Download part-1

Summary
18
4.4 Summary
We have derived the system of discretized Shockley's equations (3.9) - (3.11), we have also developed
the iterative technique (4.7). In principle, we are now able to solve this set of equations.
However, there remain a few practical problems before we can make the program work correctly
and robustly:
• An overall design of the program that links all the components together.
• The proper boundary conditions at ohmic contacts, material interfaces, and the outer boundary of
the device structure.
• A better discretization scheme that is more robust than our naive attempt.
• An efficient way to compute the function 𝑭 and more importantly the Jacobian 𝑱 .
We shall address all the above problems in the following chapters.
4.5 Exercises
4.5.1 1D MOS Capacitor ⋆
For a MOS capacitor in thermal equilibrium, the carrier concentration can be written as
𝑞𝜓
𝑘𝑇 )
𝑞𝜓
𝑝 = 𝑝0 exp (− ) ,
𝑘𝑇
𝑛 = 𝑛0 exp (
where 𝑛0 and 𝑝0 are the equilibrium carrier concentrations, and 𝜓 is the band-bending. With this, (2.1)
can be solved.
For a MOS capacitor with uniform p-type substrate doping of 𝑁𝑎 = 1 × 1017 cm−3 , and surface
potential of 𝜓𝑠 = 0.6 V,
• Discretize this 1D Poisson equation using finite-volume method. Note that the volumes of the first
and the last cell are different from the rest.
• Write down the system of non-linear equations; take care of the boundary conditions.
• Write down the Jacobian matrix of this non-linear equation. What is the bandwidth of this matrix.
• Write a program to solve this 1D MOS capacitor problem.
⋆
10 programming credits