X-ray diffraction (XRD) characterization methods for sigma=3 twin defects in cubic semiconductor (100) wafers

This patent application claims the benefit of and priority to U.S. Provisional Patent Application No. 61/877,416, entitled “X-RAY DIFFRACTION (XRD) CHARACTERIZATION METHODS FOR SIGMA=3 TWIN DEFECTS IN CUBIC SEMICONDUCTOR (100) WAFERS” filed on Sep. 13, 2013, the contents of which are hereby incorporated by reference in their entirety.

The invention described herein was made in the performance of work under a NASA contract and by employees of the United States Government and is subject to the provisions of Public Law 96-517 (35 U.S.C. §202) and may be manufactured and used by or for the Government for governmental purposes without the payment of any royalties thereon or therefore. In accordance with 35 U.S.C. §202, the contractor elected not to retain title.

Semiconductor materials are widely utilized in numerous electronic devices. An ingot/boule may be grown from a single seed crystal, and the ingot may be sliced into relatively thin (e.g. 0.75 mm thick) wafers. Various additional processing steps such as deposition, removal, patterning, cutting, doping, etc, may be performed on the wafer to fabricate an electronic device. Various crystal structure defects may be present in semiconductor materials. Such defects may adversely affect the performance of electronic devices made from semiconductor materials.

The 60° rotated twin defect on {111} planes is one of the most common crystal structure defects in many cubic semiconductors. This defect has a sigma=3 grain boundary commonly called the sigma=3 twin defect on {111} plane. It is also called a 180° rotated twin defect because every 120° rotation is identical, due to the threefold symmetry of the cubic [111] direction. Sigma=3 twin defects are also frequently found in the group IV semiconductors (Si, Ge, C) in a diamond structure and other cubic zinc blonde III-V and II-VI compound semiconductors such as GaP, InP, InGaAs, CdTe and ZnSe.

With reference to FIG. 1 a , single crystal GaAs 10 comprises gallium atoms 6 and arsenide atoms 8 . FIG. 1 a shows the single crystal GaAs 10 without defects and FIG. 1 b shows the formation of sigma=3/{111} twin defects 12 by a stacking fault 14 on {111} planes adjacent a single crystal GaAs substrate 16 . FIG. 1 b shows the cubic crystal structure of GaAs and {111} crystal plane normal vectors. The net effect of the sigma=3/{111} twin defect 12 made by a stacking fault 14 is the rotation of the crystal structure cube by 60° while it shares the common triangular {111} plane 20 with the original cube 18 as shown in FIG. 1 d.

The low stacking fault formation energy (45 mJ/m 2 for GaAs (111)), (30 mJ/m for InAs and 17 mJ/m for InP) facilitates frequent creation of sigma=3/{111} twin defects, which become the source of polymorphism between cubic zinc blende structure and hexagonal Wurtzite structure. Although there have been many nanometer-to-micrometer scale characterizations for the stacking faults and sigma=3 twins using transmission electron microscopy (TEM), only a limited number of wafer-scale macroscopic characterizations such as XRD analysis have been reported. These few reports include an XRD detection method of sigma=3/{111} twin defects on GaAs (111)B wafer and GaAs (111) pole-figure analysis of Carbon-60 induced accidental asymmetric twin defects on GaAs (100) wafer.

Si (100) wafers and GaAs (100) wafers are widely used in the micro-electronics industry. However, known defect measuring techniques (e.g. TEM and Etch-pit density test) damage or destroy the wafer, and the damaged wafer is typically useless after testing. Thus, a non-destructive test to detect/measure sigma=3/{111} defects in various materials would be beneficial.

