Emerging evidence for FFLO states in layered organic superconductors (Review Article)
In this short review, we report on the recently found growing experimental evidence for the existence of Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) states in quasi-two-dimensional organic superconductors. At high magnetic fields aligned parallel to the conducting organic layers, we observe an upturn...
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irk-123456789-1182192017-05-30T03:05:45Z Emerging evidence for FFLO states in layered organic superconductors (Review Article) Beyer, R. Wosnitza, J. К 75-летию со дня рождения И. К. Янсона In this short review, we report on the recently found growing experimental evidence for the existence of Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) states in quasi-two-dimensional organic superconductors. At high magnetic fields aligned parallel to the conducting organic layers, we observe an upturn of the upper critical field beyond the Pauli limit, as evidenced by specific-heat and torque-magnetization measurements. Inside the superconducting state a second thermodynamic transition emerges. These features appear only in a very narrow angular region close to parallel-field orientation. 2013 Article Emerging evidence for FFLO states in layered organic superconductors (Review Article) / R. Beyer, J. Wosnitza // Физика низких температур. — 2013. — Т. 39, № 3. — С. 293–300. — Бібліогр.: 40 назв. — англ. 0132-6414 PACS: 74.70.Kn, 74.25.Dw, 65.40.Ba, 74.25.Bt http://dspace.nbuv.gov.ua/handle/123456789/118219 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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К 75-летию со дня рождения И. К. Янсона К 75-летию со дня рождения И. К. Янсона |
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К 75-летию со дня рождения И. К. Янсона К 75-летию со дня рождения И. К. Янсона Beyer, R. Wosnitza, J. Emerging evidence for FFLO states in layered organic superconductors (Review Article) Физика низких температур |
| description |
In this short review, we report on the recently found growing experimental evidence for the existence of
Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) states in quasi-two-dimensional organic superconductors. At high
magnetic fields aligned parallel to the conducting organic layers, we observe an upturn of the upper critical field
beyond the Pauli limit, as evidenced by specific-heat and torque-magnetization measurements. Inside the superconducting
state a second thermodynamic transition emerges. These features appear only in a very narrow angular
region close to parallel-field orientation. |
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Beyer, R. Wosnitza, J. |
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Beyer, R. Wosnitza, J. |
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Beyer, R. |
| title |
Emerging evidence for FFLO states in layered organic superconductors (Review Article) |
| title_short |
Emerging evidence for FFLO states in layered organic superconductors (Review Article) |
| title_full |
Emerging evidence for FFLO states in layered organic superconductors (Review Article) |
| title_fullStr |
Emerging evidence for FFLO states in layered organic superconductors (Review Article) |
| title_full_unstemmed |
Emerging evidence for FFLO states in layered organic superconductors (Review Article) |
| title_sort |
emerging evidence for fflo states in layered organic superconductors (review article) |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2013 |
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К 75-летию со дня рождения И. К. Янсона |
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http://dspace.nbuv.gov.ua/handle/123456789/118219 |
| citation_txt |
Emerging evidence for FFLO states in layered organic superconductors (Review Article) / R. Beyer, J. Wosnitza // Физика низких температур. — 2013. — Т. 39, № 3. — С. 293–300. — Бібліогр.: 40 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT beyerr emergingevidenceforfflostatesinlayeredorganicsuperconductorsreviewarticle AT wosnitzaj emergingevidenceforfflostatesinlayeredorganicsuperconductorsreviewarticle |
| first_indexed |
2025-07-08T13:34:41Z |
| last_indexed |
2025-07-08T13:34:41Z |
| _version_ |
1837085943956242432 |
| fulltext |
© R. Beyer and J. Wosnitza, 2013
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 3, pp. 293–300
Emerging evidence for FFLO states in layered organic
superconductors
(Review Article)
R. Beyer and J. Wosnitza
Hochfeld-Magnetlabor Dresden (HLD),
Helmholtz-Zentrum Dresden-Rossendorf and TU Dresden, D-01314 Dresden, Germany
E-mail: j.wosnitza@hzdr.de
Received October 1, 2012
In this short review, we report on the recently found growing experimental evidence for the existence of
Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) states in quasi-two-dimensional organic superconductors. At high
magnetic fields aligned parallel to the conducting organic layers, we observe an upturn of the upper critical field
beyond the Pauli limit, as evidenced by specific-heat and torque-magnetization measurements. Inside the super-
conducting state a second thermodynamic transition emerges. These features appear only in a very narrow angu-
lar region close to parallel-field orientation.
