# Summary of Classes

• WEEK 1
• Aug. 31: An overview of numerical methods to analyze semiconductor devices.
• WEEK 2
• Sep. 5: Introduction to The Drift-Diffusion Model
• The Five Drift-Diffusion Equations and their definitions
• 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 (depletion) region.
• 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 and BCs
• 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 potential diagram
• 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 matrix
• 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 Discretization Theme
• 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, B(x)=x/(exp(x)-1).
• 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 continuity equations.
• WEEK 6
• Oct. 3: Establishing Diagonal Dominance; All Equations Restated
• Restating the SG- discretized current continuity equation--
• 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 either...
• 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 5-diagonal matrix.
• 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 diagonal dominance.
• 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
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