The present invention comprises non-destructive XRD characterization processes/methods. One aspect of the present invention is a concentration measurement process/method that provides a quality factor (ratio) that quantitatively describes the concentration of sigma=3/{111} twin defects. Another aspect of the present invention is a wafer mapping process/method for any cubic tetrahedral semiconductor wafers including, without limitation, GaAs (100) wafers and Si (100) wafers. The methods/processes of the present invention may utilize (004) pole-figures of cubic semiconductors in order to detect sigma=3/{111} twin defects which are incorporated in (100) wafers during fabrication utilizing processes such as the Vertical Gradient Freeze (VGF) growth of GaAs ingots or Czochralski growth of Silicon ingots. However, it will be understood that the present invention is not limited to semiconductor materials/devices fabricated according to these processes. The XRD methods/processes according to the present invention are applicable to any (100) wafers of other tetrahedral cubic semiconductors in the diamond structure (Si, Ge, C) and cubic zinc-blende structure (InP, InGaAs, CdTe, ZnSe, and so on) with various growth methods including Liquid Encapsulated Czochralski (LEC) growth, Molecular Beam Epitaxy (MBE), Organometallic Vapor Phase Epitaxy (OMYPE), Czochralski growth, Metal Organic Chemical Vapor Deposition (MOCVD) growth, or other processes.

The method/processes of the present invention do not require contact or treatment of the materials being tested. The methods/processes can be utilized to provide a pass/fail (quality factor) measurement of individual wafers in a very short time. Thus, the methods/processes can be utilized in connection with commercial wafer fabrication processes to ensure that the wafers that are produced meet predefined quality/defect criteria. Furthermore, the results of XRD testing/methods/processes according to the present invention can be utilized to identify problems in water fabrication processes whereby the process can be modified to reduce/eliminate defects in the wafers. Significantly, the defect measurement methods/processes of the present invention can be integrated into wafer fabrication processes to provide “real time” feedback that can be utilized to rapidly modify the wafer fabrication process and reduce the number of defective wafers that are fabricated.

The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee;

FIG. 1 a is a schematic drawing of a single crystal GaAs material;

FIG. 1 b is a schematic drawing of a single crystal GaAs material showing a stacking fault and sigma=3/{111} twin crystal GaAs;

FIG. 1 c is a schematic perspective view showing the crystal structure of GaAs;

FIG. 1 d is a schematic isometric view showing a twin crystal lattice cube rotated by 60° on a (111) plane of the original GaAs cubic lattice;

FIG. 2 is a graph showing XRD 2θ-Ω normal scan of a GaAs wafer;

FIG. 3 a is pole-figure of GaAs (004) intensity (CPS);

FIG. 3 b is a schematic isometric view showing the angles of the sigma=3/{111} twin defects with respect to the original crystal;

FIG. 4 a is a plan view of a conical semiconductor wafer specimen cut from the GaAs ingot of FIG. 4 d;

FIG. 4 b is a plan view of a cylindrical semiconductor wafer specimen cut from the GaAs ingot of FIG. 4 d;

FIG. 4 c is a plan view of a cylindrical semiconductor wafer specimen cut from the GaAs ingot of FIG. 4 d;

FIG. 4 d is a side elevational view of a GaAs ingot grown utilizing a VGF process;

FIG. 4 e is a colored image of twin defect wafer mapping results corresponding to the wafer of FIG. 4 a wherein red color has the highest defect density (0.3% by intensity ratio) through yellow and green, to blue (lowest intensity ratio);

FIG. 4 f is a colored image of twin defect wafer mapping results corresponding to the wafer of FIG. 4 h wherein red color has the highest defect density (0.3% by intensity ratio) through yellow and green, to blue (lowest intensity ratio);

FIG. 4 g is a colored image of twin defect wafer mapping results corresponding to the wafer of FIG. 4 c wherein red color has the highest defect density (0.3% by intensity ratio) through yellow and green, to blue (lowest intensity ratio);

FIG. 4 h is a side elevational view of the GaAs ingot of FIG. 4 d showing a single crystalline seed and propagation of the twin defect along the {111} direction; and

FIG. 5 is a (004) pole figure of a silicon (100) wafer, test grade, P-type 0-100 Ohm·Cm.

For purposes of description herein, the terms “upper,” “lower,” “right,” “left,” “rear,” “front,” “vertical,” “horizontal,” and derivatives thereof shall relate to the invention as oriented in FIGS. 1 a and 1 b . However, it is to be understood that the invention may assume various alternative orientations and step sequences, except where expressly specified to the contrary. It is also to be understood that the specific devices and processes illustrated in the attached drawings, and described in the following specification, are simply exemplary embodiments of the inventive concepts defined in the appended claims. Hence, specific dimensions and other physical characteristics relating to the embodiments disclosed herein are not to be considered as limiting, unless the claims expressly state otherwise.

