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Phil. Trans. R. Soc. A (2010) 368, 3503–3517
doi:10.1098/rsta.2010.0089
REVIEW
Quantum oscillations probe the normal
electronic states of novel superconductors
B Y AMALIA I. COLDEA1,2, *
1 Clarendon
Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU,
UK
2 H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue,
Bristol BS8 1TL, UK
In 2008, new classes of high-temperature superconductors containing iron have been
discovered. These iron pnictides offer a new area of exploration and understanding of
superconductivity. Quantum oscillations is a bulk probe that allows us to map out
the full Fermi surface of a superconducting system in its normal metallic state. These
oscillations are determined by the Landau quantization in high magnetic fields and are
usually observed at very low temperatures and in very clean samples. By knowing the
exact nature of the quasi-particles in the normal state and the degree of electronic
correlations, one can simplify and restrict theoretical models required to understand
the pairing mechanism in superconductors. I will discuss the current understanding of
the Fermi surface studies in iron-based superconductors as determined from quantum
oscillations.
Keywords: quantum oscillations; Fermi surface; high Tc superconductivity
1. Introduction
One of the many fascinating manifestations of strong correlations in materials
is the presence of a condensate superconducting (SC) state in which all the
electrons collapse into a unique electronic state for which the electrical current
feels no resistance. Superconductivity was found in elemental materials such as
lead and tin a long time ago, but it was not until 1986 with the discovery of
high-temperature superconductivity in copper oxides that the dream of making
these materials useful inspired and challenged many generations of physicists.
Unconventional superconductors are those in which superconductivity arises
from direct electron–electron interactions, as contrasted to the conventional
indirect interaction via phonons, found in elemental materials or more recently in
MgB2 . Direct interactions due to many body effects often favour higher (compared
with isotropic s-wave) angular momentum pairing (like p-wave or d-wave),
whereas the onsite Coulomb repulsion and spin fluctuations often play a key
*[email protected]
One contribution of 18 to a Triennial Issue ‘Visions of the future for the Royal Society’s 350th
anniversary year’.
3503
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A. I. Coldea
role in stabilizing an unconventional d-wave SC state in cuprates. A completely
new class of high-temperature superconductors containing iron coordinated with
pnictogen atoms (such as As or P), called iron pnictides, was discovered in 2008,
and their pairing symmetry is still under investigation. It is not yet clear whether
the mechanism for superconductivity in these iron-based superconductors is
closely connected to that in cuprates or whether a new route to high-temperature
superconductivity has been found.
In this paper, I will review the recent experimental developments in the study
of the electronic properties of iron pnictides, highlighting the role of the Fermi
surface topology and the effects of many-body interactions in different materials
belonging to different classes of iron pnictides. The body of the paper concentrates
on the various parameters that can be determined experimentally in the normal
metallic state stabilized at high magnetic field and which can help to understand
superconductivity in these novel materials.
(a) Iron-based superconductors
The first report of superconductivity in an iron-based material was in LaFePO,
which has a relatively low transition temperature, Tc ∼ 6 K (Kamihara et al.
2006). Based on the same crystallographic structure and replacing the pnictogen
P with As, high Tc superconductivity was found in the fluoride-doped LaFeAsO
(Tc ∼ 26 K; Kamihara et al. 2008). This work sparked intense experimental
and theoretical work to search for new materials that shared similar structural
characteristics, and the highest transition temperature of 55 K was found in
F-doped SmFeAsO (Ren et al. 2008). Stabilization of an SC phase in systems
with iron was mostly unexpected as iron, because of its strong ferromagnetic
properties, is generally expected to disrupt a conventional pairing mechanism.
The basic structural features of these materials is the presence of square planar
sheets of Fe ions coordinated tetrahedrally by pnictogens (such as As or P) or
chalcogens (Se, Te and S). A variety of compounds are obtained by sandwiching
the FeAs layers (made up of FeAs4 tetrahedra) with various ionic layers, such
as LaO (as in the case of 1111 compounds, LaO1−x Fx FeAs) or Ba (in the
case of 122 compounds, Ba1−x Kx Fe2 As2 ), as shown in figure 1b,c. Besides these
compounds, superconductivity has been found by intercalating deficient Li or Na
between FeAs layers (111 compounds) or by van der Waals gaps in chalcogenites
(11 compounds, Fe(Tex Se1−x ); the latter group do not contain arsenic and may
be the most suitable for practical applications. Interestingly, the highest Tc is
found when the FeAs4 tetrahedron has a regular shape, so it is clear that the
structural details are important in achieving the largest Tc in the iron-based
superconductors. For a general view on this new area of research, see a recent
review by Ishida et al. (2009).
A typical phase diagram for the iron pnictides is shown in figure 1a. Without
doping, most of the parent compounds are semimetallic, show universal long range
magnetic order and show a tetragonal to orthorhombic/monoclinic structural
transition at low temperatures. This is in contrast to the parent compounds
of the cuprates, which are Mott insulators owing to strong electron–electron
interactions. The magnetic order is a collinear AFM spin structure with a (p, p)
ordering wavevector in the folded Brillouin zone with two Fe ions per unit
cell. The structural distortion follows closely the magnetic transitions and is
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Review. Iron pnictides superconductors
(a) 140
LaO
1
M(µB)
120
100
T (K)
(b)
Ba(Fe1−xCox)2As2
T
0.5
0
60
O
40
0.02 0.04 0.06
x
Ta
Tb b
TTO
TAF
TSC
80
AF
FeAs
Fe
LaO
(c)
a
Ba
FeAs
20
FeAs
SC
0
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Ba
x
Figure 1. (a) Generic phase diagram of iron pnictides, as in the case of Co-doped BaFe2 As2 . Phase
labels are orthorhombic (O), tetragonal (T), SC and antiferromagnetic (AFM) (after Lester et al.
2009). The insets show the collinear AFM order of the Fe sublattice and the variation with doping,
x, of the Fe-ordered magnetic moment. The layered crystal structure (shown in the ac plane) of (b)
LaFeAsO and (c) BaFe2 As2 . The electronically active layer in the ab plane is made up of FeAs4
tetrahedra.
believed to play an important role in driving the magnetic transition. Doping
with electrons or holes in the ionic layers (figure 1a), applied pressure and
isoelectronic substitution (As for P) in the magnetic parent compounds suppress
the magnetic and structural phase transitions and stabilize superconductivity
(Ishida et al. 2009).
The origin of the magnetic order in the parent compounds is not yet clearly
understood. In one scenario, the transition has an itinerant character driven
by the nesting instability of the Fermi surface which consists of two electron
and two hole sheets with similar dxz /dyz orbital character. These sheets have
approximately compensating cross sections and can be matched reasonably well
if translated by (p, p) for an arbitrary kz value (for an illustration, see figure 3d)
(Chubukov et al. 2008; Kuroki et al. 2008; Mazin et al. 2008a,b). The fulfilment
of such a nesting condition leads to a peak in the wavevector-dependent Lindhard
susceptibility and, as the pnictides have a large density of states at the Fermi level,
N (EF ), and a strong Stoner enhancement, this results in a magnetic ordering
instability. This is a spin density wave (SDW) instability of the Fermi surface
driven by electrons at and near EF and may provide an explanation for the
experimentally observed SDW order found in most of the undoped iron pnictides.
Alternatively, a Heisenberg model with exchange interactions between localized
spin moments (where moment formation is a result of onsite interactions) is
proposed to explain the collinear AFM spin structure (Uhrig et al. 2009). Itinerant
magnetism differs from local moment magnetism in that the physics involves
the Fermi surface nesting, and understanding its exact topology is important in
understanding the SC mechanism in iron pnictides.
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3506
A. I. Coldea
The superconductivity emerging from these SDW compounds is theoretically
proposed to be unconventional and mediated by AFM spin fluctuations. Such
an SC state has an extended s-wave pairing with a sign reversal of the order
parameter between different Fermi surface sheets, as the magnetic fluctuations,
while too broad to induce a magnetic instability, are instrumental in stabilizing
superconductivity (Chubukov et al. 2008; Kuroki et al. 2008; Mazin et al.
2008a,b). So far experiments show evidence for a more complex picture where
the gap order parameter is not simply an s± -wave but may also have line nodes
(Fletcher et al. 2009) and thus the knowledge of the exact details of the Fermi
surface that may favour one type of pairing over another is essential.
(b) Quantum oscillations
An electronic gas placed in a magnetic field has its energy levels quantized
at values that depend on the strength of the magnetic field. By varying the
magnetic field strength, nearly all measurable quantities will show oscillations
periodic in 1/B (figure 2). Such quantum oscillations represent a canonical probe
of the defining aspect of a metal, its Fermi surface. By providing the precise
details of the Fermi surface in the normal metallic state and the effect of
many-body interaction on the effective masses of the quasi-particles, quantum
oscillations help to develop reliable theories for understanding superconductivity.
In iron pnictides, the degree of nesting between the various Fermi sheets and the
magnitude and anisotropy of the Fermi velocities is believed to be important for
the paring mechanism (Kuroki et al. 2008; Mazin et al. 2008a,b). Conventional
band structure calculations only include many-body effects at the mean field level;
in strongly interacting systems, there can be substantial renormalization of the
effective masses or Fermi velocities due to electron–phonon or electron–electron
interactions, and this enhancement can be determined from quantum oscillations.
Furthermore, quantum oscillations can provide unique access to the details of a
Fermi surface which suffers reconstruction by breaking up a large Fermi surface
into many small pockets due to structural or magnetic instabilities (such as charge
density waves and SDWs).
The frequencies of the observed oscillations, F , provide very accurate
measurements of the Fermi surface cross-sectional areas, Ak , via the Onsager
relation, F = (h̄/2pe)Ak . From the evolution of these frequencies as a function
of the magnetic field orientation, one can construct a detailed three-dimensional
picture of the shape and size of the Fermi surface (figure 2d).
The standard expression for the first harmonic of the oscillatory part of the
magnetization (de Haas van Alphen effect (dHvA)) for a three-dimensional Fermi
liquid is given by (Shoenberg 1984)
2pF
3/2
+4 ,
(1.1)
Mosc ∝ CB RD RT RS RSC sin
B
where F is the dHvA frequency, C = |v2 Ak /vk2 |−1/2 the curvature factor
and 4 the phase. RD , RT and RS are the damping factors from impurity
scattering, temperature and spin splitting, respectively. RT = X/sinh X , where
X = (2p2 kB m ∗ T )/(h̄eB). Another damping term is the Dingle factor, RD =
exp(−pkF /eB), where kF is the orbitally averaged Fermi wavevector and is
the inelastic quasi-particle mean free path. The effect of the electron spin is to
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Review. Iron pnictides superconductors
Fermi surface
(a)
Landau tubes
(d)
B
ky
kx
Landau levels
B2
B3
–20
(c)
30
B (T)
35
40
0
20
θ (º)
40
θ = 0º
60
a a
a1 a 2 1 2
FFT amplitude (arb. units)
torque, tosc (arb. units)
25
a2
a2
B
(b)
20
a1
a2
500
B1
b2
b1
EF (T = 0)
15
b1
1500
1000
B =0
H
b2
2000
F cos θ (T)
E
2500
45
b1
b1
b2
d1,d2e
g
0
1
2
F (kT)
q = 47º
q = 12º
3
Figure 2. (a) Landau-level quantization in a magnetic field along z as E(n, kz ) = (n + 12 )h̄uc +
(h̄ 2 kz2 /2m ∗ ). As the magnetic field increases, the Landau radius expands and Landau tubes are
pushed out of the Fermi surface. When the Landau tube coincides with the extremal area of the
Fermi surface (panel at B2 ), there is a strong enhancement in the density of states. As a function
of the magnetic field, there are oscillations in the density of states in 1/B. In reality, the Landau
levels have a finite width determined by the impurity scattering or temperature broadening effects.
(b) Quantum oscillations in superconducting LaFePO (q = 26◦ , T = 0.4 K) and (c) the Fouriertransformed spectrum which reveals the presence of various extremal orbits. (d) The quasi-twodimensionality of electron pockets, a (inner electronic sheet) and b (outer electronic sheet) in
LaFePO. A two-dimensional cylinder has an extremal area that varies like 1/ cos q and an F cos q
dependence reflects the deviation from a perfect cylinder, q being the angle between the magnetic
field direction and the c-axis. The arrow indicates the position of the Yamajii angle, where the
orbits from the neck, b1 , and the belly, b2 , coincide for a particular direction of the magnetic field
(after Coldea et al. 2008).
split each Landau level into two (spin up and spin down) with the spin splitting
term, RS , given by cos {[pgmB (1 + S )]/2me }, where 1 + S is the orbitally averaged
exchange-correlation (Stoner) enhancement factor. When RS = 0, the spin-up and
spin-down Fermi surfaces beat out of phase to produce a spin-zero minimum in
the dHvA amplitude. In an SC state, the dHvA amplitude of oscillations suffers
further exponential damping in a magnetic field, RSC , which can be understood on
the basis of the model of the quasi-particle scattering by the random vortex lattice
with large SC (vortex) fluctuation around the mean field Hc2 (Maniv et al. 2001).
Both the effect of the impurity scattering and the thermal smearing broaden the
Landau levels (figure 2a), imposing stringent conditions for the observation of
quantum oscillations only at low temperatures, in very pure samples and at high
magnetic fields (which are often large enough to allow access to the normal state
of superconductors).
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A. I. Coldea
The effective mass of the quasi-particles on the various orbits, m ∗ , is
determined by fitting the temperature-dependent amplitude of the oscillations
to the conventional Lifshitz–Kosevich formalism (Shoenberg 1984; Springford &
Wasserman 1996). The mass enhancement is given by the ratio m ∗ /mb between
the effective mass and the calculated band mass and reflects the effects of
both electron–phonon coupling and electron–electron interactions by m ∗ /me =
(1 + lel–ph )(1 + lel–el ).
Quantum oscillations studies are performed at low temperatures (below
T < 4 K) and high magnetic steady and pulsed fields (up to 55 T) using a variety
of techniques such as magnetization, torque magnetometry or magnetotransport
and tunnel diode oscillator (Shubnikov–de Haas). Torque measurements using
piezoresistive microcantilevers (Rossel et al. 1996) offer a clear advantage in
the study of extremely small single crystals, as magnetic moments as small
as 10−9 − 10−11 emu can be detected. The oscillatory part of the magnetic
torque, tosc = −dMosc /dq = −1/F (dF /dq)M B, is determined by the Fermi
surface anisotropy and thus vanishes near symmetry axes and for spherically
symmetric Fermi surfaces. Since single crystals of relatively high quality can be
grown in pnictides, especially in the 1111 and the 122 family, these materials have
become quite popular for investigation of fundamental properties of iron pnictides
including quantum oscillations studies.
2. Fermi surface in a paramagnetic phase
(a) Quasi-two-dimensional 1111 systems
The iron phosphide superconductor, LaFePO, has a low SC transition of Tc ∼ 6 K
and a correspondingly low value of the upper critical field required to suppress
superconductivity (m0 Hc2 7 for H ab plane; Coldea et al. 2008), being an ideal
candidate for quantum oscillations studies. LaFePO is isostructural to the parent
arsenide compound LaFeAsO (figure 1b; P and As are isoelectronic with As
having a larger ionic radius). At high temperatures in the tetragonal phase, the
two materials are predicted to have very similar electronic structures; however,
they have very different magnetic ground states as the phosphides are nonmagnetic and do not show any structural transitions (J. G. Analytis et al. 2008,
unpublished data). In phosphides, the pnictogen (P) ions are closer to the Fe
planes and the Fe–P bond angles depart substantially from those of a regular
FeP4 tetrahedron (Lee et al. 2008) and this may also lead to much broader bands
when compared with the arsenides, so it is expected that the effect of many-body
correlations will be reduced (Malaeb et al. 2008).
The Fermi surface of LaFePO, shown in figure 3a, consists of five different
sheets (as five Fe d bands are crossing the Fermi level and the hybridization with
pnictogen is modest); four of them have a cylindrical shape, being mostly weakly
dispersive along the c-direction; two hole cylinders are centred at the zone centre,
G, and two electron cylinders at the zone corner and are primarily derived from Fe
dxz and dyz orbitals (Lebègue 2007). In addition, there is a three-dimensional sheet
derived from a heavy hole section which consists of a distorted sphere centred at
the Z high-symmetry point. This spherical sheet corresponds to the band crossing
the Fermi level along the G–Z direction and has mostly Fe dz 2 character. The
Fermi surface of LaFeAsO in the non-magnetic phase (Mazin et al. 2008a,b)
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Review. Iron pnictides superconductors
(a)
(b)
Γ
A
Z
Γ
(c)
M
(d)
(f)
( e)
M
Γ
Γ
Γ
Q
Figure 3. The Fermi surface of (a) SC LaFePO (Tc ∼ 6 K), (b) metallic SrFe2 P2 (c/a = 3.04) and
(c) CaFe2 P2 (c/a = 2.65) (LDA calculations were performed using experimental lattice parameters,
as reported in detail by Coldea et al. (2008, 2009) and Analytis et al. (2009b). The corresponding
cross sections of the Fermi surface at kz = 0 (G point) when the magnetic field is along the [0 0 1]
direction is shown in (d–f ), respectively. The arrow indicates the nesting vector along the (p, p)
direction between the electron and hole pockets. The solid line delimits the first Brillouin zone.
has features similar to those of LaFePO except for the small three-dimensional
electron pocket centred at Z whose presence is very sensitive to the position of
the pnictogen height above the iron plane (figure 3a). LaFePO is a compensated
metal as the number of electrons is equal to the number of holes, similar to the
undoped parent arsenides.
