International Journal of Advancements in Research & Technology, Volume 5, Issue 1, January-2016
ISSN 2278-7763
6
Comparison of superconductivity in Sr2RuO4 and
La1.85Sr0.15CuO4 by simulations based on the BCS Theory
Mojtaba Mashmoola*, Artur Bohra, Florian Hetfleischa, Juan Santiago Lópeza, Hans-Peter
Roesera, Susanne Rotha, Marco Steppera, Mahdi Mottahedib, Armin Lechlerb
a
Institute of Space Systems, University of Stuttgart, Pfaffenwaldring 29, 70569 Stuttgart,
Germany. *email: mashmool@irs.uni-stuttgart.de, phone +49 (0) 711 685 62051
b
Institute for Control Engineering of Machine Tools and Manufacturing Units, University of
Stuttgart, Seidenstrasse 36, 70174 Stuttgart
Abstract
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Superconductivity in conventional superconductors can be described by using the BCS
theory, which relies on a pairing of electrons via phonon energies. In this work, we
simulate the mechanisms of the BCS theory in two unconventional superconductors,
La1.85Sr0.15CuO4 and Sr2RuO4, to find out whether their pairing mechanism works in a
similar fashion to the BCS theory. Our results show that the differences in Fermi
velocities, densities of states and spring constants of the unit cells could be a reason why
both superconductors have two different critical temperatures regardless of their same
crystal structure.
Keywords:
Simulation;
High
temperature
superconductors;
Unconventional
superconductors; Electrical properties; BCS theory.
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International Journal of Advancements in Research & Technology, Volume 5, Issue 1, January-2016
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Superconductivity in Sr2RuO4 and La1.85Sr0.15CuO4
1
Introduction
Sr2RuO4 exhibits spin-triplet superconductivity and it is therefore an unconventional
superconductor with the critical temperature 𝑇𝑐 = 1.5 K [1]. It has a tetragonal structure with
I4/mmm symmetry and lattice parameters a, b = 3.87 Å and c = 12.7 Å [2]. There are two
chemical formulas in the unit cell of Sr2RuO4. An undistorted octahedral RuO6 can be found in
its middle (Figure 1). It seems that the RuO2 planes play an important role regarding
superconductivity [3]. Its Debye temperature is about 312 K [5]. The electronic band structure
of Sr2RuO4 is quasi two dimensional because of its layered structure, which prevents the overlap
of the orbitals along the c-axis [5]. Its Fermi surface consists of three sheets corresponding to
three different types of Fermi electrons: α, β and γ (Figure 2). It is believed that the γ electrons
are responsible for the superconductivity [7, 8]. The β electrons are also relevant. Ref. [9] gives
binding energies of Δβ = 0.045 meV and Δγ = 0.15 meV for the β and γ electrons. Ref. [10]
have found a binding energy of Δγ = 0.14 meV, which corresponds to the BCS theory. A range
for the superconducting constant 𝑐 between 6.2 and 8 has been measured, whereby an
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anisotropic behavior in the superconductivity of Sr2RuO4 has been found (with the average
superconducting constant being 𝑐𝑎𝑣 = 5.5) [11].
Sr: Green
O: Red
Ru: Light brown
Figure 1: The unit cell of Sr2RuO4.
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International Journal of Advancements in Research & Technology, Volume 5, Issue 1, January-2016
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Superconductivity in Sr2RuO4 and La1.85Sr0.15CuO4
Figure 2: The Fermi surface of Sr2RuO4 with three different sheets α, β and γ.
Sr2RuO4
𝑘𝑓,𝑖 (Å−1 )
𝑚𝑒𝑓𝑓,𝑖 /𝑚𝑒
𝛼
𝛽
𝛾
0.302
0.621
0.750
1.1
2.0
2.9
m
𝑣𝑒,𝑖 [ ] /106
s
0.318
0.359
0.299
1
]
eV
𝐷2𝐷,𝑖 (𝐸𝑓 )[
0.69
1.24
1.80
Table 1: The three types of Fermi electrons α, β and γ, and their electrical properties in Sr2RuO4 [12].
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La1.85Sr0.15CuO4 is one of the high temperature superconductor cuprates with the critical
temperature 𝑇𝑐 = 38 K, a tetragonal structure with I4/mmm symmetry (Figure 3) and lattice
parameters a, b = 3.78 Å and c = 13.2 Å [13]. Its Debye temperature lies around 360 [4]. The
unit cell consists of two chemical formulas. La2CuO4 is the parent compound, which is
antiferromagnetic and a semiconductor at low temperatures (to about 30 K) [14]. Its
semiconductor properties can be due to the phase transition of the crystal structure to the
monoclinic structure which gives rise to an energy gap [14, 15]. To see the superconductivity
in La2CuO4, this parent compound must be doped by Ba or Sr atoms which bring charged holes
into the crystal structure. The critical temperature 𝑇𝑐 of the doped structure increases with
increasing doping up to an optimal doping amount 𝑥 = 0.15, at which the maximal critical
temperature 𝑇𝑐 = 38 K is reached. By overdoping, the superconductivity disappears. In contrast
to Sr2RuO4, the octahedral CuO6 of La2CuO4 located in the middle of the unit is distorted. In
cuprates it has been found that the superconductivity is induced by the CuO2 planes. The
physical properties of the cuprates change depending on the number of charge carriers on this
plane. It has been found that the doped holes are located in the oxygen-2p orbital in hole-doped
cuprates whereas the doped carriers in electron-doped cuprates sit at the cupper-3d orbitals [3].
Calculated values for the density of states 𝐷(𝐸𝑓 ) and the Fermi velocity 𝑣𝑒 for La1.85Sr0.15CuO4
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Superconductivity in Sr2RuO4 and La1.85Sr0.15CuO4
are given in Table 2. The different measured superconducting constants, binding energies and
critical temperatures are listed in Table 3.
La: Green
Cu: Blue
O: Red
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Figure 3: The unit cell of La1.85Sr0.15CuO4.
Superconductors
La1.85Sr0.15CuO4
m
𝑣𝑒 [ ] /106
s
0.314
1
]
eV
2.09
𝐷(𝐸𝑓 )[
Ref.
[16]
Table 2: Fermi velocity and density of states of La1.85Sr0.15CuO4.
𝑇𝑐 [K]
33
38
38
Δ [meV]
2.45 to 4.9
4
𝑐
8.9 ± 0.2
3 to 6
5.2
Ref.
[17]
[18]
[19]
BCS-like
BCS-like
Table 3: Critical temperature, superconducting constant and binding energy of La1.85Sr0.15CuO4.
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Superconductivity in Sr2RuO4 and La1.85Sr0.15CuO4
2
Simulation
The phonon energy required to pair two electrons is the deformation energy of the lattice due
to the Coulomb interaction between a moving electron and the ions in the unit cell. This
deformation energy can be simulated by finite elements methods. According to the BCS theory,
the binding energy Δ of a Cooper pair and the superconducting constant 𝑐 can be calculated by
the following equations:
−2
∆= ℏ𝜔𝐷 exp (
),
𝐷(𝐸𝑓 )𝑉0
𝑐=
2∆
,
𝑘𝑏 𝑇𝑐
(1)
(2)
where 𝜔𝐷 is the Debye frequency, 𝐷(𝐸𝑓 ) the density of states, 𝑉0 the deformation energy, and
𝑘𝑏 the Boltzmann constant. For the simulation of the deformation energy 𝑉0 , the FEM program
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ANSYS has been used. To this end, a simulation code containing the geometry of the unit cell
and the acting Coulomb forces on each ion was prepared for each superconductor.
2.1
Unit cell
For the construction of the unit cell, a knowledge of the spring constants between ions, the
masses of the ions and the crystal symmetry for both superconductors is required. The spring
constants can be calculated using the potential theory or measured by Raman spectroscopy.
Raman spectroscopy is based on the vibration frequency of an ionic pair; in the potential theory,
the spring constants can be calculated from the potential interactions between the ions in the
pair.
2.2
Coulomb forces
The Coulomb forces between a moving electron (with Fermi velocity) and the ions in the unit
cell depend on the positions of all species involved. The start position of the electron is assumed
to be far from the unit cell (between 10 and 30 times the lattice parameter a). The path on which
the electron moves lies on the CuO2 (RuO2) plane between oxygen and cupper (ruthenium) ion
in the [100] direction. A preliminary computation of the time-dependent Coulomb forces
between the moving electron and the different ions is performed while keeping the ion positions
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Superconductivity in Sr2RuO4 and La1.85Sr0.15CuO4
fixed, since in the final simulation their displacements should be suitably small. Due to the
different Fermi velocities and different charges of the ions in the unit cell of both
superconductors, the Coulomb forces for them are different.
3
Spring Constants
The spring constants for some ionic pairs in the unit cell of Sr2RuO4 have been calculated from
Raman spectroscopy [20]. To get other spring constants in the unit cell, a calculation with the
potential theory is required. To do that, potential constants are required which are listed in Table
4. OI stands for the oxygen ion located in the RuO2 plane whereas OII represents the apical
oxygen ion in the unit cell (Figure 1).
Pair
𝐴[eV]
Sr − OI
Sr − OII
Ru − OI
Ru − OII
OI − OI
OI − OII
OII − OII
1825
2250
2999
3874
2000
2000
2000
𝜌[Å]
0.318
0.318
0.260
0.260
0.284
0.284
0.284
𝐴[eV/Å6 ]
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-100
-100
-100
Table 4: The potential parameters for each ionic pair [21].
Ion
𝑍
𝑞𝑠ℎ𝑒𝑙𝑙
Ru
Sr
OI
OII
2.58
2
-1.52
-1.77
0.47
5.86
-3.25
-2.77
N
𝑘[ ]
m
8000/2000
3600
1800
1800
Table 5: The charge of the shell qshell, the total ion charge Z and the spring constant for each ion [21].
Due to the fact that only the values of the equilibrium distances for each ionic pair are available,
only the Coulomb interaction between the whole ions can be considered instead of the Coulomb
interactions between all parts of each ion (shells and cores). Therefore the Coulomb interaction
between ions 𝑉𝑐𝑜𝑢𝑙 , the van der Walls interaction 𝑉𝑉𝑊 and the short range interaction can be
calculated by the following equations:
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Superconductivity in Sr2RuO4 and La1.85Sr0.15CuO4
𝑍𝑖 𝑍𝑗
,
4𝜋𝜀0 𝑟𝑖𝑗
(3)
𝑉𝑉𝑊 = −𝐶𝑟𝑖𝑗 −6 ,
(4)
𝑉𝑐𝑜𝑢𝑙 =
𝑉𝑠ℎ𝑜𝑟𝑡 = 𝐴 exp (−
𝑟𝑖𝑗
).
𝜌
(5)
The total potential interaction is calculated as the sum of these interactions. Using the known
equilibrium distance 𝑟 in the second derivative of the total potential interaction (equation (6))
with respect to 𝑟𝑖𝑗 , the spring constants can be calculated with equation (7)Fehler!
Verweisquelle konnte nicht gefunden werden.. The results are given in Table 6.
𝑍𝑖 𝑍𝑗 𝑟𝑖𝑗
𝑟𝑖𝑗
𝜕 2 𝑉𝑡𝑜𝑡
𝐴
−8
=
−
−
42𝐶𝑟
+
exp
(−
),
𝑖𝑗
𝜕 2 𝑟𝑖𝑗
2𝜋𝜀0 𝑟𝑖𝑗 3
𝜌2
𝜌
𝑘𝑖𝑗 (𝑟) = −
(6)
𝑍𝑖 𝑍𝑗 𝑟
𝐴
𝑟
−8
−
42𝐶𝑟
+
exp
(−
).
2𝜋𝜀0 𝑟 3
𝜌2
𝜌
(7)
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N
Pair
𝑟[Å]
Spring constant [m]
Sr − OI
Sr − OII
Ru − OI
Ru − OII
OI − OI
OI − OII
OII − OII
2.687
2.736
1.936
2.068
2.739
2.833
3.873
64
21
173
90
57
67
21
Table 6: The equilibrium distance for each ionic pair and its spring constant calculated with the potential theory
as well as taken from Ref. [20].
The spring constants in the unit cell of La1.85Sr0.15CuO4 have been measured by Raman
spectroscopy [22] and are given in Table 7.
Copyright © 2016 SciResPub.
N
Pair
Spring constant [m]
Cu − OI
Cu − OII
La − OI
La − OII *
La − OII
La − La
La − Cu
OI − OI
OI − OII
OII − OII
85
20
160
105
50
30
10
20
4
7
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Superconductivity in Sr2RuO4 and La1.85Sr0.15CuO4
Table 7: The spring constants for each ionic pairs measured for La1.85Sr0.15CuO4 [22]. *: shifted by (a/2, a/2,
c/2).
4
4.1
Results
Simulation of the deformation energy 𝐕𝟎 and the binding energy 𝚫 in Sr2RuO4
The simulation has been performed for the three different types of Fermi electrons α, β and γ.
The results for the deformation energies are given in Table 8. The fractional volume 𝑉𝑜𝑙𝑓𝑟𝑎𝑐
corresponding to each type in the unit cell is also given in Table 8. Multiplying 𝑉𝑜𝑙𝑓𝑟𝑎𝑐 with
𝑉0,𝑖 , the amount of the deformation energy for each type is determined, which allows the
calculation of the binding energy and the superconducting constant for each type. From the
results for the superconducting constants in Table 8, it is obvious that the γ type plays the main
role regarding superconductivity, since its binding energy and superconducting constant are the
greatest. Due to the very small binding energies of the two other types, it can be assumed that
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the total binding energy and the total superconducting constant are equal to those of the γ
electrons. This result coincides with those of Ref. [7] and [8], which state that the γ electrons
should be responsible for superconductivity. The binding energy Δγ = 0.16 meV is close to the
measured values of Ref. [9] and [10]. The reasons why the binding energy Δγ is larger than the
other two in our simulations are the following: first, the Fermi velocity of the γ electrons is
lower than that of the α and β electrons; therefore, the acting time of the Coulomb force due to
the γ electrons on each ion in the unit cell is larger than for the others, which leads to a larger
deformation energy 𝑉0,𝛾 . Second, the density of states of the γ electrons is larger, increasing the
binding energy Δ due to its exponential dependence on the value of 𝐷𝑖 (𝐸𝑓 )𝑉0,𝑖 (equation (1)).
Sr2RuO4



𝑉𝑜𝑙𝑓𝑟𝑎𝑐
0.108
0.457
0.667
𝑉0,𝑖 [eV]
0.03
0.10
0.22
𝑉𝑜𝑙𝑓𝑟𝑎𝑐 × 𝑉0,𝑖 [eV]
0.003
0.046
0.147
Δi [meV]
1.13×10-40
2.08×10-6
0.16
𝑐𝑖
0
4×10-5
2.43
Table 8: Volume fraction, simulated deformation energy, binding energy and superconducting constant for each
type of Fermi electrons α, β and γ [12].
calculated
Ref. [9]
Ref. [10]
Δγ = 0.16 meV
Δγ = 0.15 meV
∆ = 0.14 meV
Table 9: Comparison between the calculated and measured binding energy.
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Superconductivity in Sr2RuO4 and La1.85Sr0.15CuO4
4.2
Simulation of the deformation energy 𝐕𝟎 and the binding energy 𝚫 in
La1.85Sr0.15CuO4
The same simulation has been performed and the simulated value of 𝑉0 has been used to
calculate the binding energy Δ with equation (1). The calculated values for the binding energy
Δ and the superconducting constant 𝑐 are smaller than the values in Table 3. This can be due to
the Fermi velocity, the density of states, or the spring constants. A small change in the first two
of these properties has a strong influence on the results of the simulation. Furthermore, the
values used correspond to theoretical calculations [16] instead of measurements. A large
inaccuracy in the results is therefore expected.
Since the potential energy of a simple harmonic oscillator depends on the spring constant, the
deformation energy of the unit cell depends on the spring constants in the unit cell, too. This
means that if the spring constants used don’t agree with their real values, the deformation energy
could differ as well. These could be a reason why the calculated superconducting constant does
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not coincide with the values from the different references in Table 3. Another reason could be,
of course, that the BCS theory doesn’t apply in this case and, if our result for the calculation of
the superconducting constant 𝑐 were correct, there would have to be a second pairing
mechanism to overcome the thermal fluctuation energy.
La1.85Sr0.15CuO4
m
𝑣𝑒 [ ] /106
s
0.314
1
]
eV
2.09
𝐷(𝐸𝑓 )[
𝑉0 [eV]
∆ [meV]
𝑐
0.31
1.5
0.91
Table 10: Electrical properties, simulated deformation energy, binding energy and superconducting constant for
La1.85Sr0.15CuO4.
5
Conclusion
It was interesting to perform the simulations for the two superconductors Sr2RuO4 and
La1.85Sr0.15CuO4, since they have the same crystal structure. But it must be said that their
electronic band structures are not similar, therefore their Fermi velocities and densities of states
are different. This leads to different deformation energies 𝑉0 for the unit cells, different binding
energies Δ and different critical temperatures 𝑇𝑐 according to the simulation. Other differences
lie in the charges and masses of the ions in the unit cell, and the different spring constants
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Superconductivity in Sr2RuO4 and La1.85Sr0.15CuO4
between the ions in each compound (compare Table 6 and Table 7), which is reflected on their
different Debye temperatures.
The results for the superconducting constants in the bottom table point out that the
superconductivity in Sr2RuO4 is BCS-like whereas the superconductivity in La1.85Sr0.15CuO4
could also correspond to the BCS theory or, on the other hand, an as yet unknown second paring
mechanism must be available.
Superconductor
Sr2RuO4
La1.85Sr0.15CuO4
𝑐
2.43
0.91
Table 11: Calculated superconducting constant in Sr2RuO4 and La1.85Sr0.15CuO4.
Acknowledgements
We would like to thank University of Stuttgart and Konrad-Adenaur-Stiftung for funding our
research.
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