Spectral Sensitivities of X-Ray Diffraction and Atomic Force Microscopy to the Roughness of Si/SiO 2 Interfaces
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=
Iamnexp(-i(mqox+nqoy+onmn), with q0o
The roughness
=
m,n
spectrum is given by: W(kx,ky)
=
IZ(k,k y)12
L
L
-L
-L
2
(ekxx+k Y)dxdy()
L
2
2
lIamn 12 (kx-mq0) 8(ky -nq0)
= m,n
We note that W(kx ,ky) is the Fourier transform of the height-height correlation function. 2 The root mean square is defined as = /< (z-) >. We define h(x,y) = z(x,y)- such that = 0, and so h (x, y) = amn exp (- i (mq0 x + nq0 y+ mn )-a00 Using our expression for z, we 2m,n
2XY>=1,
have a2 = L, no information is lost due to IL' To provide some intuition we illustrate this in one dimension. The finite Fourier transform L'
2
Z(k)
f z(x)eikxdx
=
(2)
-L1 2
can Z
be x=
written
in
reciprocal
space
as Z(k) = Z_, (k) *coh(k) where kL' sin(-) The convolution of Z,,(k) 2
j z(x)eikxdx, and coh(k)
2 with coh(k) implies that fluctuations with wave-vectors Aq
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