B. Quantum Mechanical Considerations

Solving the Schrödinger equation in one dimension the electron concentration in the channel is approximated by Fischer in [90]:

$$n(z)= N_C \cdot e^{\displaystyle \frac{E_F-E_0}{{\it k}_{\mathrm{... ...1-e^{\displaystyle - \bigg(\frac{2\cdot \pi \cdot z}{\lambda_B}\bigg)^2} \bigg]$$ (B.1)

The characteristic tunneling length in one-dimensional quantization is estimated by:

$$\frac{\lambda_B}{2\pi}= \displaystyle \frac{h}{2 \cdot \pi \cdot ... ...}_{\mathrm{B}}\cdot {\it T}_\mathrm{L}}}= \frac{1.2165}{\sqrt{m_z}} \text{[nm]}$$ (B.2)

$E_F$ represents the Fermi energy and $E_0$ the minimum energy. The following Table B.1 gives values for various material systems for ${\it T}_\mathrm{L}$= 300 K.

Table B.1: Tunneling parameter for various materials.

Material	$m_z$	$\lambda_B/2\pi$
	[-]	[nm]
GaAs	0.067	4.7
Al$_{0.2}$Al$_{0.8}$As	0.075	4.4
In$_{0.25}$Ga$_{0.75}$As	0.052	5.29
In$_{0.53}$Ga$_{0.47}$As	0.041	6.0
In$_{0.52}$Al$_{0.48}$As	0.070	4.6
GaN	0.20	2.7
Al$_{0.25}$Ga$_{0.75}$N	0.23	2.53