Features of Structural, Electrokinetic, and Energy State Characteristics of ZrNiSn 1- x Ga x Solid Solution

Features of structural, electrokinetic, and energy state characteristics of ZrNiSn 1- x Ga x semiconductive solid solution were investigated in the temperature ranges Т = 80 - 400 K and х = 0 - 0.15. Disorder of crystal structure for n -ZrNiSn compound as a result of occupation of Zr (4 d 2 5 s 2 ) atoms in 4 a sites by Ni (3 d 8 4 s 2 ) ones up to ~ 1 % was confirmed. It generated donor levels band ɛ D1 in the band gap. It was shown that introduction of Ga (4 s 2 4 p 1 ) atoms by means of substitution of Sn (5 s 2 5 p 2 ) ones ordered crystal structure. In this case acceptor defects were generated in 4 b sites and it created extended acceptor impurity band ɛ А . It was suggested that with generation of acceptor structural defects the vacancies in the Sn (4 b ) atomic sites simultaneously generated donor defects and formed deep donor band ɛ D2 (donor-acceptor pair took place).


Introduction
In studying of thermoelectric materials formed by heavy doping of n-ZrNiSn, n-HfNiSn and n-TiNiSn intermetallic semiconductors by acceptor impurity, the appearance of electrons with unknown origin mechanism was observed. A number of research results of previously obtained solid solutions based on ZrNiSn compound hasn't an adequate explanation because the emergence of donors during the substitution of Ni atoms by M = Cr, Mn, Fe, Co, etc. [1] was incomprehensible. The generation of structural defects with only acceptor nature in ZrNi 1-x M x Sn was more logical because the Ni atoms have the number of 3d-electrons more than Cr, Mn, Fe and Co. However, electrokinetic studies revealed the emergence of a significant number of donors of unknown origin, and their concentration increased with increasing number of acceptors. Structural studies have found no such defects because their concentration is outside the Xray methods accuracy [2].
Recent studies of solid solutions based on HfNiSn and TiNiSn discovered a previously unknown mechanism for generating structural defects with donor nature, which involves the appearance of vacancies in the positions of Sn (4b) atoms [3][4][5][6]. For example, in the case of doping n-TiNiSn by Ga (4s 2 4p 1 ) atoms by means of substitution of Sn (5s 2 5p 2 ) atoms it was found that in the same 4b crystallographic position as acceptor defects (Ga has a smaller number of p-electrons than Sn) so donor defects as a vacancies in the position of Sn atoms were simultaneously generated. It's noted that donor's concentration increases with increasing Ga content [6], and the semiconductor is heavy doped and compensated (HDCS) [1,7]. This unexpected result is logical, because the stability of the structure and the electrical neutrality principle of ТіNiSn 1-x Ga x crystal provided the appearance of a significant number of acceptors (N A Ga ≈3·10 21 см -3 ) ensures generation of structural defects with donor nature, effective charge of which is opposite. In this case, the formula looks like a solid solution ТіNiSn 1-x-y Ga x , where у is the concentration of vacancies in 4b positions of Sn atoms.
Today the question is open whether this mechanism works in the case of doping n-ZrNiSn? Therefore, provided the work can be regarded as the first experimental phase of the study on the availability of donor mechanism for generating a vacancy in the Sn (4b) atomic site upon the doping of n-ZrNiSn with Ga atoms which substituted Sn atoms. The calculation and analysis of the electronic structure for ZrNiSn 1-x Ga x will be the subject of further work which will establish conduction mechanisms at various concentrations and temperatures.

