# Electronic specific heat in BaFe2-xNixAs2

Dongliang Gong  Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China  Tao
Xie  Beijing National Laboratory for Condensed Matter Physics, Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China  Xingye Lu
Beijing National Laboratory for Condensed Matter Physics, Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China  Cong Ren  Beijing
National Laboratory for Condensed Matter Physics, Institute of Physics,
Chinese Academy of Sciences, Beijing 100190, China  Lei Shan  Beijing National
Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy
of Sciences, Beijing 100190, China  Collaborative Innovation Center of Quantum
Matter, Beijing, China  Rui Zhang  Department of Physics and Astronomy, Rice
University, Houston, Texas 77005-1827, USA  Pengcheng Dai  Department of
Physics and Astronomy, Rice University, Houston, Texas 77005-1827, USA  Yi-
feng Yang  Beijing National Laboratory for Condensed Matter Physics, Institute
of Physics, Chinese Academy of Sciences, Beijing 100190, China  Collaborative
Innovation Center of Quantum Matter, Beijing, China  Huiqian Luo
<EMAIL_ADDRESS>Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
Shiliang Li<EMAIL_ADDRESS>Beijing National Laboratory for Condensed Matter
Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190,
China  Collaborative Innovation Center of Quantum Matter, Beijing, China

###### Abstract

We have systematically studied the low-temperature specific heat of the
BaFe2-xNixAs2 single crystals covering the whole superconducting dome. Using
the nonsuperconducting heavily overdoped x = 0.3 sample as a reference for the
phonon contribution to the specific heat, we find that the normal-state
electronic specific heats in the superconducting samples may have a nonlinear
temperature dependence, which challenges previous results in the electron-
doped Ba-122 iron-based superconductors. A model based on the presence of
ferromagnetic spin fluctuations may explain the data between x = 0.1 and x =
0.15, suggesting the important role of Fermi-surface topology in understanding
the normal-state electronic states.

###### pacs:

74.25.Bt,74.70.Xa

##  I introduction

The normal-state electronic states of the iron-based superconductors have been
heavily studied by the specific heat technique Stewart (2011). The most
important parameter is $\gamma_{n}=C/T$, which is proportional to the density
of states at the Fermi energy and the effective electron mass. The
measurements in the iron-based superconductors have given a variety of values
of $\gamma_{n}$ for different materials Mu et al. (2009); Tan et al. (2013);
Gofryk et al. (2010a, b); Hardy et al. (2010a, b); Popovich et al. (2010);
Gofryk et al. (2011); Pramanik et al. (2011); Hu et al. (2011); Zeng et al.
(2012); Noji et al. (2012); Johnston et al. (2014); Storey et al. (2013);
Abdel-Hafiez et al. (2015). A detailed study on the Ba(Fe1-xCox)2As2 shows
that $\gamma_{n}$ becomes maximum at optimal doping Gofryk et al. (2010a);
Hardy et al. (2010b), suggesting the influence of the antiferromagnetic (AF)
order that disappears at the same doping Dai (2015). In addition to its
relation to the normal-state electronic states, $\gamma_{n}$ is also
associated with superconductivity though the ratio of $\Delta
C/\gamma_{n}T_{c}$ that is traditionally used as an estimate of the coupling
strength for a superconductor, where $\Delta C$ is the specific heat jump at
the superconducting transition. The value of $\Delta C/\gamma_{n}T_{c}$ again
peaks at the optimal doping Hardy et al. (2010b), suggesting strongest
coupling of superconductivity around the optimal doping.

