An Explanation of Miller IndicesGroup Activity #1Micah Baker & Hoble CohenFebruary 9, 20041 Introduction: Miller IndicesMiller 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 ofatoms; 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 CalculationNow that Miller indices have been defined, the method of determining specificnumbers can be described. The best way to do this is through a crystallo-1graphic example: Figure 1 shows three axes (x, y, z), and a cubic lattice. Theorientation of the desired plane on the cube is also shown–this is the greyrectangle from the top left to the bottom right. The goal is to determinethe Miller indices for this plane. First, imagine expanding the cube in alldirections. Where will the plane cross the axes? These locations are the x,y,and z intercepts, and their lengths from the origin help define the indices.In this case, length a is one unit long, as is c, however b runs parallel to they axis and will never cross it. Therefore, we define this intercept as ∞. Atthis point, our intercept lengths are (1 ∞ 1), in the same order as (x, y, z).Take the reciprocal of each length to obtain (111∞11). Now clear the fractions(1 0 1) and then reduce the numbers if possible (already there in this case)to get (1 0 1). This final result represents the Miller indices of the plane[14]. Another example follows.Figure 1: Cubic lattice with a (1 0 1) Miller indice [14].2Next, Figure 2 shows a plane from the bottom left to the middle of theright of the lattice. Just as before, imagine expanding the structure. Thistime, a is still one unit long, but in the negative direction. Again, b is ∞,and now c is half a unit long. Intercept lengths are (−1 ∞12). The reciprocalbecomes (−111∞112), clearing and reducing to (−1 0 2). The final result is(¯1 0 2), where the bar above the 1 is indicative of a negative value [15]. Somegeneralities can be made based on the Miller indices. Namely, a 0 indicatesa parallel axis to the plane [15]. Moreover, a smaller number describes anaxis that is closer to being parallel to the plane, while a larger number meansthe axis is closer to being perpendicular [15]. Lastly, the orientation of theplane stays the same when the Miller indices are multiplied or divided by aconstant [15]. Once indices have been calculated, further information can befound about the lattice.There are several brackets that can be used around Miller indices to rep-resent particular concepts. The parentheses ( ) have already been presented,and they indicate a set of parallel planes (more than one plane because ofthe multiplication/division by a constant rule). Then, braces { } representa set of equivalent planes [14]. They are equivalent in that each plane hasthe same geometry, and together the planes make up the faces of the crystallattice [14]. Then, angle brackets h i represent a set of equivalent directions.In Figure 1, the equivalent planes would be the six sides of the cube. InMiller 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 thenew brackets: {1 0 0} [14]. Likewise, a set of equivalent directions could beh1 0 0i, representing both directions of the x-axis. The final set of bracketsis square [ ], and their meaning is somewhat more complicated.Take Miller indices such as (1 0 0) and imagine it is instead the point in 3-D 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). Theline 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 setof planes that is parallel to the line. This set is known as a family, and itexists as a zone in the lattice, with the line being the zone axis [15]. Referto figure 3 for this zone axis. Actually, the zone axis can be found for twoplanes 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 thezone [0∗1 −0 ∗0, 0 ∗0−1∗1, 1∗0−0 ∗0], which equals [0¯1 0]. Note that this3Figure 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+ l2Here, h, k, and l are the Miller indices for a particular orientation, and ais a defined lattice parameter for the length of a cube–special tables contain4data on the relationship between d and a for different lattices [16]. In thenext section, applications of Miller indices are described.Figure 3: Illustration of the zone axis for two planes.3 Miller Indices in Action: SiliconAccording to [2], the lower atomic density of h1 0 0i silicon, relative to h1 1 1isilicon, causes it to oxidize slower. For short periods of time, the oxide growthrate is limited by the