Optical conductivity in superconductors with mixed symmetry order parameters
The real and imaginary part of the optical conductivity and the derived optical scattering rate of superconductors with mixed s + d or s + id symmetry order parameter is calculated for different mixing and for moderate impurity concentrations in two limiting cases, namely the Born (weak) and the...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Optical conductivity in superconductors with mixed symmetry order parameters / I. Schürrer, E. Schachinger // Condensed Matter Physics. — 1999. — Т. 2, № 3(19). — С. 463-480. — Бібліогр.: 15 назв. — англ. |
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irk-123456789-1205392017-06-13T03:05:44Z Optical conductivity in superconductors with mixed symmetry order parameters Schürrer, I. Schachinger, E. The real and imaginary part of the optical conductivity and the derived optical scattering rate of superconductors with mixed s + d or s + id symmetry order parameter is calculated for different mixing and for moderate impurity concentrations in two limiting cases, namely the Born (weak) and the resonant (unitary) scattering limit. There are pronounced differences in the optical conductivity between impurity limits and with the symmetry of the order parameter. Theoretical predictions are compared to experimental data. Розраховані дійсна і уявна складові частини оптичної провідності та інтенсивність оптичного розсіяння у надпровідниках з параметром порядку змішаної s + d та s + id симетрії для різної величини змішування і для помірних концентрацій домішок у двох граничних випадках борнівського (слабого) та резонансного (одноразого) розсіяння. Спостерігаються значні відмінності оптичної провідності між граничними випадками для розсіяння на домішках та зі зміною симетрії параметра порядку. Передбачення теорії порівнюються з експериментальними даними. 1999 Article Optical conductivity in superconductors with mixed symmetry order parameters / I. Schürrer, E. Schachinger // Condensed Matter Physics. — 1999. — Т. 2, № 3(19). — С. 463-480. — Бібліогр.: 15 назв. — англ. 1607-324X DOI:10.5488/CMP.2.3.463 PACS: 74.20.Fg, 74.25.Nf, 74.72.-h http://dspace.nbuv.gov.ua/handle/123456789/120539 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The real and imaginary part of the optical conductivity and the derived
optical scattering rate of superconductors with mixed s + d or s + id symmetry order parameter is calculated for different mixing and for moderate
impurity concentrations in two limiting cases, namely the Born (weak) and
the resonant (unitary) scattering limit. There are pronounced differences in
the optical conductivity between impurity limits and with the symmetry of
the order parameter. Theoretical predictions are compared to experimental
data. |
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Article |
| author |
Schürrer, I. Schachinger, E. |
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Schürrer, I. Schachinger, E. Optical conductivity in superconductors with mixed symmetry order parameters Condensed Matter Physics |
| author_facet |
Schürrer, I. Schachinger, E. |
| author_sort |
Schürrer, I. |
| title |
Optical conductivity in superconductors with mixed symmetry order parameters |
| title_short |
Optical conductivity in superconductors with mixed symmetry order parameters |
| title_full |
Optical conductivity in superconductors with mixed symmetry order parameters |
| title_fullStr |
Optical conductivity in superconductors with mixed symmetry order parameters |
| title_full_unstemmed |
Optical conductivity in superconductors with mixed symmetry order parameters |
| title_sort |
optical conductivity in superconductors with mixed symmetry order parameters |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120539 |
| citation_txt |
Optical conductivity in superconductors with mixed symmetry order parameters / I. Schürrer, E. Schachinger // Condensed Matter Physics. — 1999. — Т. 2, № 3(19). — С. 463-480. — Бібліогр.: 15 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT schurreri opticalconductivityinsuperconductorswithmixedsymmetryorderparameters AT schachingere opticalconductivityinsuperconductorswithmixedsymmetryorderparameters |
| first_indexed |
2025-07-08T18:06:00Z |
| last_indexed |
2025-07-08T18:06:00Z |
| _version_ |
1837103021765427200 |
| fulltext |
Condensed Matter Physics, 1999, Vol. 2, No. 3(19), pp. 463–480
Optical conductivity in
superconductors with mixed symmetry
order parameters
I.Schürrer, E.Schachinger
Institut für Theoretische Physik, Technische Universität Graz, Petersgasse
16, A-8010 Graz, Austria
Received April 21, 1998
The real and imaginary part of the optical conductivity and the derived
optical scattering rate of superconductors with mixed s+ d or s+ id sym-
metry order parameter is calculated for different mixing and for moderate
impurity concentrations in two limiting cases, namely the Born (weak) and
the resonant (unitary) scattering limit. There are pronounced differences in
the optical conductivity between impurity limits and with the symmetry of
the order parameter. Theoretical predictions are compared to experimental
data.
Key words: high- Tc superconductivity, mixed symmetry,
order-parameter, optical conductivity
PACS: 74.20.Fg, 74.25.Nf, 74.72.-h
1. Introduction
There is a number of experiments which indicate that the order parameter in
some, if not in all, high temperature superconductors (HTCS) could be of mixed
symmetry. Krishana et al. [1] investigated the electronic thermal conductivity of
Bi2Sr2CaCu2O8 in a magnetic field profile experiment. They observed at non-zero
magnetic fields a phase transition like behaviour and concluded that this could pos-
sibly be a phase transition from a simple d–wave to either a dx2
−y2 + idxy or a
dx2
−y2+is symmetric order parameter. Kouznetsov et al. [2] offers direct evidence
for a mixed order parameter in YBa2Cu3O6.95 (YBCO) using a new class of c-axis
Josephson tunnelling experiments where the conventional superconductor (Pb) is
deposited across a single twin boundary. Their results are consistent with predomi-
nantly dx2
−y2 pairing symmetry and a sign reversal of a smaller s-wave component
across the twin boundary.
