Effect of nonlinearity, magnetic and nonmagnetic impurities, and spin-orbit scattering on the nonlocal microwave response of a d-wave superconductor
By using linear response theory the low-temperature microwave response of a nonlocal and nonlinear d-wave superconductor with magnetic and nonmagnetic impurities is calculated. We will show that for the local, linear, and pure sample, penetration depth, ∆λ(T), and conductivity, ∆σ1(T), vary linear...
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| Zitieren: | Effect of nonlinearity, magnetic and nonmagnetic impurities, and spin-orbit scattering on the nonlocal microwave response of a d-wave superconductor / H. Yavari, M. Biderang, M. Kouhfar // Физика низких температур. — 2016. — Т. 42, № 12. — С. 1397-1405. — Бібліогр.: 41 назв. — англ. |
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irk-123456789-1293342018-01-19T03:04:14Z Effect of nonlinearity, magnetic and nonmagnetic impurities, and spin-orbit scattering on the nonlocal microwave response of a d-wave superconductor Yavari, H. Biderang, M. Kouhfar, M. Свеpхпpоводимость, в том числе высокотемпеpатуpная By using linear response theory the low-temperature microwave response of a nonlocal and nonlinear d-wave superconductor with magnetic and nonmagnetic impurities is calculated. We will show that for the local, linear, and pure sample, penetration depth, ∆λ(T), and conductivity, ∆σ1(T), vary linearly with temperature, consequently the resistance, ∆R(T), would change linearly with temperature in agreement with experimental results and for the nonlocal, nonlinear sample the linear temperature dependences ∆R(T) change to quadratic function. For impure samples the nonlocality and nonlinearity effects are completely hidden by impurities and the temperature dependences ∆λ(T) and ∆σ1(T) are determined by temperature interval namely the ranges of T < T* and T* << T << Tc which T* is determined by nonmagnetic impurity concentration and the strength of impurity scattering. For T < T* , ∆R(T) varies as T2, on the other hand for, T* << T << Tc, ∆R(T) varies linearly with temperature. We will also show that the temperature dependence of surface resistance is unaffected by spin-orbit interaction and magnetic impurities. 2016 Article Effect of nonlinearity, magnetic and nonmagnetic impurities, and spin-orbit scattering on the nonlocal microwave response of a d-wave superconductor / H. Yavari, M. Biderang, M. Kouhfar // Физика низких температур. — 2016. — Т. 42, № 12. — С. 1397-1405. — Бібліогр.: 41 назв. — англ. 0132-6414 PACS: 85.25.–j, 74.25.nn, 74.25.Ha http://dspace.nbuv.gov.ua/handle/123456789/129334 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| topic |
Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная |
| spellingShingle |
Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная Yavari, H. Biderang, M. Kouhfar, M. Effect of nonlinearity, magnetic and nonmagnetic impurities, and spin-orbit scattering on the nonlocal microwave response of a d-wave superconductor Физика низких температур |
| description |
By using linear response theory the low-temperature microwave response of a nonlocal and nonlinear d-wave
superconductor with magnetic and nonmagnetic impurities is calculated. We will show that for the local, linear, and
pure sample, penetration depth, ∆λ(T), and conductivity, ∆σ1(T), vary linearly with temperature, consequently the
resistance, ∆R(T), would change linearly with temperature in agreement with experimental results and for the nonlocal,
nonlinear sample the linear temperature dependences ∆R(T) change to quadratic function. For impure samples
the nonlocality and nonlinearity effects are completely hidden by impurities and the temperature dependences ∆λ(T)
and ∆σ1(T) are determined by temperature interval namely the ranges of T < T* and T* << T << Tc which T* is determined
by nonmagnetic impurity concentration and the strength of impurity scattering. For T < T*
, ∆R(T) varies as T2, on the other hand for, T* << T << Tc, ∆R(T) varies linearly with temperature. We will also show that the temperature
dependence of surface resistance is unaffected by spin-orbit interaction and magnetic impurities. |
| format |
Article |
| author |
Yavari, H. Biderang, M. Kouhfar, M. |
| author_facet |
Yavari, H. Biderang, M. Kouhfar, M. |
| author_sort |
Yavari, H. |
| title |
Effect of nonlinearity, magnetic and nonmagnetic impurities, and spin-orbit scattering on the nonlocal microwave response of a d-wave superconductor |
| title_short |
Effect of nonlinearity, magnetic and nonmagnetic impurities, and spin-orbit scattering on the nonlocal microwave response of a d-wave superconductor |
| title_full |
Effect of nonlinearity, magnetic and nonmagnetic impurities, and spin-orbit scattering on the nonlocal microwave response of a d-wave superconductor |
| title_fullStr |
Effect of nonlinearity, magnetic and nonmagnetic impurities, and spin-orbit scattering on the nonlocal microwave response of a d-wave superconductor |
| title_full_unstemmed |
Effect of nonlinearity, magnetic and nonmagnetic impurities, and spin-orbit scattering on the nonlocal microwave response of a d-wave superconductor |
| title_sort |
effect of nonlinearity, magnetic and nonmagnetic impurities, and spin-orbit scattering on the nonlocal microwave response of a d-wave superconductor |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2016 |
| topic_facet |
