Electronic structure of the noncentrosymmetric
superconductor Mg10Ir19B16
B Wiendlocha, J Tobola and S Kaprzyk
Faculty of Physics and Applied Computer Science, AGH University of Science
and Technology, Al. Mickiewicza 30, 30-059 Cracow, Poland
E-mail: bartekw@fatcat.ftj.agh.edu.pl
Abstract. Electronic structure of a novel superconducting noncentrosymmetric
compound Mg10Ir19B16 was calculated using the Korringa-Kohn-Rostoker
method. Electronic part of the electron-phonon coupling constant, McMillan-
Hopfield parameters, were calculated using the rigid-muffin-tin approximation
(RMTA). The magnitude of the electron-phonon coupling constant λ, analysing
atomic contributions, is discussed. Our results show, that superconductivity in
Mg10Ir19B16 is presumably mediated by electron-phonon interaction.
PACS numbers: 74.25.Jb
1. Introduction
The interest in noncentrosymmetric structures, exhibiting superconductivity, has
grown up in the past years. There are only a few examples of this type, which
belong to various classes of materials, e.g. an antiferromagnetic heavy fermion
system CePt3Si [1], ferromagnetic uranium compound UIr, superconducting under
pressure [2], or non-magnetic ternary borides Li2Pd3B [3] and Li2Pt3B [4]. The main
reason, why these systems are especially attracting, is related to the role of inversion
symmetry in electron pairing. The absence of inversion symmetry may suppress the
triplet pairing or mix singlet and triplet symmetry [5, 6, 7].
Very recently, Klimczuk and co-workers [8] synthesised a new type of intermetallic
light-element based compound Mg10Ir19B16, exhibiting superconductivity near 5 K.
This novel material also belongs to the rare noncentrosymmetric structures, and
crystallises in a large and rather complex bcc cell (space group I-43m) [8]. Mg10Ir19B16
is partly similar to the Li2(Pd,Pt)3B system, since these structures contain an alkali
metal (Li, Mg), boron, and heavy transition metal (Pd, Pt, Ir).
In this work we intend to start the discussion on superconductivity mechanism in
this unusual compound, analysing the electronic structure and the strength of electron-
phonon coupling (EPC). Assuming the BCS-type behaviour in Mg10Ir19B16, we study
whether superconductivity is driven by light boron sublattices, like e.g. in MgB2, or
by heavy transition metal atoms, as suggested for Li2Pd3B [9].
1.1. Computational details
Electronic structure calculations were performed using the Korringa-Kohn-Rostoker
(KKR) multiple scattering method [10, 11]. The crystal potential was constructed in
http://arxiv.org/abs/0704.1295v1
Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 2
the framework of the local density approximation (LDA), using von Barth and Hedin
formula [12] for the exchange-correlation part. For all atoms angular momentum
cut-off lmax = 3 was set, k-point mesh in the irreducible part of Brillouin zone
(IRBZ) contained over 200 points. Densities of states (DOS) were computed using the
tetrahedron k-space integration technique, generating over 600 tetrahedrons in IRBZ.
Due to the high atomic number of iridium (Z = 77) semi-relativistic calculations were
performed, but neglecting the spin-orbit coupling, which is commented at the end of
this paper.
Figure 1. Unit cell of Mg10Ir19B16. The lack of inversion centre is clearly
seen e.g. along the main diagonal, where B1 and Mg1 atoms break the inversion
symmetry. Generated by XCRYSDEN [27].
As far as the crystal structure is concerned, experimental lattice constant
a = 10.568 Å and atomic positions [8] were accounted for the computation (for clarity
also shown in table 2). Atoms in the unit cell were surrounded by muffin-tin (MT)
spheres with following radii: RMg = 2.82, RIr = 2.50, RB = 1.40 (in atomic units),
filling about 60% of the cell volume. In the primitive cell of this system, 45 atoms
occupy 7 inequivalent sites, which all are listed in table 2. The noncentrosymmetricity
of this system is important, and cannot be regarded as the effect of a lattice distortion,
as observed e.g. in UIr [9]. The inversion symmetry is broken by both boron sublattices
(B1 and B2), as well as Mg1 and Ir3 sites, thus the crystal has a half of symmetry
operations of the cubic group Oh, i.e. only 24 operations. Among of all sublattices,
the positions of iridium atoms are the closest to have inversion symmetry. Since Ir3
occupies (x, z, z) sites, with x ≃ 0.07, the full cubic symmetry is restored after shifting
this position to (0, z, z).
The analysis of superconducting properties is based on the computed McMillan-
Hopfield (MH) parameters η [13, 14] which determine electronic part of the electron-
phonon interaction, and directly enter the formula for the electron-phonon coupling
constant λ:
. (1)
Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 3
-1.0 -0.75 -0.5 -0.25 0.0 0.25
Energy (Ry)
Total DOS
-0.04 -0.02 0.0 0.02 0.04
Energy (Ry)
Total DOS
Figure 2. (a) Total DOS of Mg10Ir19B16. (b) Zoom near the Fermi level
(EF = 0) with atomic contributions.
