A Novel Method to Significantly Improve the Mechanical Properties of n-Type Bi_(1−x)Sb_x Thermoelectrics Due to Plastic Deformation

Abstract

A unique method was developed to significantly improve the strength of Bi_(1−x)Sb_x single crystals, the most effective thermoelectric (TE) materials in the temperature range from 100 to 200 K due to their plastic deformation by extrusion. After plastic deformation at room temperature under all-round hydrostatic compression in a liquid medium, n-type Bi–Sb polycrystalline solid solutions show a significant increase in mechanical strength compared to Bi–Sb single crystals in the temperature range from 300 to 80 K. The significantly higher strength of extruded alloys in comparison with Bi–Sb single crystals is associated with the development of numerous grains with a high boundary surface as well as structural defects, such as dislocations, that accumulate at grain boundaries. Significant stability of the structure of extruded samples is achieved due to the uniformity of crystal plastic deformation under all-round hydrostatic compression and the formation of the polycrystalline structure consisting of grains with the orientation of the main crystallographic directions close to the original single crystal. The strengthening of Bi–Sb single crystals after plastic deformation allows for the first time to create workable TE devices that cannot be created on the basis of single crystals that have excellent TE properties, but low strength.

1. Introduction

A significant increase in the efficiency of thermoelectric (TE) cooling at temperatures below 200 K is currently one of the urgent problems of thermoelectricity. (Bi,Sb)₂(Te,Se)₃ solid solutions have remained the preferred thermoelectric materials for cooling purposes for many decades [1]. However, at cold temperatures of T_c ≤ 200 K, thermoelectric materials based on Bi₂Te₃ compounds are not effective for refrigeration due to low thermoelectric efficiency at these temperatures [1,2,3]. Their figure of merit Z for both n- and p-type legs does not exceed 2.5 × 10⁻³ K⁻¹ at T = 200 K, restricting their practical application at cold temperatures of T_c ≤ 200 K [2,3,4,5].

An effective way to increase the efficiency of the TE device at cold temperatures is to maximize Z values. The most effective known thermoelectric (TE) materials at temperatures of T ≤ 180 K are single crystals of n-type Bi–Sb solid solutions with Sb content from 7 to 15 at.% [2,3,4,5]. Below this temperature, the Z value of these crystals is two to three times higher than that of the best n-type TE materials based on Bi₂(Te,Se)₃ solid solutions in a transverse magnetic field of B ≤ 1 T [3,5]. The thermoelectric properties of these TEs were first measured more than 100 years ago by G. Gehlhoff and F. Neumeier, as indicated by Smith and Wolfe [6], but in all of the early works on these materials, polycrystalline specimens of unknown purity were used. The ultimate bending strength σ_b of these Bi–Sb single crystals can reach 40 MPa at room temperature when the loading force is perpendicular to the trigonal axis [3,7,8,9]. To improve the bending strength of Czochralski-grown ingots, the effect of the growth parameters and the size and surface quality of the samples on σ_b has been investigated in detail by Belaya et al. [3,7]. At 77 K, the bending strength σ_b is relatively low (10–20 MPa); for example, when the bending force in a three-point test is orientated along the <10$\overline{1}$> and <1$\overline{2}$1> axes, the σ_b values of Bi_0.91Sb_0.09 single crystals were about 18–28 MPa [8]. At T < 200 K, it is promising to use n-type branches from Bi–Sb single crystals together with superconducting branches from high-temperature superconductors instead of p-type branches [8,9].

However, single crystals of Bi–Sb solid solutions have not received practical application as a material for TE legs. This is mainly due to the ease of cleavage of single-crystal branches along the (111) plane perpendicular to the direction of the maximum values of Z [2,5,7,10]. In other words, Bi–Sb single crystals cannot withstand any bending along their trigonal axis, although it is in this direction that the figure of merit is the best. Therefore, the decisive condition for the successful use of TE-legs from Bi–Sb solid solutions is the development of methods for increasing the strength properties of these TE materials.

The processing (hot and cold pressing, extrusion, etc.) of polycrystalline TE materials may significantly affect the microstructure and improve their properties. A number of works indicate that extruded Bi–Sb solid solutions are promising as n-type TE material at temperatures of T ≤ 180 K [2,5,8,9,10]. Extrusion of Bi–Sb materials may introduce a preferential orientation of the grains. From a thermoelectric point of view, the presence of a strong texture in this material would be favorable. Actually, if all the grains could be orientated so that their trigonal axes are nearly parallel, then their thermoelectric properties should approach those of the single crystals [2,3,4]. Annealing the extruded Bi–Sb alloys restores high mobility of electrons but destroys the preferred grain orientation in the <111> direction along the extrusion axis. As a result of high deformation during extrusion and subsequent annealing of Bi–Sb alloys, a textured polycrystal structure is formed with a characteristic grain size of 0.02–0.05 mm and high-angle grain boundaries [5,8].

