Electronic Band Structure and Superconducting Properties of SnAs
P. I. Bezotosnyi, K. A. Dmitrieva, A. V. Sadakov, K. S. Pervakov, A. V. Muratov, A. S. Usoltsev, A. Yu. Tsvetkov, S. Yu. Gavrilkin, N. S. Pavlov, A. A. Slobodchikov, O. Yu. Vilkov, A. G. Rybkin, I. A. Nekrasov, V. M. Pudalov
aa r X i v : . [ c ond - m a t . s up r- c on ] O c t Electronic Band Structure and Superconducting Properties of SnAs
P. I. Bezotosnyi, ∗ K. A. Dmitrieva, A. V. Sadakov, K. S. Pervakov, A. V. Muratov, A. S. Usoltsev,
1, 2
A. Yu. Tsvetkov, S. Yu. Gavrilkin, N. S. Pavlov, A. A. Slobodchikov,
3, 4
O. Yu. Vilkov, A. G. Rybkin, I. A. Nekrasov, and V. M. Pudalov
1, 2 P. N. Lebedev Physical Institute, Russian Academy of Sciences, Moscow, 119991, Russia National Research University Higher School of Economics, Moscow 101000, Russia Institute for Electrophysics, Russian Academy of Sciences, Ural Branch, Ekaterinburg, 620016, Russia Kirensky Institute of Physics, Federal Research Center KSC SB RAS, Krasnoyarsk, 660036, Russia Saint Petersburg State University, Saint Petersburg, 198504, Russia
We report comprehensive study of physical properties of the binary superconductor compoundSnAs. The electronic band structure of SnAs was investigated using both angle-resolved photoemis-sion spectroscopy (ARPES) in a wide binding energy range and density functional theory (DFT)within generalized gradient approximation (GGA). The DFT/GGA calculations were done includingspin-orbit coupling for both bulk and (111) slab crystal structures. Comparison of the DFT/GGAband dispersions with ARPES data shows that (111) slab much better describes ARPES datathan just bulk bands. Superconducting properties of SnAs were studied experimentally by specificheat, magnetic susceptibility, magnetotransport measurements and Andreev reflection spectroscopy.Temperature dependences of the superconducting gap and of the specific heat were found to be wellconsistent with those expected for the single band BCS superconductors with an isotropic s-waveorder parameter. Despite spin-orbit coupling is present in SnAs, our data shows no signatures of apotential unconventional superconductivity, and the characteristic BCS ratio 2∆ /T c = 3 . − . INTRODUCTION
Binary compounds of the SnX-family (X=Te, Se, As,S, Sb, P) attract much attention due to their unique prop-erties [1–5]. The most extensively studied SnSe [6] andSnTe [7, 8] have NaCl-structure. These compounds weretheoretically and experimentally determined as topolog-ical crystalline insulators. In this class of materialstopological properties are protected by crystal symme-tries, that differs from conventional topological insula-tors, where the time-reversal symmetry is the determin-ing factor [9]. Moreover, In-doped SnTe was found tomanifest signatures of Andreev bound states [10], whichare characteristic of unconventional superconductivity.Given this fact and the topological nature of the mate-rial, Sn − x In x Te is considered as a strong candidate fora topological superconductor [11, 12].Tin arsenide is isostructural to SnTe and SnSe. With-out doping, this compound demonstrates superconduct-ing properties, which were first reported in 1964 by Gellerand Hull [13]. For a long time there was an uncertaintyabout valence state of Sn in this compound and about itspossible influence on superconductivity. Later, Wang et.al [14] experimentally demonstrated that Sn has a singlevalence state Sn +3 . Also, from rough estimates basedon experimental data it was suggested, that SnAs is atype-I superconductor with weak-coupling. Temperaturedependences of the critical magnetic field H c and of thespecific heat in zero field are well described by the BCSmodel. Nevertheless, up to the best of our knowledge noinformation is available on the specific heat behavior innon-zero magnetic fields and on the temperature depen- dence of the superconducting gap. Furthermore, fromearlier theoretical investigation of electronic band struc-ture, electron-phonon interaction, and superconductivityperformed in [15], the calculated electron-phonon cou-pling parameter seemed to agree with the one estimatedfrom specific heat data [14], however there is a discrep-ancy between the theoretical and experimental values ofT c .Another interesting result was found in investigationsunder high-pressure. In Ref. [16] SnAs was shown toexhibit a structural phase transition from NaCl- to CsCl-structure at around 37 GPa, which confirms previouslyobtained experimental results [17]. Moreover, there isdramatic increase of T c up to 12.2 K at a critical point ofthe structural transition.On the theoretical side, in Ref. [18] electronic bandstructure of SnAs was calculated within the tight-binding approximation. Despite the experimental dataof Ref. [14] demonstrated that Sn has a single valencestate Sn +3 , this compound is considered [18] as a poten-tial candidate for the valence-skip material. The valenceskipping may lead to a negative effective Coulomb inter-action between electrons. The calculation results revealthat the Sn x state in SnAs is likely intermediate betweenvalence skipper and fixed valence compound; as a