Complex phase diagram of Ba$_{1-x}$Na$_{x}$Fe$_{2}$As$_{2}$: a multitude of phases striving for the electronic entropy

Abstract

The low-temperature electronic phase diagram of
Ba$_{1-x}$Na$_{x}$Fe$_{2}$As$_{2}$, obtained using high-resolution
thermal-expansion and specific-heat measurements, is shown to be considerably
more complex than previously reported, containing nine different phases.
Besides the magnetic $C_{2}$ and reentrant $C_{4}$ phases, we find evidence for
an additional, presumably magnetic, phase below the usual SDW transition, as
well as a possible incommensurate magnetic phase. All these phases coexist and
compete with superconductivity, which is particularily strongly suppressed by
the $C_{4}$-magnetic phase due to a strong reduction of the electronic entropy
available for pairing in this phase.

Full text

Complex phase diagram of Ba1−xNaxFe2As2: a multitude of phases striving for the
electronic entropy
L. Wang,∗F. Hardy, A. E. Böhmer,†T. Wolf, P. Schweiss, and C. Meingast‡
Institut f¨ur Festk¨orperphysik, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany
(Dated: 7/12/15)
The low-temperature electronic phase diagram of Ba1−xNaxFe2As2, obtained using high-
resolution thermal-expansion and specific-heat measurements, is shown to be considerably more
complex than previously reported, containing nine different phases. Besides the magnetic C2and
reentrant C4phases, we find evidence for an additional, presumably magnetic, phase below the usual
SDW transition, as well as a possible incommensurate magnetic phase. All these phases coexist and
compete with superconductivity, which is particularly strongly suppressed by the C4-magnetic phase
due to a strong reduction of the electronic entropy available for pairing in this phase.
High-temperature superconductivity in Fe-based sys-
tems usually emerges when a stripe-type antiferromag-
netic spin-density-wave (SDW) is suppressed by either
doping or pressure [1–3]. The SDW transition is ac-
companied, or sometimes even slightly preceeded, by
a structural phase transition from a high-temperature
tetragonal (C4) to a low-temperature orthorhombic (C2)
state, which has sparked the lively debate about elec-
tronic nematicity and the respective role of spin and
orbital physics in these materials [4–8]. In the hole-
doped compounds, Ba1−xNaxFe2As2, Ba1−xKxFe2As2,
and Sr1−xNaxFe2As2, recent studies have shown that the
C4symmetry is restored in a small pocket within the
magnetic C2phase region [9–12]. Mössbauer studies on
Sr0.63Na0.37 Fe2As2find that only half of the Fe sites carry
a magnetic moment in this phase [12], which is consistent
with the double-Q magnetic structure predicted within
the itinerant spin-nematic scenario [6, 9, 12, 13]. More-
over, neutron studies have shown that the spins flip from
in-plane in the C2phase to out of plane in the C4reen-
trant phase [14], indicating that spin-orbit interactions
cannot be neglected. In the Ba1−xKxFe2As2system, the
reentrant C4phase reverts back to the C2phase near the
onset of superconductivity, due to a stronger competition
of the C4phase with superconductivity [10]. The pres-
ence of this phase in the hole-doped systems presents
strong evidence that the physics of these Fe-based sys-
tems can be treated in an itinerant picture, and recent
theoretical studies based upon the spin-nematic scenario
can reproduce phase diagrams very similar to the exper-
imental ones [15], as well as the spin-reorientation in the
C4phase if spin-orbit interactions are included [16].
Here, we reinvestigate in greater detail the
low-temperature electronic phase diagram of
Ba1−xNaxFe2As2using high-resolution thermal-
expansion and specific- heat measurements and show
that it is considerably more complex than previously
∗liran.wang@kit.edu
†present address: The Ames Laboratory, U.S. Department of En-
ergy, Iowa State University, Ames, Iowa 50011, USA
‡christoph.meingast@kit.edu
reported, containing nine different phases. Besides the
usual C2and reentrant C4magnetic phases, we find
evidence for an additional, presumably magnetic, C2
phase, in which the orthorhombic distortion is substan-
tially reduced but still finite. These phases coexist and
compete with superconductivity, which is particularly
strongly suppressed by the reentrant C4phase. Further,
we provide indications that the SDW transition becomes
incommensurate above x = 0.22, which appears linked to
the emergence of the C4phase at this composition. The
surprising occurence of this multitude of phases near the
onset of superconductivity suggests a highly degenerate
free-energy landscape near optimal doping, which may
be related to the occurence of superconductivity in the
Fe-based systems.
Single crystals of Ba1−xNaxFe2As2were grown in alu-
mina crucibles using a self-flux method with (Ba,Na):
FeAs ratios 1:4 - 1:5. The crucibles were sealed in iron
cylinders filled with argon gas. After heating to 1150 -
1170 0Cthe furnace was cooled down slowly at rates
between 0.3 - 0.5 0C/h to minimize the amount of
flux inclusions. Near 940 - 1020 0Cthe furnace was
turned upside down to separate the remaining liquid
flux from the grown crystals and then cooled down to
room temperature with intermediate holds to in-situ an-
neal the crystals. Thermal expansion was measured us-
ing a high-resolution home-made capacitance dilatometer
[17], which is several orders of magnitude more sensitive
than traditional diffraction techniques. Heat capacity
was measured using a Physical Property Measurement
System from Quantum Design. The electronic specific
heat was obtained by subtracting an appropriate phonon
background [10, 18, 19]. Specifically, as demonstrated
for Ba1−xKxFe2As2[10, 20], the phonon background can
be approximated as the weighted sum of the individual
lattice contributions of its ’constitutents’ [21], which are
BaFe2As2and NaFe2As2for the present case. Since there
are no crystals of NaFe2As2, we determined the hypothet-
ical NaFe2As2phonon background by assuming that the
electronic component at optimal doping of Na- and K-
doped [10] systems are identical. This is quite reasonable,
since both Tcand the heat capacity jumps at optimal
doping are very similar in both systems. The Na content
arXiv:1510.03685v2 [cond-mat.supr-con] 7 Dec 2015

