Electrostatic Mechanism of Emission Enhancement in Hybrid Metal-semiconductor Light-emitting Heterostructures Page: 39
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--V2 0(z) - (z)1(z) = E (z) (4.3.3)
Due to the piezoelectric field within the QW, the value of P changes with each region of
the QW as follows:
t Ec,v z < -dqw
()e,h = -Fpz -dqw < z < 0 (4.3.4)
E- Fzd z >_ 0
Where AEc,v = 5Ec,v AEg = 5Ec,v [Eg (GaN) - Eg (InGaN)] represents the division of the
difference in bandgap energies between the conduction and valence bands. In our case
I have used 5Ec = 0.8 and 5Ev = 0.2 . Substituting Eq. 4.3.4 into 4.3.3 gives us the
set of equations which we need to solve:
V21.J I(z) + AEc,v l(z) = E I(Z) Z < -dqw
- -20ii(z) + Fzz $I(z) = E 4 )i(Z) -dqw, z < 0 (4.3.5)
-2v2III(Z) - (AEc,v - Fpzdqw) 4ii (z) = E (Z) z 0
While Schrodinger's equation can be solved analytically for very few kinds of
potential, the linear potential is one of those that can be. The solutions for a potential
Q(z) = Fz + C take the form of a linear superposition of the Airy functions Ai(z) and
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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/m1/49/: accessed February 21, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .