Electrostatic Mechanism of Emission Enhancement in Hybrid Metal-semiconductor Light-emitting Heterostructures Page: 87
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0 OdO 1 0
Ee - -Ve - -Oz e = Oz ed e = 2 0d e
e 1 4(2d + R_Sin(O)) (A.8)
4E (d + rSin(6))2 (R+Cos2(O) + (2d + R_Sin(6))2)3/2
Due to the fact that z is the variable that defines the variation of the electric field,
our gradient must be taken with respect to it. For our system, however, d determines the
position along the z-axis of the e-h pair, so we need to convert our derivative from z to
d. The ratio ad/az = 1/2 (derived from z=2d) is the term that allows for this conversion.
The same substitution used to get Oh may be used again to get Eh.
The net force acting on the center of mass of the e-h pair is just the sum over the
forces acting on the electron and hole:
Fex = eEe - eEh
e 1 4(2d + R_ Sin(6))
4((d + r_ Sin())2 (R2CoS2(O) + (2d + R_Sin(6))2)3/ z
e 1 4(2d + R_sin(6))
4E (d - r+Sin(6))2 (RCos2(O) + (2d + R_Sin(6))2)3/2 (A9)
e 1 1
4E (d + r_Sin(6))2 +(d - r+Sin(6))2
8(2d + R_Sin(0))
(RCos2(O) + (2d + R_Sin())2)3/2 Z
Finally we redefine r+,_ and R.,_ in terms of the separation between the electron
and hole rex,. It is immediately obvious that R+ = r_ + r, = rex,, furthermore it is well
known that the lengths of the vectors relative to the center of mass for a two-mass
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Llopis, Antonio. Electrostatic Mechanism of Emission Enhancement in Hybrid Metal-semiconductor Light-emitting Heterostructures, dissertation, May 2012; Denton, Texas. (digital.library.unt.edu/ark:/67531/metadc115113/m1/97/: accessed July 21, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .