X-ray diffraction device and method to measure stress with 2D detector and single sample tilt

Abstract

A method is provided for performing an X-ray diffraction stress analysis of a sample such as a thin film, a coating, or a polymer. The sample has a surface with two perpendicular axes S.sub.1, S.sub.2 within a plane of the surface, and a third axis S.sub.3 perpendicular to the sample surface plane. An X-ray beam is directed at the sample surface at a relatively low angle with regard to the surface plane. X-ray energy is diffracted from the sample and detected with a two-dimensional X-ray detector at a plurality of rotational orientations of the sample about S.sub.3. The third axis S.sub.3 is maintained at a constant tilt angle during the entire X-ray diffraction stress analysis, thereby avoiding the significant error associated to the movement of a cradle track of a goniometer used for the X-ray diffraction stress analysis and on which measurements at a low 2 angle are highly sensitive.

Claims

1. A method of performing an X-ray diffraction stress analysis of a sample having a surface with two perpendicular axes S.sub.1, S.sub.2 within a plane of the surface, a third axis S.sub.3 being perpendicular to the sample surface plane, the method comprising directing an X-ray beam at the sample surface at a relatively low angle with regard to the surface plane and detecting, at a diffraction angle 2, X-ray energy diffracted from the sample with a two-dimensional X-ray detector at a plurality of rotational orientations of the sample about S.sub.3, the third axis S.sub.3 being maintained, during the entire X-ray diffraction stress analysis, at a constant tilt angle relative to a plane formed by the X-ray beam and a normal to the two-dimensional X-ray detector.

2. The method of claim 1, wherein the sample is at least one of: a thin film, a coating, and a polymer.

3. The method of claim 2, wherein the diffraction angle is under 90.

4. The method of claim 3, wherein performing the entire X-ray diffraction stress analysis requires a number of measurements N, wherein for each of the measurements at the plurality of rotational orientations of the sample about S.sub.3 defined by an angle , the angle is incremented by an increment .

5. The method of claim 4, wherein 360 is an integer multiple of , and the number of measurements N is 360/.

6. The method of claim 5, wherein for each of the measurements, the angle is incremented by an increment =45.

7. The method of claim 1, wherein the tilt angle is selected such that a detection surface of the detector is located substantially entirely to one side of a plane defined by an incident beam from the X-ray source and S.sub.3.

8. The method of claim 7, wherein the tilt angle is selected based on an angular range, , of diffracted X-ray energy that is detectable by the detector.

9. The method of claim 8, wherein the tilt angle is about half of said angular range.

10. The method of claim 9, wherein maintaining the third axis S.sub.3 at a constant tilt angle comprises using a cradle track holding the sample and maintaining the cradle track at the tilt angle during the entire X-ray diffraction stress analysis.

11. The method of claim 1, wherein directing an X-ray beam at the sample surface at a plurality of rotational orientations of the sample about S.sub.3 is performed using a goniometer to hold and rotate the sample.

12. The method of claim 1, wherein the relatively low angle with regard to the surface plane is less than 45.

13. An apparatus for performing an X-ray diffraction stress analysis of a sample having a surface with two perpendicular axes S.sub.1, S.sub.2 within a plane of the surface, a third axis S.sub.3 being perpendicular to the sample surface plane, the apparatus comprising: a sample holder for holding the sample, the sample holder being orientable at a plurality of rotational orientations of the sample about S.sub.3; an X-ray source producing an X-ray beam and directing the X-ray beam at the sample surface at a relatively low angle with regard to the surface plane; and a two-dimensional X-ray detector detecting, at a diffraction angle 2, X-ray energy diffracted from the sample at the plurality of rotational orientations of the sample about S.sub.3, the third axis S.sub.3 being maintained, during the entire X-ray diffraction stress analysis, at a constant tilt angle relative to a plane formed by the X-ray beam and a normal to the two-dimensional X-ray detector.

14. The apparatus of claim 13, wherein the sample is at least one of: a thin film, a coating, and a polymer.

15. The apparatus of claim 14, wherein the diffraction angle is under 90.

16. The apparatus of claim 13, further comprising a goniometer to orient the sample holder.

17. The apparatus of claim 16, wherein the goniometer comprises a cradle track holding the sample holder, the cradle track being maintainable at the tilt angle during the entire X-ray diffraction stress analysis.

18. The apparatus of claim 13, wherein the tilt angle is selected such that a detection surface of the detector is located substantially entirely to one side of a plane defined by an incident beam from the X-ray source and S.sub.3.

19. The apparatus of claim 18, wherein the tilt angle is selected based on an angular range, , of diffracted X-ray energy that is detectable by the detector.

20. The apparatus of claim 19 wherein the tilt angle is about half of said angular range.

21. The apparatus of claim 13, wherein the entire X-ray diffraction stress analysis requires a number of measurements N, wherein for each of the measurements at the plurality of rotational orientations of the sample about S.sub.3 defined by an angle , the angle is incremented by an increment .

22. The apparatus of claim 21, wherein 360 is an integer multiple of , and the number of measurements N is 360/.

23. The apparatus of claim 22, wherein for each of the measurements, the angle is incremented by an increment =45.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a perspective view of a goniometer having a cradle track holding the sample;

(2) FIG. 2 is a schematic representation of diffraction cones and diffraction rings within which X-ray energy diffracted from a sample resides;

(3) FIG. 3 is a schematic representation of a diffraction cone within which X-ray energy diffracted from a sample resides;

(4) FIG. 4 is a side view of X-ray beams diffracted in a thin film sample with a variable penetration due to different incident angles;

(5) FIG. 5 is a schematic representation of X-ray diffraction cones at a shifted 2 angle from a sample under stress;

(6) FIG. 6 is a schematic representation of a diffracted X-ray collected by a point detector with a tilt angle =0 and a corresponding diffraction vector, according to the prior art;

(7) FIG. 7 is a schematic representation of a diffracted X-rays collected by a point detector with various tilt angles and corresponding diffraction vectors, according to the prior art;

(8) FIG. 8 is a schematic representation of an angle between two diffraction vectors with different values;

(9) FIG. 9 is a schematic representation of a portion of a diffraction ring collected by a 2D detector and a corresponding diffraction vector distribution with a tilt angle =0, according to an embodiment of the invention;

(10) FIG. 10 is a schematic representation of a portion of a diffraction ring collected by a 2D detector at a tilt angle =22.5, and a corresponding diffraction vector distribution covered thereby;

(11) FIG. 11 is a schematic representation of diffraction vector traces collected using a tilt angle of =22.5 and a complete rotation by 45 increments; and

(12) FIG. 12 is a graph of the results from the setting of FIG. 11, where the trace of diffraction vectors is projected on the sample surface plane.

DETAILED DESCRIPTION

(13) Goniometers can achieve a precise location and orientation for the sample on the sample holder, but this precision is not perfect. In practice, the weight of the sample and of goniometer components and the often imperfect circular shape of the cradle track used to move the sample to different tilt angles are among the most significant contributors to error. One notable error is the height error. In practice, when the sample is variably tilted by moving the sample holder on the cradle track, the location of the sample surface along the z-axis and the orientation of the sample surface change, and this change creates a shift on the diffraction angle 2 measured on the detector, as shown in FIG. 4. This undesirable effect is strong for thin films and coatings, because a low incident angle is typically used for these types of samples.

(14) One may consider detecting diffracted X-ray beams at a low 2 angle if a low incident angle is used. However, for stress measurements, the stress-induced shift in 2 for a given arc of a diffraction cone circle is proportional to 2. Therefore, stress measurements, in general, benefit from using high 2 measurements: the shift being proportional to the high value of 2 increases the sensitivity of the measurement, and the stress-induced shift in 2 stands out better from the noise-induced shift in 2 caused by the sample height error. High 2 measurements are defined as those where the angle of diffraction 2 is high enough, usually greater than 90, to ensure the height error does not affect significantly the shift in 2 compared to the stress-induced shift. The stress-induced shift is thus more accurately measured as it forms most of the total shift. If a low 2 diffraction cone is used for the measurement, the relative contribution of noise-induced shift to the total shift is larger and stress-induced shift is thus less accurately measured. As such, low 2 measurements for thin films and coatings are avoided in conventional methods. However, for many materials, especially coatings or thin films, high 2 diffraction cones may not be available or appropriate for stress evaluation.

(15) In conventional stress measurement systems, variation of the tilt angle introduces a varying mechanical error in the sample height which in turn adds an error in the measured values. The tilt angle is the angle of S.sub.3, the normal to the sample surface, relative to a plane formed by the X-ray beam and a normal to the two-dimensional X-ray detector. With the present invention, the tilt angle is kept constant during the measurement, and the only angle changing during the measurement is (the angles being defined as shown in FIG. 1). Unlike changes to the tilt angle , a rotation of the sample by an angle has little effect on the spherical error. Therefore, the stress can be more accurately determined in low 2 diffraction rings since the dominant varying shift in 2 is the stress-induced shift.

(16) When stress is a measured by a diffractometer with a point detector or 2D detector with multiple tilt angles, the X-ray measurement instrument needs to be critically aligned in different axes to avoid mechanical error on sample height during sample rotation around these axes. However, the spherical error associated with the weight of heavy mechanical components and mechanical tolerance always exists. Building and maintaining an instrument with such a small mechanical error is costly and poses a challenge to the operator who has to recalibrate the goniometer repeatedly. Another challenge results from the fact that low 2 measurements are more sensitive than high 2 measurements to vertical misalignment. Generally, high 2 peaks are preferred for stress measurement in crystals due to the more significant 2 shift and lesser sensitivity to the sample height error. The shift in the value of 2 is proportional to 2 and thus results in a greater absolute shift that can be accurately measured. But for thin films, coatings, or polymer materials, high 2 peaks may not be available or appropriate for stress measurement. With low 2 peaks, it is more difficult or even impossible to measure stress with the conventional methods, and the measurement results of these the conventional methods are extremely sensitive to the sample height error.