PACS: 74.70.Kn Organic superconductors;
74.25.Dw Superconductivity phase diagrams;
65.40.Ba Heat capacity;
74.25.Bt Thermodynamic properties.
Keywords: organic superconductors, FFLO states, thermodynamic properties.
Contents
Introduction ............................................................................................................................................. 293
Experimental ........................................................................................................................................... 294
Results and discussion ............................................................................................................................. 294
Summary ................................................................................................................................................. 299
References ............................................................................................................................................... 299
Introduction
When applying a magnetic field parallel to the supercon-
ducting planes of a material consisting of alternating super-
conducting and normal-conducting layers, the usually domi-
nating orbital Cooper-pair breaking is greatly reduced. In
that case, the orbital critical field, 0 orb ,Hμ may easily ex-
ceed the Pauli paramagnetic limit, 0 0= /( 2 ),P BHμ Δ μ
with Bμ the Bohr magneton and 0Δ the superconducting
energy gap at = 0T [1,2]. This limit occurs in spin-singlet
superconductors when the Zeeman energy becomes larger
than the superconducting condensation energy. In such a
case [3] and if the superconductor is in the clean limit with a
mean free path much larger than the coherence length, an
unconventional, spatially modulated, superconducting state
may appear at high magnetic fields and low temperatures.
This was predicted independently in 1964 by Fulde and Fer-
rell [4] as well as Larkin and Ovchinnikov [5]. Such super-
conductivity, therefore, is now called FFLO or LOFF state.
At high magnetic fields, the Zeeman-split Fermi surfaces
for up- and down-spin electrons do not allow for the usual
BCS-like pairing with zero total momentum. In the FFLO
state, however, Cooper pairing may appear with a finite cen-
ter-of-mass momentum of the order | | = 2 /( ),B Fq Hμ v
where Fv is the Fermi velocity (Fig. 1) [6,7]. This, in turn,
results in an oscillating part of the superconducting order
parameter also in real space. By that, the superconductor
“sacrifices” part of its volume to the normal state allowing to
extend superconductivity to even higher magnetic fields.
The materials of choice for searching for FFLO states are
the layered, quasi-two-dimensional (2D) molecular super-
conductors based, e.g., on the organic molecules BEDT-TTF
(bisethylenedithio-tetrathiafulvalene) or BETS (bisethylene-
dithio-tetraselenafulvalene). These well-studied materials
R. Beyer and J. Wosnitza
294 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 3
[8–12] are strong type-II spin-singlet superconductors of
high purity with long mean free pathes. During the last
years, there indeed have been a number of reports giving
growing evidence for the existence of an FFLO state in 2D
organic superconductors [13–20].
Besides the 2D organic materials, heavy-fermion super-
conductors have been suggested to show FFLO states as
well. These claims, however, later had to be revised or are
inconclusive (see [7] for an overview). For the heavy-
fermion compound CeCoIn5, clear thermodynamic evi-
dence for a field-induced phase inside the superconducting
state was found [21,22]. This phase, however, does not
show the features expected from the FFLO prediction, but
rather an antiferromagnetically ordered state [23,24]. Fur-
ther to that, recent thermodynamic experiments showed an
entropy decrease when going from the homogenous super-
conducting into the unknown phase in CeCoIn5 which dis-
agrees with the formation of an FFLO state, but fits to the
emergence of antiferromagnetic order [25].
Here, we present recent specific-heat and torque-
magnetization data obtained for two 2D organic BEDT-
TTF-based superconductors with transition temperatures,
,cT differing by more than a factor of 2 [16–19]. For both
materials, we find clear Pauli limitations of the upper criti-
cal fields for magnetic fields aligned within the planes.
Towards lower temperatures, upturns of the critical fields
beyond the Pauli limits appear. The overall shape of the
upper critical field lines agrees well with theoretical pre-
dictions for a 2D FFLO superconductor. This upturn of the
upper critical field is limited to a very narrow angular
range of a mere of about 0.5± deg around in-plane field
alignment. All this is strong evidence for the existence of
FFLO states in these organic superconductors. For smaller
out-of-plane alignment, a second phase transition within
the superconducting state emerges, the origin of which is
unclear so far.
Experimental
The investigated single crystals of κ-(BEDT-
TTF)2Cu(NCS)2 and β″-(BEDT-TTF)2SF5CH2CF2SO3
were prepared by standard electrochemical processes de-
scribed elsewhere [26,27]. The specific heat was meas-
ured by use of a continuous relaxation technique as de-
scribed in more detail in the Appendix of Ref. 28. There-
by, the sample together with the platform and Apiezon N
grease to fix the sample is heated up by a considerable
amount above the base temperature (up to 100%). Dur-
ing the relaxation back to base temperature a large num-
ber of data points is taken. As a speciality for β″-(BEDT-
TTF)2SF5CH2CF2SO3, the calorimeter was mounted on
a piezoelectric drive inside a 3He cryostat equipped with
a 20-T magnet. The drive allowed for a precise rotation of
the sample with resolution of better than 0.02 deg. The mag-
netic torque was measured by a capacitive cantilever. More
experimental details can be found in Refs. 16, 18, 19.
Results and discussion
For κ-(BEDT-TTF)2Cu(NCS)2, the upper critical field
for in-plane field alignment is somewhat above 20 T which
calls for experiments in high-field laboratories. The results
shown in Fig. 2 have been obtained at the Grenoble High
Magnetic Field Laboratory (LNCMI-Grenoble) [16]. In
this experiment, only the complete cryostat could be tilted
slightly to orient the sample. Although the crystal was
aligned as best as possible, in retrospective a small misa-
lignment on a sub-degree level cannot be excluded (see
discussion below).
In Fig. 2, the difference between the specific heat
measured in the labeled in-plane magnetic fields and the
normal-state specific heat measured for a field of 14 T ap-
plied perpendicular to the layers is shown. By that, the
specific-heat anomalies at cT become clearly visible. At
8 T, cT is reduced from 9.1 K at zero field by a mere of
0.4 K (using the maxima of the λ-like anomalies, Fig. 2).
From that, the initial critical-field slope can be estimated to
2 0 2= / = 20c cH dH dT′ −μ T/K. This, of course, is a rather
conservative estimate since the slope at lower fields most
probably is even higher. Anyway, estimating the orbital
BCS state
k
–k E EF� = F�
FFLO state
q
EF�
EF�– +k q
k
Fig. 1. (Color online) Schematic presentation of (left) the usual
BCS pairing state with zero resulting momentum and spin and
(right) the FFLO pairing state with a finite center-of-mass mo-
mentum, q. The circles represent the Fermi surfaces for spin-up
and spin-down bands.
Fig. 2. (Color online) Temperature dependence of the specific-
heat difference between the superconducting and normal state,
/ ,C TΔ of κ-(BEDT-TTF)2Cu(NCS)2 in magnetic fields applied
parallel to the superconducting layers.
�0H, T
21
21.5
22
22.5
23
24
28
0
8
14
16
18
20
�
C
T/
,
m
J/
(m
o
l·
K
)
2
�-(BEDT-TTF) Cu(NCS)2 2
T, K
2 4 6 8 10 12
60
40
20
0
–20
�0H, T
Emerging evidence for FFLO states in layered organic superconductors
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 3 295
critical field by orb 2= 0.7 c cH H T′ [29], the extraordinary
large value of about 130 T is obtained.
