Rafael Fernandes

Interplay between superconductivity, magnetism and nematic/orbital order in the iron pnictides

Blogged by Andy Schofield and Kedar Damle 

Started with in introduction of spin-fluctuations and other degrees of freedom. We have heard much about cuprates and the Mott transision We will be looking at pnictides where we are dealing with band physics. But the bands are complex yet it simplifies to some q2d electron and hole pockets ( a theorist’s view): Hole pockets in the centre of the zone (circular) and elliptical electron pockets at the X points:

Question: What about experiements? Good agreement with LDA in terms of shape though not always the mass, but see ARPES data later.

1. Unconventional Superconductivity:

A review of BCS superconductivity with phonons: The Cooper instability set as an exercise! The problem of computing Tc from first principles is difficult because of the lack of ability to predict the Coulomb pseudopotential. Rafael now moves to the blackboard. Want compute the partition function and he introduces the action for the electron phonon interaction: in the path integral formalism with coherent states. Then we get the Nambu representation of the problem where the electrons are a spinorial like combination of particle and hole. Constructing the perturbation theory gives a Dyson equation for the self-energy - and an Ansatz for developing superconductivity has a self-energy with off-diagonal parts representing the energy gap, W. It has a new dispersion E and a quasiparticle renormalization Z. BCS gets obtained in the limit that Z=1, E=dispersion and W is the gap.
Question: What is the physics of this limit? Answer: it is the limit of weak coupling.
Question: Is BCS limit applicable away from weak coupling - like in cuprates? Answer: no - but see later.
Question: Could you better define weak-coupling in terms of the parameters you introduced? Yes - I can introduce lambda which is like g^2 x density of states. There is another factor (m/M) which is small - ratio of the electron mass to ionic mass - so we can neglect vertex corrections.

So - are the iron based superconductors electron-phonon coupled? One calculation claimed maximum from e-ph is 0.8K in the pnictides - so that looks unlikely. Now let us look at the symmetries of the possible superconducting state: Crystal Lattice (X) Rotations (R) Time-reversal (T) and U(1) gauge. We look at the Cooper pair wavefunction in systems with inversion symmetry: singlet must be s- or d-wave while the triplet could be p-wve or f-wave etc. We will restrict to tetragonal singlet states with no T-breaking. Looking in 2D we have some simple possibilities:
s-wave: A_1g: no nodes - or x^2+y^2 extended s-wave with line nodes that are not fixed to symmetry lines.
g-wave: A_2g:  state unchanged by 90 degree rotation, but nodes along every 45, 90, etc degree - 4 lines of nodes
d_x^2-y^2: B_1g: state unchanged by 180 degree rotation - nodes at 90, 180 - 2 line nodes
d_xy: B_2g: line nodes
Electron-electron repulsion tends to favour nodal superconductors and we will consider pairing fluctuations by spin-fluctuations (Berk and Schrieffer, PRL 1966). We revisit these calculations and change to spin-fluctuations from bosons. The electrons now couple to a vector field and the new inverse greens function of the boson is the spin-propagator discussed in Andy Schofield's quantum criticality tutorial.
Question: what about vertex corrections? I will come back to this.
The spin-density means that the sigma_z involving the density goes to the identity which changes a sign to make the interaction effectively look repulsive not attractive.
Vertex corrections: now instead of being a small term (m/M) the spin fluctuations are also from the electrons so we cannot use Migdal's theorem. 
Question: I worked in Migdal's lab - and the theorem does not work if you use bare variables? AVC answers: there is a critical value below which this approach still works. 
Question: are we here in 2D where the boson coupling might be a relevant operator? Answer it is always relevant and we going to develop a theory here.
When visualizing the interaction it is repulsive at short range but changes sign and oscillates etc - and the diagonals it is purely repulsive. So put the nodes along the diagonal lines.
More on this in the next lecture...

Good evening blog watchers---we're back for the second installment
of Rafael Fernandes's double-header on superconductivity, magnetism,
and nematic/orbital order in the iron pnictides.


In the last lecture, Rafael gave a sketch of Eliashberg theory of phonon-mediated superconductivity (conventional)
and introduced the symmetry analysis of the SC order parameter,
focusing on singlet superconductivity with no time reversal
symmetry breaking in 2 dimensions (see description of previous lecture)
and then asked:
 
Why would the gap have nodes when the gap helps gain energy?
 
