Superconductivity in a misfit layered compound (SnSe)$_{1.16}$(NbSe$_2$)
Hua Bai Xiaohui Yang Yi Liu Meng Zhang Mengmeng Wang Yupeng Li Jiang Ma Qian Tao Yanwu Xie Guanghan Cao Zhu-An Xu






^1Department of Physics, Zhejiang University, Hangzhou 310027, P. R. China

^2State Key Laboratory of Silicon Materials, Zhejiang University, Hangzhou 310027, P. R. China

^3Zhejiang California International NanoSystems Institute, Zhejiang University, Hangzhou 310058, P. R. China

^4Collaborative Innovation Centre of Advanced Microstructures, Nanjing University, Nanjing 210093, P. R. China

zhuan@zju.edu.cn



The large size single crystals of (SnSe)_1.16(NbSe_2) misfit
layered compound were grown and superconductivity with T_c of
3.4 K was first discovered in this system. Powder X-ray
diffraction (XRD) and high resolution transmission electron
microscopy (HRTEM) clearly display the misfit feature between SnSe
and NbSe_2 subsystems. The Sommerfeld coefficient γ
inferred from specific-heat measurements is 16.73 mJ mol^-1
K^-2, slightly larger than the usual misfit compounds. The
normalized specific heat jump ΔC_e/γT_ c is
about 0.98, and the electron-phonon coupling constant
λ_e-ph is estimated to be 0.80. The estimated value of
the in-plane upper critical magnetic field, H_c2^ab(0), is
about 7.82 T, exceeding the Pauli paramagnetic limit slightly.
Both the specific-heat and H_c2 data suggest that
(SnSe)_1.16(NbSe_2) is a multi-band superconductor.



Superconductivity in a misfit layered compound (SnSe)_1.16(NbSe_2)
    Hua Bai^1, Xiaohui Yang^1, Yi Liu^1, Meng Zhang^1, Mengmeng Wang^1, Yupeng Li^1, Jiang Ma^1, Qian Tao^1, Yanwu Xie^1, Guanghan Cao^1,^3,^4  Zhu-An Xu^1,^2,^3,^4[Corresponding
author. ]
    December 30, 2023
============================================================================================================================================================================================









§ INTRODUCTION

The misfit layered compounds (MLCs) has been more attractive due to their unique layered structure recently.
The formula of MLCs is generally described as (MX)_1+y(TX_2)_n (y = 0.08-0.28, n = 1, 2, 3),
where M = Sn, Pb, Sb, Bi or a lanthanide; X = S, Se or Te; T = Ti, V, Cr, Nb or Ta.
Rocksalt structure monochalcogenides layer MX and several layers of transition metal
dichalcogenides (TMDCs) TX_2 are connected by van der Waals forces, forming natural heterojunctions.
The MX and TX_2 structural components have different  rotation symmetries - they have the same
periodicity along one in-plane direction but different periodicity along the other
in-plane direction. Therefore, when they are stacked together, incommensurate factor 1+y  appears<cit.>.
In the MLCs family, the members of Nb, Ta and Ti dichalcogenides with rocksalt Sn, Pb, Bi and La monochalcogenide layers often exhibit superconductivity<cit.>, with a T_ c ranging from 0.4 to 5.3 K. Most MLC superconductors are weak-coupling
BCS-type superconductors, and usually have low values of upper critical field (H_c2) in both inter-plane and in-plane directions<cit.>.

Similar to MLCs, the ferecrystals are another type layered compounds consisting of same MX and TX_2 layers.
The only difference is that the ferecrystals have extensive random  rotationally disorder between layers but MLCs do
not<cit.>.
Therefore the ferecrystals are not in the long-range crystalline state.
Most MLCs were prepared through traditional high-temperature synthetic techniques,
while most ferecrystals were grown using physical vapor deposition (PVD) or chemical vapor deposition (CVD) techniques.
Some ferecrystals also show superconductivity, such as the ferecrystals of [(SnSe)_1.16]_m(NbSe_2)<cit.>.
The T_ c value is 1.9 K for m = 1, and it decreases with increasing m.

Previous study on the powder sample of MLC (SnSe)_1.16(NbSe_2) did not observe superconductivity as temperature down to 4.2 K <cit.> and thus no systematic investigation on its properties has been reported.
Here we report the successful crystal growth of large size single crystals of (SnSe)_1.16(NbSe_2) and discovery of superconductivity with onset transition temperature of 3.4 K. The feature of the misfit structure is revealed by the powder X-ray diffraction (XRD) and high resolution transmission electron microscopy (HRTEM) measurements. The specific-heat and H_c2 data  can be fit within a two-band superconducting model.
The value of H_c2^ab(0) exceeds the Pauli paramagnetic limit slightly which may result from the multi-band nature.








§ EXPERIMENT DETAILS


The single crystals of (SnSe)_1.16(NbSe_2) were made via
vapor transport technique using chlorine as the transport agent.
First, Sn(99.999%), Nb(99.99%) and Se(99.999%) powders with the
total mass of 1.5 g were mixed with the ratio of 1.16:1:3.16 and
ground adequately, sealed into an evacuated quartz ampoule, heated
to 900 ^∘C and kept for 3 days. Subsequently, the mixture
was reground with 0.15 g powders of SnCl_2(99.999%), and sealed
into an evacuated quartz ampoule with a length of 16 cm. The
ampoule was heated for 7 days in a two-zone furnace, where the
temperature of source zone and growth zone were fixed at 900
^∘C and 850 ^∘C respectively. Finally, large
plate-like single crystals of (SnSe)_1.16(NbSe_2) with a
size of several millimeters in the plane direction were obtained,
and their thickness is about 0.1-0.2 mm.

The single crystal XRD and powder XRD data were collected by
a PANalytical X-ray diffractometer (Empyrean) with a Cu K_α
radiation and a graphite monochromator at room temperature. The
chemical compositions were determined by energy-dispersive X-ray
spectroscopy (EDX) with a GENESIS4000 EDAX spectrometer.
The HRTEM images were taken at room temperature with an aberration corrected
FEI-Titan G2 80-200 ChemiSTEM. The DC magnetization was measured on a
magnetic property measurement system (MPMS-XL5, Quantum Design),
The specific-heat capacity was measured on a physical properties
measurement system (PPMS-9, Quantum Design), using a relaxation technique.
The electrical properties were measured on an Oxford Instruments-15T
cryostat with a He-3 probe option.





§ RESULTS AND DISCUSSION




Figure <ref> (a) shows the EDX pattern of a
(SnSe)_1.16(NbSe_2) single crystal. According to the EDX
result, the ratio of SnSe and NbSe_2 is about 1.15(1):1, very
close to the nominal ratio. In addition, a tiny part of Se (about
2.3%) are substituted by Cl, which is unavoidable due to the
using of chlorine transport agent. We also tried to use iodine as
a transport agent but failed to obtain required crystals. Figure
<ref> (b) shows the room temperature XRD pattern of the
single crystal, where all the reflections are (00l) peaks. The
inset of Figure <ref> (b) shows a typical as-grown single
crystal. The soft and plate-like single crystals are difficult to
grind to powder. We managed to mince them to powder adequately by
blade and obtained the powder XRD data finally, which are showed
in Figure <ref> (c). The XRD patterns are almost the same as
reported in the literature<cit.>, which can be well
indexed by the SnSe and NbSe_2 subsystems. The detailed indices
of the peaks are labelled in Fig.1(c). We calculated the lattice
parameters of two subsystems subsystems respectively by using the
UnitCell program<cit.>. The lattice parameters of
two subsystems are shown in Table 1.



Table 1  The lattice parameters of (SnSe)_1.16(NbSe_2).



subsystem    a (Å)    b (Å)    c (Å)    space group

SnSe    5.940    5.968    12.340    Cm2a

NbSe_2    3.444    5.968    24.680    Fm2m




The structural parameters are very close to
the reported values in the literature<cit.>. The
incommensurate factor can be calculated by the formula: 1+y =
2a_2/a_1 ≈ 1.16, where a_2 and a_1 are lattice
parameters of a-axis for NbSe_2 and SnSe subsystems
respectively.









