@misc{oai:ir.soken.ac.jp:00000656,
 author = {RAHMAN, Mohammed Obaidur and ラーマン, モハメド オバイディール and RAHMAN, Mohammed Obaidur},
 month = {2016-02-17, 2016-02-17},
 note = {X-ray optic systems have been developed for the study of the relative lattice spacings of Si-wafers using synchrotron radiation (SR).  Since, unlike an X-ray tube, SR has no characteristic [wave1ength] spectral lines, a new tool of (+, +) high resolution channel-cut monolithic monochromators (MM) are introduced in the systems as a wavelength selective device.  Using two types of MM, two schemes are proposed and applied to the study of the lattice spacings of Si-wafers.  The lattice spacing differences are determined in the range of sub ppm level, for example in scheme-1 we obtain 0.6 ppm and 0.2 ppm in scheme-2.  One of the practical advantages of this system is that it can be applied for a fast and precise measurement of the lattice spacing changes due to the doping and defects in Si, GaAs and other single crystals.<br /><br />  Lattice spacing measurements of Si and other single crystals is of importance both for fundamental solid state physics and applications.  For decades, especially Si is routinely used in the semiconductor industry as well as commonly employed in beamline optics of synchrotron radiation facilities and other x-ray experiments.  Much of the works on lattice spacing measurement has been reported using well defined wavelength of laboratory x-ray sources.  In contrast relatively few works have been reported using the state-of the art SR x-ray source.<br />  Unlike X-ray tube data for which tables of wavelengths of the characteristic spectral lines are available, there are no lines in the synchrotron radiation (SR) spectrum.  Thus it demands to make schemes and data available for precise determinations of lattice spacing which utilize the SR source.  Thus one of our main motivation is to develop a high precision relative lattice spacing measurement system using SR for synchrotron radiation users.  One of the main advantage of these systems is that in most of the applications purposes, one does not need high accurate absolute value, but only a relative high precision value of wavelength.<br />  To this end, we have constructed two X-ray optics systems for relative high precision lattice spacing measurement of single crystal using synchrotron radiation.  In our new optics energy selective (+, +) channel-cut monolithic monochromator together with higher angle resolution goniometer with a precision of 0.36 arc sec has been introduced.  We have designed and fabricated several kinds of MM that give a fixed exit beam position and provide a convenient setting of the whole X-ray optics.  With the current set-up it is simple to make another d-spacing measurement, if need be by simply replacing the monochromator with another.  In addition to the measurements of Si crystals, measurements for other materials such as GaAs crystals can easily be performed.  The monolithic double crystal monochromator is obtained from a single perfect crystal as a means of obtaining an X-ray beam of well defined wavelength.  Monolithic monochromator (MM), is in effect a single perfect crystal where two sets of Bragg planes play the role of two separate crystals.  As the interplanar angle between the concerned diffraction plane is fixed in the MM therefore the wavelength emerging from this device is highly stable and is extremely stable against temperature variations.  The two types of MM allow us to propose two schemes for the lattice spacing measurement.  Approximately, the precision achieved in Δd/d in these systems is in the range of 10-7 to 10-8.<br /><br />1. Monolithic Monochromator:<br />  In order to obtain suitable wavelength for experiments from MM, we consider the following three equations:<br />　　　　　λ1 ＝ λ ＝ 2d1sinθ1(1-δ/sin2θ1)<br />　　　　　λ2 ＝ λ ＝ 2d2sinθ2(1-δ/sin2θ2)<br />　　　　　　　θ1 ＋ θ2 ＋ β0 ＝ π<br />where δ represents the real part of refraction index and β0 is the interplanar angle between the two planes, θ1 and θ2 are the Bragg angles for two diffraction planes d1 and d2 respectively.