I. Experimental details
The ZrNiSn 1-x Ga x samples were prepared by a direct twofold arc melting of the constituent elements under high purity Ti-gettered argon atmosphere on a watercooled copper crucible. Then the pieces of the as-cast buttons were annealed for one month at 1073 K in evacuated silica tubes and then water quenched. The lattice parameters were calculated using the powder diffraction patterns (diffractometer Guinier-Huber image plate system, CuKα 1 radiation) [2], the crystallographic parameters (atomic coordinates, isotropic displacement parameters, the site occupancies) were determined using WinPLOTR program package [8]. To increase the precision of the structural study the calculations were performed taking into account radiation of αand β-lines. The chemical and phase compositions of the obtained samples with accuracy ~ 1 at.% were examined by Scanning Electron Microscopy (SEM) using Carl Zeiss DSM 962 scanning microscope. For ZrNiSn 1-x , the temperature dependencies of electrical resistivity (ρ) and differential thermopower (α) (pure copper as a reference material) were measured in the 80 -400 K temperature range.

II. Crystal structure refinements of ZrNiSn 1-x Ga х
The X-ray analysis showed that the ZrNiSn 1-x Ga x samples are single phases, the powder patterns of the ZrNiSn 1-x Ga x samples were indexed with cubic MgAgAs-type [9]. According to microprobe analysis the compositions of the synthesized samples correspond to the initial compositions of the alloys, confirming the substitution of Sn atoms by Ga.
The crystal structure refinements of ZrNiSn 1-x Ga x by powder method including the refinements of isotropic displacement parameters and occupancy of crystallographic site Zr (4а) confirmed the result [3] concerning disordering of ZrNiSn structure (1 % (z ≈ 0.01) Ni (3d 8 4s 2 ) occupies 4a site of Zr(4d 2 5s 2 )). This leads to generate the structural defects of donor nature caused by higher number of d-electrons of Ni, and the donor impurity band appears in the band gap. Thus, the chemical formula should be written as (Zr 1-z Ni z )NiSn.
On the other hand the crystal structure study of ZrNiSn 1-x Ga x indicated, that the smallest value of final reliability factor (R Br. ≈ 2.9 %) for structure refinements was observed for structure model where the occupancy of 4а position by Zr atoms is 100% for x ≥ 0.01. Thus, introduction of Ga atoms in the structure of ZrNiSn compound leads to ordering of the crystal structure ("healing" of structural defects): the Ni atoms disappear in position Zr (4а). The similar result was obtained for ТіNiSn 1-x Ga x [6].
Taking to account that the atomic radius of Ga (r Ga = 0.141 nm) is lower than Sn (r Sn = 0.162 nm), it was expected reducing of the unit cell parameter а(х) for ZrNiSn 1-x Ga x with increasing of Ga content. However, variation of the а(х) values was not monotonic, that reflected the complex processes in the crystal. It's worth to consider the behavior of а(х) values in the 0 ≤ х ≤ 0 region. (Fig. 1).
As was noted above, the ZrNiSn structure is disordered by partial occupation of Zr positions by Ni atoms [5]. During the introduction of Ga atoms into the ZrNiSn structure in the 0 ≤ х ≤ 0.01 concentration ranges its ordering takes place by means of the displacement of small Ni atoms (r Ni = 0.124 nm) from 4а position by larger Zr atoms (r Zr = 0.160 nm). Also the displacement of Ni atoms in 4(a) site was accompanied by substitution of big Sn atoms by smaller Ga in position 4b. Taking to consideration that the difference of atomic radii of Zr and Ni (r Zr -r Ni ) = 0.036 nm, and Sn and Ga (r Sn -r Ga ) = 0.021 nm, the variation in а(х) values in the 0 ≤ х ≤ 0.01 concentration range was caused by process of Ni displacement by larger Zr atoms in 4a position. It led to some growing in а(х) dependence ( Fig. 1).
After displacement of Ni atoms from Zr position (ordering of structure) the а(х) dependence was determined by Ga atoms occupying Sn (4b) positions. It corresponded to a decrease of а(х) values at the 0.02 ≤ x ≤ 0.15 concentration range.