Figure <ref> (a) shows the HRTEM image of a
(SnSe)_1.16(NbSe_2) single crystal from zone axes [010]. The
HRTEM image clearly displays regular stacking of SnSe and NbSe_2
layers. Furthermore, the different periodicity of two subsystems
along the b-axis is revealed. The structure model of
(SnSe)_1.16(NbSe_2) is displayed in Figures <ref> (b)
and (c) in two directions. The SnSe layer is a rocksalt like
bi-layer structure and one unit cell contains one bi-layer in the
SnSe subsystem. While the NbSe_2 layer is a trigonal sandwich
structure, the same as the layer in 2Ha-NbSe_2. The stack mode
for the NbSe_2 layers along the c-axis is shown in figure
<ref> (d), one unit cell contains two layers, the green balls
represent Nb atoms and the cyan balls represent Se atoms in the
second layer, and the first layer has a offset of a/2 along the
a-axis to the second layer, constituting an orthorhombic
pattern. This special stack mode for TMDCs only exists in MLCs.




Figure <ref> (a) displays the temperature dependence of
out-of-plane and in-plane electrical resistivity for a
(SnSe)_1.16(NbSe_2) single crystal. In both directions,
normal state of (SnSe)_1.16(NbSe_2) shows a metallic
behavior. The inset shows the details of superconducting
transition. The onset superconducting transition temperature
T_ c (90% of normal state resistivity) is about 3.4 K for
both directions, with a very narrow transition width of about 0.02
K. The anisotropy of resistivity ρ_c/ρ_ab is about 128
at 300 K, and exceeds 100 over the whole temperature range. Such a
large anisotropy of resistivity should be related to the highly
anisotropic structure. Figure <ref> (b) shows the temperature
dependence of dc magnetic susceptibility under magnetic field of
10 Oe applied in the ab-plane, where both zero-field cooling
(ZFC) and field cooling (FC) were measured. The T_ c
determined from magnetic susceptibility is 3.38 K, almost the same
as T_ c^ zero in resistivity. The estimated
superconducting shielding volume fraction is very close to 100%
at 2 K, indicating completely bulk superconductivity in
(SnSe)_1.16(NbSe_2). The demagnetization factor is not
considered due to its puny influence in the ab-plane  direction
for a plate-like sample.



Figure <ref> (c) shows the low temperature specific-heat data
plotted as C/T versus T^2. The specific-heat of normal state above T_ c contains
two parts, which can be described by the equation C/T = γ+βT^2, where
 γ is the Sommerfeld coefficient for electronic specific-heat and
 β is the lattice specific-heat coefficient.  From this linear fit,
 we can obtained that γ = 16.73 mJ mol^-1 K^-2 and β = 4.79 mJ mol^-1 K^-4.
 Furthermore, the Debye temperature Θ_D can be estimated by the formula
 Θ_D = (12π^4nR/5β)^1/3 = 129.3 K, where R is the gas constant
 and n is the number of atoms per formula unit (n = 5.32).
 In addition, the electron-phonon coupling constant λ_e-ph
 can be estimated by the McMillan equation<cit.>:
 
