an explanation of miller indices

7/27/2019 An Explanation of Miller Indices

1/9

An Explanation of Miller IndicesGroup Activity #1

Micah Baker & Hoble Cohen

February 9, 2004

1 Introduction: Miller Indices

Miller indices define directional and planar orientation within a crystal lat-tice. The indices may refer to a specific crystal face, a direction, a set offaces, or a set of directions. Indices that refer to a crystal plane are enclosedin parentheses, indices that refer to a set of symmetrically equivalent planesare enclosed in braces (curly brackets), indices that represent a direction areenclosed in square brackets, and indices that represent a set of equivalentdirections 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 isapparent that Miller indices correspond with properties of the crystal whichdetermine how the material responds to chemical and mechanical processes.Processes such as oxidation and etching proceed at different rates for oneorientation versus another [6, 13]. Material used in the construction of mi-croelectromechanical systems (MEMS) might be processed with echant so-lutions that have high etching rates for particular crystal planes, and so arelargely selective of which crystal planes they attack [13]. Crystal plane align-ment can be associated with separation and handling problems when thetime comes for dice to be separated from a wafer, according to [13].

2 Notation and Calculation

Now that Miller indices have been defined, the method of determining specificnumbers can be described. The best way to do this is through a crystallo-

1


2/9


3/9


4/9

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), asit 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 final calculation isworth mentioning.

One more important concept is that of spacing between planes of Millerindices. Denoted d, this spacing is between parallel planes and defined bythe following formula [16]:

dhkl =a

h2 + k2 + l2

Here, h, k, and l are the Miller indices for a particular orientation, and ais a defined lattice parameter for the length of a cubespecial tables contain

4


5/9

data on the relationship between d and a for different lattices [16]. In the

next section, applications of Miller indices are described.

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

3 Miller Indices in Action: Silicon

According to [2], the lower atomic density of1 0 0 silicon, relative to 1 1 1silicon, causes it to oxidize slower. For short periods of time, the oxide growthrate is limited by the reaction at the silicon surface. During this short periodof time, oxide thickness, Xo, is given by [13]:

Xo(t) = (B

A)(t + )

The ratio of the 1 1 1 and 1 0 0 linear rate constants, BA

, is:

BA

(1 1 1)BA

(1 0 0)= 1.68

5


6/9

For longer times, where (t + ) A24B , t , Xo =

Bt. The parabolic rate

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 anamorphous material with a random arrangement of atoms. For an ion beamprojected straight into the lattice, the actual range of implantation couldbe twice the predicted value. As explained in [13], tilting a 1 0 0 siliconresults in a more random looking orientation, and it also makes sense thatthe greater that angle from beaming directly through the lattice, the lesschanneling.

Figure 4: From left to right: 1 0 0, 1 1 0, 1 1 1 viewing direc-tions. Modified from [9].

Anisotropic etching selectively removes crystal material based on the crys-tal orientation. Table 1 shows KOH etch rates for several planes. The etchrate 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)

and (1 1 1). The (1 0 0)(1 1 1) and(1 1 0)(1 1 1) were reported as 37 and 68 respectively.

In an ideal situation the {1 1 1} planes used in Figure 5 would make anangle of 54.74o with the surface. As [13] points out, however, that resultwould require an etchant with infinite selectivity.

6


7/9

Figure 5: Anisotropic etching of 1 0 0 Si. The 1 0 0 surface isquickly etched away to the {1 1 1} surfaces, resulting in a pyramidshaped groove and a v-shaped groove. Image borrowed from [8].

Orientation Rates (30% Concentration)(1 0 0) 0.797 (0.548)(1 1 0) 1.455 (1.000)

(1 1 1) 0.005 (0.004)Table 1: KOH etching rates versus orientation. Values taken from [6]. Nor-malized values in parentheses.

References

[1] Todd Stuckless, Chapter 1: Structure, [Online docu-ment], 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


8/9

Orientation Etching Rate ( mmin

) Etching Rate Ratio

(i j k)/(1 0 0) (i j k)/(1 1 1)(1 0 0) 0.603 1.000 37(1 1 0) 1.114 1.847 68(1 1 1) 0.017 0.027 1

Table 2: TMAH etching rates and ratios versus orientation. Values takenfrom [6].

[4] Michael H. Jones and Stephen H. Jones, Slicing Off-Axis Si, SiGe,and Ge Wafers, 2002 Aug, [cited 2004 Feb 7], Available PDF:http://www.virginiasemi.com/pdf/cutting%20off%20axis80802.pdf

[5] Khalil Najafi, Homework #1 Solutions, 2003 Sept 5, [cited 2004 Feb7], Available PDF: http://www.egr.msu.edu/nsferc/WIMS/HW%231-2003Solutions.pdf

[6] Michael H. Jones and Stephen H. Jones, Wet-Chemical Etching andCleaning of Silicon, 2003 Jan, [cited 2004 Feb 7], Available PDF:http://www.virginiasemi.com/pdf/siliconetchingandcleaning.pdf

[7] Plummer, Deal, and Griffin, Crystal Growth, Wafer Fabrication, andBasic Properties of Si Wafers- Chapter 3, 2000, [cited 2004 Feb 7],

Available PDF: http://www.eng.tau.ac.il/%7Eyosish/courses/vlsi1/II-3-Si-Wafers-growth-properties.pdf

[8] Andre Sharon, Thomas Bifano, and Michele Miller, MEMS Fabri-cation 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 semiconduc-tor), 2001 Sept 6, [cited 2004 Feb 7], Available PDF:http://www.eng.tau.ac.il/%7Eyosish/courses/vlsi1/I-3-Characteristics-of-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

8


9/9

an explanation of miller indices

Documents