Quantum oscillations were observed in the normal state of superconducting
LaFePO, as shown in figure 2b,c. The observed frequencies correspond to a fraction
varying between 2.8 and 9 per cent of the basal plane area of the Brillouin zone
(Coldea et al. 2008). To identify the exact location of the observed frequencies,
angular-dependent data provide information on the shape of the quasi-twodimensional cylinders (figure 2d). The frequencies of the extremal dHvA orbits
(figure 2c) obtained from the calculated band structure are compared with the
experimental data. The shape and the angular dispersion of the b orbits closely
resemble those expected from the outer electron cylinder in both frequency and
curvature, whereas the a orbits are similar to those of the inner electron cylinder
(figure 2d; Coldea et al. 2008).
Quantum oscillations have a net advantage over surface-sensitive probes, like
APRES, especially because, as seen in LaFePO, the cleaved plane investigated
can be electrically charged (Lu et al. 2008). In the case of complicated or a
large number of Fermi surface sheets, the quantum oscillations cannot determine
experimentally the exact k-space location of the observed Fermi surface orbits,
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A. I. Coldea
and relies on comparison with band structure calculations. In order to match
the calculated band structure predictions with the experimental data, small rigid
shifts of the electron and hole bands were necessary in LaFePO (by conserving
the number of electrons and holes in a compensated metal; Coldea et al. 2008;
Carrington et al. 2009). The band shifting leads to a shrinking of both electron and
hole sheets when compared with band structure predictions. This shrinking has
been suggested to be a natural consequence of the strong particle–hole asymmetry
of electronic bands in pnictides. It also provides indirect experimental evidence
of dominant interband over intraband scattering, as a direct consequence of the
coupling to a bosonic mode (Ortenzi et al. 2009).
A two-dimensional cut of the Fermi surface at the G point in the twodimensional conducting plane shows that in LaFePO the size of the electron
pockets at the corner of the Brillouin zone has shape and size similar to those
of the hole pockets located at the centre of the zone (figure 3d) and could be
nested with a wavevector (p, p). Nesting requires a perfect match between the
size and the Fermi surface topology of the electron and hole pockets and this
could stabilize an SDW. LaFePO is non-magnetic but may be close to fulfilling
a nesting condition. There is also a small three-dimensional pocket centred at Z
(figure 3) not observed experimentally in torque measurements and it is not clear
whether it plays any role (Coldea et al. 2008; Carrington et al. 2009). Moreover,
the symmetry of the order parameters in LaFePO is expected to have line nodes
(Fletcher et al. 2009), suggesting that the predicted s± -wave type of symmetry
(Mazin et al. 2008a,b) may not be applicable to all iron-based superconductors.
The effective masses in LaFePO have been found to be enhanced by a factor of
2 when compared with band structure calculations (Coldea et al. 2008). This
value suggests that LaFePO is a moderately correlated system with a Fermi
surface which can be described by band structure calculations. Similar mass
enhancement, when compared with the bare band mass, was found also in ARPES
and optical studies (Lu et al. 2008; Qazilbash et al. 2009). Despite the absence of
a Mott transition the electronic many-body effects are believed to be important
in iron pnictides.
(b) Quasi-three-dimensional 122 systems
The Fermi surface of the 122 phosphides of type AFe2 As2 (A = Ba, Sr and
Ca) is likely to be different from that of LaFePO; by replacing an LaO spacing
layer by a single alkali ion such as Ba, the distance between the active Fe–P
layers is expected to be much smaller (whereas the interlayer Fe–Fe distance
is approximately 2.2 Å smaller in BaFe2 As2 than in LaOFeAs, as shown in
figure 1b,c). This reduction in the two dimensionality leads to a significant c-axis
dispersion of the Fermi surface. Phosphites belonging to the 122 class of iron
pnictides are metallic and non-magnetic and their Fermi surface was measured
directly by quantum oscillations (Analytis et al. 2009b; Coldea et al. 2009). A good
qualitative agreement with band structure calculations in SrFe2 P2 (Analytis et al.
2009b) was found, but certain bands shifts were necessary to obtain an exact
agreement. The shift of the bands necessary to finely tune the band structure
calculations to the experimental data led to smaller electron and hole sheets,
as seen before in LaFePO. Shrinking of the electronic sheets was also found
approaching the maximum Tc in BaFe2 (As1−x Px )2 from quantum oscillations
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3511
measurements (Shishido et al. 2010) and also in BaFe1−x Cox As2 from ARPES
studies (Brouet et al. 2009). This tendency towards shrinking of the Fermi surface
sheets in relation to the band structure calculations is not caused by simple
band renormalization effects and seems to be a more general feature of iron
pnictides, reflecting the domination of the interband interactions over interband
ones, and/or the important role of spin fluctuations which may also be responsible
for the pairing mechanism (Ortenzi et al. 2009).
The Fermi surface characteristics of ternary phosphides with different divalent
ions between the Fe–P layers (or c/a ratio) is shown in figure 3b,c. The shape
of the electron pockets in ternary compounds deviates from that of a simple
warped cylinder along the c-axis, as found in LaFePO, and it can be parametrized
by considering that the Fermi wavevector is modulated by both a complex
in-plane and interplane dispersion, similar to what was proposed for the quasitwo-dimensional Sr2 RuO4 (Bergemann et al. 2000) and Tl2 Ba2 CuO6+d (Analytis
et al. 2007). By replacing Sr (c/a = 3.04) with a smaller Ca (c/a = 2.65), the
Fermi surface suffers a major topological change (Coldea et al. 2009), having
only one electronic band remaining, which is significantly warped, and the
system can be clearly viewed as three-dimensional (having a larger contribution
from the interlayer dispersion term, when compared with SrFe2 As2 ). The hole
pockets also change the size and shape when reducing the size of the spacing
layer (from LaO to Ca). The study of CaFe2 P2 provides access to the nonmagnetic collapsed tetragonal phase of CaFe2 As2 , which can be accessed only
under pressure, suggesting that isoelectronic substitution of As for P or chemical
pressure in iron pnictides is equivalent to the applied pressure. The Fermi surface
of CaFe2 P2 may be relevant to other systems in which, owing to the small size
of the spacer layer, the Fe–As bonding weakens and the (inter- and intraplanar)
As–As bonding gets stronger.
Quantum oscillations establish experimentally that the Fermi surface of iron
pnictides becomes more three-dimensional by reducing the spacing between the
active Fe layers; consequently, this evolution from a quasi-two-dimensional Fermi
surface, like that of LaFePO, to one that is quasi-three-dimensional in SrFe2 P2
and three-dimensional in CaFe2 P2 can explain the changes in the anisotropy of
the electronic properties. Moreover, this can also explain the anisotropy of the
upper critical field in iron pnictides (Yin et al. 2008; Singleton et al. 2009), as
the corrugations of the Fermi surface are sufficient to permit circulating currents
at all field orientations. Compared with the cuprates or organic superconductors,
iron pnictides, including the 1111 systems, are substantially less anisotropic. This
relatively low anisotropy is of potential practical importance. In particular, flux
pinning is an important issue in applications of superconductivity, and this is
greatly facilitated in low-anisotropy materials.
A comparison between cross sections of the Fermi surface at the centre of the
Brillouin zone is shown in figure 3d–f . The effect of Fermi surface corrugation
is to reduce the nesting of the cylindrical electron and hole sheets. While in
LaFePO the in-plane geometrical nesting between the electron and hole sheets
along the (p, p) direction is likely to be possible in SrFe2 P2 , this nesting is strongly
inhibited because of the variation in size of the electron and hole pockets, whereas
in CaFe2 P2 it is completely excluded. One possible avenue for good nesting could
exist only in a limited kz range, owing to the strong dispersion of the electronic
bands along this direction, and this needs to be further explored.
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The torque signal in the quantum oscillations experiments in iron phosphides
and SC As-substituted iron phosphides (Shishido et al. 2010) is dominated by
the contribution from the electronic pockets (figure 2c). In LaFePO, SrFe2 P2
and CaFe2 P2 , the mean free paths of electrons are at least a factor of 2 larger
than those of the holes, which may be caused by differences in the orbital
character between the electron and hole sheets (Analytis et al. 2009b; Coldea
et al. 2009). Furthermore, for the same electron sheet, the mean free path varies
along kz , suggesting possibly anisotropy in scattering, which may be relevant for
understanding the presence of line nodes in some of the iron pnictides (Fletcher
et al. 2009). The Hall effect data in the paramagnetic (PM) state of BaFe2 As2 also
suggest that electron bands have a higher mobility than the hole bands (Fang et al.
2009), and at low temperatures, in the magnetic state, hole mobility is suppressed
when compared with that of electrons, suggesting that hole localization may be
promoted by spin fluctuations (Fang et al. 2009).
The relevance of electron–electron interactions can be obtained by comparing
the quasi-particle mass enhancement in relation to the band mass, m ∗ /mb , and
accounting for the electron–phonon coupling. The values corresponding to the
averaged electron sheets between different compounds vary from 2.38 in SC
LaFePO (Coldea et al. 2008) to 1.85 in SrFe2 P2 (Analytis et al. 2009b) and 1.5
in both CaFe2 P2 (Coldea et al. 2009) and BaFe2 P2 (Shishido et al. 2010). It
is worth emphasizing that in the three-dimensional CaFe2 P2 the enhancement
is the same for both electron and hole sheets. The conventional electron–
phonon coupling for CaFe2 As2 in a non-magnetic, collapsed tetragonal phase is
calculated to be lel–ph ∼ 0.23 (Yildirim 2009), and it can be considered as a good
approximation for the non-magnetic phosphites that share similar structures and
are isoelectronic to arsenides. Thus, the mass enhancement, 1 + lel–ph = 1.23,
is probably due to a conventional electron–phonon coupling; the many-body
interactions, however, could be responsible for a further enhancement of the
electron–phonon interaction owing to the strong polarization of the pnictogen
ions (M. Kulic & A. Haghighirad 2009, unpublished data). While linear electron–
phonon coupling in the non-magnetic phase is calculated to be weak, strong
magneto-phonon coupling is believed to be very strong (in LaFeAsO, the Fe
moment changes at a rate of 6.8 mB /Å as the Fe–As distance is varied by changing
the As layer height) (Yin et al. 2008). Assuming that the mass enhancement of
m ∗ /mb ∼ 1.5 in non-magnetic phosphides is mainly an effect of the interaction
with the lattice, the further enhancement seen in the SC LaFePO could suggest a
significant contribution from electron–electron interactions (lel–el ∼ 0.8 − 1) and
it may reflect the large degree of nestability of this compound and its tendency
towards magnetic instabilities.
