7.1 Introduction and motivation for the use of transmission polarimetry for
7.1.2 Calculation of the transmission of light through a cross polarizer
In principle, the presence of a dislocation in the silicon crystal gives rise to a strain field.
The Neumann-Maxwell stress-optic law gives a linear dependence of the refractive index with the strains and photoelastic constants of the material. These strains will govern the photoelastic response by the Equation [327, 562]
∆(−1ij ) =pijklàkl=πijklσkl (7.1.1)
where a change in the inverse dielectric constant ∆(−1ij ) is proportional to the photoe- lastic tensor pijkl and the strain àkl, related to the piezooptic tensor πijkl and stress σkl. In a nonmagnetic material, the speed of light is c/n = 1/√
à0 and so we have a relationship between the strainà, the dielectric constant, and the refractive indexnof the material.
Assuming a single edge dislocation, the stresses à⊥/k in planes perpendicular and parallel to the Burgers vector of the edge dislocations assume the following values in polar coordinates (r, θ) [327]
à⊥=−((à0B)/r) sin(θ)cos(2θ), àk = ((à0B)/r) sin(θ)(2 +cos(2θ)).
B is the magnitude of the Burgers vector, andà0 is a strain coefficient. The magnitude of the Burgers vector is equal to the width of the extra half plane above the edge dislocation. The strain field creates compressive strain above and tensile strain below the dislocation, as shown in Figure 7.1.1. These strains are strongest near the dislocation, and are reduced by inverse proportion to the distance from the dislocation.
The strain fields will affect light transmitted through the sample by inducing a phase in the electromagnetic wave which will propagate though a gradient in the refractive index. The transmissionT of light of input amplitude√
I through the dislocated sample
Figure 7.1.2: Polar plot of normalized intensity contours 1 = cos2(2φ) cos2(φ−β) of light transmitted at an edge dislocation through a crossed polarizer instrument in the plane parallel to the Burgers vector. The angular coordinatesφare for the angle between the slip direction and the polarizer (polarization of the transmitted light). Plotted are four curves of various orientations of the edge dislocation and polarizer (β). The constant intensity contour extends outward with r2 = x2 +y2, and the transmitted intensity goes to zero for matched strain φ = nπ/2 +π/8 at multiples of n ∈ Z. Integration of the transmission intensity results in a signal through the cross polarizer instrument dependent on the defect density of the silicon. The transmitted light is extinguished for defect-free isotropic silicon by the polarization analyzer.
can be calculated from
T /I =sin2(2γ)sin2(δ/2) (7.1.3) whereδ quantifies an induced birefringence, andγ originates from the orientation of the electromagnetic field in the crystal [326]. The angleβ is defined byBullough [322] as
2γ = 2θ+ 2β. (7.1.4)
βis the angle between the polarizer and the slip direction, andθ= 12tan−1[(y2−x2)/2xy]
is the angle between the Cartesian coordinates (x, y) centered on the dislocation, and the principle coordinates of the optical indicatrix. The phase difference found from the difference in the optical pathlength is
δ = (2πt/λ)∆n (7.1.5)
for wavelengthλand pathlength t.
The difference in the refractive indices near the birefringent dislocation due to the strain field is calculated from the principle strains as
∆n= ¯C(e011−e022). (7.1.6) C¯ is the effective photoelastic constant for silicon [321, 324], and e0ii is the component in thei direction in the principle coordinate geometry. The components are calculated from the orientations of the principle strain components written ase011= 4π(1−νB
p)r2[x− (1−2νp)y] and e022 = 4π(1−νB
p)r2[x+ (1−2νp)y]. Taking the difference of the principle strains to determine the difference of the refractive indices in ∆ngives
e011−e022= 2Bx
4π(1−νp)r2 (7.1.7)
Hereνp is Poisson’s ratio for silicon [563].
