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<strong>Computational</strong> <strong>Approaches</strong> <strong>to</strong> <strong>Superconductivity</strong>, <strong>Research</strong> <strong>Report</strong><br />
2006-2012<br />
The main activity of our Minerva <strong>Research</strong> Group, estabilished in July 2009, is the study of complex solids, in<br />
particular superconduc<strong>to</strong>rs, using ab-initio DFT calculations. In collaboration with other groups, we combine<br />
these techniques with more traditional many-body methods.<br />
1 Understanding the electronic structure, magnetism and superconductivity<br />
of iron superconduc<strong>to</strong>rs<br />
<strong>Superconductivity</strong> in compounds containing iron (Fe) was considered impossible until 2008, when a critical<br />
temperature (T c ) of 26 K was reported in F-doped LaOFeAs. After this first report, dozens of related compounds<br />
were synthesized, with critical temperatures up <strong>to</strong> 56 K in SmOFeAs.[2] Fig. 1 shows the basic features that<br />
characterize these Fe superconduc<strong>to</strong>rs (FeSC):<br />
(a)-(b) A common structural motive, consisting of square planes of iron a<strong>to</strong>ms, andX 4 tetrahedra;X is either a<br />
pnic<strong>to</strong>gen - As,P - or a chalchogen - Se,Te.<br />
(c) A quasi-two-dimensional Fermi surface, comprising two hole and two electron sheets, strongly nested with<br />
each other. 1<br />
(d) A phase diagram, in which superconductivity sets in after suppressing a spin density wave (SDW) state, by<br />
means of doping or pressure.<br />
a)<br />
b)<br />
c)<br />
Γ<br />
Y<br />
M<br />
X<br />
d)<br />
Figure 1: Common features of Fe-based superconduc<strong>to</strong>rs<br />
(FeSC): FeX layers, seen from<br />
the <strong>to</strong>p (a) and side (b); Fe andX a<strong>to</strong>ms are red<br />
and green, respectively. (c) Two-dimensional<br />
Fermi surface of LaOFeAs, from Ref. [6], with<br />
the hole and electron pockets centered at ¯X, Ȳ<br />
and ¯M respectively. Here a third hole pocket at<br />
¯Γ is also present. (d) Phase diagram, showing<br />
the transition from a spin density wave (SDW)<br />
<strong>to</strong> a superconducting (SC) state;xis an external<br />
tuning parameter (doping, pressure).<br />
These properties suggest that FeSC may form <strong>to</strong>gether with the high-T c cuprates a wide class of compounds,<br />
in which the superconducting pairing is mediated by magnetic excitations (spin fluctuations). In contrast <strong>to</strong> the<br />
d-wave of the cuprates, the <strong>to</strong>pology of the Fermi surface of FeSC favours in most compounds a superconducting<br />
gap with opposite signs on the hole and electron sheets of the Fermi Surface (s ± ). [4]<br />
Performing and understanding electronic structure calculations, [5] has been the main focus of our activity in<br />
this field. We have studied in detail the formation of the non-magnetic electronic structure, explaining the weak<br />
electron-phonon (ep) interaction - Refs. [6, 7, 8, 9]. We have shown that the description of the SDW state is<br />
problematic in the standard local spin density functional theory (LSDFT) approach, due <strong>to</strong> the cohexistence<br />
of itinerant and localized magnetic moments - Refs. [6, 10]. This lead us <strong>to</strong> two different approaches <strong>to</strong> go<br />
beyond standard LSDFT calculations: combining self-consistent antiferromagnetic (AFM) and non-magnetic<br />
(NM) calculations <strong>to</strong> estimate the ep coupling in the paramagnetic (PM) state, [11] and using the Gutzwiller<br />
approximation <strong>to</strong> include strong local correlations. [12] Finally, we have used LSDFT calculations <strong>to</strong> interpret<br />
experimental data, i.e. optics, [13] phonons [14], spin susceptibilty and specific heat. [15]<br />
1.1 Non-magnetic Electronic Structure<br />
In the region of the phase diagram where superconductivity is observed, Angle Resolved Pho<strong>to</strong>emission (ARPES)<br />
and de-Haas-van-Alphen experiments [16, 17] find the typical Fermi surface <strong>to</strong>pology sketched in Fig. 1(c), with<br />
1 Here and in the following, we do not consider the 122 Fe chalchogenides, [3] which are anomalous in many respects.<br />
1
hole and electron pockets at the center and edges of the Brillouin zone. Theoretical models for superconductivity<br />
mediated by spin-fluctuations (SF) rely on this <strong>to</strong>pology, and on the modifications induced by structural deformation<br />
and three dimensional dispersions, <strong>to</strong> explain the trends in critical temperatures and gap symmetry in<br />
different FeSC. [18, 19] Non-magnetic DFT calculations reproduce the experimental Fermi surfaces quite well,<br />
except for a moderate <strong>to</strong> strong quasi-particle renormalization, and a sistematic relative shift of the hole and<br />
electron pockets. [20]<br />
Understanding the formation of this peculiar electronic structure is a highly non-trivial task. [6] We have used<br />
2<br />
xy xy xy<br />
2<br />
xz xz Xz<br />
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Γ X M Γ<br />
Γ X M Γ<br />
Γ X M Γ<br />
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z z z<br />
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x x X<br />
2<br />
y y Y<br />
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-4<br />
-4<br />
Γ X M Γ<br />
Γ X M Γ<br />
Γ X M Γ<br />
Γ X M Γ<br />
Figure 2: Band structure of the two-dimensional tigh-binding model for a single FeAs layer of<br />
LaOFeAs (2D LaOFeAs). The five Fed(xy,xz,yz,XY =x 2 −y 2 andzz=3z 2 −1) and three<br />
Asp(x,y,z) NMTO’s are shown in the insets. Thexandy axes are directed along the shortest<br />
Fe-Fe distance in Fig. 1 – From Ref. [6].<br />
the NMTO downfolding method [21] <strong>to</strong> derive an accurate model of the two-dimensional band structure of<br />
LaOFeAs (2D LaOFeAs). Using the symmetry properties of the single FeAs layer, we could exactly reduce the<br />
problem <strong>to</strong> a single Fe unit cell, and derive an accurate analytical tight-binding model, with five Fe d and three<br />
As p orbitals. Including the As p orbitals explicitely allowed us <strong>to</strong> study the material trends with tetrahedral<br />
angle [18, 22] and the effect of three-dimensional dispersion, also for those compounds in which As is replaced<br />
by a different ligandX, such as P, Se or Te.<br />
Fig. 2 shows the corresponding bands, decorated with a fatness proportional <strong>to</strong> the partial character of the orbitals<br />
shows in the small insets. This complicated electronic structure derives from a subtle interplay between the<br />
covalent bonding of Fe t 2 (xz,yz andxy) and Asporbitals, and direct metallic bonding for the Fe e states. The<br />
dashed line marks the position of the Fermi level (E F ) for the nominal electron countp 6 d 6 of most undoped Fe<br />
based superconduc<strong>to</strong>rs. The Fermi surface is shown in Fig. 1 (c); the states at E F are all oft 2 character, and the<br />
curvature of the bands that form the hole and electron pockets is strongly affected by Fe-As hybridization.<br />
One of the properties of this electronic structure, around the filling p 6 d 6 , is a weak coupling of the electronic<br />
states <strong>to</strong> phonons – the <strong>to</strong>tal electron-phonon coupling constantλ=0.2, is even lower than in elemental aluminum<br />
(λ = 0.44), which has a T c of ∼ 1 K, as we have shown in Ref. [7]. The <strong>to</strong>p panel of Fig. 3 shows the<br />
results of a linear-response calculation for LaOFeAs. [23] The phonon spectrum extends up <strong>to</strong> ∼ 500 cm −1 ,<br />
with a weak ep coupling, distributed among three peaks of Fe-As character at ω 100, 200 and 300 cm −1 . The<br />
