1 INTRODUCTION

Bi1 –xSbx alloys are a group of the best low-temperature thermoelectric and magneto-thermoelectric materials, in particular, for coolers and millivolt electronics [1]. There is a limited set of materials that have the necessary thermoelectric, mechanical, and other properties for practical applications. At present, an increase in the thermoelectric efficiency (TEE) Z = α2σ/χ is associated with new phenomena, such as the quantum-size effect and the recently discovered new state of matter known as topological insulators (TIs). Nowadays,  increased  interest  in alloys based on Bi1 –xSbx is due to a predicted increase in the thermoelectric efficiency as a result of the quantum-size effect in size-limited structures [2]. This has stimulated a large number of theoretical and experimental studies in this field [3–5]. In addition, Bi1 –xSbx alloys in the semiconductor region of concentrations with an inverted spectrum [6] are TIs [7–9] in which an increase in the TEE, when the chemical potential is located in the region of the surface band gap [10, 11], was predicted. Thus, the investigation of thermoelectric properties in low-dimensional systems is focused on objects of two types—single-crystal size-limited structures, in which the quantum-confinement effect is implemented, and bulk micro- and nanocomposites. The occurrence of a large number of interfaces, which efficiently scatter phonons, should lead to a significant decrease in the thermal conductivity, which leads to an increase in the TEE.

In this study, we present the results of experimental investigations of the thermoelectric properties of microtextured foils in semimetallic and semiconductor phases, as well as single-crystal wires based on semiconductor Bi–17 at % Sb alloys in which the TI properties manifest themselves, and the quantum-confinement effect is implemented.

2 SAMPLES AND EXPERIMENTAL

Foils from semimetallic (Bi–3 at % Sb) and semiconductor (Bi–9 at % Sb and Bi–16 at % Sb) alloys were fabricated by the method of high-speed crystallization of a melt droplet of a suitable composition on the internal polished surface of a rotating copper cylinder with a crystallization rate of ~5 × 10–5 m/s [12], which provided a uniform distribution of alloy components in the bulk. The foil thickness was t = 15–40 μm, and the X-ray-diffraction investigations indicate the formation of the (1012) texture with an average grain size of 9 μm. The grain planes are arranged parallel to the foil surface, and the symmetry axis C3 coincides with the foil-surface normal. The single-crystal wires Bi–17 at % Sb in glass insulation with diameters from 200 nm to 1000 μm were fabricated by casting from the liquid phase using the Ulitovsky method [13, 14]. The orientation of the wires determined via X-ray diffraction indicates the direction \(10\bar {1}1\) along the wire axis. In this case, the bisector axis C1 is deviated at an angle of 19.5° from the wire axis in the bisector-trigonal plane. The wire diameter was determined by a scanning electron microscope.

The galvanomagnetic and thermal-magnetic properties of the wires and the thermoelectric power of the foils were measured by the two-contact method using an InGa eutectic, which well wets the ends providing ohmic contacts. The resistance and magnetoresistance of the foils were measured by the four-contact method.

To determine the temperature dependence of the thermal-conductivity coefficient for the foil under investigation, we used the method of heat flux based on the Fourier law:

$${\mathbf{h}} = - \chi \operatorname{grad} T,$$
((1))

where h is the heat-energy flux through the sample cross section perpendicular to the flux, and χ is the thermal-conductivity coefficient.

The thermal conductivity was measured at the Institute of Low Temperature and Structural Research, Poland Academy of Sciences, Wroclaw, Poland.

The temperature dependence of the thermal-conductivity coefficient of the foil is determined in the range of 5–300 K. The total measurement error was less than 6%.

3 RESULTS AND DISCUSSION

It is known that diagrams of rotation of the transverse magnetoresistance make it possible to confirm the single-crystal structure of the samples and establish the direction of orientation of the crystallographic axes.

In Fig. 1, we show the rotation diagram for the angle θ of the transverse magnetoresistance ΔR/R = RH – R0(θ) (H ⊥ I) and the magneto-thermoelectric power α(θ), (H ⊥ gradΔT) of the Bi1 –xSbx foils and wires at 80 K, and H = 0.4 T.

Fig. 1.
figure 1

Angular diagrams of rotation of (a) the transverse magnetoresistance and (b) the magneto-thermoelectric power of foils and wires: (1) Bi–3 at % Sb foil, t = 12 μm; (2) Bi–16 at % Sb foil, t = 23 μm; and (3) Bi–17 at % Sb wire, d = 2.1 μm at θ = 0, H || C3, H = 0.4 T, and T = 80 K.

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As can be seen from Figs. 1a and 1b, the magnetoresistance of the single-crystal wires of Bi–17 at % Sb has a high anisotropy (60%) and the thermoelectric-power anisotropy Δα = 20 μV/K at H = 0.4 T (Fig. 1, curve 3), which confirms the single-crystal structure of the wires under investigation. A certain asymmetry (Fig. 1, curve 3) and the presence of extrema at the angles θ = 90° and 275° indicate the deviation of the wire axis from the bisector axis [15].

The absence of anisotropy of both the magnetoresistance and the thermoelectric power α(θ) in the foils (Fig. 1, curves 1 and 2) confirms their microtextured structure.

In the study, we carried out a set of investigations of the temperature dependences of the resistance R(T), the thermoelectric power α(T), and the thermal conductivity χ(T) in foils and wires in the temperature range of 4.2–300 K, and the calculated temperature dependences of the power factor α2σ(T) and the thermoelectric efficiency ZT(T) for the foils from the semimetal and semiconductor Bi1 –xSbx alloys.

In Figs. 2 and 3, we show the temperature dependences of the resistivity ρ(T) and the thermoelectric power α(T) for the foils from Bi–3 at % Sb, Bi–9 at % Sb, and Bi–16 at % Sb, as well as for the single-crystal wires from Bi–17 at % Sb in a glass shell with the diameters d = 200 and 900 nm. At 300 K, the resistivity increases from a value of 1.5 × 10–4 Ω cm for the semimetallic composition of the foils from Bi–3 at % Sb to a value of ~1.9 × 10–4 Ω cm for the semiconductor-alloy foils and wires and is almost independent of the wire diameter d.

Fig. 2.
figure 2

Temperature dependences of the resistivity ρ(T) of foils and wires. Foils: (1) Bi–3 at % Sb, t = 12 μm; (2) Bi–9 at % Sb, t = 27 μm; and (3) Bi–16 at % Sb, t = 23 μm. Wires: (4) Bi–17 at % Sb, d = 900 nm; and (5) Bi–17 at % Sb, d = 200 nm. In the inset, the dependence ρ(103/T) in the logarithmic scale is shown.

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Fig. 3.
figure 3

Temperature dependences of the thermoelectric power α(T) of foils and wires. Foils: (1) Bi–3 at % Sb, t = 12 μm; (2) Bi–9 at % Sb, t = 27 μm; and (3) Bi–16 at % Sb, t = 23 μm. Wires: (4) Bi–17 at % Sb, d = 900 nm; and (5) Bi–17 at % Sb, d = 200 nm. In the inset, the temperature dependence of the power factor α2σ(T) is shown.

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With decreasing temperature, the resistance increases, and there are exponential portions ρ(T) ∝ exp(ΔE/kBT) in the dependences ρ(T) (see the inset in Fig. 2). The slope of the exponential plots depends on the composition of the foil alloys and on the diameter of the wire from Bi–17 at % Sb. Using the linear dependences lnρ(103/T), we determined the thermal gap, which amounted to ΔE = (8 ± 1) meV and (16 ± 1) meV for the foils from Bi–9 at % Sb and Bi–16 at % Sb, respectively. For the single-crystal wires from Bi–17 at % Sb with d = 900 nm, the gap width is ΔE = (19 ± 1) meV and well agrees with the values for bulk samples of a similar composition. For the wires with d = 200 nm, the band-gap width increases in comparison with that for the wires with d = 900 nm to ΔE = (36 ± 1) meV. Such a dependence of ΔE on the wire diameter d is associated with manifestation of the quantum-size effect predicted in [4, 16] and observed in the wires from pure Bi and Bi–3 at % Sb alloys [13, 14]. We should pay attention to the fact that, in the temperature dependences ρ(T) for the Bi–17 at % Sb wires with d = 200 nm, the saturation of the growth resistance ρ(T) is observed in the temperature range T < 70 K, and there is a tendency toward a decrease in the resistance (curve 5) at T → 4.2 K.

As was convincingly shown in [17], such a dependence ρ(T) in semiconductor wires is associated with manifestation of the properties of a topological insulator.

In Fig. 3, we show the temperature dependences of the thermoelectric power of the samples under investigation in the temperature range of 4.2–300 K. The thermoelectric power is negative in the whole range of temperatures and, as in bulk samples, its highest values are reached in foils and wires from semiconductor alloys Bi–17 at % Sb and Bi–9 at % Sb in the temperature range of 50–100 K. With decreasing wire diameter d, the value of the highest thermoelectric power decreases, and the position of the thermoelectric-power peak shifts to higher temperatures. The decrease in the thermoelectric power in thin wires of semiconductor alloys Bi–17 at % Sb is associated with the effect of surface scattering on the electron mean free path resulting in a decrease in the electron contribution to the thermoelectric power in the low-temperature range.

The values of the power factor α2σ(T) calculated according to the experimental data on α(T) and σ(T) are listed in the inset to Fig. 3. The maximum values of 1.2 × 10–4 W/(cm K2) are achieved in Bi–9 at % Sb and Bi–16 at % Sb foils, as well as in the Bi–17 at % Sb wires with d = 900 nm in the temperature range of 70–120 K (see the inset in Fig. 3), which exceeds the values obtained on bulk samples of an appropriate composition. In the high-temperature region of T > 200 K, the highest power factor takes place in the thinnest Bi–17 at % Sb wires with d = 200 nm.