Substituting Equations 7.1.7 into 7.1.6, then 7.1.6 into 7.1.5, and Equations 7.1.5
and 7.1.4 into Equation 7.1.3 gives a final representation of the light transmission at the edge dislocation through a crossed polarizer instrument as
T /I0= sin2(2θ+ 2β)×
2BxπtC¯ 4π(1−νp)r2λ
2
. (7.1.8)
This Equation is based on the approximation that sin(θ) ≈ x− x3!3 + x5!5 ã ã ã, and since the argument (δ/2) is small, only the first term is retained.
Equation 7.1.8 may be simplified in polar coordinates (r, φ) putting φ = 0 as the direction of the polarizer (the electric field of the transmitted probe beam). Then, 2θ = 2φ−2β−π/2 and the Cartesian to polar conversion follows from the Equations cos(φ−β) =x/r, and sin(φ−β) =y/r [322]. Let K1 =BtC/2(1¯ −νp)λto collect the constant terms for the silicon sample, and the instrument. Then, the constant contours of transmission can be found from the transmission function
T /I = (K1/r)2cos2(2φ) cos2(φ−β). (7.1.9)
The contours of the Equation are shown in Figure 7.1.2 for a value of K1 = 1, and in Figure 7.1.3 it can be seen that the majority of the transmitted light originates near the center of the edge dislocation, and falls of as the square of the distance from the dislocation. These calculations were performed with the Burgers vector oriented parallel to the wafer as shown in Figure 7.2.1(A). Similar findings result for various orientations of the Burgers vector in the plane transverse to transmitted light.
7.1.3 Determining the dislocation density in a raw silicon wafer using a transmission polarimeter
The transmission of light through the multicrystalline silicon wafer as expressed in Equa- tion 7.1.9 is thus proportional to the strength of the Burgers vector of dislocations, the dislocation density, and inversely proportional to the wavelength of transmitted light. In the absence of dislocations, a signal will be extinguished to the noise level of the detector.
This assumes an effective photoelastic constant governs a small refractive index change
Figure 7.1.3: Intensity contours of the transmitted light in arbitrary units. The ma- jority of the transmitted intensity can be seen to originate near the edge dislocation.
A transmission signal occurs for various angular orientations of the polarization of the input light and edge dislocation shown between plots A to D. The angle β is marked in the plot and the coordinates in units are marked in the upper left Figure A, which serves similarly for all plots.
in the material due to the presence of the dislocation. A sum of transmitted light from Equation 7.1.9 over an area bounding a silicon wafer may assume a merit to determine the average defect density of the wafer. This can then be used to achieve a raw wafer sorting instrument, which collects the entire transmitted signal through the sample and instrument, and equates it to a dislocation density in a linear fashion.
The situation is complex considering a real dislocation network, or for dislocations forming as so-called sub-grain boundaries (SAGB) or dislocation networks [482, 546]
which is of interest in silicon wafer characterization, since the presence of multiple strain distributions lead to interference. For example, two closely spaced edge dislocations will cancel the strain field between themselves. Comparatively, dislocation networks will develop strain fields distributed through combinations of their Burgers vectors, which are subject to reduction of energy in the crystal during dynamic solidification [564, 565].
Secondly, due to the interference of strains by neighboring dislocations, the total intensity which results from measurement on multicrystalline silicon wafers is expected to be
reduced from that of a model which sums the edge dislocations.
Modeling of dislocation networks in multicrystalline silicon can be performed, and the strain fields can be generated so that a transmission function can be determined for these samples. This is necessary when completing a theoretical calculation of the defect density from the signal achieved from a transmission polarimeter. Alternatively, statistical methods may be used by measuring the transmission of light through the raw silicon wafers in the polarimeter, and then compared to the dislocations densities counted after a chemical etch is applied. It may be assumed that dislocation networks will still be detectable due to the presence of the strains in the crystal, and the effect on the transmitted light will give an increasing signal of the defect density which is monotonic.