An Explanation of Miller Indices

Group Activity #1 Micah Baker & Hoble Cohen February 9, 2004

Introduction: Miller Indices

Miller indices dene directional and planar orientation within a crystal lattice. The indices may refer to a specic crystal face, a direction, a set of faces, or a set of directions. Indices that refer to a crystal plane are enclosed in parentheses, indices that refer to a set of symmetrically equivalent planes are enclosed in braces (curly brackets), indices that represent a direction are enclosed in square brackets, and indices that represent a set of equivalent directions are enclosed in angle brackets [9, 10]. Each plane oriented within a lattice corresponds with an arrangement of atoms; one plane might have a higher atomic density than another. It is apparent that Miller indices correspond with properties of the crystal which determine how the material responds to chemical and mechanical processes. Processes such as oxidation and etching proceed at dierent rates for one orientation versus another [6, 13]. Material used in the construction of microelectromechanical systems (MEMS) might be processed with echant solutions that have high etching rates for particular crystal planes, and so are largely selective of which crystal planes they attack [13]. Crystal plane alignment can be associated with separation and handling problems when the time comes for dice to be separated from a wafer, according to [13].

Notation and Calculation

Now that Miller indices have been dened, the method of determining specic numbers can be described. The best way to do this is through a crystallo1

graphic example: Figure 1 shows three axes (x, y, z ), and a cubic lattice. The orientation of the desired plane on the cube is also shownthis is the grey rectangle from the top left to the bottom right. The goal is to determine the Miller indices for this plane. First, imagine expanding the cube in all directions. Where will the plane cross the axes? These locations are the x,y , and z intercepts, and their lengths from the origin help dene the indices. In this case, length a is one unit long, as is c, however b runs parallel to the y axis and will never cross it. Therefore, we dene this intercept as . At this point, our intercept lengths are (1 1), in the same order as (x, y, z ). 1 1 1 ). Now clear the fractions Take the reciprocal of each length to obtain ( 1 1 (1 0 1) and then reduce the numbers if possible (already there in this case) to get (1 0 1). This nal result represents the Miller indices of the plane [14]. Another example follows.

Figure 1: Cubic lattice with a (1 0 1) Miller indice [14].

Next, Figure 2 shows a plane from the bottom left to the middle of the right of the lattice. Just as before, imagine expanding the structure. This time, a is still one unit long, but in the negative direction. Again, b is , 1 and now c is half a unit long. Intercept lengths are (1 2 ). The reciprocal 1 1 1 becomes ( 1 1 ), clearing and reducing to (1 0 2). The nal result is 2 ( 1 0 2), where the bar above the 1 is indicative of a negative value [15]. Some generalities can be made based on the Miller indices. Namely, a 0 indicates a parallel axis to the plane [15]. Moreover, a smaller number describes an axis that is closer to being parallel to the plane, while a larger number means the axis is closer to being perpendicular [15]. Lastly, the orientation of the plane stays the same when the Miller indices are multiplied or divided by a constant [15]. Once indices have been calculated, further information can be found about the lattice. There are several brackets that can be used around Miller indices to represent particular concepts. The parentheses ( ) have already been presented, and they indicate a set of parallel planes (more than one plane because of the multiplication/division by a constant rule). Then, braces { } represent a set of equivalent planes [14]. They are equivalent in that each plane has the same geometry, and together the planes make up the faces of the crystal lattice [14]. Then, angle brackets represent a set of equivalent directions. In Figure 1, the equivalent planes would be the six sides of the cube. In Miller indices, these planes are (1 0 0), (0 1 0), (0 0 1), ( 1 0 0), (0 1 0), and (0 0 1) [14]. To represent the set, just select one of the planes and use the new brackets: {1 0 0} [14]. Likewise, a set of equivalent directions could be 1 0 0 , representing both directions of the x-axis. The nal set of brackets is square [ ], and their meaning is somewhat more complicated. Take Miller indices such as (1 0 0) and imagine it is instead the point in 3D space (1, 0, 0). Now, draw a line from the origin (0, 0, 0) through the point (1, 0, 0). This creates a line that is perpendicular to the plane (1 0 0). The line is represented as [1 0 0], which is one meaning for the square brackets [15]. Once this perpendicular line has been found, it also represents a set of planes that is parallel to the line. This set is known as a family, and it exists as a zone in the lattice, with the line being the zone axis [15]. Refer to gure 3 for this zone axis. Actually, the zone axis can be found for two planes as follows: if the planes are (h k l) and (p q r ), the zone axis is [kr lq, lp hr, hq kp] [15]. For example, (1 0 0) and (0 0 1) are in the zone [0 1 0 0, 0 0 1 1, 1 0 0 0], which equals [0 1 0]. Note that this 3

Figure 2: Cubic lattice with a ( 1 0 2) Miller indice. zone axis is parallel to the two given planes, but perpendicular to (0 1 0), as it should be. In addition, any Miller indices that are a linear combination of (1 0 0) and (0 0 1) are also in the same zone [15]. One nal calculation is worth mentioning. One more important concept is that of spacing between planes of Miller indices. Denoted d, this spacing is between parallel planes and dened by the following formula [16]: a dhkl = 2 h + k 2 + l2 Here, h, k , and l are the Miller indices for a particular orientation, and a is a dened lattice parameter for the length of a cubespecial tables contain 4

data on the relationship between d and a for dierent lattices [16]. In the next section, applications of Miller indices are described.

Figure 3: Illustration of the zone axis for two planes.

Miller Indices in Action: Silicon

According to [2], the lower atomic density of 1 0 0 silicon, relative to 1 1 1 silicon, causes it to oxidize slower. For short periods of time, the oxide growth rate is limited by the reaction at the silicon surface. During this short period of time, oxide thickness, Xo , is given by [13]: B Xo (t) = ( )(t + ) A The ratio of the 1 1 1 and 1 0 0 linear rate constants,
B (1 A B (1 A B , A

is:

1 1) = 1.68 0 0) 5

 A2 , t , X = For longer times, where (t + ) Bt. The parabolic rate o 4B constants, B , for 1 1 1 and 1 0 0 silicon, [13] implies, are the same. As illustrated in Figure 4, there is a lot of open space in a crystal lattice. A theory designed to predict the depth of ion implantation may assume an amorphous material with a random arrangement of atoms. For an ion beam projected straight into the lattice, the actual range of implantation could be twice the predicted value. As explained in [13], tilting a 1 0 0 silicon results in a more random looking orientation, and it also makes sense that the greater that angle from beaming directly through the lattice, the less channeling.

Figure 4: From left to right: 1 0 0 , 1 1 0 , 1 1 1 viewing directions. Modied from [9]. Anisotropic etching selectively removes crystal material based on the crystal orientation. Table 1 shows KOH etch rates for several planes. The etch rate for the (1 1 1) plane is much lower than the etch rates for (1 0 0) and (1 1 0) [6, 13]. Table 2 shows TMAH eching rate ratios relative to (1 0 0) (1 0 0) 1 0) and (1 1 1). The (1 and (1 were reported as 37 and 68 respectively. 1 1) (1 1 1) In an ideal situation the {1 1 1} planes used in Figure 5 would make an angle of 54.74o with the surface. As [13] points out, however, that result would require an etchant with innite selectivity.

Figure 5: Anisotropic etching of 1 0 0 Si. The 1 0 0 surface is quickly etched away to the {1 1 1} surfaces, resulting in a pyramid shaped groove and a v-shaped groove. Image borrowed from [8]. Orientation (1 0 0) (1 1 0) (1 1 1) Rates (30% Concentration) 0.797 (0.548) 1.455 (1.000) 0.005 (0.004)

Table 1: KOH etching rates versus orientation. Values taken from [6]. Normalized values in parentheses.