distribution of theep spectral functionα 2 F(ω) (Eliashberg function) is uniform, indicating no strong dormantep<br />
interaction. [24, 25]. The lower panel of the same figure shows a similar calculation for the analogous compound,<br />
in which Fe is replaced by nickel - LaONiAs. This compound is also superconducting, but with a much lower<br />
T c 5 K, and the nominal filling is p 6 d 8 . Here, the Fermi level sits ∼ 1 eV higher in Fig. 2, crossing the<br />
states derived from the directed x 2 − y 2 orbitals. Single-band Migdal Eliashberg theory for phonon-mediated<br />
superconductivity is sufficient <strong>to</strong> explain the experimental T c ’s, and several experimental observations such as<br />
the specific heat jump at T c and the size of the superconducting gap – Refs. [7, 8]. Similar results are found for<br />
other iron and nickel compounds. [26]<br />
2
DOS<br />
α 2 F(ω)<br />
500<br />
400<br />
λ q<br />
ν<br />
La<br />
Fe<br />
As<br />
O<br />
ω (cm -1 )<br />
300<br />
200<br />
100<br />
0<br />
Γ X M Γ Z R A<br />
0 0.2<br />
DOS<br />
0.2<br />
α 2 F(ω)<br />
500<br />
400<br />
λ q<br />
ν<br />
La<br />
Ni<br />
As<br />
O<br />
ω (cm -1 )<br />
300<br />
200<br />
100<br />
0<br />
Γ X M Γ Z R A 0 0.2<br />
0.4 0.8<br />
Figure 3: Phonon dispersion, density of states<br />
(DOS) andep spectral function –α 2 F(ω) – calculated<br />
in density functional perturbation theory<br />
[23] for LaOFeAs (<strong>to</strong>p) and LaONiAs (bot<strong>to</strong>m).<br />
The dashed line represents λ(ω) =<br />
2 ∫ ω<br />
0 α2 F(Ω)dΩ – From Refs. [7, 8].<br />
If E F is shifted further up, until an electron count p 6 d 10−x , the ep matrix elements are even higher, due <strong>to</strong> the<br />
strong hybridization with the X p states. This is realized in BiOCu 1−x S, where the Cu-S layers play the role<br />
of the Fe-X layers – Ref. [9]. For this material, we have studied the competition of spin fluctuations and ep<br />
coupling, using linear response calculations and the local spin-density functional version of the random-phase<br />
approximation (RPA) for spin fluctuations. We have shown that BiOCu 1−x S is a quite unique compound where<br />
both a conventional phonon-driven and an unconventional triplet superconductivity are possible, and compete<br />
with each other. We argued that, in this case, it may be possible <strong>to</strong> switch from conventional <strong>to</strong> unconventional<br />
superconductivity by varying such parameters as doping or pressure.<br />
1.2 The SDW state: problems and hints from DFT calculations<br />
In a large part of their phase diagram – red in Fig. 1 (d) – many FeSC exhibit long-range magnetic order; the most<br />
common pattern is aQ = (0,π) SDW, in which the the Fe spins are aligned aligned ferromagnetically along one<br />
of the edges of the Fe squares, and antiferromagnetically along the other, forming parallel stripes. For this reason,<br />
this pattern is usually called AFM stripe. In this AFM state, the lattice often displays a small orthorhombic<br />
dis<strong>to</strong>rtion. 11 and 122 chalchogenides exhibit a double-stripe and a block-AFM pattern respectively. Local spin<br />
density functional calculations encounter several problems in describing the formation and suppression of this<br />
magnetic order. In Ref. [10] we have shown for the first time that it is impossible <strong>to</strong> reproduce, within a single<br />
DFT calculation, the correct value of electronic (fermiology, magnetic moment) and structural (tetrahedral angle,<br />
B 1g phonon frequency) properties. We have interpreted this as an indication that in actual FeSC two types of<br />
magnetic moments cohexist. [30]<br />
The problem is the following: The magnetic moments measured byµSR and inelastic neutron scattering typically<br />
range from 0.5-0.6 – 1.0µ B in the pnictides, <strong>to</strong>∼2.0µ B for the 11 chalchogenides. The magnetic ground state<br />
