Corresponding author: Kirill D. Shcherbachev ( chterb@mail.ru ) © 2018 Vladimir T. Bublik, Marina I. Voronova, Kirill D. Shcherbachev.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Bublik VT, Voronova MI, Shcherbachev KD (2018) Capabilities of Xray diffuse scattering method for study of microdefects in semiconductor crystals. Modern Electronic Materials 4(4): 125134. https://doi.org/10.3897/j.moem.4.4.47197

The capabilities of Xray diffuse scattering (XRDS) method for the study of microdefects in semiconductor crystals have been overviewed. Analysis of the results has shown that the XRDS method is a highly sensitive and information valuable tool for studying early stages of solid solution decomposition in semiconductors. A review of the results relating to the methodological aspect has shown that the most consistent approach is a combination of XRDS with precision lattice parameter measurements. It allows one to detect decomposition stages that cannot be visualized using transmission electron microscopy (TEM) and moreover to draw conclusions as to microdefect formation mechanisms. TEMinvisible defects that are coherent with the matrix and have smeared boundaries with low displacement field gradients may form due to transmutation doping as a result of neutron irradiation and relaxation of disordered regions accompanied by redistribution of point defects and annihilation of interstitial defects and vacancies. For GaP and InP examples, a structural microdefect formation mechanism has been revealed associated with the interaction of defects forming during the decomposition and residual intrinsic defects. Analysis of XRDS intensity distribution around the reciprocal lattice site and the related evolution of lattice constant allows detecting different decomposition stages: first, the formation of a solution of Frenkel pairs in which concentration fluctuations develop, then the formation of matrixcoherent microdefects and finally coherency violation and the formation of defects with sharp boundaries. Fundamentally, the latter defects can be precipitating particles. Study of the evolution of diffuse scattering isointensity curves in GaP, GaAs(Si) and Si(O) has allowed tracing the evolution of microdefects from matrixcoherent ones to microdefects with smeared coherency resulting from microdefect growth during the decomposition of nonstoichiometric solid solutions heavily supersaturated with intrinsic (or impurity) components.
microdefects, Xray diffuse scattering, single crystals, semiconductors
Among the problems faced by the technologies of semiconductor single crystals with preset properties that are determined mainly by the structural perfection and homogeneity of the crystals, there are important issues relating to the origins and properties of microdefects in their crystal lattice as well as microdefect study and nondestructive control methods. Hereinafter we will consider microdefects to be local violations of crystal lattice periodicity caused by clusters of point defects (intrinsic or impurity), dislocation loops or dispersed phase precipitates having submicron or micron sizes. All these violations are first type defects [
Below we will dwell upon more detailed aspects of the problem in question: capabilities of the XRDS method paired with precision lattice parameter measurements on a diffractometer with a laboratorygrade Xray source for the study of microdefects in semiconductor single crystals.
The theory of Xray diffuse scattering (XRDS) from defects near Bragg reflections was put forward by Dederichs [
The XRDS intensity distribution I (q) is determined by the Fourier image of the defect’s displacement field u(q)
$I\left(\mathbf{q}\right)\sim \mathbf{Q}\cdot \mathbf{u}\left(\mathbf{q}\right){}^{2}$, (1)
where q = Q – H is the vector describing the deviation of the diffraction vector Q from the reciprocal lattice site described by the vector H.