3. Fermi surface in an AFM phase
In an AFM phase, the lattice periodicity is changed owing to the presence of
an additional periodicity introduced by the magnetic order which leads to a
Fermi surface reconstruction. Band structure calculations considering the AFM
Brillouin zone have been reported by Analytis et al. (2009a,b) using the LDA + U
methodology with a negative value for U in order to suppress the value of
the magnetic moment, as LDA significantly overestimated the value of the
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Review. Iron pnictides superconductors
(a)
3513
(b)
Figure 4. The Fermi surface of BaFe2 As2 in the (a) PM orthorhombic phase and (b) the
reconstructed Fermi surface due to the collinear AFM order. The method of calculations of the
reconstructed Fermi surface is presented in detail in Analytis et al. (2009a,b). Solid lines delimit
the PM Brillouin zone and the dashed lines the AFM Brillouin zone.
moment. Although most of the Fermi surface is gapped by the SDW in the
magnetic pnictides, there are remaining carriers; this is in contrast to cuprates,
where the undoped materials are AFM Mott insulators. The Fermi surface
of the AFM BaFe2 As2 is represented by electron and hole ellipsoids and two
tiny electron and hole pockets originated from Dirac-like crossing of the bands
(Analytis et al. 2009a,b).
ARPES data in the SDW phase on BaFe2 As2 (P. Richard et al. 2009,
unpublished data) confirm the presence of these tiny pockets as originating from
an anisotropic Dirac cone. This cone is located away from high-symmetry points
and formed by bands that are gapped away from the cone. As the cone’s apex is
located very near the Fermi level this will lead to the formation of tiny FS pockets
which are very sensitive to structural and magnetic effects, less than 0.05 per
cent of the PM Brillouin zone (Analytis et al. 2009a,b). Quantum oscillations
studies (Harrison & Sebastian 2009) suggest the existence of these tiny pockets
as a signature for nodal SDW. It is not clear at the moment whether these tiny
pockets with high velocity play a major role in the superconductivity of the iron
pnictides as their presence is strongly affected by the changes in the lattice and
magnetism and the details of theoretical models.
Quantum oscillations studies in (Ba/Sr/Ca)Fe2 As2 (Sebastian et al. 2008;
Analytis et al. 2009a,b; Harrison et al. 2009) only observe small pockets, which
occupy 3 per cent of the PM Brillouin zone when compared with the large
sheets of the non-reconstructed Fermi surface, which occupies 15 per cent of
the PM Brillouin (figure 4). These results are in agreement with the presence of
a Fermi surface reconstruction and opening of an SDW gap, but the size of the
largest observed pocket is half the size of that predicted for the larger ellipsoid
areas. However, as the band structure calculations in the magnetic phase of the
iron pcnitides cannot clearly account for the experimentally observed magnetic
moments at this point, any direct association should be made with caution until
all the predicted pockets are observed.
As in other new areas of research, the quality of the sample always requires
time to be improved. In this respect, the research on iron pnictides has been done
at such a high pace because various tools and experimental techniques developed
Phil. Trans. R. Soc. A (2010)
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3514
A. I. Coldea
for cuprates were already available. In the AFM phase, only small pockets were
observed, which would be consistent with a Fermi surface reconstruction, and,
at this point, it is not clear whether possible crystal twining may impede the
observation of other pockets in the underdoped region.
4. Future prospects
The discovery of new materials based on iron tetrahedrally coordinated with
pnictides or chalcogenites has raised the expectation that novel materials with
even higher transition temperatures than iron-based superconductors or cuprates
may be in our reach. One possible scenario relies on identifying structures in which
the electronically active layers are isolated to provide enough density of carriers at
the Fermi level to drive the systems towards magnetic instability, which can then
be easily tuned to suppress the magnetism and stabilize a superconductive state.
One of the key ingredients to understanding high-temperature superconductivity in pnictides and in cuprates could be linked to understanding the evolution
of the Fermi surface topology from a reconstructed Fermi surface in the AFM
(underdoped) phase to a large Fermi surface (overdoped phase). By varying
an external parameter (pressure, doping or a magnetic field), a second-order
quantum phase transition, or a quantum critical point under the SC dome, may
separate a PM Fermi liquid regime from the magnetically ordered state. Quantum
oscillations measurements are ideally suited to access a quantum critical point
as a function of a tuning parameter as they provide the effective mass of the
elementary fermionic excitations that can be traced across the quantum critical
point. In pnictides, a possible way to access such a critical point under the SC
dome is through an isovalent substitution of phosphorus at the arsenic site (Dai
et al. 2009) in BaFe2 As2 and quantum oscillations indeed suggest an increase
in the effective masses and nesting between the electron and hole pockets upon
approaching the maximum Tc (Shishido et al. 2010; J. G. Analytis, J.-H. Chu, R.
D. McDonald, S. C. Riggs & I. R. Fisher 2010, unpublished data). Further studies
to follow the detailed topological changes of the Fermi surface across the whole
phase diagram will establish the relevant role played by the itinerant electrons
in iron pnictides and will provide additional experimental guides for the further
theoretical models concerning these materials.
In conclusion, the quantum oscillations studies on iron phosphides provide a
unique access to the Fermi surface of two classes of iron-based superconductors.
These systems do not show magnetic order, despite the fact that they have
similar calculated Fermi surfaces in the tetragonal phase at high temperatures.
While nesting may be important in stabilizing an SC state with low Tc (like
LaFePO), the high Tc values observed in arsenides in the proximity of magnetic
order suggest that spin fluctuations are also essential (Dai et al. 2009). Lower
dimensional systems with an even larger spacer between Fe–As layers may
further enhance these fluctuations and may possibly stabilize superconductivity
at temperatures even higher than 55 K.
I would like to acknowledge the Royal Society, EPSRC and EuroMagnet II for financial support.
I would like to thank my colleagues and collaborators A. Carrington, J. Analytis, I.R. Fisher,
M. Johanness, I. Mazin, A.F. Bangura, R.D. McDonald, J.-H. Chu, J. Fletcher, H. Shisido, K.
Yashimoto, S. Kasahara, S. Hayden and E. Yelland and my family for their immense support.