In equation (1), ηi is the MH parameter for each nonequivalent atom i which is
characterised by the atomic massMi and averaged squared vibrational frequency 〈ω
This equation divides the electron-phonon coupling constant into site-dependent parts,
thus it allows to justify which sublattice gives the most important contribution to the
total λ. Calculations of MH parameters were preformed using the rigid muffin tin
approximation (RMTA) [15, 16, 17, 18], which gives the following expression for η at
each atomic site i:
(2l+ 2)ni
(EF )n
(EF )
(2l + 1)(2l+ 3)N(EF )
dr r2Ril(r)
dVi(r)
Ril+1(r)
. (2)
Here, l is the angular momentum number, nil(EF ) is the partial (angular-decomposed)
density of states per spin at the Fermi energy (EF ), R
(r) is a regular solution of the
radial Schrödinger equation, normalised to unity inside the MT sphere of radius RiMT ,
and Vi(r) is the self-consistent, spherically-symmetric potential. N(EF ) is the total
DOS at EF per spin and per cell. For more detailed discussion of approximations
involved in this method, see e.g. [19, 20] and references therein.
2. Results and discussion
The calculated total and atomic densities of states functions are presented in figure 2.
As one could expect, electronic structure is dominated by iridium 5d states. The total
DOS curve may be characterised as a collection of numerous van Hove singularities,
reflecting the large number of atoms in the unit cell and various interatomic distances.
The site-decomposed DOS are shown in figure 3, and their values at the Fermi level
are gathered in table 1. The total DOS at Fermi level is about N(EF ) ≃ 150 Ry
−1 per
formula unit. The densities at EF , calculated per atom, are rather low (average value
per atom: 3.3 Ry−1), and generally EF is located outside of dominating peaks of DOS,
on a small decreasing slope of Ir DOS. Strong hybridisation of d states of Ir and p
states of B and Mg is manifested in the separation of bonding and anti-bonding states,
Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 4
Table 1. Densities of states n(EF ), l-decomposed DOS nl (Ry
−1/spin per atom)
McMillan-Hopfield parameters η, MH parameters for each scattering channel
ηl,l+1 (mRy/a.u.
2 per atom), and summary MH parameters for each site wi × ηi
(mRy/a.u.2 per site). wi is the number of atoms occupying site in primitive cell.
Atom wi n(EF ) ns np nd nf ηi η
wi × ηi
Ir1 1 2.77 0.15 0.88 1.86 0.03 2.8 0.1 1.4 1.3 2.8
Ir2 6 3.16 0.07 0.34 2.71 0.03 3.3 0.0 0.8 2.5 19.8
Ir3 12 2.41 0.06 0.34 1.97 0.03 2.6 0.0 0.6 2.0 31.2
B1 4 0.40 0.03 0.35 0.01 0.00 1.2 0.0 1.2 0.0 4.8
B2 12 0.27 0.04 0.22 0.01 0.00 0.6 0.0 0.6 0.0 7.2
Mg1 4 0.98 0.17 0.69 0.10 0.03 0.7 0.5 0.2 0.0 2.8
Mg2 6 0.67 0.09 0.41 0.14 0.03 0.2 0.1 0.1 0.0 1.2
and EF is located in the DOS valley (figure 2 and figure 3). The interatomic distances,
listed in table 2, supports the enhanced p-d hybridisation, especially between Ir and
B atoms (the smallest distances, 2.1 - 2.2 Å).
It is also interesting to compare the computed site-decomposed Ir densities, to
the DOS of metallic fcc iridium, which is presented in figure 3(d). The shape of lower
part of DOS in Ir2 and Ir3 is quite similar to the case of Ir-fcc. This is probably due
to the fact, that the Ir2-Ir3 and Ir3-Ir3 coordination, as well as interatomic distances,
are very close to the fcc phase (Ir-Ir distance is ∼ 2.7 Å in the aforementioned cases,
see table 2). However, the n(EF ) values on Ir atoms are much lower, comparing
to fcc structure (n(EF ) ≃ 6.3 Ry
−1/spin in Ir-fcc), being the effect of enhanced
hybridisation near EF . Noteworthy, Ir1 has a quite different atomic coordination,
with respect to Ir2 and Ir3 positions, being surrounded practically only by 4 boron
atoms (B1). It is clearly reflected by the apparently different DOS shape below EF .