Свеpхпpоводимость, в том числе высокотемпеpатуpная |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/129334 |
| citation_txt |
Effect of nonlinearity, magnetic and nonmagnetic impurities, and spin-orbit scattering on the nonlocal microwave response of a d-wave superconductor / H. Yavari, M. Biderang, M. Kouhfar // Физика низких температур. — 2016. — Т. 42, № 12. — С. 1397-1405. — Бібліогр.: 41 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT yavarih effectofnonlinearitymagneticandnonmagneticimpuritiesandspinorbitscatteringonthenonlocalmicrowaveresponseofadwavesuperconductor AT biderangm effectofnonlinearitymagneticandnonmagneticimpuritiesandspinorbitscatteringonthenonlocalmicrowaveresponseofadwavesuperconductor AT kouhfarm effectofnonlinearitymagneticandnonmagneticimpuritiesandspinorbitscatteringonthenonlocalmicrowaveresponseofadwavesuperconductor |
| first_indexed |
2025-07-09T11:09:30Z |
| last_indexed |
2025-07-09T11:09:30Z |
| _version_ |
1837167412420542464 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12, pp. 1397–1405
Effect of nonlinearity, magnetic and nonmagnetic impurities,
and spin-orbit scattering on the nonlocal microwave response
of a d-wave superconductor
H. Yavari, M. Biderang, and M. Kouhfar
Department of Physics, University of Isfahan, Isfahan 81744, Iran
E-mail: h.yavary@sci.ui.ac.ir
Received October 23, 2015, revised June 25, 2016, published online October 24, 2016
By using linear response theory the low-temperature microwave response of a nonlocal and nonlinear d-wave
superconductor with magnetic and nonmagnetic impurities is calculated. We will show that for the local, linear, and
pure sample, penetration depth, ∆λ(T), and conductivity, ∆σ1(T), vary linearly with temperature, consequently the
resistance, ∆R(T), would change linearly with temperature in agreement with experimental results and for the non-
local, nonlinear sample the linear temperature dependences ∆R(T) change to quadratic function. For impure samples
the nonlocality and nonlinearity effects are completely hidden by impurities and the temperature dependences ∆λ(T)
and ∆σ1(T) are determined by temperature interval namely the ranges of T < T* and T* << T << Tc which T* is de-
termined by nonmagnetic impurity concentration and the strength of impurity scattering. For T < T*, ∆R(T) varies as
T2, on the other hand for, T* << T << Tc, ∆R(T) varies linearly with temperature. We will also show that the temper-
ature dependence of surface resistance is unaffected by spin-orbit interaction and magnetic impurities.
PACS: 85.25.–j Superconducting devices;
74.25.nn Surface impedance;
74.25.Ha Magnetic properties including vortex structures and related phenomena.
Keywords: surface impedance, d-wave superconductor, linear response, impurities.
1. Introduction
The origins of the growing interest in the study of high-
temperature superconducting (HTSC) microwave resona-
tors to many scientific communities mainly are their ap-
plicability to a wide range of devices and providing a bet-
ter understanding of the physics of these materials.
Superconducting resonators have many important applica-
tions such as photon detection, SQUID multiplexed read
out for astronomy [1,2], quantum computation [3–6] and
nanomechanical resonator read out for motion sensing [7].
Today the absorption and noise due to tunneling two-level
systems (TLS) [8–11] which are often found at surfaces or
in dielectric materials [1,12–14] and tunneling barriers [15]
are important problems in improving the performance of
these devices. Despite the recent experimental surge inves-
tigating the nature of excess noise in these resonators, a
full theoretical characterization of noise due to these TLS
defects still remains largely a mystery and a deeper under-
standing is extremely needed.
In developing passive superconducting devices theoret-
ical and experimental analysis are very important for gen-
erating sufficient design data or for devising good design
models. The high-temperature superconductors are attrac-
tive for use in microwave circuits because of their low sur-
face resistance as compared to those of normal metals [16].
The key development of quality microwave resonators
filters and delay lines is the low loss HTSC epitaxial thin
films on appropriate dielectric substrates [17–19]. To ac-
cess the ideal optimum device performance, a knowledge
of absolute values of the specimen's microwave surface
resistance R and penetration depth λ is necessary. Simul-
taneously valuable information about intrinsic and extrin-
sic properties of the superconductor can be deduced from
these quantities [20]. Another important issue is the estab-
lishment of a standard characterization technique for
HTSC thin films for microwave applications [21].
Measurements of the temperature behavior of the sur-
face impedance Z(T) = R(T) + iX(T) of HTSC, give infor-
mation about the nature of quasiparticles in the supercon-
ducting state, their scattering, density of states, and about
the superconducting pairing mechanism in these materials.
The real part of the surface impedance, i.e., the surface
© H. Yavari, M. Biderang, and M. Kouhfar, 2016
H. Yavari, M. Biderang, and M. Kouhfar
resistance R, arise due to normal carriers is proportional to
the loss of the microwave power. The imaginary part of the
surface impedance, i.e., the surface reactance X, is largely
determined by the response of superconducting carriers
and characterizes the nondissipating energy stored in the
superconductor surface layer.
