# Out-of-plane impurities induced the deviation from the monotonic d-wave
superconducting gap in cuprate superconductors

Zhi Wang and Shiping Feng∗ Department of Physics, Beijing Normal University,
Beijing 100875, China

###### Abstract

The electronic structure of cuprate superconductors is studied within the
kinetic energy driven superconducting mechanism in the presence of out-of-
plane impurities. With increasing impurity concentration, although both
superconducting coherence peaks around the nodal and antinodal regions are
suppressed, the position of the leading-edge mid-point of the electron
spectrum around the nodal region remains at the same position, whereas around
the antinodal region it is shifted towards higher binding energies, this leads
to a strong deviation from the monotonic d-wave superconducting gap in the
out-of-plane impurity-controlled cuprate superconductors.

###### pacs:

74.62.Dh, 74.20.Rp, 74.25.Jb, 74.20.Mn

##  I Introduction

The superconducting gap is a fundamental property of superconductors
schrieffer , and the nature of its anisotropy has played a crucial role in the
testing of the microscopic theory of superconductivity in cuprate
superconductors anderson . Experimentally, by virtue of systematic
measurements tsuei , particularly using the angle-resolved photoemission
spectroscopy (ARPES) technique shen , the d-wave nature of the superconducting
gap has been well established by now. In particular, this d-wave
superconducting symmetry remains one of the cornerstones of our understanding
of the physics in cuprate superconductors shen ; tsuei ; perali ; sangiovanni
; zhang . The early ARPES measurements on the cuprate superconductor
Bi2Sr2CaCu2O8+δ shi  showed that in the real space the gap function and the
pairing force have a range of one lattice spacing, and then the
superconducting gap function is of the monotonic d-wave form $\Delta_{\bf
k}=\Delta[{\rm cos}k_{x}-{\rm cos}k_{y}]/2$. Later, the ARPES measurements on
the cuprate superconductor Bi2Sr2CaCu2O8+δ mesot  indicated that the
superconducting gap significantly deviates from this monotonic d-wave form.
Furthermore, it was argued that this deviation should be attributed to an
increase of the electron correlation, which may increase the intensity of the
higher order of the harmonic component in the monotonic d-wave gap function
mesot . However, recent ARPES measurements kondo ; hashimoto  on the cuprate
superconductors (Bi,Pb)2(Sr,La)2CuO6+δ and Bi2Sr1.6$Ln$0.4CuO6+δ ($Ln$-Bi2201)
with $Ln$=La, Nd, and Gd showed that a much stronger deviation from the
monotonic d-wave superconducting gap form is unlikely to be due to the strong
correlation effect.

The cuprate superconductors have a layered structure consisting of the two-
dimensional CuO2 layers separated by insulating layers bednorz ; kastner . The
single common feature is the presence of the CuO2 plane shen ; kastner , and
it seems evident that the unusual behaviors of cuprate superconductors are
dominated by this CuO2 plane anderson . It has been well established that the
Cu2+ ions exhibit an antiferromagnetic long-range order in the parent
compounds of cuprate superconductors, and superconductivity occurs when the
antiferromagnetic long-range order state is suppressed by doped charge
carriers kastner . Since these doped charge carriers are induced by the
replacement of ions by those with different valences or the addition of excess
oxygens in the block layer, therefore in principle, all cuprate
superconductors have naturally impurities (or disorder). However, for the
cuprate superconductors (Bi,Pb)2(Sr,La)2CuO6+δ and $Ln$-Bi2201, the mismatch
in the ionic radius between Bi and Pb or Sr and $Ln$ causes the out-of-plane
impurities eisaki , where the concentration of the out-of-plane impurities is
controlled by varying the radius of the Pb or $Ln$ ions, and then the
superconducting transition temperature $T_{c}$ is found to be decreasing with
increasing impurity concentration. These cuprate superconductors
(Bi,Pb)2(Sr,La)2CuO6+δ and $Ln$-Bi2201 are often referred to as the out-of-
plane impurity-controlled cuprate superconductors. Recently, the electronic
structure of the out-of-plane impurity-controlled cuprate superconductors and
the related superconducting gap function have been investigated experimentally
by using ARPES kondo ; hashimoto . It was shown that although the effect of
the out-of-plane impurity scattering around the antinodal region is much
stronger than that around the nodal region, both superconducting coherence
peaks around the nodal and antinodal regions are suppressed. Furthermore, the
magnitude of the deviation from the monotonic d-wave superconducting gap form
increases with increasing impurity concentration kondo ; hashimoto . The
appearance of the strong deviation from the monotonic d-wave superconducting
gap form observed recently in the out-of-plane impurity-controlled cuprate
superconductors (Bi,Pb)2(Sr,La)2CuO6+δ and $Ln$-Bi2201 is the most remarkable
effect kondo ; hashimoto , however, its full understanding is still a
challenging issue. To the best of our knowledge, this strong deviation from
the monotonic d-wave superconducting gap form in the out-of-plane impurity-
controlled cuprate superconductors has not been treated starting from a
microscopic superconducting theory yet.

In the absence of out-of-plane impurity scattering, the electronic structure
of cuprate superconductors in the superconducting state has been discussed guo
; feng  within the framework of the kinetic energy driven superconductivity
feng1 , where the superconducting gap function has a monotonic d-wave form,
and the main features of the ARPES experiments shen  on cuprate
superconductors have been reproduced. In this paper, we study the electronic
structure of the out-of-plane impurity-controlled cuprate superconductors in
the superconducting state and the related superconducting gap function along
with this line. We employ the $t$-$J$ model by considering the out-of-plane
impurity scattering, and then show explicitly that the strong deviation from
the monotonic d-wave superconducting gap form occurs due to the presence of
the impurity scattering. Although both sharp superconducting coherence peaks
around the nodal and the antinodal regions are suppressed, the effect of the
impurity scattering is stronger in the antinodal region than that in the nodal
region. Our results also show that the electronic structure of the out-of-
plane impurity-controlled cuprate superconductors in the superconducting state
can be understood within the framework of the kinetic energy driven
superconducting mechanism with the out-of-plane impurity scattering taken into
account.