The experimental dependences χ(T) of the thermal conductivity in the temperature range of 4.2–300 K for the foils of the Bi–3 at % Sb and Bi–16 at % Sb compositions are shown in Fig. 4 (curves 1 and 2). For comparison, we show also χ(T) from [18] in the bulk samples.

Fig. 4.
figure 4

Temperature dependence of the thermal conductivity χ(T) foils: (1) Bi–3 at % Sb, t = 12 μm; (2) Bi–16 at % Sb, t = 23 μm; (3, 4, 5) the experimental curves of the bulk samples: (3) Bi, (4) Bi–1 at % Sb, and (5) Bi–12 at % Sb [18]. In the inset, the temperature dependences of the thermoelectric efficiency factor ZT(T). The dashed curve corresponds to the Bi–12 at % Sb bulk sample [18].

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In the temperature range of 200–300 K, the value of the foil’s thermal conductivity practically coincides with that of solid samples. In the bulk samples at temperatures of T < 200 K, the curves χ(T) vary significantly—a sharp increase by two orders of magnitude in the thermal conductivity of Bi–3 at % Sb semimetallic samples and one order of magnitude for Bi–12 at % Sb semiconductor (Fig. 4, curves 4 and 5, respectively).

In the foils under investigation with a similar composition, the sharp growth of the thermal conductivity (at T < 100 K) is suppressed, and the so-called “dielectric peak” manifesting itself in bulk samples at temperatures of 3–4 K shifts to the temperature range of 10–30 K (Fig. 4, curves 1 and 2). A decrease in the thermal conductivity by two orders of magnitude in comparison with Bi–Sb bulk samples is observed at T < 200 K for semimetal Bi–3 at % Sb foils and by one order of magnitude for semiconductor Bi–16 at % Sb foils.

It is known that the peak of the thermal conductivity is observed in the temperature dependence χ(T) for bulk samples from pure Bi at Tmax = 3.5 K. At temperatures above the peak temperature, an exponential temperature dependence of the thermal conductivity of Bi is observed due to the presence of umklapp processes (U processes) [19]. Exponential growth of the thermal conductivity with decreasing temperature <100 K is induced by exponential growth of the mean free path of phonons (l ∝ eΘ/bT) (Fig. 4, curve 3), where Θ = 120 K is the Debye temperature. At temperatures of T < 15 K, the free-path length of phonons in bismuth increases exponentially only until it becomes comparable with the transverse sample size [20]. As a result, a peak arises in the temperature dependence of the thermal conductivity with a further decrease in the temperature due to phonon scattering at the sample boundaries.

In the foils which have a granular structure with a grain diameter of ~9 μm and with the presence of twins, a medium is implemented with a large number of interfaces, which leads to the additional scattering of phonons at the boundaries of grains and, consequently, to a significant decrease in the thermal conductivity.

The results of investigation of the temperature dependences of the resistivity ρ(T), the thermoelectric power α(T), and the thermal conductivity χ(T) enabled us to calculate the temperature dependences of the thermoelectric efficiency ZT(T) = α2σ/χ(T) (see the inset in Fig. 4).

As can be seen from Fig. 4, the peak values of the thermoelectric efficiency ZT are observed in the foils from the Bi–16 at % semiconductor alloys with n-type conductivity in the temperature range of 120–220 K; it amounts to ZT = 0.5, which exceeds almost two times the peak values obtained for the Bi–12 at % Sb semiconductor n-type alloys and in the films of similar compositions and similar crystallographic orientations [21–23].

Thus, the foils under investigation from the Bi–16 at % Sb semiconductor alloy with a thermoelectric figure of merit of ZT ≈ 0.5 in the region of 80 K < T < 150 K can be used as n-type legs of low-temperature miniature thermoelectric energy converters.

4 CONCLUSIONS

The comprehensive investigation of the temperature dependences of the thermoelectric power α(T), the resistivity ρ(T), and the thermal conductivity χ(T) of microtextured foils and single-crystalline wires on the basis of semimetal and semiconductor Bi1 –xSbx alloys was carried out. It was found that the thermal conductivity is two orders of magnitude lower in the semimetal Bi–3 at % Sb foils at low temperatures and one order of magnitude in the semiconductor Bi–16 at % Sb foils than in bulk samples of a similar composition. This is related to an increase in the scattering of phonons at the surface and grain boundaries and leads to an increase in the thermoelectric efficiency in the temperature region of 120–200 K.

It is established that the band gap in thin Bi–17 at % Sb (d = 200 nm) semiconductor wires grows due to manifestation of the quantum-dimensional effect and, in the region of low temperatures, a decrease in the resistivity is detected due to manifestation of the TI properties. It is shown that a decrease in the wire diameter d of the Bi1 –xSbx semiconductor alloys leads to an increase in the power factor at T > 200 K.

FUNDING

This study was supported by the D. Ghitu IEEN Institutional Project no. 15.817.02.09A.