Mechanical alloying of the elemental materials as reported by Martin-Lopez et al. [10] is also a powerful method for the synthesis of homogeneous powders of the Bi_(1−x)Sb_x alloy at x = 0.07, 0.1, 0.12, 0.15, 0.22, with grain sizes about 10 μm, using relatively short milling times (4–15 h). The mechanical properties of the Bi_0.85Sb_0.15 alloy prepared by mechanical alloying and consolidated by hot extrusion were investigated at two fixed temperatures (77 and 293 K). The modulus of rupture of the polycrystalline material was roughly the same at these temperatures, with a value of about 100 MPa compared to 10 and 20 MPa for a Bi_0.85Sb_0.15 single crystal [10]. However, sintering methods offer the advantage of improving the mechanical strength of alloys but, unfortunately, degrade the thermoelectric performance as a result of random orientation. Moreover, impurities such as oxygen during power elaboration could be factors hindering improvement of the figure of merit [2].

However, the application of known extrusion methods [11] to Bi–Sb solid solutions does not make it possible to obtain a material possessing high strength and thermoelectric characteristics simultaneously. The effect of plastic deformation during extrusion on the structural, mechanical, and TE properties of Bi–Sb solid solutions has been little studied. This work is devoted to the development of a method for cold deformation of Bi–Sb single crystals at high hydrostatic pressure, which will improve the strength characteristics of the material without substantial decrease in the TE efficiency. A detailed study of the structure, texture, and mechanical properties of deformed Bi–Sb alloy was carried out in the temperature range from 300 to 80 K.

2. Materials and Methods

2.1. Growth of Single Crystals

The initial Bi_0.91Sb_0.09 single crystals were obtained using the Czochralski method [3,5,9,12,13,14], which is currently used in the production of more than half of the technically important crystals grown from the melt; at the same time, about 60% of all single crystals are used to obtain semiconductors [13. The single crystal was grown on a single crystal seed, and the melt was fed with solid antimony during the growth without additional alloying. The purity of the starting materials was 99.9999%. The single crystals 25–35 mm in diameter and 40–50 mm in length were grown in an atmosphere of high-purity helium at 140–150 kPa at the crystal pulling speed of 1.7 × 10⁻⁶ m s⁻¹.

2.2. Extrusion of Single Crystals

To increase the strength of Bi_0.91Sb_0.09 single crystals, the method for their plastic deformation in a liquid medium at high hydrostatic pressure was used [5,8,9], for which the scheme and view of the extruder are shown in Figure 1. Here, castor oil used as a working fluid was first compressed to a pressure P, and the TE material was then extruded. Bi–Sb single-crystal samples for plastic deformation 12 mm in diameter and length of 25–30 mm were cut on an electric discharge machine and etched in concentrated HNO₃ to remove the damaged layer of 40–50 μm depth. The cylindrical axis was oriented along the trigonal axis [111] of the single crystal.

For extrusion, blanks with a smooth surface without visible cracks were used. Quality control of the workpiece surfaces was carried out using an MBS-9 binocular microscope. The pressure of the working fluid P required for the extrusion was selected based on the microhardness HV of Bi_0.91Sb_0.09 single crystals, which was measured using a PMT-3 microhardness tester on the cleavage plane (111) at room temperature and ranged from 460 to 500 MPa. Considering the plastic deformation of a crystal with an internal crack placed in a medium with a high hydrostatic pressure P, a criterion P > HV was developed for preventing crystal cracking during plastic deformation, where HV is the microhardness of the crystalline material. To perform plastic deformation of Bi_0.91Sb_0.09 single crystals at room temperature (T = 300 K), the pressure P of the surrounding liquid was chosen as 600–1100 MPa.

The extrusion process in a hydrostat consists of two stages: preparatory and extrusion. In the preparatory phase, the working fluid is compressed as the plunger moves inside the pressure chamber, prior to extrusion of the TE material. The preparatory stage ends when the hydrostat plunger contacts the extruder plunger. Extrusion ratio, showing how many times the length increased or the cross-sectional area of the product decreased in one pass of extrusion, was 1.2, 3, 5, and 10.

2.3. Study of the Crystal Structure of Single Crystals and Extruded Specimens

The deformation of the material in the surface layer occurs under conditions that differ significantly from the conditions of deformation of the bulk of the crystal. Therefore, the crystal structure of the surface layer differs from the crystal structure of the bulk of the crystal. The 10–15 μm thick surface layer was removed by etching in 70% nitric acid; then, the sample was washed sequentially in concentrated hydrochloric acid, distilled water, and isopropyl alcohol and dried.