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References
Analytis, J. G., Abdel-Jawad, M., Balicas, L., French, M. M. J. & Hussey N. E. 2007 Angledependent magnetoresistance measurements in Tl2 Ba2 CuO6+d and the need for anisotropic
scattering. Phys. Rev. B 76, 104523. (doi:10.1103/PhysRevB.76.104523)
Analytis, J. G., McDonald, R. D., Chu, J.-H., Riggs, S. C., Bangura, A. F., Kucharczyk,
C., Johannes, M. & Fisher, I. R. 2009a Quantum oscillations in the parent pnictide
BaFe2 As2 : itinerant electrons in the reconstructed state. Phys. Rev. B 80, 064507. (doi:10.1103/
PhysRevB.80.064507)
Analytis, J. G., Andrew, C. M. J., Coldea, A. I., McCollam, A. & Carrington, A. 2009b Fermi
surface of SrFe2 P2 determined by the de Haas–van Alphen effect. Phys. Rev. Lett. 103, 076401.
(doi:10.1103/PhysRevLett.103.076401)
Bergemann, C., Julian, S. R., Mackenzie, A. P., NishiZaki, S. & Maeno, Y. 2000 Detailed
topography of the Fermi surface of Sr2 RuO4 . Phys. Rev. Lett. 84, 2662–2665. (doi:10.1103/
PhysRevLett.84.2662)
Brouet, V. et al. 2009 Nesting between hole and electron pockets in Ba(Fe1−x Cox )2 As2
(x = 0–0.3) observed with angle-resolved photoemission. Phys. Rev. B 80, 165115. (doi:10.1103/
PhysRevB.80.165115)
Carrington, A. et al. 2009 Quantum oscillation studies of the Fermi surface of LaFePO. Physica C
469, 459–468. (doi:10.1016/j.physc.2009.03.045)
Chubukov, A. V., Efremov, D. & Eremin, I. 2008 Magnetism, superconductivity, and
pairing symmetry in iron-based superconductors. Phys. Rev. B 78, 134512. (doi:10.1103/
PhysRevB.78.134512)
Coldea, A. I. et al. 2008 Fermi surface of superconducting LaFePO determined from quantum
oscillations. Phys. Rev. Lett. 101, 216402. (doi:10.1103/PhysRevLett.101.216402)
Coldea, A. I., Andrew, C. M. J., Analytis, J. G., McDonald, R. D., Bangura, A. F., Chu, J.-H.,
Fisher, I. R. & Carrington, A. 2009 Topological change of the Fermi surface in ternary iron
pnictides with reduced c/a ratio: a de Haas–van Alphen study of CaFe2 P2 . Phys. Rev. Lett.
103, 026404. (doi:10.1103/PhysRevLett.103.026404)
Dai, J., Si, Q., Zhu, J.-X. & Abrahams, E. 2009 Iron pnictides as a new setting for quantum
criticality. Proc. Natl Acad. Sci. USA 106, 4118–4121. (doi:10.1073/pnas.0900886106)
Fang, L. et al. 2009 Roles of multiband effects and electron-hole asymmetry in the superconductivity
and normal-state properties of Ba(Fe1−x Cox )2 As2 . Phys. Rev. B 80, 140508. (doi:10.1103/
PhysRevB.80.140508)
Fletcher, J. D., Serafin, A., Malone, L., Analytis, J., Chu, J.-H., Erickson, A. S., Fisher, I. R. &
Carrington, A. 2009 Evidence for a nodal-line superconducting state in LaFePO. Phys. Rev.
Lett. 102, 147001. (doi:10.1103/PhysRevLett.102.147001)
Harrison, N. & Sebastian, S. E. 2009 Dirac nodal pockets in the antiferromagnetic parent phase of
FeAs superconductors. Phys. Rev. B 80, 224512. (doi:10.1103/PhysRevB.80.224512)
Harrison, N., McDonald, R. D., Mielke, C. H., Bauer, E. D., Ronning, F. & Thompson, J. D. 2009
Quantum oscillations in antiferromagnetic CaFe2 As2 on the brink of superconductivity. J. Phys.
Cond. Matter 21, 322202. (doi:10.1088/0953-8984/21/32/322202)
Ishida, K., Nakai, Y. & Hosono, H. 2009 To what extent iron–pnictide new superconductors have
been clarified: a progress report. J. Phys. Soc. Jpn 78 062001. (doi:10.1143/JPSJ.78.062001)
Kamihara, Y., Hiramatsu, H., Hirano, M., Kawamura, R., Yanagi, H., Kamiya, T. & Hosono,
H. 2006 Iron-based layered superconductor: LaOFeP. J. Am. Chem. Soc. 128, 10012–10013.
(doi:10.1021/ja063355c)
Kamihara, Y., Watanabe, T., Hirano, M. & Hosono, H. 2008 Iron-based layered superconductor
La[O1−x Fx ]FeAs (x = 0.05–0.12) with Tc = 26 K. J. Am. Chem. Soc. 130, 3296–3297.
(doi:10.1021/ja800073m)
Kuroki, K., Onari, S., Arita, R., Usui, H., Tanaka, Y., Kontani, H. & Aoki H. 2008 Unconventional
superconductivity originating from disconnected Fermi surfaces in the iron-based oxypnictide.
J. Phys. Soc. Jpn. 77, 96–98.
Lebègue, S. 2007 Electronic structure and properties of the Fermi surface of the superconductor
LaOFeP. Phys. Rev. B 75, 035110. (doi:10.1103/PhysRevB.75.035110)
Phil. Trans. R. Soc. A (2010)
Downloaded from http://rsta.royalsocietypublishing.org/ on June 14, 2017
3516
A. I. Coldea
Lee, C.-H. et al. 2008 Effect of structural parameters on superconductivity in fluorine-free
LnFeAsO1−y (Ln = La, Nd). J. Phys. Soc. Jpn. 77, 083704. (doi:10.1143/JPSJ.77.083704)
Lester, C., Chu, J.-H., Analytis, J. G., Capelli, S. C., Erickson, A. S., Condron, C. L., Toney,
M. F., Fisher, I. R. & Hayden, S. M. 2009 Neutron scattering study of the interplay between
structure and magnetism in Ba(Fe1−x Cox )2 As2 . Phys. Rev. B 79, 144523. (doi:10.1103/
PhysRevB.79.144523)
Lu, D. H. et al. 2008 Electronic structure of the iron-based superconductor LaOFeP. Nature 455,
81–84. (doi:10.1038/nature07263)
Malaeb, W. et al. 2008 Photoemission study of the electronic structure of LaFeAsO1–x Fx and
LaFePO1–x Fx . J. Phys. Soc. Jpn Suppl. 77 (Suppl. C), 69.
Maniv, T., Zhuravlev, V., Vagner, I. & Wyder, P. 2001 Vortex states and quantum magnetic
oscillations in conventional type-II superconductors. Rev. Mod. Phys. 73, 867. (doi:10.1103/
RevModPhys.73.867)
Mazin, I. I., Singh, D. J. & Johannes, M. D. 2008a Unconventional superconductivity with a sign
reversal in the order parameter of LaFeAsO1−x Fx . Phys. Rev. Lett. 101, 057003. (doi:10.1103/
PhysRevLett.101.057003)
Mazin, I. I., Johannes, M. D., Boeri, L., Koepernik, K. & Singh, D. J. 2008b Problems with
reconciling density functional theory calculations with experiment in ferropnictides. Phys.
Rev. B 78, 085104. (doi:10.1103/PhysRevB.78.085104)
Ortenzi, L., Cappelluti, E., Benfatto, L. & Pietronero, L. 2009 Fermi-surface shrinking and
interband coupling in iron-based pnictides. Phys. Rev. Lett. 103, 046404. (doi:10.1103/
PhysRevLett.103.046404)
Qazilbash, M. M., Hamlin, J. J., Baumbach, R. E., Zhang, L., Singh, D. J., Maple, M. B. & Basov,
D. N. 2009 Electronic correlations in the iron pnictides. Nat. Phys. 5, 647–650. (doi:10.1038/
nphys1343)
Ren, Z.-A. et al. 2008 Canonical entropy and phase transition of rotating black hole. Chin. Phys.
Lett. 25 2215. (doi:10.1088/0256-307X/25/7/015)
Rossel, C., Bauer, P., Zech, D., Hofer, J., Willemin, M. & Keller. H. 1996 Active microlevers as
miniature torque magnetometers. J. Appl. Phys. 79, 8166. (doi:10.1063/1.362550)
Sebastian, S. E., Gillett, J., Harrison, N., Lau, P. H. C., Singh, D. J., Mielke, C.
H. & Lonzarich, G. G. 2008 Quantum oscillations in the parent magnetic phase of
an iron arsenide high temperature superconductor. J. Phys. Cond. Matter 20, 322202.
(doi:10.1088/0953-8984/20/42/422203)
Shishido, H. et al. 2010 Evolution of the Fermi surface of BaFe2 (As1−x Px )2 on entering the
superconducting dome. Phys. Rev. Lett. 104, 057008. (doi:10.1103/PhysRevLett.104.057008)
Shoenberg, D. 1984 Magnetic oscillations in metals. Cambridge, UK: Cambridge University Press.
Singleton, J., Balakirev, F. F., Baily, S. A., Chen, G. F., Luo, J. L. & Wang, N. L. 2009 Nearly
isotropic superconductivity in (Ba, K)Fe2 As2 . Nature 457, 565–568. (doi:10.1038/nature07676)
Springford, M. & Wasserman, A. 1996 De Haas–van Alphen experiments in the superconducting
state. J. Low Temp. Phys. 105, 273–283. (doi:10.1007/BF00768401)
Uhrig, G. S., Holt, M., Oitmaa, J. P., Sushkov, O. P. & Singh, R. R. P. 2009 Pnictides as frustrated
quantum antiferromagnets close to a quantum phase transition. Phys. Rev. B 79, 092416.
(doi:10.1103/PhysRevB.79.092416)
Yildirim, T. 2009 Strong coupling of the Fe-spin state and the As–As hybridization in iron–
pnictide superconductors from first-principle calculations. Phys. Rev. Lett. 102, 037003.
(doi:10.1103/PhysRevLett.102.037003)
Yin, Z. P., Lebègue, S., Han, M. J., Neal, B. P., Savrasov, S. Y. & Pickett, W. E. 2008 Electron–hole
symmetry and magnetic coupling in antiferromagnetic LaFeAsO. Phys. Rev. Lett. 101, 047001.
(doi:10.1103/PhysRevLett.101.047001)
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AUTHOR PROFILE
Amalia Coldea
Born in Transylvania, Romania, Amalia Coldea read physics at the Babes-Bolyai
University, Cluj-Napoca, and graduated with an MSc with first class honours. She
continued her studies at the University of Oxford, where she investigated colossal
magnetoresistive materials and obtained her PhD in 2001. Later she continued as
a postdoctoral researcher in Oxford, working on quantum oscillations in organic
superconductors. In September 2005, she took up a Royal Society Dorothy
Hodgkin Fellowship at the University of Bristol and, since January 2010, has
been based at the University of Oxford. Her current field of research is quantum
oscillations in iron-based superconductors and metallic systems on magnetically
frustated geometries. Amalia is married to Radu Coldea and they have two
children, Victor and Sandra.
Phil. Trans. R. Soc. A (2010)