Despite the abundant results shown above, the effect of spin fluctuations on
the specific heat has been little studied. In a quasi-two-dimensional system
close to an AF quantum critical point (QCP), $C/T$ may diverge logarithmically
with decreasing temperature Petrovic et al. (2001); Stewart (2001). The iron-
based superconductors undoubtedly exhibit strong AF fluctuations Dai (2015).
Particularly, in the electron-doped BaFe${}_{2-x}T_{x}$As2 ($T$ = Ni or Co)
system, it has been demonstrated that the long-range AF order is totally
suppressed near optimal doping level, whereas the AF spin fluctuations survive
in much higher doped samples, as shown in inelastic neutron scattering and
nuclear magnetic resonance (NMR) measurements Luo et al. (2013); Ning et al.
(2010); Zhou et al. (2013). While the presence of an AF QCP in
BaFe${}_{2-x}T_{x}$As2 is still under debate Zhou et al. (2013); Luo et al.
(2012); Lu et al. (2013), the normal-state transport properties can be
affected by the AF spin fluctuations around the optimal doping May et al.
(2013); Arsenijevi$\acute{c}$ et al. (2013); Rullier-Albenque et al. (2013).
Recently, ferromagnetic (FM) spin fluctuations have also been observed in the
iron pnictides by the NMR technique Wiecki et al. (2015). These results
clearly indicate that the effect of spin fluctuations should not be ignored in
dealing with the specific-heat data.

For the specific heat in the iron-based superconductors, phonon contribution
is often dominated above $T_{c}$. To reveal the electronic specific heat, one
may try to fit the data above $T_{c}$ with certain functions Mu et al. (2009);
Tan et al. (2013) or find a reference sample without superconductivity by
assuming that the specific heat of phonons change little with doping Gofryk et
al. (2010a, b). In the latter case, the specific heat of the phonons of the
reference sample may be simply adjusted as aC${}_{phonon}(bT)$ with a and b as
the tuning parameters (a-b method) to account for the change of the phonon
spectra with doping Hardy et al. (2010a, b); Popovich et al. (2010); Pramanik
et al. (2011); Hu et al. (2011); Zeng et al. (2012); Noji et al. (2012);
Johnston et al. (2014); Abdel-Hafiez et al. (2015). In most cases, a linear
temperature dependence of the electronic specific heat below $T_{c}$ is
assumed to account for the entropy conservation, i.e., $\gamma_{n}$ is
temperature-independent. An artificial linear or quadratic temperature
dependence of $\gamma_{n}$ may also be chosen Gofryk et al. (2010a); Storey et
al. (2013). Apparently, the non-linear temperature-dependent contribution to
the specific heat from spin fluctuations is neglected.

In this paper, we systematically study the low-temperature electronic specific
heat of the electron-doped BaFe2-xNixAs2 to address the role of spin
fluctuations. Similar to other iron-based superconductors, the BaFe2-xNixAs2
system shows a superconducting dome with the total suppression of static AF
order near the optimal doping level x $\sim$ 0.1 Luo et al. (2012); Lu et al.
(2013). We find that using the non-superconducting heavily overdoped x = 0.3
sample as a reference for the phonon part of the specific heat without any
artificial adjustment is sufficient to obtain the electronic specific heat in
this system. For the samples with x between 0.1 and 0.15, $\gamma_{n}$
increases with decreasing temperature, which may be explained by the spin-
fluctuation theory. Our results initiate that extra caution is necessary in
studying the specific heat near the AF instability.

##  II experiments

High-quality single crystals of BaFe2-xNixAs2 and BaFe2-x-yNixCryAs2 were
grown by the self-flux method as reported previously Chen et al. (2011); Zhang
et al. (2014). The superconductivity can be easily suppressed by a few percent
of Cr doping Zhang et al. (2014). In the following text, the pure Ni doped
sample is denoted by only the value of x. To further simplify the description,
the exact value of y will only be labeled in the figure captions for the non-
superconducting Cr-doped samples. The actual and nominal doping levels of both
Ni and Cr have linear relationships with the ratios of about 0.8 and 0.7,
respectively. We will use the nominal values to be consistent with our
previous reports Luo et al. (2012); Lu et al. (2013); Zhang et al. (2015). The
specific heat was measured by the Physical Properties Measurement System
(PPMS) from Quantum Design with or without He-3 option. The magnetic
susceptibility was measured by the SQUID.