If the local displacements around the defects are small and Q.u(q) << 1, the differential crosssection of diffuse scattering S_{dif}(q) is written in the form [
${S}_{df}\left(\mathbf{q}\right)=N{\left{F}_{H}\right}^{2}[\genfrac{}{}{0.1ex}{}{{H}^{2}{C}^{2}}{{q}^{2}}{\displaystyle \sum _{1}^{3}}{\pi}_{i}{\gamma}_{i}+\genfrac{}{}{0.1ex}{}{{\left(HC\right)}^{\genfrac{}{}{0.1ex}{}{5}{2}}}{{v}_{0}^{2}q}\sqrt{{\pi}_{1}{\gamma}_{1}}]$, (2)
where F_{H} is the structural amplitude, N is the number of scattering centers, C is the defect power that characterizes the crystal volume change ΔV caused by the defect and, for a specific microdefect type, can be expressed through defect parameters (shape, size and deformation sign), and ν_{0} is the unit cell volume. For example, a defect with a Coulomb type displacement field u(r) >> C(r_{0})/r^{2} has the power C(r_{0}) which depends on the direction of the unit vector r_{0} and its sign is similar to that of ΔV. If ΔV > 0, then С > 0 too, and vice versa. The parameters π_{i} and γ_{i} are determined (see e.g. [
${\pi}_{1}=\genfrac{}{}{0.1ex}{}{\left(Tr{P}_{mn}\right)}{3}$;
${\pi}_{2}={\displaystyle \sum _{n\to m}}\genfrac{}{}{0.1ex}{}{{P}_{nn}{P}_{mm}}{6}$;
${\pi}_{3}=2{\displaystyle \sum _{n>m}}\genfrac{}{}{0.1ex}{}{{\left({P}_{nv}\right)}^{2}}{3}$, (3)
where TrP_{mn} is the trace of the doubleforce tensor P_{mn}. The factors γ_{i} depend on the elastic constants of the material and the directions of the vectors q and H. For highsymmetry directions of q and H in cubic crystals, e.g., for 100, 110 and 111 type sites, the γ_{i} expressions are relatively simple (see, table 8 [
During point defect association in the course of structural transformation caused by postcrystallization cooling and process heat treatments, the sizes of the strong distortion regions around defects increase. As a result, the range of q for which the relationship I (q) ~ q^{2} is true (the Huang scattering region) is narrowed. For heavily distorted regions (Q.u(q) >> 1) the diffuse scattering intensity is described by the Stokes–Wilson asymptotic approximation (${S}_{diff}^{ADS}\left(q\right)$). In the q >> q_{0}(QC)^{–1/2} range the asymptotic diffuse scattering crosssection expression takes on as follows [
${S}_{diff}^{ADS}\left(q\right)=N{\left{F}_{H}\right}^{2}\genfrac{}{}{0.1ex}{}{CH}{{\mathrm{\nu}}_{0}{q}^{4}}\Psi (\genfrac{}{}{0.1ex}{}{\mathbf{H}}{H},\genfrac{}{}{0.1ex}{}{\mathbf{q}}{q})$; (4)
here the function Ψ ~ 2 depends on the angle between the vectors H and q.
The components of the tensor P_{nm} for dislocation loops are described as follows:
${P}_{mn}=({C}_{12}Tr{\Omega}_{mn}+{\Omega}_{m}d){\delta}_{nm}+2{C}_{44}{\Omega}_{mm}$, (5)
where Ω_{mn} = 0.5(F_{n}b_{m} + F_{m}b_{n}), d = C_{11} + C_{12} + 2C_{44}; С_{ij} are the components of the elasticity tensor for the cubic symmetry crystal; F and b are the vector characterizing the dislocation loop plane and its Burgers vector, respectively.
The first term in brackets of Eq. (2) corresponds to Huang scattering which is symmetrical relative to the reciprocal lattice site (q = 0). The second term characterizes the asymmetrical part of the diffuse scattering intensity and the shift of the diffuse scattering intensity distribution towards positive (at ΔV > 0) or negative (at ΔV < 0) q_{z} (q_{z} is the projection of the vector q parallel to the reciprocal lattice vector). The XRDS intensity near reciprocal lattice sites can be conveniently represented in the form of equal intensity surfaces (isointensity curves) or their sections by the diffraction plane. In the Huang scattering region at small q the equation of the diffuse scattering isointensity contours is
${q}^{2}=\mathrm{const}{\displaystyle \sum _{i=1}^{3}}{\pi}_{i}{\gamma}_{i}$
The shape of the diffuse scattering isointensity contours is determined by the type of the defects and the symmetry of their displacement fields. The shape of the diffuse scattering isointensity curves allows one to determine the symmetry of the displacement field and choose between their possible configurations. It is sometimes sufficient to analyze the XRDS intensity distribution along with directions parallel or perpendicular to the reciprocal lattice vector of the respective site. With an increase in q length the experimental dependences should exhibit changes in the law of XRDS intensity decrease. One can separate three q regions:
${q}_{0}=\genfrac{}{}{0.1ex}{}{1}{\sqrt{HC}}$; (6)
The expression of diffuse scattering at thermal oscillations [
${S}_{diff}^{\mathrm{T}}=N{\left{F}_{H}\right}^{2}\genfrac{}{}{0.1ex}{}{kT{H}^{2}}{{v}_{0}{q}^{2}}K(\mathbf{Q},\mathbf{q})$
where K (Q, q) is the Christoffel determinant that depends on the directions of the vectors Q and q [
A convenient expression was reported [
$\genfrac{}{}{0.1ex}{}{{s}^{H}\left({q}_{H}\right)}{{S}^{\mathrm{T}}\left({q}_{\mathrm{T}}\right)}=\genfrac{}{}{0.1ex}{}{{n}_{\mathrm{d},\mathrm{f}}{\left(C\right)}^{2}}{kT}\genfrac{}{}{0.1ex}{}{{\left{q}_{\mathrm{T}}\right}^{2}}{{\left{q}_{H}\right}^{2}}\genfrac{}{}{0.1ex}{}{\Psi (\mathbf{Q},\mathbf{q})}{K(\mathbf{Q},\mathbf{q})}$, (7)
where q_{H}, q_{T} are the wave vectors for the angular deviations from the reciprocal lattice site where Huang or thermal diffuse scattering dominate.
This expression allows measuring the XRDS intensity from microdefects in absolute units without allowance for the scattering layer volume, the structural amplitude, the DebyeWaller factor and the solid scattering angle. Thus such parameters as the symmetry, volume and concentration of the microdefects can be determined directly from the experimental data on the diffuse scattering intensity for the test crystal.
The above q dependences of differential scattering crosssections are somewhat different from the experimental intensity vs q dependences. To provide for the required light power when a synchrotron radiation source is used, one has to impart a certain vertical divergence to the beams (this divergence is perpendicular to the diffraction plane). In this case the slopes of the XRDS intensity vs q dependences I (q) are not 2 and 4, but 1 and 3, respectively.
Huang scattering is efficient if all the microdefects have the same sign of power C. However, if the test crystal contains microdefects with different power signs, i.e., vacancy and interstitial type microdefects are present, and especially if there are microdefects smaller than ~1 mm in size, the asymptotic scattering region is more experimentally accessible. In this region the vectors q satisfy the condition (CQ)^{0.5} << q << CQR_{0}^{3}, where R_{0} is the characteristic microdefect size [
$C=\genfrac{}{}{0.1ex}{}{1}{H{({q}_{{z}_{2}}^{2/3}{q}_{{z}_{1}}^{2/3})}^{3}}$, (8)
where q_{zi} (i = 1, 2) are the coordinates of the two adjacent maxima in the Iq_{z}^{3} = f (q_{z}) dependence. The number of maxima in the region (CQ)^{0,5} << q << (CQ)R_{0}^{3} is as follows [
${n}_{L}=Q{b}_{L}$, (9)
where b_{L} is the Burgers vector of the loop;
${n}_{cl}=\genfrac{}{}{0.1ex}{}{4}{3}\pi Q{R}_{cl}$.
Thus, the XRDS intensity is determined by the doubleforce tensor components. However, the actually measured XRDS intensity is that for the near region of the reciprocal lattice sites. Clearly, this situation refers to defects far greater than the atomic sizes. The use of synchrotron sources and low temperature for experiments allow studying displacement fields for discrete point defects by measuring XRDS intensity distribution over the entire Brillouin zone around the reciprocal lattice site, when the intensity is low or (at low defect concentrations) even lower than the room temperature thermal diffuse scattering intensity.