Since low-temperatures surface resistance of s-wave
superconductors at low-frequency involves mechanisms of
very weak dissipation instrumental for many applications
such as decoherence in superconducting qubits [22], it has
recently attracted much attention. For example, niobium
superconducting radio-frequency (SRF) cavities for parti-
cle accelerators exhibit fantastic high-quality factors due to
a very low averaged surface resistance.
By developing a fully microscopic theory, the surface
impedance of s-wave superconductors with magnetic im-
purities was investigated by M. Kharitonov et al. [23].
They explicitly demonstrated that, in the regime of gapless
superconductivity, the system exhibits saturation of the
surface resistance at zero temperature.
Nonlocal effect on the low-temperature surface re-
sistance of d-wave superconductors was calculated as a
function of frequency assuming normal state quasiparticle
mean free paths in excess of the penetration depth [24]. It
was shown that the nonlocal effects can be observed in the
surface resistance of high-temperature superconductors if
the order parameter has nodes.
The effect of magnetic impurities on the surface imped-
ance of superconducting multilayer and unconventional su-
perconductors has been investigated experimentally [25–27].
Using a simple two-fluid model the results on impedance
[25–28] were analyzed theoretically. This model consider a
simplified frequency-independent Drude-type dissipative
conductivity σ, while the influence of magnetic impurities
on the electron spectrum should lead to a nontrivial σ(ω)
dependence on the frequency ω.
The effects of both nonmagnetic and magnetic impuri-
ties on the complex conductivity of a superconductor was
studied by Skalski et al. [29]. Their general results can be
used for weak magnetic scattering; however, equivalently
to the Abrikosov and Gor’kov theory, scattering off mag-
netic impurities was considered in the Born limit, which
did not allow to treat the effect of localized states on the
impedance.
By employing the quasiclassical approach, the effects
of frequency, the density and strength of magnetic impuri-
ties, and the density and temperature of quasiparticles on
the dissipative part of the surface impedance for conven-
tional superconductors have been studied [30]. In the limit
of small temperatures and small frequencies they have
shown that at equilibrium, the dissipation is always propor-
tional to the density of thermally excited quasiparticles and
thus exponentially suppressed due to gapped character of
the spectrum [30].
To the best of our knowledge, all of the pervious theo-
retical researches did not take in to account the effects of
nonlinearity, spin orbit and magnetic impurities on the sur-
face impedance of a d-wave superconductor. In this paper
by using linear response theory we will determine the low-
temperature behavior of the real part of the conductivity
and consequently surface impedance of a nonlocal and
nonlinear d-wave superconductor in the presence of mag-
netic and nonmagnetic impurities.
2. Formalism
In the Bardeen–Cooper–Schrieffer (BCS) formalism the
Hamiltonian is (throughout the paper we use the units
1Bk= = )
( ) ( )† † †
,
k kk k kkk
k k
H k c c c c c cαα − ↓ ↑− ↓↑
α
= ε − ∆ + +∑ ∑
( ) †
imp( ) e ii
kk
kk i
U c c− − ′
αβ β′α
αβ′
+ +∑ k k R
( ) ( ) †
2 e ,iiso
kk
kk iF
v
i c c
k
− − ′
αβ β′α
αβ′
+ × ′ ∑ k k R k k σ (1)
where ( )kε is the energy spectrum of an electron in the
normal state, †
kc α ( )kc α is the creation (distraction) opera-
tor, cos (2 )k∆ = ∆ ϕ is the gap for a d-wave superconduc-
tor 2 2(
x y
d
−
symmetry), imp( )U αβ is the potential due to
nonmagnetic and magnetic impurities (with randomly ori-
ented magnetic impurities which does not preserves the
electron spin), and sov refer to spin-orbit coupling to the
impurities ( Fk is the Fermi wave vector).
The general expression for the surface impedance of a
uniformly disordered system reads [31,32]
2
2
2
8
4 ( , , , , )s
i dqZ
c T T
q
c
∗
ω
= −
π ω
+
∫ q νΚ
. (2)
The response function ( , , , , )s T T∗ω q νΚ defines the rela-
tion
1( , ) ( , , , , ) ,sq T T q
c
∗ω = − ω ωJ q A( )νΚ (3)
in the Fourier representation between the electric current
( , )q ωJ and the electro-magnetic field, described by the
vector potential , .q ωA( ) The ohmic dissipation is deter-
mined by the imaginary part of the response function.