##  III results

###  III.1 Validity of the a-b method

Fig. 1(a) shows the whole specific heats of the BaFe2-xNixAs2 samples up to 30
K. Fig. 2(b) further shows the low-temperature data with $T^{2}$ as x axis.
All the low-temperature data can be well fitted by a cubic equation as
$C_{tot}/T=\gamma_{0}+BT+CT^{2}$ except for the x = 0.3 sample, where
$C_{tot}$, $\gamma_{0}T$ correspond to the total and residual electronic
specific heats, respectively. The slight deviation at very low temperature for
the x = 0.3 sample may come from Schottky anomaly, which is absent in other
samples. Fig. 1(c) and 1(d) show the doping dependence of coefficients B and
C. The contribution of B$T^{2}$ has been attributed to line nodes or deep
minima in the energy gap Gofryk et al. (2011); Mu et al. (2011); Zeng et al.
(2012). The values of C around optimal doping are much larger than those in
the nonsuperconducting underdoped and overdoped samples, suggesting that part
of it should come from the electronic contribution such as point nodes in the
superconducting gaps or some kind of bosonic mode Gofryk et al. (2011). In the
x = 0.18 sample, the value of C is much smaller than that of other samples,
most likely due to its low $T_{c}$, which makes the current upper bound of
fitting temperature too high. Fitting the data with smaller temperature range
results in larger value of C. Our results at low temperature are consistent
with previous results in the electron-doped BaFe2As2 materials Gofryk et al.
(2011); Mu et al. (2011); Zeng et al. (2012), suggesting the good quality of
our samples. In this paper, we will focus on the study of the normal-state
electronic specific heat.

Since the x = 0.3 sample shows neither superconductivity nor static AF order,
it may be used as a reference to remove the phonon contribution. The presence
of a large ( 50 meV) spin gap in the x=0.3 sample Wang et al. (2013) means
that one can safely ignore the spin-fluctuation contribution to the specific
heat in this material. In the following analysis, the phonon specific heat of
the x = 0.3 sample is calculated as CT3 where C is the fitted parameter as
shown in Fig. 1(d). The phonon specific heat for the whole temperature range
can be thus derived by subtracting $\gamma_{n}T$ from the total specific heat.
Using $C_{e}^{x}=C_{tot}^{x}-aC_{phonon}^{0.3}(bT)$ where x and 0.3 represent
doping levels, we can get the electronic specific heats $C_{e}^{x}$ of all the
superconducting samples meeting the requirement of the entropy conservation by
adjusting parameters a and b, as shown in Fig. 1(c). No adjustment is needed
if both a and b are equal to one.

Figure 1:  (a) Temperature dependence of the total specific heat of
BaFe2-xNixAs2. (b) The $T^{2}$ dependence of low-temperature specific heat of
BaFe2-xNixAs2. The solid lines are fitted by a cubic function as described in
the text. (c) and (d) give the doping dependence of the fitting parameters B
and C as discussed in the text, respectively. (e) The electronic specific heat
of selected samples obtained by the a-b method. (f) The doping dependence of
the parameters a and b in the a-b method.

The doping dependence of parameters a and b is shown in Fig. 1(d). At first
glance, both a and b show only slight deviation from 1, suggesting the phonons
do not change much with Ni doping as expected. Usually, it means that the data
treatment is satisfactory. However, we note that these two parameters do not
change monotonically with doping. Specially, the value of b is supposed to
only depend on the Debye temperature which should show monotonic dependence on
Ni doping, whereas it changes abruptly around the optimal doping. Checking the
details of the a-b method, the requirement of entropy conservation in the data
treatment actually assumes that $\gamma_{n}$ is temperature independent. The
nomonotonic Ni doping dependence of a and b suggests that such assumption may
not be valid. Here we suggest that the a-b method assuming a temperature-
independent $\gamma_{n}$ is indeed incorrect in this system based on the
following arguments.

