1 Introduction

Recent progresses in silicon carbide (SiC) crystal, film’s synthesis [1,2,3,4,5,6,7,8,9,10], and investigations of nano-polytype properties are highly important for applications in harsh environments [11], in optoelectronics [12, 13] and spintronics [14, 15], in quantum information processing [16,17,18], in biotechnology [19, 20] and medicine [21, 22]. Silicon carbide [23] can be a suitable prototype material for investigations of the polytypism phenomena in wide band semiconductors and binary materials. The fundamental difference between polytypes of semiconductors is only in the number and sequence of packing of atomic layers in a cubic (ABCABC) or hexagonal (ABAB) position relative to neighboring layers. Investigation of the growth process, phase transitions and defects of SiC are important to control the synthesis processes of nanostructure’s creations [24,25,26] in bulk, in porous [27], nanowires [28], quantum rings [29], nano and quantum dots [3, 9, 30]. Usage techniques such as X-ray [31] and TEM combined with photoluminescence [25, 32] allows investigation of binary compounds with a large number of possible configurations (polytypes). One of the well-developed research methods which is used in biology [33] and therapy [30] is luminescence analysis. This has rekindled scientific interest in the study of such materials.

Earlier we have introduced the LTPL spectra of SiC crystals and films with different impurity concentrations. In particular we reported on high purity [34,35,36], lightly doped [37, 38] and doped SiC [39,40,41,42]. The research results have shown that there are two types of the spectra stacking fault (SFi) and deep lever (DLi) associated with formation of intermediate metastable phases on 3C↔6H transitions. The SiC polytypes can be described by different stacking of Si–C layers perpendicular to the direction of the closed-packed plane {111} in cubic or {0001} in hexagonal SiC. The stacking sequences of the two close-packed lattices are as fcc: ABCABC, hcp: ABABAB. The agglomeration of vacancies on the {111} planes removes part of {111} plane and causes a stacking fault because the stacking sequence of ABCABC has changed to the faulty sequence of ABCABABC. SFi and DLi spectra follow the structure transformations hand-in-hand. SFi and DLi spectra comprise two parts which are superposed on each other and each one consists of two parts SFi (SFI and SFII) and DLi (DLI(X) and DLII(Y). The SFi and DLi reflect the fundamental logic of SiC polytypic structure [42]. However, in these articles, the relationship between the low-temperature spectra of binary crystals and phase transitions in crystals was not disclosed.

2 Experiment and discussion

2.1 Method of decoding

Low-temperature photoluminescence spectra (LTPL) were registered by the DFS-12 spectrograph with the photodetector (FEU-79). The luminescence spectra were obtained by using excitation by the lasers such as nitrogen, He–Cd, Ar-lasers and lamps such as mercury ultrahigh-pressure lamp (SVDSh-1000) and the xenon lamp (DKSSh-1000).

Crystals were grown by sublimation (Lely method) and were selected according to the phase composition and degree of structural disorder. Crystal structural researches were carried out using X-ray diffraction (Laue method) and electron diffraction method. Thermal treatment of crystals (high-temperature annealing) was performed in a resistance furnace with graphite heater on argon atmosphere in T = 2000–2100°C for 1–10 h. Plastic deformation of the samples was carried out in a resistance furnace at the argon atmosphere by three points bending at 2000°C during 15–30 min. The photoluminescence spectra were measured using samples contained in liquid helium or nitrogen cryostat, which provided a temperature range from 4.2 to 330 K.

In order to find the relationship between the photoluminescent and the lattice parameters of polytypism were chosen SiC crystals as wideband semiconductors with the most distinctive structure in the LTPL spectra and with a similar internal spectral structure which is not dependent on the location of binding to this or that metastable matrix in same energy scale.

The LTPL study was based on simultaneous control of similar structural imperfections in the SiC crystals selected after high-temperature annealing and plastic deformation. In addition to the above methods, the selection was carried out by a combination of methods such as by intensity of the excitation light, the decay time of intensity of the luminescent and the polarization. Due to the above approach to selected SiC crystals, LTPL spectra (at 4.2 K) had the same character (algorithm) of their internal structure and the location maxima, which were independent of the location on a metastable matrix at one energy scale. It is precisely such an integrated approach and measurement statistics made it possible to obtain a key approach to deciphering the algorithm for the internal construction of SFi and DLi spectra. These are the key points that facilitate decoding.