For structural study of the perfection of the grown crystals, a method with reflection recording was used. In this case, the cleavage plane (111) was investigated. For the topographic study, the Schultz method was used with a point source of white X-ray radiation [15,16]. The cleavage plane of the sample was located at an angle of 20–25° to the source. On radiographs, the surface protrusions are displayed as light stripes, and the depressions are dark. The width of these stripes was used to determine the angle of disorientation of the substructural components. Additionally, the divergent beam X-ray technique [17] has been used for precision measurement of the lattice period of perfect single crystals and for studying structural changes in deformed crystals. By changing the shape and width of the lines, it is possible to determine the angles of rotation of the substructural components of the crystal and the perfection of a small region with a length of 10–50 μm.

To determine the quantitative characteristics of the preferential structural orientation, we used a model in which the texture of a real crystal is considered as a combination of perfectly oriented substructures corresponding to the observable crystalline textures. In this model, a technique for magnetometric analysis was developed, which is almost independent of the state of grain boundaries, the presence of micropores at grain boundaries, as well as inevitable fluctuations in the concentration of impurities and composition [18]. This method is based on the magnetic properties of Bi–Sb crystals, which are anisotropic diamagnets at room temperature, with a weak dependence of the magnetic susceptibility on composition.

The rotating moment which acts on a textured diamagnetic sample of the mass m in a magnetic field can be determined through the free energy of the sample F as [18]

$$M(\varnothing )=-\partial \mathrm{F}/(\partial \varnothing )=-M{{\displaystyle \sum }}_i{w}_i\partial f_i/\partial \varnothing$$

(1)

where i is one of the textures, w_i = m_i/m is the specific content of the ith texture; f_i is the specific (per unit mass) free energy of the ith texture in the magnetic field, and ∅ is the angle between the chosen direction in the sample, which is usually the one with the highest symmetry.

The magnetic properties of single crystals of diamagnetic Bi–Sb alloys are determined by two main values of the specific magnetic susceptibility tensor, namely k₃ along the (111) trigonal axis and k₁ perpendicular to the (111) direction. If the (111) axis in the ith texture is tilted at an angle ∅_i to the direction of the uniform magnetic field H, then

$$f_i=- \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right. H^2\left[k_{3 }-\left(k_1-k_3\right)si{n}^2{\varnothing }_i\right].$$

(2)

By varying the orientation of poly- and monocrystalline samples with respect to the magnetic field so that the rotating moments are maximal for the given field, we obtain the specific content of the prevailing texture

$$w_{1= }{m}_0/m\times {\left(H_0/H\right)}^2\times \overline{M}/{\overline{M}}_0$$

(3)

where $\overline{M}$ and ${\overline{M}}_0$ are the maximal rotating moments for a textured polycrystal in the field H and for a single crystal of the same composition in the field H₀, respectively.

Since Bi–Sb alloys have a predominant <111> texture at a degree of deformation over 90%, the {111} axial texture prevails, and in this case, the {111} basal planes of the grains are oriented along the extrusion axis. Using this method, which does not require direct measurements of the magnetic susceptibility, the content of texture <111> (W_<111>) was obtained by changing the orientation of the polycrystalline samples with respect to the magnetic field so that the torques were maximum for a given field.

2.4. Three-Point Bending Test of Bi_0.91Sb_0.09 Alloy

A schematic diagram of the rhombohedral and hexagonal lattice of Bi–Sb solid solutions is shown in Figure 2a. A hexagonal lattice system is conventionally used with $\overrightarrow{a}$₁, $\overrightarrow{a}$₂, $\overrightarrow{a}$₃, and $\overrightarrow{c}$ lattice vectors. The angles between the $\overrightarrow{a}$ vectors equal 120°, and they lie in the plane normal to the $\overrightarrow{c}$ vector. A Cartesian coordinate system is also shown with axes denoted binary (OX), bisectrix (OY), and trigonal (OZ). The trigonal axis is aligned with the $\overrightarrow{c}$ direction in the lattices. The bisectrix axis is defined as being orthogonal to the trigonal and binary axes.

For n-type branches from Bi–Sb crystals, the most dangerous deformation is bending perpendicular to the direction of maximum TE efficiency, which coincides with the trigonal axis for single crystals or with the extrusion axis for crystals after plastic deformation. Therefore, from the point of view of their use as a TE material, the most important strength parameter for crystals of Bi–Sb solid solutions is the ultimate bending strength σ_b, when a bending force F_b is applied perpendicular to the trigonal or extrusion axis. Bending tests of samples by the three-point test method (Figure 2b) were carried out on an Instron 37-10-16 at temperatures of 77–300 K in a special chamber cooled with a mixture of gaseous and liquid nitrogen [9].

The surfaces of the samples undergoing bending were carefully mechanically ground and then chemically polished in a solution composed of 250 mL of H₂O + 70 mL of HCl + 83 g of CrO₃. The thickness of the removed layer was 5–10 μm. The average grain size in polycrystals of extruded samples was determined on the basis of the number N of grains observed in a given area S = 0.5 × 0.5 mm², containing at least 100 grains. Then, the diameter d of a circle with an area equal to the area of one grain was determined. The value d was taken as the average grain size:

d = (4/π)^0.5(S/N)^0.5

(4)

In bending tests, Bi–Sb crystals of rectangular cross-section were used with dimensions h = (1 ± 0.1) mm, b = (4 ± 0.1) mm, l = 10 ± 0.1 mm, for which l / h ≈ 10, which justifies the application of the Equations (5) and (6) for calculation of the maximum tensile deformation ε_b and strength σ_b during bending [19]:

ε_b = 6·Δ_b·h/l²,

(5)

σ_b = l.5·F_b·l/(b∙h²),

(6)

where Δ_b is the maximum deflection of the specimen and F_b is the bending force.

3. Results

3.1. Crystal Structure of Bi_0.91Sb_0.09 Single and Polycrystalline Crystals

The dislocations within and at the boundaries of the blocks are responsible for the total rotation field of the crystal lattice. In Figure 3a, an X-ray diffraction pattern of Bi–Sb crystals is presented. The shape and width of the diffraction lines indicate the high perfection of the grown crystals. Figure 3b shows the images of diffraction spots from the crystal grown using the Schultz method. The images show images of substructural components with boundaries parallel and perpendicular to the growth axis. The size of the reflexes from individual substructural components and the width of the gaps between them were used to determine the sizes of substructural components and the angles of misorientation between them.

It was shown by the Schultz topographic method that, first, the blocks are predominantly elongated in the direction of crystal elongation; second, the transverse dimensions of the blocks are from 0.1 to 2.0 mm; third, the angles of rotation of the blocks relative to each other are in the range from several minutes to 1°. Note that a substructure in the form of weakly misoriented single-crystal blocks was also observed earlier in Bi–Sb crystals grown by the Czochralski method [5,9,12,13,14]. The formation of blocks is associated with the fact that the growth of Bi–Sb crystals is accompanied by the appearance of local concentration overcooling regions at the crystallization front. The results obtained using these methods indicate the presence of weakly misoriented blocks, relatively free of dislocations, with boundaries consisting of dislocation networks. The formation of blocks is due to local concentration of overcooling regions at the crystallization front. Blocks are regions that are relatively free of dislocations, and their boundaries consist of dislocation networks. The length and width of single-crystal blocks along the direction of crystal growth and perpendicular to it were 0.7–1.4 mm and 0.1–0.2 mm, respectively, at an angle of rotation of the blocks relative to each other of 3–5 arc min.

In Figure 4, the typical diffraction pattern of the sample after deformation is compared with the isotropic polycrystalline structure of the reference sample. This indicates the predominant orientation of the axes of trigonal crystallites in the direction of the extrusion axis, namely, the predominance of the <111> texture. As noted above, to quantify the degree of this texture, a magnetometric method was used in which the W_<111> content in the direction of the extrusion axis was calculated using Equation (3). With an increase in the extrusion ratio from 1.2 to 10, the value of W_<111> reduces from 88 to 50% (

Table 1. Influence of the extrusion ratio and annealing temperature for 8 h on the content of the crystallographic texture (W_<111> × 100%) in the direction of the extrusion axis.

Ke	No Annealing	Annealing
		150 °C	180 °C	210 °C
1.2	88 ± 4	83 ± 5	80 ± 5	72 ± 5
3	72 ± 5	62 ± 5	55 ± 5	38 ± 4
5	60 ± 5	48 ± 4	41 ± 4	23 ± 3
10	50 ± 4	33 ± 4	26 ± 4	14 ± 2

). The W_<111> values were determined by averaging five identically prepared samples over the results of magnetometric measurements. Table 1 also shows the root-mean-square errors of the reduced values of W_<111>. The values of pole density P_hkl for extruded samples before and after additional annealing are presented in

Table 2. Pole density P_hkl of the normals to reflecting planes (hkl) in the direction of the extrusion axis for Bi_0.91Sb_0.09 crystals, before and after annealing at 180 °C for 8 h.