    λ_e-ph=μ^* ln(Θ_D/1.45Tc)+1.04/ ln(Θ_D/1.45Tc)(1-0.62μ^*)-1.04

where μ^* is Coulomb pseudopotential which is often set to 0.15 as the empirical value.
The estimated λ_e-ph = 0.80, suggesting that (SnSe)_1.16(NbSe_2) is in an intermediate coupling range.
The λ_e-ph value is also similar to other MLC superconductors<cit.>.
An obvious specific-heat jump appears around  T_ c,
which is also a characteristic for superconducting transition.
Figure <ref> (d) shows the normalized electronic specific-heat
C_e/γT, as a function of reduced temperature t = T/T_ c.
The C_e data cannot be fitted with the one-gap BCS model: C_e = C_0exp(-Δ/k_BT).
 Therefore, we adopted a two-gap BCS model which is often applied to multi-band superconductors<cit.>
. In this model, C_e is the sum of contributions
from two bands where each band follows one-gap BCS model. By this fit, we obtained that 2Δ_1 = 4.52 k_BT_ c
while 2Δ_2 = 1.4 k_BT_ c, and the weight γ_1/γ_2 = 60%:40%.
Therefore, one gap exceed the BCS value of 3.53 k_BT_ c, while the other gap is smaller than it.
2H-NbSe_2 is also a multi-band superconductor. In 2H-NbSe_2,
2Δ_1 = 4.5 k_BT_ c, 2Δ_2 = 2.6 k_BT_ c,
and the weight γ_1/γ_2 = 80%:20%<cit.>.
We can regard (SnSe)_1.16(NbSe_2) as an intercalated NbSe_2.
Therefore,  it is reasonable that (SnSe)_1.16(NbSe_2) and 2H-NbSe_2 have the similar Δ_1,
but the former has a smaller Δ_2. The weights for two gaps are also different.
MLCs always have charge transfer from MX layers to TX_2 layers<cit.>.
We speculate that the charge transfer changes the Fermi surface of NbSe_2
in (SnSe)_1.16(NbSe_2), leading to the differences of gaps between the two compounds.
Moreover, the normalized specific heat jump ΔC_e/γT_ c is estimated
to be 0.98, close to the value in (SnS)_1.15(TaS_2)(0.812)<cit.> and
(SnSe)_1.18(TiSe_2)_2 (0.88)<cit.>, but lower than
(Pb_0.8Sn_0.2Se)_1.16(TiSe_2)_2 (1.38)<cit.>.


Figures <ref> (a) and (b) show the temperature dependence of
in-plane electrical resistivity around T_ c under different magnetic fields of a
(SnSe)_1.16(NbSe_2) single crystal with field applied parallel to the
c-axis and in the ab-plane respectively. For H//c, the superconducting
transition width increases significantly with increasing fields, while for H//ab,
the resistivity curves shift parallel down towards the low temperature with increasing fields.




We summarize the temperature dependent H_c2 determined by
T_ c^ mid (50% of normal state resistivity) in Figure
<ref>. For both directions, the H_c2 curves exhibit a
positive curvature close to T_ c. The similar phenomenon
were also observed in other multi-band
superconductors<cit.>.
Therefore we adopt a two-band model for
H_c2<cit.>:




    a_0[ lnt+U(h/t)][ lnt+U(η h/t)] + 
    
    a_1[ lnt+U(h/t)] + a_2[ lnt+U(η h/t)] = 0



where t=T/T_ c is the reduced temperature.
a_0 = 2(λ_11λ_22-λ_12λ_21),
a_1 = 1+(λ_11-λ_22)/λ_0,
a_2 = 1-(λ_11-λ_22)/λ_0,
λ_0 = [(λ_11-λ_22)^2+4λ_12λ_21]^1/2,
h = H_c2D_1/2ϕ_0T, η = D_1/D_2,
U(x) = Ψ(x+1/2)-Ψ(x), the Ψ(x) is digamma function.
λ_11 and λ_22 are the intraband BCS
coupling constants, where λ_12 and λ_21 are the interband BCS coupling constants,
and D_1 and D_2 are the in-plane diffusivity of each band. The fit parameters are listed in Table 2.
The fitting parameters reveal the differences between two bands.
For the first band, the intraband coupling constant is greater than 1,
which is in a strong-coupling range. Furthermore,
the intraband coupling is much stronger than the interband coupling.
While in the second band, the intraband coupling constant is still in
a weak-coupling range, and is similar to interband coupling.
This fitting results are consistent with the two-gap fit in specific-heat,
proving the multi-band superconductivity in (SnSe)_1.16(NbSe_2) qualitatively.
In addition, this fit yields μ_0H_c2^c(0) = 4.53 T and
μ_0H_c2^ab(0) = 7.82 T. The Pauli paramagnetic limit (H_P)
for the upper critical field is μ_0H_P = 1.84 T_ c =
6.26 T. Therefore the value of μ_0H_c2^ab(0) exceed H_P
slightly. As mentioned above, most MLC superconductors have low values of
H_c2 in both directions which are below H_P, hence the values of H_c2 in
(SnSe)_1.16(NbSe_2) are unusually large. We suggest that the multi-band effects may enhance H_c2<cit.>.
The anisotropy factor γ = H_c2^ab(0)/H_c2^c(0) ≈ 1.73,
which is within the range of others MLC superconductors (1.4 - 5.29)
<cit.>.