<br />(1) Solving the above equations the equations yield after simple algebra:<br />　　　　　　　　　　　　　　2d1sinβ0       <br />　　　　　λ ＝ ---------------------------------------------------------------    (1)<br />　　　　　　　　　[(√(h2 2＋k2 2＋l2 2/h2 1＋k2 1＋l2 1)-cosβ0)2＋sin2β0]1/2<br />(2) if d1 ＝ d2 ＝ d Eq. 1 further reduces to:<br />　　　　　　　　　　2d sinβ0<br />　　　　　λ ＝ ----------------------------                          (2)<br />　　　　　　　　　[(1-cosβ0)2＋sin2β0]1/2<br />  Using Eqs. 1 and 2 a simulation code MMCD [Monolithic Monochromator Crystal Design] has been realized which can generate different wavelengths using (h1, k1, l1) and (h2, k2, l2) combinations.  Our code relies on the Deslatte's (1973) lattice spacing of silicon wafer of d220＝1.9201715±0.O000006 angstrom at 25℃.<br />  Two types of Mono1ithic monochromator [MM] have been designed and fabricated on the basis of Eqs. 1 and 2.  These are notated by Type-1 and Type-2.  Type-1 is used in the experimental set-up, referred to as scheme-1 and similarly Type-2 is utilized in the set-up called scheme-2.<br />  Figs. 1 and 2 show the MM that have been made for the wavelength defined by Eqs. 1 and 2 respectively.  Some of the simulated wavelengths from MMCD code is given in Table I for both types of MM with refraction correction δ, given by δ＝ 4.48 x 10-6 n0λ2.  Here n0 is the number density, for example for Si n0 ＝ 699 nm-3, and λ is the wavelength of x-rays.<br /><br />  As mentioned above, two relative lattice spacing measurement methods using two types (i.e.  scheme-1 and scheme-2 based respectively on the diffraction plane conditions d1 ≠ d2 and d1 ＝ d2) of (＋, ＋) energy selective MM with SR have been developed.  Results of the two schemes are summarized in this section.<br />  In scheme-1 we applied our method to several MM.  The novelty of the method is that in each case MM and sample can be changed.  For the Si sample grown by FZ and for the plane (444) a MM wavelength of 0.1410 nm is utilized.  Nine measurements were taken at different position of Si wafer and the measured d444 value obtained on the average from nine measurement points is 0.078390564 nm ± 2 x 10-8.  The average value of Δd/d is 6.2 x 10-7.  Table III shows the results of our lattice spacing measurement for Si grown by CZ, and FZ methods where the planes (800) and (444) are considered in addition to GaAs grown by CZ method for the plane (800).  We can see that the average value of Δd/d is higher for GaAs compared to that of Si.  Temperature and refraction corrections have been taken into account.  In scheme-2 two equivalent planes and a few arc sec rotation (D) of the samples provide two quasi-simultaneous Bragg diffraction.  The results of seventy measurements taken at different positions of Si(153) FZ prepared wafer showed that the approximate average value of Δd/d is 1.1 x 10-7.<br />  Temperature variation has been carefully monitored.  The true value of d is given by dobs ＋ Δdr ＋ Δdt, where Δdr and Δdt are the refraction and temperature corrections respectively.  The refraction correction is calculated for wavelength .16126 nm as 0.00000749Å and temperature correction was 0.00000291Å to the d-value.  The measured d153 value obtained on the average from seventy measurement points is 0.091801632 nm ± 2 x 10-8.  For 70 measurement the Δd/d obtained from Δd is within 0.2 ppm level.  The standard deviation calculated from 100 measurement of differential peak difference D at one point was 2 x 10-8.<br />  In type-2 monochromating there is a possibility of third beam diffraction which may modify the intensity.  Detailed calculation shows in our present study that there could be a Bragg-peak shift when we consider the 3-beam case and the shift is around 12 micro-radians with respect to the 2-beam.  Table II shows some of the monochromator designed together with their characteristics parameter.<br />  Further, it is hoped that the Monolithic monochromator and the systems developed will result in a more widespread use in the condensed matter research.<br />  This work has been carried out under the Graduate University student beam time (internal) approval of proposal PF-02, 99PF-26., application/pdf, 総研大甲第576号},
 title = {Design, Fabrication and Performance of Monochromators and its application for Si and GaAs Lattice Spacing Measurement Using Synchrotron Radiation},
 year = {}
}