# Doping and energy dependent microwave conductivity of kinetic energy driven
superconductors with extended impurities

Zhi Wang  Department of Physics, Beijing Normal University, Beijing 100875,
China  Huaiming Guo  Department of Physics, Capital Normal University, Beijing
100037, China  Shiping Feng∗ Department of Physics, Beijing Normal University,
Beijing 100875, China

###### Abstract

Within the framework of the kinetic energy driven superconducting mechanism,
the effect of the extended impurity scatterers on the quasiparticle transport
of cuprate superconductors in the superconducting state is studied based on
the nodal approximation of the quasiparticle excitations and scattering
processes. It is shown that there is a cusplike shape of the energy dependent
microwave conductivity spectrum. At low temperatures, the microwave
conductivity increases linearly with increasing temperatures, and reaches a
maximum at intermediate temperature, then decreases with increasing
temperatures at high temperatures. In contrast with the dome shape of the
doping dependent superconducting gap parameter, the minimum microwave
conductivity occurs around the optimal doping, and then increases in both
underdoped and overdoped regimes.

###### pacs:

74.25.-h, 74.25.Fy, 74.25.Nf

##  I Introduction

After over 20 years extensive studies, it has become clear that
superconductivity in doped cuprates results when charge carriers pair up into
Cooper pairs tsuei  as in the conventional superconductors bcs , then these
charge carrier Cooper pairs condensation reveals the superconducting (SC)
ground-state. However, as a natural consequence of the unconventional SC
mechanism that is responsible for the high SC transition temperatures anderson
, the charge carrier Cooper pairs in cuprate superconductors have a dominated
d-wave symmetry tsuei ; shen . In particular, this d-wave SC state remains one
of the cornerstones of our understanding of the physics in cuprate
superconductors. However, in spite of the unconventional SC mechanism, the
angle-resolved photoemission spectroscopy (ARPES) experimental results have
unambiguously established the Bogoliubov-quasiparticle nature of the sharp SC
quasiparticle peak in cuprate superconductors matsui ; campuzano1 , then the
SC coherence of the low energy quasiparticle peak is well described by the
simple Bardeen-Cooper-Schrieffer (BCS) formalism bcs  with the d-wave SC gap
function $\Delta({\bf k})=\Delta({\rm cos}k_{x}-{\rm cos}k_{y})/2$. In this
d-wave case, the characteristic feature is the existence of four nodal points
$[\pm\pi/2,\pm\pi/2]$ (in units of inverse lattice constant) in the Brillouin
zone tsuei ; shen , where the SC gap function vanishes, then the most physical
properties of cuprate superconductors in the SC state are controlled by the
quasiparticle excitations around the nodes turner . In particular, the key
signature of the nodal quasiparticle transport appears in the microwave
conductivity $\sigma(\omega,T)$ turner , which is essentially electromagnetic
absorption by the quasiparticles excited out of the condensation (either
thermally excited quasiparticles or excitations created by the impurity
scattering).

Understanding the role of impurities in cuprate superconductors has taken many
years of great effort palee ; hirschfeld . This follows from that the physical
properties of cuprate superconductors in the SC state are extreme sensitivity
to the impurity effect than the conventional superconductors due to the finite
angular-momentum charge carrier Cooper pairing hirschfeld1 . Experimentally,
By virtue of systematic studies using the microwave conductivity measurements,
some essential features of the evolution of the quasiparticle transport of
cuprate superconductors with energy and temperature in the SC state have been
established turner ; bonn ; lee ; hosseini ; harris : (1) at low temperatures,
the experimental results show the existence of the very long-live excitation
deep in the SC state, as evidenced by the sharp cusplike energy dependent
microwave conductivity spectrum, where the width of the sharp peak is nearly
temperature independent, and main behaviors of the microwave conductivity are
governed by thermally excited quasiparticles being scattered by impurities or
other defects turner ; harris ; (2) at low energies, the temperature dependent
microwave conductivity increases linearly with increasing temperatures at low
temperatures, and reaches a maximum (a large broad peak) around intermediate
temperature, then decreases with increasing temperatures at high temperatures
bonn ; hosseini ; harris . In particular, this broad peak shifts to higher
temperatures as the energy is increased. Theoretically, an agreement has
emerged that the BCS formalism with the d-wave SC gap function is useful in
the phenomenological description of the quasiparticle transport of cuprate
superconductors in the SC state palee ; hirschfeld ; quinlan ; hettler ; durst
; duffy ; nunner , although the SC pairing mechanism is beyond the
conventional electron-phonon mechanism. In this case, the microwave
conductivity of cuprate superconductors has been phenomenologically discussed
in the zero temperature and energy by including the contributions of the
vertex corrections durst . Recently, these discussions have been generalized
to study the temperature and energy dependence of the quasiparticle transport
of cuprate superconductors in the SC state nunner . To the best of our
knowledge, the microwave conductivity of cuprate superconductors has not been
treated starting from a microscopic SC theory, and no explicit calculations of
the doping dependence of the microwave conductivity has been made so far.

In this paper, we start from the kinetic energy driven SC mechanism feng1 ,
and study the effect of the extended impurity scatterers on the microwave
conductivity of cuprate superconductors. We evaluate explicitly the microwave
conductivity of cuprate superconductors within the nodal approximation of the
quasiparticle excitations and scattering processes, and qualitatively
reproduced some main features of the microwave conductivity measurements on
cuprate superconductors in the SC state turner ; bonn ; lee ; hosseini ;
harris . It is shown that there is a cusplike shape of the energy dependent
microwave conductivity spectrum. At low temperatures, the microwave
conductivity increases linearly with increasing temperatures, and reaches a
maximum at intermediate temperature, then decreases with increasing
temperatures at high temperatures. In contrast with the dome shape of the
doping dependent SC gap parameter, the minimum microwave conductivity occurs
around the optimal doping, and then increases in both underdoped and overdoped
regimes.

This paper is organized as follows. We present the basic formalism in Sec. II,
and then discuss the energy, temperature, and doping dependence of the
quasiparticle transport of cuprate superconductors in the SC state in Sec.
III, where we show that the quasiparticle transport of cuprate superconductors
in the SC state can be qualitatively understood within the framework of the
kinetic energy driven SC mechanism by considering the effect of the extended
impurity scatterers. Finally, we give a summary in Sec. IV.

##  II Formalism

In cuprate superconductors, the single common feature is the presence of the
two-dimensional CuO2 plane, and it is believed that the unconventional
physical properties of cuprate superconductors is closely related to the doped
CuO2 planes shen ; kastner . It has been argued that the $t$-$J$ model on a
square lattice captures the essential physics of the doped CuO2 plane anderson
; shen ,

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

where $\hat{\eta}=\pm\hat{x},\pm\hat{y}$, ${\bf
S}_{i}=\hat{C}^{\dagger}_{i}{\boldsymbol{\tau}}\hat{C}_{i}/2$ is spin operator
with ${\boldsymbol{\tau}}=(\tau_{1},\tau_{2},\tau_{3})$ as Pauli matrices, the
constrained electron operator $\hat{C}_{i\sigma}=C_{i\sigma}(1-n_{i-\sigma})$
with $n_{i\sigma}=C^{\dagger}_{i\sigma}C_{i\sigma}$, and $\mu$ is the chemical
potential. In the constrained electron operator, the operators
$C^{\dagger}_{i\sigma}$ and $C_{i\sigma}$ are to be thought of as operating
within the full Hilbert space, while the constrained electron operator
$\hat{C}^{\dagger}_{i\sigma}$ ($\hat{C}_{i\sigma}$) does not create (destroy)
any doubly occupied sites, and therefore represents physical creation
(annihilation) operator acting in the restricted Hilbert space without double
electron occupancy rice ; anderson1 . The strong electron correlation in the
$t$-$J$ model manifests itself by the restriction of the motions of the
electrons in the restricted Hilbert space without double electron occupancy
anderson , which can be treated properly in analytical calculations within the
charge-spin separation (CSS) fermion-spin theory feng2 , where the constrained
electron operators are decoupled as
$\hat{C}_{i\uparrow}=h^{\dagger}_{i\uparrow}S^{-}_{i}$ and
$\hat{C}_{i\downarrow}=h^{\dagger}_{i\downarrow}S^{+}_{i}$, with 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 hole itself (dressed holon),
while the spin operator $S_{i}$ describes the spin degree of freedom (spin),
then the motions of electrons are restricted in the restricted Hilbert space
without double electron occupancy in analytical calculations. In particular,
it has been shown that under the decoupling scheme, this CSS fermion-spin
representation is a natural representation of the constrained electron defined
in the restricted Hilbert space without double electron occupancy feng3 .
Moreover, these dressed holon and spin are gauge invariant feng2 , and in this
sense, they are real and can be interpreted as the physical excitations
laughlin . In this CSS fermion-spin representation, the low-energy behavior of
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) $\displaystyle-$
$\displaystyle\mu\sum_{i\sigma}h^{\dagger}_{i\sigma}h_{i\sigma}+J_{{\rm
eff}}\sum_{i\hat{\eta}}{\bf S}_{i}\cdot{\bf S}_{i+\hat{\eta}},$

with $J_{{\rm eff}}=(1-\delta)^{2}J$, and $\delta=\langle
h^{\dagger}_{i\sigma}h_{i\sigma}\rangle=\langle h^{\dagger}_{i}h_{i}\rangle$
is the hole doping concentration. As an important consequence, the kinetic
energy term in the $t$-$J$ model has been transferred as the dressed holon-
spin interaction, which reflects that even the kinetic energy term in the
$t$-$J$ Hamiltonian has strong Coulombic contribution due to the restriction
of the motions of electrons in the restricted Hilbert space without double
electron occupancy.

Recently, we have developed a kinetic energy driven SC mechanism feng1  based
on the CSS fermion-spin theory feng2 , where the dressed holon-spin
interaction from the kinetic energy term in the $t$-$J$ model (2) induces the
dressed holon pairing state with the d-wave symmetry by exchanging spin
excitations, then the electron Cooper pairs originating from the dressed holon
pairing state are due to the charge-spin recombination, and their condensation
reveals the d-wave SC ground-state. Moreover, this d-wave SC state is
controlled by both SC gap function and quasiparticle coherence, then the
maximal SC transition temperature occurs around the optimal doping, and
decreases in both underdoped and overdoped regimes. In particular, we have
shown that this SC state is the conventional BCS like with the d-wave symmetry
feng3 ; guo1 , so that the basic BCS formalism with the d-wave SC 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 ; arai  is beyond the
BCS formalism. Following our previous discussions feng1 ; feng3 ; guo1 , the
full dressed holon Green’s function in the SC state can be obtained in the
Nambu representation as,

$\displaystyle\tilde{g}({\bf k},\omega)$ $\displaystyle=$ $\displaystyle
Z_{hF}{1\over\omega^{2}-E^{2}_{h{\bf
k}}}\left(\begin{array}[]{cc}{\omega+\bar{\xi}_{{\bf
k}}}&{\bar{\Delta}_{hZ}({\bf k})}\\\ {\bar{\Delta}_{hZ}({\bf
k})}&{\omega-\bar{\xi}_{{\bf k}}}\end{array}\right)$ (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, the renormalized dressed holon excitation
spectrum $\bar{\xi}_{{\bf k}}=Z_{hF}\xi_{\bf k}$, with the mean-field (MF)
dressed holon excitation spectrum $\xi_{{\bf k}}=Zt\chi\gamma_{{\bf k}}-\mu$,
the spin correlation function $\chi=\langle
S_{i}^{+}S_{i+\hat{\eta}}^{-}\rangle$, $\gamma_{{\bf
k}}=(1/Z)\sum_{\hat{\eta}}e^{i{\bf k}\cdot\hat{\eta}}$, $Z$ is the number of
the nearest neighbor sites, the renormalized dressed holon d-wave pair gap
function $\bar{\Delta}_{hZ}({\bf k})=Z_{hF}\bar{\Delta}_{h}({\bf k})$, where
the effective dressed holon 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 dressed holon quasiparticle spectrum
$E_{h{\bf k}}=\sqrt{\bar{\xi}^{2}_{{\bf k}}+\mid\bar{\Delta}_{hZ}({\bf
k})\mid^{2}}$, while the dressed holon quasiparticle coherent weight $Z_{hF}$
and effective dressed holon gap parameters $\bar{\Delta}_{h}$ are determined
by the following two equations feng1 ; feng3 ; guo1 ,