This paper is organized as follows. In Sec. II we present the basic formalism
of the electronic structure calculation in the presence of the out-of-plane
impurities. Within this theoretical framework, we discuss the electronic
structure of the out-of-plane impurity-controlled cuprate superconductors in
the superconducting state and the related superconducting gap function in Sec.
III, where we show that the well pronounced deviation from the monotonic
d-wave superconducting gap form is mainly caused by the out-of-plane impurity
scattering. Finally, we give a summary in Sec. IV.

##  II Formalism

It has been shown that the essential physics of cuprate superconductors is
properly accounted by the two-dimensional $t$-$J$ model on a square lattice
anderson ,

$\displaystyle H$ $\displaystyle=$
$\displaystyle-t\sum_{i\hat{\eta}\sigma}C^{\dagger}_{i\sigma}C_{i+\hat{\eta}\sigma}+t^{\prime}\sum_{i\hat{\tau}\sigma}C^{\dagger}_{i\sigma}C_{i+\hat{\tau}\sigma}+\mu\sum_{i\sigma}C^{\dagger}_{i\sigma}C_{i\sigma}$
(1) $\displaystyle+$ $\displaystyle J\sum_{i\hat{\eta}}{\bf S}_{i}\cdot{\bf
S}_{i+\hat{\eta}},$

acting on the Hilbert subspace with no doubly occupied site, i.e.,
$\sum_{\sigma}C^{\dagger}_{i\sigma}C_{i\sigma}\leq 1$, where
$\hat{\eta}=\pm\hat{x},\pm\hat{y}$, $\hat{\tau}=\pm\hat{x}\pm\hat{y}$,
$C^{\dagger}_{i\sigma}$ ($C_{i\sigma}$) is the creation (annihilation)
operator of an electron with spin $\sigma$, ${\bf
S}_{i}=(S^{x}_{i},S^{y}_{i},S^{z}_{i})$ are spin operators, and $\mu$ is the
chemical potential. To deal with the constraint of no double occupancy in
analytical calculations, the charge-spin separation fermion-spin theory feng2
has been developed, where the constrained electron operators $C_{i\uparrow}$
and $C_{i\downarrow}$ are decoupled as
$C_{i\uparrow}=h^{\dagger}_{i\uparrow}S^{-}_{i}$ and
$C_{i\downarrow}=h^{\dagger}_{i\downarrow}S^{+}_{i}$, respectively, here the
spinful fermion operator $h_{i\sigma}=e^{-i\Phi_{i\sigma}}h_{i}$ describes the
charge degree of freedom together with some effects of spin configuration
rearrangements due to the presence of the doped charge carrier itself, while
the spin operator $S_{i}$ describes the spin degree of freedom, then the
electron on-site local constraint for the single occupancy,
$\sum_{\sigma}C^{\dagger}_{i\sigma}C_{i\sigma}=S^{+}_{i}h_{i\uparrow}h^{\dagger}_{i\uparrow}S^{-}_{i}+S^{-}_{i}h_{i\downarrow}h^{\dagger}_{i\downarrow}S^{+}_{i}=h_{i}h^{\dagger}_{i}(S^{+}_{i}S^{-}_{i}+S^{-}_{i}S^{+}_{i})=1-h^{\dagger}_{i}h_{i}\leq
1$, is satisfied in analytical calculations. In particular, it has been shown
that under the decoupling scheme, this charge-spin separation fermion-spin
representation is a natural representation of the constrained electron defined
in the Hilbert subspace without double electron occupancy feng . Furthermore,
these charge carrier and spin are gauge invariant feng2 , and in this sense
they are real and can be interpreted as physical excitations laughlin . In
this charge-spin separation fermion-spin representation, the $t$-$J$ model (1)
can be expressed as,

$\displaystyle H$ $\displaystyle=$ $\displaystyle
t\sum_{i\hat{\eta}}(h^{\dagger}_{i+\hat{\eta}\uparrow}h_{i\uparrow}S^{+}_{i}S^{-}_{i+\hat{\eta}}+h^{\dagger}_{i+\hat{\eta}\downarrow}h_{i\downarrow}S^{-}_{i}S^{+}_{i+\hat{\eta}})$
(2) $\displaystyle-$ $\displaystyle
t^{\prime}\sum_{i\hat{\tau}}(h^{\dagger}_{i+\hat{\tau}\uparrow}h_{i\uparrow}S^{+}_{i}S^{-}_{i+\hat{\tau}}+h^{\dagger}_{i+\hat{\tau}\downarrow}h_{i\downarrow}S^{-}_{i}S^{+}_{i+\hat{\tau}})$
$\displaystyle-$
$\displaystyle\mu\sum_{i\sigma}h^{\dagger}_{i\sigma}h_{i\sigma}+J_{{\rm
eff}}\sum_{i\hat{\eta}}{\bf S}_{i}\cdot{\bf S}_{i+\hat{\eta}},$

with $J_{{\rm eff}}=(1-\delta)^{2}J$, and $\delta=\langle
h^{\dagger}_{i\sigma}h_{i\sigma}\rangle=\langle h^{\dagger}_{i}h_{i}\rangle$
being the charge carrier doping concentration. This $J_{{\rm eff}}$ is similar
to that obtained in Gutzwiller approach zhang . As an important consequence,
the kinetic energy term in the $t$-$J$ model has been transferred as the
interaction between charge carriers and spins, which reflects that even the
kinetic energy term in the $t$-$J$ Hamiltonian has a strong Coulombic
contribution due to the restriction of no double occupancy of a given site.
This interaction from the kinetic energy term is quite strong, and it has been
shown feng1  in terms of the Eliashberg’s strong coupling theory eliashberg
that in the case without an antiferromagnetic long-range order, this
interaction can induce a charge carrier pairing state (then the electron
Cooper pairing state) with d-wave symmetry by exchanging spin excitations in
the higher power of the charge carrier doping concentration $\delta$. In this
case, the electron Cooper pairs originating from the charge carrier pairing
state are due to the charge-spin recombination, and their condensation reveals
the d-wave superconducting ground-state. Furthermore, this d-wave
superconducting state is controlled by both the superconducting gap function
and the quasiparticle coherence, which leads to the fact that the maximal
superconducting transition temperature occurs around the optimal doping, and
then decreases in both underdoped and overdoped regimes feng1 . Moreover, it
has been shown guo ; feng  that this superconducting state is the conventional
Bardeen-Cooper-Schrieffer (BCS) like schrieffer ; bcs  with the d-wave
symmetry, so that the basic BCS formalism with the d-wave superconducting gap
function is still valid in quantitatively reproducing all main low energy
features of the ARPES experimental measurements on cuprate superconductors,
although the pairing mechanism is driven by the kinetic energy by exchanging
spin excitations, and other exotic magnetic scattering dai  is beyond the BCS
formalism. Following previous discussions guo ; feng ; feng1 , the full charge
carrier Green’s function in the superconducting state with a monotonic d-wave
gap function can be obtained in the Nambu representation as wang ,

