Bulk superconducting phase with a full energy gap in the doped topological insulator Cu_xBi_2Se_3

Abstract

The superconductivity recently found in the doped topological insulator
Cu_xBi_2Se_3 offers a great opportunity to search for a topological
superconductor. We have successfully prepared a single-crystal sample with a
large shielding fraction and measured the specific-heat anomaly associated with
the superconductivity. The temperature dependence of the specific heat suggests
a fully-gapped, strong-coupling superconducting state, but the BCS theory is
not in full agreement with the data, which hints at a possible unconventional
pairing in Cu_xBi_2Se_3. Also, the evaluated effective mass of 2.6m_e (m_e is
the free electron mass) points to a large mass enhancement in this material.

Full text

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arXiv:1103.4750v1 [cond-mat.supr-con] 24 Mar 2011
Bulk superconducting phase with a full energy gap in the doped topological insulator
CuxBi2Se3
M. Kriener, Kouji Segawa, Zhi Ren, Satoshi Sasaki, and Yoichi Ando
Institute of Scientific and Industrial Research, Osaka University, Osaka 567-0047, Japan
(Dated: March 25, 2011)
The superconductivity recently found in the doped topological insulator CuxBi2Se3 offers a great
opportunity to search for a topological superconductor. We have successfully prepared a single-
crystal sample with a large shielding fraction and measured the specific-heat anomaly associated
with the superconductivity. The temperature dependence of the specific heat suggests a fully-
gapped, strong-coupling superconducting state, but the BCS theory is not in full agreement with
the data, which hints at a possible unconventional pairing in CuxBi2Se3. Also, the evaluated effective
mass of 2.6me (me is the free electron mass) points to a large mass enhancement in this material.
PACS numbers: 74.25.Bt; 74.70.Ad; 74.62.Dh; 74.25.Op
In the past two years, the three-dimensional (3D) topo-
logical insulator (TI) is attracting a lot of interest as a
new state of matter [1–3]. It is characterized by the ex-
istence of a gapless surface state that emerges because
of the non-trivial Z2 topology of the insulating bulk
state and is protected against backscattering by time-
reversal symmetry. The discovery of the 3D TI stimu-
lated the search for a superconducting (SC) analogue,
a time-reversal-invariant topological superconductor [4–
9], which is characterized by a fully-gapped, odd-parity
pairing state that leads to the emergence of gapless Ma-
jorana surface states. Such a SC phase has implications
on topological quantum computing [4, 10–12] because of
the non-Abelian Majorana bound state expected to ap-
pear in the vortex core. However, a concrete example of
such a topological superconductor is currently unknown.
In this context, the superconductivity recently found
[15] in CuxBi2Se3is very interesting. Bi2Se3is a “second-
generation” TI that has a relatively large (∼0.3 eV) band
gap and a simple surface-state structure [13, 14]. Surpris-
ingly, when Cu is introduced to this system with the nom-
inal formula CuxBi2Se3, superconductivity with a maxi-
mum transition temperature Tcof 3.8 K was observed for
the doping range 0.10 ≤ x ≤ 0.30, even though the bulk
carrier density n was only ∼ 1020cm−3[15]. Note that
this Tcis uncharacteristically high for such a low n [16].
Furthermore, the topological surface state was found to
be well-separated from the doped bulk conduction band
in CuxBi2Se3[17]. So far, apparent SC shielding fractions
of only up to 20% have been achieved and the resistivity
always remained finite [15, 17], leaving some doubt about
the bulk nature of the superconductivity in this system.
Nonetheless, if this superconductivity is indeed a bulk
property of carrier-doped Bi2Se3, it has a profound im-
plication on the search of topological superconductors,
because (i) it is a potential candidate [18] to realize a 3D
topological superconductor, and (ii) if its bulk turns out
to be an ordinary s-wave superconductor, the topological
surface state may turn into a 2D topological supercon-
ductor as a result of a SC proximity effect [4]. Therefore,
it is important to confirm whether the superconductivity
is really occurring in the bulk of CuxBi2Se3and, if so, to
elucidate the fundamental nature of its SC state.