in LSDFT is the same SDW order observed by experiments, but the magnitude of the magnetic moment (∼ 2.0<br />
µ B ) is almost independent on the compound, and often much larger than the experimental one. Even worse,<br />
for a large range of dopings for which experiments see no trace of long-range magnetic order, LSDFT equally<br />
predicts an AFM ground state, with m ∼ 2.0 µ B . The puzzling thing is that suppressing this large ordered<br />
moment in the calculations spoils the almost perfect agreement with experiment for the structural and vibrational<br />
properties, introducing errors as large as 25 % for some phonon frequencies, and for the internal position of the<br />
X a<strong>to</strong>ms. [11, 28, 29]<br />
In order <strong>to</strong> understand the origin and implications of this surprising resut, in Ref. [6] we have studied a simplified<br />
model for AFM in FeSC. This is an extension of the so-called extended S<strong>to</strong>ner theory for itinerant ferromagnets<br />
[31], which we have recently used <strong>to</strong> study spin fluctuations in Ni 3 Al, [32] and is usually a very good<br />
approximation <strong>to</strong> the full self-consistent DFT result. [31, 33] A staggered magnetic field ∆ Q , with the same<br />
3
m/∆(µ B<br />
eV -1 )<br />
2<br />
1.8<br />
1.6<br />
1.4<br />
1.2<br />
1<br />
Q=(0,π)<br />
I=0.59 eV<br />
I=0.82 eV<br />
x=0.00<br />
x=0.05<br />
x=0.10<br />
x=0.15<br />
x=0.20<br />
0 0.5 1 1.5 2 2.5<br />
m(µ B<br />
)<br />
Y<br />
Γ<br />
Y<br />
Γ<br />
M<br />
X<br />
M<br />
X<br />
Figure 4: Non-interacting, static spin susceptibilities<br />
χ(m)=m/∆ for 2D LaOFeAs from a<br />
S<strong>to</strong>ner model for AFM stripes m is the magnetization,<br />
I the S<strong>to</strong>ner parameter, ∆ the applied<br />
staggered magnetic field, andxis doping in the<br />
rigid band approximation. For a given value<br />
of the S<strong>to</strong>ner interaction parameter, I, the selfconsistent<br />
moment is given by χ(m) = 1/I.<br />
I = 0.59 and I = 0.82 yield the experimental<br />
and LSDFT value of the magnetic moment,<br />
respectively. The two corresponding Fermi surfaces<br />
are also shown. From Ref. [6].<br />
periodicityQof the SDW, is applied <strong>to</strong> a system, coupling the k and k+Q blocks of the corresponding tightbinding<br />
Hamil<strong>to</strong>nian. The resulting magnetizationm is related <strong>to</strong>∆via the self-consistency condition∆ = mI,<br />
where I is the S<strong>to</strong>ner interaction parameter, which in LSDFT calculations is set by the choice of the exchange<br />
and correlation functional. Using the 2D tight-binding Hamil<strong>to</strong>nian for LaOFeAs of Ref. [6] and an orbitalindependent<br />
exchange field for the Fedorbitals, we obtained the non-interacting static spin susceptibilityχ(m)<br />
plotted in Fig. 4, as: χ(m) = m/∆, for several rigid-band dopings x. The curve shown in the figure is for an<br />
AFM stripe – Q = (0,π). This function is uniquely determined by the non-interacting band structure of Fig. 2.<br />
Its shape reflects two different regimes for magnetism: (a) a small moment regime (m ≤ 1.0), dominated by the<br />
strong nesting between hole and electron Fermi pockets, where magnetism is very feeble, and rapidly suppressed<br />
by doping; and a more robust large moment regime (m ≥ 1.0µ B ), which persists up <strong>to</strong> very high dopings. The<br />
two values ofI represented by dashed lines were chosen <strong>to</strong> reproduce the ordered moment of LaOFeAs detected<br />
by experiments (m ∼ 0.4µ B ,I = 0.59eV ), [27] and the value found in self-consistent LSDFT calculations (m<br />
∼ 2.0 µ B , I = 0.82 eV). At the side of the figure, we also show the two corresponding Fermi surfaces. These<br />
are qualitatively very different: the small-moment one is reminiscent of the non-magnetic (NM) Fermi surface<br />