result,moderate charge fluctuations (and electron-phonon inter-action) can be responsible for superconductivity.The band dispersion of SnAs resembles that for SnTe,the known crystalline topological insulator. Moreover,due to the similarity of SnAs to SnTe, it was suggested,that the mixed compound Sn(As,Te) may manifest topo-logical superconductivity, if it is superconducting. Pre-liminary experimental studies of electronic band struc-ture of SnAs by ARPES technique [19] have shown thepresence of features that can be interpreted as a mani-festation of the spin-orbit coupling (SOC): energy bandsplitting in the vicinity of the Γ point, which might beimportant in context of possible topological nature ofthe mixed compound Sn(As, Te). The ARPES data [19]was obtained for the (111) sample surface, whereas bandstructure calculations are available for bulk dispersionsolely. This impedes comparison of the data with theory;obviously, one needs band structure calculated within thecorresponding cross-section of the Brillouin zone.Despite the existence of previous investigations, thereis still lack of knowledge of the origin of superconductiv-ity in SnAs, and the question about the impact of SOCis still opened. In the current paper we report detailedexperimental study of the normal and superconductingproperties of SnAs. For the normal state we studied byARPES technique electronic band structure in a widebinding energy interval. The measured superconductingproperties include specific heat and magnetic suscepti-bility in various magnetic felds, and the superconduct-ing gap by Andreev reflection spectroscopy. Also, wereport refined calculations of the SnAs band structurewithin DFT/GGA for bulk and (111) slab crystal struc-tures, with and without SOC, and compare them withthe ARPES spectra. SAMPLES AND EXPERIMENTAL DETAILS
Figure 1. a) Image of the SnAs crystal. b) Crystal structureof SnAs.
The SnAs samples were synthesized from pure ele-ments: Sn and As with purity of 99.99% and 99.9999%,respectively. The sealed quartz ampoule with elementstaken in a stoichiometric ratio of 1:1 was slowly heatedto 600 ◦ C, held for 48 hours to form the phase, thenheated to 800 ◦ C and held for 24 hours to homogenize themelt. The crystals were then grown by a modified Bridg-man method at a melt cooling rate of 1 ◦ C/hour downto 550 ◦ C. The grown crystals were further annealed ata temperature of 550 ◦ C for 24 hours to remove growthdefects.We studied two crystals (No. 1 and No. 2) obtained intwo different growths. Being nominally identical, the (111)
Figure 2. XRD spectrum of the SnAs (sample No. 2). Theabscissa represents the angle 2Θ in degrees, the ordinate isthe reflection intensity.
Inset : LEED pattern for SnAs crystalwith electron beam energy E = 120 eV. samples however had slightly different T c values. ForARPES measurements both SnAs samples were used, forresistivity and Andreev spectroscopy we used sample No.1, for specific heat and magnetic measurements – sampleNo. 2.The obtained SnAs crystals were cleaved into the smallpieces of about 3 mm size, which had a pyramidal shapewith triangular base (Fig. 1(a)). Structural investiga-tions of both SnAs crystals were done with Rigaku Mini-Flex 600 X-ray powder diffractometer. The measureddiffraction pattern of SnAs is shown in Fig. 2. It con-tains only reflexes related to the SnAs phase, and noreflections intrinsic to other phases. From XRD analysiswe confirmed the NaCl-type structure (Fig. 1(b)) andfound the lattice parameter a = 5 .
723 ˚A to be in agree-ment with earlier reported data [14]. The quality of thesample surface was investigated in situ by LEED (low–energy electron diffraction). A regular hexagon with zeroreflection at the center in the LEED pattern is shown inFig. 2 (inset). Since SnAs has the NaCl structure, theobtained pattern of reflections corresponds to the (111)plane. Clear reflection pattern also proves the singlephase composition at the investigated surface.Elemental composition of the sample was measured byEDS (Energy–Dispersive X-ray Spectroscopy) techniqueusing SEM; the spectrum contains only lines related tothe Sn- and As- atoms. Quantitative analysis shows Snand As content of 48.9% and 51.1%, respectively, thatcorresponds to the ratio of 0.978 : 1.022.ARPES measurements were performed using a semi-spherical VG Scienta R4000 energy analyzer with en-ergy resolution of 17 meV and angular resolution of 0 . ◦ .The ultraviolet helium lamp VUV 5k was used as a lightsource with photon energy 21 . − × − mbar. The sam-ples were cleaved along the (111) plane (see the LEEDpattern on Fig. 2, inset, as described above). Before mea-suring photoelectron spectra, the surface of the sampleswas cleaned by Ar-ion beam and subsequent heating attemperature of 150 ◦ C under ultrahigh vacuum.Resistivity, magnetic and thermal properties of SnAssample were measured using PPMS-9 (Quantum Design).Specific heat measurements of SnAs crystals were carriedout by thermal relaxation technique with He calorime-ter, in the range of temperatures 0.4 – 6 K and magneticfields 0 – 500 Oe. Low-frequency magnetic susceptibilitymeasurements of SnAs crystals were