2
FIG. 1. (a) Relative length change, ∆L/L, versus temperature of the orthorhombic lattice parameters a and b of
Ba1−xNaxFe2As2for Na doping levels of x = 0, 0.221, 0.265 obtained using high-resolution capacitance dilatometry (see
text for details). (b) Temperature dependence of the orthorhombic distortion δ= (a−b)/(a+b)inferred from the data in
(a). The inset presents an expanded view of the data at higher doping levels. Vertical arrows indicate the location of the
superconducting transition at Tc, the C4-reentrant transition at T1, and the stripe-type SDW transition at Ts,N .
of seven single crystals (x = 0.093(4), 0.182(2), 0.221(2),
0.283(2), 0.320(2), 0.360(3), and 0.401(4)) used for the
thermal-expansion and specific-heat measurements was
accurately determined by 4-circle single crystal x-ray re-
finement of a small piece of the measured crystals. The
Na content of the other crystals were interpolated be-
tween these fixed points using the SDW transition tem-
perature as a reference. The values of the structural pa-
rameters from our x-ray refinement are in good agree-
ment with previous results [22].
Fig. 1a presents the relative thermal expansion, ∆L/L,
measured along the a- and b-axes for three representative
Na doping levels. As we have demonstrated previously
[10, 23], the shorter b-axis in the low-temperature or-
thorhombic phase can be obtained directly by measuring
the expansion of the crystal along the [110]Tdirection of
the original tetragonal cell, because in this configuration
the small force from the dilatometer detwins the crystal.
The larger a-axis, on the other hand, is obtained by com-
bining a ’twinned’ measurement (along [100]T) with the
’detwinned’ data [10, 23]. The expected orthorhombic
splitting of the a- and b-lattice parameters at the SDW
transition at Ts,N is clearly observed for all three con-
centrations and reduces in magnitude with increasing Na