shown in Fig. 1 (c), with small SDW gaps; in the large moment one, there is an almost complete reconstruction,<br />
due <strong>to</strong> a 10 times larger exchange splitting (∆ ∼ 2 eV). ARPES experiments in magnetic samples show Fermi<br />
surfaces similar <strong>to</strong> the small-moment one; the second <strong>to</strong>pology is never observed. The two Fermi surfaces give a<br />
pic<strong>to</strong>rial representation of the problems encountered by LSDFT in the pnictides: the fermiology and the suppression<br />
of the magnetic moment with doping is well reproduced by small moment calculations, which correspond<br />
<strong>to</strong> an exchange field of ∼ 2 − 300 meV; in order <strong>to</strong> reproduce the structural properties, instead, much larger<br />
exchange fields∼ 2 eV are required.<br />
What we proposed in Ref. [10] is that these two regimes cohexist in actual FeSC samples, with large, localized<br />
moments that extend in the whole phase diagram, even the regions where experiments see no trace of static<br />
magnetism. [30] The superconducting samples are thus strongly paramagnetic, i.e. very distant from the nonmagnetic<br />
DFT picture.<br />
1.3 Magnetism, going beyond Density Functional Theory<br />
We proposed a model the paramagnetic state in DFT in Ref. [11], where we studied the effect of magnetism<br />
on phonon dispersions and electron-phonon coupling. The three <strong>to</strong>p panels of fig. 5 show the results of<br />
three self-consistent calculations of phonon dispersions and ep coupling for BaFe 2 As 2 , with a ground state<br />
which is non-magnetic (NM), anti-ferromagnetic with a checkerboard pattern (AFMc), and anti-ferromagnetic<br />
stripe (AFMs). The thick, colored lines represent the phonon densities of states (DOS), the dashed lines give<br />
the same-spin component of the Eliashberg function: α 2 F σσ (ω). The corresponding ep coupling constants<br />
λ σσ = 2 ∫ dωα 2 F σσ (ω)/ω are 0.18, 0.33 and 0.18 respectively. The average ep matrix elements of the two<br />
AFM calculations (λ σσ /N σ (0)=0.25) are about 1.5 time larger than the NM result (λ/N(0)=0.15), due <strong>to</strong> magne<strong>to</strong>elastic<br />
coupling. We obtained an estimate of the coupling in the paramagnetic phase, combining the phonon<br />
frequencies andep matrix elements of the AFM solution, with the Fermi surface given by the non-magnetic calculati<br />
ons. The resultingep spectal functionα 2 F(ω) is plotted in blue in Fig. 5 <strong>to</strong>gether with the non-magnetic<br />
one; the increased weight around 20 meV yields a 50 % increase in the <strong>to</strong>tal ep coupling. The corresponding<br />
phonons are the B 1g Raman-active modes that dis<strong>to</strong>rt the tetrahedra. From this result, <strong>to</strong>gether with a rigid-band<br />
approximation for doping, we estimated a new upper bound for the ep coupling in the paramagnetic phase of<br />
FeSC, λ PM = 0.35. This value, which is twice as large as the one estimated in the NM phase [7], is still <strong>to</strong>o<br />
4
F(ω)<br />
0.2<br />
0.1<br />
0<br />
0.2<br />
0.1<br />
0<br />
0.2<br />
0.1<br />
NM<br />
λ=0.18<br />
N σ<br />
(0)=1.18<br />
AFMc<br />
λ=0.33<br />
N σ<br />
(0)=1.36<br />
AFMs<br />
λ=0.18<br />
N σ (0)=0.68<br />
<strong>to</strong>t<br />
Fe<br />
As<br />
0<br />
0 10 ω (meV) 20 30<br />
10 ω (meV) 20 30<br />
0.3<br />
α 2 F(ω)<br />
0.2 λ(ω)<br />
0.1<br />
PM<br />
NM<br />
Figure 5: Top panel: Phonon DOS (full lines)<br />
and same-spin ep spectral function α 2 F σσ(ω)<br />
(dashed line) for non-magnetic (NM), antiferromagnetic<br />
checkerboard (AFMc) and<br />
antiferromagnetic stripe (AFMs) BaFe 2As 2.<br />
Lower panel: α 2 F σσ(ω) and λ σσ(ω) for<br />
non-magnetic (NM) and paramagnetic (PM)<br />
BaFe 2As 2. – From Ref. [11].<br />
m/µ B<br />
2.5<br />
2.0<br />
1.5<br />
1.0<br />
0.5<br />
J=0.10U<br />
J=0.075U<br />
J=0.05U<br />
S<strong>to</strong>ner<br />
0.0<br />
