6.5 Optimization Algorithm
7.1.2 Numerical method for solving state equations
1. Stationary solutions as the limit of time-dependent solutions
diffusion equations defined in Section 6.2. Note that, without the exciton equation, it becomes the standard drift-diffusion model for semiconductor devices. The numerical solutions to the drift-diffusion model have been a challenging task, mostly because of the convection terms in both equations for the electrons and the holes. Nonetheless, much has been known about its numerical solution. There are two perspectives on how to solve the stationary drift-diffusion equations.
• Iterative methods
Most iterative methods are built upon two basic methods: the Gummel’s map and the Newton’s method. For both methods, one needs to have some initial guess of the solution and hope the solution converges to a faithful numerical approximation of an analytical solution.
Gummel’s map is known to be less sensitive to the initial guess but also has a relatively slow rate of convergence. By contrast, Newton’s map has better convergence rate, but it’s much more sensitive to the initial guess. Note that, before convergence, any intermediate step of the iterative process has no physical meaning. Details of the Gummel’s map and the Newton’s method can be found in [21, 22].
• Limiting solution of the transient drift-diffusion model.
Recall that the stationary drift-diffusion model is in fact obtained from the time-dependent drift-diffusion model by setting the time derivative ∂t∂
to be zero [22]. Therefore, it is natural to obtain the stationary solution by solving the time-dependent drift-diffusion equations for a sufficiently large time period. This approach is arguably more “physical” in that it in fact simulates the evolution of electrical potentials and particle densities in time, and can be compared to experimental results.
In our work, we adopt the second perspective and solve the following time dependent drift-diffusion equations
−λ2∇ ·(∇ψ) =p−n (7.1.1) ∂n ∂t =−∇ ·Fn+ (ku−γnp) (7.1.2) ∂p ∂t =−∇ ·Fp + (ku−γnp) (7.1.3) ∂u ∂t =−∇ ·Fu+Q−duu−(ku−γnp) (7.1.4) (7.1.5)
Letting t→ ∞, we obtain the stationary solutions.
2. Spatial discretization
To illustrate the spatial discretization of the divergence operator and gradient operator, we pick an arbitrary interior nodePi,j = (xi, yj), We also letPi+12,j =
(xi+1
2, yj) be the mid-point betweenPi,jandPi+1,j. Thereforexi+ 1
2 = (i+ 1 2)hx.
Similar definitions hold for Pi,j+1 2, Pi−
1
2,j, and Pi,j− 1 2.
• Approximation of divergence operator (∇ ·F)ij
standard central difference scheme to discretize divergence operator (∇ ·F)i,j = Fx i+12,j −F x i−1 2,j hx + Fi,jy+1 2 −Fi,jy+1 2 hy (7.1.6) Note thatFx i,j+12 and F y i+1 2,j
are not needed for our purpose, since all we care about is the divergence of a flux at a grid point.
• Approximation of diffusion flux F=−µ∇y
If the flux is a simple diffusive flux, we again apply central difference scheme to discretize the gradient operator
Fix+1 2,j =−µi+1 2,j ui+1,j−ui,j hx (7.1.7) Fy i,j+12 =−µi,j+12 ui,j+1−ui,j hy (7.1.8)
• Approximation of convection-diffusion flux F=−µ(∇y+y∇ψ)
It is well-known that, for the convection-diffusion flux, an ordinary cen- tral difference scheme causes numerical instability. The Scharfetter- Gummel discretization scheme was invented to ensure the stability of convection-diffusion fluxes in drift-diffusion models, and it can be im- plemented on either a structured mesh (such as the one we use) or an unstructured finite element mesh. We refer the readers to [21] and [29] for details of the formulation. Below we simply state the formula for the Scharfetter-Gummel flux approximation for the electron flux
ing theψ in Fn to−ψ.
Fnx(i+1 2,j)
=−µn(i+1 2,j)
ni+1,jB(ψi+1,j−ψi,j)−ni,jB(ψi,j−ψi+1,j)
hx
(7.1.9) Fy
n(i,j+12)=−µn(i,j+12)
ni,j+1B(ψi,j+1−ψi,j)−ni,jB(ψi,j−ψi,j+1)
hy
(7.1.10) where B(z) is the Bernoulli function
B(z) = z ez−1 z 6= 0 1 z = 0 (7.1.11)
Note that when there is no convection, i.e. ∇ψ = 0, we effectively have a pure diffusion, and this Scharfetter-Gummel flux recovers exactly the central-difference scheme for diffusive fluxes.
3. Time discretization
To numerically solve the time-dependent equations, we need to discretize “time” and transform differentiation into a difference. We pick a sufficiently small ∆t >0 as our time step for solving (7.1.2-7.1.4). Specifically, let
tl =l∆t, l= 0,1, ...
lutions in time −λ2∇ ·(∇ψ(l+1)) = p(l)−n(l) (7.1.12) n(l+1)−n(l) ∆t =−∇ ·F (l+1) n + (ku−γnp) (l) (7.1.13) p(l+1)−p(l) ∆t =−∇ ·F (l+1) p + (ku−γnp) (l) (7.1.14) u(l+1)−u(l) ∆t =−∇ ·F (l+1) u + (ku−γnp) (l) (7.1.15)
where the superscript of y(l) is the numerical approximation of y(t) at time
tl=l∆t.
Note that the equations (7.1.13-7.1.15) are all reaction-diffusion equations. To ensure numerical stability, we use a backward Euler scheme for the divergence of fluxes;forward Euler schemeis applied for the reaction terms for simplicity.
We also add a note on the numerical implementation of the fluxes. In fact, the fluxes Fn, Fp, andFu are not fully evaluated at timetl+1. We takeF
(l+1) n
as an example; F(pl+1) andF(ul+1) have similar expression. To have a numerical
as below Fnx,((i+l+1)1
2,j)
=−µ(nl)(i+1 2,j)
ni(+1l+1),jB(ψi(+1l) ,j−ψi,j(l))−n(i,jl+1)B(ψi,j(l)−ψi(+1l) ,j) hx (7.1.16) Fy,(l+1) n(i,j+12) =−µ (l) n(i,j+12)
ni,j(l+1)+1B(ψi,j(l)+1−ψi,j(l))−ni,j(l+1)B(ψi,j(l)−ψi,j(l)+1) hy
(7.1.17) In other words, in the expression of F(nl+1), only the unknown n is evaluated
at time tl+1, and all the other terms are evaluated at timetl.
For each time step, we solve the equations (7.1.12-7.1.15) successively. The time marching process is terminated when the rate of change for all of {ψ(l),
n(l), p(l), u(l)} is smaller than some chosen tolerance >0.