$\displaystyle\tilde{g}({\bf k},\omega)$ $\displaystyle=$ $\displaystyle
Z_{hF}{1\over\omega^{2}-E^{2}_{h{\bf
k}}}\left(\begin{array}[]{cc}{\omega+\bar{\xi}_{{\bf
k}}}&{\bar{\Delta}_{hZ}({\bf k})}\\\ {\bar{\Delta}_{hZ}({\bf
k})}&{\omega-\bar{\xi}_{{\bf k}}}\end{array}\right)$ (5) $\displaystyle=$
$\displaystyle Z_{hF}{\omega\tau_{0}+\bar{\Delta}_{hZ}({\bf
k})\tau_{1}+\bar{\xi}_{{\bf k}}\tau_{3}\over\omega^{2}-E^{2}_{h{\bf k}}},$ (6)

where $\tau_{0}$ is the unit matrix, $\tau_{1}$ and $\tau_{3}$ are the Pauli
matrices, the renormalized charge carrier excitation spectrum $\bar{\xi}_{{\bf
k}}=Z_{hF}\xi_{\bf k}$, with the mean-field charge carrier excitation spectrum
$\xi_{{\bf k}}=Zt\chi_{1}\gamma_{{\bf
k}}-Zt^{\prime}\chi_{2}\gamma^{\prime}_{{\bf k}}-\mu$, the spin correlation
functions $\chi_{1}=\langle S_{i}^{+}S_{i+\hat{\eta}}^{-}\rangle$ and
$\chi_{2}=\langle S_{i}^{+}S_{i+\hat{\tau}}^{-}\rangle$, $\gamma_{{\bf
k}}=(1/Z)\sum_{\hat{\eta}}e^{i{\bf k}\cdot\hat{\eta}}$, $\gamma^{\prime}_{{\bf
k}}=(1/Z)\sum_{\hat{\tau}}e^{i{\bf k}\cdot\hat{\tau}}$, $Z$ is the number of
the nearest neighbor or next nearest neighbor sites, the renormalized charge
carrier monotonic d-wave pair gap function $\bar{\Delta}_{hZ}({\bf
k})=Z_{hF}\bar{\Delta}_{h}({\bf k})$, where the effective charge carrier
monotonic d-wave pair gap function $\bar{\Delta}_{h}({\bf
k})=\bar{\Delta}_{h}\gamma^{(d)}_{{\bf k}}$ with $\gamma^{(d)}_{{\bf k}}=({\rm
cos}k_{x}-{\rm cos}k_{y})/2$, and the charge carrier quasiparticle spectrum
$E_{h{\bf k}}=\sqrt{\bar{\xi}^{2}_{{\bf k}}+\mid\bar{\Delta}_{hZ}({\bf
k})\mid^{2}}$. The charge carrier quasiparticle coherent weight $Z_{hF}$ and
effective charge carrier gap parameter $\bar{\Delta}_{h}$ are determined by
the following two equations guo ; feng ; feng1 ,

$\displaystyle 1$ $\displaystyle=$ $\displaystyle{1\over N^{3}}\sum_{{\bf
k,p,p^{\prime}}}\Lambda^{2}_{{\bf p+k}}\gamma^{(d)}_{{\bf
k-p^{\prime}+p}}\gamma^{(d)}_{{\bf k}}{Z^{2}_{hF}\over E_{h{\bf k}}}{B_{{\bf
p}}B_{{\bf p^{\prime}}}\over\omega_{{\bf p}}\omega_{{\bf
p^{\prime}}}}\left({F^{(1)}_{1}({\bf k,p,p^{\prime}})\over(\omega_{{\bf
p^{\prime}}}-\omega_{{\bf p}})^{2}-E^{2}_{h{\bf k}}}-{F^{(2)}_{1}({\bf
k,p,p^{\prime}})\over(\omega_{{\bf p^{\prime}}}+\omega_{{\bf
p}})^{2}-E^{2}_{h{\bf k}}}\right),$ (7a) $\displaystyle{1\over Z}_{hF}$
$\displaystyle=$ $\displaystyle 1+{1\over N^{2}}\sum_{{\bf
p,p^{\prime}}}\Lambda^{2}_{{\bf p}+{\bf k}_{0}}Z_{hF}{B_{{\bf p}}B_{{\bf
p^{\prime}}}\over 4\omega_{{\bf p}}\omega_{{\bf
p^{\prime}}}}\left({F^{(1)}_{2}({\bf p,p^{\prime}})\over(\omega_{{\bf
p}}-\omega_{{\bf p^{\prime}}}-E_{h{\bf
p-p^{\prime}+k_{0}}})^{2}}+{F^{(2)}_{2}({\bf p,p^{\prime}})\over(\omega_{{\bf
p}}-\omega_{{\bf p^{\prime}}}+E_{h{\bf p-p^{\prime}+k_{0}}})^{2}}\right.$ (7b)
$\displaystyle+$ $\displaystyle\left.{F^{(3)}_{2}({\bf
p,p^{\prime}})\over(\omega_{{\bf p}}+\omega_{{\bf p^{\prime}}}-E_{h{\bf
p-p^{\prime}+k_{0}}})^{2}}+{F^{(4)}_{2}({\bf p,p^{\prime}})\over(\omega_{{\bf
p}}+\omega_{{\bf p^{\prime}}}+E_{h{\bf p-p^{\prime}+k_{0}}})^{2}}\right),$

