Boron atom segregation near a ninety degree dislocation
(a) What is the pattern of Boron atom segregation near a
60 degree dislocation in silicon?
(Show occupation probabilities of Boron atoms in the form of various color
contours. Smaller than silicon atom, boron atoms segregate in a
compressional side of the dislocation.
(b) Calculate the break-away stress for the dislocation.
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Answer
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(a) Figure 1
(a) Figure 1 shows the atomisitc configuration
of a ninety degree dislocation in Silicon
(1/2 <110>{111} | <110>).
The Burger's vector is 1/2<110>. The dislocation is along
<110> type direction. The dislocation line and the Burger's vector
lie in {111} type plane.
The dislocation line is perpendicular to the plane of the screen.
At the center of the cell is the core of the ninety degree dislocation.
It consists of an eight atom ring.
The color contours show the occupation probabilities of Boron atoms
at Si sites near the core of this dislocation.
The color changes for the atoms indicate relative increases
in occupation probabilities for each site. The chart on the right shows the
respective increase in the occupation probability/site
indicated by each color.
The segregation of boron atoms near the dislocation in silicon
takes place because of the differences in sizes of Boron and silicon
(size effect).
The smaller boron atom reduced the strain energy at the compressional side
of the dislocation. Hence you can observe the segregation of
Boron atoms primarily above the dislocation line, which is the compressional
part of the strain field of the dislocation.
(b) Figure 2
Because of this "cloud" of boron atoms segregated near
the compressional side of the dislocation, the dislocation movement
is somewhat restricted. The dislocation needs to overcome the
"pinning" effect of the segregated boron atoms
(The so called Binding Energy Barrier) before it
can break free and start moving. This break away stress is
calculated using the maximum slope of the curve in Figure 2.
The break away stress for a ninety degree dislocation in silicon
was 4.751 dynes/cm*cm.
Procedure to Calculate the Binding Energy Barrier
and Break Away Stress
(Atmosphere of Dopants Near the Dislocation Core):
Another contribution to the activation energy is from the binding energy of the
dopants (or foreign atoms) and the dislocation. This energy is calculated as the size
interaction energy. The difference between the sizes of silicon and dopant atoms
relieves the stresses near the dislocations thus reducing the overall energy of the
system. The energy (commonly referred to as the first order size effect elastic
interaction) was calculated by the expression:
'Delta'V is the change in volume of the lattice caused by substitutional dopant, 'mu' the
shear modulus of Si, 'nu' Poisson's ration, b the Burgers vector of the dislocation
(3.84 A), and 'theta' and r are cylindrical coordinates of the dopant with respect to the
dislocation plane and line respectively (Johnson - 1979).
Hence the first order size interaction energies for various atom sites near
the dislocation line were calculated. Using these energies, probability of occupancy
for these sites was also calculated using the Fermi-Dirac distribution function
(Nandedkar and Johnson - 1982). The probabilities depend on the concentration of
dopants (we used 0.0001 fraction sites or 1 dopant in 10000 silicon atoms),
dislocation density (100 cm/cm3). Using the probabilities, an atmosphere of dopant
atoms near the dislocation line was calculated. In order for the dislocation to
migrate, it will have to break away from this 'atmosphere' (or segregated dopants).
By holding the atmosphere fixed, and moving the dislocation in its plane, a
breakaway stress was calculated. The interaction energies were recalculated and
the sum was obtained by multiplying the new energies with unchanged occupation
probabilities. The plot of Energy vS Displacement of the dislocation was
constructed. The minimum shear stress required to break the dislocation away
from its atmosphere was proportional to the maximum slope of this curve.
REFERENCES
(1) A. S. Nandedkar and R. A. Johnson, Acta Metall. Mater., Vol. 30, P. 2055, 1982.
(2) R. A. Johnson, J. Appl Phys., Vol. 50 (3), P. 1263, (1979).
Questions? Comments? Suggestions?
Send a note to Anjali Nandedkar at CASA Engineering
info@casaengineering.com