Ke	As Extruded				After Annealing
	1.2	3	5	10	1.2	3	5	10
P003	29	21	15.2	10.1	25	16	9	5
P102	0.1	0.3	0.4	0.5	0.2	0.4	0.2	0.4
P104	0.1	0.2	0.4	0.6	0.2	0.2	0.4	0.3
P105	0.3	0.6	0.7	0.8	0.5	0.4	0.9	1.3
P022	1.4	1.6	1.3	1.2	1.7	1.9	1.1	1.6
P107	0.3	0.6	0.5	0.7	0.4	0.7	1.0	1.1
P116	0.2	0.3	0.3	0.5	0.2	0.3	0.6	0.4
P212	0.1	0.5	0.6	0.8	0.3	0.4	0.9	1.4
P108	0.8	1.0	0.9	0.8	1.2	1.3	1.2	1.0
P214	0.1	0.4	0.4	0.3	0.2	0.6	0.5	0.5

. Annealing was carried out at a temperature of 180 °C for 8 h in an inert gas atmosphere at a pressure of 150 kPa. In calculating P_hkl, only the most intense diffraction lines were taken into account. According to the data presented in Table 2, the values of the pole density P₀₀₃ with Miller indices in multiples of (001) significantly exceed P_hkl of other observed diffraction lines. This indicates the predominant orientation of the trigonal axes of the crystallites along the extrusion axis or, in other words, the dominance of the <111> texture.

Annealing of Bi–Sb extruded crystals causes a further decrease in the W_<111> values. As shown in Figure 5 and listed in Table 1, annealing of extruded samples causes intense recrystallization, leading to reorientation and coarsening of polycrystalline grains. In this case, the scattering of the <111> texture occurs more intensely for the samples obtained at higher values of the K_e (Figure 5).

Figure 6 shows examples of the structure of cleavages of Bi–Sb extruded samples at K_e = 1.2, 3, and 5. After annealing, on the cleavages perpendicular to the extrusion axis, there are triangular etching pits typical for the etched surface of cleavages along the (111) plane of single-crystal samples (Figure 6a). This indicates that during the extrusion process, the initial crystallographic orientation of the crystals is largely retained. Elongated grains coincide with the direction of extrusion (Figure 6b). At high magnification, the features of the cleavage of the sample extruded at K_e 5 show that the structure was practically isotropic in the plane perpendicular to the extrusion axis (Figure 6c). In Figure 6d, the scanning electron image of the fracture surface for the representative extruded Bi_0.91Sb_0.09 sample shows the excellent densification and uniform distribution of grains. No pores or cracks were identified on the surface of the sample, which is in agreement with the high density of the extruded samples, ~99% of the theoretical density. The estimated average grain size ranges from 20 to 50 μm, and the size of the largest grains does not exceed ~80 μm.

As a result of structural studies, it is shown that a strongly textured polycrystalline Bi–Sb at K_e ≤ 3 is formed with a predominant orientation of the crystallographic axes of grains with an average size varying from 20 to 50 μm, coinciding with the orientation of the corresponding crystallographic axes of the single crystal before extrusion. At the extrusion ratio K_e ≥ 5, the structure was practically isotropic in the plane perpendicular to the extrusion axis. For K_e = 10, as a result of extrusion, the axial texture <$11\overline{1}$> dominates in the direction of the extrusion axis, and the structure of the extruded polycrystal is practically independent of the orientation of a single crystal before extrusion.

3.2. Mechanical Properties of Bi_0.91Sb_0.09 Single Crystals and Polycrystalline Material

After homogenizing annealing at 453 K for 8 h, bending tests were performed on Bi_0.91Sb_0.09 single crystals at 77–300 K. The tested crystals were made so that the direction OZ (Figure 2b), coinciding with the trigonal axis of the crystal, was perpendicular to the direction of the bending force F_b. In a plane perpendicular to the trigonal axis, the bending force F_b was oriented along the bisectrix or binary axes (OY or OX), that is, F_b || <1$\overline{2}$1> or F_b || <10$\overline{1}$>, respectively. Ultimate bending strength σ_b of Bi–Sb extruded samples was much higher than that of single crystals and increased significantly with an increase in the extrusion ratio (Figure 7) and insignificantly with a decrease in the test temperature (

Table 3. Bending strength of extruded Bi–Sb polycrystals depending on the test temperature.

Ke	σb, MPa
	300 K	200 K	150 K	80 K
1.2	27 ± 3	27 ± 3	29 ± 3	31 ± 3
3	35 ± 3	36 ± 3	38 ± 3	42 ± 3
5	48 ± 4	52 ± 4	55 ± 4	57 ± 4
10	65 ± 4	68 ± 4	70 ± 4	72 ± 4

). After extrusion at K_e ≥ 3, the bending strength of the extruded crystals was practically independent of F_b orientation in the plane perpendicular to the direction of the extrusion axis.

