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4.4  Discretization

Various authors  [85, 86, 87, 88, 89] reported numeric instabilities, when directly applying the finite volume method to Equation (4.2) using drift diffusion and thus developed various discretization schemes for DG and developed higher resolution schemes. An almost complete list of schemes is given in  [9]. In this work only two discretization schemes, suggested by  [9], will be discussed. The first discretization is called the simplified scheme, which simply neglects the second order term in Equation (4.2) and exhibits numeric stability. The ‘full scheme’, is the second discretization of Equation (4.2). In this scheme Equation (4.4) is used instead of Equation (4.2). In the course of this thesis, both schemes, have been implemented in ViennaSHE  [65]. The simplified and full scheme have been implemented into the device simulator MinimosNT by  [90]. In all implementations convergence of the non-linear solver has been achieved without any problems for the full and the simplified scheme. Although the simplified scheme often lead to faster convergence for the non-linear solver, when used in a Gummel-loop with the spherical harmonics expanded BTE.

4.4.1  The Simple Scheme

Following  [9], the second order terms in Equation (4.2) are neglected yielding,

γ(x,t) = ℏ2 ---------*- 12λkBTLm( ) ∇2xφ(x,t)+ ∇2xγ(x, t), (4.17)
in a homogeneous semiconductor. Directly applying a finite volume discretization gives
γi = 2 -----ℏ----- 12 λkBTLm *1- Vi ∑ jAij dij(φi + γi - φj - γj), (4.18)
where V i is the volume of the box around vertex i, Aij are the interfaces and dij are the distances between vertices i and j. Additionally it has been assumed that the semiconductor is homogeneous.

4.4.2  The Full Scheme

In the full scheme, Equation (4.2) is replaced by Equation (4.4). Thus the equations for electrons and holes to be discretized are

γp(x,t) = - 2 ----ℏ------ 6λnkBTLm *n 2∘ ------ ∇∘x--n(x,t) n (x,t), (4.19)
γn(x,t) = 2 ----ℏ-----* 6λpkBTLm p∇2 ∘p-(x,t) -∘x-------- p(x, t). (4.20)
Applying the finite volume method yields
γn,i = - 2 ----ℏ------ 6λnkBTLm *n-1 Vi ∑ jAij dij(√ --- ) √-nj - 1 ni, (4.21)
γp,i = ℏ2 ----------* 6λpkBTLm p 1 -- Vi ∑ jAij --- dij(√pj- ) -√-- - 1 pi, (4.22)
which can result in convergence problems, since changes in the electrostatic potential result in exponential changes in the charge carrier concentrations. To mitigate this one can approximate the charge carrier concentrations using Boltzmann statistics and thus obtains
γn,i = - 2 ----ℏ------ 6λnkBTLm *n1- Vi ∑ jAij dij( ( ) ) exp φi-+-γi --φj---γj - 1 2, (4.23)
γp,i = ℏ2 ----------* 6λpkBTLm p1 -- Vi ∑ jAij --- dij( ( φi + γi - φj - γj) ) exp ---------------- - 1 2, (4.24)
where it was assumed that the semiconductor is homogeneous.
PICT
Figure 4.1: Calibration results for a density gradient model for quantum corrected drift diffusion, implemented in MinimosNT, to a solution of the Schrödinger-Poisson equation, using VSP  [51]. In this case the fit has been obtained using a 1D NMOS structure, Robin boundary conditions at the silicon-silicon-dioxide-interface, an acceptor doping of 3⋅1017cm-3, an oxide thickness of 1nm and a uniform grid spacing (orthonormal grid) of 0.1nm. A second fit using a grid spacing of 1nm and 2nm are shown to illustrate the grid spacing dependence of density gradient. The parameters have been previously published in  [90].

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