References
[1] Todd Stuckless, Chapter 1: Structure, [Online document], 2003 Sept 19, [cited 2004 Feb 7], Available PDF: http://www.chem.ubc.ca/personnel/faculty/stuckless/chem410/ch1a.pdf [2] Marc Madou, Miller Indices, [cited 2004 Feb 7], Available PDF: http://mmadou.eng.uci.edu/Classes/MSE621/MSE62101(3).pdf [3] K.J. Hemker, Crystallography, [cited 2004 Feb 7], Available PDF: http://www2.ece.jhu.edu/faculty/andreou/487/2003/LectureNotes/Handout2a.pdf 7

Orientation (1 0 0) (1 1 0) (1 1 1)

m Etching Rate ( min )

0.603 1.114 0.017

Etching Rate Ratio (i j k )/(1 0 0) (i j k )/(1 1 1) 1.000 37 1.847 68 0.027 1

Table 2: TMAH etching rates and ratios versus orientation. Values taken from [6]. [4] Michael H. Jones and Stephen H. Jones, Slicing O-Axis Si, SiGe, and Ge Wafers, 2002 Aug, [cited 2004 Feb 7], Available PDF: http://www.virginiasemi.com/pdf/cutting%20o%20axis80802.pdf [5] Khalil Naja, Homework #1 Solutions, 2003 Sept 5, [cited 2004 Feb 7], Available PDF: http://www.egr.msu.edu/nsferc/WIMS/HW%2312003Solutions.pdf [6] Michael H. Jones and Stephen H. Jones, Wet-Chemical Etching and Cleaning of Silicon, 2003 Jan, [cited 2004 Feb 7], Available PDF: http://www.virginiasemi.com/pdf/siliconetchingandcleaning.pdf [7] Plummer, Deal, and Grin, Crystal Growth, Wafer Fabrication, and Basic Properties of Si Wafers- Chapter 3, 2000, [cited 2004 Feb 7], Available PDF: http://www.eng.tau.ac.il/%7Eyosish/courses/vlsi1/II3-Si-Wafers-growth-properties.pdf [8] Andre Sharon, Thomas Bifano, and Michele Miller, MEMS Fabrication Process, 2002 Feb 21, [cited 2004 Feb 7], Available PDF: http://mle2.bu.edu/mn500/pdf/class11.pdf [9] Nathan Cheung, Characteristics of Si (a semiconductor), 2001 Sept 6, [cited 2004 Feb 7], Available PDF: http://www.eng.tau.ac.il/%7Eyosish/courses/vlsi1/I-3-Characteristicsof-Si.pdf [10] Bruce Gale, Fundamentals of Micromachining, 2002 Jan 17, [cited 2004 Feb 7], Available PDF: http://www.eng.utah.edu/%7Egale/mems/Lecture%2006%20Materials %20Science%20for%20MEMS.pdf

[11] Simon J. Garrett, 2.1. The Structure of Solids and Surfaces, 2001 Jan 16, [cited 2004 Feb 7], Available PDF: http://www.cem.msu.edu/%7Ecem924sg/Topic03.pdf [12] Silicon Electrical Properties (Si), [cited 2004 Feb 7], Available HTML: http://www.ai.mit.edu/people/tk/tks/silicon-electrical.html [13] Richard C. Jaeger, Introduction to Microelectronic Fabrication, New Jersey: Prentice Hall, 2002. [14] Jessey, David, Crystallography IV, [Online document], [cited 2004 Feb 10], Available HTTP: http://geology.csupomona.edu/drjessey/class/gsc215/minnotes5.htm [15] Dutch, Steve, Miller Indices, 1997, [cited 2004 Feb 10], Available HTTP: http://www.uwgb.edu/dutchs/symmetry/Millerdx.htm [16] Brown, W., ed., Advanced Electronic Packaging. New York: The Institute of Electrical and Electronics Engineers, Inc., 1999.