In yet another experiment, using tunnelling spectroscopy, Covington et al. [3]
observed surface-induced broken time-reversal symmetry in YBCO tunnelling junc-
c© I.Schürrer, E.Schachinger 463
I.Schürrer, E.Schachinger
tions. This manifested itself by a split zero bias conductance peak. Such a result
has been predicted theoretically by Fogelström et al. [4] as an Andreev bound state
which is shifted to finite energy and this results in such a split zero bias conductance
peak. This broken time-reversal symmetry state is identified as a (d+is) symmetric
state which develops in YBCO close to the boundary with an insulator.
Optical conductivity experiments seem to be becoming more and more important
as a diagnostic tool in the field of high temperature superconductivity [5]. Palumbo
and Graf [6] considered the case of pure dx2
−y2 symmetry, extended s-wave and
anisotropic s-wave, and the effect of resonance impurity scattering on these. They
find significant differences in the real part of the conductivity and discuss how to
obtain information on the gap symmetry from data.
Here we will be interested in the optical conductivity of superconductors having
an order parameter of either (s+ d) or (s+id)-symmetry as a function of increasing
contributions of the s-wave symmetric part with the view of determining such an
admixture from the experiment. We also consider and compare Born and resonant
impurity scattering and concentrate on qualitative effects. Therefore, our calcula-
tions are restricted to the framework of the BCS-theory of superconductivity.
2. Order parameter and crystal symmetry
The dominant feature of HTCS are the CuO2 planes which are connected via
atomic chains. These planes are commonly regarded to be most important for su-
perconductivity. They are of tetragonal symmetry and for optimally doped systems
the Fermi surface (FS) is a circle in the (kx, ky)-plane. (An example is, for instance,
the material Tl2Sr2CuO6+δ (Tl2201).) The existence of CuO chains in YBCO means
that it is orthorhombic and that large in-plane anisotropies are observed between
the x and y (along the chains) axis in the optical conductivity [7].
The symmetry operations on the tetragonal lattice are as follows (besides the
trivial identity operation): a) rotations by 90◦ and 180◦, and b) reflections along
the x and y–axis as well as along the diagonal of the unit cell. Thus we find five
irreducible representations (table 1). An isotropic s–wave order parameter is the
trivial choice because it is invariant under all operations and its FS average 〈∆ s〉 =
const. Another possibility is the dx2
−y2 symmetry which requires the order parameter
to change its sign along the FS which results in nodes on the FS and a 〈∆d〉 = 0.
The combination (s + id)-symmetry – there is a 90◦ phase shift between the s-
and the d-wave component of the order parameter – is also permitted. It has to be
noted though that the (s+ id)-symmetric order parameter breaks the time-reversal
symmetry and cannot be expected to exist in the clean limit as far as pairing channels
of different symmetry compete with each other and the dominant channel prevents
the appearance of the subdominant one. On the other hand, due to pair breaking
defects [8], like impurities in a d–wave superconductor, the dominant symmetry
channel can be sufficiently suppressed to permit for a phase transition to either a
mixed (s + id)-symmetric or even to a pure s-wave state. As we are not dealing
with the clean limit in this paper it is quite realistic to study the possibility of an
464
Optical conductivity
Table 1. Character table of the three dimensional space group of the tetragonal
lattice.
C4v E 2C4 C2 2σv 2σd
A1 1 1 1 1 1 x2 + y2, z2 s–wave
A2 1 1 1 -1 -1
B1 1 -1 1 1 -1 x2 − y2, z2 dx2
−y2–wave
B2 1 -1 1 -1 1 xy dxy–wave
E 2 0 -2 0 0
(s+id)-symmetric order parameter in a superconductor of tetragonal symmetry and
how it affects the optical conductivity.
The situation changes drastically for orthorhombic systems with their (x, y)-an-
isotropy. The rotation by 90◦ and the reflection along the diagonal are no longer
permitted as symmetry operations. Therefore, it becomes impossible to distinguish
between the isotropic s and a dx2
−y2-symmetric order parameter from a group theo-
retical point of view. Thus, both are in the same irreducible representation and can
of course mix freely leading to an (s+d) symmetry. Moreover, the simplest approach
to band anisotropy in orthorhombic systems is a free electron infinite band model
but with different effective masses along the x- and y-directions. This results in an
ellipsoidal FS as opposed to the circular FS in systems of tetragonal symmetry.
3. Formalism
3.1. Tetragonal symmetry
We define the anisotropic, mixed symmetry order parameter ∆s+id(ϕ) in the
clean limit by
∆s+id(ϕ) = ∆s + i∆d(ϕ), (1)
with ∆s the s-wave contribution and ∆d(ϕ) = ∆d
√
2 cos 2ϕ the d-wave component
of dx2−y2-symmetry. Furthermore, we concentrate on the low temperature regime
which makes it unnecessary to solve the BCS gap equation explicitly. Instead, we
choose the clean limit, zero temperature order parameter ∆0 by either using the BCS
relation 2∆0/kBTc = 3.57 (or 2
√
2∆0/kBTc = 4.29 for pure d-wave) or by choosing
an appropriate experimental value.