0.4 0.5 0.6 0.7 0.8 0.9<br />
I/eV, χDFT −1 /eV<br />
Figure 6: Ordered stripe magnetic moment<br />
m of undoped, 2D LaOFeAs as a function of<br />
I eff = U −0.017J, in the Gutzwiller approximation<br />
(red, green, blue symbols), <strong>to</strong>gether<br />
with the inverse of the non-interacting spin susceptibility<br />
χ −1 (m), shown in fig. 4. – From<br />
Ref. [12].<br />
low <strong>to</strong> explain the superconducting T c alone, but is high enough <strong>to</strong> yield visible effects in realistic models for<br />
the pairing.<br />
The cohexistence of localized and itinerant moments in FeSC can only be captured by methods that include local<br />
fluctuations, such as the dynamical mean field theory (DMFT). The solution of multiband Hubbard models, in<br />
which the kinetic part is given by realistic models of the DFT band structure (LDA+DMFT), reproduces a state<br />
with small ordered moments and large local fluctuating moments. [34] A very similar picture is given by the<br />
simpler Gutzwiller approximation. [35]. In Ref. [12], we solved a multiband Hubbard Hamil<strong>to</strong>nian, in which<br />
the kinetic part is given by the 8-band model of Ref. [6], and the full a<strong>to</strong>mic interaction between the electrons<br />
in the iron orbitals is parametrized by the Hubbard interactionU and an average Hund’s-rule interaction J. We<br />
found that for a wide range of (U,J) parameters the ground state is a metallic spin density wave (SDW), with<br />
a small magnetic moment. The local moment (< S 2 >∼ 2.0µ B ) is much larger than the ordered one, but still<br />
far from the fully polarized a<strong>to</strong>mic value (m = 4.0µ B ). Consistently with the DMFT results, we found that the<br />
value of the magnetic moment is mostly determined by the Hund’s coupling J. Indeed, Fig. 6 shows that the<br />
results for different sets of calculations collapse on<strong>to</strong> a single curve, if the moment is plotted as a function of the<br />
effective S<strong>to</strong>ner parameterI = J −0.017U. For comparison, the inverse of the static susceptibility at x = 0 is<br />
also shown; the good agreement between the two curves indicates that the magnetically ordered phase is a stripe<br />
spin-density wave of quasiparticles.<br />
1.4 Interpretation of experimental spectra<br />
In Ref. [13] we compared model calculations of the intraband optical spectra, treated in a multi-band Eliashberg<br />
model, experimental data [36] and first-principles calculations, <strong>to</strong> show that interband transitions give a non<br />
negligible contribution already in the infrared region of the spectrum. This leads <strong>to</strong> a substantial failure of the<br />
extended-Drude-model analysis on the measured optical data without subtraction of interband contributions.<br />
5
As a consequence the electron-boson coupling gets dramatically overestimated. This effect was also recently<br />
measured by the Keimer Department. [37]<br />
In Ref. [14], in collaboration with the group of Bernhard Keimer, we have analyzed the anomalous dependence<br />
of the c-axis B 1g phonon in FeSe/Te samples, using Raman scattering and first-principles calculations.<br />
The local density calculations, including magnetic polarization and Fe nons<strong>to</strong>ichiometry in the virtual crystal<br />
approximation, reproduce well the experimental data in the Fe-deficient samples, while the effects of Fe excess<br />
are poorly reproduced. This may be due <strong>to</strong> excess Fe-induced local magnetism and low-energy magnetic<br />
fluctuations that cannot be treated accurately within these approaches. In Ref. [15], we have investigated the<br />
specific heat and upper critical fields of KFe 2 As 2 . The measured Sommerfeld coefficient γ n =94(3) mJ/mol K 2<br />
of KFe 2 As 2 is anomalously large, but this is due <strong>to</strong> extrinsic effects. Taking this in<strong>to</strong> account, KFe 2 As 2 should<br />