Ir1a)
Ir2b)
Ir3c)
fcc Ird)
-0.6 -0.3 0.0 0.3
Energy (Ry)
-0.6 -0.3 0.0 0.3
Energy (Ry)
-0.6 -0.3 0.0 0.3
Energy (Ry)
Mg1g)
-0.6 -0.3 0.0 0.3
Energy (Ry)
Mg2h)
Figure 3. Site-decomposed DOS of Mg10Ir19B16 (EF = 0). The total DOS of
fcc iridium metal is given for comparison in panel (d).
Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 5
Table 2. The smallest interatomic distances between atoms (in Å) and atomic
positions in Mg10Ir19B16. The Ir-Ir distance in Ir-fcc is about 2.7 Å, as between
Ir2-Ir3 and Ir3-Ir3.
Ir1 Ir2 Ir3 B1 B2 Mg1 Mg2 Atomic position Site
Ir1 9.1 5.9 3.9 2.1 5.0 3.1 3.6 (0, 0, 0) (2a)
Ir2 3.8 2.7 4.5 2.2 4.0 3.1 (0, 0.25, 0.5) (12d)
Ir3 2.7 2.1 2.2 4.0 2.9 (0.0702, 0.2525, 0.2525) (24g)
B1 3.4 3.3 3.0 3.0 (0.3331, 0.3331, 0.3331) (8c)
B2 3.3 2.6 2.5 (0, 0, 0.3473) (24g)
Mg1 5.0 3.1 (0.1127, 0.1127, 0.1127) (8c)
Mg2 3.1 (0.1639, 0.1639, 0.4140) (12e)
Electronic dispersion curves near the Fermi level are presented in figure 4. The
bands located below −0.1 Ry (partly showed) are very flat, and form the narrow large
DOS peaks, seen already in figure 2. Conversely, the bands that cross EF are quite
dispersive, which results in the low DOS in this energy range. Noteworthy, there is an
energy gap along the P-N direction (parallel to the kz axis in the reciprocal space).
This may suggest some anisotropic transport properties of this compound.
The calculated McMillan-Hopfield factors for all sites, with contributions from
each scattering channel (l → l+1), are presented in table 1. Estimation of the electron-
phonon coupling constant λ, using MH parameters, requires also the knowledge
of average phonon frequencies. For such a large structure, containing 90 atoms
in the cubic unit cell, phonon spectra calculations are difficult to be carried out.
Nevertheless we can try to investigate the strength of electron-phonon interaction
assuming reasonable values of 〈ω2i 〉 and studying their influence on estimated λ values.
Debye frequencies Θ of monoatomic crystals of iridium (fcc), boron (rhomboedral)
and magnesium (hcp) may be helpful for choosing sensible phonon frequency range
for our discussion. At first we assume, that the same type of atoms, occupying
different sites in the cell, have similar average vibrational frequencies. Iridium, as
the heaviest element, is expected to have the lowest 〈ω2i 〉, Debye frequency of metallic
Figure 4. Dispersion curves near the EF = 0 in Mg10Ir19B16.
Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 6
15 20 25 30 35 40
(meV)
0.3 (a) Ir1
40 50 60 70 80 90 100
(meV)
15 20 25 30 35 40
(meV)
2 ΘB/
2 ΘMg/
Figure 5. Contributions to the electron-phonon coupling constant λ in
Mg10Ir19B16 from iridium (a), boron (b) and magnesium (c) as a function of a root
of average square frequency. Top curve on each graph is a sum of contributions
from particular sites. Vertical lines mark the value of Θi/
iridium is about ΘIr ≃ 36 meV (420 K) [21]. In contrast, the lightest boron is
certainly expected to have the highest phonon frequencies, and for crystalline boron
ΘB ≃ 100 meV (1200 K) [22]. Finally, magnesium average frequencies are expected
to locate between the values of iridium and boron. Debye frequency of crystalline Mg
is rather low: ΘMg ≃ 34 meV (400 K) [21]. We may also recall, that average square
phonon frequency is often estimated as 〈ω2〉 ≃ 1
Θ2, which is a good approximation
in monoatomic structures. In our case, these values may also be helpful for choosing
feasible range of 〈ω2i 〉.
Figure 5 presents estimated electron-phonon coupling constant λ, associated with
particular crystal sites, plotted as a function of average square phonon frequency. For
each type of atoms, a wide frequency range was chosen, to illustrate the changeability
of λ. In the case of iridium, the largest contribution to λ comes from Ir3 sublattice,
due to the large population of this site. Among boron and magnesium sites, the B2
and Mg1 atoms provide the largest contributions. However, one has to remember that
this comparison is valid only if we assume identical frequencies for the same atoms
at different sites. In this simplified analysis, we may also plot the overall coupling
constant for constituent atoms, by adding the contributions from each site, which is
illustrated by solid lines in figure 5.
As we can see, the obtained partial coupling constants are not high for each atom.