And his answer was: repulsive interactions.
 
To see how that works, he took the example of interactions mediated by
spin fluctuations and repeated Eliashberg theory, this
time for an interaction which couples the electron spin density to antiferromagnetic spin fluctuations, and showed
how the final gap equation gets an additional minus sign.

This minus sign comes from
the fact that the antiferromagnetic spin fluctuations mediate
a bonafide repulsive interaction.

Now in the second lecture, he's promised to start with this repulsive
interaction and show us how it leads naturally to a gap function with nodes when we look for solutions to the gap equation. So here we go with the
ball by ball commentary, a la cricinfo, of Rafael's second innings...

To begin, Rafael notes that the effective interaction in real space is proportional to \chi(r).
The gap would like to vanish across diagonals to avoid unfavourable
part of \chi(r), and that is the origin of the diagonal nodes in the gap.
 
For the detailed analysis, it is better to go back to momentum space and think
in terms of different parts of the Fermi surface connected by Q (Q is \pi,0
or 0,\pi). 

Simplification:  We can assume
V_{k-k'} = \chi(Q) \delta_{k-k',Q}
since the  real interaction has a strong peak at Q.

So we take two Deltas, \Delta_1 and Delta_2, on the two pockets
that are connected by Q, and get a two by two system of coupled
BCS equations for \Delta_1 and \Delta_2. Further, we can linearize
the gap equations (near T_c) and get two solutions:
One solution has an eigenvector with same sign for \Delta_1 and \Delta_2
The other has a relative minus sign. 

The equal sign solution requires
*attractive* interactions---this is the s++ state of Kontani and Onari,
which would need *orbital* fluctuations to mediate attraction.
The solution s+-, with opposite relative sign, is the one picked
by repulsive interactions, since the repulsive interaction looks effectively
attractive for this eigenvector.

Question: What is the cutoff to use in the momentum sum in the BCS eqn?
Ans: Bands are very shallow, so size of the pockets is the appropriate cutoff

Question: What about Coulomb repulsion (local)?
Ans:  Can show that this state can survive with little reduction in T_c

With this sketch of the mechanism for superconductivity, Rafael switches
focus to magnetism in the normal state:

The ordering is along two equivalent wavevectors Q_1 = \pi,0
and Q_2=0,\pi.
 
First question to address is: How do we describe the magnetically ordered state?
With an itinerant description or a localized picture of local moments ordering?

To explain this, Rafael now digresses to an elementary discussion
of antiferromagnetism in the Mott insulating state of the Hubbard
model, and explains how virtual charge fluctuations give antiferromagnetic
exchange interactions between the local moments of the Mott insulator.
This is contrasted with the prototypical example of an itinerant magnet, a Stoner
ferromagnet: 
Polarizing a free Fermi gas costs kinetic energy.
But a polarized free fermi gas pays less Coulomb repulsion energy
when we turn on interactions. At some critical value of the
Coulomb repulsion, this leads to ferromagnetism. This critical
value of repulsion is inversely proportional to the DOS at the Fermi
energy (Stoner criterion). Generalizing to ordering at some
other wavevector, say Q, the Stoner criterion becomes:
When U_Q \chi_0(Q) >1, itinerant SDW state is favoured.
Further, \chi_0(Q), the non-interacting susceptibility, is enhanced
if there is nesting with nesting wavevector Q, so it
is easier to have SDW instabilities when there is nesting.

These are the two limiting ways of thinking about magnetism.
Even in simple magnets like Ni and Fe, *neither* picture
applies literally, and we have to just make do with whichever picture
is closer to reality.

After this digression, Rafael makes his main point:
The pnictides are metallic, so it makes sense to choose the itinerant
picture to describe magnetism, even if they have significant correlations.

Question: When there is nesting, doesn't a gap open and destroy the Fermi surface?
Ans: That only happens if there is perfect nesting. Here we are not
thinking of perfect nesting.
In the pnictides, Q_1 and Q_2 are actually approximate nesting wavevectors
that connect the hole and electron parts of the Fermi surface. This nesting
gradually goes away on doping. And that is when the magnetism
goes away too. So the itinerant picture seems to hang together all right.
Of course, since the nesting is imperfect, there is still a threshold interaction
without which we would not have the SDW.