be classified as a weakly or intermediately coupled superconduc<strong>to</strong>r with a <strong>to</strong>tal electron-boson coupling constant<br />
λ <strong>to</strong>t =1 (including a calculated weak electron-phonon couplingλ ph =0.17).<br />
2 <strong>Superconductivity</strong> in intercalated picene<br />
Studying carbon, with its many allotropes and compounds, is one of the most fascinating problems in science.<br />
To mention only physics, before graphene, which was awarded the Nobel prize in 2010 for opening exciting perspectives<br />
for fundamental and applied research, many other systems such as nanotubes, fullerenes, intercalated<br />
graphites have played an important role in several fields, including superconductivity.<br />
5<br />
0 1 2 3 800 4 1600 2400 6 3200<br />
ω (cm -1 )<br />
1 2 3<br />
4 5 6<br />
Figure 7: Top: Structure of a picene<br />
molecule, and of the corresponding crystal.<br />
Bot<strong>to</strong>m: Phonon spectrum of crystalline picene<br />
in DFPT [23], showing representative vibration<br />
patterns. The modes 4 and 5, in red, give the<br />
highest contribution <strong>to</strong> ep coupling in our rigid<br />
band calculations. – From Ref. [45].<br />
The most recent discovery is a new class of aromatic superconduc<strong>to</strong>rs, comprising molecular crystals doped<br />
with alkali or alkaline earths metals. These are polycyclic aromatic hydrocarbons, i.e. planar molecules formed<br />
by a number of juxtaposed hexagonal benzene rings (C 6 H 6 ). The first report of superconductivity in this class of<br />
materials concerned picene doped with potassium (K) and rubidium (Rb) with a maximum critical temperature<br />
(T c ) of 18 K [38]. After that, superconductivity was also found in phenanthrene and coronene doped with K<br />
(T c =5K and 15 K), [39, 40] phenanthrene doped with strontium (Sr) and barium (Ba) (T c =5.6-5.8 K) [41], and<br />
1,2:8,9-dibenzopentacene(C 30 H 18 ) doped with K (T c =33 K) [42]. Intercalated hydrocarbons are strongly related<br />
<strong>to</strong> two families of carbon superconduc<strong>to</strong>rs: graphite intercalation compounds (GICs), which are conventional,<br />
BCS-like superconduc<strong>to</strong>rs [43] and alkali-doped fullerenes (A 3 C 60 ), which instead have a rich phase diagram<br />
determined by a non-trivial interplay between strong localep interactions and on-site Coulomb correlations in a<br />
highly non-adiabatic regime [44].<br />
In Ref. [45] we performed first-principles calculations of the electronic structure, phonon spectra and electronphonon<br />
(ep) interaction of doped picene in the rigid band approximation (RBA), using density functional perturbation<br />
theory (DFPT). Based on our results for the vibrational properties, and on previous studies on electronic<br />
properties and correlations, we showed that picene, and most likely other intercalated hydrocarbons, belong <strong>to</strong><br />
a class of strongly correlatedep superconduc<strong>to</strong>rs, for which a local approach that includes both the ep coupling<br />
6
and Coulomb correlations on an equal footing may be more appropriate. The main results of our study are summarized<br />
in Fig. 7. The <strong>to</strong>p panel shows the structure of the isolated picene molecules, and their arrangement in<br />
a crystal; the lower panel shows the spectrum of crystalline picene calculated in LDA; in the inset, typical displacements<br />
are shown. The modes indicated in red are those which give a larger contribution <strong>to</strong> theep coupling.<br />
The molecularep matrix element, isV ep = 150±20 meV for holes andV ep = 110±5 meV for electrons. These<br />
values are comparable <strong>to</strong> the bandwidth of the conduction band (W∼300 meV) and typical phonon frequencies<br />
(ω ph ∼ 200 meV).<br />
For a rigid-band doping of picene with ∼ 3 electrons per molecule, N(E F ) = 7.06 states spin −1 meV −1 , λ =<br />