As discussed in more detail below, one aspect of the present invention is a process or method for determining a quality factor comprising a ratio as defined in equations 1.0, 1.1, and 1.2 below. The method includes determining the intensity of an original cubic substrate's (004) peak utilizing an XRD process. The XRD process is also utilized to measure the intensity of sigma=3/{111} peaks or spots, as also described in more detail below. The intensity ratio of the sigma=3/{111} defect spots and the original cubic crystals (004) peak defines a quality factor of the semiconductor wafer which is independent from X-ray intensity, slit size and detector sensitivity. This is because the instrumental parameters are compensated in the ratio equation. Thus, the numerical quantity (ratio) provides a quality factor of the wafers in terms of sigma=3/{111} twin defects, in which a lower number signifies fewer sigma=3/{111} defects.

As also described in more detail below, another aspect of the present invention involves an XRD wafer mapping process whereby twin defect density maps of a wafer are developed. The XRD wafer mapping may be utilized to generate a color image using an array/arrangement of one or more colors, each color corresponding to a measured sigma=3/{111} twin defect density, where, for example, red corresponds to a high defect density ( FIGS. 4 e -4 g ), and blue represents a low twin defect density. The XRD wafer mapping process can be utilized to determine the propagation of sigma=3/{111} twin defects in an ingot formed utilizing a VGF growth process.

In an exemplary embodiment of the present invention, a GaAs ingot 52 ( FIGS. 4 d and 4 h ) was grown utilizing a Vertical Gradient Freezing (VGF) process. The GaAs ingot 52 was sliced to produce multiple 3-inch (100) wafers 50 A, 50 B, and 50 C of 500 micrometer thickness. Each wafer 50 A, 50 B, 50 C was labeled from the conical region 54 adjacent to the single crystal GaAs seed 64 ( FIG. 4 h ) at the bottom 70 to the straight cylindrical upper region 56 where the commercial GaAs (100) wafers are produced. This particular GaAs ingot 52 showed a small portion of a hazy area and a few line defects which propagated through multiple wafers.

A PANalytical X'Pert Pro MRI) X-ray diffractometer (not shown) with a 4-circle high resolution goniometer in the Bragg-Brentano configuration was used to characterize the GaAs wafers 50 A, 50 B, and 50 C. The X-ray source was Cu Kα lines with an average wavelength of 1.54187 Å which were filtered by a parabolic X-ray mirror crystal monochrometer. The intensity ratio of Cu Kα/Cu kα was 0.5. A line X-ray source with a parabolic mirror was used for the 2θ-Ω scan and a point X-ray source with a beam mask (not shown) was used for the pole figure measurement and the defect wafer mapping. In the 2θ-Ω scan, a 0.02 mm nickel filter and ¼° divergence slit were used for the incidence beam optics and 1/16° receiving slit and ¼° anti-scatter slit were used for the diffracted beam optics.

For the pole figure measurement, a Soller slit of 0.04 radian with a 10 mm beam mask and 2° divergence slit were used for the incidence beam optics and ¼° receiving slit and ½° anti-scatter slit were used for the diffracted beam optics. For the (004) pole figure, 2θ was set to 66.0987° and Ω was set to 33.1141° for the maximum intensity. The in-plane rotation (angle Φ) scan was made in the range of 0°-360° with 3° step and the tilt angle (angle Ψ) scan was made from 0° to 90″ with 3° steps.

Wafer defect mapping ( FIGS. 4 e -4 g ) was made with XY movement of the sample stage in 0.5 mm steps. A 5 mm beam mask and 1° divergence slit were used for the incidence beam optics and 0.04 radian Soller slit with ½° receiving slit and 1° anti-scatter slit were used for the diffracted beam optics. PANalytical X'pert Data Collector software was used for acquisition of the X-ray diffraction data. The pole figure and wafer mapping were analyzed with X'pert Texture software and X'pert Epitaxy software, respectively.