With this background, Rafael moves on to a more detailed
theory for the itinerant magnetism, basically doing a Hertz-Millis
theory for two simultaneous SDW instabilities (along Q_1 and along
Q_2). The final punch-line (details went by too quickly for a
ball-by-ball account) is

F_mag = \frac{a}{2} (\vec{m}_1^2 +\vec{m}_2^2) +\frac{u}{4}
(\vec{m}_1^2 +\vec{m}_2^2)^2 - \frac{g}{4}(\vec{m}_1^2-\vec{m}_2^2)^2

From the details that went by too quickly, 
Rafael concludes:  g>0

This g>0 picks stripes along one or the other direction from the plethora
of states that minimize the g=0 free energy. And these are exactly
the states seen in experiment. This gives qualitative agreement.
But does it work quantitatively?
The answer seems to be yes(!)
[the details again went by a bit too quickly for this blogger]

Q: What happen when g gets very large?
Ans: First order transition. But g is never that large in
the pnictides, so we don't need to worry about this possibility.

With magnetism in the bag, Rafael now moves on to discuss nematicity
in the normal state (already mentioned in earlier talks this week).

 Basic observation: The structural transition in the pnictides closely follows
the SDW transition, being slightly above the SDW transition (at slightly
higher temperatures). 

In the orthorhombic phase (below the structural
transition), experiments see strong anisotropies that cannot be attributed
to lattice distortions alone: A very tiny uniaxial pressure (of order
mega, not giga pascals) immediately leads to a resistivity anisotropy.
Now, the maximum resistivity anisotropy is at finite doping and
almost 100% in magnitude, while the maximum orthorhombic distortion
is at zero doping and tiny by comparison. So there must be
something electronic about the symmetry breaking of tetragonal
symmetry, and the lattice follows in the wake of the electrons.
This is what people mean when they talk of electronic nematicity.

Question: Are these measurements (of resistivity anisotropy) done above or below the SDW transition?
Ans: Both above and below SDW transition. Of course, the anisotropy
is largest below the SDW transition, but it is very big even above the
SDW transition (in comparison with the orthorhombic distortion).

Take-II on nematic order, starting from F_mag:

Ordering at Q_1 and Q_2 are both equally favoured.
To pick one of them, system has to break O(3) spin symmetry
and break the Z_2 symmetry (the symmetry that says Q_1 and Q_2
are equivalent).

Are both these symmetries necessarily broken simultaneously?
No!
In fact, the discrete symmetry breaking is less susceptible
to fluctuations, and we can imagine that the Z_2 symmetry
breaks first without O(3) symmetry breaking. This state,
with Z_2 symmetry breaking but no O(3) symmetry breaking, is
to be identified with electronic nematicity above the SDW transition.

In this picture, the nematicity comes from the spin physics, and other
things follow because they are coupled to the spin physics.

To put some flesh on this, in the last part of the lecture, Rafael moves on to consider the role of spatial fluctuations in \vec{m}_1 and \vec{m}_2
starting with F_mag derived earlier:
 
To do this, we decouple both quartic terms (g and u)  by two
separate Hubbard Stratanovich fields. The HS field dual
to the g term is the nematic order parameter. Integrating
out \vec{m}_1 and \vec{m}_2, one gets an
action functional for the two HS fields---one of these, \phi,
is the nematic order parameter, and the other \psi, gives
the magnitude of the tendency to magnetic order.
Next, we look for the saddle point of this action.
The saddle-point condition gives two coupled equations
for \psi and \phi at the saddle point.

Punchline: \phi is non-zero at the saddle-point *without* any magnetic
order developing---this happens because of
terms in the action coming from magnetic fluctuations (which
we integrated out to get the action for the HS fields). This is the transition to the nematic.  In this picture, this nematic order parameter (in the spin physics)
then couples to orbital degrees of freedom and induces
orbital order and everything else follows.
This is a nice way to rationalize, if not actually explain, the essential
features of what is seen experimentally.

And this concludes Rafael's very nice overview of the theory
for pnictide superconductors.
 
 


 

Friday, 10th August,

Hai Hu Wen (Nanjing University, China)
Materials and pairing mechanism in iron pnictides/chalcogenides:  
what we have learned and have still to learn.