Given the current, we proceed to calculate the corre-
sponding current-current correlation function. This correla-
tion function can be expressed diagrammatically as a
fermionic bubble with fully dressed propagators and a fully
dressed vertex. Assuming that the impurity scattering po-
tential is isotropic in k space, the correction to the vertex
1398 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12
Effect of nonlinearity, impurities and spin-orbit scattering on the microwave response of a d-wave superconductor
vanishes. Thus, in this approximation, the correlation func-
tion can be obtained from the calculation of bubble with
dressed propagators (i.e., Green's functions with self-
energy included, but have vertices or bare bubble dia-
gram). With this replacement the response function be-
comes
( ), , , ,s mT T∗ ω =q νΚ
2
2
11
,
2 ˆ1 [ ( , ) ( , )]n n m
n k
ne T k k k
mc m + −
π
= − + ℑ ω ℑ ω + ω
∑ ,
(4)
where /2,k k q± = ± (2 1)n n Tω = + π is the Matsubara
fermionic frequencies, 11k̂ is the direction of the supercur-
rent and . . .〈 〉 represents a Fermi surface average.
In the presence of impurities, the bare Green's function
is dressed via scattering from the impurities and obtains a
Matsubara self-energy imp ( )iΣ ω due to impurities. Thus
the dressed Matsubara Green's function is given by (clearly
the ( , )n mkℑ ω + ω is obtained from ( , )nkℑ ω with replac-
ing niω by )n mi iω + ω
( )
( )2 2
imp
1, n
n s F k
i
i E
ℑ ω = ×
ω − ∑ + −
k
kν
imp
imp
n s F k k
k n s F k
i
i
ω − ∑ + + ξ ∆
× ∆ ω − ∑ + + ξ
k
k
ν
ν
, (5)
where k kξ = ε − µ is the normal state quasiparticle energy
and 2 2 2 .k k kE = ξ + ∆ All quasiparticles Matsubara energies
modified by the semiclassical Doppler shift .s F⋅kν The
basic idea is that, like impurity effects, the magnetic field
itself may serve as a Cooper pair breaking that creates
nodal quasiparticles, leading to a quadratic temperature-
dependent penetration depth at temperature 2( ( ) )T T∆λ ∝
below the scale for nonlinear electrodynamics
nonlin s FE v k≈ with a typical supercurrent velocity sν
and the Fermi wave vector Fk .
We consider an impurity potential combining the non-
magnetic and the magnetic scattering,
( ) ( ) ( )imp nonmag magU U U− = − + −′ ′ ′k k k k k k , (6)
where the nonmagnetic potential is often assumed to be
completely local nonmag 0 0( ) ( )U U− = δ −′r r r r with the
impurity at 0r and mag ( )U J= ,r Sα with S is the spin
operator of the conduction electron; 0 0( ) ( )J J= δ −r r r is
the exchange interaction between the local spin on the
impurity site and the conduction electrons, =α
3 3 3 3[(1 ) (1 ) ]/2,= + τ + − τ σ σσ σ iσ are the Pauli matrices
acting in spin space, 3τ are the Pauli matrices in the par-
ticle-hole space, and i iτ σ denotes a direct product of the
matrices operating in the 4-dimensional Nambu space.
We assume that the impurity spins are oriented arbitrari-
ly. Difference of the random-spin case from the spin-
polarized case is the absence of the average Zeeman en-
ergy. Thus the presence of magnetic impurities with ran-
domly oriented can result in the spin-flip scattering.
The self-energy in the second order (Born approximation)
can be written as
( )
( )
( ) ( ) ( )
3
imp imp 3, ,
2
dn U U′∑ ω = − ℑ ω −′ ′ ′
π
∫
kk k k k k k ,
(7)
where
( ) [ ]imp 2
ˆ, so
F
v
U U i z
k
= + ×′ ′k k k k , (8)
here the spin-orbit coupling to the impurities is described
as impsov U g= ∆ with g∆ is the shift of the g factor
which, for cuprates, is of order 0.1.
Using Eq. (8) into Eq. (7) we obtain
3
imp imp imp imp3( , ) ( ) ( , ) ( )
(2 )
dn U U′∑ ω = − ℑ ω − +′ ′ ′
π∫
kk k k k k k
2
2
imp 4 ( , )so
kF
v
n
k ′
+ × ℑ ω′ ′∑ k k k . (9)
The impedance can be written as ( ) ( ) ( ),Z T R T i T= + Χ
where R and X are the surface resistance and reactance, re-
spectively [33]. The complex conductivity can be written as
( ) ( )1 2, , , ,iq T i q Tσ ω = σ − σ = ω
ω
Κ . (10)
The dissipation is determined by the real part of conductivity
1( , , , ) Re ( , , , )s sT T T T∗ ∗σ = σ =q qν ν
1 Im ( , , , )s T T∗= −
ω
q νΚ . (11)
The imaginary part of the conductivity which is deter-
mined by the real part of the response function
(Re ( , , , )),s T T∗q νΚ in terms of a frequency and tempera-
ture-dependent penetration depth can be written as
2
2 2( , , , )
4 ( , , , )
s
s
cT T
T T
∗
∗σ =
πωλ
q
q
ν
ν
. (12)
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12 1399
H. Yavari, M. Biderang, and M. Kouhfar
For superconductors 2 1( ),σ >> σ we have
2 2 3
14
8R
c
≈ π ω λ σ , (13)
2
4
c
π
χ ≈ ωλ . (14)
According to Eq. (13), to determine the effects of
nonlocality, nonlinearity, impurity and spin-orbit scattering
on the surface resistance we need to calculate the penetra-
tion depth and the real part of the conductivity. Following
the method of Ref. 34, first we evaluate the temperature
behavior of magnetic penetration depth.
By using Eq. (5) into Eq. (4) we have
____________________________________________________
( )
( )
2 2
2
2
2
imp
2 2
2 2
imp /2 imp /2
2 ˆ( , , , ) 1
2
2 2
s
n
n F s
n s k q n s k q
ne T d kT T k
mc m
i
i E i E
∗
+ − + −
+ −
π
= + × π
ω − ∑ + + ξ ξ + ∆ ∆
×
ω − ∑ + + − ω − ∑ + − −
∑∫q
k
q qk k
ν
ν
ν ν
Κ
, (15)
where /2 .k k q± × ±ξ = ξ = ε − µ
In the absence of magnetic impurities and spin-orbit interaction which are included in the self-energy, Eq. (15) reduces
to Eq. (6) of Ref. 34.
We separate out the 0T = local, linear, and pure response as
( , , , ) 0,0,0,0 ( , , , )s sT T T T∗ ∗= ( ) + δq qν νΚ Κ Κ , (16)
where 2 2
00,0,0,0 /(4 )c( ) = πλΚ .
By summing over the Matsubara frequencies in Eq. (15) we get [34]
( ) ( )
( ) ( )
22
2
22 20 2 2
imp imp
1
2
2
2
2
2
2
2 ˆ( , , , ) Re
2
sinh
2ˆ1 2
1
2 2
1
2
ˆ Ln
s F s F k
s
F
k k
F
k
F F
k k
F
k
f fneT T k d
mc
i i
m
mne k
mc
m m
m
k
∞
+ −∗
−
ε − − −ε − ∆ δ = − ε =
ω − ∑ − ∆ ∆ + − ω − ∑
∆ = − − +
+ ∆ ∆
∆ + −
+
∫
k k
q
qk
qk
qk qk
qk
ν ν
νΚ
( ) ( )
2 2
2 2
imp imp
2 22
2 2 2
imp imp
2
1
2 2 2
1 1
2 2 2 2
ˆ2 Re
F F F
k
k k k
F F F F
k k
k k k
s F s F
m m m
m m m m
f f
k d + −
+ ∑ − + ∑ −∆ ∆ ∆ ∆ +
∆ + + + ∑ − + ∑ −∆ ∆ ∆ ∆
ε − + ε + + ε
ω −
qk qk qk
qk qk qk qk
k k
ν ν
( ) ( )
22 20 2 2
imp imp
,
2
F
k ki i
m
∞
∑ − ∆ ∆ + − ω − ∑
∫
qk (17)
where /2.±ε = ε ± ω
1400 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12
Effect of nonlinearity, impurities and spin-orbit scattering on the microwave response of a d-wave superconductor
The first two terms in Eq. (17) represent the nonlocal
correction to the London penetration depth and the third
represents the nonlocal and impure renormalization of the
response while the forth combined nonlocal, nonlinear, and
impure corrections to the temperature dependence. Note
that the third and fourth terms also include the effect of
spin-orbit correction.
As mentioned in Ref. 34, the resulting response func-
tion includes nonlinearity, nonlocality, and impurity, from
which a three parameter scaling function of the penetration
depth is obtained. The well-known linear T-dependence of
the penetration depth in the pure system should be ex-
pected at (T >> Enonloc, Enonlin). But in the opposite limit,
i.e., at extremely low temperature, either nonlinear or non-
local effects play a crucial role in modifying the linear
temperature behavior to quadratic behavior.
For the impure sample, as shown in Ref. 35 in the Born
limit, the contribution of spin-orbit scattering is 2g∆ times
smaller than that of nonmagnetic impurity scattering, while
in the unitarity limit (strong scattering), the spin-orbit scat-
tering leads to a renormalization of the same order of the
impurity scattering. Thus, it is necessary to treat the impu-
rity and spin-orbit interactions on the same level by gener-
alizing the usual t-matrix approach for the impurity scatter-
ing also to the spin-orbit contribution. To characterize the
strength of the impurity scattering for the nonmagnetic and
the magnetic components we introduce 01/( )n nC N U= π
and 01/( ).( 1)mC N J s s= π +
Following the notation of Ref. 35 for arbitrary strength
of the scattering potential the renormalized frequency and
gap equations, respectively, become
0
2 2 2
0
n
n n
n m n
g
i i
C C g
ω = ω + Γ +
+ −
2 2
2 2
2
22 2
2 2
2
( )
( ) 4
n m
n n
n
n m
n n n n
C C
f g
g
g
C C
f g f g
g
+
+ −
∆
+ Γ
+
+ − −
∆
, (18)
2 2
2 2
2
22 2
2 2
2
( )
( ) 4
n m
n n
n
n m
n n n n
C C
f g
g
f
C C
f g f g
g
+
− −
∆
∆ = ∆ + Γ
+
− − −
∆
, (19)
2 2
22 2
2 2
2
2
( ) 4
n m
n n
n
n m
n n n n
C C
f g
g
f
C C
f g f g
g
+
∆
Ω = − Γ
+
− − −
∆
, (20)
where
2 2
0
,
1 1
i
n m
n m
n
NC C
Γ Γ
Γ = Γ + Γ = + Γ = π + +
2
2 2 2
sin ,n
n
f ∆ ϕ
=
∆ + Ω + ω
2
2 2 2
sin
,n
n
n
i
g
ω ϕ
=
∆ + Ω + ω
0 2 2 2
n
n
n
i
g
ω
=
∆ + Ω + ω
. (21)
By considering the self-consistent equation for the order
parameter (Eq. (19)) the following expression is obtained
for the critical temperature
0
1 1Ln
2 2 2
c n n so
c c
T
T T
Γ + Γ + Γ = Ψ − Ψ + π
, (22)
where
2
2 2 2
2 2
3(2 / ) 1 ( )
[(2 / ) 1]
so n m
C g C C C
C g
∆ +
Γ = Γ = +
∆ +
.
In the absence of magnetic impurities Eq. (22) reduces
to Eq. (20) of Ref. 35.
In the Born limit ( , 1)n mC C >> Eq. (22) becomes
0
1 1 1 1 1 3Ln
2 2 2 2 2 4
c
c c n m so
T
T T
= Ψ − Ψ + + + π τ τ τ
,
(23)
here
( )
2
22
0 0
1 1 1, 1 , and
2 2n n m
n m so
gn N U n N s s J
C
∆ = π = π + = Γ τ τ τ
(not that, nτ and mτ are spin-independent and spin-depen-
dent relaxation times due to nonmagnetic and magnetic im-
purities, respectively).
By solving the gap equation in the Ginzburg–Landau
region, the transition temperature in the presence of mag-
netic and nonmagnetic impurities was also computed in
Ref. 36 and the obtained result in the Born limit (Eq. (31))
is the same as Eq. (23) of our paper in the absence of spin-
orbit coupling ( ).soτ → ∞ On the other hand, Eq. (23)
without magnetic impurities ( )mτ → ∞ reduces to Eq. (31)
of Ref. 37.
In the unitarity limit ( , 1)n mC C << Eq. (23) becomes
0
1 1 1 1 1 1Ln ,
2 2 2 2 2 2
c
c c n m so
T
T T
= Ψ − Ψ + + + π τ τ τ
(24)
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12 1401
H. Yavari, M. Biderang, and M. Kouhfar
0
1 1 1where
2 2 2
i
n m so
n
N
= = =
τ τ τ π
.
In the unitarity limit, the superconductivity is destroyed for
the critical value 00.88n m so cT∗ ∗ ∗ ∗Γ = Γ + Γ + Γ = , i.e., to
destroy the superconductivity one needs the impurity con-
centration about the third value of the impurity concentra-
tion in the unitarity limit without magnetic impurities and
spin-orbit scattering.
As can be seen from Eqs. (23) and (24), the spin-orbit
scattering rate negligible with respect to magnetic and
nonmagnetic impurity scattering rates in the Born limit,
while close to the unitarity limit it has the same order of
magnitude than these relaxation rates and cannot be ne-
glected.
Now the kernel ( , , , )s T T∗δ q νΚ can be written as
____________________________________________________
( ) ( ) ( )
2
2
11 22 2 22 2 2 2
ˆ( , , , ) 2 k
s
n n F s k n F s k F
T T T d k
i i
∗ ∆
δ = π ϕ
ω + + Ω + ∆ ω + + ∆ + Ω +
∑∫q
k k q
ν
ν ν ν
Κ . (25)
_______________________________________________
In the impurity dominated gapless regime, the renor-
malized frequency ω takes the limiting form ,iω → ω + γ
where imp (0).γ = ∑ For dilute impurity concentration
( 0)γ → the gapless behavior restricted to a vanishingly
small range below T T ∗< ≈ γ and in this range of temper-
ature we can replace the normalized frequency everywhere
by its low-frequency limiting form and after integration
over frequency we find 2( , , ) .T T T∗δ ∝ ∆λ ∝qΚ
In the impure sample below the crossover temperatures,
,T T ∗< ≈ γ the impurity effect determines the temperature
dependence of the penetration depth, and nonlocality and
nonlinearity behavior should be hidden by the impurity
effects.