First, we find that different results will be achieved in some samples if we
choose different temperature range to subtract the phonon contribution. Fig.
2(a) gives the subtracted $C/T$ of x = 0.15 sample obtained by considering
different temperature ranges using the a-b method. The zero value of $C/T$
usually means that the adjustment by tuning a and b correctly captures the
change of the specific heat from the phonon contribution. Large deviation from
zero is found above 30 K if we only consider the data from 16 K to 30 K. On
the other hand, while the fit looks better for the temperature range from 30 K
to 60 K, the entropy does not conserve at $T_{c}$ as shown in the inset of
Fig. 2(a). The inconsistency between the above two results suggests that the
a-b method is not reliable.

Second, the change of the phonon contribution to specific heat due to Ni
dopings should be smaller than that suggested by Fig. 1(d). At low
temperatures, the effect of doping on the specific heat of phonons may be
directly associated with the atomic mass if the lattice structure is not
changed. Fig. 2(b) gives the electronic specific heat of the x = 0.3, y = 0.3
sample using zero Cr doped sample as the reference without any adjustment. The
actual atomic mass change is about 0.4%, whereas that between the x = 0.3 and
x = 0 samples is about 0.17%. With much larger change of the atomic mass, the
$C/T$ of the Cr-doped sample above 15 K suggests that no further adjustment is
needed. The low temperature upturn below 15 K may be attributed to the Cr
impurities Zhang et al. (2015). Treated as a diluted spin system, we may fit
the low-temperature data with the spin-fluctuation theory as follows Trainor
et al. (1975); Stewart et al. (1984),

$C=AT+BT^{3}+DT^{3}lnT,$ (1)

as shown by the solid line in Fig. 2(b). The physical meanings of the
parameters will be discussed later. We note that the $T_{SF}$ of the heavily
Cr-doped sample in Fig. 2(b) is about 15 K, which is much smaller than the
$T_{SF}$ shown in Fig. 4(a).

Third, the specific heat of slightly Cr-doped nonsuperconducting samples
suggests that there is also low-temperature upturn of $C/T$ in the normal
state of the superconducting samples. Fig. 2(c) gives the electronic specific
heat of the x = 0.15 sample using the Cr-doped sample as a reference without
any adjustment. The entropy is conserved as shown in the inset of Fig. 2(c),
which suggests the reliability of the data treatment. While it indeed gives
the correct electronic specific heat below $T_{c}$, the information on
temperature dependence of the normal-state specific heat is missing since the
zero value of $C_{e}/T$ above $T_{c}$ only suggests that the superconducting
and nonsuperconducting samples have the same electronic specific heat assuming
that the phonon specific heat changes negligibly with slight Cr doping.

Figure 2:  (a) The electronic specific heat of the x = 0.15 sample using the x
= 0.3 sample as a reference adjusted by the a-b method, where the range of
normal-state data is selected either from 30 K to 60 K (black triangles) or
from 16 K to 30 K (red triangles). The entropy differences between the normal
and superconducting states are shown in the inset. (b) The electronic specific
heat of the x = 0.3, y = 0.3 sample using the x = 0.3 sample as the reference.
The low temperature data are fitted by Eq. 1 as shown by the solid line. (c)
The electronic specific heat of the x = 0.15 sample using the
nonsuperconducting x = 0.15, y = 0.03 sample as the reference. The inset shows
that the entropy is conserved. (d) The electronic specific heat of the Cr-
doped nonsuperconducting BaFe1.85Ni0.1Cr0.05As2 (squares) and
BaFe1.82Ni0.15Cr0.03As2 (circles) using the x = 0.3 sample as the reference.
The solid line is fitted by Eq. 1.