respectively, where ${\bf k}_{0}=[\pi,0]$, $\Lambda_{\bf k}=Zt\gamma_{\bf
k}-Zt^{\prime}\gamma^{\prime}_{\bf k}$, $B_{{\bf
p}}=2\lambda_{1}(A_{1}\gamma_{{\bf
p}}-A_{2})-\lambda_{2}(2\chi^{z}_{2}\gamma_{{\bf p}}^{\prime}-\chi_{2})$,
$\lambda_{1}=2ZJ_{{\rm eff}}$, $\lambda_{2}=4Z\phi_{2}t^{\prime}$,
$A_{1}=\epsilon\chi^{z}_{1}+\chi_{1}/2$,
$A_{2}=\chi^{z}_{1}+\epsilon\chi_{1}/2$, $\epsilon=1+2t\phi_{1}/J_{{\rm
eff}}$, the charge carrier’s particle-hole parameters $\phi_{1}=\langle
h^{\dagger}_{i\sigma}h_{i+\hat{\eta}\sigma}\rangle$ and $\phi_{2}=\langle
h^{\dagger}_{i\sigma}h_{i+\hat{\tau}\sigma}\rangle$, the spin correlation
functions $\chi^{z}_{1}=\langle S_{i}^{z}S_{i+\hat{\eta}}^{z}\rangle$ and
$\chi^{z}_{2}=\langle S_{i}^{z}S_{i+\hat{\tau}}^{z}\rangle$, $F^{(1)}_{1}({\bf
k,p,p^{\prime}})=(\omega_{{\bf p^{\prime}}}-\omega_{{\bf
p}})[n_{B}(\omega_{{\bf p}})-n_{B}(\omega_{{\bf
p^{\prime}}})][1-2n_{F}(E_{h{\bf k}})]+E_{h{\bf k}}[n_{B}(\omega_{{\bf
p^{\prime}}})n_{B}(-\omega_{{\bf p}})+n_{B}(\omega_{{\bf
p}})n_{B}(-\omega_{{\bf p^{\prime}}})]$, $F^{(2)}_{1}({\bf
k,p,p^{\prime}})=(\omega_{{\bf p^{\prime}}}+\omega_{{\bf
p}})[n_{B}(-\omega_{{\bf p^{\prime}}})-n_{B}(\omega_{{\bf
p}})][1-2n_{F}(E_{h{\bf k}})]+E_{h{\bf k}}[n_{B}(\omega_{{\bf
p^{\prime}}})n_{B}(\omega_{{\bf p}})+n_{B}(-\omega_{{\bf
p^{\prime}}})n_{B}(-\omega_{{\bf p}})]$, $F^{(1)}_{2}({\bf
p,p^{\prime}})=n_{F}(E_{h{\bf p-p^{\prime}+k_{0}}})[n_{B}(\omega_{{\bf
p^{\prime}}})-n_{B}(\omega_{{\bf p}})]-n_{B}(\omega_{{\bf
p}})n_{B}(-\omega_{{\bf p^{\prime}}})$, $F^{(2)}_{2}({\bf
p,p^{\prime}})=n_{F}(E_{h{\bf p-p^{\prime}+k_{0}}})[n_{B}(\omega_{{\bf
p}})-n_{B}(\omega_{{\bf p^{\prime}}})]-n_{B}(\omega_{{\bf
p^{\prime}}})n_{B}(-\omega_{{\bf p}})$, $F^{(3)}_{2}({\bf
p,p^{\prime}})=n_{F}(E_{h{\bf p-p^{\prime}+k_{0}}})[n_{B}(\omega_{{\bf
p^{\prime}}})-n_{B}(-\omega_{{\bf p}})]+n_{B}(\omega_{{\bf
p}})n_{B}(\omega_{{\bf p^{\prime}}})$, $F^{(4)}_{2}({\bf
p,p^{\prime}})=n_{F}(E_{h{\bf p-p^{\prime}+k_{0}}})[n_{B}(-\omega_{{\bf
p^{\prime}}})-n_{B}(\omega_{{\bf p}})]+n_{B}(-\omega_{{\bf
p}})n_{B}(-\omega_{{\bf p^{\prime}}})$, $n_{B}(\omega_{{\bf p}})$ and
$n_{F}(E_{h{\bf k}})$ are the boson and fermion distribution functions,
respectively, and the mean-field spin excitation spectrum,

$\displaystyle\omega^{2}_{{\bf p}}$ $\displaystyle=$
$\displaystyle\lambda_{1}^{2}\left[\left(A_{4}-\alpha\epsilon\chi^{z}_{1}\gamma_{{\bf
p}}-{1\over 2Z}\alpha\epsilon\chi_{1}\right)(1-\epsilon\gamma_{{\bf
p}})+{1\over 2}\epsilon\left(A_{3}-{1\over
2}\alpha\chi^{z}_{1}-\alpha\chi_{1}\gamma_{{\bf
p}}\right)(\epsilon-\gamma_{{\bf
p}})\right]+\lambda_{2}^{2}\left[\alpha\left(\chi^{z}_{2}\gamma_{{\bf
p}}^{\prime}-{3\over 2Z}\chi_{2}\right)\gamma_{{\bf p}}^{\prime}\right.$ (8)
$\displaystyle+$ $\displaystyle\left.{1\over 2}\left(A_{5}-{1\over
2}\alpha\chi^{z}_{2}\right)\right]+\lambda_{1}\lambda_{2}\left[\alpha\chi^{z}_{1}(1-\epsilon\gamma_{{\bf
p}})\gamma_{{\bf p}}^{\prime}+{1\over 2}\alpha(\chi_{1}\gamma_{{\bf
p}}^{\prime}-C_{3})(\epsilon-\gamma_{{\bf p}})+\alpha\gamma_{{\bf
p}}^{\prime}(C^{z}_{3}-\epsilon\chi^{z}_{2}\gamma_{{\bf p}})-{1\over
2}\alpha\epsilon(C_{3}-\chi_{2}\gamma_{{\bf p}})\right],~{}~{}~{}~{}~{}$

