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Theoretical description of the SrPt3P superconductor in the strong-coupling limit
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2014 Phys. Scr. 89 125701
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Royal Swedish Academy of Sciences
Physica Scripta
Phys. Scr. 89 (2014) 125701 (6pp)
doi:10.1088/0031-8949/89/12/125701
Theoretical description of the SrPt3P
superconductor in the strong-coupling limit
R Szczȩ ś niak1,2, A P Durajski1 and Ł Herok2
1
Institute of Physics, Czȩstochowa University of Technology, Ave. Armii Krajowej 19, 42–200
Czȩstochowa, Poland
2
Institute of Physics, Jan Długosz University, Ave. Armii Krajowej 13/15, 42–200 Czȩstochowa, Poland
E-mail: [email protected]
Received 17 May 2014, revised 31 July 2014
Accepted for publication 8 September 2014
Published 18 November 2014
Abstract
The thermodynamic properties of a SrPt3P compound in the superconducting state have been
investigated by taking the Eliashberg approach. The anti-perovskite SrPt3P is identiﬁed as a
strong-coupling (λ = 1.33) s-wave superconductor with 2Δ (0) k B TC = 4.35, where the zerotemperature energy gap (2Δ (0)) obtained from the Eliashberg calculations is 3.15 meV. The
critical temperature TC was recently reported to be 8.4 K and this paper conﬁrms the Coulomb
pseudopotential repulsions ( μ⋆ ) to be equal to 0.123. Moreover, the free energy difference
between the superconducting and the normal state has been calculated. On the basis of the
obtained results, the speciﬁc heat for the superconducting and the normal state, as well as
the thermodynamic critical ﬁeld, have been determined. It has been also proved that the
electron effective mass is high and reaches its maximum equal to 2.46me for the critical
temperature.
Keywords: superconductors, anti-perovskite compound, thermodynamic properties
(Some ﬁgures may appear in colour only in the online journal)
Iron pnictides, like cuprates, represent a fascinating class of
materials that exhibit unconventional superconductivity
properties. In these compounds, not only is the highest TC
interesting, but also the ratio between the superconducting
energy gap and transition temperature 2Δ k B TC is anomalously large and well above the predictions of classical
Bardeen–Cooper–Schrieffer (BCS) theory [1]. Pnictides
without iron can also exhibit superconductivity, however their
TC values are signiﬁcantly lower than those of iron-based
superconductors [2]. Recently, the platinum-based superconductor family with chemical composition APt3P (where
A = Sr, Ca and La) and experimental values of critical temperatures TC equal to 8.4 K, 6.6 K and 1.5 K, respectively, has
been reported [3]. The crystal structures of these pnictides
were experimentally determined to be anti-perovskite and
have tetragonal space group P4/nmm [4].
In this paper, we determine the basic thermodynamic
properties of the superconducting state in a SrPt3P compound.
The strong-coupling Eliashberg formalism has been
employed due to a large value of electron–phonon coupling
0031-8949/14/125701+06$33.00
parameter (λ = 1.33). This model has been found to be highly
accurate for materials with λ ⩾ 1 [5, 6]; for small coupling
constants this approach has led to practically the same results
for thermodynamic properties of superconductors as the BCS
theory [7].
In particular, we have calculated the exact value of
Coulomb pseudopotential, the ratio of the energy gap to the
critical temperature (R Δ ≡ 2Δ (0) k B TC ), the value of the
parameter RC ≡ ΔC ( TC ) C N ( TC ) (where ΔC denotes the
speciﬁc heat jump and CN is the speciﬁc heat of the normal
state), the ratio R H ≡ TC C N ( TC ) HC2 (0) (where HC represents
the thermodynamic critical ﬁeld) and the electron effective
mass (me⋆ ). In general, analytically solving the Eliashberg
equations is impossible therefore the exact results for the halfﬁlled electron band have been obtained by means of the
iteration method, which were described in [9] and successfully used in recent works [6, 10].
