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Breaking Translational Invariance by Population Imbalance 361

Fig. 9. Phase diagram of λ-(BETS)

FeCl

for a magnetic ﬁeld aligned paral-

lel to the conducting layers (H||c). The triangles depict the transition from the

antiferromagnetic-insulating (AFI) to the paramagnetic (PM) state, solid circles

correspond to the onset of superconductivity, H

, and open circles are extracted

from the dip positions in the second derivative of the resistance (solid arrows in the

inset). The squares show estimated values of the order-parameter wavelength in

the assumed FFLO states at T = 0.7 K (see text). The solid lines are guides to the

eye. The dotted arrows in the inset correspond to broader features at H

. After

Ref. 68.

the internal exchange ﬁeld of the iron atoms acting on the π electrons of the

BETS molecules becomes compensated in the applied external ﬁeld.

Since λ-(BETS)

FeCl

is a clean-limit superconductor, as evidenced by

the observation of magnetic quantum oscillations, and the Pauli limit is esti-

mated at 13 T,

the FFLO state may be expected. When Uji et al. measured

the interplane resistance for in-plane ﬁelds, anomalous dip structures were

observed.

These features appear much clearer in the second derivative

of the ﬁeld-dependent resistance curves (inset of Fig. 9) which allowed to

construct the phase diagram shown in Fig. 9. The Jaccarino–Peter com-

pensation ﬁeld is about 32 T. Therefore, critical ﬁelds appear both below

as well as above this value with two regions for possible FFLO states. The

multiple dip structures observed are suggested to be caused by a commensu-

rability eﬀect.

According to Bulaevskii et al.

a possible Josephson-vortex

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362 G. Zwicknagl and J. Wosnitza

lattice (with lattice constant l) may get pinned at the oscillating FFLO order

parameter (with wavelength λ

FFLO

). This eﬀect may lead to the observed

dip structures whenever m = l/λ

FFLO

becomes an integer. Whether this

assumption is correct requires further studies.

Another recent report suggested a possible FFLO state in the 2D organic

superconductor β

-(BEDT-TTF)

This was based on

in-plane penetration-depth measurements by using a tunnel-diode-oscillator

technique. Some features in the second derivative of the observed frequency

changes were attributed to transitions into the FFLO phase and to H

. Be-

low about 1.23 K, H

exceeds the Pauli-limiting ﬁeld and the H

line be-

comes steeper. At the same time the second feature at lower ﬁelds suggests

an emerging FFLO state.

The resulting phase diagram needs, however,

further support.

Further to that, there has recently been a report suggesting an FFLO

state in the quasi-one-dimensional organic superconductor (TMTSF)

ClO

where TMTSF stands for tetramethyltetraselenafulvalene.

Constructing

the magnetic phase diagrams from resistivity measurements, one low-quality

sample revealed a saturation of H

reminiscent of Pauli limiting, whereas a

better-quality sample showed a clear upturn in H

at lowest temperatures.

The resulting H

data are very similar to the phase diagrams shown in

Fig. 8 here for λ-(BETS)

GaCl

In all of the above cases no thermodynamic proof for an FFLO state in or-

ganic superconductors was given. Only for κ-(BEDT-TTF)

Cu(NCS)

such

evidence has been presented recently.

For magnetic ﬁelds aligned exactly

parallel to the 2D layers spike-like anomalies in the speciﬁc heat are observed

for ﬁelds above 21 T and temperatures below 3 K [Fig. 10(a)]. These features

can be much better resolved after subtraction of the phonon contribution to

the speciﬁc heat. For that, data in the normal state measured at 14 T ap-

plied perpendicular to the layers have been used.

The resulting electronic

contribution to the speciﬁc heat, C

, divided by temperature is shown in

Fig. 10(b). At about 21 T, a sharpening of the speciﬁc-heat anomaly and

at 21.5 T a spike in C

occurs. This is clear evidence for a ﬁrst-order na-

ture of the phase transition from the normal into the superconducting state.

For higher ﬁelds, a second very sharp anomaly appears within the super-

conducting state. This proves the existence of an additional thermodynamic

phase. The transition at lower temperatures shows a well-resolvable hystere-

sis, while the main superconducting transition reveals a smaller hysteresis

[inset of Fig. 10(b)]. Above 23 T the transition left the temperature window

accessible in the experiment.

September 14, 2010 15:18 World Scientiﬁc Review Volume - 9.75in x 6.5in ch14

Breaking Translational Invariance by Population Imbalance 363

Fig. 10. (a) Temperature dependence of the speciﬁc heat divided by temperature,

C/T , of κ-(BEDT-TTF)

Cu(NCS)

for magnetic ﬁelds applied parallel to the su-

perconducting layers. (b) Low-temperature part of the electronic part of the speciﬁc

heat, C

/T . Data taken in 14 T applied perpendicular to the layers were used to

separate the phonon contribution. See Ref. 62 for more details.

0 2 4 6 8 10

T (K)

H (T)

κ-(BEDT-TTF)

Cu(NCS)

2 3 4 5

FFLO up

FFLO dn

Fig. 11. Phase diagram of κ-(BEDT-TTF)

Cu(NCS)

for ﬁelds applied parallel to

the superconducting layers. The solid line represents the calculated H

using the

known band-structure parameters. After Ref. 62.

From these data the magnetic phase diagram shown in Fig. 11 was ex-

tracted. From the very steep initial increase of H

an orbital critical ﬁeld of

roughly 130 T can be estimated. This large value results from the strongly

reduced orbital currents for the ﬁeld orientation parallel to the layers.

Towards higher ﬁelds, however, H

levels oﬀ towards saturation at lower

temperatures. This clearly signals the crossover to Pauli-paramagnetic

September 14, 2010 15:18 World Scientiﬁc Review Volume - 9.75in x 6.5in ch14

364 G. Zwicknagl and J. Wosnitza

limitation. The Pauli limit, H

, can be determined quite accurately from

the known gap value, ∆

, extracted in earlier speciﬁc-heat studies.

72,73

Using ∆

= 2.4,

leads to H

= 23 T which ﬁts nicely with

the ﬁeld of ∼ 21 T where the sharp ﬁrst-order phase transition emerges

[Fig. 10(b)].

Above this ﬁeld, the curvature of the H

line clearly changes. Together

with an increased slope of this line, the second phase transition appears.

This is favorably in line with the expectations for the existence of an FFLO

state. A comparison with the calculated phase diagram is shown by the

solid line in Fig. 11. For that, the well-known electronic band-structure

parameters of κ-(BEDT-TTF)

Cu(NCS)

have been considered and s-wave

superconductivity has been assumed.

72–74

The latter, however, is not an

essential ingredient. The calculated line and the experimental data agree

very well except at lower temperatures. This as well as the unexpected small

area of the FFLO phase may be caused by a small misalignment of the sample

in the experiment. Also, the theoretical description might be improved when

taking more detailed information on the electronic parameters into account.

For the 2D organic superconductors based on BEDT-TTF further evi-

dence for the existence of the FFLO state has been gathered. For κ-(BEDT-

TTF)

Cu(NCS)

, the results from magnetic-torque studies are shown in

Fig. 12.

With decreasing temperature the magnetic torque, which is pro-

portional to the magnetization, shows an increasingly sharp transition at

Fig. 12. (a) Field dependence of the magnetic-torque signal of κ-(BEDT-

TTF)

Cu(NCS)

for in-plane magnetic ﬁelds at diﬀerent temperatures. (b) Mag-

netic phase diagram constructed from the data. After Ref. 75.

September 14, 2010 15:18 World Scientiﬁc Review Volume - 9.75in x 6.5in ch14

Breaking Translational Invariance by Population Imbalance 365

. This indicates that the Pauli limit, H

, is reached. Nevertheless, su-

perconductivity survives even beyond this ﬁeld for lower temperatures. In

addition, a second sharp feature appears in the torque signal below H

signaling the evolution of the FFLO phase. The magnetic phase diagram

constructed from the torque data is shown in the right panel of Fig. 12.

These results nicely conﬁrm the speciﬁc-heat data for the H

line. There is,

however, a discrepancy in the extension of the FFLO phase. This might be

explained by the much better in-plane alignment of the magnetic ﬁeld in the

torque measurements which was made possible by an in situ rotation of the

torque platform. The alignment in the speciﬁc-heat experiment is estimated

to be not better than about 1

◦

4.2. Heterogenous systems

Besides the still partially controversial evidences for the FFLO state in the

homogeneous systems, a quasi-one-dimensional FFLO-like state may exist

in artiﬁcially layered superconducting/ferromagnetic superlattices.

Due

to the proximity eﬀect, Cooper pairs from the superconductor can enter the

ferromagnetic region. In the spin-split bands of the ferromagnet, however,

the wavevectors of the spin-up and spin-down electrons must be largely dif-

ferent resulting in a Cooper pair with ﬁnite momentum.

Consequently,

the real-space pairing function oscillates spatially inside the ferromagnet.

Of course, on top of that oscillation the pairing function decays rapidly in-

side the ferromagnet. However, if the ferromagnetic layer is thin enough,

the pairing-function wave may be reﬂected at the opposite ferromagnetic-

layer surface, similar as in a Fabry–P´erot interferometer. Upon changing

the ferromagnetic layer thickness, d

, the pairing functions running oppo-

sitely may interfere either constructively or destructively. This eventually

may lead to an oscillatory proximity eﬀect, i.e. T

might oscillate as function

of d

78,79

For optimum parameters, this could even result in a reentrant

superconductivity.

A ﬁrst report on the observation of reentrant superconductivity in

Fe/V/Fe trilayers as a function of the iron thickness was given in

Ref. 80. The T

oscillation length is given by the magnetic coherence length

F O

= ~v

, where E

is the exchange-splitting energy. For the strong

ferromagnet iron, this length scale is very short (0.7–1.0 nm). Indeed, su-

perconductivity in the Fe/V/Fe system vanished at this iron-layer thickness,

but the number of experimental points is low and shows a large scatter.

A much clearer evidence for such a reentrant behavior has been given by

Zdravkov et al. for the bilayer system Nb/Cu

1−x

Alloying the strong

October 6, 2010 13:37 World Scientiﬁc Review Volume - 9.75in x 6.5in ch14

366 G. Zwicknagl and J. Wosnitza

ferromagnet nickel by copper reduces E

strongly which led to ξ

F O

= 8.6 nm

for the chosen Cu

layer. This largely reduces the inﬂuence of surface

roughness on the interference eﬀects. The thicknesses of the superconducting

niobium layers were chosen so that T

was in the right range to be strongly

susceptible on the thickness of the Cu

layer, d

CuNi

(inset of Fig. 13). In

order to produce samples with diﬀerent d

CuNi

, but exactly the same niobium

thickness, a wedge-shape ferromagnetic Cu

layer was deposited on top

of the niobium layer (right panel of Fig. 13). After evaporation of a silicon

cap layer, samples of equal width were cut perpendicular to the wedge.

The resistively determined T

s as a function of d

CuNi

for two diﬀerent nio-

bium layer thicknesses are shown in the left panel of Fig. 13. For the thicker

niobium layer a clear minimum in T

is resolved. For the 7.3 nm thick nio-

bium layer, T

vanishes at d

CuNi

≈ 5 nm and reappears at about 12 nm. The

data can be well described using a model for a clean ferromagnet, i.e. a fer-

romagnet where the electronic mean-free path is larger than ξ

F O

(solid lines

in Fig. 13).

82,83

Consequently, superconducting/ferromagnetic superlattices

seem to be a good realization for studying FFLO-like behavior.