The ∆s to ∆d admixture in equation (1) can be defined by introducing a param-
eter γ in such a way that
∆s = γ∆0, ∆d =
√
1− γ2∆0, (2)
which ensures that
∆0 =
√
〈
∆2
s+id(ϕ)
〉
ϕ
, (3)
with 〈· · ·〉ϕ the FS-average. Thus, 0 6 γ 6 1 with γ = 0 the pure d-wave and γ = 1
465
I.Schürrer, E.Schachinger
the pure s-wave case. An even more convenient parameter is
x =
γ
γ +
√
1− γ2
, (4)
which gives, when multiplied by 100, the percentage of s-wave gap ∆s, contained in
∆s+id(ϕ).
Following the standard BCS theory, the quasiparticle propagator G(k, ν) is de-
termined by Dyson’s equation
G−1(k, ν) = ντ0 + εkτ3 − ΣBCS − Σimp (5)
in a Nambu notation. Here, τ0 is the unity matrix, τi, i = 1, 2, 3 are the Pauli matri-
ces which take care of the particle hole symmetry, εk is the quasiparticle dispersion
relation, ΣBCS is the BCS self energy contribution describing the correlation of quasi-
particles in Cooper pairs. Included is also Σ imp, the self energy which accounts for
the quasiparticle impurity scattering. According to Ambegaokar [9] ΣBCS and Σimp
can be combined into one renormalized BCS self energy contribution
Σ̃BCS = ν − ω̃ − (∆1 − ∆̃1)τ1 + (∆2 − ∆̃2)τ2, (6)
with ∆1 and ∆2 the real and imaginary part of the clean limit order parameter,
and ω̃, ∆̃1, and ∆̃2 yet unknown complex functions. If we identify ∆1 with ∆s and
∆2 with ∆d(ϕ) we find the following renormalization equations in the weak (Born)
impurity scattering approximation for dilute impurity concentrations n I [10]:
ω̃(ν) = ν + iπt+
〈
ω̃(ν)
√
ω̃2(ν)− ∆̃2
s(ν)− ∆̃2
d(ϕ)
〉
ϕ
, (7)
∆̃s(ν) = ∆s + iπt+
〈
∆̃s(ν, ϕ)
√
ω̃2(ν)− ∆̃2
s(ν)− ∆̃2
d(ϕ)
〉
ϕ
, ∆̃d(ϕ) = ∆d(ϕ), (8)
with ∆̃d(ϕ) = ∆̃d
√
2 cos 2ϕ. t+ = nIN(0)|V (kF)|2, N(0) is the normal state quasi-
particle density of states at the Fermi energy, and V (kF) is the scattering potential
evaluated at the Fermi momentum kF. By the square root is meant the branch of
the complex function which corresponds to a positive imaginary part.
In the more general case of arbitrary impurity scattering, strength t+ is to be
replaced by Γ+ = nI/N(0)π2 and the renormalization equations are modified:
ω̃(ν) = ν + iπΓ+ Ω(ν)
c2 + Ω2(ν) +D2(ν)
, (9)
∆̃s(ν) = ∆s + iπΓ+ D(ν)
c2 + Ω2(ν) +D2(ν)
, ∆̃d(ϕ) = ∆d(ϕ), (10)
with c = cotδ0 and δ0 the T-matrix phase shift. For very large values of c the results
(7) and (8) for Born scattering are recovered and for c = 0 we are in the so-called
466
Optical conductivity
unitary (resonant) scattering limit. We also introduced the abbreviations
Ω(ν) =
〈
ω̃(ν)
√
ω̃2(ν)− ∆̃2
s(ν)− ∆̃2
d(ϕ)
〉
ϕ
, (11)
D(ν) =
〈
∆̃s(ν)
√
ω̃2(ν)− ∆̃2
s(ν)− ∆̃2
d(ϕ)
〉
ϕ
. (12)
3.2. Orthorhombic symmetry
The order parameter of a two dimensional Fermi liquid which is of (s+ d) sym-
metry can be described by [11]:
∆(ϕ) = ∆0(cos 2ϕ+ β), (13)
where β describes the s-wave contribution to the dominant d-wave part which is
modeled according to dx2
−y2 symmetry; ϕ is the polar angle in the two dimensional
Brillouin zone. The Fermi energy εF of the free electron infinite band model is given
by
εF =
k2
F,x
2mx
+
k2
F,y
2my
, (14)
with kF,(x,y) the Fermi momenta in x and y-direction respectively, and m (x,y) the
corresponding effective masses.
This ellipsoidal Fermi surface can be transformed into a circular one through the
transformation from (ki, ϕ) to (pi, φ) space
pi = ki
√
mx +my
2mi
, i = x, y, (15)
tanφ =
√
my
mx
tanϕ. (16)
We introduce the mass anisotropy parameter
α =
mx −my
mx +my
, (17)
with α > 0, needed to describe YBCO because the chain direction has the smaller
effective mass and is along y. In this new system of coordinates the gap (13) trans-
forms into
∆(φ) = ∆0(f(φ) + β)
= ∆0
(
cos 2φ+ α
1 + α cos 2φ
+ β
)
, (18)
which is no longer of d-wave symmetry. Nevertheless, it is possible to transform (18)
into a form which corresponds to an (s+d)-model with one component ∆d(φ) which
467
I.Schürrer, E.Schachinger
contains all the angular dependence and which also has the property 〈∆d(φ)〉φ = 0
and a second component ∆s which has no angular dependence. We introduce
f̄ =
1
2π
2π
∫
0
dφ f(φ) =
1−
√
1− α2
α
. (19)
This results in the two components of the order parameter:
∆d(φ) =
∆0√
N
(
f(φ)− f̄
)
, (20)
∆s =
∆0√
N
(
f̄ + β
)
, (21)
N =
〈
(f(φ) + β)2
〉
φ
. (22)
The normalization factor N ensures that ∆0 =
√
〈
∆2
s+d(φ)
〉
φ
. In this notation equa-
tions (9) and (10) are again valid but now with
Ω(ν) =
〈
ω̃(ν)
√
ω̃2(ν)− ∆̃2
s+d(ν, φ)
〉
φ
, (23)
D(ν) =
〈
∆̃s+d(ν, φ)
√
ω̃2(ν)− ∆̃2
s+d(ν, φ)
〉
φ
. (24)
and ∆̃s+d(ν, φ) = ∆̃s(ν) + ∆̃d(ν, φ). The addition of an s-wave component (β 6= 0)
to the gap or the inclusion of mass anisotropy (mx 6= my, α 6= 0) can be referred to
as orthorhombic d-wave symmetry.