To calculate the real part of the conductivity we need to
determine the imaginary part of the response function
( ).ni iω = ε + δ From Eq. (15) we obtain
____________________________________________________
( ) ( )
( )( ) ( ) ( )
( )( ) ( ) ( )
( )( )
2
2
11
2 2 2
2 22 22 2 2
ˆIm ( , , , )
2
Im
s F s F s
F s F s F s F kF s
k F F s k kF s F s
F s F s
ne dT T k d f f
mc
i i i i
ii i
i i
∗ ϕ = ε ε + ω − − ε − × π
ε − + δ ε − + δ + ε − + δ − − ∆′ ε − + δ× + − ∆ − − ε − + δ + ∆ − ∆′ε − + δ ε − + δ
ε − + δ ε − + δ +′
+
∫ ∫q k k
k k k q k
q kk k
k k
ν ν ν
ν ν ν ν ν
ν νν ν
ν ν
Κ
( ) ( )
( )( ) ( ) ( )
( )( ) ( ) ( )
( )( ) ( ) ( )
2 2 2
2 22 22 2 2
2 2 2
2 22 22 2
F s F kF s
k F F s k kF s F s
F s F s F s F kF s
k F F s kF s F s
i i
ii i
i i i q i
ii i
ε − + δ + − ∆′ ε − + δ′
−
− ∆ + − ε − + δ + ∆ − ∆ε − + δ ε − + δ′ ′
ε − + δ ε − − δ + ε − + δ − − ∆′ ε − + δ
−
− ∆ − − ε − − δ + ∆ −′ε − + δ ε − + δ
k q k
q kk k
k k k k
q kk k
ν ν ν
ν νν ν
ν ν ν ν ν
ν νν ν
( )( ) ( ) ( )
( )( ) ( ) ( )
2
2 2 2
2 22 22 2 2
.
,
k
F s F s F s F kF s
k F F s k kF s F s
i i i i
ii i
−
∆
ε − + δ ε − − δ + ε − − δ − − ∆′ ′ ε − − δ′ − − ∆ + − ε − + δ + ∆ − ∆ε − − δ ε − − δ′ ′
k k k q k
q kk k
ν ν ν ν ν
ν νν ν
(26)
_______________________________________________
where ε = ε − ω′ and imp ( ).ε = ε − Σ ε
In the absence of magnetic impurities ( ),mτ → ∞ spin-
orbit coupling ( 0)sov → and nonlinear effect ( 0),sv →
Eq. (26) reduces to Eq. (6) of Ref. 38. In the local limit
they obtained the Drude-like form of the conductivity of a
d-wave superconductor in the range of temperature
.cT T T∗ << << In this limit the quasiparticle density var-
ies linearly with temperature, and if the average lifetime
were constant, 1σ would vary linearly with temperature.
For ,T T ∗< they found that the real part of the conductivi-
ty varies as 2T in the unitarity limit.
The real part of the conductivity in the microwave re-
gime can be written as
1402 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12
Effect of nonlinearity, impurities and spin-orbit scattering on the microwave response of a d-wave superconductor
( ) ( )
( ) ( )
( )( ) ( ) ( )
( )
2
2
1
2 2 2
22 2 22 2 2
2
Im , , ˆlim, , , 0 0 2
2
Im
2
F s
s
F s F F s k
F s k F F s k F s k
F s F F s
fT T, ne dT T, k d
mc
i i
i i i
i i
∗
∗ ∂ ε −( ω) ϕ
σ ω → = − = − ε × ω → ω π ∂ε
ε − + δ − ε − + δ − ∆× + ε − + δ − ∆ − − ε − + δ + ∆ ε − + δ − ∆
ε − + δ + ε − +
+
∫ ∫
kqq
k q k
k q k k
k q k
ν
ν
ν ν ν
ν ν ν ν
ν ν ν
Κ
( )
( )( ) ( ) ( )
( )( ) ( ) ( )
( )( ) ( ) ( )
( )( )
2 2
22 2 22 2 2
2 2 2
22 2 22 2 2
k
F s k F F s k F s k
F s F s F s F F s k
F s k F F s k F s k
F s F s
i i i
i i i i
i i i
i i
δ − ∆
−
ε − + δ − ∆ + − ε − + δ + ∆ ε − + δ − ∆
ε − + δ ε − − δ + ε − + δ − ε − + δ − ∆
− −
ε − + δ − ∆ − − ε − − δ + ∆ ε − + δ − ∆
ε − + δ ε − − δ + ε −
−
k q k k
k k k q k
k q k k
k k k
ν ν ν ν
ν ν ν ν ν
ν ν ν ν
ν ν ( ) ( )
( )( ) ( ) ( )
2 2 2
22 2 22 2 2
.F s F F s k
F s k F F s k F s k
i i
i i i
− δ − ε − − δ − ∆
ε − − δ − ∆ + − ε − + δ + ∆ ε − − δ − ∆
q k
k q k k
ν ν ν
ν ν ν ν
(27)
_______________________________________________
Equation (27) is a general expression which contains all
effect (temperature, nonlocality, nonlinearity, impurity,
and spin-orbit coupling) on the real part of the conductivity
and may be evaluated numerically. Here we focus on the
low-temperature behavior of surface resistance which can
be derived an approximated result.
In the normal state ( 0),k∆ = from Eq. (27) the normal
state conductivity
2
1
3 1
4 F
ne
mc qv
π
σ =
can be obtained.
According to Eq. (13) we can write
( ) ( ) ( )1
1
3T T
R T
∆σ ∆λ
∆ ∝ + σ λ
. (28)
For the local, linear, and pure sample, ( )T∆λ and
1( )T∆σ vary linearly with temperature, consequently
( )R T∆ would change linearly with temperature in agree-
ment with experimental results for high quality sample [39].