(17) The term low 2 should be viewed in the context of the present invention, since in other applications outside of the context of the invention, the 2 angle may not be considered as low even though it is under 90. The measurements with relatively low 2 angles are intended to encompass those angles under 90 for which there is significant sample height error. Furthermore, the incident angle of the X-ray beam may be as large as the 2 angle but is typically half the 2 angle, and if both angles have an exemplary value of 30, the incident angle of 30 will not be considered as low in many X-ray applications. However, the 2 angle of 30 is considered as relatively low in the present context because significant sample height error is usually present at this 2 angle. Since the incident angle of the X-ray is typically half the 2 angle, it should be considered relatively low if it less than 45.

(18) There is disclosed below a device, more precisely a diffractometer with a 2D detector, and a method to operate the device to measure stress with a single sample tilt angle using low 2 measurements.

(19) With two-dimensional X-ray diffraction, i.e., using a 2D detector, stress measurement is based on a direct relationship between a stress tensor and a degree of distortion in a measured diffraction cone that results therefrom. The fundamental equation for stress measurement is developed with the matrix transformation defined for the two-dimensional diffraction. For each single measurement, the diffraction vectors cover a wide range of directions and sufficient angular coverage can be achieved with a single tilt angle during the entire stress measurement. Therefore, the data collection is performed at a fixed angle with only rotation, if needed, as described more thoroughly below.

(20) FIG. 9 illustrates the diffraction vector coverage of the diffraction pattern collected with a 2D detector. The diffraction vector trace on the hemisphere is equally distributed on both sides of the sample normal S.sub.3. Based on Equation (3), the range of this coverage is determined by both the 2 angle and the angular coverage of the detector, . The value of 2 is determined by the crystal structure of the material and the wavelength of the X-ray beam. is determined by the size of the detector and the sample-to-detector distance. Since the sample can be rotated to be at various angles, the redundancy of the diffraction vector coverage above the sample normal direction is not necessary. In order to expand the diffraction vector coverage, the sample can be tilted to an angle 0.

(21) FIG. 10 shows the diffraction vector distribution with a single tilt angle 0. For instance, with a tilt angle of =22.5, the diffraction vector covers a range similar to the four diffraction vector orientations for a 0D detector with taking the following values: 0, 15, 30 and 45 (shown in FIG. 10 for comparison). For low 2 diffraction rings at proper detector distance, it is possible to cover sufficient angular range for stress evaluation with a single tilt angle. The complete data set for the stress tensor can be collected at several angles, for instance by performing a 360 scan with 45 steps. Therefore, the complete data set is collected by scanning along a plurality of values (i.e., varying , preferably with different discrete values along a range) while maintaining a fixed angle. The plurality of rotational orientations (i.e., ) are provided by rotating the sample about S.sub.3, which is the direction normal to the S.sub.1S.sub.2 plane forming the sample surface. The direction of S.sub.3 varies with the tilt angle and is therefore kept constant during the measurements of the entire X-ray diffraction stress analysis.

(22) The tilt angle can be chosen to maximize coverage for determining the trace of diffraction vectors. In the embodiment of FIG. 10, the tilt angle is chosen to allow measurement of diffraction angles corresponding to diffraction vectors ranging from the normal to the sample (e.g., H.sub.0) to the diffraction vector that is angularly the most distant from the normal while still allowing detection by the detector. In an exemplary embodiment of the invention, the diffraction vector that is angularly the most distant from the normal forms an angle of 45 with the normal. As such, a tilt angle of is chosen to be 22.5 is used, which represents half of the maximum angle. Equation (3), depending on and , can be applied to determine this maximum range. Therefore, for a given measurement, and depending on the sample and the physical characteristics of the detector and the goniometer, the maximum value of the diffraction vector with respect to the normal from the sample surface may be determined.

(23) Avoiding changes in the tilt angle can significantly reduce the sample height variation during the data collection, improving the accuracy of measurements as the sample height error is minimized and kept constant. The constant error implies that there is no variable error-induced shift in 2 that could be confused with the stress-induced one (both being combined into the total shift that is actually measured).

(24) The diffraction vector is in the normal direction of the measured crystalline planes. The stress components within the sample surface plane are calculated using the elasticity theory from the measured strain in other directions. The final stress measurement results can be considered as an extrapolation from the measured values. The trace of diffraction vectors shown in FIG. 10 follows a single curve on the hemisphere that is transverse to the diffraction vectors that extend in the longitudinal direction from the origin O and pointing to the hemisphere where the trace is formed. In order to get more precise stress or stress tensor results, rotation is necessary to expand the coverage.