3.3. Optical conductivity
The optical conductivity describes the linear response of a material which is
exposed to an electromagnetic field. This field induces shielding currents
Jµ(q, ν) = −
∑
ρ
Kµρ(q, ν)Aρ(q, ν), ρ, µ = 0, 1, 2, 3, (25)
with ν the photon energy, Aρ(q, ν) the Fourier transform of the covariant vector
potential and Kµρ(q, ν) the response kernel which depends only on the properties of
the material. It can be expressed as a product of quasiparticle propagators G(k, ν)
and once this is known, the optical conductivity follows from
σ(ν) =
i
ν
K(ν) (26)
after a q-integration. The quasiparticle propagator is calculated from equation (5)
using either equations (9–12) for the (s+id) symmetry or equations (9,10,23,24) for
468
Optical conductivity
the orthorhombic d-wave case. The response kernel in (25) has first been calculated
by Lee et al. [12] for superconductors with arbitrary impurity concentrations and it
has been modified by Schürrer et al. [10,11] to encompass mixed symmetry order
parameters.
We find the isotropic conductivity in a tetragonal system by
σ(ν) =
i
ν
e2N(0)v2F
2
〈S(ν, ϕ)〉ϕ , (27)
with e the charge on the electron, vF the Fermi velocity, and S(ν, ϕ) according to
[10] equation (19). For the orthorhombic system Schürrer et al. [11] quote
σxx′(ν) =
i
ν
e2Nn(0)
v2F
1 + α
〈
cos2 φS(ν, φ)
〉
φ
, (28)
σyy′(ν) =
i
ν
e2Nn(0)
v2F
1− α
〈
sin2 φS(ν, φ)
〉
φ
, (29)
for the anisotropic optical conductivity.
4. Results
Optical conductivity experiments study the excitation of quasiparticles from en-
ergies below the Fermi level into empty states above this level by photons of energy
ν. Therefore, the quasiparticle density of states (QDOS) N(ν) is reflected in σ(ν)
and this makes the optical conductivity such a successful probe of the QDOS of
complex quasiparticle systems.
We present in this chapter the results of numerical calculations for supercon-
ductors with Born impurity scattering (for instance dilute concentrations of impu-
rities caused by radiation damage) and for superconductors which contain resonant
scatterers (for instance Zn atoms substituted on Cu-sites). The impurity concen-
tration is either determined by the parameter t+/∆0 = 0.0083 (Born limit) or by
Γ+/∆0 = 0.0083 (unitary limit). This ensures the same Tc in all systems. All calcu-
lations have been performed for a sample temperature of 10 K.
4.1. (s + id) symmetry
The quasiparticle density of states N(ν) normalized to its normal state value at
the Fermi level N(0) is calculated from
N(ν)
N(0)
∝
〈
ℜe
ω̃(ν + i0+)
√
ω̃2(ν + i0+)− ∆̃2
s(ν + i0+)− ∆̃2
d(ν + i0+, ϕ)
〉
ϕ
, (30)
with ω̃(ν + i0+), ∆̃s(ν + i0+), and ∆̃d(ν + i0+, ϕ) determined from equations (9–12)
for Born impurity scattering (top frame) or by equations (23) and (24) for resonant
scatterers (bottom frame) respectively. It is obvious from equation (30) that the
469
I.Schürrer, E.Schachinger
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
∆d
2∆
0.67
N
( ν
)/N
(0
)
ν/∆
0
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
∆d
2∆
0.24
2∆
0.5
1∆
0.5
1∆
0.24
ν
o
ν
o
ν/∆0
N
( ν
)/N
(0
)
Figure 1. The superconducting QDOS N(ν) normalized to its normal state value
at the Fermi level N(0), as a function of the reduced energy ν/∆0 for a system
with an (s + id)-symmetric order parameter, for various values of x. The top
frame is for a system with Born impurity scattering and the bottom frame is for
a system with resonant impurity scattering. The various x values are: x = 0 (pure
d-wave, solid line), x = 0.24 (dash-dotted line), x = 0.5 (dash-double-dotted line),
x = 0.67 (dashed line), and x = 1 (thin, dotted line).
QDOS has to have a gap for all x > 0 because the real and imaginary parts of
∆̃2
s and ∆̃2
d never vanish simultaneously. Results are presented in figure 1 where the
QDOS N(ν) normalized to its normal state value N(0) is presented as a function of
the reduced photon energy ν/∆0. Both frames contain as a reference the QDOS of
a pure s-wave (dotted) and of a pure d-wave (solid) superconductor. The top frame
of figure 1 presents the results of a superconductor which contains Born impurity
scattering. We observe one peak in the QDOS which can be found at 1 < ν/∆0 <
√
2
for 0 < x < 1 and there is always a gap in the QDOS in the region 0 6 ν/∆0 6 α
with a sharp rise in the QDOS at ν = νo, the onset energy.