Fig. 13. (left) Critical temperature, T

, of an (a) 8.3 and (b) 7.3 nm thick niobium

layer as a function of the thickness of a ferromagnetic Cu

layer. The solid and

dashed lines are calculations for the clean and dirty cases, respectively. The inset

shows T

as a function of niobium-layer thickness for a 56 nm thick Cu

layer

with the solid line calculated for the clean case. (right) Sketch of the investigated

superconductor-ferromagnet bilayer. After Ref. 81.

September 14, 2010 15:18 World Scientiﬁc Review Volume - 9.75in x 6.5in ch14

Breaking Translational Invariance by Population Imbalance 367

5. Summary and Outlook

The central focus of the present work are the Fulde–Ferrell–Larkin–

Ovchinnikov states which, in addition to oﬀ-diagonal long-range order break

spatial symmetries. These superﬂuid phases are suggested by mean-ﬁeld

analyses to occur at low temperatures in multi-component fermionic super-

ﬂuids with population imbalance. The theoretical considerations apply to a

wide variety of systems including metals, ultra-cold atoms, nuclear matter,

and dense quark systems. The discussion presented here refers to the weakly

interacting BCS regime where the systems mentioned are all described by es-

sentially the same microscopic theory. This fact implies that the knowledge

and insights gained by studying superconducting metals in high magnetic

ﬁelds or ultra-cold atoms can provide explanations for phenomena observed

in nuclear or astrophysical contexts.

There are indications that FFLO phases are realized in some highly

anisotropic superconductors. Especially, for the low-dimensional organic

superconductors, a number of experimental facts point to the existence of

such an inhomogeneous state, whereas the situation is much more contro-

versial for heavy-fermion materials. The main criteria employed to identify

perspective candidate systems have been the upturn of the Pauli-limited

critical ﬁeld and the appearance of a ﬁrst-order transition from the normal

to the superconducting state at low temperatures as well as the existence

of transitions within the superconducting phase. Besides these homoge-

neous systems, FFLO-like behavior may exist in artiﬁcial superconduct-

ing/ferromagnetic superlattices that show an oscillating or even reentrant

superconducting phase transition as a function of the ferromagnetic layer

thickness.

An unambiguous identiﬁcation of FFLO states requires a better qualita-

tive understanding of superﬂuids with periodically textured order parame-

ters. The investigations should concentrate on characteristic features which

are tied to the quantum nature of the order parameter and reﬂect its peri-

odicity. Prominent candidates are Andreev bound-states which should form

bands in a superconductor with a periodically modulated order parame-

ter. The spatial variation of the latter should give rise to characteristic van

Hove-type structures in the corresponding spectral density. The latter can be

observed by tunneling spectroscopy. In addition, it should aﬀect transport

properties.

The interpretation of experimental data needs detailed quantitative ref-

erence calculations which account for realistic quasiparticle dispersion and

include quasiparticle interactions. An appropriate theoretical treatment will

September 14, 2010 15:18 World Scientiﬁc Review Volume - 9.75in x 6.5in ch14

368 G. Zwicknagl and J. Wosnitza

require new eﬀective calculational tools. In particular, it is highly desirable

to have methods which allow for direct “unbiased” minimization of the free

energy. Finally, the role of (quantum) ﬂuctuations should be considered.

We hope that this article will stimulate further work on the exciting

subject of fermionic superﬂuids with population imbalance and the FFLO

phases.

Acknowledgments

We have beneﬁtted from discussions with many colleagues, especially we

would like to thank P. Fulde, R. Lortz, A. D. Bianchi, B. Bergk, G. Eilen-

berger, M. Sigrist, L. N. Bulaevskii, M. R. Beasley and D. J. Scalapino for

stimulating discussions. G. Zwicknagl acknowledges the hospitality of the

Kavli Institute for Theoretical Physics where part of the manuscript was

completed. This work was supported in part by the National Science Foun-

dation under Grant No. PHY05-51164 and by EuroMagNET (EU contract

No. 228043).

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