On the other hand, a change in the lattice parameter caused by point defects in the crystal is also related to the doubleforce tensor components. For example, in cubic crystals [
$3\genfrac{}{}{0.1ex}{}{\Delta a}{a}=n\genfrac{}{}{0.1ex}{}{\Delta V}{V}=\genfrac{}{}{0.1ex}{}{n\sqrt{3{\pi}_{1}}}{{v}_{0}({C}_{11}+2{C}_{12})}$, (10)
where ΔV/V is the relative change in the crystal volume caused by one defect and C_{ij} are the elastic moduli. If point defects produce a displacement field with a cubic symmetry, then P_{11} = P_{22} = P_{33} and π_{1} = 3P_{11}^{2}, whence
$3\genfrac{}{}{0.1ex}{}{\Delta a}{a}=\genfrac{}{}{0.1ex}{}{3n{\pi}_{11}}{{v}_{0}({C}_{11}+2{C}_{12})}$, (11)
where
${P}_{11}=({C}_{11}+2{C}_{12}){v}_{0}\genfrac{}{}{0.1ex}{}{\Delta r}{r}$. (12)
Equations (11) and (12) give the relaxation volume for one point defect which can be evaluated from the respective change in the lattice parameter:
$3{n}^{1}\genfrac{}{}{0.1ex}{}{\Delta a}{a}=3\genfrac{}{}{0.1ex}{}{\Delta r}{r}$, (13)
where Δr/r is the relative difference in the atomic radii of the matrix and the point defect. If the lattice contains point defects with positive and negative Δr, then the overall change Δa/a will be the superposition of these solid solutions:
$3\genfrac{}{}{0.1ex}{}{\Delta a}{a}=\genfrac{}{}{0.1ex}{}{1}{{v}_{0}({C}_{11}+2{C}_{12})}[{n}_{i}Tr\left({P}_{jj}^{i}\right){n}_{v}Tr\left({P}_{jj}^{v}\right)]$, (14)
where n_{i}, n_{v} are the concentrations of interstitial and vacancy type defects.
Thus, precision lattice parameter measurements allow evaluating the differential point defect concentration if the relaxation volumes are known. The symmetrical part of Huang scattering does not depend on the defect power sign, and the intensities of the radiation scattered from defects with opposite power signs are superimposed. However, at somewhat greater q for which the asymptotic approximation is valid (see Eq. (4)), ${I}_{diff}^{ADS}\sim \Delta V/{q}^{4}$ here ΔV is the volume change per one defect. As a result, the scattering intensity for vacancy and interstitial type defects is distributed at different sides of the reciprocal lattice site.
Thus, precision lattice parameter and diffuse scattering measurements are complementary methods allowing one to analyze structural changes in the state of point defects at early stages of solid solution decomposition in semiconductors. We will demonstrate this below with some examples.
Early precipitation stages in Si(O) solid solution containing (7–8) ∙ 10^{17} cm^{3} oxygen in silicon was studied [
XRDS scattering intensity distribution near the [[400]] site for silicon single crystal after homogenizing annealing at 1000 °C and annealing at 450 °C (16 h): (a) experimental isointensity curve, (b) symmetrical component and (c) asymmetrical component. XRDS intensity for the isointensity curves varies in the 0.5–9.5 cps range with a 1.0 cps step.
TEMundetectable defects that are coherent with the matrix and have smeared boundaries with small displacement field gradients can also be produced by neutron irradiation and relaxation of disordered regions accompanied by redistribution of point defects and annihilation of interstitial and vacancy type defects. The dependence of the increase in the indium antimonide lattice parameter on the neutron fluence showed that, according to calculations, only ~20 Frenkel pairs out of 600 per one fast reactor neutron (Е > 0.1 MeV) remained after annihilation [
XRDS scattering intensity distribution along the q_{z} crosssection of the diffuse scattering isointensity curve for the [[224] site of fast neutron irradiated InSb crystals at different fluencies φ_{t}, cm^{–2}: (1) initial crystal; (2) fluence 5 × 10^{16} cm^{–2}; (3) 5 × 10^{17} cm^{–2}; (4) 1.3 × 10^{18} cm^{–2}.
