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EE 341 LAB 1
Spring 1996
Helen Kantes and Prof. S. M. Lord
Please bring a ruler with mm to lab!!!
Please hand in your work for the example on pg. 5 in lab on Tuesday, February 6, 1996. This is not optional!!
For lab on Tuesday, please come to Dana 347.
In this lab, you will learn how to determine the lattice parameters from X-ray diffraction data. After the pre-lab discussion about the precession camera, you will interpret the precession photographs. You will also learn how to calculate the lattice parameters for an elemental semiconductor or one that is composed of a compound or an alloy.
BACKGROUND
A crystalline material is one in which the atoms are arranged in a repetitive three-dimensional pattern. There are many different crystal structures; however, they all have a small portion that repeats itself called a unit cell. To describe the structures of crystals, a three dimensional lattice must be considered.
The unit cell geometry can be completely defined in terms of six parameters called lattice parameters. The six parameters are the three edge lengths a, b, and c, and the three interaxial angles a, b, and g. There are seven different possible combinations of these parameters, each of which represent a different crystal system. The seven systems are: cubic, hexagonal, tetragonal, rhombohedral, orthorhombic, monoclinic, and triclinic. APPENDIX 1, TABLE A, lists the relationships among the lattice parameters for each of these seven systems. Seven additional systems can be obtained depending on whether the atoms are positioned as in a regular cubic, a face-centered (fcc), or a body-centered configuration (bcc).
The principal semiconductors crystallize in a cubic lattice structure. In Si and Ge, the lattice structure is known as a diamond lattice . The diamond lattice is really composed of two interpenetrating fcc lattices. The corner and face atoms of the unit cell can be viewed as belonging to one fcc lattice, while the atoms totally contained within the cell belong to the second fcc lattice. (See FIGURE 1, pg. 2).
Most of the compound semiconductors crystallize in a zincblende structure. The zincblende structure is essentially identical to the diamond lattice, except that lattice sites are apportioned equally between two atoms. For instance, with the compound GaAs, Ga occupies sites on one of the two interpenetrating fcc sublattices while As occupies the other. (See FIGURE 1).
Max von Laue found that the regular array of atoms in a crystal could act as a three-dimensional diffraction grating for x-rays. FIGURE 2 shows a schematic diagram of the technique used to observe the diffraction of x-rays by a single crystal. The array of spots formed on the film by the strongly diffracted beams is called a Laue pattern.
W. L. Bragg noticed the similarity of diffraction to ordinary reflection and deduced a simple equation treating diffraction as "reflection" from planes in the lattice. The relationship he derived is known as Bragg's Law.
(1)
or
(2)
where n is an integer and l is the wavelength of the incident radiation. Copper (l = 1.5418 è) radiation is most commonly used. Bragg's Law indicates an inverse relationship between the angle of diffraction (q) and the distance between repeating lattice planes in real space (d). Interpretation of X-ray diffraction patterns is easier if the inverse relation between sin q and d is direct. This can be achieved by constructing a reciprocal lattice based on 1/d. This way, there is a direct relationship between the interplanar distance (1/d) and the angle of diffraction (sin q).
In reciprocal space, a single point represents an entire family of planes. A network of such points forms a lattice in reciprocal space called the reciprocal lattice. FIGURE 4 compares the real lattice with the reciprocal lattice.
In general, the intensity I of a beam after passing through a thickness x of material is given by
(3
where I0 is the intensity of the incident beam and a is the linear absorption coefficient. The heavier the atom in the sample, the more intense the spot appears on the precession photograph.
Bragg's Law and the concept of similar triangles yields the following relationship that is used to calculate lattice parameters:
(4)
60 mm
where d* is the reciprocal lattice parameter, l is the wavelength of radiation, and Fd is the distance between reflections in mm. It is best to measure the total distance between multiple reflections of a row or column and divide by the number of intervals to obtain Fd.