At the moment, we are not aware of any experimental findings of the EPC constant
in this compound. However, we can try to estimate the range of ”experimental” λ,
analysing the magnitude of the observed critical temperature. If we assume, that
we are dealing with BCS-type superconductivity, we may substitute the experimental
value of TC = 4.5 K into the McMillan formula for TC [13]:
1.04(1 + λ)
λ− µ⋆(1 + 0.62λ)
. (3)
However, because the Debye temperature Θ of Mg10Ir19B16 is not known yet, we
plot λ in figure 6 as a function of Θ, for typical values of Coulomb pseudopotential
parameter µ⋆. The resulting EPC constant λ varies between 0.5 and 0.75, and for
e.g. Θ = 250 K we obtain λ ≃ 0.60, whereas for Θ = 350 K we get λ ≃ 0.55
(for µ⋆ = 0.11), as one can see in the figure 6. Thus, if Mg10Ir19B16 is treated as
Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 7
200 250 300 350 400 450 500
Debye temperature (K)
= 0.11
= 0.13
= 0.15
Figure 6. ”Experimental” value of the electron-phonon coupling constant λ,
evaluated from the McMillan formula, using observed TC = 4.5 K, plotted as a
function of Debye temperature Θ for three values of Coulomb pseudopotential
parameters µ⋆.
a conventional superconductor, the lower limit of λ is expected to be 0.5. In order
to get this value from our RMTA calculations, relatively low average frequencies for
all constituent atoms are required. If we take the following
〈ω2i 〉: 14 meV for Ir,
50 meV for B and 20 meV for Mg, we obtain λIr ≃ 0.3, λB ≃ 0.1 and λMg ≃ 0.1,
which gives the expected lower limit of EPC constant (λ ≃ 0.5). If higher values of λ
are experimentally observed, it will indicate either additional contributions, neglected
within the RMTA framework, or even lower phonon frequencies, than the values used
in the aforementioned estimations. Especially iridium contribution is sensitive to the
change of
〉 parameter, i.e. increasing it to 25 meV (probably an upper limit)
results in decrease of λIr to 0.1, that is to the value found for other sublattices.
Generally, our calculations indicates, that iridium sublattices seem to be the most
important for the onset of superconductivity in Mg10Ir19B16.
Finally, we shortly comment the possible influence of the lack of inversion
symmetry and spin-orbit (SO) interaction on the electronic structure and
superconductivity. The absence of a centre of inversion in a superconductor affects
the symmetry of superconducting state, allowing for an admixture of singlet and
triplet components [6, 7]. Because the singlet pairing is based on the time-reversal
symmetry [23], which is present as long as the compound is not magnetic, mainly
the triplet channel is affected by the lack of inversion, and the superconductivity may
be even suppressed, see e.g. [5, 6, 7]. The SO interaction, which mixes the initial
spin-up and spin-down electronic states, plays an additional role. It was found, that
it may control the mixing of parity of the superconducting state [7]. This seems to be
the case of Li2Pd3B and Li2Pt3B compounds, where specific heat [24] and NMR [25]
measurements strongly support phonon-mediated isotropic superconductivity, while
penetration depth measurements suggests an admixture of spin-singlet and triplet
components in the superconducting energy gap [26], with larger triplet component in
the Pt-case. This kind of experimental study for Mg10Ir19B16 should be prior to the
theoretical discussion of gap symmetry in this compound.
As far as the band structure of Mg10Ir19B16 is concerned, the modifications due
to the SO interaction are not expected to significantly affect the obtained values of
DOS and MH parameters. For metallic fcc iridium, our value of MH parameter,
Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 8
calculated also neglecting SO interaction, ηIr ≃ 135 mRy/(a.u.)
2, gives the correct
magnitude of the electron-phonon coupling constant λ = 0.32, comparing to the
observed λ = 0.34 [13]. Here, the average square frequency was estimated from the
formula 〈ω2〉 ≃ 1
Θ2, using the Debye temperature Θ = 420 K.
3. Summary and conclusions
The results of LDA electronic structure calculations of new Mg10Ir19B16
superconductor were presented. The main contributions to densities of states
near EF are provided by iridium atoms. The electron-phonon coupling constant
λ was roughly estimated, using the calculated McMillan-Hopfield parameters and
qualitative discussion of average phonon frequencies. We discussed the relation of
the experimental transition temperature and the magnitude of EPC coupling. Within
the rigid-muffin-tin approximation, the main contribution to λ comes from iridium,
with smaller contributions from boron and magnesium. If any information about
the dynamic properties of atoms in this compound become available, our analysis
presented of figure 5 will allow to find better theoretical estimation of λ. The location
of EF on the slope of Ir DOS peak leads to rough suggestion, that hole doping on
iridium sites, e.g. with rhodium, may increase the densities and MH parameters.
Acknowledgments
We would like to thank dr Tomasz Klimczuk for helpful discussions. This work was
partly supported by the Polish Ministry of Science and Higher Education (PhD grant).