Fig. 2(d) gives temperature dependence of the specific heat of the Cr-doped
samples. For the Cr-doped x = 0.15 sample, $C/T$ increases quickly with
decreasing temperature. Such increase of $C/T$ at low temperature cannot be
compromised by the a-b method since the latter mainly affect the specific heat
above 10 K. As shown in Fig. 2(c), the entropy conservation and the zero value
of $C/T$ above $T_{c}$ suggests that $C/T$ of the x = 0.15 sample should also
show upturn at low temperature if no superconductivity is present. In other
words, the normal-state electronic specific heat of the x = 0.15 sample cannot
be simply described by a linear temperature dependence. Similar to the heavily
Cr-doped sample (Fig. 2(b)), we can also fit the low temperature data of the
Cr-doped x = 0.15 sample with the spin-fluctuation theory.

Unfortunately , using Cr-doped nonsuperconducting samples to obtain the
normal-state electronic specific heat of the corresponding superconducting
samples does not work for the samples below the optimal doping level. For
example, the entropy is not conserved for the x = 0.1 sample using the x = 0.1
Cr-doped sample as the reference, suggesting that the electronic specific
heats of the two samples are not the same. As shown in Fig. 2(d), $C/T$ of the
x = 0.1 Cr-doped sample shows a kink around 25 K, which may be due to the
enhancement of the AF order upon Cr doping Zhang et al. (2015).

###  III.2 Electronic specific heat of BaFe2-xNixAs2

Figure 3:  (a) - (h) The temperature dependence of the electronic specific
heat of pure Ni-doped samples with Ni doping levels as labeled in the figures
using the x = 0.3 sample as reference without any adjustment. The dashed lines
are fitted by Eq. 1. The dotted dashed lines are straight lines with constant
value of $C/T$.

The above results suggest that no adjustment is needed in deducting the phonon
specific heat of x = 0.3 sample. Fig. 3 shows the electronic specific heat of
superconducting samples with both a and b equal to one. A slightly decrease of
$C/T$ with decreasing temperature above $T_{c}$ is found for the x = 0.08
sample, probably due to the presence of long-range AF order Luo et al. (2012);
Lu et al. (2013). For the samples from x = 0.1 to 0.15, $C/T$ shows clear sign
of upturn. While the normal-state electronic specific heat above $T_{c}$ can
be obtained just by this simple subtraction, we may further obtain its value
below $T_{c}$ without the presence of superconductivity using the spin-
fluctuation theory with the constraint of entropy conservation, as shown by
the dashed lines in Fig. 3.

Fig. 4(a) and 4(b) shows the doping dependence of parameter $T_{SF}$ and $D$,
where $T_{SF}=e^{-B/D}$ is the characteristic spin-fluctuation temperature
Trainor et al. (1975); Stewart et al. (1984). The weak upturns of $C/T$ in the
x = 0.092, 0.096 and 0.16 samples cause unrealistic fitting parameters, such
as negative $D$ or very large value of $T_{SF}$ ( about 1024 K for the x =
0.16 sample). For the x = 0.1, 0.12, and 0.15 samples, while $T_{SF}$ shows
weak doping dependence, $D$ increases linearly with doping. According to the
spin-fluctuation theory, $D=\alpha\gamma/T_{SF}^{2}$ where $\alpha$ and
$\gamma$ are associated with the Stoner-enhancement factor and the electronic
specific-heat coefficient determined from the band-structure density of
states. It will be interesting to see which factor dominates in determining
the doping dependence of $D$.

Figure 4:  (a) Doping dependence of $T_{SF}$. (b) Doping dependence of $D$.
(c) Doping dependence of $\gamma_{n}$ and $\gamma_{0}$. At zero K, two values
of $\gamma_{n}$ obtained from the a-b method (red downward triangles) and our
method (red triangles, $\gamma_{n}^{0}$) are given at each doping.
$\gamma_{n}$ at $T_{c}$ is shown by the green squares ($\gamma_{n}^{T_{c}}$).
(d) Doping dependence of $\Delta
C_{eff}/T_{c}/(\gamma_{n}^{T_{c}}-\gamma_{0})$ where $\gamma_{n}^{T_{c}}$ is
obtained from either the a-b method (red downward triangles) or our method
(green squares).

If there is indeed a non-linear temperature dependence of the normal-state
electronic specific heat, one may ask how significant the effect is in
determining the relevant physical properties. Fig. 4(c) gives the doping
dependence of $\gamma_{n}$ and $\gamma_{0}$. The value of $\gamma_{0}$ gives
the residual electronic specific heat coefficient at zero K, obtained by
fitting the raw data as shown in Fig. 1(b). The very small values of
$\gamma_{0}$ compared to $\gamma_{n}$ around the optimal doping level suggest
the good quality of our samples. For the $\gamma_{n}$ at zero K, no distinct
difference is found between the a-b method and our method. However,
$\gamma_{n}$ at $T_{c}$ from x = 0.1 to 0.15 is significantly smaller than
that of $\gamma_{n}$ at zero K. This results in large increases of $\Delta
C_{eff}/T_{c}\gamma_{n}^{T_{c}}$ within the above doping range, where $\Delta
C_{eff}=\Delta C\times\gamma_{n}/(\gamma_{n}-\gamma_{0})$ assuming
$\gamma_{0}$ comes from the non-superconducting part of the sample Hardy et
al. (2010a). The maximum of the normalized superconducting jump shifts from
the optimal doping level ( x $\approx$ 0.1 ) to slightly overdoped doping
level ( x $\approx$ 0.12). The large deviation at x = 0.18 indicates that the
large residual specific heat in heavily overdoped regime cannot be explained
by phase separation, suggesting an inhomogeneously gapped superconducting
state or pair breaking effect Hardy et al. (2010a). On the other hand, the
quick increase of $\gamma_{0}$ with decreasing Ni for x $<$ 0.09 may come from
the non-superconducting long-range antiferromagnetism. It should be noted that
the determination of $\gamma_{n}^{T_{c}}$ is independent of any model
including the spin-fluctuation theory discussed in the following section.

###  III.3 Magnetic properties of BaFe2-xNixAs2

Fig. 5(a)-(f) show the magnetic susceptibility of BaFe2-xNixAs2 at various
doping levels, where linear backgrounds at high fields have been subtracted.
This is done by fitting the data between $\pm 3000$ Oe to $\pm 10000$ Oe as
A$H$ $\pm\Delta M$, respectively, where A and $\Delta M$ are positive
constants. Clearly ferromagnetic behavior at low fields is found for the
samples from x = 0.1 to 0.15. Surprisingly, little change is seen between 25 K
and 300 K. Fig. 5(g) gives the doping dependence of the saturate magnetic
susceptibility at 25 K from the above results, which clearly suggests that the
ferromagnetic moment becomes much stronger between x = 0.1 to 0.15. While
these results are consistent with those of specific heat, it is the FM spin
fluctuations that contribute to the electronic specific heat. Therefore,
whether the observed ferromagnetism is directly associated with the
enhancement of specific heat at low temperature is arguable. Fig. 5(h) further
shows the temperature dependence of magnetic susceptibility at 7 Tesla for
various samples, where an upturn at low temperature shows up in the x = 0.3
sample. However, such upturn seems to have negligible contribution to the
specific heat as suggested by the nice fitting in Fig. 1(b). Similarly, the Cr
impurities also introduce upturn in the magnetic susceptibility as shown in
the inset of Fig. 5(h), but the electronic specific heats of the x = 0.15
sample with or without Cr doping are the same as discussed above.

Figure 5:  (a)-(f) Magnetic susceptibility of BaFe2-xNixAs2 samples with
doping levels labeled in the figures after subtracting linear backgrounds
measured at high fields as described in the text. (g) Doping dependence of
saturated magnetic susceptibility at 25 K. (h) Temperature dependence of
magnetic susceptibility for the pure Ni-doped samples at 7$\times$104 Oe. The
inset shows the low temperature magnetic susceptibility for the x = 0.15
sample and the x = 0.15, y = 0.03 sample.

##  IV discussions

From the above results of section III.A, we find that the a-b method cannot
give the correct normal-state electronic specific heat in BaFe2-xNixAs2. In
some of the studies Hardy et al. (2010a, b); Pramanik et al. (2011); Johnston
et al. (2014); Abdel-Hafiez et al. (2015), only parameter a is introduced to
account for the experimental uncertainties such as the error of sample mass.
Since such process does not involve changing the temperature dependence of the
phonon specific heat, it may not introduce some of the artificial results
discussed above. However, we emphasis that the most important factor that
determines applicability of the a-b or only-a method is whether the electronic
specific heat has a linear temperature dependence, which is crucial in judging
the entropy conservation. Moreover, if only a narrow range of normal-state
specific heat is considered, the non-linear temperature dependence of the
electronic specific heat can be easily neglected as shown in Fig. 2(a).

It is rather surprising that we cannot observe the contribution to the
specific heat from AF spin fluctuations. As shown by the neutron scattering
and NMR experiments Luo et al. (2013); Ning et al. (2010); Zhou et al. (2013),
AF spin fluctuations should be dominant in this regime. Specially, the quantum
critical spin fluctuations around the AF QCP can result in significant effect
on specific heat Petrovic et al. (2001); Stewart (2001). However, as suggested
by both NMR and neutron scattering measurements Dioguardi et al. (2013); Lu et
al. (2014), the long-rang AF order in the electron-doped
BaFe${}_{2-x}T_{x}$As2 system evolves into a cluster spin-glass state without
the presence of AF QCP, which may explain the negligible contribution of AF
spin fluctuations to the specific heat. On the other hand, the zero-$Q$ nature
of FM spin fluctuations could have significant influence on specific heat.

While the upturn of $C/T$ at low temperature for the samples with x from 0.1
to 0.15 can be reasonably explained by the spin-fluctuation theory, the origin
of FM spin fluctuations needs to be further studied. Recently, the NMR
measurement has found strong FM spin fluctuations in the Ba(Fe1-xCox)2As2
system, which may be associated with our results. However, it seems to
contradict with the observation that the low-temperature upturn of $C/T$
suddenly disappears above x = 0.16, as shown in Fig. 3. It should be noted
that while $C/T$ of the x = 0.16 shows upturn in Fig. 3(g), the unrealistic
fitting values of $T_{SF}$ and $D$ suggest that the effect of FM spin
fluctuations should be weak and uncertainties such as masses of the samples
may have to be considered. According to angle-resolved photo emission
spectroscopy (ARPES) measurements Liu et al. (2011); Ideta et al. (2013), such
doping level corresponds to a topological change of the Fermi surfaces where
the hole pocket at $\Gamma$ point disappears. The FM spin fluctuations
contributed to the specific heat may thus come from the Stoner instability of
the hole pocket at $\Gamma$ point, which should be suppressed by the AF order
below x = 0.1 where the nesting of the hole and electron pockets sets in.
Based on theoretical calculation Singh (2008), the mass enhancement around
optimal doping is between 2 - 3. While the decrease of $\gamma_{n}$ at in the
underdoped regime seems to be related to the Fermi surfaces reconstruction due
to the presence of AF order Hardy et al. (2010b), the weakening of electronic
correlation with Ni doping may cause the reduction of $\gamma_{n}$ in the
overdoped regime Ye et al. (2014). The spin-fluctuation contribution to the
mass enhancement is 1 + $\frac{9}{2}$ln($S$/3) Trainor et al. (1975), which
gives value of the Stoner exchange-enhancement factor S between 3.7 to 4.7
around optimal doping level. It should be noted that the above analysis may be
too simplified considering the multi-band and correlated nature of iron-based
superconductors.