3
FIG. 2. (a)-(f) In-plane thermal expansion coefficients in ’twinned’ (solid lines) and ’detwinned’ (dashed lines) orientations
versus T for Na concentrations of x = 0.182, 0.265, 0.302, 0.318, 0.36, and 0.401. The location of the various phase transitions
is marked by vertical arrows. The breaking of the C4symmetry at Ts,N in (a)-(e) is clearly indicated by the anisotropy of the
’twinned’ and detwinned’ expansion coefficients below Ts,N. On the other hand, the reentrant C4phase is characterized by
equivalent expansion coefficients below T1in (b) and between T2and Tcin (c). The near optimally doped sample in f ) exhibits
only a well-defined jump at Tc.
content. For the x = 0.265 sample, this splitting suddenly
disappears, within the accuracy of the measurements, at
a first-order transition at T1= 45K, which we identify
with the C4magnetic phase [9, 10].
In order to study the doping evolution of these tran-
sitions in greater detail, we present in Fig. 1b the or-
thorhombic distortion, δ= (a−b)/(a+b), inferred from
our thermal-expansion data for a number of compositions
between x = 0 and x = 0.36. We detect clear signatures of
the structural distortion associated with the SDW tran-
sition at Ts,N all the way to x = 0.36, which is consider-
ably higher than observed previously by neutron diffrac-
tion [9, 22]. We note, however, that the orthorhombic
splitting becomes extremely small in this high-doping re-
gion (see inset of Fig. 1b), which is probably why it
was missed previously. The presence of the reentrant C4
phase is signaled by a sudden disappearance of δat T1,
which we observe for 0.22 ≤x≤0.29. The behavior
of the lattice parameters changes dramatically for x =
0.302, where we observe a more gradual reduction of δat
T1, indicative of a second-order transition, followed by
a previously unobserved transition at T2. Upon further
doping, the transition at T2disappears and the transi-
tions at Ts,N and T1appear to merge together. The well-
known reduction of δat the superconducting transition
in the C2SDW phase due to the competition between
superconductivity and magnetism [23–25] is clearly ob-
served for the crystal with x = 0.221, whereas the effect
of superconductivity on the in-plane lattice parameters
in the C4phase is too small to be seen in these curves.
The small anomalies associated with the onset of Tc,
as well as the other phase transitions, are more clearly
observed in the thermal-expansion coefficients, α(T) =
1/L ·dL(T)/dT , for the ’twinned’ and ’detwinned’ direc-

4
FIG. 3. Phase diagrams of Na- and K-doped systems. (a), (c) Extrapolated (to T = 0) maximum orthorhombic distortion versus
Na and K doping, respectively. (b) Electronic phase diagram of Ba1−xNaxFe2As2obtained from thermal-expansion (squares)
and specific-heat (circles) data revealing nine different phases (see text for details). (d) Phase diagram of Ba1−xKxFe2As2from
Ref. [10] for comparison. The kinks in (a) and (c), as well as the inflection points of Ts,N in (b) and (d), near x = 0.22 -
0.23 are interpreted as marking a possible transition from a commensurate (C2-C) phase to an incommensurate (C2-IC) phase.
This transition is indicated by the vertical dashed lines and the color transition of the Ts,N line from blue to red. ’S’ stands
for superconductivity.
tions, which are presented as α/T versus T for repre-
sentative Na contents in Fig.2. Fig.2a displays data for
the crystal with x = 0.182, which becomes orthorhom-
bic below Ts,N = 112 K and superconducting below Tc
= 6.5 K. The clear anisotropy of the in-plane expansion
below Ts,N , as well as the anisotropic response at Tc,
are indicative of the expected orthorhombic state at this
doping level. We note that the small anisotropic tail
above Ts,N results from the small, but finite, uniaxial
pressure we apply in our dilatometer [10]. In contrast to
the behavior for x = 0.182, the anisotropy of the expan-
sivity vanishes nearly completely below the transition at
T1for the x = 0.265 sample (see Fig.2b), indicating the
reentrant tetragonal state below T1. As expected at the
onset of superconductivity, small jump-like anomalies at
Tcare observed for both directions. The behavior of the
x = 0.302 crystal is more complicated (see Fig.2c). Here
the crystal clearly becomes orthorhombic at Ts,N , then
δ(T)decreases gradually between T1and T2(see inset
of Fig.1b), but remains orthorhombic. The expansivi-
ties for both orientations are equal below T2, suggesting
that the system again enters a tetragonal state. The
curves below Tc, however, again exhibit an anisotropic
response, suggesting that the C4phase reverts back to
the C2’ phase below Tc, in analogy to what has been
observed in K-doped BaFe2As2[10]. There is an addi-
tional sharp anomaly at TL=10 K for both orientations,
which is however observed only upon heating, possibly
indicating another phase transition with a large thermal
hysteresis. Nearly identical behavior was observed in an-
other crystal with a similar composition. Our expansion
data thus clearly show that the reentrant C4phase exists
only in a limited temperature range between T2and Tc
for x = 0.302. The transitions at T2and TLboth disap-
pear for the next higher Na content (see Fig.2d), and this
sample also clearly displays strongly anisotropic thermal
expansivities below Ts,N , which is incompatible with a
C4symmetry. The x = 0.36 crystal (Fig. 2e) exhibits
only very small effects at Ts,N and T1. Finally, any sig-
nature of the anomaly at Ts,N has disappeared in the
crystal with x = 0.401, which only has a clear anomaly
at Tc= 35 K.
The transition temperatures Ts,N ,Tc,T1,T2and TL
obtained by the thermal -expansion data shown in Figs.
1 and 2 allow us to construct a detailed phase diagram
(see Fig.3b). Here, we have also included the transition
temperatures extracted from the heat-capacity data (see
Fig.4). The phase diagram exhibits a remarkable de-
gree of complexity, with a surprising number of additional
(other than the usual C2-magnetic and superconductiv-
ity) phases emerging as magnetism is suppressed by Na
doping. We note that these phases appear to emerge at

5
FIG. 4. (a)-(o) Temperature dependence of thermal expansion (αa/T) (a-e), electronic heat capacity Ce/T (f-j) and electronic
entropy (Se/T ) (k-o) for crystals with x = 0.182, 0.265, 0.296, 0.302 and 0.401. The different shades of gray represents the
step-wise reduction of Se/T from the high temperature paramagnetic phase to the low temperature superconducting state.
the point where Ts,N changes curvature from concave
to convex near x = 0.22. This change is indicated by
the changing color of the line from blue to red. The
doping dependence of the extrapolated zero-temperature
orthorhombic distortion of the C2phase (see Fig. 3a)
illustrates this change even more clearly, with a very
distinct kink near x = 0.22. We interpret the inflec-
tion point of Ts,N (x) as a sign for a commensurate-to-
incommensurate transition as expected in a simple mean-
field SDW picture [26–28]. Previously, clear evidence
for incommensurability has only been reported in elec-
tron doped BaFe2As2[29–31]. Since we do not observe
a splitting of the Ts,N line into two transitions above x
= 0.22, we postulate the vertical dashed line to indicate
the proposed commensurate-to-incommensurate transi-
tion. Such a vertical line implies a first-order transition,
evidence of which is provided by the jumps of both T1
and Tcat x = 0.22. In Fig. 3c and d we compare the
present results to those of K-doped BaFe2As2[10]. Sim-
ilar to the Na-doped system, we also find an inflection
point in its phase diagram (see Fig. 3 d), as well as a kink
in the T = 0 orthorhombic distortion (see Fig. 3c), at a K
content at which the C4phase emerges (see Fig. 3c and
d). This strongly suggests that these are both common
features in hole doped BaFe2As2. In Ba1−xNaxFe2As2we
observe, in addition to the magnetic C4phase, a previ-
ously unobserved phase (labeled C2’ in Fig. 3 b), which
has a reduced, but finite, orthorhombic distortion. A
similar phase in not observed in the K-doped system. Al-
though we can not examine the microscopic order in this
phase using our macroscopic probes, the smooth doping
variations of both Ts,N and δ, suggest that this phase is
probably also of magnetic origin, although some kind of
charge [13] order cannot be excluded. Preliminary μSR
measurements on a crystal with x=0.33 provide evidence
for a magnetic C2’ phase [32]. Detailed investigations
of the magnetic structure using e.g. neutron scattering
are highly desirable once large enough crystals become
available.
In order to gain more insight into the different phases,
we present the electronic heat capacity for several Na
concentrations in Fig.4 together with thermal expansion
of the a-axis for comparison. As demonstrated in Fig. 3
and 4, the transition temperatures from the heat capacity
(solid gray circles in Fig.3) closely match those from the
thermal expansion. With increasing Na-doping the step-
like anomalies in Ce/T associated with superconductivity
generally increase in size, whereas the anomalies associ-
ated with the magnetic transitions weaken, indicating the
well-known competition between magnetism and super-
conductivity in the Fe-based systems [10, 23–25]. This
trend is made even more transparent in Fig.4 k-o, where

6
we plot Se/T = (´Ce/T dT )/T , i.e. the electronic en-
tropy divided by T, which for a Fermi liquid is expected
to be constant. Upon entering the C4phase at 45 K for
the x = 0.265 Na sample we observe a particularly large
additional reduction of Se/T at T1, which is more promi-
nent than the anomaly at Ts,N , and apparently results
in a large suppression of Tcand the condensation energy
(equal to the black shaded area in Fig. 4 k)-o)) in the C4
phase. This highlights the much stronger competition of
superconductivity with the double-Q C4magnetic phase
than with the usual magnetic C2phase, which was also
observed in Ba1−xKxFe2As2[10]. However, in contrast
to Ba1−xKxFe2As2[10], we find no evidence for a reemer-
gence of the usual stripe-type C2phase below Tc. For
the crystals with x = 0.296 and 0.302, the largest (non
superconducting) anomalies in Ce/T and Se/T occur not
at T1, but rather upon entering the C4phase at T2. In-
terestingly, the Se/T plot for both these samples (Fig. 4
m and n) provide evidence for a pseudogap-like behavior
above T2- i.e. a gradual loss of density-of-states as the
temperature is lowered. The competition of supercon-
ductivity with the C2’ phase appears to be much weaker
than with the C4-magnetic phase, as evidenced by the
increase of the superconducting condensation energy , as
well as the rise of Tcseen in Fig. 3 within the C2’ phase.
Finally, we note that the negligible residual Ce/T values
of all of our samples (except for x = 0.265) demonstrate
that our samples are of high quality and that doping away
from the FeAs layer does not introduce pair breaking, as
it does in Co-doped BaFe2As2[33].
In summary, our detailed thermodynamic studies of
Ba1−xNaxFe2As2show that the phase diagram of this
system exhibits a surprising degree of complexity. As
stripe-type magnetism is suppressed by Na-doping, two
additional magnetic phases emerge, which coexist and
compete with superconductivity. The emergence of these
additional phases is shown to be possibly triggered by
a doping-induced commensurate-incommensurate tran-
sition near x = 0.22, which would provide further evi-
dence for electronic itinerancy in these systems. There
are many similarities between the phase diagrams of K-
and Na-doped BaFe2As2, and the differences are likely re-
lated to chemical pressure, since our previous studies on
the K-doped system have shown that the phase bound-
aries are extremely pressure dependent [10]. Importantly,
the presently observed complexity of these phase diagram
suggests a high degree of degeneracy of several energy
scales as the optimally-doped state is approached, which
may also be related to the superconducting pairing mech-
anism.
We acknowledge fruitful discussions with Christian
Bernhard, Markus Braden, Rafael Fernandes, Maria Gas-
tiasoro, Benjamin Mallett, Jörg Schmalian, and Florian
Waßer.
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