References
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Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 9
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[27] A. Kokalj 1999 J. Mol. Graphics Modelling 17, 176. Code available from
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http://www.xcrysden.org/
	Introduction
	Computational details
	Results and discussion
	Summary and conclusions

  Electronic structure of a novel superconducting noncentrosymmetric compound
Mg10Ir19B16 was calculated using the Korringa-Kohn-Rostoker method. Electronic
part of the electron-phonon coupling constant, McMillan-Hopfield parameters,
were calculated using the rigid-muffin-tin approximation (RMTA). The magnitude
of the electron-phonon coupling constant, analysing atomic contributions, is
discussed. Our results show, that superconductivity in Mg10Ir19B16 is
presumably mediated by electron-phonon interaction.

Introduction
The interest in noncentrosymmetric structures, exhibiting superconductivity, has
grown up in the past years. There are only a few examples of this type, which
belong to various classes of materials, e.g. an antiferromagnetic heavy fermion
system CePt3Si [1], ferromagnetic uranium compound UIr, superconducting under
pressure [2], or non-magnetic ternary borides Li2Pd3B [3] and Li2Pt3B [4]. The main
reason, why these systems are especially attracting, is related to the role of inversion
symmetry in electron pairing. The absence of inversion symmetry may suppress the
triplet pairing or mix singlet and triplet symmetry [5, 6, 7].
Very recently, Klimczuk and co-workers [8] synthesised a new type of intermetallic
light-element based compound Mg10Ir19B16, exhibiting superconductivity near 5 K.
This novel material also belongs to the rare noncentrosymmetric structures, and
crystallises in a large and rather complex bcc cell (space group I-43m) [8]. Mg10Ir19B16
is partly similar to the Li2(Pd,Pt)3B system, since these structures contain an alkali
metal (Li, Mg), boron, and heavy transition metal (Pd, Pt, Ir).
In this work we intend to start the discussion on superconductivity mechanism in
this unusual compound, analysing the electronic structure and the strength of electron-
phonon coupling (EPC). Assuming the BCS-type behaviour in Mg10Ir19B16, we study
whether superconductivity is driven by light boron sublattices, like e.g. in MgB2, or
by heavy transition metal atoms, as suggested for Li2Pd3B [9].
1.1. Computational details
Electronic structure calculations were performed using the Korringa-Kohn-Rostoker
(KKR) multiple scattering method [10, 11]. The crystal potential was constructed in
http://arxiv.org/abs/0704.1295v1
Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 2
the framework of the local density approximation (LDA), using von Barth and Hedin
formula [12] for the exchange-correlation part. For all atoms angular momentum
cut-off lmax = 3 was set, k-point mesh in the irreducible part of Brillouin zone
(IRBZ) contained over 200 points. Densities of states (DOS) were computed using the
tetrahedron k-space integration technique, generating over 600 tetrahedrons in IRBZ.
Due to the high atomic number of iridium (Z = 77) semi-relativistic calculations were
performed, but neglecting the spin-orbit coupling, which is commented at the end of
this paper.
Figure 1. Unit cell of Mg10Ir19B16. The lack of inversion centre is clearly
seen e.g. along the main diagonal, where B1 and Mg1 atoms break the inversion
symmetry. Generated by XCRYSDEN [27].
As far as the crystal structure is concerned, experimental lattice constant
a = 10.568 Å and atomic positions [8] were accounted for the computation (for clarity
also shown in table 2). Atoms in the unit cell were surrounded by muffin-tin (MT)
spheres with following radii: RMg = 2.82, RIr = 2.50, RB = 1.40 (in atomic units),
filling about 60% of the cell volume. In the primitive cell of this system, 45 atoms
occupy 7 inequivalent sites, which all are listed in table 2. The noncentrosymmetricity
of this system is important, and cannot be regarded as the effect of a lattice distortion,
as observed e.g. in UIr [9]. The inversion symmetry is broken by both boron sublattices
(B1 and B2), as well as Mg1 and Ir3 sites, thus the crystal has a half of symmetry
operations of the cubic group Oh, i.e. only 24 operations. Among of all sublattices,
the positions of iridium atoms are the closest to have inversion symmetry. Since Ir3
occupies (x, z, z) sites, with x ≃ 0.07, the full cubic symmetry is restored after shifting
this position to (0, z, z).
The analysis of superconducting properties is based on the computed McMillan-
Hopfield (MH) parameters η [13, 14] which determine electronic part of the electron-
phonon interaction, and directly enter the formula for the electron-phonon coupling
constant λ:
. (1)
Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 3
-1.0 -0.75 -0.5 -0.25 0.0 0.25
Energy (Ry)
Total DOS
-0.04 -0.02 0.0 0.02 0.04
Energy (Ry)
Total DOS
Figure 2. (a) Total DOS of Mg10Ir19B16. (b) Zoom near the Fermi level
(EF = 0) with atomic contributions.
In equation (1), ηi is the MH parameter for each nonequivalent atom i which is
characterised by the atomic massMi and averaged squared vibrational frequency 〈ω
This equation divides the electron-phonon coupling constant into site-dependent parts,
thus it allows to justify which sublattice gives the most important contribution to the
total λ. Calculations of MH parameters were preformed using the rigid muffin tin
approximation (RMTA) [15, 16, 17, 18], which gives the following expression for η at
each atomic site i:
(2l+ 2)ni
(EF )n
(EF )
(2l + 1)(2l+ 3)N(EF )
dr r2Ril(r)
dVi(r)
Ril+1(r)
. (2)
Here, l is the angular momentum number, nil(EF ) is the partial (angular-decomposed)
density of states per spin at the Fermi energy (EF ), R
(r) is a regular solution of the
radial Schrödinger equation, normalised to unity inside the MT sphere of radius RiMT ,
and Vi(r) is the self-consistent, spherically-symmetric potential. N(EF ) is the total
DOS at EF per spin and per cell. For more detailed discussion of approximations
involved in this method, see e.g. [19, 20] and references therein.
2. Results and discussion
The calculated total and atomic densities of states functions are presented in figure 2.
As one could expect, electronic structure is dominated by iridium 5d states. The total
DOS curve may be characterised as a collection of numerous van Hove singularities,
reflecting the large number of atoms in the unit cell and various interatomic distances.
The site-decomposed DOS are shown in figure 3, and their values at the Fermi level
are gathered in table 1. The total DOS at Fermi level is about N(EF ) ≃ 150 Ry
−1 per
formula unit. The densities at EF , calculated per atom, are rather low (average value
per atom: 3.3 Ry−1), and generally EF is located outside of dominating peaks of DOS,
on a small decreasing slope of Ir DOS. Strong hybridisation of d states of Ir and p
states of B and Mg is manifested in the separation of bonding and anti-bonding states,
Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 4
Table 1. Densities of states n(EF ), l-decomposed DOS nl (Ry
−1/spin per atom)
McMillan-Hopfield parameters η, MH parameters for each scattering channel
ηl,l+1 (mRy/a.u.
2 per atom), and summary MH parameters for each site wi × ηi
(mRy/a.u.2 per site). wi is the number of atoms occupying site in primitive cell.
Atom wi n(EF ) ns np nd nf ηi η
wi × ηi
Ir1 1 2.77 0.15 0.88 1.86 0.03 2.8 0.1 1.4 1.3 2.8
Ir2 6 3.16 0.07 0.34 2.71 0.03 3.3 0.0 0.8 2.5 19.8
Ir3 12 2.41 0.06 0.34 1.97 0.03 2.6 0.0 0.6 2.0 31.2
B1 4 0.40 0.03 0.35 0.01 0.00 1.2 0.0 1.2 0.0 4.8
B2 12 0.27 0.04 0.22 0.01 0.00 0.6 0.0 0.6 0.0 7.2
Mg1 4 0.98 0.17 0.69 0.10 0.03 0.7 0.5 0.2 0.0 2.8
Mg2 6 0.67 0.09 0.41 0.14 0.03 0.2 0.1 0.1 0.0 1.2
and EF is located in the DOS valley (figure 2 and figure 3). The interatomic distances,
listed in table 2, supports the enhanced p-d hybridisation, especially between Ir and
B atoms (the smallest distances, 2.1 - 2.2 Å).
It is also interesting to compare the computed site-decomposed Ir densities, to
the DOS of metallic fcc iridium, which is presented in figure 3(d). The shape of lower
part of DOS in Ir2 and Ir3 is quite similar to the case of Ir-fcc. This is probably due
to the fact, that the Ir2-Ir3 and Ir3-Ir3 coordination, as well as interatomic distances,
are very close to the fcc phase (Ir-Ir distance is ∼ 2.7 Å in the aforementioned cases,
see table 2). However, the n(EF ) values on Ir atoms are much lower, comparing
to fcc structure (n(EF ) ≃ 6.3 Ry
−1/spin in Ir-fcc), being the effect of enhanced
hybridisation near EF . Noteworthy, Ir1 has a quite different atomic coordination,
with respect to Ir2 and Ir3 positions, being surrounded practically only by 4 boron
atoms (B1). It is clearly reflected by the apparently different DOS shape below EF .
Ir1a)
Ir2b)
Ir3c)
fcc Ird)
-0.6 -0.3 0.0 0.3
Energy (Ry)
-0.6 -0.3 0.0 0.3
Energy (Ry)
-0.6 -0.3 0.0 0.3
Energy (Ry)
Mg1g)
-0.6 -0.3 0.0 0.3
Energy (Ry)
Mg2h)
Figure 3. Site-decomposed DOS of Mg10Ir19B16 (EF = 0). The total DOS of
fcc iridium metal is given for comparison in panel (d).
Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 5
Table 2. The smallest interatomic distances between atoms (in Å) and atomic
positions in Mg10Ir19B16. The Ir-Ir distance in Ir-fcc is about 2.7 Å, as between
Ir2-Ir3 and Ir3-Ir3.
Ir1 Ir2 Ir3 B1 B2 Mg1 Mg2 Atomic position Site
Ir1 9.1 5.9 3.9 2.1 5.0 3.1 3.6 (0, 0, 0) (2a)
Ir2 3.8 2.7 4.5 2.2 4.0 3.1 (0, 0.25, 0.5) (12d)
Ir3 2.7 2.1 2.2 4.0 2.9 (0.0702, 0.2525, 0.2525) (24g)
B1 3.4 3.3 3.0 3.0 (0.3331, 0.3331, 0.3331) (8c)
B2 3.3 2.6 2.5 (0, 0, 0.3473) (24g)
Mg1 5.0 3.1 (0.1127, 0.1127, 0.1127) (8c)
Mg2 3.1 (0.1639, 0.1639, 0.4140) (12e)
Electronic dispersion curves near the Fermi level are presented in figure 4. The
bands located below −0.1 Ry (partly showed) are very flat, and form the narrow large
DOS peaks, seen already in figure 2. Conversely, the bands that cross EF are quite
dispersive, which results in the low DOS in this energy range. Noteworthy, there is an
energy gap along the P-N direction (parallel to the kz axis in the reciprocal space).
This may suggest some anisotropic transport properties of this compound.
The calculated McMillan-Hopfield factors for all sites, with contributions from
each scattering channel (l → l+1), are presented in table 1. Estimation of the electron-
phonon coupling constant λ, using MH parameters, requires also the knowledge
of average phonon frequencies. For such a large structure, containing 90 atoms
in the cubic unit cell, phonon spectra calculations are difficult to be carried out.
Nevertheless we can try to investigate the strength of electron-phonon interaction
assuming reasonable values of 〈ω2i 〉 and studying their influence on estimated λ values.
Debye frequencies Θ of monoatomic crystals of iridium (fcc), boron (rhomboedral)
and magnesium (hcp) may be helpful for choosing sensible phonon frequency range
for our discussion. At first we assume, that the same type of atoms, occupying
different sites in the cell, have similar average vibrational frequencies. Iridium, as
the heaviest element, is expected to have the lowest 〈ω2i 〉, Debye frequency of metallic
Figure 4. Dispersion curves near the EF = 0 in Mg10Ir19B16.
Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 6
15 20 25 30 35 40
(meV)
0.3 (a) Ir1
40 50 60 70 80 90 100
(meV)
15 20 25 30 35 40
(meV)
2 ΘB/
2 ΘMg/
Figure 5. Contributions to the electron-phonon coupling constant λ in
Mg10Ir19B16 from iridium (a), boron (b) and magnesium (c) as a function of a root
of average square frequency. Top curve on each graph is a sum of contributions
from particular sites. Vertical lines mark the value of Θi/
iridium is about ΘIr ≃ 36 meV (420 K) [21]. In contrast, the lightest boron is
certainly expected to have the highest phonon frequencies, and for crystalline boron
ΘB ≃ 100 meV (1200 K) [22]. Finally, magnesium average frequencies are expected
to locate between the values of iridium and boron. Debye frequency of crystalline Mg
is rather low: ΘMg ≃ 34 meV (400 K) [21]. We may also recall, that average square
phonon frequency is often estimated as 〈ω2〉 ≃ 1
Θ2, which is a good approximation
in monoatomic structures. In our case, these values may also be helpful for choosing
feasible range of 〈ω2i 〉.
Figure 5 presents estimated electron-phonon coupling constant λ, associated with
particular crystal sites, plotted as a function of average square phonon frequency. For
each type of atoms, a wide frequency range was chosen, to illustrate the changeability
of λ. In the case of iridium, the largest contribution to λ comes from Ir3 sublattice,
due to the large population of this site. Among boron and magnesium sites, the B2
and Mg1 atoms provide the largest contributions. However, one has to remember that
this comparison is valid only if we assume identical frequencies for the same atoms
at different sites. In this simplified analysis, we may also plot the overall coupling
constant for constituent atoms, by adding the contributions from each site, which is
illustrated by solid lines in figure 5.
As we can see, the obtained partial coupling constants are not high for each atom.