where $A_{3}=\alpha C_{1}+(1-\alpha)/(2Z)$, $A_{4}=\alpha
C^{z}_{1}+(1-\alpha)/(4Z)$, $A_{5}=\alpha C_{2}+(1-\alpha)/(2Z)$, and the spin
correlation functions
$C_{1}=(1/Z^{2})\sum_{\hat{\eta},\hat{\eta^{\prime}}}\langle
S_{i+\hat{\eta}}^{+}S_{i+\hat{\eta^{\prime}}}^{-}\rangle$,
$C^{z}_{1}=(1/Z^{2})\sum_{\hat{\eta},\hat{\eta^{\prime}}}\langle
S_{i+\hat{\eta}}^{z}S_{i+\hat{\eta^{\prime}}}^{z}\rangle$,
$C_{2}=(1/Z^{2})\sum_{\hat{\tau},\hat{\tau^{\prime}}}\langle
S_{i+\hat{\tau}}^{+}S_{i+\hat{\tau^{\prime}}}^{-}\rangle$,
$C_{3}=(1/Z)\sum_{\hat{\tau}}\langle
S_{i+\hat{\eta}}^{+}S_{i+\hat{\tau}}^{-}\rangle$, and
$C^{z}_{3}=(1/Z)\sum_{\hat{\tau}}\langle
S_{i+\hat{\eta}}^{z}S_{i+\hat{\tau}}^{z}\rangle$. In order to satisfy the sum
rule of the correlation function $\langle S^{+}_{i}S^{-}_{i}\rangle=1/2$ in
the case without the antiferromagnetic long-range order, an important
decoupling parameter $\alpha$ has been introduced in the above calculation guo
; feng ; feng1 , which can be regarded as the vertex correction. These two
equations (4a) and (4b) must be solved simultaneously with other self-
consistent equations, then all order parameters, the decoupling parameter
$\alpha$, and the chemical potential $\mu$ are determined by the self-
consistent calculation guo ; feng ; feng1 . In this sense, the calculations in
this kinetic energy driven superconductivity scheme are controllable without
using any adjustable parameters. We emphasize that the Green’s function (3) is
obtained within the kinetic energy driven superconducting mechanism, although
the similar phenomenological expression has been used to discuss the impurity
effect in cuprate superconductors haas ; graser .

With the charge carrier BCS formalism (3) under the kinetic energy driven
superconducting mechanism, we can now introduce the effect of impurity
scatterers into the electronic structure. In the presence of impurities, the
unperturbed charge carrier Green’s function in Eq. (3) is dressed by impurity
scattering wang ,

$\displaystyle\tilde{g}_{I}({\bf k},\omega)^{-1}=\tilde{g}({\bf
k},\omega)^{-1}-Z_{hF}^{-1}\tilde{\Sigma}({\bf k},\omega),$ (9)

with the self-energy function $\tilde{\Sigma}({\bf
k},\omega)=\sum_{\alpha}\Sigma_{\alpha}({\bf k},\omega)\tau_{\alpha}$. In this
case, the charge carrier Green’s function in Eq. (6) can be explicitly
rewritten as,

$\displaystyle\tilde{g}_{I}({\bf k},\omega)=\sum_{\alpha}g_{I\alpha}({\bf
k},\omega)\tau_{\alpha}=Z_{hF}{[\omega-\Sigma_{0}({\bf
k},\omega)]\tau_{0}+[\bar{\Delta}_{hZ}({\bf k})+\Sigma_{1}({\bf
k},\omega)]\tau_{1}+[\bar{\xi}_{{\bf k}}+\Sigma_{3}({\bf
k},\omega)]\tau_{3}\over[\omega-\Sigma_{0}({\bf
k},\omega)]^{2}-[\bar{\xi}_{{\bf k}}+\Sigma_{3}({\bf
k},\omega)]^{2}-[\bar{\Delta}_{hZ}({\bf k})+\Sigma_{1}({\bf k},\omega)]^{2}}.$
(10)

Based on this Green’s function (7), we wang  have discussed the effect of the
extended impurity scatterers on the quasiparticle transport of cuprate
superconductors in the superconducting state within the nodal approximation of
the quasiparticle excitations and scattering processes, where the main effect
on the quasiparticle transport comes from the extended impurity forward (or
diagonal) scatterers, and therefore the component of the self-energy function
$\Sigma_{1}({\bf k},\omega)$ has been neglected, while the components of
$\Sigma_{0}({\bf k},\omega)$ and $\Sigma_{3}({\bf k},\omega)$ have been
treated within the framework of the T-matrix approximation. However, it has
been demonstrated that the superconducting transition temperature is
considerably affected by the out-of-plane impurity scattering in spite of a
relatively weak increase of the residual resistivity eisaki . This reflects
the fact that the superconducting pairing is very sensitive to the out-of-
plane impurity scattering, and then the effect of the out-of-plane impurity
scattering is always accompanied by breaking of the superconducting pairs. In
this case, the out-of-plane impurities can be described as the elastic off-
diagonal scatterers or pairing impurity scatterers. In particular, the
modulation of the out-of-plane impurity scattering potential observed in
scanning tunneling microscopy experiments pan  has a characteristic wavelength
of a few lattice spacings, this may arise because the impurities give rise to
an atomic-scale modulation of the charge carrier pairing potential which
causes larger, coherence length size fluctuations in the out-of-plane impurity
scattering potential nunner . Furthermore, the crude effect of the order
parameter modulations on the quasiparticle scattering by allowing the order
parameter to be modulated on the four bonds around the impurity has been
estimated graser  by adding the off-diagonal scattering potential,

$\displaystyle\hat{V}$ $\displaystyle=$ $\displaystyle\sum_{{\bf k},{\bf
k}^{\prime}}[V({\bf k})+V({\bf k}^{\prime})]\tau_{1}$ (11) $\displaystyle=$
$\displaystyle{1\over 2}V_{0}\sum_{{\bf k},{\bf k}^{\prime}}[({\rm
cos}k_{x}-{\rm cos}k_{y})+({\rm cos}k^{\prime}_{x}-{\rm
cos}k^{\prime}_{y})]\tau_{1},~{}~{}~{}$

to the phenomenological d-wave BCS Hamiltonian, then it was shown that the
scattering rate is largest at the antinode.