Blogged by Leni Bascones and Piers Coleman















Hai Hu will start presenting the FeAs and FeSe based materials and structures. He talks about the discovery of superconductivity at 26 K in doped LaOFeAs by Hosono's group. He calls attention to  the large resisitivity of this material and the resistivity anomaly which was observed and that was later known to be an spin density wave transition. He notes that many of the parent materials had been synthesized some time before but were not studied in detailed.  Quickly after the discovery of superconductity above 50 K was found in related materials of the same family, so called 1111. Later on superconductivity was discovered in other families, like 122 or 111, what also contained FeAs layers, and in 11 famlily with simple structure based in FeSe layers. Nowdays there are seven related  families showing superconducting. Superconductivity is also found in Ni-compounds.

In most cases parent materials are non-superconducting. Superconductivity can be induced by electron or hole doping, by non-dopant substitution or by the application of pressure. Parent materials show an antiferromagnetic instability . It was early proposed that antiferromagnetism was due to Fermi nesting. Later on a different interpretation in terms of local moments was put forward. Magnetic order is called stripe or co-linear and show antiferromagnetic order along one direction and ferromagnetic along the other direction. Through the phase diagram, plotted as a function of pressure, the superconducting critical temperature shows a dome shape. A similar phase diagram is found when doping with electrons or holes. Antiferromagnetism disappears and superconductivity shows up.

Hai Hu compares the phase diagram of iron superconductors with that of cuprates. Cuprates are single band antiferromagnetic Mott insulators . with an exchange constant of order 150 meV while pnictides are multi-band metallic materials with an estimated second nearest neighbor exchange constant of order 50 meV.
He focuses on the transition towards antiferromagnetism and presents experimental results which show that at temperatures slightly larger than the Neel temperature there is also an structural transition. The gap in temperatures between the  structural and Neel temperatures depends on the family. He says that there are  two main proposals to explain the structural transitions  one based in spin nematicity and the other one in orbital ordering. Andrey Chubukov points that so far there are no microscopic calculations to support the  orbital ordering proposal. Structural details like As height or Fe-As.Fe angle seem to determine the optimal superconductivity. Theoretical input on why this could happen was given by Kuroki et al in 2009.

Hai-Hu wonders whether nematicity is important for superconductivy or not and whether it is just a consequence of magnetic or orbital fluctuations. He talks about transport measurements in detwinned samples by I. R. Fisher group which showed resistivity anisotropy with conductivity larger in the antiferromagnetic direction. Anisotropy was observed even above the structural transition and it has been discussed in terms of nematicity. Rafael Fernandes points out that because strain is applied to detwin the samples, above the structural transition one cannot talk about symmetry breaking. Other experiments showing anisotropic features where  published by Davis's and L. Green's group. Recent experiments in Matsuda's group suggest that the nematicity starts at temperatures much higher than the structural transition.

Hai-Hu suggests to try all kinds all of possibilities to make new materials. In particular he suggests to try systems with FeB and FeGe layers. So far only Tc ~5K have  been achived in these compoung but he hopes that Tc could become larger. To try new possibilities for materials lead to the observation of superconductivity in hole doped 1111 compounds, oxygen free 1111 materials and in more complex FeAs based materials. More recently superconductivity was observed in KFe2Se2 this compound is heavily electron doped. Some related compounds show Fe vacancies and are antiferromagnetic insulators. This family is called 245. Two different superconducting phases have been found in this family as a function of pressure. Q: Rafael Fernandes asks whether superconductivity in this materials is a bulk property . A:Hai Hu notes that the superconducting volume in not very large. Q: Andrey Chubukov asks whether there is phase separation A: yes it seems so.   There are current efforts to avoid Fe vacancies.

He concludes the talk saying that there is still some space to be explored for making new superconductors.

Q:  About Sr4V2O6Fe2As2 at high magnetic fields, whether a 2D vortex pancakes has been observed. A: Not directly.
Q: Why is iron special? A: Because of a possible balance between correlations and Drude weight.
Hai-Hu believes that many other superconductors without Cu or Fe will be found
Q: About the mechanism of superconductivity. A: Phonons seem not to mediate superconductivity, which seems related .to antiferromagnetic or orbital fluctuations.
Some other questions were asked but I could not hear too well the participants from my place.