Variation а(х) also was calculated for ZrNiSn 1-x Ga х (ordered structure variant) with suggestion that the ZrNiSn compound is ordered (Fig. 1, curve 2). The calculated value of the lattice parameter а(х) for ZrNiSn is larger than experimental value because in the real compound, as mentioned above, ~ 1% Zr atoms displaced by smaller Ni atoms. Comparison of the calculated and experimental values of the lattice parameter а(х) for ZrNiSn 1-x Ga х (Fig. 1) showed that the more complex changes appeared in the crystal structure. In particular, the fact that calculated values of unit cell parameter а(х) for ZrNiSn 1-x Ga х at x = 0.15 were higher than obtained experimentally was unclear and required Features of Structural, Electrokinetic, and Energy State Characteristics … additional study (Fig. 1).
The ZrNiSn 1-x Ga х crystal_structure ordering in addition to structural changes was accompanied by a restructuring of the electronic structure in the semiconductor. For example, if the n-ZrNiSn in the band gap has donor band ɛ D 1 as a result of the substitution of up to ~ 1% Zr atoms by Ni atoms [3], then the ordering of the ZrNiSn 1-x Ga х structure (displacement of Ni atoms from a Zr (4a) site) results in decreasing of donor number and elimination of donor band ɛ D 1 . At the same time, since the Ga (4s 2 4p 1 ) atom has one р-electron less than Sn (5s 2 5p 2 ), then the substitution of Sn atom by Ga generated acceptor defect in the 4b site that led to an appearance in the band gap impurity acceptor level, which in a significant number of impurities formed extended impurity acceptor band ɛ А . The presence of a large number of donors and acceptors affect the ZrNiSn 1-x Ga х band structure [1,3], that will be observed in the study of the electrokinetic and energy state characteristics.

III. Investigations of electrokinetic and energy state characteristics of ZrNiSn 1-x Ga х
The temperature and concentration dependences of electrical resistivity ρ and thermopower coefficient α for ZrNiSn 1-x Ga x are shown in Fig. 2-4. The lnρ(1/T) and α(1/T) dependences for ZrNiSn 1-x Ga x samples (Fig. 2) are typical for heavily doped and highly compensated semiconductors, and existing activation regions indicate several mechanisms of charge transport [1,3]. These mechanisms are activation of current carriers from the Fermi level ε F to continuous energy band levels (high temperature) and hopping conduction (except for x = 0.15) within the energy states close to ε F (low temperature). From the activation regions of the lnρ(1/T) dependences plots the activation energies from the Fermi level ε F to the percolation levels of the conduction band (valence band) ε 1 ρ and hopping conduction energy ε 3 ρ were calculated. From the activation regions of α(1/T) dependences plots the values of activation energy ε 1 α and ε 3 α were obtained; they give the values of the modulation amplitude of continuous energy bands, and small-scale HDCS fluctuations, respectively [1,7].
The presence of high-temperature activation parts in the lnρ(1/T) dependences for all ZrNiSn 1-x Ga x samples shows that the Fermi level ε F is located in the band gap, from which the thermal activation of carriers to the percolation levels takes place. Assuming that introduction of Ga impurity atoms into ZrNiSn crystal generates, as expected, only structural defects with acceptor nature, then at the Ga concentration, for example, x = 0.15 (N A Ga ≈3·10 21 сm -3 ), the Fermi level ε F would cross the percolation level in valence band and insulator-metal transition of conductivity would occur (Anderson transition) [1,7]. However, the shapes of dependencies lnρ(1/T) for ZrNiSn 1-x Ga x at high temperatures show that the conductivity metallization is absent.
This behavior of the semiconductor at appropriate impurity concentration is possible only if along with the generation of the acceptors occurs the simultaneous generation of the donors of unknown origin, which compensate acceptors, forcing the Fermi level ε F to stay within the energy band gap and reflect the compensation degree of ZrNiSn 1-x Ga x .