Pole Figure Analysis

FIG. 2 is a logarithmic Y-scale plot of the 2θ-Ω XRD normal scan of a GaAs (100) wafer which shows (hkl) peaks in the [001] direction (i.e. the surface normal direction). The strongest (004) peak 30 is located at 2θ=66.039° with an intensity of 1,742,878 counts per second (cps) with the overlap of the 2 nd order peak of quasi-forbidden (002) plane. The first order (002) peak 32 is located at 2θ=31.618° with an intensity of 99,217 cps and the third order (002) peak 34 is located at 2θ=109.670° with an intensity of 18,505 cps. The third order (002) peak 34 is often called the (006) peak although there is no actual atomic plane at ⅙ of the vertical lattice constant. The quasi-forbidden GaAs (002) peaks appear in many XRD reports as a result of the lattice strain and defects. The 2θ-Ω XRD normal scan of FIG. 2 with the very strong (004) peak 30 , the weak (002) 32 , 34 and no other peaks shows that this VGF grown GaAs wafer exhibits commercial grade mono-crystalline quality.

The pole figure of GaAs (004) plane diffraction is plotted in FIG. 3 a . The pole figured 40 is made in a logarithmic intensity scale with a polar coordinate (radius ψ=tilt angle) of the in-plane rotation angle Φ for the water rotation (0° to 360°) and the radius ψ for the wafer tilt angle (0° to 90°) in order to reveal the weak twin defect peaks. The single crystal GaAs (004) peak 42 is located at the center of the pole-figure with a very strong intensity of 1,289,770 cps. At the tilt angle ψ=48.2°, eight small spots 1 A, 2 A which are usually called peaks in XRD-scans appear in a generally symmetric pattern. At another tilt ψ=78.5°, four weak peaks 3 A appear every 90°. These 12 peaks, i.e. 8 peaks at ψ=48.2° and 4 peaks at ψ=78.5° are {004} peaks of sigma=3/{111}) twin defects. The angular relationships of the crystal planes are shown in FIG. 3 b . Three inter-planar angles of 1 A(ψ), 2 A(ψ) and 3 A (φ) with respect to the vertical c-axis direction in FIG. 3 b are assigned to three twin defect peaks, 1 A and 2 A at the same angle ψ=48.2° and 34 A at v=78.5° from the center 42 of the pole FIG. 40 , i.e. [004] direction in FIG. 3 a.

The vertical tilt angles and projected in-plane rotation angles in the XY plane between the twin's [004] plane and the original single crystal GaAs [100] and [110] directions are listed below.

For angle 1 A in FIG. 3 b

In-plane rotation angle (ΔΦ) between twin's [004] direction and original single crystal's [100] direction=−26.57°, Vertical tilt angle (Δψ) between twin's [004] direction and original single crystal's [004] direction=48.2°.

For angle 2 A in FIG. 3 b

In-plane rotation angle (ΔΦ) between twin's [004] direction and original single crystal's [010] direction=+26.57°, Vertical tilt angle (Δψ) between twin's [004] direction and original single crystal's [004] direction=48.2°.

For angle 3 A in FIG. 2 b,

In-plane rotation angle (ΔΦ′) between twin's [004] 3 A direction and original single crystal's [110] direction=180°, Vertical tilt angle (Δψ) between twin's [004] 3 A direction and original single crystal's [004] direction=78.5°.

Therefore, four {±1, ±1, 1} corner planes on a (100) wafer makes (4 planes under 90° rotation)×(3 twin peaks per plane)=12 twin defect peaks, of which 8 peaks are at ψ=48.2 and 4 peaks at ψ′=78.5° in the (004) pole FIG. 40 of FIG. 3 a . The intensity of the twin defect peaks 1 A and 2 A at ψ=48.2° are 3,459 cps and 4,276 cps, respectively and that of the third peak 3 A is 817 cps. The intensity of the peaks in the pole Figure decreases as the tilt angle ψ increases because the X-ray beam passing through and returning from the material is strongly attenuated due to the longer beam path near the glancing exit angle at the higher tilt angle. The ratio of averaged height intensity (magnitude) of twin's {004} peaks 1 A and 2 A, divided by the height intensity (magnitude) of the original single crystals (004) peak is

( 3 ⁢ , ⁢ 459 + 4 ⁢ , ⁢ 276 ) / 2 1 ⁢ , ⁢ 289 ⁢ , ⁢ 770 = 0.0030 = 0.30 ⁢ % ,

which means that the concentration of sigma=3/{111} twin defects is small but detectable with XRD methods/processes according to the present invention.

XRD Wafer Mapping

A wafer mapping XRD scan was made using the twin defect's (004) peak 1 A in FIG. 3 a . After aligning the wafer angles (Ω, ψ, Φ) and the detector angle (2θ) to the twin defect's (004) peak 1 A with a beam mask, the sample stage was moved in the XY direction in 0.5 mm steps. FIGS. 4 a , 4 b , and 4 c show GaAs wafers 50 A, 50 B, and 50 C, respectively. With further reference to FIG. 4 d , the wafers 50 A, 50 B, and 50 C were cut from different sections of a VGF grown GaAs ingot 52 . Wafer 50 A comprises a conical sample cut from conical portion 54 of GaAs ingot 52 ( FIG. 4 d ), and wafers 50 B and 50 C comprise cylindrical wafers cut from straight cylindrical upper region 56 of GaAs ingot 52 .

FIGS. 4 e -4 g are twin defect density maps (color) corresponding to FIGS. 4 a -4 c , respectively. Regions 58 A- 58 C ( FIGS. 4 a -4 c ) have relatively high sigma=3/{111} twin defects. Regions 58 A- 58 C generally correspond to the red regions 49 , 49 B, 49 C, respectively, of FIGS. 4 c -4 g . In FIGS. 4 e -4 g , the color red has the highest defect density (0.3% by intensity ratio) followed by yellow and green, to blue (lowest intensity ratio). However, it will be understood that this is merely an example of a suitable mapping arrangement and the present invention is not limited to this example.

The conical wafer 50 A was measured using the planar bottom surface 66 A which is close to the single crystal GaAs seed 64 ( FIG. 4 h ) utilized in the VGF growth process. The other wafers 50 B and 50 C were measured using the top surfaces 68 B and 68 C, respectively. The drawing and wafer mapping result ( FIG. 4 c ) of the bottom surface 66 A of conical wafer 50 A is flipped horizontally in order to provide the same orientation with respect to other wafers' top surfaces. Because the conical wafer 50 A has a slope with a tall thickness, the XRD wafer mapping ( FIG. 4 e ) shows a background tail area 48 in the boundary where the wafer height deviates from the XRD focal point. The flat circular (center) area 49 ( FIG. 4 e ) shows the correct XRD twin defect mapping result corresponding to the flat circular bottom surface 66 A ( FIG. 4 a ). The orientation of the pole figure in FIG. 3 a and the XRD twin defect wafer mapping results in FIGS. 4 e -4 g are aligned in the same direction. The red color ( FIG. 4 e ) shows that there is high density of twin defects in the left side ( FIGS. 4 a and 4 e ) of the conical wafer 50 A. The high defect region 76 ( FIG. 4 h ) extends/propagates to the top left corners of upper wafers 50 B and 50 C, which is the [111] direction of the GaAs wafer in FIG. 3 a.

With reference to FIG. 4 h , propagation of the sigma=3/{111} twin defects along [111] direction can be explained as follows. In the VGF growth of a GaAs ingot 52 , a small single crystal seed 64 is positioned at the bottom 70 under the conical region 54 . Very careful thermal controls are applied in order to regulate the crystallization velocity as the GaAs ingot 52 is formed. Arrows 72 and 74 represent the (111) and (100) planes, respectively of the crystal seed 64 . During the vertical freezing process, {111} facet planes can be created accidentally or natively from the seed crystal's {111} facets. Also, the VGF growth inside the conical region 54 requires the expansion of the GaAs crystal into the side directions including <111> directions. Therefore, it is very easy to create sigma=3/{111} twin defects on the {111} facets due to the low formation energy in such a growth condition. Once the twin defect is created, it propagates to the upper wafer regions vertically as the GaAs ingot 52 grows to form a high defect region 76 having high twin defect density/frequency. A boundary 78 extends between high defect region 76 and low defect region 80 .

According to another exemplary embodiment of the present invention, a Czochralski grown commercial grade Silicon (100) wafer was tested utilizing substantially the same X-ray diffraction methods as described above in connection with FIGS. 4 a -4 h . The silicon wafer (not shown) was mounted on the XRD sample holder with a slightly different in-plane angle from GaAs wafer alignment. The pole-figure analysis for the silicon (100) wafer is shown in FIG. 5 . Substantially, the same sigma=3/{111} twin defect peaks are shown for the silicon (100) wafer ( FIG. 5 ) as for the GaAs wafer ( FIG. 3 a ). Thus, the positions of peaks 1 B, 2 B, and 3 B ( FIG. 5 ) are substantially the same as peaks 1 A, 2 A, 3 A, respectively ( FIG. 3 a ) of the GaAs (100) wafer case described above. The vertical tilt angle of sigma=3/Si(100) peaks 1 B and 2 B are 48.2°, which is the same as the peaks 1 A and 2 A of GaAs (100) wafer's sigma=3 defect. The vertical tilt angle of peak 3 B in FIG. 5 is 780, which is the same vertical tilt angle as the peak 3 A of GaAs (100) case in FIG. 3 . The Si(100) wafer sigma=3 defect's in-plane angles between the 1 B, 2 B and 3 B peaks in FIG. 5 are the same as those of the 1 A, 2 A and 3 A peaks, respectively, of the GaAs (100) wafer described above ( FIG. 3 a ). This particular Si(100) wafer contains many smaller peaks which are different from the sigma=3/{111} twin defect peaks. These smaller peaks may represent other polycrystalline defects, such as low angle twin defects on (110) plane.

The important eight strong spots at 48.2° vertical tilt angle and four weak spots at 78.5° vertical tilt angle are detected in both GaAs (100) wafer and Si(100) wafer. These total 12 spots in FIG. 3 and FIG. 5 came from [4-upper corners, i.e. {111} planes of Si/GaAs (100) cubic crystal]×[3 facets of sigma=3/{111} defects per corner]=12 peaks in the XRD pole figures of FIGS. 3 a and 5 .

Thus, according to the exemplary embodiments described above, the present invention comprises at least two systematic X-ray diffraction (XRD) processes/methods that may be utilized to characterize sigma=3/{111} twin defects on VGF grown mono-crystalline GaAs (100) wafers and Czochralski grown Si (100) wafers. The XRD analysis of GaAs and Si (004) pole figures reveals information about the total concentration and orientation distribution of the twin defects. The XRD wafer mapping method shows the spatial distribution of the twin defects. XRD analysis of multiple sequential wafers from the same ingot reveals the defect formation and propagation mechanisms.

XRD methods/processes according to the present invention are applicable to all mono-crystalline tetrahedral cubic semiconductor wafers including group IV semiconductors in a diamond structure and group III-V & II-VI semiconductors in a cubic zinc-blende structure. The fabrication of mono-crystalline semiconductor wafers and epitaxial thin films in various fields of industry may be improved utilizing methods/processes according to the present invention.

The methods/processes of the present invention provide unique solutions that can be utilized to characterize sigma=3 twin defects in (100) wafers and ingots. For example, the intensity of sigma=3/{111} spots, such as peak volume (height×tilt-angle)×in-plane angle), peak area (height)×tilt angle or height×in-plane angle), or peak height can be measured, and the numerical data can be used as a standard parameter to evaluate the quality of a wafer. If the intensity of the original cubic substrate's (004) peak is also measured, the intensity ratio of sigma=3/{111} defect spots and original cubic crystal's (004) peak may comprise a quality factor of the wafer Which is independent from X-ray intensity, slit size and detector sensitivity because the instrumental parameters are compensated in the ratio equation. Therefore, the following numerical quantity (intensity ratio) may serve as a quality factor of the wafers in terms of sigma=3 twin defects, in which a lower number indicates that there are fewer sigma=3 defects. This number (intensity ratio) can be used as an industrial standard to indicate the quality of a wafer.