The exact form of the out-of-plane impurity scattering potential is very
important for a better understanding of the electronic structure of the out-
of-plane impurity-controlled cuprate superconductors. In the following
discussions, we determine the form of the out-of-plane impurity scattering
potential in terms of the calculation of Dyson’s equation. The potential which
scatters the electron is taken as summation of impurity potentials
$\tilde{V}=\sum_{l}V({\bf r}_{i}-{\bf R}_{l})$, where the summation is over
all impurity sites $l$, and then its Fourier transform is obtained kohn ;
mahan  as $\tilde{V}({\bf q})=\rho_{i}V({\bf q})\rho({\bf q})$, where

$\displaystyle\rho({\bf q})$ $\displaystyle=$ $\displaystyle\sum_{{\bf
k}}h^{\dagger}_{{\bf k}+{\bf q}}h_{{\bf k}},$ (12) $\displaystyle\rho_{i}({\bf
q})$ $\displaystyle=$ $\displaystyle\sum_{l}e^{i{\bf q}\cdot{\bf R}_{l}},$
(13)

are the charge carrier density in the Nambu representation and the impurity
density, respectively. In the calculation of the self-energy function induced
by the impurity scattering, usually it is assumed that the impurities are
randomly located and that there is no correlation between their positions kohn
; mahan . In this case, the self-energy function can be obtained as
$\tilde{\Sigma}({\bf k},\omega)=\tilde{\Sigma}^{(1)}({\bf
k},\omega)+\tilde{\Sigma}^{(2)}({\bf k},\omega)$ within the Born
approximation, with the corresponding first-order and second-order self-energy
functions are evaluated as kohn ; mahan ,

$\displaystyle\tilde{\Sigma}^{(1)}({\bf k},\omega)$ $\displaystyle=$
$\displaystyle\rho_{i}\sum_{\bf k^{\prime}}\delta_{{\bf k^{\prime}}=0}V({\bf
k^{\prime}})=\rho_{i}V(0),$ (14a) $\displaystyle\tilde{\Sigma}^{(2)}({\bf
k},\omega)$ $\displaystyle=$ $\displaystyle\rho_{i}\sum_{{\bf k^{\prime}},{\bf
k^{\prime\prime}}}\delta_{{\bf k^{\prime}}+{\bf k^{\prime\prime}}=0}V({\bf
k^{\prime}})\tilde{g}_{I}({\bf k}+{\bf k^{\prime}},\omega)V({\bf
k^{\prime\prime}})$ (14b) $\displaystyle=$ $\displaystyle\rho_{i}\sum_{{\bf
k^{\prime}}}V({\bf k^{\prime}})\tilde{g}_{I}({\bf k}+{\bf
k^{\prime}},\omega)V(-{\bf k^{\prime}}),$

where $\rho_{i}$ is the impurity concentration. As we have mentioned above,
the out-of-plane impurities are the off-diagonal scatterers. Although their
scattering has a very weak effect on the residual resistivity for cuprate
superconductors, a heavy effect on the d-wave SC state is observed
experimentally eisaki . With these considerations, we introduce the following
out-of-plane impurity scattering potential,

$\displaystyle\tilde{V}=\sum_{{\bf k^{\prime}}}V({\bf
k^{\prime}})\tau_{1}=V_{0}\sum_{{\bf k^{\prime}}}[{\rm cos}k^{\prime}_{x}-{\rm
cos}k^{\prime}_{y}]\tau_{1}.$ (15)

In this case, $V(0)=V_{0}[{\rm cos}(0)-{\rm cos}(0)]=0$ (then
$\tilde{\Sigma}_{1}({\bf k},\omega)=0$), and $\tilde{\Sigma}({\bf
k},\omega)=\tilde{\Sigma}^{(2)}({\bf k},\omega)$. This form of the out-of-
plane impurity scattering potential in Eq. (12) is very similar to that in Eq.
(8) used in Ref. graser, , and the scattering rate is also largest at the
antinode. This is indeed confirmed by the quantitative characteristics
presented in the following section. With the help of the impurity scattering
potential in Eq. (12), the components of the charge carrier self-energy
function $\tilde{\Sigma}({\bf k},\omega)$ are obtained explicitly as,

$\displaystyle\Sigma_{0}({\bf k},\omega)$ $\displaystyle=$
$\displaystyle\rho_{i}{1\over N}\sum_{{\bf k^{\prime}}}|V({\bf
k^{\prime}})|^{2}{g}_{I0}({\bf k^{\prime}+k},\omega)$ (16a) $\displaystyle=$
$\displaystyle\rho_{i}{1\over N}\sum_{{\bf k^{\prime}}}|V({\bf
k^{\prime}-k})|^{2}{g}_{I0}({\bf k^{\prime}},\omega),$
$\displaystyle\Sigma_{3}({\bf k},\omega)$ $\displaystyle=$
$\displaystyle-\rho_{i}{1\over N}\sum_{{\bf k^{\prime}}}|V({\bf
k^{\prime}})|^{2}{g}_{I3}({\bf k^{\prime}+k},\omega)$ (16b) $\displaystyle=$
$\displaystyle-\rho_{i}{1\over N}\sum_{{\bf k^{\prime}+k}}|V({\bf
k^{\prime}-k})|^{2}{g}_{I3}({\bf k^{\prime}},\omega),~{}~{}~{}~{}$
$\displaystyle\Sigma_{1}({\bf k},\omega)$ $\displaystyle=$
$\displaystyle\rho_{i}{1\over N}\sum_{{\bf k^{\prime}}}|V({\bf
k^{\prime}})|^{2}{g}_{I1}({\bf k^{\prime}+k},\omega)$ (16c) $\displaystyle=$
$\displaystyle\rho_{i}{1\over N}\sum_{{\bf k^{\prime}}}|V({\bf
k^{\prime}-k})|^{2}{g}_{I1}({\bf k^{\prime}},\omega).$