The starting point of the Eliashberg model is the Fröhlich
Hamiltonian which describes the interaction of electron gas
with phonons. In the second quantization it takes the
1
© 2014 The Royal Swedish Academy of Sciences
Printed in the UK
R Szczȩ śniak et al
Phys. Scr. 89 (2014) 125701
following form [11]:
H≡
representation [14]:
∑εk ck†σ ckσ + ∑ωq bq† bq + ∑gk (q) ck† + qσ ckσ ϕq .
kσ
q
(1)
ϕ ( ω + iδ ) =
kqσ
The ﬁrst two terms represent the non-interacting electrons and phonons, respectively. The third term describes the
electron–phonon interaction. Moreover, εk ≡ εk − μ; εk and μ
denote the electron band energy and the chemical potential,
respectively. For a two-dimensional square lattice and the
nearest-neighbour hopping integral t, we have: εk = −tγ (k ),
where γ (k ) ≡ 2 ⎡⎣ cos ( k x ) + cos k y ⎤⎦. Symbols ckσ and ck†σ
denote the electron annihilation and creation operator in the
Bloch state with momentum k and spin σ. Symbol ωq stands
for the energy of phonons and bq (bq†) is the phonon annihilation (creation) operator. The matrix element gk (q ) describes
the electron–phonon coupling function and ϕq ≡ b−†q + bq
[11]. Let us notice that using a canonical transformation,
during the elimination of the phonon degrees of freedom in
Fröhlich Hamiltonian, it is possible to obtain the Hamiltonian
of the BCS theory [7]. In order to bring out the Eliashberg
equations the Fröhlich Hamiltonian should be rewritten with
the use of the Nambu spinors [12]. Next, by using the matrix
Matsubara functions the Dyson equations are determined. In
the last step, the Eliashberg set is determined in a self-consistent way [7].
Equations of the Eliashberg theory can be formulated in
terms of both real and imaginary frequency axes. On the
imaginary axis, the equations for the order parameter function
ϕn ≡ ϕ ( iωn ) and the wave function renormalization factor
Z n ≡ Z ( iωn ) take the following form [13]:
ϕn =
(
)
⋆
M λ iω − iω
( n
m ) − μ θ ωc − ωm
π
ϕm ,
∑
β m =−M
ωm2 Z m2 + ϕm2
λ ( iωn − iωm )
1 π
ωm Z m ,
∑
ωn β m =−M ω 2 Z 2 + ϕ 2
m m
m
∫0
Ω max
dΩ
Ω
α 2F (Ω).
Ω − z2
2
⎡ λ ( ω − iω ) − μ ⋆ θ ω − ω ⎤
m
c
m ⎦
⎣
(
)
m =−M
+ ϕm2
dω′α 2F ( ω′) ⎡⎣ ⎡⎣ N ( ω′) + f ( ω′ − ω) ⎤⎦
+ iπ
∫0
×
⎤
⎥
⎥
2 2
( ω − ω′) Z ( ω − ω′ + iδ ) − ϕ2 ( ω − ω′ + iδ ) ⎥⎦
+ iπ
∫0
×
⎤
⎥
⎥ (5)
2 2
2
( ω + ω′) Z ( ω + ω′ + iδ ) − ϕ ( ω + ω′ + iδ ) ⎥⎦
ϕ ( ω − ω′ + iδ )
+∞
dω′α 2F ( ω′) ⎡⎣ ⎡⎣ N ( ω′) + f ( ω′ + ω) ⎤⎦
ϕ ( ω + ω′ + iδ )
and
Z ( ω + iδ ) = 1 +
+
iπ
ω
+∞
∫0
i π
ωβ
M
∑ λ ( ω − iω m )
m =−M
⎡
dω′α 2F ( ω′) ⎢ ⎡⎣ N ( ω′)
⎣
ωm Z m
ωm2 Z m2 + ϕm2
+ f ( ω′ − ω) ⎤⎦
⎤
( ω − ω′) Z ( ω − ω′ + iδ )
⎥
⎥
2 2
( ω − ω′) Z ( ω − ω′ + iδ ) − ϕ2 ( ω − ω′ + iδ ) ⎥⎦
×
+
iπ
ω
+∞
∫0
⎡
dω′α 2F ( ω′) ⎢ ⎡⎣ N ( ω′) + f ( ω′ + ω) ⎤⎦
⎣
⎤
(2)
×
(3)
( ω + ω′) Z ( ω + ω′ + iδ )
⎥
⎥,
2 2
( ω + ω′) Z ( ω + ω′ + iδ ) − ϕ2 ( ω + ω′ + iδ ) ⎥⎦
(6)
where symbols N (ω) and f (ω) are the Bose and Fermi distributions, respectively.