$\displaystyle 1$ $\displaystyle=$ $\displaystyle{(Zt)^{2}\over
N^{3}}\sum_{{\bf k,p,p^{\prime}}}\gamma^{2}_{{\bf p+k}}\gamma^{(d)}_{{\bf
k-p^{\prime}+p}}\gamma^{(d)}_{{\bf k}}{Z^{2}_{hF}\over E_{h{\bf k}}}{B_{{\bf
p}}B_{{\bf p^{\prime}}}\over\omega_{{\bf p}}\omega_{{\bf
p^{\prime}}}}\left({F^{(1)}_{1}({\bf k,p,p^{\prime}})\over(\omega_{{\bf
p^{\prime}}}-\omega_{{\bf p}})^{2}-E^{2}_{h{\bf k}}}-{F^{(2)}_{1}({\bf
k,p,p^{\prime}})\over(\omega_{{\bf p^{\prime}}}+\omega_{{\bf
p}})^{2}-E^{2}_{h{\bf k}}}\right),$ (7a) $\displaystyle{1\over Z}_{hF}$
$\displaystyle=$ $\displaystyle 1+\left({Zt\over N}\right)^{2}\sum_{{\bf
p,p^{\prime}}}\gamma^{2}_{{\bf p}+{\bf k}_{0}}Z_{hF}{B_{{\bf p}}B_{{\bf
p^{\prime}}}\over 4\omega_{{\bf p}}\omega_{{\bf
p^{\prime}}}}\left({F^{(1)}_{2}({\bf p,p^{\prime}})\over(\omega_{{\bf
p}}-\omega_{{\bf p^{\prime}}}-E_{h{\bf
p-p^{\prime}+k_{0}}})^{2}}+{F^{(2)}_{2}({\bf p,p^{\prime}})\over(\omega_{{\bf
p}}-\omega_{{\bf p^{\prime}}}+E_{h{\bf p-p^{\prime}+k_{0}}})^{2}}\right.$ (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 ${\bf k}_{0}=[\pi,0]$, $B_{{\bf
p}}=\lambda[(2\epsilon\chi^{z}+\chi)\gamma_{{\bf
p}}-(2\chi^{z}+\epsilon\chi)]$, $\lambda=2ZJ_{{\rm eff}}$,
$\epsilon=1+2t\phi/J_{{\rm eff}}$, the dressed holon’s particle-hole parameter
$\phi=\langle h^{\dagger}_{i\sigma}h_{i+\hat{\eta}\sigma}\rangle$, the spin
correlation function $\chi^{z}=\langle S_{i}^{z}S_{i+\hat{\eta}}^{z}\rangle$,
$F^{(1)}_{1}({\bf k,p,p^{\prime}})=(\omega_{{\bf p^{\prime}}}-\omega_{{\bf
p}})[n_{B}(\omega_{{\bf p}})-n_{B}(\omega_{{\bf
p^{\prime}}})][1-2n_{F}(E_{h{\bf k}})]+E_{h{\bf k}}[n_{B}(\omega_{{\bf
p^{\prime}}})n_{B}(-\omega_{{\bf p}})+n_{B}(\omega_{{\bf
p}})n_{B}(-\omega_{{\bf p^{\prime}}})]$, $F^{(2)}_{1}({\bf
k,p,p^{\prime}})=(\omega_{{\bf p^{\prime}}}+\omega_{{\bf
p}})[n_{B}(-\omega_{{\bf p^{\prime}}})-n_{B}(\omega_{{\bf
p}})][1-2n_{F}(E_{h{\bf k}})]+E_{h{\bf k}}[n_{B}(\omega_{{\bf
p^{\prime}}})n_{B}(\omega_{{\bf p}})+n_{B}(-\omega_{{\bf
p^{\prime}}})n_{B}(-\omega_{{\bf p}})]$, $F^{(1)}_{2}({\bf
p,p^{\prime}})=n_{F}(E_{h{\bf p-p^{\prime}+k_{0}}})[n_{B}(\omega_{{\bf
p^{\prime}}})-n_{B}(\omega_{{\bf p}})]-n_{B}(\omega_{{\bf
p}})n_{B}(-\omega_{{\bf p^{\prime}}})$, $F^{(2)}_{2}({\bf
p,p^{\prime}})=n_{F}(E_{h{\bf p-p^{\prime}+k_{0}}})[n_{B}(\omega_{{\bf
p}})-n_{B}(\omega_{{\bf p^{\prime}}})]-n_{B}(\omega_{{\bf
p^{\prime}}})n_{B}(-\omega_{{\bf p}})$, $F^{(3)}_{2}({\bf
p,p^{\prime}})=n_{F}(E_{h{\bf p-p^{\prime}+k_{0}}})[n_{B}(\omega_{{\bf
p^{\prime}}})-n_{B}(-\omega_{{\bf p}})]+n_{B}(\omega_{{\bf
p}})n_{B}(\omega_{{\bf p^{\prime}}})$, $F^{(4)}_{2}({\bf
p,p^{\prime}})=n_{F}(E_{h{\bf p-p^{\prime}+k_{0}}})[n_{B}(-\omega_{{\bf
p^{\prime}}})-n_{B}(\omega_{{\bf p}})]+n_{B}(-\omega_{{\bf
p}})n_{B}(-\omega_{{\bf p^{\prime}}})$, $n_{B}(\omega_{{\bf p}})$ and
$n_{F}(E_{h{\bf k}})$ are the boson and fermion distribution functions,
respectively, and the MF spin excitation spectrum,

$\displaystyle\omega^{2}_{{\bf p}}$ $\displaystyle=$
$\displaystyle\lambda^{2}[(A_{1}-\alpha\epsilon\chi^{z}\gamma_{{\bf
p}}-{1\over 2Z}\alpha\epsilon\chi)(1-\epsilon\gamma_{{\bf p}})$ (8)
$\displaystyle+$ $\displaystyle{1\over 2}\epsilon(A_{2}-{1\over
2}\alpha\chi^{z}-\alpha\chi\gamma_{{\bf p}})(\epsilon-\gamma_{{\bf p}})],$

where $A_{1}=\alpha C^{z}+(1-\alpha)/(4Z)$, $A_{2}=\alpha C+(1-\alpha)/(2Z)$,
and the spin correlation functions
$C=(1/Z^{2})\sum_{\hat{\eta},\hat{\eta^{\prime}}}\langle
S_{i+\hat{\eta}}^{+}S_{i+\hat{\eta^{\prime}}}^{-}\rangle$ and
$C^{z}=(1/Z^{2})\sum_{\hat{\eta},\hat{\eta^{\prime}}}\langle
S_{i+\hat{\eta}}^{z}S_{i+\hat{\eta^{\prime}}}^{z}\rangle$. 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 MF calculation feng1 ; feng3 ; guo1 , which can be regarded as the
vertex correction. These two equations in Eqs. (4a) and (4b) must be solved
simultaneously with other self-consistent equations, then all order
parameters, decoupling parameter $\alpha$, and chemical potential $\mu$ are
determined by the self-consistent calculation.