An interesting situation arises when the superconductor contains resonant impu-
rity scattering. These impurities create localized states which cause a finite QDOS
at ν = 0 in the pure d-wave superconductor (solid line, bottom frame of figure 1). In
470
Optical conductivity
0 1 2 3 4 5
0.000
0.001
0.002
2ν
o
2ν
o
2|2∆
0.67
|
σ 1( ν
) (
ar
b.
u
ni
ts
)
ν/∆
0
0 1 2 3 4 5
0.000
0.002
0.004
0.006
2|1∆
0.5
|
|1∆
0.24
+2∆
0.24
|
2|1∆
0.24
|
ν/∆0
σ 1( ν
) (
ar
b.
u
ni
ts
)
Figure 2. The real part of the optical conductivity σ1(ν) in arbitrary units as a
function of the reduced energy ν/∆0 for a system with (s+ id)-symmetric order
parameter. Various values of the parameter x are considered, namely x = 0 (pure
d-wave, solid line), x = 0.24 (dash-dotted line), x = 0.5 (dash-double-dotted line),
and x = 0.67 (dashed line). The top frame is for Born impurity scattering and
the bottom frame is for resonant scatterers.
the system with an (s+ id)-symmetric order parameter such localized states cannot
be created at ν = 0 because of the (s + id) symmetry requirements and, there-
fore, they “pile up” at the onset frequency νo creating a sharp peak in the QDOS
quite similar to the well known square root singularity at the gap of a standard s-
wave superconductor. A second peak in the QDOS is again found at the frequencies
1 < ν/∆0 <
√
2 for 0 < x < 1.
The real part of the optical conductivity σ1(ν) is presented in figure 2 as a func-
tion of the reduced frequency ν/∆0. As a reference both frames contain the results
for a d-wave superconductor (solid lines). In the case of Born impurity scattering
the QDOS has no gap and, therefore, even photons of infinitesimal energy will be
capable of exciting some quasiparticles from the occupied to the unoccupied parts of
the QDOS. We observe a very narrow peak structure of finite width around ν = 0.
In addition there is a δ-function peak at ν = 0 which corresponds to the condensate
471
I.Schürrer, E.Schachinger
and gives the infinite d.c. conductivity of the superconductor. This is not shown
here. The absorptive part of the conductivity stays finite for all energies above the
Drude peak centered at ν = 0 and shows a second maximum at ν = 2|∆d| = 2
√
2∆0
(although it is not very pronounced). For even higher energies, σ1(ν) approaches its
normal state value.
As far as the QDOS for x > 0 always has got a gap, the peak in σ1(ν) around
ν = 0 is very small at T = 10 K because few quasiparticles are thermally activated
at such temperatures and so this peak can hardly be seen in the figure which shows
a region of nearly zero conductivity, i.e.: σ1(ν) exhibits a gap for frequencies smaller
than two times the QDOS onset frequency νo. In superconductors which contain
Born impurity scattering, σ1(ν) increases with increasing frequency for ν > νo until
a peak is reached near or at twice the frequency at which the QDOS has got its
main peak. (For instance at ν = 2|2∆0.67| for x = 0.67, dashed line in the top frame
of figure 2.)
Superconductors which contain resonant impurity scattering behave quite differ-
ently as is demonstrated by our results presented in the bottom frame of figure 2.
For the pure d-wave case there is now a finite QDOS at ν = 0 (see bottom frame
of figure 1, solid line). These electrons are easily excited and can absorb a photon
directly just as in a normal metal. The result is a broad Drude like peak (solid line
in the bottom frame of figure 2) centered around ν = 0. Direct absorption from the
ν ≈ 0 region to the peak in the QDOS will now lead to the structure at the peak fre-
quency rather than twice this value as has been seen in the Born impurity scattering
case. For x > 0 σ1(ν) develops a steep rise at ν = νo followed by either a pronounced
peak or a shoulder indicating the availability of large numbers of free states because
of the peak structure in the QDOS at the onset frequency νo. See for instance the
peak at ν = 2|1∆0.24| (x = 0.24, dash-dotted line) or the shoulder at ν = 2|1∆0.5|
(x = 0.5, dash-double-dotted line). This first peak (shoulder) is then followed by a
second, less pronounced peak which corresponds to excitation processes from states
at the onset frequency below the Fermi level into free states at the main peak of
the QDOS above the Fermi level. For x = 0.24, this peak is found at the frequency
ν = |1∆0.24 +
2∆0.24|.
A comparison of experimental data for an optimally doped Tl2201 single crystal
with theoretical predictions for a superconductor of (s+ id)-symmetry is presented
in figure 3. What is shown is the optical scattering rate of the superconducting state
1
τS(ν)
=
ωp
4π
ℜe
[
1
σ(ν)
]
(31)
normalized to its frequency independent normal state value τ −1
N as a function of the
reduced energy ν/∆0. (ωp is the plasma frequency.) Experimental data are indicated
by solid squares. They were read off figure 9 of [14] and rescaled in energy to match
the main rise in the theoretical data. These are presented for x = 0 (pure d-wave,
solid line), x = 0.24 (dash-dotted line), and x = 0.5 (dashed line). We conclude from
this comparison that the order parameter of Tl2201 is most likely of pure d-wave
symmetry because no gap opens up in the τ−1
S data as it would be required if there
were a finite s-wave contribution to the order parameter.