The Eliashberg equations have been solved for 2201
Matsubara frequencies (M = 1100). In the considered case, the
convergence of the solutions has been obtained for T ⩾ T0 ,
where T0 = 0.5 K.
The Coulomb pseudopotential, apart from the Eliashberg
function, is the second input parameter in the Eliashberg
equations. The physical value of the Coulomb pseudopotential μC⋆ has been chosen in order to reproduce the experimental value of critical temperature. On the basis of the
expression ⎡⎣ Δm = 1 ⎤⎦ μ = μ ⋆ = 0 for TC, where the critical tem-
where ωn stands for the Matsubara frequency:
ωm ≡ (π β ) (2 m − 1) and symbol β is associated with the
−1
Boltzmann constant kB in the following way: β ≡ ( k B T ) .
The order parameter is deﬁned by the formula: Δn ≡ ϕn Z n .
The pairing kernel for the electron–phonon interaction can be
deﬁned as:
λ (z ) ≡ 2
ωm2 Z m2
+∞
M
Zn = 1 +
M
∑
ϕm
×
( )
π
β
(4)
Our calculation is based on the Eliashberg spectral function
α 2F (Ω ) determined in [8], where the value of the maximum
phonon frequency (Ωmax ) is equal to 52.35 meV. Symbol μ⋆
represents the Coulomb pseudopotential, which models the
depairing interaction between the electrons. The quantity θ
denotes the Heaviside function, and ωc is a cut-off frequency,
chosen three times the maximum phonon frequency:
ωc = 3Ωmax .
The form of the functions ϕ and Z on the real axis has
been determined using the Eliashberg equations in the mixed
C
perature has been taken from the experimental measurement
exp
of resistive transition (⎡⎣ TC ⎤⎦
= 8.4 K [3]), we obtained
SrPt 3P
⎡⎣ μ ⋆ ⎤⎦
C SrPt 3P = 0.123. In ﬁgures 1(a) and (b), the dependency
of the order parameter on the number m for the
selected values of the Coulomb pseudopotential and the full
dependency of Δm = 1 (μ⋆ ) have been presented, respectively.
The high value of the Coulomb pseudopotential
suggests that the critical temperature of SrPt3P cannot be
2
R Szczȩ śniak et al
Phys. Scr. 89 (2014) 125701
The re-parametrized Allen–Dynes expression takes the
following form:
k B TC = f1 f2
⎡
⎤
ω ln
− 1.172(1 + λ) ⎥
exp ⎢
,
⎢⎣ λ − μ* (1 + 0.1λ) ⎥⎦
1.44
(7)
where the strong-coupling correction function (f1) and the
shape correction function (f2) are given by the expressions:
⎛ ω2
⎞
1
− 1⎟ λ2
3 ⎤3
⎜
⎡
⎛ λ ⎞2
⎝ ω ln
⎠
f1 ≡ ⎢ 1 + ⎜ ⎟ ⎥ and f2 ≡ 1 +
. (8)
2
2
⎢
⎥
Λ
⎝
⎠
+
λ
Λ
1
2
⎣
⎦
Symbol λ denotes the electron-phonon coupling constant:
Figure 1. (a) The order parameter on the imaginary axis for the
λ≡2
selected values of the Coulomb pseudopotential (the ﬁrst 50 values
of Δm have been presented). (b) The full dependency of the
maximum value of the order parameter on the Coulomb
pseudopotential.
