Summary of Classes
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- WEEK 1
- Aug. 31: An overview of numerical methods to analyze
- WEEK 2
- Sep. 5: Introduction to The Drift-Diffusion Model
- The Five Drift-Diffusion Equations and their
- Simple Example 1: An Uniformly-Doped n-Type
Semiconductor Bar, under no bias. The Mass-Action Law.
- Simple Example 2: A Non-uniformly Doped n-Type
Semiconductor Bar, under no bias. Displays built-in potential due to
nonuniform doping. Derived the equations that relate the potential
profile to the carrier profile under such a case. Derived the
Non-Linear Poisson Equation (NL PE)
- Sep. 7: Another Example
- Simple Example 3: A Uniformly-Doped pn-Junction
- Textbook (segmentation) approach: Region-by-region
analytic solution, with different approximations at each region. The
quasi-neutrality (charge-neutrality) approximation for p and n
regions. The depletion approximation for the junction
- First discretization example: Discretizing the
linear PE that had been obtained in the depletion
regions. Creating an algebraic matrix equation that can be solved by
standard matrix manipulation methods.
- WEEK 3
- Sep. 12: Review and Newton's Method
- Quick review of D-D Equations, the NL PE and the
L PE by depletion approximation.
- Newton's Method: A root-finding method for
nonlinear equations that is based on making an initial guess and
iteratively approaching the root by using Taylor Theorem repeatedly.
Extending the method to two functions with two unknowns.
- Sep. 14: Applying NM to the whole system
- First discretization; review of how to write
derivatives as differences
- NLPE discretized and rewritten as a function f=0,
represented as a set of equations with a Jacobian matrix, an update
(delta phi) vector and an f vector for each iteration
- Boundary condition (BC) specification
- Introduction to the MOS capacitor potential profile
- WEEK 4
- Sep. 19: More on MOS capacitor: Non-equilibrium case
- Review of band diagrams, E_f, E-c, E_v, E_i,
definition of potential
- Band diagram for a MOS capacitor, the corresponding
- Effects of having nonzero current: No longer possible
to reduce the DD system to the single PE
- Reducing the system to differential equations for the
charge densities instead, that will be discretized and solved
- Sep. 21: Gaussian elimination; tentative
discretization on non-eq. DD eqns
- Gaussian elimination algorithm for a tridiagonal
- Initial Discretization of the nonequilibrium case DD
eqns: define mesh, discretize all five equations; assume linear
interpolation for d(phi)/dx and dn/dx to obtain a matrix equation...
- ...which is not a very sound approach, because--
- WEEK 5
- Sep. 26: An Interlude in C and Variation on The
- Basics of C--program files, how to compile,
including system libraries, program structure, defining functions,
manipulating files and declaring (dynamic or static) arrays
- Back to the discretization problem: ---because n (or
p) is varying exponentially, the linear interpolation is really not valid.
- Assume generation/recombination occurs only at the
mesh points, and hence the current is constant between mesh
points; start solving the current equations with this assumption
- Further assume that potential changes linearly, so
electric field is constant between mesh points; use this assumption for an
integration trick and obtain a form for the current expression that can be
written in terms of Bernoulli functions...
- ...which were also defined. To wit,
- Sep. 28: Review of Scherfetter-Gummel Discretization
- A review of solving for the current density
between mesh points to get a result in terms of Bernoulli functions...
- ...obtaining a discretization of the current
- WEEK 6
- Oct. 3: Establishing Diagonal Dominance; All
- Restating the SG- discretized current continuity
- This equation does ensure diagonal dominance, as
demonstrated by looking at the coefficients that would result if it was
written as a matrix equation for all possible cases
- We now have discretized forms of the Poisson
equation and of the two current continuity equations.
- Oct. 5: Setting up Matrices and Iteration Schemes
- We have three discretized equations. We can
- a) ...write three matrix equations
for these, start with an initial guess, and solve them one by one by
iteration, putting the solution of each iteration as the updated value to
the next matrix equation, or...
- b) ...put all three into a single
matrix equation, whose matrix will be 3nx3n, and solve it by either
Newton's Method or with a block iteration scheme.
- WEEK 7
- Oct. 10: Establishing Boundary
Conditions; Introduction to Mobility
- For the three 2nd order DEs we have, we need 6
boundary conditions, which are set with charge-neutrality assumption.
- Mobility--approximated as a constant in low fields,
but for high fields it is proportional to 1/(electric field). Its
derivation starts with considering all the forces on an electron in a
field, where scattering mechanisms also plays a role.
- Oct. 12: Interlude on Iterations, Continue on
Mobility, Introduction to 2D Simulations
- Suggested program structure for project
- When you're trying to solve for a given bias, use the
results from a close bias as initial guess
- Mobility: Proportional to 1/(scattering rate), which
is a sum of scattering rates of several mechanisms: Ion scattering,
acoustic phonon scattering, optical phonon scattering, ...
- Ion scattering and optical phonon scattering are
low-field mobility components, and they are almost field-independent.
- The field dependent part is about proportional to
1/E; there are empirical fits.
- Working this into simulators is just a matter of
defining mu, or D, as a function of field.
- 2D Simulations : Same principle. more
algebra, worse-conditioned matrices. Same method of discretization yields
- WEEK 8
- Oct. 17: MOSFET modeling
- How to set up the BCs
- Effects of surface electron mobility on IV curves
- Oct. 19: Transient Simulation
- Transient Semiconductor Equations: The
current-cont. equations have time-dependent terms now.
- Displacement current and particle current
- Discretization of a transient equation, establishing
- WEEK 9
- Oct. 24: Introduction to Transport Physics
- Survival solid-state physics: Schroedinger's
Wave Equation (SWE)
- The potential components in a crystal
- Bloch's Theorem
- Oct. 26: Band structures from SWE
- Solving SWE with crystal potential yields another
eigenvalue equation that yields eigenvalues of energy for each k
- Which is what we use to construct band diagrams
- What can be learned from band diagrams (E
vs. k): Electron velocity, effective mass, scattering rate