Part II

Well, we're off to the races again!  After a short break, Hai Hu is begining his second lecture. He mentions that in his opinion, the iron based superconductors are intermediate coupling in character -
lying between the two extremes of spin fluctuation theory and the RVB theory of superconductivity.  He writes down the BCS formula, noting that over regions where the interaction is strongly repulsive, the gap likes to have different signs. Since the strongest interaction is betwen the electron and hole pockets, this lead to the proposal(by Igor Mazin and others), that the order parameter must have s+- symmetry, changing sign between the electron and hole pockets, but without a node crossing the Fermi surfaces. Proving that this is the nature of the pairing proves to be difficult, he says, and we don't have much direct evidence for the change of sign of the OP at this stage, though we have lots of evidence for a fully gapped Fermi surface.

Measurement by ARPES, shows that there are in fact two gaps, a feature of multiband superconductivity.  (I was not sure which part has the largest gap - the electrons or the holes? ) 

Using Hall probes, it becomes possible to measure the superfluid density.  There is an interesting temperature dependence, which can be fit to a two-component form, in which rho_a and rho_b both have BCS temperature dependence, with gaps

Deltaa= 1.6 meV,  Delta_b = 9.2 meV

rho = x rho_a + (1-x) rho_b

with a fit value of x =

Now the Hall constant is also consistent with multi-band physics. The strong temperature dependnece of the Hall constant is said to be a consequence of the multi-band character. This is contraversial, and some interpret this as a strong correlation effect, as a result of temperature dependent Z. I prefer to keep the interpretation simple, he says.

Now Hai Hu turns to NMR measurements. He notes the relationship

1/T_1T ~ Sum A(q)^2 chi''(q,omega)/omega at \omega

at omega = 0.  The upturn in 1/(T1 T) is consistent with antiferromagnetic spin fluctuations.

Now RPA spin fluctuation theory is predicted to have a sharp peak  below Tc - a resonant peak that is indeed observed in experiments.  This is consistent with s+-.   A second  measurement is using Fourier transformed STM measurements, carried out by Haneguri et al.  The magnetic field enhances sign-changing scattering, and this is what is observed in Fe(Se,Te).  Unfortunately so far, we have been unable to reproduce this scattering.

Q: what about sc loop experiments with half flux by Tsui et al?
HHW: Unfortunately, this also has not been reproduced and I am very sceptical.

Now he turns to the debate on gap structure.  There is a table -


ARPES - sees isotropic gap

SUPERFLUID density - most time shows powerlaw dependence, evidence of nodal gap.

NMR low T  1/ T1T shows power law, consistent with nodal gap or the s+- under impurity scattering

THERMODYNAMICS : also shows specific heat consistent with nodal gap


Now the spin fluctuation models the iron pnictides are found to give rise to gap anisotropy, but ARPES has not see it.

Q: Isn't this apples and oranges, doesn't a gap anisotropy exist in the 111
A: yes - (but blogger didn't understand the details of the answer...)

Linear dependence of the superfluid density (from penetration depth) is consistent with line-nodes in some materials, but with full gap on other materials.  Now many samples have lambda ~ (T)^n, with n =2 . Not consistent with ARPES data.

Thermal conductivity in Ba(Fe(1-x)Ni_xAs)_2, for current along the c-axis, there is evidence for a node in this direction. Now in Ba(FeAs1-xPx)_2, there is evidence from four fold oscllation in the thermal conductivity, that there may be nodal lines in two directions (see figure below).  Yet Zhang et al on the same material have data consistent with a different gap assignment.

Hai Hu also sees a four-fold oscillation of the C/T in a field in Fe-Se. (see Nature Comm, 1, 157 (2010). This has been phenomenologically fitted by Vorontosv and Vekhter, PRL 105, 187003 (2010) and Chubukov and Eremin (reference not gotten).

Continuing the discussion about gap anisotropy, in the sf theory, intrapocket scattering will enhance gap symmetry.  This remains a very contraversial issue and many groups are working hard on this topic.

Quantum Critical Point:  Much circumstantial evidence for a hidden quantum critical point.

(1) Entropy S(T) ~ T ln T
(2) Resistivity rho(T)= rho_0 + AT^n, where n is found to go down to n=1 at x = 0.4, but rises towards n=2 either side.
(3) Penetration depth seen to have a divergence.