For the nonlocal, nonlinear, and pure sample, according
to Eqs. (19) and (20), 2( )T T∆λ ∝ and 2
1( ) ,T T∆σ ∝
consequently the linear temperature dependence of ( )R T∆
change to quadratic function.
For impure samples without magnetic impurities, as we
mentioned before the nonlocal and nonlinear effects are
completely hidden by impurities and the temperature de-
pendences ( )T∆λ and 1 ( )T∆σ are determined by temper-
ature interval namely the ranges of T T ∗< and
.cT T T∗ << << The energy range between zero and the
gap edge may be partitioned crudely into two regimes,
separated by a crossover energy or temperature ,T ∗ de-
pendent on the nonmagnetic impurity concentration and
phase shift. It can be shown that T ∗ to be of order of
imp (0).γ = ∑ Since in the Born limit the effect of spin-
orbit interaction is small and in the unitarity limit it has a
considerable effect, the crossover temperature in this two
limits, respectively, is n iT n• ≈ γ ≈ Γ and
( )/ .n so iT n• ≈ γ ≈ Γ + Γ For T T ∗< both ( )T∆λ and
1( )T∆σ vary as 2 ,T on the other hand, for cT T T∗ << <<
both ( )T∆λ and 1( )T∆σ vary linearly with temperature.
In Fig. 1 it is shown the real part of the conductivity as a
function of temperature at several different relaxation times
(the imaginary part of the self-energy in Eq. (27)
1
imp ( )n i−τ ∝ Σ ω imp( ( )))ε = ε − Σ ε due to scattering by non-
magnetic impurities. As can be seen from Fig. 1, the 2T
behavior of 1σ below the crossover temperature 0.34 cT T∗ ≈
is changed to the linear above T ∗ in agreement with previ-
Fig. 1. (Color online) Real part of the complex conductivity
σ1(T), normalized to the Drude conductivity vs reduced tempera-
ture T/Tc for various impurity concentrations.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12 1403
H. Yavari, M. Biderang, and M. Kouhfar
ous results [38,40]. In Fig. 2 we also present the real part of
the conductivity as a function of impurity relaxation time for
different values of temperature.
It is recognized that the quadratic dependence ( )R T∆
is largely due to the presence of defects in samples (extrin-
sic origin), and the linear temperature dependences ( )R T∆
is due to intrinsic microscopic properties of unconventional
superconductors. This conjecture was confirmed by sys-
tematic research of YBCO thin films [41] as their quality
improved, the quadratic dependence in the low-tem-
perature range was replaced by a linear function.
Magnetic impurities do not show the same scheme of
resonant scattering that seems to give a reasonable descrip-
tion of the nonmagnetic impurity effects. The magnetic
impurity concentration which needs to suppresses cT
about the nonmagnetic impurities is nearly two times and
provides at least as much scattering as the nonmagnetic
impurities, but does not produce any quadratic, gapless
behavior in either ( )T∆λ or 1 ( ).T∆σ It is attractive to
suggest that magnetic impurities might be a nonresonant
scatterer and that results for the Born limit rather than the
unitary limit are more appropriate. In the Born limit
0 /
0e ,NT −∆ Γ• ≈ γ ≈ ∆ where 0∆ is the gap maximum
over the Fermi surface and 2/(1 )N m mCΓ = Γ + gives a
much lower crossover temperature for a given scattering
rate NΓ than does the resonant scattering result.
In the presence of magnetic impurities the gapless super-
conductor exhibits saturation of the surface resistance at zero
temperature and at low temperatures the linear temperature
dependence in the magnetic impurity-doped sample does not
change which is in agreement with experiment results [35].
3. Conclusions
The effect of magnetic and nonmagnetic impurities and
spin-orbit scattering on the low-temperature surface im-
pedance of a nonlocal and nonlinear d-wave superconduc-
tor in the framework of linear response theory was investi-
gated. At first, the general expressions for penetration
depth and the real part of the conductivity were obtained.
For arbitrary strength of the scattering potentials the
renormalized frequency and gap equations were derived.
For the impure sample, in the Born limit, the contribution
of spin-orbit scattering is smaller than that of nonmagnetic
impurity scattering, while in the unitarity limit (strong scat-
tering), the spin-orbit scattering leads to a renormalization
of the same order of the impurity scattering. The magnetic
impurity concentration which needs to suppresses cT
about the nonmagnetic impurities is nearly two times and
provides at least as much scattering as the nonmagnetic
impurities, but does not produce any quadratic, gapless
behavior in either ( )T∆λ or 1 ( ).T∆σ
For local, linear, and pure sample the surface resistance
change linearly with temperature which is consist with the
experimental result for high quality sample. For impure
sample the nonlocal and nonlinear effects are completely
hidden by impurities and surface resistance varies as 2T
below the crossover temperature ( )T • which is deter-
mined by the impurity concentration and phase shift.
Our results show that the quadratic dependence of
( )R T∆ is due to nonlocality, nonlinearity and impurity
effects (extrinsic origin), and the linear temperature de-
pendence ( )R T∆ is due to intrinsic microscopic properties
of unconventional superconductors.
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12 1405
1. Introduction
2. Formalism
3. Conclusions
|