(a) XRDS scattering intensity distribution along the q_{z} crosssection of reciprocal space for fast neutron irradiated InSb(Te) crystal at 1.3·10^{18} cm^{2} fluence (1) before and (2) after heat treatment at different temperatures and (b) lattice parameter of InSb single crystals with different initial carrier concentration irradiated with neutrons at different fluencies F as a function of subsequent heat treatment temperature: (a) (1) before heat treatment, (2) after heat treatment at 150 °C, (3) at 200 °C and (4) at 300 °C; (b) (1) InSb(Te), n_{0} = 4·10^{14} cm^{3}, F_{fn} = 1.3·10^{18} cm^{2}; (2) InSb(Mn), p_{0} =5·10^{13} cm^{3}, F_{fn} = 5·10^{17} cm^{2}; (3) InSb(Te), n_{0} = 1.7·10^{15} cm^{3}, F_{sn} = 4.3·10^{17} cm^{2}.
An interesting example demonstrating the efficiency of combined lattice parameter and XRDS measurements is a study of InP crystal structure after transmutation doping by reactor neutron irradiation [
InP lattice parameter as a function of major carrier concentration (1) before and (2) after neutron irradiation at 2.3·10^{19} cm^{–2} fluence (thermal and rapid neutron fluence ratio is 1).
InP lattice parameter as a function of rapid neutron fluence for crystals with different initial impurity content n, cm^{–3}: (1) major carrier concentration (1.6–2.3)·10^{18} ; (2) 3.5·10^{16}; (3) (3–4.1)·10^{16} ; (4) 2·10^{17} ; (5) (4.2–5.2)·10^{16} ; (6) 8·10^{17}; (7) 3.9·10^{18}.
We consider another typical application example of the methods. Postgrowth cooling of nonstoichiometric GaAs and GaP crystals with significant quantities of excess cations caused precipitation of excess Ga resulting in the formation of interstitial Ga (Ga_{i}) and Ga_{i} complexes with residual interstitial phosphorus atoms (Р_{i}) [
Typical XRDS scattering intensity distribution for GaP and GaAs crystals grown from Ga excess melt: (a) ingot beginning, (b) middle and (c) end.
Fig.
XRDS scattering intensity distribution along q_{x} crosssection: (a) ingot beginning and (b) end.
Thus, analysis of the XRDS pattern shown in Figs
We consider another interesting XRDS method application: a study of dislocation loops forming at a certain stage of point defect structural state evolution in nonequilibrium solid solutions. Dislocation loops are described by their strength (diameter and Burgers vector) and planes. Typically the loop plane and its Burgers vector are determined by comparing the calculated and experimental diffuse scattering isointensity curves. If the loops have a homogeneous size distribution, their linear parameters can be determined using two methods. The average loop radius R_{0} can be evaluated from the vector q_{0} corresponding to the knee point in the qx crosssection of the diffuse scattering isointensity curve using the formula
,
where H is the diffraction vector and b is the Burgers vector.
The other method is based on analysis of the dependence. Fig.
Analysis of the results showed that the XRDS method is a highly sensitive and information valuable tool for studying early stages of solid solution decomposition in semiconductors. A review of the results relating to the methodological aspect showed that the most consistent approach is a combination of XRDS with precision lattice parameter measurements. It allows one to detect decomposition stages that cannot be visualized using TEM and to draw conclusions as to microdefect formation mechanisms.
For the GaP and InP examples, a structural microdefect formation mechanism was revealed associated with the interaction of defects forming during the decomposition and residual intrinsic defects.
The mechanism of sulfur polytropy in gallium phosphide was observed for the first time.
Study of the evolution of diffuse scattering isointensity contours in GaP, GaAs(Si) and Si(O) allows tracing the evolution of microdefects from matrixcoherent ones to microdefects with smeared coherency resulting from microdefect growth.