0.78 andω ln = 1021 cm −1 . Usingµ ∗ = 0.12 in the McMillan formula, we obtain a critical temperatureT c = 56.5<br />
K. This is more than three times the experimental value of T c , and we need <strong>to</strong> use a larger value of µ ∗ = 0.23<br />
<strong>to</strong> reproduce the experimental T c . However, it is important <strong>to</strong> stress that the estimates of T c based on Migdal-<br />
Eliashberg theory must be taken with care in the case of picene. In fact, the bandwidth of the conduction bands<br />
(W ∼ 300 meV for electrons and ∼ 1 eV for holes), the frequencies of strongly coupled phonons (ω ph ∼ 200<br />
meV), theep coupling strength (V ep ∼ 110-150 eV), all have similar magnitudes, close also <strong>to</strong> that of Coulomb<br />
repulsion (U ∼ 1.2 eV), estimated by other authors [46]. In this range of parameters, the two most important<br />
approximations in Migdal-Eliashberg theory of superconductivity, Migdal’s theorem and the Morel-Anderson<br />
scheme for the screening of the Coulomb repulsion, are invalid. In fullerenes, this regime of parameters gives<br />
rise <strong>to</strong> a variety of interesting phenomena, the most spectacular being the occurrence of ep superconductivity<br />
near a Mott insulating phase, well captured by theoretical studies of local models of interacting phonons and<br />
electrons in the non-adiabatic regime. [44]<br />
This first study motivated us <strong>to</strong> start a collaboration on electronic and vibrational properties of intercalated<br />
hydrocarbons, with the high-pressure and optics group of Università la Sapienza, in Rome. [47]<br />
3 Iridates<br />
Recently, Ohgushi et al. reported resonant x-ray diffraction study of CaIrO 3 that indicates this material exhibits<br />
a Mott insulating J eff = 1/2 state [48]. Previously, it was found that spin-orbit coupling can play an important<br />
role in the electronic properties even for a 4d 5 system such as Sr 2 RhO 4 [49, 50]. Since CaIrO 3 is a 5d 5 system,<br />
this material provides a more oppurtune platform <strong>to</strong> study the interplay between spin-orbit coupling and onsite<br />
Coulomb repulsion. Furthermore, this material proffers a playground for a recent theory that suggests<br />
that systems in the J eff = 1/2 state, depending on bond geometry, lead <strong>to</strong> interesting varieties of low-energy<br />
Hamil<strong>to</strong>nians [51]. The density functional calculations within the local density approximation showed that the<br />
spin-orbit coupling splits the Ir t 2g bands in<strong>to</strong> fully filled quartet of J eff = 3/2 bands and half-filled doublet of<br />
J eff = 1/2 bands [52]. This metallic state is contrary <strong>to</strong> the experiments that show that CaIrO 3 is an insula<strong>to</strong>r.<br />
However, the half-filled J eff = 1/2 bands have a band width of ∼1 eV and a value for the on-site Coulomb<br />
repulsion ofU = 2.75 eV yields a Mott insulating state. The LDA+SO+U calculations give an antiferromagnetic<br />
ground state for CaIrO 3 along the c axis with <strong>to</strong>tal moments aligning antiparallel along the c axis, in agreement<br />
with the experiment. For U = 2.75 eV, the <strong>to</strong>tal magnetic moment is 0.67 µ B , with an orbital contribution of<br />
0.28 µ B and a spin contribution of 0.38 µ B . These values differ from what is expected for the ideal J eff = 1/2<br />
state, and this deviation might be explained by the mixing of J eff =1/2 bands with Ir e g bands due <strong>to</strong> the tilting<br />
of IrO 6 octahedra.<br />
Note: Our Minerva <strong>Research</strong> Group was estabilished in July 2009; before this, L. Boeri was a staff member in<br />
the Andersen Department (Electronic Structure Theory). For consistency, all the activity on fe-based superconduc<strong>to</strong>rs<br />
is reported here.<br />
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9