[next] [prev] [prev-tail] [tail] [up]

4.2  The First-Order Quantum Corrected Drift Diffusion Model

The density gradient model is first applied to the drift diffusion model (cf. Section 2.3.1), where in thermal equilibrium the electron concentration is given by

n = NC exp( ) Ec---φ---γn kBTL. (4.3)
Inserting the above relation into Equation (4.2) yields
γn = - 2 -----ℏ-----* 12λnkBTLm n( ) ∇2xln(n2)+ 1(∇x ln(n))2 2 = - 2 ----ℏ----*- 6λnkBTLm n 2√ -- ∇√x--n n, (4.4)
which gave the model its name ‘density gradient’.

Applying the model to the drift diffusion equations, the quantum corrected drift diffusion equations now read:

∇⋅(ε∇ φ) = ∥q∥(n - p + C), (4.5)
∥q∥∂tn -∇Jn = ∥q∥R, (4.6)
∥q∥∂tp + ∇Jp = -∥q∥R, (4.7)
Jn = -∥q∥μn∇x(φ + γn) + ∥q∥Dn∇n, (4.8)
Jp = -∥q∥μp∇x(φ + γp) -∥q∥Dp∇p, (4.9)
γn = 2 -----ℏ------ 12λnkBTLm *n( ) ∇2 φ + ∇2γn - ---1--(∇x φ + ∇x γn)2 x 2kBTL, (4.10)
γp = ℏ2 12λ-k-T--m*- p B L p( 2 2 1 2) ∇ xφ + ∇ γp - 2k-T--(∇x φ + ∇x γp) B L. (4.11)

[next] [prev] [prev-tail] [front] [up]