At the moment, we are not aware of any experimental findings of the EPC constant
in this compound. However, we can try to estimate the range of ”experimental” λ,
analysing the magnitude of the observed critical temperature. If we assume, that
we are dealing with BCS-type superconductivity, we may substitute the experimental
value of TC = 4.5 K into the McMillan formula for TC [13]:
1.04(1 + λ)
λ− µ⋆(1 + 0.62λ)
. (3)
However, because the Debye temperature Θ of Mg10Ir19B16 is not known yet, we
plot λ in figure 6 as a function of Θ, for typical values of Coulomb pseudopotential
parameter µ⋆. The resulting EPC constant λ varies between 0.5 and 0.75, and for
e.g. Θ = 250 K we obtain λ ≃ 0.60, whereas for Θ = 350 K we get λ ≃ 0.55
(for µ⋆ = 0.11), as one can see in the figure 6. Thus, if Mg10Ir19B16 is treated as
Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 7
200 250 300 350 400 450 500
Debye temperature (K)
= 0.11
= 0.13
= 0.15
Figure 6. ”Experimental” value of the electron-phonon coupling constant λ,
evaluated from the McMillan formula, using observed TC = 4.5 K, plotted as a
function of Debye temperature Θ for three values of Coulomb pseudopotential
parameters µ⋆.
a conventional superconductor, the lower limit of λ is expected to be 0.5. In order
to get this value from our RMTA calculations, relatively low average frequencies for
all constituent atoms are required. If we take the following
〈ω2i 〉: 14 meV for Ir,
50 meV for B and 20 meV for Mg, we obtain λIr ≃ 0.3, λB ≃ 0.1 and λMg ≃ 0.1,
which gives the expected lower limit of EPC constant (λ ≃ 0.5). If higher values of λ
are experimentally observed, it will indicate either additional contributions, neglected
within the RMTA framework, or even lower phonon frequencies, than the values used
in the aforementioned estimations. Especially iridium contribution is sensitive to the
change of
〉 parameter, i.e. increasing it to 25 meV (probably an upper limit)
results in decrease of λIr to 0.1, that is to the value found for other sublattices.
Generally, our calculations indicates, that iridium sublattices seem to be the most
important for the onset of superconductivity in Mg10Ir19B16.
Finally, we shortly comment the possible influence of the lack of inversion
symmetry and spin-orbit (SO) interaction on the electronic structure and
superconductivity. The absence of a centre of inversion in a superconductor affects
the symmetry of superconducting state, allowing for an admixture of singlet and
triplet components [6, 7]. Because the singlet pairing is based on the time-reversal
symmetry [23], which is present as long as the compound is not magnetic, mainly
the triplet channel is affected by the lack of inversion, and the superconductivity may
be even suppressed, see e.g. [5, 6, 7]. The SO interaction, which mixes the initial
spin-up and spin-down electronic states, plays an additional role. It was found, that
it may control the mixing of parity of the superconducting state [7]. This seems to be
the case of Li2Pd3B and Li2Pt3B compounds, where specific heat [24] and NMR [25]
measurements strongly support phonon-mediated isotropic superconductivity, while
penetration depth measurements suggests an admixture of spin-singlet and triplet
components in the superconducting energy gap [26], with larger triplet component in
the Pt-case. This kind of experimental study for Mg10Ir19B16 should be prior to the
theoretical discussion of gap symmetry in this compound.
As far as the band structure of Mg10Ir19B16 is concerned, the modifications due
to the SO interaction are not expected to significantly affect the obtained values of
DOS and MH parameters. For metallic fcc iridium, our value of MH parameter,
Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 8
calculated also neglecting SO interaction, ηIr ≃ 135 mRy/(a.u.)
2, gives the correct
magnitude of the electron-phonon coupling constant λ = 0.32, comparing to the
observed λ = 0.34 [13]. Here, the average square frequency was estimated from the
formula 〈ω2〉 ≃ 1
Θ2, using the Debye temperature Θ = 420 K.
3. Summary and conclusions
The results of LDA electronic structure calculations of new Mg10Ir19B16
superconductor were presented. The main contributions to densities of states
near EF are provided by iridium atoms. The electron-phonon coupling constant
λ was roughly estimated, using the calculated McMillan-Hopfield parameters and
qualitative discussion of average phonon frequencies. We discussed the relation of
the experimental transition temperature and the magnitude of EPC coupling. Within
the rigid-muffin-tin approximation, the main contribution to λ comes from iridium,
with smaller contributions from boron and magnesium. If any information about
the dynamic properties of atoms in this compound become available, our analysis
presented of figure 5 will allow to find better theoretical estimation of λ. The location
of EF on the slope of Ir DOS peak leads to rough suggestion, that hole doping on
iridium sites, e.g. with rhodium, may increase the densities and MH parameters.
Acknowledgments
We would like to thank dr Tomasz Klimczuk for helpful discussions. This work was
partly supported by the Polish Ministry of Science and Higher Education (PhD grant).
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	Introduction
	Computational details
	Results and discussion
	Summary and conclusions
