Miller IndicesNavid FanaeianChristian SchneiderENEE 416: IC FabricationSeptember 18, 2007What is it?● Planes in lattice structure● Method for describing the orientation of planes● Set of numbers in vector notation● (hkl) similar to xyz axisWhy do we care?● Czochralski Method (seed)− Choose orientation● Laser cutting− Keeps uniformity− Know plane of waferHow to Calculate for Planes●Set up coordinate axes along the edges of the unit cell.●Note where the plane of interest intersects the axes.●Normalize the intercepts by dividing each of them by the number of unit cell lengths along each coordinate axis and record the numbers in the order x, y, z.●Take the inverse of each of the normalized intercept values.●Multiply the inverted intercepts by the constant that would result in the smallest possible set of whole numbers.●Insert the whole numbers in parentheses ( ).Plane Calculations continued● If a plane is not intersected, use infinity● If on negative axis, put a bar above the number● Families of equivalent indices (due to symmetry):− (100), (010), (001), (100), (010), (001)− {100} familyHow to Calculate for Directions● Draw a vector of any length in the direction of interest. ● Project the vector onto the coordinate axes to find its components in the x, y, and z directions. ● Turn the projection values into the smallest possible whole numbers by multiplying them with a constant. ● Insert the whole numbers in square brackets [ ]. ● Similar to Miller indices for planes, an equivalent set of directions is designated using triangular brackets < >.Problem 1● Given a unit cell length of 7, what is the Miller index for the plane that intersects the coordinate axes at x = -7, y = 14, and z = 7?Problem 1 Solution● We are given the intercepts with the coordinate axes as -7, 14, and 7, so we begin by normalizing these:− x: -7/7 = -1− y: 14/7 = 2− z: 7/1 = 1● Then we take the inverse of each:− 1/x = -1/1 = -1− 1/y = 1/2− 1/z = 1/1 = 1Problem 1 Solution continued● We need to multiply the inverted intercepts to obtain the smallest possible set of whole numbers---the multiplier 2 achieves this:− 2*(1/x) = -2− 2*(1/y) = 1− 2*(1/z) = 2● Insert the resulting whole numbers in parentheses to get the answer (underline should be read as “over bar”):− (212)Problem 2● Find the family of equivalent planes to (200).Problem 2 Solution● Due to crystal symmetry, the following planes are indistinguishable from (200) because each one is parallel to two axes and intersects the third at 2:− (020)− (002)− (200)− (020)− (002)● These 6 equivalent planes are referred to using curved brackets { } as the {200} family of planes.Problem 3● Find the Miller index for the vector that falls in the x-y plane halfway between the x- and y-directions (i.e., at a 45 degree angle starting at the x-axis and rotating counter-clockwise) with a magnitude of 3*sqrt(2).Problem 3 Solution● Begin by decomposing the vector into its constituent parts by projecting it onto the coordinate axes:− x direction: 3− y direction: 3− z direction: 0● This is already the smallest possible set of whole numbers, so we simply insert them in square brackets to get the answer:− [330]Problem 4● Find the Miller index for the plane that is parallel to the z axis but which crosses the xand y axes at 3 and -9, respectively. Assume the unit cell length is 3.Problem 4 Solution● We are given the intercepts with two of the coordinate axes (x,y) as 3 and -9, but the plane is parallel with the z axis so it does not cross it Æ intersect is at infinity− x: 3/3 = 1− y: -9/3 = -3− z: ∞● Then we take the inverse of each:− 1/x = 1/1 = 1− 1/y = -1/3− 1/z = 1/∞ = 0Problem 4 Solution continued● We need to multiply the inverted intercepts to obtain the smallest possible set of whole numbers---the multiplier 3 achieves this:− 3*(1/x) = 3− 3*(1/y) = -1− 3*(1/z) = 0● Insert the resulting whole numbers in parentheses to get the answer (underline should be read as “over bar”):− (310)Conclusion● High numbers are rarely encountered− Most are 0's or 1's● Necessary to have method for describing orientations● Can tell angles between