The fact that the atoms of Ga, introduced into n-ZrNiSn, generate structural defects with acceptor nature, can be seen from the dependences of changes in the values of resistivity ρ(х,Т) and thermoelectric coefficient α(х,Т) in all concentration and temperature ranges (Figs.  3 and 4). First, we analyze the dependences ρ(х) and α(х) at 80 K.
For example, the introduction of the smallest Ga concentration in the experiment rapidly increases the value of the resistivity ρ(х) at T = 80 K from ρ(х = 0) = 4751.1 μΩ•m to ρ(х = 0.01) = 10677.7 μΩ•m. We can assume that the sample with the concentration x = 0.01 is highly compensated, since the number of generated acceptors is close to the number of donors in n-ZrNiSn [1]. The rapid growth of ρ(х) dependence at the x = 0 -0.01 results from two processes: The maxima on the ρ(х) dependence at introducing into the n-ZrNiSn semiconductor with the electron-type conductivity the Ga acceptor impurity reflects the equilibration of the competing processes which determine the mechanisms of conductivity. For example, at doping of n-ZrNiSn with the acceptor impurity Y the dependence ρ(х) within the concentration range х = 0 -0.02 also increased drastically, passed through its maxima at х≈0.02 and decreased at х>0.02, accompanied by the change of the conductivity type for Zr 1-x Y x NiSn, х ≥ 0.02, from electron-type to hole-type [1]. It is caused by the passing by the Fermi level of the middle of the Features of Structural, Electrokinetic, and Energy State Characteristics … band gap and its drift to the valence band, which increases the number of free holes and, as a result, the sign of the thermopower coefficient becomes positive, and the dependence ρ(х) decreases drastically.
On the other hand, assuming that in ZrNiSn structural defects of donor nature are absent, and the semiconductor is intrinsic (donor levels band ɛ D 1 is absent), then at introducing of the Ga atoms the values of the resistivity ρ(х) also decreased within all temperature and concentration ranges due to appearance and increasing of the free holes number in the valence band upon ionization of acceptors in the impurity acceptor band ɛ А . It is clear that the sign of the thermoelectric coefficient in this case will also be positive.
It was predicted that as in the case of Zr 1-x Y x NiSn [1], replacing Sn atoms by Ga atoms would be accompanied by the generation of structural defects with acceptor nature in 4 b crystallographic position and appearance of impurity acceptor band ɛ А close to the valence band. At the concentration of Ga atoms, when the Fermi level ε F crosses the mid gap and begin to drift to percolation levels of the valence band, the free holes become main charge carriers. Such an assumption is logical, because at concentration of Ga, x > 0.01, the number of generated acceptors ɛ А already exceeds the number of donors with the energy ɛ D 1 for n-ZrNiSn (i.e. number of Ni atoms in Zr (4a) position).
However, as seen in Figs. 2 and 4, at 80 K sign of the thermoelectric coefficient for ZrNiSn 1-x Ga х remains negative for all concentrations, and electrons are the main carriers. And despite the fact that the concentration of generated structural defects with acceptor nature in a sample, for example, ZrNiSn 1-x Ga х , x = 0.10, is in one order higher than the concentration of donors in n-ZrNiSn. It is possible only provided that the depth of the acceptor band is such that 80 K is not sufficient for hole to overcome the energy barrier between the percolation level in the valence band and acceptor level ɛ А .
With increasing temperature (T > 80 K) the ρ(х) dependence for ZrNiSn 1-x Ga х transformed, reflecting changes in the electronic structure of the semiconductor. For example, in the dependence ρ(х) at T = 160 K and concentration in the vicinity of x ≈ 0.06 a step first appears, which gradually develops into extreme (T = 300 K). At higher temperature, T = 380 K, it is shifted in the region of higher concentrations х ≈ 0.08. At the same time the maximum in the dependence ρ(х) at concentration about х≈0.01 disappears.
It can be stated that at low concentrations of Ga acceptor impurity the maximum in the dependence ρ(х) for ZrNiSn 1-x Ga х at concentration about х ≈ 0.01 (Fig. 3) is due to the existence of the donor level ɛ D 1 in the band gap, generated by structural defects with donor nature (Ni atoms in 4a positions of Zr atoms). At acceptor concentration, which corresponds to the concentration of generated donors ( х ≈ 0.01) the depletion of donor occurs, the electrical resistivity gains maximum values and semiconductor is highly compensated. Since the values of thermopower coefficient at 80 K are negative for all concentrations, then the temperature is insufficient to complete ionization of acceptors (thermal transport of hole to percolation level in the valence band). In this context it should be noted that in the semiconductor ZrNiSn 1-x Ga х , х=0.01, concentration of donors with energy ɛ D 1 will be much smaller than the number of generated acceptors with energy ɛ А . At the least concentration of Ga impurity atoms it is caused by the process of structure ordering that rapidly reduces the concentration of donors with energy ɛ D 1 . At concentrations х ≥ 0.02, when the number of generated acceptors exceeds number of donors in n-ZrNiSn, with the temperature increasing from T = 80 K to Т= T 1 inv sign of the thermopower coefficient changes from negative to positive (Fig. 5). Namely for samples ZrNiSn 1-x Ga х , х ≈ 0.02 and х ≈ 0.05, the temperature T ~ 93 K is sufficient for hole to overcome the energy barrier between the percolation level in the valence band and acceptor levels band ɛ А . Conduction mechanism in this temperature range corresponds to that in Zr 1-x Y x NiSn [1].
With further increasing of temperature in the samples ZrNiSn 1-x Ga х , х ≈ 0.02 and х ≈ 0.05, 0,10, at temperatures T ≈ 156 K and T≈216 K, respectively, sign of the thermopower coefficient abruptly changes from positive to negative at T 2 inv. (Fig. 5), and electrons again become the main charge carriers. And despite the fact that the concentration of generated acceptors in ZrNiSn 1-x Ga х , х > 0.01 is higher than the number of donors with energy ɛ D 1 in the n-ZrNiSn (number of Ni atoms in Zr(4a) position). That behavior of thermopower coefficient for ZrNiSn 1-x Ga х at Т = T 2 inv. is possible provided that in the semiconductor, along with the acceptor impurity band ɛ А the donors with ɛ D 2 are generated, energy levels of which form a donor band ɛ D 2 , deeper than ɛ D 1 . It is noted that for the ionization of donors with ɛ D 2 and overcoming barriers to the percolation level of the conduction band needed higher energy. It seems that in the semiconductor donoracceptor pairs are generated simultaneously, energy levels of which are in the band gap of semiconductor.
The analysis of the ρ(х) behavior for ZrNiSn 1-x Ga х at different temperatures leads to the same conclusion. As an extremum on ρ(х) dependence of ZrNiSn 1-x Ga х represents equilibration of the competing processes in the electronic structure of the semiconductor, so the disappearance of the maximum on ρ(х) with increasing temperature at low concentration of Ga impurity (х ≈ 0.01) and appearance of new extremum on ρ(х) at the concentrations х ≈ 0.06 indicate an existence of donor band ɛ D 2 , the depth of which more than ɛ D 1 . Indeed, at the concentrations х ≥ 0.02 and the temperatures, values of which are insufficient for ionisation of donor ɛ D 2 , the concentration of acceptors in ZrNiSn 1-x Ga х prevails the concentration of donors, and the sign of thermopover coefficient is positive. With increasing temperature Т ≥ T inv ionising of donors begins, the number of free electrons which become the main carriers of current grows headily. It's indicated by the negative values of thermopower coefficient (Fig. 4).
On the other hand, as the higher number of the acceptors are generated in the ZrNiSn 1-x Ga х semiconductor, so higher energies (temperatures) are needed to prevail the number of ionized donors ɛ D 2 over the number of the ionized acceptors ɛ А . It means that the defects with donor nature appear in the crystal simultaneously with the structural defects with acceptor nature in position 4b at the substitution of Sn atoms by Ga.