In the CSS fermion-spin theory feng2 ; feng3 , the electron Green’s function
is a convolution of the spin Green’s function and dressed holon Green’s
function. Following our previous discussions feng1 ; feng3 ; guo1 , we can
obtain the electron diagonal and off-diagonal Green’s functions in the SC
state as,

$\displaystyle G({\bf k},\omega)$ $\displaystyle=$ $\displaystyle{1\over
N}\sum_{{\bf p}}Z_{hF}{B_{{\bf p}}\over 4\omega_{{\bf p}}}\left\\{{\rm
coth}[{1\over 2}\beta\omega_{{\bf p}}]\left({U^{2}_{h{\bf
p+k}}\over\omega+E_{h{\bf p+k}}-\omega_{{\bf p}}}+{U^{2}_{h{\bf
p+k}}\over\omega+E_{h{\bf p+k}}+\omega_{{\bf p}}}\right.\right.$ (9a)
$\displaystyle+$ $\displaystyle\left.{V^{2}_{h{\bf p+k}}\over\omega-E_{h{\bf
p+k}}+\omega_{{\bf p}}}+{V^{2}_{h{\bf p+k}}\over\omega-E_{h{\bf
p+k}}-\omega_{{\bf p}}}\right)+{\rm tanh}[{1\over 2}\beta E_{h{\bf
p+k}}]\left({U^{2}_{h{\bf p+k}}\over\omega+E_{h{\bf p+k}}+\omega_{{\bf
p}}}\right.$ $\displaystyle-$ $\displaystyle{U^{2}_{h{\bf
p+k}}\over\omega+E_{h{\bf p+k}}-\omega_{{\bf p}}}+\left.\left.{V^{2}_{h{\bf
p+k}}\over\omega-E_{h{\bf p+k}}-\omega_{{\bf p}}}-{V^{2}_{h{\bf
p+k}}\over\omega-E_{h{\bf p+k}}+\omega_{{\bf p}}}\right)\right\\},$
$\displaystyle\Gamma^{\dagger}({\bf k},\omega)$ $\displaystyle=$
$\displaystyle{1\over N}\sum_{{\bf p}}Z_{hF}{\bar{\Delta}_{hZ}({\bf p+k})\over
2E_{h{\bf p+k}}}{B_{{\bf p}}\over 4\omega_{{\bf p}}}\left\\{{\rm coth}[{1\over
2}\beta\omega_{{\bf p}}]\left({1\over\omega-E_{h{\bf p+k}}-\omega_{{\bf
p}}}+{1\over\omega-E_{h{\bf p+k}}+\omega_{{\bf p}}}\right.\right.$ (9b)
$\displaystyle-$ $\displaystyle\left.{1\over\omega+E_{h{\bf p+k}}+\omega_{{\bf
p}}}-{1\over\omega+E_{h{\bf p+k}}-\omega_{{\bf p}}}\right)+{\rm tanh}[{1\over
2}\beta E_{h{\bf p+k}}]\left({1\over\omega-E_{h{\bf p+k}}-\omega_{{\bf
p}}}\right.$ $\displaystyle-$ $\displaystyle\left.\left.{1\over\omega-E_{h{\bf
p+k}}+\omega_{{\bf p}}}-{1\over\omega+E_{h{\bf p+k}}+\omega_{{\bf
p}}}+{1\over\omega+E_{h{\bf p+k}}-\omega_{{\bf p}}}\right)\right\\},$

respectively, where the dressed holon quasiparticle coherence factors
$U^{2}_{h{\bf k}}=(1+{\bar{\xi}_{{\bf k}}/E_{h{\bf k}}})/2$ and $V^{2}_{h{\bf
k}}=(1-{\bar{\xi}_{{\bf k}}/E_{h{\bf k}}})/2$. These convolutions of the spin
Green’s function and dressed holon diagonal and off-diagonal Green’s functions
reflect the charge-spin recombination anderson2 . Since the spins center
around the $[\pi,\pi]$ point in the MF level feng1 ; feng3 ; guo1 , then the
main contributions for the spins comes from the $[\pi,\pi]$ point. In this
case, the electron diagonal and off-diagonal Green’s functions in Eqs. (6a)
and (6b) can be approximately reduced as the BCS formalism with the d-wave SC
gap function in terms of $\omega_{{\bf p}=[\pi,\pi]}\sim 0$ and the equation
feng1 ; feng2  $1/2=\langle S_{i}^{+}S_{i}^{-}\rangle=(1/N)\sum_{{\bf
p}}B_{{\bf p}}{\rm coth}(\beta\omega_{{\bf p}}/2)/(2\omega_{{\bf p}})$,

$\displaystyle G({\bf k},\omega)$ $\displaystyle\approx$ $\displaystyle
Z_{F}\left({U^{2}_{{\bf k}}\over\omega-E_{{\bf k}}}+{V^{2}_{{\bf
k}}\over\omega+E_{{\bf k}}}\right),$ (10a) $\displaystyle\Gamma^{\dagger}({\bf
k},\omega)$ $\displaystyle\approx$ $\displaystyle Z_{F}{\bar{\Delta}_{Z}({\bf
k})\over 2E_{{\bf k}}}\left({1\over\omega-E_{{\bf k}}}+{1\over\omega+E_{{\bf
k}}}\right),$ (10b)

where the electron quasiparticle coherent weight $Z_{F}=Z_{hF}/2$, the
electron quasiparticle coherence factors $U^{2}_{{\bf k}}\approx V^{2}_{h{\bf
k+k_{A}}}=(1+{\bar{\varepsilon}_{{\bf k}}/E_{{\bf k}}})/2$ and $V^{2}_{{\bf
k}}\approx U^{2}_{h{\bf k+k_{A}}}=(1-{\bar{\varepsilon}_{{\bf k}}/E_{{\bf
k}}})/2$, with ${\bar{\varepsilon}_{{\bf k}}=Z_{F}\varepsilon}_{{\bf k}}$,
$\varepsilon_{{\bf k}}=Zt\chi\gamma_{{\bf k}}+\mu$, and ${\bf
k_{A}}=[\pi,\pi]$, $\bar{\Delta}_{Z}({\bf k})=\bar{\Delta}_{hZ}({\bf k})/2$,
and electron quasiparticle spectrum $E_{{\bf k}}\approx E_{h{\bf
k+k_{A}}}=\sqrt{\bar{\varepsilon}^{2}_{{\bf k}}+\mid\bar{\Delta}_{Z}({\bf
k})\mid^{2}}$, i.e., the hole-like dressed holon quasiparticle coherence
factors $V_{h{\bf k}}$ and $U_{h{\bf k}}$ and hole-like dressed holon
quasiparticle spectrum $E_{h{\bf k}}$ have been transferred into the electron
quasiparticle coherence factors $U_{{\bf k}}$ and $V_{{\bf k}}$ and electron
quasiparticle spectrum $E_{{\bf k}}$, respectively, by the convolutions of the
spin Green’s function and dressed holon Green’s functions due to the charge-
spin recombination. This means that within the kinetic energy driven SC
mechanism, the dressed holon pairs condense with the d-wave symmetry in a wide
range of the hole doping concentration, then the electron Cooper pairs
originating from the dressed holon pairing state are due to the charge-spin
recombination, and their condensation automatically gives the electron
quasiparticle character. For the convenience in the following discussions,
these electron Green’s functions in Eq. (7) in the SC state can be expressed
in the Nambu representation as,