How strong is the pairing: do we have a pairing group?  In the K-doped BaFe2As2, the jump in C/T is about   100mJ/mol/K^2.  This is consistent with delta C/C_n ~ 2, like strong coupling.  Canfield gorup has found C/Tc ~ T_c^2,  which has been interpreted by different groups in different ways.
(impurities + s+- one group, quantum criticality another group).

STM -

Hai Hu talks about STM in BaK Fe2As2.  His group has seen a dip-hump structure in the STM measurements. Giving an Eliashberg style interpretation, they can read-off

Delta = 7.5meV
Omega = 14meV
Delt + Omega = 21.5 meV

Hai Hu compares with the classic electron-phonon example, Rowell et al, and others used d^2I/dV^2 to read off the phonon spectrum. This is the classic example from lead.  HHW compares the two -
On the left hand side, dI/dV,  x=1 corresponds to omega = gap + resonant frequency.  Now shows results for the irons.

Now compares with Eliashberg simulation, puts in a strong mode at 14.5meV, for inter-pocket scattering. Comparing the blue simulation with the red expt, the two look qualitatively similar. 
Similar calculations for cuprates,

To convince you that the 14meV mode is not phonons, we have done this on more than one material - now on NaFeCoAs- still see a wiggle feature similar to the 122. Gap energy 5meV, mode energy 7-8 meV, consistent with the neutron resonance feature. Putting them all together,  neutron resonance energy versus STM wiggle data, find that

mode energy/ Tc ~ 4.5

Emerging Challenges

(1) Why does superconductivity survive with enormous impurity centers.  ON site doping induces stron impurity scattering, but superconductivity survives.  25% doping still gives superconductivity.
This is not consistent with an Abrikosov-Gorkov theory of splusminus.  A glaring inconsistency!

Remark:  Pressure effect and doping effect give comparable Tcs, even though one adds disorder, one does.

Q: What happens if you dope zinc onto the Fe site.
HHW:  No effect, but maybe on overdoped side.

Further data - Co doped Ba122 has an enormous gamma, but Tc is not effected.
Further data,  Cu and Mn doping on the Fe site,  exactly smae Tc reduction.

Rafael - one of the difficulties is that we don't know how to model the detailed impurity potentials.
HHW - still very hard to understand - one should still get large momentum transfer.

Further data - STM shows that Co doping does not affect the DOS very much. One would expect an ingap state forming from cobalt, but not seen. Small scattering potential or small momentum transfer scattering?   (The blogger is very sceptical about all of this - it all sounds like adding epicycles!)

(2) KyFe2-ySe2

arXiv1012.5164 

Absence of hole pocket - yet still pairing. The absence of the pocket is different to DFT, and it is not seen in ARPES. My group found that these samples are not homogenious, a dip in the field dependent magnetization. Signs of phase separation. Big debate in china

(1) I think the sample is inhomgenious. Second phase K2Fe4Se5
(2) Many disagree.

But I think now we know that the sample is in homogenious. 

Just recently STM data  - white area is 245, only the grey area is the superconducting phase - islands? Perhaps consistent with small sc fraction.

But with this sample, K0.8Fe1.6 Se2 - Keimers group sees a resonance - which they suggest corresponds to coupling between 0,pi and pi,0 electron pockets.  This experiment claims to reconcile the basic picture based on the AF SF mediated pairing.

Q:  Why is the sc volume so small still see this feature in the neutron scattering. ?

(3) Drude Weight. Slide from Basov. If you want high Tc, need strong correlation, but mobile carriers.

Concluding remarks.
s+- model - support from many experiments. Multiband widely observed.

(2) In most vases, the gap is nodeless. While gap anisotropy should ne an event with high probbablity, whcih so far disagrees with ARPES>

(3) The weak coupling picture with the Splus minus can account for some results, but it is challenged by some recent expts - the pair breaking effect induced by nm disorder is weak and the missing of hole pockets in KyFe2-ySe2.

(4) QCP, electronic nematicity, orbital physics have been observed in some systems. It needs more efforts to resolve how they relate to superconductivity.




Q: Is there any evidence for superconducting fluctuations.
HHW:  They are weak. No Nernst effect like in the cuprates.

Q:  Given you have multi-band, you can have diversity of band structure, and then you probably don't need a node to fit the data.
HHW:  At low T, the small gap will still dominate, but we don't go to low enough temperature.