]
νν′ν′′
(q,p) = nB(ωνq)+nB(ων′p)+2nB(ωνq)nB(ων′p),
νν′ν′′(k,q,p) = [2nF (Ehν′′k) − 1][ων′p −
ωνq][nB(ωνq) − nB(ων′p)]/Ehν′′k, F
νν′ν′′(q,p) =
1 + nB(ωνq) + nB(ων′p) + 2nB(ωνq)nB(ων′p),
νν′ν′′
(k,q,p) = [2nF (Ehν′′k) − 1][ων′p + ωνq][1 +
nB(ωνq) + nB(ων′p)]/Ehν′′k, H
νν′ν′′(q,p) =
nF (Ehν′′p−q+k0)[nB(ων′p) − nB(ωνq)] + nB(ωνq)[1 +
nB(ων′p)], H
νν′ν′′(q,p) = nF (Ehν′′p−q+k0)[nB(ωνq) −
nB(ων′p)] + nB(ων′p)[1 + nB(ωνq)], H
νν′ν′′(q,p) =
[1 − nF (Ehν′′p−q+k0)][1 + nB(ωνq) + nB(ων′p)] +
nB(ωνq)nB(ων′p), H
νν′ν′′
(q,p) = nF (Ehν′′p−q+k0)[1 +
nB(ωνq)+nB(ων′p)]+nB(ωνq)nB(ων′p), and k0 = [π, 0].
These four equations must be solved self-consistently
in combination with other equations as in the single
layer case18,19, then all order parameters, decoupling
parameter α, and chemical potential µ are determined
by the self-consistent calculation.
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http://arxiv.org/abs/cond-mat/0205385
http://arxiv.org/abs/cond-mat/0107002