taken in the range oftemperatures 2 – 6 K and magnetic fields 0 – 200 Oe. Itis worth of noting, that magnetic susceptibility at “H=0”is actually measured at a finite low value (2–5 Oe) of theAC-magnetic field while the external DC-magnetic fieldis set to zero. Magnetization loop was measured usingVibrating Sample Magnetometer (VSM) at temperatureof 2 K. For thermal and magnetic measurements we useda sample with mass of 6.66 mg.In order to implement the Andreev reflection (AR)spectroscopy, the SnS contacts were prepared in situ us-ing the break-junction technique [20, 21]. In this tech-nique, a flat thin crystal is mounted on a flexible sub-strate and then is finely bent until it cracks at the pre-liminary notched narrowing. As a result, a cleft at thedesignated location is formed in clean cryogenic environ-ment. After cleaving, using a fine micrometric screw, weadjusted the cryogenic cleavage until a desired Andreev-or Josephson-type I − V characteristic was obtained.During the cryogenic experiment we tuned mechanicallythe cleavage, and probed, respectively, different micro-contacts in order to select a high-transparency contactwith the required type I − V characteristic. In orderto take four-contact transport measurements the sam-ple was placed on top of the thin printed circuit board(PCB) made of FR-4 textolite; the PCB, in turn, wasfixed on a flat Be-bronze springy holder, made of 0.1 mmthick foil. The sample was fixed to PCB with In-clampingcontacts, reinforced with silver paste. The springy holderwas finely bent by a tip of the micrometric screw push-ing the holder from the side opposite to the sample. Thisdesign enabled us to cleave the sample and precisely varythe distance between the cleft banks during the cryogenicexperiment.It is a common knowledge, that the dI/dV characteris-tics of high transparency superconductor - normal metal- superconductor (SnS) microcontacts exhibit a so calledsubharmonic gap structure [22], comprising characteris-tic dips at bias voltages V n = 2∆ /ne with n being thesubharmonic number. The subharmonic order n equalsto the number of Andreev reflections the carriers experi-ence when moving in the normal constriction of the SnSjunction. Therefore, the superconducting gap value ∆may be determined directly from the measured V n ver- sus 1 /n dependence.The dI/dV measurements were made in the tempera-ture range from 1.5 to 4 K. The measurements were takenby four probe technique. In order to bias the sample weused a voltage controlled current source fed with a sum ofthe DC bias and AC modulation voltages. The runningcurrent and voltage were measured independently usinga precise digitizer; dI , and dV were obtained using twolock-in amplifiers SR-830. RESULTS AND DISCUSSIONDFT/GGA calculations
Theoretical band structure for SnAs was calculatedwithin density functional theory (DFT) with the full-potential linear augmented plane-wave framework as im-plemented in WIEN2k [23] together with the generalizedgradient approximation (GGA) by Perdew, Burke andErnzerhof [24] to the exchange-correlation functional.These calculations were converged self-consistently on agrid of 1000 k -points in the irreducible Brillouin zone.Calculations were performed both, with and withoutspin-orbit coupling (SOC). The SOC was considered ina second-variational procedure [25]. Presented belowDFT/GGA results for bulk SnAs agree well with thosereported earlier in Refs. [15, 18]The DFT calculations for bulk SnAs were based onthe space symmetry group F m m (No. 225) and latticeconstant obtained in this work. Also, to consider real ex-perimental geometry of the samples we have constructed(111) slab with topmost layer of Sn (vacuum gap betweenslabs is 25 ˚A). The (111) slab crystal structure is found tohave monoclinic C /m (No. 12) space symmetry group.Calculations were performed for slabs with thickness of5 and 2 unit cells (UC). No difference between the bandstructure was found for 5UC and 2UC slabs. Correspond-ingly, all discussed below results were obtained for 2UC(111) slab. Also, we have done structural relaxation for2UC SnAs (111) slab, but have not seen any significantatomic position changes. The relaxation causes only mi-nor modification of the electronic structure and is notexploited furthermore.The DFT/GGA results for SnAs are presented on Fig.3. Panel a) shows the total, and panels b) and c) - partialdensity of states for bulk SnAs. A mixture of various elec-tronic states (Sn-5s, Sn-5p and As-4p) is observed nearFermi level with a slight predominance of As-4p states.As a result, the Fermi level lies in the recess of densityof states.SOC impact on the DOS is most pronounced for As-4pstates in the interval between -0.5 eV and -2 eV as fol-lows from the dotted line on panels c) and g). Also, SOCslightly affects the Sn-5p states. The same effect is vis-ible for dispersions along X-Γ-L directions on panels d) E n e r gy , e V total GGAtotal GGA+SO DOS, states/eV/f.u.
Sn-5s GGASn-5p GGASn-5s GGA+SOSn-5p GGA+SO
As-4s GGAAs-4p GGAAs-4s GGA+SOAs-4p GGA+SO
K X L X W L U-8-6-4-20
GGAGGA+SO
Γ Γ a) b) c) d) E n e r gy , e V total GGAtotal GGA+SO DOS, states/eV/f.u.
Sn-5s GGASn-5p GGASn-5s GGA+SOSn-5p GGA+SO
As-4s GGAAs-4p GGAAs-4s GGA+SOAs-4p GGA+SO
K K
GGAGGA+SO
M M'-8-6-4-20
GGAGGA+SO
Γ Γ e) f) g) h)
Figure 3. DFT/GGA calculated total and partial densities of states [panels (a),b), c), e), f), g)] and band dipersions for thebulk [panel d)] and for the slab [panel h)] crystal structures. Dotted lines denote the DFT/GGA calculations with SOC andsolid lines – without SOC. Zero energy corresponds to the Fermi level. and h) of Fig. 3. Other bands are practically not affectedby SOC. The strongest manifestation of SOC can be ob-served near Γ-point at about -1 eV and provides lifting ofthe bands degeneracy with the splitting by a few tens ofeV, in a qualitative agreement with earlier calculations[18, 19]. Concerning possible link between SOC and su-perconductivity in SnAs, one can see that SOC doesn’tmanifest considerably in the vicinity of the Fermi level.Therefore, one should not expect strong influence of SOCon superconductivity.Several observations based on the DFT/GGA resultsshould be mentioned regarding the superconductivity character in SnAs. Firstly, bulk DFT/GGA band struc-ture of SnAs (Fig. 3 (d)) has many bands crossing theFermi level. It supposes SnAs to be a multiband su-perconductor and, in the normal state, a good three-dimensional metal. The shape of the calculated Fermisurface presented on Fig. 4 confirms the last two state-ments. Thirdly, the total density of states at the Fermilevel is rather small – 0.81 states/eV/f.u. (f.u. - for-mula unit) which might lead to experimentally observedlow T c .From Fig. 3 (d) and Fig. 4 one can see that electronicspectra have a complicated anisotropic character. There Figure 4. SnAs DFT/GGA calculated Fermi surface. are 10 different Fermi level crossings, each has its par-ticular Fermi momentum k F and effective electron mass m ∗ . Since SnAs with a large Fermi surface is a prettygood metal, for furher analysis we employ the Fermi liq-uid type description and deduce from DFT/GGA calcu-lations several parameters important for the supercon-ducting state. Firstly, we estimated the effective massenhancement through the specific heat linear coefficient γ calc = π N A k B N ( E F )=1.89 mJ/(mol × K ), where N A is the Avogadro constant, k B – Boltzmann constant and N ( E F ) – the DFT/GGA value of total DOS at the Fermilevel. Then γ n γ calc = m ∗ m e with γ n =2.67 mJ/(mol × K ) –electronic specific heat in the normal state (see below)gives m ∗ m e =1.41. This ratio agrees well with Ref. [18] de-spite both calculated γ calc and experimental γ n exceedby a factor of ∼ k F value again using N ( E F ) – the calculated value of the total DOS at theFermi level. Then k F = N ( E F ) π ~ V m ∗ me m e = 0 . − , where V is elementary cubic cell volume. Also it may be foundas k F = (3 π n ) / , where n = N/V with N – numberof valence electrons in the unit cell. From partial DOSone can see that there are 2.1 of As-4p electrons, 0.1Sn-5s electrons and 0.72 Sn-5p electrons; hence, in total N per formula unit is 2.92. There are 4 formula unitsper cubic cell with V =187.5 ˚A . It gives k F =1.23 ˚A − .Such discrepancy between k F values comes from multi- band anisotropic nature of SnAs spectra while all defini-tions above are for the isotropic single band Fermi liquid.There is also theoretical conjecture for concentration inRef. [13] n=2.14 × cm − , which gives k F =0.86 ˚A − .Previously these values were estimated in the Ref. [14]to be m ∗ = 1 . m e and k F = 1 .
24 ˚A − respectivelywithin a straightforward single-sheet spherical Fermi sur-face approach. Here, from the DFT/GGA calculationswe see that the Fermi surface has many sheets and israther anisotropic. Although the values of m ∗ and k F obtained here on the basis of DFT/GGA calculations aresurprisingly quite similar to those of the Ref. [14].Lower panels e) to h) in Fig. 3 present DFT/GGAresults for (111) slab. One can see that the total andpartial DOS are practically the same as those for bulkSnAs. The value of total DOS at the Fermi level also isnearly the same as the bulk one. Manifestations of SOCare weak, however the band dispersions are rather differ-ent. First of all, the difference comes from different pathsin k -space for bulk and (111) slab as shown on Fig. 5(a).Secondly, there are much more bands for the slab since k z translation invariance is broken for the surface. As aresult, all atoms even of the same sort become inequiva-lent and produce their own set of bands. Nevertheless, ingeneral SnAs bulk band shape to some extent resemblesthat for (111) slab crystal structure. ARPES vs. DFT/GGA
Earlier ARPES results obtained by our group were re-ported in Ref. [19]. We noted there the observed bandbundle spliting at the Γ-point not captured by the existedDFT calculations [15, 18]. We suggested this disagree-ment to be caused either by SOC, or by a band struc-ture reconstruction on the crystal surface [19]. We nowcompare ARPES data and DFT/GGA calculated banddispersions to clarify these issues and conclude that thereason of the band splitting is indeed the reconstructionof the band structure at the crystal surface [19].Bulk calculations for k z other than 0 (the k z directioncoincides with the [111] direction) show, that by varying k z it is not possible to improve agreement between thecalculated and experimental spectra, similar to a numberof studies [26, 27]. In this regard, the work presents thecalculation results for the case k z = 0.In Fig. 5 panels (b) and (e) show ARPES data andDFT/GGA bands for bulk SnAs without and with SOC,respectively. The SnAs bulk DFT/GGA bands resem-ble ARPES data very remotely. There is a bundle ofbands around 1.1 eV (predominantly of As-4p character)in calculated and ARPES data. However, DFT/GGAshows only three bands in the bundle plus one band by1.5eV lower, whereas ARPES clearly demonstrate fourbands in the vicinity of the Γ-point. Also energy posi-tions of those bands are quite different in ARPES and Figure 5. a) The first Brillouin zone for the NaCl-type structure with the projection to the (111) plane. b), e) calculated bulkband structure (solid lines) and ARPES spectra (contour plots) for SnAs along K − Γ − K direction. c), d), f), g) ARPESspectra with DFT/GGA calculated band structure of (111) slab along high symmetry directions ¯K − ¯Γ − ¯K and ¯M − ¯Γ − ¯M ′ from panel (a). The lower row corresponds to DFT/GGA calculations with SOC. Dashed lines near Fermi level show maximaof the ARPES data maps. Zero energy corresponds to the Fermi level. DFT/GGA. Somehow ARPES does not resolve bandscrossing the Fermi level, the most intensive signal is inthe vicinity of the Γ-point (the momenta interval from-0.2 to 0.2 ˚A − of the ARPES spectra, Fig. 5).Inclusion of SOC splitting slightly improves the situa-tion. The wings around 0.9 and 1.3 eV agrees a bit betterwith experiment but ARPES does not show clear sepa-ration of those bands. One should note also that bulkDFT/GGA bands along K − Γ − K direction do not re-produce ARPES bands near Fermi level shown by reddashed curves on ARPES spectra.In case of (111) slab calculations, as panels (c,d,f,g)of Fig. 5 show, the calculations demonstrate the emer-gence of electronic dispersion which much better repro-duce ARPES bands near the Fermi level shown with reddashed lines for both ¯K − ¯Γ − ¯K and ¯M − ¯Γ − ¯M ′ direc-tions. We conclude, the Fermi level crossing positions arewell captured by DFT/GGA calculations in the vicinity of the Fermi level. Wings at about 0.9 eV are well re-produced here too. Dark ARPES data region at Γ-pointabout 1.1 eV below E F is qualitatively represented bybunch of DFT/GGA “spaghetti”. Strictly speaking notall of these “spaghetti” may be seen by ARPES sinceat a given incident beam energy the penetration depthis about 7˚A. It corresponds to one, maximum two topmost surface layers. The layers below should not pro-vide significant ARPES signal. SOC has almost no ef-fect on the shape of the DFT/GGA bands and does notbring about agreement with ARPES data. Based on theabove comparison we conclude, that the ARPES spectrain general in the vicinity of E F are in good agreementwith the DFT/GGA calculated bands for (111) slab ofSnAs. However, the only one band with maximum atabout 3 eV (highlighted with light blue dashed curves inFig. 5) isn’t reproduced by any of the performed calcu-lations. We suppose, that this feature might be a maintheme of further studies. Resistivity, specific heat and magnetic susceptibility
For more detailed information about superconductingproperties of SnAs we performed resistivity, specific heatand magnetic susceptibility measurements. Temperaturedependence of resistivity is shown in Fig. 6 and demon-strates the metallic-type conduction of the sample, whichis consistent with the band structure calculations andARPES data (Fig. 5). At a temperature of about 3.6 K(for sample No. 1), the resistivity shows a transition tothe superconducting state (Fig. 6, inset). This agreeswith the earlier data [14].Temperature dependence of the magnetic AC suscep-tibility χ of SnAs crystal is presented in Fig. 7 (for sam-ple No. 2). The susceptibility for this sample demon-strates superconducting (SC) transition near 3.8 K inzero field, slightly higher than the SC transition in trans-port (for sample No. 1). As the applied DC magneticfield increases, the superconducting transition tempera-ture gradually decreases and an additional positive peakappears in the χ ( T ) dependence near T c . This peak cor-responds to the so called differential paramagnetic ef-fect (DPE) in superconductors [28–30]. DPE (positive χ ac ) corresponds to the existence of a region with stronggrowth in M ( H ) dependence (and, as a consequence, pos-itive derivative ∂ M/ ∂ H). This effect is observed for mixedstate of type-I superconductors or in type-II supercon-ductors with close H c and H c values. Figure 6. Temperature dependence of resistivity for SnAs(sample No. 1). The inset shows the zoomed-in region of su-perconducting transition.
Figure 8 shows isothermal magnetization loop M ( H )at T = 2 K. The shape of the curve is nearly typical fortype-I superconductor, however it reveals a small hys-teresis. Such hysteresis is observed in other compounds Figure 7. AC–Magnetic susceptibility of the SnAs (sampleNo. 2). considered to be type-I superconductors and, generally,is associated with sample shape effects, domain or grainwalls etc [29, 31–34].
Figure 8. Isothermal magnetization loop M ( H ) at 2 K (sam-ple No. 2). Temperature dependences of the specific heat in var-ious magnetic fields are shown on Fig. 9. The sharpjump in specific heat at H = 0 for temperature near4 K confirms the bulk superconducting transition. Fromthe local entropy conservation we find the critical tem-perature T c = 3 . ± .
05 K, consistent with susceptibil-ity measurements. The total width of the SC transition∆ T c = 0 .
15 K is another evidence of the high quality ofthe studied crystal. In general, our data for zero fieldspecific heat is reasonably consistent with [14].Specific heat tends to zero at T −→ C ( T ) /T = βT + γ r ( β =0,317 ± × K )) gives residual electronic specific heat γ r = 0 .
015 mJ/(mol × K ). Figure 9. Temperature dependences C ( T ) /T for the SnAscrystal (sample No. 2) in magnetic fields 0 – 500 Oe. Magnetic field suppresses the superconducting transi-tion and shifts it gradually to lower temperatures. Atfields above 300 Oe, the superconductivity is fully sup-pressed, as follows from the equality of the specific heatat H = 300 Oe and higher fields, for example, 500 Oe.Thus, the electronic component of the specific heat canbe defined as C e = C ( H ) − C ( H ) + γ n T for H < H ,(where H = 300 Oe exceeds the critical magnetic field),taking into account the entropy conservation. The elec-tronic specific heat in the normal state γ n at tempera-tures above T c was found to be γ n = 2 .
67 mJ/(mol × K ). Figure 10. Electronic specific heat C e ( T ) /T in fields 0 –500 Oe (sample No. 2). Inset : the difference between the en-tropy in superconducting state at zero field and normal state(at H = 300 Oe) ∆ S = S s − S n . The residual term γ r is much less than the electronicspecific heat in the normal state γ n . The ratio γ r /γ n ≈ Figure 11. Normalized electronic specific heat of the super-conducting condensate C e T γ n and its best fit with the s- andd- wave α -model (sample No. 2). .
5% quantifies the relative amount of unpaired carri-ers; it confirms the bulk character of the superconductingstate, and evidences for a high crystal purity.Temperature dependence of the electronic contributionto specific heat C e ( T ) /T in various fields is shown inFig. 10.Inset of Fig. 10 shows difference between entropy insuperconducting and normal states. We now estimatecritical magnetic field value from the obtained data. Itcould be done from equation w = | ∆ F |· NV = H c π , where w is the density of energy, ∆F is the difference between freeenergies in superconducting and normal states, N is themolar quantity, and V is the sample volume. Numericalintegration give us ∆ F = R T c ∆ SdT = – 9.00 mJ/mol,where ∆ S is the difference between the entropy in super-conducting state at zero field and in normal state. Takinginto account N=m/ µ =3.44 · − mol (here µ is the for-mula mass of SnAs, m= 6.66 g is the sample mass) V=m/ ρ = 0.971 · − cm ( ρ =6.86 g/cm is the mass den-sity) we have for critical magnetic field H c =283 Oe.To analyze thermal properties of superconducting con-densate, it is convenient to consider the normalized elec-tronic specific heat C en = C e T γ n . For the superconduct-ing condensate it may be calculated within the frame-work of the BCS theory using the so-called alpha model[35]. The α -model may be generalized in case of possi-ble in-plane anisotropy (d-wave α -model) as presented in[36], with two adjustable parameters α = 2∆(0) k B T c , (∆(0)is the superconducting gap at zero temperature), and m ( ϕ ) = 1 + µ cos(2 ϕ ) – angular dependence of the gap.Normalized electronic specific heat of the supercon-ducting condensate C e T γ n versus ( T /T c ) and its best fitwith the s- and d- wave α -model [35] are presentedin Fig. 11. The simplest s-wave model (isotropic gap, m ( ϕ ) = 1) has only one free parameter α . The best de-scription of the experimental data was obtained with α = 3.73 which is very close to the characteristic BCS ratio( α = 3.52) and corresponds to the superconducting gapof ∆ = 0 . m ( ϕ ) = cos 2 ϕ ) is at odd with theexperimental data of Fig. 11. Fitting with the extendeds-wave gap symmetry ( m ( ϕ ) = 1 + µ cos(2 ϕ )) leads to µ →
0, and therefore reduces to the simple s-wave modelwith an isotropic gap. The same result is obtained for thetwo-band α -model which corresponds to the two-bandsuperconductivity [35]: C ( T ) = ϕ C ( T ) + ϕ C ( T ). Inthis case from the fitting procedure, the contribution ofone of the gaps ϕ becomes equal to zero, whereas angu-lar modulation for the second one vanishes ( µ −→ α = 2∆(0) k B T c = 3 .
73. Despite SnAsas shown above has essentially multi band electronic sys-tem application of single band BCS model can be justifiedin case matrix of coupling constant consists of approxi-mately equal values for all Fermi surface sheets [37].Figures 9 and 10 show temperature dependence ofspecific heat in magnetic field. Suppression of the spe-cific heat jump at H=500 Oe was also reported in Ref.[14], however, the temperature dependence of specificheat in lower magnetic fields wasn’t explored there. Fornonzero fields, beside the conventional jump at T c ( H ),the electronic specific heat demonstrates an additionalsharp peak near the SC transition (see Figs. 9 and 10).Commonly, such feature is considered as a transformationof the 2nd- to the 1st- order (in the presence of a magneticfield) phase transition in type-I superconductors, such as,e.g., thallium and aluminum [38, 39] as well as for type-I compound superconductors [29, 31, 33]. The increasein the specific heat near the superconducting transitionsignifies, that an additional energy is required to realizethe superconducting transition in magnetic field.Similar behavior may be also observed for other first-order phase transitions: e.g., for melting transition,where the specific heat of a crystal changes to the specificheat of a liquid right at the melting point, however, thetemperature does not change during melting despite theheat enters the system; as a result, the specific heat ex-hibits a sharp peak. In our case, similarly, the transitionfrom superconducting to normal state occurs with theabsorption of latent heat. The finite width of the peakis due to the intermediate state (shape effect). Thus, thespecific heat in magnetic fields demonstrates features in-trinsic to the first-order phase transition and indicatesSnAs to be the type-I superconductor.Figure 12 shows magnetic field dependence of the su-perconducting transition temperature, determined fromthe AC susceptibility and specific heat data. The dashed Figure 12. Critical temperature T c vs magnetic field, deter-mined from the AC susceptibility and specific heat; solid linerepresents the BCS parabolic fit H c ( T ) = H c (0)(1 − ( T /T c ) )(sample No. 2). lines depict the corridor between the beginning and end-ing of the transition (red lines and dots correspond tothe specific heat data and blue ones – to the AC sus-ceptibility). The points correspond to the mean val-ues. The mean T c ( H ) data obtained from the specificheat coincides with that obtained from the susceptibilitymeasurements and slightly exceeds the resistivity data.This dependence is well described by the BCS depen-dence H c ( T ) = H c (0)(1 − ( T /T c ) ) that extrapolates to H c (0) = 275 Oe; the latter value is reasonably consis-tent with H c = 283 Oe, estimated from the difference inentropy between superconducting and normal states.The data obtained enables one to estimate supercon-ducting parameters of the SnAs compound. Electronicpart of specific heat in normal state together with k F and m ∗ found from DFT/GGA results described abovegive possibility to estimate London penetration depth λ L , coherence length ξ , and Ginzburg-Landau parame-ter κ . For calculations we take DFT/GGA based Fermimomenta k F from 0.6 to 1.23 ˚A − , while m ∗ =1.41m e israther safely obtained from DFT/GGA and specific heat.Then London penetration depth may be estimated us-ing formula λ L (0) = (3 π m ∗ / [ µ k F e ]) / . It gives λ L (0)in the range from 25.7 to 75.4 nm. Corespondingly,coherence length ξ (0) = 0 . ~ k F / ( k B T c m ∗ ) equals to171.0 – 350.5 nm [40] and Ginzburg–Landau parameter κ = λ L (0) /ξ (0) = 0.07 – 0.4 < / √
2; the latter confirmsour conclusion on the type I superconductivity for SnAs.For a particular choice k F =1.23 ˚A − , our results are closeto those of Ref. [14].0 Andreev reflections spectroscopy
In order to have a deeper insight into the supercon-ducting properties of SnAs and, particularly, tempera-ture dependence of the superconducting energy gap ∆,we measured Andreev reflection spectra. The supercon-ductor - normal metal - superconductor (SnS) Andreevreflection spectroscopy is a powerful tool to determine theenergy gap value and its temperature evolution. Unlikeseveral other intimate techniques, such as scanning tun-neling spectroscopy (STS), and point-contact Andreevreflections (PCAR), this technique does not require anydata fitting [41, 42] or a model for data interpreting.Recently, Andreev reflections spectroscopy on symmet-ric SnS contacts was successfully used to quantify thegap structure for several multi-gap superconducting ar-senides [43–45]. a)b)
Figure 13. Dynamic conductances, measured at T=1.5K fora single SnS contact (a) and for a 4-contact stack (b) (sampleNo. 1).
