4.1  The Density Gradient Model

A method, other than IMLDA, to describe carrier confinement in arbitrary potential wells is called ‘density gradient’ model (DG)  [8158283], which is a first-order quantum-correction model. Density gradient uses, as its names suggests, the gradient of the carrier densities to describe carrier confinement by locally modifying the electrostatic potential through a correction potential γ. Nevertheless, neither the charge carrier wave functions nor the sub band structure can be obtained by this approach, although it might be possible to assess direct tunneling using DG  [83]. The equations for the correction potential are derived from Wigner’s equation, where it is assumed that any effects associated with Fermi-Dirac statistics and many-body effects can be safely neglected. Additionally, the effective mass and parabolic band approximations are used and it is assumed that the described electron gas has an infinite extent. The correction potential γ(x,t) reads  [8283]

γ(x,t) =      2
----ℏ------
12λkBTLm *(                              )
 ∇2 φ (x,t)- --1---(∇x φ(x,t))2
   x         2kBTL, (4.1)
where λ is a fitting parameter, which is determined by comparing the carrier density in a MOS structure to the carrier density obtained by the solution of Equation (2.56). Since the electric field E = -∇xφ can be undefined at abrupt potential steps, the electrostatic potential φ is replaced, in the above equation, by φ + γ, yielding
γ(x,t) =      ℏ2
12λk--T-m-*
     B L(                                                     )
    2          2           1                         2
  ∇ xφ(x,t)+ ∇ xγ (x, t) - 2k-T--(∇x φ(x,t)+ ∇x γ(x,t))
                           B  L, (4.2)
which is fine for this first order model, since the error in γ is of second order. This correction potential has to be calculated separately for each carrier type and added to the electrostatic potential for and only for the respective charge carrier transport equation.