Towards higher magnetic fields, cT shifts much more
rapidly to lower temperatures (Fig. 2). This is a clear sign
for Pauli limitation of the upper critical field. Indeed, the
Pauli-limiting field can be determined quite reliably from
specific-heat data. As mentioned, PH is directly propor-
tional to 0Δ . This superconducting energy gap at zero
temperature is given by the jump height of the specific-
heat anomaly at zero field. Assuming a BCS-like tempera-
ture dependence of the energy gap [30] the complete tem-
perature dependence of the specific-heat difference,
( )C TΔ can be described by the so-called α model [31].
Thereby, the temperature dependence of the energy gap,
0 ( ),TΔ is scaled by an appropriate parameter
0= ( )/ .B cT k Tα Δ For κ-(BEDT-TTF)2Cu(NCS)2, α va-
lues between 2.4 [32] and 2.8 [33] have been found. Using
the smaller value results in 0 = 23PHμ T. This nicely
agrees with ∼21 T estimated from the specific-heat and
magnetic-torque data discussed below.
At 21 T, the specific-heat anomaly sharpens indicating
a possible first-order transition. For fields above 21 T, a
second, very sharp anomaly just below the first one ap-
pears. This can be seen much better in Fig. 3(a). Since the
relaxation technique allows to record data during heating
and cooling, this was utilized at these high magnetic fields
to search for hysteresis. Indeed, as seen in Fig. 3(b), a well-
resolved hysteresis evolves for the lower-temperature tran-
sition, whereas for the main superconducting transition a
small hysteresis might be present, but the effect is barely
above the resolution limit. Anyway, the existence of an
additional anomaly and the upturn of the upper critical
field at low temperatures are strong evidence for the exis-
tence of an FFLO state in κ-(BEDT-TTF)2Cu(NCS)2.
Further support for the occurrence of the FFLO state
was found in magnetic-torque data. The magnetic torque is
proportional to the perpendicular component of the magne-
tization with respect to the magnetic field. Using a capaci-
tive cantilever allows for a highly sensitive detection of
magnetization changes. In the experiment discussed below,
the cantilever could be rotated allowing an alignment of
the sample with precision of about 0.01 deg in the applied
magnetic field. For in-plane field, the magnetic-torque data
shown in Fig. 4(a) have been obtained [18]. Upon lowering
the temperature, an increasingly sharp transition appears at
0 2.cHμ This may be taken as an indication that the Pauli
limit is reached. Indeed, this sharp transition occurs at
about 21 T fitting nicely to the estimated Pauli limit and
the specific-heat data.
When lowering the temperature further, superconduc-
tivity survives even beyond this field. In addition, a second
dip-like feature appears in the torque data below 0 2cHμ
[vertical arrows in Fig. 4(a)], signaling the emergence of
the FFLO phase. These results nicely confirm the specific-
heat data for the upper critical field. There is, however, a
distinctive difference in the extension of the phase between
the two anomalies. Whereas the specific-heat data suggest
a very narrow region for the additional thermodynamic
phase, the dip-like feature in the torque data stays almost
constant in magnetic field at about the Pauli limit. This
might be related to the much better in-plane alignment of
the magnetic field in the torque measurements as for the
specific-heat experiment. This will be discussed in more
detail below.
Fig. 3. (Color online) (a) High-field data shown in Fig. 2 in an
expanded scale. (b) Specific-heat data measured at 22 T during
warming and cooling.
�0H, T
21
21.5
22
22.5
23
24
28
�
C
T/
,
m
J/
(m
o
l·
K
)
2
�-(BEDT-TTF) Cu(NCS)2 2
T, K
60
40
20
0
–20
2.0 2.5 3.0 3.5
�0H = 22 T
�
C
T/
,
m
J/
(m
o
l·
K
)
2
T, K
100
80
60
40
20
2.4 2.6 2.8 3.0
(a)
(b)
R. Beyer and J. Wosnitza
296 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 3
A schematic magnetic phase diagram constructed from
the specific-heat and magnetic-torque data is shown in
Fig. 4(b). After the very steep initial increase of 0 2cHμ
this slope quickly levels off at higher fields upon reaching
the Pauli limit. At lower temperatures, a strong upturn of
the 0 2cHμ line occurs. Together with this upturn, the
second phase transition appears. This agrees favorably well
with the FFLO scenario. Recent NMR data have given
further strong support for the existence of an additional
superconducting phase above the Pauli limit in κ-(BEDT-
TTF)2Cu(NCS)2 [20].
In an effort to find further superconductors revealing an
FFLO phase and to study this phase in more detail, we
investigated the specific heat of the organic superconductor
β″-(BEDT-TTF)2SF5CH2CF2SO3 [19]. This superconduc-
tor has a modest critical temperature of 4.3 K and a corres-
ponding low Pauli-limiting field of 9.73 T (see below).
This allows for a thorough investigation of the phase dia-
gram in readily available commercial superconducting
magnets.
The specific heat of the ′′β material in zero field and
with 10 T applied perpendicular to the BEDT-TTF planes,
i.e., along *,c is shown in Fig. 5(a) on a double-logarithmic
scale. Due to the strong phonon contribution to the specific
heat the anomaly at = 4.3cT K (arrow) is hardly visible in
Fig. 4. (Color online) (a) Magnetic-torque data of κ-(BEDT-
TTF)2Cu(NCS)2 at various temperatures for in-plane magnetic
fields. For T = 1.7 K, data for both up and down field sweeps,
otherwise only down sweeps are shown. The vertical arrows
mark the small dip-like features, which are associated with the
transition into the FFLO state. (b) Schematic phase diagram
deduced from the data shown in (a) and Fig. 2 [18].
�0H, T
(a)
T, K
0.6
0.8
1.3
1.7
2.4
2.9
3.4
4.2
5.4
8.0
10 15 20 25 30
�0 2Hc
Normal state
FFLO
�-(BEDT-TTF) Cu(NCS)2 2
(b)
T, K
30
20
10
0
2 4 6 8 10
�
0
H
,
T
Superconducting
N
o
rm
al
iz
ed
to
rq
u
e
Fig. 5. (a) Temperature dependence of the specific heat of β″-
(BEDT-TTF)2SF5CH2CF2SO3 in a double-logarithmic scale.
Data in the superconducting state in zero field and in the normal
state at 10 T applied perpendicular to the BEDT-TTF planes are
shown. The solid line is a fit to the 10-T data below 2 K using a
linear and cubic term. (b) Temperature dependence of the specif-
ic-heat difference between the superconducting and normal state.
The lines show the BCS behavior for weak coupling (dashed)
[30] and moderately strong coupling (solid) [31]. The inset shows
the electronic part of the specific heat, ,eC divided by cTγ as
function of / .cT T The solid line shows the exponential vanishing
of eC towards low T.
10
1
0.1
0.01
10
T, K
(a)
C
,
J/
(m
o
l·
K
)
1
�0H c|| *
C
,
J/
(m
o
l·
K
)
T, K
0.2
0.1
0
420
�exp (–1.9 / )T Tc
2 3 4
1
0.1
0.01
T Tc /
� = 2.18
�BCS = 1.76
C
T
e
c
/
(b)
Tc
0 T
10 T
Emerging evidence for FFLO states in layered organic superconductors
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 3 297
Fig. 5(a). Below 2 K, the data in the normal state at 10 T can
be ell described by 3= ,C T Tγ +β resulting in a Sommer-
feld coefficient = 19.0(5)γ mJ/(mol·K2) and a Debye con-
tribution with = 12.8(4)β mJ/(mol·K4) (corresponding to a
Debye temperature of 218(3) K). This is in excellent agree-
ment with previous results [34]. We used the normal-state
data set in 10 T applied along the *c direction to determine
all the specific-heat differences, ,CΔ discussed in the fol-
lowing.
The specific-heat difference between the superconduct-
ing (0 T) and normal (10 T) state is shown in Fig. 5(b).