Ω max
∫0
dΩ
α 2F (Ω)
.
Ω
(9)
The quantity ω2 represents the second moment of the normalized weight function:
ω2 ≡
2
λ
∫0
Ω max
dΩα 2F (Ω) Ω
(10)
and ω ln is the logarithmic average of the phonon frequencies:
⎡2
ω ln ≡ exp ⎢
⎣λ
∫0
Ω max
dΩ
⎤
α 2F (Ω)
ln(Ω) ⎥ .
Ω
⎦
(11)
In particular, for SrPt3P the following values have been
obtained: λ = 1.33, ω 2 = 9.76 meV , and ω ln = 6.63 meV.
The ﬁtting functions Λ1 and Λ2 are deﬁned as:
Λ1 ≡ 1.2 1 + 4.75μ*
and
(
)
(
)(
Λ2 ≡ 0.09 1 − 0.55μ*
)
ω 2 ω ln .
Based on the solutions of Eliashberg equations deﬁned
on the imaginary axis, we have calculated the free energy
difference between the superconducting and the normal state
( ΔF ) using the following formula [7]:
Figure 2. The dependency of the critical temperature on the values of
the Coulomb pseudopotential determined with the use of the selected
methods. The arrow shows the experimental value of TC for
μC⋆ = 0.123.
ΔF
2π
=−
ρ (0)
β
M
∑
m =1
(
ωm2 + Δm2 − ωm
⎛
ωm
× ⎜ Z mS − Z mN
⎜
ωm2 + Δm2
⎝
correctly evaluated by means of the original and modiﬁed by
the Allen and Dynes McMillan equation for TC [15, 16]. In
particular, we have: 7.96 K and 7.19 K, respectively when the
physical value of the critical temperature amounts to
8.4 K [3].
The strict dependency of the critical temperature on the
Coulomb pseudopotential is presented in ﬁgure 2. We can see
that the critical temperature calculated in the analytical way is
underestimated, especially in the range of the higher values of
μ⋆. In the considered case, we have re-parametrized the
Allen–Dynes expression using the least square method and
300 values of T μ⋆ for μ⋆ from the range 0.1–0.2. The exact
results have been obtained based on the condition
⎡⎣ Δm = 1 ⎤⎦
= 0.
T =T
)
⎞
⎟.
⎟
⎠
(12)
Moreover, the thermodynamic critical ﬁeld, deviation
function of the thermodynamic critical ﬁeld, and speciﬁc heat
difference between the superconducting and the normal state
have been determined using the following expressions:
HC
− 8π [ΔF ρ (0) ] ,
(13)
⎡
⎛ T ⎞2 ⎤
Hc (T ) ⎢
− 1−⎜ ⎟ ⎥,
⎢⎣
Hc (0)
⎝ Tc ⎠ ⎥⎦
(14)
ρ (0)
( )
D=
and
C
3
=
R Szczȩ śniak et al
Phys. Scr. 89 (2014) 125701
Figure 4. The speciﬁc heat for the superconducting (open circles)
and the normal state (full circles) as a function of the temperature.
At TC the characteristic jump has been marked by a vertical line.
The dependency of the speciﬁc heats of the superconducting and the normal state as a function of the temperature is presented in ﬁgure 4. The speciﬁc heat in normal
state CN is given as: C N k B ρ (0) = γ β , where the Sommerﬁeld constant is deﬁned as: γ ≡ (2 3) π 2 (1 + λ ). The
jump in the speciﬁc heat at the transition temperature ﬁnds its
origin in the fact that the superconducting gap opens up
exponentially fast [17].
The above results enable the determination of the fundamental dimensionless parameters:
Figure 3. (a) The free energy difference between the super-
conducting and normal state (full circles) and the thermodynamic
critical ﬁeld (open circles) as a function of the temperature.
(b) Deviation of the thermodynamic critical ﬁeld as a function of
reduced temperature.
ΔC
1 d 2 [ΔF ρ (0) ]
=−
.
k B ρ (0)
β d ( k B T )2
RH ≡
(15)
TC C N ( TC )
HC2 (0)
and
RC ≡
ΔC ( TC )
C N ( TC )
.
(16)
In the framework of the weak-coupling limit, their values
are universal and equal to 0.168 and 1.43, respectively
[20, 21]. For the SrPt 3P compound these thermodynamic
parameters of the superconducting state signiﬁcantly deviate
from the prediction of the BCS theory, in particular:
R H = 0.136 and RC = 2.57. It is connected with the fact that
the Eliashberg formalism, in contrast to the BCS model, does
not omit the strong-coupling and retardation effects.
The Eliashberg equations in the mixed representation
have been solved for an identical range of temperatures and
Coulomb pseudopotential as in the case of the imaginary axis.
In ﬁgures 5 and 6 the form of the order parameter and the
wave function renormalization factor for the range of the
frequencies from 0 to Ωmax and for the selected temperatures
are shown, respectively.
We can notice that for low frequencies, the non-zero
values are taken only by the real part of the order parameter
and the wave function renormalization factor. The obtained
result indicates that in the considered range of frequencies the
damping effects related with the imaginary part of these
functions do not exist. Moreover, it has been stated that the
shapes of the calculated functions are strongly correlated with
the complicated form of the Eliashberg function. Let us notice
that the dependency of the order parameter and the wave
function renormalization factor on temperature can be traced
In ﬁgure 3(a), the dependencies of ΔF ρ (0) and
HC ρ (0) functions on the temperature are plotted. From a
physical point of view, the negative values of ΔF prove that
the superconducting state is stable to the critical temperature.
It should be emphasized that the strong-coupling
Eliashberg theory of superconductivity was designed to
explain anomalous properties of superconducting, which
weak-coupling BCS theory cannot describe. In particular, the
BCS theory was developed for materials with spherical Fermi
surfaces. The results presented in [3] point to the fact that the
charge carriers, accommodated in the multiple Fermi surface
pockets, couple very strongly with low lying phonons and
that strong-coupling superconductivity is realized in the
SrPt3P compound. In this material, the strong-coupling effects
and deviation from the BCS theory can be observed in the
deviation function of the thermodynamic critical ﬁeld, which
is presented in ﬁgure 3(b). The positive values of the D
function can be observed for the strong electron-phonon
coupling (λ > 1) and D is negative for weak coupling (λ < 1)
[18]. We can see that for SrPt3P the coupling is strong.
Results predicted by BCS theory are presented with a dashed
line [19]. Moreover the shaded area denotes the weak coupling region.
4
R Szczȩ śniak et al
Phys. Scr. 89 (2014) 125701
Figure 5. The real and imaginary part of the order parameter on the real axis for the selected temperature values. The rescaled Eliashberg
function 2α2F(Ω) is plotted in the background.
Figure 6. The dependency of the real and imaginary part of the wave function renormalization factor on the frequency for the selected
temperature values. The rescaled Eliashberg function 2α2F(Ω) is plotted in the background.
most conveniently after plotting functions Δ (T ) and Z 0 (T ),
where Z 0 = Re [Z (0) ] (see ﬁgure 7). Our results for Δ (T ) and
Z 0 (T ) functions can be parametrized using the expressions:
⎛ T ⎞β
Δ (T ) = Δ (0) 1 − ⎜ ⎟
⎝ TC ⎠
and
⎛ T ⎞β
⎡
⎤
Z 0 (T ) = ⎣ Z 0 ( TC ) − Z 0 ( T0 ) ⎦ ⎜ ⎟ + Z 0 ( T0 )
⎝ TC ⎠
where Δ (0) = 1.575, Z 0 ( T0 ) = 2.245, Z 0 ( TC ) = 2.460 and
β = 3.8. It is worth noting that, in the framework of the BCS
model, the parameter β is equal to 3.
The maximum value of the order parameter has been
achieved for temperature equal to 0.5 K. For that reason the
order parameter for T0 = 0.5 K can be used in the calculations
of the energy gap at the temperature of zero Kelvin
(2Δ (0) = 3.15 meV). It is worth emphasizing that this result
reproduces recent experimental reports very well, where
Δ (0) = 1.58(2) meV [22].
Figure 7(a) presents the reduced temperature dependency
of normalized energy gap for SrPt3P in comparison with the
Figure 7. (a) The normalized energy gap at the reduced temperature.
(b) The dependency of the wave function renormalization factor on
the temperature.
5
R Szczȩ śniak et al
Phys. Scr. 89 (2014) 125701
Acknowledgments
BCS weak-coupling limit normalized energy gap (red line).
On ﬁrst sight, the ﬁgure reveals deviations from the weakcoupling BCS limit. However, for weak-coupling tin (Sn) the
relation Δ (T ) Δ (0) as a function of T TC takes almost identical results, like in the case of SrPt3P [23]. On the other hand,
for strong-coupling lead (Pb), this relation takes a BCS-like
shape [24]. So the deviation in ﬁgure 7(a) cannot be a decisive one, requiring the use of Eliashberg analysis. From a
physical point of view, the strong-coupling deviation effects
are described satisfactorily in terms of the parameter
k B TC ω ln , which in the weak-coupling limit, one can assume
to be zero. In the case of SrPt3P and Pb, we have obtained:
0.11 and 0.13, respectively. On the other hand, for Sn this
value is equal to 0.04 which is close to the weak-coupling
limit [16, 23].
The value of the dimensionless ratio R Δ ≡ 2Δ (0) k B TC
equals 4.35 and is clearly larger than the BCS value of 3.53.
This situation is a consequence of the strong-coupling electron–phonon interactions and retardation effects existing in
the investigated compound. Similar results were observed in
strong-coupling simple elements like lead (4.24) and mercury
(4.41), for tin the value close to BCS prediction (3.6) was
observed [24, 25].
The ratio of the electron effective mass (me⋆ ) to the band
electron mass (me) has been obtained on the basis of the data
presented in ﬁgure 7(b). We have used the expression:
me⋆ me = Re [Z (0) ]. It has been stated that me⋆ takes a high
value in the whole range of the temperature, where the
superconducting state exists, and its maximum is equal to
2.46me at the critical temperature.
To summarize, the superconducting property of the
SrPt3P compound has been investigated using the strongcoupling Eliashberg theory of superconductivity. The Elishberg phonon spectral function has been essential in determining: (i) the physical value of Coulomb pseudopotential
(μ⋆ =0.123); (ii) the thermodynamic critical ﬁeld and the
speciﬁc heat jump at TC; (iii) the energy gap at the temperature of zero Kelvin (2Δ (0) = 3.15 meV) and the electron
effective mass at TC (me⋆ = 2.46me ); (iv) the dimensionless
ratios R H = 0.136, RC = 2.57 and R Δ = 4.35. On the basis of
obtained results we can state that the properties of the
superconducting state of SrPt3P differ markedly from the
predictions of BCS theory which, in contrast to Eliashberg
theory, omits a very important role of the strong-coupling and
phonon retardation effects.
All numerical calculations were based on the Eliashberg
function sent to us by Dr Alaska Subedi to whom we are very
thankful.
Additionally, we are grateful to the Czȩstochowa University of Technology—MSK CzestMAN for granting access
to the computing infrastructure built in the project No.
POIG.02.03.00–00-028/08 ‘PLATON—Science Services
Platform’.
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