(1) Quality Factors with Instrumental Dependence Such as X-Ray Intensity, Slit Size, Detector Sensitivity:

1. Absolute intensity of sigma=3/{111} spots in XRD scan including (004) pole figure, tilt-angle vs. intensity scan, in-plane angle vs. intensity scan, tilt-angle vs. in-plane angle vs. intensity, omega-scan around 48.2° or 78.5° tilt angle and in-plane angle, detector angle (2-theta) scan around 48.2° or 78.5° tilt angle and in-plane angle and two-theta—omega scan around 48.2° or 78.5° tilt angle and in-plane angle.

(2) Quality Factors Independent From XRD Instrument Parameters:

Sigma=3 spots/peaks can be selected from eight spots/peaks at 48.2° tilt angle or four spots/peaks at 78.5°. Eight spots/peaks at 48.2° tilt angle are stronger than those at 78.5°. Therefore, it may be preferable to use the eight strong spots/peaks at 48.2° tilt angle to measure the quality factor (ratio). The magnitude of the intensities for the ratios 1-3 above can be measured with a conventional X-ray diffraction machine with one or two scanning detectors and a rotating sample goniometer.

The quality factor(s) (Ratios 1-3) can also be measured with multiple fixed detectors which are installed at predefined angles rather than scanning and rotating the wafers. If the wafer is loaded with the same in-plane angle every time, the detectors located at predefined angles are capable of measuring the quality factor much faster than scanning the angles with one detector. The actual density of sigma=3/{111} defect can be calculated from the quality factor (Ratios 1-3) with a proportional coefficient.

In general, all three Ratios may be utilized to define a quality factor. Alternatively, a single ratio may be utilized to define a quality factor, or any combination of Ratios 1-3 may be utilized to define a quality factor.

(3) Wafer Mapping Method for Sigma=3/{111} Twin Defect on (100) Wafers

The detector and sample angles are aligned with 48.2° tilt angle and one of the eight peaks/spots' in-plane angles. For the best spatial resolution, a beam mask is inserted in front of the X-ray source to form a narrow focused beam. The wafer is moved in the X-Y directions (i.e. the X-Y plane), and the instrument measures the intensity of the diffracted beam resulting from the sigma=3 twin defects. The instrument creates a map of the concentration of sigma=3 twin defects by showing the intensity of the refracted beam at each (X, Y) coordinate.

All cited patents, patent applications, and other references are incorporated herein by reference in their entirety. However, if a term in the present application contradicts or conflicts with a term in the incorporated reference, the term from the present application takes precedence over the conflicting term from the incorporated reference.

All ranges disclosed herein are inclusive of the endpoints, and the endpoints are independently combinable with each other. Each range disclosed herein constitutes a disclosure of any point or sub-range lying within the disclosed range.

The use of the terms “a” and “an” and “the” and similar referents in the context of describing the invention (especially in the context of the following claims) are to be construed to cover both the singular and the plural, unless otherwise indicated herein or clearly contradicted by context. “Or” means “and/or.” As used herein, the term “and/or” includes any and all combinations of one or more of the associated listed items. As also used herein, the term “combinations thereof” includes combinations having at least one of the associated listed items, wherein, the combination can further include additional, like non-listed items. Further, the terms “first,” “second,” and the like herein do not denote any order, quantity, or importance, but rather are used to distinguish one element from another. The modifier “about” used in connection with a quantity is inclusive of the stated value and has the meaning dictated by the context (e.g., it includes the degree of error associated with measurement of the particular quantity).

Reference throughout the specification to “another embodiment”, “an embodiment”, “exemplary embodiments”, and so forth, means that a particular element (e.g., feature, structure, and/or characteristic) described in connection with the embodiment is included in at least one embodiment described herein, and can or cannot be present in other embodiments. In addition, it is to be understood that the described elements can be combined in any suitable manner in the various embodiments and are not limited to the specific combination in which they are discussed.