Here, the anomaly at cT is well resolved now, although it
is rather broad due to fluctuation effects. The idealized
specific-heat jump at cT and the temperature dependence
of ( )C TΔ does not follow the weak-coupling BCS beha-
vior [dashed line in Fig. 5(b)] [30]. However, by use of the
mentioned α model [31] a reasonable description of the
data is possible assuming a moderately strong coupling
with a gap ratio = (0)/ = 2.18,B ck Tα Δ where α = 1.76 in
the weak-coupling limit. From that, we can extract the Pau-
li limiting field 0 = 9.73(3)PHμ T.
The inset of Fig. 5(b) shows the exponential vanishing
of exp ( 1.9 / )e cC T T∝ − without any indication of a resi-
dual electronic contribution in the superconducting state.
This fit line agrees perfectly with the α-model description
of the data shown in the main panel. Such exponential de-
pendences have as well been found for three other organic
superconductors [32,35,36] in line with reports by other
groups [33,37]. These specific-heat results, therefore,
prove the existence of a complete superconducting gap and
are strictly against any unconventional nodal order parame-
ter. A recent report indicating a 2T dependence of the
electronic specific-heat contribution at low temperatures is
questionable since different Sommerfeld coefficients in the
normal and superconducting state have been used to de-
scribe the data [38].
The differences between the specific heat for selected in-
plane fields and the normal-state specific heat is shown in
Fig. 6. By using an equal-entropy construction the field-
dependent transition temperatures have been extracted
(Fig. 7). Starting at = 4.3cT K for zero field, the specific-
heat anomaly first shifts only slightly with increasing field.
From the initial slope of the critical field, 2 25cH ′ ≈ T/K, we
obtain the orbital critical field 0 orb 2= 0.7 c cH T H ′μ − ≈ 75 T.
As for κ-(BEDT-TTF)2Cu(NCS)2, this field is much larger
than 0 .PHμ Consequently, the Pauli-limiting effect be-
comes dominant at higher magnetic fields. This leads to the
rapid decrease of the critical-field slope. Then again, below
about 1.6 K (above 9.3 T), 2cH rises steeply to values con-
siderably larger than the Pauli limit (dashed line in Fig. 7).
As for the κ-phase material, this is a clear indication for the
emergence of the FFLO phase.
There are, however, also remarkable differences to the
results obtained for κ-(BEDT-TTF)2Cu(NCS)2 (Figs. 2
and 3). For the κ-phase material, the anomaly from the
normal to the superconducting state becomes very sharp,
whereas it remains rather broad for the β″-phase supercon-
ductor. In addition, in the latter material no indication for a
second phase transition is found, in sharp contrast to the
clear first-order transition in the κ-phase compound.
A remarkable feature for Pauli-limited superconductors
is the restoration of the electronic part of the specific heat,
,eC with applied magnetic field. As was analyzed by Ma-
chida and Ichioka [39], a concave shape of the field-
dependent Sommerfeld coefficient, ( )Hγ , is expected.
Fig. 6. (Color online) Difference between the specific heat for
selected in-plane magnetic fields and the normal-state specific
heat divided by temperature as a function of temperature for
β″-(BEDT-TTF)2SF5CH2CF2SO3. The inset shows the field-
dependent evolution of the electronic part of the specific heat,
,eC divided by temperature.
T, K
�
C
T/
,
m
J/
(m
o
l·
K
)
2
�0H, T
�0H, T
20
10
0
4 8 12
20
0
–20
0 1 2 3 4 5
H || layer
0
5
7
8
8.5
9
9.5
10
10.5
11
C
T
e
/
|
,
m
J/
(m
o
l·
K
)
0
.7
K
2
1 2 3 4
0
2
4
6
8
10
12
0 deg
0.23 deg
0.31 deg
0.8 1.0 1.2 1.4 1.6
–0.5
0
0.5
A
n
g
le
,
d
eg
Tc (9.5 T)
Tc (10 T)
T*(10 T)
T*
T, K
�
0
H
,
T
Fig. 7. (Color online) Superconducting phase diagram of β″-(BEDT-
TTF)2SF5CH2CF2SO3 for fields aligned parallel to and by 0.23 and
0.31 deg out of the superconducting layers. The data of the second
anomaly observed at 0.23 deg [Fig. 9(b)] are labeled by *T (open
blue triangles). The dashed line is a rough extrapolation of the data
between 2 and 3 K to the Pauli limit of 9.73 T. The dotted line
represents the calculated 2cH (see text for details). The inset shows
the angular dependence of cT and *T at 9.5 and 10 T.
