An Explanation of Miller Indices Group Activity 1 Micah Baker Hoble Cohen February 9 2004 1 Introduction Miller Indices Miller indices define directional and planar orientation within a crystal lattice The indices may refer to a specific 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 different 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 2 Notation and Calculation Now that Miller indices have been defined the method of determining specific 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 shown this 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 define 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 define this intercept as At this point our intercept lengths are 1 1 in the same order as x y z 1 1 Now clear the fractions Take the reciprocal of each length to obtain 11 1 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 2 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 and now c is half a unit long Intercept lengths are 1 21 The reciprocal 1 1 clearing and reducing to 1 0 2 The final result is becomes 1 1 1 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 h i 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 h1 0 0i representing both directions of the x axis The final 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 figure 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 final 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 defined 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 defined lattice parameter for the length of a cube special tables contain 4 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 of h1 0 0i silicon relative to h1 1 1i 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 …