In the charge-spin separation fermion-spin theory feng2 , the electron
diagonal and off-diagonal Green’s functions are the convolutions of the spin
Green’s function guo ; feng ; feng1  $D^{-1}({\bf
p},\omega)=(\omega^{2}-\omega_{{\bf p}}^{2})/B_{{\bf p}}$ and the charge
carrier diagonal and off-diagonal Green’s functions in Eq. (7), respectively.
These convolutions reflect the charge-spin recombination anderson1 . Following
the previous discussions guo ; feng ; feng1 , we can obtain the electron
diagonal and off-diagonal Green’s functions in the present case. Then the
electron spectral function from the electron diagonal Green’s function is
found explicitly as,

$\displaystyle A({\bf k},\omega)$ $\displaystyle=$ $\displaystyle{1\over
N}\sum_{\bf p}{B_{\bf p}\over 2\omega_{\bf p}}\\{[n_{B}(\omega_{\bf
p})+n_{F}(\omega_{\bf p}-\omega)]A_{h}({\bf p}-{\bf k},\omega_{\bf p}-\omega)$
(17) $\displaystyle-$ $\displaystyle[n_{B}(-\omega_{\bf p})+n_{F}(-\omega_{\bf
p}-\omega)]A_{h}({\bf p}-{\bf k},-\omega_{\bf p}-\omega)\\},~{}~{}~{}~{}~{}$

where $A_{h}$ is the charge carrier spectral function, which can be expressed
as $A_{h}=-2{\rm Im}g^{dia}_{I}({\bf k},\omega)$, with $g^{dia}_{I}$ obtained
from Eq. (7) as,

$\displaystyle g^{dia}_{I}({\bf k},\omega)=Z_{hF}{\omega-\Sigma_{0}({\bf
k},\omega)+\bar{\xi}_{{\bf k}}+\Sigma_{3}({\bf
k},\omega)\over[\omega-\Sigma_{0}({\bf k},\omega)]^{2}-[\bar{\xi}_{{\bf
k}}+\Sigma_{3}({\bf k},\omega)]^{2}-[\bar{\Delta}_{hZ}({\bf
k})+\Sigma_{1}({\bf k},\omega)]^{2}}.$ (18)

##  III Electronic structure of the out-of-plane impurity-controlled cuprate
superconductors

Experimentally, it has been shown that the average of the next-nearest
neighbor hopping $t^{\prime}$ is not appreciably affected by the out-of-plane
impurities hashimoto . In this case, the commonly used parameters in this
paper are chosen as $t/J=2.5$ and $t^{\prime}/t=0.3$. We are now ready to
discuss the electronic structure of the out-of-plane impurity-controlled
cuprate superconductors and the related superconducting gap. In cuprate
superconductors, the information revealed by ARPES experiments shen  has shown
that around the nodal [$\pi/2,\pi/2$] and antinodal [$\pi,0$] points of the
Brillouin zone contain the essentials of the whole low energy quasiparticle
excitation spectrum. In this case, we have performed a calculation for the
electron spectral function $A({\bf k},\omega)$ in Eq. (14) at both nodal and
antinodal points. The results at (a) the nodal point and (b) the antinodal
point with the impurity concentration $\rho_{i}=0.001$ (solid line),
$\rho_{i}=0.002$ (dashed line), and $\rho_{i}=0.003$ (dotted line) under the
impurity scattering potential with $V_{0}=50J$ for the charge carrier doping
concentration $\delta=0.15$ are plotted in Fig. 1. For comparison, the
corresponding ARPES experimental results hashimoto  for the out-of-plane
impurity-controlled cuprate superconductors $Ln$-Bi2201 in the superconducting
state are also presented in Fig. 1 (inset). Our results show that the
quasiparticle peak is strongly dependent on the impurity concentration, and
the peaks at both nodal and antinodal points are suppressed due to the
presence of impurity scattering. At the nodal point, there is a sharp
superconducting quasiparticle peak near the Fermi energy, however, although
the peak at the high impurity concentration is dramatically reduced compared
to that at the low impurity concentration, the position of the leading-edge
mid-point of the electron spectral function remains almost unchanged. In
particular, the position of the leading-edge mid-point of the electron
spectral function reaches the Fermi level, indicating that there is no
superconducting gap. On the other hand, the spectral intensity from the Fermi
energy down to $\sim-1.1J$ decreases as the impurity concentration increases
at the antinodal point, this is the same case as that at the nodal point.
However, the position of the leading-edge mid-point of the electron spectral
function is shifted towards higher binding energies with increasing impurity
concentration, this is in contrast with the behavior observed at the nodal
point, and indicates the presence of the superconducting gap. The present
results also show that the effect of the out-of-plane impurity scattering is
stronger at the antinodal point than at the nodal one, in qualitative
agreement with the experimental results kondo ; hashimoto .

Figure 1: The electron spectral function at (a) the nodal point and (b) the
antinodal point with $\rho_{i}=0.001$ (solid line), $\rho_{i}=0.002$ (dashed
line), and $\rho_{i}=0.003$ (dotted line) for $V_{0}=50J$ at $\delta=0.15$.
Inset: the corresponding experimental results taken from Ref. hashimoto, .
Figure 2: The superconducting gap as a function of the Fermi surface angle
$\phi$ with $\rho_{i}=0$ (dashed line) and $\rho_{i}=0.001$ (solid line) for
$V_{0}=50J$ at $\delta=0.15$. Inset: the corresponding experimental results
taken from Ref. kondo, .

The behavior of the electron spectrum in Fig. 1 indicates an enhancement of
the superconducting gap in the antinodal region by the impurity scattering. To
show this point clearly, we have calculated the electron spectral function
$A({\bf k},\omega)$ along the direction $[\pi,0]\rightarrow[\pi/2,\pi/2]$, and
then employed the shift of the leading-edge mid-point as a measure of the
magnitude of the superconducting gap at each momentum just as it has been done
in the experiments kondo ; hashimoto . The results for the extracted
superconducting gap as a function of the Fermi surface angle $\phi$, defined
in the inset, with the impurity concentration $\rho_{i}=0$ (dashed line) and
$\rho_{i}=0.001$ (solid line) under the impurity scattering potential with
$V_{0}=50J$ for the charge carrier doping concentration $\delta=0.15$ are
plotted in Fig. 2 in comparison with the corresponding ARPES experimental
results for the out-of-plane impurity-controlled cuprate superconductor
(Bi,Pb)2(Sr,La)2CuO6+δ in the superconducting state kondo  (inset). It is
clearly shown that the superconducting gap $\Delta$ increases with the Fermi
surface angle decreasing from 45o (node) to 0o (antinode). Although the
superconducting gap in the presence of the impurity scattering is basically
consistent with the d-wave symmetry, it is obvious that there is a strong
deviation from the monotonic d-wave form around the antinodal region. In
particular, this strong deviation is mainly caused by a remarkable enhancement
of the superconducting gap value around the antinodal region, in qualitative
agreement with the experimental results kondo ; hashimoto . In other words,
the superconducting gap around the antinodal region is strongly enhanced by
the impurity scattering, whereas around the nodal region its value remains the
same. As a consequence, the well pronounced deviation from the monotonic
d-wave superconducting gap form in the out-of-plane impurity-controlled
cuprate superconductors is mainly caused by the effect of the out-of-plane
impurity scattering. This is also the reason why the superconducting gap
function for very high quality samples of the cuprate superconductor
La1-xSrxCuO4 has a monotonic d-wave form shi1 .

Figure 3: The superconducting gap as a function of $[{\rm cos}k_{x}-{\rm
cos}k_{y}]/2$ with $\rho_{i}=0001$ (solid line), $\rho_{i}=0.002$ (dashed
line), and $\rho_{i}=0.003$ (dotted line) for $V_{0}=50J$ at $\delta=0.15$.
Inset: the corresponding experimental results taken from Ref. hashimoto, .

For a better understanding of the impurity concentration dependence of the
deviation from the monotonic d-wave superconducting gap function, we have made
a series of calculations for the superconducting gap at different impurity
concentration levels, and the results of the superconducting gap as a function
of the monotonic d-wave function $[{\rm cos}k_{x}-{\rm cos}k_{y}]/2$ with the
impurity concentration $\rho_{i}=0001$ (solid line), $\rho_{i}=0.002$ (dashed
line), and $\rho_{i}=0.003$ (dotted line) under the impurity scattering
potential with $V_{0}=50J$ for the charge carrier doping concentration
$\delta=0.15$ are plotted in Fig. 3 in comparison with the corresponding ARPES
experimental results for the out-of-plane impurity-controlled cuprate
superconductors Ln-Bi2201 hashimoto  (inset). Obviously, our results show that
the magnitude of the deviation from the monotonic d-wave superconducting gap
form around the antinodal region increases with increasing impurity
concentration, in qualitative agreement with the experimental results kondo ;
hashimoto . This strong out-of-plane impurity effect in the antinodal region
is also consistent with scanning tunneling spectroscopy results sugimoto ,
where the average of the superconducting gap size, which corresponds to the
antinodal superconducting gap in the ARPES spectra, increases with increasing
impurity concentration.

Within the framework of the kinetic energy driven superconducting mechanism
feng1  in the presence of the out-of-plane impurities, our present results
show that the out-of-plane impurity scattering potential (12) in which the
impurities modulate the pair interaction locally give qualitative agreement
with respect to the main features observed in the ARPES measurements on the
out-of-plane impurity-controlled cuprate superconductors in the
superconducting state. Although this out-of-plane impurity effect in cuprate
superconductors can also be discussed starting directly from a
phenomenological d-wave BCS formalism graser ; nunner , in this paper we are
primarily interested in exploring the general notion of the effects of the
out-of-plane impurity scatterers in the kinetic energy driven cuprate
superconductors in the superconducting state. The qualitative agreement
between the present theoretical results and ARPES experimental data also
indicates that the presence of the out-of-plane impurities has a crucial
impact on the electronic structure of cuprate superconductors. On the other
hand, we emphasize that the quasiparticle scattering rate in the antinodal
region is strongly increased by the impurity scattering potential (12), while
the nodal quasiparticles are very weakly scattered by the impurity scattering
potential (12), this is why the superconducting transition temperature is
considerably affected by the out-of-plane impurity scattering in spite of a
relatively weak increase of the residual resistivity eisaki , since the
transport properties are mainly governed by the quasiparticles in the nodal
region.

##  IV Summary

In conclusion, we have shown very clearly in this paper that if the out-of-
plane impurity scattering is taken into account within the framework of the
kinetic energy driven d-wave superconductivity feng1 , the quasiparticle
spectrum of the $t$-$J$ model calculated based on the off-diagonal impurity
scattering potential (12) per se can correctly reproduce some main features
found in the ARPES measurements on the out-of-plane impurity-controlled
cuprate superconductors in the superconducting state kondo ; hashimoto . In
the presence of the out-of-plane impurities, although both sharp
superconducting coherence peaks around the nodal and antinodal regions are
suppressed, the effect of the impurity scattering is stronger in the antinodal
region than that in the nodal region, this leads to a strong deviation from
the monotonic d-wave superconducting gap form in the out-of-plane impurity-
controlled cuprate superconductors.

Finally, we have noted that within a phenomenological BCS approach, the
electron spectral properties of the underdoped cuprates as resulting from a
momentum dependent pseudogap in the normal state have been discussed
sangiovanni , where a normal state pseudogap function deviating from the
monotonic d-wave pseudogap form has been used to fit the ARPES experimental
data in the normal state. It has been shown kastner ; shen95  that there are
some subtle differences for different families of underdoped cuprates in the
normal state, and therefore it is possible that the pseudogap in the normal
state is effected by the impurity scattering as well.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
under Grant No. 10774015, and the funds from the Ministry of Science and
Technology of China under Grant Nos. 2006CB601002 and 2006CB921300.

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