R. Beyer and J. Wosnitza
298 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 3
Since we could not determine ( ),Hγ i.e., /eC T extrapo-
lated to = 0T for all magnetic fields, we extracted /eC T
at = 0.7T K as a function of field (inset of Fig. 6). The
clear concave shape of the data is obvious proving the Pau-
li limitation once again. A similar field-dependent /eC T
appears for κ-(BEDT-TTF)2Cu(NCS)2 as well. The ap-
pearance of the maximum in /eC T at about 11 T beyond
the normal-state = 19.0(5)γ mJ/(mol·K2) is caused by the
determination of /eC T at finite temperature.
By use of the highly accurate rotation mechanism, we
intentionally rotated the sample by 0.31 deg out of the in-
plane field orientation. For this field alignment, we meas-
ured the temperature dependence of the specific heat close
to cT for a number of different magnetic fields around the
Pauli limit. The specific-heat differences are shown in
Fig. 8 in comparison to the data for accurate in-plane field
orientation. At 8.8 T, there is hardly any difference in the
specific heat for the two orientations discernible. At higher
fields, the superconducting phase transition shifts notably
to lower temperatures and, concomitantly, the anomaly
becomes considerably sharper for the data at 0.31 deg. This
might indicate a first-order-like transition for slight out-of-
plane alignments. However, we could not resolve any clear
latent heat in our relaxation data. The critical temperatures
extracted from the data at 0.31 deg are plotted as well in
Fig. 7. Here, it becomes clear that the upturn of the 2cH
line is highly sensitive to the precise in-plane alignment of
the magnetic field; the upturn at 0.31 deg is much less pro-
nounced than for 0 deg. Indeed, when rotating the crystal
at 9.5 T to out-of-plane angles of 0.4 deg, the phase transi-
tion could not be resolved any more in the available tem-
perature window (see Fig. 2 in Ref. 19).
The dotted line in Fig. 7 is the calculated phase-transition
line assuming s-wave superconductivity, which, however, is
not an essential ingredient, and modeling the system by a
stack of 2D superconducting planes with negligibly small
interplane conductivity [40]. In Fig. 7, we used the phase line
calculated for κ-(BEDT-TTF)2Cu(NCS)2 [16] by simply
rescaling for the different critical temperatures and Pauli-
limiting fields. Very good agreement between the experimen-
tal data for 0 deg and the calculation is obtained which might
become even better when the more correct electronic para-
meters for β″-(BEDT-TTF)2SF5CH2CF2SO3 would be used.
At 10 T, we as well measured the specific heat close to
the phase transition at constant magnetic fields by rotating
the sample in small steps around in-plane ( = 90Θ deg)
alignment. Similar as for 9.5 T, cT reduces quickly and the
anomaly sharpens clearly as seen in Fig. 9(a). In addition, a
second sharp anomaly, just below ,cT emerges when the
magnetic field is rotated out of plane by a mere of
±0.2 deg. The anomalies move out of the available temper-
ature window for slightly larger angles. The angular de-
pendences of the observed anomalies for 9.5 (Fig. 2 in
Ref. 19) and 10 T are shown in the inset of Fig. 7. The
second anomaly below cT is labeled *.T Thereby, as the
criterion determining *,T we used the maximum of the
specific-heat anomaly in the /C TΔ data.
We investigated this second anomaly in more detail by
measuring the specific heat for different magnetic fields
close to and above the Pauli limit at the out-of-plane angle
of 0.23 deg [Fig. 9(b)]. The *T anomaly appears at about
9.4 T. With increasing field, the two anomalies move al-
most parallel to each other towards lower temperatures
keeping a distance of about 0.3 K. At 10.5 T, the *T ano-
maly is moved out of the accessible temperature window.