$\displaystyle\tilde{G}({\bf k},\omega)$ $\displaystyle=$ $\displaystyle
Z_{F}{1\over\omega^{2}-E^{2}_{{\bf
k}}}\left(\begin{array}[]{cc}{\omega+\bar{\varepsilon}_{{\bf
k}}}&{\bar{\Delta}_{Z}({\bf k})}\\\ {\bar{\Delta}_{Z}({\bf
k})}&{\omega-\bar{\varepsilon}_{{\bf k}}}\end{array}\right)$ (13)
$\displaystyle=$ $\displaystyle Z_{F}{\omega\tau_{0}+\bar{\Delta}_{Z}({\bf
k})\tau_{1}+\bar{\varepsilon}_{{\bf k}}\tau_{3}\over\omega^{2}-E_{{\bf
k}}^{2}}.$ (14)

With the helps of this BCS formalism under kinetic energy driven SC mechanism,
now we can discuss the effect of the extended impurity scatterers on the
quasiparticle transport in cuprate superconductors. In the presence of
impurities, the unperturbed electron Green’s function in Eq. (8) is dressed
via the impurity scattering durst ; nunner ,

$\displaystyle\tilde{G}_{I}({\bf k},\omega)^{-1}=\tilde{G}({\bf
k},\omega)^{-1}-\tilde{\Sigma}({\bf k},\omega),$ (15)

with the self-energy function $\tilde{\Sigma}({\bf
k},\omega)=\sum_{\alpha}\Sigma_{\alpha}({\bf k},\omega)\tau_{\alpha}$. It has
been shown that all but the scalar component of the self-energy function can
be neglected or absorbed into $\bar{\Delta}_{Z}({\bf k})$ durst ; nunner . In
this case, the dressed electron Green’s function in Eq. (9) can be explicitly
rewritten as,

$\displaystyle\tilde{G}_{I}({\bf k},\omega)=Z_{F}{[\omega-\Sigma_{0}({\bf
k},\omega)]\tau_{0}+\bar{\Delta}_{Z}({\bf k})\tau_{1}+[\bar{\varepsilon}_{{\bf
k}}+\Sigma_{3}({\bf k},\omega)]\tau_{3}\over[\omega-\Sigma_{0}({\bf
k},\omega)]^{2}-\bar{\varepsilon}^{2}_{{\bf k}}-\bar{\Delta}^{2}_{Z}({\bf
k})}.$ (16)

Based on the phenomenological d-wave BCS-type electron Green’s function durst
, the energy and temperature dependence of the microwave conductivity of
cuprate superconductors has been fitted nunner , where the electron self-
energy functions $\Sigma_{0}({\bf k},\omega)$ and $\Sigma_{3}({\bf k},\omega)$
have been treated within the framework of the T-matrix approximation.
Following their discussions durst ; nunner , the electron self-energy function
$\tilde{\Sigma}({\bf k},\omega)$ can be obtained approximately as,

$\displaystyle\tilde{\Sigma}({\bf k},\omega)=\rho_{i}\tilde{T}_{{\bf k}{\bf
k}}(\omega),$ (17)

with $\rho_{i}$ is the impurity concentration, and $\tilde{T}_{{\bf k}{\bf
k}}(\omega)$ is the diagonal element of the T-matrix,

$\displaystyle\tilde{T}_{{\bf k}{\bf k}^{\prime}}(\omega)=V_{{\bf k}{\bf
k}^{\prime}}\tau_{3}+\sum_{{\bf k}^{\prime\prime}}V_{{\bf k}{\bf
k}^{\prime\prime}}\tau_{3}\tilde{G}_{I}({\bf
k}^{\prime\prime},\omega)\tilde{T}_{{\bf k}^{\prime\prime}{\bf
k}^{\prime}}(\omega),$ (18)

where $V_{{\bf k}{\bf k}^{\prime}}$ is the impurity scattering potential. As
mentioned in Sec. I, there is no gap to the quasiparticle excitations at the
four nodes for the d-wave SC state of cuprate superconductors, therefore the
quasiparticles are generated only around these four nodes. It has been shown
durst  that this characteristic feature is very useful when considering the
impurity scattering, since the initial and final momenta of a scattering event
must always be approximately equal to the ${\bf k}$-space location of one of
the four nodes in the zero temperature and zero energy, while the impurity
scattering potential $V_{{\bf k}{\bf k}^{\prime}}$ varies slowly over the area
of a node. In this case, a general scattering potential $V_{{\bf k}{\bf
k}^{\prime}}$ need only be evaluated in three possible case: the intranode
impurity scattering $V_{{\bf k}{\bf k}^{\prime}}=V_{1}$ (${\bf k}$ and ${\bf
k}^{\prime}$ at the same node), the adjacent-node impurity scattering $V_{{\bf
k}{\bf k}^{\prime}}=V_{2}$ (${\bf k}$ and ${\bf k}^{\prime}$ at the adjacent
nodes), and the opposite-node impurity scattering $V_{{\bf k}{\bf
k}^{\prime}}=V_{3}$ (${\bf k}$ and ${\bf k}^{\prime}$ at the opposite nodes),
then the impurity scattering potential $V_{{\bf k}{\bf k}^{\prime}}$ in the
T-matrix can be effectively reduced as durst ,

$\displaystyle V_{{\bf k}{\bf
k}^{\prime}}\rightarrow\underline{V}=\left(\begin{array}[]{cccc}V_{1}&V_{2}&V_{3}&V_{2}\\\
V_{2}&V_{1}&V_{2}&V_{3}\\\ V_{3}&V_{2}&V_{1}&V_{2}\\\
V_{2}&V_{3}&V_{2}&V_{1}\end{array}\right)\,.$ (23)

At the zero temperature and zero energy, these nodes reduce to points. In this
case, this nodal approximation for the impurity potential can reproduce any
impurity potential. However, at finite temperatures and energies, there is a
limitation on the forward scattering character of the impurity potential
because this nodal approximation assumes the Brillouin zone quadrant around a
particular node durst . It has been shown nunner  that although the strict
forward scattering limit can therefore not be reached at finite temperatures
and energies, this nodal approximation is still appropriate to treat the
intermediate range scatters. Therefore in the following discussions, we employ
the simplified impurity scattering potential in Eq. (13) to study the impurity
scattering effect on the quasiparticle transport of cuprate superconductors.
Substituting Eq. (13) into Eq. (12), the T-matrix can be obtained as a
$4\times 4$-matrix around the nodal points,

$\displaystyle\tilde{T}_{jj^{\prime}}(\omega)=V_{jj^{\prime}}\tau_{3}+\tilde{I}_{G}(\omega)\tau_{3}\sum_{j^{\prime\prime}}V_{jj^{\prime\prime}}\tilde{T}_{j^{\prime\prime}j^{\prime}}(\omega),$
(24)

where $\tilde{I}_{G}(\omega)$ is the integral of the electron Green’s
function, and can be obtained as,

$\displaystyle\tilde{I}_{G}(\omega)={1\over N}\sum_{{\bf k}}\tilde{G}_{I}({\bf
k},\omega)\approx\tilde{G}_{I0}(\omega)\tau_{0}+\tilde{G}_{I3}(\omega)\tau_{3},$
(25)

with $\tilde{G}_{I0}(\omega)$ and $\tilde{G}_{I3}(\omega)$ are given by,

$\displaystyle\tilde{G}_{I0}(\omega)$ $\displaystyle=$ $\displaystyle{1\over
N}\sum_{{\bf k}}Z_{F}{\omega-\Sigma_{0}(\omega)\over\omega^{2}-E^{2}_{{\bf
k}}},$ (26a) $\displaystyle\tilde{G}_{I3}(\omega)$ $\displaystyle=$
$\displaystyle{1\over N}\sum_{{\bf k}}Z_{F}{\bar{\varepsilon}_{{\bf
k}}+\Sigma_{3}(\omega)\over\omega^{2}-E^{2}_{{\bf k}}},$ (26b)

and the self-energy functions $\Sigma_{0}(\omega)$ and $\Sigma_{0}(\omega)$
are evaluated as,

$\displaystyle\Sigma_{0}(\omega)$ $\displaystyle=$
$\displaystyle{\rho_{i}\over
4}\left({2\tilde{G}_{I0}(\omega)V_{13}^{2}\over[1-\tilde{G}_{I3}(\omega)V_{13}]^{2}-[\tilde{G}_{I0}(\omega)V_{13}]^{2}}+{\tilde{G}_{I0}(\omega)(V^{-}_{123})^{2}\over[1-\tilde{G}_{I3}(\omega)V^{-}_{123}]^{2}-[\tilde{G}_{I0}(\omega)V^{-}_{123}]^{2}}\right.$
(27a) $\displaystyle+$
$\displaystyle\left.{\tilde{G}_{I0}(\omega)(V^{+}_{123})^{2}\over[1-\tilde{G}_{I3}(\omega)V^{+}_{123}]^{2}-[\tilde{G}_{I0}(\omega)V^{+}_{123}]^{2}}\right),$
$\displaystyle\Sigma_{3}(\omega)$ $\displaystyle=$
$\displaystyle{\rho_{i}\over
4}\left({2V_{13}[1-\tilde{G}_{I3}(\omega)V_{13}]\over[1-\tilde{G}_{I3}(\omega)V_{13}]^{2}-[\tilde{G}_{I0}(\omega)V_{13}]^{2}}+{V^{-}_{123}[1-\tilde{G}_{I3}(\omega)V^{-}_{123}]\over[1-\tilde{G}_{I3}(\omega)V^{-}_{123}]^{2}-[\tilde{G}_{I0}(\omega)V^{-}_{123}]^{2}}\right.$
(27b) $\displaystyle+$
$\displaystyle\left.{V^{+}_{123}[1-\tilde{G}_{I0}(\omega)V^{+}_{123}]\over[1-\tilde{G}_{I3}(\omega)V^{+}_{123}]^{2}-[\tilde{G}_{I0}(\omega)V^{+}_{123}]^{2}}\right),$

where $V_{13}=V_{1}-V_{3}$, $V^{-}_{123}=V_{1}-2V_{2}+V_{3}$, and
$V^{+}_{123}=V_{1}+2V_{2}+V_{3}$.

In the framework of the linear response theory, the microwave conductivity of
cuprate superconductors can be calculated by means of the Kubo formula as
mahan ,

$\displaystyle\sigma(\omega,T)=-{{\rm Im}\Pi(\omega,T)\over\omega},$ (28)

with $\Pi(\omega,T)$ is the electron current-current correlation function in
the SC state, and can be expressed as,

$\displaystyle\Pi(\tau-\tau^{\prime})=-<T_{\tau}{\bf J}(\tau)\cdot{\bf
J}(\tau^{\prime})>.$ (29)

In the CSS fermion-spin representation feng2 ; feng3 , the electron
polarization operator can be evaluated as,

$\displaystyle{\bf P}=\sum_{i}{\bf R}_{i}\hat{n}_{i}=\sum_{i,\sigma}{\bf
R}_{i}\hat{C}^{{\dagger}}_{i,\sigma}\hat{C}_{i,\sigma}={1\over
2}\sum_{i,\sigma}{\bf R}_{i}h_{i,\sigma}h^{{\dagger}}_{i,\sigma},~{}~{}~{}$
(30)

then within the $t$-$J$ model (2), the current density of electrons is
obtained by the time derivation of this polarization operator using the
Heisenberg’s equation of motion as,

$\displaystyle{\bf J}=ie[H,{\bf P}]=i{1\over
2}et\sum_{i\hat{\eta}}\hat{\eta}(h^{\dagger}_{i+\hat{\eta}\uparrow}h_{i\uparrow}S_{i}^{+}S^{-}_{i+\hat{\eta}}+h^{\dagger}_{i+\hat{\eta}\downarrow}h_{i\downarrow}S^{\dagger}_{i}S^{-}_{i+\eta})=-i{1\over
2}et\sum_{i\eta\sigma}\hat{\eta}\hat{C}^{\dagger}_{i\sigma}\hat{C}_{i+\hat{\eta}\sigma}\approx{ev_{f}\over\sqrt{2}}\sum_{{\bf
k}\sigma}{\bf k}\hat{C}^{\dagger}_{{\bf k}\sigma}\hat{C}_{{\bf k}\sigma},$
(31)

with $v_{f}=\sqrt{2}t$ is the electron velocity at the nodal points. According
to this current density (21), the current-current correlation function in Eq.
(19) can be obtained as,

$\displaystyle\Pi(i\omega_{n},T)$ $\displaystyle=$
$\displaystyle{e^{2}v_{f}^{2}\over 2}{1\over N}\sum_{{\bf
k}}{1\over\beta}\sum_{i\omega_{n}^{\prime}}{\rm Tr}[\tilde{G}({\bf
k},i\omega_{n}^{\prime})$ (32) $\displaystyle\times$
$\displaystyle\tilde{G}({\bf
k},i\omega_{n}^{\prime}+i\omega_{n})\tilde{\Gamma}({\bf
k},i\omega_{n}^{\prime},i\omega_{n})],$

with $\tilde{\Gamma}({\bf k},\omega^{\prime},\omega)$ is the vertex function,
and can be evaluated explicitly as the sum of ladder diagrams,

$\displaystyle\Pi(i\omega_{n})=e^{2}v^{2}_{f}{1\over\beta}\sum_{i\omega_{n}^{\prime}}J(i\omega_{n},i\omega_{n}^{\prime}),$
(33)

where the kernel function $J(\omega,\omega^{\prime})$ is expressed as,

$\displaystyle
J(\omega,\omega^{\prime})={I^{(0)}_{0}+L_{A}[I^{(0)}_{0}I_{3}^{(3)}+I_{0}^{(3)}I_{3}^{(0)}]\over[1-(L_{A}I_{0}^{(0)}+L_{B}I^{(0)}_{3})][1-(L_{A}I_{3}^{(3)}+L_{B}I^{(3)}_{0})]-[L_{A}I_{0}^{(3)}+L_{B}I^{(3)}_{3}][L_{A}I_{3}^{(0)}+L_{B}I^{(0)}_{0}]},$
(34)

with the functions,

$\displaystyle L_{A}(\omega,\omega^{\prime})$ $\displaystyle=$
$\displaystyle[T_{11}^{(0)}(\omega)T_{11}^{(0)}(\omega+\omega^{\prime})+T_{11}^{(3)}(\omega)T_{11}^{(3)}(\omega+\omega^{\prime})-T_{13}^{(0)}(\omega)T_{13}^{(0)}(\omega+\omega^{\prime})-T_{13}^{(3)}(\omega)T_{13}^{(3)}(\omega+\omega^{\prime})],$
(35a) $\displaystyle L_{B}(\omega,\omega^{\prime})$ $\displaystyle=$
$\displaystyle[T_{11}^{(0)}(\omega)T_{11}^{(3)}(\omega+\omega^{\prime})+T_{11}^{(3)}(\omega)T_{11}^{(0)}(\omega+\omega^{\prime})-T_{13}^{(0)}(\omega)T_{13}^{(3)}(\omega+\omega^{\prime})-T_{13}^{(3)}(\omega)T_{13}^{(0)}(\omega+\omega^{\prime})],$
(35b)

while the functions $I^{(0)}_{0}(\omega,\omega^{\prime})$ and
$I^{(0)}_{3}(\omega,\omega^{\prime})$ are give by,

$\displaystyle I^{(0)}_{0}(\omega,\omega^{\prime})\tau_{0}$ $\displaystyle+$
$\displaystyle I^{(3)}_{0}(\omega,\omega^{\prime})\tau_{3}={1\over
N}\sum_{{\bf k}}\tilde{G}_{I}({\bf k},\omega)\tilde{G}_{I}({\bf
k},\omega+\omega^{\prime}),$ (36a) $\displaystyle
I^{(0)}_{3}(\omega,\omega^{\prime})\tau_{0}$ $\displaystyle+$ $\displaystyle
I^{(3)}_{3}(\omega,\omega^{\prime})\tau_{3}={1\over N}\sum_{{\bf
k}}\tilde{G}_{I}({\bf k},\omega)\tilde{\tau}_{3}\tilde{G}_{I}({\bf
k},\omega+\omega^{\prime}).$ (36b)

Substituting Eq. (23) into Eq. (18), the microwave conductivity of cuprate
superconductors is obtained explicitly as,

$\displaystyle\sigma(\omega,T)$ $\displaystyle=$ $\displaystyle
e^{2}v^{2}_{f}\int^{\infty}_{-\infty}{d\omega^{\prime}\over
2\pi}{n_{F}(\omega^{\prime})-n_{F}(\omega^{\prime}+\omega)\over\omega^{\prime}}[{\rm
Re}J(\omega^{\prime}-i\delta,\omega^{\prime}+\omega+i\delta)-{\rm
Re}J(\omega^{\prime}+i\delta,\omega^{\prime}+\omega+i\delta)].$ (37)

We emphasize that based on the nodal approximation of the quasiparticle
excitations and scattering processes durst ; nunner , this microwave
conductivity of cuprate superconductors in Eq. (27) is obtained within the
kinetic energy driven SC mechanism, although its expression is similar to that
obtained within the phenomenological BCS formalism with the d-wave SC gap
function durst ; nunner .

##  III Microwave conductivity of cuprate superconductors

In cuprate superconductors, although the values of $J$ and $t$ is believed to
vary somewhat from compound to compound, however, as a qualitative discussion,
the commonly used parameters in this paper are chosen as $t/J=2.5$, with an
reasonably estimative value of $J\sim 1000$K. We are now ready to discuss the
doping, energy, and temperature dependence of the quasiparticle transport of
cuprate superconductors in the SC state with extended impurities. We have
performed a calculation for the microwave conductivity $\sigma(\omega,T)$ in
Eq. (27), and the results of $\sigma(\omega,T)$ as a function of energy with
temperature $T=0.002J=2$K (solid line), $T=0.004J=4$K (dashed line),
$T=0.008J=8$K (dash-dotted line), and $T=0.01J=10$K (dotted line) under the
slightly strong impurity scattering potential with $V_{1}=58J$,
$V_{2}=49.32J$, and $V_{3}=40.6J$ at the impurity concentration
$\rho=0.000014$ for the doping concentration $\delta=0.15$ are plotted in Fig.
1 in comparison with the corresponding experimental results of cuprate
superconductors in the SC state harris  (inset). Obviously, the energy
evolution of the microwave conductivity of cuprate superconductor harris  is
qualitatively reproduced. In particular, a low temperature cusplike shape of
the microwave conductivity is obtained for cuprate superconductors in the
presence of the impurity scattering. At the low energy regime
($\omega<0.0002J$), this low temperature microwave conductivity
$\sigma(\omega,T)$ rises rapidly from the universal zero-temperature limit to
a much larger microwave conductivity. However, this low temperature microwave
conductivity $\sigma(\omega,T)$ becomes smaller and varies from weakly energy
dependence at the intermediate energy regime ($0.0002J<\omega<0.0008J$), to
the almost energy independence at the high energy regime ($\omega>0.0008J$).

Figure 1: The microwave conductivity as a function of energy with
$T=0.002J=2$K (solid line), $T=0.004J=4$K (dashed line), $T=0.008J=8$K (dash-
dotted line), and $T=0.01J=10$K (dotted line) at $\rho=0.000014$ for
$t/J=2.5$, $V_{1}=58J$, $V_{2}=49.32J$, and $V_{3}=40.6J$ in $\delta=0.15$.
Inset: the corresponding experimental result of cuprate superconductors in the
SC state taken from Ref. [14]. Figure 2: The microwave conductivity as a
function of doping with $T=0.002J=$K and $\omega=0.000087J\approx 1.81$GHz at
$\rho=0.000014$ for $t/J=2.5$, $V_{1}=58J$, $V_{2}=49.32J$, and $V_{3}=40.6J$.
The dashed line is the corresponding result of the superconducting gap
parameter.

For a better understanding of the physical properties of the microwave
conductivity $\sigma(\omega,T)$ in cuprate superconductors, we have studied
the doping evolution of the microwave conductivity, and the result of
$\sigma(\omega,T)$ as a function of doping with temperature $T=0.002J=2$K and
energy $\omega=0.000087J\approx 1.81$GHz under the slightly strong impurity
scattering potential with $V_{1}=58J$, $V_{2}=49.32J$, and $V_{3}=40.6J$ at
the impurity concentration $\rho=0.000014$ is plotted in Fig. 2 (solid line).
For comparison, the corresponding result of the SC gap parameter of cuprate
superconductors is also shown in the same figure (dashed line). Our result
shows that in contrast to the dome shape of the doping dependent SC gap
parameter huefner ; feng1 ; feng3 ; guo1 , the microwave conductivity
$\sigma(\omega,T)$ decreases with increasing doping in the underdoped regime,
and reaches a minimum in the optimal doping, then increases in the overdoped
regime. This doping dependent behavior of the low energy microwave
conductivity $\sigma(\omega,T)$ at low temperatures is also qualitatively
consistent with the universal microwave conductivity limit palee
$\sigma\propto 1/\Delta$ at low energy as temperature $T\rightarrow 0$, if
this SC gap parameter $\Delta$ in the phenomenological BCS formalism palee
has the similar dome shape doping dependence.

Figure 3: The microwave conductivity as a function of temperature with energy
$\omega\approx 1.14$GHz (solid line), $\omega\approx 2.28$GHz (long dashed
line), $\omega\approx 13.4$GHz (dash-dotted line), $\omega\approx 22.8$GHz
(dotted line), and $\omega\approx 75.3$GHz (short dashed line) at
$\rho=0.000014$ for $t/J=2.5$, $V_{1}=58J$, $V_{2}=49.32J$, and $V_{3}=40.6J$
in $\delta=0.15$. Inset: the corresponding experimental result of cuprate
superconductors in the SC state taken from Ref. [14].

In the above discussions, we mainly study the effect of the extended impurity
scatterers on the quasiparticle transport at low temperatures ($T\ll T_{c}$)
and low energies ($\omega\ll\Delta$). Now we discuss the temperature
dependence of the quasiparticle transport of cuprate superconductors, where
$T$ may approach to $T_{c}$ from low temperature side. In this case, it has
been shown nunner  that the inelastic scattering process should be considered
at higher temperatures, such as the quasiparticle-quasiparticle scattering.
This is followed from the fact that the inelastic quasiparticle-quasiparticle
scattering process is suppressed at low temperatures due to the large SC gap
parameter in the quasiparticle excitation spectrum. However, the contribution
from this inelastic quasiparticle-quasiparticle scattering process is
increased rapidly when $T$ approaches to $T_{c}$ from low temperature side,
since there is a small SC gap parameter near $T_{c}$. In particular, it has
been pointed out walker  that the contribution from the quasiparticle-
quasiparticle scattering process to the transport lifetime is exponentially
suppressed at low temperatures, and therefore the effect of this inelastic
quasiparticle-quasiparticle scattering can be considered by adding the inverse
transport lifetime duffy  $\tau^{-1}_{{\rm inel}}(T)$ to the imaginary part of
the self-energy function $\Sigma_{0}(\omega)$ in Eq. (17a), then the total
self-energy function $\Sigma^{{\rm tot}}_{0}(\omega)$ can be expressed as
walker ; nunner ,

$\displaystyle\Sigma^{{\rm tot}}_{0}(\omega)=\Sigma_{0}(\omega)-i[2\tau_{{\rm
inel}}(T)]^{-1},$ (38)

with $\tau_{{\rm inel}}(T)$ has been chosen as $[2\tau_{{\rm
inel}}(T)]^{-1}=91.35(T-0.005)^{4}J$. Using this total self-energy function
$\Sigma^{{\rm tot}}_{0}(\omega)$ to replace $\Sigma_{0}(\omega)$ in Eq. (27),
we have performed a calculation for the microwave conductivity
$\sigma(\omega,T)$, and the results of $\sigma(\omega,T)$ as a function of
temperature $T$ with energy $\omega=0.0000547J\approx 1.14$GHz (solid line),
$\omega=0.0001094J\approx 2.28$GHz (long dashed line),
$\omega=0.0006564J\approx 13.4$GHz (dash-dotted line),
$\omega=0.001094J\approx 22.8$GHz (dotted line), and $\omega=0.0036102J\approx
75.3$GHz (short dashed line) under the slightly strong impurity scattering
potential with $V_{1}=58J$, $V_{2}=49.32J$, and $V_{3}=40.6J$ at the impurity
concentration $\rho=0.000014$ for the doping concentration $\delta=0.15$ are
plotted in Fig. 3 in comparison with the corresponding experimental results of
cuprate superconductors in the SC state harris  (inset). Our results show that
the temperature dependent microwave conductivity $\sigma(\omega,T)$ increases
rapidly with increasing temperatures to a large broad peak around temperature
$T\approx 0.25J\approx 25$K for energy $\omega=0.0000547J\approx 1.14$GHz, and
then falls roughly linearly. However, this broad peak in the microwave
conductivity spectrum resulting from the temperature dependent impurity
scattering rate in Eq. (28) is energy dependent, and moves to higher
temperatures with increasing energies, in qualitative agreement with the
experimental data harris .

Within the framework of the kinetic energy driven d-wave cuprate
superconductivity feng1 , our present results of the energy and temperature
dependence of the microwave conductivity by considering the effect of the
extended impurity scatterers are qualitatively similar to the earlier attempts
to fit the experimental data by using a phenomenological d-wave BCS formalism
palee ; hirschfeld ; quinlan ; hettler ; durst ; duffy ; nunner . Establishing
this agreement is important to confirming the nature of the SC phase of
cuprate superconductors as the d-wave BCS-like SC state within the kinetic
energy driven SC mechanism. It has been shown turner ; bonn ; lee ; hosseini ;
harris  that there are some subtle differences for different families of
cuprate superconductors, and these subtle differences may be induced by the
other effects except the impurity scattering. However, we in this paper are
primarily interested in exploring the general notion of the effect of the
extended impurity scatterers on the kinetic energy driven cuprate
superconductors in the SC state. The qualitative agreement between the present
theoretical results and experimental data also show that the presence of
impurities has a crucial effect on the microwave conductivity of cuprate
superconductors.

##  IV Summary

In conclusion we have shown very clearly in this paper that if the effect of
the extended impurity scatterers is taken into account in the framework of the
kinetic energy driven d-wave superconductivity, the microwave conductivity of
the $t$-$J$ model calculated based on the nodal approximation of the
quasiparticle excitations and scattering processes per se can correctly
reproduce some main features found in the microwave conductivity measurements
on cuprate superconductor in the SC state turner ; bonn ; lee ; hosseini ;
harris , including the energy and temperature dependence of the microwave
conductivity spectrum. The theory also predicts a V-shaped doping dependent
microwave conductivity, which is in contrast with the dome shape of the doping
dependent SC gap parameter, and therefore should be verified by further
experiments.

###### 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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