2.2 Parameters of LTPL spectra

To demonstrate the decoding method the fine structure spectra SF1 and DL1 have been selected. SF1 refers to the metastable phase SiC < 34 > which corresponds to the clearest fine structure. The analysis of the measurements and statistics allowed us to decrypt an algorithm of the internal construction of the spectra SFi and DLi and to decode the spectra into SFi = SFI + SFII and of DLi = DLI(X) + DLII(Y). The experimental results allowed to establish the spectral range boundaries of SF1 spectrum (2.853–2.793) eV and their components SFI (2.853–2.819) eV and SFII (2.827–2.793) eV [36, 42].

The decoding method is established by determining the spectral overlap SFI and SFII which is 8.6 meV. This value of the spectral overlap of the two similar structures SFI and the SFII attracted our attention and provided “the original hook” for further evaluation. According to the fact that in the observed spectral overlap is referred to the two structures SFI and SFII, therefore half number values of their spectral overlap was chosen as a new unit “i-unit”. The new spectral unit of the measurement is 1/2 of the number of the spectral overlap of 8.6 meV and thus i = 4.3 meV.

To build new spectra with the our new i-scale on the x-axis we need to uniformly expand the energy scale left and right by using the i-unit. As a result of such scaling using the i-unit a new abscissa axis—“i“-axis” coordinate is obtained (Fig. 1). The maxima of the LTPL spectra perfectly match to the i-axis, i.e. the i-unit on i-axis coincides perfectly with all the fine structures of the LTPL spectra. The main spectral parameters all correspond to exact multiples of the i-unit. This helps to clarify the value of the parameters to tenths of meV. It is significant that all the main maxima of the fine structures of the LTPL spectra all coincided with multiple numbers of i-unit. Thus, the new scale using i-units allows us to move to a simple and more convenient method of spectral analysis of fine structures in SiC polytypes.

Fig. 1
figure 1

The zero-phonon spectra SF1 of the different SiC samples (T = 4.2 K)

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2.3 LTPL characteristics and the lattice parameters

Using i-units determined in this way allow us to associate the relation between the luminescence characteristics and the lattice parameters of SiC (Table 1). The i-axis gives the exact multiples of the basic spectral features, and this helps to clarify the value of the features with high precision (to tenths of meV).

Table 1 The relation between the spectrum parameters and the lattice parameters of SiC
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Curves (1) and (2) are related to the extended stacking faults that have arisen during the growth process. Curve (3) demonstrates the stacking faults after high-temperature annealing of β-SiC crystal in the area of the phase transformation 3C → 6H via metastable phases (Fig. 1). Actually, the zero-phonon spectra are not observed because of indirect energy transitions in SiC crystals. Therefore, in Fig. 1, this structure is represented by the transition of TA phonons in the non-phonon part area [36]. Finally, curve (4) is the zero-phonon spectrum, which is really manifested in its spectral range. Perhaps in this case, the direct observation of electron–hole transitions without saving the wave number in covalent crystals is associated with the localization of radiative centers in the region of maximum plastic deformation of the bent crystal. But this is an exception under this specific condition.

The use of the i- unit allows us to create an i- axis which organizes all the fine structures of the spectra allowing a clear correlation of the parameters. That led us to the search for analogies in the elements of the tetrahedron structure themselves. Above all, this raised the possibility of introducing a similar i-axis along the axis < 111 > which defines the possible direction of building a hexagonal structure. Considering that the i = 4.3 meV determined above is a unit of measurement which also equals to 0.315 Å, i.e. 1/2 of the distance between adjacent Si–C layers (0.63 Å).

The basic parameters of the tetrahedron can thus be presented in the i-axis units as follows: (Si–C)1 = 0.63 Å = 2i, (Si–C)2  = “b” = 1.89 Å = 6i, (C–C) and (Si–Si) = h = 2.52 Å = 8i (Fig. 2).

Fig. 2
figure 2

The basic parameters of the tetrahedron of SiC in the i-scale: (Si–C)1 = 2i, (Si–C)2 = 6i, (C–C) and (Si–Si) = 8i

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The parameter equal to (Si–C) = 14i is found only in binary tetrahedron structures such as SiC, CdS, etc., and covers the atoms of the second coordination sphere (Fig. 2). It is useful to note that the tetrahedron lattice parameters represented in i-units are as follows:

ah = 3.08–3.15 Å = 10i, ac = 4.36–4.41 Å = 14i.