472
Optical conductivity
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Tl2201 Basov et al.
x = 0
x = 0.24
x = 0.5
τ-1
S
( ν
)/
τ-1
N
ν/∆
0
Figure 3. The superconducting state optical scattering rate τ−1
S (ν) normalized to
its normal state value τ−1
N as a function of the reduced energy ν/∆0. Theoretical
predictions for an (s + id)-symmetric order parameter are compared to experi-
mental data for a nominally clean, optimally doped Tl2201 single crystal (solid
squares) [13]. The various x values are: x = 0 (pure d-wave, solid line), x = 0.24
(dashed line), and x = 0.5 (dash-dotted line). (The data points have been read
off figure 9 of [13].) The energy scale of the experimental data was rescaled to
match the main rise with theoretical predictions.
4.2. Orthorhombic d -wave
The function f(φ), equation (18), is central to understanding the effect of ef-
fective mass anisotropy on the resulting superconducting properties. The maxima
and minima of f(φ) fall at 0 and π and π/2 and 3π/2 respectively as they would
in the pure d-wave case. The absolute magnitude of f(φ) is the same at both ex-
trema. Therefore, the introduction of an effective mass anisotropy will not result
in a splitting of the van Hove singularity in the QDOS positioned in energy at the
value of the gap extrema. Only for β 6= 0 a splitting of the van Hove singularity can
be expected.
In figure 4 (top and bottom frame) we consider the QDOS defined by
N(ν)
N(0)
∝
〈
ℜe
ω̃(ν + i0+)
√
ω̃2(ν + i0+)− ∆̃2
s+d(ν + i0+, φ)
〉
φ
, (32)
as a function of the reduced energy ν/∆0 for four different values of α in the top
frame, and for α = 0.4 and three different values of β in the bottom frame. The
impurity scattering is in the resonant scattering limit. In both frames dotted, solid,
dashed and dash-dotted curves are presented. In the top frame only effective mass
anisotropy is included and α = 0, 0.2, 0.4, and 0.6 respectively. In the bottom frame
the s-wave admixture represented by the parameter β of equation (13) is β = 0.4,
473
I.Schürrer, E.Schachinger
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.5
1.0
1.5
2.0
2.5
α = 0
α = 0.2
α = 0.4
α = 0.6
N s( ν)
/N
n(0
)
ν/∆
0
0.0 0.1 0.2 0.3
0.1
0.2
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.5
1.0
1.5
2.0
2.5
N s( ν)
/N
n(0
)
ν/∆0
α = β = 0
β = 0.4
β = 0.6
β = 0.8
0.0 0.1 0.2 0.3 0.4
0.1
0.2
0.3
0.4
Figure 4. The superconducting state QDOS N(ν) normalized to its normal state
value at the Fermi energy N(0) as a function of the reduced energy ν/∆0 in a
system with resonant impurity scattering. The top frame is for a pure d-wave
gap but with orthorhombic band structure. The dotted curve is for α = 0, no
band anisotropy, solid is for α = 0.2, dashed is for α = 0.4, and dash-dotted is
for α = 0.6. Band structure anisotropy, i.e. α 6= 0, rapidly degrades the finite
value of N(ν)/N(0) in the ν → 0 regime seen in the dotted curve and in the
insert of the top frame which shows an expanded version of the low ν part of
the QDOS. The bottom frame includes a s-wave component (β 6= 0) into the gap
function before transformation. The dotted line (α = β = 0) is unchanged and it
is for comparison. All other curves have a α = 0.4. The solid curve is for β = 0.4,
dashed is for β = 0.6, and dash-dotted is for β = 0.8. The splitting of the van
Hove singularity not seen in the top frame is clearly visible and is traced directly
to β 6= 0.
0.6, and 0.8 respectively. The dotted curve has α = β = 0 and is a pure d-wave
case with no band mass anisotropy and is presented for comparison. The other three
curves have α = 0.4 as well as a finite value of β. It is obvious that for α = β = 0 we
recover the well known result that the QDOS acquires a finite value at ν = 0 when
resonant scattering is included.
Our results for the absorptive part of the optical conductivity σ1(ν) are presented
in figure 5 as a function of the reduced photon energy ν/∆0. In the top frame
the impurity scattering is in Born approximation and in the bottom frame it is in
the resonant scattering limit. The gray curve is the (yy ′)-component and the short
dashed curve is the (xx′)-component for α = 0.2, solid and dotted are the same for
α = 0.4 and dashed and dash-dotted are for α = 0.6. The inserts give an expanded
view of the low frequency part of the results.
The most important idea to keep in mind in analyzing our results for the op-
474
Optical conductivity
0 1 2 3 4 5
0.000
0.001
0.002
0.003
σ1,yy' σ1,xx', α = 0.2
σ1,yy' σ1,xx', α = 0.4
σ1,yy' σ1,xx', α = 0.6
σ 1( ν)
(a
rb
. u
ni
ts
)
ν/∆0
0.00 0.02 0.04
0.1
0.2
0.3
0.4
0 1 2 3 4 5
0.000
0.005
0.010
0.015
0.020
ν/∆0
σ 1( ν)
(a
rb
. u
ni
ts
)
σ1,yy' σ1,xx', α = 0.2
σ1,yy' σ1,xx', α = 0.4
σ1,yy' σ1,xx', α = 0.6
0.0 0.1 0.2
0.1
0.2
Figure 5. The top frame presents the absorptive part of the infrared conductivity
σ1(ν) as a function of the reduced energy ν/∆0 for the pure d-wave case (β = 0)
with orthorhombic band structure (α 6= 0). The gray line gives σ1,xx′ and the
short dashed line gives σ1,yy′ for α = 0.2, solid and dotted lines correspond to
σ1,xx′ and σ1,yy′ for α = 0.4, and the dashed and dash-dotted ones correspond to
σ1,xx′ and σ1,yy′ for α = 0.6. The insert shows an expanded plot of the low energy
part. The scattering is treated in Born approximation. The bottom frame is the
same as the top frame but now the scattering is treated in the unitary limit.