Experimental stress–strain σ_b–ε_b dependencies (Figure 8) illustrate the main stress state during bending of samples. Fracture of the single crystals upon bending in the direction of the bisectrix axis (Figure 2, F_b || <1$\overline{2}$1>) occurs almost simultaneously with the appearance of the first stress breakdown at practically zero plastic deformation. In the meantime, at F_b, directed along the binary axis (F_b || <10$\overline{1}$>), single crystals can undergo plastic deformation about 0.08%. In the directions of the binary axes of single crystals (Figure 2a, F_b || <10$\overline{1}$>), just after the yield point at 22.7 MPa, the stress-strain curves show progressive plastic deformation with sharp drops and rises in stress (Figure 8a).

For the samples after extrusion at K_e 3, by analogy with the case of bending of single crystals, the dependences σ_b–ε_b were investigated for two variants of orientation of the bending force F_b, namely (a) F_b || <10$\overline{1}$>, that is, parallel to the predominant direction of the binary axes of crystallites (Figure 2a), which coincides with the direction of these axes of the initial single crystal and (b) F_b || <1$\overline{2}$1>, that is, parallel to the predominant direction of the bisectrix axis OY. Both orientations of the bending force F_b were found to have little effect on bending strength of extruded samples at 80 K. For example, for F_b || <10$\overline{1}$> or <1$\overline{2}$1> directions, the bending strength was 31 ± 3 and 28 ± 3 MPa at K_e 1.2 as well as 42 ± 3 and 41 ± 3 MPa at K_e 3, respectively.

For samples extruded at K_e ≥ 3, the dependences σ_b–ε_b were investigated at an arbitrary orientation of the bending force F_b in the plane perpendicular to the extrusion axis. For specimens extruded at K_e = 1.2, the bending force was oriented along the predominant direction of the binary axes. As a result of the extrusion of Bi–Sb crystals at K_e ≥ 5, a TE material with an axially symmetric structure is formed. The extruded samples were investigated in the temperature range 80–300 Κ before and after annealing at 453 K for 8 h. For samples extruded at K_e = 10, the stress–strain dependences at room temperature had a relatively pronounced nonlinear region without significant disruptions of the σ_b values with increasing ε_b up to the destruction of the sample. With decreasing temperature, a decrease in the nonlinear section on the σ_b–ε_b curves was observed. At the lowest temperature of 80 K, the nonlinear section on these curves corresponding to plastic deformation was practically absent (Figure 8b). As a result of extrusion at K_e ≥ 5, a polycrystalline structure with a high ultimate bending strength was formed, reaching values over 70 MPa in the temperature range 80–300 K.

Thus, the data obtained on the effect of extrusion on ultimate strength to bending show that plastic deformation of Bi–Sb single crystals under conditions of high hydrostatic pressure, even at low extrusion ratios, makes it possible to obtain a thermoelectric material with a significantly higher ultimate strength to bending compared to the original single crystals in the entire temperature range 80–300 Κ. In addition, in the same temperature range at different extrusion ratios, elongation-to-fracture (plastic deformation) of the extruded crystals increases two to three times compared to single-crystal samples, remaining rather low (Figure 9).

4. Discussion

As follows from Figure 7, Figure 8 and Figure 9, plasticity (elongation-to-fracture) and the strength of extruded samples at high hydrostatic pressure can reach values several times higher than those for single crystals. This is consistent with the data of compression tests of Fe₃Si single crystals [20] and AISI 4330 steel [21,22]. As reported by Lorrek and Pawelski [20], brittle materials are usually successfully deformed by additional applying high hydrostatic pressure, which leads to an increase in the plasticity of single crystals of iron silicide by several times, especially at a certain crystallographic orientation of the sample relative to the loading axis (Figure 10a). In the Tadano and Hagihara model [21], the calculated flow stresses under uniaxial compression are very close to the experimental results obtained by Spitzig et al. [22] at hydrostatic pressures p = 0.552, and 1104 MPa for AISI 4330 steel (Figure 10b). In this analysis, the hydrostatic pressure is applied as the initial stress, which means that σ₁₁ = σ₂₂ = σ₃₃ = −p at the initial state. In general, the ultimate plastic deformability of a material is a function of stress configuration, strain rate, and temperature. The effect of the stress configuration is the most significant and characterized by the mean normal stress σ_m = 1/3 (σ₁ + σ₂ + σ₃), which may not be low enough to avoid cracking or fracture during the deformation process of a brittle material. In this case, σ_m can be reduced by applying hydrostatic pressure.

The significant stability of the structure of the samples obtained by extrusion of single crystals in a high pressure liquid medium at K_e ≤ 3 is explained by the fact that the all-round hydrostatic compression used in the process of extrusion increases the uniformity of crystal plastic deformation, which leads to the formation of a more stable polycrystalline structure of extruded samples, consisting of blocks with an orientation main crystallographic directions close to the original single crystal.

As follows from Figure 5, annealing of the extruded samples causes a decrease in the specific content of the crystallographic texture W_<111>, which reflects intense recrystallization leading to reorientation and coarsening of polycrystalline grains. However, such heat treatment improves the TE properties of the alloy [1,2]. Of course, the value of W_<111> reaches the maximum if the trigonal axis of the single crystals is directed along the extrusion axis.

At low values of extrusion ratio (K_e ≤ 3), Bi–Sb polycrystalline alloy is strongly textured and has a preferred orientation of the crystallographic axes of grains coinciding with the orientation of the corresponding crystallographic axes of the single crystal before extrusion. However, a higher plastic deformation of single crystals (K_e ≥ 5) leads to practically isotropic grains in the plane perpendicular to the extrusion axis (Figure 6c). At the highest plastic deformation of the single crystal at K_e = 10, the axial texture <111> dominates in the direction of the extrusion axis, and the structure of the polycrystalline extruded material is practically independent of the orientation of a single crystal before extrusion.

At room temperature, elastic elongation of Bi_0.91Sb_0.09 alloy depends on applied stress, which can be calculated from Young’s modulus and is about 0.08% (Figure 8). Gopinathan and Padmini [23] observed anomalous variations in elastic properties in the Sb concentration range from 2.2 to 14.89 at.%, in which, for example, Young’s modulus (E) varied nonmonotonically from 25 to 37.5 GPa. In this defined region of critical concentrations, this is interpreted in terms of changes in lattice parameter and carrier concentration. In accordance to Woodcox et al. [24], the E values for the Bi_0.91Sb_0.09 alloy are in the range from 26 to 32 GPa. Taking the modulus value equal to 30 GPa, and stress at the transition point of the linear dependence σ_b–ε_b into the nonlinear one (yield point) equal to 23 MPa, we find elastic deformation is 23 × 100%/30,000 = 0.077%. Thus, the plastic deformation (elongation-to-fracture) of extruded samples at 300 K, obtained at extrusion ratios of 3 and 10, is 0.08 and 0.28%, respectively (Figure 8b and Figure 9). At 80 K, the plastic deformation of the extruded samples is zero and 0.09% for the K_e values of 3 and 10, respectively.

As shown above in Figure 8a, at bending force directed along the binary axis OZ (F_b || <10$\overline{1}$>), the stress–strain curves of single crystals show sharp drops and rises after the yielding point at 22.7 MPa. The occurrence of sharp sawtooth stress drops or serrations during compression and tension deformations of bismuth and antimony single crystals due to twinning was indicated earlier [25,26,27,28]. Klassen-Neklyudova [25] summarized that elastic and residual twinning is formed under loading of bismuth and antimony crystals. Slonaker and Smutz [26] found that the transition from slip to twinning as the predominant deformation mechanism of 99.99% Bi was observed at tensile deformation when the angle Θ between the normal to the true cleavage plane (111), and the major axis of the single crystal equals 70°. The value of the critical resolved shear stress for slip on the (111) plane in bismuth at room temperature, obtained by Steegmuller and Daniel [27], is in agreement with the results of Slonaker and Smutz [26]. Deformation modes observed after compression tests at the strain rate of 1.7 × 10⁻⁴ on bismuth crystals of various orientations, mainly at the temperature −80 °C (193 K) and more, are slip on (111) plane and twinning along the (011), (101), and (110) planes [27]. In the tensile test, twinning easily occurs at a strain rate of 1 × 10⁻³ s⁻¹ [28] and is suppressed at a strain rate of 1 × 10⁻⁴ s⁻¹ or lower [29,30].

The destruction of single crystals upon bending when F_b || <1$\overline{2}$1> occurs almost simultaneously with the appearance of the first breakdown of stress at a deformation of about 0.05%, while with F_b directed along the binary axis (F_b || <10$\overline{1}$>) single crystals can undergo plastic deformation three times greater. Thus, when the bending force F_b is directed along the binary or bisectrix axis of the single crystal, differences in σ_b values are apparently determined by the conditions for the formation of microcracks μ (Figure 11) in the vicinity of the B–C line (Figure 2b), where maximum tensile forces T and stresses act upon the bending of single crystals [19,31].

Under the action of sufficient compression or tension stresses directed along one of the equivalent <100> axes, twins can be formed in Bi–Sb single crystals with a rhombohedral lattice symmetry in the form of thin interlayers with twinning planes {110} and twinning directions <100> [25]. With this bending scheme, one of the {110} twinning planes makes an angle of 71° with the (111) plane, and one of the twinning directions <100> is perpendicular to the binary or bisector axis coinciding with the B–C line, depending on the orientation F_b. When a single crystal is bent, tensile forces T act in the vicinity of the B–C line. As a result, conditions are realized for the formation of twinning layers along the {110} plane. The symmetry of the tensile forces upon bending of single crystals with respect to the middle cross-section ABCD (Figure 11), which coincides with the (111) plane, leads to the formation of a twin structure symmetric with respect to the ABCD plane. As a result, twin intersection regions appear in the vicinity of the BC line.

In the region of intersection of twins, there is a high density of dislocations and the appearance of cavities—Rose channels [32]—that have been actually reported as early as 1868 by Rose, who described these channels, or voids, formed by intersecting twins in calcite [24,31]. These were later described as microcracks in body-centered cubic (BCC) metals by Priestner [33], and by Sleeswyk [34] to explain ductile–brittle fracture transition in BCC iron through a complex emissary dislocation proposed mechanism. These cavities can penetrate the crystal in the direction of the binary axis at macroscopic distances and are the nuclei of microcracks. Therefore, when a single crystal is bent, a microcrack μ will most likely appear along the direction of the binary axis <10$\overline{1}$>. As a result, firstly, microcracks μ most likely arise along the B–C line, which coincides with the direction of one of the binary axes, when F_b || <1$\overline{2}$1>, and secondly, when F_b || <10$\overline{1}$>, then the line B–C coincides with one of the bisectrix axes <1$\overline{2}$1> (Figure 11c) in this case, while microcracks μ most likely occur along a binary axis oriented at an angle of 30° to the B–C line.

Obviously, in the case of F_b || <1$\overline{2}$1>, the action of tensile stresses during bending of a single crystal, which form and open a microcrack μ, is more effective than in the case of F_b || <10$\overline{1}$>. This explains the higher values of the ultimate bending strength observed during bending of single crystals, when F_b || <10$\overline{1}$>, compared with bending of single crystals, when F_b || <1$\overline{2}$1>.

Plastic deformation at compression by an all-round load can develop both due to sliding and due to twinning depending on the orientation of single crystals, temperature, and other factors. For example, for Bi single crystals deformed at a rate of ~10⁻⁴ s⁻¹, Steegmuller and Daniel [27] found a slip mode at room temperature for the crystal orientation {$\overline{1}$11}, (111), and {100} and twinning mode for orientations (011), (101), and (110).

In the process of bending of extruded samples, the increased density of dislocations and other structural defects and the development of grain boundaries (GBs) determine a significant resistance to twinning. The twins that appear in individual grains are small in size, and in the areas of their intersection, extended microcracks—which can become nuclei of the destruction of the entire polycrystal—cannot form. This apparently explains the absence of sharp drops in σ_b during bending of extruded specimens, regardless of the orientation of F_b. In addition, GBs, dislocation pile-ups and other defects that prevent the development of microcracks increase the ultimate strength σ_b.

Thus, as previously reported [9], the TE properties of the Bi_0.91Sb_0.09 alloy obtained by extrusion at high hydrostatic pressure are close to those of single crystals, while having several times higher strength (Figure 7 and Figure 8). For example, the dimensionless thermoelectric figure of merit ZT at 80 K decreases from 0.42 to 0.36 with an increase in the extrusion ratio Ke from 1.2 to 10, compared to 0.44 for single crystals measured without a transverse magnetic field. Recently, a unique thermoelectric solid-state multistage cooler for operating temperatures up to ~140 K was developed. In this case, the two-stage TE module for 140–200 K temperature range was developed on the n-type Bi_0.91Sb_0.09 crystal after plastic deformation and p-type Bi_1.6Sb_0.4Te₃ extruded crystal [35].

5. Conclusions

(1)

A new method was developed to significantly improve the mechanical properties of such effective thermoelectric (TE) materials as Bi_(1−x)Sb_x single crystals due to their plastic deformation by extrusion under all-round hydrostatic compression at room temperature in a liquid medium.

(2)

A detailed study of n-type Bi–Sb solid solutions in a wide temperature range shows a significant increase in the mechanical strength of Bi–Sb crystals after extrusion in comparison with high-quality Bi–Sb single crystals. The increase in the strength of the Bi–Sb single crystals after plastic deformation is associated with the development of numerous grains with a high boundary surface as well as structural defects, such as dislocations that accumulate at grain boundaries.

(3)

The significant stability of the structure of the extruded samples can be explained by the uniformity of the plastic deformation of the crystal under all-round hydrostatic compression, which leads to the formation of a more stable polycrystalline structure of the thermoelectrics, consisting of blocks with the orientation of the main crystallographic directions close to the original single crystal.

(4)

Strengthening of Bi–Sb crystals after plastic deformation under all-round compression, as reported earlier, makes it possible to develop a unique thermoelectric solid-state multistage cooler for operating temperatures up to T~140 K for the first time.