2.4 Energy and crystal parameters

If we compare the scale of the energy parameters of the spectrum with crystal structure parameters using the corresponding i-units one easily finds the similarity between their parameters: h = aT = (a′T′), ac = aa′, aT/aa′ = h/ac = 0.571 = Sin35°16′.

The SFi spectrum is associated with the dislocation center which is formed by the participation of Shockley Partial Dislocations (SPD) with a Burger’s vector (bs = 1/6 < 112 > , │bs│ = 1.788 Å) and Frank partial dislocation (FPD) (bF = 1/3 < 111 > , │bF│ = 2.52 Å). Since the dislocation core area covers no more than two atomic coordination spheres, where the atoms are characterized by the same potential energy and surrounding environment, it is proof that all SFi spectra have the same spectral range and the same internal fine structure characteristics. This result has also been proven by multiple experiments in different samples.

It is important to note that the decoding of the DL spectrum confirms that the DL1 spectra are also associated with a metastable matrix < 34 > . The researchers [25, 31, 37, 38, 42] have shown that the DL1 spectrum with (2.73–2.625) eV is a complicated compound, which has two parts DLI(X) (2.73–2.67) eV and DLII(Y) (2.685–2.625) eV (hereinafter DL(X) + DL(Y)) with similar structure and the spectral ranges.

The energy ranges of every DL(X) and DL(Y) are the same as the energy ranges of all the SFi. DL(X) and DL(Y) have a region of overlap, and as a whole they are displaced relative to one another. The application of i-scale (as in the case of SF-spectrum) firstly produced, the same i-scale step structure and spectral position within the ranges (X) and (Y) and, secondly, the presence of similar benchmark markers as in the case of SF-spectrum. In general, the spectral ranges of DL(X) and DL(Y) are the same as the energy spectral intervals of SF-spectra. This fact indicates the same pattern in the mutual positions of the fine-structure elements, which, in turn, may indicate a general source of such laws. A clear shift between these two parts equal to 4i can be captured in this common i-axis for X + Y as well as the whole structure of X and Y offset from each other by exactly 10i = 43 meV (Fig. 3).

Fig. 3
figure 3

The non-phonon part of spectra DL1 for two different samples with impurity concentration: 2·1016 cm−3 (solid line) and 3·1016 cm−3 (dotted line) at T = 4.2 K [37, 38]

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Comparing the values of the width of the non-phonon DL line, it is not difficult to see that the entire picture of the DL spectra is reproduced inside three maxima (Fig. 3). Actually (SF) should be considered in two tetrahedra, while (DL) are a three tetrahedra. The linear scale in the angular expression allowed us to identify the directions in the crystal lattice (see above Table 1). Due to this, the spectra of the non-phonon part can be traced to the position of atoms with an accuracy of 0.0787 Å which is i/4 = 1.075 meV.

The total spectral ranges (of the non-phonon part DL1 spectrum is denoted “ab” and equals: ab = 103.2 meV or ab = 24i. Wherein ab = 3aT = 4ef, ef + cd = 68.8 meV = 2 h = 16i. Within DL(X) and DL(Y) (inside DL(X) and inside the DL(Y)) are relevant to actual spectral parameters, similar to the case of the SF spectrum: aT = a′T′ = 34.4 meV = 8i, aa′ = bb′ = 60.2 meV = 14i. Here the similarity principle is observed. It is the similarity of the value of the spectral parameters within parts DL = 14i, (within structures DL(X) and DL(Y)) and the whole spectral range of DL (ab = 24i). Those similarities of total spectrum range DL equal to 24i and each part equal to 14i.

aT/aa′ = 0.571 8i/14i = 0.571

aa′/ab = 0.583 14i/24i = 0.583

Mean value equals 0.577, which is equal to Sin 35°16′. The 35°16′ is a real tetrahedron angle and perfect angle is 36°. aT/aa′ = aa′/ab is exactly equal to Sin 35°16′. It is more interesting to note the correlation of spectral parameters such as ef = 25.8 meV = 6i and cd = 43 meV = 10i, which seem to correspond to the so-called “Golden Section” rule: ef/cd = cd/ef + cd = 0.612.