Another possible origin of FM spin fluctuations is the presence of magnetic
impurities Vojta (2010). Both Ni and Cr dopants may act as magnetic impurities
as shown in Fig. 5(h) for the x = 0.3 and x = 0.15, y = 0.03 samples probably
due to incomplete charge transfer Teague et al. (2011). The temperature
dependence of magnetic susceptibility may be fitted by a Curie-Weiss-like plus
a linear functions, the latter of which has already been observed in BaFe2As2
Wang et al. (2009). The fitted mean-field transition temperatures for both
samples are negative, suggesting an AF coupling. The low-temperature upturn of
magnetic susceptibility has already been found in BaFeNiAs2 Sefat et al.
(2009) and BaFe2-xCrxAs2 system Sefat et al. (2009). Both Ni and Cr magnetic
impurities have negligible effect on the specific heat as discussed
previously. The doping evolution of the low-temperature upturn of magnetic
susceptibility due to the Ni or Cr dopants does not shows a direct connection
to that of the specific heat either. At current stage, we are unable to rule
out the possibility that a very tiny amount of magnetic impurities may give
rise to the enhancement of the specific heat at low temperature. The fact that
the large ferromagnetic moment is only seen for the samples from x = 0.1 to
0.15 (Fig. 5(g)) also suggests that the magnetic impurities, if any, may not
come from growth of the samples since they were grown with same process.

In the end, we give a brief discussion on spin glass and its effect. It has
been shown by neutron scattering experiments Luo et al. (2012); Lu et al.
(2014) that a spin-glass-like short-range incommensurate antiferromagnetism
presents above x = 0.09 in BaFe2-xNixAs2. While the spin glass may give a FM-
like hysteresis Binder and Young (1986), it should show strong temperature
dependence considering the $T_{N}$ measured by neutron scattering is just
about 30-40 K around optimal doping, which contradicts our results in Fig. 5.
Moreover, the spin glass state should die out quickly with increasing Ni
doping above the optimal doping level, as shown in the similar Co-doped Ba-122
system Dioguardi et al. (2013), but both the FM-like hysteresis and specific
heat enhancement persist up to x = 0.15. Since the spin glass present at lower
doping levels, it gives a natural explanation on the decrease of
$\gamma_{n}^{0}$ below x = 0.12. The peak of $\Delta
C_{eff}/T_{c}/(\gamma_{n}^{T_{c}}-\gamma_{0})$ may thus be related to the
phase fluctuations of the antiferromagnetic order, which shows glassy behavior
around optimal doping.

##  V conclusion

In this paper, we have systematically studied the specific heat of the
electron-doped iron-based superconductors BaFe2-xNixAs2. Our detailed analysis
shows that the a-b method that is frequently used in the literatures cannot
give the correct electronic specific heat for samples with x from about 0.1 to
0.15. The temperature dependence of the normal-state electronic specific heat
within this doping range can be described by the spin-fluctuation theory,
where the FM fluctuations may come from the Stoner instability of the hole
pocket at $\Gamma$ point or a tiny amount of magnetic impurities. Our results
suggest that the a-b method should be carefully used around the AF
instability.

###### Acknowledgements.

This work is supported by the ”Strategic Priority Research Program (B)” of the
Chinese Academy of Sciences (XDB07020300), the Ministry of Science and
Technology of China (973 project No. 2012CB821400, 2011CBA00110) and the
National Science Foundation of China (No. 11374011, 11374346, 91221303). The
work at Rice University is supported by the U.S. NSF-DMR-1362219, DMR-1436006,
and in part by the Robert A. Welch Foundation Grant Nos. C-1839 (P.D.).

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