PRECESSION PHOTOGRAPHS
Polaroid films are used in the precession camera. The film measures X-ray radiation. The special advantage of the precession camera is that it gives an undistorted image of the reciprocal lattice. The undistorted image results because the crystal moves and the film moves in such a way that only one reciprocal lattice level is mapped onto one sheet of film. Each film can contain a plane net of spots characterized by three indices (hkl). Since the grid spacings on these films are proportional to the reciprocals of the real crystal spacings, it is easy to calculate the unit cell constants from just two photographs and the value of the angle through which the crystal was rotated between photographs.
Crystallography data is generally collected from the orientation, zero level, and upper level precession photographs. This data is used to determine the lattice structure and then the space group of an unknown crystal. Crystallographers rely on the use of systematic absences to help identify space groups. Systematic absences appear as discernible patterns when comparing different level photographs. TABLE 1, pg. 7, lists these systematic absences. The presence of each symmetry element listed in column one in TABLE 1 of the table causes those reflections of the classes in column two to vanish when the indices are as specified in column three. For example, if the crystal has a 2-fold screw along the a axis, the h00 reflection will disappear when h = 2n + 1 = odd.
Since the space group is already known for semiconductor samples(cubic), the symmetry is already known. The systematic absences appearing in the precession photographs will always be the same. Thus, it is a much easier task to figure out the lattice parameters using (4). Angles a, b, and g will always be 90°. Edges a, b, and c will always be equal. In addition, the only conversion formulas needed are those listed in (5).
To solve for a real cell constant in a cubic system, the following direct/reciprocal relationships must be used:
a* = 1/a (5)
b* = 1/b
c* = 1/c
where the starred letters represent the reciprocal lattice parameters. There are more complicated conversions for the other systems. (See APPENDIX I, TABLE B). Thus, the lattice parameters that characterize semiconductor crystals can easily be determined from just one precession photograph and equations (4) and (5).
Elemental crystals such as Si and Ge form a
diamond lattice structure and belong to the space group of Fd3m (227)
found in the International Tables for Crystallography. Semiconductors
composed of compounds such as GaAs and AlAs, as well as alloys such
as
, form a zincblende lattice. This corresponds to the
space group F43m (216). The number in parenthesis indicates which
space group the crystal belongs to out of the possible 230 space
groups.
EXAMPLE
Once the crystal is aligned in the diffractometer, filtered radiation is used to take zero-level and upper level photographs, which are a representation of a cross-section of the three dimensional network of reflections that make up reciprocal space. FIGURE 5, pg. 6, is an example of actual precession photographs from an organic crystal composed of carbon, nitrogen and hydrogen.
First, measure the distances between reflections in all 3 directions and solve for the reciprocal parameters a*, b*, and c*. Using (4) with copper radiation yields the following results:
a* = 0.060 è-1
b* = 0.078 è-1
c* = 0.048 è-1
In this particular example, then, angles a = b = g = 90° (all axes are perpendicular to each other) and a* ¹ b* ¹ c*. APPENDIX I, TABLE A, indicates that this is an orthorhombic system. Thus, the direct/reciprocal conversions for the orthorhombic system are needed. The same conversions are used for an orthorhombic system as for the cubic. Thus, using (5), the final lattice parameters for this crystal are as follows:
a = 16.63 è a = 90°
b = 12.68 è b = 90°
c = 21.00 è g = 90°
For this example, the hk0 net photo (photograph 1) has every other dot missing in both the k direction and the h direction. This is notated as "h + k = 2n + 1". This corresponds to an n-glide perpendicular to c in TABLE 1. The 0kl photo has k odds missing along the k axis. This is denoted as "k = 2n + 1" and corresponds to a b-glide perpendicular to side a. (Because the a parameter or h index is zero in this particular photo). In semiconductor lattices, which are fcc, h, k, and l are always either all odd or all even.
For a crystallographer, the final step is to go to the International Tables for Crystallography. Using the orthorhombic tables, and through the process of elimination, possible space groups can be deduced. A sample portion from these tables is contained in APPENDIX I, TABLE D. The space group for this crystal has been determined to be Pbcn (60) or Pbnn (52).
**"h + k" is the same as "h + k = 2n + 1", odds missing
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