Bi2Se3 has a layered crystal structure (R¯3m, space
group 166) consisting of stacked Se-Bi-Se-Bi-Se quintu-
ples that are only weakly van-der-Waals bonded to each
other. We call the rhombohedral [111] direction the c
axis and the (111) plane the ab plane. When Cu is intro-
duced into Bi2Se3, it may either intercalate as Cu1+into
the van-der-Waals gaps and act as a donor, or replace Bi
as a substitutional impurity and act as an acceptor [19];
hence, Cu is an ambipolar dopant [20, 21]. The nominal
formula of CuxBi2Se3 suggests that most Cu atoms in
this SC material occupy the intercalation sites; however,
the reported carrier density of ∼1020cm−3[15] corre-
sponds to only ∼1% of electron doping, which is much
smaller than that expected from the x value. This dis-
crepancy suggests either that most of the intercalated
Cu ions remain inactive as donors, or that substitution
of Bi with Cu also occurs in this material and it almost
compensates the electrons doped by the intercalated Cu.
Partly related to such an uncontrollability of the Cu
atoms in CuxBi2Se3, the quality of the SC samples has
been poor as mentioned above, and improvements in the
sample quality are indispensable for a solid understand-
ing of the SC state in this material.
In this Letter, we report a comprehensive study of the
basic SC properties of a Cu-intercalated Bi2Se3 single
crystal by means of resistivity, magnetization, and spe-
cific heat measurements. For the first time in this mate-
rial, we observed zero-resistivity and a specific-heat jump
at the SC transition. The apparent shielding fraction of
our sample exceeds 40%, and the specific-heat data con-
firms the bulk nature of the superconductivity.
importantly, the temperature dependence of the specific
heat suggests a fully-gapped, strong-coupling SC state,
but the data do not fully agree with the strong-coupling
BCS calculation. This suggests that the pairing symme-
try may not be simple isotropic s-wave. Furthermore, the
effective mass is found to be 2.6me(meis the free electron
Most

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mass), suggesting a change in the bulk band curvature.
Single crystals of Bi2Se3 were grown by melting stoi-
chiometric amounts of elemental shots of Bi (99.9999%)
and Se (99.999%) in sealed evacuated quartz glass tubes
at 800◦C for 48 h, followed by a slow cooling to 550◦C
over 48 h and keeping at that temperature for 24 h. The
crystals were cleaved and cut into rectangular pieces, and
then the Cu intercalation was done by an electrochemi-
cal technique under inert atmosphere inside a glove box,
using CuI reagent in CH3CN solvent. The sample was
wound with a Cu wire, and a Cu stick was used as counter
and reference electrode. The current was fixed at typi-
cally 10 µA. The concentration of intercalated Cu was de-
termined from the weight change before and after the in-
tercalation process, and the sample was briefly annealed
afterward. The particular sample used in the present
study was 3.9×1.6 mm2in the ab plane with a thick-
ness of 0.40 mm, and its Cu concentration was x = 0.29.
In fact, by employing electrochemical intercalation, we
found that samples with x of up to ∼0.5 become super-
conducting; the precise phase diagram is currently under
investigation.
The magnetization data were measured with a
commercial SQUID magnetometer (Quantum Design,
MPMS). The resistivity ρxxand the Hall resistivity ρyx
were measured by a standard six-probe technique, where
the electrical current was applied in the ab plane. The
specific heat cp data were taken by a relaxation-time
method using a commercial system (Quantum Design,
PPMS); the addenda signal was measured before mount-
ing the sample and was duly subtracted from the mea-
sured signal. The cpmeasurement was done in zero-field
(for the SC state) and in 2 T applied along the c axis (for
the normal state), and the change of the addenda signal
between the two were found to be negligible.
Figure 1 shows the results of the transport measure-
ments. In zero field, the onset of the SC transition oc-
curs at 3.6 K and zero-resistivity was observed at 2.8 K
[Fig. 1(a)]. From the magnetic-field (B) dependence of
ρxx[Figs. 1(c) and (d)], the upper critical field Bc2(T)
defined as the midpoint of the resistivity transition [22]
is obtained for the two principal field directions [Fig.
1(e)], and the extrapolation to 0 K using the Werthamer-
Helfand-Hohenberg (WHH) theory gives Bc2,⊥(0) and
Bc2,?(0) (perpendicular and parallel to the ab plane) of
1.71 and 3.02 T, respectively [Fig. 1(e)]. As shown in the
inset of Fig. 1(b), ρyxis completely linear in B, suggest-
ing the dominance of only one type of bulk carriers. The
Hall coefficient RHwas found to be only weakly tempera-
ture dependent [Fig. 1(b)] and gives the electron density
n = 1.3×1020cm−3.
