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[anonimizat] 3D topological insulator BiSb

Fiodor M. Muntyanu1, Andrzej Gilewski2, Andrzej J. Zaleski3, Vitalie Chistol4, and

Krzysztof Rogacki3

[anonimizat] 3D topological insulator (TI) BiSb at temperatures 1,8-100K and magnetic fields up 400 kOe. The similar Fermi surface (FS) consisting of layer components of interfaces (CIs) and bulk crystallites was found. [anonimizat]. [anonimizat]. A [anonimizat], therby indicating that the flow of Dirac fermions is sensitive to the field orientation. [anonimizat] (thickness ~ 60nm) and two adjacent layers (thickness ~ 20nm each) of the CIs for different values ​​[anonimizat]. Two/one superconducting phases with the onset of transition ≤ 36K [anonimizat] 3DTI Bi1-xSbx are diamagnetic and does not show superconductivity. In large crystallite disorientation angle Bi1-xSbx interfaces it was found superconductivity and weak ferromagnetism.

Keywords: bicrystals, [anonimizat]-antimony, [anonimizat]

1. Introduction

Semimetalic Bi and Sb are one of most intensively studied materials in solid state physics. Remarkable agreement between the calculations of the band structure of these materials and almost all available experimental methods has been reached. It was found that FS of Bi consists of one hole ellipsoid of revolution at the T point and three equivalent electron pockets at the L points of the Brillouin zone. [anonimizat] a direction close to bisector axes of a crystal. FS of bulk single crystalline Sb consists of three electronic pockets at the L points and six hole isoenergetic surfaces at the H points of the Brillouin zone. These features of FS greatly enhance the interest in electron transport research at low temperatures. [anonimizat]. [1-4].

Bi1-xSbx (0.07 ≤x ≤ 0.2) alloys are typical representative of the 3D TIs [1, 2, 4], [anonimizat] [2, 3]. The 3D TIs are insulators in the bulk but have topologically protected metallic surface states with an unconventional spin texture [3, 5] and electron dynamics. It is useful to note that the first 3D TI, [anonimizat] [1, 4] [anonimizat].

The energy spectrum of Bi1-xSbx (0 ≤x ≤ 0.2) alloys derives from that of bismuth. An increase in Sb content x [anonimizat]ure; especially at L points of the bulk Brillouin zone (see Figure 1(c)).

So, for the x = 0.04, the gap between La and Ls pockets in energy spectrum of Bi closes and a 3D Dirac point takes place. If x increases again, the gap at L points re-opens with an inverted symmetry ordering. At x ≈ 0.07 the overlapping of bismuth specific bands (L-T45) disappears and the Bi1-xSbx alloys becomes an inverted-band semiconductors up to x = 0.2 dominated by a spin-orbit coupled Dirac particle at L points. Subsequently at x > 0.2, the alloys recovered semimetalic properties, possessing characteristics of a strong Z2 (υ0 = 1) topological insulator, similar to those of Sb. Numerous studies show that the topological phase in Bi1-xSbx is due to an odd number of band inversions, asymmetric Dirac surface state, and absence of back-scattering [5].

The surface electronic structure of Bi1-xSbx alloys, consisting of one electron pocket centered around the Г point and six hole pockets on the line between Г and M points of the 111 surface Brillouin zone [3, 4], is very similar with that of semimetals Bi and Sb.

Unusual opportunities occur when using bulk and low-dimensional combined structures of these materials because of large possibilities of studying various phenomena due to specific manifestations of Dirac electrons, Majorana excitations in the presence of magnetic fields, and unveiling novel effects with respect to spin and charge transport. In this context a very special role can be played the bicrystals of 3D topological insulators, consisting of bulk single crystalline blocks (crystallites) and the perfect nano-width (~ 100nm) crystallite interfaces (CIs). The bicrystal interfaces play a significant role in the manifestation of numerous anomalies of physical properties associated with the phonon spectrum modification and the band structure reconstruction. The CIs having a high degree of perfection (compared only to a surface cleaved in ultrahigh vacuum) and similar properties to grain boundaries of crystalline nano-objects are more affordable in experimental research. On the other hand, the nano-width CIs are a low – dimensional structure, which can be superconducting under certain conditions (high concentration of charge carriers, changes in stoichiometry, microstructural defects, stress states and high degree of deformations, disorder with large areas of poor fit, diverse forms of amorphous or even liquid layers, etc). Numerous new phenomena [6, 7] were predicted in low-dimensional samples (interfaces, surfaces, quantum structures, etc.) and hybrid structures with a proximity-induced superconducting and ferromagnetic order. A wonderful feature of Bi and Bi1-xSbx [8-10] interfaces is that they are superconducting with critical temperature Tc ≤ 21 K, while bulk rhombohedral Bi and its alloys with Sb are diamagnetic and non-superconducting.

The investigation of Bi and Bi1-xSbx bicrystals by using the quantum oscillations in high magnetic fields found out [11, 12] the same consisting of the Fermi surface of the CIs and the crystallites as well as considerable distorted charge carrier pockets at interfaces.

It prefigures a vast ability to manipulate the superconducting and magnetic states at crystallite interfaces by use the proximity effect and varying the spin splitting of the density of states which arises from the effect of magnetic field or the exchange field of a ferromagnetic insulator. In this regard, the crystallite interfaces as superconductor/ferromagnetic hybrid structures may exhibit the large thermoelectric effects, generation and propagation of highly spin polarized current etc, opening new perspectives in the development of various devices based on interplay between superconductivity and weak magnetism in interfaces of 3D topological insulator Bi1-xSbx (0 < x < 0.2) such as spin-injection devices, superconductor/ferromagnetic hybrid structures, thermoelectric devices etc.

Here we report on the quantum transport and Fermi-surface rearrangement in Bi and BiSb crystallite interfaces, the high-field peculiarities of the galvanomagnetic effects in bicrystal with superconducting nano-width crystallite interfaces, the superconductivity and weak ferromagnetism at the bicrystal interfaces.

2. Samples and experimental procedure

High quality of Bi and Bi1-xSbx (0 ≤x ≤ 0.2) bicrystals (the Dingle temperature of charge carriers Td ≤ 1K) were prepared by the horizontal zone recrystallization method using the double seed technique. The samples for measurements were prepared in the form of rectangular bars (1 x 2 x 4 mm3), where the fraction of the interface volume in the overall volume of the bicrystal was ~ 10-4.

Two types (see Fig. 1b) of bicrystals were made: inclination type (the crystallographic axes of crystallites are revolved in a single plane) and twisting type (the crystallographic axes of crystallites are revolved in two planes).

Depending on disorientation angle of cristallites and the type of conductivity of their interfaces, the bicrystals were distributed in two groups: small crystallite disorientation angle (SDA) bicrystals with θ1 < 9o and large crystallite disorientation angle (LDA) bicrystals with θ1 > 26o. In our case SDA bicrystals had n-type conductivity; LDA bicrystals had p-type.

The composition of the samples was controlled by a scanning electron microscopy (SEM) instrument equipped with Oxford and PV 9800 energy-dispersive X-ray (EDX) analyzers (examples are given in Figure 1(a)). The magnetic impurity concentration was evaluated by the of optical emission spectrometry methods. To change the carrier concentration, the initial ingot for the preparation of bicrystals was lightly doped (≤ 0.01 at. %) with either tin or tellurium. The width of CIs was estimated by means of SEM and from the quantum oscillations [11]. It turned out that the bicrystal interfaces are complicate systems composed of a solitary central part (the thickness ~60nm) and two similar adjacent layers (~ 20nm thick each) on both sides of it.

Figure 1. (a) EDS spectrum of Bi0.882 Sb0.118 bicrystal, Inset: Foto of Bi0.925 Sb0.075 bicrystal. The arrows indicate the CI location. (b) Schematic representation of bicrystal: θ1, relative CI disorientation angle; θ2, rotation angle in an interface plane. (c) Energy diagram of the spectrum reconstruction in Bi-Sb alloys at low Sb concentrations.

The magnetic and galvanomagnetic properties of the bicrystals were studied in temperature range 1.6 – 100 K using very sensitive measurement systems such as a Quantum Design SQUID magnetometer and a Physical Property Measuring System (PPMS) with a 140 kOe induction magnet. Electrical resistance was measured in a standard four-probe circuit. The longitudinal Hall effect ( measured in longitudinal configuration: magnetic field H parallel to the Hall voltage E and both perpendicular to the current I, H‖E┴ I) was selected as the prferential research method for investigating the features of the quantum oscillation in bicrystals. Contact electrodes for resistivity studies and Hall effect (ρij (Hi)) were soldered directly on the CIs by electrospark welding. The quantum oscillations of ρij (Hi) were registered in stationary (up to 180 kOe) and pulse magnetic fields (up to 400 kOe) after transition of CI in the normal state (converted by magnetic field and/or a current).

3. Results and discussion

3.1 Fermi-surface rearrangement in Bi and Bi1-xSbx (x< 0.18) bicrystals

Finding the Fermi surface shape features is one of the most important steps in predicting the basic electronic properties in condenced matter phisics and for a better understanding of all the properties of metals, semimetals and doped seiconductors.

So far as, the FS of single crystalline Bi and Bi1-xSbx (x< 0.18) alloys is highly anisotropic and has complex structure [13], in the spectrum of transport quantum oscillations the some harmonics are shown, are observed beating of oscillations, suppression of harmonics by spin splitting, displacement of oscillations peaks in the magnetic field, etc. On the other hand, the anisotropy of isoenergetic surfaces results in carried on magnetic field oscillations and their spectrum can rather easily be separated on harmonics for some field orientations. By the fact that, most of experimental techniques of FS study, practically, are discovered/ tested (or can be applicable) to Bi, its FS is known in the finest detail. This considerably facilitates research of band structure of Bi and BiSb bicrystals.

3.1.1 Small crystallite disorientation angle interfaces of Bi twisting bicrystals

As a rule, in SDA bicrystals of bismuth there are shown quantum oscillations of ρij (H) describing FS of crystallites (single crystalline blocks) and layer components of crystallite interface. Examples of such oscillations at the magnetic field parallel to the trigonal axes C3 of crystallites and at its rotation in the binary plane are given in Figure 2. We reliably registered quantum oscillations of ρij(H) for crystallites of SDA bicrystals. The periods of these oscillations correspond to well-known [13, 14]  electron and hole FS of single crystalline Bi (see Figure 4, curves 1, 2, 3). These data testify that the number of electrons and holes in crystallites of bicrystals as well as in rhombohedral Bi is almost equal (n, p ~ 3  1023 m-3).

Besides the mentioned above frequencies of quantum oscillations in SDA bicrystals a number of new frequencies appear in large angular intervals of the magnetic field rotation. For example, two new harmonics in the spectrum of quantum oscillations of ρij (H) in bicrystals of Bi are revealed at H || CI. The frequencies of these harmonics essentially differ from the values characteristic of single-crystalline samples. In [14 ] was observed, that at the magnetic field orientation close to the trigonal axes (H || C3) the magnetoresistance of single crystalline bismuth has monotone increasing up to 450 kOe and in the magnetic fields H > 100 kOe the Shubnikov-de Haas oscillations disappear. On the other hand, in SDA bicrystals at the same orientation of the magnetic field the quantum oscillations of the Hall resistance are observed at the H > 100 kOe (see Figure 3), but their frequencies are higher approximately by 5 and 10 times than frequencies typical for extremal cross-sectional areas of the FS of Bi. Usually, the Hall resistance oscillations of first frequency appear at H > (25-35) kOe, of the second – at H > 90 kOe. Calculated cyclotron orbit diameter corresponding to fields of the first component appearance (depending on crystallites disorientation angles of bicrystals) is ~ (100 -140) nm and correlates with the CI width determined by means of the SEM. The diameter of cyclotron orbits of charge carriers corresponding to field of the second harmonic appearance is ~ (60 – 80) nm. We believe that this value characterizes the thickness of central layer of interfacial region, resulting from a well known fact, that the CI is a complex system consisting of a solitary central part and two similar adjacent layers on both sides of it. Hence, our experimental results on SDA bicrystals of Bi testify that the width of the CI central part with high density of states makes ~ (60 – 80) nm and of two adjacent layers from which the area of a spatial charge is distributed in the crystal volume takes value ~ (20 – 30) nm each. Figure 2 (b) displays the examples of angular dependences of the periods of quantum oscillations of ρij(H) for adjacent layers of CI of SDA bismuth bicrystals.

Figure 2. (a) The quantum oscillations of the Hall resistance in Bi SDA bicrystals with θ1 = 50 and θ2 = 5.50. 1, H  C3, T = 1.5 K; 2, H  C1, T = 4.2 K; 3, [H, C3]  10, T = 4.2 K; 4, H  C2, T = 4.2 K (the monotonic part has been subtracted).C1, C2, C3-bisectrix, binary, and trigonal axes, respectively. Inset: The quantum oscillations of the Hall resistance in Bi SDA bicrystal with θ1 = 8.60 and θ2 = 40 .1, H  C3; 2, [H, C3]  310; 3, [H, C3]  200; 4, [H, C3]  170; 5, [H, C3] 80; 6, [H, C3]  110 (the monotonic part has been subtracted). (b) The angular dependences of oscillation periods in the crystallite binary plane of SDA Bi bicrystals. 1, 2, 3, crystallites; 4, 6, CI with θ1 = 50and θ2 = 5.50; 5, 7, CI with θ1 = 8. 60 and θ2 = 40.

The structure of the electron Fermi surface of these layers is same as in crystallites (three electron pockets located in L points of the Brillouin zone and generated from the principal ones by ± 1200 rotations around the trigonal axis C3). The isoenergetic surfaces are less anisotropic and are much more in volume, than electronic pockets of crystallites. The ratio of semiaxes of electron pocket in adjacent layers considerably differs from the similar ratio in pure Bi (for example, the ratio of semiaxes of electronic quasiellipsiod of the adjacent layers of bicrystal with θ1 = 50 and θ2 = 5. 50 makes 1:1.34:12.6 while in rhombohedral bismuth it makes 1:1.40:14.8 [13]). Hence, in SDA bicrystals of Bi not only the electron concentrations grow as, for instance, at bismuth doped with donor impurity of Te or Se, but also the shape of FS changes. The density of electrons in adjacent layers of SDA bicrystals makes ~ 0.67  1020 m-2 , that is three order higher than in bismuth thin films of similar sizes.

Quantum oscillations of the Hall resistance from the central layer of СI of the Bi bicrystals, connected with electron FS, were clearly observed at the magnetic field orientation along the internal surface plane, at its deviation by an angle > 100 they disappeared. Nevertheless, the obtained data testify that the density of electrons in CI central part is much more, than in the adjacent layers and makes ~ 1.5  1020 m-2.

3.1.2 Large crystallite disorientation angle interfaces of twisting Bi bicrystals

Typical feature of superconducting LDA bicrystals is observation of the quantum oscillation of ρij (H), connected with hole FS of the crystallite interface. Oscillations from electron FS of the CI and FS of the single crystalline blocks were not detected.

As well as in the case of SDA bicrystals, at the magnetic field orientation along the CI plane the spectrum of the quantum oscillations of ρij(H) of LDA bicrystals contains two new harmonics (see Figure 3(a), Inset), which start at magnetic fields H > (20 ÷ 25) kOe (first frequency) and H > 100 kOe (second frequency). This specifies that the width of the central part and of the adjacent layers of the CI in SDA and LDA bicrystals are close in values.

It was found from frequencies of the quantum oscillations of ρij (H) that the hole concentration (bicrystals had р-type of conductivity) in the CI central part of LDA bicrystals makes ~ 2.5  1020 m-2 and in the adjacent layers ~ 1.2  1020 m-2. That is considerably higher, than in SDA bicrystals. Probably due to low carrier concentration in SDA bicrystals only one superconducting phase is shown, at that phase with Tc ~ 4.3 K is located in the central part of the CI.

Figure 3. (a) The quantum oscillations of the magnetoresistance in LDA Bi bicrystals with θ1 = 620 and θ2 = 20 at 4.2 K. The magnetic field rotation in the binary plane: 1, [H, C3]  600; 2, [H, C3]  1200; 3, [H, C3]  1400; 4, [H, C3]  400; 5, [H, C3]  200. Inset: The quantum oscillations of the Hall resistance at the magnetic field directed along the inner boundary plane (the monotonic part has been subtracted). (b) The angular dependences of hole oscillation periods in the crystallite binary plane of LDA Bi bicrystals. 1, Single crystalline Bi; 2, 3, CI with θ1 = 620 and θ2 = 20; 4, CI with θ1 = 270 and θ2 = 100; 5, CI with θ1 = 330 and θ2 = 90; 2, adjacent layers; 3, central part of bicrystals.

The quantum oscillations of ρij(H) in the adjacent layers of the CI (see Figure 3 (a)) of LDA bicrystals were registered in large angular intervals of magnetic field rotation. At the same time quantum oscillations from the CI central part were detected in the limited angular intervals (± 400 near H || С3). The angular dependences of the oscillation periods of ρij(H) at the magnetic field rotation in the binary plane of crystallites are given in Figure  3 (b). Apparently from the figure, the hole FS of the central and adjacent layers of the CI of LDA bicrystals consists of one weakly anisotropic pocket, the shape of it essentially differing from the hole FS of single crystalline Bi (the ellipsoid of revolution extended along the trigonal axis С3, the ratio of cross-sectional area S1/ S3 = 3.327 [13]). Moreover, in some bicrystals the elongation of FS in adjacent layers changes (FS is rotated in comparison with the ones in crystallites) and the hole ellipsoid of revolution in pure Bi turns to a deformed hole pocket with all three main cross-sectional areas different on size. The hole FS of the central part of LDA bicrystals in the investigated angular intervals of the magnetic field rotation is almost similar to FS of the adjacent layers and both considerably surpass in volume the hole isoenergetic surface of single crystalline Bi. Thus, the Fermi surface consisting of layer components of bicrystal interfaces is similar to those of single crystalline Bi, but the shape, elongation and volume of isoenergetic surfaces at CI undergo essential changes that finally stimulates electron pair correlation.

The same FS restructuring features have been detected [12] at the interfaces of twisting bicrystals of Bi1-x Sbx (0 ≤ x ≤ 0.2) alloys.

3.2 High-field quantum transport in Bi and BiSb bicrystals

The studing of bicrystals of Bi and Bi1-x Sbx (0 ≤ x ≤ 0.2) alloys in high-magnetic fields is aimed of revealing the specific features of the interaction of charge carriers in systems of different dimensions, including the boundary of a quasi- two-dimensional superconductor and a 3D TI.

Figure 4 shows the magnetoresistance and the longitudinal Hall quantum oscillations of SDA Bi and BiSb bicrystals of inclination and twisting types. The spectrum of oscillations is complex and contains frequencies from the Fermi surface of crystallites and interfaces. At 20 kOe, the oscillation peaks take an unusual configuration (see Figure 4(a)) and their position is essentially shifted [15] from of the quantum oscillation maxima in Bi single crystalline samples [13]. Also, between the peaks, along with minima in magnetoresistance, a number of Hall quasi-plateaus (~ 30 kOe, ~ 60 kOe, ~ 150 kOe) were found that are indicator of bulk electron fractionalization.

It should be noted, that the Hall plateaus and changing of the shapes of oscillation peaks are clearly developed only in SDA bicrystals of inclination type. Compared to SDA bicrystals of inclination type, the all others investigated bicrystals have an increased disorder strength and higher dislocation density; therefore, the Hall plateaus are not found and the localized states in the spectrum of Landau levels do not appear. According to our data [11], the FS of the majority carriers of LDA bicrystals considerably exceeds in volume [13] the holes isoenergetic surface of single crystalline bismuth. Thus, the presence of non-interacting carriers that contributes to fractionalization seems to be problematic.

Figure 4. The quantum oscillations of magnetoresistance ρii(Hi) and longitudinal Hall effect ρij(Hi )in Bi (a,b) and Bi 1-x Sb x (c,d) bicrystals of inclination (a) and twisting (b,c,d) types. (a) (1) ρij(Hi +), 4,2 K, θ1 = 4o; (2) ρij(Hi), 2,1 K, θ1 = 5o; (3) ρij(Hi – ), 4,2 K, θ1 = 4o; (4) ρii(H), 2,1 K, θ1 = 5o; (5) ρij(Hi), 4,2 K, θ1 = 7o; (6) ρij(Hi), 4,2 K, θ1 = 8o. The marks indicate the position of oscillation peaks in Bi single crystals. (b) (1) ρij(Hi), 1,5 K, θ1 = 5o, θ2 = 5,5o; (2) ρij(Hi), 4,2 K, θ1 = 62o, θ2 =2o; (3) ρij(Hi), 1,6 K, θ1 = 3o, θ2 =5o; (4) ρii(H), 4,2 K, θ1 = 29o, θ2 =11o.The monotonic part of ρij(H) and ρij(Hi) has been subtracted. Inset: The H-1 position of the Hall peaks versus their Landau level index: (1, 2) for the oscillation curve 4; (3, 4) for the oscillation curve 2. (c) (1) single crystal, x = 0.08; (2) and (3) bicrystals: (2) x = 0.08, Θ1 = 4o, Θ2 = 2o; (3) x = 0.09, Θ1 = 12o, Θ2 = 2o. Arrows point the fields at which the semiconductor-semimetal transitions appear. Inset: Band evolution in high magnetic fields directed along the trigonal axis C3 in bulk (crystallites) Bi1-xSbx (0.07 < x < 0.15). BC1 is the critical magnetic field for the semiconductor-semimetal transition. (d) Twisting LDA bicrystals of Bi1-xSbx (x = 0.08, 0.12, 0.15) and Bi0.93Sb0.07Te at 4.2 K: (1) ρii(H), x = 0.08, Θ1 = 15o, Θ2 = 3o; (2) ρij(Hi), x = 0.12, Θ1 = 12o, Θ2 = 2o; (3) ρii(H), Bi0.93Sb0.07Te, Θ1 = 19o, Θ2 = 2o; (4) ρii(H), x = 0.15, Θ1 = 15o, Θ2 = 3o. The magnetic field is directed along CI plane (near C3 axis of crystalline bloks). Inset: Landau level index, n, versus the Hn-1 position of the oscillation peaks.

Another interesting feature of the development of Hall plateaus is they vanish by magnetic field reverse, thereby indicating that at least in SDA bicrystals the flow of Dirac fermions along the CI plane is sensitive to the field orientation and as well as the localization process runs only in a specific direction of the magnetic field.

The two new harmonics in the spectrum of transport quantum oscillations are registered (see Figure 4 (b)) practically in all investigated Bi [11] bicrystals. At least one of the harmonics defines extremal cross-sectional areas of Fermi surface which is approximately 70 times higher than that of crystallites.

The same harmonics in the spectrum of quantum oscillations of the resistance ii(B) and the longitudinal Hall-effect resistance ij(Bi) has been observed also in Bi1-xSbx bicrystals ( see Figure 4 (d)).

The oscillation spectra of the Bi1-xSbx bicrystals we analyse graphically based on Hn-1 position of the oscillation peaks versus the Landau level index, n, where Hn is the field at which the Fermi level falls between two Landau levels. The relation between Hn and n is described by the formula: 1/Hn = ne/(hNs), where: e is the elementary charge, h is the Planck’s constant, and Ns is the surface density of states. The Hn-n relations obtained for three bicrystals are shown in the inset of Figure 4 (d). The results show that the interface electronic states are of the Schrödinger type, since n takes integer values and for Dirac electrons nD = n+½. By the temperature dependencies of the oscillation amplitudes and the extreme cross-sectional areas of the isoenergetic surface of the CI layers were evaluated carriers cyclotron masses. They exceed several times the respective crystallite parameters [13]. For example, in magnetic field oriented along the interface plane in the Bi0,93Sb0,07Te bicrystal, the cyclotron mass of carriers mc/me (where me is the free electron mass) in crystallites and CI adjacent and central layers is 0.05, 0.25 and 0.5, respectively. The density of states, Ns, of the CI layers of these bicrystals also has been estimated from the quantum oscillations [11]. The following values for Ns are obtained: (0.2-0.3)1019 m-2 for adjacent, and (1.5-2.5)1019 m-2 for central layers. That’s several orders of magnitude higher than for similarly thick films of Bi1-xSbx alloys [16].

The single crystalline Bi1-xSbx (0.07 < x < 0.15) alloys provide a remarkable opportunity to obtain ultra quantum limit (UQL) in lower magnetic fields than in Bi, where the electrons (or holes) occupy the lowest (j = 0) Landau level. Beyond UQL the band edge displacement Δε takes place (see inset of Figure 4(c)), depending on the ratio of spin Δεs and orbital Δεo level splitting: Δε = ½hω(1 – Δεs/Δεo). Consequently, the various electronic phase transitions occur in high magnetic fields, including the semiconductor-semimetal transition, semimetal-semiconductor transition, etc.

Figure 4(c) shows the field dependency of the magnetoresistance Δρ/ρ in a single crystal and twisting SDA bicrystals of Bi1-xSbx (0.07 < x < 0.15) alloys. For magnetic field oriented along the CI plane, regular Shubnikov-de Haas oscillations are observed up to 20 kOe. In higher fields charge carriers are in the UQL and at H ~85 kOe (see the maxima in Figure 1(a) the semiconductor-semimetal transition takes place both in the single crystal (curve No 1) and in the SDA bicrystals (curves No 2, 3). The first maximum in the Δρ/ρ versus field dependency is detected at the same field as for the single crystal, which means that the semiconductor-semimetal transition is induced in crystallites. For fields higher than 110 kOe, an additional maximum is observed in the bicrystals only. We believe that the second maximum reflects the semiconductor-semimetal transition in the adjacent layers of CI. For fields higher than 250 kOe, third maximum appears which can be attributed to the semiconductor-semimetal transition in the central layer of the CI of the SDA bicrystals. Both of these maxima (second and third) are manifested at different fields, for the bicrystals with different disorientation angles Θ1 and Θ2.

It is interesting that the transitions appears at quite different values of the field, so the cyclotron mass of the relevant charge carriers and the ratio of spin and orbital level splitting Δεs/Δεo in crystallites and in the CI region of the SDA bicrystals should differ considerably. This implies a significant growth of the spin-orbit interaction at the interfaces and the existence of gapless electronic states at the bicrystal boundary.

3.3 Superconductivity and weak ferromagnetism at the interface of bicrystals of Bi and 3D topological insulator BiSb

We will remind that, in the normal state bulk single crystalline Bi and 3D topological insulator Bi1-xSbx (0 ≤ x ≤ 0.2) are diamagnetics and do not exhibit superconductivity above 30mK. On the other hand, pressurized samples, disordered thin films, amorphous specimens, nanowires and other nanometric objects show superconductivity at temperatures up to (4 – 7) K.

Using contact and non-contact signal recording methods (including transport effects, magnetic moment, magnetic susceptibility, specific heat, etc) we detect [17] at the bicrystal interfaces of Bi and BiSb bicrystals two new superconducting transitions with critical temperature Tc1 ~ (3.7 – 4.6) K , Tc2 ~ (8.3 – 21) K [8]), while Tonset takes values up to 36 K.

Figure 5 shows the examples of temperature dependences of the static magnetic moment of Bi bicrystals with the CI of the twisting type.

The two superconducting phases have been observed at CI of LDA bicrystals. On the other hand in SDA bicrystals only one superconducting phase (Tc ~ 4.3 K) is revealed and its superconducting parameters differ from values of LDA bicrystals. For example, in bicrystal with θ1 = 50 and θ2 = 5.50, the slope of the Hc2(T) dependence makes dHc2/ dT ~ 2.97 kOe/ K, upper critical field Hc2(0) ~ 8.8 kOe and coherence length ξ (0) ~ 19 nm, while in LDA bicrystals these parameters have the following values: dHc2 / dT ~ 3 kOe/ K, Hc2(0) ~ 25 kOe, ξ (0) ~ 12 nm (for the first superconducting phase with Tc ~ 8. 4 K) and dHc2/ dT ~ 5.5 kOe/ K, Hc2(0) ~ 16.6 kOe, ξ (0) ~ 14 nm (for the second phase with Tc ~ 4.3 K). The evaluation of Hc2(0) were carried out by using well-known WHH formula [18]:

. (1)

Figure 5. (a) Temperature dependences of a magnetic moment and magnetic hysteresis loop in Bi SDA bicrystals with θ1 = 50 and θ2 = 5. 50; 1- 50 Oe, 2- 100 Oe, 3- 200 Oe, 4- 400 Oe, 5- 800 Oe. Inset: Magnetic hysteresis loop at T = 3 K. (b) Temperature dependences of magnetic moment and magnetic hysteresis loop in Bi LDA bicrystals. 1 – θ1 = = 290, θ2 = 110; 2 – θ1 = 620, θ2 = 20; 1 – 10 Oe; 2 – 20 Oe. Inset: (b) Magnetic hysteresis loop at T = 2 K of Bi bicrystal with θ1 = 290, θ2 = 110. The magnetic field is orientated along the inner boundary plane.

The coherence length (0) in LDA and SDA superconducting phases are much smaller than the width of the central part of CI (d1 ~ 60 nm), so the adjacent layers do not improve conditions for correlation of the Cooper pairs in the central part of the crystallite interface.

The magnetic hysteresis loop of LDA and SDA interfaces (see Figure 5), clearly shows the behavior typical for strong type-II superconductor and becomes reversible above 2.5 kOe at the irreversibility field, which is a few times lower than the upper critical field Hc2 of both superconducting phases at this temperature. The magnetisation curve at T ~ 2 K exhibits a large hysteresis and they are almost symmetrical, the remnant moment is significant at zero applied field and the screening effects in the interfacial plane is expected to be rather weak, since the interfacial thickness is somewhat less than 2λ (λ is the penetration depth).

As noted above the interface of twisting bicrystals have the modified shape of the isoenergetic surfaces and higher carrier densities, which are beneficial for superconductivity as this favor [11] electron pairing and substantially enhances the transition temperature more than up to 10.4 K. The topological features of FS of bicrystal interfaces indirectly reflect changes in displacement of atoms from the equilibrium positions characteristic for crystalline Bi. Also, intensification of the bonding of electrons with the lattice takes place. Based on superconducting transition temperature Tc ~ 10.4 K and bulk Debye energy of 10 meV for Bi, using the approximate McMillan formula [19], were extracted a coupling constant of 0.71, which shows that the electron phonon interaction in CI of Bi bicrystals is strong enough. The magnetic hysteresis loops and the current-voltage characteristics [8] of Bi superconducting interfaces clearly testify the behavior typical for strong type-II superconductors with the possible value of energy gap 2Δ(0) ≈ 3.3 meV.

In spite of the fact, that the carriers concentration in CI of inclination type almost on the order is lower than in twisting CI, in them the greatest Tc is observed. For example, Tc of the first superconducting phase in one of the pure Bi bicrystal of inclination type with Θ = 5o achieves ~ 21K (see Figure 7, curve 2), while the transition temperature of the second superconducting phase is nearly same as in twisting bicrystals [8].

At low temperatures (T < 30 K) in studied bicrystals, two kinds of dependences m(T) are observed – diamagnetic and paramagnetic. Estimations of carrier density Npara (dia) from the Hall effect and Shubnikov-de Haas oscillations show that N para at the paramagnetic bicrystal interfaces is almost 1.5 – 2 times higher than Ndia at the diamagnetic CI; therefore, the main reason of paramagnetism is higher carrier concentration.

For a more detailed identification of magnetization anomalies, will be examined the results of magnetic field studies employing zero-field-cooled (ZFC) and field-cooled (FC) measurements. Figure 6(a) shows ZFC and FC temperature dependences of magnetic moment of bicrystals of Bi1-x – Sbx (x ≤ 0.2) alloys, which in magnetic field H = 50 Oe applied perpendicular to the CI plane (H ┴ CI) branch in all investigated samples at branching temperatures Tb ~36 K.

It should be noted, that in diamagnetic samples below Tb, especially at T < 9 K, in the course of both the ZFC and FC measurements, the enhancement of the diamagnetic signal typical for the Meissner effect and the magnetic flux expulsion are observed. Detection of the well pronounced Meissner effect confirms the fact that on CI of bicrystals there is the superconducting material in the amount sufficient for the influence on the magnetic moment value. In this connection, unusually high Tc superconductibility of twisting interfaces, surraunded by a nonsuperconductiving similar material is an unwonted manifestation.

Figure 6. Temperature dependences of ZFC and FC magnetic moment of bicrystals of Bi1-x – Sbx (x ≤ 0.2) alloys: (a) Bi 0.94 Sb 0.06 Te , Θ1=9o, 50 Oe , (H ┴ CI). Inset: (H׀׀ CI); (b). Bi 0.85 Sb 0.15, Θ1=15o , Θ2 = 3o , 50 Oe , (H ┴ CI). Inset: Temperature dependence of specific heat of Bi 0.85 Sb 0.15 , Θ1=15o, Θ2 = 30

No sign of branching of ZFC and FC curves in the field (H = 50 Oe) parallel to the CI plane (H׀׀ CI) at T > 9 K could be detected. Also, below 9 K, the superconducting signal is considerably lower and the diamagnetic moment for (H׀׀ CI) is smaller than for (H ┴ CI).

As is known, the point of branching of ZFC and FC curves in superconductors determines the transition temperature Tc. Therefore, in these bicrystals the temperature of the superconducting transition onset can achieve a value of 36 K, being the highest for the group-VB semimetals and their alloys.

Typical examples of FC and ZFC curves of paramagnetic bicrystals of Bi1-x – Sbx (x ≤ 0.2) alloys in the low applied magnetic field H ┴ CI are shown in Figure 6 (b). It is observed an abrupt downturn of ZFC magnetic moment (showing also the presence of diamagnetic contribution). The difference between ZFC and FC curves increases sharply in a narrow temperature range (35–28 K) and then goes to saturation. In the same temperature range, the specific heat has a well expressed jump (see Figure 6 (b), inset), indicating the second-order phase transition, which is caused by superconductivity.

Figure 7 show the hysteresis loops and temperature dependences of magnetic moment of the bicrystals of Bi and Bi1-xSbx (0 < x < 0.2) alloys. At low temperatures (≤ 36 K), the CIs exhibit one (for various samples Tc ~ 3.7 – 4.6 K) or two (Tc1 ~ 8.3 – 21 K, Tc2 ~ 3.7 – 4.6 K) superconducting transitions. At one transition (Figure 7(b)), the magnetic hysteresis loops are typical for weak ferromagnetic materials and develop against a paramagnetic background. In most samples with two superconducting transitions (Figure 7(a)), the hysteresis loops are symmetric and typical for strong type-II superconductors, but there are also those (especially in specimens of inclination type with higher Sb content), who exhibit

Figure 7. Temperature dependences of magnetic moment and magnetic hysteresis loops in bicrystals with (a) two or (b) one superconducting transitions. (a) (1) Bi0.93Sb0.07Sn, θ1 = 4.60 , θ2 = 1o , (2) Bi, θ =5o, scale for m(T) 1:4, (3) Bi0.85Sb0.15, θ1 = 40 , θ2 = 1o, (4) Bi0.93Sb0.07Sn, θ1 = 40 , θ2 = 1o , scale for m(T) 1:2, (5) Bi0.94Sb0.06Te, θ1 = 90 , θ2 = 2o , scale for m(T) 1:2; Inset: Bi0.93Sb0.07Sn, θ1 = 4.60, (1) 2K, (2) 5K; (b) (1) Bi0.93Sb0.07Sn, θ1 = 750 , θ2 = 4o, scale for m(T) 1:4 (2) Bi0.93Sb0.07Sn, θ1 = 680 , θ2 = 2o, scale for m(T) 1: 10 , (3) Bi0.85Sb0.15Te, θ1 = 190 , θ2 = 3o , scale for m(T) 20:1, (4) Bi0.94Sb0.06Te, θ1 = 690 , θ2 = 2o, (5) Bi0.82Sb0.18Sn, θ1 = 180 , θ2 = 5o , scale for m(T) 300:1, (6) Bi0.85Sb0.15, θ1 = 120 , θ2 = 5o; Inset: Bi0.85Sb0.15, θ1 = 12o , θ2 =5o ,(1) 10K, (2) 300K, (3) 1.8K; (c)

feromagnetic loops (see Figures 8 (a, b)). Note that weak ferromagnetic loops not registered in the CI of Bi bicrystals. Consequently, their manifestation is specific only for interfaces of

Figure 8. Temperature dependences of resistiviti in magnetic fields and magnetic hysteresis loops in bicrystals of inclination type. (a) Bi0.85Sb0.15Sn, θ = 190 ; 1 – 0; 2 – 0.4 kOe; 3 – 1.5 kOe; 4 – 2 kOe; 5 – 4 kOe; 6 – 6 kOe; 7 – 8 kOe; 8 – 10 kOe; 9 – 12 kOe; 10 – 15 kOe; 11 – 20 kOe, Inset: Magnetic hysteresis loops at 1.8K. (b) ) Bi0.85Sb0.15Te, θ = 160 ; 1 – 0; 2 – 0.1 kOe; 3 – 0.2 kOe; 4 – 0.5 kOe; 5 – 1 kOe; 6 – 1.5 kOe, Inset: Magnetic hysteresis loops at 1.9K.

Bi1-xSbx (0 < x < 0.2) alloys which are 3D TI.

The typical resistivity temperature dependencies ρ(T,H) of SDA bicrystals of Bi1-xSbx (0 < x < 0.2) alloys with two superconducting transitions at magnetic fields perpendicular and parallel to the interface planes are shown in Figures 9(a, c). The effect of increasing magnetic field is attributed to expanded transitions in higher fields and their shift in the same way toward lower temperature without significant broadening at H ≤ 2 kOe. The perpendicular H┴c2(T) and parallel H∥c2(T) upper critical fields both determined at the onset and the midpoint of the transition are presented in Figures 9(b, d).

The H┴c2(T) dependences in the immediate vicinity of Tc exhibit a quadratic behavior; further characteristics are linear in T in the entire temperature range, which is specific for weakly anisotropic superconductors. The parallel upper critical field exhibits the same behavior as H┴c2(T), except that at low temperatures (see Figure 9(d), insert) H∥c2(T) manifest tendency to saturation making it possible to estimate the experimental value of H∥c2(0) ≈ 20.5 kOe. It was found that, for various samples of SDA bicrystals of Bi1-xSbx (0 < x < 0.2) alloys, Horbc2 (0) lies (see also [10]) within the range of 24–27 kOe (for the first superconducting phase with higher Tc) and 11–16 kOe (for the second phase with lower Tc).

Critical paramagnetic field Hc2p(0) estimated from the relation µB Hc2p(0) = 1.84 kTc (µB is the Bohr’s magneton) gives a value of 232 kOe ( for the first phase) and 120 kOe (for second phase), which is an order of magnitude higher than the upper orbital critical field. Consequently, the Maki parameter α = √2 Horbc2 (0) / Hp c2(0) in our CIs is very small (α ≈ 0.1 – 0.14), the spin-paramagnetic effect is unimportant, and the conventional orbital upper critical field at zero temperature fully determines the magnitude of Hc2(0).

Figure 9. Temperature dependences of the resistance and upper critical magnetic field Hc2 (T) in bicrystal of Bi0,93Sb0,07 alloy with a low (~ 0,01 at.%) content of Sn, θ1 = 4.60 , θ2 = 1o. (a), (b) H perpendicular; (c), (d) H paralel; (a) (1) 5kOe,( 2) 4kOe,( 3) 3kOe,( 4) 2kOe, (5) 1kOe, (6) 0.5kOe; (b) (1, 3) onset, (2) midpoint; Inset: transition 1, midpoint; (c) (1) 7kOe,( 2) 4 kOe, (3) 2kOe, (4) 1 kOe, 5) 0.5kOe; 1),(3) onset, (2) midpoint; Inset: transition 1, midpoint

The critical field anisotropy γ = H∥c2(0)/ H┴c2(0) at CIs of bicrystals of Bi1-xSbx (0 < x < 0.2) alloys is relatively weak; it decreases from γ ≈ 1.3 – 1.5 (near Tc ) up to γ ≈ 1.0 – 1.1 (at T ≈ 0 K), and insignificantly deviates from the temperature-independent behavior of a one-band superconductor.

The Ginzburg-Landau coherense lengths can be estimated using the formula ξ2 = ϕ0/2π H┴c2(0), where ϕ0 is the flux quantum. It is found, that in the first superconducting phase of CIs of our bicrystals ξ1 ≈ 11 – 12nm, whereas in the second phase ξ2 ≈ 14 – 17nm.

Note that the hysteresis loops of SDA bicrystals lead to lower critical field values of Hc1 ~ (100 – 130) Oe. They are roughly symmetric and exhibit nearly reversible behavior for fields higher than 2 kOe. The shape of the loops does not change essentially with temperature; some of them do not exhibit any initial diamagnetic magnetization peak. These features suggest that interaction between Dirac fermions in a topological insulator may be coherently controlled by the superconducting phase.

The temperature dependences of magnetic moment and upper critical fields of LDA interfaces of Bi1-xSbx (0 < x < 0. 2) alloys are given in Figure 7 and References [9,10]. The values of Hc2 (T) and dHc2/dT are somewhat lower [17] than the values at interfaces with two superconducting transition. For instance, dHc2/dT, depending on the sample composition, take a value of – (0.9– 1.5) kOe/K, while Hc2 (0) ≈ 2.6 – 3.7 kOe. This value corresponds to the zero-temperature coherence length ξ ≈ 30 – 35nm and a superconducting layer thickness of about 100 – 120nm. The data on superconducting layer thickness are in good agreement with SEM data (see Figure 1a).

The ferromagnetic hysteresis loops in LDA interfaces clearly stand out against the paramagnetic background of m(T) with a slightly temperature–dependent saturation moment ms ≈ (0.7 – 1.2) x 10-5 emu/g at H ≤ ±2 kOe. The loops exhibit ferromagnetic properties in the entire temperature interval studied; their form is slightly modified; their width decreases with increasing temperature, despite of diamagnetic response at Tc < 5 K.

Thus, the LDA interfaces of 3D topological insulator Bi1-x-Sbx (0 ≤ x ≤ 0.2) exhibit simultaneously ferromagnetic hysteresis loops and brings out specific characteristics of superconducting layer with the thickness comparable to the total CI thickness, despite the fact that the quantum oscillations data clearly indicate the presence of several homogeneous layers.

Coexistence of magnetic and superconducting states at various interfaces in 3D and 2D superconductivity systems has been reported recently [20]. Typically, this coexistence is associated with the different contribution of the charge carriers in the interface phenomena [21]; in addition, magnetism is due to rather confined electrons around vacancies, while the superconductivity is caused by electrons in paired state [22]. Therefore, we assume [9] that the manifestation of magnetism at Bi-Sb interfaces is attributed to the effect of a strongly pronounced structural disorder (dislocations, local distortions, vacancies etc.), as evidenced by charge carriers mean-free path estimations from quantum oscillations [10]. An increase in the structural disorder leads to the breaking of electron pairs in different areas of the CIs (apparently, some CIs layer components) and to the formation of a ferromagnetic underlying electronic structure, which are in constant competition or coupling with Cooper paired electrons.

4. Conclusions

It has been found that the Fermi surface consisting of layer components of bicrystal interfaces of Bi and BiSb alloys is similar to those of single crystalline specimens, while the shape, elongation and volume of isoenergetic surfaces at CI undergo essential changes that finally stimulates electron pair correlation.

In Bi and 3D TI Bi1-xSbx (0 ≤ x ≤ 0.2) bicrystals, two new frequencies of quantum oscillation of longitudinal Hall effect belonging to the FS of components of the CIs were found. The Hall quasi-plateaus in SDA bicrystals of inclination type was detected, which disappeared at magnetic field reversal, indicating that the flow of Dirac fermions along the CI plane is sensitive to the field orientation and as well as the localization process runs only in a specific direction of the magnetic field.

At the interfaces of Bi and BiSb alloys two new superconducting transitions with critical temperature Tc ~ (3.7 – 4.6) K , Tc ~ (8.3 – 21) K was observed , just as ordinary rombohedral Bi and 3D TI BiSb alloys are not a superconductors.

The interfaces of 3D topological insulator Bi1-xSbx (0 ≤ x ≤ 0.2) exhibit simultaneously ferromagnetic hysteresis loops and brings out specific characteristics of superconductivity, per contra the CI of Bi bicrystals show only superconducting features and do not fully satisfy the conditions of topological insulator development.

Asknowledgements

The authors express their thanks to Prof. V. Nizhankovskii, Prof. T. Palewski and Dr. K.Nenkov for assistance in carrying out of these investigations.

Author details

Fiodor Muntyanu1,2*, Andrzej Gilewski2, Andrzej J. Zaleski3, Vitalie Chistol4, and Krzysztof Rogacki2,3

1 Institute of Electronic Engineering and Nanotechnologies, Academy of Sciences of Moldova, 2028 Chisinau, Moldova

2 Magnet, 50421 Wroclaw, Poland

3 Institute of Low Temperatures and Structural Research, Polish Academy of Sciences, 50-950 Wroclaw, Poland

4 Technical University of Moldova, 2004 Chisinau, Moldova

*Corresponding author: E-mail address: muntean_teodor @yahoo.com (F. M. Muntyanu).

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