If the properties and behavior of the SFi spectrum parameters reflect the effect of Shockley Partial Dislocations (SPD) as the origin of the growth under non-equilibrium conditions, and also after high-temperature annealing and plastic deformation, then the spectrum of the DLi type spectra shows different property responses to external impacts.

There are two shift mechanisms in the crystals, resulting in phase changes caused by the deformation: shifting of the atoms in the direction of the Burgers vector of Shockley Partial Dislocations (as a result of SPD) and by diffusion: changing the orientation of the layer due to the displacement by diffusion. The latter implementation is as a result of the Frank partial dislocation FPD, but the first one is as a result of the thermal activation process, which requires additional energy. Comparing the lengths of the Burgers vectors SPD (bs= 1/6 < 112 > , │bs│ = 1.778 Å) with FPD (bF = 1/3 < 111 > , │bF│ = 2.52 Å), it becomes clear that FPD has twice the energy of SPD. The difference in energy can be reduced by cleavage of the FPD on SPD and “stair-rod” dislocation. Thus, it is a reason to consider DL1 spectra as a radiation of the dislocation center involving FPD, namely as the so-called “stair-rod” dislocation. This is supported by the absence of any plastic deformation impact on the structure of the spectrum.

Studies of the luminescence intensity dependence on temperature give the value of the thermal activation energy quenching E = 0.21 eV for DL(X) and E = 0.253 eV for DL(Y). These match with sufficient accuracy with the difference in the energies DL (X) and the DL(Y), equal to 0.043 eV [37, 38]. Taking into account the spectral energy parameters of SF and DL obtained from LTPL spectra associated with dislocation centers in phase transformations in SiC phase, a diagram of the interlayer scheme radiative recombination with resulting dislocation levels was made (Fig. 4).

Fig. 4
figure 4

The diagram of the interlayer scheme radiative recombination with resulting dislocation levels

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This is explained if we consider the fact that in covalent crystals the edge dislocation centers act as acceptors. On the one hand metastable structures are formed due to recombination of excitons localized in shallow states near Egx. On the other hand, the SF has the same shallow states near the conduction band and it presents SF in all its complexity. Finite DL states are represented by two deep acceptor levels near the valence band which differ by 0.043 eV. This overall picture reflects a link between these parameters.

3 Summary

The relationship between LTPL spectra and phase transitions in binary crystals such as SiC is revealed. The new atomic parameter (i-unit) of energy scale which allows a deeper understanding of the physics of phase states in binary crystals SiC was introduced. The introduction of i-unit using SiC as an example allows clearer explanation of the fine structure of the low temperature photoluminescence spectra through a correlation of the parameters of the photoluminescence spectra and lattice parameters. The i-unit of energy scale allows the simplifying method of spectral analysis of fine structures in SiC polytypes.

The value of the i-unit is calculated from the spectral overlap of the two structures of non-phonon SFi spectra (8.6 meV) and i = 4.3 meV. All the fine structures of the spectra occur in exact multiples of the i-unit and enables evaluation of the basic spectral parameters to tenths of meV.

The relationship of the LTPL with the lattice parameters in SiC crystals is also considered using this new atomic parameter of energy i-scale. The application of the i-scale to the DL-spectrum (same as in the case of SF-spectrum) revealed the same structure of the ranges within parts (X) and (Y) in the zero-phonon parts of DL spectrum and the same markers as in the case of the SF-spectrum.

The parameter of i-unit was introduced along the axis < 111 > with the unit of measurement equals to 0.315 Å (4.3 meV) and corresponds to half the distance between the nearest Si–C (0.63 Å) atoms. The main parameters of the tetrahedral structure are expressed in i-units as follows: (Si–C)1 = 2i, (Si–C)2=“b” = (1.89 Å) = 6i, (C–C) and (Si–Si) = h (2.52 Å) = 8i, (Si–C)3 = 14i.

Tetrahedron lattice parameters are shown in i-units and are the following:

ah = 3.08–3.15 Å = 10i, ac = 4.36–4.41 Å = 14i.

The proposed new atomic i-scale allows us to monitor the processes of phase transformations up to 0.0787 Å. It is shown that phase transformations in SiC are related to dislocation in the crystal and implemented by the deformation and diffusion mechanisms.

Due to calculations of spectral energy parameters obtained from LTPL spectra the diagram of the interlayer scheme radiative recombination with resulting dislocation levels was made.