Figure 2 summarizes the magnetization M measure-
ments. To minimize the effect of the demagnetization
factor, those measurements were made for B ? ab [see
the sketch in Fig. 2(a)]. The temperature dependence
of the diamagnetic shielding fraction for the zero-field-
T (K)
0100
–50
–40
–30
–20
–10
0
RH (10−3 cm3/C)
024
0
100
200
ρxx (µΩcm)
B (T)
024
B (T)
–202
–10
0
10
B (T)
ρyx (µΩcm)
T = 5 K
(b)
0100 200300
0
100
200
300
400
T (K)
ρxx (µΩcm)
0123
0
1
2
3
T (K)
Bc2 (T)
23
T (K)
45
0
100
200
ρ (µΩcm)
B || ab
B ⊥ ab
(a)
(e)
B || ab
1.86 K
2.36 K
T = 2.92 K
B ⊥ ab
1.82 K
2.34 K
T = 2.91 K
(c)(d)
FIG. 1. (color online) (a) ρxx(T) data of the Cu0.29Bi2Se3
sample. (b) Temperature dependence of RH; inset shows the
ρyx(B) data at 5 K. (c,d) ρxx(B) data for B ? ab and B ⊥ ab,
respectively. (e) Bc2 vs. T phase diagram determined from
the midpoint in ρxx(B) at various temperatures; dashed lines
show the WHH behavior. The midpoint definition for Bc2
gives Tc = 3.2 K consistent with the M(T) data.
0.00.5
B (mT)
1.0
1.8 K
2.6 K
2.8 K
3.0 K
(d)
Bc1,||
∆M
012
T (K)
34
–40
–20
0
shield. frac. (%)
FC
ZFC
BDC || ab
(a)
0123
0.0
0.2
0.4
T (K)
(e)
Bc1 (mT)
M (emu/mol)
B (mT)
–100 –500 50100
–10
0
10(b)
T = 1.8 K
T = 2.4 K
0.00.5
B (mT)
1.0
–15
–10
–5
0
M (emu/mol)
T = 1.8 K
2.6 K
2.8 K
3.0 K
(c)
Bc1,||
FIG. 2. (color online) (a) Temperature dependence of the ap-
parent shielding fraction of Cu0.29Bi2Se3measured in B = 0.2
mT ? ab. (b) M(B) curves at 1.8 and 2.4 K after subtracting
the diamagnetic background. (c) Initial M(B) behavior af-
ter ZFC to various temperatures. Arrows mark the position
of Bc1,?. Note the very small magnetic-field scale. (d) Plots
of ∆M ≡ M − aB, where a is the initial slope, and the de-
termination of Bc1,?shown by arrows. (e) Bc1,?vs. T phase
diagram; the solid line is a fit within the local dirty limit.

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3
01234
0
1
2
3
T (K)
cel/T (mJ / molK2)
(b)
∆0/Tc = 1.9
∆0/Tc = 2.3
γn
γs
γres
01234
0
10
20
30
40
50
T (K)
cp/T (mJ / molK2)
B = 0 T
B = 2 T
Debye Fit
(a)
BDC ⊥ ab
FIG. 3. (color online) (a) cp(T)/T data measured in 0 and 2
T applied along the c axis; the latter represent the normal-
state behavior. The dashed line is a fit to the 2-T data using
the standard Debye formula. (b) Electronic term cel/T in 0
T obtained after subtracting the phonon term determined in
2 T. The dotted line shows the calculated cel/T curve given
by strong-coupling BCS theory with α = 1.9; the dashed line
is the BCS curve for α = 2.3, which is obtained from Bc, N0,
and Tc. The horizontal dash-dotted line denotes the value of
γn, and its breakdown to γs and γres is indicated.
cooled (ZFC) and field-cooled (FC) measurements are
shown in Fig. 2(a), where the onset of the Meissner signal
occurs at Tc= 3.2 K and the apparent shielding fraction
reaches 43% at 1.8 K [23]. Note that this Meissner Tc
corresponds to the midpoint of the resistivity transition.
