Energy Transport Equation (2.41)

U_i−1,j

− µ 1

hi+1

+ 1

h_i−1+ hi

¶

Ui,j + 1 hi+ hi+1

Ui+1,j

4.7 Energy Transport Equation (2.41)

Finally, we have

−∇ ·¡

ρ²_p∇g^p¢

− f⁷(gp) = 0, (4.116)

f7(gp) = δ²J_p· E + pW^p, (4.117) The finite element method and the Scharfetter-Gummel exponential fitting scheme of [5] are used to approximate the boundary value problem (2.41) and yield a system of nonlinear algebraic equations in matrix form as

A7U7 = F7(U7). (4.118) We know that the partial derivative of f7 with respect to gp is all positive on the domain, so we know (4.66) satisfy Theorem3.1. The maximum norm a posteriori error estimate of Theorem 3.1. implies that

e :=°

°U₄^B− u°

° ≤Cη,^˜ (4.119)

where

Here η_l and M^(l), involve discrete analogues of lth-order derivatives and

D_x²Uij = 2 [hi+1U_i−1,j − (hⁱ+ hi+1)Uij+ hiUi+1,j] (hi+ hi+1) hi+1hi

, (4.130)

D_y²Uij = 2 [kj+1U_i,j−1− (k^j + kj+1)Uij+ kjUi,j+1] (kj+ kj+1) kj+1kj

, (4.131)

D_x²U_i−1,j = 2 [hiU_i−2,j − (hi−1+ hi)U_i−1,j + h_i−1Ui,j] (h_i−1+ hi) h_i−1hi

, (4.132)

D²_yU_i,j−1 = 2 [kjU_i,j−2− (kj−1+ kj)U_ij−1+ k_j−1Ui,j] (k_j−1+ kj) k_j−1kj

(4.133)

D_x⁻D_x²Uij = − 2

h_i−1hi(hi+ hi+1)U_i−2,j (4.134)

+2

µ 1

h_i−1hi(hi+ hi+1) + 1 hi+1

¶ U_i−1,j

− µ 1

hi+1

+ 1

h_i−1+ hi

¶

Ui,j + 1 hi+ hi+1

Ui+1,j

5 An Adaptive Algorithm for the QCET Model

The main ingredients of the algorithm solving the QCET model are adap-tive finite element approximation of the model, node-by-node and monotone iterative solution of the resulting nonlinear algebraic systems, and Gummel’s iteration consecutively on the PDEs as described in [5] for the ET model.

For the sake of clearness, we briefly illustrate the algorithm and refer to [5, 6] for more details on the adaptive finite element formulation, monotone convergence analysis, and practical implementation issues.

Here we use the notation l as Gummel’s (outer) iteration index and m as the monotone (inner) iteration index.

Step 1. Initial Mesh: Create a coarse and structured mesh for which the number of nodes can be chosen as small as possible.

Step 2. Preprocessing: See [5].

Step 3. Gummel and Monotone Iterations on (44)-(48).

Step 3.0. Set l := 0

Step 3.1. Solve the potential equation in (44).

Step 3.1.1. Set m := 0 and set the initial guess φ^(m)_j =

⎧⎨

⎩

φe_j or bφ_j if l = 0, φ^(l)_j otherwise,

for all (xj, yj)∈ Ω^h,

where eφ_j and bφ_j are constant values that can be easily verified to be an upper and lower solution of φ, respectively, and Ω^h denotes the set of mesh points on the closure of the domain.

Step 3.1.2 If l = 0, set u^(l) and v^(l) by the charge neutrality condition.

Step 3.1.3. Compute φ^(m+1)_j by solving the discrete potential system of (44)

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

ξ_jφ^(m+1)_j + γ_j(φ) φ^(m+1)_j =P

k∈V (j)ξ_kφ^(m)_k

−F (φ^(m)j , u^(l)_j , v_j^(l), ζ^(l)_n , ζ^(l)_p ) + γ_j(φ) φ^(m)_j , ∀(x^j, yj)∈ Ω^h, φ^(m+1)_j = VO+ Vb, ∀(x^j, yj)∈ ∂Ω^hD,

∂φ^(m+1)_j

∂n = 0, ∀(x^j, yj)∈ ∂Ω^hN ,

(1)

where

γ_j(φ) = max

½∂F (φ_j)

∂φ ; ˆφ_j ≤ φj ≤ ˜φ_j

¾

, (2)

ξ_k are the matrix elements of the discretization, and Ω^h, ∂Ω^h_D, and

∂Ω^h_N represent the sets of mesh points in the interior, Dirichlet part, and Neumann part of the domain, respectively.

Step 3.1.4. Set φ^(m)_j := φ^(m+1)_j ∀ j and m := m + 1. Go to Step 3.1.3 until the stopping criteria of the inner iteration are satisfied.

Step 3.1.5. Set φ^(l+1)_j := φ^(m+1)_j ∀ j.

Step 3.2. Solve the electron continuity equation (45).

Step 3.2.1. Set m := 0 and set the initial guess u^(m)_j =

⎧⎨

⎩ e

uj or bu^j if l = 0, u^(l)_j otherwise, for all (xj, yj) ∈ Ω^h, where uej and buj are constant values for all (xj, yj)∈ Ω^h that can be easily verified to be an upper and lower solution of u, respectively.

Step 3.2.2. Compute u^(m+1)_j by solving the discrete electron system of (45).

Step 3.2.3. Set u^(m)_j := u^(m+1)_j ∀ j and m := m + 1. Go to Step 3.2.2 until the stopping criteria of the inner iteration are satisfied.

Step 3.2.4. Set u^(l+1)_j := u^(m+1)_j ∀ j.

Step 3.3. Solve the hole continuity equation (46) similarly to that in Step 3.2.

Step 3.4. Solve the DG equation (47).

Step 3.4.1. Set m := 0 and set the initial guess [ζ_n]^(m)_j =

⎧⎨

⎩

[ζg_n]j or d[ζ_n]j if l = 0, [ζ_n]^(l)_j otherwise, for all (xj, yj) ∈ Ω^h, where [ζ_n]j ≈ ζn(xj, yj) and g[ζ_n]j and d[ζ_n]j

are constant values for all (xj, yj)∈ Ω^h that can be easily verified to be an upper and lower solution of ζ_n, respectively.

Step 3.4.2. Compute [ζ_n]^(m+1)_j by solving the discrete system of (47).

Step 3.4.3. Set [ζ_n]^(m)_j := [ζ_n]^(m+1)_j ∀ j and m := m + 1. Go to Step 3.4.2 until the stopping criteria of the inner iteration are satisfied.

Step 3.4.4. Set [ζ_n]^(l+1)_j := [ζ_n]^(m+1)_j ∀ j.

Step 3.5. Solve the DG (48) similarly to that in Step 3.4.

Step 3.6. Update [φ_qn]^(l+1)_j and [φ_qp]^(l+1)_j by the equations (57)-(58).

Step 3.7. Set l := l + 1 and go to Step 3.1 until the stopping criteria of the outer iteration are satisfied.

Step 4. Monotone Iteration on (49) and (50).

Step 4.1. Solve the energy equation (49) for gnsimilarly to that in Step 3.2.

Step 4.2. Solve the energy equation (50) for gp similarly to that in Step 3.2.

Step 5. Error Estimation: See below.

Step 6. Refinement: See below.

Step 7. Postprocessing: All computed solutions are then postprocessed for further analysis of physical phenomena.

Note that, in each one of Steps 3.1-3.5, 4.1, and 4.2, a Jacobi (node-by-node) type of solution is performed for the corresponding discrete system

(63), for example, in which the monotone parameters (64) can be easily obtained by means of lower and upper solutions. Two important factors that guarantee a global convergence with this kind of simple solutions as initial guesses are the diagonally dominant property of the matrices due to the self-adjoint formulation and the monotonicity of the parameters by the special nonlinearity of the formulation. The diagonally dominant property for (44)-(50) can proved in exactly the same manner as that given in [5, 6].

It can also be easily shown that each one of the nonlinear functionals in (44)-(50) is monotone in its respective state variable. It is thus a straightforward generalization from our previous theoretical analysis that all the nonlinear algebraic systems generated by this algorithm preserve these two factors. We thus summarize these important results in the following theorem.

Theorem. For each one of the PDEs (44)-(50) with associated boundary conditions, the matrices resulting to the adaptive finite element approxima-tion are diagonally dominant. Moreover, starting with suitable lower and upper solutions of the corresponding PDE, the Jacobi iteration in each of Steps 3.1-3.5, 4.1, and 4.2 generates a pair of lower and upper sequences which converge monotonically to the exact solution of the nonlinear alge-braic system of equations of that PDE.

5.1 Error Estimation

For each element, for example the ith element, the error indicartor is defined as

ei := max

1≤k≤7

½

0≤lmaxk≤3h²_iM_1,i^(lk), h²_iM_1,i+1^(lk)

¾

(5.1)

Error indicators are obtained on an element-by-element basis according to (5.1). A set of criteria on such as global error estimators of approximated solutions. inner iteration, and outer iteration. etc. will be verified. If none of the stopping criteria is satisfied, the adaptive process will continue to Step 6. otherwise it will go to Step 7 for postprocessing the computed solutions.

5.2 Refinement

Each one of the elements that are associated with error indicators greater than a preset error tolerance is divided into two subelements. We then move to Step 3.

6 Numerical Results

For the 1D diode model problem [4], the error indicators given by (5.1) give satisfactory results as shown on the right panels of Figs. 1-7 as compared with the original results of [4] shown on the left panels in these figures.

In Table 3, we present the maximum error indicator for each equation.

For example, for the first equation F1(φ) = 0, the largest error indicator is 6.6 x 10⁴ at the 86th element with 121 total elements.

Fig. 1. Potential (V)

Fig. 2. Electron Concentration

Fig. 3. Hole Concentration

Fig. 4. u

Fig. 5. v

Fig. 6. Gn (K)

Fig. 7. Gp (K)

Table 3. The maximum error indicator for QCET

equation total elements maximum element maximum error indicator

F1(φ) 121 86 6.06E + 04

F2(ζ_n) 121 114 1.29E + 05

F3(ζ_p) 104 61 1.86E + 05

F4(u) 90 37 3.66E + 06

F5(v) 119 69 8.57E + 05

F6(gn) 81 38 1.87E + 06

F7(gp) 140 119 1.10E + 06

7 Conclusion

We have extended the a posteriori error theory developed by Kopteva [11] to our singularly perturbed quantum-corrected energy transport model which consists of seven semilinear PDEs with the scaled Debye length, intrinsic carrier density, Planck constant, and Thermal conductivity as the singular

perturbation parameters. For these seven equations, we present explicit for-mulas for computing the error indicators which are indispensable for the adaptive computations of the semiconductor device simulation for advanced nano-devices. Our numerical experiments on the a posteriori error estima-tion for the 1D diode model problem have shown good results of the proposed error indicators for all seven PDEs of the QCET model.

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