tical conductivity is, that in Born scattering, the number of states at the chemical
potential goes to zero linearly with decreasing frequency ν (except at very low tem-
peratures and frequencies not of interest here) while for unitary scattering there is
a finite density of states at ν = 0 which provides the possibility of transition from
the region of ν = 0 to all energies in the unoccupied density of states and so these
transitions will reflect directly structures in the density of states at the same ener-
gies. Such transitions are progressively suppressed, however, as the orthorhombicity
α is increased or as the s-wave admixture β is increased or the scattering strength
is reduced towards Born scattering. In such circumstances these transitions can be
greatly suppressed and may not lead to significant structures in optical properties
at the critical points for the density of states. The structure at twice the gap energy
scale reflecting transitions from the main peak in the density of occupied states to
the main peak in the unoccupied states can now dominate the optical characteris-
475
I.Schürrer, E.Schachinger
0 1 2 3 4 5
0.000
0.001
0.002
0.003
0.004
σ
1,yy'
σ
1,xx'
, β = 0.4
σ
1,yy'
σ
1,xx'
, β = 0.5
σ
1,yy'
σ
1,xx'
, β = 0.6
σ 1( ν)
(a
rb
. u
nit
s)
ν/∆
0
0 1 2 3 4 5
0.000
0.002
0.004
0.006
0.008
0.010
σ 1( ν)
(a
rb
. u
nit
s)
ν/∆
0
σ
1,yy'
σ
1,xx'
, β = 0.4
σ
1,yy'
σ
1,xx'
, β = 0.5
σ
1,yy'
σ
1,xx'
, β = 0.6
Figure 6. Top frame: the real part of the infrared conductivity σ1(ν) vs. the
reduced energy ν/∆0 for α = 0.4 and various values of the s-wave admixture β.
The solid, dotted, and gray curves are for σ1,yy′ (along the chains) with β = 0.4,
0.5, and 0.6 respectively. The dashed, dash-dotted, and short dashed curves are
for σ1,xx′ . We see in σ1,yy′ the 2∆ structure at low energies corresponding to the
lower energy van Hove peak in the QDOS (see bottom frame of figure 4; this curve
applies to resonant scattering, the results for Born scattering are not shown here
but are quite similar). In σ1,xx′ we see the high energy van Hove structure of the
QDOS at a reduced energy near 3.0. Bottom frame: same as for the top frame but
now the impurity scattering is treated in the resonant limit. As before, structures
are seen in the σ1,xx′ curves but now around a reduced energy of 1.8.
tics. As far as we are dealing with an orthorhombic system, an extra factor of the
x- and y-components of the Fermi velocity will enter the conductivity but not the
QDOS. This leads to different characteristics in x- and y-directions and the struc-
tures expected on the basis of the QDOS alone get shifted somewhat by the extra
Fermi velocity weighting.
The top and bottom frame of figure 6 present results for the real part of the
infrared conductivity with finite band structure orthorhombicity included with α =
0.4 while at the same time some s-wave part is included through β. Three values
are considered, namely β = 0.4, 0.5, and 0.6. There are 6 curves, solid, dotted, and
gray for σ1,yy′(ν), dashed, dash-dotted, and short dashed for σ1,xx′(ν) in order to
increase β. The top frame is for Born impurity scattering. On comparing solid and
dotted curves for α = 0.4 with the top frame of figure 5 we see that including a
476
Optical conductivity
finite value of β has not changed the conductivity in the (xx′)-direction by much,
although, the peak around ν/∆0 = 2.8 has been shifted a little towards higher values
(∼ 3.0). In comparison, the effect of finite β has had a drastic effect on the σ1,yy′(ν)
characteristic which now shows new structures at low energies just above the rapid
Drude like rise at ν → 0. These structures are traced to the low energy van Hove
singularities seen in the bottom frame of figure 4 and are a result of the admixture of
a s-wave component to the gap function. They are not present when band structure
anisotropy alone is accounted for and thus offer the possibility of distinguishing
between the two effects and measuring the amount of s-wave admixture through
optical methods.
If resonant impurity scattering is considered instead of Born scattering we get
the results shown in the bottom frame of figure 6. Again, the structure in the curves
for σ1,xx′(ν) has moved from ν/∆0 = 3.0 or so to 1.7 or so just as in the bottom
frame of figure 5. The curves in the (yy ′)-direction show little sign of the low energy
van Hove structure and this case is not as favourable as the Born scattering case for
measuring the value of β. On the other hand, they show much more spectral weight
at low ν.