Neither ZFC nor FC data saturate at 1.8 K.
Magnetization M(B) curves are shown in Figs. 2(b)
and (c); each data set was obtained after cooling to its
respective temperature from above Tcin zero field, and
the background diamagnetism, which can be easily de-
termined at B > Bc2,?, is subtracted from the data. As
already noted by Hor et al. [15], the lower critical field
Bc1is very small: Using the deviation of the M(B) curve
from its initial linear behavior as a measure of Bc1,?[Fig.
2(d)], we obtained the Bc1,?(T) data shown in Fig. 2(e).
To determine the 0-K limit, we used Bc1 ∝ 1/λ2
[(∆(T)/∆(0))tanh(∆(T)/2kBT)] for the local dirty limit
[25] to fit the extracted data points (λeff is the effective
penetration depth and ∆ is the SC gap [26]), and ob-
tained Bapp
c1,?(0) = 0.43 mT. For the quantitative analysis
discussed later, this apparent value was corrected for the
demagnetization effect, though it is small for B ? ab: Us-
ing the approximation given for the slab geometry [24],
we obtain Bc1,?(0) = Bapp
where b/a = 3.9/0.40 in our case. Note that the flux pin-
ning in the present system is weak as evidenced by the
low irreversibility field of ∼0.1 T at 1.8 K [Fig. 2(b)].
The temperature dependence of cp is shown in Fig.
3(a) as cp/T vs. T for the SC state (B = 0 T) and
the normal state achieved by applying B ⊥ ab of 2 T
(> Bc2,⊥). As shown by the dotted line in Fig. 3(a),
a conventional Debye fit to the normal-state data below
eff∝
c1,?(0)/tanh?0.36b/a = 0.45 mT,
4 K using cp = cel+ cph = γnT + A3T3+ A5T5, with
the normal-state specific-heat coefficient γnand the co-
efficients of the phononic contribution A3and A5, yields
a good description of the data. The obtained parame-
ters are γn=1.95 mJ/molK2, A3= 2.22 mJ/molK4[27],
and A5 = 0.05 mJ/molK6.
contribution from the zero-field data gives the electronic
specific heat celin the SC state plotted in Fig. 3(b), re-
vealing a clear jump around Tc. This provides compelling
evidence for bulk superconductivity in CuxBi2Se3. In
passing, we note that our cpdata in 2 T do not exhibit
any Schottky anomaly related to electron spins, suggest-
ing that there is no local moment possibly associated with
Cu2+ions.
From the above results, one can estimate various basic
parameters. Assuming a single spherical Fermi surface,
one obtains the Fermi wave number kF = (3π2n)1/3=
1.6 nm−1. The effective mass m∗is evaluated as m∗=
(3?2γn)/(Vmolk2
BkF) = 2.6me, with the molar volume of
Bi2Se3Vmol≈ 85 cm3/mol. Note that the effective mass
of pristine Bi2Se3is ∼0.2me[28], so there is an order-of-
magnitude mass enhancement in CuxBi2Se3 [29]. Since
electron correlations are weak in Bi2Se3, the origin of
this enhancement is most likely a change in the band
curvature near the Fermi level. From Bc2,⊥ = 1.71 T,
the coherence length ξab=?Φ0/(2πBc2,⊥) = 13.9 nm is
obtained, while from Bc2,?we use ξabξc= Φ0/(2πBc2,?)
and obtain ξc = 7.9 nm. Since we have the Bc1 value
only for B ? ab, we define the effective GL parameter
κab≡
?λabλc/ξabξcand use Bc1,?= Φ0lnκab/(4πλabλc)
together with Bc2,?/Bc1,?= 2κ2
κab≈ 128. We then obtain the thermodynamic critical
field Bc=?Bc1,?Bc2,?/lnκab= 16.7 mT.
To analyze cel/T in the SC state shown in Fig. 3(b),
we tried to fit the BCS-type temperature dependence to
the data. Since the simple weak-coupling BCS model
does not describe the cel/T data (not shown), we use
the modified BCS model applicable to strong-coupling
superconductors as proposed in Ref.
called “α model” with α = ∆0/Tcand ∆0is the SC gap
size at 0 K. We note that strong coupling means α >
αBCS= 1.764, and that this model still assumes a fully-
gapped isotropic s-wave pairing. Using the theoretical
curve cBCS
el
of the α model [31], we tried to reproduce the
experimental data with cel(T)/T = γres+ cBCS
Note that the parameter γresis necessary for describing
the contribution of the non-SC part of the sample [32];
also, the theoretical term cBCS
el
γn− γres) at T > Tc.
It turned out that with α = 1.9, γres= 0.6 mJ/molK2,
and γs= 1.185 mJ/molK2, the experimental data is rea-
sonably well reproduced and the entropy balance is sat-
isfied, as shown in Fig. 3(b) by the dotted line and the
dash-dotted line [33]. This result strongly suggests that
the SC state of CuxBi2Se3is fully gapped. The resulting
γn(= γres+ γs) value of 1.785 mJ/molK2slightly devi-
Subtracting the phononic
ab/lnκab [30] to obtain
[31], where it is
el
(T)/T.
/T is set to yield γs (=

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4
ates from the γn value estimated from the Debye fit to
the normal-state data in 2 T, 1.95 mJ/molK2. This slight
difference (∼9%) might be the result of a possible field
dependence of the normal-state Sommerfeld parameter,
which has to be clarified in future studies.
To gain further insight into the nature of the SC state
in CuxBi2Se3, we examine the implication of the obtained
γs value: The density of states (DOS) N0 is calculated
from γsvia N0= γs/(π2k2
B/3) = 1.51 states/eV per unit
cell, which is large for a low-carrier-density system and
is in accord with the “high” Tc. This value allows us to
calculate ∆0through the expression for the SC conden-
sation energy1
already calculated, we obtain ∆0 = 7.3 K which gives
the coupling strength α = ∆0/Tc = 2.3. This exceeds
the BCS value of 1.764 and hence CuxBi2Se3is a strong-
coupling superconductor, as was already inferred in our
analysis of the cel/T data. More importantly, the α value
of 2.3 obtained from γsis too large to explain the cel(T)
data within the strong-coupling BCS theory: As shown in
Fig. 3(b) with the dashed line, the expected BCS curve
for α = 2.3 does not agree with the data at all. This
probably means that the actual temperature dependence
of ∆ in CuxBi2Se3is different from that of the BCS the-
ory, which suggests that the pairing symmetry may not
be the simple isotropic s-wave. Obviously, a direct mea-
surement of ∆0and ∆(T) is strongly called for. On the
other hand, the low-temperature behavior of cel(T) ro-
bustly indicates the absence of nodes and points to a
fully-gapped state. It will be interesting to see if the fully-
gapped, time-reversal-invariantp-wave state proposed for
CuxBi2Se3[18] would provide a satisfactory explanation
of our data.
2N0∆2
0= (1/2µ0)B2
c. With Bc≈ 16.7 mT
In summary, we report a comprehensive study of the
superconductivity in CuxBi2Se3by means of resistivity,
magnetization, and specific-heat measurements on a sin-
gle crystal with x = 0.29 that shows, for the first time
in this material, zero-resistivity and a shielding fraction
of more than 40%. An analysis in the framework of a
generalized BCS theory leads to the conclusion that the
superconductivity in this system is fully gapped with a
possibly non-BCS character. The fully-gapped nature
qualifies this system as a candidate for a topological su-
perconductor: Since this system hosts a topological sur-
face state above Tc[17], depending on whether the parity
of the bulk SC state is even or odd, either the surface or
the bulk should realize the topological SC state associ-
ated with intriguing Majorana edge states.
We acknowledge S. Wada for technical assistance.
We thank L. Fu, S. Kuwabata, K. Miyake, and Y.
Tanaka for helpful discussions. This work was supported
by JSPS (KAKENHI 19674002 and Next-Generation
World-Leading Researchers Progaram), MEXT (Inno-
vative Area “Topological Quantum Phenomena” KAK-
ENHI 22103004), and AFOSR (AOARD 10-4103).
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[32] In a homogeneous nodal superconductor with 100% vol-
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even at T = 0. However, the sample used here consists
of superconducting and non-superconducting (metallic)
parts, and the volume fraction of the latter (∼ 60%) en-
tirely accounts for the observed residual DOS, which is
33% of the total DOS.
[33] We allowed γres and γs to be tuned independently.
Dwith the number of atoms