These data resemble the specific-heat anomalies ob-
served in κ-(BEDT-TTF)2Cu(NCS)2 (Figs. 2 and 3). The
temperature difference between the two anomalies there
has a similar value of about 0.2 K. This indicates that for
the experiment on the κ-phase material the magnetic field
probably had a small out-of-plane component.
All thermodynamically determined phase-transition
points are shown in Fig. 7. The hypothetical low-
temperature upper critical field without an FFLO state is
visualized by the dashed line. For that, we extrapolated the
data between 2 and 3 K to the Pauli limit of 9.73 T at
= 0.T For in-plane field alignment (0 deg), an upturn of
the 2cH line almost as predicted (dotted line in Fig. 7) is
found. For the out-of-plane angles 0.23 and 0.31 deg, the
upturn is strongly reduced. In addition, for 0.23 deg the
second *T anomaly appears. For angles larger than about
0.5 deg, no upturn of 2 ,cH i.e., no FFLO state appears.
A puzzle is still the origin of the *T anomaly. It most
likely is not related to the transition from the FFLO to the
homogeneous state since that should appear at the Pauli-
limiting field (dashed line in Fig. 7). It may be speculated
that the *T anomaly is caused by a commensurability ef-
fect of the wavelength of the superconducting order para-
meter in the FFLO state and the distance of a line parallel
T, K
20
0
–20
–40
�0H, T
10.4
10.2
10.0
9.8
9.6
9.4
9.2
9.0
8.8
0.5 1.0 1.5 2.0 2.5
�
C
T/
,
m
J/
(m
o
l·
K
)
2
Fig. 8. (Color online) Specific-heat differences divided by
temperature of β″-(BEDT-TTF)2SF5CH2CF2SO3 measured in
different magnetic fields aligned parallel (open symbols) and
0.31 deg out of the conducting plane (closed symbols). The data
are plotted offset for clarity.
Emerging evidence for FFLO states in layered organic superconductors
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 3 299
to the applied field crossing two superconducting layers.
By that, zeros of the order parameter could accommodate
in the non-superconducting anion layers.
In the specific-heat measurements no feature at the
dashed line in Fig. 7, i.e., at the expected transition from
the FFLO state to the homogeneous superconducting state
was observed. However, our kind of specific-heat mea-
surements are not well suited to detect such a transition
line. When sweeping the temperature at constant magnetic
field, in the best case, one only would cross the transition
line at a very shallow angle. Here, magnetocaloric-effect
experiments, i.e., sweeping the magnetic field at constant
temperature while measuring the heat tone of the sample,
would be the method of choice. Such experiments are,
however, rather challenging due to the small heat capacity
of the samples.
Summary
High-resolution specific-heat experiments as well as
torque-magnetization data have allowed to extract reliable
thermodynamic phase diagrams for two 2D organic super-
conductors based on BEDT-TTF. The very steep initial
critical-field slopes for in-plane magnetic fields reflect the
very short out-of-plane coherence lengths and very high
orbital critical fields. Towards lower temperatures, the up-
per critical fields become Pauli limited. Reducing the tem-
perature even further, clear upturns of the critical-field
lines appear, giving evidence, for the emergence of FFLO
states. The occurrence of a second phase transition inside
the superconducting state in a very narrow angular range
close to optimum in-plane alignment is puzzling so far.
Further experiments are under way to search for a micro-
scopic evidence of modulated FFLO order parameters.
Acknowledgments
This article is dedicated to Igor Yanson whom JW had
the pleasure to meet at many occasions. We thank our co-
workers B. Bergk, R. Lortz, A. Demuer, I. Sheikin, S. Ya-
sin, Y. Wang, Y. Nakazawa, J.A. Schlueter, and G. Zwick-
nagl which contributed substantially to the work presented
here. Part of this work was supported by EuroMagNET II
(EU contract No. 228043).
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T, K
�0H, T
�
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