It is now, of course, quite interesting to compare theoretical predictions of this
model with experimental data for the orthorhombic superconductor YBCO. Ac-
cording to Schürrer et al. [11] the anisotropy parameter α = 0.41 is found from
zero-temperature penetration depth measurements by Bonn et al. [15] and the value
β = −0.25 was found to allow the best possible fit to the temperature dependence
of the low temperature dependence of the London penetration depth. These pa-
rameters are used to calculate the optical conductivity of an orthorhombic d-wave
superconductor and figure 7 presents a comparison of these results with experimen-
tal data reported by Wang et al. [14] for a nominally clean, detwinned YBCO single
crystal. On the horizontal scale these data have been scaled to a value of ∆0 which
corresponds to a ratio of 2∆0/kBTc ≈ 10. The arrow indicates the data point used
to fit the experimental data on the vertical scale.
The first thing to be noted is the maximum at ν/∆0 slightly above 2 in the dashed
curve for σ1,xx′ . It is not present in the solid curve for σ1,yy′(ν). The energy scale
defined in the dashed curve is twice the value of the low energy van Hove structure
seen in the QDOS shown in the bottom frame. There is no discernible structure
(corresponding to the second, high energy van Hove structure in N(ν)/N(0)) seen in
the conductivity curve so that this set of parameters is not favourable for measuring
the s-wave admixture in the gap function. We note several aspects of the top frame
of figure 7. First the peak in σ1,yy′(ν) has moved to lower frequency (ν/∆0 ∼ 1.5)
as compared with the σ1,xx′-case (ν/∆0 ∼ 2). This softening is observed in the
data of Wang et al.[14] on untwinned pure single crystals of YBCO shown. Also
the depth of the dip below ν = ∆0 is more pronounced in our work in the (xx′)
than in the (yy′) direction and this is in qualitative agreement with the experiment.
While the qualitative agreement with the data is good for the (xx ′)-component it is
less good for the (yy′)-component. In particular the large peak in the data around
ν/∆0 = 0.6 is not in our theoretical calculations. Nevertheless, we can conclude that
477
I.Schürrer, E.Schachinger
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.5
1.0
1.5
N(
ν)/
N(
0)
ν/∆
0
0 1 2 3 4 5
0
2000
4000
6000
8000
σ 1( ν)
( Ω
-1
cm
-1
)
ν/∆
0
E||a E||b Exp.
σ1,xx' σ1,yy'
Figure 7. Top frame: the real part of the infrared conductivity σ1(ν) vs. reduced
frequency ν/∆0 for the case α = 0.41 and β = −0.25. The dotted curve is for
σ1,xx′(ν), the solid one is for σ1,yy′ . The structure slightly above ν/∆0 = 2 reflects
the low energy van Hove structure in the QDOS shown in the bottom frame. The
position of this structure is displaced slightly upwards in the σ1,xx′(ν) curve.
It is not seen in the σ1,yy′(ν) curve which has got a peak at lower frequencies.
Also included are experimental results reported by Wang et al.[14] for untwinned
YBCO single crystals. The solid squares show σ1(ν) data in a-direction and the
solid triangles σ1(ν) show data in b-direction. The arrow indicates the data point
used to fit our theoretical results. Bottom frame: the superconducting state QDOS
N(ν) normalized to its normal state value at the Fermi level N(0) as a function
of reduced energy ν/∆0.
the orthorhombic d-wave model is in a rather good qualitative agreement with the
experiment.
5. Conclusion
It is one of the main results of this study that the properties of the optical con-
ductivity in superconductors with a mixed symmetry order parameter are explained
by the fact that in the (s+id)-symmetric case there is always a gap in the QDOS for
all possible s-wave admixtures while the orthorhombic d-wave case is characterized
by an order parameter with nodes on the Fermi surface for all reasonable values of
the mass anisotropy parameter α. Additional splitting of the van Hove singulari-
ties which is reflected by additional structure in the optical conductivity occurs for
further s-wave contributions to the order parameter which are characterized by a
478
Optical conductivity
β 6= 0.
We also considered both the effect of Born and of resonant impurity scatter-
ing. The main difference between these two limits is that while for Born impurity
scattering the structure in the electromagnetic response due to the corresponding
structures in the QDOS are to be found on a 2∆0 scale, in the resonant case they are
to be found on the scale of ∆0 instead. For orthorhombic systems, however, these
structures can get shifted in energy or might even not appear because of the extra
Fermi velocity weighting factors in the conductivity. Still, structures can be iden-
tified which correspond to the van Hove structures. Also, the x-axis conductivity
is seen to show a more “s-wave”-like behaviour as compared to the y-axis which is
more “d-wave”-like.
Acknowledgements
Work supported in part by Fonds zur Förderung der wissenschaftlichen Forschung
(FWF), Vienna, Austria under contract No. P11890-NAW.
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Оптична провідність у надпровідниках з
параметром порядку займаної симетрії
І.Шюррер, Е.Шахінґер
Інститут теоретичної фізики, Технічний університет м. Грац,
Австрія, A-8010 Грац, вул. Петерсгассе, 16
Отримано 21 квітня 1998 р.
Розраховані дійсна і уявна складові частини оптичної провідності та
інтенсивність оптичного розсіяння у надпровідниках з параметром
порядку змішаної s+ d та s+ id симетрії для різної величини змішу-
вання і для помірних концентрацій домішок у двох граничних випад-
ках борнівського (слабого) та резонансного (одноразого) розсіяння.
Спостерігаються значні відмінності оптичної провідності між гранич-
ними випадками для розсіяння на домішках та зі зміною симетрії па-
раметра порядку. Передбачення теорії порівнюються з експеримен-
тальними даними.
Ключові слова: високотемпературна надпровідність, змішана
симетрія, параметр порядку, оптична провідність
PACS: 74.20.Fg, 74.25.Nf, 74.72.-h
480
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