Table 2  The fit parameters of two-band model.



direction    λ_11    λ_22    λ_12    λ_21    η    D_1

c    1.09    0.44    0.17    0.65    20.4    0.31

ab    1.18    0.60    0.08    0.62    20.4    0.27








§ CONCLUSION

In summary, we have successfully grown large single crystals of
MLC (SnSe)_1.16(NbSe_2) and discovered superconductivity
with  T_ c of 3.4 K. The details of the misfit structure
were revealed by the powder XRD and HRTEM measurements.
Superconducting shielding volume fraction is close to 100%,
confirming the bulk superconductivity. The Sommerfeld coefficient
γ is 16.73 mJ mol^-1 K^-2, and
 H_c2(0) is estimated to be 4.53 T and 7.82 T for the out-plane and in-plane directions respectively. Especially for
the in-plane direction, the value of H_c2(0) exceeding the
Pauli limit H_P slightly. The electron-phonon coupling constant
λ_e-ph = 0.80. Both the specific-heat and H_c2
data suggest that (SnSe)_1.16(NbSe_2) should be a multi-band
BCS superconductor.






§ ACKNOWLEDGMENTS

 This work was supported by the National Basic Research Program of China
(Grant Nos. 2014CB92103) and the National Key R&D Projects of
China (Grant No. 2016FYA0300402 and 2016FYA0300204), the National Science Foundation
of China (Grant No. 11774305), and the Fundamental
Research Funds for the Central Universities of China.




§ REFERENCES


10
url<#>1#1urlprefixURL 



Wiegers1996Misfit
Wiegers G A 1996 Prog. Solid State Chem. 24 1

Reefman1990Superconductivity
Reefman D, Baak J, Brom H B and Wiegers G A 1990 Solid State Commun.
  75 47

Hamersm1992Anomalies
Hamersm M A, Muller C J, Reefman D and van Ruitenbeek J M 1992 Physica C:
  Superconductivity 197 106

Oosawa1992Three
Oosawa Y, Gotoh Y, Akimoto J, Tsunoda T, Sohma M and Onoda M 1992 Jpn. J.
  Appl. Phys. 31 L1096

Roesky1993ChemInform
Roesky R, Meerschaut A, Rouxel J and Chen J 1993 Z. Anorg. Allg. Chem.
  619 117

Rouxel1995Chalcogenide
Rouxel J, Meerschaut A and Wiegers G A 1995 J. Alloys Compd. 229
  144

Nader1997Superconductivity
Nader A, Briggs A, Meerschaut A and Lafond A 1997 Solid State Commun.
  102 401

Nader1997Superconductivity2
Nader A, Briggs A and Gotoh Y 1997 Solid State Commun. 101 149

Nader1998Structural
Nader A, Lafond A, Briggs A, Meerschaut A and Roesky R 1998 Synth. Met.
  97 147

Gressier2006Misfit
Gressier P, Rabu P, Meerschaut A, Guemas L and Rouxel J 1991 Phase
  Transitions 30 39

Giang2010Superconductivity
Giang N, Xu Q, Hor Y S, Williams A J, Dutton S E, Zandbergen H W and Cava R J
  2010 Phys. Rev. B 82 024503

Luo2016Superconductivity
Luo H X, Yan K, Pletikosic I, Xie W W, Phelan B F, Valla T and Cava R J 2016
  J. Phys. Soc. Jpn. 85 064705

Song2016Superconducting
Song Y J, Min J K, Wan G J, Kim B J and Rhyee J S 2016 Phys. Status Solidi
  B 253 1517

Sankar2018Superconductivity
Sankar R, Peramaiyan G, Muthuselvam I P, Wen C Y, Xu X F and Chou F C 2018 Chem. Mater. 30 1373

Reefman1991Superconductivity
Reefman D, Koorevaar P, Brom H B and Wiegers G A 1991 Synth. Met. 43 3775

Nader1996Critical
Nader A, Briggs A, Monceau P, Meerschaut A and Lafond A 1996 Czech. J.
  Phys. 46 639

Nader1998Critical
Nader A, Lafond A, Briggs A and Meerschaut A 1998 Phys. Scr. 57
  310

Atkins2013Synthesis
Atkins R, Wilson J, Zschack P, Grosse C, Neumann W and Johnson D C 2012 Chem. Mater. 24 4594

Beekman2014Ferecrystals
Beekman M, Heideman C L and Johnson D C 2014 Semicond. Sci. Technol.
  29 064012

Westover2014Synthesis
Westover R D, Atkins R A, Ditto J J and Johnson D C 2014 Chem. Mater.
  26 3443

Alemayehu2015Structural
Alemayehu M B, Falmbigl M, Grosse C, Ta K, Fischer S F and Johnson D C 2015
  J. Alloys Compd. 619 861

Grosse2017Superconducting
Grosse C, Alemayehu M B, Mogilatenko A, Chiatti O, Johnson D C and Fischer S F
  2017 Cryst. Res. Technol. 52 1700126

0953-2048-31-6-065006
Trahms M, Grosse C, Alemayehu M B, Hite O K, Chiatti O, Mogilatenko A, Johnson
  D C and Fischer S F 2018 Supercond. Sci. Technol. 31 065006

Wiegers1991The
Wiegers G A and Zhou W Y 1991 Mater. Res. Bull. 26 879

Holland1997Unit
Holland T J B and Redfern S A T 1997 Mineral. Mag. 61 65

Mcmillan1968Transition
Mcmillan W L 1968 Phys. Rev. 167 331

Wang2013Multiband
Wang H D, Dong C H, Mao Q H, Khan R, Zhou X, Li C X, Chen B, Yang J H, Su Q P
  and Fang M H 2013 Phys. Rev. Lett. 111 207001

Bhoi2016Interplay
Bhoi D, Khim S, Nam W, Lee B S, Kim C, Jeon B G, Min B H, Park S and Kim K H
  2016 Sci. Rep. 6 24068

Bouquet2001Phenomenological
Bouquet F, Wang Y, Fisher R A, Hinks D G, Jorgensen J D, Junod A and Phillips
  N E 2001 Europhys. Lett. 56 856

Yang2012Comparative
Yang H D, Huang C L, Lin J Y, Sun C P, Lee T K and Lee S I 2006 Phys. Rev.
  B 73 012502

Huang2007Experimental
Huang C L, Lin J Y, Chang Y T, Sun C P, Shen H Y, Chou C C, Berger H, Lee T K
  and Yang H D 2007 Phys. Rev. B 76 212504

Yao2018Charge
Yao Q, Shen D W, Wen C, Hua C Q, Zhang L Q, Wang N Z, Niu X H, Chen Q Y, Dudin
  P and Lu Y H 2018 Phys. Rev. Lett. 120 106401

Shulga1998Upper
Shulga S V, Drechsler S L, Fuchs G and Müller K H 1998 Phys. Rev.
  Lett. 80 1730

S2001Magnetoresistivity
Bud'ko S L, Petrovic C, Lapertot G, Cunningham C E, Canfield P C, Jung M H and
  Lacerda A H 2001 Phys. Rev. B 63 220503

Hunte2008Two
Hunte F, Jaroszynski J, Gurevich A, Larbalestier D C, Jin R, Sefat A S, Mcguire
  M A, Sales B C, Christen D K and Mandrus D 2008 Nature 453 903

Gurevich2003Enhancement
Gurevich A 2003 Phys. Rev. B 67 184515