(25) The rotation is performed by incrementing the value of by an increment between each measurement. This increment between measurements needs to be added between measurements N times. Preferably, a complete set of measurements provides discrete increments of covering a 360 range. By selecting a value of the increment where 360 is an integer multiple of , the 360 range can be covered with N=360/ measurements. For example, a value of =45 can be selected, thereby requiring 8 measurements in total to cover the 360 range of values.

(26) The complete measurement may also involve a smaller number of measurements if precision is not critical or if shearing stresses do not need to be measured. For example, it may be possible to perform a complete measurement with a single tilt angle and with a single value of . Using the equations of stress determination, the stress in one axis of the sample surface will be determined. By performing an additional measurement with the opposite value of (involving a 180 rotation from the initial value of ), the stress in the same axis will be determined, but with greater accuracy as the 180 rotation of causes some error to cancel out. By performing an additional measurement with another value of involving a 90 rotation from the initial value of , the stress in the other axis of the sample surface will be determined. Optionally a 180 rotation of with respect to this second value of can also be performed to ensure greater accuracy in the stress determined along this other axis. If 45 increments of between measurements are chosen, shearing stresses can be determined. Again, the opposite value of can be used for each measurement to ensure greater accuracy in the determination of each stress component. Therefore, eight measurements using 45 increments of are sufficient to collect information for accurate stress determination for most tensor components of the stress.

(27) Since the resulting diffraction vector distribution is symmetric on either side of the value =0, the portions of the two-dimensional detector that detect, respectively, diffracted X-ray energy to either side of the value =0 will measure data giving the same information, albeit with greater accuracy overall. With the tilt angle chosen so that the trace of diffraction vectors determined to only one side of the normal to the sample, the angular coverage from the normal to the sample is maximized. For example, a trace of diffraction vectors covering an angular range from 0 from the normal to 45 provides data more useful for stress determination than a trace of diffraction vectors covering an angular range from 22.5 from the normal to +22.5.

(28) FIG. 11 illustrates a system arranged with a tilt angle of =22.5 with eight different values of angles at 45 intervals. In this data collection scheme, eight frames are collected to produce a comprehensive coverage in a symmetric distribution. The data set collected with this strategy can be used to calculate the complete biaxial stress tensor components and shear stress (.sub.11, .sub.12, .sub.22, .sub.13, .sub.23). As can be expected by a person skilled in the field, a different combination of angle or number of angles and steps could also be chosen to achieve the desired angular coverage.

(29) Exemplary results of this measurement are shown in FIG. 12, which illustrates a projection on the S.sub.1S.sub.2 plane of the continuously varying intersection points of the diffraction vector H along a spherical surface like those shown in previous figures. Each branch in the graph of FIG. 12 corresponds to all intersection points of the diffraction vector H with the sphere for a measurement with a given value of , that is, the eight branches correspond to the eight measurements with eight different values of .

(30) These collected data can then be analyzed for the determination of the stress tensor. Maintaining the cradle track of the goniometer at a single value of the tilt angle during all measurements provides the necessary minimization of sample height error to be able to perform low 2 measurements (those where the 2 angle is low enough to cause significant sample height error) with high accuracy. Calibration time and efforts are also minimized. The number of measurements and duration of each measurement are thus reduced too.

(31) In contrast, with the present invention, prior art methods using point detector would require each branch in the graph to be determined by making four measurements (H.sub.0 to H.sub.3) which requires varying the tilt angle each time, as shown in FIG. 7. Thus, to have a complete data set with eight branches for stress tensor determination, one would need to perform 32 measurements instead of 8 as in the present invention. The duration of the complete data set acquisition would thus be significantly longer and the final result would be less accurate due to errors introduced by varying the tilt angle between each one of the 32 measurements.

(32) For thin films, coatings, polymers or other polycrystalline materials, when the diffraction cones at high 2 angles are not available or appropriate, a low 2 measurement can be performed for stress evaluation. With diffraction rings at a low 2 angle, the diffraction vector distribution can satisfy the angular coverage needed for stress measurement at a fixed tilt angle . Without changes of the tilt angle during data collection and with rotation along only, the sample height is accurately maintained. The error that would be introduced in the sample height by varying the tilt angle is avoided and is not passed on to the accuracy of stress measurement. Therefore the single tilt method with a 2D detector system can measure residual stress with high accuracy and high speed for thin films, coatings and polymers.

(33) While the invention has been shown and described with reference to specific embodiments thereof, it will be recognized that various changes in form and detail may be made herein without departing from the spirit and scope of the invention as defined by the appended claims.