Figure 13(a) shows the dynamic conductance dI/dV of a single Andreev-type contact, measured at T =1.5 K.The two symmetric dips in dI/dV marked with label n in Fig. 13(a) correspond to bias voltages V = ± /n e .No other features are seen at higher bias, and we con- clude that the dips correspond to the major n = 1 res-onance. From this representative data we obtain an es-timate ∆ ≈ .
55 meV. Andreev reflection dips of higherorder may not be seen on this contact due to the followingreasons. According to Ref. [42], the number of possibleAndreev reflections is limited by the ratio 2 a/l , where a - is characteristic dimension of the constriction, l - is themean free path. Another possible reason is that the sec-ond and higher-order reflection dips may be masked bythe sharply growing “foot” (excess conductance at lowbiases, which is intrinsic to Andreev contacts).By mechanical tuning the sample bending and thus,the break junctions area one can realize Andreev con-tacts with different 2 a/l ratio and also stacks of sequen-tially connected S-n-S-n...-S contacts [20]. For the chainof m contacts, evidently, the resonant bias voltage V n = m × (2∆ /ne ) is increased by a factor of m . Figure 13(b)shows dynamic conductance for such stack of several con-tacts. One can see 3 features, corresponding to 3 reso-nances. Having the preceding single-contact spectrum,as a reference, we immediately disentangle the spectrumFig. 13(b) and find that it is produced by a chain of 3 se-quential contacts. Indeed, the observed dips at 4.1, 2,15and 1.35 meV may be fitted with n = 1 , n = 2 , n = 3and the consistency with the single contact spectrum isobtained for m = 3. Finally, based on the above data werefine the gap value, ∆ = 0 . ± .
04 meV and the ex-trapolated value ∆( T →
0) = 0 .
54 meV. We stress, thatfor all studied contacts we didn’t observe signatures ofthe second gap.
Figure 14. Temperature evolution of dynamic conductancefor 1 contact stack a single Andreev contact (the peak at zerobias is cut off).
In order to find ∆( T ) temperature dependence, wemeasured the dynamic conductance of a single contactin the range 1.5 – 4 K. Figure 14 shows that with tem-perature rising, the dips in dI/dV shrink towards zerobias and the dynamic conductance finally linearizes at1temperature ≈ . T clocal . Figure 15. Energy gap temperature dependence deducedfrom the measured Andreev spectra. ∆ = 0 . ± .
04 meVis found from extrapolation of ∆( T ) to T = 0, obtained withthe BCS fitting dependence. Evolution of the gap value with temperature ∆( T ) isplotted in Fig. 15. One can see that temperature de-pendence of the energy gap is well approximated by thesingle-gap BCS theory, and the ratio 2∆(0) /k B T c = 3 . CONCLUSIONS
Parameters Values Values from Ref.[14]a, ˚A 5.723 5.72513(4) ρ , g/cm T c , K 3.6 (sample No. 1) 3.583.8(sample No. 2)H c , Oe 283 (entropy) 178 (extrapolation)275 (extrapolation)∆(0), meV 0.6 (specific heat) –0.53 ± /T c γ r , mJ/(mol × K ) 0.015 – γ n , mJ/(mol × K ) 2.67 2.18 β , mJ/(mol × K ) 0.317 0.30Table I. Structural and thermodynamic data for SnAs in thenormal and superconducting states. Parameters H c , γ r , γ n , β , and ∆(0) (specific heat) were obtained for sample No. 2,∆(0) (AR) – for sample No. 1. Interval of 2∆(0) /T c includeparameters for both samples. In summary, we performed comprehensive study of theband structure and superconducting properties of theSnAs binary compound. DFT/GGA band structure cal-culations were carried out for both bulk and (111) slabcrystal structures with and without SOC. Our calculated spectra for bulk SnAs are in a good agreement with pre-vious theoretical results. Experimental investigation ofSnAs band structure was done by ARPES technique.Comparison of the ARPES results and band structurecalculation shows that the calculated (111) slab bandstructure much better agrees with ARPES data than thebulk band structure. SOC does not have a strong influ-ence on electronic structure of (111) slab. From our datawe deduced quantitative parameters and the type of thesuperconducting state in SnAs. More specifically,(i) we confirmed the type I superconductivity in SnAs byfeatures of the specific heat in non-zero magnetic fields,and by the relationship between the estimated quantita-tive superconducting parameters;(ii) we found that the temperature dependences of criticalmagnetic field, specific heat jump, and superconductingenergy gap are consistent with the conventional weak-coupling BCS model. Our data shows no signature ofthe unconventional superconductivity, and the supercon-ducting state in SnAs is likely to have the s -type sym-metry;(iii) using two independent techniques (specific heat andAndreev reflection spectroscopy) we determined the su-perconducting energy gap value ∆(0) = 0 . . ± .
04 meV - fromAndreev reflection spectroscopy. These values also sat-isfy the weak coupling BCS relationship 2∆(0) /T c =3.52.Table I summaries our obtained values of the SnAsparameters and compare them with earlier data. ACKNOWLEDGEMENTS
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