Based on obtained results, similar to the case of ТіNiSn 1-x Ga x [6], now we can only assume that in ZrNiSn 1-x Ga х to ensure stability of the structure and electrical neutrality principle both the structural acceptor defects and defects with donor nature as the vacancies in position 4b are simultaneously generated (an effective charges of which are opposite) and the concentration of which increases with increasing of Ga content. In this case the formula of the semiconductive solid solution can be expressed as ZrNiSn 1-x-y Ga x , where у -concentration of vacancies in 4b position of Sn atoms.
The noted above suggestion about appearance of the vacancies in (4b) position of Sn atoms explains the fact that at х=0.10 the calculated values of the lattice parameter а(х) are higher than values obtained from the experiment (Fig. 1). Since only the substitution of the Sn atoms by Ga was taken into account at calculations and appearance of vacancies is not taken into account, then the presence of the last would result in the certain "compression of the structure" and decreasing of а(х) values in the real crystal.
Analysis of the behaviour of the energy state characteristics of ZrNiSn 1-x Ga х , in particular, changes of the values of activation energy ε 1 ρ (х) from the Fermi level ε F to the percolation level of conduction band and the modulation amplitude of continuous energy band ε 1 α in HDCS [1,7] also shows that the acceptors and donors are simultaneously generated in the semiconductor. For example, in n-ZrNiSn the value of energy ε 1 ρ (х = 0) = 97.6 mеV represents a energy gap between position of the Fermi level ε F and percolation level of conduction band. Doping of the semiconductor with ntype conductivity by least concentrations of Ga acceptor impurity increases the compensation degree, and the Fermi level ε F go deep into the band gap in the intervals At the same time, simultaneous generation of donors and acceptors in ZrNiSn 1-x Ga х changes the compensation degree and modulation amplitude value of continuous energies band in HDCS [1,7]. The variation of the values of activation energy ε 1 α (х), which is proportional to modulation amplitude of continuous energy band in ZrNiSn 1-x Ga х , is shown in Fig. 6. It's seen that in the case of n-ZrNiSn the value of modulation amplitude is ε 1 α (х = 0) = 83.8 mеV. The introduction of Ga (х = 0.01) acceptor impurity with the smallest concentration into the semiconductor with electronic type conductivity is accompanied by increasing of the compensation degree, confirmed by increasing of modulation amplitude up to  (sign inversion from positive to negative).
of the localized states of donor band ɛ D 2 are recovered, the metallization of conductivity within this band take place, confirmed by absence of the low temperature activation region on lnρ(1/T) dependence (Fig. 2).

Conclusions
Thus, as a result of comprehensive study of structural, kinetic and energy state characteristics of ZrNiSn 1-x Ga x semiconductor solid solution it was shown that introduction of Ga atoms (4s 2 4p 1 ) by substituting for Sn (5s 2 5p 2 ) orders the crystal structure of ZrNiSn 1-x Ga x generating the structural defects of acceptor nature, which form the acceptor impurity band ɛ А . It was suggested that simultaneously with acceptors structural defects with donor nature were generated as vacancies in position 4b of Sn atoms (donor-acceptor pair), which formed extended donor band ɛ D 2 in the band gap. For a final conclusion concerning to the mechanism of generation of structural defects with donor nature upon heavy doping of n-ZrNiSn by Ga acceptor impurity it is necessary to perform the electronic structure calculations of ZrNiSn 1-x Ga x for the different crystal structure models and to find such variant of atomic positions arrangement when the calculated behavior of Fermi level ε F will coincide with experimental. The obtained results will give an answer for correctness of suggestion concerning to appearance of vacancies in position 4b of Sn atoms for ZrNiSn 1-x Ga x and extended deep donor band ɛ D 2 .
The paper was supported by the Ministry of Education and Science of Ukraine (grants No. 0115U003257 and No. 0116U004142).